{ "name": "Math and Theory", "url": "https://dvgyl.github.io/math-and-theory/", "author": "Dev D. Goyal", "sources": [ "catalogue.json", "CONTENTS.md" ], "results": [ { "id": "keller-klein", "title": "The small-coefficient upper half of the Keller–Klein conjecture", "url": "https://dvgyl.github.io/math-and-theory/papers/the-small-coefficient-upper-half-of-the-kellerklein-conjecture/", "pdf": "https://dvgyl.github.io/math-and-theory/papers/the-small-coefficient-upper-half-of-the-kellerklein-conjecture/manuscript.pdf", "discipline": "Mathematics", "area": "Probability and normal approximation", "subject": "Discrete analysis, tilings, probability and entropy", "subject_url": "https://dvgyl.github.io/math-and-theory/subjects/discrete-analysis-tilings-probability-and-entropy/", "href": "/math-and-theory/papers/the-small-coefficient-upper-half-of-the-kellerklein-conjecture/", "tone": "partial", "queries": [ "Keller–Klein conjecture", "Keller-Klein conjecture", "Keller-Klein conjecture proof", "Berry-Esseen bound Rademacher sums Phi(a) - 1/2", "Kolmogorov distance Rademacher sum normal approximation", "randomly signed sums normal approximation conjecture", "Tomaszewski conjecture Keller Klein Berry-Esseen", "Ivanisvili Keller-Klein conjecture small coefficients" ], "rank": 10, "result_type": "Theorem completing an open range and reproducible proof package", "formal_proof": "standalone", "standout": true, "year": "2026", "keywords": [ "Keller–Klein conjecture", "Keller-Klein conjecture", "Rademacher sums", "Berry–Esseen", "normal approximation", "Kolmogorov distance", "Prawitz smoothing", "randomly signed sums", "Keller–Klein conjecture on Berry–Esseen bounds for Rademacher sums", "Keller-Klein conjecture proof", "Berry-Esseen bound Rademacher sums Phi(a) - 1/2", "Kolmogorov distance Rademacher sum normal approximation", "randomly signed sums normal approximation conjecture", "Tomaszewski conjecture Keller Klein Berry-Esseen", "Ivanisvili Keller-Klein conjecture small coefficients" ], "abstract": "Let X = Σ vᵢεᵢ with independent uniform signs, real coefficients, Σ vᵢ² = 1 and |vᵢ| ≤ a, where 0 < a ≤ 7/100. For every x ≥ 0, P(|X| ≤ x) ≤ 2Φ(x) + 2Φ(a) − 2, equivalently P(X ≤ x) − Φ(x) ≤ Φ(a) − 1/2. Keller and Klein conjectured the two-sided bound |P(X ≤ x) − Φ(x)| ≤ Φ(a₁) − 1/2 with a₁ the largest coefficient; Ivanisvili's computer-assisted manuscript of 21 October 2026 proves the lower half and the full conjecture for a₁ ≥ 0.07, and leaves this range open. The proof follows the analytic route his manuscript proposes, using Prawitz smoothing, coefficient profiles and a compensated alias estimate. Together with his results it implies the full conjecture, conditionally on those unrefereed external results.", "scope": "The theorem for every finite vector of real coefficients, with the exact count of sign vectors and Mathlib's standard normal distribution function, is formalized in 62 Lean modules, including Prawitz smoothing, the characteristic-function envelopes and both analytic zones. Ivanisvili's Theorems 1.1 and 10.9, needed for the full conjecture, are external computer-assisted results and are not formalized.", "problem": "Keller–Klein conjecture on Berry–Esseen bounds for Rademacher sums", "posed_by": "Keller and Klein, Advances in Mathematics 407 (2022), 108558, Conjecture 9.3 (in their proof of Tomaszewski's conjecture)", "problem_url": "https://doi.org/10.1016/j.aim.2022.108558", "status": "Partially resolved", "claim": "For X = Σ vᵢεᵢ with independent uniform signs, Σ vᵢ² = 1 and |vᵢ| ≤ a ≤ 7/100, P(X ≤ x) − Φ(x) ≤ Φ(a) − 1/2 for every x ≥ 0: the upper half of the Keller–Klein conjecture in this small-coefficient range." }, { "id": "strong-gradient", "title": "The exact last-gradient bound for constant-step gradient descent in the strongly convex case", "url": "https://dvgyl.github.io/math-and-theory/papers/the-exact-last-gradient-bound-for-constant-step-gradient-descent-in-the-strongly-convex-case/", "pdf": "https://dvgyl.github.io/math-and-theory/papers/the-exact-last-gradient-bound-for-constant-step-gradient-descent-in-the-strongly-convex-case/manuscript.pdf", "discipline": "Mathematics", "area": "Convex optimization", "subject": "Optimization", "subject_url": "https://dvgyl.github.io/math-and-theory/subjects/optimization/", "href": "/math-and-theory/papers/the-exact-last-gradient-bound-for-constant-step-gradient-descent-in-the-strongly-convex-case/", "tone": "proved", "queries": [ "Taylor Hendrickx Glineur conjecture 3", "Taylor-Hendrickx-Glineur conjecture gradient norm", "exact worst-case gradient norm gradient descent strongly convex", "last iterate gradient norm gradient descent tight bound", "performance estimation gradient descent strongly convex exact rate", "gradient descent final gradient initial distance worst case", "gradient descent step size 2/L last iterate gradient bound" ], "rank": 20, "result_type": "Standalone theorem and reproducible proof package", "formal_proof": "standalone", "standout": true, "year": "2026", "keywords": [ "gradient descent", "strongly convex optimization", "worst-case analysis", "performance estimation", "Taylor–Hendrickx–Glineur conjecture", "last-iterate gradient norm", "Taylor–Hendrickx–Glineur Conjecture 3: last-iterate gradient norm of gradient descent", "Taylor Hendrickx Glineur conjecture 3", "Taylor-Hendrickx-Glineur conjecture gradient norm", "exact worst-case gradient norm gradient descent strongly convex", "last iterate gradient norm gradient descent tight bound", "performance estimation gradient descent strongly convex exact rate", "gradient descent final gradient initial distance worst case", "gradient descent step size 2/L last iterate gradient bound" ], "abstract": "The exact final-gradient guarantee holds on every real Hilbert space for L > 0, 0 ≤ μ ≤ L, normalized step 0 ≤ h ≤ 2 and every natural horizon. Actual one-dimensional Huber and quadratic functions attain its two branches.", "scope": "The actual-function upper bound and normalized actual-function attainment are formalized, including every horizon and closed parameter boundary. Sharpness for arbitrary L > 0 and radius R > 0 is proved by rescaling in the manuscript, without a separate Lean wrapper.", "problem": "Taylor–Hendrickx–Glineur Conjecture 3: last-iterate gradient norm of gradient descent", "posed_by": "Taylor, Hendrickx and Glineur, Mathematical Programming 161 (2017), 307–345, Conjecture 3 (Section 4.1.3)", "problem_url": "https://doi.org/10.1007/s10107-016-1009-3", "status": "Proved", "claim": "Gradient descent with step h/L, 0 ≤ h ≤ 2, on L-smooth μ-strongly convex f on any real Hilbert space satisfies ‖∇f(x_N)‖ ≤ L‖x₀ − x*‖ max{p_N(μ/L, h), |1 − h|^N} for every N, with equality attained in one dimension." }, { "id": "admm-multiplier-step", "title": "A narrow interval for the universal multiplier step of two-block ADMM", "url": "https://dvgyl.github.io/math-and-theory/papers/a-narrow-interval-for-the-universal-multiplier-step-of-two-block-admm/", "pdf": "https://dvgyl.github.io/math-and-theory/papers/a-narrow-interval-for-the-universal-multiplier-step-of-two-block-admm/manuscript.pdf", "discipline": "Mathematics", "area": "Optimization", "subject": "Optimization", "subject_url": "https://dvgyl.github.io/math-and-theory/subjects/optimization/", "href": "/math-and-theory/papers/a-narrow-interval-for-the-universal-multiplier-step-of-two-block-admm/", "tone": "partial", "queries": [ "Glowinski open question ADMM", "ADMM dual step size golden ratio", "ADMM dual step size larger than 1.618 convergence", "ADMM multiplier step tau up to 2 counterexample", "alternating direction method of multipliers dual step length", "Fortin Glowinski golden ratio step ADMM", "ADMM periodic orbit nonconvergence convex polytopes", "relaxed multiplier update ADMM convergence threshold", "ADMM dual step size scalar convex problems 0 < tau < 2", "ADMM step size threshold 1.8868" ], "rank": 30, "result_type": "Convergence theorem, periodic counterexample and reproducible proof package", "formal_proof": "standalone", "standout": true, "year": "2026", "keywords": [ "ADMM", "alternating direction method of multipliers", "dual step size", "Glowinski golden ratio", "multiplier relaxation", "convergence threshold", "Lyapunov certificate", "Glowinski's open question on the ADMM dual step size beyond the golden ratio", "Glowinski open question ADMM", "ADMM dual step size golden ratio", "ADMM dual step size larger than 1.618 convergence", "ADMM multiplier step tau up to 2 counterexample", "alternating direction method of multipliers dual step length", "Fortin Glowinski golden ratio step ADMM", "ADMM periodic orbit nonconvergence convex polytopes", "relaxed multiplier update ADMM convergence threshold", "ADMM dual step size scalar convex problems 0 < tau < 2", "ADMM step size threshold 1.8868" ], "abstract": "Two-block ADMM with relaxation only in the multiplier update converges for every constant step 0 < τ ≤ 1.886795 on finite-dimensional convex problems whose proximal subproblems attain their minima and which have a saddle point. Both primal blocks, the multiplier and the objective values converge, with no smoothness, strong convexity or bounded-iterate assumption. Fixed compact polytopes in R¹²⁸ have a nonconstant periodic ADMM orbit for every τ ≥ 1.8868840…, so the largest step with universal convergence lies in an interval of width below 0.00009, about a tenth of the version 1.0 interval. For scalar problems the universal range is exactly 0 < τ < 2. Affine-constraint versions are also proved. The earlier assertion of convergence up to τ = 2 by Gao et al. (2022) is shown to be false.", "scope": "The consensus convergence theorem for 0 < τ ≤ 377359/200000, the arbitrary-rank and full-rank affine-constraint theorems, all seven finite-memory interval certificates and the periodic orbit in R^128 for τ ≥ 1.8868840… are formalized in 301 Lean modules. The scalar theorem, individual objective-value limits, the reduction from proper closed convex functions to proximal attainment, and penalty rescaling are conventional.", "problem": "Glowinski's open question on the ADMM dual step size beyond the golden ratio", "posed_by": "Fortin and Glowinski proved convergence for 0 < τ < (1+√5)/2 (1983); Tao and Yuan, J. Optim. Theory Appl. 179 (2018), 163–196, call the general convex case Glowinski's open question", "problem_url": "https://doi.org/10.1007/s10957-018-1338-x", "status": "Partially resolved", "claim": "Two-block ADMM with multiplier step τ converges for every 0 < τ ≤ 1.886795 on finite-dimensional convex problems with attained proximal steps and a saddle point; fixed polytopes in R^128 cycle for every τ ≥ 1.8868840…; scalar problems converge exactly for 0 < τ < 2" }, { "id": "upper-ppt-hierarchy", "title": "No universal finite collapse of the upper PPT entanglement-cost hierarchy", "url": "https://dvgyl.github.io/math-and-theory/papers/no-universal-finite-collapse-of-the-upper-ppt-entanglement-cost-hierarchy/", "pdf": "https://dvgyl.github.io/math-and-theory/papers/no-universal-finite-collapse-of-the-upper-ppt-entanglement-cost-hierarchy/manuscript.pdf", "discipline": "Mathematical physics", "area": "Quantum information", "subject": "Quantum information", "subject_url": "https://dvgyl.github.io/math-and-theory/subjects/quantum-information/", "href": "/math-and-theory/papers/no-universal-finite-collapse-of-the-upper-ppt-entanglement-cost-hierarchy/", "tone": "refuted", "queries": [ "Lami Mele Regula hierarchy collapse", "Lami-Mele-Regula PPT entanglement cost hierarchy", "exact PPT entanglement cost single-letter formula", "computable entanglement cost under PPT operations hierarchy", "kappa hierarchy entanglement cost noncollapse", "PPT entanglement cost semidefinite hierarchy finite level", "multinegativity upper hierarchy open question Brinster" ], "rank": 40, "result_type": "Theorem resolving an open question and reproducible proof package", "formal_proof": "standalone", "standout": true, "year": "2026", "keywords": [ "entanglement cost", "PPT operations", "semidefinite hierarchy", "partial transpose", "exact PPT entanglement cost", "Bell-diagonal states", "Lami–Mele–Regula", "Finite collapse of the Lami–Mele–Regula upper PPT entanglement-cost hierarchy", "Lami Mele Regula hierarchy collapse", "Lami-Mele-Regula PPT entanglement cost hierarchy", "exact PPT entanglement cost single-letter formula", "computable entanglement cost under PPT operations hierarchy", "kappa hierarchy entanglement cost noncollapse", "PPT entanglement cost semidefinite hierarchy finite level", "multinegativity upper hierarchy open question Brinster" ], "abstract": "For every integer q ≥ 1 there is a bipartite density matrix ρ_q with κ_{q+1}(ρ_q) < κ_q(ρ_q) in the upper hierarchy of semidefinite programs of Lami, Mele and Regula, whose logarithms converge to the exact entanglement cost under PPT operations. The witnesses