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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 |
1704.00029#1 | 1704.00029 | https://arxiv.org/abs/1704.00029 | yesno | For Wright's equation y'(t) = -alpha y(t-1)[1+y(t)] with alpha > pi/2, call y a slowly oscillating periodic solution (SOPS) if, after a time translation, there are t_-, t_+ > 1 and a minimal period L = t_- + t_+ such that y(t) > 0 for t in (0,t_+), y(t) < 0 for t in (-t_-,0), and y(t+L) = y(t) for every t. The continuo... | Yes means every SOPS for every alpha > pi/2 lies on the Hopf-bifurcation branch. | null | SOLVED_EXACT | true | Yes | high | https://arxiv.org/abs/1801.09806 | J. Jaquette, A proof of Jones' conjecture, J. Differential Equations 266 (2019) 3818-3859 (arXiv:1801.09806) | Theorem 1.1 (plus the abstract's corollary: 'there are no isolas of periodic solutions to Wright's equation; all periodic orbits arise from Hopf bifurcations') | Theorem 1.1 (Jones' conjecture). For every α > π/2 there exists a unique slowly oscillating periodic solution to (2). | By [1704.00029] + prior work, the principal (Hopf) branch of SOPS is globally parametrized by α>π/2, so for every α>π/2 it contains a SOPS y_α. Theorem 1.1: for each α>π/2 the SOPS is unique up to time translation. So any SOPS y at parameter α is a translate of y_α and lies on the Hopf branch; hence no SOPS component i... | Jaquette writes the periodic orbits of Wright's equation as zeros of a map F on Fourier coefficients, with α treated as a parameter (all estimates hold uniformly in α) and a phase condition fixing the time translation. He first bounds a set containing the Fourier coefficients of every SOPS (Section 5, turning pointwise... | null | null | [
"COMPUTATIONAL"
] | Refereed (JDE 2019). The proof is computer-assisted (interval arithmetic, INTLAB). The question is Jones' geometric conjecture part (ii), no isolas of SOPS; Theorem 1.1 implies it immediately, and the abstract says so. No AI involvement declared. | [{"id_or_ref": "arXiv:1801.09806 / doi:10.1016/j.jde.2018.09.017", "title": "A proof of Jones' conjecture (J. Jaquette, J. Differential Equations 266 (2019))", "verified": true}, {"id_or_ref": "arXiv:1704.00029", "title": "A proof of Wright's conjecture (J. B. van den Berg, J. Jaquette) -- no folds near Hopf point; pri... | 6 |
1704.00029#4 | 1704.00029 | https://arxiv.org/abs/1704.00029 | yesno | For Wright's delay equation y'(t)=-αy(t-1)[1+y(t)] with α∈(π/2,1.94), does every slowly oscillating periodic solution belong to the solution branch emanating from the Hopf bifurcation at α=π/2? Here a slowly oscillating periodic solution is a periodic solution y for which, after a time translation, there are t_-,t_+>1 ... | Yes means every slowly oscillating periodic solution in the stated parameter range belongs to the Hopf-bifurcation branch. | null | SOLVED_EXACT | true | Yes | high | https://arxiv.org/abs/1801.09806 | J. Jaquette, A proof of Jones' conjecture, J. Differential Equations 266 (2019) 3818-3859 (arXiv:1801.09806) | Theorem 1.1; the computer-assisted part proves uniqueness of the SOPS for α∈(π/2,1.9] (proof at end of Section 6) | Theorem 1.1 (Jones' conjecture). For every α > π/2 there exists a unique slowly oscillating periodic solution to (2). | (π/2,1.94) ⊂ (π/2,1.9] ∪ [1.9,6.0]. Jaquette proves the SOPS is unique up to translation for α∈(π/2,1.9], and JLM 2017 proves it for α∈[1.9,6.0] (the paper combines both in its proof of Theorem 1.1). The Hopf branch has a SOPS at every α>π/2, so on this range every SOPS is a translate of the branch solution and there i... | Jaquette writes the periodic orbits of Wright's equation as zeros of a map F on Fourier coefficients, with α treated as a parameter (all estimates hold uniformly in α) and a phase condition fixing the time translation. He first bounds a set containing the Fourier coefficients of every SOPS (Section 5, turning pointwise... | null | null | [
"COMPUTATIONAL"
] | The screen's reason ('uniqueness stops at 1.9') is misleading. Jaquette's own computation covers (π/2,1.9], and [1.9,1.94) is covered by Jaquette-Lessard-Mischaikow's α∈[1.9,6.0], so the whole gap is closed. Refereed; computer-assisted proof. | [{"id_or_ref": "arXiv:1801.09806 / doi:10.1016/j.jde.2018.09.017", "title": "A proof of Jones' conjecture (J. Jaquette, J. Differential Equations 266 (2019))", "verified": true}, {"id_or_ref": "arXiv:1704.00029", "title": "A proof of Wright's conjecture (J. B. van den Berg, J. Jaquette) -- no folds near Hopf point; pri... | 6 |
1704.00350#1 | 1704.00350 | https://arxiv.org/abs/1704.00350 | numerical | Let n >= 1, let v_1,...,v_n be real numbers with sum_{i=1}^n v_i^2 <= 1, let a_1,...,a_n be independent uniform random signs, and set S = sum_{i=1}^n a_i v_i. What is the best universal constant c such that Pr(|S| <= 1) >= c for every n and every such choice of v_1,...,v_n? The proposed improvement seeks a value greate... | The answer is the infimum of Pr(|S| <= 1) over all permitted n and v_1,...,v_n. | null | SOLVED_EXACT | true | 1/2 | high | https://arxiv.org/abs/2006.16834 | N. Keller, O. Klein, Proof of Tomaszewski's conjecture on randomly signed sums, Advances in Mathematics 407 (2022) 108558 (arXiv:2006.16834) | Theorem 1.2 | Theorem 1.2. Let X = sum_{i=1}^n a_i x_i, where sum_{i=1}^n a_i^2 = 1 and {x_i} are independent and uniformly distributed in {-1,1}. Then Pr[|X| <= 1] >= 1/2. | (INFERRED scaling step) Let s = sum v_i^2 ≤ 1. If s=0 then S≡0 and Pr=1. Otherwise X = S/√s has unit variance, and {|S|≤1} = {|X|≤1/√s} ⊇ {|X|≤1}, so Pr(|S|≤1) ≥ 1/2 by Theorem 1.2. Tightness (noted in the paper): v_1=v_2=1/√2 gives S∈{-√2,0,0,√2}, so Pr(|S|≤1)=1/2. The infimum is attained, and the best constant is c=1... | Keller and Klein split the proof by the size of the largest coefficient and the structure of the coefficients. They use (i) new local concentration inequalities (segment comparison, Theorem 1.3, proved by an explicit injection) comparing Pr[X∈I] with Pr[X∈J] for intervals; (ii) a sharper Berry-Esseen bound for Rademach... | null | null | [
"HAS_INFERRED_STEPS"
] | The question allows sum v_i^2 ≤ 1 rather than = 1; the reduction is simple scaling (INFERRED, not stated in the paper). The v3 comment on the source reports 0.406259. Refereed (Adv. Math. 2022); no AI involvement. | [{"id_or_ref": "arXiv:2006.16834 / doi:10.1016/j.aim.2022.108558", "title": "Proof of Tomaszewski's Conjecture on Randomly Signed Sums", "verified": true}, {"id_or_ref": "arXiv:1704.00350", "title": "Tomaszewski's Problem on Randomly Signed Sums: Breaking the 3/8 Barrier (source)", "verified": true}] | 6 |
1704.00851#1 | 1704.00851 | https://arxiv.org/abs/1704.00851 | numerical | For integers n and k with 0 ≤ k < (1/2)·binom(n,2), let W_n be the weak order on S_n, let (W_n)_j be its permutations with j inversions, and set ν_w = 𝔖_w(1,1,…,1). Define the square matrix D̃(n,k), with rows indexed by u ∈ (W_n)_k and columns indexed by v ∈ (W_n)_{binom(n,2)−k}, by D̃(n,k)_{uv} = ν_{vu⁻¹} when u ≤ v ... | The answer is the smallest n at which the stated determinant identity fails for an admissible k, or −1 if it holds for all admissible n and k. | -1 | SOLVED_EXACT | true | -1 | high | https://arxiv.org/abs/1812.00321 | Z. Hamaker, O. Pechenik, D. E. Speyer, A. Weigandt, 'Derivatives of Schubert polynomials and proof of a determinant conjecture of Stanley', arXiv:1812.00321; Algebraic Combinatorics 3(2) (2020) 301-307, doi:10.5802/alco.93 | Theorem 2 | Theorem 2 (Conjectured by Stanley [Sta17, Conjecture 2.2]). For ℓ ≤ binom(n,2) − ℓ, det M̃(ℓ) = ± ∏_{k=0}^{ℓ} ((ℓ − k + 1)(ℓ − k + 2) · · · (binom(n,2) − ℓ − k))^{|S_n(k)| − |S_n(k−1)|}, where M̃(ℓ) = M_{ℓ+1} M_{ℓ+2} · · · M_{binom(n,2)−ℓ} and M_ℓ[u,v] = k if v = u s_k, 0 if u^{-1}v ∉ S. | M̃(ℓ) is the matrix of U^{binom(n,2)−2ℓ} from rank ℓ to rank binom(n,2)−ℓ, i.e. Stanley's D(n,ℓ); by Macdonald's identity (Stanley eq. (2.1)) D(n,k) = (binom(n,2)−2k)! · D̃(n,k), so det D̃(n,k) = det M̃(k)/((binom(n,2)−2k)!)^{#(W_n)_k}. With N=binom(n,2), a_i=#(W_n)_i: ∏_{j=0}^{k}[(N−k−j)!/(k−j)!]^{a_j−a_{j−1}} / ((N−2... | Key identity (Proposition 1): with ∇ = Σ ∂/∂x_i, ∇(S_w) = Σ_{ℓ(ws_k)<ℓ(w)} k S_{ws_k}; so Stanley's M-matrices are the matrices of ∇ in the Schubert basis of the span W of monomials x^a with 0 ≤ a_j ≤ n−j (∇ commutes with divided differences, check at w_0). Schubert and monomial bases of each graded piece differ by a u... | null | null | [
"HAS_INFERRED_STEPS"
] | HPSW state the result for M̃(ℓ)=D(n,ℓ) rather than D̃; equivalence with Stanley's normalized form is the factorial rescaling stated in the source and verified numerically. Invertibility (strong Sperner) was earlier proven by Gaetz-Gao 2018. | [{"id_or_ref": "arXiv:1812.00321", "title": "Derivatives of Schubert polynomials and proof of a determinant conjecture of Stanley", "verified": true}, {"id_or_ref": "arXiv:1704.00851", "title": "Some Schubert shenanigans (Stanley) - source, Conjecture 2.2", "verified": true}] | 5 |
1704.01172#6 | 1704.01172 | https://arxiv.org/abs/1704.01172 | numerical | What is the smallest nonnegative integer m for which there exists an undirected simple loopless planar graph G with |E(G)|=m and no connected component isomorphic to K_2 such that every bijection w:E(G)→{1,...,m} gives some edge uv with σ_w(u)=σ_w(v), where σ_w(u)=∑_{x∈N(u)}w(ux)? Return -1 if no such m exists. For con... | Give the smallest counterexample edge count, or -1 if every graph in the stated family admits the required bijection. | -1 | SOLVED_EXACT | true | -1 | high | https://arxiv.org/abs/1705.09957 | J. Haslegrave, Proof of a local antimagic conjecture, Discrete Mathematics & Theoretical Computer Science 20:1 (2018) #18 (arXiv:1705.09957) | Theorem 12 | Theorem 12. For every graph G with m edges, none of which is isolated, labelling the edges of G according to a random permutation of [m] produces a local antimagic labelling with positive probability. Further, this probability is Ω(1/m). | 'None isolated' means no K_2 component (the introduction states he proves Bensmail et al.'s stronger form: every graph without isolated edges, disconnected graphs included). A local antimagic labelling is exactly a bijection w:E→{1..m} with σ_w(u)≠σ_w(v) for every edge uv. So every graph with no K_2 component, planar o... | Label the edges by a uniformly random permutation of [m]. For an edge xy, S_x - S_y has the same distribution as D_{a,b} = (sum of a random labels) - (sum of b other random labels), with a=d(x)-1 and b=d(y)-1; the label of xy cancels. Haslegrave shows P(D_{a,b}=0) ≤ 1/m in three regimes: degrees differing by 1 or 2 (Th... | null | null | [] | The screen said the result is for connected graphs and applied it per component. In fact Theorem 12 covers disconnected graphs directly: a single bijection on all m edges. Per-component application would be wrong, since each component would receive a non-initial set of labels. Refereed (DMTCS 2018); no AI involvement. | [{"id_or_ref": "arXiv:1705.09957 / doi:10.23638/DMTCS-20-1-18", "title": "Proof of a local antimagic conjecture", "verified": true}, {"id_or_ref": "arXiv:1704.01172", "title": "On a combination of the 1-2-3 Conjecture and the Antimagic Labelling Conjecture (source)", "verified": true}] | 6 |
1704.06536#2 | 1704.06536 | https://arxiv.org/abs/1704.06536 | numerical | What is the smallest integer t >= 3 for which no integer bound B exists such that every graph with no K_{3,t} minor has a vertex 4-colouring in which every monochromatic component has at most B vertices? Answer -1 if there is no such t. | The answer is the smallest such t, or -1 if a suitable integer-valued bound f(t) exists for every integer t >= 3. | -1 | SOLVED_EXACT | true | -1 | high | https://arxiv.org/abs/2306.06224 | V. Dujmović, L. Esperet, P. Morin, D. R. Wood, 'Proof of the Clustered Hadwiger Conjecture', arXiv:2306.06224; FOCS 2023, pp. 1921-1930, doi:10.1109/FOCS57990.2023.00116 | Theorem 2 (corollary of Theorem 4) | Theorem 2. There is a function f such that every K_{s,t}-minor-free graph is (s + 1)-colourable with clustering at most f(s, t). | Apply Theorem 2 with s=3 and any t≥3 (the paper's convention t ≥ s): every K_{3,t}-minor-free graph is 4-colourable with every monochromatic component of at most f(3,t) vertices. So a bound B=f(3,t) exists for every t≥3 and no failing t exists: -1. The paper notes this with the lower bound of van den Heuvel-Wood gives ... | The paper proves a single common generalization (Theorem 4): for J_{s,t} = {K_s ⊕ T : T a t-vertex tree}, every J_{s,t}-minor-free graph is (s+1)-colourable with clustering f(s,t); since K_{s,t} ⊆ every graph in J_{s,t}, Theorem 2 follows (and K_h via J_{h−2,2}={K_h}). The proof builds on Liu-Wood's result that K_{s,t}... | null | null | [] | Direct and clean match; 4 colours = s+1 with s=3. | [{"id_or_ref": "arXiv:2306.06224", "title": "Proof of the Clustered Hadwiger Conjecture", "verified": true}, {"id_or_ref": "arXiv:1704.06536", "title": "Improper Colourings inspired by Hadwiger's Conjecture (van den Heuvel, Wood) - source", "verified": true}] | 5 |
1704.06686#1 | 1704.06686 | https://arxiv.org/abs/1704.06686 | yesno | For every integer r >= 2, do the germs of sufficiently small saturated neighborhoods of the unique singular fiber in a four-dimensional almost-toric integrable system with connected compact fibers, where that fiber contains exactly r focus-focus points (multiplicity r), admit a complete and realizable semiglobal invari... | Yes means such an invariant and symmetry action exist for every integer r >= 2 and have the stated realizability and isomorphism properties. | null | SOLVED_EXACT | true | Yes | high | https://arxiv.org/abs/1803.00998 | Á. Pelayo, X. Tang, 'Vũ Ngọc's conjecture on focus-focus singular fibers with multiple pinched points', arXiv:1803.00998; J. Fixed Point Theory Appl. 26 (2024), art. 6, doi:10.1007/s11784-023-01089-1 | Theorem 3.9 (Main Theorem) and Corollary 3.13 | Theorem 3.9 (Main Theorem). There is a bijection Φ : M^k_{ff,→} → I^k_{ff,→} = {((s_j)_{j∈Z_k}, (g_{j,ℓ})_{j,ℓ∈Z_k}) ∈ R_{2πX}^k × R_+^{k²} | (3.3)}, [(M, ω, F, (α1, α2), m0)] ↦ (s_0, …, s_{−1}, g_{0,0}, …, g_{−1,−1}). [Equivalently, via the one-to-one projection to (s_0, g_{0,1}, …, g_{k−2,k−1}) ∈ R_{2πX} × R_+^{k−1}:... | For r = k ≥ 2 focus-focus points on the singular fiber, Corollary 3.13 gives a complete (injective) and realizable (surjective) invariant — k formal power series (s_0; g_{0,1},…,g_{k−2,k−1}) — classifying germs of saturated neighborhoods up to fiberwise symplectomorphism, modulo an effective Z_2 × D_k action. Remark 3.... | Choose Eliasson local charts at each of the k focus-focus points m_j compatible with a chosen direction (α1, α2) and a marked point m_0. The desingularized action integral, Taylor-expanded in each chart, gives action Taylor series s_j; the transition maps between Eliasson charts at m_j and m_ℓ give 'transition Taylor s... | null | null | [] | Resolver verified (Pelayo & Tang; arXiv v3 Oct 2022; Crossref shows J. Fixed Point Theory Appl. vol 26, art. 6, published Jan 2024, not 2023 as in input). Proves Vũ Ngọc's 2002/2003 conjecture. The symmetry group is Z_2 × D_k (order 4k), which is the natural analogue of K4 = Z_2 × D_1. Related smooth classification: Bo... | [{"id_or_ref": "arXiv:1803.00998 / doi:10.1007/s11784-023-01089-1", "title": "Vũ Ngọc's conjecture on focus-focus singular fibers with multiple pinched points", "verified": true}] | 6 |
1704.07041#8 | 1704.07041 | https://arxiv.org/abs/1704.07041 | numerical | Let X range over aspherical Kähler surfaces with Euler characteristic e(X)>0, let G=pi_1(X), let S(G)=(H^1(G;R)\{0})/R_{>0}, and let Sigma^2(G) be the homotopical second Bieri–Neumann–Strebel–Renz invariant. What is the smallest e(X) for which Sigma^2(G) is nonempty? Return -1 if no such X exists. When X has Albanese d... | Return the smallest positive integer e(X) for a counterexample, or -1 if Sigma^2(G) is empty for every X in the stated family. | -1 | SOLVED_EXACT | true | -1 | high | https://arxiv.org/abs/2401.05545 | S. Hughes, D. Kielak, 'BNSR invariants and ℓ²-homology', arXiv:2401.05545 (2024) | Corollary C (via Theorem A/B) | Corollary C. Let M be a closed connected 2n-manifold or (more generally) a finite PD2n(Q)-complex. If χ(M) ≠ 0, then Σ^n(M) = Σ^n(M; Q) = ∅. In particular, if M is additionally aspherical, then Σ^n(π1 M) = Σ^n(π1 M; Q) = ∅. | Take M = X, a closed aspherical Kähler surface (real dimension 4, so n = 2) with χ(X) = e(X) > 0 ≠ 0. Corollary C gives Σ^2(π1 X) = ∅, where Σ^n(G) is defined in the paper (Def. 2.15) as the HOMOTOPICAL BNSR invariant Σ^n(BG) — exactly the invariant in the question. Hence no X in the family has nonempty Σ^2, so no smal... | Theorem A: if G is of type FP_n(Q) and the n-th ℓ²-Betti number b_n^(2)(G) ≠ 0, then Σ^n(G;Q) = ∅; Theorem B is the analogue for CW complexes. The proof (Prop. 5.1) shows that if a free QG-chain complex has vanishing Novikov homology (Sikorav's characterization of Σ^n(·;Q)) up to degree n for some character φ, then its... | null | null | [
"HAS_INFERRED_STEPS"
