ResearchMath-14K: Scaling Research-Level Mathematics via Agents
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0704.0028#0 | 0704.0028 | https://arxiv.org/abs/0704.0028 | numerical | For a symmetric real 2m×2m matrix C=(c_{i,j}), m≥1, define haf C = (1/(m!2^m)) times the sum, over all permutations π of {1,…,2m}, of the product c_{π(1),π(2)}c_{π(3),π(4)}⋯c_{π(2m−1),π(2m)}. Let n≥1 and p≥1 be integers, let A=(a_{i,j}) be a real symmetric positive semi-definite n×n matrix, and let B be the 2pn×2pn mat... | The answer is the smallest such n, or −1 if the inequality and equality characterization hold for every permitted n, p, and A. | -1 | SOLVED_EXACT | true | -1 | high | https://arxiv.org/abs/2609.20234 | Frédéric Ouimet and Dylan Greaves, 'A proof of the strong Gaussian product inequality conjecture', arXiv:2609.20234 (2026 preprint) | Corollary 4 (Hafnian inequalities), derived from Theorem 1 (Strong GPI) | Corollary 4 (Hafnian inequalities). Let Σ = (σ_ij)_{1≤i,j≤n} be a positive-semidefinite matrix, let m = (m_1,...,m_n), where m_1,...,m_n ∈ {1,2,...}, and let M = Σ_{i=1}^n m_i. Let Σ[2m] denote the 2M × 2M matrix obtained by repeating each index i exactly 2m_i times as both a row index and a column index. Then haf(Σ[2m... | Take Σ = A and m_i = p for all i. B (the 2p×2p block array of copies of A) is A[2m] after a simultaneous permutation of rows and columns (INFERRED: the index of B at block position (k,i) maps to the k-th copy of i). The hafnian is invariant under simultaneous row and column permutation (INFERRED, standard), so haf B = ... | By Wick's formula haf(Σ[2m]) = E[Π X_i^{2m_i}] for X ~ N(0,Σ), so the claim is the even-exponent Gaussian product inequality. The paper proves the stronger statement for all positive real exponents. After normalizing to a correlation matrix R (assumed positive definite first), it studies the kernel K_R(c) = E exp(−½ Σ ... | null | null | [
"AI_ASSISTED_RESOLVER"
] | This is a very recent preprint and not yet refereed. Its AI-use statement says the early proof was generated by ChatGPT and formalized in Lean (github dylgre/gaussian-product-inequality); the equality characterization was added later by Ouimet. 2606.02567 (weighted strong polarization) does not prove the hafnian inequa... | [{"id_or_ref": "arXiv:2609.20234", "title": "A proof of the strong Gaussian product inequality conjecture (Ouimet, Greaves)", "verified": true}, {"id_or_ref": "arXiv:0704.0028", "title": "Pfaffians, hafnians and products of real linear functionals (Frenkel)", "verified": true}, {"id_or_ref": "arXiv:2606.02567", "title"... | 4a |
0704.0542#2 | 0704.0542 | https://arxiv.org/abs/0704.0542 | numerical | What is the smallest nonnegative integer d for which the following fails; answer -1 if it never fails? For every algebraically closed field k of characteristic different from 2, every nondegenerate symmetric bilinear space V over k with dimension n=2d or n=2d+1, every Borel subgroup B of SO(V), every reduced closure X ... | Give the smallest d where the stated property fails, or -1 if it holds for every d. | -1 | SOLVED_EXACT | true | -1 | high | https://arxiv.org/abs/0710.2950 | K. N. Raghavan and Shyamashree Upadhyay, 'Initial ideals of tangent cones to Schubert varieties in orthogonal Grassmannians', arXiv:0710.2950 (v2, March 2008); the paper announced as ref. [16] in the source 0704.0542 ('If all goes well, the computation will soon appear [16]'). Journal venue not verified (arXiv page lis... | Theorem 1.8.1 (with term orders of Section 1.6 and Remark 1.9.1 for the non-Groebner-basis phenomenon) | Setting (Sec. 1.8): Let k be a field, algebraically closed and of characteristic not 2. Let d be a positive integer and M_d(V) the (even) orthogonal Grassmannian over k. Let v \le w elements of I(d), X(w) the Schubert variety in M_d(V) corresponding to w, and e_v the torus fixed point in M_d(V) corresponding to v. Let ... | 1. (Paper, Sec. 1.4-1.5) At a T-fixed point e_v, the affine patch A_v has coordinates X_(r,c), (r,c) in OR, and is identified with T_{e_v} M_d(V). The ideal I of Y(w)=X(w)\cap A_v is generated by the Pfaffians f_tau, tau not<= w (1.5.1); these are homogeneous so Y(w) is a cone and I is exactly the ideal of the tangent ... | The paper builds on the authors' earlier result (0704.0542, Theorem 2.3.1), obtained via standard monomial theory, that the Hilbert function of the tangent cone to X(w) at e_v in degree m equals the number of degree-m monomials in OR that are O-dominated by w. Since P/I and P/in(I) have the same Hilbert function, it su... | null | ROUTINE_STEPS_INFERRED | [] | Follow-up computes initial ideals at T-fixed points; extension to all points via B-translation is the agent's inference. | [] | 1-2 |
0704.0918#0 | 0704.0918 | https://arxiv.org/abs/0704.0918 | numerical | For each positive integer n, let R = C[sigma_ij : 1 <= i <= j <= n], where Sigma = (sigma_ij) is symmetric and its distinct entries are indeterminates. For a directed acyclic graph G on [n] whose edges go from smaller to larger labels, introduce indeterminates a_i for vertices and lambda_ij for edges i -> j. Let phi_G ... | The answer is the smallest number of vertices of a counterexample, or -1 if the equality holds for every such DAG at every positive n. | -1 | SOLVED_EXACT | true | -1 | high | https://arxiv.org/abs/2404.04024 | T. Boege, K. Kubjas, P. Misra, L. Solus, 'Colored Gaussian directed acyclic graphical models', arXiv:2404.04024 (J. R. Stat. Soc. B 2025, doi:10.1093/jrsssb/qkaf068) | Theorem 4.7(1) (with Conjecture 4.6 = Sullivant's Conjecture 3.3); also Theorem 4.11 / Theorem D.1 | Theorem 4.7. Let S_V = { prod_{A subseteq V} |Sigma_A|^{k_A} : k_A in N } subseteq R[Sigma] and S'_V = {|Sigma|^k : k in N} subseteq R[Sigma]. (1) P_{G,c} = I_{G,c} : S_V for any colored DAG (G,c). (2) P_{U,c} = I_{U,c} : S'_V for any colored undirected graph (U,c). (3) P_A = I_A : S_V for any directed ancestral graph ... | Take the injective (trivial) coloring, so P_{G,c}=P_G (vanishing ideal = kernel of the trek-rule pullback) and I_{G,c}=I_G (local Markov ideal). Since I_G subseteq J_G subseteq P_G (J_G = ideal of (|K|+1)-minors of Sigma_{IK,JK} over d-separation statements, Lemma 2.3 = Sullivant's C_G) and P_G is prime and disjoint fr... | By global rational identifiability of Gaussian DAG models (Lemma 3.4), the parameters are rational functions of Sigma whose denominators are principal minors: omega_i = |Sigma_{pa(i) cup i}|/|Sigma_{pa(i)}|, lambda_ij = lambda_{ij|pa(j)}(Sigma). This gives a ring map psi*: R[Lambda,Omega] -> S^{-1}R[Sigma] left-inverse... | null | null | [
"HAS_INFERRED_STEPS"
] | Both candidates claim the result. Roozbehani-Polyanskiy (1401.5551, Theorem 1(b) with remark (a)) was the first claimed proof of Sullivant's Conjecture 3.3, but Boege et al. state 'there are inaccuracies in the details of the proof' of [RP14] and give a short rigorous proof (Thm 4.7) plus a patched version of the RP14 ... | [{"id_or_ref": "arXiv:2404.04024", "title": "Colored Gaussian directed acyclic graphical models", "verified": true}, {"id_or_ref": "arXiv:1401.5551", "title": "Algebraic Methods of Classifying Directed Graphical Models (Roozbehani, Polyanskiy)", "verified": true}, {"id_or_ref": "arXiv:0704.0918", "title": "Algebraic ge... | 5 |
0704.1282#2 | 0704.1282 | https://arxiv.org/abs/0704.1282 | numerical | For every integer q >= 2, let S(q) be the least positive integer k such that q divides k!, and let P(q) be the largest prime divisor of q. What is the value of lim_{N -> infinity} N^{-1} #{q in {2,...,N} : q^2 < S(q)!}? | The answer is the value of the displayed limit, if it exists. | 1 | SOLVED_EXACT | true | 1 | high | https://arxiv.org/abs/1907.00370 | Xiumei Li and Min Sha, 'A proof of Sondow's conjecture on the Smarandache function', arXiv:1907.00370, Amer. Math. Monthly 128 (2021), doi:10.1080/00029890.2020.1820789 | Theorem 1.2 | For any real k > 1 and x > 1, denote by N_k(x) the number of positive integers n such that n <= x and S(n)! <= n^k. Theorem 1.2. For any fixed number k > 1 and any sufficiently large x, we have N_k(x) <= x exp( -sqrt(2 log x log log x) (1 + O(log log log x / log log x)) ). | Take k=2. Then #{q in {2,...,N}: q^2 >= S(q)!} <= N_2(N) = o(N) by Theorem 1.2 (the paper states N_k(x)/x -> 0). So #{q in {2,...,N}: q^2 < S(q)!} = (N-1) - o(N), and dividing by N shows the limit exists and equals 1. [Only the step from the complement count to the limit is INFERRED, and it is trivial.] | Split the n <= x with S(n)! <= n^k into two groups: S(n) != P(n) and S(n) = P(n), where P(n) is the largest prime factor. For the first group, Ivic's theorem gives #{n <= x: S(n) != P(n)} = x exp(-sqrt(2 log x log log x)(1+o(1))), which is o(x).
For the second group, if S(n) = P(n) >= 7, then Stirling's bound (P(n)/e)... | null | null | [] | The second candidate (1907.00370) is the right paper. The first candidate ('... Kempner function', no ID) appears to be the same result under the other name for S. The arXiv title says 'Smarandache function'; arXiv lists authors Xiumei Li and Min Sha. | [{"id_or_ref": "arXiv:1907.00370", "title": "A proof of Sondow's conjecture on the Smarandache function (Li, Sha)", "verified": true}, {"id_or_ref": "arXiv:0704.1282", "title": "A geometric proof that e is irrational and a new measure of its irrationality (Sondow) - Conjecture 1", "verified": true}] | 4a |
0704.1747#0 | 0704.1747 | https://arxiv.org/abs/0704.1747 | numerical | What is the smallest integer n >= 2 for which there is no constant c(n), depending only on n, such that every polynomial f in C[X_1,...,X_n] having degree d_i in X_i for each 1 <= i <= n has at most c(n)d_1...d_n isolated torsion points in H(f) = {x in (C^*)^n : f(x)=0}? A torsion point has coordinates that are roots o... | Give the smallest such n, or -1 if the bound exists for every n >= 2. | -1 | SOLVED_EXACT | true | -1 | high | https://arxiv.org/abs/1509.05898 | C. Martinez, 'The number of maximal torsion cosets in subvarieties of tori', arXiv:1509.05898 (J. reine angew. Math. 2019, doi:10.1515/crelle-2017-0029) | Corollary 1.5 (from Theorem 1.4), proving Conjecture 1.1 (Ruppert) | Corollary 1.5. Let f in Q[X_1,...,X_n] and let Delta subset R^n be a convex body such that supp(f) subset Delta. Then the number of isolated torsion points on the hypersurface Z(f) subset G_m^n is bounded by c~_n vol_n(Delta), where c~_n = ((2n-1)(n-1)(2^{2n}+2^{n+1}-2))^{n(n-1)} 2^n n^{2n} omega_{n-1}. 'Given f a poly... | Take Delta = box [0,d_1]x...x[0,d_n], which contains supp(f), vol_n(Delta) = d_1...d_n; Corollary 1.5 gives #isolated torsion points <= c~_n d_1...d_n with c~_n depending only on n. This holds for every n >= 2, so no failing n exists: answer -1. Note on field: the arXiv text of Cor. 1.5 prints 'Q[X]' (likely an extract... | Martinez combines the Ruppert / Aliev-Smyth (Beukers-Smyth) Galois-conjugation approach with the Amoroso-Viada induction for small points. Proposition 3.2: for an irreducible V not a torsion coset, an explicit V' (built from V via a Galois automorphism acting on roots of unity, e.g. zeta->zeta^2, composed with multipli... | null | null | [] | Isolated torsion points = 0-dimensional part V^0_tors of the torsion closure; deg_Delta(V^0_tors) = card, as noted before Cor. 1.5. | [{"id_or_ref": "arXiv:1509.05898", "title": "The number of maximal torsion cosets in subvarieties of tori", "verified": true}, {"id_or_ref": "arXiv:0704.1747", "title": "Solving algebraic equations in roots of unity (Aliev, Smyth)", "verified": true}] | 5 |
0704.1747#1 | 0704.1747 | https://arxiv.org/abs/0704.1747 | numerical | What is the smallest integer n >= 2 for which no constant c(n), depending only on n, has the property that for every polynomial f(X_1,...,X_n) in C[X_1,...,X_n], the number of isolated torsion points on H(f) = {x in (C^*)^n : f(x)=0} is at most c(n) vol_n(Newt(f))? Return -1 if there is no such n. A torsion point has c... | Answer with the smallest such integer n, or -1 if the bound holds for every integer n >= 2. | -1 | SOLVED_EXACT | true | -1 | high | https://arxiv.org/abs/1509.05898 | C. Martinez, 'The number of maximal torsion cosets in subvarieties of tori', arXiv:1509.05898 (J. reine angew. Math. 2019, doi:10.1515/crelle-2017-0029) | Corollary 1.5 (from Theorem 1.4), proving Conjecture 1.2 (Aliev-Smyth) | Corollary 1.5. Let f in Q[X_1,...,X_n] and let Delta subset R^n be a convex body such that supp(f) subset Delta. Then the number of isolated torsion points on the hypersurface Z(f) subset G_m^n is bounded by c~_n vol_n(Delta), where c~_n = ((2n-1)(n-1)(2^{2n}+2^{n+1}-2))^{n(n-1)} 2^n n^{2n} omega_{n-1}. '... Moreover, ... | Take Delta = Newt(f); Corollary 1.5 gives the bound c~_n vol_n(Newt(f)) for every n >= 2, so no failing n exists: answer -1. INFERRED (consistency check): if Newt(f) is lower-dimensional (vol 0), f is, up to a monomial, invariant under a positive-dimensional subtorus, so Z(f) is a union of its cosets and has no isolate... | Martinez combines the Ruppert / Aliev-Smyth (Beukers-Smyth) Galois-conjugation approach with the Amoroso-Viada induction for small points. Proposition 3.2: for an irreducible V not a torsion coset, an explicit V' (built from V via a Galois automorphism acting on roots of unity, e.g. zeta->zeta^2, composed with multipli... | null | null | [