are full-rank product-Bell-diagonal states in local dimension 2^((2q+4)4^q), and the auxiliary matrices are unrestricted. Hence no fixed finite upper level is exact for all states. The proof builds arbitrarily long strictly positive reverse orbits of the absolute partial transpose by a weighted-sign argument and converts full support into a quantitative gap for every shorter chain. Brinster et al. had proved the corresponding all-depth statement for the lower χ hierarchy and left the upper question open. The theorem does not give one fixed state or dimension with infinitely many strict levels.", "scope": "The strict separation at every adjacent level, for the actual semidefinite programs with unrestricted complex auxiliaries and positive-definite trace-one states with positive Bell coefficients, and the absence of a universal finite collapse are formalized in 17 Lean modules. The operational exact-cost and tensor-multiplicativity consequences use published theorems of Lami, Mele and Regula.", "problem": "Finite collapse of the Lami–Mele–Regula upper PPT entanglement-cost hierarchy", "posed_by": "Lami, Mele and Regula, Physical Review Letters 134, 090202 (2025), Supplemental Material (the finite-collapse question accompanying Conjecture S33); left open for the upper κ hierarchy by Brinster, Kondra, Kampermann, Bruß and Wyderka (arXiv:2609.20698)", "problem_url": "https://arxiv.org/abs/2405.09613", "status": "Answered negatively", "claim": "For every q ≥ 1 a full-rank product-Bell-diagonal state in local dimension 2^((2q+4)4^q) has κ_{q+1}(ρ) < κ_q(ρ), so no fixed finite level of the upper PPT entanglement-cost hierarchy is exact for all states." }, { "id": "artin-schreier-depth", "title": "Depth-one Artin–Schreier defect towers are composita", "url": "https://dvgyl.github.io/math-and-theory/papers/depth-one-artinschreier-defect-towers-are-composita/", "pdf": "https://dvgyl.github.io/math-and-theory/papers/depth-one-artinschreier-defect-towers-are-composita/manuscript.pdf", "discipline": "Mathematics", "area": "Valuation theory", "subject": "Valuation theory", "subject_url": "https://dvgyl.github.io/math-and-theory/subjects/valuation-theory/", "href": "/math-and-theory/papers/depth-one-artinschreier-defect-towers-are-composita/", "tone": "proved", "queries": [ "Nart–Novacoski conjecture", "Nart-Novacoski conjecture", "depth of Artin-Schreier defect towers", "Artin-Schreier defect extension depth one compositum", "Mac Lane-Vaquie depth defect extensions", "Okutsu sequences depth Artin-Schreier", "Hahn series Artin-Schreier tower henselian field" ], "rank": 50, "result_type": "Theorem resolving a published conjecture and reproducible proof package", "formal_proof": "standalone", "standout": true, "year": "2026", "keywords": [ "Artin–Schreier extensions", "defect extensions", "depth of valued field extensions", "Okutsu sequences", "Mac Lane–Vaquié chains", "Hahn series", "Krasner lemma", "Nart–Novacoski conjecture", "Nart–Novacoski conjecture on depth-one Artin–Schreier defect towers", "Nart-Novacoski conjecture", "depth of Artin-Schreier defect towers", "Artin-Schreier defect extension depth one compositum", "Mac Lane-Vaquie depth defect extensions", "Okutsu sequences depth Artin-Schreier", "Hahn series Artin-Schreier tower henselian field" ], "abstract": "Let k be algebraically closed of characteristic p > 0 and K a perfect henselian subfield of the Hahn field k((y^ℚ)) that contains k(y) and whose elements have no finite accumulation points in their supports. If a nontrivial finite Artin–Schreier tower M/K has depth one, then M is a compositum of Artin–Schreier extensions of K. Nart and Novacoski proved the reverse implication and conjectured this one. For every θ in M outside K, the truncation of θ at the first finite accumulation point of its support generates a nontrivial elementary abelian subextension of K(θ)/K; the extraction adapts an Ore-relation argument of Kedlaya.", "scope": "Both implications are formalized in two Lean projects. The Lean 4.35 project states the conjecture for every finite Artin–Schreier tower over a base whose valuation ring is henselian in Mathlib's sense, with depth one defined as the least length of a complete Mac Lane–Vaquié chain over all primitive generators; it compares these chains with Okutsu sequences at length one only. The Lean 4.19 project states depth one through Okutsu sequences in the algebraic closure of K inside the Hahn field.", "problem": "Nart–Novacoski conjecture on depth-one Artin–Schreier defect towers", "posed_by": "Nart and Novacoski, 'Depth of Artin–Schreier defect towers', J. Pure Appl. Algebra 230 (2026), 108184 (arXiv:2503.18827), Conjecture 4.1", "problem_url": "https://doi.org/10.1016/j.jpaa.2026.108184", "status": "Proved", "claim": "Over a perfect henselian K ⊆ k((y^ℚ)) containing k(y) without finite accumulation points (k algebraically closed, char. p), each nontrivial finite Artin–Schreier tower of depth one is a compositum of Artin–Schreier extensions." }, { "id": "matrix-pencil-completion", "title": "Matrix-pencil rank distance as an explicit completion problem", "url": "https://dvgyl.github.io/math-and-theory/papers/matrix-pencil-rank-distance-as-an-explicit-completion-problem/", "pdf": "https://dvgyl.github.io/math-and-theory/papers/matrix-pencil-rank-distance-as-an-explicit-completion-problem/manuscript.pdf", "discipline": "Mathematics", "area": "Matrix theory", "subject": "Matrix theory", "subject_url": "https://dvgyl.github.io/math-and-theory/subjects/matrix-theory/", "href": "/math-and-theory/papers/matrix-pencil-rank-distance-as-an-explicit-completion-problem/", "tone": "proved", "queries": [ "Dodig–Stošić conjecture", "Dodig-Stosic conjecture", "bounded rank perturbations matrix pencils completion problem", "matrix pencil completion problem", "is the bounded rank perturbations problem for matrix pencils just a completion problem", "rank perturbation matrix pencils Kronecker canonical form", "strict equivalence matrix pencils normal rank distance" ], "rank": 60, "result_type": "Theorem resolving a published conjecture and reproducible proof package", "formal_proof": "standalone", "standout": true, "year": "2026", "keywords": [ "matrix pencils", "bounded rank perturbations", "matrix pencil completion", "strict equivalence", "normal rank", "Kronecker canonical form", "block minimal bases", "Dodig–Stošić conjecture", "Dodig–Stošić conjecture on bounded rank perturbations of matrix pencils", "Dodig-Stosic conjecture", "bounded rank perturbations matrix pencils completion problem", "matrix pencil completion problem", "is the bounded rank perturbations problem for matrix pencils just a completion problem", "rank perturbation matrix pencils Kronecker canonical form", "strict equivalence matrix pencils normal rank distance" ], "abstract": "Let A and B be m × n matrix pencils over an infinite field and δ(A, B) the least normal rank of A′ − B′ over pencils A′, B′ strictly equivalent to A, B. For distinct c, d and every x ≥ 0, δ(A, B) ≤ x exactly when an explicit (2m+n) × (2n+m) dilation K_{c,d}(A) is obtained from K_{c,d}(B) ⊕ (t − d)I_x by one constant full-row-rank and one constant full-column-rank matrix. No regularity, rank or spectral condition is imposed, and the anchors may be eigenvalues. Over an algebraically closed field and at anchors outside the spectra, a known block minimal bases theorem turns this into Conjecture 1 of Dodig and Stošić: the bounded rank perturbation problem for matrix pencils is a completion problem.", "scope": "The explicit equivalence over every infinite field, for arbitrary rectangular pencils, every border size and distinct anchors including eigenvalues, and the common-summand cancellation are formalized in two Lean projects. The identification with the barred pencils of the conjecture uses a published linearization theorem and is not formalized.", "problem": "Dodig–Stošić conjecture on bounded rank perturbations of matrix pencils", "posed_by": "Dodig and Stošić, 'Is the bounded rank perturbations problem for matrix pencils just a completion problem?', Electron. J. Linear Algebra 41 (2025), 491–510, Conjecture 1", "problem_url": "https://doi.org/10.13001/ela.2025.9593", "status": "Proved", "claim": "Over any infinite field, pencils A, B have δ(A,B) ≤ x exactly when K_{c,d}(A) is a constant full-rank compression of K_{c,d}(B) ⊕ (t − d)I_x; a cited linearization theorem then gives Conjecture 1 over algebraically closed fields." }, { "id": "nonnegative-rank-kronecker", "title": "Counterexamples to the Kronecker-square lower bound for nonnegative rank", "url": "https://dvgyl.github.io/math-and-theory/papers/counterexamples-to-the-kronecker-square-lower-bound-for-nonnegative-rank/", "pdf": "https://dvgyl.github.io/math-and-theory/papers/counterexamples-to-the-kronecker-square-lower-bound-for-nonnegative-rank/manuscript.pdf", "discipline": "Mathematics", "area": "Matrix theory", "subject": "Matrix theory", "subject_url": "https://dvgyl.github.io/math-and-theory/subjects/matrix-theory/", "href": "/math-and-theory/papers/counterexamples-to-the-kronecker-square-lower-bound-for-nonnegative-rank/", "tone": "refuted", "queries": [ "Vandaele Gillis Glineur Tuyttens conjecture", "Vandaele-Gillis-Glineur-Tuyttens conjecture counterexample", "nonnegative rank Kronecker product conjecture", "nonnegative rank tensor product lower bound", "rank+(M kron M) >= rank+(M) rank(M)", "exact nonnegative matrix factorization Kronecker product", "nonnegative rank multiplicativity Kronecker counterexample" ], "rank": 70, "result_type": "Counterexample to a published conjecture and reproducible proof package", "formal_proof": "standalone", "standout": true, "year": "2026", "keywords": [ "nonnegative rank", "nonnegative matrix factorization", "exact NMF", "Kronecker product", "tensor product", "Vandaele–Gillis–Glineur–Tuyttens conjecture", "Vandaele–Gillis–Glineur–Tuyttens conjecture rank₊(M ⊗ M) ≥ rank₊(M) rank(M)", "Vandaele Gillis Glineur Tuyttens conjecture", "Vandaele-Gillis-Glineur-Tuyttens conjecture counterexample", "nonnegative rank Kronecker product conjecture", "nonnegative rank tensor product lower bound", "rank+(M kron M) >= rank+(M) rank(M)", "exact nonnegative matrix factorization Kronecker product", "nonnegative rank multiplicativity Kronecker counterexample" ], "abstract": "Vandaele, Gillis, Glineur and Tuyttens conjectured in 2016 that every nonnegative matrix M satisfies rank₊(M ⊗ M) ≥ rank₊(M) rank(M). For every k ≥ 0 an explicit rational nonnegative matrix C of order k + 11, rank k + 10 and nonnegative rank k + 11 has rank₊(C ⊗ C) ≤ (k + 10)² + 9, so the conjecture fails by at least k + 1. The matrix is the direct sum of a classical 4 × 4 matrix of rank three and nonnegative rank four with a small diagonal perturbation of an all-ones matrix; in the mixed Kronecker blocks a positive rank-one term absorbs a small real perturbation of rank q with only q + 1 nonnegative terms. At k = 0 an exact rational factorization of width 109 beats the conjectured bound 110.", "scope": "The whole family is formalized in Lean, with nonnegative rank minimized over all real nonnegative factorizations, together with a literal negation of the conjecture over all finite rectangular nonnegative real matrices. The width-109 factorization of the order-11 witness is also checked in exact rational arithmetic. The exact nonnegative rank of the squares is not determined.", "problem": "Vandaele–Gillis–Glineur–Tuyttens conjecture rank₊(M ⊗ M) ≥ rank₊(M) rank(M)", "posed_by": "Vandaele, Gillis, Glineur and Tuyttens, 'Heuristics for exact nonnegative matrix factorization', J. Global Optim. 