] | Resolver verified: arXiv 2401.05545, Sam Hughes & Dawid Kielak, 'BNSR invariants and ℓ²-homology' (v1, 10 Jan 2024; no journal ref on arXiv or found on Crossref, so treat as preprint, not a 2026 one). Paper explicitly states (Question 1.2) the Friedl–Vidussi/Llosa Isenrich–Py question 'closed aspherical Kähler 2n-manif... | [{"id_or_ref": "arXiv:2401.05545", "title": "BNSR invariants and ℓ²-homology (Hughes, Kielak)", "verified": true}, {"id_or_ref": "arXiv:1704.07041", "title": "Virtual algebraic fibrations of Kähler groups (Friedl, Vidussi)", "verified": true}] | 6 |
1704.07041#9 | 1704.07041 | https://arxiv.org/abs/1704.07041 | numerical | Among all aspherical Kähler surfaces X with e(X)>0 for which Σ²(π₁(X)) is nonempty, what is the smallest value of e(X)? Answer -1 if there is no such surface. For context, when X has Albanese dimension two, Σ²(π₁(X)) is known to be a proper subset of Σ¹(π₁(X)). | The answer is the smallest Euler characteristic of a counterexample, or -1 if every surface in the stated family has empty Σ². | -1 | SOLVED_EXACT | true | -1 | high | https://arxiv.org/abs/2401.05545 | S. Hughes, D. Kielak, 'BNSR invariants and ℓ²-homology', arXiv:2401.05545 (2024) | Corollary C (via Theorem A/B) | Corollary C. Let M be a closed connected 2n-manifold or (more generally) a finite PD2n(Q)-complex. If χ(M) ≠ 0, then Σ^n(M) = Σ^n(M; Q) = ∅. In particular, if M is additionally aspherical, then Σ^n(π1 M) = Σ^n(π1 M; Q) = ∅. | Take M = X, a closed aspherical Kähler surface (real dimension 4, so n = 2) with χ(X) = e(X) > 0 ≠ 0. Corollary C gives Σ^2(π1 X) = ∅, where Σ^n(G) is defined in the paper (Def. 2.15) as the HOMOTOPICAL BNSR invariant Σ^n(BG) — exactly the invariant in the question. Hence no X in the family has nonempty Σ^2, so no smal... | Theorem A: if G is of type FP_n(Q) and the n-th ℓ²-Betti number b_n^(2)(G) ≠ 0, then Σ^n(G;Q) = ∅; Theorem B is the analogue for CW complexes. The proof (Prop. 5.1) shows that if a free QG-chain complex has vanishing Novikov homology (Sikorav's characterization of Σ^n(·;Q)) up to degree n for some character φ, then its... | null | null | [
"HAS_INFERRED_STEPS"
] | Resolver verified: arXiv 2401.05545, Sam Hughes & Dawid Kielak, 'BNSR invariants and ℓ²-homology' (v1, 10 Jan 2024; no journal ref on arXiv or found on Crossref, so treat as preprint, not a 2026 one). Paper explicitly states (Question 1.2) the Friedl–Vidussi/Llosa Isenrich–Py question 'closed aspherical Kähler 2n-manif... | [{"id_or_ref": "arXiv:2401.05545", "title": "BNSR invariants and ℓ²-homology (Hughes, Kielak)", "verified": true}, {"id_or_ref": "arXiv:1704.07041", "title": "Virtual algebraic fibrations of Kähler groups (Friedl, Vidussi)", "verified": true}] | 6 |
1704.08472#0 | 1704.08472 | https://arxiv.org/abs/1704.08472 | numerical | What is the smallest integer k >= 4 for which no constant c(k) satisfies f(n,k) <= c(k) sqrt(n) for every integer n >= 1? Answer -1 if there is no such k. Here f(n,k) = max{f_k(G) : |G| <= n}, where f_k(G) is the minimum number of vertices whose deletion leaves an induced subgraph H that either has fewer than k vertice... | The answer is -1 if the bound holds for every integer k >= 4, and otherwise is the smallest integer k >= 4 for which it fails. | -1 | SOLVED_EXACT | true | -1 | medium | https://arxiv.org/abs/2609.01681 | Zhen Liu, Qinghou Zeng, 'Large induced subgraphs with k vertices of maximum degree', arXiv:2609.01681 (Sep 2026) | Theorem 1.2 (together with the bound s_k(G) ≤ A_k√Δ(G) for every graph G established in its proof) | Theorem 1.2. For every integer k ≥ 2 there exists a constant A_k such that every graph G on at least R(k, k) vertices satisfies f_k(G) ≤ A_k √Δ(G). | The question's f_k(G) (deletions leaving fewer than k vertices or ≥k vertices of degree Δ(H)) is exactly the paper's s_k(G) ('successful set'). The proof of Theorem 1.2 (via Lemma 2.13, valid for every graph) shows s_k(G) ≤ max{r_0, 1+√(64c_2(k)R(k,k))}·√Δ(G) for every graph G with Δ≥1, and s_k(G)=0 when Δ=0; the hypot... | Suppose s_k(G) > r. Choose S of size |V|−⌊r/2⌋ minimizing e(G[S]), D the rest, sort D by degree into S and cut into blocks of size R(k,k). A Ramsey argument (Lemma 2.4 on hypergraphs, Lemma 2.5/Cor. 2.11) shows each degree value into S repeats boundedly often, else a small successful set exists. Lemma 2.12 (based on a ... | null | null | [
"RECENT_PREPRINT_2026",
"HAS_INFERRED_STEPS",
"AI_ASSISTED_RESOLVER"
] | Resolver is a very recent (1 Sep 2026) unrefereed preprint (authors declare use of ChatGPT for discussing/checking proofs); theorem and definitions match, proof read at sketch level only, hence medium confidence. It proves the stronger Caro-Lauri-Zarb √Δ form for all k≥2. | [{"id_or_ref": "arXiv:2609.01681", "title": "Large induced subgraphs with k vertices of maximum degree (Liu, Zeng)", "verified": true}, {"id_or_ref": "arXiv:1704.08472", "title": "Equating two maximum degrees (Caro, Lauri, Zarb) - source", "verified": true}] | 5 |
1704.08472#4 | 1704.08472 | https://arxiv.org/abs/1704.08472 | numerical | For every integer k>=2, let f_k(G) be the minimum number of vertices that must be deleted from a finite simple graph G so that the remaining induced subgraph H either has fewer than k vertices or has at least k vertices of degree Delta(H). For every integer Delta>=0, define g(Delta,k)=max{f_k(G): Delta(G)<=Delta}; in p... | Give the smallest failing k, or -1 if the bound holds for every fixed k>=2. | -1 | SOLVED_EXACT | true | -1 | medium | https://arxiv.org/abs/2609.01681 | Zhen Liu, Qinghou Zeng, 'Large induced subgraphs with k vertices of maximum degree', arXiv:2609.01681 (2026) | Theorem 1.2 | Theorem 1.2. For every integer k >= 2 there exists a constant A_k such that every graph G on at least R(k,k) vertices satisfies f_k(G) <= A_k sqrt(Delta(G)). | Same as 1704.08472#7: the proof gives s_k(G)=f_k(G) <= A_k sqrt(Delta(G)) for all graphs G, so g(Delta,k) <= A_k sqrt(Delta) for every Delta>=0 (Delta=0: g(0,k)=0). Proof of Theorem 1.2 (paper): 'If Delta(G)=0, then the empty set is successful, so s_k(G)=0 ... Therefore in all cases, s_k(G) <= max{r0, 1+sqrt(64 c2(k) R... | Fix t=floor(r/2) and choose the (n-t)-subset S minimizing e(G[S]); D=V\S. The exchange inequality d_S(v)-d_S(u)-1_{uv}>=0 (v in D, u in S) controls degrees. Lemma 2.3 (via a Caro-Shapira-Yuster zero-sum lemma on {-1,0,1}^{k-1} vectors) shows k vertices inducing a regular graph whose degrees differ by <= b can be equali... | g(Delta,k)=Theta_k(sqrt(Delta)); exact g(3,k) for general k still undetermined (only g(3,2)=ceil((-3+5)/2)=1 from the k=2 formula, and g(3,k)<=3(k-1)). | null | [
"RECENT_PREPRINT_2026",
"AI_ASSISTED_RESOLVER"
] | The reformatted question asks only the sqrt(Delta) part, which is settled. The source question's other part (exact value of g(3,k) for all k) is NOT addressed by the resolver and remains open. | [{"id_or_ref": "arXiv:2609.01681", "title": "Large induced subgraphs with $k$ vertices of maximum degree", "verified": true}, {"id_or_ref": "arXiv:1704.08472", "title": "Equating two maximum degrees (Caro, Lauri, Zarb)", "verified": true}] | 5 |
1704.08472#5 | 1704.08472 | https://arxiv.org/abs/1704.08472 | numerical | For every integer k >= 4, let f_k(G) be the minimum number of vertices that must be deleted from a finite simple graph G so that the remaining induced subgraph H either has fewer than k vertices or has at least k vertices whose degrees equal its maximum degree Delta(H), and let f(n,k)=max{f_k(G): |G| <= n}. What is the... | The answer is the smallest such k, or -1 if every k >= 4 satisfies at least the square-root bound or the sublinear bound. | -1 | SOLVED_EXACT | true | -1 | medium | https://arxiv.org/abs/2609.01681 | Zhen Liu, Qinghou Zeng, 'Large induced subgraphs with k vertices of maximum degree', arXiv:2609.01681 (2026) | Theorem 1.2 | Theorem 1.2. For every integer k >= 2 there exists a constant A_k such that every graph G on at least R(k,k) vertices satisfies f_k(G) <= A_k sqrt(Delta(G)). | Every graph G with |G|<=n has Delta(G)<=n-1<n. By the proof of Theorem 1.2, f_k(G)=s_k(G) <= A_k sqrt(Delta(G)) <= A_k sqrt(n) for every G (Delta=0 case gives 0). Proof of Theorem 1.2 (paper): 'If Delta(G)=0, then the empty set is successful, so s_k(G)=0 ... Therefore in all cases, s_k(G) <= max{r0, 1+sqrt(64 c2(k) R(k... | Fix t=floor(r/2) and choose the (n-t)-subset S minimizing e(G[S]); D=V\S. The exchange inequality d_S(v)-d_S(u)-1_{uv}>=0 (v in D, u in S) controls degrees. Lemma 2.3 (via a Caro-Shapira-Yuster zero-sum lemma on {-1,0,1}^{k-1} vectors) shows k vertices inducing a regular graph whose degrees differ by <= b can be equali... | (1-o(1))sqrt(n) <= f(n,2) (Caro-Yuster); f(n,k)=Theta_k(sqrt(n)). | null | [
"RECENT_PREPRINT_2026",
"HAS_INFERRED_STEPS",
"AI_ASSISTED_RESOLVER"
] | Abstract of resolver: 'This confirms a conjecture of Caro and Yuster in strong form.' Unrefereed 2026 preprint. | [{"id_or_ref": "arXiv:2609.01681", "title": "Large induced subgraphs with $k$ vertices of maximum degree", "verified": true}, {"id_or_ref": "arXiv:1704.08472", "title": "Equating two maximum degrees (Caro, Lauri, Zarb)", "verified": true}] | 5 |
1704.08472#7 | 1704.08472 | https://arxiv.org/abs/1704.08472 | numerical | Let f_k(G) be the minimum number of vertices that must be deleted from a graph G to obtain an induced subgraph H that either has fewer than k vertices or has at least k vertices of degree Delta(H). Let g(Delta,k) = max{f_k(G) : Delta(G) <= Delta}. What is the smallest integer k >= 2 for which there is no constant g(k) ... | The answer is the smallest such k, or -1 if no such k exists. | -1 | SOLVED_EXACT | true | -1 | medium | https://arxiv.org/abs/2609.01681 | Zhen Liu, Qinghou Zeng, 'Large induced subgraphs with k vertices of maximum degree', arXiv:2609.01681 (2026) | Theorem 1.2 (and its proof, which bounds s_k(G) for every graph G) | Theorem 1.2. For every integer k >= 2 there exists a constant A_k such that every graph G on at least R(k,k) vertices satisfies f_k(G) <= A_k sqrt(Delta(G)). | Source Conjecture 5.1 asks for g(k) with g(Delta,k)=max{f_k(G):Delta(G)<=Delta} <= g(k)sqrt(Delta) for all Delta>=0. The paper's proof bounds s_k(G) <= A_k sqrt(Delta(G)) for every graph G (the R(k,k) hypothesis is used only to identify s_k with the Ramsey-restricted f_k), and s_k equals the source's f_k. Proof of Theo... | Fix t=floor(r/2) and choose the (n-t)-subset S minimizing e(G[S]); D=V\S. The exchange inequality d_S(v)-d_S(u)-1_{uv}>=0 (v in D, u in S) controls degrees. Lemma 2.3 (via a Caro-Shapira-Yuster zero-sum lemma on {-1,0,1}^{k-1} vectors) shows k vertices inducing a regular graph whose degrees differ by <= b can be equali... | Omega_k(sqrt(Delta)) <= g(Delta,k) <= O_k(sqrt(Delta)); exact for k=2: ceil((-3+sqrt(8Delta+1))/2). | null | [
"RECENT_PREPRINT_2026",
"HAS_INFERRED_STEPS",
"AI_ASSISTED_RESOLVER"
] | Resolver (Sep 2026 preprint, v1, unrefereed; authors disclose ChatGPT assistance) explicitly states it fully confirms Caro-Lauri-Zarb Conjecture 1.1 (= source Conjecture 5.1). Proof read end-to-end at the level of lemma statements and the final inversion; looks coherent. Constant A_k is huge (Ramsey/hypergraph-Ramsey). | [{"id_or_ref": "arXiv:2609.01681", "title": "Large induced subgraphs with $k$ vertices of maximum degree", "verified": true}, {"id_or_ref": "arXiv:1704.08472", "title": "Equating two maximum degrees (Caro, Lauri, Zarb)", "verified": true}] | 5 |
1705.00164#0 | 1705.00164 | https://arxiv.org/abs/1705.00164 | numerical | For each integer k >= 1, define the following formal power series in t with polynomial coefficients in x. For n >= 1, let S_n(132) be the permutations σ=σ_1⋯σ_n avoiding the classical pattern 132. For a,b,c,d in ℕ ∪ {∅}, let mmp^(a,b,c,d)(σ) count the entries σ_i for which, relative to (i,σ_i) in the permutation graph,... | Give the smallest such k, or -1 if the two series are equal for every integer k >= 1. | -1 | SOLVED_EXACT | true | -1 | medium | https://arxiv.org/abs/2608.18461 | S. Cao, S. Kitaev, Y. Wu, 'Wilf Equivalence for Length-Three Patterns and Flat POPs, and a Conjecture of Qiu and Remmel', arXiv:2608.18461 (Aug 2026) | Theorem 4.1 | Theorem 4.1. For all k ≥ 1, we have Q_132^{(0,k,∅,0)}(t, q) = Q_132^{(1,k−1,∅,0)}(t, q). | Theorem 4.1 is exactly Qiu–Remmel Conjecture 1 (1705.00164, eq. (57)) with q in place of x. Since the two series are equal for every k ≥ 1, there is no k at which they differ; by the answer convention the answer is -1. | Let A_k = Q_132^{(0,k,∅,0)} and B_k = Q_132^{(1,k−1,∅,0)}. Qiu–Remmel's Theorem 13 gives A_k = (1/(1−tA_0))(1 + t Σ_{j=0}^{k−2} C_j t^j (A_{k−1−j} − A_0)). Forming the generating function P(z) = Σ_{k≥1} A_k z^{k−1} and solving with the Catalan generating function C(z), the authors show P(z) = (1 − A_0 z t C(zt))/((1 − ... | null | null | [
"RECENT_PREPRINT_2026"
] | Resolver is an UNREFEREED 2026 preprint (submitted 19 Aug 2026); no AI involvement declared. The proof depends on the authors' correction of a typo (missing factor t) in Qiu–Remmel Thm 14, so confidence is medium. Source title is about 123-avoiding perms but Conjecture 1 (eq. 57) is indeed about Q_132, as stated in the... | [{"id_or_ref": "arXiv:2608.18461", "title": "Wilf Equivalence for Length-Three Patterns and Flat POPs, and a Conjecture of Qiu and Remmel (Cao, Kitaev, Wu)", "verified": true}, {"id_or_ref": "arXiv:1705.00164", "title": "Quadrant marked mesh patterns in 123-avoiding permutations (Qiu, Remmel), Conjecture 1", "verified"... | 6 |
1705.00575#3 | 1705.00575 | https://arxiv.org/abs/1705.00575 | numerical | Let p,q≥1, let G be the complete bipartite graph with vertex partition A⊔B=[p+q], where |A|=p and |B|=q, and let S=K[x_1,…,x_{p+q},y_1,…,y_{p+q}] have the Z^{p+q}-grading deg(x_r)=deg(y_r)=e_r. Set J_G=(x_i y_j−x_j y_i : {i,j} is an edge of G), J=gin(J_G) for the Z^{p+q}-graded generic initial ideal, and m=(x_1,…,x_{p+... | The answer is the smallest counterexample size p+q, or −1 if the equality holds for all stated parameters. | -1 | SOLVED_EXACT | true | -1 | high | https://arxiv.org/abs/1901.08645 | J. Alvarez Montaner, Local cohomology of binomial edge ideals and their generic initial ideals (arXiv:1901.08645); the general case is Conca-Varbaro, Square-free Groebner degenerations (arXiv:1805.11923, Invent. Math. 2020), Theorem 1.3 with Remark 2.5 | Theorem 4.5 (Alvarez Montaner, arXiv:1901.08645) | Theorem 4.5. Let A = K[x1,...,xn,y1,...,yn] be a polynomial ring with coefficients over a field K and m be its homogeneous maximal ideal. Let J_G ⊆ A be the binomial edge ideal associated to a graph G in the set of vertices [n]. Then we have dim_K H^r_m(A/J_G)_a = dim_K H^r_m(A/gin(J_G))_a for every r ∈ N and every a ∈... | Theorem 4.5 covers every graph G on [n], with the Z^n-graded gin of Conca-De Negri-Gorla [11, Theorem 2.1]. Take n = p+q and G = K_{p,q} (INFERRED specialization, immediate). Then the equality holds for all p,q >= 1, all i and all a in Z^{p+q}. So no counterexample exists and, by the answer convention, the answer is -1... | Alvarez Montaner uses the local cohomology spectral sequences of Alvarez Montaner-Boix-Zarzuela, attached to the poset Q_{J_G} of sums of the minimal primes P_S(G) of J_G. For binomial edge ideals every ideal in this poset is again of the form P_S(H) = (x_i,y_i : i in S) + J_{H~1} + ... + J_{H~c}, with each H~i complet... | null | null | [
"HAS_INFERRED_STEPS"
] | The question is CDG Conjecture 1.14 specialized to complete bipartite graphs. The candidate is correct and states the result for all graphs. The first proof is Conca-Varbaro (arXiv May 2018, Invent. Math. 221 (2020)). Their Theorem 1.3 with Remark 2.5 covers multigraded gradings, and they say CDG Conjectures 1.13 and 1... | [{"id_or_ref": "arXiv:1901.08645", "title": "Local cohomology of binomial edge ideals and their generic ideals (J. Alvarez Montaner)", "verified": true}, {"id_or_ref": "arXiv:1805.11923", "title": "Square-free Groebner degenerations (A. Conca, M. Varbaro)", "verified": true}, {"id_or_ref": "arXiv:1705.00575", "title": ... | 6 |
1705.02954#8 | 1705.02954 | https://arxiv.org/abs/1705.02954 | yesno | For every abelian group A and every subgroup H ≤ A, is H uniformly fully inert—that is, does there exist n ∈ N such that |(H + φ(H))/H| ≤ n for every endomorphism φ of A—if and only if H is commensurable with a fully invariant subgroup K of A, meaning that φ(K) ≤ K for every φ ∈ End(A) and H ∩ K has finite index in bot... | Yes means the stated equivalence holds for every abelian group A and every subgroup H ≤ A. | null | SOLVED_EXACT | true | Yes | high | https://arxiv.org/abs/2307.15932 | A. R. Chekhlov, P. V. Danchev, Solution to the Uniformly Fully Inert Subgroups Problem for Abelian Groups, arXiv:2307.15932; Rocky Mountain J. Math. 55 (2025), doi:10.1216/rmj.2025.55.1575 | Theorem 2.4 | Theorem 2.4 Each group lies in U, that is, any uniformly fully inert subgroup H of a group G is commensurable with some fully invariant subgroup of G. | Forward direction: Theorem 2.4 (all groups are abelian in the paper). Converse (INFERRED, elementary, also standard in Dardano-Dikranjan-Salce 2020): let H ~ K with K fully invariant, m = |H/(H∩K)| and l = |K/(H∩K)|. For any endomorphism φ, φ(H∩K) ⊆ K and φ(H)/φ(H∩K) is a quotient of H/(H∩K), so it has at most m elemen... | (1) Lemma 2.1: every characteristic subgroup of G⊕G is fully invariant. End(G⊕G) = M_2(End G), and by Henriksen's theorem every 2x2 matrix over any ring is a sum of three units, so every endomorphism of G⊕G is a sum of automorphisms.
(2) Proposition 2.2: let C be uniformly fully inert in G⊕G. Then C is uniformly chara... | null | null | [
"HAS_INFERRED_STEPS"
] | Candidate correct. Published in Rocky Mountain J. Math. (a legitimate refereed venue) and checked by P. Keef per the acknowledgements. I checked each step of the short proof: Henriksen's sum-of-three-units theorem for M_n(R), n>=2; DDR Cor. 1.9; Facts 1 and 2 are elementary. It holds up. The paper proves only the forwa... | [{"id_or_ref": "arXiv:2307.15932", "title": "Solution to the Uniformly Fully Inert Subgroups Problem for Abelian Groups (Chekhlov, Danchev)", "verified": true}, {"id_or_ref": "doi:10.1216/rmj.2025.55.1575", "title": "same, Rocky Mountain Journal of Mathematics 55 (2025)", "verified": true}, {"id_or_ref": "arXiv:1705.02... | 6 |
1705.04303#1 | 1705.04303 | https://arxiv.org/abs/1705.04303 | yesno | For every integer n >= 1 and each prime p, let s_p(n) be the sum of the base-p digits of n. Define \(\mathfrak p_n^+=\prod_{p>\sqrt n,\ s_p(n)\ge p}p\), and let \(\omega(m)\) denote the number of prime divisors of a positive integer m. Does there exist a constant \(\kappa>1\) such that \(\omega(\mathfrak p_n^+)\sim\kap... | Yes means such a constant κ > 1 exists. | null | SOLVED_EXACT | true | Yes | high | https://arxiv.org/abs/1705.06128 | Olivier Bordellès, 'On a conjecture of B. C. Kellner', arXiv:1705.06128 (2017) | Theorem 1 and Corollary 2 | Theorem 1. For any integer n >= 1 sufficiently large, omega(p_n^+) = n E_1(log sqrt(n)) + O(n^{1/2} delta(sqrt(n))), where delta(x) := exp(-0.2098 (log x)^{3/5} (log log x)^{-1/5}) and E_1(x) = int_x^infty e^{-t}/t dt is the exponential integral. Corollary 2. For n sufficiently large and any N>=1, omega(p_n^+) = 2 sqrt... | Corollary 2 with N=1: omega(p_n^+) = 2 sqrt(n)/log n + O(sqrt(n)/(log n)^2), so omega(p_n^+) ~ 2 sqrt(n)/log n. Thus kappa = 2 > 1 exists; answer Yes (INFERRED only the trivial N=1 specialization). | Kellner's identity omega(p_n^+) = #{sqrt(n) < p <= n : floor((n-1)/(p-1)) > floor(n/p)} is rewritten as a sum of floor(n/p + (n-p)/(p(p-1))) - floor(n/p). Split at p = sqrt(n)/delta(sqrt(n)): the upper range is O(sqrt(n) delta(sqrt(n))) by a short-interval divisor-count bound (Lemma 3/Cor. 4). In the lower range, write... | null | null | [
"HAS_INFERRED_STEPS"
] | Kellner (source, Conjecture 2) conjectured kappa>1 and numerically suggested kappa ~ 1.8 up to n=10^7; Bordellès shows kappa = 2, the numerical 1.8 being explained by the negative second-order term -4 sqrt(n)/(log n)^2. Read v3 (28 May 2017). | [{"id_or_ref": "arXiv:1705.06128", "title": "On a conjecture of B. C. Kellner", "verified": true}, {"id_or_ref": "arXiv:1705.04303", "title": "On a product of certain primes (B. C. Kellner)", "verified": true}] | 5 |
1705.04596#0 | 1705.04596 | https://arxiv.org/abs/1705.04596 | numerical | For each integer n >= 4, let A_n = {1 < 2 < ... < n}, let ≡_plac identify two words in A_n* exactly when Schensted's algorithm gives them the same Young tableau, and let plac_n = A_n*/≡_plac. What is the smallest integer n >= 4 for which there are no distinct words u and v in a free monoid on finitely many variables su... | Give the smallest such n, or -1 if no such n exists. | -1 | SOLVED_EXACT | true | -1 | high | https://arxiv.org/abs/1906.03991 | M. Johnson, M. Kambites, Tropical representations and identities of plactic monoids, arXiv:1906.03991; Trans. Amer. Math. Soc. 374 (2021) 4423-4447, doi:10.1090/tran/8355 | Theorem 3.1 (via Theorem 2.8; refined in Corollary 3.3) | Theorem 3.1. The plactic monoid of each finite rank satisfies a non-trivial semigroup identity. | Theorem 2.8 gives a faithful representation rho_n: P_n -> M_{2^[n]}(T) into upper triangular tropical matrices; upper triangular tropical matrix semigroups of each dimension satisfy non-trivial semigroup identities (Izhakian; Okninski; Taylor, refs [16,36,43] of the paper), so P_n = plac_n satisfies a non-trivial ident... | Johnson and Kambites build, for every n, a faithful monoid morphism from plac_n to square upper triangular matrices over the tropical (max-plus) semiring, with rows and columns indexed by subsets of [n]. The entries count maximal lengths of certain scattered (weakly increasing) subwords of a word representing the table... | null | null | [] | Candidate correct. TAMS 374 (2021) 4423-4447, refereed. It settles the Kubat-Okninski conjecture. An earlier proof proposed by Okninski was retracted (the paper says so), so cite Johnson-Kambites as the first valid proof. The paper declares no AI involvement. Cain-Klein-Kubat-Malheiro-Okninski (the source) had shown th... | [{"id_or_ref": "arXiv:1906.03991", "title": "Tropical representations and identities of plactic monoids (Johnson, Kambites)", "verified": true}, {"id_or_ref": "doi:10.1090/tran/8355", "title": "same, Trans. Amer. Math. Soc. 374 (2021) 4423-4447", "verified": true}, {"id_or_ref": "arXiv:1705.04596", "title": "A note on ... | 6 |
1705.04596#1 | 1705.04596 | https://arxiv.org/abs/1705.04596 | numerical | For each integer n >= 4, let plac_n be the quotient of the free monoid on the ordered alphabet {1<2<...<n} by the congruence identifying two words exactly when they have the same Schensted insertion Young tableau. A non-trivial identity is an equality u=v of distinct words in a free monoid that holds after every substi... | The answer is the smallest qualifying n >= 4, or -1 if every plac_n with n >= 4 satisfies a non-trivial identity. | -1 | SOLVED_EXACT | true | -1 | high | https://arxiv.org/abs/1906.03991 | M. Johnson, M. Kambites, Tropical representations and identities of plactic monoids, arXiv:1906.03991; Trans. Amer. Math. Soc. 374 (2021) 4423-4447, doi:10.1090/tran/8355 | Theorem 3.1 (via Theorem 2.8; refined in Corollary 3.3) | Theorem 3.1. The plactic monoid of each finite rank satisfies a non-trivial semigroup identity. | Theorem 2.8 gives a faithful representation rho_n: P_n -> M_{2^[n]}(T) into upper triangular tropical matrices; upper triangular tropical matrix semigroups of each dimension satisfy non-trivial semigroup identities (Izhakian; Okninski; Taylor, refs [16,36,43] of the paper), so P_n = plac_n satisfies a non-trivial ident... | Johnson and Kambites build, for every n, a faithful monoid morphism from plac_n to square upper triangular matrices over the tropical (max-plus) semiring, with rows and columns indexed by subsets of [n]. The entries count maximal lengths of certain scattered (weakly increasing) subwords of a word representing the table... | null | null | [] | Candidate correct. TAMS 374 (2021) 4423-4447, refereed. It settles the Kubat-Okninski conjecture. An earlier proof proposed by Okninski was retracted (the paper says so), so cite Johnson-Kambites as the first valid proof. The paper declares no AI involvement. Cain-Klein-Kubat-Malheiro-Okninski (the source) had shown th... | [{"id_or_ref": "arXiv:1906.03991", "title": "Tropical representations and identities of plactic monoids (Johnson, Kambites)", "verified": true}, {"id_or_ref": "doi:10.1090/tran/8355", "title": "same, Trans. Amer. Math. Soc. 374 (2021) 4423-4447", "verified": true}, {"id_or_ref": "arXiv:1705.04596", "title": "A note on ... | 6 |
1705.05634#0 | 1705.05634 | https://arxiv.org/abs/1705.05634 | numerical | For integers r,s>=1 and n>=2, let H(r,n,s)=<x_0,...,x_{n-1} | product_{j=0}^{r-1} x_{i+j}=product_{j=0}^{s-1} x_{i+j+r} for 0<=i<n>, with all subscripts read modulo n. What is the smallest n for which there exist r,s>=1, neither congruent to 0 modulo n, such that H(r,n,s)^ab is trivial? Answer -1 if no such n exists. I... | Give the smallest qualifying integer n, or -1 if there is none. | -1 | SOLVED_EXACT | true | -1 | high | https://arxiv.org/abs/2010.13000 | I. Chinyere and B. O. Bainson, Perfect Prishchepov groups, J. Algebra 588 (2021) 515–532; arXiv:2010.13000 | Theorem A (and Corollary B) | Theorem A. For integers r, s >= 1 and n >= 2, the following are equivalent: 1. The group H(r, n, s) is perfect; 2. The group H(r, n, s) is trivial; 3. |r - s| = 1 and either r ≡ 0 mod n or s ≡ 0 mod n. Corollary B. Let r, s >= 1 and n >= 2. Then the group H(r, n, s) is a connected LOG group if and only if one of the fo... | By Theorem A, H(r,n,s)^ab trivial (i.e. H perfect) ⇔ H trivial ⇔ |r−s|=1 and (r≡0 mod n or s≡0 mod n). Under the hypothesis r,s ≢ 0 mod n, H(r,n,s) is never perfect, so H(r,n,s)^ab ≠ 1 for all n >= 2 (the paper states 'Conjecture 1.1 follows from Theorem A', Conjecture 1.1 being Williams' conjecture exactly as asked). ... | Relation matrix of H(r,n,s)^ab is circulant with representer polynomial f(t)=sum_{i<r} t^i - t^r sum_{i<s} t^i, so |H^ab| = |prod_{λ^n=1} f(λ)| (0 meaning infinite). Perfectness forces |f(1)|=|r-s|=1 (Lemma 2.1), so WLOG r=s+1. Grouping conjugate roots, Theorem 3.1 shows |P^ab| = prod_{λ in S(n)} |F(λ)+1| where for non... | null | null | [] | Theorem A is unconditional; the 'modulo a certain conjecture' in the abstract concerns only Conjecture 1.3, which is about general Prishchepov groups P(r,n,k,s,q), not H(r,n,s). Independent check (INFERRED, not in paper): brute-force resultant computation of |prod f(λ)| for all n<=18, 1<=s<=r<=2n with n∤r, n∤s found no... | [{"id_or_ref": "arXiv:2010.13000", "title": "Perfect Prishchepov groups (Chinyere, Bainson), J. Algebra 588 (2021) 515-532, doi:10.1016/j.jalgebra.2021.09.005", "verified": true}, {"id_or_ref": "arXiv:1705.05634", "title": "Generalized Fibonacci groups H(r,n,s) that are connected Labelled Oriented Graph groups (G. Will... | 5 |
1705.05634#1 | 1705.05634 | https://arxiv.org/abs/1705.05634 | numerical | For integers a,b>=1 and m>=2, let H(a,m,b)=<x_0,...,x_{m-1} | product_{j=0}^{a-1}x_{i+j}=product_{j=0}^{b-1}x_{i+j+a} for 0<=i<m>, with subscripts read modulo m. What is the smallest integer n>=2 for which there exist integers r,s>=1 such that gcd(r,n,s)=2, |r-s|=2, {r,s}!={4,2}, gcd(n,r+s)=2, neither r nor s is congru... | The answer is the smallest qualifying n, or -1 if there is none. | -1 | SOLVED_EXACT | true | -1 | high | https://arxiv.org/abs/2010.13000 | I. Chinyere and B. O. Bainson, Perfect Prishchepov groups, J. Algebra 588 (2021) 515–532; arXiv:2010.13000 | Theorem A (with proof of Corollary B) | Theorem A. For integers r, s >= 1 and n >= 2, the following are equivalent: 1. The group H(r, n, s) is perfect; 2. The group H(r, n, s) is trivial; 3. |r - s| = 1 and either r ≡ 0 mod n or s ≡ 0 mod n. Corollary B. Let r, s >= 1 and n >= 2. Then the group H(r, n, s) is a connected LOG group if and only if one of the fo... | Suppose n,r,s satisfy the conditions, in particular n∤r, n∤s, gcd(r,n,s)=2 and H(r/2,n/2,s/2) perfect. If n=2, gcd(r,2,s)=2 forces r even, i.e. r≡0 mod n, excluded; so n/2 >= 2 and Theorem A applies to H(r/2,n/2,s/2): perfect ⇒ |r/2−s/2|=1 and r/2≡0 or s/2≡0 mod n/2 ⇒ n|r or n|s, contradiction. (This is exactly the pap... | Relation matrix of H(r,n,s)^ab is circulant with representer polynomial f(t)=sum_{i<r} t^i - t^r sum_{i<s} t^i, so |H^ab| = |prod_{λ^n=1} f(λ)| (0 meaning infinite). Perfectness forces |f(1)|=|r-s|=1 (Lemma 2.1), so WLOG r=s+1. Grouping conjugate roots, Theorem 3.1 shows |P^ab| = prod_{λ in S(n)} |F(λ)+1| where for non... | null | null | [] | Theorem A is unconditional; the 'modulo a certain conjecture' in the abstract concerns only Conjecture 1.3, which is about general Prishchepov groups P(r,n,k,s,q), not H(r,n,s). Independent check (INFERRED, not in paper): brute-force resultant computation of |prod f(λ)| for all n<=18, 1<=s<=r<=2n with n∤r, n∤s found no... | [{"id_or_ref": "arXiv:2010.13000", "title": "Perfect Prishchepov groups (Chinyere, Bainson), J. Algebra 588 (2021) 515-532, doi:10.1016/j.jalgebra.2021.09.005", "verified": true}, {"id_or_ref": "arXiv:1705.05634", "title": "Generalized Fibonacci groups H(r,n,s) that are connected Labelled Oriented Graph groups (G. Will... | 5 |
1705.05634#2 | 1705.05634 | https://arxiv.org/abs/1705.05634 | numerical | What is the smallest integer n≥2 for which there exist r,s≥1 with n∤r and n∤s such that the abelianization of H(r,n,s)=⟨x_0,…,x_{n−1} | ∏_{j=0}^{r−1}x_{i+j}=∏_{j=0}^{s−1}x_{i+j+r} for 0≤i<n⟩ is trivial? All subscripts are read modulo n. Answer −1 if no such n exists. | The answer is the smallest such n, or −1 if none exists. | -1 | SOLVED_EXACT | true | -1 | high | https://arxiv.org/abs/2010.13000 | I. Chinyere and B. O. Bainson, Perfect Prishchepov groups, J. Algebra 588 (2021) 515–532; arXiv:2010.13000 | Theorem A (and Corollary B) | Theorem A. For integers r, s >= 1 and n >= 2, the following are equivalent: 1. The group H(r, n, s) is perfect; 2. The group H(r, n, s) is trivial; 3. |r - s| = 1 and either r ≡ 0 mod n or s ≡ 0 mod n. Corollary B. Let r, s >= 1 and n >= 2. Then the group H(r, n, s) is a connected LOG group if and only if one of the fo... | By Theorem A, H(r,n,s)^ab trivial (i.e. H perfect) ⇔ H trivial ⇔ |r−s|=1 and (r≡0 mod n or s≡0 mod n). Under the hypothesis r,s ≢ 0 mod n, H(r,n,s) is never perfect, so H(r,n,s)^ab ≠ 1 for all n >= 2 (the paper states 'Conjecture 1.1 follows from Theorem A', Conjecture 1.1 being Williams' conjecture exactly as asked). ... | Relation matrix of H(r,n,s)^ab is circulant with representer polynomial f(t)=sum_{i<r} t^i - t^r sum_{i<s} t^i, so |H^ab| = |prod_{λ^n=1} f(λ)| (0 meaning infinite). Perfectness forces |f(1)|=|r-s|=1 (Lemma 2.1), so WLOG r=s+1. Grouping conjugate roots, Theorem 3.1 shows |P^ab| = prod_{λ in S(n)} |F(λ)+1| where for non... | null | null | [] | Theorem A is unconditional; the 'modulo a certain conjecture' in the abstract concerns only Conjecture 1.3, which is about general Prishchepov groups P(r,n,k,s,q), not H(r,n,s). Independent check (INFERRED, not in paper): brute-force resultant computation of |prod f(λ)| for all n<=18, 1<=s<=r<=2n with n∤r, n∤s found no... | [{"id_or_ref": "arXiv:2010.13000", "title": "Perfect Prishchepov groups (Chinyere, Bainson), J. Algebra 588 (2021) 515-532, doi:10.1016/j.jalgebra.2021.09.005", "verified": true}, {"id_or_ref": "arXiv:1705.05634", "title": "Generalized Fibonacci groups H(r,n,s) that are connected Labelled Oriented Graph groups (G. Will... | 5 |
1705.05634#4 | 1705.05634 | https://arxiv.org/abs/1705.05634 | numerical | Let H(r,n,s)=<x_0,...,x_{n-1} | product_{j=0}^{r-1} x_{i+j} = product_{j=0}^{s-1} x_{i+j+r} for every 0 <= i < n>, with subscripts read modulo n. What is the smallest integer n >= 2 for which there exist r,s >= 1, neither congruent to 0 modulo n, such that H(r,n,s)^ab is trivial? Return -1 if no such n exists. This inc... | The answer is the smallest qualifying n, or -1 if no qualifying triple exists. | -1 | SOLVED_EXACT | true | -1 | high | https://arxiv.org/abs/2010.13000 | I. Chinyere and B. O. Bainson, Perfect Prishchepov groups, J. Algebra 588 (2021) 515–532; arXiv:2010.13000 | Theorem A (and Corollary B) | Theorem A. For integers r, s >= 1 and n >= 2, the following are equivalent: 1. The group H(r, n, s) is perfect; 2. The group H(r, n, s) is trivial; 3. |r - s| = 1 and either r ≡ 0 mod n or s ≡ 0 mod n. Corollary B. Let r, s >= 1 and n >= 2. Then the group H(r, n, s) is a connected LOG group if and only if one of the fo... | By Theorem A, H(r,n,s)^ab trivial (i.e. H perfect) ⇔ H trivial ⇔ |r−s|=1 and (r≡0 mod n or s≡0 mod n). Under the hypothesis r,s ≢ 0 mod n, H(r,n,s) is never perfect, so H(r,n,s)^ab ≠ 1 for all n >= 2 (the paper states 'Conjecture 1.1 follows from Theorem A', Conjecture 1.1 being Williams' conjecture exactly as asked). ... | Relation matrix of H(r,n,s)^ab is circulant with representer polynomial f(t)=sum_{i<r} t^i - t^r sum_{i<s} t^i, so |H^ab| = |prod_{λ^n=1} f(λ)| (0 meaning infinite). Perfectness forces |f(1)|=|r-s|=1 (Lemma 2.1), so WLOG r=s+1. Grouping conjugate roots, Theorem 3.1 shows |P^ab| = prod_{λ in S(n)} |F(λ)+1| where for non... | null | null | [] | Theorem A is unconditional; the 'modulo a certain conjecture' in the abstract concerns only Conjecture 1.3, which is about general Prishchepov groups P(r,n,k,s,q), not H(r,n,s). Independent check (INFERRED, not in paper): brute-force resultant computation of |prod f(λ)| for all n<=18, 1<=s<=r<=2n with n∤r, n∤s found no... | [{"id_or_ref": "arXiv:2010.13000", "title": "Perfect Prishchepov groups (Chinyere, Bainson), J. Algebra 588 (2021) 515-532, doi:10.1016/j.jalgebra.2021.09.005", "verified": true}, {"id_or_ref": "arXiv:1705.05634", "title": "Generalized Fibonacci groups H(r,n,s) that are connected Labelled Oriented Graph groups (G. Will... | 5 |
1705.05703#1 | 1705.05703 | https://arxiv.org/abs/1705.05703 | numerical | Let \(\mathcal K(r)=\int_0^{\pi/2}(1-r^2\sin^2 t)^{-1/2}\,dt\) for \(0<r<1\). For each real \(p\), define \(f_p(x)=(1-x)^p\mathcal K(\sqrt{x})\) for \(0<x<1\). What is the exact value of \(p_* = \inf\{p\in\mathbb R: \log f_p\text{ is concave on }(0,1)\}\)? | Give the exact real value of the critical parameter \(p_*\). | 7/32 | SOLVED_EXACT | true | 7/32 | high | https://arxiv.org/abs/2407.14547 | M. Bouali, Convexity and concavity of a class of functions related to the elliptic functions, arXiv:2407.14547 (2024) | Theorem 1.3 (proof in Section 3.3) | Theorem 1.3. For p ∈ R, the function h_p(x) = (1 − x)^p K(x) is log-concave on (0, 1) if and only if p ≥ 7/32 and log-convex if and only if p ≤ 0. [Here K(x) denotes \mathcal K(\sqrt x); abstract: 'h_p is strictly log-concave on (0,1) if and only if p ≥ 7/32'.] | h_p is exactly f_p from the question (the paper sets K(x)=\mathcal K(\sqrt x)). By Theorem 1.3 the set {p : log f_p concave on (0,1)} = [7/32, ∞), so p_* = inf = 7/32 (attained). Consistency check (INFERRED, from the paper's own expansion): (log f_p)''(x) = -(p+G(x))/(1-x)^2 with G(0^+) = -7/32, so (log f_p)''(0^+) = 7... | Write g = log h_p = p log(1-x) + log K(x). Using the standard derivative formulas for K and E, one gets −(1−x)^2 g''(x) = p + G(x) with G(x) = [(2x−1)(1−x)K^2 − 2xKE + E^2]/(4x^2K^2).
