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ResearchMath-2-6Sol-Rewrite

uid stringlengths 11 16	arxiv_id stringlengths 9 14	paper_url stringlengths 31 36	primary_category stringclasses 118 values	signal_type stringclasses 5 values	question_category stringclasses 8 values	format stringclasses 1 value	route stringclasses 7 values	question stringlengths 93 7.2k	answer_convention stringlengths 21 275	distinguished_value stringclasses 349 values	source_question stringlengths 65 7.21k	engine stringclasses 1 value
2603.03386#2	2603.03386	https://arxiv.org/abs/2603.03386	math.AG	conjecture	universality	numerical	B	Let Q=(I,Ω) range over finite quivers with no edge-loops. Define its double Q̄=(I,Ω̄), where Ω̄=Ω∪{e:e∈Ω} and e:j→i for e:i→j. Let A=(a_{i,j}) be the symmetric generalized Cartan matrix with a_{i,i}=2 and, for i≠j, a_{i,j} equal to minus the number of edges of Ω joining i and j. Set T_max={(γ_e)∈(C)^Ω̄:γ_eγ_{e}=γ_f...	Give the smallest failing edge multiplicity n≥2, or -1 if the Serre relations follow from (1)–(5) for every finite loopless quiver over its full equivariant parameter ring.	-1	Setting. Let Q = (I, Omega) be a quiver with I and Omega finite and with no edge-loops, and let Qbar = (I, Omegabar) be its double, so Omegabar = Omega union {e* : e in Omega}, where e* : j -> i whenever e : i -> j. Let A = (a_{i,j})_{i,j in I} be the associated symmetric generalized Cartan matrix: a_{i,i} = 2 and, for...	openai/gpt-6-sol
2604.26158#1	2604.26158	https://arxiv.org/abs/2604.26158	math.CO	conjecture	universality	numerical	B	For every graph G with no induced subgraph isomorphic to the claw K_(3,1), let X_G(x)=sum_C product_(v in V(G)) x_(C(v)), where the sum is over all proper colorings C of G. What is the smallest number of vertices of such a graph G for which [s_lambda]X_G(x)<0 for some partition lambda? Answer -1 if no such graph exists...	The answer is the minimum number of vertices in a counterexample, or -1 if every such graph is Schur-positive.	-1	For every graph G having no induced subgraph isomorphic to the claw K_(3,1), let X_G(x)=sum_C product_(v in V(G)) x_(C(v)), where the sum is over all proper colorings C of G. Is X_G Schur-positive; that is, does [s_lambda]X_G(x) >= 0 for every partition lambda?	openai/gpt-6-sol
2505.11779#0	2505.11779	https://arxiv.org/abs/2505.11779	math.GT	conjecture	universality	numerical	B	Work in S^3. For a hyperbolic link L, let M=S^3−N(L), where N(L) is an open tubular neighborhood, and let c(L) be the minimum crossing number of a diagram of L. A diagram D is a projection to a 2-sphere with over/under information at each double point. Checkerboard-color its complementary regions; the black and white c...	The answer is the minimum crossing number of a hyperbolic link with no taut diagram, or −1 if every hyperbolic link admits one.	-1	Work in the 3-sphere. Let L be a hyperbolic link in S^3 and let M = S^3 - N(L) be the complement of an open tubular neighborhood of L. Let D be a diagram of L, i.e. a projection of L to a 2-sphere with over/under information at each double point, and color the complementary regions of D in the 2-sphere in checkerboard ...	openai/gpt-6-sol
2510.03835#1	2510.03835	https://arxiv.org/abs/2510.03835	math.AP	conjecture	characterization	numerical	A	Consider small smooth perturbations f(t,x,v) of μ(v)=(2π)^{-3/2}e^{-\|v\|²/2} on T³×R³ in the weakly collisional Vlasov–Poisson–Boltzmann system, with ν>0, 0<s<1, max{−3,−3/2−2s}<γ<−2s, and κ=1/(1+2s). The established nonzero-spatial-mode decay is bounded by min{exp[−δ_N(ν^κt)^{s/(s−γ)}],exp[−δ_N(νt)^{1/(1−γ−2s)}]} in th...	Give the single sharp exponent a; the proposed answer is 2/5.	2/5	Consider small smooth perturbations f(t,x,v) of the normalized Maxwellian μ(v)=(2π)^{-3/2}e^{-\|v\|²/2} on x∈T³ and v∈R³ for the weakly collisional Vlasov–Poisson–Boltzmann system with collision strength ν>0, non-cutoff soft-potential parameters 0<s<1 and max{−3,−3/2−2s}<γ<−2s, and κ=1/(1+2s). The established nonzero-spa...	openai/gpt-6-sol
1703.05362#4	1703.05362	https://arxiv.org/abs/1703.05362	math.AG	conjecture	existence	numerical	E	Let F be a perfect field with char(F) != 2, and let B_gm O(1) = B_et O(1) be the classifying space of étale-locally trivial O(1)-torsors. Among the two Chow--Witt twist classes L in Pic(B_gm O(1)) = Z/2Z, how many have a noninjective canonical, always-surjective map from the L-twisted Chow--Witt groups to the fiber pro...	Give the number of twist classes with a noninjective map; the original question has answer Yes exactly when this number is positive.	null	Let F be a perfect field with char(F) != 2, and let B_gm O(1) = B_et O(1) be the classifying space of étale-locally trivial O(1)-torsors. Since Pic(B_gm O(1)) = Z/2Z, consider each of its two Chow--Witt twist classes L. For a fixed twist L, does the canonical, always-surjective map from the L-twisted Chow--Witt groups ...	openai/gpt-6-sol
1707.08372#0	1707.08372	https://arxiv.org/abs/1707.08372	math.CO	conjecture	universality	numerical	B	What is the smallest vertex count n for which there exists a hypergraph H=(V,E), with n=\|V\| and every edge a subset of V, such that any two distinct edges intersect in at most one vertex, every edge e satisfies \|e\| != 1, and the edges cannot be colored with at most n colors so that intersecting distinct edges receive d...	Give the smallest such n, or -1 if no such hypergraph exists.	-1	For every hypergraph H=(V,E) with n=\|V\| vertices, in which each edge is a subset of V, any two distinct edges intersect in at most one vertex, and every edge e satisfies \|e\| != 1, is it true that the edges of H can be colored with at most n colors so that any two intersecting distinct edges receive different colors?	openai/gpt-6-sol
2602.02342#5	2602.02342	https://arxiv.org/abs/2602.02342	math.QA	conjecture	universality	numerical	B	What is the smallest integer n>=1 for which there exist an associative algebra H, an invertible R in a suitable completion of H widehat{tensor} H satisfying R_{12}R_{13}R_{23}=R_{23}R_{13}R_{12}, and a transitive sign matrix epsilon=(epsilon_{i,j})_{1<=i,j<=n} such that R^{(epsilon)} fails the QYBE when regarded as an ...	Return the smallest n admitting a counterexample, or -1 if there is no counterexample.	-1	Let n >= 1, let H be an associative algebra, and let R be an invertible element of a suitable completion H widehat{tensor} H satisfying R_{12}R_{13}R_{23}=R_{23}R_{13}R_{12}. For i<j, let R_{i,j} denote R placed in tensor legs i and j of H^{widehat{tensor} 2n}, and set R_{i,j}^{(1)}=R_{i,j} and R_{i,j}^{(-1)}=R_{j,i}^{...	openai/gpt-6-sol
2409.18019#1	2409.18019	https://arxiv.org/abs/2409.18019	math.AG	conjecture	universality	numerical	B	Let \underline{\Omega}_X^p:=\operatorname{Gr}_F^p\underline{\Omega}_X^\bullet[p], and say that X is pre-r-Du Bois if \mathcal H^i\underline{\Omega}_X^p=0 for every i>0 and every integer 0\le p\le r, with the pre-(-1) condition vacuous. What is the smallest integer k\ge 0 for which there exist a complex algebraic variet...	The answer is the smallest such k, or -1 if the proposed vanishing holds in every case.	-1	Let X be a complex algebraic variety, let k\ge 0 be an integer, and write \underline{\Omega}_X^p:=\operatorname{Gr}_F^p\underline{\Omega}_X^\bullet[p] for the p-th Du Bois complex; write \mathcal H^i\underline{\Omega}_X^p for its cohomology sheaves. Say that X is pre-r-Du Bois if \mathcal H^i\underline{\Omega}_X^p=0 fo...	openai/gpt-6-sol
2410.09830#3	2410.09830	https://arxiv.org/abs/2410.09830	math.CO	conjecture	universality	numerical	B	For every real p >= 2 and every connected simple, undirected, unweighted graph G on n vertices, let λ₁,…,λₙ be the eigenvalues of its adjacency matrix and define its positive p-energy by Eₚ⁺(G)=∑_{λᵢ>0}λᵢᵖ. What is the smallest positive integer n for which there exist such a p and G with Eₚ⁺(G)<Eₚ⁺(Pₙ)=∑_{k=1}^{⌊(n+1)/...	Give the smallest such n, or -1 if no such n exists.	-1	For every real p >= 2 and every connected simple, undirected, unweighted graph G on n vertices, let \lambda_1,\ldots,\lambda_n be the eigenvalues of its adjacency matrix and define its positive p-energy by \mathcal{E}_p^+(G)=\sum_{\lambda_i>0}\lambda_i^p. Is it true that \mathcal{E}_p^+(G) >= \mathcal{E}_p^+(P_n)=\sum_...	openai/gpt-6-sol
2604.07142#0	2604.07142	https://arxiv.org/abs/2604.07142	math.NT	conjecture	universality	numerical	B	Let \mathcal{L}_1 be the positive integers, let \mathcal{L}_2=\{2\}\cup\{2k+1:k\geq1\}, and, for each integer m\geq2, form \mathcal{L}_{m+1} by deleting from the increasing sequence (\ell_{m,j})_{j\geq1} of elements of \mathcal{L}_m every term whose index is divisible by \ell_{m,m}. Define \ell_n=\ell_{n,n} for every i...	Give the smallest failing integer n, or -1 if the sharper inequality holds for every integer n\geq4.	-1	Let \mathcal{L}_1 be the positive integers, let \mathcal{L}_2=\{2\}\cup\{2k+1:k\geq1\}, and, for each integer m\geq2, form \mathcal{L}_{m+1} by deleting from the increasing sequence (\ell_{m,j})_{j\geq1} of elements of \mathcal{L}_m every term whose index is divisible by \ell_{m,m}; define \ell_n=\ell_{n,n} for every i...	openai/gpt-6-sol
2512.13865#14	2512.13865	https://arxiv.org/abs/2512.13865	math.DS	conjecture	universality	numerical	B	Among all compact symplectic manifolds (Q,omega) and probability measures mu supported on smooth symplectomorphisms of Q satisfying the following assumptions, what is the smallest value of dim Q for which (Q,omega) is not locally homogeneous? Let Gamma_mu be the subgroup of Diff(Q) generated by supp(mu). Assume that fo...	The answer is the smallest counterexample dimension, or -1 if every instance satisfying the assumptions is locally homogeneous.	-1	Let (Q,omega) be a compact symplectic manifold, let mu be a probability measure supported on smooth symplectomorphisms of Q, and let Gamma_mu be the subgroup of Diff(Q) generated by supp(mu). Assume that mu is uniformly expanding on every isotropic subspace of T_qQ for every q in Q, meaning that for each isotropic dime...	openai/gpt-6-sol
1712.01037#2	1712.01037	https://arxiv.org/abs/1712.01037	math.CO	conjecture	universality	numerical	B	Let P be a finite poset, P* an induced subposet containing every minimal element of P, and lambda:P->R order-preserving. Write tilde P=P\P. A covering relation p≺q is non-redundant if, for every a,b in P* with a≤q and p≤b, either a=b or lambda(a)<lambda(b). Call (P,lambda) regular if every covering relation is non-re...	Give the smallest cardinality \|P\| of a counterexample, or -1 if regularity suffices for every finite marked poset.	-1	Let P be a finite poset, let P* be an induced subposet containing every minimal element of P, and let lambda:P->R be order-preserving; write \tilde P=P\setminus P. Call a covering relation p\prec q non-redundant if, for every a,b in P* with a\le q and p\le b, either a=b or lambda(a)<lambda(b), and call (P,lambda) reg...	openai/gpt-6-sol
1705.05816#2	1705.05816	https://arxiv.org/abs/1705.05816	math.CO	conjecture	universality	numerical	B	Let M be an essential Z-matroid on a finite ground set [n], meaning an assignment A ↦ M(A) of finitely generated abelian groups such that, for every A ⊆ [n] and b,c ∈ [n]\A, there are x,y ∈ M(A) with M(A∪{b}) ≅ M(A)/(x), M(A∪{c}) ≅ M(A)/(y), and M(A∪{b,c}) ≅ M(A)/(x,y). Write M(A) ≅ Z^{d(A)} × G_A, set r=d(∅), m(A)=#G_...	Give the smallest such n, or −1 if there is no counterexample.	-1	Let M be an essential Z-matroid on a finite ground set [n], meaning an assignment A \|-> M(A) of finitely generated abelian groups such that, for every A subseteq [n] and b,c in [n] minus A, there are x,y in M(A) with M(A union {b}) congruent to M(A)/(x), M(A union {c}) congruent to M(A)/(y), and M(A union {b,c}) congru...	openai/gpt-6-sol
1711.04806#0	1711.04806	https://arxiv.org/abs/1711.04806	math.AT	conjecture	universality	numerical	B	For every prime p and height n≥1, let E_n denote Morava E-theory at height n and EO_n the corresponding higher real K-theory, with EO_{p-1}=E_{p-1}^{hC_p}, where C_p is cyclic of order p. The known suspension-generated calculations are pic(KO)≅Z/8 and pic(EO_{p-1})≅Z/(2p²). What is the smallest height n≥1 for which pic...	Return the smallest such height, or -1 if pic(EO_n) is cyclic for every prime p and height n≥1.	-1	For every prime p and height n≥1, with E_n denoting Morava E-theory at height n and EO_{p-1}=E_{p-1}^{hC_p} (C_p the cyclic group of order p), is the Picard group of the corresponding higher real K-theory EO_n cyclic? This should extend the known suspension-generated calculations pic(KO)≅Z/8 and pic(EO_{p-1})≅Z/(2p^2);...	openai/gpt-6-sol
1707.08178#7	1707.08178	https://arxiv.org/abs/1707.08178	math.NT	conjecture	universality	numerical	B	Let \(\mathcal H\) be the \(\mathbb Q\)-algebra of motivic multiple zeta values with increasing depth filtration \(\mathfrak D\), and let \(\zeta^\mathfrak m_{\mathfrak D}(n_1,\ldots,n_r)\) denote the image of \(\zeta^\mathfrak m(n_1,\ldots,n_r)\) in \(\operatorname{gr}^{\mathfrak D}\mathcal H=\bigoplus_{r\ge1}\mathfra...	Give the smallest such k, or -1 if the identity holds for all positive integers k and r.	-1	Let \(\mathcal H\) be the \(\mathbb Q\)-algebra of motivic multiple zeta values, equipped with its increasing depth filtration \(\mathfrak D\), and write \(\zeta^\mathfrak m_{\mathfrak D}(n_1,\ldots,n_r)\) for the image of \(\zeta^\mathfrak m(n_1,\ldots,n_r)\) in \(\operatorname{gr}^{\mathfrak D}\mathcal H=\bigoplus_{r...	openai/gpt-6-sol
2412.14891#7	2412.14891	https://arxiv.org/abs/2412.14891	math.CO	conjecture	universality	numerical	B	What is the smallest odd integer k ≥ 5 for which the following assertion fails? The asymptotic edge density of the recursive n-vertex k-uniform hypergraph construction—partition the vertices into parts X and Y of distinct sizes, include every k-edge meeting X in an odd number of vertices, and apply the same constructio...	The answer is the smallest odd k ≥ 5 where the assertion fails, or -1 if there is none.	-1	For every odd integer k >= 5, consider the recursive construction of an n-vertex k-uniform hypergraph obtained by partitioning its vertex set into two parts X and Y of distinct sizes, including every k-edge that meets X in an odd number of vertices, and applying the same construction recursively inside Y. Is the result...	openai/gpt-6-sol
2509.08564#0	2509.08564	https://arxiv.org/abs/2509.08564	math.DG	conjecture	universality	numerical	B	Let ι:(M^m,g) → N^n(c) be a connected Riemannian submanifold, meaning a smooth isometric immersion into a real space form of constant sectional curvature c. Let τ(ι)=Tr_g(∇dι)=mH, where H is the mean-curvature vector. For a local orthonormal frame {e_i}_{i=1}^m, define Δ^ιX=-∑_{i=1}^m(∇^ι_{e_i}∇^ι_{e_i}X-∇^ι_{∇_{e_i}e_...	The answer is the smallest counterexample dimension m, or -1 if every immersion in the question is minimal.	-1	Let iota:(M^m,g) -> N^n(c) be a connected Riemannian submanifold, meaning a smooth isometric immersion, where N^n(c) is a real space form of constant sectional curvature c. Let tau(iota)=Tr_g(nabla diota)=mH be the tension field of iota and H its mean-curvature vector. For a local orthonormal frame {e_i}_{i=1}^m on M, ...	openai/gpt-6-sol
2510.19417#1	2510.19417	https://arxiv.org/abs/2510.19417	math.NT	conjecture	universality	numerical	B	What is the smallest positive prime power q for which the following property fails for some quadratic irrational Θ(t) ∈ F_q((t^{-1})) and some irreducible P(t) ∈ F_q[t]? Here a quadratic irrational is an irrational Laurent series satisfying a quadratic equation over F_q[t]. For each k ∈ N, let A_1^[ΘP^k](t), …, A_{ℓ_{Θ...	Return the smallest qualifying q, or −1 if the property holds for every permitted q, Θ, and P.	-1	Let q be a positive prime power, let F_q((t^{-1})) be the field of formal Laurent series, and let Theta(t) in F_q((t^{-1})) be a quadratic irrational, meaning an irrational Laurent series satisfying a quadratic equation with coefficients in F_q[t]. Let P(t) in F_q[t] be irreducible. For each k in N, write the eventuall...	openai/gpt-6-sol
2407.20994#0	2407.20994	https://arxiv.org/abs/2407.20994	math.CA	conjecture	universality	numerical	B	Let a,b,c∈ℝ\{0} and W(x)=e^(−x²) [[e^(2bx)+a²x², ax, acx²],[ax,1,cx],[acx²,cx,c²x²+1]] on ℝ. For any sequence {P_n}_{n≥0} of 3-by-3 matrix-valued orthogonal polynomials for W, let 𝒟(W) be the ℂ-algebra of right-acting differential operators D=∑_{j=0}^s ∂^j F_j(x), with F_j(x)∈Mat₃(ℂ[x]), such that P_n·D=Λ_n(D)P_n for ...	The answer is the smallest counterexample order across all allowed parameter triples, or −1 if every element belongs to the generated algebra for every allowed triple.	-1	Let \(a,b,c\in\mathbb R\setminus\{0\}\) and let \(W(x)=e^{-x^2}\begin{pmatrix}e^{2bx}+a^2x^2&ax&acx^2\\ax&1&cx\\acx^2&cx&c^2x^2+1\end{pmatrix}\) on \(\mathbb R\). For any sequence \(\{P_n\}_{n\ge0}\) of 3-by-3 matrix-valued orthogonal polynomials for \(W\), let \(\mathcal D(W)\) be the \(\mathbb C\)-algebra of right-ac...	openai/gpt-6-sol
0705.0427#4	0705.0427	https://arxiv.org/abs/0705.0427	math-ph	conjecture	universality	numerical	B	Fix 0<x<1, a generic r with Re(r)>0, r^=r-1, and a fixed parameter alpha. On the bosonic Fock representation with [P,iQ]=2, put hat{pi}=sqrt(r r^)P and let B_m=B_m^1 satisfy [B_n,B_m]=n([r^*n]/[rn])(x^n+x^{-n})delta_{n+m,0}. With normal ordering of the bosons, define Lambda_1^{DV}(z)=x^{-hat{pi}}:exp(sum_{k!=0}(x^{rk...	Return the smallest such n, or -1 if the commutator vanishes for every pair of positive integers m,n.	-1	Fix 0<x<1, a generic r with Re(r)>0, r^=r-1, and a fixed parameter alpha. On the bosonic Fock representation with [P,iQ]=2, put hat{pi}=sqrt(r r^)P and let B_m=B_m^1 satisfy [B_n,B_m]=n([r^*n]/[rn])(x^n+x^{-n})delta_{n+m,0}. With normal ordering of the bosons, define Lambda_1^{DV}(z)=x^{-hat{pi}}:exp(sum_{k!=0}(x^{rk...	openai/gpt-6-sol
2412.06068#1	2412.06068	https://arxiv.org/abs/2412.06068	math.CO	conjecture	quantity	numerical	A	For each integer n >= 3, let M_n = max{psr(G) : G is a maximal planar graph on n vertices}. Such a graph has e(G) = 3n - 6. Define psr(G) as the minimum of e(H)/e(G) over all plane graphs H that are subgraphs of G with n vertices and whose drawing is plane-saturated: adding any edge to the drawing of H either creates a...	The answer is the value of the limit, if it exists.	1/2	For each integer n >= 3, let M_n = max{psr(G) : G is a maximal planar graph on n vertices}, where e(G)=3n-6 and psr(G) is the minimum of e(H)/e(G) over all plane graphs H that are subgraphs of G with n vertices and whose drawing is plane-saturated: adding any edge to the drawing of H either creates a crossing or makes ...	openai/gpt-6-sol
2606.06352#3	2606.06352	https://arxiv.org/abs/2606.06352	math.CO	conjecture	universality	numerical	B	Fix integers n >= 2 and 1 <= k <= n-1. Let P_{k,n} be the set of partitions lambda = (lambda_1 >= ... >= lambda_k >= 0) with lambda_1 <= n-k. Let y_1,...,y_n, z_1,...,z_n and q be independent indeterminates, set A = Z[y_1,...,y_n,z_1,...,z_n], and let V be the free A[q]-module with basis v_1,...,v_n. Identify wedge^k V...	The answer is the smallest n admitting a failure, or -1 if the stated positivity holds for every permitted n, k, lambda, mu, nu and d.	-1	Fix integers n >= 2 and 1 <= k <= n-1, and let P_{k,n} be the set of partitions lambda = (lambda_1 >= lambda_2 >= ... >= lambda_k >= 0) whose Young diagram fits into the k x (n-k) rectangle, i.e. lambda_1 <= n-k. Let y_1,...,y_n, z_1,...,z_n and q be independent indeterminates, put A = Z[y_1,...,y_n,z_1,...,z_n], let V...	openai/gpt-6-sol
2504.15180#1	2504.15180	https://arxiv.org/abs/2504.15180	math.AP	conjecture	universality	numerical	A	In R^2, let K(x)=(1-\|x\|)chi_{B(1)}(x). For each j>=1, let k_j(x)=j^2 M_0(j\|x\|)/(2pi), where M_0(r)=integral_0^infinity exp(-r cosh s) ds, so k_j is the Green function of -j^{-2}Delta+1. Define A(r)=(2r/pi)log(1/r+sqrt(1/r^2-1))chi_{B(1)}(r) for r>=0 and h(lambda)=A(-log lambda)/lambda for lambda in (0,1]. For every N>=...	Give the limit as one number; a value of 0 means the convergence asked about in the original question holds.	0	In R^2, let K(x)=(1-\|x\|)chi_{B(1)}(x), and for each j>=1 let k_j(x)=j^2 M_0(j\|x\|)/(2pi), where M_0(r)=integral_0^infinity exp(-r cosh s) ds, so k_j is the Green function of -j^{-2}Delta+1. Define A(r)=(2r/pi)log(1/r+sqrt(1/r^2-1))chi_{B(1)}(r) for r>=0 and h(lambda)=A(-log lambda)/lambda for lambda in (0,1]. For every ...	openai/gpt-6-sol
2502.18895#0	2502.18895	https://arxiv.org/abs/2502.18895	math-ph	conjecture	universality	numerical	B	Let Omega be a homogeneous CohFT on a finite-dimensional state space (F,eta), with conformal dimension delta and Euler vector field E. For tau in F, let Omega^tau be its shifted CohFT and assume it has a vacuum vector vac^tau(z) in F[[z]], meaning that pi_bullet^* Omega^tau_{g,n}(v_1,...,v_n)=Omega^tau_{g,n+1}(vac^tau(...	The answer is the smallest failing genus, or -1 if every stated Virasoro constraint holds in every genus.	-1	Let Omega be a CohFT on a finite-dimensional state space (F,eta), let Omega^tau be its shifted CohFT at tau in F, and assume it has a vacuum vector vac^tau(z) in F[[z]], meaning that pi_bullet^* Omega^tau_{g,n}(v_1,...,v_n)=Omega^tau_{g,n+1}(vac^tau(psi_bullet),v_1,...,v_n) for every stable (g,n), where pi_bullet forge...	openai/gpt-6-sol
1708.08044#0	1708.08044	https://arxiv.org/abs/1708.08044	math.AP	conjecture	universality	numerical	B	What is the smallest spatial dimension d>=2 for which there exist mu>0 and beta>1 such that, for the Cauchy problem u_tt-Delta u+mu(1+t)^(-beta)u_t=±\|u\|^p on [0,T)×R^d with initial data (u(0),u_t(0))=(u_0,u_1), the critical exponent p_c separating small-data global existence from small-data blow-up for suitable (u_0,u_...	The answer is the smallest such dimension, or -1 if the equality holds for every d>=2, mu>0, and beta>1.	-1	For each spatial dimension d>=2, mu>0, and beta>1, consider the Cauchy problem u_tt-Delta u+mu(1+t)^(-beta)u_t=±\|u\|^p on [0,T)xR^d, with initial data (u(0),u_t(0))=(u_0,u_1). Since b(t)^(-1) is not in L^1(0,infinity), does the critical exponent p_c—defined as the threshold separating small-data global existence from sm...	openai/gpt-6-sol
2408.08457#8	2408.08457	https://arxiv.org/abs/2408.08457	math.PR	conjecture	universality	numerical	B	Let G range over locally finite connected simple graphs equipped with Bernoulli bond percolation, where each edge e is independently open with its assigned probability p_e, and let a and b range over distinct vertices. For each integer k>=1, let ab^{square k} be the event that there exist k nonintersecting open paths f...	The answer is the smallest counterexample index k, or -1 if the inequality lambda_{k+1}<=lambda_k always holds.	-1	Let G be a locally finite connected simple graph equipped with Bernoulli bond percolation, in which each edge e is independently open with its assigned probability p_e, and let a and b be distinct vertices. For each integer k>=1, let ab^{square k} be the event that there exist k nonintersecting open paths from a to b, ...	openai/gpt-6-sol
2512.02661#6	2512.02661	https://arxiv.org/abs/2512.02661	math.PR	conjecture	quantity	numerical	A	For the cluttered rectangular configuration of snapping-out Brownian motion illustrated in Figure 7, consider Brownian motion in a bounded planar domain with orthogonal reflection at an impermeable outer boundary and at disjoint smooth simple closed semipermeable barriers, crossing each barrier according to its side-de...	Give the single exponent a for which t_mix has order Delta^a as the illustrated domain grows.	2	For the cluttered rectangular configuration of snapping-out Brownian motion illustrated in Figure 7—namely, Brownian motion in a bounded planar domain with orthogonal reflection at an impermeable outer boundary and at disjoint smooth simple closed semipermeable barriers, crossing a barrier according to its side-depende...	openai/gpt-6-sol
2505.00501#3	2505.00501	https://arxiv.org/abs/2505.00501	hep-th	conjecture	characterization	numerical	B	Consider all real forms g of all complex simple Lie algebras g_C, ordered by the rank of g_C. What is the smallest rank at which the following proposed characterization fails? A constant antisymmetric real r in g wedge g solves the split-type modified classical Yang–Baxter equation [r_12,r_23]+[r_23,r_31]+[r_31,r_12]=-...	Give the smallest counterexample rank, or -1 if there is no counterexample.	-1	Setting. Consider factorizing, at the classical (Poisson) level, the phase space of three-dimensional Chern-Simons gauge theory with gauge group G across a codimension-one entangling surface. Here 'factorizing' means constructing a surjective Poisson map (a gluing map) that assembles two one-sided phase spaces into the...	openai/gpt-6-sol
2412.00914#16	2412.00914	https://arxiv.org/abs/2412.00914	math.AG	conjecture	construction	numerical	B	For every integer r >= 1, let Sigma be the prismatization stack, i.e. the fpqc sheaf on animated p-nilpotent rings sending R to the Cartier--Witt divisors on R, and let Sigma'_r be the prismatization stack encoding the r-Nygaard data. The r-divided Frobenius and the inclusion give two morphisms Sigma -> Sigma'_r; after...	The answer is the smallest failing r, with -1 meaning that the construction and identification hold for every integer r >= 1.	-1	For every integer r >= 1, let Sigma be the prismatization stack, i.e. the fpqc sheaf on animated p-nilpotent rings sending R to the Cartier--Witt divisors on R, and let Sigma'_r be the prismatization stack encoding the r-Nygaard data. The r-divided Frobenius and the inclusion give two morphisms Sigma -> Sigma'_r; after...	openai/gpt-6-sol
1801.00983#3	1801.00983	https://arxiv.org/abs/1801.00983	math.AP	conjecture	universality	numerical	B	What is the smallest dimension of a smooth compact Riemannian manifold (M,g) for which there exist m∈L^∞(M) with m≥0 and a∈L^∞(M) with a≥0 such that the following two statements are not equivalent? (i) There exist C,c>0 such that, for every initial datum (u₀,u₁)∈H¹(M)×L²(M), the solution of (∂ₜ²−Δ_g+a(x)∂ₜ+m)u=0 satisf...	The answer is the smallest counterexample dimension, or −1 if the equivalence holds for every manifold and admissible m,a.	-1	Let (M,g) be a smooth compact Riemannian manifold, let m in L^infty(M) satisfy m >= 0, and let a in L^infty(M) be a nonnegative damping. For every initial datum (u_0,u_1) in H^1(M) x L^2(M), consider (partial_t^2 - Delta_g + a(x)partial_t + m)u=0 and E_m(u)(t)=integral_M (\|nabla_g u\|_g^2+\|partial_tu\|^2+m\|u\|^2) dvol_g. ...	openai/gpt-6-sol
2406.19499#7	2406.19499	https://arxiv.org/abs/2406.19499	math.DS	conjecture	existence	numerical	B	For each integer N >= 2, consider the deterministic chain of rotators with q_j in T, p_j in R, Hamiltonian H(p,q) = sum_{j=1}^N p_j^2/2 + sum_{j=1}^{N-1} V_j(q_j-q_{j+1}), and dynamics dot q_j = p_j, dot p_j = -partial_{q_j}H - delta_{1j}p_1. Require each V_j:T -> R to be C^3, with V_j >= 1 and (V_j'(x))^2+(V_j''(x))^2...	The answer is the smallest N where the stated sharpness example fails, or -1 if such examples exist for every N >= 2.	-1	For every N >= 2, consider the deterministic chain of rotators with q_j in T, p_j in R, Hamiltonian H(p,q) = sum_{j=1}^N p_j^2/2 + sum_{j=1}^{N-1} V_j(q_j-q_{j+1}), and dynamics dot q_j = p_j, dot p_j = -partial_{q_j}H - delta_{1j}p_1. Assume each V_j:T -> R is C^3, V_j >= 1, and (V_j'(x))^2+(V_j''(x))^2 != 0 for every...	openai/gpt-6-sol
1706.09941#9	1706.09941	https://arxiv.org/abs/1706.09941	hep-th	conjecture	universality	numerical	E	For n>0, let alpha^(vac)(theta) denote the vacuum eigenvalue alpha_+^(vac)(theta) of the alpha_+(theta) connection-coefficient operator, normalized by a_+(theta)=i exp(-i pi p_2) alpha_+^(vac)(theta-i pi n/2). Using the associated ODE [-d^2/dy^2+p_2^2 e^y/(1+e^y)+(1/4+p_1^2)e^y/(1+e^y)^2+e^(2 theta)(1+e^y)^(-n-2)] Psi_...	Count the regimes for which at least one such zero exists.	0	Fix n>0, and let alpha^(vac)(theta) denote the vacuum eigenvalue alpha_+^(vac)(theta) of the alpha_+(theta) connection-coefficient operator, normalized by a_+(theta)=i exp(-i pi p_2) alpha_+^(vac)(theta-i pi n/2). Prove, directly from the associated ODE [-d^2/dy^2+p_2^2 e^y/(1+e^y)+(1/4+p_1^2)e^y/(1+e^y)^2+e^(2 theta)(...	openai/gpt-6-sol
1801.00238#2	1801.00238	https://arxiv.org/abs/1801.00238	math.LO	conjecture	existence	numerical	E	Let \(\stick\) be the least size of a family \(\mathcal X\subseteq[\omega_1]^\omega\) such that every uncountable \(Y\subseteq\omega_1\) contains some \(X\in\mathcal X\), and let \(\mathfrak b\) be the least cardinality of an unbounded family in \(\omega^\omega\) under eventual domination. Among the two hypotheses \(\s...	Count each of the two hypotheses once if it does not guarantee both colourings.	0	Let \(\stick\) be the least size of a family \(\mathcal X\subseteq[\omega_1]^\omega\) such that every uncountable \(Y\subseteq\omega_1\) contains some \(X\in\mathcal X\), and let \(\mathfrak b\) be the least cardinality of an unbounded family in \(\omega^\omega\) under eventual domination. Under each of the hypotheses ...	openai/gpt-6-sol
2409.16390#7	2409.16390	https://arxiv.org/abs/2409.16390	math.GT	conjecture	extension	numerical	B	What is the smallest number of components of a link K for which an isomorphism I#(K) ≅ B(K) extending the known isomorphism for two-component links with non-zero linking number fails to exist? Answer -1 if there is no such link.	The answer is the smallest component count of a counterexample, or -1 if the proposed extension exists for every link.	-1	For every link K, does the framed instanton invariant I#(K) admit an isomorphism with Bloom's invariant B(K), extending the isomorphism known when K has two components with non-zero linking number?	openai/gpt-6-sol
2604.27066#0	2604.27066	https://arxiv.org/abs/2604.27066	hep-th	conjecture	universality	numerical	B	What is the smallest rank r_G of a counterexample to the following assertion? Let a gauge^3-anomaly-free 4d N=1 gauge theory with reductive compact gauge group G of rank r_G be compactified on R^3 x S^1 of radius R_1, with R_1 sufficiently small that an asymptotically free parent theory is weakly coupled at 1/R_1. Writ...	Give the smallest counterexample rank, or -1 if the assertion holds throughout the stated setting.	-1	Let a gauge^3-anomaly-free 4d N=1 gauge theory with reductive compact gauge group G of rank r_G be compactified on R^3 x S^1 of radius R_1, with R_1 sufficiently small that an asymptotically free parent theory is weakly coupled at 1/R_1. Write the Cartan holonomies as x=(x_1,...,x_{r_G}) in (-1/2,1/2]^{r_G} modulo Weyl...	openai/gpt-6-sol
1703.06725#0	1703.06725	https://arxiv.org/abs/1703.06725	math.CO	conjecture	universality	numerical	B	What is the smallest integer r>=1 for which there exist integers g>=0 and n>=1 with 2g-2+n>0 and positive integers mu_1,...,mu_n such that b=(2g-2+n+sum_i mu_i)/r is a nonnegative integer, but the following equality fails? The connected r-spin Hurwitz number h^{circ,1,r}_{g;mu_1,...,mu_n}, counting covers of P^1 with r...	The answer is the least r admitting a counterexample, or -1 if the equality holds for every admissible choice.	-1	For every integer r>=1, g>=0, n>=1 with 2g-2+n>0, and positive integers mu_1,...,mu_n such that b=(2g-2+n+sum_i mu_i)/r is a nonnegative integer, let h^{circ,1,r}_{g;mu_1,...,mu_n} be the connected r-spin Hurwitz number counting covers of P^1 with ramification profile (mu_1,...,mu_n) over 0 and b completed (r+1)-cycles...	openai/gpt-6-sol
1708.08129#5	1708.08129	https://arxiv.org/abs/1708.08129	math.AG	conjecture	universality	numerical	B	Let S be a nonsingular projective surface and V a K-theory class on S with H=det V. For each integer r, define the universal power series a_r(z) and b_r(z) by the Hilbert-scheme Verlinde identity ∑_{n≥0} z^n χ(S^[n],H_n⊗E^r)=f_r(z)^{χ(𝒪_S)/2}g_r(z)^{χ(H)}a_r(z)^{H·K_S−K_S²/2}b_r(z)^{K_S²}, where H_n is induced by H on...	The answer is the smallest failing r, or −1 if all the stated identities hold.	-1	Let S be a nonsingular projective surface and let V be a K-theory class on S with H=det V. For each integer r, define the universal power series a_r(z) and b_r(z) by the Hilbert-scheme Verlinde identity \sum_{n\geq0}z^n\chi(S^{[n]},H_n\otimes E^r)=f_r(z)^{\chi(\mathcal O_S)/2}g_r(z)^{\chi(H)}a_r(z)^{H\cdot K_S-K_S^2/2}...	openai/gpt-6-sol
2605.26916#18	2605.26916	https://arxiv.org/abs/2605.26916	math.CO	conjecture	universality	numerical	B	For each positive integer k, let τ_{n,k} be the preorder on a ground set E partitioned into n blocks B_1,…,B_n of cardinality k, with B_1 ≺ ⋯ ≺ B_n (elements in the same block are equivalent, and every element of B_i is below every element of B_j when i<j). For any preorder τ on E, let Q_τ={x∈R^E : x_e≥0 for all e∈E an...	Give the smallest such k, or -1 if the identity holds for every positive integer k.	-1	For every positive integer k, let tau_{n,k} be the preorder on a ground set E partitioned into n blocks B_1,...,B_n of cardinality k, whose vertices form the chain B_1 prec B_2 prec ... prec B_n (equivalently, elements in the same B_i are equivalent and every element of B_i is below every element of B_j for i<j). For a...	openai/gpt-6-sol
2605.12712#1	2605.12712	https://arxiv.org/abs/2605.12712	math.AP	conjecture	existence	numerical	C	What is the best universal constant C in the inequality phi(z)^2 <= C diam(K)^2 ∫_{Sigma_z} \|A_z(Tgrad(f_{x_1}),Tgrad(f_{x_2}))\| dH^2, for every compact K ⊂ R^3 with nonempty interior, every f ∈ C^2(R^3) supported in K, and every z ∈ R? Here Sigma_z={f_{x_3}=z}; Z_hat is the set of nonzero regular values of f_{x_3} in ...	Give the infimum of admissible positive constants, or +∞ if there are none.	null	Let K be a compact subset of R^3 with nonempty interior, and let f be a C^2(R^3) function whose support is contained in K (so f and all its partial derivatives vanish outside K). Write points of R^3 as x = (x_1, x_2, x_3), let p_1, p_3 : R^3 -> R denote the projections onto the first and third coordinates, and write f_...	openai/gpt-6-sol
1707.01823#2	1707.01823	https://arxiv.org/abs/1707.01823	math.CO	conjecture	universality	numerical	B	What is the smallest positive integer n for which there exist a positive integer r, an integer k>=3, an assignment of x_{i,1},...,x_{i,k} to pairwise distinct members of {c_1,...,c_r} in each row i (with assignments allowed to repeat variables across rows), and alpha in Z_{>=0}^r such that the coefficient of product_{t...	The answer is the smallest n admitting a violation, or -1 if the bound holds in every case.	-1	Let n,r be positive integers and let k>=3. For each i in {1,...,n}, assign the k indeterminates x_{i,1},...,x_{i,k} to members of {c_1,...,c_r} so that x_{i,1},...,x_{i,k} are pairwise distinct, while assignments in different rows i may repeat variables. After these substitutions, set S=product_{i=1}^n(x_{i,1}+...+x_{i...	openai/gpt-6-sol
2411.08848#0	2411.08848	https://arxiv.org/abs/2411.08848	math.PR	conjecture	extension	numerical	B	What is the smallest dimension d≥1 for which there exist a stationary random measure X on R^d with diagonal intensity λ_D and truncated correlation measure K of finite total variation, and bounded open Lipschitz domains A,B, such that at least one of the following assertions fails under its stated hypotheses? Define X_...	Give the smallest failing dimension, or −1 if both assertions hold in every dimension.	-1	Let X be a stationary random measure on R^d with diagonal intensity λ_D and truncated correlation measure K of finite total variation, and define X_L(A):=X(LA) for every L≥1. If A and B are bounded open Lipschitz domains, does Theorem 3.1 extend to give L^{-d} Cov(X_L(A),X_L(B)) → (λ_D+∫_{R^d}K(dz))\|A∩B\| and, whenever ...	openai/gpt-6-sol
2501.17806#5	2501.17806	https://arxiv.org/abs/2501.17806	math.GR	conjecture	universality	numerical	B	Let G be a finite group. Call G mixable if there exist k >= 1, elements g_1,...,g_k in G, and independent Bernoulli random variables epsilon_i with P(epsilon_i=1)=p_i for p_i in [0,1], such that g_1^{epsilon_1}...g_k^{epsilon_k} is exactly uniformly distributed on G. For an irreducible complex representation rho:G -> G...	The answer is the smallest counterexample group order, or -1 if the implication holds for every finite group.	-1	Let G be a finite group. Call G mixable if there exist k >= 1, elements g_1,...,g_k in G, and independent Bernoulli random variables epsilon_i with P(epsilon_i=1)=p_i for p_i in [0,1], such that g_1^{epsilon_1}...g_k^{epsilon_k} is exactly uniformly distributed on G. For an irreducible complex representation rho:G -> G...	openai/gpt-6-sol
2412.05855#3	2412.05855	https://arxiv.org/abs/2412.05855	math.AP	conjecture	universality	numerical	B	What is the smallest n in {3,4,5} for which at least one of the following assertions fails for some bounded smooth domain Ω⊂R^n and some 2≤p<(n+2)/(n−2)? Set X=L_q(Ω) for each q∈(n(p−1)/2,∞], and let u(t;u₀) be the maximal mild solution, with maximal existence time T_max(u₀), of u_t−Δu=(∫_Ω \|u(y,t)\|^p/\|x−y\|^(n−2) dy)\|u...	The answer is the smallest dimension in {3,4,5} admitting a failure, or −1 if the original assertions hold throughout the stated range.	-1	Let Omega be a bounded smooth domain in R^n, with 3<=n<=5, let p satisfy 2<=p<p_S:=(n+2)/(n-2), and consider the Dirichlet problem u_t-Delta u=(integral_Omega \|u(y,t)\|^p/\|x-y\|^(n-2) dy)\|u(x,t)\|^(p-2)u(x,t) in Omega x (0,T), u=0 on partial Omega x (0,T), u(.,0)=u_0. For every q in (n(p-1)/2,infinity] and u_0 in X:=L_q(O...	openai/gpt-6-sol
2410.21733#0	2410.21733	https://arxiv.org/abs/2410.21733	math.CO	conjecture	universality	numerical	B	What is the smallest integer n >= 70 for which, with r = floor((n-1)/2)-1, there exists an n-vertex r-uniform hypergraph H with minimum vertex degree delta(H) >= binom(floor((n-1)/2),r-1)+1 that lacks a Berge cycle of some length ell with 3 <= ell <= n? A Berge cycle of length ell consists of ell distinct vertices v_1,...	The answer is the smallest such n, or -1 if no such n exists.	-1	For every integer n >= 70, let r = floor((n-1)/2)-1. Does every n-vertex r-uniform hypergraph H with minimum vertex degree delta(H) >= binom(floor((n-1)/2),r-1)+1 contain Berge cycles of every length ell with 3 <= ell <= n, where a Berge cycle of length ell consists of ell distinct vertices v_1,...,v_ell and ell distin...	openai/gpt-6-sol
2511.16342#4	2511.16342	https://arxiv.org/abs/2511.16342	math.AG	conjecture	universality	numerical	B	What is the smallest integer n≥2 for which the following equality fails for some smooth projective curve C of genus n, line bundle L on C with deg L>2n−2, semisimple group G, and semisimple element g∈G? Write H=C_G(g), H^circ for its neutral component, and Gamma_g=pi_0(H). In the moduli stack CH^{ss}_{H,L} of L-twisted...	The answer is the smallest counterexample genus n, or −1 if the equality holds for every admissible C, L, G, and g.	-1	Let C be a smooth projective curve with g(C) >= 2, let L be a line bundle on C with deg L > 2g(C)-2, let G be a semisimple group, and let g in G be semisimple. Write H = C_G(g), H^circ for its neutral component, and Gamma_g = pi_0(H). In the moduli stack CH^{ss}_{H,L} of L-twisted semistable H-Higgs bundles, let overli...	openai/gpt-6-sol
2505.11087#0	2505.11087	https://arxiv.org/abs/2505.11087	math.DG	conjecture	existence	numerical	B	Among all one-parameter polarised meromorphic degenerations X → D_t^* of n-dimensional projective Calabi–Yau manifolds defined over the convergent Laurent series, with a relatively ample line bundle L → X and a nowhere-vanishing holomorphic volume form Ω, let ω_CY,t be the Calabi–Yau metric on X_t in c_1(L). Let Sk(X) ...	The answer is the smallest failing m across all eligible degenerations and dimensions n, or -1 if none fails.	-1	Let X -> D_t^* be a one-parameter polarised meromorphic degeneration of n-dimensional projective Calabi-Yau manifolds, defined over the convergent Laurent series and equipped with a relatively ample line bundle L -> X and a nowhere-vanishing holomorphic volume form Omega. For each t in D_t^*, let omega_CY,t be the Cala...	openai/gpt-6-sol
2412.12946#0	2412.12946	https://arxiv.org/abs/2412.12946	math.DG	conjecture	universality	numerical	B	Let p,q be relatively prime positive integers, r a positive integer, and lambda a real number. Choose one of the ranges -(2p+q)/2 < k < (p-q)/2 or k > (p+2q)/2. Set m_1=(-2p-q+k)/3, m_2=(p-q+k)/3, m_3=(p+2q+k)/3, a>0 by a^2=(k-m_1)(k-m_2)(k-m_3), and b=m_1m_2+m_1m_3+m_2m_3-lambda^2. From the plane wave z(x,t)=a exp(-i(...	The answer is the smallest r at which the knot-type invariance fails, or -1 if it never fails.	-1	Let p,q be relatively prime positive integers and let r be a positive integer. For either of the two closed-curve families obtained as follows, is the knot type of the resulting closed transverse curve in S^3 unchanged when (p,q) is replaced by (rp,rq)? Fix lambda in R and choose k in either - (2p+q)/2 < k < (p-q)/2 or...	openai/gpt-6-sol
1710.09400#0	1710.09400	https://arxiv.org/abs/1710.09400	quant-ph	conjecture	universality	numerical	B	What is the smallest integer m≥1 for which the following assertion fails; answer -1 if it never fails? Let M_1=Q_1^{-1}Λ_1Q_1 and M_2=Q_2^{-1}Λ_2Q_2 be m×m self-adjoint matrices. After changing basis, write M=M_1+M_2=Λ_1+Q_s^{-1}Λ_2Q_s, where Q_s=Q_2Q_1^{-1}. Let dν_1 and dν_2 be the eigenvalue densities of M_1 and M_2...	Give the smallest m at which the assertion fails, or -1 if it holds for every m≥1.	-1	Let m>=1, let M_1=Q_1^{-1}Lambda_1Q_1 and M_2=Q_2^{-1}Lambda_2Q_2 be m x m self-adjoint matrices, and, after changing basis, write M=M_1+M_2=Lambda_1+Q_s^{-1}Lambda_2Q_s with Q_s=Q_2Q_1^{-1}. Let dnu_1 and dnu_2 be the eigenvalue densities of M_1 and M_2, define dnu^c=dnu_1 boxplus_c dnu_2 as the eigenvalue density of ...	openai/gpt-6-sol
2412.01732#6	2412.01732	https://arxiv.org/abs/2412.01732	quant-ph	conjecture	universality	numerical	B	Let d>=1 and consider the classical Ising-spin system S={-1,1} on Z^d with interaction J={J_E}_{E\Subset Z^d}, Hamiltonian H_V(omega)=-\sum_{E:E\cap V\ne\emptyset}J_E\prod_{x\in E}omega(x) for every finite V\Subset Z^d, and finite-volume Gibbs measures mu_V^tau for boundary conditions tau in Omega={-1,1}^{Z^d}. Suppose...	The answer is the smallest dimension with a counterexample, or -1 if there is none.	-1	Let D>=1 and consider the classical Ising-spin system S={-1,1} on Z^D with interaction J={J_A}_{A\Subset Z^D}, Hamiltonian H_V(omega)=-\sum_{A:A\cap V\ne\emptyset}J_A\prod_{x\in A}\omega(x) for every finite V\Subset Z^D, and finite-volume Gibbs measures mu_V^tau for boundary conditions tau in Omega={-1,1}^{Z^D}. Suppos...	openai/gpt-6-sol