65 (2016), 369–400, Conjecture 1 (Section 6.1)", "problem_url": "https://doi.org/10.1007/s10898-015-0350-z", "status": "Disproved", "claim": "For every k ≥ 0 an explicit rational nonnegative matrix C of order k + 11, rank k + 10 and nonnegative rank k + 11 has rank₊(C ⊗ C) ≤ (k + 10)² + 9, at least k + 1 below the conjectured bound rank₊(C) rank(C)." }, { "id": "q-permanent-finite", "title": "Finite real counterexamples and spectral obstructions for q-permanent monotonicity", "url": "https://dvgyl.github.io/math-and-theory/papers/finite-real-counterexamples-and-spectral-obstructions-for-q-permanent-monotonicity/", "pdf": "https://dvgyl.github.io/math-and-theory/papers/finite-real-counterexamples-and-spectral-obstructions-for-q-permanent-monotonicity/manuscript.pdf", "discipline": "Mathematics", "area": "Matrix theory", "subject": "Matrix theory", "subject_url": "https://dvgyl.github.io/math-and-theory/subjects/matrix-theory/", "href": "/math-and-theory/papers/finite-real-counterexamples-and-spectral-obstructions-for-q-permanent-monotonicity/", "tone": "other", "queries": [ "Bapat-Lal q-permanent conjecture real symmetric", "q-permanent monotonicity real counterexample", "MI-18 Bapat q-permanent monotonicity", "q-permanent decreasing positive definite matrix (0,1)", "inversion-weighted permanent derivative at q = 1", "da Fonseca mu-permanent conjecture interval [-1,1]", "Bapat q-permanent conjecture counterexample real matrix", "q-permanent Gram matrix rank three counterexample" ], "rank": 75, "result_type": "Real-symmetric counterexample with an exact certificate, a Lean reduction and asymptotic obstructions", "formal_proof": "standalone", "standout": true, "year": "2026", "keywords": [ "q-permanent", "Bapat–Lal conjecture", "permanent monotonicity", "positive definite matrices", "Da Fonseca conjecture", "exact certificate", "matrix permanents", "Real-symmetric case of the Bapat–Lal q-permanent monotonicity conjecture on [−1,1]", "Bapat-Lal q-permanent conjecture real symmetric", "q-permanent monotonicity real counterexample", "MI-18 Bapat q-permanent monotonicity", "q-permanent decreasing positive definite matrix (0,1)", "inversion-weighted permanent derivative at q = 1", "da Fonseca mu-permanent conjecture interval [-1,1]", "Bapat q-permanent conjecture counterexample real matrix", "q-permanent Gram matrix rank three counterexample" ], "abstract": "An explicit integer Gram matrix B of order 896 and rank three, with row coordinates at most 32 in absolute value, has a negative derivative of its inversion-weighted q-permanent at q = 1, by the exact integer comparison 1000U > 1001·C(896,2)·per(B). The integer matrix A = MB + I is positive definite, its q-permanent decreases between explicit rationals 0 < q₁ < q₂ < 1, and identity padding gives such counterexamples in every order N ≥ 896. This answers the real-symmetric case of the Bapat–Lal monotonicity conjecture on [−1,1] negatively and refutes da Fonseca's conjectures inside the original interval. For bounded-row fixed-rank perturbations I + (t/N)VVᵀ the endpoint derivative is eventually positive for every ordering when tλ_max(Σ) < 1 or tλ₃(Σ) ≤ 1 < tλ₁(Σ), and correlation counterexamples exist with condition number tending to 227.", "scope": "Lean 4.19.0 proves, for the inversion-weighted q-permanent of ordered real matrices, the endpoint identity 2D₁(VVᵀ) = C(n,2)‖f‖² − ‖G‖² for every real n × 3 matrix, soundness of the four-row integer certificate for every type list, the reduction of a negative endpoint to a rational positive-definite matrix with rational 0 < q₁ < q₂ < 1, identity padding, rank three of the frozen witness and the all-orders statement from its certificate inequality; a strict audit covers all 749 project declarations other than code-generation auxiliaries, with no native evaluation. The finite comparison for the 896-row witness is checked outside Lean by two independent exact programs, one of which determines per(B) and the wedge norms through the Chinese remainder theorem. The explicit regularization and the asymptotic and spectral theorems are conventional.", "problem": "Real-symmetric case of the Bapat–Lal q-permanent monotonicity conjecture on [−1,1]", "posed_by": "Bapat and Lal, Linear Algebra Appl. 197–198 (1994); extended by da Fonseca (2010, Conjecture 1; 2018, Conjecture 2)", "problem_url": "https://doi.org/10.1016/0024-3795(94)90497-9", "status": "Disproved for real symmetric positive-definite matrices", "claim": "An explicit integer positive-definite matrix of order 896 has a q-permanent that strictly decreases between two explicit rationals in (0,1); counterexamples exist in every order N ≥ 896." }, { "id": "sharp-eighth", "title": "The sharp level-one Fourier weight at density one eighth", "url": "https://dvgyl.github.io/math-and-theory/papers/the-sharp-level-one-fourier-weight-at-density-one-eighth/", "pdf": "https://dvgyl.github.io/math-and-theory/papers/the-sharp-level-one-fourier-weight-at-density-one-eighth/manuscript.pdf", "discipline": "Mathematics", "area": "Discrete analysis", "subject": "Discrete analysis, tilings, probability and entropy", "subject_url": "https://dvgyl.github.io/math-and-theory/subjects/discrete-analysis-tilings-probability-and-entropy/", "href": "/math-and-theory/papers/the-sharp-level-one-fourier-weight-at-density-one-eighth/", "tone": "proved", "queries": [ "level-1 Fourier weight density 1/8", "level one Fourier weight Boolean function mean 1/8 3/64", "Lei Yu problem W(1/8) level-1 Fourier weight", "average Hamming distance subset density 1/8 hypercube", "minimum average distance binary code size 2^(n-3)", "level-1 inequality Chang's lemma sharp constant", "Fang Huang average pairwise Hamming distance hypercube" ], "rank": 80, "result_type": "Sharp finite-cube Fourier bound", "formal_proof": "standalone", "standout": true, "year": "2026", "keywords": [ "Fourier analysis of Boolean functions", "level-one Fourier weight", "level-one inequality", "Boolean cube", "subcubes", "sharp inequality", "Lei Yu's problem on the level-1 Fourier weight W(1/8) of Boolean functions of density 1/8", "level-1 Fourier weight density 1/8", "level one Fourier weight Boolean function mean 1/8 3/64", "Lei Yu problem W(1/8) level-1 Fourier weight", "average Hamming distance subset density 1/8 hypercube", "minimum average distance binary code size 2^(n-3)", "level-1 inequality Chang's lemma sharp constant", "Fang Huang average pairwise Hamming distance hypercube" ], "abstract": "For every Boolean indicator of mean 1/8 on a finite uniform cube, the level-one Fourier weight is at most 3/64. A codimension-three subcube attains the bound in every dimension at least three.", "scope": "The formal proof includes arbitrary dimensions, the exact moment identities, the Hamming-distance bound, and sharpness witnesses.", "problem": "Lei Yu's problem on the level-1 Fourier weight W(1/8) of Boolean functions of density 1/8", "posed_by": "Lei Yu, 'On Average Distance, Level-1 Fourier Weight, and Chang's Lemma', arXiv:2504.02593 (v2, 2026 revision), Section VII, Problem 2", "problem_url": "https://arxiv.org/abs/2504.02593", "status": "Determined exactly", "claim": "Every Boolean function of mean 1/8 on {−1,1}^n has level-one Fourier weight at most 3/64, attained by a codimension-three subcube for n ≥ 3; equivalently, density-1/8 sets have average Hamming distance ≥ (n − 3)/2." }, { "id": "nesterov-subcritical", "title": "Escaping Nesterov flows and iterates below critical damping", "url": "https://dvgyl.github.io/math-and-theory/papers/escaping-nesterov-flows-and-iterates-below-critical-damping/", "pdf": "https://dvgyl.github.io/math-and-theory/papers/escaping-nesterov-flows-and-iterates-below-critical-damping/manuscript.pdf", "discipline": "Mathematics", "area": "Convex optimization", "subject": "Optimization", "subject_url": "https://dvgyl.github.io/math-and-theory/subjects/optimization/", "href": "/math-and-theory/papers/escaping-nesterov-flows-and-iterates-below-critical-damping/", "tone": "partial", "queries": [ "Nesterov ODE subcritical damping bounded trajectories", "Su Boyd Candes ODE r < 3 trajectory unbounded", "Su-Boyd-Candes differential equation subcritical", "vanishing damping r/t convex unbounded trajectory", "Nesterov accelerated gradient momentum k/(k+r) unbounded iterates", "Chambolle-Dossal momentum alpha < 3 iterates diverge", "Nesterov flow alpha less than 3 counterexample" ], "rank": 90, "result_type": "Sharp thresholds, explicit counterexamples and reproducible proof package", "formal_proof": "standalone", "standout": true, "year": "2026", "keywords": [ "Nesterov accelerated gradient", "Nesterov ODE", "vanishing damping", "subcritical damping", "unbounded trajectories", "FISTA momentum", "Su–Boyd–Candès", "Boundedness of Nesterov ODE trajectories and iterates below critical damping r < 3", "Nesterov ODE subcritical damping bounded trajectories", "Su Boyd Candes ODE r < 3 trajectory unbounded", "Su-Boyd-Candes differential equation subcritical", "vanishing damping r/t convex unbounded trajectory", "Nesterov accelerated gradient momentum k/(k+r) unbounded iterates", "Chambolle-Dossal momentum alpha < 3 iterates diverge", "Nesterov flow alpha less than 3 counterexample" ], "abstract": "For zero-velocity solutions of X″ + (r/t)X′ + ∇F(X) = 0 and for the Nesterov iteration with momentum k/(k+r), with F convex, its gradient 1-Lipschitz and its minimum attained, boundedness holds for r ≥ 3. On the Hilbert space ℓ²(ℕ;ℝ²) this threshold is sharp for both: for every 0 < r < 3 an explicit block sum of smoothed valleys has a trajectory, and another has iterates, with norms tending to infinity. In the plane, the potential h(√(x²+y²)) − x with the Huber function h has an escaping trajectory from (2,1) for every 0 < r ≤ 2 and bounded trajectories for r > 2. The finite-dimensional range 2 < r < 3 is open.", "scope": "The continuous-time Hilbert-space threshold, the discrete threshold and the planar escape for 1 < r ≤ 2, including actual gradients, global solutions from rest and norm divergence, are formalized. The planar escape of the same potential for 0 < r ≤ 1, its boundedness for r > 2 and existence of solutions for r ≤ 1 are conventional.", "problem": "Boundedness of Nesterov ODE trajectories and iterates below critical damping r < 3", "posed_by": "Recorded as open by Jang and Ryu, 'Point convergence of Nesterov's accelerated gradient method: an AI-assisted proof', arXiv:2510.23513 (version 2, 2026); the equation is due to Su, Boyd and Candès", "problem_url": "https://arxiv.org/abs/2510.23513", "status": "Partially resolved", "claim": "On ℓ²(ℕ;ℝ²), Nesterov flows from rest and iterates with momentum k/(k+r) stay bounded for all convex objectives with 1-Lipschitz gradient and attained minimum exactly when r ≥ 3; a planar potential escapes for 0 < r ≤ 2." }, { "id": "universal-embezzlement", "title": "Universal entanglement embezzlement with inverse-quadratic logarithmic infidelity", "url": "https://dvgyl.github.io/math-and-theory/papers/universal-entanglement-embezzlement-with-inverse-quadratic-logarithmic-infidelity/", "pdf": "https://dvgyl.github.io/math-and-theory/papers/universal-entanglement-embezzlement-with-inverse-quadratic-logarithmic-infidelity/manuscript.pdf", "discipline": "Mathematical physics", "area": "Quantum information", "subject": "Quantum information", "subject_url": "https://dvgyl.github.io/math-and-theory/subjects/quantum-information/", "href": "/math-and-theory/papers/universal-entanglement-embezzlement-with-inverse-quadratic-logarithmic-infidelity/", "tone": "proved", "queries": [ "universal embezzling family rate", "entanglement embezzlement universal catalyst", "van Dam-Hayden embezzling state fidelity", "Leung Wang universal embezzling families", "embezzlement of entanglement 1/log^2 n infidelity", "embezzling states convergence rate", "universal entanglement transformations without communication" ], "rank": 100, "result_type": "Universal catalyst with a quantitative error bound", "formal_proof": "standalone", "standout": true, "year": "2026", "keywords": [ "entanglement embezzlement", "embezzling states", "quantum catalysts", "catalytic transformations", "van Dam–Hayden", "Leung–Wang question on the rate of general universal entanglement embezzling families", "universal embezzling family rate", "entanglement embezzlement universal catalyst", "van Dam-Hayden embezzling state fidelity", "Leung Wang universal embezzling families", "embezzlement of entanglement 1/log^2 n infidelity", "embezzling states convergence rate", "universal entanglement transformations without communication" ], "abstract": "A target-independent catalyst with local dimension 2^(4L) gives root infidelity at most 125m/L² for every Schmidt probability vector on m entries.", "scope": "The proof constructs local basis permutations and establishes their finite complex Hilbert-space interpretation. Arbitrary changes to an external Schmidt basis are not separately formalized.", "problem": "Leung–Wang question on the rate of general universal entanglement embezzling families", "posed_by": "Leung and Wang, 'Characteristics of universal embezzling families', Physical Review A 90, 042331 (2014), Section 6", "problem_url": "https://arxiv.org/abs/1311.6842", "status": "Answered positively", "claim": "For each L ≥ 1 one target-independent catalyst of local dimension n = 2^(4L) lets local basis permutations reach root fidelity at least 1 − 125m/L² = 1 − 2000m/(log₂ n)² for every Schmidt probability vector on m entries." }, { "id": "keller-klein-sharp-stability", "title": "Sharp stability and exact cap optima for Rademacher sums", "url": "https://dvgyl.github.io/math-and-theory/papers/sharp-stability-and-exact-cap-optima-for-rademacher-sums/", "pdf": "https://dvgyl.github.io/math-and-theory/papers/sharp-stability-and-exact-cap-optima-for-rademacher-sums/manuscript.pdf", "discipline": "Mathematics", "area": "Probability and normal approximation", "subject": "Discrete analysis, tilings, probability and entropy", "subject_url": "https://dvgyl.github.io/math-and-theory/subjects/discrete-analysis-tilings-probability-and-entropy/", "href": "/math-and-theory/papers/sharp-stability-and-exact-cap-optima-for-rademacher-sums/", "tone": "proved", "queries": [ "Rademacher sum minimum signed sum stability", "support gap Rademacher sum inverse theorem", "Kolmogorov distance Rademacher sum coefficient cap", "Keller-Klein Berry-Esseen exact optimum large coefficients", "Hollom-Portier anti-concentration Rademacher application", "sqrt(2+sqrt(2)) stability constant signed sums", "capped Rademacher sums normal approximation extremizers" ], "rank": 105, "result_type": "Sharp inverse and stability theorems, exact cap optima and reproducible proof package", "formal_proof": "standalone", "standout": false, "year": "2026", "keywords": [ "Rademacher sums", "support gap", "minimum signed sum", "stability", "inverse theorem", "Kolmogorov distance", "Keller–Klein conjecture", "Berry–Esseen", "normal approximation", "Hollom–Portier", "Exact Kolmogorov-distance cap optimum and support-gap stability for Rademacher sums", "Rademacher sum minimum signed sum stability", "support gap Rademacher sum inverse theorem", "Kolmogorov distance Rademacher sum coefficient cap", "Keller-Klein Berry-Esseen exact optimum large coefficients", "Hollom-Portier anti-concentration Rademacher application", "sqrt(2+sqrt(2)) stability constant signed sums", "capped Rademacher sums normal approximation extremizers" ], "abstract": "For normalized Rademacher sums X = Σ bᵢεᵢ with |bᵢ| ≤ a, if every signed sum has magnitude at least x > a/2, the core {i : |bᵢ| ≥ x} is odd and the coefficients lie within squared distance 2 − 2√(1 − (a − x)²) of the flat vector on it, with equality exactly for a saturated core and one residual. At every cap, near-maximizers of the exact support-gap optimum lie within √(2+√2) times the deficit of an admissible vector of one of the two optimal shapes, with an optimal constant, and the exact nonlinear modulus on the saturated branch is determined. Conditionally on the completed Keller–Klein comparison, the maximal Kolmogorov distance K(a) is determined with all optimizers for 1/√5 ≤ a ≤ 1/√3, extending an earlier formula for a ≥ 1/√3.", "scope": "Lean 4.19.0 proves, for actual finite signed coefficient vectors and their actual minimum signed sum, the inverse theorem with its equality classification, stability at every cap with an explicit admissible comparison vector and its optimality, the exact saturated modulus with its equality cases and attainment, the fixed-flat inequalities and the sharpness witnesses: 67 audited statements in 15 new modules, replayed with nine inherited modules and the 62 earlier modules. A conditional Lean bridge derives the four-shift envelope with the analytic tail inputs as hypotheses. The probability optima have conventional proofs using the Keller–Klein, Hollom–Portier and Tomaszewski inputs and validated interval certificates on 27,972 rational boxes.", "problem": "Exact Kolmogorov-distance cap optimum and support-gap stability for Rademacher sums", "posed_by": null, "problem_url": null, "status": "Determined exactly", "claim": "A large support gap forces an odd flat core with the sharp distance bound 2 − 2√(1 − (a − x)²); support-gap near-optimizers are stable with optimal constant √(2+√2) at every cap; and, conditionally on the completed Keller–Klein comparison, the maximal Kolmogorov distance is exact for 1/√5 ≤ a ≤ 1/√3." }, { "id": "asymmetric-tutte", "title": "The exact asymmetric Merino–Welsh region for finite matroids", "url": "https://dvgyl.github.io/math-and-theory/papers/the-exact-asymmetric-merinowelsh-region-for-finite-matroids/", "pdf": "https://dvgyl.github.io/math-and-theory/papers/the-exact-asymmetric-merinowelsh-region-for-finite-matroids/manuscript.pdf", "discipline": "Mathematics", "area": "Matroid theory", "subject": "Matroid theory", "subject_url": "https://dvgyl.github.io/math-and-theory/subjects/matroid-theory/", "href": "/math-and-theory/papers/the-exact-asymmetric-merinowelsh-region-for-finite-matroids/", "tone": "proved", "queries": [ "Merino–Welsh conjecture matroids", "Merino-Welsh conjecture", "asymmetric Merino-Welsh inequality", "multiplicative Merino-Welsh inequality threshold", "Tutte polynomial T(x,0)T(0,y) >= T(1,1)^2", "Merino-Welsh conjecture false for matroids", "Tutte polynomial forests orientations inequality matroid", "Merino-Welsh sharp threshold matroids" ], "rank": 110, "result_type": "Exact asymmetric parameter region", "formal_proof": "standalone", "standout": false, "year": "2026", "keywords": [ "Merino–Welsh conjecture", "Merino-Welsh", "Tutte polynomial", "matroids", "forests and orientations", "matroid inequalities", "Asymmetric multiplicative Merino–Welsh inequality T_M(x,0)T_M(0,y) ≥ T_M(1,1)²", "Merino–Welsh conjecture matroids", "Merino-Welsh conjecture", "asymmetric Merino-Welsh inequality", "multiplicative Merino-Welsh inequality threshold", "Tutte polynomial T(x,0)T(0,y) >= T(1,1)^2", "Merino-Welsh conjecture false for matroids", "Tutte polynomial forests orientations inequality matroid", "Merino-Welsh sharp threshold matroids" ], "abstract": "The theorem characterizes all nonnegative pairs (x,y) for which T_M(x,0) T_M(0,y) ≥ T_M(1,1)² holds for every finite matroid without loops or coloops.", "scope": "The statement uses actual Mathlib matroids and the literal subset rank sum. It includes empty ground sets and finite ground sets in arbitrary ambient types.", "problem": "Asymmetric multiplicative Merino–Welsh inequality T_M(x,0)T_M(0,y) ≥ T_M(1,1)²", "posed_by": null, "problem_url": null, "status": "Determined exactly", "claim": "T_M(x,0)T_M(0,y) ≥ T_M(1,1)² for every finite matroid without loops or coloops exactly when x > 1, y > 1, xy^k ≥ (√(x−1) + k√(y−1))² and yx^k ≥ (√(y−1) + k√(x−1))² for every integer k ≥ 2." }, { "id": "critical-hardcore", "title": "Critical hard-core influence bounds via nonbacktracking collisions", "url": "https://dvgyl.github.io/math-and-theory/papers/critical-hard-core-influence-bounds-via-nonbacktracking-collisions/", "pdf": "https://dvgyl.github.io/math-and-theory/papers/critical-hard-core-influence-bounds-via-nonbacktracking-collisions/manuscript.pdf", "discipline": "Theoretical computer science", "area": "Sampling algorithms and spectral independence", "subject": "Sampling algorithms and information theory", "subject_url": "https://dvgyl.github.io/math-and-theory/subjects/sampling-algorithms-and-information-theory/", "href": "/math-and-theory/papers/critical-hard-core-influence-bounds-via-nonbacktracking-collisions/", "tone": "partial", "queries": [ "hard-core model spectral independence uniqueness threshold", "critical hardcore model Glauber dynamics mixing", "hardcore model at criticality influence matrix bound", "spectral independence critical two-spin systems sqrt n", "Chen Jiang improved mixing of critical hardcore model", "nonbacktracking random walk self-intersection spectral independence", "independent sets Glauber dynamics critical fugacity mixing time", "hard-core model lambda_c total influence sqrt(n) log n" ], "rank": 120, "result_type": "Theorem and reproducible proof package", "formal_proof": "standalone", "standout": false, "year": "2026", "keywords": [ "hard-core model", "spectral independence", "uniqueness threshold", "Glauber dynamics", "independent sets", "nonbacktracking walks", "Spectral independence of the hard-core model at the uniqueness threshold λ_c(Δ)", "hard-core model spectral independence uniqueness threshold", "critical hardcore model Glauber dynamics mixing", "hardcore model at criticality influence matrix bound", "spectral independence critical two-spin systems sqrt n", "Chen Jiang improved mixing of critical hardcore model", "nonbacktracking random walk self-intersection spectral independence", "independent sets Glauber dynamics critical fugacity mixing time", "hard-core model lambda_c total influence sqrt(n) log n" ], "abstract": "For each fixed Δ ≥ 3, every row of the hard-core influence matrix has absolute sum O_Δ(√n log n) on every n-vertex graph of maximum degree at most Δ, at every activity up to the uniqueness threshold λ_c(Δ) and under every feasible pinning, with an explicit constant. The proof combines exact vertex-deletion identities, a known local squared-message contraction, a self-avoiding-walk transfer in the style of Sinclair, Srivastava, Štefankovič and Yin, and a finite form of the nonbacktracking collision argument of Bencs et al. It recovers, up to a logarithm, an O(√n) bound that was stated in 2025 and withdrawn in 2026. The constant is large, and the mixing-time consequence for Glauber dynamics is conditional on a published proposition.", "scope": "The influence bound for the actual hard-core Gibbs measure on every finite simple graph of maximum degree at most Δ, all activities up to λ_c(Δ) and all feasible pinnings is formalized in 32 Lean modules, including the regular completion and the nonbacktracking collision estimate. The Glauber-dynamics mixing corollary and the numerical observations on the constant are not formalized.", "problem": "Spectral independence of the hard-core model at the uniqueness threshold λ_c(Δ)", "posed_by": "Bencs, Goldberg, Jenssen, Jerrum, Pete, Regts and Yin, arXiv:2608.09729, Section 5 (critical two-spin question)", "problem_url": "https://arxiv.org/abs/2608.09729", "status": "Partially resolved", "claim": "For every Δ ≥ 3, every n-vertex graph of maximum degree at most Δ, every activity 0 < λ ≤ λ_c(Δ) and every feasible pinning, each absolute row sum of the hard-core influence matrix is O_Δ(√n log n), with an explicit constant." }, { "id": "gyires-circulant-five", "title": "Gyires's permanent inequality for structured matrices", "url": "https://dvgyl.github.io/math-and-theory/papers/gyires-s-permanent-inequality-for-structured-matrices/", "pdf": "https://dvgyl.github.io/math-and-theory/papers/gyires-s-permanent-inequality-for-structured-matrices/manuscript.pdf", "discipline": "Mathematics", "area": "Matrix theory", "subject": "Matrix theory", "subject_url": "https://dvgyl.github.io/math-and-theory/subjects/matrix-theory/", "href": "/math-and-theory/papers/gyires-s-permanent-inequality-for-structured-matrices/", "tone": "partial", "queries": [ "Gyires conjecture permanent", "Gyires permanent inequality doubly stochastic matrices", "Cheon Wanless Conjecture 20 permanent", "Minc's list Conjecture 20 Gyires", "permanent inequality per(A)^2 per(AA^T) per(A^2)", "permanent of circulant doubly stochastic matrix inequality", "Gyires ratio inequality special cases proof", "permanent conjectures doubly stochastic Lean formalization" ], "rank": 130, "result_type": "Structured matrix inequalities and related polynomial bounds", "formal_proof": "comparator", "standout": false, "year": "2026", "keywords": [ "Gyires conjecture", "Gyires permanent inequality", "doubly stochastic matrices", "circulant matrices", "permanents", "rank-two matrices", "Gyires's permanent inequality for doubly stochastic matrices (Cheon–Wanless Conjecture 20)", "Gyires conjecture permanent", "Gyires permanent inequality doubly stochastic matrices", "Cheon Wanless Conjecture 20 permanent", "Minc's list Conjecture 20 Gyires", "permanent inequality per(A)^2 per(AA^T) per(A^2)", "permanent of circulant doubly stochastic matrix inequality", "Gyires ratio inequality special cases proof", "permanent conjectures doubly stochastic Lean formalization" ], "abstract": "The preprint proves Gyires's permanent inequality for several structured matrix classes. These include nonnegative circulants of orders five through seven, all-order permutation polynomials A=aI+bQ+cQ², and all-order four-block doubly stochastic matrices. Here Q is any permutation matrix and a,b,c are nonnegative. The four-block result permits unequal block sizes and zero entries. Further results cover one exceptional column or row and every rank-at-most-two doubly stochastic matrix through order fifteen. The supporting sections prove structural, scalar, deletion, balancing, local polynomial, and counting results. Exact counterexamples identify the limits of auxiliary assertions. The principal order-five claims include equality and a sharp sorted-gap bound with coefficient 484. The principal order-five and order-six claims, the all-order permutation-polynomial and four-block theorems, and the exceptional-vector theorem have complete Lean proofs. The rank-two result has a complete Lean proof for orders 2 through 8. The general order-seven circulant result and rank-two orders 9 through 15 remain incomplete in Lean. The unrestricted conjecture remains outside these results.", "scope": "The selected Comparator statements cover order-five circulants, the all-order four-block theorem, and the exceptional-vector theorem for n ≥ 2. Separate Lean packages prove the general order-six and all-order permutation-polynomial theorems. General permanent comparisons and direct-sum closure are also proved in Lean. The rank-two theorem has a complete Lean proof