The author shows G is strictly increasing on (0,1): 4(1−x)x^3K^3 G'(x) is rewritten as (xK−E)(E^2−(1−x)K^2) + (1−x)K^2((2−x)K−2E), and ... | null | null | [
"HAS_INFERRED_STEPS"
] | Bouali states this conjecture is due to Yang–Tian [24] (the source paper) and remarks that the log-convexity part was also proved independently by Wang et al. by another method. The proof is short and elementary; I checked it against the full text. Its threshold agrees with the second-order Taylor coefficient: log K(√x... | [{"id_or_ref": "arXiv:2407.14547", "title": "Convexity and concavity of a class of functions related to the elliptic functions (M. Bouali)", "verified": true}, {"id_or_ref": "arXiv:1705.05703", "title": "Convexity and monotonicity for the elliptic integrals of the first kind and applications (Z.-H. Yang, J. Tian)", "ve... | 5 |
1705.06232#1 | 1705.06232 | https://arxiv.org/abs/1705.06232 | yesno | Let M be a closed, connected, oriented, aspherical topological manifold of dimension d with nontrivial centre C(pi_1(M)). Let pi:E->B\widetilde{TOP}_h^+(M) be the universal oriented topological M-block bundle whose block-homeomorphism monodromy is homotopic to the identity, let T_v^s(pi):E->BSTOP be its stable vertical... | Yes means the stated Gysin class vanishes for every ring R and every specified class c. | null | SOLVED_EXACT | true | No | high | https://arxiv.org/abs/2609.06345 | M. Bustamante, Aspherical manifolds with nonvanishing tautological classes, arXiv:2609.06345 (2026) | Theorem B (proof in Section 5.2) and Theorem A; the Introduction states explicitly that the HLLRW general conjecture is false | Theorem A. For every even integer m ≥ 2, there exists a (2m + 1)-dimensional closed, oriented, smooth aspherical manifold M whose fundamental group has nontrivial center and for which κ_{v_m^{2r+1} Sq^1 v_m} ≠ 0 ∈ H^{2mr}(B Diff_h(M); F_2) for infinitely many r ≥ 0. Moreover, for each such r, the class κ_{v_m^{2r+1} Sq... | Take R = F_2 and d = 5. Let M be the mapping torus of the Davis–Januszkiewicz hyperbolization of complex conjugation on CP^2. This M is closed, oriented, smooth and aspherical, and Lemma 3.1 shows π1(M) has nontrivial centre. Section 5.2 computes κ_{w2w3} = 1 ∈ H^0(B Diff_h(M); F_2), i.e. ⟨w2(M)w3(M),[M]⟩ = 1. Now take... | Apply Davis–Januszkiewicz hyperbolization equivariantly to an orientation-preserving finite-order diffeomorphism φ of N that has a fixed point (complex conjugation on CP^m, m even). Functoriality gives an aspherical W with an order-k map ρ that fixes a point. The mapping torus T(ρ) is aspherical, and t^k is central in ... | null | null | [
"RECENT_PREPRINT_2026",
"HAS_INFERRED_STEPS"
] | The question is the HLLRW Conjecture (Intro, p.6) verbatim, with no restriction on the degree of c. The degree-0 (characteristic-number) case already refutes it. Theorem A also gives nonvanishing in arbitrarily high positive degrees for smooth bundles, which map to the block-bundle classes. The resolver is recent (Sep ... | [{"id_or_ref": "arXiv:2609.06345", "title": "Aspherical manifolds with nonvanishing tautological classes (M. Bustamante)", "verified": true}, {"id_or_ref": "arXiv:1705.06232", "title": "A vanishing theorem for tautological classes of aspherical manifolds (Hebestreit, Land, Lück, Randal-Williams), Geom. Topol. 25 (2021)... | 5 |
1705.10353#1 | 1705.10353 | https://arxiv.org/abs/1705.10353 | yesno | For every (3+1)-avoiding poset P (a poset containing no subposet consisting of a three-element chain together with an element incomparable to all three), let G be its incomparability graph, with edges joining incomparable pairs. Is the chromatic symmetric function X_G, the sum of the monomials of all proper colorings o... | Yes means that every (3+1)-avoiding poset has an e-positive incomparability-graph chromatic symmetric function. | null | SOLVED_EXACT | true | Yes | high | https://arxiv.org/abs/2410.12758 | T. Hikita, A proof of the Stanley–Stembridge conjecture, arXiv:2410.12758; J. Amer. Math. Soc. (2025), DOI 10.1090/jams/1090 | Theorem 1.6, Corollary 1.7, Corollary 1.8 (with Guay-Paquet's reduction, Theorem 2.2) | Corollary 1.7. For any unit interval graph Γ, the e-expansion coefficients c_λ(Γ; q) of the chromatic quasisymmetric function X_Γ(x; q) evaluated at q ∈ R_{>0} are nonnegative. ... By the result of Guay-Paquet [53] (see Theorem 2.2), the Stanley–Stembridge conjecture has been reduced to the case of unit interval graphs... | Set q = 1 in Corollary 1.7. Then X_Γ(x;1) = X_Γ(x) is e-positive for every unit interval graph, i.e. for the incomparability graph of every (3+1)- and (2+2)-free poset. Guay-Paquet's Theorem 2.2 extends this to all (3+1)-free posets. That is exactly the question's statement (Conjecture 2.1 in the paper), so the answer ... | For a unit interval graph Γ with conjugate Hessenberg function e, comparing coefficients of x1...xn shows that Σ_λ c_λ(Γ;q)/Π_i[λ_i]_q! = 1. Hikita therefore reads P_Γ(λ;q) = c_λ/Π[λ_i]_q! as a candidate probability distribution on partitions of n.
He builds an explicit time-inhomogeneous Markov chain on standard Youn... | null | null | [] | The question states the Stanley–Stembridge conjecture exactly. The published version is JAMS; the DOI resolves to the JAMS early-view page jams1090. I did not separately open the Guay-Paquet arXiv ID; the reduction is cited as Theorem 2.2 inside Hikita. | [{"id_or_ref": "arXiv:2410.12758", "title": "A proof of the Stanley–Stembridge conjecture (T. Hikita)", "verified": true}, {"id_or_ref": "doi:10.1090/jams/1090", "title": "J. Amer. Math. Soc. publication of Hikita's proof", "verified": true}, {"id_or_ref": "Guay-Paquet, arXiv:1306.2400", "title": "A modular relation fo... | 5 |
1706.00738#3 | 1706.00738 | https://arxiv.org/abs/1706.00738 | yesno | Let D={z∈C:|z|<1}, let dμ(z)=(1−|z|²)^(−2) dx dy/π for z=x+iy, and let H² be the analytic Hardy space on D with norm ||f||_{H²}=sup_{0<r<1}(∫₀^{2π}|f(re^{iθ})|² dθ/(2π))^(1/2). Does μ({z∈D:|f(z)|²(1−|z|²)>λ||f||_{H²}²})≤1/λ−1 hold for every f∈H² and every 0<λ<1? Answer "Yes" or "No". | Yes means the inequality holds for every specified f and λ. | null | SOLVED_EXACT | true | Yes | high | https://arxiv.org/abs/2203.12349 | A. Kulikov, Functionals with extrema at reproducing kernels, arXiv:2203.12349; Geom. Funct. Anal. 32 (2022), DOI 10.1007/s00039-022-00608-5 | Theorem 3.1 | Theorem 3.1. Let f ∈ H^p have norm 1 and put u(z) = |f(z)|^p (1 − |z|^2). Then for all t ∈ (0, ∞) we have μ(t) ≤ max(1/t − 1, 0). Equality in (3.1) holds for all 0 < t < ∞ if f(z) ≡ 1. Note that this theorem verifies Conjecture 2 from [3]. [Here μ(t) = m({z : u(z) > t}) and dm(z) = (1−|z|^2)^{-2} dxdy/π.] | Take p = 2. Kulikov's m is exactly the question's μ. For f ≠ 0, apply Theorem 3.1 to f/||f||_{H^2} with t = λ ∈ (0,1). This gives μ({|f|^2(1−|z|^2) > λ||f||^2}) ≤ max(1/λ − 1, 0) = 1/λ − 1 (INFERRED normalization step). For f ≡ 0 the set is empty. So the inequality holds for all f ∈ H^2 and 0 < λ < 1, and the answer is... | Let u(z) = |f(z)|^a(1−|z|^2)^b, which is bounded and tends to 0 at the boundary, with superlevel sets A_t of hyperbolic measure μ(t). The core result (Theorem 2.1) says that a normalized function of μ(t) is monotone. In the Hardy case (a = p, b = 1), g(t) = t(μ(t)+1) is nonincreasing. The proof uses the hyperbolic isop... | null | null | [
"HAS_INFERRED_STEPS"
] | The source's Conjecture 2 matches the question verbatim. Kulikov's Theorem 3.1 covers all H^p, which is more general than p = 2. | [{"id_or_ref": "arXiv:2203.12349", "title": "Functionals with extrema at reproducing kernels (A. Kulikov)", "verified": true}, {"id_or_ref": "doi:10.1007/s00039-022-00608-5", "title": "GAFA publication of Kulikov's paper", "verified": true}, {"id_or_ref": "arXiv:1706.00738", "title": "Contractive inequalities for Hardy... | 5 |
1706.00839#0 | 1706.00839 | https://arxiv.org/abs/1706.00839 | numerical | For the formal perturbative expansion in g of the O(N)-invariant rank-3 tensor model with real field phi^{abc}, a,b,c=1,...,N, fully symmetric and traceless (phi^{abc}=phi^{(abc)} and sum_a phi^{aac}=0), and quartic interaction V_4=(3g/2) phi^{abc} phi^{ade} phi^{fbe} phi^{fdc}, what is the smallest order in g at which... | The answer is the smallest perturbative order in g where the stated property fails, or -1 if it never fails. | -1 | SOLVED_EXACT | true | -1 | medium | https://arxiv.org/abs/1712.00249 | D. Benedetti, S. Carrozza, R. Gurau, M. Kolanowski, The 1/N expansion of the symmetric traceless and the antisymmetric tensor models in rank three, arXiv:1712.00249; Commun. Math. Phys. (2019), DOI 10.1007/s00220-019-03551-z | Theorem 1 and Theorem 2 | Theorem 1. We have (in the sense of perturbation series): F_P(λ) = Σ_{ω∈N/2} N^{−ω} F_P^{(ω)}(λ). Theorem 2. The leading order contribution F_P^{(0)}(λ) is a sum over melonic stranded maps. [P = S the symmetric traceless projector; interaction (λ/(4N^{3/2})) Σ T_{a1a2a3}T_{a3a4a5}T_{a5a2a6}T_{a6a4a1}; abstract: 'This p... | The model with the S projector and tetrahedral coupling λ/N^{3/2} is the Klebanov–Tarnopolsky model, with g ∝ λ/N^{3/2}, i.e. g^2N^3 fixed. Theorem 1 gives, at every order of the perturbation series, an expansion in N^{−ω} with ω ∈ N/2, ω ≥ 0. Theorem 2 identifies the ω = 0 term as a sum of melonic graphs. So all non-m... | Expand the free energy in stranded Feynman maps, where the propagator is the symmetric-traceless projector S (including its −2/(N+2) trace terms). Each map scales as N^{−ω} with degree ω = 3 + 3V/2 − F. Naively ω can be negative because of 'bad tadpoles' and melon-decorated tadpole chains.