2503.14007#3	2503.14007	https://arxiv.org/abs/2503.14007	math.DS	conjecture	existence	numerical	B	In the hyperplane absolute game on R^d, let beta range over (0,1/3). Bob starts with a closed ball B_0 of radius rho_0. After Bob chooses B_i of radius rho_i, Alice removes the rho_i'-neighbourhood of an affine hyperplane, where rho_i' is at most beta times rho_i; Bob chooses a closed ball B_{i+1} in the remainder with...	The answer is the minimum product dimension among counterexamples, or -1 if the slice-to-product principle holds for every such M, N, and S.	null	Recall the hyperplane absolute game on R^d with parameter beta in (0, 1/3) and target set S contained in R^d: Bob begins by choosing a closed ball B_0 of radius rho_0; after Bob chooses a closed ball B_i of radius rho_i, Alice chooses the rho_i'-neighbourhood of an affine hyperplane L_i in R^d with rho_i' at most beta ...	openai/gpt-6-sol
2408.03803#0	2408.03803	https://arxiv.org/abs/2408.03803	math.NT	conjecture	universality	numerical	D	Let ε* be the infimum of the ε > 0 for which there exists δ > 0 such that, for every B ≥ 1 and every nonzero integer a, the sum over integers m ≤ x^(1−ε) with P⁺(m) ≤ x^δ and (m,a) = 1 of \|π(x;m,−a) − π(x)/φ(m)\| is ≪_(B,ε,a) x/(log x)^B. Here P⁺(m) is the largest prime factor of m, with P⁺(1) = 0; π(x) counts primes p ...	Give the infimum as one number; ε* = 0 is equivalent to Hypothesis Z(1) holding.	0	Does Hypothesis Z(1) hold: for every epsilon > 0, does there exist delta > 0 such that, for every B >= 1 and every nonzero integer a, one has sum over integers m <= x^(1-epsilon) with P^+(m) <= x^delta and (m,a)=1 of \|pi(x;m,-a) - pi(x)/phi(m)\| <<_(B,epsilon,a) x/(log x)^B, where P^+(m) is the largest prime factor of m...	openai/gpt-6-sol
2411.08939#0	2411.08939	https://arxiv.org/abs/2411.08939	math.DS	conjecture	universality	numerical	E	Work with Wang tilesets over the colors white and black. A tile assigns a color to each edge of the unit square, and X_T ⊆ T^(Z^2) consists of configurations in which neighboring cells have the same color on their common edge. A tile is even if each color occurs on an even number of its four edges. The 8 even tiles are...	The answer is the number of such tilesets whose subshift belongs to L^1.	0	Work with Wang tilesets over the two colors white and black: a tile assigns a color to each edge of the unit square, and a tileset T induces the Z^2-subshift of finite type X_T contained in T^(Z^2) consisting of the configurations in which two neighboring cells carry the same color on their common edge. Call a tile eve...	openai/gpt-6-sol
2408.10183#4	2408.10183	https://arxiv.org/abs/2408.10183	math.NT	conjecture	universality	numerical	B	Let P=theta^4+tP_1(theta)+...+t^rP_r(theta) in Q<t,theta>, where theta=t d/dt and each P_i is a degree-four polynomial, be a Calabi–Yau operator of order 4 with a maximal-unipotent-monodromy point at t=0, arising from a family of Calabi–Yau threefold fibres X_t. For each good prime p and regular t in F_p, form the Frob...	The answer is the smallest counterexample prime, or -1 if the assertion holds for every specified operator and every eligible prime and t.	-1	Let P=theta^4+tP_1(theta)+...+t^rP_r(theta) in Q<t,theta>, with theta=t d/dt and each P_i a degree-four polynomial, be a Calabi–Yau operator of order 4 having a maximal-unipotent-monodromy point at t=0 and arising from a family of Calabi–Yau threefold fibres X_t. For every prime p and every regular t in F_p for which p...	openai/gpt-6-sol
2602.16580#2	2602.16580	https://arxiv.org/abs/2602.16580	math.AG	conjecture	universality	numerical	B	For nonnegative integers m and s, let z=(z_0,...,z_m) be homogeneous coordinates on P^m. Define gamma:P^m dashrightarrow P^{m+s} on P^m minus V(z_0) by y_0=z_0^2, y_i=z_0z_i for 1<=i<=m, and y_i=gamma_i(z) for m+1<=i<=m+s, where gamma_{m+1},...,gamma_{m+s} are general homogeneous quadratic polynomials in z_0,...,z_m. W...	The answer is the smallest m admitting a failure for some s; -1 means the stated degree formulas hold for every nonnegative m and s.	-1	For all nonnegative integers m and s, let z=(z_0,...,z_m) be homogeneous coordinates on P^m, and let gamma:P^m dashrightarrow P^{m+s} be the rational map defined on P^m minus V(z_0) with homogeneous output coordinates y=(y_0,...,y_{m+s}) given by y_0=z_0^2, y_i=z_0z_i for 1<=i<=m, and y_i=gamma_i(z) for m+1<=i<=m+s, wh...	openai/gpt-6-sol
2502.10747#1	2502.10747	https://arxiv.org/abs/2502.10747	math.AP	conjecture	extension	numerical	B	For every d in N, sigma in (0,infinity)\N, and f in S(R^d), let u be the bounded solution on R^{d+1}_+ = R^d x (0,infinity) of div(y^{1-2sigma} grad u)=0 with u(x,0)=f(x). Writing [sigma] for the integer part of sigma, the ordinary Dirichlet-to-Neumann map is not well-defined, whereas Chang–González obtain lim_{y->0+} ...	The answer is the smallest failing dimension d, or -1 if the formula admits such a derivation or reinterpretation for every specified d, sigma, and f.	-1	For every d in N, sigma in (0,infinity)\N, and f in S(R^d), let u be the bounded solution on R^{d+1}_+ = R^d x (0,infinity) of div(y^{1-2sigma} grad u)=0 with u(x,0)=f(x). Writing [sigma] for the integer part of sigma, the ordinary Dirichlet-to-Neumann map is not well-defined, whereas Chang-González obtain lim_{y->0+} ...	openai/gpt-6-sol
2501.07406#3	2501.07406	https://arxiv.org/abs/2501.07406	math-ph	conjecture	universality	numerical	B	Work on R^4 identified with the quaternions H, and on S^4 = H union {infinity}. Identify the conformal group of R^4 with SL(2,H), the group of 2x2 quaternionic matrices [[A,B],[C,D]] satisfying \|A C^{-1} D C - B C\| = 1 (read as \|A D\| = 1 when C = 0, with conjugation by 0 taken to be the identity). Its action on S^4 is ...	Give the smallest instanton number k of a counterexample, or -1 if the containment holds for every non-flat instanton.	-1	Work on R^4 identified with the quaternions H, and on S^4 = H union {infinity}. Identify the conformal group of R^4 with SL(2,H), the group of 2x2 quaternionic matrices [[A,B],[C,D]] satisfying \|A C^{-1} D C - B C\| = 1 (the condition being read as \|A D\| = 1 when C = 0, with the convention that conjugation by 0 is the i...	openai/gpt-6-sol
2506.18788#13	2506.18788	https://arxiv.org/abs/2506.18788	math.CO	conjecture	universality	numerical	B	Let M(G) be the cycle matroid of a graph G, and write its Speyer polynomial as g_{M(G)}(t)=t\sum_{i=0}^{\operatorname{rk}(M(G))-1}N_i(G)(1+t)^i. For every integer n≥5, let C^{2n}_{1,n-1} be the graph on vertices {1,…,2n}, with labels read modulo 2n, obtained from the 2n-cycle by adding the edges {i,i+n−1} for 1≤i≤2n. W...	The answer is the smallest counterexample n, or −1 if the equality holds for every integer n≥5.	-1	Let M(G) be the cycle matroid of a graph G, and write its Speyer polynomial as g_{M(G)}(t)=t\sum_{i=0}^{\operatorname{rk}(M(G))-1} N_i(G)(1+t)^i; thus N_2(G) is the coefficient N_2(M(G)) in this expansion. For every integer n >= 5, let C^{2n}_{1,n-1} be the graph on vertices {1,...,2n}, with vertex labels read modulo 2...	openai/gpt-6-sol
2502.12727#0	2502.12727	https://arxiv.org/abs/2502.12727	math.CO	conjecture	existence	numerical	C	For every finite set P⊆ℝ² with N=\|P\|≥2, let Λ(P)={p·q:p,q∈P}, where · is the Euclidean dot product. What is the sharp constant c=inf_{P⊆ℝ² finite, \|P\|≥2} \|Λ(P)\|log\|P\|/\|P\| in the inequality \|Λ(P)\|≥cN/log N?	Give one real number; the original question has answer Yes exactly when c*>0.	null	For every finite set P subseteq R^2 with N=\|P\|>=2, let Lambda(P)={p dot q : p,q in P}, where dot denotes the Euclidean dot product. Is there an absolute constant c>0 such that \|Lambda(P)\|>=cN/log N for every such P? Equivalently in the paper's asymptotic terminology, can the currently proved bound \|Lambda(P)\| gtrsim N^...	openai/gpt-6-sol
1704.06349#5	1704.06349	https://arxiv.org/abs/1704.06349	math.DS	conjecture	universality	numerical	B	What is the smallest cardinality \|A\| of a finite set A for which there exist a countable group Γ and a continuous Γ-equivariant map φ:A^Γ→A^Γ that is injective but not surjective? Here A^Γ has the product topology and Γ acts by the left shift (g·x)(f)=x(g^{-1}f). Answer -1 if no such cardinality exists.	The answer is the smallest such \|A\|, or -1 if injectivity implies surjectivity in every case.	-1	For every finite set A, every countable group Γ, and every continuous Γ-equivariant map φ: A^Γ → A^Γ, where A^Γ is the product-topological space of functions x: Γ → A and Γ acts by the left shift (g·x)(f)=x(g^{-1}f), must injectivity of φ imply surjectivity of φ?	openai/gpt-6-sol
2411.14308#6	2411.14308	https://arxiv.org/abs/2411.14308	math.NT	conjecture	universality	numerical	B	Let N={0,1,2,...}. What is the smallest integer n>51 for which there do not exist w,x,y,z in N such that n=w(5w-1)/2+x(5x-1)/2+y(5y+1)/2+z(5z+1)/2? Answer -1 if no such n exists.	The answer is the smallest integer n>51 for which the representation fails, or -1 if it never fails.	-1	Let N={0,1,2,...}. Is it true that, for every integer n>51, there exist w,x,y,z in N such that n=w(5w-1)/2+x(5x-1)/2+y(5y+1)/2+z(5z+1)/2?	openai/gpt-6-sol
1709.01169#5	1709.01169	https://arxiv.org/abs/1709.01169	math.GR	conjecture	universality	numerical	B	What is the smallest odd prime p for which constructing a lookup table for the inverse map from a black-box copy of the prime field to F_p is computationally feasible, and for which there exist a finite field F of characteristic p, a sufficiently rich algebraic structure A(F) functorially defined over F (for example, o...	The answer is the smallest such prime p, or -1 if there is none.	-1	For every small odd prime p, where “small” means that it is computationally feasible to construct a lookup table for the inverse map from a black-box copy of the prime field to F_p, every finite field F of characteristic p, and every algebraic structure A(F) functorially defined over F that is sufficiently rich—for exa...	openai/gpt-6-sol
1710.05274#4	1710.05274	https://arxiv.org/abs/1710.05274	math.AG	conjecture	universality	numerical	B	What is the smallest length n of a finite ordered collection (E_1,...,E_n) in D^b(coh(X)) for some smooth projective nonrational surface X over an algebraically closed field k of characteristic 0 such that Ext^m(E_i,E_i)=0 for every i and every m != 0, Ext^0(E_i,E_i)=k for every i, Ext^m(E_j,E_i)=0 for all m and all j>...	The answer is the smallest such n, or -1 if no counterexample exists.	-1	Let k be an algebraically closed field of characteristic 0, and let X be a smooth projective surface over k. Suppose there is a finite ordered collection (E_1,...,E_n) of objects of D^b(coh(X)) such that Ext^m(E_i,E_i)=0 for every i and every m != 0, Ext^0(E_i,E_i)=k for every i, Ext^m(E_j,E_i)=0 for all m and all j>i,...	openai/gpt-6-sol
1801.07094#5	1801.07094	https://arxiv.org/abs/1801.07094	math.AG	conjecture	universality	numerical	E	Let F be a non-archimedean local field with ring of integers O_F and residue field k, let G be a connected reductive F-group, let Gcal=Gcal_f be the parahoric O_F-group scheme attached to a facet f, and let chi: G_m,O_F -> Gcal be an O_F-cocharacter. Write Fl_Hcal for the affine flag k-ind-scheme of an O_F-group scheme...	Give one integer from 0 to 3, counting each map type once if it fails for any permitted (G,Gcal,chi).	0	Let F be a non-archimedean local field with ring of integers O_F and residue field k, let G be a connected reductive F-group, let Gcal=Gcal_f be the parahoric O_F-group scheme attached to a facet f, and let chi: G_m,O_F -> Gcal be an O_F-cocharacter. Write Fl_Hcal for the affine flag k-ind-scheme of an O_F-group scheme...	openai/gpt-6-sol
2407.16093#9	2407.16093	https://arxiv.org/abs/2407.16093	math.CO	conjecture	universality	numerical	B	What is the smallest integer n≥1 for which there exists an irreducible weighted directed graph G=(X,E,r) with positive edge weights, every edge reversible and r(+e)=r(-e) for each pair of oppositely oriented edges, and n distinct pinned unoriented edges 1,...,n, such that dim span{τ_x:x∈X}≠n+1? For each root x∈X, defin...	The answer is the smallest n admitting a counterexample, or -1 if the equality holds for every permitted n and graph.	-1	For every integer n ≥ 1, let G=(X,E,r) be an irreducible weighted directed graph with positive edge weights, assume that every edge is reversible (r(+e)=r(-e) for each pair of oppositely oriented edges), and choose n distinct pinned unoriented edges 1,...,n. For each root x∈X, let T_x range over spanning trees whose ed...	openai/gpt-6-sol
0705.3605#4	0705.3605	https://arxiv.org/abs/0705.3605	math.RT	conjecture	universality	numerical	B	What is the smallest prime power q for which the following assertion fails for some admissible (alpha;beta)? Let k=F_q, let U(k) be the compact group of infinite upper triangular matrices over k with all diagonal entries equal to 1, and let alpha=(alpha_1,alpha_2,...) and beta=(beta_1,beta_2,...) be nonnegative weakly ...	The answer is the smallest prime power q admitting an admissible (alpha;beta) for which the assertion fails, or -1 if there is none.	-1	Let k=F_q be a finite field, let U(k) be the compact group of infinite upper triangular matrices over k with all diagonal entries equal to 1, and let (alpha;beta), where alpha=(alpha_1,alpha_2,...) and beta=(beta_1,beta_2,...), be nonnegative weakly decreasing sequences satisfying sum_i alpha_i+sum_i beta_i<=1. For the...	openai/gpt-6-sol
2412.19799#6	2412.19799	https://arxiv.org/abs/2412.19799	math.AC	conjecture	universality	numerical	B	What is the smallest prime p for which there exist a field k of characteristic p and a smooth cubic surface X over k such that, writing F:X→X for the absolute Frobenius morphism, there is no indecomposable coherent sheaf M on X with F_O_X ≅ O_X ⊕ M and F_^e M indecomposable for every integer e ≥ 0? Answer -1 if no su...	The answer is the smallest such prime p, or -1 if the stated property holds for every field of positive characteristic and every smooth cubic surface over it.	-1	Let k be a field of characteristic p > 0, let X be a smooth cubic surface over k, and let F:X -> X be its absolute Frobenius morphism. Is it true that there is an indecomposable coherent sheaf M on X such that F_O_X is isomorphic to O_X ⊕ M and, for every integer e >= 0, F_^e M is indecomposable? Equivalently in the ...	openai/gpt-6-sol
1712.01243#0	1712.01243	https://arxiv.org/abs/1712.01243	math.CO	conjecture	quantity	numerical	A	For each integer n >= 1, let B_n be the set of sign vectors delta=(delta_0,...,delta_n) in {-1,1}^{n+1} satisfying sum_{k=0}^n delta_k binom(n,k)=0. Call a member of B_n trivial if, when n is even, delta_k=epsilon(-1)^k for every 0 <= k <= n for some epsilon in {-1,1}, and, when n is odd, delta_{n-k}=-delta_k for every...	Answer with the value of the limit, if it exists.	5/6	For each integer n >= 1, let B_n be the set of sign vectors delta=(delta_0,...,delta_n) in {-1,1}^{n+1} satisfying sum_{k=0}^n delta_k binom(n,k)=0. Call a member of B_n trivial if, when n is even, delta_k=epsilon(-1)^k for every 0 <= k <= n for some epsilon in {-1,1}, and, when n is odd, delta_{n-k}=-delta_k for every...	openai/gpt-6-sol
1704.04510#0	1704.04510	https://arxiv.org/abs/1704.04510	math.RT	conjecture	universality	numerical	B	For every integer i>0, let M_{2i} be the braid matroid on the ordered pairs (a,b) of distinct elements of [2i], whose bases are the oriented spanning trees of the complete graph on [2i]. Write P_{M_{2i}}(t)=sum_j C_{M_{2i},j}t^j for its non-equivariant Kazhdan–Lusztig polynomial. Its largest possible degree is i-1, sin...	The answer is the smallest counterexample i, or -1 if the equality holds for every integer i>0.	-1	For every integer i>0, let M_{2i} be the braid matroid on the ordered pairs (a,b) of distinct elements of [2i], whose bases are the oriented spanning trees of the complete graph on [2i], and write P_{M_{2i}}(t)=sum_j C_{M_{2i},j}t^j for its non-equivariant Kazhdan-Lusztig polynomial. Since the degree of P_{M_{2i}}(t) i...	openai/gpt-6-sol
1708.09700#3	1708.09700	https://arxiv.org/abs/1708.09700	math.CO	conjecture	universality	numerical	B	Let G be a simple graph with n_G vertices and adjacency matrix A. Call G walk-regular if, for every integer ell >= 1, every vertex of G is contained in the same number of closed walks of length ell. For every beta >= 0, define S^V(G,beta)=-sum_{i=1}^{n_G} p_i(beta) log p_i(beta), where p_i(beta)=[exp(beta A)]_{ii}/Tr(e...	Give the smallest such vertex count, or -1 if there is no counterexample.	-1	Let G be a simple graph with n_G vertices and adjacency matrix A. Call G walk-regular if, for every integer ell >= 1, every vertex of G is contained in the same number of closed walks of length ell. For every beta >= 0, define S^V(G,beta)=-sum_{i=1}^{n_G} p_i(beta) log p_i(beta), where p_i(beta)=[exp(beta A)]_{ii}/Tr(e...	openai/gpt-6-sol
1712.04411#17	1712.04411	https://arxiv.org/abs/1712.04411	math.AC	conjecture	universality	numerical	B	For every integer n >= 2, let I_n=(a^(2n)b^(2n)c^(2n), b^(4n)c^(2n), a^(3n)c^(3n), a^(6n-1)b) be the equigenerated degree-6n monomial ideal in k[a,b,c]. Let Stab(I_n) be the least integer D such that, for every d >= D and all i,j, beta_(i,j+6nd)(I_n^d) != 0 if and only if beta_(i,j+6nD)(I_n^D) != 0. What is the smalles...	The answer is the smallest counterexample n, or -1 if the equality holds for every n >= 2; in the latter case the stabilization indices are arbitrarily large.	-1	For every integer n >= 2, let I_n=(a^(2n)b^(2n)c^(2n), b^(4n)c^(2n), a^(3n)c^(3n), a^(6n-1)b) be the equigenerated degree-6n monomial ideal in k[a,b,c]. Is its stabilization index Stab(I_n) equal to 12n-13, where Stab(I_n) is the least integer D such that, for every d >= D and all i,j, beta_(i,j+6nd)(I_n^d) != 0 if and...	openai/gpt-6-sol
1704.01327#6	1704.01327	https://arxiv.org/abs/1704.01327	math-ph	conjecture	quantity	numerical	A	Let H be a real three-dimensional Hilbert space, and let T(H) be the space of third-order tensors A whose components in orthonormal bases transform by a_ijk=p_iq p_jr p_ks b_qrs. Suppose A is symmetric, meaning that a_ijk is unchanged under every permutation of i, j, and k. How many independent scalar invariants does A...	Give the number of independent scalar invariants.	7	Let H be a real three-dimensional Hilbert space, and let T(H) be the space of third-order tensors A whose components in orthonormal bases transform by a_ijk=p_iq p_jr p_ks b_qrs. Suppose A is symmetric, meaning that, in an orthonormal basis, a_ijk is unchanged under every permutation of i, j, and k. Does A have exactly...	openai/gpt-6-sol
2603.29082#1	2603.29082	https://arxiv.org/abs/2603.29082	math.RT	conjecture	universality	numerical	B	Fix integers r>=3 and m>=2. For each j_0 in {-r+1,...,-2}, define P_{j_0,n}(c) for n>=-2r by P_{j_0,j_0}(c)=1, P_{j_0,j}(c)=0 for every j in {-2r,...,-1} with j!=j_0, and, for every k>=0, (2r+m+km)P_{j_0,k}(c)=2c(r+(1+k-r)m)P_{j_0,k-r}(c)-(k-(2r-1))mP_{j_0,k-2r}(c). What is the smallest j_0 in {-r+1,...,-2} for which t...	The answer is the smallest failing j_0, or -1 if every specified family satisfies such an equation.	-1	Fix integers r>=3 and m>=2. For each j_0 in {-r+1,...,-2}, define polynomials P_{j_0,n}(c), for n>=-2r, by P_{j_0,j_0}(c)=1, P_{j_0,j}(c)=0 for every j in {-2r,...,-1} with j!=j_0, and, for every k>=0, (2r+m+km)P_{j_0,k}(c)=2c(r+(1+k-r)m)P_{j_0,k-r}(c)-(k-(2r-1))mP_{j_0,k-2r}(c). Do the resulting r-2 genuinely new type...	openai/gpt-6-sol
2501.17016#2	2501.17016	https://arxiv.org/abs/2501.17016	math.AP	conjecture	universality	numerical	B	What is the smallest complex dimension n for which the following counterexample exists? Let (M,omega_0) be a closed Hermitian manifold of complex dimension n; let chi be a smooth real (1,1)-form with lambda[chi](x) in Gamma for every x in M; and let Gamma subset R^n be an open symmetric convex cone containing the posit...	The answer is the smallest complex dimension admitting a counterexample, or -1 if none exists.	-1	Let (M,omega_0) be a closed Hermitian manifold of complex dimension n, let chi be a smooth real (1,1)-form with lambda[chi](x) in Gamma for every x in M, and let Gamma subset R^n be an open symmetric convex cone containing the positive orthant and contained in {lambda: sum_i lambda_i>0}. Let f:Gamma->R_+ be smooth, sym...	openai/gpt-6-sol
2507.12579#2	2507.12579	https://arxiv.org/abs/2507.12579	math.CO	conjecture	universality	numerical	B	For every integer l >= 0, let ILT_0(K_2)=K_2 and obtain ILT_l(K_2) recursively by simultaneously adding, for every vertex v of ILT_{l-1}(K_2), a new clone v' whose neighborhood is the closed neighborhood N[v] in ILT_{l-1}(K_2). Let Z(G) be the minimum size of an initial forced set from which the zero-forcing rule—a for...	Give the smallest such l, or -1 if no such l exists.	-1	For every integer l >= 0, let ILT_0(K_2)=K_2 and obtain ILT_l(K_2) recursively by, at each step, simultaneously adding for every vertex v of ILT_{l-1}(K_2) a new clone v' whose neighborhood is the closed neighborhood N[v] in ILT_{l-1}(K_2). If Z(G) denotes the minimum size of an initial forced set from which the zero-f...	openai/gpt-6-sol
2506.09840#0	2506.09840	https://arxiv.org/abs/2506.09840	math.DG	conjecture	universality	numerical	B	What is the smallest integer n≥1 for which there exist θ∈(0,π) and a properly embedded, smooth, compact, strictly convex hypersurface Σ in the closed half-space overline{R^{n+1}_+}, with boundary ∂Σ contained in ∂R^{n+1}_+, such that, writing ν for its outward unit normal, e=−E_{n+1}, X for the position vector, and K f...	Return the smallest such n, or −1 if every hypersurface satisfying the stated conditions equals C_θ.	-1	For every n >= 1 and theta in (0,pi), let Sigma be a properly embedded, smooth, compact, strictly convex hypersurface in the closed half-space overline{R^{n+1}_+}, with boundary partial Sigma contained in partial R^{n+1}_+, and let nu be its outward unit normal. Put e=-E_{n+1}, let X denote the position vector, and let...	openai/gpt-6-sol
2501.00394#0	2501.00394	https://arxiv.org/abs/2501.00394	math.AG	conjecture	universality	numerical	E	Let Q=(Q_f⊂Q_0,Q_1,W) be a cluster quiver with potential (no loops or 2-cycles), decorated by positive integers (r_u)_{u∈Q_0}, and let G=∏_{u∈Q_0\Q_f}GL(r_u). Choose a stability character θ(g)=∏_{u∈Q_0\Q_f}det(g_u)^{σ_u} such that semistability equals stability for both the representation space and the critical locus Z...	Give the number of gauge nodes for which the identity corresponding to its strict inequality fails; nodes with N_f(v)=N_a(v) are not counted.	0	Let Q=(Q_f subset Q_0,Q_1,W) be a cluster quiver with potential (no loops or 2-cycles), decorated by positive integers (r_u)_{u in Q_0}, and let G=product_{u in Q_0\Q_f} GL(r_u). Choose a stability character theta(g)=product_{u in Q_0\Q_f} det(g_u)^{sigma_u} such that semistability equals stability for both the represe...	openai/gpt-6-sol
2511.07316#0	2511.07316	https://arxiv.org/abs/2511.07316	math.CO	conjecture	universality	numerical	B	What is the smallest integer n >= 3 for which there exists a simple graph G subset of binom([n],2) with no universal vertex that is copious but not topologically copious? Return -1 if no such n exists. Here [n]={1,...,n}, and a universal vertex is one adjacent to every other vertex. Copious means that, for L_G(x)=sum_{...	Give the smallest such n, or -1 if every graph in the stated family that is copious is topologically copious.	-1	For every integer n >= 3 and every simple graph G subset of binom([n],2) with no universal vertex (that is, no vertex adjacent to all other vertices), is G topologically copious whenever it is copious? Here [n]={1,...,n}; copious means that, for the graphical scattering potential L_G(x)=sum_{ij in G} s_{ij} log(x_i-x_j...	openai/gpt-6-sol
2410.24124#10	2410.24124	https://arxiv.org/abs/2410.24124	math.AT	conjecture	existence	numerical	G	Let D(BU(1)) = ⨆_{n≥0} D_n(BU(1)), where D_n(BU(1)) = E(Σ_n) ×_{Σ_n} BU(1)^n = E(Σ_n) ×_{Σ_n} (CP^∞)^n, equipped with its underlying E_2-structure (the classical and Atiyah-induced E_2-structures are homotopic). For every n≥0, let F_n be the subspace of D_n(BU(1)) consisting of classes (p_1,…,p_n) ×_{Σ_n} (x_1,…,x_n) w...	Give one cardinal number; the original question has answer Yes exactly when this cardinality is uncountable.	null	Let D(BU(1)) = ⨆_{n>=0} D_n(BU(1)), where D_n(BU(1)) = E(Sigma_n) ×_{Sigma_n} BU(1)^n = E(Sigma_n) ×_{Sigma_n} (CP^infinity)^n, equipped with its underlying E_2-structure (the classical and Atiyah-induced E_2-structures are homotopic). For every n >= 0, let F_n be the subspace of D_n(BU(1)) consisting of classes (p_1,....	openai/gpt-6-sol
2504.03341#4	2504.03341	https://arxiv.org/abs/2504.03341	math.RT	conjecture	universality	numerical	B	Let \mathbb{F} be an algebraically closed field of prime characteristic p, let G be a reductive group over \mathbb{F} in the paper's conventions, let B^-\subset G be the Borel opposite to a fixed Borel B, and let X be the fixed smooth complete curve over \mathbb{F}. Let \overline{\mathrm{Bun}}_{B^-} be Drinfeld's compa...	The answer is the smallest rank exhibiting a failure, or -1 if the dimensions are independent of coefficient characteristic for every G, i, and x in the stated setting.	-1	Let \mathbb{F} be an algebraically closed field of prime characteristic p, let G be a reductive group over \mathbb{F} in the paper's conventions, let B^-\subset G be the Borel opposite to a fixed Borel B, and let X be the fixed smooth complete curve over \mathbb{F}. Let \overline{\mathrm{Bun}}_{B^-} be Drinfeld's compa...	openai/gpt-6-sol
2512.04641#12	2512.04641	https://arxiv.org/abs/2512.04641	math.NT	conjecture	extension	numerical	B	What is the smallest degree d=[F_w:Q_p] for which there are data satisfying all the conditions below such that at least one of the two stated Hecke-eigenspace assertions fails? Answer -1 if no such degree exists. Fix a prime p>=5, an imaginary quadratic field F_0 in which p splits, and a Galois totally real field F^+....	Return the least failing degree d among admissible data, or -1 if both assertions hold for every admissible choice.	-1	Fix a prime p >= 5, an imaginary quadratic field F_0 in which p splits, and a Galois totally real field F^+; put F := F_0 F^+ and Phi := Gal(F/F_0). Fix an isomorphism between a fixed algebraic closure of Q_p and C and use it to identify Hom(F,C) with Hom(F,Qbar_p) = Phi. Let v be the p-adic place of F_0 determined by ...	openai/gpt-6-sol
2507.11013#5	2507.11013	https://arxiv.org/abs/2507.11013	math.CO	conjecture	universality	numerical	B	What is the smallest dimension n in which there is a polytope K in R^n, with finite set H of outer unit normal vectors, for which k > max_{H' subseteq H} h(H')? If there is no such dimension, answer -1. For any finite set A of vectors, an A-convex set is an intersection of halfspaces {x : <a,x> <= b} with a in A; conv_...	Give the smallest counterexample dimension, or -1 if the stated upper bound holds for every such polytope.	-1	Let K be a polytope in R^n with finite set H of outer unit normal vectors. Call a set H-convex if it is an intersection of halfspaces {x : <a,x> <= b} with a in H, and call a set K-strongly convex if it is an intersection of translates of K. For every closed X subset of R^n, let conv_H X be the minimal H-convex set con...	openai/gpt-6-sol
1705.01819#2	1705.01819	https://arxiv.org/abs/1705.01819	math.AG	conjecture	construction	numerical	B	For each integer n >= 2, let IG(2,2n) be the smooth projective variety of 2-dimensional isotropic subspaces of a 2n-dimensional complex symplectic vector space, and assume its genus-zero Gromov--Witten potential converges on an open neighbourhood of the origin after setting q=1. What is the smallest such n for which no...	The answer is the smallest n at which the specified construction fails, or -1 if it succeeds for every n >= 2 under the stated convergence assumption.	-1	For every integer n >= 2, let IG(2,2n) be the smooth projective variety of 2-dimensional isotropic subspaces of a 2n-dimensional complex vector space equipped with a symplectic form. Assume that the genus-zero Gromov--Witten potential of IG(2,2n), after setting q=1, converges on an open neighbourhood of the origin. Can...	openai/gpt-6-sol
1709.00547#0	1709.00547	https://arxiv.org/abs/1709.00547	math.RT	conjecture	universality	numerical	B	For every integer n >= 2, let g = sl_n(C), let W be its Weyl group, and let O_0 be the principal block of its BGG category O. For w in W, write L(w)=L(w·0) for the simple highest-weight module of highest weight w·0 under the dot action w·lambda=w(lambda+rho)-rho. Let theta_w be the indecomposable projective endofunctor...	-1 means theta_x L(y) is either indecomposable or zero for every x,y in W and every integer n >= 2.	-1	For every integer n >= 2, let g = sl_n(C), let W be its Weyl group, and let O_0 be the principal block of its BGG category O. For w in W, write L(w)=L(w·0) for the simple highest-weight module of highest weight w·0 under the dot action w·lambda=w(lambda+rho)-rho, and let theta_w be the indecomposable projective endofun...	openai/gpt-6-sol
2504.02223#3	2504.02223	https://arxiv.org/abs/2504.02223	math.AG	conjecture	universality	numerical	B	What is the smallest integer q≥0 for which the following assertion fails for some perfect field k of characteristic p>0? For every log-smooth Q-modulus pair \mathcal X=(X,D), with X smooth over k, D an effective Q-simple-normal-crossings divisor, and \mathcal X^\circ=X-\|D\|, the Brylinski--Kato-filtration sheaf \underli...	Give the smallest failing q, or -1 if there is none.	-1	Let k be a perfect field of characteristic p>0. For every q>=0 and every log-smooth Q-modulus pair \mathcal X=(X,D), where X is smooth over k, D is an effective Q-simple-normal-crossings divisor, and \mathcal X^\circ=X-\|D\|, let H^{q+1}_{ur} be the Nisnevich sheafification on Sm_k of U\mapsto H^{q+1}_{et}(U,Q/Z(q)), and...	openai/gpt-6-sol
1802.01793#1	1802.01793	https://arxiv.org/abs/1802.01793	math.NT	conjecture	universality	numerical	B	Let T_p(x)=2T_p^{classical}(x/2), where T_p^{classical} is the Chebyshev polynomial of the first kind. For each integer n>=3 such that n!=T_p(j) for every prime p and every integer j>=3, define s_0(n)=1, s_1(n)=n+1, and s_{k+2}(n)-n s_{k+1}(n)+s_k(n)=0 for every k>=0. Let k_N be the index of the Nth prime term, in incr...	Give the smallest qualifying integer n where the property fails, or -1 if no such n exists.	-1	For each integer n>=3 such that n != T_p(j) for every prime p and every integer j>=3, let (s_k(n))_{k>=0} be defined by s_0(n)=1, s_1(n)=n+1, and s_{k+2}(n)-n s_{k+1}(n)+s_k(n)=0 for every k>=0, where T_p(x)=2T_p^{classical}(x/2) is the dilated Chebyshev polynomial of the first kind. If k_N is the index of the Nth prim...	openai/gpt-6-sol
2410.00720#3	2410.00720	https://arxiv.org/abs/2410.00720	math.QA	conjecture	universality	numerical	B	What is the smallest integer m≥1 for which there exist a simply connected compact simple Lie group K, a number 0<q<1, pairwise distinct μ_1,...,μ_m∈P^+, positive a_1,...,a_m, and λ∈P^+ such that π_{μ_1}⊕⋯⊕π_{μ_m} is faithful, {μ_1,...,μ_m} is invariant under μ↦−w_0μ, a_l=a_k whenever −w_0μ_l=μ_k, the case m=1 with μ_1=...	Give the smallest such m, or −1 if no counterexample exists.	-1	Let K be a simply connected compact simple Lie group, let 0 < q < 1, and let P^+ be the dominant integral weights of its complexified Lie algebra. For pairwise distinct μ_1,...,μ_m in P^+ and a_1,...,a_m > 0, assume that π_{μ_1} ⊕ ... ⊕ π_{μ_m} is faithful, that {μ_1,...,μ_m} is invariant under μ ↦ -w_0μ (with w_0 the ...	openai/gpt-6-sol

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ResearchMath-2-6Sol-Rewrite

97,497 research-level mathematics questions reformatted into benchmark-shaped items, derived from amphora/ArXivOpenProblems.

Two configs:

config	rows	the answer is
`numerical`	35,274	a single number
`yesno`	62,223	Yes or No

⚠️ There are no answer keys

Every item derives from a future-work or open-problem statement in an arXiv paper. The answers are, by construction, not known — not to the source paper and in most cases not to the literature. This is a corpus of well-posed unanswered questions, not a scoreable benchmark. Do not use it to compute accuracy without sourcing answers independently.

How a research question becomes a number

Most source questions ask whether something holds, not for a value. The conversion uses seven routes; route records which was applied.

route	construction	share of `numerical`
B	minimal counterexample: "smallest p at which the property fails; −1 if none"	80.8%
E	size of the exceptional set (0 = the original's "yes")	6.6%
A	the question already asked for a quantity	6.0%
C	the sharp constant in an inequality	2.5%
D	the critical value of a continuous parameter	2.4%
G	count of a finite classification	1.4%
F	minimal/maximal multiplicity or order (1 = the original's "yes")	0.3%

Route B dominates. distinguished_value gives the number corresponding to the original question's "yes" — usually -1.

Route B strengthens the problem. "Is there a counterexample?" and "what is the smallest counterexample?" are different questions; the second is strictly harder. The reformulation is faithful in that the distinguished value is equivalent to the original's affirmative answer, but the item is not of equal difficulty to its source.

Fields

field	description
`uid`	`<arxiv_id>#<index>`, joins to the source dataset
`arxiv_id`, `paper_url`	provenance
`primary_category`	arXiv category of the source paper
`signal_type`	source statement type: `conjecture`, `limitation`, `open_problem`, `natural_extension`, `announced_forthcoming`
`question_category`	logical shape of the original question (see below)
`format`	`numerical` or `yesno`
`route`	`A`–`G` for numerical items, `null` for yes/no
`question`	the reformatted item
`answer_convention`	one sentence on what the answer means
`distinguished_value`	number equal to the original's "yes", or `null`
`source_question`	the unmodified original
`engine`	`openai/gpt-6-sol`

question_category is one of universality, existence, extension, characterization, construction, comparison, classification, quantity — assigned during conversion, so no re-classification is needed.

What converts, and what doesn't

Numerical yield by logical shape of the source question:

question_category	→ numerical
universality ("does P hold for every X")	~65%
quantity	~59%
classification	~42%
existence	~29%
extension	~21%
construction	~19%
characterization	~13%
comparison	~9%

By source statement type, measured on the full run:

signal_type	n	→ numerical
conjecture	39,444	50%
open_problem	8,756	37%
announced_forthcoming	897	31%
limitation	41,265	25%
natural_extension	7,197	24%

Universality statements convert because negating "for all n" yields a minimal counterexample. Questions asking for a construction, a characterization, or a qualitative comparison have no number in them, and no reformulation produces one — those became yes/no items.

Construction and filtering

Converted with openai/gpt-6-sol via the OpenRouter batch API, temperature 0.2. 98,000 source questions in, 97,497 kept. Removed: 441 unparseable responses, 55 items where a yes/no answer had been encoded numerically (indicator variables, 2a+b), 4 degenerate short items, 3 duplicate uids.

The conversion prompt forbade encoding booleans as numbers; the residual rate was 0.16% before filtering and 0 after.

Caveats

No answer keys. Restating the warning above because it determines what this dataset is for.
yesno items are unverified claims, not labelled true/false.
Route B items are harder than their sources.
A small number of arxiv_id values are wrong, inherited from the source dataset: for part of that corpus the id came from a model-written field rather than the filename, so a few questions are attached to the wrong paper.
signal_type is an extraction label, not a verified claim that a problem is still open.

Licensing

Questions and derived fields: CC-BY-4.0. source_question is inherited from amphora/ArXivOpenProblems. Unlike that dataset, no verbatim paper excerpts are included here.

Citation

@article{son2026researchmath,
  title={ResearchMath-14K: Scaling Research-Level Mathematics via Agents},
  author={Son, Guijin and Yi, Seungyeop and Gwak, Minju and Ko, Hyunwoo and Jang, Wongi and Yu, Youngjae},
  journal={arXiv preprint arXiv:2605.28003},
  year={2026}
}

Collaborations

I'm interested in creating larger datasets to train open models for research-level math. If you are interested let me know. (guijin.son@snu.ac.kr)

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Paper for amphora/ResearchMath-2-6Sol-Rewrite

ResearchMath-14K: Scaling Research-Level Mathematics via Agents

Paper • 2605.28003 • Published May 27 • 47