Datasets:

ulamai
/

SOTA-Math

difficulty stringclasses 2 values	domain stringlengths 17 49	expected_output stringclasses 3 values	inspiration stringlengths 33 216	license stringclasses 1 value	milestones listlengths 0 5	problem_id stringlengths 9 18	prompt stringlengths 472 2.76k	quality_signals listlengths 3 3	rationale stringlengths 119 1.56k	research_status stringclasses 3 values	rl_ready bool 2 classes	schema_version stringclasses 1 value	source_record_index int64 1 149	stream stringclasses 3 values	task_type stringclasses 3 values	title stringlengths 33 73	verification stringlengths 69 134
frontier_research	Additive combinatorics	A rigorous proof, disproof, or mathematically substantive partial result, with explicit hypotheses, gap control, and a current literature boundary.	Erdős Problem #1: if an n-element set A contained in {1,...,N} has all subset sums distinct, must N be bounded below by a positive constant times 2^n?	MIT	[]	erdos_001	Bounded-defect rigidity for distinct subset sums. Let A={a_1<...<a_n} be positive integers such that all 2^n subset sums are distinct, and define D_2(A) = sum_{i=1}^n a_i^2 - (4^n-1)/3. For every fixed integer D>=0, does there exist r=r(D) such that, whenever D_2(A)<=D, one has a_i=2^{i-1} for every i>r? More preci...	[ "sharp equality case", "elementary variance lower bound", "finite-defect inverse-theorem target" ]	It studies the same dissociated or distinct-subset-sum condition as #1, but replaces the extremal maximum element by a quadratic energy and asks for a finite-defect inverse theorem. There is an exact elementary lower bound behind the normalization. If Z is the sum of a uniformly random subset of A, then Var(Z)=(1/4)sum...	candidate_open_problem_requires_expert_signoff	false	1.0.0	1	erdos_variant	prove_or_refute	Bounded-defect rigidity for distinct subset sums	Specialist mathematical review; exact computations must be reproducible.
frontier_research	Extremal set theory	A rigorous proof, disproof, or mathematically substantive partial result, with explicit hypotheses, gap control, and a current literature boundary.	Erdős Problem #20, the sunflower conjecture: for each fixed k, is the threshold f(n,k) for a k-sunflower in an n-uniform family at most c_k^n?	MIT	[]	erdos_003	Profile-balanced sunflower threshold. Fix integers k>=3 and r>=2. Color the ground set of an n-uniform family with r colors. Call a k-sunflower S_1,...,S_k with core C profile-balanced if the r-vectors ( \|(S_i minus C) intersect color j\| )_{j=1}^r are the same for all i. Let f_bal(n,k,r) be the least M such that ever...	[ "classical sunflower connection", "rigorous polynomial-factor baseline", "clean asymptotic target" ]	It retains the same sunflower configuration but imposes an adversarially colored, equitable-petal condition. The question is not about the exponential growth rate alone, but whether the extra requirement costs asymptotically nothing even at the sharp threshold. There are immediate bounds. Monochromatically coloring the...	candidate_open_problem_requires_expert_signoff	false	1.0.0	3	erdos_variant	prove_or_refute	Profile-balanced sunflower threshold	Specialist mathematical review; exact computations must be reproducible.
frontier_research	Analytic and combinatorial number theory	A rigorous proof, disproof, or mathematically substantive partial result, with explicit hypotheses, gap control, and a current literature boundary.	Erdős Problem #208, which asks for sharp upper bounds on gaps between consecutive squarefree numbers, including the conjectural scale (π²/6)·log x/log log x.	MIT	[]	erdos_025	Second-order certificate complexity of a squarefree-free interval. Let κ(h) be the smallest cardinality of a set Q of primes for which there exists an integer M such that every one of M+1,M+2,…,M+h is divisible by q² for at least one q∈Q. Thus Q is a square-divisor certificate that the whole interval contains no squa...	[ "proved first-order term", "explicit conjectural second-order constant", "certificate-complexity interpretation" ]	Similarity: a gap between consecutive squarefree numbers is exactly an interval in which every integer has a square prime divisor. Erdős #208 studies the length of such an interval at a given location; the new problem studies the minimum number of distinct prime-square obstructions needed to manufacture an interval of ...	candidate_open_problem_requires_expert_signoff	false	1.0.0	25	erdos_variant	prove_or_refute	Second-order certificate complexity of a squarefree-free interval	Specialist mathematical review; exact computations must be reproducible.