The authors add and subtract... | null | null | [] | Confidence is medium only because of the question's wording. The theorem holds for the order-by-order sum (the 1/N expansion of the free energy), and the paper stresses that individual stranded maps with bad tadpoles can have negative degree. Only after resummation and cancellation is every non-melonic contribution sup... | [{"id_or_ref": "arXiv:1712.00249", "title": "The 1/N expansion of the symmetric traceless and the antisymmetric tensor models in rank three (Benedetti, Carrozza, Gurau, Kolanowski)", "verified": true}, {"id_or_ref": "doi:10.1007/s00220-019-03551-z", "title": "Commun. Math. Phys. publication", "verified": true}, {"id_or... | 5 |
1706.01157#0 | 1706.01157 | https://arxiv.org/abs/1706.01157 | numerical | What is the smallest integer n for which there exists a graph G of order n, with every component having at least three vertices, such that gamma_tg(G) > 3n/4? Here gamma_tg(G) is the length under optimal play of the total domination game: Dominator and Staller alternately select vertices, Dominator moves first and seek... | Return the smallest counterexample order, or -1 if the conjectured bound holds for every such graph. | -1 | SOLVED_EXACT | true | -1 | high | https://arxiv.org/abs/2211.16432 | J. Portier, L. Versteegen, A proof of the 3/4-conjecture for the total domination game, SIAM J. Discrete Math. (2024), DOI 10.1137/23M1551584; arXiv:2211.16432 | Theorem 1.2 | Theorem 1.2. For every graph G without isolated vertices or edges, γtg(G) ≤ 3/4 n. | 'Every component has at least three vertices' is the same as 'no isolated vertices or isolated edges'. Theorem 1.2 gives gamma_tg(G) <= 3n/4 for all such G, so no graph with gamma_tg(G) > 3n/4 exists; per the convention the answer is -1. (Bound is tight: disjoint copies of P4.) | Dominator follows a multi-phase strategy. Vertices are painted white/black (black only if totally dominated) and marked depleted (unplayed with no white neighbor, hence unplayable) or dependent. Since the number of totally dominated vertices is at most n and every vertex is either played or not, it suffices to show Dom... | null | null | [] | Full text verified; the conjecture statement (Conjecture 1.1, Henning-Klavzar-Rall) matches the source exactly. The paper also proves gamma_tg(G) <= 5(n+1)/7 when min degree >= 2. | [{"id_or_ref": "arXiv:2211.16432", "title": "A proof of the 3/4 conjecture for the total domination game", "verified": true}, {"id_or_ref": "arXiv:1706.01157", "title": "On the game total domination number", "verified": true}] | 6 |
1706.01365#2 | 1706.01365 | https://arxiv.org/abs/1706.01365 | yesno | Does there exist a function G such that, for every pair of integers 1<k<n with n≥G(k), the following implication holds? Let J(n,k) be the association scheme on the k-subsets of an n-set, with relations indexed by i=1,...,k according to k−|A∩B|=i. If J(n,k) is non-separating—meaning that for some nonempty proper subset ... | Yes means such a function G exists. | null | SOLVED_EXACT | true | Yes | high | https://arxiv.org/abs/2605.30092 | Cherkashin, Shubin, 'Short proofs of three combinatorial results in the Johnson scheme', arXiv:2605.30092 (2026), DOI 10.47443/dml.2026.157 | Theorem 1 (Cherkashin-Shubin; proof in Section 2) | Theorem 1. Let L be a non-empty proper subset of {0, ..., k-1}. If alpha(J[n,k,L]) * omega(J[n,k,L]) = binom(n,k) and n is sufficiently large in terms of k, then there exists t such that either L or {0, ..., k-1} \ L is equal to {0, 1, ..., t-1}. [Abstract: '...if n > n_0(k) and there are an (n,k,L)-system and an (n,k... | Relation index i = k-|A cap B| corresponds to intersection size k-i, so I subset {1..k} nonempty proper <-> L = {k-i: i in I} nonempty proper subset of {0..k-1}, and Gamma_I = J(n,k,L). Non-separating means omega*alpha = binom(n,k) for some such L. For n >= n_0(k), Theorem 1 gives L or its complement = {0,...,t-1} with... | Write L={l_1<...<l_r} and S=[0,k-1]\L={s_1<...<s_{k-r}}. A maximum clique F of J(n,k,L) is an (n,k,L)-system and a maximum coclique G is an (n,k,S)-system. Deza-Erdos-Frankl (n>=n_0(k)): (ii) |F| <= prod_i (n-l_i)/(k-l_i), |G| <= prod_i (n-s_i)/(k-s_i); since the product over all j in [0,k-1] of (n-j)/(k-j) equals bino... | null | null | [
"RECENT_PREPRINT_2026"
] | This is ABC Conjecture 2.2 verbatim; Cherkashin-Shubin call it the Aljohani-Bamberg-Cameron conjecture and prove it. ABC had proved k<=4 (n>=10 for k=4). | [{"id_or_ref": "arXiv:2605.30092 / DOI 10.47443/dml.2026.157", "title": "Short proofs of three combinatorial results in the Johnson scheme", "verified": true}, {"id_or_ref": "arXiv:1706.01365", "title": "Synchronization and separation in the Johnson schemes (Aljohani, Bamberg, Cameron) - source", "verified": true}] | 5 |
1706.01365#3 | 1706.01365 | https://arxiv.org/abs/1706.01365 | numerical | What is the smallest integer k>0 for which there are arbitrarily large integers n>k such that the following equivalence fails: the Johnson association scheme J(n,k), with vertices the k-subsets of {1,...,n} and relations determined by intersection size, is non-separating if and only if there is an integer t with 0<t<k ... | The answer is the smallest such k, or -1 if the equivalence holds for all sufficiently large n for each k. | -1 | SOLVED_EXACT | true | -1 | high | https://arxiv.org/abs/2605.30092 | Cherkashin, Shubin, 'Short proofs of three combinatorial results in the Johnson scheme', arXiv:2605.30092 (2026), DOI 10.47443/dml.2026.157 | Theorem 1 (Cherkashin-Shubin; proof in Section 2) combined with Keevash's existence-of-designs theorem | Theorem 1. Let L be a non-empty proper subset of {0, ..., k-1}. If alpha(J[n,k,L]) * omega(J[n,k,L]) = binom(n,k) and n is sufficiently large in terms of k, then there exists t such that either L or {0, ..., k-1} \ L is equal to {0, 1, ..., t-1}. [Abstract: '...if n > n_0(k) and there are an (n,k,L)-system and an (n,k... | (=>) If J(n,k) is non-separating with n >= n_0(k), Theorem 1 and its proof (tightness of the DEF bound) give an S(t,k,n) with 0<t<k; the divisibilities binom(k-i,t-i) | binom(n-i,t-i), 0<=i<=t-1, are necessary for any S(t,k,n) (count blocks through an i-set). (<=) INFERRED (combination stated in Section 1 of the resolv... | Write L={l_1<...<l_r} and S=[0,k-1]\L={s_1<...<s_{k-r}}. A maximum clique F of J(n,k,L) is an (n,k,L)-system and a maximum coclique G is an (n,k,S)-system. Deza-Erdos-Frankl (n>=n_0(k)): (ii) |F| <= prod_i (n-l_i)/(k-l_i), |G| <= prod_i (n-s_i)/(k-s_i); since the product over all j in [0,k-1] of (n-j)/(k-j) equals bino... | null | null | [
"RECENT_PREPRINT_2026",
"HAS_INFERRED_STEPS"
] | The source (ABC 2017) itself notes Conjecture 2.3 = Conjecture 2.2 + Keevash; Cherkashin-Shubin prove Conj. 2.2 (ABC conjecture; also BCC30 Problem 30.6). Resolver is a 2026 short note (Discrete Math. Letters); proof is a short, checkable application of Deza-Erdos-Frankl. | [{"id_or_ref": "arXiv:2605.30092 / DOI 10.47443/dml.2026.157", "title": "Short proofs of three combinatorial results in the Johnson scheme", "verified": true}, {"id_or_ref": "arXiv:1401.3665", "title": "The existence of designs (Keevash)", "verified": true}, {"id_or_ref": "arXiv:1706.01365", "title": "Synchronization a... | 5 |
1706.01586#0 | 1706.01586 | https://arxiv.org/abs/1706.01586 | yesno | Does there exist a constant c > 0 such that, for every pair of nonisomorphic elliptic curves E_1 and E_2 over C, each equipped with its quotient double-cover map pi_i: E_i -> E_i/{x mapsto -x} isomorphic to P^1, the cardinality of pi_1(E_1[infinity]) intersection pi_2(E_2[infinity]) is less than c, where E_i[infinity] ... | Yes means that one constant c > 0 satisfies the stated bound for every such pair. | null | SOLVED_EXACT | true | Yes | high | https://arxiv.org/abs/2212.13215 | DeMarco, Mavraki, 'Dynamics on P^1: preperiodic points and pairwise stability', arXiv:2212.13215, Compositio Math. (2024), DOI 10.1112/S0010437X23007546; also Garcia-Fritz, Pasten, 'Intersecting the torsion of elliptic curves', arXiv:2308.05708, Bull. Aust. Math. Soc., DOI 10.1017/S000497272300134X (Theorem 1.3 with d=... | Corollary 8.2 (DeMarco-Mavraki); Theorem 1.3 (Garcia-Fritz-Pasten) | Corollary 8.2. There exists a constant M > 0 such that for every pair of elliptic curves E1 and E2 over C, equipped with degree-two projections pi_i : E_i -> P^1 ramified at the 2-torsion points E_i[2], we have |pi_1(E_1^tors) cap pi_2(E_2^tors)| <= M, if and only if pi_1(E_1[2]) != pi_2(E_2[2]). [GFP Theorem 1.3: Let... | The quotient map E_i -> E_i/<-1> = P^1 is the degree-2 even map ramified exactly at E_i[2], with branch locus pi_i(E_i[2]) (4 points). INFERRED standard fact: the branch locus determines the curve (E_i is the double cover of P^1 branched at those 4 points; j is a function of their cross-ratio), so E_1 not isomorphic to... | DeMarco-Mavraki prove Theorem 1.2: for each d there is M_d such that for every elliptic curve E and every f in Rat_d, either |Preper(f) cap x(E_tors)| <= M_d or Preper(f) = x(E_tors), via arithmetic intersection theory/equidistribution (Yuan-Zhang, Gauthier-Vigny, Mavraki-Schmidt) on the parameter space of pairs, with ... | null | null | [
"HAS_INFERRED_STEPS"
] | Source Conjecture 2 (EFC-I) of Bogomolov-Fu-Tschinkel. Both candidates correct; the conjecture was first fully settled by Poineau (arXiv:2207.01574) and also follows from Uniform Manin-Mumford (Kuhne, Gao-Ge-Kuhne); DeMarco-Krieger-Ye (2020) did the Legendre case. No explicit value of c is known from these proofs. | [{"id_or_ref": "arXiv:2212.13215", "title": "Dynamics on P^1: preperiodic points and pairwise stability", "verified": true}, {"id_or_ref": "arXiv:2308.05708", "title": "Intersecting the torsion of elliptic curves", "verified": true}, {"id_or_ref": "arXiv:2207.01574", "title": "Dynamique analytique sur Z. II : Ecart uni... | 5 |
1706.03447#5 | 1706.03447 | https://arxiv.org/abs/1706.03447 | numerical | What is the smallest integer d for which there exist an integer r with 3 <= r < floor(d/2) and a convex centrally symmetric simplicial d-polytope P subset R^d such that g_r(P)=g_r(C_d^*) but g_{r+1}(P)≠g_{r+1}(C_d^*)? If no such d exists, answer -1. Here P=-P and every proper face of P is a simplex; for each s, g_s(P)=... | Answer the smallest such integer d, or -1 if no counterexample exists. | -1 | SOLVED_EXACT | true | -1 | high | https://arxiv.org/abs/2008.12503 | I. Novik, H. Zheng, 'The stresses on centrally symmetric complexes and the lower bound theorems', Algebraic Combinatorics 4(3) (2021) 541-549, doi:10.5802/alco.168, arXiv:2008.12503 | Theorem 3.6(2) (proving Conjecture 1.3 = Klee-Nevo-Novik-Zheng Conjecture 8.5) | Theorem 3.6. ... 2. Let d and 1 ≤ i < d/2 be integers. If ∆ = ∂P for some cs simplicial d-polytope P and g_i(∆) = binom(d,i) − binom(d,i−1), then g_j(∆) = binom(d,j) − binom(d,j−1) for all i ≤ j ≤ d/2. | Take i=r with 3 ≤ r < floor(d/2) (so r < d/2). If g_r(P)=g_r(C_d^*)=binom(d,r)-binom(d,r-1), Theorem 3.6(2) gives g_j(P)=binom(d,j)-binom(d,j-1) for all r ≤ j ≤ d/2; since r+1 ≤ floor(d/2), j=r+1 is included, and binom(d,r+1)-binom(d,r)=g_{r+1}(C_d^*). The question's f-vector formula for g_s equals h_s-h_{s-1} (INFERRE... | Use Lee's stress spaces: for a cs simplicial d-polytope, g_i(∂P) is the dimension of the space of affine i-stresses (w.r.t. the canonical linear forms Θ̃), and g_i = binom(d,i)-binom(d,i-1) iff every affine i-stress is symmetric (invariant under the antipodal involution) (Corollary 2.2). The main Theorem 3.5 shows, for... | null | null | [
"HAS_INFERRED_STEPS"
] | Candidate had arxiv=null; correct arXiv ID is 2008.12503 (verified on abs page). Paper proves the conjecture in full generality (all 1 ≤ i < d/2), including every r with 3 ≤ r < floor(d/2). | [{"id_or_ref": "arXiv:2008.12503", "title": "The stresses on centrally symmetric complexes and the lower bound theorems (Novik, Zheng)", "verified": true}, {"id_or_ref": "arXiv:1706.03447", "title": "A lower bound theorem for centrally symmetric simplicial polytopes (Klee, Nevo, Novik, Zheng) [source]", "verified": tru... | 5 |
1706.04533#0 | 1706.04533 | https://arxiv.org/abs/1706.04533 | yesno | For every commutative ring R with 1 carrying a reflexive, transitive, total relation ≼ such that, for all x,y,z∈R, (QR1) 0≺1, (QR2) xy≼0 implies x≼0 or y≼0, (QR3) x≼y and 0≼z imply xz≼yz, (QR4) x≼y and z≁y imply x+z≼y+z, and (QR5) 0≺z and xz≼yz imply x≼y (where x∼y means x≼y and y≼x), and for a valuation v:R→Γ∪{∞}, can... | Yes means that such a notion of compatibility and corresponding Baer–Krull theorem can be developed for the stated setting. | null | SOLVED_EXACT | true | Yes | high | https://arxiv.org/abs/1707.08410 | Salma Kuhlmann and Simon Mueller, 'Compatibility of quasi-orderings and valuations; A Baer-Krull Theorem for quasi-ordered rings', arXiv:1707.08410 (v4, Sept 2018); journal venue not verified from arXiv page. Sequel to S. Mueller, 'Quasi-ordered rings' (arXiv:1706.04533, Comm. Algebra 46 (2018) 4979-4985), the source p... | Definition 3.2 (compatibility), Theorem 3.12 (characterisation of compatibility), Theorem 4.10 / Corollary 4.11 (Baer-Krull for quasi-ordered rings, Manis v), Theorems 4.12 and 4.17 (general / special* v) | Definition 3.2. Let (R, \preceq) be a quasi-ordered ring. A valuation v on R is said to be compatible with \preceq (or \preceq-compatible), if for all y, z \in R: 0 \preceq y \preceq z \Rightarrow v(z) \le v(y).
Theorem 4.10 (Baer-Krull Theorem for quasi-ordered rings I). Let R be a commutative ring with 1 and v a Man... | 1. (Paper) The quasi-ordered ring axioms (QR1)-(QR5) in the question are verbatim Definition 2.6 of 1707.08410 (= [11, Definition 3.2], Mueller's source paper), and the dichotomy quoted in the question is recalled as [11, Theorem 4.6].
2. (Paper) Definition 3.2 supplies the notion of compatibility between a valuation v... | The paper first isolates the right notion of compatibility (Definition 3.2) and shows that, unlike for fields, the classical equivalent conditions (compatibility, convexity of R_v and I_v, I_v < 1, induced residue order) come apart for arbitrary valuations on rings (Example 3.3 and the implication table); restricting t... | null | null | [] | Kuhlmann-Muller: compatibility + Baer-Krull for quasi-ordered rings. | [] | 1-2 |
1706.05230#3 | 1706.05230 | https://arxiv.org/abs/1706.05230 | yesno | Let N_0 be the non-negative integers. For each integer g >= 0, let n_g be the number of numerical semigroups Lambda subseteq N_0 of genus g, where a numerical semigroup contains 0, is closed under addition, has finite complement in N_0, and has genus #(N_0\Lambda). Is it true that (n_{g-1}+n_{g-2})/n_g tends to 1 as g ... | Yes means that both stated limits hold. | null | SOLVED_EXACT | true | Yes | high | https://arxiv.org/abs/1111.3142 | A. Zhai, 'Fibonacci-like growth of numerical semigroups of a given genus', Semigroup Forum 86 (2013) 634-662, arXiv:1111.3142 (proof outlined in N. Kaplan, 'Counting Numerical Semigroups', Amer. Math. Monthly 124 (2017) 862-875, arXiv:1707.02551) | Zhai, Theorem 1 (= Kaplan's survey Theorem 1 (Zhai)) | Theorem 1. Let n_g be the number of numerical semigroups of genus g. Then, lim_{g→∞} n_g/φ^g = S, where S is a constant. [Kaplan's restatement adds: 'where S is a constant that is at least 3.78.'] | S is finite (Zhai proves sup t_g φ^{-g} < ∞) and positive (n_g = Ω(φ^g) was already known, cited in Zhai's intro; Kaplan states S ≥ 3.78). Then n_g/n_{g-1} = (n_g φ^{-g})/(n_{g-1} φ^{-(g-1)}) · φ → φ. And (n_{g-1}+n_{g-2})/n_g → S(φ^{-1}+φ^{-2})/S = 1 since φ^{-1}+φ^{-2}=1 (INFERRED arithmetic; Zhai's intro explicitly ... | Let t_g count genus-g semigroups with Frobenius number f < 3m (m = multiplicity). Zhao had shown lim t_g φ^{-g} = S as a (possibly divergent) sum and conjectured t_g/n_g → 1 and sup t_g φ^{-g} < ∞. Zhai proves both. Using Bras-Amorós's semigroup tree, he splits semigroups by the relation between their effective generat... | null | null | [
"HAS_INFERRED_STEPS"
] | The candidate (Kaplan 2017) is an expository survey that outlines the proof; the original resolution is Zhai (arXiv 2011, Semigroup Forum 2013), which predates the 2017 arXiv posting of the 2008 source paper. Flag BIBLIOGRAPHIC for that reason. Bras-Amorós's other conjecture n_g ≥ n_{g-1}+n_{g-2} remains open, but it i... | [{"id_or_ref": "arXiv:1111.3142", "title": "Fibonacci-like growth of numerical semigroups of a given genus (Zhai)", "verified": true}, {"id_or_ref": "arXiv:1707.02551", "title": "Counting Numerical Semigroups (Kaplan)", "verified": true}, {"id_or_ref": "arXiv:1706.05230", "title": "Fibonacci-Like Behavior of the Number... | 5 |
1706.07023#0 | 1706.07023 | https://arxiv.org/abs/1706.07023 | yesno | Let G be a complex Lie group whose Lie algebra is \mathfrak g=\mathfrak g_R\otimes\mathbb C, where \mathfrak g_R is a compact simple real Lie algebra with invariant metric \langle\ ,\ \rangle. Let \kappa\in\operatorname{Sym}^{1,1}(\mathfrak g) be the Hermitian tensor corresponding to \langle\ ,\ \rangle^{-1}\in\operato... | Yes means the stated convergence holds for every solution on its maximal interval. | null | SOLVED_EXACT | true | No | high | https://arxiv.org/abs/2112.09344 | J. Stanfield, Positive Hermitian Curvature Flow on special linear groups and perfect solitons, arXiv:2112.09344 (2021) | Theorem 1.3 (with Theorem 3.5, Remark 3.6, Lemma 3.7) | Theorem 1.3. Let n >= 2. Then on SL_{n+1}(C), there exists a left-invariant solution (g_t)_{t in [0,T_max)} to (1) with g_0 chosen arbitrarily close to g_can in the C^infty topology such that as t -> T_max, (G, J, (T_max - t)^{-1} g_t) converges in the Cheeger-Gromov sense to (G, J, g-bar), which is locally isometric t... | Stanfield explicitly states the paper is 'a counterexample to Conjecture 1.2 on SL_n(C) for n>=3', where Conjecture 1.2 is cited as [24, Conjecture 5.4] = Ustinovskiy 1706.07023 Conjecture 5.4 (the question). Concretely: take g_R = su(n+1), n>=2 (compact simple), G = SL_{n+1}(C). Stanfield's ansatz metric is <sigma_{x,... | Restrict HCF+ on SL_{n+1}(C) to the 3-parameter family of left-invariant, Ad(SU(n))-invariant metrics given by scaling the three Ad(SL_n)-invariant summands sl_n(C), CI, s of sl_{n+1}(C) (Lemma 3.1 computes the torsion-twisted Chern-Ricci operator explicitly), reducing the flow to the Riccati-type ODE (11). All solutio... | null | null | [
"HAS_INFERRED_STEPS"
] | Counterexample on G = SL_{n+1}(C), n>=2 (g_R = su(n+1)); holds for SL_2(C) per Ustinovskiy's Example 5.2 but fails in general. Stanfield's abstract: canonical scale-static metrics on special linear groups are dynamically unstable, 'This disproves a conjecture of Ustinovskiy.' Both uids ask the same universal statement ... | [{"id_or_ref": "arXiv:2112.09344", "title": "Positive Hermitian Curvature Flow on special linear groups and perfect solitons (J. Stanfield)", "verified": true}, {"id_or_ref": "arXiv:1706.07023", "title": "Hermitian curvature flow on complex homogeneous manifolds (Y. Ustinovskiy) [source; Conjecture 5.4]", "verified": t... | 5 |
1706.07023#2 | 1706.07023 | https://arxiv.org/abs/1706.07023 | yesno | Let G be the complexification of a compact simple real Lie group with compact real Lie algebra g_R, let g=g_R tensor C, and let kappa in Sym^{1,1}(g) be the Hermitian form corresponding to the inverse of an invariant metric on g_R. For a positive-definite Hermitian initial form h(0)=h_0 in Sym^{1,1}(g), let h(t) solve ... | Yes means the stated convergence holds for every positive-definite Hermitian initial form h_0 with some g in G and lambda in R. | null | SOLVED_EXACT | true | No | high | https://arxiv.org/abs/2112.09344 | J. Stanfield, Positive Hermitian Curvature Flow on special linear groups and perfect solitons, arXiv:2112.09344 (2021) | Theorem 1.3 (with Theorem 3.5, Remark 3.6, Lemma 3.7) | Theorem 1.3. Let n >= 2. Then on SL_{n+1}(C), there exists a left-invariant solution (g_t)_{t in [0,T_max)} to (1) with g_0 chosen arbitrarily close to g_can in the C^infty topology such that as t -> T_max, (G, J, (T_max - t)^{-1} g_t) converges in the Cheeger-Gromov sense to (G, J, g-bar), which is locally isometric t... | Stanfield explicitly states the paper is 'a counterexample to Conjecture 1.2 on SL_n(C) for n>=3', where Conjecture 1.2 is cited as [24, Conjecture 5.4] = Ustinovskiy 1706.07023 Conjecture 5.4 (the question). Concretely: take g_R = su(n+1), n>=2 (compact simple), G = SL_{n+1}(C). Stanfield's ansatz metric is <sigma_{x,... | Restrict HCF+ on SL_{n+1}(C) to the 3-parameter family of left-invariant, Ad(SU(n))-invariant metrics given by scaling the three Ad(SL_n)-invariant summands sl_n(C), CI, s of sl_{n+1}(C) (Lemma 3.1 computes the torsion-twisted Chern-Ricci operator explicitly), reducing the flow to the Riccati-type ODE (11). All solutio... | null | null | [
"HAS_INFERRED_STEPS"
] | Counterexample on G = SL_{n+1}(C), n>=2 (g_R = su(n+1)); holds for SL_2(C) per Ustinovskiy's Example 5.2 but fails in general. Stanfield's abstract: canonical scale-static metrics on special linear groups are dynamically unstable, 'This disproves a conjecture of Ustinovskiy.' Both uids ask the same universal statement ... | [{"id_or_ref": "arXiv:2112.09344", "title": "Positive Hermitian Curvature Flow on special linear groups and perfect solitons (J. Stanfield)", "verified": true}, {"id_or_ref": "arXiv:1706.07023", "title": "Hermitian curvature flow on complex homogeneous manifolds (Y. Ustinovskiy) [source; Conjecture 5.4]", "verified": t... | 5 |
1706.08411#0 | 1706.08411 | https://arxiv.org/abs/1706.08411 | yesno | For every separable operator system S, let A = C*(S). Suppose every irreducible *-representation rho: A -> B(H_rho) is the only completely positive map A -> B(H_rho) whose restriction to S equals rho|_S. Must every unital *-representation pi: A -> B(H_pi) likewise be the only completely positive map A -> B(H_pi) whose ... | Yes means the stated implication holds for every separable operator system S. | null | SOLVED_EXACT | true | No | high | https://arxiv.org/abs/2404.05018 | B. Bilich, A. Dor-On, Arveson's hyperrigidity conjecture is false, arXiv:2404.05018 (2024) | Theorem 1.1 | Theorem 1.1. There is an operator T on Hilbert space such that the unital operator algebra A generated by T is not hyperrigid in B = C*(A), while the restriction to A of all irreducible *-representations of B have unique completely positive extensions to B. In fact, T is a rank-one perturbation of a unitary, so that wi... | Pass from A to the separable operator system S = A + A*. Every representation of A extends uniquely to a UCP map on S (Section 2), so S generates the same C*-algebra B, the UEP for S and A agree, and so does hyperrigidity (Arveson, Thm 2.1). So S is a separable operator system in which every irreducible representation ... | Take T to be a rank-one perturbation of a unitary; the generated C*-algebra B is type I. They check that every irreducible *-representation of B is a boundary representation, meaning its restriction to A is maximal in the Dritschel-McCullough sense and hence has the UEP. They then exhibit a (reducible) *-representation... | null | null | [] | Full text verified: the paper's statement of Arveson's conjecture (Conj. 4.3) matches the source question. I did not confirm journal publication (v2, April 2024), but the result is widely accepted and followed up, e.g. Scherer's finite-dimensional counterexample (Bull. LMS 2025) and Clouatre arXiv:2509.19238. | [{"id_or_ref": "arXiv:2404.05018", "title": "Arveson's hyperrigidity conjecture is false", "verified": true}, {"id_or_ref": "arXiv:1706.08411", "title": "Unperforated pairs of operator spaces and hyperrigidity of operator systems", "verified": true}, {"id_or_ref": "doi:10.1112/blms.70120", "title": "A finite-dimensiona... | 6 |
1707.02010#0 | 1707.02010 | https://arxiv.org/abs/1707.02010 | numerical | For each pair of integers 0 ≤ k ≤ n, let Gr_{≥0}(k,n) be the set of k-planes in R^n whose Plücker coordinates are all nonnegative, decomposed into cells by prescribing each Plücker coordinate to be strictly positive or zero. What is the smallest integer n for which, for some k with 0 ≤ k ≤ n, this decomposition fails t... | The answer is the smallest such n, or -1 if the stated property holds for every pair 0 ≤ k ≤ n. | -1 | SOLVED_EXACT | true | -1 | high | https://arxiv.org/abs/1904.00527 | P. Galashin, S. N. Karp, T. Lam, Regularity theorem for totally nonnegative flag varieties, J. Amer. Math. Soc. 35 (2022), DOI 10.1090/jams/983; arXiv:1904.00527 | Corollary 1.2 (of Theorem 1.1) | Corollary 1.2. The decomposition of Gr≥0(k, n) into positroid cells forms a regular CW complex. Thus the closure of each positroid cell is homeomorphic to a closed ball. | The cells obtained by prescribing each Plucker coordinate as positive or zero are exactly Postnikov's positroid cells. Corollary 1.2 holds for all 0 <= k <= n: the decomposition is a regular CW complex and every cell closure is a closed ball. Gr≥0(k,n) itself is a closed ball (GKL 2017, the top cell's closure). So no n... | Theorem 1.1 proves Williams' conjecture that Rietsch's cell decomposition of (G/P)≥0 is a regular CW complex; the Grassmannian case gives Postnikov's positroid decomposition via (9.16). The proof uses a topological criterion (Section 3): a stratified space is a regular CW complex if the links of the strata are suitably... | null | null | [] | The candidate (Bao-He, Invent. Math. 2024) is correct but not the first proof: it generalizes to J-total positivity and Kac-Moody groups and explicitly credits Galashin-Karp-Lam [12] with the regularity of the partial-flag case. I cite GKL 1904.00527 (JAMS 2022) as the first and cleanest resolution of Postnikov's conje... | [{"id_or_ref": "arXiv:1904.00527", "title": "Regularity theorem for totally nonnegative flag varieties", "verified": true}, {"id_or_ref": "arXiv:2203.02137", "title": "Product structure and regularity theorem for totally nonnegative flag varieties", "verified": true}, {"id_or_ref": "arXiv:1707.02010", "title": "The tot... | 6 |
1707.04411#0 | 1707.04411 | https://arxiv.org/abs/1707.04411 | numerical | For every dimension d and every finite set U={u_1,...,u_k} of non-zero vectors generating the additive group Z^d, let G_U be the directed graph with vertex set Z^d and edges (x,x+u) for every x in Z^d and u in U. For finite S⊆Z^d, define ∂_{G_U}(S)=|{(x,y)∈E(G_U): x∈S, y∉S}| and ∂*_{G_U}(n)=min{∂_{G_U}(S): S⊆Z^d, |S|=n... | Answer -1 if no such dimension exists; otherwise answer the smallest such dimension. | -1 | SOLVED_EXACT | true | 1 | high | https://arxiv.org/abs/2402.14087 | Joseph Briggs, Chris Wells, 'Phase transitions in isoperimetric problems on the integers', arXiv:2402.14087 (v1 Feb 2024, v2 14 Jan 2026) | Theorem 1.2 (case i = e), via Theorem 2.4 | Theorem 1.2. For every integer N and for each i ∈ {e, v}, there is a subset B ⊆ Z and integers n1, n2 satisfying • N ≤ n1 ≤ n2 − N, and • B is finite, symmetric and generates Z, and • With G = Cay(Z, B), no element of Opt_{i,G}(n1) is a subset of any element of Opt_{i,G}(n2). [Theorem 2.4: Fix an integer b ≥ 2 and set ... | The paper's ∂_{e,G}A = {(u,v)∈E : u∈A, v∉A} on the directed Cay(Z,B) equals the source's ∂_{G_U}; B = ±{1,b} is a finite set of nonzero integers generating Z. Proof of Thm 1.2 (Sec 2.4): even b ≥ max{N,10}, k=(b−2)/2, n1=k^2, n2=(b+1)(k−1); Opt_e(n1) = translates of [k]+b[k], Opt_e(n2) = intervals of length n2, and tra... | Cay(Z, ±{1,b}) is a 'not-quite-isomorphic' copy of the l1 cylinder Z×(Z/bZ) via x=qb+r ↦ (q,r). Prop 2.3 shows that if A meets every residue class mod b, then |∂e A| ≥ 2(b+1) with equality iff A is an interval; so an optimal non-interval must miss a residue class, and cutting along it embeds A into the l1 grid Z^2 pres... | null | null | [
"HAS_INFERRED_STEPS"
] | Verified title/authors and full text. Source question = Barber–Erde Question 1, first bullet (edge boundary). Briggs–Wells give a d = 1 counterexample, so the smallest dimension is 1. The paper also mentions Strachan–Swanepoel's d ≥ 2 edge examples (irrelevant to the minimum). Resolver is an arXiv preprint (v2 Jan 2026... | [{"id_or_ref": "arXiv:2402.14087", "title": "Phase transitions in isoperimetric problems on the integers (Briggs, Wells)", "verified": true}, {"id_or_ref": "arXiv:1707.04411", "title": "Isoperimetry in integer lattices (Barber, Erde), Discrete Analysis 2018:7 - source, Question 1", "verified": true}, {"id_or_ref": "Str... | 6 |
1707.04411#1 | 1707.04411 | https://arxiv.org/abs/1707.04411 | numerical | What is the smallest integer d ≥ 1 for which there exists a finite set U = {u_1,...,u_k} of nonzero vectors in Z^d that generates Z^d as a group, but no ordering w_1,w_2,... of Z^d satisfies ∂_{v,G_U}({w_1,...,w_n}) = ∂^*_{v,G_U}(n) for every integer n ≥ 1? Here G_U has edges (x,x+u) for every x ∈ Z^d and u ∈ U; ∂_{v,G... | Give the smallest such dimension d, or -1 if no such dimension exists. | -1 | SOLVED_EXACT | true | 1 | high | https://arxiv.org/abs/2402.14087 | Joseph Briggs, Chris Wells, 'Phase transitions in isoperimetric problems on the integers', arXiv:2402.14087 (v1 Feb 2024, v2 14 Jan 2026) | Theorem 1.2 (case i = v), via Theorem 2.8 | Theorem 1.2. For every integer N and for each i ∈ {e, v}, there is a subset B ⊆ Z and integers n1, n2 satisfying • N ≤ n1 ≤ n2 − N, and • B is finite, symmetric and generates Z, and • With G = Cay(Z, B), no element of Opt_{i,G}(n1) is a subset of any element of Opt_{i,G}(n2). [Theorem 2.8: Fix a positive integer b ≥ 3 ... | The paper's ∂_{v,G}A = {v ∉ A : ∃u∈A, (u,v)∈E} on the directed Cayley graph Cay(Z,B) (edges (u,v) iff v−u∈B) is exactly the source's ∂_{v,G_U}, and B = ±{1,b−1,b,b+1} (b ≥ 3) is a finite set of nonzero integers generating Z. Paper's proof of Thm 1.2 (Sec 2.4): take even b ≥ max{N,10}, k=(b−2)/2, n1=k^2, n2=(b+1)(k−1); ... | The idea is that Cay(Z, ±{1,b−1,b,b+1}) mimics the l∞ cylinder Z×(Z/bZ) via x=qb+r ↦ (q,r). For small n (√n < (b−1)/2), an optimal set that is not an interval must miss two consecutive residues mod b (Thm 2.7), so one can 'cut' along a missing residue class and embed it into the l∞ grid on Z^2 without changing its vert... | null | null | [
"HAS_INFERRED_STEPS"
] | Verified title/authors on arXiv abs page and full text (v2). Barber–Erde's Question 1 (second bullet) is exactly the source question; Briggs–Wells answer it negatively already in d = 1 with symmetric generating sets. Their Theorem 1.3 shows nestedness holds for all large n on Z (so the failure is only for finitely many... | [{"id_or_ref": "arXiv:2402.14087", "title": "Phase transitions in isoperimetric problems on the integers (Briggs, Wells)", "verified": true}, {"id_or_ref": "arXiv:1707.04411", "title": "Isoperimetry in integer lattices (Barber, Erde), Discrete Analysis 2018:7 - source, Question 1", "verified": true}] | 6 |
1708.00024#2 | 1708.00024 | https://arxiv.org/abs/1708.00024 | numerical | What is the smallest ambient dimension n for which there exists a projective subvariety X⊆P^{n−1} over C, not contained in Q={x_1^2+⋯+x_n^2=0}, such that, for a general hyperplane H⊆P^{n−1}, EDdegree(X)≠(−1)^{dim X}χ_Ma(X∖(Q∪H))? Here EDdegree(X) is the number of critical points at nonsingular points of the affine cone... | Give the smallest ambient dimension in which the proposed extension fails, or −1 if it holds for all such X. | -1 | SOLVED_EXACT | true | -1 | high | https://arxiv.org/abs/1901.05550 | L. G. Maxim, J. I. Rodriguez, B. Wang, Euclidean distance degree of projective varieties, arXiv:1901.05550; Int. Math. Res. Not. IMRN 2021 (doi:10.1093/imrn/rnz266) | Theorem 1.3 | Theorem 1.3. Let X be any irreducible closed subvariety of P^n. Then for a general beta = (beta_0, beta_1, ..., beta_n) in C^{n+1} \ {0} we have EDdeg_proj(X) = (-1)^{dim X} chi(Eu_X|_{U_beta}), where U_beta := P^n \ (Q \cup H_beta), with Q denoting the isotropic quadric and H_beta := {beta_0 x_0 + beta_1 x_1 + ... + b... | Apply Thm 1.3 with P^n = P^{n-1} of the question, X irreducible and X not in Q; a general beta gives a general hyperplane H. Since U = X \ (Q \cup H) is Zariski open in X, Eu_U = Eu_X|_U (property (vii), analytic invariance). By MacPherson's theorem, the degree of c_*(phi) equals the weighted Euler characteristic chi(p... | The proof is Morse-theoretic. Prop. 3.1: for X irreducible, X not in Q, and general beta, the ED degree of the cone C(X) equals the number of degeneration points on X_reg of the 1-form d log h_beta, where h_beta = (sum beta_i x_i)^2/(sum x_i^2). Two facts give this: no critical point of the distance function lies on H_... | null | null | [
"HAS_INFERRED_STEPS"
] | The source's conjecture is in Aluffi-Harris Section 3 after Prop. 3.1, and MRW Theorem 1.3 answers it affirmatively. MRW assume X irreducible, which is the standard meaning of 'subvariety' in Aluffi-Harris (they use the irreducible conormal/ED correspondence). Caveat (my own check): if reducible, non-equidimensional X ... | [{"id_or_ref": "arXiv:1901.05550", "title": "Euclidean distance degree of projective varieties (Maxim, Rodriguez, Wang)", "verified": true}, {"id_or_ref": "arXiv:1708.00024", "title": "The Euclidean distance degree of smooth complex projective varieties (Aluffi, Harris) - source", "verified": true}] | 5 |
1708.00024#3 | 1708.00024 | https://arxiv.org/abs/1708.00024 | yesno | Let X be a smooth complex projective subvariety of P(C^n), let Q⊂P(C^n) be the isotropic quadric defined by x_1^2+...+x_n^2=0, and let H be a general hyperplane. Define Edd(X) as the number of critical points, at nonsingular points of the affine cone over X, of (x_1-u_1)^2+...+(x_n-u_n)^2 for general u∈C^n. When X meet... | Yes means such a direct derivation exists for every smooth X without the transversality assumption. | null | SOLVED_EXACT | true | Yes | medium | https://arxiv.org/abs/1901.05550 | L. G. Maxim, J. I. Rodriguez, B. Wang, Euclidean distance degree of projective varieties, Int. Math. Res. Not. IMRN 2021 (arXiv:1901.05550) | Theorem 1.3 (with the smooth-case reduction stated right after it) | Theorem 1.3. Let X be any irreducible closed subvariety of P^n. Then for a general beta = (beta_0, beta_1, ..., beta_n) in C^{n+1} \ {0} we have EDdeg_proj(X) = (-1)^{dim X} chi(Eu_X|_{U_beta}), where U_beta := P^n \ (Q u H_beta), with Q denoting the isotropic quadric and H_beta := {beta_0 x_0 + ... + beta_n x_n = 0}. ... | For smooth X, Eu_X = 1_X, so Theorem 1.3 gives Edd(X) = (-1)^{dim X} chi(X \ (Q u H)) with no transversality hypothesis. (INFERRED, but it is the standard step stated in Aluffi-Harris Sec. 3, '(3.2) is equivalent to (1.1)') Additivity of the compactly supported / algebraic Euler characteristic gives chi(X \ (Q u H)) = ... | Proposition 3.1: for general beta, critical points of the squared distance function f_beta on the smooth part of the affine cone C(X) correspond one-to-one to degeneration points of the logarithmic 1-form d log h_beta restricted to X_reg, where h_beta = (beta.x)^2/(x.x). The key step is showing that no critical point l... | null | null | [
"HAS_INFERRED_STEPS"
] | The source (Aluffi-Harris) already proves the formula for all smooth X not contained in Q (Thm 8.1), but through heavy characteristic-class and Segre-class machinery. It asks for a simpler, more direct derivation. MRW explicitly answer this: they give a short Morse-theoretic proof (a double cover plus the count of crit... | [{"id_or_ref": "arXiv:1901.05550", "title": "Euclidean distance degree of projective varieties (Maxim, Rodriguez, Wang)", "verified": true}, {"id_or_ref": "arXiv:1708.00024", "title": "The Euclidean distance degree of smooth complex projective varieties (Aluffi, Harris) - source; Thm 8.1 proves the formula via characte... | 5 |
1708.00024#4 | 1708.00024 | https://arxiv.org/abs/1708.00024 | numerical | What is the smallest integer n for which there exists a projective subvariety X⊆P^{n−1}, with X⊄Q where Q is the isotropic quadric x₁²+⋯+xₙ²=0, such that for a general hyperplane H⊆P^{n−1}, EDdeg(X)≠(−1)^{dim X}∫c_*(Eu_{X\(Q∪H)})? Give −1 if no such n exists. | The answer is the smallest n admitting a failure, or −1 if the equality holds for every X in the stated setting. | -1 | SOLVED_EXACT | true | -1 | high | https://arxiv.org/abs/1901.05550 | L. G. Maxim, J. I. Rodriguez, B. Wang, Euclidean distance degree of projective varieties, arXiv:1901.05550; Int. Math. Res. Not. IMRN 2021 (doi:10.1093/imrn/rnz266) | Theorem 1.3 | Theorem 1.3. Let X be any irreducible closed subvariety of P^n. Then for a general beta = (beta_0, beta_1, ..., beta_n) in C^{n+1} \ {0} we have EDdeg_proj(X) = (-1)^{dim X} chi(Eu_X|_{U_beta}), where U_beta := P^n \ (Q \cup H_beta), with Q denoting the isotropic quadric and H_beta := {beta_0 x_0 + beta_1 x_1 + ... + b... | Eu_{X \ (Q \cup H)} = Eu_X restricted to U_beta (Zariski open; property (vii)). The integral of c_*(Eu_U) equals chi(Eu_X|_U) (MacPherson; INFERRED standard fact). Thm 1.3 then gives EDdeg(X) = (-1)^{dim X} times the integral of c_*(Eu_{X \ (Q \cup H)}) for every irreducible X in P^{n-1}. It needs neither smoothness no... | The proof is Morse-theoretic. Prop. 3.1: for X irreducible, X not in Q, and general beta, the ED degree of the cone C(X) equals the number of degeneration points on X_reg of the 1-form d log h_beta, where h_beta = (sum beta_i x_i)^2/(sum x_i^2). Two facts give this: no critical point of the distance function lies on H_... | null | null | [
"HAS_INFERRED_STEPS"
] | Same resolution as 1708.00024#2. The same irreducibility caveat applies: a non-equidimensional reducible X (a line plus a point in P^2) would break the formula, but 'subvariety' is irreducible in the source's conventions. | [{"id_or_ref": "arXiv:1901.05550", "title": "Euclidean distance degree of projective varieties (Maxim, Rodriguez, Wang)", "verified": true}, {"id_or_ref": "arXiv:1708.00024", "title": "The Euclidean distance degree of smooth complex projective varieties (Aluffi, Harris) - source", "verified": true}] | 5 |
1708.01305#1 | 1708.01305 | https://arxiv.org/abs/1708.01305 | yesno | For every integer t≥4 and integers t=n₁=n₂=n₃≤n₄≤⋯≤nₜ, let G=∏ᵢ₌₁ᵗ K_{nᵢ}, with vertices given by t-tuples and two vertices adjacent exactly when they differ in every coordinate. A set D⊆V(G) is dominating if every vertex is in D or adjacent to a vertex of D, and γ(G) is the minimum size of such a set. Is γ(G)=t+2 for ... | Yes means γ(G)=t+2 for every tuple in the stated range. | null | SOLVED_EXACT | true | No | high | https://arxiv.org/abs/1908.02445 | H. Vemuri, Domination in direct products of complete graphs, Discrete Appl. Math. (2020), doi:10.1016/j.dam.2020.06.015 (arXiv:1908.02445) | Theorem 3.4 (and Theorem 1.5(3)) | Theorem 3.4. Let G = prod_{i=1}^t K_{n_i}, where 2 <= n_1 <= n_2 <= ... <= n_t, t >= 4, and n_1 = n_2 = n_3 = t. Then gamma(G) = t + 2 if and only if n_4 >= t + 2. | Here n_2 = t >= 4 >= 3, so the hypotheses hold. By Theorem 3.4, if n_4 is t or t+1 (for example n_4 = ... = n_t = t), then gamma(G) > t+2. So gamma(G) = t+2 does not hold for every choice of n_4, ..., n_t, and the answer is No. The exact characterization is gamma(G) = t+2 if and only if n_4 >= t+2. | If n_4 >= t+2: the Defant-Iyer bound gives gamma >= t+2. An explicit set of t+2 vertices, d_0 = (0,...,0), d_1 = (0,1,1,1,...,1), d_2 = (1,0,1,2,...,2), d_3 = (1,1,0,3,...,3), d_{j} = (j-2, j-2, j-2, j, ..., j) for 4 <= j <= t+1, is shown to be (totally) dominating by a case analysis.
If n_4 <= t+1: suppose D is domin... | null | null | [] | The source question asked for the characterization. The reformatted yes/no asks whether gamma = t+2 for all n_4..n_t, which is false because n_4 in {t, t+1} gives gamma > t+2. Vemuri's full classification (Thm 1.5) combines Defant-Iyer Thm 2.8, Prop 3.2 and Thm 3.4. | [{"id_or_ref": "arXiv:1908.02445", "title": "Domination in direct products of complete graphs (Vemuri)", "verified": true}, {"id_or_ref": "arXiv:1708.01305", "title": "Domination and Upper Domination of Direct Product Graphs (Defant, Iyer) - source", "verified": true}] | 5 |
1708.01305#5 | 1708.01305 | https://arxiv.org/abs/1708.01305 | numerical | What is the smallest positive integer J for which there is no integer n > 1 with omega(n) >= J and gamma(X_{Z/nZ}) <= g(n)-2? Here omega(n) is the number of distinct prime divisors of n; X_{Z/nZ} has the residue classes modulo n as vertices, with x and y adjacent exactly when x-y is relatively prime to n; gamma(X_{Z/nZ... | The answer is the smallest failing positive integer J, or -1 if the stated inequality holds for a suitable n for every positive integer J. | -1 | SOLVED_EXACT | true | -1 | high | https://arxiv.org/abs/1809.04769 | A. Burcroff, Domination parameters of the unitary Cayley graph of Z/nZ, arXiv:1809.04769 (2018) [first proof]; strengthened by H. Vemuri, Domination in direct products of complete graphs, Discrete Appl. Math. (2020), doi:10.1016/j.dam.2020.06.015 (arXiv:1908.02445) | Burcroff Theorem 3.1; Vemuri Theorem 1.10 (via Theorem 1.9 and Lemma 4.4) | Burcroff, Theorem 3.1: 'The set omega(M_{c,2}) is unbounded', where M_{c,j} is the set of positive integers n with g(n) - gamma_c(X_{Z/nZ}) >= j and gamma_c >= gamma_t >= gamma. (Construction: q = 1 mod 3 prime, k = (2q-2)/3, primes 2q+10 < p_1 < ... < p_k, n = 6q p_1...p_k.) Vemuri, Theorem 1.10: 'There exist n with a... | For any graph, gamma <= gamma_t <= gamma_c. Burcroff's family has omega(n) = k + 3 = (2q-2)/3 + 3, which is unbounded as q ranges over primes = 1 mod 3. Each member satisfies gamma(X_n) <= gamma_c(X_n) <= g(n) - 2. Hence for every J there is n > 1 with omega(n) >= J and gamma(X_n) <= g(n) - 2, so no failing J exists an... | Burcroff: take n = 6q p_1...p_k. Using the CRT, choose residues so that g(n) >= 2q + 8 (a run of 2q+7 consecutive integers not coprime to n). Then build a dominating cycle of length at most g(n) - 2 from a block of consecutive integers plus 2 extra vertices. These extra vertices cover the residues left over mod 6 and m... | null | null | [] | The candidate (Vemuri) is correct, but the first proof of the gap >= 2 with unbounded omega(n) is Burcroff 2018 (arXiv:1809.04769, Thm 3.1), which Vemuri cites as Theorem 1.7. Both bound gamma_t (Burcroff even gamma_c), and gamma <= gamma_t. | [{"id_or_ref": "arXiv:1809.04769", "title": "Domination Parameters of the Unitary Cayley Graph of Z/nZ (Burcroff)", "verified": true}, {"id_or_ref": "arXiv:1908.02445", "title": "Domination in direct products of complete graphs (Vemuri)", "verified": true}] | 5 |
1708.01781#1 | 1708.01781 | https://arxiv.org/abs/1708.01781 | numerical | What is the smallest integer k≥5 for which the following assertion fails: every connected graph G with chromatic number χ(G)=k has at most k!(k−1)^{|V(G)|−k} proper k-colourings, with equality if and only if G is obtained from K_k by adding a tree rooted at each vertex of K_k? Answer −1 if no such k exists. | Return the smallest such k, or −1 if the assertion holds for every integer k≥5. | -1 | SOLVED_EXACT | true | -1 | high | https://arxiv.org/abs/1712.06067 | Jacob Fox, Xiaoyu He, Freddie Manners, 'A proof of Tomescu's graph coloring conjecture', J. Combin. Theory Ser. B 136 (2019) 204–221, doi:10.1016/j.jctb.2018.10.005, arXiv:1712.06067 | Theorem 1 | Theorem 1. If G is a connected graph on n vertices with χ(G) = k ≥ 4, then P_G(k) ≤ k!(k − 1)^{n−k}, with equality if and only if the 2-core of G is a k-clique. | P_G(k) is the number of proper k-colourings. The paper notes that 'the 2-core of G is a k-clique' is equivalent to G being formed by attaching a tree to each vertex of K_k. Theorem 1 covers all k ≥ 4, in particular every k ≥ 5, with exactly the bound and equality characterization in the question. So the assertion never... | Take a minimal counterexample (fewest vertices, then edge-minimal). Using Brooks' theorem and structural lemmas, one shows it is essentially k-critical with controlled degrees, and small cases |V| ≤ 2k−2 are checked directly as base cases. For large graphs, the 'Overprediction Lemma' (an entropy / cross-entropy argumen... | null | null | [] | Refereed (JCTB 2019). The source is Knox–Mohar Part I (k=4), which announces the k=5 case (Part II). The first proof for general k ≥ 5 (k ≥ 6 being new) is Fox–He–Manners. The equality case in the question ('adding a tree rooted at each vertex of K_k') matches 'the 2-core is a k-clique'. No mistranscription. | [{"id_or_ref": "arXiv:1712.06067 / doi:10.1016/j.jctb.2018.10.005", "title": "A proof of Tomescu's graph coloring conjecture (Fox, He, Manners)", "verified": true}, {"id_or_ref": "arXiv:1708.01781", "title": "Maximum number of colourings. I. 4-chromatic graphs (Knox, Mohar) - source", "verified": true}, {"id_or_ref": "... | 6 |
1708.01916#2 | 1708.01916 | https://arxiv.org/abs/1708.01916 | numerical | For each integer n >= 1, let G range over all connected graphs on n vertices. Let a_k(G) be the number of connected induced subgraphs of G with k vertices, define Phi_G(x) = sum_{k=1}^n a_k(G)x^k, and set M_G = Phi_G'(1)/Phi_G(1). What is the smallest n for which some such G satisfies M_G < M_{P_n}, where P_n is the pa... | Give the smallest counterexample order n, or -1 if the inequality holds for every connected graph of every order n >= 1. | -1 | SOLVED_EXACT | true | -1 | high | https://arxiv.org/abs/2103.15174 | Andrew Vince, 'A lower bound on the average size of a connected vertex set of a graph', J. Combin. Theory Ser. B 152 (2022) 153–170, doi:10.1016/j.jctb.2021.09.008, arXiv:2103.15174 (28 Mar 2021); independently John Haslegrave, 'The path minimises the average size of a connected induced subgraph', Discrete Math. 345 (2... | Vince, Theorem 1.1; Haslegrave, Theorem 1 | Vince Theorem 1.1: If G is a connected graph of order n, then A(G) ≥ (n+2)/3, with equality if and only if G is a path. In particular, D(G) > 1/3 for all connected graphs G. (Haslegrave Theorem 1: Let G be a connected graph of order n. Then A(G) ≥ (n + 2)/3, with equality if and only if G is a path.) | A(G) = (sum over nonempty connected vertex sets U of |U|)/(number of such sets) = Σ k a_k(G) / Σ a_k(G) = Φ_G'(1)/Φ_G(1) = M_G. For the path, A(P_n) = (n+2)/3 (the equality case; INFERRED check: P_n has n−k+1 connected sets of size k, giving (n+2)/3). Hence M_G ≥ M_{P_n} for every connected G of every order n (n = 1 tr... | Haslegrave's short proof inducts on n. Pick a non-cut vertex v such that the number N(G;v) of connected sets containing v is at least 2/(n+1) of all connected sets (strictly more unless G is a path; Lemma 3). The local mean satisfies A(G;v) ≥ (n+1)/2 (Lemma 2, the h=1 case of Vince's Thm 3.1 A(G,H) ≥ (n+h)/2). By induc... | null | null | [
"HAS_INFERRED_STEPS"
] | Both candidates are correct and refereed. Vince appeared on arXiv first (28 Mar 2021) and Haslegrave two days later; Haslegrave acknowledges Vince's independent proof. The source conjecture (end of 1708.01916) is exactly that P_n minimizes the mean among all connected graphs of order n. No mistranscription. | [{"id_or_ref": "arXiv:2103.15174 / doi:10.1016/j.jctb.2021.09.008", "title": "A Lower Bound on the Average Size of a Connected Vertex Set of a Graph (Vince)", "verified": true}, {"id_or_ref": "arXiv:2103.16491 / doi:10.1016/j.disc.2022.112799", "title": "The path minimises the average size of a connected induced subgra... | 6 |
1708.02317#0 | 1708.02317 | https://arxiv.org/abs/1708.02317 | yesno | For every α∈(0,1), let N_α(n) be the maximum number of equiangular lines in R^n with common angle arccos α, and set λ=(1−α)/(2α). Let k(λ) be the smallest number of vertices of a graph whose adjacency-matrix spectral radius is λ, with k(λ)=∞ if no such graph exists. As n→∞ through positive integers, is N_α(n)=[k(λ)/(k(... | Yes means both stated asymptotic claims hold for every α∈(0,1), with an O(1) constant allowed to depend on α. | null | SOLVED_EXACT | true | No | high | https://arxiv.org/abs/2302.12230 | Carl Schildkraut, Equiangular lines and large multiplicity of fixed second eigenvalue, arXiv:2302.12230 (2023) | Theorem 1.4 (with Corollary 6.3 showing k(λ)=∞ for the angles used) | Theorem 1.4. For infinitely many α ∈ (0, 1), the function Nα(n) satisfies n + Ω(log log n) ≤ Nα(n) ≤ n + O(n/ log log n). | The question is Jiang–Polyanskii's Conjecture B (1708.02317). Its k(λ)<∞ half is TRUE: Jiang–Tidor–Yao–Zhang–Zhao (arXiv:1907.12466, Annals 2021, Thm 1.2) prove Nα(n)=⌊k(n−1)/(k−1)⌋ for all large n, which is k n/(k−1)+O(1). Its k(λ)=∞ half is FALSE: in the proof of Theorem 1.4 Schildkraut takes α=1/(4√(u²+1)−1) with in... | Schildkraut constructs bounded-degree d-regular graphs whose second eigenvalue is exactly λ with multiplicity Ω(log log n) (Theorem 1.5/5.8). Starting from a k-partite graph whose parts are joined by regular bipartite graphs, he alternates Marcus–Spielman–Srivastava Ramanujan 2-lifts (which add no large eigenvalues and... | null | null | [] | Candidate's why_direct is slightly off: Schildkraut's angles all have k(λ)=∞; the k<∞ half of Conjecture B is actually proved (JTYZZ). The 'No' comes solely from the disproof of the k=∞ half (Schildkraut explicitly states he disproves Conjecture 1.3 = [JP20, Conjecture B] k=∞ part). | [{"id_or_ref": "arXiv:2302.12230", "title": "Equiangular lines and large multiplicity of fixed second eigenvalue (Schildkraut)", "verified": true}, {"id_or_ref": "arXiv:1907.12466 / doi:10.4007/annals.2021.194.3.3", "title": "Equiangular lines with a fixed angle (Jiang, Tidor, Yao, Zhang, Zhao)", "verified": true}, {"i... | 5 |
1708.02370#2 | 1708.02370 | https://arxiv.org/abs/1708.02370 | numerical | What is the smallest positive integer s for which there exists a positive integer t with s ≤ t and t ≥ max{s,3} such that the clustered chromatic number of M_{K_{s,t}} is not s+1? Here K_{s,t} is the complete bipartite graph with parts of sizes s and t; M_{K_{s,t}} is the class of finite simple graphs containing no K_{... | Give the smallest such s, or -1 if the equality holds for every permitted s and t. | -1 | SOLVED_EXACT | true | -1 | high | https://arxiv.org/abs/2306.06224 | Vida Dujmović, Louis Esperet, Pat Morin, David R. Wood, Proof of the Clustered Hadwiger Conjecture, arXiv:2306.06224; FOCS 2023 (doi:10.1109/FOCS57990.2023.00116) | Theorem 2 (with the van den Heuvel–Wood lower bound, Proposition 9) | Theorem 2. There is a function f such that every Ks,t-minor-free graph is (s + 1)-colourable with clustering at most f(s, t). [Following text:] Combined with the above-mentioned lower bound, Theorem 2 shows that the clustered chromatic number of the class of Ks,t-minor-free graphs equals s+1 for t ⩾ max{s, 3}. This res... | Theorem 2 gives upper bound s+1 for all 1 ≤ s ≤ t; van den Heuvel–Wood's standard-example lower bound gives s+1 for t ≥ max{s,3}. So equality holds for every permitted (s,t); no exceptional s exists; answer −1. | Both Theorem 1 (Clustered Hadwiger) and Theorem 2 are corollaries of a single theorem (Theorem 4) about J_{s,t}-minor-free graphs. The proof uses the Graph Minor Structure Theorem for k-apex-minor-free pieces, a layered-partition/'curtain' decomposition of the near-embedded parts, and bounded-treewidth lemmas showing t... | null | null | [] | Verified on full text (v2, Sep 2026). Paper explicitly states equality s+1 for t ≥ max{s,3}, exactly the question's parameter range. | [{"id_or_ref": "arXiv:2306.06224", "title": "Proof of the Clustered Hadwiger Conjecture (Dujmović, Esperet, Morin, Wood)", "verified": true}, {"id_or_ref": "arXiv:1708.02370", "title": "Clustered Colouring in Minor-Closed Classes (Norin, Scott, Seymour, Wood) - source", "verified": true}] | 5 |
1708.02381#0 | 1708.02381 | https://arxiv.org/abs/1708.02381 | numerical | Let τ lie in the upper half-plane, q=exp(2πiτ), and η_m=q^(m/24)∏_{k≥1}(1−q^(mk)). Define f by f²=16(η_1η_4²/η_2³)^8 and set R(f)=2(2f/(1+f²))²−1. Define A(n) for every integer n≥0 by the Fourier expansion φ(τ)=(1/2)(η_1η_2)^4R(f)=−∑_{n≥0}(n+1/2)A(n)q^(n+1/2). The function φ is the weight-4 modular form considered here... | Answer −1 if A(n)∈Z for every integer n≥0; otherwise answer the smallest integer n≥0 for which A(n)∉Z. | -1 | SOLVED_EXACT | true | -1 | high | https://arxiv.org/abs/1808.04157 | Yingkun Li, Michael Neururer, A magnetic modular form, arXiv:1808.04157; Int. J. Number Theory (doi:10.1142/S1793042119500489) | Conjecture 1.1 proved via Theorems 1.2 and 1.4 (Proposition 3.5) | Conjecture 1.1 (Conjecture 1 of [4]). The meromorphic modular form φ(z) satisfies (1.0.1) [n | a(n) for all n ∈ N]. ... 'From the definition of the map Φ, it is clear that Theorems 1.2 and 1.4 together imply Conjecture 1.1, i.e., n|a(n).' Theorem 1.4: the free Z-module M!_{5/2,ϱ} has a canonical basis {G_d} ... Suppose... | Li–Neururer's φ(z) = (η(2z)η(4z))^4 (1−96ψ(2z)+256ψ(2z)²)/(1+16ψ(2z))², ψ=η1^8η4^16/η2^24, = Σ_{n odd} a(n) q_z^n. INFERRED normalization: with τ=2z, 16ψ(τ)=f², and R(f)=−(1−6f²+f⁴)/(1+f²)², so LN φ(z) = −(η1η2)^4 R(f) = −2·φ_BZ(τ) = Σ (2n+1)A(n) q_z^{2n+1}; hence a(2n+1)=(2n+1)A(n) (check: a(3)=−132=3·(−44), a(5)=5630... | Realize −4φ as a Borcherds regularized theta lift Φ(z,G1) of a weakly holomorphic weight-5/2 vector-valued form G1 for a 3-dimensional subrepresentation ϱ of a Weil representation (Theorem 1.2, matching polar parts at the double poles z± and using that M_{4,χ}(Γ0+(8)) = 0). The lift's coefficients are Σ_{r|n} r·c(G,(n/... | null | null | [
"HAS_INFERRED_STEPS"
] | The factor −4 relating Φ(z,G1) and φ is harmless since n is odd. Refereed journal publication. | [{"id_or_ref": "arXiv:1808.04157 / doi:10.1142/S1793042119500489", "title": "A magnetic modular form (Li, Neururer)", "verified": true}, {"id_or_ref": "arXiv:1708.02381 / doi:10.1017/S1446788718000186", "title": "A magnetic double integral (Broadhurst, Zudilin) - source, Conjecture 1", "verified": true}] | 5 |
1708.03267#0 | 1708.03267 | https://arxiv.org/abs/1708.03267 | numerical | For every integer n≥1, let Irr(S_n) be the set of irreducible characters of S_n. Let μ range over the partitions of n, and let χ(μ) be the value of χ∈Irr(S_n) on the conjugacy class with cycle lengths μ. When (χ,μ) is chosen uniformly from Irr(S_n)×{partitions of n}, what is liminf_{n→∞} Prob(χ(μ) is even)? | The answer is a single number; it equals 1 exactly when the probability in the original question tends to 1. | 1 | SOLVED_EXACT | true | 1 | high | https://arxiv.org/abs/2007.06652 | Sarah Peluse, On even entries in the character table of the symmetric group, arXiv:2007.06652 (2020); also implied by arXiv:2010.12410 and arXiv:2301.02203 | Theorem 1.1 of arXiv:2007.06652 | Theorem 1.1. The density of odd entries in the character table of Sn goes to 0 as n → ∞. | Uniform (χ,μ) ∈ Irr(S_n)×{partitions of n} is a uniform entry of the p(n)×p(n) character table. Density of odd entries → 0 means Prob(χ(μ) even) → 1, so the limit, hence the liminf, equals 1. | By Murnaghan–Nakayama, χ^λ_μ = 0 whenever μ has a part t and λ is a t-core, and most λ are t-cores when t is substantially larger than the typical largest part (√6/(2π))√n log n. Modulo 2, replacing two equal parts m of μ by one part 2m does not change χ^λ_μ mod 2. For almost every μ, repeatedly merging pairs of equal ... | null | null | [] | Candidate implies the result (k=2 in Corollary 1.2) but the original resolution of Miller's parity conjecture is Peluse 2007.06652, which explicitly says 'This resolves a conjecture of Miller.' | [{"id_or_ref": "arXiv:2007.06652", "title": "On even entries in the character table of the symmetric group (Peluse)", "verified": true}, {"id_or_ref": "arXiv:2301.02203 / doi:10.2140/ant.2025.19.365", "title": "Divisibility of character values of the symmetric group by prime powers (Peluse, Soundararajan)", "verified":... | 5 |
1708.03267#1 | 1708.03267 | https://arxiv.org/abs/1708.03267 | numerical | For each fixed prime p and each n, choose uniformly an entry χ(μ) of the ordinary irreducible-character table of S_n, where χ ranges over its irreducible characters and μ over the partitions of n indexing its conjugacy classes. What is the smallest prime p for which Prob(p divides χ(μ)) does not tend to 1 as n tends to... | Give the smallest such prime, or -1 if there is none. | -1 | SOLVED_EXACT | true | -1 | high | https://arxiv.org/abs/2010.12410 | Sarah Peluse, Kannan Soundararajan, Almost all entries in the character table of the symmetric group are multiples of any given prime, arXiv:2010.12410 (J. reine angew. Math.); strengthened to prime powers in arXiv:2301.02203 (Algebra & Number Theory 19 (2025)) | Theorem 1 of arXiv:2010.12410 (also Theorem 1.1 / Corollary 1.2 of arXiv:2301.02203) | Theorem 1. Let p(N) denote the number of unrestricted partitions of N, so that p(N)² denotes the number entries in the character table for S_N. Let N be large, and let q be a prime with q ≤ (log N)/(log log N)². The number of entries in the character table of S_N that are not divisible by q is at most O(p(N)² N^{−1/(12... | For fixed prime p, once N is large p ≤ (log N)/(log log N)², so the proportion of entries not divisible by p is O(N^{−1/(12p)}) → 0; uniform choice of (χ,μ) is uniform over the p(N)² table entries. Hence Prob(p | χ(μ)) → 1 for every prime p; no exceptional prime; answer −1. | Two facts: (i) if μ has a part substantially larger than the typical largest part (√(6N)/(2π))·log N, then χ^λ_μ = 0 for almost all λ, since most λ are t-cores for such t (Murnaghan–Nakayama); (ii) merging p equal parts m of μ into one part pm gives ν with χ^λ_μ ≡ χ^λ_ν mod p. For almost every μ, iterating the merging ... | null | null | [] | Candidate 2301.02203 is valid (Corollary 1.2: divisibility by any fixed k) but the first proof for all primes is 2010.12410; Peluse 2007.06652 had p ≤ 13. | [{"id_or_ref": "arXiv:2010.12410", "title": "Almost all entries in the character table of the symmetric group are multiples of any given prime (Peluse, Soundararajan)", "verified": true}, {"id_or_ref": "arXiv:2301.02203 / doi:10.2140/ant.2025.19.365", "title": "Divisibility of character values of the symmetric group by... | 5 |
1708.03982#2 | 1708.03982 | https://arxiv.org/abs/1708.03982 | numerical | Let n range over integers n>=1. For each n, let W be a smooth, uniformly convex body containing the origin in R^{n+1}, and let Omega be an open bounded convex body in R^{n+1}. Define d_W(x,y)=inf{r>0:y in x+rW}, d_W(Omega,y)=inf{d_W(x,y):x in Omega}, and let p_W(Omega,y) be the unique point of the boundary of Omega att... | Give the smallest such integer n, or -1 if no such n exists. | -1 | SOLVED_EXACT | true | -1 | high | https://doi.org/10.6094/UNIFR/167349 | Daniel Hug, 'Measures, curvatures and currents in convex geometry', Habilitationsschrift, Albert-Ludwigs-Universität Freiburg, December 1999, doi:10.6094/UNIFR/167349 (predates the 2017 conjecture); see also D. Hug & M. Santilli, 'Curvature measures and soap bubbles beyond convexity', arXiv:2204.04909, Theorem B / Theo... | Hug 1999, Theorem 2.43 (Section 2.7 'A characterization of gauge bodies') | Theorem 2.43 Let K ∈ K^d_o, B ∈ K̃^d [standing assumption of Secs 2.6–2.7: B a convex body of class C^2_+ containing the origin as an interior point; not necessarily symmetric], and assume that there are constants λ_0, …, λ_{d−2} ≥ 0 such that C_{d−1}(K, B; ·) = Σ_{j=0}^{d−2} λ_j C_j(K, B; ·). Then K is homothetic to B... | Hug's relative distance d(K,B,x) = min{r ≥ 0 : x ∈ K + rB} equals Andrews–Wei's d_W(Ω,y) with B = W, and the Minkowski curvature measures C_j(K,B;·), defined from the relative local Steiner formula (Thm 2.7), agree with the AW measures C_j^W(Ω,·) up to positive normalizing constants depending only on j and d = n+1 (INF... | Hug develops relative (Minkowski) curvature theory for arbitrary convex bodies K with a C^2_+ gauge body B: generalized relative principal curvatures k_i^B on the normal bundle, a Lebesgue decomposition of C_r(K,B;·) (Thm 2.44), a relative Sangwine-Yager volume formula, and a relative Minkowski integral formula (Thm 2.... | null | null | [
"HAS_INFERRED_STEPS"
] | The question is exactly Andrews–Wei Conjecture 8.2 (checked in the source full text). The candidate arXiv:2108.01476 (Santilli) does state this result for arbitrary convex bodies. However, its arXiv comment says it 'is not going to be published' because the conjecture was already solved in Hug's 1999 habilitation, Thm ... | [{"id_or_ref": "doi:10.6094/UNIFR/167349", "title": "Measures, curvatures and currents in convex geometry (Hug, Habilitationsschrift 1999), Thm 2.43", "verified": true}, {"id_or_ref": "arXiv:2204.04909", "title": "Curvature measures and soap bubbles beyond convexity (Hug, Santilli)", "verified": true}, {"id_or_ref": "a... | 6 |
1708.05298#0 | 1708.05298 | https://arxiv.org/abs/1708.05298 | numerical | Let G=(V,E) be a finite undirected simple Laman graph, meaning that |E|=2|V|-3 and every subgraph H=(V_H,E_H) satisfies |E_H|<=2|V_H|-3. For edges e,e' in E, write e~e' if there is a sequence of edges from e to e' in which each two consecutive edges lie together in a triangle subgraph of G; call G triangle-connected if... | The answer is the smallest number of vertices in such a graph, or -1 if none exists. | -1 | SOLVED_EXACT | true | -1 | high | https://arxiv.org/abs/2412.16018 | K. Clinch, D. Garamvölgyi, J. Haslegrave, T. Huynh, J. Legerský, A. Nixon, 'Stable cuts, NAC-colourings and flexible realisations of graphs', arXiv:2412.16018, J. Graph Theory (2025), doi:10.1002/jgt.70064 | Theorem 1.2 (equivalently Theorem 3.4) | Theorem 1.2. A minimally 2-rigid graph on at least two vertices has a NAC-colouring if and only if it is not a 2-tree. | Source Conjecture 5.1 of 1708.05298v2 (symbol rendered as '4-connected' in pdftotext is the triangle-connected symbol): every Laman graph that is not triangle-connected has a NAC-coloring. Laman graphs = minimally 2-rigid graphs (Pollaczek-Geiringer/Laman); every Laman graph has |V|>=2 since |E|=2|V|-3>=0. INFERRED (ea... | By Grasegger-Legerský-Schicho (Lemma 3.1 in the paper, [13, Thm 4.4]), any connected graph with a stable cut (an independent vertex set whose removal disconnects the graph) has a NAC-colouring. Le and Pfender (with a gap fixed by Rauch and Rautenbach) characterised graphs with exactly 2|V|-3 edges and no stable cut: th... | null | null | [
"HAS_INFERRED_STEPS"
] | Verified full text (v2, 'to appear in J Graph Theory'). The paper phrases the GLS conjecture as 'every minimally rigid graph that is not a 2-tree has a NAC-colouring'; the source phrases it via triangle-connectivity. The equivalence (triangle-connected Laman <=> 2-tree) is a short inferred step. Proof relies on Le-Pfen... | [{"id_or_ref": "arXiv:2412.16018", "title": "Stable cuts, NAC-colourings and flexible realisations of graphs", "verified": true}, {"id_or_ref": "arXiv:1708.05298", "title": "Graphs with Flexible Labelings (source; Conjecture 5.1)", "verified": true}] | 6 |
1708.07464#5 | 1708.07464 | https://arxiv.org/abs/1708.07464 | numerical | For every depth l >= 0, integers s_1,...,s_l >= 1, and integers r_1,...,r_l >= 0, define the bi-bracket [s_1,...,s_l; r_1,...,r_l] = sum over u_1>...>u_l>0 and v_1,...,v_l>0 of (product from i=1 to l of u_i^{r_i}/r_i! times v_i^{s_i-1}/(s_i-1)!) q^{u_1v_1+...+u_lv_l} in Q[[q]]. What is the smallest depth l >= 0 for whi... | The answer is the smallest depth of a counterexample, or -1 if every such bi-bracket has the stated representation. | -1 | SOLVED_EXACT | true | -1 | high | https://arxiv.org/abs/2605.28584 | Hirose, Maesaka, Watanabe, 'A unified proof of conjectures on the spaces of multiple q-zeta values', arXiv:2605.28584 (2026) | Theorem 1.1 (implies Z_q^o = Z_q, Bachmann's conjecture = Conjecture 3.2 (B2) of the source), combined with source Theorem 2.3 | Abstract: 'We prove two conjectures on the spaces generated by multiple q-zeta values. More precisely, we show that the spaces Z_q^o and Z_{q,1}^o already generate the larger spaces Z_q and Z_{q,1}, respectively.' Theorem 1.1: For r >= 0 and k_1,...,k_r, l_1,...,l_r in Z_{>=1}, we have zeta_q^dagger(1bar,...,1bar [l_1-... | The question is Conjecture 3.2 (B1) of Bachmann-Kuehn (unfiltered form; not the filtered (B1*)). Source Theorem 2.3(i),(ii): the bi-brackets span Z_q and the brackets [s_1..s_l]=[s;0..0] (s_j>=1) span Z_q^o (source's Z_q, Z_q^o use the same definition as HMW's, with index order reversed). HMW Theorem 1.1 gives Z_q^o = ... | HMW introduce a 'dagger' model zeta_q^dagger(k) of q-MZVs with formal entries 1bar, and show via the Schlesinger-Zudilin model (Lemmas 2.2-2.4) that Z_q is spanned by dagger-values with arbitrary entries while Z_q^o is spanned by those with no 1bar entries. They construct explicit words E_q(l_1,k_1,...,l_r,k_r) with in... | null | null | [
"RECENT_PREPRINT_2026",
"HAS_INFERRED_STEPS"
] | Source Conjecture 3.2 (B1); positive bi-brackets Z_q^+ = span{[s;r] : s_j > r_j}. HMW prove (B2) Z_q^o = Z_q, which implies (B1) since brackets are positive bi-brackets. The filtered refinement (B1*) Fil^{W,L}(Z_q^+) = Fil^{W,L}(Z_q) is NOT addressed; the question as phrased is unfiltered. HMW is a 2026 arXiv preprint ... | [{"id_or_ref": "arXiv:2605.28584", "title": "A unified proof of conjectures on the spaces of multiple q-zeta values", "verified": true}, {"id_or_ref": "arXiv:1708.07464", "title": "A dimension conjecture for q-analogues of multiple zeta values", "verified": true}] | 5 |
1708.07464#7 | 1708.07464 | https://arxiv.org/abs/1708.07464 | numerical | Let q be a formal variable. For every depth l >= 0, integers s_1,...,s_l >= 1 and r_1,...,r_l >= 0, define the bi-bracket in Q[[q]] by [s_1,...,s_l; r_1,...,r_l] = sum_{u_1>...>u_l>0, v_1,...,v_l>0} product_{j=1}^l (u_j^{r_j}/r_j!)(v_j^{s_j-1}/(s_j-1)!) q^{u_jv_j}, with the depth-zero series equal to 1. Define the brac... | Give the smallest such depth, or -1 if no such depth exists. | -1 | SOLVED_EXACT | true | -1 | high | https://arxiv.org/abs/2605.28584 | Hirose, Maesaka, Watanabe, 'A unified proof of conjectures on the spaces of multiple q-zeta values', arXiv:2605.28584 (2026) | Theorem 1.1 (implies Z_q^o = Z_q), combined with source Theorem 2.3 | Abstract: 'We prove two conjectures on the spaces generated by multiple q-zeta values. More precisely, we show that the spaces Z_q^o and Z_{q,1}^o already generate the larger spaces Z_q and Z_{q,1}, respectively.' Theorem 1.1: For r >= 0 and k_1,...,k_r, l_1,...,l_r in Z_{>=1}, we have zeta_q^dagger(1bar,...,1bar [l_1-... | This is Conjecture 3.2 (B2) of Bachmann-Kuehn ('Every bi-bracket equals a linear combination of brackets, i.e. Z_q^o = Z_q'), first stated as Conjecture 4.3 in Bachmann's 'The algebra of bi-brackets'. Source Theorem 2.3(i),(ii) identify span(bi-brackets) = Z_q and span(brackets [t_1..t_m], t_j>=1) = Z_q^o. HMW Theorem ... | HMW reduce Z_q and Z_q^o to spans of their dagger model (Z_q: entries may be the formal symbol 1bar; Z_q^o: no 1bar), via the Schlesinger-Zudilin model and binomial transformation formulas (Lemmas 2.2-2.4). They define explicit integer words E_q in Z<x,y> recursively and prove zeta^dagger({1bar}^{l_1-1},k_1,...,{1bar}^... | null | null | [
"RECENT_PREPRINT_2026"
] | Definitions match: source's Z_q, Z_q^o (sum over n_1>...>n_l, Q_1 in tQ[t]) equal HMW's (sum over n_1<...<n_r, Q_r in XQ[X]) up to index reversal. HMW is a 2026 arXiv preprint, not yet refereed; authors established. | [{"id_or_ref": "arXiv:2605.28584", "title": "A unified proof of conjectures on the spaces of multiple q-zeta values", "verified": true}, {"id_or_ref": "arXiv:1708.07464", "title": "A dimension conjecture for q-analogues of multiple zeta values", "verified": true}] | 5 |
1708.09261#7 | 1708.09261 | https://arxiv.org/abs/1708.09261 | numerical | Fix a prime p. What is the smallest integer n>0 for which there exists a Landweber exact BP_*-algebra E_* of height n such that pi_*: Stable_{E_*E}^omega -> Mod_{pi_*E_*} is faithful? Here height n means E_*/I_n is nonzero and E_*/I_k=0 for every k>n, with I_k=(p,v_1,...,v_{k-1}); Landweber exactness means extension of... | Answer -1 if no such n and E_* exist; otherwise answer the smallest such n. | -1 | SOLVED_EXACT | true | -1 | high | https://arxiv.org/abs/2609.15253 | Sihao Ma, Zhouli Xu, 'Generating hypotheses along the motivic deformation', arXiv:2609.15253 (2026) | Theorem 4.11 (with Section 4.1 remark on E_*E-comodules); also Theorem 1.2 / Theorem 4.15 for the K(h)-local version | Theorem 4.11. Let E be an even, p-local Landweber exact theory with a p-typical orientation, and assume E_*/p != 0. For every r, n >= 1, there is an m >= 1 such that the ghost f = 1 - Theta_{1+p^r} : M(m,r) -> M(m,r) satisfies p^r . id_{M(m,r)} = 0, p^{r-1} . f^j != 0 (1 <= j <= n). Thus each of f, f^2, ..., f^n has ex... | Fix n >= 1. Take E = E(n) (Johnson-Wilson), which is even, p-local, Landweber exact of height n with E_*/p != 0. Theorem 4.11 gives a nonzero ghost f between objects of D_fin(E_*E) = Thick{shifts of E_*}, which are compact in Stable_{E_*E}; 'ghost' means Hom(shifts of E_*, f) = 0, i.e. pi_*(f) = 0 with pi_*M = pi_*^st ... | For r >= 1 and u = 1 (mod p^r), multiplication [u]_F on the formal group of E over E_*/p^r is a strict automorphism, so it is classified by a map theta_u : E_*E/p^r -> E_*/p^r and induces a natural automorphism Theta_u of every E_*E/p^r-comodule (Lemma 4.2). Theta_u acts as the identity on Ext_{E_*E}(E_*, L) (Propositi... | null | null | [
"RECENT_PREPRINT_2026",
"HAS_INFERRED_STEPS"
] | Source (Barthel-Heard Remark 5.15) only vaguely poses the 'local algebraic generating hypothesis as in [Bar17]' and suspects it fails at all positive heights. The reformatted question places it in Stable_{E_*E} (the L_n-local category); Ma-Xu Theorem 4.11 covers exactly D_fin(E_*E) for even E, and Theorem 4.15 covers t... | [{"id_or_ref": "arXiv:2609.15253", "title": "Generating hypotheses along the motivic deformation", "verified": true}, {"id_or_ref": "arXiv:1708.09261", "title": "Algebraic chromatic homotopy theory for BP_*BP-comodules", "verified": true}] | 5 |
1708.09700#1 | 1708.09700 | https://arxiv.org/abs/1708.09700 | numerical | Give one explicit real number β₀>0 such that, for every simple graph G with n_G vertices and adjacency matrix A, G is walk-regular—meaning that for every integer ℓ≥1, every vertex is contained in the same number of closed walks of length ℓ, equivalently all diagonal entries of A^ℓ are equal—if and only if its walk entr... | Any one explicit positive real number satisfying the stated equivalence is an acceptable answer. | null | SOLVED_EXACT | true | beta_0 = 1 (indeed any positive algebraic number, e.g. any positive rational, works) | high | https://arxiv.org/abs/2007.08956 | Francesco Ballini, Nikita Deniskin, 'Vertex distinction with subgraph centrality: a proof of Estrada's conjecture and some generalizations', Linear Algebra Appl. (2021), doi:10.1016/j.laa.2020.11.006, arXiv:2007.08956 | Theorem 3 and Theorem 4(1) | Theorem 3. Let G be a directed graph with adjacency matrix A, and assume that A is diagonalizable. Let beta != 0 be an algebraic number. If two vertices i, j are beta-subgraph equivalent, then they are cospectral. Theorem 4 (Main Result). Let beta > 0 be an algebraic number and let G be a connected undirected graph wit... | Take beta_0 = 1 (algebraic). S^V(G,1) = log n_G iff p_i = 1/n_G for all i iff all [exp(A)]_ii are equal (G is 1-subgraph regular). By Theorem 3 (applied to all vertex pairs of the symmetric, hence diagonalizable, A) this implies all vertices are cospectral, i.e. G walk-regular; conversely walk-regular implies equal dia... | With A = QDQ^{-1} and Q algebraic, [exp(beta A)]_ii = sum_h C_h^i e^{beta mu_h} with algebraic C_h^i and distinct algebraic exponents beta mu_h. Equality of two such diagonal entries is an algebraic linear relation among distinct exponentials, which Lindemann-Weierstrass forces to be trivial: C_h^i = C_h^j for all h, g... | null | null | [
"HAS_INFERRED_STEPS"
] | Source open problem (Kloster-Kral-Sullivan) asked for an explicit beta_0; Ballini-Deniskin show every positive algebraic beta works, in particular beta_0 = 1 (Estrada's original conjecture). Only transcendental beta can fail (KKS counterexamples). | [{"id_or_ref": "arXiv:2007.08956", "title": "Vertex distinction with subgraph centrality: a proof of Estrada's conjecture and some generalizations", "verified": true}, {"id_or_ref": "arXiv:1708.09700", "title": "Walk entropy and walk-regularity", "verified": true}] | 5 |
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