"HAS_INFERRED_STEPS"
] | Source 0704.1747 is Aliev-Smyth; this is their strengthening of Ruppert's conjecture. | [{"id_or_ref": "arXiv:1509.05898", "title": "The number of maximal torsion cosets in subvarieties of tori", "verified": true}, {"id_or_ref": "arXiv:0704.1747", "title": "Solving algebraic equations in roots of unity (Aliev, Smyth)", "verified": true}] | 5 |
0704.2556#0 | 0704.2556 | https://arxiv.org/abs/0704.2556 | numerical | What is the smallest value of dim Y for which there exist a smooth family f: X -> Y of canonically polarized complex varieties over a smooth complex base Y with Var(f) = dim Y, and a smooth projective compactification \bar Y with simple-normal-crossings boundary D = \bar Y \setminus Y, such that K_{\bar Y}+D is not big... | The answer is the smallest dimension in which the stated failure occurs, or -1 if it never occurs. | -1 | SOLVED_EXACT | true | -1 | high | https://arxiv.org/abs/1508.02456 | F. Campana, M. Paun, 'Foliations with positive slopes and birational stability of orbifold cotangent bundles', arXiv:1508.02456 (Publ. Math. IHES 129 (2019)); also C. Wei, L. Wu, 'Isotriviality of smooth families of varieties of general type', arXiv:2001.08360 (Manuscripta Math., doi:10.1007/s00229-022-01394-y); M. Pop... | Campana-Paun Theorem 8.1 (arXiv numbering); Wei-Wu Theorem 1.0.2 | Campana-Paun, Theorem 8.1: Let f : V -> B be as above [a projective submersion with connected fibres between quasi-projective connected manifolds; B-bar a smooth projective compactification with D = B-bar - B snc]. Assume that the fibres of f all have an ample canonical bundle and that Var(f) = dim(B). Then the base B ... | Canonically polarized fibers have ample K, hence are of general type. With Var(f) = dim Y: Campana-Paun Thm 8.1 gives kappa(Y-bar, K+D) = dim Y directly, i.e. K_{Y-bar}+D big. Via Wei-Wu Thm 1.0.2: case (2) is impossible since it would force dim Y > Var(f) = dim Y; so kappa-bar(Y) > -infinity and kappa-bar(Y) >= Var(f)... | Viehweg-Zuo showed that for a family of canonically polarized manifolds with maximal variation there is, for some m>0, a big line subsheaf L of Sym^m Omega^1_{Y-bar}(log D) (a Viehweg-Zuo sheaf). Campana-Paun's Theorem 7.11 (birational stability of orbifold cotangent bundles, proved via a generalized Bogomolov-McQuilla... | null | null | [
"HAS_INFERRED_STEPS"
] | Candidate 2001.08360 is correct but not the first proof: Wei-Wu themselves state Viehweg's hyperbolicity conjecture for canonically polarized families 'was proved by Campana and Paun [CP15]' (Popa-Schnell extended to general-type fibers). The relative-good-minimal-model assumption in the Wei-Wu abstract applies to the ... | [{"id_or_ref": "arXiv:1508.02456", "title": "Foliations with positive slopes and birational stability of orbifold cotangent bundles", "verified": true}, {"id_or_ref": "arXiv:2001.08360", "title": "Isotriviality of smooth families of varieties of general type", "verified": true}, {"id_or_ref": "arXiv:1511.00294", "title... | 5 |
0704.2734#0 | 0704.2734 | https://arxiv.org/abs/0704.2734 | numerical | Let R range over all commutative noetherian Cohen–Macaulay local rings, and let S_0(R) be the set of isomorphism classes of finitely generated R-modules C such that the homothety map R → Hom_R(C,C) is an isomorphism and Ext_R^i(C,C)=0 for every i ≥ 1. What is the smallest Krull dimension of a ring R for which S_0(R) is... | The answer is the smallest such Krull dimension, with -1 meaning that S_0(R) is finite for every ring in the stated class. | -1 | SOLVED_EXACT | true | -1 | high | https://arxiv.org/abs/1201.0037 | S. Nasseh, S. Sather-Wagstaff, Geometric aspects of representation theory for DG algebras: answering a question of Vasconcelos, J. London Math. Soc. (2) 96 (2017), doi:10.1112/jlms.12055 (arXiv:1201.0037) | Theorem A (proved in 4.2), with Remark 4.3 | Theorem A. If R is a local ring, then the set of shift-isomorphism classes of semidualizing R-complexes in the derived category D(R) is finite. -- Remark 4.3. Note that Theorem A provides a positive answer to Question 1.1 since one has S0(R) ⊆ S(R); see Definition A.11. | Semidualizing modules (Hom_R(C,C)≅R via homothety, Ext^i_R(C,C)=0 for i≥1) are exactly the elements of S_0(R), and S_0(R) embeds in S(R) (semidualizing complexes up to shift-isomorphism; two modules that are shift-isomorphic complexes are isomorphic modules, both concentrated in degree 0). Theorem A gives |S(R)|<∞ for ... | Reduce to R complete with algebraically closed residue field k (flat local extension; S(R)→S(R') injective). Replace R by the Koszul complex K on a minimal generating set of m (C↦K⊗C is a bijection on semidualizing objects), then by a finite-dimensional positively graded commutative DG k-algebra U quasi-isomorphic to K... | null | null | [] | Full text (v4, final version for JLMS) checked: Question 1.1 is exactly Vasconcelos' question (R local CM, finitely many semidualizing modules?) and Theorem A answers it without the CM hypothesis. The 'smallest Krull dimension' framing is just a numerical wrapper; -1 = finite for all rings. | [{"id_or_ref": "arXiv:1201.0037", "title": "Geometric aspects of representation theory for DG algebras: answering a question of Vasconcelos", "verified": true}, {"id_or_ref": "arXiv:0704.2734", "title": "A Cohen-Macaulay algebra has only finitely many semidualizing modules", "verified": true}] | 6 |
0705.1653#0 | 0705.1653 | https://arxiv.org/abs/0705.1653 | numerical | Let (S,omega_K) be a K3 surface with Kahler form. For every nonzero beta in Pic(S) with integral_beta omega_K>0 and every g>=0, define R_{g,beta}=integral_[overline{M}_g(S,beta)]^{red} (-1)^g lambda_g, where the reduced virtual class is obtained by removing the trivial holomorphic-symplectic obstruction direction and l... | The answer is the smallest genus admitting such a pair, or -1 if no pair exists in any genus. | -1 | SOLVED_EXACT | true | -1 | high | https://arxiv.org/abs/1404.6698 | R. Pandharipande, R. P. Thomas, The Katz-Klemm-Vafa conjecture for K3 surfaces, Forum Math. Pi 4 (2016) e4, arXiv:1404.6698 (surveyed in the candidate: Pandharipande-Thomas, Notes on the proof of the KKV conjecture, arXiv:1411.0896, Surveys in Differential Geometry 21 (2016)) | Theorem 1 of arXiv:1404.6698 (Conjecture 1 + Conjecture 2 there); stated as property (4.1) + formula (4.2) in the survey arXiv:1411.0896 | Theorem 1. The BPS count r_{g,beta} depends upon beta only through <beta,beta> = 2h - 2, and the Katz-Klemm-Vafa formula holds: sum_{g>=0} sum_{h>=0} (-1)^g r_{g,h} (y^{1/2} - y^{-1/2})^{2g} q^h = prod_{n>=1} 1/((1-q^n)^20 (1-y q^n)^2 (1-y^{-1} q^n)^2). | In arXiv:1404.6698, R_{g,beta} is defined as the integral of (-1)^g lambda_g over the reduced class of Mbar_g(S,beta) (eq. 0.1), and r_{g,m alpha} is defined by eq. (0.5): F_alpha = sum R_{g,m alpha} lambda^{2g-2} v^{m alpha} = sum r_{g,m alpha} lambda^{2g-2} sum_{d>0} (1/d)(sin(d lambda/2)/(lambda/2))^{2g-2} v^{d m al... | Pandharipande-Thomas turn the reduced Gromov-Witten invariants of S into fibre-class invariants of a K3-fibred 3-fold, called T (an algebraic approximation to the twistor family). The global GW/pairs correspondence of Pandharipande-Pixton and the Maulik-Pandharipande GW/Noether-Lefschetz correspondence are combined wit... | null | null | [] | The candidate is a survey, not the original proof. The survey states property (4.1), 'n_{g,beta}(S) depends only on beta^2, not the divisibility of beta', and says it is proved in [23] = arXiv:1404.6698 (checked on its abs page). Theorem 1 of 1404.6698 states the result with exactly the question's R/r normalisation. Th... | [{"id_or_ref": "arXiv:1411.0896", "title": "Notes on the proof of the KKV conjecture (Pandharipande, Thomas)", "verified": true}, {"id_or_ref": "arXiv:1404.6698", "title": "The Katz-Klemm-Vafa conjecture for K3 surfaces (Pandharipande, Thomas)", "verified": true}, {"id_or_ref": "arXiv:0705.1653", "title": "Gromov-Witte... | 5 |
0705.1653#1 | 0705.1653 | https://arxiv.org/abs/0705.1653 | numerical | Let (S,ω_K) be a K3 surface and let β∈Pic(S) be nonzero with ∫_βω_K>0. Define R_{g,β}=∫_[Mbar_g(S,β)]^red(-1)^gλ_g, where [Mbar_g(S,β)]^red is the reduced virtual class and λ_g is the top Chern class of the Hodge bundle. For every primitive positive-degree α∈Pic(S), define r_{g,mα} for g≥0 and m>0 by Σ_{g≥0,m>0}R_{g,mα... | The answer is the smallest failing h, or -1 if the formal identity holds and thus uniquely determines all r_{g,h}. | -1 | SOLVED_EXACT | true | -1 | high | https://arxiv.org/abs/1404.6698 | R. Pandharipande, R. P. Thomas, 'The Katz-Klemm-Vafa conjecture for K3 surfaces', arXiv:1404.6698 (Forum Math. Pi 4 (2016) e4, doi:10.1017/fmp.2016.2); survey: 'Notes on the proof of the KKV conjecture', arXiv:1411.0896 | Theorem 1 of arXiv:1404.6698 | Theorem 1. The BPS count r_{g,beta} depends upon beta only through <beta,beta> = 2h-2, and the Katz-Klemm-Vafa formula holds: sum_{g>=0} sum_{h>=0} (-1)^g r_{g,h} (y^{1/2} - y^{-1/2})^{2g} q^h = prod_{n>=1} 1/((1-q^n)^20 (1-yq^n)^2 (1-y^{-1}q^n)^2). | The BPS counts r_{g,m alpha} in the question are defined by exactly the paper's equation (0.5) from the same reduced Hodge integrals R_{g,beta} (0.1). Theorem 1 proves both the assumed independence (Conjecture 1) and the identity (Conjecture 2) for all g and all beta, so the q^h coefficients of both sides agree for eve... | Apply the GW/Pairs (MNOP) correspondence, proven by Pandharipande-Pixton for suitable projective K3-fibred 3-folds, together with the GW/Noether-Lefschetz correspondence (Maulik-Pandharipande) and a new Pairs/Noether-Lefschetz correspondence, to an anticanonical K3-fibred hypersurface in Bl_p(P^2 x P^1) x P^1 chosen so... | null | null | [
"HAS_INFERRED_STEPS"
] | Candidate 1411.0896 is a survey of the proof (by the same authors), not the proof itself; the primary resolver is arXiv:1404.6698 Theorem 1. The question's assumption (dependence only on beta^2) is itself proven as part of Theorem 1. | [{"id_or_ref": "arXiv:1404.6698", "title": "The Katz-Klemm-Vafa conjecture for K3 surfaces", "verified": true}, {"id_or_ref": "arXiv:1411.0896", "title": "Notes on the proof of the KKV conjecture", "verified": true}, {"id_or_ref": "arXiv:0705.1653", "title": "Gromov-Witten theory and Noether-Lefschetz theory (Maulik, P... | 5 |
0705.2168#0 | 0705.2168 | https://arxiv.org/abs/0705.2168 | numerical | What is the smallest integer n >= 1 for which there exist a non-archimedean local field F of characteristic 0 and a decomposition W = V direct-sum U, with dim_F(V) = n and dim_F(U) = 1, such that at least one of the following statements fails? Embed GL(V) in GL(W) using the decomposition, and choose a basis of V and a ... | The answer is the smallest n at which either statement fails for some F and decomposition, or -1 if both hold for every n and F. | -1 | SOLVED_EXACT | true | -1 | high | https://arxiv.org/abs/0707.2363 | A. Aizenbud, D. Gourevitch, A proof of the multiplicity one conjecture for GL(n) in GL(n+1), arXiv:0707.2363 (2007); published as part of Aizenbud-Gourevitch-Rallis-Schiffmann, Multiplicity one theorems, Ann. of Math. 172 (2010) | Theorem 1.1 and Theorem 1.2 | Let F be a non-archimedean local field of characteristic zero. Theorem 1.1: Every GL(n,F)-invariant distribution on GL(n+1,F) is invariant with respect to transposition. Theorem 1.2: Let pi be an irreducible smooth representation of GL(n+1,F) and rho be an irreducible smooth representation of GL(n,F). Then dim Hom_{GL(... | Theorem 1.1 is statement (1) of the question (GL(n,F) acts on GL(n+1,F) by conjugation through the standard embedding). It holds for every n and every non-archimedean F of characteristic 0. Theorem 1.2 is statement (2). Irreducible smooth representations of p-adic GL_m are admissible (Jacquet), so 'smooth' and 'admissi... | Rallis-Schiffmann reduce Theorem 1.1 to Theorem 1.3: any (G~, chi)-equivariant distribution on X = sl(V) x V x V* vanishes, where G~ = GL(V) x| S2 with S2 acting by transposition and chi is the sign character. The proof is by induction on n. By the Rallis-Schiffmann results (localization, Frobenius reciprocity, Harish-... | null | null | [
"HAS_INFERRED_STEPS"
] | The abs page shows the title with 'GL(n)'. The paper covers every n with no restriction, including n >= 9, which the source left open. The Archimedean analogue was later proved separately (Aizenbud-Gourevitch; Sun-Zhu), but the question is only about p-adic fields. | [{"id_or_ref": "arXiv:0707.2363", "title": "A proof of the multiplicity one conjecture for GL(n) in GL(n+1) (Aizenbud, Gourevitch)", "verified": true}, {"id_or_ref": "arXiv:0705.2168", "title": "Multiplicity one Conjectures (Rallis, Schiffmann) [source]", "verified": true}] | 5 |