for orders 2 through 8. The general order-seven circulant theorem and rank-two orders 9 through 15 remain incomplete in Lean.", "problem": "Gyires's permanent inequality for doubly stochastic matrices (Cheon–Wanless Conjecture 20)", "posed_by": "Gyires (1978); listed as Conjecture 20 in Cheon and Wanless (2005)", "problem_url": "https://doi.org/10.1016/j.laa.2005.02.030", "status": "Partially resolved", "claim": "Proves Gyires's permanent inequality for nonnegative circulants of orders 5, 6 and 7, permutation polynomials aI+bQ+cQ² and four-block matrices in all orders, one-exceptional-column/row matrices, and rank ≤ 2 through order 15." }, { "id": "chambolle-pock-cycles", "title": "Periodic obstructions to small-extrapolation Chambolle–Pock", "url": "https://dvgyl.github.io/math-and-theory/papers/periodic-obstructions-to-small-extrapolation-chambollepock/", "pdf": "https://dvgyl.github.io/math-and-theory/papers/periodic-obstructions-to-small-extrapolation-chambollepock/manuscript.pdf", "discipline": "Mathematics", "area": "Optimization", "subject": "Optimization", "subject_url": "https://dvgyl.github.io/math-and-theory/subjects/optimization/", "href": "/math-and-theory/papers/periodic-obstructions-to-small-extrapolation-chambollepock/", "tone": "refuted", "queries": [ "Chambolle–Pock convergence small extrapolation theta", "Chambolle-Pock theta less than 1/2 convergence", "PDHG step size condition tau sigma ||L||^2 counterexample", "Chambolle-Pock nonconvergence periodic orbit", "primal-dual hybrid gradient extrapolation parameter counterexample", "Chambolle-Pock 4/(1+2 theta) step size bound", "Banert Upadhyaya Giselsson Chambolle-Pock converges weakly", "PDHG three-cycle counterexample" ], "rank": 140, "result_type": "Counterexample families, exact threshold and reproducible proof package", "formal_proof": "standalone", "standout": false, "year": "2026", "keywords": [ "Chambolle–Pock", "Chambolle-Pock", "PDHG", "primal-dual hybrid gradient", "extrapolation parameter", "periodic orbits", "Small-extrapolation convergence region of the Chambolle–Pock (PDHG) method", "Chambolle–Pock convergence small extrapolation theta", "Chambolle-Pock theta less than 1/2 convergence", "PDHG step size condition tau sigma ||L||^2 counterexample", "Chambolle-Pock nonconvergence periodic orbit", "primal-dual hybrid gradient extrapolation parameter counterexample", "Chambolle-Pock 4/(1+2 theta) step size bound", "Banert Upadhyaya Giselsson Chambolle-Pock converges weakly", "PDHG three-cycle counterexample" ], "abstract": "For the unmodified primal-first Chambolle–Pock iteration with extrapolation θ and step product κ = τσ‖L‖², every κ > 2 admits fixed compact polytopes in R⁴ whose iteration cycles without converging for all sufficiently small θ ≥ 0. Any universal sufficient condition κ < c(θ) therefore satisfies limsup c(θ) ≤ 2 as θ ↓ 0, while bilinear and quadratic problems allow κ < 4/(1+2θ). A nonconstant three-cycle of actual proximal updates exists exactly when κ²(1+θ)² − 2κ(1+3θ) − 3 ≥ 0, attained by compact triangles, and a rational seven-cycle has an open set of nonconvergent starting points. The obstructing θ-intervals are very small and the bound two is not shown to be sharp.", "scope": "The universal three-cycle threshold and its attainment, the seven-cycle with its open basin, the four-dimensional family for every κ > 2 and the quadratic eigenmode comparison are formalized in 21 Lean modules. The limsup consequence, numerical observations of larger θ-intervals and basin sizes, and the finite-dimensional matrix-convergence bridge are conventional or observational.", "problem": "Small-extrapolation convergence region of the Chambolle–Pock (PDHG) method", "posed_by": "Left open in the conclusion of Upadhyaya, arXiv:2604.06423v2 (improving the small-θ region)", "problem_url": "https://arxiv.org/abs/2604.06423", "status": "Counterexample", "claim": "For unmodified Chambolle–Pock, every step product κ = τσ‖L‖² > 2 admits polytopes in ℝ⁴ with periodic nonconvergent iterates for all small θ ≥ 0, so any universal condition κ < c(θ) has limsup c(θ) ≤ 2 as θ ↓ 0." }, { "id": "harmonic-boundary-five", "title": "Graphs with five harmonic boundary vertices, subcubic graphs with six, and minimum-boundary two-trees", "url": "https://dvgyl.github.io/math-and-theory/papers/graphs-with-five-harmonic-boundary-vertices-subcubic-graphs-with-six-and-minimum-boundary-two-trees/", "pdf": "https://dvgyl.github.io/math-and-theory/papers/graphs-with-five-harmonic-boundary-vertices-subcubic-graphs-with-six-and-minimum-boundary-two-trees/manuscript.pdf", "discipline": "Mathematics", "area": "Graph theory", "subject": "Graph theory and graph polynomials", "subject_url": "https://dvgyl.github.io/math-and-theory/subjects/graph-theory-and-graph-polynomials/", "href": "/math-and-theory/papers/graphs-with-five-harmonic-boundary-vertices-subcubic-graphs-with-six-and-minimum-boundary-two-trees/", "tone": "other", "queries": [ "graphs with five boundary vertices", "Steinerberger boundary of a graph classification", "Chiem Dudarov Lee Liu boundary vertices problem", "graph boundary average distance neighbours", "two-connected graph boundary diameter two", "minimum boundary two-tree", "subcubic graphs six boundary vertices", "boundary stability number graph" ], "rank": 145, "result_type": "Computer-assisted complete classification with exact certificates and partial Lean proofs", "formal_proof": "standalone", "standout": false, "year": "2026", "keywords": [ "graph boundary", "Steinerberger boundary", "harmonic boundary vertices", "graph classification", "subcubic graphs", "two-trees", "five-point metrics", "Classify the connected graphs with exactly five Steinerberger boundary vertices", "graphs with five boundary vertices", "Steinerberger boundary of a graph classification", "Chiem Dudarov Lee Liu boundary vertices problem", "graph boundary average distance neighbours", "two-connected graph boundary diameter two", "minimum boundary two-tree", "subcubic graphs six boundary vertices", "boundary stability number graph" ], "abstract": "Every two-connected graph with at most five Steinerberger boundary vertices has diameter at most two, and a connected graph has exactly five boundary vertices if and only if it is a bridge subdivision of one of 227 reduced graphs of order at most fourteen; this answers Problem 6.1 of Chiem, Dudarov, Lee, Lee and Liu. The proof reduces boundary metrics of every diameter, through the three cones of five-point metrics and a cap preserving local configurations, to an exact certificate that leaves fifteen metrics, excluded by hand. Connected graphs of maximum degree three with exactly six boundary vertices are the bridge subdivisions of 201 reduced graphs. For n ≥ 6 every two-tree on n vertices has at least ⌈n/2⌉ + 2 boundary vertices, with three extremal two-trees at each even n ≥ 8 and, at odd n ≥ 9, a period-eight quartic quasi-polynomial number of them, asymptotic to n⁴/192.", "scope": "Lean 4.35.0 proves, for finite simple graphs with certified shortest-path distances and the original strict boundary inequality, that a nonboundary degree-three vertex whose deletion leaves the graph connected forces six boundary vertices; the one-point gluing and block-accounting identities; the 14-vertex sharpness example; all 835 integer infeasibility systems of the local certificate; the bipartite subcubic boundary characterization; a sharp rooted metric truncation theorem; and the grammar weight bounds and polynomial identities for two-trees. A strict audit covers all 15,771 project declarations. The cone coverage, capping reduction, diameter-three argument, subcubic interior reduction, block decomposition, grammar correspondence and catalogue enumerations are conventional proofs and exact computations, replayed by the package checkers and by independent enumerations.", "problem": "Classify the connected graphs with exactly five Steinerberger boundary vertices", "posed_by": "Chiem, Dudarov, Lee, Lee and Liu, A characterization of graphs with at most four boundary vertices, J. Combin. 15 (2024), Problem 6.1; boundary defined by Steinerberger, Discrete Appl. Math. 338 (2023)", "problem_url": "https://arxiv.org/abs/2209.04438", "status": "Solved with a computer-assisted proof: 227 reduced graphs; subcubic six-boundary graphs and minimum-boundary two-trees also classified", "claim": "A connected graph has exactly five boundary vertices if and only if it is a bridge subdivision of one of 227 reduced graphs; two-connected graphs with at most five boundary vertices have diameter at most two." }, { "id": "structural-selfadhesivity", "title": "Structural selfadhesivity does not imply a numerically selfadhesive realization", "url": "https://dvgyl.github.io/math-and-theory/papers/structural-selfadhesivity-does-not-imply-a-numerically-selfadhesive-realization/", "pdf": "https://dvgyl.github.io/math-and-theory/papers/structural-selfadhesivity-does-not-imply-a-numerically-selfadhesive-realization/manuscript.pdf", "discipline": "Mathematics", "area": "Entropy and conditional independence", "subject": "Discrete analysis, tilings, probability and entropy", "subject_url": "https://dvgyl.github.io/math-and-theory/subjects/discrete-analysis-tilings-probability-and-entropy/", "href": "/math-and-theory/papers/structural-selfadhesivity-does-not-imply-a-numerically-selfadhesive-realization/", "tone": "refuted", "queries": [ "structural selfadhesivity numerical selfadhesivity", "selfadhesive polymatroid conditional independence", "Boege selfadhesivity Question 4.18", "selfadhesivity in Gaussian conditional independence structures", "self-adhesive polymatroid five variables counterexample", "Matúš selfadhesivity polymatroid", "Matus selfadhesive polymatroid copy lemma", "structural semigraphoid selfadhesive realization" ], "rank": 150, "result_type": "Counterexample answering a published question and reproducible proof package", "formal_proof": "standalone", "standout": false, "year": "2026", "keywords": [ "conditional independence", "polymatroids", "selfadhesivity", "adhesive extensions", "entropy region", "Shannon inequalities", "copy lemma", "Boege", "Boege's selfadhesivity question: structural vs numerical selfadhesive polymatroids", "structural selfadhesivity numerical selfadhesivity", "selfadhesive polymatroid conditional independence", "Boege selfadhesivity Question 4.18", "selfadhesivity in Gaussian conditional independence structures", "self-adhesive polymatroid five variables counterexample", "Matúš selfadhesivity polymatroid", "Matus selfadhesive polymatroid copy lemma", "structural semigraphoid selfadhesive realization" ], "abstract": "An explicit conditional-independence model on five variables, with 26 of the 80 elementary statements, admits for every overlap an adhesive polymatroid gluing whose two restrictions both induce it, yet no real polymatroid inducing it is selfadhesive in Matúš's numerical sense. This answers negatively Question 4.18 of Boege, who asked whether every structurally selfadhesive structural semigraphoid is induced by a selfadhesive polymatroid; his computation gave a positive answer for four variables. The positive part is certified by 32 integer polymatroids, the negative part by an exact identity among 51 nonnegative conditional mutual informations valid for every real inducing profile.", "scope": "Both parts are formalized in Lean for arbitrary real polymatroids, with actual subset and relabeling semantics, all 32 overlap witnesses and the exact separating identity. The minimality suggested by Boege's four-variable computation is a literature consequence and is not formalized.", "problem": "Boege's selfadhesivity question: structural vs numerical selfadhesive polymatroids", "posed_by": "Boege, IJAR 2023 (arXiv:2205.07667), Question 4.18 (4.14 in the PDF version)", "problem_url": "https://doi.org/10.1016/j.ijar.2023.109027", "status": "Answered negatively", "claim": "An explicit five-variable conditional-independence model with 26 of the 80 elementary CI statements is structurally selfadhesive at every overlap, yet no real polymatroid inducing it is selfadhesive in Matúš's numerical sense." }, { "id": "berge-path-saturation", "title": "Exact three-uniform Berge-path saturation numbers and a counterexample to a conjectured formula", "url": "https://dvgyl.github.io/math-and-theory/papers/exact-three-uniform-berge-path-saturation-numbers-and-a-counterexample-to-a-conjectured-formula/", "pdf": "https://dvgyl.github.io/math-and-theory/papers/exact-three-uniform-berge-path-saturation-numbers-and-a-counterexample-to-a-conjectured-formula/manuscript.pdf", "discipline": "Mathematics", "area": "Extremal hypergraph theory", "subject": "Graph theory and graph polynomials", "subject_url": "https://dvgyl.github.io/math-and-theory/subjects/graph-theory-and-graph-polynomials/", "href": "/math-and-theory/papers/exact-three-uniform-berge-path-saturation-numbers-and-a-counterexample-to-a-conjectured-formula/", "tone": "other", "queries": [ "Berge path saturation number", "saturation of Berge hypergraphs conjecture 2.4", "sat_3(n, Berge P_m)", "Berge-P_m saturated three-uniform hypergraph", "English Graber Kirkpatrick Methuku Sullivan Berge saturation", "Berge path saturation counterexample loose cycles isolated vertices", "exact saturation number Berge paths 3-uniform", "hypergraph saturation Berge path conjecture" ], "rank": 160, "result_type": "Counterexample to a published conjecture and exact formula with reproducible proof package", "formal_proof": "standalone", "standout": false, "year": "2026", "keywords": [ "Berge hypergraphs", "Berge paths", "saturation numbers", "three-uniform hypergraphs", "loose cycles", "isolated vertices", "extremal hypergraph theory", "Berge-path saturation conjecture (English–Graber–Kirkpatrick–Methuku–Sullivan, Conjecture 2.4)", "Berge path saturation number", "saturation of Berge hypergraphs conjecture 2.4", "sat_3(n, Berge P_m)", "Berge-P_m saturated three-uniform hypergraph", "English Graber Kirkpatrick Methuku Sullivan Berge saturation", "Berge path saturation counterexample loose cycles isolated vertices", "exact saturation number Berge paths 3-uniform", "hypergraph saturation Berge path conjecture" ], "abstract": "English, Graber, Kirkpatrick, Methuku and Sullivan conjectured that the three-uniform Berge-P_m saturation number is ⌈(n − ⌊n/b⌋)/2⌉ for all m ≥ 10 and n ≥ b = 2a_m + 1. Two disjoint loose 8-cycles with two isolated vertices form a Berge-P₁₀-saturated hypergraph on 34 vertices with 16 hyperedges, one below the conjectured 17. The preprint determines the saturation number on the whole conjectured range: it is one less than the conjectured value exactly when b < n < 2b and n ≡ 2 (mod 2(m − 2)), and the extremal hypergraph at those orders is unique.", "scope": "Lean 4.35.0 proves, for actual three-uniform hypergraphs, Berge paths and saturation under every absent triple, that t disjoint loose d-cycles with two isolated vertices are Berge-P_{d+2}-saturated for every d ≥ 3, that the explicit 34-vertex hypergraph is Berge-P₁₀-saturated with 16 hyperedges, and that the three-uniform case of the conjecture fails for every m ≥ 10. The matching lower bound, the exact formula and the uniqueness of extremal hypergraphs have conventional proofs that use two results of the source paper.", "problem": "Berge-path saturation conjecture (English–Graber–Kirkpatrick–Methuku–Sullivan, Conjecture 2.4)", "posed_by": "English, Graber, Kirkpatrick, Methuku and Sullivan, Saturation of Berge hypergraphs, arXiv:1710.03735v1, Conjecture 2.4", "problem_url": "https://arxiv.org/abs/1710.03735v1", "status": "Disproved for three-uniform hypergraphs; exact formula given", "claim": "Two loose 8-cycles plus two isolated vertices give a Berge-P₁₀-saturated 3-graph on 34 vertices with 16 edges, not 17; the exact 3-uniform formula holds for all m ≥ 10, n ≥ 2a_m + 1." }, { "id": "berge-subdivided-star", "title": "Exact Berge saturation numbers of the subdivided star in every uniformity and its minimality among trees", "url": "https://dvgyl.github.io/math-and-theory/papers/exact-berge-saturation-numbers-of-the-subdivided-star-in-every-uniformity-and-its-minimality-among-trees/", "pdf": "https://dvgyl.github.io/math-and-theory/papers/exact-berge-saturation-numbers-of-the-subdivided-star-in-every-uniformity-and-its-minimality-among-trees/manuscript.pdf", "discipline": "Mathematics", "area": "Extremal hypergraph theory", "subject": "Graph theory and graph polynomials", "subject_url": "https://dvgyl.github.io/math-and-theory/subjects/graph-theory-and-graph-polynomials/", "href": "/math-and-theory/papers/exact-berge-saturation-numbers-of-the-subdivided-star-in-every-uniformity-and-its-minimality-among-trees/", "tone": "other", "queries": [ "Berge saturation number of trees", "sat_r(n, Berge-T)", "Berge saturation subdivided star", "Berge-saturated hypergraph loose stars", "minimum saturation number among trees hypergraph", "Faudree Gould Jacobson saturation numbers for trees Berge analogue", "exact Berge saturation number uniform hypergraph", "saturation of Berge hypergraphs trees" ], "rank": 165, "result_type": "Exact formula with a complete Lean proof and a minimality theorem among trees", "formal_proof": "standalone", "standout": false, "year": "2026", "keywords": [ "Berge hypergraphs", "saturation numbers", "trees", "subdivided star", "loose stars", "uniform hypergraphs", "extremal hypergraph theory", "Berge saturation numbers of trees and the Berge analogue of the minimum-tree theorem of Faudree, Faudree, Gould and Jacobson", "Berge saturation number of trees", "sat_r(n, Berge-T)", "Berge saturation subdivided star", "Berge-saturated hypergraph loose stars", "minimum saturation number among trees hypergraph", "Faudree Gould Jacobson saturation numbers for trees Berge analogue", "exact Berge saturation number uniform hypergraph", "saturation of Berge hypergraphs trees" ], "abstract": "Let B_a be the star K_{1,a} with one edge subdivided. For all a, r ≥ 3 and n ≥ (r − 1)a + 1 the r-uniform Berge saturation number, in which any absent r-set may be added, is sat_r(n, Berge-B_a) = ⌊a(n + 1)/((r − 1)a + 1)⌋, attained by disjoint loose stars; it is a for max(a + 2, r + 1) ≤ n < (r − 1)a + 1. Saturated hypergraphs within k hyperedges of the minimum consist of loose stars with at least a spokes and at most (r − 1)((r − 1)a + 1)(k + 1) − 1 further vertices once n ≥ ((r − 1)a + 1)((r − 1)(k + 1) + 1), sharply for k = 0. For a ≥ 5 every tree T ≇ B_a on a + 2 vertices has sat₃(n, Berge-T) ≥ ⌈(a + 1)(n − 2)/(2a + 3)⌉, so B_a is the unique tree of its order with the least asymptotic lower three-uniform saturation coefficient, a/(2a + 1), and the unique minimizer for n ≥ 3(a + 1)(2a + 1); this is the Berge analogue of the theorem of Faudree, Faudree, Gould and Jacobson for graphs.", "scope": "Lean 4.35.0 proves, for actual r-uniform hypergraphs on Fin n, Berge copies of B_a with injective vertex and hyperedge maps, and saturation under every absent r-set, that the saturation number, defined as the least size of a saturated hypergraph on Fin n, equals ⌊a(n + 1)/((r − 1)a + 1)⌋ for all a, r ≥ 3 and n ≥ (r − 1)a + 1, together with the near-minimum structure theorem; a strict audit covers all 570 project declarations. The smaller orders, the sharpness constructions and the minimality among trees have conventional proofs.", "problem": "Berge saturation numbers of trees and the Berge analogue of the minimum-tree theorem of Faudree, Faudree, Gould and Jacobson", "posed_by": "English, Graber, Kirkpatrick, Methuku and Sullivan, Saturation of Berge hypergraphs, arXiv:1710.03735; graph case: Faudree, Faudree, Gould and Jacobson, Saturation numbers for trees, Electron. J. Combin. 16 (2009), R91", "problem_url": "https://arxiv.org/abs/1710.03735", "status": "Exact formula proved in every uniformity; minimality among trees proved for three-uniform hypergraphs", "claim": "sat_r(n, Berge-B_a) = ⌊a(n + 1)/((r − 1)a + 1)⌋ for a, r ≥ 3 and n ≥ (r − 1)a + 1; for a ≥ 5, B_a is the unique tree on a + 2 vertices with the least three-uniform saturation coefficient." }, { "id": "upb-tensor-rank", "title": "Strong minimum-rank multiplicativity from unextendible product constraints", "url": "https://dvgyl.github.io/math-and-theory/papers/strong-minimum-rank-multiplicativity-from-unextendible-product-constraints/", "pdf": "https://dvgyl.github.io/math-and-theory/papers/strong-minimum-rank-multiplicativity-from-unextendible-product-constraints/manuscript.pdf", "discipline": "Mathematical physics", "area": "Quantum information", "subject": "Quantum information", "subject_url": "https://dvgyl.github.io/math-and-theory/subjects/quantum-information/", "href": "/math-and-theory/papers/strong-minimum-rank-multiplicativity-from-unextendible-product-constraints/", "tone": "proved", "queries": [ "Schmidt number tensor power bound entangled state", "Tiles UPB bound entangled state Schmidt number", "unextendible product basis minimum rank tensor product", "Schmidt number multiplicativity PPT entangled states", "zero-error entanglement cost Tiles state", "minimum Schmidt rank of UPB complement subspace", "rank-four two-qutrit PPT entangled state Schmidt number", "Schmidt number of N copies 2^N" ], "rank": 170, "result_type": "Theorem, exact all-copy formula and reproducible proof package", "formal_proof": "standalone", "standout": false, "year": "2026", "keywords": [ "unextendible product bases", "UPB", "Schmidt number", "bound entanglement", "Tiles state", "PPT entangled states", "zero-error entanglement cost", "minimum rank", "Schmidt number of tensor powers of the Tiles UPB bound entangled state", "Schmidt number tensor power bound entangled state", "Tiles UPB bound entangled state Schmidt number", "unextendible product basis minimum rank tensor product", "Schmidt number multiplicativity PPT entangled states", "zero-error entanglement cost Tiles state", "minimum Schmidt rank of UPB complement subspace", "rank-four two-qutrit PPT entangled state Schmidt number", "Schmidt number of N copies 2^N" ], "abstract": "A complex matrix subspace annihilated by an unextendible family of product functionals doubles minimum rank under tensor products: if every nonzero member of T has rank at least r, every nonzero member of S ⊗ T has rank at least 2r, and the minimum rank is exactly doubled when S contains a rank-two matrix. For the Tiles bound entangled state ρ, this gives Schmidt number SN(ρ^⊗N) = 2^N for every N ≥ 1, so N copies cost exactly N ebits to prepare by LOCC with zero error. The two-copy case and nonsharp exponential growth were known.", "scope": "The rank-doubling theorem with an arbitrary partner, strong minimum-rank multiplicativity, the Tiles support and normalized density operators, and the exact all-copy Schmidt-number equivalence are formalized in 22 Lean modules. The zero-error LOCC cost, the rank-four two-qutrit PPT class and the channel interpretation combine the formal theorem with published results.", "problem": "Schmidt number of tensor powers of the Tiles UPB bound entangled state", "posed_by": null, "problem_url": null, "status": "Determined exactly", "claim": "If a complex matrix subspace S is annihilated by an unextendible family of product functionals, every nonzero element of S ⊗ T has rank at least 2m(T); for the Tiles bound entangled state ρ, SN(ρ^⊗N) = 2^N for all N ≥ 1." }, { "id": "quartic-rankings", "title": "A quartic entropy deficit for random rankings", "url": "https://dvgyl.github.io/math-and-theory/papers/a-quartic-entropy-deficit-for-random-rankings/", "pdf": "https://dvgyl.github.io/math-and-theory/papers/a-quartic-entropy-deficit-for-random-rankings/manuscript.pdf", "discipline": "Mathematics", "area": "Probability and combinatorics", "subject": "Discrete analysis, tilings, probability and entropy", "subject_url": "https://dvgyl.github.io/math-and-theory/subjects/discrete-analysis-tilings-probability-and-entropy/", "href": "/math-and-theory/papers/a-quartic-entropy-deficit-for-random-rankings/", "tone": "proved", "queries": [ "Leighton Moitra entropy conjecture", "Leighton-Moitra conjecture rankings entropy", "Acan Devlin Kahn proof of an entropy conjecture of Leighton and Moitra", "entropy of random permutations with biased pairwise comparisons", "permutation entropy bound pairwise comparison bias", "entropy of linear extensions posets quantitative bound", "ranking distribution entropy log n! deficit epsilon^4" ], "rank": 180, "result_type": "Entropy deficit from pairwise comparison bias", "formal_proof": "standalone", "standout": false, "year": "2026", "keywords": [ "random rankings", "permutation entropy", "pairwise comparisons", "entropy bounds", "ranking distributions", "Leighton–Moitra entropy conjecture for rankings: an explicit quartic ε⁴ entropy deficit", "Leighton Moitra entropy conjecture", "Leighton-Moitra conjecture rankings entropy", "Acan Devlin Kahn proof of an entropy conjecture of Leighton and Moitra", "entropy of random permutations with biased pairwise comparisons", "permutation entropy bound pairwise comparison bias", "entropy of linear extensions posets quantitative bound", "ranking distribution entropy log n! deficit epsilon^4" ], "abstract": "A distribution on rankings of n labels has entropy at most (1 − ε⁴/144) log(n!) if every pairwise comparison has probability at distance at least ε > 0 from 1/2. The proof includes all finite sizes and actual position-valued permutations. It assumes neither independent comparisons nor transitive preferred directions.", "scope": "The proof includes all finite sizes and actual position-valued permutations. It assumes neither independent comparisons nor transitive preferred directions.", "problem": "Leighton–Moitra entropy conjecture for rankings: an explicit quartic ε⁴ entropy deficit", "posed_by": "Leighton and Moitra (qualitative conjecture, proved by Acan, Devlin and Kahn); a Cε⁴ deficit is proposed in Devlin's research statement", "problem_url": null, "status": "Sharpened", "claim": "Every probability distribution on rankings of n labels in which each pairwise comparison has probability at distance at least ε > 0 from 1/2 has entropy at most (1 − ε⁴/144) log(n!), for every n." }, { "id": "width-two-order-polynomial", "title": "Order-polynomial positivity for width-two posets with a chain of at most five elements", "url": "https://dvgyl.github.io/math-and-theory/papers/order-polynomial-positivity-for-width-two-posets-with-a-chain-of-at-most-five-elements/", "pdf": "https://dvgyl.github.io/math-and-theory/papers/order-polynomial-positivity-for-width-two-posets-with-a-chain-of-at-most-five-elements/manuscript.pdf", "discipline": "Mathematics", "area": "Integer sequences and combinatorics", "subject": "Integer sequences and combinatorics", "subject_url": "https://dvgyl.github.io/math-and-theory/subjects/integer-sequences-and-combinatorics/", "href": "/math-and-theory/papers/order-polynomial-positivity-for-width-two-posets-with-a-chain-of-at-most-five-elements/", "tone": "partial", "queries": [ "Ferroni Morales Panova conjecture width two", "order polynomial positive coefficients width two poset", "order polynomial nonnegative coefficients two chains", "Ehrhart positivity order polytope width two", "weak order polynomial positivity", "linear coefficient order polynomial lower bound", "non-nesting rook placements descents width two poset", "Skew shapes Ehrhart positivity and beyond Conjecture 7.1" ], "rank": 190, "result_type": "Partial resolution of a conjecture for an infinite family, with exact certificates and reproducible proof package", "formal_proof": "standalone", "standout": false, "year": "2026", "keywords": [ "order polynomial", "width-two posets", "Ferroni–Morales–Panova conjecture", "Ehrhart positivity", "order polytope", "non-nesting rook placements", "sum-of-squares certificate", "posets", "Ferroni–Morales–Panova conjecture on positivity of order polynomials of width-two posets", "Ferroni Morales Panova conjecture width two", "order polynomial positive coefficients width two poset", "order polynomial nonnegative coefficients two chains", "Ehrhart positivity order polytope width two", "weak order polynomial positivity", "linear coefficient order polynomial lower bound", "non-nesting rook placements descents width two poset", "Skew shapes Ehrhart positivity and beyond Conjecture 7.1" ], "abstract": "For every finite poset P that is the union of two chains A and B with |A| ≤ 5, the chain B being arbitrarily long and the comparabilities between the chains arbitrary, every coefficient of |P|! Ω_P(t) is a nonnegative integer, where Ω_P is the order polynomial; this is a case of Conjecture 7.1 of Ferroni, Morales and Panova. The linear coefficient is at least (a − 1)!(b − 1)! m/(a + b)!, with m the number of elements of B comparable with some element of A, so it is positive exactly when the comparability graph is connected. The proof reduces the linear coefficient on each of 64 chambers to a polynomial in the gaps between boundaries, certified nonnegative by exact square certificates and 194,112 point identities, and lifts it to all coefficients by the ideal–filter reduction of Ferroni, Morales and Panova.", "scope": "Lean 4.35.0 checks the chamber certificates for a short chain of size at most four: for each of the 22 chambers the kernel (decide +kernel, no native_decide) verifies that squares and a nonnegative remainder add up exactly to the certified gap polynomial, and Lean proves that each such polynomial is nonnegative on the closed orthant of any linearly ordered field and dominates (a − 1)! times the sum of the comparable gaps, and that the chambers are exactly the Dyck words. The identification of the gap polynomials with the linear coefficient of the order polynomial (descent–rook correspondence, degree bound, determining set and point identities), the 42 chambers for a = 5 and the reduction to the linear coefficient have conventional proofs or certificates checked by an independent standard-library program.", "problem": "Ferroni–Morales–Panova conjecture on positivity of order polynomials of width-two posets", "posed_by": "Ferroni, Morales and Panova, Skew shapes, Ehrhart positivity and beyond, arXiv:2503.16403v3, Conjecture 7.1; Proceedings of the London Mathematical Society 133 (2026), e70204", "problem_url": "https://arxiv.org/abs/2503.16403v3", "status": "Partially resolved", "claim": "If a finite poset P is the union of two chains, one with at most five elements and the other arbitrary, then |P|! Ω_P(t) has nonnegative coefficients, and [t]Ω_P(t) ≥ (a − 1)!(b − 1)! m/(a + b)!." }, { "id": "berge-c4-saturation", "title": "Three-uniform Berge-C₄ saturation: linear bounds with coefficients 5/8 and 3/4, the cone spectrum and exact small values", "url": "https://dvgyl.github.io/math-and-theory/papers/three-uniform-berge-c4-saturation-linear-bounds-with-coefficients-5-8-and-3-4-the-cone-spectrum-and-exact-small-values/", "pdf": "https://dvgyl.github.io/math-and-theory/papers/three-uniform-berge-c4-saturation-linear-bounds-with-coefficients-5-8-and-3-4-the-cone-spectrum-and-exact-small-values/manuscript.pdf", "discipline": "Mathematics", "area": "Extremal hypergraph theory", "subject": "Graph theory and graph polynomials", "subject_url": "https://dvgyl.github.io/math-and-theory/subjects/graph-theory-and-graph-polynomials/", "href": "/math-and-theory/papers/three-uniform-berge-c4-saturation-linear-bounds-with-coefficients-5-8-and-3-4-the-cone-spectrum-and-exact-small-values/", "tone": "other", "queries": [ "Berge C4 saturation number", "sat_3(n, Berge C4)", "saturation number of Berge cycles in 3-uniform hypergraphs", "Berge-C4-saturated hypergraph", "unrestricted Berge four-cycle saturation", "saturation of Berge hypergraphs English Graber Kirkpatrick Methuku Sullivan", "Berge C4 saturation spectrum cones", "hypergraph saturation lower bound Berge cycle" ], "rank": 200, "result_type": "Two-sided asymptotic bounds, structural classification and exact small values with reproducible proof package", "formal_proof": "standalone", "standout": false, "year": "2026", "keywords": [ "Berge hypergraphs", "Berge cycles", "Berge-C4", "saturation numbers", "three-uniform hypergraphs", "saturation spectrum", "extremal hypergraph theory", "Three-uniform Berge-C₄ saturation number (English–Graber–Kirkpatrick–Methuku–Sullivan)", "Berge C4 saturation number", "sat_3(n, Berge C4)", "saturation number of Berge cycles in 3-uniform hypergraphs", "Berge-C4-saturated hypergraph", "unrestricted Berge four-cycle saturation", "saturation of Berge hypergraphs English Graber Kirkpatrick Methuku Sullivan", "Berge C4 saturation spectrum cones", "hypergraph saturation lower bound Berge cycle" ], "abstract": "The three-uniform Berge-C₄ saturation number s(n), in which any absent triple may be added, satisfies max(⌈(n − 1)/2⌉, ⌈(5n − 3623)/8⌉, ⌈(5n + 89)/8 − √(270n + 1891)/4⌉) ≤ s(n) ≤ ⌈(3n − 9)/4⌉ for every n ≥ 9, so 5/8 ≤ liminf s(n)/n ≤ limsup s(n)/n ≤ 3/4; the previous explicit upper bound was n − 2. The preprint classifies the saturated cones for n ≥ 5, whose sizes are exactly the integers from ⌈(3n − 5)/4⌉ to n − 2 together with n − 1 when n ≡ 1 (mod 3), proves that a vertex reached from every other vertex by paths of lengths one and two forces 3(n − 1)/4 hyperedges, and gives s(n) for n ≤ 13 from exhaustive certificates.", "scope": "Lean 4.35.0 proves, for actual three-uniform hypergraphs on Fin n, Berge-C₄s with distinct vertices and distinct hyperedges, and saturation under every absent triple, that for every n ≥ 9 a saturated hypergraph with ⌈(3n − 9)/4⌉ hyperedges exists, together with the general rooted-gluing theorem, the exact values for n ≤ 5 and the upper bounds for n = 6, 7, 8; explicit pieces are kernel-checked without native_decide. The lower bound, the cone classification and spectrum and the flexible-root theorem have conventional proofs, and the values for 6 ≤ n ≤ 13 rest on exhaustive certificates checked by an independent C++ verifier.", "problem": "Three-uniform Berge-C₄ saturation number (English–Graber–Kirkpatrick–Methuku–Sullivan)", "posed_by": "English, Graber, Kirkpatrick, Methuku and Sullivan, Saturation of Berge hypergraphs, arXiv:1710.03735v1, Theorem 5.7 (upper bound n − 2)", "problem_url": "https://arxiv.org/abs/1710.03735v1", "status": "Linear coefficient bounded between 5/8 and 3/4; exact values for n ≤ 13", "claim": "For n ≥ 9, (5n − 3623)/8 ≤ sat₃(n, Berge-C₄) ≤ ⌈(3n − 9)/4⌉; saturated cones are classified and the values for n ≤ 13 are exact." }, { "id": "galvin-zhang", "title": "Ultra-log-concavity of domination sequences of powers of paths and cycles", "url": "https://dvgyl.github.io/math-and-theory/papers/ultra-log-concavity-of-domination-sequences-of-powers-of-paths-and-cycles/", "pdf": "https://dvgyl.github.io/math-and-theory/papers/ultra-log-concavity-of-domination-sequences-of-powers-of-paths-and-cycles/manuscript.pdf", "discipline": "Mathematics", "area": "Graph theory", "subject": "Graph theory and graph polynomials", "subject_url": "https://dvgyl.github.io/math-and-theory/subjects/graph-theory-and-graph-polynomials/", "href": "/math-and-theory/papers/ultra-log-concavity-of-domination-sequences-of-powers-of-paths-and-cycles/", "tone": "proved", "queries": [ "Galvin Zhang conjecture domination polynomial", "Galvin-Zhang conjecture 1.8 ultra-log-concave", "domination polynomial powers of paths and cycles log-concave", "ultra-log-concavity domination sequence", "domination polynomial unimodal powers of cycles", "number of dominating sets log-concave path power", "Galvin–Zhang domination polynomial proof" ], "rank": 210, "result_type": "Ultra-log-concavity of domination sequences", "formal_proof": "standalone", "standout": false, "year": "2026", "keywords": [ "domination polynomial", "ultra-log-concavity", "powers of paths", "powers of cycles", "Galvin–Zhang", "dominating sets", "Galvin–Zhang conjecture on ultra-log-concave domination sequences of path and cycle powers", "Galvin Zhang conjecture domination polynomial", "Galvin-Zhang conjecture 1.8 ultra-log-concave", "domination polynomial powers of paths and cycles log-concave", "ultra-log-concavity domination sequence", "domination polynomial unimodal powers of cycles", "number of dominating sets log-concave path power", "Galvin–Zhang domination polynomial proof" ], "abstract": "The domination counts of every natural power of a path or cycle form an ultra-log-concave sequence. The proof includes all natural graph orders and powers, the binomial normalization, and the boundary graph cases.", "scope": "Actual finite graph domination counts for powers of paths and cycles. No theorem for arbitrary graphs or real-rootedness is asserted.", "problem": "Galvin–Zhang conjecture on ultra-log-concave domination sequences of path and cycle powers", "posed_by": "Galvin and Zhang, arXiv:2408.12731, Conjecture 1.8; also Zhang's 2025 dissertation, Conjecture 4.1.8", "problem_url": "https://arxiv.org/abs/2408.12731", "status": "Proved", "claim": "For every n ≥ 0 and power ℓ ≥ 1, the number d_k of size-k dominating sets of P_n^ℓ and of C_n^ℓ satisfies (k+1)(n−k+1)d_{k−1}d_{k+1} ≤ k(n−k)d_k², so d_k/binom(n,k) is log-concave." }, { "id": "multimarginal-minty", "title": "The multi-marginal converse of Minty's theorem fails in the plane", "url": "https://dvgyl.github.io/math-and-theory/papers/the-multi-marginal-converse-of-minty-s-theorem-fails-in-the-plane/", "pdf": "https://dvgyl.github.io/math-and-theory/papers/the-multi-marginal-converse-of-minty-s-theorem-fails-in-the-plane/manuscript.pdf", "discipline": "Mathematics", "area": "Optimization", "subject": "Optimization", "subject_url": "https://dvgyl.github.io/math-and-theory/subjects/optimization/", "href": "/math-and-theory/papers/the-multi-marginal-converse-of-minty-s-theorem-fails-in-the-plane/", "tone": "refuted", "queries": [ "multi-marginal Minty theorem", "multi-marginal maximal monotonicity and convex analysis", "Bartz Bauschke Phan Wang Remark 2.6", "maximally c-monotone set sum image not surjective", "Minty theorem converse multi-marginal counterexample", "c-monotonicity multi-marginal optimal transport cost sum of inner products", "partition of the identity firmly nonexpansive maps extension", "Bartz-Bauschke-Phan-Wang multimarginal Minty question" ], "rank": 220, "result_type": "Counterexample answering a published question and reproducible proof package", "formal_proof": "standalone", "standout": false, "year": "2026", "keywords": [ "multi-marginal optimal transport", "c-monotonicity", "maximal monotone operators", "Minty's theorem", "firmly nonexpansive mappings", "partition of the identity", "convex analysis", "Multi-marginal converse of Minty's theorem (Bartz–Bauschke–Phan–Wang, Remark 2.6)", "multi-marginal Minty theorem", "multi-marginal maximal monotonicity and convex analysis", "Bartz Bauschke Phan Wang Remark 2.6", "maximally c-monotone set sum image not surjective", "Minty theorem converse multi-marginal counterexample", "c-monotonicity multi-marginal optimal transport cost sum of inner products", "partition of the identity firmly nonexpansive maps extension", "Bartz-Bauschke-Phan-Wang multimarginal Minty question" ], "abstract": "Bartz, Bauschke, Phan and Wang asked whether the converse half of Minty's theorem extends to several marginals: does maximal c-monotonicity for the cost Σ⟨xᵢ, xⱼ⟩ force the sum image to be the whole space? For three marginals on ℝ², three integer points form a c-monotone set to which no tuple with sum zero can be added, because four compatibility inequalities have a positive weighted sum equal to a sum of squares plus 23/26. Every maximal extension, which exists by Zorn's lemma, therefore has a sum image that omits the origin and is not convex.", "scope": "The whole argument is formalized in Lean for the actual two-point swap condition and maximality among all subsets of (ℝ²)³: the finite obstruction over arbitrary real tuples, the Zorn extension and the nonconvexity of the sum image. The reformulation through firmly nonexpansive maps is proved in the text only.", "problem": "Multi-marginal converse of Minty's theorem (Bartz–Bauschke–Phan–Wang, Remark 2.6)", "posed_by": "Bartz, Bauschke, Phan and Wang, Math. Program. 185 (2021), Remark 2.6", "problem_url": "https://doi.org/10.1007/s10107-019-01433-9", "status": "Answered negatively", "claim": "For three marginals on ℝ² with cost Σ⟨xᵢ, xⱼ⟩, an explicit three-point c-monotone set admits no compatible zero-sum tuple, so every maximally c-monotone set containing it has a nonconvex sum image omitting 0." }, { "id": "berge-triangle-spectrum", "title": "The upper saturation spectrum of Berge triangles in triple systems", "url": "https://dvgyl.github.io/math-and-theory/papers/the-upper-saturation-spectrum-of-berge-triangles-in-triple-systems/", "pdf": "https://dvgyl.github.io/math-and-theory/papers/the-upper-saturation-spectrum-of-berge-triangles-in-triple-systems/manuscript.pdf", "discipline": "Mathematics", "area": "Graph theory", "subject": "Graph theory and graph polynomials", "subject_url": "https://dvgyl.github.io/math-and-theory/subjects/graph-theory-and-graph-polynomials/", "href": "/math-and-theory/papers/the-upper-saturation-spectrum-of-berge-triangles-in-triple-systems/", "tone": "partial", "queries": [ "Berge triangle saturation spectrum", "saturation spectrum Berge K3 3-uniform hypergraph", "Berge-triangle-free triple systems stability", "triangle-free triple systems paired product", "Bushaw English Heath Johnston Rombach Berge triangle", "edge spectrum of Berge-triangle-saturated hypergraphs", "Győri n²/8 Berge triangle near-extremal structure", "sharp stability threshold triangle-free 3-graphs (n−1)²/8" ], "rank": 230, "result_type": "Sharp structural threshold and reproducible proof package", "formal_proof": "standalone", "standout": false, "year": "2026", "keywords": [ "Berge triangle", "saturation spectrum", "triple systems", "3-uniform hypergraphs", "Berge-triangle-free", "paired product", "stability", "extremal hypergraph theory", "Saturation spectrum of the Berge triangle in triple systems (Bushaw–English–Heath–Johnston–Rombach)", "Berge triangle saturation spectrum", "saturation spectrum Berge K3 3-uniform hypergraph", "Berge-triangle-free triple systems stability", "triangle-free triple systems paired product", "Bushaw English Heath Johnston Rombach Berge triangle", "edge spectrum of Berge-triangle-saturated hypergraphs", "Győri n²/8 Berge triangle near-extremal structure", "sharp stability threshold triangle-free 3-graphs (n−1)²/8" ], "abstract": "For every n ≥ 8, a Berge-triangle-free triple system on n vertices with more than ⌊(n−1)²/8⌋ edges lies inside a paired product F(n,p): p disjoint pairs, the other n − 2p vertices, and all triples formed by one pair and one other vertex. Saturated systems above this threshold are therefore paired products, and the saturation spectrum there consists exactly of the sizes p(n − 2p) up to Győri's maximum ⌊n²/8⌋. The threshold is sharp, since a saturated system with exactly ⌊(n−1)²/8⌋ edges that is not a paired product exists for every n ≥ 8. About n/4 sizes lie in the window and n/4 − O(√n) of them are missing, so the missing sizes near the top are not finitely many uniformly in n.", "scope": "Lean 4.35.0 proves that paired products are Berge-triangle-free and saturated with p(n−2p) edges, that for every n ≥ 8 the explicit system J(⌊n/4⌋, n−2⌊n/4⌋) is saturated, is not a paired product and has exactly ⌊(n−1)²/8⌋ edges, and the arithmetic of the window, including the explicit deficits and at least ⌊n/4⌋ − ⌊√(2n)⌋/2 − 1 sizes there that are not of the form p(n−2p). It derives the upper spectrum and the maximum ⌊(n−1)²/8⌋ for saturated nonproducts from the containment theorem, which enters as a hypothesis and has a conventional proof, checked computationally for small orders.", "problem": "Saturation spectrum of the Berge triangle in triple systems (Bushaw–English–Heath–Johnston–Rombach)", "posed_by": "Bushaw, English, Heath, Johnston and Rombach, The saturation spectrum of Berge stars, arXiv:2502.17686 (2025), Section 6", "problem_url": "https://arxiv.org/abs/2502.17686", "status": "Partially resolved", "claim": "For n ≥ 8 every saturated Berge-triangle-free triple system with more than ⌊(n−1)²/8⌋ edges is a paired product F(n,p), so the spectrum above this sharp threshold is exactly {p(n−2p)}." }, { "id": "singer-orbit-blocking", "title": "Blocking sets of PG(2,13) invariant under a fixed-point-free Singer subgroup meet some line in five points", "url": "https://dvgyl.github.io/math-and-theory/papers/blocking-sets-of-pg-2-13-invariant-under-a-fixed-point-free-singer-subgroup-meet-some-line-in-five-points/", "pdf": "https://dvgyl.github.io/math-and-theory/papers/blocking-sets-of-pg-2-13-invariant-under-a-fixed-point-free-singer-subgroup-meet-some-line-in-five-points/manuscript.pdf", "discipline": "Mathematics", "area": "Finite geometry", "subject": "Finite geometry", "subject_url": "https://dvgyl.github.io/math-and-theory/subjects/finite-geometry/", "href": "/math-and-theory/papers/blocking-sets-of-pg-2-13-invariant-under-a-fixed-point-free-singer-subgroup-meet-some-line-in-five-points/", "tone": "proved", "queries": [ "blocking sets PG(2,13) line intersections", "Erdős problem 1159 blocking set bounded intersection", "Singer cycle invariant blocking set", "blocking set meeting every line in at most C points", "Blokhuis Brouwer Wilbrink 21-point blocking set PG(2,13)", "fixed-point-free collineation order three blocking set", "projective plane order 13 blocking set maximum line intersection", "kernel-checked exhaustive search Lean blocking set" ], "rank": 240, "result_type": "Exact finite classification and reproducible proof package", "formal_proof": "standalone", "standout": false, "year": "2026", "keywords": [ "blocking sets", "projective planes", "PG(2,13)", "Singer cycle", "line intersections", "Erdős problem 1159", "finite geometry", "kernel-checked search", "Blocking sets with bounded line intersections in finite projective planes (Erdős Problem #1159), Singer-symmetric case in PG(2,13)", "blocking sets PG(2,13) line intersections", "Erdős problem 1159 blocking set bounded intersection", "Singer cycle invariant blocking set", "blocking set meeting every line in at most C points", "Blokhuis Brouwer Wilbrink 21-point blocking set PG(2,13)", "fixed-point-free collineation order three blocking set", "projective plane order 13 blocking set maximum line intersection", "kernel-checked exhaustive search Lean blocking set" ], "abstract": "Let H be the subgroup of order three of a Singer cycle group of PG(2,13); it fixes no point. Every H-invariant blocking set of PG(2,13) meets some line in at least five points, and the 21-point blocking set of Blokhuis, Brouwer and Wilbrink, invariant under a conjugate of H, meets every line in at most five points. So five is the least possible largest line intersection of an H-invariant blocking set, for every fixed-point-free collineation of order three. This settles a symmetric case of the plane of order 13 in Erdős Problem 1159; blocking sets without this symmetry are not covered.", "scope": "Lean 4.35.0 models PG(2,13) with Mathlib's projectivization of F₁₃³ and proves that the Singer cycle has order 183, that its subgroup H of order three fixes no point, that the 21-point witness is an H-invariant minimal blocking set with largest line intersection five, the reduction of every H-invariant set to 61 orbit conditions, and the nonexistence of a solution with intersections at most four, by a kernel-checked search with a soundness proof; the main theorem is IsLeast admissibleCaps 5. The conjugacy of all fixed-point-free subgroups of order three is conventional.", "problem": "Blocking sets with bounded line intersections in finite projective planes (Erdős Problem #1159), Singer-symmetric case in PG(2,13)", "posed_by": "P. Erdős; Erdős Problem #1159, erdosproblems.com", "problem_url": "https://www.erdosproblems.com/1159", "status": "Determined exactly", "claim": "Every blocking set of PG(2,13) invariant under the fixed-point-free subgroup of order three of a Singer group meets some line in at least five points, and the 21-point Blokhuis–Brouwer–Wilbrink set attains five." }, { "id": "unit-squareclasses", "title": "A rank bound for represented totally positive unit squareclasses", "url": "https://dvgyl.github.io/math-and-theory/papers/a-rank-bound-for-represented-totally-positive-unit-squareclasses/", "pdf": "https://dvgyl.github.io/math-and-theory/papers/a-rank-bound-for-represented-totally-positive-unit-squareclasses/manuscript.pdf", "discipline": "Mathematics", "area": "Quadratic forms and number theory", "subject": "Quadratic forms and number theory", "subject_url": "https://dvgyl.github.io/math-and-theory/subjects/quadratic-forms-and-number-theory/", "href": "/math-and-theory/papers/a-rank-bound-for-represented-totally-positive-unit-squareclasses/", "tone": "refuted", "queries": [ "totally positive unit square classes quadratic lattice", "Daans Kala Man Widmer Yatsyna Question 3.4", "most totally real fields do not have universal forms or the Northcott property", "universal quadratic forms totally real fields unit square classes", "nonclassical quadratic lattice represented units", "rank of universal lattice totally positive units", "unit squareclasses represented by quadratic form" ], "rank": 250, "result_type": "Theorem answering a published question and reproducible proof package", "formal_proof": "standalone", "standout": false, "year": "2026", "keywords": [ "quadratic lattices", "totally real fields", "unit square classes", "universal quadratic forms", "nonclassical lattices", "reflection groups", "Daans–Kala–Man–Widmer–Yatsyna", "Daans–Kala–Man–Widmer–Yatsyna Question 3.4 on represented unit square classes", "totally positive unit square classes quadratic lattice", "Daans Kala Man Widmer Yatsyna Question 3.4", "most totally real fields do not have universal forms or the Northcott property", "universal quadratic forms totally real fields unit square classes", "nonclassical quadratic lattice represented units", "rank of universal lattice totally positive units", "unit squareclasses represented by quadratic form" ], "abstract": "Let K be a totally real algebraic field, possibly of infinite degree, and L a totally positive definite quadratic lattice over its ring of integers with Q(L) ⊆ O_K, not necessarily classical. Then every family of pairwise distinct classes of totally positive units modulo squares of units represented by L has at most rank L members, and the bound is attained when the field has enough classes. This answers negatively Question 3.4 of Daans, Kala, Man, Widmer and Yatsyna, who proved the bound 2n − 2 and asked whether n + 1 classes can occur. The bound is a short consequence of standard facts on finite reflection groups: unit-valued vectors give integral reflections, and conjugate reflections carry the same unit class. As a corollary, a universal lattice has rank at least the number of totally positive unit classes.", "scope": "The bound for every algebraic totally real field, including infinite extensions, every O_K-submodule with integral norms in a finite-dimensional totally positive definite space, and actual units of the ring of integers is formalized in 11 Lean modules. The derivation from finite reflection groups, sharpness and the universality corollary are conventional.", "problem": "Daans–Kala–Man–Widmer–Yatsyna Question 3.4 on represented unit square classes", "posed_by": "Daans, