HorizonMath: Measuring AI Progress Toward Mathematical Discovery with Automatic Verification
Paper • 2603.15617 • Published • 6
id string | prompt string | output_type string | domain string | evaluation_mode string | solvability int64 | numeric_value string | source_url string | source_note string | test_points null | metric_key null | optimization_direction null | baseline_value null | baseline_note null |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
w4_watson_integral | Consider the following research problem in mathematics.
**Closed Form for the 4-Dimensional Lattice Green's Function ($W_4$)**
**Definition:** The Watson integrals $W_d$ represent the Green's function at the origin for the hypercubic lattice Green's function constant at the origin. They are defined by the integral: \... | constant | lattice_models | ground_truth_computable | 0 | 0.3098667804621204281696744162147501775383222672904396642383504626790703346638908327580983261838473482149795083 | https://arxiv.org/pdf/1801.02182 | Zhou, 'On Laporta's 4-loop sunrise formulae' (2018) - Laporta (2018) conjectures a closed-form, and Zhou proves it, giving a hypergeometric/Gamma expression | null | null | null | null | null |
w5_watson_integral | Consider the following research problem in mathematics.
**Closed Form for the 5-Dimensional Lattice Green's Function ($W_5$)**
**Definition:** The Watson integrals $W_d$ represent the Green's function at the origin for the hypercubic lattice Green's function constant at the origin. They are defined by the integral: \... | constant | lattice_models | ground_truth_computable | 2 | 0.23126162496804623574142702438771339710908546970102847765391320224201754069413746234473308609901834330534861291 | https://arxiv.org/abs/1004.1435 | Guttmann, 'Lattice Green functions in all dimensions' (2010) - covers Watson integrals W_d for arbitrary d-dimensional hypercubic lattices | null | null | null | null | null |
w6_watson_integral | Consider the following research problem in mathematics.
**Closed Form for the 6-Dimensional Lattice Green's Function ($W_6$)**
**Definition:** The Watson integrals $W_d$ represent the Green's function at the origin for the hypercubic lattice Green's function constant at the origin. They are defined by the integral: \... | constant | lattice_models | ground_truth_computable | 2 | 0.18616056220444530728094072199476887544269877039883875411399992156674267940911681325387509047530591295459637041 | https://arxiv.org/abs/1004.1435 | Guttmann, 'Lattice Green functions in all dimensions' (2010) - comprehensive treatment of lattice Green functions and Watson integrals in all dimensions | null | null | null | null | null |
bessel_moment_c5_0 | Consider the following research problem in mathematics.
**Closed Form for the Bessel Moment $c_{5,0}$**
**Definition:** The Bessel function moments are defined by the integral $c_{n,k} = \int_0^{\infty} t^k K_0(t)^n \, dt$, which arise in $(n-1)$-loop Feynman diagram calculations. For $n=5, k=0$, the value is approxi... | constant | integrals | ground_truth_computable | 2 | 135.26830258086883759422627964619220742030588935942352678469351371045888711773849131554701138246193550710196669 | https://arxiv.org/abs/0801.0891 | Bailey, Borwein, Broadhurst, Glasser, 'Elliptic integral evaluations of Bessel moments' (2008) - closed forms for c_{n,k} Bessel moments with progress on c_{5,0} | null | null | null | null | null |
bessel_moment_c6_0 | Consider the following research problem in mathematics.
**Closed Form for the Bessel Moment $c_{6,0}$**
**Definition:** The Bessel function moments are defined by $c_{n,k} = \int_0^{\infty} t^k K_0(t)^n \, dt$. For the case $n=6, k=0$, the numerical value is approximately $809.62...\dots$. Here $c_{n,k}$ means exactl... | constant | integrals | ground_truth_computable | 2 | 809.62084822486627594007354000392747913008434556749563772879133821833933609599367021661064055934872732418948686 | https://arxiv.org/abs/0801.0891 | Bailey, Borwein, Broadhurst, Glasser, 'Elliptic integral evaluations of Bessel moments' (2008) - formulae for integrals of products of six or fewer Bessel functions | null | null | null | null | null |
bessel_moment_c5_1 | Consider the following research problem in mathematics.
**Closed Form for the Bessel Moment $c_{5,1}$**
**Definition:** The Bessel function moments are defined by $c_{n,k} = \int_0^{\infty} t^k K_0(t)^n \, dt$. This problem concerns the first moment ($k=1$) with $n=5$ Bessel functions. The numerical value is approxim... | constant | integrals | ground_truth_computable | 2 | 2.4965992507497653561840017811514997432406114327981162232729101382421014141270463045039463065513848490719149810 | https://arxiv.org/abs/0801.0891 | Bailey, Borwein, Broadhurst, Glasser, 'Elliptic integral evaluations of Bessel moments' (2008) - substantial progress on c_{5,2k+1} odd moments | null | null | null | null | null |
box_integral_b6_1 | Consider the following research problem in mathematics.
**Closed Form for the 6D Box Integral $B_6(1)$**
**Definition:** The box integral $B_n(s)$ measures the $s$-th moment of the Euclidean distance from the origin to a point in the unit hypercube $[0,1]^n$: \[ B_n(s) = \int_{[0,1]^n} |\mathbf{x}|^s \, d\mathbf{x} \... | constant | integrals | ground_truth_computable | 2 | 1.388574084457347842530254073030788815910945088782207029758933139762637896937682885791843577 | https://www.osti.gov/biblio/964379 | Bailey, Borwein, Crandall, 'Higher-dimensional box integrals' (2010) - first nontrivial closed forms for six-dimensional box integrals and also provides closed forms for $n$ up to 5. | null | null | null | null | null |
box_integral_b7_1 | Consider the following research problem in mathematics.
**Closed Form for the 7D Box Integral $B_7(1)$**
**Definition:** The box integral $B_n(s)$ measures the $s$-th moment of the Euclidean distance from the origin to a point in the unit hypercube $[0,1]^n$: \[ B_n(s) = \int_{[0,1]^n} |\mathbf{x}|^s \, d\mathbf{x} \... | constant | integrals | ground_truth_computable | 2 | 2.1031677468737035517164242261635051336191256398255234438587726962237281589021474209489946038383277181415894854 | https://www.osti.gov/biblio/964379 | Bailey, Borwein, Crandall, 'Higher-dimensional box integrals' (2010) - first nontrivial closed forms for six-dimensional box integrals and also provides closed forms for $n$ up to 5. | null | null | null | null | null |
box_integral_b5_neg2 | Consider the following research problem in mathematics.
**Closed Form for the Box Integral $B_5(-2)$**
**Definition:** The box integral $B_n(s) = \int_{[0,1]^n} |\mathbf{x}|^s \, d\mathbf{x}$ generally becomes harder for negative $s$. For $n=5$ and $s=-2$, the value is approximately $0.76560...\dots$. This represents... | constant | integrals | ground_truth_computable | 0 | 0.76560088060035042048313592041746790597916235131578395215189528953020852443035092982996181509585989486734309034 | https://www.osti.gov/biblio/964379 | Bailey, Borwein, Crandall, 'Higher-dimensional box integrals' (2010) - first nontrivial closed forms for six-dimensional box integrals and also provides closed forms for $n$ up to 5. | null | null | null | null | null |
elliptic_k_moment_3 | Consider the following research problem in mathematics.
**Third Moment of the Complete Elliptic Integral $K(k)$**
**Definition:** This problem asks for the closed form of the moment integral $\int_0^1 K(k)^3 \, dk$, where $K(k)$ is the complete elliptic integral of the first kind. The numerical value is approximately... | constant | integrals | ground_truth_computable | 0 | 7.0902270048462694609898023700595492524524185476584179865587158041145846347861787736244562389891764350266529514 | https://arxiv.org/abs/1303.2259 | Rogers, Wan, Zucker: 'Moments of elliptic integrals and critical L-values'. Ramanujan J. 37 (2015), 113-130. Provides a closed form for the third moment of K(k) expressible via gamma functions | null | null | null | null | null |
elliptic_k_moment_4 | Consider the following research problem in mathematics.
**Fourth Moment of the Complete Elliptic Integral $K(k)$**
**Definition:** This problem asks for the closed form of the moment integral $\int_0^1 K(k)^4 \, dk$, where $K(k)$ is the complete elliptic integral of the first kind and $K(k)=\int_{0}^{\pi/2} \frac{d h... | constant | integrals | ground_truth_computable | 2 | 15.611523683715693929074704703647595914409260699418022257962398941624312278709557178035465062471152754769332293 | https://arxiv.org/abs/1303.2259 | Rogers, Wan, Zucker: 'Moments of elliptic integrals and critical L-values'. Ramanujan J. 37 (2015), 113-130. Derives closed forms for elliptic integral moments expressible via gamma functions | null | null | null | null | null |
elliptic_k2_e_moment | Consider the following research problem in mathematics.
**Mixed Moment of Elliptic Integrals $K(k)^2 E(k)$**
**Definition:** This problem concerns the integral of the product of the square of the complete elliptic integral of the first kind $K(k)$ and the complete elliptic integral of the second kind $E(k)$: $\int_0^... | constant | integrals | ground_truth_computable | 0 | 4.7268180032308463073265349133730328349682790317722786577058105360763897565241191824163041593261176233511019676 | https://arxiv.org/abs/0801.0891 | Wan: 'Moments of products of elliptic integrals'. (2018). Develops closed forms for Bessel moments with connections to elliptic integrals | null | null | null | null | null |
airy_moment_a4 | Consider the following research problem in mathematics.
**Fourth Moment of the Airy Function ($a_4$)**
**Definition:** The Airy power moments are defined by $a_n = \int_0^\infty \mathrm{Ai}(x)^n \, dx$. These moments appear in random matrix theory. The fourth moment $a_4$ has the numerical value approx.\ $0.0046380..... | constant | integrals | ground_truth_computable | 0 | 0.0046380290604946057287443641210015069017195022230366911564643170644289766133364996131025023047197563677273764507 | https://dlmf.nist.gov/9.11 | DLMF Section 9.11: Products of Airy Functions. The closed form is ln(3)/(24*pi^2) | null | null | null | null | null |
airy_moment_a5 | Consider the following research problem in mathematics.
**Fifth Moment of the Airy Function ($a_5$)**
**Definition:** The Airy power moments are defined by $a_n = \int_0^\infty \mathrm{Ai}(x)^n \, dx$. For $n=5$, the value is approximately $0.0013493...\dots$.
**Task:** Find a symbolic closed-form expression for the... | constant | integrals | ground_truth_computable | 2 | 0.0013493589835177305394535748997338260553653997404797424839336973256901140935986288565766973541821804238164374932 | https://link.springer.com/article/10.1007/BF00942815 | Laurenzi, B.J. 'Moment integrals of powers of airy functions.' Z. angew. Math. Phys. 44, 891-908 (1993. Studies powers of the Airy function Ai(z) and its derivative Ai'(z). | null | null | null | null | null |
central_binomial_s5 | Consider the following research problem in mathematics.
**Closed Form for Central Binomial Sum $S_5$**
**Definition:** The series is defined as $S_k = \sum_{n=1}^\infty \frac{1}{n^k \binom{2n}{n}}$. Known results exist for $k=1, 2, 3, 4$ involving $\pi$, Clausen functions, and polylogarithms. The case $k=5$ (approx.\... | constant | mathematical_constants | ground_truth_computable | 0 | 0.50542947468351924164245048190843214918866901456826286498266471287573347337617590682716453318150013661960285541 | https://arxiv.org/abs/hep-th/0004153 | Borwein, Broadhurst, Kamnitzer: 'Central Binomial Sums, Multiple Clausen Values and Zeta Values', Exper. Math. 10 (2001), 25-34. Finds relationships between zeta values and central binomial sums | null | null | null | null | null |
autocorr_upper | Consider the following optimization problem.
**Improve Upper Bound on Autocorrelation Constant $C$**
**Definition:** The autocorrelation constant $C$ is defined as $C = \inf_f \frac{\max_{t} (f * f)(t)}{(\int f(x)\, dx)^2}$ where the infimum is over all non-negative, not identically zero functions $f$ supported on $[... | construction | combinatorics | benchmark_best_known | 1 | null | https://arxiv.org/abs/2601.16175 | Yuksekgonul et al. (2025) 'Learning to Discover at Test Time' (arXiv:2601.16175) - achieves C ≤ 1.50286 via a 30000-piece step function, and Cloninger & Steinerberger (2017) 'On Suprema of Autoconvolutions with an Application to Sidon sets' (Proc. AMS 145(8):3191-3200, arXiv:1403.7988) | null | null | null | null | null |
autocorr_signed_upper | Consider the following optimization problem.
**Signed Autocorrelation Constant $C'$ Upper Bound**
**Definition:** The signed autocorrelation constant $C'$ is defined as $C' = \inf_f \max_t (f * f)(t) / (\int f)^2$, where the infimum is over all not identically zero functions $f$ (which may take negative values) suppo... | construction | combinatorics | benchmark_best_known | 2 | null | https://arxiv.org/abs/1205.0626 | Jedwab, Katz & Schmidt (2013) 'Advances in the merit factor problem for binary sequences' - establishes asymptotic merit factor bounds and addresses signed autocorrelation properties | null | null | null | null | null |
resultant_chebyshev | Consider the following research problem in mathematics.
**Resultant of Chebyshev and Legendre Polynomials**
**Definition:** Let $T_n(x) = \cos(n \arccos x)$ be the Chebyshev polynomial of the first kind of degree $n$, and let $P_m(x)$ be the Legendre polynomial of degree $m$, defined by $(1 - 2xt + t^2)^{-1/2} = \sum... | constant | mathematical_constants | ground_truth_computable | 0 | 3.50250188617129022035975427961480421661370306852776070285584178979291528698154779416561876786842808192139e+146 | https://en.wikipedia.org/wiki/Chebyshev_polynomials | Resultant Res_x(T_30, P_20) of Chebyshev T_30 and Legendre P_20 polynomials. While Res(T_n, T_m) and Res(T_n, U_m) have known closed forms (Gishe-Ismail 2008), no general closed-form formula is known for cross-family Res(T_n, P_m). | null | null | null | null | null |
quartic_oscillator_lambda | Consider the following open problem in mathematical physics.
**Eigenvalues of a Quartic Oscillator with Quadratic Parameter**
**Definition:** In units where \(\hbar=m=1\), define \(\varepsilon_n(\lambda)\) as the \(n\)-th eigenvalue of
\[ -\tfrac12\,\psi''(x) + \Big(\tfrac{x^4}{4} - \tfrac{\lambda x^2}{2}\Big)\psi(x)... | function | continuum_physics | ground_truth_computable | 2 | null | https://dft.uci.edu/pubs/OB20.pdf | Problem definition (Schr\u00f6dinger equation and potential v_\u03bb) follows Okun & Burke (2020). Published 40-digit eigenvalue benchmarks are in the Supplemental Information Table S1: https://dft.uci.edu/pubs/OB20s.pdf. The paper explicitly notes the quartic oscillator lacks a simple analytic solution, supporting the... | null | null | null | null | null |
spheroidal_eigenvalue_lambda_m0 | Consider the following open problem in spectral theory / special functions.
**Angular Prolate Spheroidal Eigenvalues (order m = 0)**
Let \(c \ge 0\) be a real parameter. Consider the Sturm-Liouville eigenvalue problem on \((-1,1)\):
\[
-\frac{d}{dx}\Big((1-x^2)\,y'(x)\Big) + c^2 x^2\,y(x) = \lambda\,y(x),\qquad -1<x... | function | continuum_physics | ground_truth_computable | 2 | null | https://arxiv.org/abs/math-ph/0212051 | Falloon, Abbott, Wang (2003). Journal of Physics A: Mathematical and General. 'Theory and computation of spheroidal wavefunctions.' Background: spheroidal eigenvalues are typically computed numerically (continued fractions / tridiagonal-matrix truncations) and only limited analytic identities/special cases are availabl... | null | null | null | null | null |
feigenbaum_delta | Consider the following research problem in mathematics.
**Closed Form for the Feigenbaum Constant $\delta$**
**Definition:** The Feigenbaum constant $\delta$ is the limiting ratio of consecutive bifurcation intervals in the period-doubling route to chaos for unimodal maps. For the logistic map $f(x) = rx(1-x)$, if $r... | constant | mathematical_constants | ground_truth_computable | 3 | 4.6692016091029906718532038204662016172581855774757686327456513430041343302113147371386897440239480138171659848 | https://oeis.org/A006890 | OEIS decimal expansion of Feigenbaum bifurcation velocity constant delta = 4.669201609102990671853...; no closed form known | null | null | null | null | null |
feigenbaum_alpha | Consider the following research problem in mathematics.
**Closed Form for the Feigenbaum Constant $\alpha$**
**Definition:** The Feigenbaum constant $\alpha$ governs the geometric scaling of the attractor in period-doubling bifurcations. It is defined as the limit $\alpha = \lim_{n \to \infty} d_n / d_{n+1}$ (quadrat... | constant | mathematical_constants | ground_truth_computable | 3 | 2.50290787509589282228390287321821578638127137672714997733619205677923546317959020670329964974643383412959 | https://oeis.org/A006891 | OEIS decimal expansion of Feigenbaum reduction parameter alpha = 2.502907875095892822283...; no closed form known | null | null | null | null | null |
fransen_robinson_constant | Consider the following research problem in mathematics.
**Closed Form for the Fransén-Robinson Constant**
**Definition:** The Fransén-Robinson constant $F$ is defined by the integral $F = \int_0^{\infty} \frac{1}{\Gamma(x)}\,dx$, where $\Gamma$ is the Euler gamma function. Its numerical value begins $2.8077...\dots$ ... | constant | mathematical_constants | ground_truth_computable | 2 | 2.8077702420285193652215011865577729323080859209301982912200548095971008891219016655101853081681966381418741643 | https://oeis.org/A058655 | OEIS A058655: Decimal expansion of the Fransén-Robinson constant; no closed form known | null | null | null | null | null |
nested_radical_kasner | Consider the following research problem in mathematics.
**Closed Form for the Nested Radical Constant**
**Definition:** The nested radical constant (also called Kasner's number) is defined as the limit of the nested radical expression $\sqrt{1 + \sqrt{2 + \sqrt{3 + \sqrt{4 + \cdots}}}}$. Its numerical value begins $1... | constant | mathematical_constants | ground_truth_computable | 2 | 1.7579327566180045327088196382181385276531999221468377043101355003851102326744467575723445540002594529709324718 | https://oeis.org/A072449 | OEIS A072449: Decimal expansion of Kasner's number sqrt(1+sqrt(2+sqrt(3+...))); no closed form known. Herschfeld (1935) in 'On Infinite Radicals' says Kasner suggested investigation of “infinite radicals” and introduces K as the 'Kasner number.' | null | null | null | null | null |
mrb_constant | Consider the following research problem in mathematics.
**Closed Form for the MRB Constant**
**Definition:** The MRB constant (named after Marvin Ray Burns) is defined as the alternating sum $M = \sum_{n=1}^{\infty} (-1)^n (n^{1/n} - 1)$. Its numerical value begins $0.18785...\dots$. The constant arises in the study ... | constant | mathematical_constants | ground_truth_computable | 2 | 0.18785964246206712024851793405427323005590309490013878617200468408947723156466021370329665443310749690384234586 | https://oeis.org/A037077 | OEIS A037077: Decimal expansion of the MRB constant sum((-1)^n*(n^(1/n)-1)); no closed form known. There are known forms that are not closed-form, such as an infinite series involving derivatives of the Dirichlet eta function and an integral representation according to MathWorld's article, 'https://mathworld.wolfram.co... | null | null | null | null | null |
torsional_rigidity_square | Consider the following research problem in mathematics.
**Closed Form for the Torsional Rigidity Ratio of a Square**
**Definition:** The torsional rigidity of a prismatic bar with a full side length $b$ is characterized by the dimensionless ratio $J/b^4$, where $J$ is the torsion constant. Using Saint-Venant's classi... | constant | continuum_physics | ground_truth_computable | 2 | 0.140577014955153715588468730737731115267593118830092268073958148912875912876 | https://oeis.org/A180309 | OEIS entry for the decimal expansion of the torsional rigidity constant for a square shaft. MathWorld confirms the numerical value, 'Torsional Rigidity'. | null | null | null | null | null |
bernstein_constant | Consider the following research problem in mathematics.
**Closed Form for Bernstein's Constant**
**Definition:** Let $P^*_n$ denote the polynomial of degree $\le n$ that minimizes $\sup_{x \in [-1,1]} ||x| - P^*_n(x)|$. Define $E_n = \sup_{x \in [-1,1]} ||x| - P^*_n(x)|$. Bernstein's constant is $\beta = \lim_{n \to ... | constant | mathematical_constants | ground_truth_computable | 2 | 0.28016949902386913303643649123067200004248213981236 | https://oeis.org/A073001 | Varga & Carpenter, Constr. Approx. 1 (1985) 333-348; Lubinsky (2003) integral representation | null | null | null | null | null |
townes_soliton | Consider the following research problem in mathematics.
**Townes Soliton Critical Mass (2D Cubic NLS Ground State Norm)**
**Definition:** Let $Q(r)$ be the unique positive radial solution of the ODE $Q''(r) + (1/r)Q'(r) - Q(r) + Q(r)^3 = 0$ for $r > 0$, with $Q'(0) = 0$ and $Q(r) \to 0$ as $r \to \infty$ (uniqueness:... | constant | continuum_physics | ground_truth_computable | 2 | 11.70089652455965387865397 | https://math.unm.edu/~plushnik/publications/LushnikovVladimirovaOptLett2014.pdf | Lushnikov and Vladimirova (2014). Optics Letters, v.39, 3429-3432, 'Nonlinear combining of laser beams.' They define the Townes soliton and provide N_c up to 1.7008965... | null | null | null | null | null |
mahler_1_x_y_z_w | Consider the following research problem in mathematics.
**Mahler Measure of $1+x+y+z+w$**
**Definition:** The logarithmic Mahler measure of the 4-variable polynomial $P(x,y,z,w) = 1+x+y+z+w$ is defined by the integral over the unit torus, and $m(P) = \int_0^1 \cdots \int_0^1 \log |P(e^{2\pi i t_1}, \dots, e^{2\pi i t... | constant | number_theory | ground_truth_computable | 2 | 0.54441256175218558519587806274502767666605280202852627449556789488000645997738563329065126658200759562393248342 | https://dms.umontreal.ca/~mlalin/surveyMahlerfinal-revised.pdf | Bertin & Lalin survey on Mahler measure of multivariable polynomials. The Mahler measure m(1+x+y+z+w) extends Smyth's results to 4 variables with connections to L-functions | null | null | null | null | null |
mahler_elliptic_product | Consider the following research problem in mathematics.
**Mahler Measure of $(x+y+1)(x+1)(y+1)-xy$**
**Definition:** This problem concerns the logarithmic Mahler measure $m(P) = \frac{1}{(2\pi)^2} \int_0^{2\pi} \int_0^{2\pi} \log |P(e^{i\theta}, e^{i\phi})| \, d\theta \, d\phi$ of the two-variable Laurent polynomial ... | constant | number_theory | ground_truth_computable | 2 | 0.66422509302916593526284646964035380327719614159380234519653938087512261465036362537617710889395147153204690603639639539212919594553663512901466775635 | https://arxiv.org/abs/1012.3036 | Rogers and Zudilin: 'From L-series of elliptic curves to Mahler measures'. Studies genus-one Mahler-measure families of product-of-linear-factors type via regulators and q-series methods | null | null | null | null | null |
mzv_reduction_zeta_3_3_3 | Consider the following research problem in mathematics.
**Reduction of $\zeta(3,3,3)$**
**Definition:** The Multiple Zeta Value $\zeta(3,3,3)$ is a depth-3, weight-9 value defined by $\sum_{n_1 > n_2 > n_3 \geq 1} (n_1 n_2 n_3)^{-3}$. The problem is to determine if and how this value can be expressed in terms of lowe... | constant | number_theory | ground_truth_computable | 0 | 0.012034182574412003861599684421693740505784954499279660274108607505043368975229731321242723660408603557091175883 | https://arxiv.org/abs/math/0309425 | Hoffman: 'Algebraic Aspects of Multiple Zeta Values'. Establishes algebraic framework for reducing MZVs like zeta(3,3,3) using shuffle/stuffle algebra relations | null | null | null | null | null |
stieltjes_gamma_1 | Consider the following research problem in mathematics.
**Closed Form for Stieltjes Constant $\gamma_1$**
**Definition:** The Stieltjes constants $\gamma_n$ are the coefficients in the Laurent series expansion $\zeta(1+s) = \frac{1}{s} + \sum_{n \geq 0} \frac{(-1)^n}{n!} \gamma_n s^n$ of the Riemann zeta function $\z... | constant | number_theory | ground_truth_computable | 0 | -0.072815845483676724860586375874901319137736338334337952599006559741401433571511484878086928244844014604077207279 | https://oeis.org/A082633 | OEIS provides an entry for the decimal expansion of the 1st negated Stieltjes constant gamma_1. It also cites Maślanka, K., & Koleżyński, A. (2022). The High Precision Numerical Calculation of Stieltjes Constants. Simple and Fast Algorithm. Computational Methods in Science & Technology, 28(2) to provide 0.072815... | null | null | null | null | null |
closed_form_ramanujan_soldner | Consider the following open problem.
**Closed-Form Expression for the Ramanujan-Soldner Constant (μ)**
**Definition:** μ is the unique positive real number satisfying li(μ)=0, where li is the non-offset logarithmic integral (Cauchy principal value). Equivalently, li(x)=Ei(log x) for x>0.
**Task:** Find a finite expl... | constant | number_theory | ground_truth_computable | 2 | 1.45136923488338105028396848589202744949303228 | https://oeis.org/A070769 | OEIS A070769: Ramanujan-Soldner constant μ, the unique positive zero of li(x). See also MathWorld and Wikipedia for definition and properties. | null | null | null | null | null |
schur_6 | Let S(k) be the Schur number: the largest n such that {1,2,...,n} can be partitioned into k sum-free sets. A subset A of positive integers is sum-free if there do not exist x,y in A with x+y in A (x and y may be equal).
Task: Construct a 6-coloring of {1,2,...,N} with no monochromatic solution to x+y=z (equivalently, ... | construction | combinatorics | benchmark_best_known | 2 | null | https://www.combinatorics.org/ojs/index.php/eljc/article/view/v7i1r32 | Fredricksen & Sweet (2000) give explicit constructions proving S(6)≥536. Later work notes only bounds are 536≤S(6)≤1836, so the optimum is unknown. | null | null | null | null | null |
euler_mascheroni_closed_form | Consider the following research problem in mathematics.\n\n**Closed-Form Expression for the Euler-Mascheroni Constant**\n\n**Definition:** The Euler-Mascheroni constant is \(\gamma = \lim_{n\to\infty}(\sum_{k=1}^n 1/k - \log n)\). Although many representations are known (limits, integrals, series), no closed-form expre... | constant | number_theory | ground_truth_computable | 3 | 0.5772156649015328606065120900824024310421593359399235988057672348848677267776646709369470632917467495 | https://www.ams.org/bull/2013-50-04/S0273-0979-2013-01423-X/ | Lagarias (Bull. AMS, 2013) surveys Euler's constant and modern developments; key arithmetic questions and the absence of a known closed-form expression remain open. Decimal expansion is standard; see OEIS A001620. | null | null | null | null | null |
elliptic_curve_rank_30 | Consider the following optimization problem.
**Elliptic Curve with Rank at Least 30**
**Definition:** The rank of an elliptic curve $E$ over $\mathbb{Q}$ measures the number of independent rational points of infinite order. An elliptic curve with rank at least 29 is known; and under GRH the rank is exactly 29, achiev... | construction | number_theory | benchmark_best_known | 2 | null | https://arxiv.org/abs/0709.2908 | Noam Elkies, 'Three lectures on elliptic surfaces and curves of high rank' (2007). Documents the rank 28 record from 2006; note that Elkies-Klagsbrun found rank 29 in 2024 (no single arXiv paper yet, but announced August 2024). | null | null | null | null | null |
elliptic_curve_rank_torsion_z7z | Consider the following optimization problem.
**High-Rank Elliptic Curve with Torsion $\mathbb{Z}/7\mathbb{Z}$**
**Definition:** For elliptic curves over $\mathbb{Q}$ with torsion subgroup $\mathbb{Z}/7\mathbb{Z}$, the current rank record is 6. Finding curves with higher rank and prescribed torsion is a major challeng... | construction | number_theory | benchmark_best_known | 2 | null | https://arxiv.org/abs/2003.00077 | Elkies and Klagsbrun, 'New Rank Records For Elliptic Curves Having Rational Torsion' (2020). Presents rank-record breaking elliptic curves with torsion subgroups including Z/7Z (current record rank >= 6 by Klagsbrun). | null | null | null | null | null |
sum_three_cubes_114 | **Sum of Three Cubes for $n = 114$**
**Definition:** The equation $x^3 + y^3 + z^3 = n$ asks whether an integer $n$ can be expressed as a sum of three integer cubes. After the solutions for 33 and 42 were found in 2019, only seven integers below 1000 remain unsolved: 114, 390, 627, 633, 732, 921, and 975. No solutions... | construction | number_theory | new_construction | 1 | null | https://oeis.org/A060464 | OEIS A060464: Integers that potentially can be represented as sums of three cubes. After solving 33 and 42, 114 is the smallest remaining unsolved case as of 2025. References Booker-Sutherland computations. | null | null | null | null | null |
sum_three_cubes_390 | **Sum of Three Cubes for $n = 390$**
**Definition:** The equation $x^3 + y^3 + z^3 = n$ asks whether an integer $n$ can be expressed as a sum of three integer cubes. The integer 390 is one of seven remaining unsolved cases below 1000. Since $390 \equiv 3 \pmod 9$, a solution is not ruled out by congruence conditions.
... | construction | number_theory | new_construction | 1 | null | https://arxiv.org/pdf/2007.01209 | Booker and Sutherland (2020). 'On a question of Mordell.' Lists 390 among unresolved values ≤1000 at that time, and describes very large searches for solutions (including ruling out solutions with small “min(|x|,|y|,|z|)” up to huge bounds) | null | null | null | null | null |
sum_three_cubes_627 | **Sum of Three Cubes for $n = 627$**
**Definition:** The integer 627 is one of seven remaining integers below 1000 for which no representation as a sum of three cubes is known. Since $627 \equiv 6 \pmod 9$, congruence conditions do not rule out a solution.
**Task:** Find integers $x, y, z$ such that $x^3 + y^3 + z^3 ... | construction | number_theory | new_construction | 1 | null | https://arxiv.org/abs/1903.04284 | Booker (2019). 'Cracking the problem with 33.' Lists 390 among the seven remaining unsolved cases under 1000 (114, 390, 627, 633, 732, 921, 975). No representation as sum of three cubes is known. | null | null | null | null | null |
sum_three_cubes_primitive_192 | **Primitive Sum of Three Cubes for $n = 192$**
**Definition:** While $192=4^3+4^3+4^3$ admits a non-primitive solution with $\text{gcd}(x,y,z)=4,$, no primitive solution (where $\gcd(x,y,z) = 1$) is known for $x^3 + y^3 + z^3 = 192$.
**Task:** Find integers $x, y, z$ with $\gcd(x, y, z) = 1$ such that $x^3 + y^3 + z^... | construction | number_theory | new_construction | 1 | null | https://oeis.org/A060464 | OEIS sequence on sums of three cubes; references Elsenhans & Jahnel (2009) showing 192, 375, 600 have no known primitive solutions with gcd(x,y,z)=1 | null | null | null | null | null |
mahler_x_3_y_3_1_5xy | Consider the following research problem in mathematics.
**Mahler Measure of $x^3+y^3+1-5xy$**
**Definition:** This problem concerns the logarithmic Mahler measure of the polynomial $Q_5(x, y) = x^3 + y^3 + 1 - 5xy$. This polynomial belongs to the Hesse family $Q_k(x, y) = x^3 + y^3 + 1 - kxy$, whose Mahler measures a... | constant | number_theory | ground_truth_computable | 0 | 1.5923685610864577552648762016584343966931986506568980628466025871066531426921883851477685159655913223305979340 | https://arxiv.org/abs/math/0308041 | Rogers (2010), 'Hypergeometric formulas for lattice sums and Mahler measures.' Provides a general hypergeometric formula for $Q_k(x, y)=x^3+y^3+1-kxy. | null | null | null | null | null |
c5_ising_susceptibility | Consider the following research problem in mathematics.
**Closed Form for the 5th Ising Susceptibility Integral ($C_5$)**
**Definition:** The integrals $C_n$ appear in the susceptibility expansion of the 2D Ising model and are defined as: $C_n = \frac{2^n}{n!} \int_0^\infty t K_0(t)^n dt$ where $K_0(t)$ is the modifi... | constant | lattice_models | ground_truth_computable | 2 | 0.66575980019993742831573380830706659819749638207949765953944270353122704376721234786771901508036929308584399492431185604034925933005075368056386687474090556074714047548823410663129381029978766539289878 | https://www.davidhbailey.com/dhbpapers/ising.pdf | Bailey, Borwein, Crandall, 'Integrals of the Ising class' (2006) - provides a definition for these Ising integrals and high-precision numerical results | null | null | null | null | null |
c6_ising_susceptibility | Consider the following research problem in mathematics.
**Closed Form for the 6th Ising Susceptibility Integral ($C_6$)**
**Definition:** The integrals $C_n$ appear in the susceptibility expansion of the 2D Ising model and are defined as: $C_n = \frac{2^n}{n!} \int_0^\infty t K_0(t)^n dt$ where $K_0(t)$ is the modifi... | constant | lattice_models | ground_truth_computable | 2 | 0.64863420903100707526314984345035169088977250948162799561505088718478178178800557923682516243508678874630577856026398027701536062285107772881321904645186423022491587784838301747 | https://www.davidhbailey.com/dhbpapers/ising.pdf | Bailey, Borwein, Crandall, 'Integrals of the Ising class' (2006) - provides a definition for these Ising integrals and high-precision numerical results | null | null | null | null | null |
c7_ising_susceptibility | Consider the following research problem in mathematics.
**Closed Form for the 7th Ising Susceptibility Integral ($C_7$)**
**Definition:** The integrals $C_n$ appear in the susceptibility expansion of the 2D Ising model and are defined as: $C_n = \frac{2^n}{n!} \int_0^\infty t K_0(t)^n dt$ where $K_0(t)$ is the modifi... | constant | lattice_models | ground_truth_computable | 2 | 0.63997304682795750054991340799259099278899717666159325886302862532801001076106427 | https://www.davidhbailey.com/dhbpapers/ising.pdf | Bailey, Borwein, Crandall, 'Integrals of the Ising class' (2006) - provides a definition for these Ising integrals and high-precision numerical results | null | null | null | null | null |
calabi_yau_c5 | Consider the following research problem in mathematics.
**Structural Identification of the Calabi-Yau Variety for $C_5$**
**Definition:** The Ising susceptibility integral $C_5$ is conjectured to be a period of a specific Calabi-Yau 3-fold. This structural connection suggests that $C_5$ can be represented via the geo... | constant | continuum_physics | ground_truth_computable | 2 | 9586.9411228790989677465668396217590140439479019447662973679749308496694302478578092951538171573178204361535269 | https://arxiv.org/abs/1007.0535 | Bostan et al., 'The Ising model: from elliptic curves to modular forms and Calabi-Yau equations' (2010) - Calabi-Yau differential equations emerging in Ising susceptibility analysis | null | null | null | null | null |
mzv_decomposition_c5 | Consider the following research problem in mathematics.
**Multiple Zeta Value Decomposition of $C_5$**
**Definition:** The Ising susceptibility integrals are believed to belong to the algebra of Multiple Zeta Values (MZVs). While the structure is known for small $n$, the specific weight and depth decomposition for $C... | constant | number_theory | ground_truth_computable | 2 | 0.6657598001999374283157338083070665981974963820794976595394427035312270437672123478677190150803692930858440 | https://arxiv.org/abs/0907.2557 | Blumlein, Broadhurst, Vermaseren, 'The Multiple Zeta Value Data Mine' (2009) - proven MZV reductions relevant to physics integrals including Ising-class | null | null | null | null | null |
feynman_3loop_sunrise | Consider the following research problem in mathematics.
**3-Loop Sunrise Diagram at Threshold**
**Definition:** This problem concerns the 3-loop sunrise (banana) Feynman diagram with 4 equal-mass propagators evaluated at threshold $s = 16m^2$. In the position-space Bessel representation, the integral is $B(4) = \int_... | constant | continuum_physics | ground_truth_computable | 2 | 2.27729529146683223972828877133800817650258821452965244985120378395321356945250809311211331151764131842932 | https://link.springer.com/content/pdf/10.1007/JHEP05%282021%29066.pdf | Bönisch, Fischbach, Klemm, Nega, Safari (2021). 'Analytic structure of all loop banana integrals' - Eq. (2.10) gives the D=2 Bessel representation. | null | null | null | null | null |
feynman_4loop_banana | Consider the following research problem in mathematics.
**4-Loop Banana Diagram at Threshold**
**Definition:** This problem concerns the 4-loop banana graph with equal masses at the corresponding threshold, $$B(5) = \int_0^{\infty} r \, I_0(5r) \, K_0(r)^5 \, dr,$$ where $I_0$ and $K_0$ are modified Bessel functions ... | constant | continuum_physics | ground_truth_computable | 2 | 3.5649669441225491856098202100926563331364799751675362407992703859275965557517521603709835573861024583018782717 | https://link.springer.com/content/pdf/10.1007/JHEP05%282021%29066.pdf | Bönisch, Fischbach, Klemm, Nega, Safari (2021). 'Analytic structure of all loop banana integrals' - Eq. (2.10) gives the D=2 Bessel representation. Eq. (2.10) with their notation gives a prefactor of 16, while our numeric value matches the integral without the prefactor 16 evaluated at threshold. | null | null | null | null | null |
elliptic_kernel_f2_001 | Consider the following open problem in mathematical physics.
**Elliptic-Kernel Log-Moment Constant f2(0,0,1)**
We define the complete elliptic integral of the first kind K(m) for complex parameter m by
K(m) = ∫_{0}^{π/2} dθ / sqrt(1 - m sin^2 θ),
using the principal branch of the square root and analytic continuati... | constant | continuum_physics | ground_truth_computable | 2 | 30.7476526736391709896774235351358778861783865155459326024781812950213971132375910461620684439641407962420702403407811170933205901539809821596 | https://pos.sissa.it/290/077/pdf | Several other sources reference this quantity: https://pos.sissa.it/303/073/pdf, https://arxiv.org/pdf/1704.06996, and https://arxiv.org/pdf/1910.01248. See equations 23 to 24 in the source_url paper. | null | null | null | null | null |
tracy_widom_f2_mean | Consider the following research problem in mathematics.
**Mean of the Tracy-Widom $F_2$ Distribution**
**Definition:** The Tracy-Widom distribution $F_2$ is the cumulative distribution function (CDF) of a real-valued random variable $X$ describing the fluctuations of the largest eigenvalue of GUE random matrices (aft... | constant | continuum_physics | ground_truth_computable | 2 | -1.77108680741160162612693822832370833445514095085934616781672203 | https://arxiv.org/abs/0804.2543 | Folkmar Bornemann, 'On the Numerical Evaluation of Fredholm Determinants' (2010). Math. Comp. 79(270):871-915. Provides accurate algorithms for numerical evaluation of Tracy-Widom distributions including mean (approx -1.7711) and variance for F2 (GUE). | null | null | null | null | null |
tracy_widom_f2_variance | Consider the following research problem in mathematics.
**Variance of the Tracy-Widom $F_2$ Distribution**
**Definition:** The variance of the Tracy-Widom $F_2$ distribution is: \[ \mathrm{Var}[X] = \mathbb{E}[X^2] - \mathbb{E}[X]^2 = 0.81319... \] where $X \sim F_2$ with the random-matrix limit definition and standa... | constant | continuum_physics | ground_truth_computable | 2 | 0.8131947928329 | https://arxiv.org/abs/0904.1581 | Folkmar Bornemann, 'On the Numerical Evaluation of Distributions in Random Matrix Theory' (2010). Provides algorithms to compute variance (approx 0.8132) and other moments of Tracy-Widom F2 distribution. | null | null | null | null | null |
tracy_widom_f1_mean | Consider the following research problem in mathematics.
**Mean of the Tracy-Widom $F_1$ Distribution (GOE)**
**Definition:** Let $q(s)$ be the Hastings--McLeod solution of Painlev\'e II, $q\''(s)=s q(s)+2 q(s)^3$ with $q(s)\sim\mathrm{Ai}(s)$ as $s\to+\infty$. Define
\[ F_2(s)=\exp\!\left(-\int_s^{\infty}(x-s)q(x)^2\... | constant | continuum_physics | ground_truth_computable | 2 | -1.206533574582093757882324561830899612811508928919795846796986046439531871428069093892948158498295831217412832146379216871 | https://arxiv.org/abs/0904.1581 | Bornemann, 'On the Numerical Evaluation of Distributions in Random Matrix Theory: A Review' (2009), Example 8.4.1 tabulates the mean of $F_1$ as approximately -1.2065335745820; higher-precision digits here are computed offline using Painlev\'e/Fredholm-determinant methods following Bornemann. | null | null | null | null | null |
monomer_dimer_entropy | Let \Lambda_{m,n} be the m\times n rectangular subgraph of the 2D square lattice with free boundary. A configuration is a matching: a set of disjoint dimers (edges), with all uncovered vertices treated as monomers. Assign weight z to each monomer and weight 1 to each dimer. Define the finite-volume partition function
... | constant | lattice_models | ground_truth_computable | 2 | 0.662798972834 | https://arxiv.org/abs/cond-mat/0610690 | Kong (2006) estimates the square-lattice monomer-dimer constant as h2 = 0.662798972834 (claimed 11 correct digits) and brackets it near 0.662798972831 < h2 < 0.662798972845. Butera et al. (2012, arXiv:1206.0872) summarize tight bounds 0.66279897190 ≤ h2 ≤ 0.662798972844913 and a best estimate h2 = 0.6627989727(1). | null | null | null | null | null |
hard_square_entropy | Consider the following research problem in mathematics.
**Hard Square Entropy Constant**
**Definition:** The hard square model (also called the hard-core lattice gas on $\mathbb{Z}^2$) counts independent sets on the square lattice. Let $F(m,n)$ be the number of $m \times n$ binary matrices with no two adjacent 1s (ho... | constant | lattice_models | ground_truth_computable | 2 | 1.5030480824753322643220663294755536893857810 | https://oeis.org/A085850 | OEIS A085850: Decimal expansion of hard square entropy constant kappa = 1.503048082475... References Baxter's 'Planar Lattice Gases with Nearest-Neighbour Exclusion' and Finch's 'Mathematical Constants' (2003). | null | null | null | null | null |
saw_square_lattice | Consider the following research problem in mathematics.
**Connective Constant for Square Lattice Self-Avoiding Walks**
**Definition:** A self-avoiding walk (SAW) on a lattice is a path that visits each lattice site at most once. The number of $n$-step SAWs starting from the origin on the square lattice $\mathbb{Z}^2$... | constant | lattice_models | ground_truth_computable | 3 | 2.63815853032790 | https://arxiv.org/pdf/1607.02984 | Jacobsen, Scullard, Guttmann. (2016). Provides a high-precision estimate for the growth constant for square-lattice self-avoiding walks. The best conjecture from Jacobsen-Scullard-Guttmann provide $t = \sqrt{\frac{7 + \sqrt{30261}}{26}} = 2.6381585303417408684\dots$ as their estimate, but it only matches 11 significant... | null | null | null | null | null |
saw_triangular_lattice | Consider the following research problem in mathematics.
**Connective Constant for Triangular Lattice Self-Avoiding Walks**
**Definition:** The connective constant $\mu = \lim_{n \to \infty} c_n^{1/n}$ for self-avoiding walks on the triangular lattice governs the exponential growth rate of $n$-step walks: $c_n \sim A ... | constant | lattice_models | ground_truth_computable | 3 | 4.15079722 | https://arxiv.org/abs/cond-mat/0409039 | Iwan Jensen, “Self-avoiding walks and polygons on the triangular lattice,” J. Stat. Mech. (2004) P10008. Reports the estimate as $\mu = 4.150797226(26)$. | null | null | null | null | null |
saw_simple_cubic | Consider the following research problem in mathematics.
**Connective Constant for Simple Cubic Lattice Self-Avoiding Walks**
**Definition:** The connective constant $\mu=\lim_{n \to \inf} c_n^{1/n}$ for self-avoiding walks on the three-dimensional simple cubic lattice $\mathbb{Z}^3$ has been computed via the pivot al... | constant | lattice_models | ground_truth_computable | 3 | 4.684039931 | https://arxiv.org/abs/1302.2106 | Clisby (2013) 'Calculation of the connective constant for self-avoiding walks on the simple cubic lattice'; mu = 4.684039931(27) | null | null | null | null | null |
madelung_nacl | Consider the following research problem in mathematics.
**Closed Form for the NaCl Madelung Constant**
**Definition:** The Madelung constant $M$ for a crystal structure quantifies the electrostatic energy of an ion in the lattice. For the rock salt (NaCl) structure with alternating positive and negative ions on a cub... | constant | lattice_models | ground_truth_computable | 2 | 1.7475645946331821906362120355443974034851614366247417581528 | https://oeis.org/A085469 | OEIS decimal expansion of negated Madelung constant for NaCl structure; value approximately 1.7475645946...; no closed form known (Bailey et al. 2006) | null | null | null | null | null |
madelung_cscl | Consider the following research problem in mathematics.
**Closed Form for the CsCl Madelung Constant**
**Definition:** The Madelung constant for the cesium chloride (CsCl) structure, where each ion is surrounded by 8 nearest neighbors of opposite charge in a body-centered cubic arrangement, is $M = 1.7626...$. The la... | constant | lattice_models | ground_truth_computable | 2 | 1.76267477307098839793567332063864429117052861958858528064941843772796622376934083047150945811216988908569 | https://oeis.org/A181152 | OEIS decimal expansion of the (magnitude of the) CsCl Madelung constant; OEIS describes it as 'negated' under a common sign convention, but this benchmark uses the positive magnitude $M \approx 1.7627$. | null | null | null | null | null |
madelung_zns | Consider the following research problem in mathematics.
**Closed Form for the Zincblende (ZnS) Madelung Constant**
**Definition:** The Madelung constant for the zincblende (sphalerite) structure, adopted by ZnS and many III-V semiconductors, is $M = 1.6380...$. In this structure, each ion has 4 nearest neighbors in a... | constant | lattice_models | ground_truth_computable | 2 | 1.638055053388789423750034776358619465360179663136657883957644623927706812837223137698546420043494665161 | https://oeis.org/A182566 | OEIS decimal expansion of negated Madelung constant for zincblende (sphalerite) ZnS; value 1.6380550533887894... | null | null | null | null | null |
site_percolation_square | Consider the following research problem in mathematics.
**Site Percolation Threshold on the Square Lattice**
**Definition:** Consider independent nearest-neighbor site percolation on $\mathbb{Z}^2$ (the infinite square lattice): each vertex is independently declared 'open' with probability $p$ and 'closed' with proba... | constant | lattice_models | ground_truth_computable | 2 | 0.59274605079210 | https://iopscience.iop.org/article/10.1088/1751-8113/48/45/454003/pdf | Jacobsen 2015 J. Phys. A: Math. Theor. 48 454003 'Critical points of Potts and O(N) models from eigenvalue identities in periodic Temperley-Lieb algebras'. Approximately 14 reliable digits. No closed form or conjecture known. | null | null | null | null | null |
knot_volume_6_3 | Consider the following research problem in mathematics.
**Hyperbolic Volume of the $6_3$ Knot**
**Definition:** The complement of the knot $6_3$ in the 3-sphere is a hyperbolic 3-manifold with a finite volume (approximately $5.7760...\dots$). The volume is known to be expressible as a sum of Bloch\u2013Wigner dilogar... | constant | discrete_geometry | ground_truth_computable | 3 | 5.693021091281300765112483277481222926944301733006880037850870699995476072590906707654919542407040036141224456802400770331855359928066927002673172155677 | https://katlas.org/wiki/6_3 | R.M. Kashaev's 1996 paper 'The hyperbolic volume of knots from quantum dilogarithm' (arXiv:q-alg/9601025, Lett. Math. Phys.). Establishes the fundamental connection between hyperbolic volumes of knot complements and quantum dilogarithm expressions. Provides the mathematical framework for understanding why finding expli... | null | null | null | null | null |
lattice_packing_dim10 | Consider the following optimization problem.
**Improve a 10D Lattice Packing (Λ10 Baseline)**
**Definition:** A lattice in $\mathbb{R}^{10}$ is $L = \{ z^T B : z \in \mathbb{Z}^{10}\}$ where $B$ is a $10\times 10$ basis matrix (rows are basis vectors). Let $\lambda_1(L)$ be the shortest nonzero vector length and $\op... | construction | discrete_geometry | benchmark_best_known | 2 | null | https://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/LAMBDA10.html | One can also compute this by noting that the laminated lattice Λ10 has Gram matrix determinant 768, so covolume = sqrt(768) = 16√3, shortest vector length 2, packing radius 1, and density Vol(B_10(1))/(16√3) = π^5/(1920√3) ≈ 0.09202111843130556. This is from RWTH Aachen “Catalogue of Lattices” entry for LAMBDA10. Brouw... | null | null | null | null | null |
periodic_packing_dim10 | Consider the following optimization problem.
**Improve a 10D Periodic Packing (P10c Baseline)**
**Definition:** A periodic packing is a finite union of lattice translates:
\[P = \bigcup_{i=1}^k (L + s_i),\]
where $L\subset\mathbb{R}^{10}$ is a lattice and $s_1,\dots,s_k\in\mathbb{R}^{10}$ are shift vectors (with $s_1... | construction | discrete_geometry | benchmark_best_known | 2 | null | https://ir.cwi.nl/pub/6831/6831D.pdf | Best constructs a (10,40,4) binary code; applying Construction A yields a 10D periodic packing with center density 40/1024 = 5/128 and packing density (5/128)*Vol_10(1) ≈ 0.0996157828077088. | null | null | null | null | null |
lattice_packing_dim12 | Consider the following optimization problem.
**Dense Lattice Packing in Dimension 12 ($LPD-12$)**
**Definition:** The sphere packing problem in $\mathbb{R}^{12}$. The current best known lattice is $K_{12}$ with packing density 0.0494.
**Task:** Construct a lattice in $\mathbb{R}^{12}$ with a packing density strictly... | construction | discrete_geometry | benchmark_best_known | 2 | null | https://arxiv.org/abs/math/0503446 | Nebe (2005) 'Low dimensional strongly perfect lattices I: The 12-dimensional case' - proves Coxeter-Todd lattice K12 is unique strongly perfect lattice in dimension 12 with densest known packing | null | null | null | null | null |
kissing_number_dim5 | Consider the following optimization problem.
**Kissing Number in Dimension 5**
**Definition:** The kissing number is the maximum number of unit spheres that can touch a central unit sphere. In 5 dimensions, the known bounds are $40 \le \tau_5 \le 44$. The exact value is unknown.
**Task:** Construct a valid kissing c... | construction | discrete_geometry | benchmark_best_known | 1 | null | https://arxiv.org/abs/2412.00937 | Cohn & Rajagopal (2024) 'Variations on five-dimensional sphere packings' - analyzes kissing configurations achieving the bound of 40 in dimension 5, presents fourth known construction | null | null | null | null | null |
kissing_number_dim9 | Consider the following optimization problem.
**Kissing Number in Dimension 9**
**Definition:** The kissing number in 9 dimensions has bounds $306 \le \tau_9 \le 363$. The gap is significant.
**Task:** Construct a valid kissing configuration in $\mathbb{R}^9$ with strictly more than 306 spheres.
**Current State-of-t... | construction | discrete_geometry | benchmark_best_known | 1 | null | https://arxiv.org/abs/2412.00937 | Cohn & Rajagopal (2024) 'Variations on five-dimensional sphere packings' - also constructs new kissing configuration in dimension 9 | null | null | null | null | null |
kissing_number_dim11 | Consider the following optimization problem.
**Kissing Number in Dimension 11**
**Definition:** The kissing number in 11 dimensions has bounds $593 \le \tau_{11} \le 868$.
**Task:** Construct a valid kissing configuration in $\mathbb{R}^{11}$ with strictly more than 593 spheres.
**Current State-of-the-Art:**
- Metr... | construction | discrete_geometry | benchmark_best_known | 1 | null | https://arxiv.org/abs/1507.03631 | Novikov et al. (2025) 'AlphaEvolve: A coding agent for scientific and algorithmic discovery' - Improves the lower bound to 593 | null | null | null | null | null |
kakeya_finite_field | Consider the following optimization problem.
**Smaller Kakeya Set in $\mathbb{F}_p^3$**
**Definition:** A Kakeya set in $\mathbb{F}_p^d$ contains a line in every direction. For $d=3$ and primes $p \equiv 1 \pmod 4$, the current best construction has size approx $p^3/4 + 7p^2/8$.
**Task:** Construct an explicit Kakey... | construction | discrete_geometry | benchmark_best_known | 1 | null | https://arxiv.org/abs/1609.01048 | Lund, Saraf & Wolf (2016) 'Finite field Kakeya and Nikodym sets in three dimensions' - improved lower bounds on Kakeya sets over F_q^3 | null | null | null | null | null |
spherical_9_design_s2 | Consider the following optimization problem.
**Minimal Spherical 9-Design on $S^2$**
**Definition:** A spherical 9-design on the 2-sphere ($S^2 \subset \mathbb{R}^3$) is a finite set of points such that the average of any polynomial of degree $\le 9$ over the points equals the average value over the sphere. The Delsa... | construction | discrete_geometry | benchmark_best_known | 2 | null | https://arxiv.org/abs/math/0207211 | Hardin & Sloane (1996) 'McLaren's Improved Snub Cube and Other New Spherical Designs in Three Dimensions' - provides spherical t-design constructions on S^2, including a 48-point 9-design. The DGS lower bound is 30 points. | null | null | null | null | null |
spherical_7_design_minimal | Consider the following optimization problem.
**Spherical 7-Design with Minimal Points**
**Definition:** Construct a spherical $t$-design for $t=7$ on $S^3$ (dimension 4) with the minimum possible number of points.
**Task:** Construct an explicit spherical 7-design in dimension 4 with fewer points than the current be... | construction | discrete_geometry | benchmark_best_known | 2 | null | https://www.researchgate.net/publication/4021411_Spherical_designs_in_four_dimensions | Hardin, Sloane, and Cara (2004), 'Spherical Designs in Four Dimensions', Table 1. The best known spherical 7-design on S^3 (4D) uses 48 points. The DGS lower bound is 40 points. | null | null | null | null | null |
keich_thin_triangles_128 | ### Thin-Triangle Kakeya (128 slopes): Minimize Union Area
This benchmark is a *discrete, thickened* Kakeya-type construction in the style of Schoenberg/Keich.
Fix N=128 and δ = 1/128.
For each i=0,1,...,127 you must specify a unit line segment
l_i = {(x, a_i x + b_i) : x in [0,1]} with slope a_i = i/128.
From... | construction | discrete_geometry | benchmark_best_known | 1 | null | https://arxiv.org/abs/2506.13131 | Baseline from AlphaEvolve (Google DeepMind, 2025). The AlphaEvolve triangles conv{(x_i, 0), (x_i + i/128, 0), (x_i + (i+1)/128, 1)} map exactly to our triangles conv{(0, b_i - 1/128), (0, b_i), (1, b_i + i/128)} by swapping coordinates (x, y) → (y, x) and setting b_i = x_i + i/128, an area-preserving transformation. | null | null | null | null | null |
tammes_n15 | Consider the following optimization problem.
**Tammes Problem for $n=15$**
**Definition:** The Tammes problem asks to maximize the minimum distance between any pair of $n$ points on a sphere. For $n=15$, the optimal configuration is not rigorously proven.
**Task:** Construct a configuration of 15 points on $S^2$ ach... | construction | discrete_geometry | benchmark_best_known | 1 | null | https://cohn.mit.edu/spherical-codes/ | Cohn et al., Spherical Codes database. Best known configuration for n=15 on S^2 has cosine of minimal angle 0.59260590292507377809642492233276 (minimal polynomial 13x^5 - x^4 + 6x^3 + 2x^2 - 3x - 1). Not proven optimal. | null | null | null | null | null |
heilbronn_n12 | Consider the following optimization problem.
**Heilbronn Configuration for $n=12$**
**Definition:** Place $n$ points in a unit square to maximize the minimum area of any triangle formed by three of the points. For $n=12$, the exact optimal value and configuration are unknown.
**Task:** Construct a configuration of 1... | construction | discrete_geometry | benchmark_best_known | 1 | null | https://www.combinatorics.org/ojs/index.php/eljc/article/view/v9i1r6/pdf | Baseline lower bound for the unit square Heilbronn number at n=12 from Comellas & Yebra (2002): explicit 12-point configuration with minimum triangle area ≈ 0.032599 (rounded). This is a best-known published construction, not a proven optimum. Global-optimization context: Monji, Modir, Kocuk (arXiv:2512.14505) certifie... | null | null | null | null | null |
dts_7_5_min_scope | Consider the following optimization problem.
**Minimum-Scope Difference Triangle Set (7,5)**
An (n,k)-DTS is an nx(k+1) array A with entries a[i][j] such that each row is strictly increasing and normalized:
0 = a[i][0] < a[i][1] < ... < a[i][k]
Define the set of positive within-row differences:
D = { a[i][j] - a... | construction | combinatorics | benchmark_best_known | 1 | null | https://doi.org/10.1002/jcd.22009 | Shehadeh, M., Kingsford, W., & Kschischang, F. R. (2026). 'New Difference Triangle Sets by a Field-Programmable Gate Array-Based Search Technique.' Journal of Combinatorial Designs, 34(1), 37-50. https://doi.org/10.1002/jcd.22009, Table I reports best-known upper bound m(7,5) ≤ 112. | null | null | null | null | null |
kissing_number_dim6 | Consider the following optimization problem.
**Kissing Number in Dimension 6**
**Definition:** The kissing number $\tau_6$ is the maximum number of non-overlapping unit spheres that can touch a central unit sphere in 6 dimensions. The known bounds are $72 \le \tau_6 \le 77$. The lower bound is achieved by the $E_6$ r... | construction | discrete_geometry | benchmark_best_known | 1 | null | https://arxiv.org/abs/2404.18794 | D. de Laat, N. Leijenhorst, and W. H. H. de Muinck Keizer, 'Optimality and uniqueness of the D4 root system' (2024). Proves upper bound tau_6 <= 77 via exact semidefinite programming. Lower bound of 72 from E6 root system due to A. Korkine and G. Zolotareff (1873). | null | null | null | null | null |
knot_volume_7_2 | Consider the following research problem in mathematics.
**Hyperbolic Volume of the $7_2$ Knot**
**Definition:** The complement of the knot $7_2$ in the 3-sphere is a hyperbolic 3-manifold with a finite volume (approximately $3.3317...\dots$). The volume is known to be expressible as a sum of Bloch\u2013Wigner dilogar... | constant | discrete_geometry | ground_truth_computable | 2 | 3.3317442316411148239145691080297127955469579091860049212216044555987413728423665155788622603487862838857647164 | https://katlas.org/wiki/7_2 | Knot Atlas 7_2 page gives 3.33174, and Wakelin (2023)'s 'A hyperbolic perspective on the Dehn surgery characterisation problem' lists 3.3317442316. | null | null | null | null | null |
diff_basis_upper | Consider the following optimization problem.\n\n**Improve Upper Bound on Difference Basis Constant**\n\n**Definition:** For any natural number $n$, let $\Delta(n)$ denote the size of the smallest set $B$ of integers such that every natural number $k \in \{1,\dots,n\}$ is expressible as a difference of two elements of $... | construction | combinatorics | benchmark_best_known | 1 | null | https://arxiv.org/abs/2103.15850 | Balogh, Furedi & Roy (2021) 'An upper bound on the size of Sidon sets' - proves maximum Sidon set size is at most sqrt(n) + 0.998n^(1/4), directly related to difference basis bounds | null | null | null | null | null |
diff_basis_optimal_10000 | Consider the following optimization problem.
**Restricted Difference Basis (Sparse Ruler) for n=10000**
**Definition:** A set B ⊆ {0,1,...,9999} is a restricted difference basis for n=10000 if every integer d in {1,...,9999} can be written as |a-b| for some a,b ∈ B.
**Task:** Construct an explicit B with |B| smaller... | construction | combinatorics | benchmark_best_known | 2 | null | https://oeis.org/A046693 | Sparse ruler / minimal complete ruler context; excess discussion also in OEIS A326499 and Wolfram references. | null | null | null | null | null |
vdw_W72_ap7 | Consider the following optimization problem.
**2-Coloring with No Monochromatic 7-Term Arithmetic Progression**
**Definition (certificate format):** A candidate solution is a list `c[0..n-1]` with entries in {0,1}, interpreted as a 2-coloring of the integers {0,1,...,n-1}.
A **7-term arithmetic progression** in {0,.... | construction | combinatorics | benchmark_best_known | 1 | null | https://arxiv.org/abs/1603.03301 | Monroe (2019) compiles lower bounds from explicit constructions; reports W(7,2) > 3703 (baseline). | null | null | null | null | null |
general_diff_basis_algo | Consider the following optimization problem.
**General Algorithm for Difference Bases**
**Definition:** Construct a deterministic algorithm or formula that generates difference bases for any range $n$ with size close to the theoretical lower bound, replacing sporadic search-based results.
**Task:** Find a universal ... | formula_discovery | combinatorics | benchmark_best_known | 2 | null | https://en.wikipedia.org/wiki/Difference_set | Wikipedia article on difference sets. Singer (1938) proved perfect difference sets exist mod (q^2+q+1) when q is prime power. General algorithmic construction for difference bases not found in verified sources. | null | null | null | null | null |
ramsey_asymptotic | Consider the following optimization problem.
**Asymptotic Upper Bound Constant for Diagonal Ramsey Numbers**
**Definition:** The diagonal Ramsey numbers satisfy classical bounds of the form $2^{n/2} \lesssim R(n,n) \lesssim 4^n$.
**Goal:** Improve the best known exponential **upper bound base** $c$ in $R(k,k) \le c^... | construction | combinatorics | benchmark_best_known | 1 | null | https://arxiv.org/abs/2407.19026 | Gupta, Ndiaye, Norin, Wei (2024) 'Optimizing the CGMS upper bound on Ramsey numbers'. Baseline c = 4·exp(-0.14/e) = 3.7992… from Theorem 1. Arbitrary-degree polynomial correction p(λ) = a1·λ + … + ad·λ^d (no constant term). Split validator with rigorous interval arithmetic: on (0, 10^-3] it uses fixed analytic M(λ)=λe^... | null | null | null | null | null |
crossing_number_kn | Consider the following optimization problem.
**Rectilinear Crossing Number $\overline{\mathrm{cr}}(K_n)$ (Straight-Line Drawings)**
**Definition:** A *rectilinear drawing* of the complete graph $K_n$ is obtained by placing $n$ points in the plane in general position (no three collinear) and drawing each edge as the s... | formula_discovery | combinatorics | benchmark_best_known | 2 | null | https://www.sciencedirect.com/science/article/pii/S0166218X09003734 | Baseline is an explicit published rectilinear drawing of K_99 with 1404552 crossings (Ábrego et al. (2010). 'How to construct a drawing of K_99 with 1404552 crossings'). | null | null | null | null | null |
kcore_threshold_c3 | Consider the following research problem in mathematics.\n\n**3-Core Emergence Threshold Constant in G(n, c/n)**\n\n**Definition:** Let G(n,p) be the Erd\u0151s\u2013R\u00e9nyi random graph. The 3-core of a graph is its largest induced subgraph with minimum degree at least 3. There exists a sharp threshold at p = c_3/n ... | constant | mathematical_constants | ground_truth_computable | 2 | 3.35091887151167277315681440498709807619062659090935600532811122807017749104521799074756363155452191680828276744801164941414782014826348832037202660117572096525917495822458142281358203481658555212080736970109895 | https://cs.nyu.edu/~spencer/papers/k-core.pdf | Pittel, Spencer, Wormald (1996) define the k-core threshold for G(n,m) as c_k = min_{\u03bb>0} \u03bb/\u03c0_k(\u03bb), with \u03c0_k(\u03bb)=P(Poisson(\u03bb)\u2265 k-1), and state c_3\u22483.35. Later work quotes the more precise value qc\u22483.35091887 for k=3 (e.g. Baxter et al., Phys. Rev. X 5, 031017 (2015)). | null | null | null | null | null |
turan_petersen | Consider the following optimization problem.
**Petersen Graph Tur\'an Problem**
**Definition:** Find the maximum number of edges in a graph on $n=50$ vertices that does not contain the Petersen graph as a subgraph.
**Task:** Construct an explicit graph on 50 vertices with no Petersen subgraph achieving a higher edge... | construction | combinatorics | benchmark_best_known | 1 | null | https://arxiv.org/pdf/2508.12070 | Fang, Lin, Zhai (2025), 'The spectral Turan problem: Characterizing spectral-consistent graphs.' For n=50: T_2(48)=K_{24,24} has 24*24=576 edges, and joining two universal vertices adds 2*48=96 edges, plus the edge between them adds 1. Total is 576+96+1=673. | null | null | null | null | null |
ramsey_coloring_k5 | Consider the following optimization problem.
**2-Coloring of $K_n$ Without Monochromatic $K_5$**
**Definition:** The Ramsey number $R(5,5)$ is unknown (bounds: 43-48). Constructing a coloring for a specific $n$ (e.g., $n=43$) without a monochromatic $K_5$ would improve the lower bound.
**Task:** Construct an explici... | construction | combinatorics | benchmark_best_known | 1 | null | https://arxiv.org/abs/2212.12630 | Study of Exoo's lower bound for R(5,5) - analyzes the 2-coloring of K_42 with no monochromatic K_5, establishing R(5,5) >= 43. Current bounds: 43 <= R(5,5) <= 46. | null | null | null | null | null |
merit_factor_6_5 | Consider the following research problem in mathematics.
**Polynomial with Maximum Merit Factor**
**Definition:** The merit factor of a binary polynomial $p(z) = \sum_{i=0}^{n-1} a_i z^i$ with coefficients $a_i \in \{-1, 1\}$ is:
$$F(p) = \frac{n^2}{2 \sum_{k=1}^{n-1} C_k^2}$$
where $C_k = \sum_{i=0}^{n-1-k} a_i a_{... | construction | coding_theory | benchmark_best_known | 2 | null | https://ieeexplore.ieee.org/document/8247176/ | Brest, J., & Bošković, B. (2018). A heuristic algorithm for a low autocorrelation binary sequence problem with odd length and high merit factor. IEEE Access, 6, 4127-4134. | null | null | null | null | null |
parametric_spherical_codes | Consider the following optimization problem.
**Parametric Family of Spherical Codes**
**Definition:** Discover a parametric family of spherical codes (depending on dimension $d$ and size $N$) that produces configurations with high minimum distance, generalizing isolated optimal codes.
**Task:** Find a universal form... | formula_discovery | coding_theory | benchmark_best_known | 2 | null | https://arxiv.org/abs/2008.10728 | Miyamoto, Costa, Sa Earp, 'Constructive Spherical Codes by Hopf Foliations' (2021). Parametric family construction in dimensions 2^k using Hopf foliations. O(n) storage, O(n log n) encoding. Published in IEEE Trans. Inf. Theory 67(12):7925-7939. | null | null | null | null | null |
bklc_68_15 | Consider the following optimization problem.
**Improve Minimum Distance of a Binary Linear [68,15] Code**
**Definition:** A binary linear [n,k,d] code is a k-dimensional subspace of F_2^n. Its minimum distance d is the minimum Hamming weight among all nonzero codewords.
**Task:** Construct an explicit binary linear ... | construction | coding_theory | benchmark_best_known | 2 | null | https://www.codetables.de/BKLC/BKLC.php?k=15&n=68&q=2 | Grassl BKLC lists lower bound 24 and upper bound 26 for binary linear codes with (n,k)=(68,15), so 24 is best-known but not proven optimal. | null | null | null | null | null |
covering_C13_k7_t4 | Consider the following optimization problem.
**Covering Design $C(13,7,4)$ With Fewer Blocks**
**Definition:** A candidate solution is a list of blocks (each block is a 7-element subset of {0,1,...,12}). The solution is valid if every 4-element subset of {0,1,...,12} is contained in at least one block.
**Task:** Out... | construction | coding_theory | benchmark_best_known | 2 | null | https://ljcr.dmgordon.org/cover/show_cover.php?k=7&t=4&v=13 | Baseline uses LJCR explicit cover for C(13,7,4), currently giving 28 ≤ C(13,7,4) ≤ 30. | null | null | null | null | null |
A21_10_binary_code | Consider the following optimization problem.
**Binary Code A(21,10)**
**Definition:** Let A(n,d) be the maximum possible size of a binary code C \subseteq {0,1}^n such that the Hamming distance between any two distinct codewords is at least d. In this problem, n=21 and d=10.
**Task:** Construct an explicit binary co... | construction | coding_theory | benchmark_best_known | 3 | null | https://aeb.win.tue.nl/codes/binary-1.html | Lower bound A(21,10) >= 42 attributed to M.K. Kaikkonen (IEEE Trans. Inf. Theory 35 (1989) p. 1344). Upper bound A(21,10) <= 47 given by Gijswijt-Mittelmann-Schrijver via semidefinite programming. | null | null | null | null | null |
cwcode_29_8_5 | Consider the following optimization problem.\n\n**Constant-Weight Code A(29,8,5): Pack Pairs by Quintuples**\n\n**Definition:** Let A(n,d,w) be the maximum size of a binary constant-weight code of length n, weight w, and minimum Hamming distance at least d. Here n=29, w=5, d=8. Equivalently, represent each codeword as ... | construction | coding_theory | benchmark_best_known | 2 | null | https://aeb.win.tue.nl/codes/Andw.html | Brouwer's table lists A(29,8,5) in the A(n,8,5) section as 36^{Bl}-39 and cites Bluskov (ENDM 65 (2018), 31-36) for the lower bound 36. | null | null | null | null | null |
End of preview. Expand in Data Studio
Citation
@article{wang2026horizonmathmeasuringaiprogress,
title={HorizonMath: Measuring AI Progress Toward Mathematical Discovery with Automatic Verification},
author={Erik Y. Wang and Sumeet Motwani and James V. Roggeveen and Eliot Hodges and Dulhan Jayalath and Charles London and Kalyan Ramakrishnan and Flaviu Cipcigan and Philip Torr and Alessandro Abate},
year={2026},
eprint={2603.15617},
archivePrefix={arXiv},
primaryClass={cs.LG},
url={https://arxiv.org/abs/2603.15617},
}
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