0706.2141#1 | 0706.2141 | https://arxiv.org/abs/0706.2141 | yesno | Let omega be a translation-invariant state on a one-dimensional quantum spin chain, let Phi be a translation-invariant finite-range interaction with local Hamiltonians H_[1,n](Phi), and let p^alt_omega(Phi) be the alternative pressure functional discussed in the paper. It agrees with p_omega(Phi)=lim_{n->infinity} n^{-... | Yes means that the conjecture implies the stated probability-measure representation in the precise form required. | null | SOLVED_EXACT | true | Yes | high | https://arxiv.org/abs/1107.4875 | Herbert R. Stahl, 'Proof of the BMV Conjecture', arXiv:1107.4875 (2011; v-final 2012); published Acta Mathematica 211 (2013) 255-290. The implication BMV => moment-generating-function representation is spelled out in Hiai-Mosonyi-Ohno-Petz, 'Free energy density for mean field perturbation of states of a one-dimensional... | Stahl, Theorem 1 (with Theorem 2, support inclusion (1.15)); applied via Hiai-Mosonyi-Ohno-Petz arXiv:0706.4148, Remark 2.8 | Stahl, Theorem 1: "If A and B are two Hermitian matrices with B positive semidefinite, then there exists a unique positive measure µ_{A,B} on [0, ∞) such that (1.3) holds for t ≥ 0. In other words: the BMV conjecture holds true." [the referenced identity is f(t) := Tr e^{A−tB} = ∫ e^{−ts} dµ_{A,B}(s), labelled (1.2) in... | (1) SOURCE (0706.2141, Sec. 7): 'In [20] a different version of the pressure functional is introduced, which coincides with our definition (4) when the local densities of the reference state commute with the local Hamiltonians. An advantage of that definition is that a variational principle can be established between t... | Stahl's proof (his outline, Sec. 1.5): by a unitary change of basis (Lemma 1) B is diagonal, B = diag(b_1 <= ... <= b_n), with a_ij = 0 when b_i = b_j, and by the shift B -> B + eps I (translation of the measure, eq. (2.4)) B may be taken positive definite. The eigenvalues lambda_j(t) of A - tB are the n branches of th... | null | ROUTINE_STEPS_INFERRED | [] | BMV proved by Stahl; link to the source's claim is inferred. | [] | 1-2 |
0706.2777#0 | 0706.2777 | https://arxiv.org/abs/0706.2777 | yesno | Let (M,J) be a compact closed Kähler manifold, let Ω be a Kähler class, and let H_Ω be the set of Kähler forms representing Ω. Assume that H_Ω contains a constant scalar curvature Kähler metric. For every initial form ω_0=ω in H_Ω, does there exist a sequence {ω_k}_{k≥0} in H_Ω such that, for every k≥0, ω_{k+1}=ω_k+H_{... | Yes means that every initial form admits such a sequence and that the iteration converges as stated. | null | SOLVED_EXACT | true | Yes | high | https://arxiv.org/abs/2311.15524 | K. Zhang, The Ricci iteration towards cscK metrics, arXiv:2311.15524, Adv. Math. (2025), doi:10.1016/j.aim.2025.110340 | Theorem 1.6 (confirming Conjecture 1.3 = Rubinstein [34, Conjecture 2.1]) | Theorem 1.6. Assume that there exists a cscK metric in {omega}. Then for any tau > 0 the iteration sequence omega_i exists, and, up to biholomorphic automorphisms, converges to a cscK metric smoothly. | Zhang's iteration (1.2) is (omega_{i+1} - omega_i)/tau = -Ric(omega_{i+1}) + H Ric(omega_{i+1}), with H Ric the harmonic part of Ric(omega_{i+1}) with respect to omega_{i+1}. Setting tau = 1 gives omega_{i+1} = omega_i + H_{i+1}Ric(omega_{i+1}) - Ric(omega_{i+1}), which is the time-one iteration in the question. Theore... | Zhang first shows (Theorem 1.4) that the iteration exists for all steps with each omega_i uniquely determined, for small tau. For general tau, existence follows once a cscK metric exists. Each step is solved variationally as a minimiser of a twisted K-energy-type functional on the metric completion of the space of Kahl... | null | null | [] | Convergence is 'up to biholomorphic automorphisms', which is the natural 'appropriate sense' when Aut is non-discrete. Existence holds for all tau > 0 under the cscK hypothesis, and in particular for the time-one step tau = 1. | [{"id_or_ref": "arXiv:2311.15524", "title": "The Ricci iteration towards cscK metrics (Kewei Zhang)", "verified": true}, {"id_or_ref": "arXiv:0706.2777", "title": "The Ricci iteration and its applications (Rubinstein) [source]", "verified": true}] | 5 |
0707.0299#0 | 0707.0299 | https://arxiv.org/abs/0707.0299 | yesno | Call an integer y-smooth if all its prime factors are at most y. For x,y >= 2, a modulus q, and a residue a with gcd(a,q)=1, let Psi(x,y;q,a) count the y-smooth integers n <= x with n congruent to a modulo q, and let Psi_q(x,y) count those with gcd(n,q)=1. Fix a real A with 0 < A < 4 sqrt(e). Is it true that, for all s... | Yes means the stated asymptotic holds in the full specified range for every fixed A with 0 < A < 4 sqrt(e). | null | SOLVED_EXACT | true | Yes | high | https://arxiv.org/abs/1103.2106 | A. J. Harper, On a paper of K. Soundararajan on smooth numbers in arithmetic progressions, J. Number Theory 132 (2012), arXiv:1103.2106 | Theorem 1 (proving Conjecture 1 of Soundararajan for A < 4 sqrt(e)) | Theorem 1. Let delta > 0, and suppose that y <= x, that 2 <= q <= y^{4 sqrt(e) - delta}, and (a,q) = 1. If y is large enough depending on delta, then as log x / log q -> infinity we have Psi(x,y;q,a) ~ (1/phi(q)) Psi_q(x,y). | Fix A with 0 < A < 4 sqrt(e) and set delta = 4 sqrt(e) - A > 0. Then q <= y^A means q <= y^{4 sqrt(e) - delta}, and Theorem 1 gives the asymptotic for y large enough in terms of delta (that is, of A), with no further restriction on x. Uniformity in a: the paper proves a smoothed version with an explicit error term O_Ph... | Harper follows Soundararajan: expand Psi(x,y;q,a) over characters mod q and bound the smoothly weighted character sums Psi(x,y;chi,Phi) through truncated Euler products L(s,chi;y). He uses Soundararajan's statistical bounds over all characters mod q (the Rodosskii bounds and a character-sum bound, taken as black boxes)... | null | null | [
"HAS_INFERRED_STEPS"
] | Harper's abstract and introduction say this proves Soundararajan's conjecture in the range A < 4 sqrt(e), removing the restriction e^{y^{1-eps}} >= x >= y^{(log log y)^4}. Theorem 2 of the paper gives the coset version for general A. | [{"id_or_ref": "arXiv:1103.2106", "title": "On a paper of K. Soundararajan on smooth numbers in arithmetic progressions (Harper)", "verified": true}, {"id_or_ref": "arXiv:0707.0299", "title": "The distribution of smooth numbers in arithmetic progressions (Soundararajan) [source]", "verified": true}] | 5 |
0707.1130#1 | 0707.1130 | https://arxiv.org/abs/0707.1130 | numerical | For every oriented knot or link K ⊂ S³, let B_K be its set of oriented closed braid diagrams. For D ∈ B_K, let b_D be its number of braid strands, w_D its number of positive crossings minus its number of negative crossings, and β_D=w_D−b_D. Set b_K=min{b_D:D∈B_K} and β_K=max{β_D:D∈B_K}. Let w_K be the distinguished min... | The answer is the smallest failing braid index, or −1 if the property holds for every oriented knot or link. | -1 | SOLVED_EXACT | true | -1 | high | https://arxiv.org/abs/1206.0898 | Ivan Dynnikov and Maxim Prasolov, "Bypasses for rectangular diagrams. Proof of Jones' conjecture and related questions", arXiv:1206.0898 (v2, 24 Mar 2013); published in Trans. Moscow Math. Soc. 2013 (journal details recalled from memory, not re-verified). Independent confirmation: Douglas J. LaFountain and William W. M... | Theorem 9 (generalized Jones' conjecture), with Corollary 4 (Section 5.3) giving the self-linking/Bennequin form directly; independently LaFountain–Menasco Theorem 1.2 | Dynnikov–Prasolov, §5.2: 'For every n ⩾ 2 we denote by c the homomorphism from B_n to Z defined by c(σ_1) = 1. In other words, c(β) is the algebraic number of crossings of a braid diagram representing β.'
Theorem 9. Let braids β_1 ∈ B_m and β_2 ∈ B_n close up to equivalent oriented links, with β_1 having minimal possib... | 1. INFERRED (dictionary): a closed braid diagram D ∈ B_K is the closure of a braid word β ∈ B_{b_D}; its writhe w_D equals the algebraic crossing number c(β) (DP's c, LM's ℓ), and β_D = w_D − b_D = c(β) − n, which DP call the self-linking number sl(T_β) (§5.3). B_K is nonempty and b_K is well defined by Alexander's the... | Dynnikov–Prasolov work with rectangular (grid) diagrams, which present Legendrian links. Their Key Lemma, on shortening 'bypasses' through complexity-preserving elementary moves (proved with Birman–Menasco braid-foliation techniques adapted to arc presentations), yields Theorem 7, a commutation result. If two rectangul... | null | null | [] | Generalised Jones conjecture proved. | [] | 1-2 |
0707.1975#2 | 0707.1975 | https://arxiv.org/abs/0707.1975 | numerical | Let n be a positive integer, G a finite abelian group of exponent n, and A a nonempty subset of {1,2,...,n}. For sequences over G, with repetition allowed, let d_A(G) be the least positive t such that every length-t sequence has a nonempty subsequence whose elements, multiplied by coefficients in A, sum to 0. Let ZS_A(... | Return the smallest failing exponent n, or −1 if the equality holds for every permitted G and A. | -1 | SOLVED_EXACT | true | -1 | high | https://arxiv.org/abs/0903.2810 | D. J. Grynkiewicz, L. E. Marchan, O. Ordaz, A weighted generalization of two theorems of Gao, Ramanujan J. 28 (2012), arXiv:0903.2810 | Corollary 1.2 (of Theorem 1.1) | Corollary 1.2. Let G be a finite abelian group, let S in F(G), let n >= |G|, let A subset Z be nonempty, and let W in F(Z) be a sequence with supp(W) = A. If |S| >= n + D_A(G) - 1, then n.G cap Sigma_n(W^n, S) != empty. In particular, E_A(G) = |G| + D_A(G) - 1. | The paper's D_A(G) is the least length forcing a nontrivial subsequence s_1...s_r with sum w_i s_i = 0 for some w_i in A. That is the question's d_A(G). E_A(G) is the least length forcing a length-|G| subsequence with an A-weighted zero sum, which is the question's ZS_A(G). Corollary 1.2 holds for every finite abelian ... | The lower bound E_A(G) >= |G| + D_A(G) - 1 comes from appending |G|-1 zeros to an extremal D_A-sequence. For the upper bound, the authors first reduce to gcd(A) = 1 by dividing the weights by d' and multiplying the sequence by d'. They then prove the structural Theorem 1.1 by induction, in a translated setting inside a... | null | null | [] | The abstract says the result confirms a conjecture of Thangadurai (the source). Earlier partial results: Yuan-Zeng for the cyclic case, and Adhikari-Chen when gcd(A - a_0) = 1. | [{"id_or_ref": "arXiv:0903.2810", "title": "A Weighted Generalization of Two Theorems of Gao (Grynkiewicz, Marchan, Ordaz)", "verified": true}, {"id_or_ref": "arXiv:0707.1975", "title": "A variant of Davenport's constant (Thangadurai) [source]", "verified": true}] | 5 |
0707.3417#3 | 0707.3417 | https://arxiv.org/abs/0707.3417 | numerical | What is the smallest integer k>2 for which the following assertion fails? Let u_1,...,u_k be nonzero integers with gcd(u_1,...,u_k)=1, and set f(x_1,...,x_k)=sum_{i=1}^k u_i x_i. For each positive integer N, let I_N={0,...,N}, form A by including each n in I_N independently with probability p(N), where p:N→(0,1) satisf... | The answer is the smallest integer k>2 for which the assertion fails for some permitted coefficients and probability function, or -1 if there is no such k. | -1 | SOLVED_EXACT | true | 3 | high | https://arxiv.org/abs/2309.01801 | R. Jeong, S. J. Miller, 'Phase Transitions for Sparse Random Sets Under Linear Forms', arXiv:2309.01801 (v3, Jan 2026) | Theorem 1.1 (with the discussion after it, which resolves [HM09, Conjecture 4.2]) | Theorem 1.1. Let p : N → (0,1) be a function satisfying (1.6) [p(N) ≪ 1 ≪ Np(N)]. Fix an integer h ≥ 2 and a linear form L : Z^h → Z with coefficients u1,...,uh ∈ Z≠0 such that gcd(u1,...,uh) = 1. Let A ⊆ I_N be a random subset where each element of I_N is independently included in A with probability p(N). The followin... | The conjecture (Hegarty–Miller Conj. 4.2) holds for k = 2 (their Theorem 3.1; N^{-1/2} = N^{-(k-1)/k}). Take k = 3, u = (1,1,1) (gcd 1, θ_f = 6) and p(N) = N^{-1/2}. This satisfies N^{-1} = o(p) and p = o(1), and p = o(N^{-1/3}), so part (i) would give D_f ∼ (Np)^3/6 = N^{3/2}/6. But f(A) ⊆ [0, 3N], so D_f ≤ 3N + 1 = o... | Let E_k be the essentially distinct L-expressions (injective h-tuples modulo Aut(L) and, for balanced forms, reversal). The expected count of generated expressions is ∼ (Np)^h/|Aut(L)|, while the feasible interval has length about mN. This puts the global threshold at p ≍ N^{-(h-1)/h}. Below it, a first-moment and Mark... | null | null | [
"HAS_INFERRED_STEPS"
] | I checked the source text: Conjecture 4.2 of 0707.3417 does use the threshold N^{-1/k}, so the reformatted question is faithful. For k ≥ 3, part (i) is already false by the trivial bound D_f ≤ (Σ|u_i|)N + 1. Jeong–Miller say they settle the conjecture 'after correcting its formulation': the threshold becomes N^{-(k-1)/... | [{"id_or_ref": "arXiv:2309.01801", "title": "Phase Transitions for Sparse Random Sets Under Linear Forms (Jeong, Miller)", "verified": true}, {"id_or_ref": "arXiv:0707.3417", "title": "When almost all sets are difference dominated (Hegarty, Miller)", "verified": true}] | 5 |
0707.4256#3 | 0707.4256 | https://arxiv.org/abs/0707.4256 | numerical | What is the smallest integer m for which there exists a simple connected graph G whose cover pebbling number is m but whose cover rubbling number differs from m? Answer -1 if no such graph exists. The cover pebbling number is the least integer m such that, from every nonnegative pebble distribution of total size m on V... | Report the smallest such integer m, or -1 if every simple connected graph has equal cover rubbling and cover pebbling numbers. | -1 | SOLVED_EXACT | true | -1 | high | https://doi.org/10.1016/j.disc.2020.112080 | T. W. Haynes, R. Keaton, 'Cover rubbling and stacking', Discrete Math. 343(11) (2020) 112080; author preprint http://faculty.etsu.edu/keatonr/documents/CoverRubbling.pdf | Theorem 2 (via Lemma 5 and the Cover Pebbling Theorem, Theorem 1 [Sjöstrand]) | Theorem 2. If w is a strictly positive weight function on a connected graph G, then π_w(G) = ρ_w(G). [π_w = w-cover pebbling number, ρ_w = w-cover rubbling number; the paper states: 'We first answer the question of Belford and Sieben in the affirmative by proving a rubbling analogue of the Cover Pebbling Theorem.'] | Take w ≡ 1. Then π_w(G) is the cover pebbling number and ρ_w(G) is the cover rubbling number, and Theorem 2 says they are equal for every connected graph G. So no simple connected graph has cover rubbling number different from its cover pebbling number, and the answer is −1. [Specialization to w ≡ 1 INFERRED, but immed... | Trivially ρ_w ≤ π_w, since rubbling allows more moves. By the Cover Pebbling Theorem, π_w(G) = st_w(G) = max_v st_w(v), the worst single-vertex stack. Lemma 5 shows that from a stack on v, rubbling moves never help: by induction on distance, placing a pebble on a vertex at distance d costs at least 2^d pebbles from v. ... | null | null | [
"HAS_INFERRED_STEPS"
] | The resolver has no arXiv version. I read the full text from the author's ETSU preprint, because ScienceDirect was blocked; title and authors match Crossref. The source's Section 8 asks: 'Is the cover rubbling number the same as the cover pebbling number for every graph?' Lemma 5's written proof handles reaching a sing... | [{"id_or_ref": "doi:10.1016/j.disc.2020.112080", "title": "Cover rubbling and stacking (Haynes, Keaton)", "verified": true}, {"id_or_ref": "arXiv:0707.4256", "title": "Rubbling and Optimal Rubbling of Graphs (Belford, Sieben)", "verified": true}] | 5 |
0707.4499#0 | 0707.4499 | https://arxiv.org/abs/0707.4499 | numerical | Theorem 1 establishes the property below for C=1/320. What is the maximum positive real constant C such that, for every real ε with 0<ε<C, there exists n₀=n₀(ε) for which every integer n≥n₀ and every graph G of order n with largest adjacency-matrix eigenvalue μ(G)>√⌊n²/4⌋, the graph G contains a cycle of every integer ... | The answer is the maximum positive constant C satisfying the stated property. | null | SOLVED_EXACT | true | (3-sqrt(5))/2 | high | https://arxiv.org/abs/2607.24361 | Bo Ning, Mingqing Zhai, "Nikiforov's spectral consecutive cycle problem and the connected-matching method", arXiv:2607.24361v1 [math.CO], 27 Jul 2026 (preprint; no journal version known) | Theorem 1.1 (lower bound via Theorems 4.1 and 5.1; sharpness is Theorem 3.3) | Theorem 1.1. Denote $C_0 := \frac12(3-\sqrt5)$. For every $\varepsilon>0$, there exists $n_0=n_0(\varepsilon)$ such that for all $n\ge n_0$, any $n$-vertex graph $G$ satisfying (1) contains a cycle $C_g$ for every integer length $3\le g\le (C_0-\varepsilon)n$. Furthermore, the largest constant in Problem 1.1 is $C=C_0$... | 1. The question's property P(C) is exactly the property in Nikiforov's Problem 1.1 (paper's Problem 1.1, with mu(G)=rho(G)); the question quantifies over 0<eps<C instead of all eps>0, but for eps>=C the interval [3,(C-eps)n] is empty so the condition is vacuous -- the two formulations are equivalent (INFERRED, trivial;... | Sharpness (Theorem 3.3) comes from the split graph S_{n,k}=K_k v (complement of K_{n-k}) with k ~ (3-sqrt5)n/4: its spectral radius, the largest root of x^2-(k-1)x-k(n-k), exceeds n/2 once k/n > C0/2, while every cycle uses at most k independent-set vertices so the longest cycle has length 2k ~ C0 n. The 1/320 (Nikifor... | null | null | [
"AI_ASSISTED_RESOLVER"
] | Ning-Zhai 2026 determine the sharp constant; abstract confirms. | [] | 1-2 |
0708.1919#1 | 0708.1919 | https://arxiv.org/abs/0708.1919 | yesno | Let p=p(n)=n^{-o(1)} and let C>0 be constant. Suppose (G_n) is a sequence of finite graphs with |G_n|=n such that, for every finite graph F, s_p(F,G_n):=emb(F,G_n)/(p^{e(F)}n_(|F|)) converges to c_F with 0<=c_F<=C^{e(F)}. Here emb(F,G_n) is the number of injective graph homomorphisms F into G_n, e(F) is the number of e... | Yes means every sequence satisfying the stated assumptions admits such a kernel. | null | SOLVED_EXACT | true | No | high | https://arxiv.org/abs/2003.05272 | A. Sah, M. Sawhney, J. Tidor, Y. Zhao, A counterexample to the Bollobás–Riordan conjectures on sparse graph limits, Combin. Probab. Comput. 30 (2021); arXiv:2003.05272 | Theorem 1 | There exists a sequence of graphs G_n with |G_n| → ∞ and edge density p_n = |G_n|^{−o(1)} such that for every graph F, writing △_F for the number of triangles in F, t_{p_n}(F, G_n) → e^{−△_F} as n → ∞. Moreover, there is no kernel W satisfying t(F, W) = e^{−△_F} for all graphs F. | The question is Bollobás–Riordan Conjecture 3.3 (arXiv:0708.1919v3, p.20), which the paper lists explicitly and refutes. Take G_n from Theorem 1 with p = p_n (the edge density), which is |G_n|^{-o(1)}. Limits c_F = e^{-△_F} lie in [0,1], so 0 <= c_F <= C^{e(F)} with C = 1. Since there is no kernel at all (bounded or un... | Take G = K_n^{⊗ n^2}, the n^2-th tensor power of K_n: vertices are n^2-tuples over [n], adjacent iff they differ in every coordinate. Edge density p = (1 − 1/n)^{n^2} = |G|^{-o(1)}. Since hom(F, K_n) counts proper n-colourings, inclusion–exclusion gives hom(F,K_n) = n^{|F|} − e_F n^{|F|−1} + (C(e_F,2) − △_F) n^{|F|−2} ... | null | null | [
"HAS_INFERRED_STEPS"
] | Source conjecture is exactly BR Conjecture 3.3. Counterexample satisfies it with C=1; normalizing p as edge density is allowed since the question lets p be any n^{-o(1)} function. | [{"id_or_ref": "arXiv:2003.05272", "title": "A counterexample to the Bollobás-Riordan conjectures on sparse graph limits", "verified": true}, {"id_or_ref": "arXiv:0708.1919", "title": "Metrics for sparse graphs (Bollobás–Riordan), Conjecture 3.3", "verified": true}] | 5 |
0708.1919#2 | 0708.1919 | https://arxiv.org/abs/0708.1919 | yesno | Let p=p(n)=n^{-o(1)}, and let (G_n) be finite graphs with |G_n|=n. For every finite graph F, define s_p(F,G_n)=emb(F,G_n)/(p^{e(F)}n_{(|F|)}), where emb(F,G_n) is the number of injective graph homomorphisms from F to G_n, e(F) is the number of edges of F, and n_{(k)}=n(n-1)...(n-k+1) for k=|F|. Suppose that, for every ... | Yes means every graph sequence satisfying the stated assumptions admits such a kernel. | null | SOLVED_EXACT | true | No | high | https://arxiv.org/abs/2003.05272 | A. Sah, M. Sawhney, J. Tidor, Y. Zhao, 'A counterexample to the Bollobás–Riordan conjectures on sparse graph limits', Combin. Probab. Comput. (2021), arXiv:2003.05272 | Theorem 1 (refuting [Bollobás–Riordan, Conjecture 3.4]) | Theorem 1. There exists a sequence of graphs G_n with |G_n| → ∞ and edge density p_n = |G_n|^{−o(1)} such that for every graph F, writing △_F for the number of triangles in F, t_{p_n}(F, G_n) → e^{−△_F} as n → ∞. Moreover, there is no kernel W satisfying t(F, W) = e^{−△_F} for all graphs F. | The question is exactly Bollobás–Riordan Conjecture 3.4, which SSTZ list explicitly and refute. Take p(n) to be the edge density of the blown-up SSTZ graphs with |G_n| = n (Remark: 'slowly blowing-up'). Then p = n^{−o(1)}, and for every finite F, s_p(F,G_n) ∼ t_p(F,G_n) → c_F = e^{−△_F}, which is finite. The source not... | G = K_n^{⊗n^2} (tuples in [n]^{n^2} adjacent iff they differ in every coordinate), so hom(F,G) = hom(F,K_n)^{n^2}. Inclusion–exclusion gives hom(F,K_n) = n^{|F|}(1 − e_F/n + (C(e_F,2) − △_F)/n^2 + O(n^{−3})). With p = (1−1/n)^{n^2}, the normalized density is (1 − △_F/n^2 + O(n^{−3}))^{n^2} → e^{−△_F}. The limit has c_{... | null | null | [
"HAS_INFERRED_STEPS"
] | I checked the source: 0708.1919 Conjecture 3.4 matches the question (p = n^{−o(1)}, |G_n| = n, s_p(F,G_n) → c_F < ∞ for every F ⇒ there is a kernel κ with c_F = s(F,κ)). SSTZ state this conjecture verbatim in their list and refute it. The same example also satisfies c_F ≤ 1, so it refutes the bounded Conjecture 3.3 as ... | [{"id_or_ref": "arXiv:2003.05272", "title": "A counterexample to the Bollobás–Riordan conjectures on sparse graph limits (Sah, Sawhney, Tidor, Zhao)", "verified": true}, {"id_or_ref": "arXiv:0708.1919", "title": "Metrics for sparse graphs (Bollobás, Riordan)", "verified": true}] | 5 |
0708.1919#7 | 0708.1919 | https://arxiv.org/abs/0708.1919 | yesno | Let p=p(n) tend to 0 and satisfy np(n)>=n^alpha for all sufficiently large n for some alpha>0. Let t>=1 be the smallest integer for which n^(t-1)p(n)^t>=n^(-o(1)), meaning that the inequality holds eventually with o(1) replaced by some epsilon_n->0. Let A=T union F_{>=t}, where T is the set of finite trees and F_{>=t} ... | Yes means such a kernel exists for every choice satisfying the stated assumptions. | null | SOLVED_EXACT | true | No | medium | https://arxiv.org/abs/2003.05272 | A. Sah, M. Sawhney, J. Tidor, Y. Zhao, 'A counterexample to the Bollobás–Riordan conjectures on sparse graph limits', Combin. Probab. Comput. (2021), arXiv:2003.05272 | Theorem 1 (with the following Remark) | Theorem 1. There exists a sequence of graphs G_n with |G_n| → ∞ and edge density p_n = |G_n|^{−o(1)} such that for every graph F, writing △_F for the number of triangles in F, t_{p_n}(F, G_n) → e^{−△_F} as n → ∞. Moreover, there is no kernel W satisfying t(F, W) = e^{−△_F} for all graphs F. [Remark: the normalized kern... | The question is Bollobás–Riordan Conjecture 5.5 (L ⊂ s(K)). Take p(n) to be the edge density of the (blown-up) Sah–Sawhney–Tidor–Zhao graphs G_n = K_m^{⊗m^2}. Then p = n^{−o(1)} → 0 and np ≥ n^{1/2}, so the condition np ≥ n^α holds with α = 1/2. Here n^0·p^1 ≥ n^{−o(1)}, so t = 1. F_{≥1} is then every finite simple gra... | Let G = K_n^{⊗n^2}, the tensor power on [n]^{n^2} where tuples are adjacent iff they differ in every coordinate. Then hom(F,G) = hom(F,K_n)^{n^2}, and hom(F,K_n) counts proper n-colourings: n^{|F|} − e_F n^{|F|−1} + (C(e_F,2) − △_F) n^{|F|−2} + O(n^{|F|−3}). The edge density is p = (1−1/n)^{n^2}. Normalizing gives t_p(... | null | null | [
"HAS_INFERRED_STEPS"
] | The item is Conjecture 5.5 of 0708.1919 (A = T ∪ F_{≥t}, Assumptions 4.1 and 5.3). SSTZ say their single example 'refutes all conjectures in [1]' and note that Conjecture 5.5 would imply Conjecture 3.3. Their example has p → 0 and t = 1, so it lies inside the Conj 5.5 regime. I rate confidence medium rather than high o... | [{"id_or_ref": "arXiv:2003.05272", "title": "A counterexample to the Bollobás–Riordan conjectures on sparse graph limits (Sah, Sawhney, Tidor, Zhao)", "verified": true}, {"id_or_ref": "arXiv:0708.1919", "title": "Metrics for sparse graphs (Bollobás, Riordan)", "verified": true}] | 5 |
0708.2160#0 | 0708.2160 | https://arxiv.org/abs/0708.2160 | numerical | Let ko and ku be connective real and complex topological K-theory spectra, let KO and KU be their periodic counterparts, and let K(-) denote algebraic K-theory. Complexification KO -> KU is a Z/2-Galois extension of commutative S-algebras, with Z/2 acting on KU by complex conjugation, and induces K(KO) -> K(KU)^{hZ/2}.... | Give the smallest failing degree, or -1 if the comparison is a rational equivalence. | -1 | SOLVED_EXACT | true | -1 | high | https://arxiv.org/abs/1606.03328 | D. Clausen, A. Mathew, N. Naumann, J. Noel, Descent in algebraic K-theory and a conjecture of Ausoni–Rognes, J. Eur. Math. Soc. 22 (2020); arXiv:1606.03328 | Theorem 5.10(1) (with Example 5.9) | Theorem 5.10. (1) The fiber of the comparison map K(KO) → K(KU)^{hC2} admits the structure of an F2-module spectrum. ... (3) More generally, if R → R′ is a G-Galois extension and the wrong-way map K0(R′) → K0(R) has image containing a prime number p, then the fiber of the map K(R) → K(R′)^{hG} admits the structure of a... | Example 5.9: [KU] = 2 in K0(KO) by Wood's theorem KO ⊗ Σ^{-2}CP^2 ≃ KU, so 2 lies in the image of the transfer; Theorem 5.10(3) with p=2 gives that the fiber of K(KO) → K(KU)^{hC2} is an F2-module spectrum, hence has zero rational homotopy, so the map is an isomorphism on rational π_n for every n. (Example 5.9 also rec... | CMNN work in the stable ∞-category of noncommutative A-linear motives. For a G-Galois extension R → R′, the fiber of U(Perf R) → U(Perf R′)^{hG} is a module over the cofiber of the transfer; if the transfer K0(R′) → K0(R) hits a prime p, this E∞ cofiber is an Fp-algebra, so the fiber is an Fp-module (Theorem 4.5). Appl... | null | null | [
"HAS_INFERRED_STEPS"
] | Caveat on the premise: CMNN Example 5.30 shows K(ko) → K(ku)^{hC2} is NOT a rational equivalence if one rationalizes after taking homotopy fixed points (K(Z) → K(Z)^{hC2} fails rationally); it is an equivalence only when rationalizing first, which is what Ausoni–Rognes' computation π*(K(KU)^{hZ/2})⊗Q ≅ [K*(KU)⊗Q]^{Z/2}... | [{"id_or_ref": "arXiv:1606.03328", "title": "Descent in algebraic K-theory and a conjecture of Ausoni-Rognes", "verified": true}, {"id_or_ref": "arXiv:1402.6038", "title": "Regularity of structured ring spectra and localization in K-theory (Barwick–Lawson)", "verified": true}, {"id_or_ref": "arXiv:0708.2160", "title": ... | 5 |
0708.2539#0 | 0708.2539 | https://arxiv.org/abs/0708.2539 | yesno | For a set S of positive integers and a real number t >= 0, let S(t) = |{s in S : 1 <= s <= t}|. Say that S has positive lower asymptotic density if liminf_{x -> infinity} S(x)/x > 0. For every pair of subsets A and B of the positive integers, with no further structural or arithmetic assumptions, does the existence of a... | Yes means the implication holds for every such pair A and B. | null | SOLVED_EXACT | true | No | high | https://arxiv.org/abs/2205.08511 | Yuchen Ding, A counterexample of two Romanov type conjectures, C. R. Math. Acad. Sci. Paris (2023), doi:10.5802/crmath.425; arXiv:2205.08511 | Section 2 (Counterexample); no numbered theorem — disproves Conjecture 1.1 | Conjecture 1.1 [disproved]. Let A and B be two sets of positive integers. If there exists a constant c > 0 such that A(log x/log 2)B(x) > cx for all sufficiently large x, then the set {2^a + b : a ∈ A, b ∈ B} has a positive lower asymptotic density. — Section 2 constructs A = N, B = ∪_t B_t, B_t = {n : d_t | n} ∩ [2^{2... | The hypothesis holds with c = 1 for all large x (eq. 2.5), while C(x) = o(x), so the lower (indeed upper) density of 2^A + B is 0. Hence the implication fails; answer No. If N is read as including 0, replacing A by the positive integers changes A(log_2 x) by at most 1 and only shrinks C, so the example stays valid (INF... | Take A = N (all exponents) and B = ∪_t B_t with B_t = {n : d_t | n} ∩ [2^{2^{t^2}}, 2^{2^{(t+1)^2}}), where d_t = p_1⋯p_t is the product of the first t odd primes. For x with 2^{2^{j^2}} <= x < 2^{2^{(j+1)^2}}, Chebyshev gives d_j <= exp(3√(log log x)·log log log x) = (log x)^{o(1)}, so B(x) >> x/(log x)^{o(1)} and A(l... | null | null | [
"HAS_INFERRED_STEPS"
] | Ding's Conjecture 1.1 matches source Conjecture 1 verbatim. | [{"id_or_ref": "arXiv:2205.08511", "title": "A counterexample of two Romanov type conjectures", "verified": true}, {"id_or_ref": "arXiv:0708.2539", "title": "Source (Pan–Li, The Romanoff theorem revisited), Remark: Chen's Conjectures 1 and 2", "verified": true}] | 5 |
0708.2539#1 | 0708.2539 | https://arxiv.org/abs/0708.2539 | yesno | Let A and B be arbitrary subsets of the positive integers. For every real y≥1, write A(y)=|{a∈A:1≤a≤y}| and B(y)=|{b∈B:1≤b≤y}|. Suppose there is a constant c>0 such that A(log x/log 2)B(x)>cx for infinitely many positive integers x. Must S={2^a+b:a∈A, b∈B} satisfy limsup_{x→∞}|S∩[1,x]|/x>0? Answer "Yes" or "No". | Yes means the stated hypothesis implies positive upper asymptotic density for every such pair A,B. | null | SOLVED_EXACT | true | No | high | https://arxiv.org/abs/2205.08511 | Yuchen Ding, A counterexample of two Romanov type conjectures, C. R. Math. Acad. Sci. Paris (2023), doi:10.5802/crmath.425; arXiv:2205.08511 | Section 2 (Counterexample); no numbered theorem — disproves Conjecture 1.2 | Conjecture 1.2 [disproved]. Let A and B be two sets of positive integers. If there exists a constant c > 0 such that A(log x/log 2)B(x) > cx for infinitely many x, then the set {2^a + b : a ∈ A, b ∈ B} has a positive upper asymptotic density. — The same Section 2 example gives A(log x/log 2)B(x) > x for all large x (he... | The hypothesis for all large x implies it for infinitely many integers x. Since C(x) = o(x), limsup C(x)/x = 0, so the upper density is 0. Answer No. | Take A = N (all exponents) and B = ∪_t B_t with B_t = {n : d_t | n} ∩ [2^{2^{t^2}}, 2^{2^{(t+1)^2}}), where d_t = p_1⋯p_t is the product of the first t odd primes. For x with 2^{2^{j^2}} <= x < 2^{2^{(j+1)^2}}, Chebyshev gives d_j <= exp(3√(log log x)·log log log x) = (log x)^{o(1)}, so B(x) >> x/(log x)^{o(1)} and A(l... | null | null | [] | One construction disproves both conjectures at once. | [{"id_or_ref": "arXiv:2205.08511", "title": "A counterexample of two Romanov type conjectures", "verified": true}, {"id_or_ref": "arXiv:0708.2539", "title": "Source (Pan–Li, The Romanoff theorem revisited), Remark: Chen's Conjecture 2", "verified": true}] | 5 |
0708.2632#1 | 0708.2632 | https://arxiv.org/abs/0708.2632 | numerical | What is the smallest dimension n for which the following property fails for some finite full-rank multiset X of nonzero vectors in R^n with X subset Z^n and span_Z(B)=Z^n for every basis B subset X of R^n? Define Z(X)={sum_{x in X} t_x x : 0<=t_x<=1}, Z_-(X)=int(Z(X)) cap Z^n, and let M_X be the box spline obtained by ... | The answer is the smallest dimension admitting a counterexample, or -1 if there is none. | -1 | SOLVED_EXACT | true | -1 | high | https://arxiv.org/abs/1211.1187 | Matthias Lenz, Interpolation, box splines, and lattice points in zonotopes, Int. Math. Res. Not. IMRN 2014(20), 5697–5712, doi:10.1093/imrn/rnt142; arXiv:1211.1187 | Theorem 2 (Main Theorem) | Theorem 2 (Main Theorem). Let X ⊆ Λ ⊆ U ≅ R^d be a list of vectors that is totally unimodular. Let f be a real valued function on Z_−(X), the set of interior lattice points of the zonotope defined by X. Then there exists a unique polynomial p ∈ P_−(X) ⊆ R[s_1,...,s_d], s.t. p(D)B_X equals f on Z_−(X). | The question is Holtz–Ron Conjecture 1.8 (arXiv:0708.2632). Lenz's P_−(X) := ∩_{x∈X} P(X\x) equals ker I_−(X), with I_−(X) = Ideal{p_{η_H}^{m(H)−1}} exactly as in the question, by Holtz–Ron Theorem 5.3. 'Every basis in X is a Z^n-basis' is Lenz's total unimodularity with Λ = Z^n. Lenz's B_X differs from the convolution... | Define γ_X : P_−(X) → Ξ(X) = {f : Λ → R, supp f ⊆ Z_−(X)}, p ↦ (z ↦ p(D)B_X(z)); Proposition 1 (via the wall-crossing formula) ensures p(D)B_X is continuous so point evaluation makes sense. The theorem is equivalent to γ_X being an isomorphism.
Proof by deletion–contraction (Proposition 19): for x ∈ X that is not a co... | null | null | [
"HAS_INFERRED_STEPS"
] | The candidate Ardila–Postnikov (0809.2143) claimed Conjecture 1.8 as Corollary 4.19, deduced from Theorem 4.18 (= Holtz–Ron Conjecture 6.1). Their own corrigendum (arXiv:1211.1368; Trans. AMS 367 (2015) 3759–3762) shows Theorem 4.18 / Conj 6.1 is FALSE (counterexample in C^4) and that their proof had an error, so the c... | [{"id_or_ref": "arXiv:1211.1187", "title": "Interpolation, box splines, and lattice points in zonotopes (Lenz)", "verified": true}, {"id_or_ref": "arXiv:1211.1368", "title": "Two counterexamples for power ideals of arrangements (Ardila–Postnikov corrigendum)", "verified": true}, {"id_or_ref": "arXiv:0809.2143", "title"... | 5 |
0708.3201#0 | 0708.3201 | https://arxiv.org/abs/0708.3201 | numerical | What is the smallest positive integer n for which there exist a positive integer m and symmetric real n×n matrices A_1,...,A_m such that (sum_{r=1}^m ||A_r||^2)^2 < 2 sum_{1<=r<s<=m} ||[A_r,A_s]||^2? Here ||A|| is the Frobenius norm and [A,B]=AB-BA. Answer -1 if no such n exists. | The answer is the smallest dimension admitting a counterexample, or -1 if the inequality holds for every permitted n, m, and collection of matrices. | -1 | SOLVED_EXACT | true | -1 | high | https://arxiv.org/abs/0801.0650 | J. Ge, Z. Tang, A proof of the DDVV conjecture and its equality case, Pacific J. Math. 237 (2008) 87-95, arXiv:0801.0650 (independently Z. Lu, Normal scalar curvature conjecture and its applications, arXiv:0803.0502, J. Funct. Anal. 2011) | Theorem 1.1 | Theorem 1.1. Let B1, ..., Bm be (n x n) real symmetric matrices. Then sum_{r,s=1}^m ||[Br, Bs]||^2 <= (sum_{r=1}^m ||Br||^2)^2, where the equality holds if and only if under some rotation all Br's are zero except 2 matrices which can be written as P [[0,mu,0,...],[mu,0,0,...],0...] P^t, P [[mu,0,...],[0,-mu,...],0...] ... | INFERRED rewriting: in sum_{r,s=1}^m the diagonal terms r=s vanish, and ||[Bs,Br]|| = ||[Br,Bs]||, so sum_{r,s} = 2 sum_{r<s}. Theorem 1.1 then reads (sum ||Br||^2)^2 >= 2 sum_{r<s} ||[Br,Bs]||^2 for all n, m >= 1 and all real symmetric B_r. So no n admits a strict violation (sum)^2 < 2 sum_{r<s}, and the answer is -1. | Ge-Tang expand the family in an orthonormal basis {E_alpha} of the N = n(n+1)/2 dimensional space of symmetric matrices, (B1,...,Bm) = (E)B with B in M(N,m). A second-compound (Plucker) map phi, which is multiplicative (Lemma 2.5), turns the commutator sum into sum_{r,s} ||[Br,Bs]||^2 = sum_{alpha,beta} x_alpha x_beta ... | null | null | [] | The candidate DOI matches Ge-Tang (Pacific J. Math. 237 (2008) 87-95), whose arXiv ID is 0801.0650. This is the classical DDVV conjecture, proved independently by Lu and by Ge-Tang. | [{"id_or_ref": "arXiv:0801.0650", "title": "A proof of the DDVV conjecture and its equality case (Ge, Tang); DOI 10.2140/pjm.2008.237.87", "verified": true}, {"id_or_ref": "arXiv:0803.0502", "title": "Normal scalar curvature conjecture and its applications (Lu)", "verified": true}, {"id_or_ref": "arXiv:1006.5326", "tit... | 4 |
0708.3201#2 | 0708.3201 | https://arxiv.org/abs/0708.3201 | numerical | For each integer n ≥ 1 and all nonnegative integers m₁,m₂, let A₁,…,A_{m₁} be real symmetric n × n matrices and A_{m₁+1},…,A_{m₁+m₂} be real skew-symmetric n × n matrices. Write [Aᵣ,Aₛ]=AᵣAₛ−AₛAᵣ and ‖A‖²=∑ᵢ,ⱼ aᵢⱼ². What is the smallest n for which some choice of m₁,m₂ and these matrices violates (∑ᵣ ‖Aᵣ‖²)² ≥ 2∑ᵣ<ₛ ‖[... | Give the smallest dimension admitting a violation, or −1 if the inequality holds for every permitted choice. | -1 | SOLVED_EXACT | true | 2 | high | https://arxiv.org/abs/0801.0650 | J. Ge, Z. Tang, A proof of the DDVV conjecture and its equality case, Pacific J. Math. 237 (2008) 87-95, arXiv:0801.0650 | Remark and Example following Theorem 1.1 (Section 1); also Ge-Li-Zhou arXiv:1807.07307, Remark 2.4 | Remark. By the same method, one can see that Conjecture 1 also holds for anti-symmetric matrices. However, the following example shows that Conjecture 1 fails when there're both symmetric and anti-symmetric matrices in {B1, ..., Bm}, which was conjectured in [Lu1]. Example. Let B1 = [[1,0],[0,-1]], B2 = [[0,1],[1,0]], ... | Ge-Tang's Conjecture 1 has the form sum_{r,s=1}^m ||[B_r,B_s]||^2 <= (sum ||B_r||^2)^2, with the sum over ordered pairs. Diagonal terms vanish and ||[B_s,B_r]|| = ||[B_r,B_s]||, so this is exactly (sum ||A_r||^2)^2 >= 2 sum_{r<s} ||[A_r,A_s]||^2, i.e. Lu's P(n,m1,m2). Their example is a mixed family with n=2, m1=2 (B1,... | The mixed conjecture P(n,m1,m2) was proposed by Lu (source, Conjecture 4) as a common generalization of DDVV and Bottcher-Wenzel. Ge and Tang, right after proving the symmetric DDVV inequality (Theorem 1.1), note that the inequality with constant 1 (ordered-sum form) also holds for purely skew-symmetric families but fa... | Optimal constant for arbitrary real (hence mixed symmetric/skew) families with m>=3 is 4/3 in ordered-sum form (Ge-Li-Zhou Thm 2.3), versus 1 conjectured. | null | [] | The answer is the opposite of the distinguished value -1: the mixed conjecture is false. The candidate is correct in that it contains the disproof, but in a remark and example, not in its main theorem. Ge-Li-Lu-Zhou (arXiv 1908.06624, with Lu as coauthor) also records the constant 4/3 for arbitrary real matrices. | [{"id_or_ref": "arXiv:0801.0650", "title": "A proof of the DDVV conjecture and its equality case (Ge, Tang); Pacific J. Math. 237 (2008) 87-95, DOI 10.2140/pjm.2008.237.87", "verified": true}, {"id_or_ref": "arXiv:1807.07307", "title": "Some generalizations of the DDVV and BW inequalities (Ge, Li, Zhou)", "verified": t... | 4 |
0708.3431#2 | 0708.3431 | https://arxiv.org/abs/0708.3431 | yesno | Let a mass-action network be a finite directed graph G=(V,E) on vertices {1,...,n}, with vertex i labeled by c^{y_i} for y_i∈Z^s_{>=0} and each edge (i,j) assigned a positive rate constant κ_{ij}. Write Ψ(c)=(c^{y_1},...,c^{y_n}); let A_κ have off-diagonal entry κ_{ij} on each edge and row sums zero; and let Y have row... | Yes means that at least one of the three stated universal global-attraction assertions holds. | null | SOLVED_EXACT | true | Yes | high | https://arxiv.org/abs/0903.0901 | D. F. Anderson, A. Shiu, The dynamics of weakly reversible population processes near facets, SIAM J. Appl. Math. 70(6) (2010) 1840–1858 (arXiv:0903.0901) | Corollary 4.7 (GAC for two-dimensional P), from Theorem 4.6 | Corollary 4.7 (GAC for two-dimensional P). The Global Attractor Conjecture holds for all complex-balancing (and in particular, detailed-balancing or weakly reversible zero deficiency) chemical reaction systems whose positive stoichiometric compatibility classes are two-dimensional. | Class (ii) = detailed balancing systems with dim S = 2 and unbounded P. Detailed balancing ⊂ complex balancing, and dim S = 2 means every positive stoichiometric compatibility class is two-dimensional. Corollary 4.7 has no boundedness (conservativity) hypothesis; the authors state explicitly (Sec. 4.2) that it eliminat... | Trajectories of complex-balanced systems are bounded and converge to the equilibrium set (Horn–Jackson Lyapunov function), so GAC is equivalent to showing no boundary equilibrium is an omega-limit point of an interior trajectory (persistence). Boundary equilibria can only lie in faces F_W with W a semilocking set (siph... | Refereed: GAC for dim S ≤ 2 (Anderson–Shiu 2010), dim S = 3 (Pantea, SIAM J. Math. Anal. 2012, arXiv:1103.0603, Thm 6.3), three species (Craciun–Nazarov–Pantea 2013, arXiv:1010.3050), single linkage class (Anderson 2011, arXiv:1101.0761). Full GAC claimed by Craciun arXiv:1501.02860 (unrefereed; v3 posted 23 Sep 2026 a... | null | [] | Answer is Yes, but the candidate is not the right citation. Craciun's 1501.02860 claims the full GAC, which would cover all three classes, but as far as I could check it is still unrefereed, and its v3 (23 Sep 2026) is described as a major revision that completes the general n-dimensional case. So it should not be the ... | [{"id_or_ref": "arXiv:0903.0901", "title": "The dynamics of weakly reversible population processes near facets", "verified": true}, {"id_or_ref": "arXiv:1103.0603", "title": "On the persistence and global stability of mass-action systems", "verified": true}, {"id_or_ref": "arXiv:1501.02860", "title": "Toric Differentia... | 6 |
0708.3625#0 | 0708.3625 | https://arxiv.org/abs/0708.3625 | numerical | What is the smallest integer N≥1 for which the following equality fails for at least one noncritical nearest-neighbour Ising coupling with s=sinh(2K_c)>0 and s≠1 and at least one pair of physical, normalized transfer-matrix eigenstates on the N-site periodic square lattice? If it never fails, answer −1. Define γ(k)>0 b... | Give the smallest failing N; give −1 if the equality holds for every specified coupling and pair of physical states at every N≥1. | -1 | SOLVED_EXACT | true | -1 | high | https://arxiv.org/abs/0711.0457 | G. von Gehlen, N. Iorgov, S. Pakuliak, V. Shadura, Yu. Tykhyy, 'Form-factors in the Baxter-Bazhanov-Stroganov model II: Ising model on the finite lattice', J. Phys. A 41 (2008) 095003, arXiv:0711.0457 (first proof); re-derived in von Gehlen-Iorgov-Pakuliak-Shadura, 'Factorized finite-size Ising model spin matrix elemen... | arXiv:0904.2265, Section 7.4 eq. (128) and Section 7.5 eq. (129) (main result of the paper, Sect. 7); details in arXiv:0711.0457. | 0904.2265 Sect. 7.5: 'In [18] the following formula for the square of the matrix element of spin operator for the finite-size Ising model was conjectured: |NS<q1,...,qK|sigma^z_n|p1,...,pL>_R|^2 = xi xi_T prod_{k=1}^K [prod^{NS}_{q!=q_k} sinh((gamma(q_k)+gamma(q))/2) / (n prod^R_p sinh((gamma(q_k)+gamma(p))/2))] prod_{... | The SoV computation (eq. (127)-(128)) yields the squared spin matrix element between arbitrary NS and R eigenstates of the finite periodic transfer matrix, for both parities of the lattice width n, for anisotropic couplings K_x,K_y; (129) follows. INFERRED: for the isotropic nearest-neighbour coupling of the question, ... | The Ising model is treated as the N=2 degenerate case of the cyclic Baxter-Bazhanov-Stroganov tau^(2) model. Using the Sklyanin-Kharchev-Lebedev separation of variables adapted to cyclic models, the auxiliary (open/fixed boundary) problem is solved, then periodic eigenvectors are built from solutions of Baxter equation... | null | null | [
"HAS_INFERRED_STEPS"
] | Candidate verified (authors von Gehlen, Iorgov, Pakuliak, Shadura). It is a follow-up that summarizes/re-derives the proof; the first proof is arXiv:0711.0457 (abstract: 'we derive for the first time the factorized formula ... conjectured previously by A. Bugrij and O. Lisovyy'). Proof is of the squared modulus (overal... | [{"id_or_ref": "arXiv:0904.2265", "title": "Factorized finite-size Ising model spin matrix elements from Separation of Variables", "verified": true}, {"id_or_ref": "arXiv:0711.0457", "title": "Form-factors in the Baxter-Bazhanov-Stroganov model II: Ising model on the finite lattice", "verified": true}, {"id_or_ref": "a... | 5 |
0709.0084#0 | 0709.0084 | https://arxiv.org/abs/0709.0084 | yesno | Let X be a nonvoid set, T:X→X a mapping, x∈X, and N_+={1,2,3,...}. For every finite partition Δ of X, define Δ(x)={A∈Δ:{n∈N_+:T^n(x)∈A} is infinite}. Order finite partitions by Δ≤Δ′ when every cell of Δ′ is contained in a cell of Δ, and for Δ≤Δ′ let ψ_{Δ′,Δ,x}:Δ′(x)→Δ(x) send each cell to its unique containing cell. Fo... | Yes means the stated equivalence holds for every nonvoid X, mapping T:X→X, and x∈X. | null | SOLVED_EXACT | true | No | high | https://arxiv.org/abs/0710.2497 | D. Litt, Z. Abel, S. D. Kominers, 'A Categorical Construction of Ultrafilters', Rocky Mountain J. Math. 40 (2010) 1611, arXiv:0710.2497 | Corollary 5 (via Theorem 4) | Corollary 5. Let X be a set and T : X -> X be a function. For Delta in FP(X), let Delta(x) := {D in Delta : {n in N : T^n(x) in D} is infinite}. Then for each x in X, we have lim_{<- Delta in FP(X)} Delta(x) != emptyset. ... Corollary 5 negatively answers the conjecture Rosinger posed in [4]. | By Corollary 5 the inverse limit is nonempty for every X, T, x. Take any non-periodic point, e.g. X=N, T(n)=n+1, x=0 (INFERRED example): the limit is nonempty but no n>=1 has T^n(x)=x, so the 'only if' direction fails; the equivalence does not hold for every X,T,x. Answer: No. (Whether N or N_+ indexes the orbit does n... | Theorem 3 identifies the set of ultrafilters on I with the inverse limit over finite partitions Delta of I (each ultrafilter picks the unique cell of each partition that it contains), and free ultrafilters with the inverse limit of the 'infinite cells'. Theorem 4: for f: I -> X with I infinite, a free ultrafilter U on ... | null | null | [
"HAS_INFERRED_STEPS"
] | The paper's proof uses the free ultrafilter theorem (choice), but the conjecture already fails without choice for eventually-periodic non-periodic points (e.g. constant map, x != c). Rosinger's own Example 2.2 (eq. 23) claims the limit is empty for constant T and x!=c, which contradicts his eq. (8); that error underlie... | [{"id_or_ref": "arXiv:0710.2497", "title": "A Categorical Construction of Ultrafilters", "verified": true}, {"id_or_ref": "arXiv:0709.0084", "title": "A Fixed Point Conjecture", "verified": true}] | 5 |
0709.1458#0 | 0709.1458 | https://arxiv.org/abs/0709.1458 | numerical | For every g>=0 and h>=1 with (g,h) not equal to (0,1) or (0,2), let H_g(x_1,...,x_h)=sum_{mu: ell(mu)=h} z_mu H_{g,mu} m_mu(x)/(2g-2+h+|mu|)!, where H_{g,mu} is the connected simple Hurwitz number, mu is a partition with length ell(mu)=h and size |mu|, z_mu=|Aut(mu)| product_i mu_i, and m_mu(x)=|Aut(mu)|^{-1} sum_{sigm... | Give the smallest genus with a failure, or -1 if every stated identity holds. | -1 | SOLVED_EXACT | true | -1 | high | https://arxiv.org/abs/1307.4729 | P. Dunin-Barkowski, M. Kazarian, N. Orantin, S. Shadrin, L. Spitz, 'Polynomiality of Hurwitz numbers, Bouchard-Marino conjecture, and a new proof of the ELSV formula', Adv. Math. 279 (2015) 67-103, arXiv:1307.4729. Earlier proofs: Eynard-Mulase-Safnuk arXiv:0907.5224; Borot-Eynard-Mulase-Safnuk arXiv:0906.1206 (matrix ... | Theorem 3.1 (Bouchard-Marino conjecture) | Theorem 3.1 (Bouchard-Marino conjecture). The polynomials W_{g,n} can be determined by the either of the following recursive formulas W_{g,n}(t_1,t_{L'}) = - res_{z=0} K(z,t_1) W~_{g,n}(1/z, 1/z; t_{L'}) = res_{z=0} K(z,t_1) W~_{g,n}(1/z, 1/sigma(z); t_{L'}) = - res_{z=0} K(z,t_1) W~_{g,n}(1/sigma(z), 1/sigma(z); t_{L'... | Theorem 3.1 holds for all (g,n) outside the unstable cases, with the unstable data W_{0,1}=0 and W_{0,2} = Bergman kernel minus the pullback of dx1dx2/(x1-x2)^2 (eqs. (3-6),(3-8),(3-9)), which are the g=0 identities of the question, on the Lambert curve x=y e^{-y} with y=1+z (Sect. 3.2). INFERRED: the question's normal... | First, (quasi-)polynomiality of connected simple Hurwitz numbers h_{g;mu} = prod mu_i^{mu_i}/mu_i! x P_{g,n}(mu) is proved without ELSV, refining Okounkov-Pandharipande's operator formalism (Theorem 2.1). Polynomiality makes the generating functions H_{g,n}, after x=(1+1/t)e^{-1-1/t}, polynomials in t_i. Then the Lapla... | null | null | [
"HAS_INFERRED_STEPS"
] | Candidate correct but not first: EMS 0907.5224 (Publ. RIMS 2011) gave the first rigorous proof (using ELSV); 1307.4729 gives an ELSV-independent proof. Normalization matching to BM's H_g is inferred, not re-checked term by term. | [{"id_or_ref": "arXiv:1307.4729", "title": "Polynomiality of Hurwitz numbers, Bouchard-Marino conjecture, and a new proof of the ELSV formula", "verified": true}, {"id_or_ref": "arXiv:0907.5224", "title": "The Laplace transform of the cut-and-join equation and the Bouchard-Marino conjecture on Hurwitz numbers", "verifi... | 5 |
0709.1458#3 | 0709.1458 | https://arxiv.org/abs/0709.1458 | numerical | Let X be any toric Calabi–Yau threefold, and let Σ={H(x,y)=0}⊂(C*)² be its mirror curve. Choose a projection x:Σ→C* corresponding to the local open modulus of a toric brane L of topology C×S¹ with b₁(L)=1. Assume every ramification point q_i of x is simple, and let q̄ be the distinct local point near q_i with x(q)=x(q̄... | Return the smallest failing genus, or −1 if equality holds for every genus, number of boundary components, and permitted choice of data. | -1 | SOLVED_EXACT | true | -1 | medium | https://arxiv.org/abs/1205.1103 | B. Eynard, N. Orantin, 'Computation of open Gromov-Witten invariants for toric Calabi-Yau 3-folds by topological recursion, a proof of the BKMP conjecture', Comm. Math. Phys. 337 (2015) 483-567, arXiv:1205.1103. Independent complete proof (including orbifolds): B. Fang, C.-C. M. Liu, Z. Zong, 'On the Remodeling Conject... | Theorem 4.8 (proof of Conjecture 4.1, BKMP) | Theorem 4.8 The BKMP conjecture holds true. In other words, the invariants W_{g,n} of the mirror curve S do coincide with the Gromov-Witten invariants: for all (g,n) in N^2 \ {(0,0),(1,0)}, W_{g,n}(S; x_1,...,x_n) = W_{g,n}(X, x_1,...,x_n) dx_1 (x) ... (x) dx_n. [Conjecture 4.1: If X is a toric Calabi-Yau 3-fold, for ... | Theorem 4.8 covers every genus g and every number of boundaries n (n=0 gives closed F_g, g>=2; (0,1),(0,2) are the disk/annulus initial data), for every smooth toric CY3 and toric brane on a half-edge with framing. Hence there is no smallest failing genus: -1. INFERRED: the question's kernel (log y - log ybar) dx/x is ... | Both sides are written as sums over weighted graphs. On the A-side, torus localization expresses open/closed GW invariants as sums over graphs with vertex weights given by triple Hodge integrals (framed vertices) and edge propagators. On the B-side, the topological recursion invariants of a spectral curve with several ... | null | null | [
"HAS_INFERRED_STEPS"
] | Medium confidence because EO's proof defines open GW invariants by localization and has been regarded by some as needing more detail; Fang-Liu-Zong 1604.07123 gives an independent full proof of the all-genus open/closed remodeling (BKMP) conjecture for toric CY3 (orbifolds too), so the answer -1 stands. | [{"id_or_ref": "arXiv:1205.1103", "title": "Computation of open Gromov-Witten invariants for toric Calabi-Yau 3-folds by topological recursion, a proof of the BKMP conjecture", "verified": true}, {"id_or_ref": "arXiv:1604.07123", "title": "On the Remodeling Conjecture for Toric Calabi-Yau 3-Orbifolds", "verified": true... | 5 |
0709.3350#0 | 0709.3350 | https://arxiv.org/abs/0709.3350 | yesno | Let H and G be semisimple groups defined over Q, with associated Hermitian symmetric spaces X_H and X_G. Let F:H→G be a Q-defined homomorphism and f:X_H→X_G an F-equivariant holomorphic embedding. For suitable arithmetic subgroups Δ⊂H(Q) and Γ⊂G(Q), set S_H=Δ\X_H and S_G=Γ\X_G, and let g:S_H→S_G be induced by f. Suppos... | Yes means the stated conclusions follow from the André–Oort conjecture under all the given hypotheses. | null | SOLVED_EXACT | true | Yes | high | https://arxiv.org/abs/2109.08788 | J. Pila, A. N. Shankar, J. Tsimerman (with appendix by H. Esnault, M. Groechenig), Canonical heights on Shimura varieties and the Andre-Oort conjecture, 2021; combined with B. Moonen, Linearity properties of Shimura varieties I, J. Algebraic Geom. 7 (1998) | Pila-Shankar-Tsimerman Theorem 1.1 (the Andre-Oort statement) + Moonen (1998) characterization: special subvariety <=> totally geodesic and contains a special point. The implication itself is asserted in the source (Clozel, Introduction p.318). | Theorem 1.1. Let S be a Shimura variety. Let V ⊂ S be a subvariety. Then there are only finitely many maximal special subvarieties contained in V. [PST, arXiv:2109.08788]. Source text (Clozel 0709.3350, p.318): "A conjecture of André [1] and Oort [7] then implies that g(S_H) is a totally geodesic submanifold of S_G (an... | (1) V := g(S_H) is an irreducible algebraic subvariety of S_G (given; Borel) whose CM points are dense in the complex topology, hence Zariski dense (INFERRED, trivial). (2) Andre-Oort (PST Thm 1.1): V contains finitely many maximal special subvarieties; every CM point is a 0-dimensional special subvariety, so lies in o... | The implication is the standard 'Andre-Oort => special => totally geodesic' chain that Clozel invokes as motivation. Special points of g(S_H) are dense because H(Q) acts on X_H and moves one CM point to a dense set of CM points; AO (Pila-Shankar-Tsimerman via o-minimal point counting plus height bounds for CM points, e... | null | null | [] | Caveat: requires that S_G is (a connected component of) a Shimura variety so that 'special subvariety' makes sense; the source assumes this implicitly ('suitable' arithmetic groups, CM points). The yes/no is about an implication, which holds; moreover AO is now proved (PST 2021) and Clozel's own theorem gives total geo... | [{"id_or_ref": "arXiv:0709.3350", "title": "Equivariant embeddings of Hermitian symmetric spaces, L. Clozel", "verified": true}, {"id_or_ref": "arXiv:2109.08788", "title": "Canonical Heights on Shimura Varieties and the Andre-Oort Conjecture, Pila-Shankar-Tsimerman (+Esnault-Groechenig)", "verified": true}, {"id_or_ref... | 3 |
0709.3591#3 | 0709.3591 | https://arxiv.org/abs/0709.3591 | numerical | What is the smallest prime p >= 5 for which there exist N >= 1 with (N,p)=1 and p not dividing phi(N), and r >= 1, such that the following property fails? Put F_r=Q(mu_{Np^r}); let S consist of the primes of F_r above Np and its real places; and let X_1^r(N)=X_1(Np^r) have cusp set C_1^r(N). On the +1 eigenspace for th... | Give the smallest such prime p, or -1 if the property holds for every permitted p, N, and r. | -1 | SOLVED_EXACT | true | -1 | high | https://arxiv.org/abs/2011.07241 | T. Fukaya, K. Kato, 'On conjectures of Sharifi', Kyoto J. Math. 64 (2024), DOI 10.1215/21562261-2023-0018 (preprint 2012; not on arXiv). Status recorded in Sharifi-Venkatesh arXiv:2011.07241 (§4.3) and Lecouturier-Wang arXiv:2208.06921 = Canad. Math. Bull. 66 (2023) 1194-1212 (the candidate). | Fukaya-Kato [FuKa, Theorem 5.3.5] (as cited in Sharifi-Venkatesh, stated there as Theorem 4.3.6); integral refinements: Sharifi-Venkatesh Theorem 4.3.7 (T_l part) and Lecouturier-Wang Theorem 1.2 (U_l part). | Sharifi-Venkatesh (arXiv:2011.07241v2, p.37): 'Part a of Conjecture 4.3.5 is a stronger form of an earlier conjecture [Sha1, Conjecture 5.8] that the tensor product of Π_N with the identity on Z_p for a prime p | N is Eisenstein. The earlier conjecture was proven by Fukaya and Kato in [FuKa, Theorem 5.3.5]. In fact, th... | The question is verbatim Sharifi's Conjecture 5.8 in the source (arXiv:0709.3591, p.35): restriction of ϖ_r to H_1(X_1^r(N);Z_p)^+ satisfies ϖ_r(T_l x)=(1+lε(<l>))ϖ_r(x) for l∤Np and ϖ_r(U_l x)=ϖ_r(x) for l|Np. Sharifi-Venkatesh state explicitly that [Sha1, Conjecture 5.8] was proven by Fukaya-Kato; their Theorem 4.3.6... | Fukaya-Kato write Sharifi's map as the composition of a Hecke-equivariant 'zeta map' z_N sending Manin symbols to cup products of Siegel units (Beilinson-Kato elements) with a specialization map given by pulling back at the cusp 0. Hecke equivariance of z_N is proved through Iwasawa- and Hida-theoretic constructions an... | null | null | [] | Candidate (Lecouturier-Wang, arXiv:2208.06921, same paper as the DOI; the item listed arxiv null) is correct in conclusion but is not the primary resolver: for the question's exact p-adic setting (p≥5, p | level Np^r), both the T_l and U_l relations were already proven by Fukaya-Kato; Lecouturier-Wang's Remark 1.2(ii) ... | [{"id_or_ref": "arXiv:2208.06921 / doi:10.4153/S0008439523000267", "title": "Level compatibility in Sharifi's conjecture (Lecouturier, Wang)", "verified": true}, {"id_or_ref": "arXiv:2011.07241", "title": "Eisenstein cocycles in motivic cohomology (Sharifi, Venkatesh)", "verified": true}, {"id_or_ref": "doi:10.1215/215... | 5 |
1109.2168#0 | 1109.2168 | https://arxiv.org/abs/1109.2168 | numerical | Let Y be a closed oriented three-manifold and let (L,p) be an oriented l-component link in Y with components L_1,...,L_l and one basepoint p_i on each L_i. Work over F_2 with CFL(Y,L,p), the naturality-defined associated graded minus link Floer chain complex over F_2[U_1,...,U_l], considered up to U_i-equivariant chain... | The answer is the smallest number of link components admitting a failure, or -1 if the automorphisms agree in every case. | -1 | SOLVED_EXACT | true | -1 | high | https://arxiv.org/abs/1604.04316 | I. Zemke, Quasi-stabilization and basepoint moving maps in link Floer homology, Algebr. Geom. Topol. 17 (2017) 3461–3518, doi:10.2140/agt.2017.17.3461 (arXiv:1604.04316) | Theorem B (proved in Section 10.2) | Theorem B. Suppose that L = (L, w, z) is a multibased link in an arbitrary 3-manifold Y and K is a component of L. Suppose that the basepoints on K are z1, w1, ..., zn, wn. Letting ς denote the diffeomorphism resulting from a finger move around a link component K, the induced map ς_* on CFL^∞_UV(Y, L, σ, P, s) has the ... | Zemke's paper states that Theorem B is Sarkar's conjecture (Conj. 4.7 of 1109.2168: ρ(σ_i) = Id + Ψ_iΦ_i on CFL(Y,L,p)). It holds for arbitrary Y and on the full complex. Zemke also notes that the formula for other flavors (e.g. CFL^-, and the associated graded version) follows by setting variables to zero. Sarkar's Th... | Zemke first builds quasi-stabilization maps S^±_{w,z}, which add or remove an adjacent pair of basepoints on a link component. He proves they are natural (Theorem A) and derives commutation relations with the maps Φ_w and Ψ_z, which are formal derivatives of the differential in U_w and V_z.
The key lemma (10.1) comput... | null | null | [
"HAS_INFERRED_STEPS"
] | The source's Conjecture 4.7 matches the question. Sarkar proved it for S^3 (Thm 1.1); Zemke proves it for general Y. | [{"id_or_ref": "arXiv:1604.04316", "title": "Quasi-stabilization and basepoint moving maps in link Floer homology", "verified": true}, {"id_or_ref": "arXiv:1109.2168", "title": "Moving basepoints and the induced automorphisms of link Floer homology", "verified": true}] | 6 |
1603.09589#0 | 1603.09589 | https://arxiv.org/abs/1603.09589 | numerical | For positive integers d, a, and b, let λ=δ_d(b^a)=(((d−1)b)^a,((d−2)b)^a,…,b^a), obtained from the staircase δ_d=(d−1,d−2,…,1) by replacing each cell with an a-by-b block. For a permutation w, let c_i(w)=#{j>i:w(i)≥w(j)}, and let its shape be the weakly decreasing rearrangement of its Lehmer code. What is the smallest ... | Give the smallest such d, or −1 if the stated property holds for every d, a, b, and w in the specified ranges. | -1 | SOLVED_EXACT | true | -1 | high | https://arxiv.org/abs/1811.02404 | Sam Hopkins, The CDE property for skew vexillary permutations, J. Combin. Theory Ser. A 168 (2019) | Theorem 1.3 (= Theorem 5.18 / Corollary 5.19) of arXiv:1811.02404, together with RTY Theorem 1.1(c) for [∅,λ] and [∅,λ]* | Theorem 1.3. Let σ = λ/ν be a balanced shape of height a and width b and w ∈ S_n a skew vexillary permutation of shape σ. Then [e, w] is CDE with edge density ab/(a + b). [Also, Sec. 3: 'Another observation about this collection: it is closed under duality. This is because, as mentioned at the beginning of Section 3.1... | (1) The source's property is RTY Conjecture 1.2 (arXiv:1603.09589): 'When λ = δ_d(b^a), the conclusion in Theorem 1.1(c) holds for all vexillary w of shape λ.' That conclusion is that [∅,λ], [e,w], [∅,λ]*, [e,w]* are all CDE with E(X)=E(Y)=(d−1)ab/(a+b). Hopkins states that his main result proves this conjecture. (2) [... | Hopkins works with the 'toggle' perspective of Chan-Haddadan-Hopkins-Moci, extended to semidistributive lattices using Barnard's edge labeling by join-irreducibles. Intervals [e,w] of weak order are semidistributive, and toggles T_(i,j) are indexed by inversions (boxes of the inverse Rothe diagram). A distribution is t... | null | null | [] | Hopkins's restatement of RTY Conj. 1.2 mentions only [e,w]. The dual [e,w]* follows from his explicit duality remark in Sec. 3 plus Theorem 1.3 (inferred step), and [∅,λ], [∅,λ]* are already in RTY Thm 1.1(c). The question defines c_i(w) with '>=' where the source has '>'. Since j>i this makes no difference for permuta... | [{"id_or_ref": "arXiv:1603.09589", "title": "Reiner, Tenner, Yong - Poset edge densities, nearly reduced words, and barely set-valued tableaux (JCTA 158, 2018)", "verified": true}, {"id_or_ref": "arXiv:1811.02404", "title": "Hopkins - The CDE property for skew vexillary permutations (JCTA 168, 2019)", "verified": true}... | 3 |
1603.09589#2 | 1603.09589 | https://arxiv.org/abs/1603.09589 | numerical | For d >= 2 and a,b >= 1, let lambda=delta_d(b^a) be the Ferrers shape obtained from delta_d=(d-1,d-2,...,1) by replacing each cell with an a-by-b rectangle. Let w be the unique dominant (132-avoiding) permutation with Lehmer code c(w)=lambda, and put ell=ell(w)=|lambda|=binom(d,2)ab. In the type-A 0-Hecke monoid genera... | The answer is the least d admitting a counterexample, or -1 if the claim holds for every d >= 2 and a,b >= 1. | -1 | SOLVED_EXACT | true | -1 | high | https://arxiv.org/abs/1807.08292 | Neil J.Y. Fan, Peter L. Guo, Sophie C.C. Sun, Proof of a Conjecture of Reiner-Tenner-Yong on Barely Set-valued Tableaux, 2018 | Theorem 3.4 (together with Theorem 3.1) of arXiv:1807.08292 | Theorem 3.4. Let w be a dominant permutation whose Lehmer code is a balanced shape λ with r rows and c columns, and let ℓ = ℓ(w). Then we have FK(w, ℓ+1)/FK(w, ℓ) = binom(ℓ+1, 2) (2xrc/(ℓ(r+c)) + 1). (3.6) [Theorem 3.1. For any positive integer k and a balanced shape λ with r rows and c columns, we have |BSSYT(λ,k)| =... | (1) The claim in the question is exactly RTY Conjecture 6.4 (arXiv:1603.09589), stated with the same FK(w,L) of Definition 6.3. (2) The paper states that δ_d(b^a) is balanced (as observed by Chan-Haddadan-Hopkins-Moci) with r = a(d−1) rows and c = b(d−1) columns. (3) INFERRED: substituting ℓ = binom(d,2)ab = d(d−1)ab/2... | RTY (Theorem 6.8, via Fomin-Kirillov/Fomin-Stanley and Grothendieck-polynomial ideas) express FK(w,L) for vexillary w as a sum of flagged column-strict set-valued tableau counts times j!S(L,j). For dominant w (flag (1,2,3,...)) and L = ℓ, ℓ+1, this gives FK(w,ℓ+1)/FK(w,ℓ) = binom(ℓ+1,2) + (ℓ+1)·|BSSYT(λ,k)|/|SYT(λ,k)| ... | null | null | [] | RTY Conjecture 6.4 exactly matches the source question. RTY proved it is equivalent to Conjecture 6.4' (Cor. 6.11) and proved the case d=2 (Cor. 6.12). Fan-Guo-Sun proved it for all balanced shapes; their Conjecture 1.2 is RTY's 6.4 verbatim. The journal publication of Fan-Guo-Sun was not checked. In the derivation, on... | [{"id_or_ref": "arXiv:1603.09589", "title": "Reiner, Tenner, Yong - Poset edge densities, nearly reduced words, and barely set-valued tableaux (JCTA 158, 2018)", "verified": true}, {"id_or_ref": "arXiv:1807.08292", "title": "Fan, Guo, Sun - Proof of a Conjecture of Reiner-Tenner-Yong on Barely Set-valued Tableaux", "ve... | 3 |
1610.03873#3 | 1610.03873 | https://arxiv.org/abs/1610.03873 | numerical | What is the smallest l for which there exists an r-uniform hyperwheel ^rW_l^{r+1} in the paper's hyperwheel regime (with center l and rim vertices 1,...,l-1 cyclically ordered, whose r-edges are exactly those for which every r consecutive rim vertices together with l form an (r+1)-vertex r-uniform clique) whose Turán p... | Report the smallest such l, or -1 if no hyperwheel in the stated regime is a counterexample. | -1 | SOLVED_EXACT | true | -1 | high | https://d-nb.info/1066163421 | Annie Raymond, "Polyhedral Methods Applied to Extremal Combinatorics Problems", Dr. rer. nat. dissertation, Technische Universität Berlin (Fakultät II), defended 23 June 2014 (advisor M. Grötschel). Full text: https://d-nb.info/1066163421 (PDF at /34). | Theorem 1.3.16 (Section 1.3.5 'Hyperwheel Facets') | Theorem 1.3.16. We have that
$$T({}^rW_l^{r+1}, r+1, r) = \{x \in \mathbb{R}^{|E({}^rW_l^{r+1})|} \mid x(Q^{r+1}) \le r \ \ \forall Q^{r+1} \in \mathcal{Q}^{r+1}_{K_n^r},\ \ x({}^rW_l^{r+1}) \le r\cdot(l-1) - \lceil \tfrac{l-1}{2} \rceil,\ \ 0 \le x_e \le 1 \ \ \forall e \in E({}^rW_l^{r+1})\} = \{x \in \mathbb{R}^{|E(... | 1. (Source, Def. 1.3.11 with a=r+1) The question's ^rW_l^{r+1} is exactly the dissertation's hyperwheel with a=r+1: every a-1=r consecutive rim vertices together with centre l span an (r+1)-vertex r-uniform clique, and the edge set is the union of these l-1 cliques (Section 1.3.5 counts (l-1)*C(r,r-1)=r(l-1) edges).
2.... | Validity of the hyperwheel inequality (Thm 1.3.13) is a Chvátal–Gomory cut: sum the l-1 (r+1)-clique inequalities with weight 1/(a-r+1)=1/2 and add edge inequalities x_e<=1 with weight (β-r+1)/2 for an edge spanning β rim vertices (it lies in a-β cliques), so every edge gets total weight 1; the right-hand side simplifi... | null | null | [] | Proof in Raymond's TU Berlin 2014 dissertation (d-nb.info/1066163421), not on arXiv. | [] | 1-2 |
1703.05510#2 | 1703.05510 | https://arxiv.org/abs/1703.05510 | yesno | For every reduced analytic plane curve germ (C,z) in C^2 that is invariant under complex conjugation and has a real singular point z, do there exist a Milnor ball B_{C,z}, a number zeta>0, and an equivariant analytic family C_t={f_t(x,y)=0}, t in [0,zeta), with C_0=C, such that its complexification to t in a disc D_zet... | Yes means that such a family exists for every germ satisfying the stated conditions. | null | SOLVED_EXACT | true | Yes | medium | https://arxiv.org/abs/2608.07212 | Pablo Portilla Cuadrado, 'Real morsifications via the trace map', arXiv:2608.07212 (Aug 2026) | Theorem A (= Theorem 6.8) | Theorem A. Every reduced real plane curve germ (C, 0) admits a real morsification. Equivalently, there is a real nodal deformation (C_s) of a representative of (C, 0) in a Milnor ball such that, for every s > 0 small enough, the curve C_s has exactly δ(C, 0) − imbr(C, 0) real hyperbolic nodes and no other singularities... | The question is Conjecture 1 of Leviant-Shustin (source), with their definitions of Milnor ball, (equivariant) nodal deformation and real morsification. The paper's Definition 2.12 (real nodal deformation: C_s transverse to and smooth along ∂B, only nodes for s≠0, constant node count, equivariant real power series F(x,... | Key new tool: the trace map. For a conjugate pair of nonreal branches Q, Q̄ with distinct tangents and primitive parametrizations γ_Q, γ_Q̄, define on the smoothing uv=s of the nodal source the equivariant map (u,v,s) ↦ γ_Q(u)+γ_Q̄(v); the fixed circle (re^{iθ}, re^{-iθ}) maps into the real plane and, for small r, its ... | null | null | [
"RECENT_PREPRINT_2026",
"HAS_INFERRED_STEPS"
] | Resolver is a very recent (Aug 2026) single-author preprint, not yet refereed; statement and definitions checked against full text, proof not independently verified, hence medium confidence. It handles the Leviant-Shustin example (4,6,7) pair explicitly (Example 5.24). | [{"id_or_ref": "arXiv:2608.07212", "title": "Real morsifications via the trace map (Portilla Cuadrado)", "verified": true}, {"id_or_ref": "arXiv:1703.05510", "title": "Morsifications of real plane curve singularities (Leviant, Shustin) - source, Conjecture 1", "verified": true}] | 5 |
1703.07251#0 | 1703.07251 | https://arxiv.org/abs/1703.07251 | numerical | What is the smallest integer n >= 10 for which there exist real coefficients a_1,...,a_n with sum_{i=1}^n a_i^2=1 such that, for independent Rademacher random variables epsilon_1,...,epsilon_n, P{|sum_{i=1}^n a_i epsilon_i| <= 1} < 1/2? Answer -1 if no such n exists. | -1 means the original inequality holds for every integer n >= 10 and every permitted choice of coefficients. | -1 | SOLVED_EXACT | true | -1 | high | https://arxiv.org/abs/2006.16834 | N. Keller, O. Klein, Proof of Tomaszewski's conjecture on randomly signed sums, Adv. Math. 407 (2022) 108558, doi:10.1016/j.aim.2022.108558 (arXiv:2006.16834) | Theorem 1.2 | Theorem 1.2. Let X = Σ_{i=1}^n a_i x_i, where Σ_{i=1}^n a_i^2 = 1 and {x_i} are independent and uniformly distributed in {−1, 1}. Then Pr[|X| ≤ 1] ≥ 1/2. | Theorem 1.2 holds for every n and every real unit vector a. So for every n ≥ 10 there is no choice of coefficients with P{|Σ a_i ε_i| ≤ 1} < 1/2. No such n exists, and the answer is -1. | Order |a_1| ≥ |a_2| ≥ ... and split into cases by the sizes of the a_i. If a_1 + a_2 ≥ 1, a semi-inductive (stopping-time) argument applies. If max|a_i| ≤ 0.31, a refined Berry–Esseen inequality for Rademacher sums (via Prawitz's smoothing inequality) gives the bound. The intermediate cases use a 'segment comparison' i... | null | null | [] | The proof includes a light computer-aided verification in two cases, but it is not AI/LLM-assisted. Refereed (Adv. Math. 2022). | [{"id_or_ref": "arXiv:2006.16834", "title": "Proof of Tomaszewski's Conjecture on Randomly Signed Sums", "verified": true}, {"id_or_ref": "arXiv:1703.07251", "title": "Linear combinations of Rademacher random variables", "verified": true}] | 6 |
End of preview. Expand in Data Studio
ResearchMath-2-6Sol-Rewrite
97,497 research-level mathematics questions reformatted into benchmark-shaped items, derived
from amphora/ArXivOpenProblems,
plus an answer key of 154 literature-verified answers.
| config | rows | contents |
|---|---|---|
numerical |
35,274 | answer is a single number; labelled with math_intent |
yesno |
62,223 | answer is Yes or No |
answers |
270 (154 verified) | literature search results, with link, theorem, derivation and solution sketch |
⚠️ Most items have no answer key
Every item derives from a future-work or open-problem statement in an arXiv paper, so for the
vast majority the answer is not known. The answers config covers only the small subset where
a later paper resolved the question — see below. Do not compute accuracy on numerical/yesno
without an answer source.
How a research question becomes a number
route records the construction used for numerical items:
| route | construction | share |
|---|---|---|
| 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 | sharp constant in an inequality | 2.5% |
| D | critical value of a continuous parameter | 2.4% |
| G | count of a finite classification | 1.4% |
| F | minimal/maximal multiplicity or order | 0.3% |
distinguished_value is the number equal to the original question's "yes" (usually -1).
Route B strengthens the problem: "what is the smallest counterexample?" is strictly harder than
"is there one?".
math_intent (numerical config)
What each question is about — orthogonal to question_category (its logical shape). Assigned by
openai/gpt-6-sol from the source question; every one of the 35,274 rows is labelled.
| math_intent | n | share |
|---|---|---|
| structural_characterization | 7,423 | 21.0% |
| existence_construction | 6,802 | 19.3% |
| generalization | 6,749 | 19.1% |
| bound_improvement | 6,374 | 18.1% |
| hypothesis_weakening | 2,545 | 7.2% |
| classification | 1,590 | 4.5% |
| asymptotic_rate | 1,535 | 4.4% |
| uniqueness | 971 | 2.8% |
| algorithmic | 677 | 1.9% |
| other | 608 | 1.7% |
math_intent_secondary gives a second label where one clearly applies.
The answers config
270 questions were searched against the literature; 154 have a verified answer
(is_verified_answer = true). Each verified row has:
| field | meaning |
|---|---|
answer |
the answer, in the item's own convention |
answer_link |
the paper containing the answer |
resolving_paper / resolving_theorem |
authors, title; exact theorem number |
theorem_statement |
the theorem, quoted verbatim |
derivation |
how that theorem answers the question as literally phrased; steps not stated in the paper are marked INFERRED |
solution_sketch |
1–3 paragraphs on how the paper proves it, from the paper's own outline |
confidence, provenance, notes |
caveats |
Unanswered rows keep their status (STILL_OPEN, IMPROVED_BOUNDS, SOLVED_DIFFERENT, …) and,
where known, current_bounds.
Answer mix: −1 (83), Yes (34), No (17), exact values (20) — including 7/32, (3-sqrt(5))/2,
κ = 2, finite minimal counterexamples such as 6, 3 and 2, and two +infinity.
Filter by provenance
124 answers are "clean": high confidence with none of these tags.
provenance tag |
n | meaning |
|---|---|---|
RECENT_PREPRINT_2026 |
19 | resolving paper not yet refereed |
AI_ASSISTED_RESOLVER |
9 | resolving paper declares AI/LLM involvement |
COMPUTATIONAL |
4 | rests on a computer(-assisted) check |
NON_FINITE_ANSWER |
2 | answer is +infinity; handle specially when grading |
HAS_INFERRED_STEPS |
79 | derivation contains an inferred step — usually the routine "proved for all n, hence −1" |
How it was built
- Citation graph. Semantic Scholar citations for all 12,019 pre-2020 source papers.
- Candidate filter. Citing papers whose title/abstract announces a proof, disproof or resolution.
- Abstract screen.
openai/gpt-6-soljudged each (question, citing-paper) pair asDIRECT,SPECIAL_CASE,SIBLING,BOUNDS_ONLYorUNRELATED— most candidates resolve a different conjecture from the same source paper. - Full-text verification. Claude Code subagents read each
DIRECTcandidate in full, checked the main theorem against the exact question, traced priority to the first proof, and wrote the entry. - Citation check. Every resolver arXiv ID was matched against arXiv author metadata.
Verification changed many screen verdicts: disproofs mistaken for proofs, special cases, retracted or corrected proofs replaced by valid later ones, and "proofs" from predatory venues rejected.
Mis-transcriptions in the source data
19 of 270 searched questions (7%) are flagged MISTRANSCRIPTION: the source paper misstated its
conjecture — a dropped hypothesis, a typo, a reversed sign, a wrong constant, or a degenerate edge
case (e.g. n = 1) — so the literal question is trivially answerable while the real problem is
different. These are excluded from the verified answers. They cluster by source paper; a
paper with one confirmed mis-transcription should be treated as suspect.
13 further questions are flagged AMBIGUOUS_QUESTION (vacuous, about a proof method, or mixing an
empirical and an asymptotic claim) and are also excluded.
Fields (numerical / yesno)
uid, arxiv_id, paper_url, primary_category, signal_type, question_category, format,
route, question, answer_convention, distinguished_value, source_question, engine, and for
numerical also math_intent, math_intent_secondary.
Caveats
- No answer key for most items — see the warning above.
yesnoitems are unverified claims, not labelled true/false.- Route B items are harder than their sources.
- A small number of
arxiv_idvalues are wrong, inherited from the source dataset. signal_typeis an extraction label, not a verified claim that a problem is still open.
Licensing
Questions, labels and answer-key fields: CC-BY-4.0. source_question is inherited from
amphora/ArXivOpenProblems; theorem_statement quotes short excerpts from the resolving papers,
with attribution via resolving_paper and answer_link.
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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