frontier_research	Discrete geometry	A rigorous proof, disproof, or mathematically substantive partial result, with explicit hypotheses, gap control, and a current literature boundary.	Erdős Problem #604: must every n-point set in the plane contain at least one point from which there are n^{1-o(1)} distinct distances, perhaps as many as a constant multiple of n/sqrt(log n)?	MIT	[]	erdos_075	Median pinned-distance conjecture. Let A be a set of n distinct points in the Euclidean plane, and for x in A write d_A(x) = \|{\|\|x-y\|\| : y in A, y != x}\|. Define the upper median pinned-distance count by q(A) = max{D : at least ceil(n/2) points x in A satisfy d_A(x) >= D}, and define q(n) = min_{\|A\|=n} q(A). Is q(...	[ "robust pinned-distance formulation", "lattice upper-bound model", "incidence-geometric proof route" ]	It uses exactly the pinned-distance statistic from #604, but replaces one exceptional good pin by a positive proportion of good pins. Thus it is a distributional or robust version rather than a change of exponent. The m by m integer grid, with n=m^2, gives the expected upper scale for every pin: every squared distance ...	candidate_open_problem_requires_expert_signoff	false	1.0.0	75	erdos_variant	prove_or_refute	Median pinned-distance conjecture	Specialist mathematical review; exact computations must be reproducible.
frontier_research	Harmonic analysis and polynomial inequalities	A rigorous proof, disproof, or mathematically substantive partial result, with explicit hypotheses, gap control, and a current literature boundary.	Erdős Problem #1150, which asks whether there is an absolute c>0 such that every sufficiently high-degree polynomial with coefficients in {−1,1} has max_{\|z\|=1}\|P(z)\|>(1+c)√n. Parseval gives only the baseline √(n+1).	MIT	[]	erdos_149	RMS-superlevel concentration for Littlewood polynomials. Fix 0<η<1. For a Littlewood polynomial P(z)=∑_{j=0}^n ε_j z^j, ε_j∈{−1,1}, define μ_η(P)= (1/2π) · meas{θ∈[0,2π] : \|P(e^{iθ})\| ≥ (1−η)√(n+1)} and μ_η(n)=min_P μ_η(P), where the minimum is over all degree-n Littlewood polynomials. Determine the order of μ_...	[ "Parseval-scale formulation", "rigorous exponent gap", "explicit Dirichlet-kernel construction" ]	Similarity: both problems compare the size of a Littlewood polynomial on the unit circle with its Parseval or root-mean-square scale √(n+1). Erdős #1150 asks for one point with a fixed excess above that scale. The new problem asks how much of the circle must remain near the RMS scale, even when the polynomial is allowe...	candidate_open_problem_requires_expert_signoff	false	1.0.0	149	erdos_variant	prove_or_refute	RMS-superlevel concentration for Littlewood polynomials	Specialist mathematical review; exact computations must be reproducible.
frontier_research	Enumerative algebraic geometry	claim_status: proved, disproved, partial, or inconclusive a precise main result with hypotheses and quantifiers a complete proof or reproducible computation for every claimed milestone an adversarial check or counterexample search an explicit list of unresolved gaps and dependencies a literature boundary separating kno...	AIM-AG research-conjecture stream	MIT	[ "m1 (10%): Verify that the top Chern number on Gr(4,9) equals 321489. [exact_symbolic]", "m2 (25%): For the displayed normal jets, prove that the 20-by-20 multiplication map has rank 19, identify its kernel, and verify a nonzero quadratic obstruction. [exact_linear_algebra_and_CAS]", "m3 (20%): Prove smoothness...	aim_ag_001	There exists a smooth complex cubic sevenfold X in P^8 whose Fano scheme F_3(X) is finite of length 321489, reduced except at exactly one 3-plane Lambda, where the completed local ring is C[[t]]/(t^2). Consequently the geometric monodromy of the 321489 three-planes on a general cubic sevenfold is the full symmetric gro...	[ "exact enumerative target", "explicit local algebra", "clear monodromy consequence" ]	A concrete high-degree monodromy problem with an exact local model, a symbolic enumerative check, and a sharply isolated global transversality step.	candidate_open_problem	true	1.0.0	1	aim_ag_rl	prove_or_refute_with_milestone_credit	A single simple branch among 321,489 planes	High for the enumerative and local-algebra milestones; specialist review for global transversality.
frontier_research	Hodge theory and Fano geometry	claim_status: proved, disproved, partial, or inconclusive a precise main result with hypotheses and quantifiers a complete proof or reproducible computation for every claimed milestone an adversarial check or counterexample search an explicit list of unresolved gaps and dependencies a literature boundary separating kno...	AIM-AG research-conjecture stream	MIT	[ "m1 (20%): Prove irreducibility and codimension d+1-h of the line-incidence component for a fixed line-Hilbert component H. [formal_algebraic_geometry_review]", "m2 (15%): Verify line-regularity and the expected dimension h=iota in at least one nontrivial Fano family. [symbolic_or_literature_verified_computation]...	aim_ag_002	Let Y be a general line-regular Picard-rank-one smooth complex Fano threefold with Pic(Y)=Z[A], A a very ample primitive generator, and -K_Y=iota A. For all sufficiently large d, every component of the Noether-Lefschetz locus of smooth surfaces in \|dA\| has codimension at least d-iota+1, and equality occurs exactly for ...	[ "sharp codimension target", "proved incidence component", "family-by-family computational route" ]	A sharp extremal-classification problem combining incidence geometry with equality-sensitive Hodge-theoretic estimates.	candidate_open_problem	true	1.0.0	2	aim_ag_rl	prove_or_refute_with_milestone_credit	Lines as the largest Noether-Lefschetz loci on Fano threefolds	Moderate: incidence and family-specific algebra are checkable; the universal equality classification requires specialist proof review.
frontier_research	Commutative algebra and projective geometry	claim_status: proved, disproved, partial, or inconclusive a precise main result with hypotheses and quantifiers a complete proof or reproducible computation for every claimed milestone an adversarial check or counterexample search an explicit list of unresolved gaps and dependencies a literature boundary separating kno...	AIM-AG research-conjecture stream	MIT	[ "m1 (20%): Prove the Mayer-Vietoris short exact sequence, the regularity bound, and the cokernel description of the nonlinear Tor strand. [formal_commutative_algebra_review]", "m2 (20%): Reproduce the benchmark of two quadric surfaces meeting in a conic and its unique beta_(2,4)=1. [Macaulay2_or_exact_Betti_verif...	aim_ag_003	Fix a nonempty irreducible characteristic-zero parameter family of clean tree arrangements X=union X_v in projective space, with fixed tree, Hilbert polynomials, span dimensions, and incidence data, such that every component X_v and every edge overlap D_e is a variety of minimal degree in its span. For a general member...	[ "proved Tor reduction", "explicit Betti prediction", "natural computer-algebra verifier" ]	A highly verifiable syzygy problem with an exact reduction, explicit Betti-table benchmarks, and determinantal rank tests.	candidate_open_problem	true	1.0.0	3	aim_ag_rl	prove_or_refute_with_milestone_credit	Generic maximal-rank edge maps for tree-glued varieties of minimal degree	High for reductions and bounded examples; specialist review for the generic dominance theorem.
frontier_research	Toric geometry and numerical semigroups	claim_status: proved, disproved, partial, or inconclusive a precise main result with hypotheses and quantifiers a complete proof or reproducible computation for every claimed milestone an adversarial check or counterexample search an explicit list of unresolved gaps and dependencies a literature boundary separating kno...	AIM-AG research-conjecture stream	MIT	[ "m1 (25%): Prove that free-map classes are exactly the integral curve classes nonnegative on all effective torus-invariant divisors, under the stated multiple-cover convention. [formal_toric_geometry_review]", "m2 (20%): Implement a reproducible pipeline from fan data to the Hilbert basis, normalized degree semig...	aim_ag_004	Let X be a smooth projective toric Fano variety of dimension n over an algebraically closed characteristic-zero field. Let M_X be the monoid of integral numerical curve classes nonnegative on every effective divisor, let g_X be the gcd of the positive anticanonical degrees -K_X.beta for beta in M_X, and normalize those...	[ "dimension-only quantitative target", "finite threefold benchmark", "deterministic fan-to-semigroup pipeline" ]	A finite fan-combinatorics program that turns a frontier existence bound into exact Hilbert-basis and numerical-semigroup computations.	candidate_open_problem	true	1.0.0	4	aim_ag_rl	prove_or_refute_with_milestone_credit	A conductor bound for free anticanonical degrees on toric Fano varieties	High for fixed dimensions and classified fan lists; specialist review for the uniform dimension bound.
frontier_research	Arithmetic monodromy and Picard-Lefschetz theory	claim_status: proved, disproved, partial, or inconclusive a precise main result with hypotheses and quantifiers a complete proof or reproducible computation for every claimed milestone an adversarial check or counterexample search an explicit list of unresolved gaps and dependencies a literature boundary separating kno...	AIM-AG research-conjecture stream	MIT	[ "m1 (15%): Reconstruct the finite-index implication from a complete integral vanishing lattice in the symplectic case. [formal_lattice_theory_review]", "m2 (25%): Prove integral generation and saturation of the vanishing lattice for one new ambient family. [expert_proof_review]", "m3 (30%): Realize the pairing-...	aim_ag_005	Let Z be a smooth simply connected complex projective variety of dimension n+1 at least 2 and A an ample line bundle. For d sufficiently large, let U_d be the smooth-divisor locus in \|A^d\| and let Lambda_d be the saturated orthogonal complement of the ambient middle cohomology inside the torsion-free H^n of a smooth di...	[ "rigorous finite-index reduction", "clear thinness adversary", "explicit odd/even lattice split" ]	A long-horizon proof task where integral lattice completeness, geometric vanishing cycles, and arithmetic-group criteria can be graded separately.	candidate_open_problem	true	1.0.0	5	aim_ag_rl	prove_or_refute_with_milestone_credit	Arithmeticity of vanishing cohomology in high-power linear systems	Moderate in explicit families; specialist review is required for the full arbitrary-ambient theorem.
frontier_research	Algebraic topology and toric geometry	claim_status: proved, disproved, partial, or inconclusive a precise main result with hypotheses and quantifiers a complete proof or reproducible computation for every claimed milestone an adversarial check or counterexample search an explicit list of unresolved gaps and dependencies a literature boundary separating kno...	AIM-AG research-conjecture stream	MIT	[ "m1 (20%): Construct the based Cox-coordinate discriminant resolution over the relevant Picard or Jacobian base and verify its augmentation. [formal_topology_and_geometry_review]", "m2 (25%): Control orientations, sign local systems, and the first nontrivial integral differentials associated with overlapping prim...	aim_ag_006	Let C be a fixed smooth projective complex curve of genus g and X a smooth projective toric variety. For every i there is B(i,g,Sigma) such that, if a curve class beta satisfies beta.D_rho >= B for every invariant prime divisor, then the inclusion of the based algebraic mapping space Mor^*_beta(C,X) into the correspond...	[ "classical boundary cases", "explicit low-degree torsion test", "spectral-sequence milestones" ]	A synthesis task connecting classical mapping-space stability to integral configuration-space and Cox-coordinate calculations.	candidate_open_problem	true	1.0.0	6	aim_ag_rl	prove_or_refute_with_milestone_credit	An integral Segal theorem for toric targets	Moderate: low-degree spectral-sequence pages and benchmark targets are computable; full integral stability needs specialist review.
frontier_research	Algebraic cycles and motives	claim_status: proved, disproved, partial, or inconclusive a precise main result with hypotheses and quantifiers a complete proof or reproducible computation for every claimed milestone an adversarial check or counterexample search an explicit list of unresolved gaps and dependencies a literature boundary separating kno...	AIM-AG research-conjecture stream	MIT	[ "m1 (15%): Prove the strict-filtration nilpotence lemma and state precisely the faithfulness and filtration hypotheses. [formal_algebra_review]", "m2 (25%): Compute or sharply bound the geometric kernel ideal for conics, Severi-Brauer surfaces, or another explicit low-dimensional homogeneous variety. [exact_corre...	aim_ag_007	Let X be a d-dimensional projective homogeneous variety under a semisimple algebraic group over a field k, and let p be a prime. In Chow motives with F_p coefficients, the ideal I_X,p of endomorphisms of M(X) that vanish after base change to an algebraic closure satisfies I_X,p^((d+1)!)=0. Definitions: I_X,p is the ke...	[ "simple universal bound", "proved filtration lemma", "explicit Severi-Brauer benchmark" ]	A quantitative motivic nilpotence problem with a clean universal exponent, low-dimensional test cases, and an abstract filtration lemma.	candidate_open_problem	true	1.0.0	7	aim_ag_rl	prove_or_refute_with_milestone_credit	A factorial bound for Rost nilpotence	Moderate for low-dimensional motives and abstract filtration lemmas; specialist review for the uniform factorial estimate.
frontier_research	Logarithmic enumerative geometry	claim_status: proved, disproved, partial, or inconclusive a precise main result with hypotheses and quantifiers a complete proof or reproducible computation for every claimed milestone an adversarial check or counterexample search an explicit list of unresolved gaps and dependencies a literature boundary separating kno...	AIM-AG research-conjecture stream	MIT	[ "m1 (25%): Define the labelled tree-type stacks, rigidification, matching lattice, deck action, and source and target of every virtual K-class without ambiguity. [expert_definitional_review]", "m2 (20%): Prove the one-edge contact-m regular-representation formula over an etale trivialization. [formal_local_log_ge...	aim_ag_008	For a labelled rigid genus-zero tropical type tau in a projective log-smooth simple-normal-crossings degeneration over characteristic zero, let B_tau be the derived fiber product of the vertex stable-map moduli along evaluation diagonals, let M_tau be the moduli of basic logarithmic maps of that type, and let mu_tau fo...	[ "character-level refinement", "six-character benchmark", "explicit rigidification checks" ]	A representation-valued refinement of logarithmic degeneration formulas with exact finite-group benchmarks and stack-theoretic failure modes.	candidate_open_problem	true	1.0.0	8	aim_ag_rl	prove_or_refute_with_milestone_credit	A character-valued logarithmic gluing formula	Moderate for local labelled tree types; specialist review for global stack descent and obstruction-theory compatibility.
frontier_research	Cyclotomic homotopy theory and K3 geometry	claim_status: proved, disproved, partial, or inconclusive a precise main result with hypotheses and quantifiers a complete proof or reproducible computation for every claimed milestone an adversarial check or counterexample search an explicit list of unresolved gaps and dependencies a literature boundary separating kno...	AIM-AG research-conjecture stream	MIT	[ "m1 (15%): Reconstruct the fiberwise identification of cyclotomic degree -2 with H^2(W O_X) and the finite-height Frobenius criterion. [formal_literature_and_proof_review]", "m2 (25%): Construct relative finite V-quotients with base change on an explicit smooth chart of polarized K3 moduli. [derived_geometry_expe...	aim_ag_009	Let p be odd, let p not divide 2d, and let M be the moduli stack of primitively polarized K3 surfaces of degree 2d over F_p. The degree -2 cyclotomic homotopy object of THH globalizes to a base-change-compatible V-complete Cartier sheaf for the universal K3 family. For h=1,...,10, its successive V-adic Frobenius obstru...	[ "known fiberwise detector", "finite-height local tests", "decisive supersingular multiplicity" ]	A relative comparison problem linking cyclotomic homotopy, formal Brauer height, higher Hasse sections, and scheme-theoretic multiplicity.	candidate_open_problem	true	1.0.0	9	aim_ag_rl	prove_or_refute_with_milestone_credit	A cyclotomic Hasse tower for K3 moduli	Moderate for fiberwise and finite-Witt milestones; specialist review for relative cyclotomic descent and terminal multiplicity.
frontier_research	p-adic homotopy theory and logarithmic geometry	claim_status: proved, disproved, partial, or inconclusive a precise main result with hypotheses and quantifiers a complete proof or reproducible computation for every claimed milestone an adversarial check or counterexample search an explicit list of unresolved gaps and dependencies a literature boundary separating kno...	AIM-AG research-conjecture stream	MIT	[ "m1 (15%): Derive the relative logarithmic differential module B/eB generated by dlog(t) for the Kummer chart and verify tame vanishing. [formal_log_algebra_review]", "m2 (25%): Prove Galois descent on every complete log-prismatic or Hodge-Tate associated graded in a tame control family. [filtered_derived_review]...	aim_ag_010	Let L/K be a finite Galois extension of p-adic local fields with group G, and give their valuation rings the divisorial log structures. The p-completed log-TP descent map TP^log(O_K;Z_p) -> TP^log(O_L;Z_p)^{hG} is an equivalence if and only if L/K is tamely ramified. For a totally ramified Kummer extension pi_K=pi_L^e ...	[ "clean tame/wild criterion", "proved log-differential calculation", "explicit first wild class" ]	An if-and-only-if ramification detector with a computable logarithmic differential, filtered descent milestones, and a concrete wild Kummer test.	candidate_open_problem	true	1.0.0	10	aim_ag_rl	prove_or_refute_with_milestone_credit	Logarithmic TP as a tame-ramification detector	High for logarithmic differentials and associated-graded tests; specialist review for Tate-totalization interchange and permanence.
frontier_computational_search	Computational number theory	An exact witness with every requested invariant, plus an independently checkable certificate for feasibility and global optimality.	C114 — Giuga's primality conjecture	MIT	[]	counterexample_114	Let P_30 be the set of the first 30 odd primes. Choose a 9-element subset S of P_30 and put n=product_{p in S} p. For p in S define G_p=1 when n/p is congruent to 1 modulo p, and C_p=1 when n is congruent to 1 modulo p-1. Put G(n)=sum_p G_p, C(n)=sum_p C_p, and B(n)=sum_p G_p C_p. Find S maximizing the lexicographic sc...	[ "near-counterexample score", "small exact modular checks", "pseudo-Boolean optimality certificate" ]	For a squarefree composite n, the equations p \| n/p-1 for every p \| n are the local Giuga conditions, while p-1 \| n-1 for every p \| n are Korselt's local conditions for being Carmichael. A composite counterexample to Giuga's primality criterion would satisfy both systems at every prime factor. The search is restricted ...	finite_search_task_existence_or_optimum_not_preasserted	false	1.0.0	114	counterexample_variant	find_and_certify_optimal_object	Closest bounded Giuga–Carmichael local fit	Exact witness checks plus a machine-checkable exhaustive certificate.
frontier_computational_search	Structural and computational graph theory	An exact witness with every requested invariant, plus an independently checkable certificate for feasibility and global optimality.	C128 — Hajós's conjecture for t=5 and t=6	MIT	[]	counterexample_128	Let G range over simple graphs on thirteen vertices that are 5-critical, meaning chi(G)=5 while chi(G−e)=4 for every edge e, and require omega(G)=4. Let tau_5(G) be the number of edge subsets whose edge-subgraph is homeomorphic to K_5, with each subdivided K_5 counted once as an edge set. Find G minimizing tau_5(G). Su...	[ "first open Hajós chromatic boundary", "zero optimum would be a genuine counterexample", "DRAT and isomorph-free enumeration certificates" ]	This searches for the most subdivision-poor positive instance at the first open chromatic boundary, rather than directly demanding a counterexample. If the optimum were zero, the winner would be a genuine 5-chromatic graph with no K_5 subdivision. Otherwise the exact minimum measures how deeply the required subdivision...	finite_search_task_existence_or_optimum_not_preasserted	false	1.0.0	128	counterexample_variant	find_and_certify_optimal_object	Subdivision-poor five-critical graph	Exact witness checks plus a machine-checkable exhaustive certificate.
frontier_computational_search	Spectral graph theory and discrete geometry	An exact witness with every requested invariant, plus an independently checkable certificate for feasibility and global optimality.	C131 — Hot Spots conjecture for convex planar domains	MIT	[]	counterexample_131	Let P range over two-dimensional convex lattice polygons contained in [0,8]^2, normalized so that min{x:(x,y) in P}=min{y:(x,y) in P}=0, and such that S(P)=P intersect Z^2 has exactly 31 points. Join two points of S(P) when their Euclidean distance is one, obtaining the unit-grid graph G(P), and require G(P) to be conn...	[ "counterexample-oriented discrete analogue", "exact algebraic eigenvalue certificate", "finite convex-polygon search" ]	This is a discrete Neumann hot-spots problem on lattice-convex planar domains. A positive value of h(P) means that both signs of the first nonconstant mode have all their hottest vertices strictly inside the discrete domain; h(P)=0 means that at least one extremum reaches the boundary. It is not a continuum counterexam...	finite_search_task_existence_or_optimum_not_preasserted	false	1.0.0	131	counterexample_variant	find_and_certify_optimal_object	Interior hot spots on a convex lattice domain	Exact witness checks plus a machine-checkable exhaustive certificate.
frontier_computational_search	Diophantine approximation and dynamics	An exact witness with every requested invariant, plus an independently checkable certificate for feasibility and global optimality.	C134 — Lonely Runner conjecture, first unresolved case of 14 runners	MIT	[]	counterexample_134	Let V={v_1<...<v_13} be a set of thirteen positive integers with v_13<=210 and gcd(v_1,...,v_13)=1. Define F_V(t)=min_{1<=i<=13} \|\|t v_i\|\| for t in R/Z, where \|\|.\|\| is distance to the nearest integer, and put Delta(V)=max_t F_V(t). Find V minimizing Delta(V). Subject to that, minimize v_13, then maximize the number of ...	[ "first unresolved runner count", "exact rational verification", "counterexample-or-near-miss outcome" ]	After moving to the frame of the slowest of fourteen runners, the remaining thirteen relative speeds can be taken to be distinct positive integers, and the conjectural loneliness threshold is 1/14. This variant asks for the most dangerous primitive speed set below a fixed height, rather than for an arbitrary counterexa...	finite_search_task_existence_or_optimum_not_preasserted	false	1.0.0	134	counterexample_variant	find_and_certify_optimal_object	Most dangerous bounded fourteen-runner instance	Exact witness checks plus a machine-checkable exhaustive certificate.
frontier_computational_search	Algebraic combinatorics and representation theory	An exact witness with every requested invariant, plus an independently checkable certificate for feasibility and global optimality.	C135 — King–Tollu–Toumazet positivity conjecture	MIT	[]	counterexample_135	Let lambda, mu, and nu be partitions with at most five parts, largest part at most 12, \|lambda\|+\|mu\|=\|nu\|<=36, lambda lexicographically no larger than mu, and gcd of all their positive parts equal to one. Require the Littlewood–Richardson coefficient c_{lambda,mu}^{nu} to exceed one and the stretching polynomial P(t)=c...	[ "negative optimum would refute a positivity conjecture", "exact hive-polytope enumeration", "bounded exhaustive search" ]	For partitions of length at most five, the hive polytope has dimension at most six, so seven exact Littlewood–Richardson counts determine the entire stretching polynomial. A negative optimum would be a KTT counterexample inside a small box; a nonnegative optimum would identify the closest coefficient-level near-miss in...	finite_search_task_existence_or_optimum_not_preasserted	false	1.0.0	135	counterexample_variant	find_and_certify_optimal_object	Bounded King–Tollu–Toumazet stretching-positivity search	Exact witness checks plus a machine-checkable exhaustive certificate.

SOTA Math: Ulam.ai Research Problem Showcase

This public release contains 20 internally authored, research-grade mathematics tasks selected to demonstrate the style of problem design and evaluation developed at ulam.ai:

5 new problems inspired by Erdős problems;
all 10 problems in the AIM-AG research sample, with milestone-based RL task definitions; and
5 bounded counterexample or near-counterexample searches inspired by major conjectures.

These are candidate research problems, not a collection of known-answer exercises. A finite literature search cannot establish global novelty, and mathematical status can change. Proofs, disproofs, and novelty claims require current specialist review.

Why this sample

The sample favors problems with crisp statements, nontrivial but checkable baseline arguments, multiple routes to progress, and explicit standards for computational evidence. It is intended to show prospective research and training partners what a larger private Ulam.ai collection can support: long-horizon reasoning, exact tool use, counterexample search, milestone credit, critic training, and expert-gated evaluation.

Configuration	Rows	Purpose
`showcase`	20	Uniform, buyer-friendly schema across all three streams
`erdos_variants`	5	Full-fidelity selected Erdős-inspired records
`aim_ag_tasks`	10	Full research context, references, milestones, and failure conditions
`counterexample_variants`	5	Full-fidelity finite search and certification tasks
`aim_ag_rl_tasks`	10	Policy-visible RL task prompts
`aim_ag_rl_episodes`	53	Type-stable public curriculum: 18 train, 14 validation, 21 test
`aim_ag_exact_benchmarks`	33	Public exact-computation prompts: 11 per split
`aim_ag_frontier_eval`	10	Public full-task evaluation prompts

The rl/ directory also includes the full-fidelity public curriculum file, public fixtures, schemas, and reward/curriculum configuration. The Hugging Face configuration uses a lossless type-stable view because the original file represents exact prompts as strings and frontier prompts as objects. It records the original field inventory and JSON-encodes mixed or structured values, so every production-format public episode can be reconstructed. It intentionally excludes hidden grader records, hidden targets, hidden fixtures, calibration answers, and expert reviews.

The full AIM-AG research records and RL prompts are two public views of the same ten tasks. Their conjecture and definitions are synchronized. RL instructions, required deliverables, and allowed-tool wording intentionally remain production-oriented and may differ from the client research record. All included prompts and fixtures are public; train/dev/eval labels are organizational splits, not secrecy or holdout claims.

Selected problem families

Erdős-inspired variants

Bounded-defect rigidity for distinct subset sums
Profile-balanced sunflower threshold
Second-order certificate complexity of a squarefree-free interval
Median pinned-distance conjecture
RMS-superlevel concentration for Littlewood polynomials

AIM-AG research tasks

A single simple branch among 321,489 planes
Lines as the largest Noether-Lefschetz loci on Fano threefolds
Generic maximal-rank edge maps for tree-glued varieties of minimal degree
A conductor bound for free anticanonical degrees on toric Fano varieties
Arithmeticity of vanishing cohomology in high-power linear systems
An integral Segal theorem for toric targets
A factorial bound for Rost nilpotence
A character-valued logarithmic gluing formula
A cyclotomic Hasse tower for K3 moduli
Logarithmic TP as a tame-ramification detector

Counterexample-oriented variants

A closest joint Giuga-Carmichael local fit
A subdivision-poor five-critical graph at the open Hajós boundary
Interior hot spots on a convex lattice domain
The most dangerous bounded 14-runner instance
A bounded King-Tollu-Toumazet coefficient search

Loading

from datasets import load_dataset

showcase = load_dataset("ulamai/SOTA-Math", "showcase", split="train")
aim_ag = load_dataset("ulamai/SOTA-Math", "aim_ag_tasks", split="train")
episodes = load_dataset("ulamai/SOTA-Math", "aim_ag_rl_episodes")

The default showcase configuration uses one stable schema. The full-fidelity configurations preserve stream-specific metadata and should be loaded separately.

Uniform showcase schema

Each of the 20 showcase rows has:

stable identifiers and stream labels;
title, domain, task type, and difficulty;
a complete prompt and provenance-oriented inspiration field;
rationale and expected output;
research-status and verification caveats;
RL readiness, milestones, and quality signals; and
per-record MIT license metadata.

The machine-readable schema is in schema/showcase.schema.json.

Evaluation policy

Exact finite subproblems may be checked with reproducible computation and machine-verifiable certificates. Frontier claims are expert-gated. Numerical evidence, finite-field experiments, dimension counts, or literature summaries are not silently upgraded to complete proofs. Strong partial progress is a valid outcome when its scope and remaining gaps are explicit.

The release contains no canonical solutions to the 20 headline problems. Public exact episodes are scaffolding for tool and evidence discipline, not answer keys to unresolved terminal conjectures.

Intended uses

evaluate research-agent decomposition and gap control;
train or assess tool-augmented mathematical reasoning;
prototype milestone-based process rewards;
study exact-certificate generation; and
assess fit for a larger private Ulam.ai research-problem program.

Out-of-scope uses

treating a model response as a verified mathematical result without review;
claiming that every problem is globally novel or still open without a current literature refresh;
using public episodes as a secret held-out benchmark; or
inferring performance on all of mathematics from this deliberately small showcase.

Creation and provenance

The problem statements and task packaging were authored internally at Ulam.ai. Inspiration labels identify the mathematical lineage of variants; they do not claim that the new task is the canonical statement of the cited conjecture. See DATA_STATEMENT.md and MANIFEST.json for selection, provenance, and integrity details.

License

Bibliographic citations and names of third-party mathematical results remain attribution facts about their respective sources; the MIT grant covers the Ulam.ai-authored dataset content and packaging.

Citation

@dataset{ulamai_sota_math_2026,
  author    = {{ulam.ai}},
  title     = {SOTA Math: Ulam.ai Research Problem Showcase},
  year      = {2026},
  publisher = {Hugging Face},
  url       = {https://huggingface.co/datasets/ulamai/SOTA-Math},
  version   = {0.1.0}
}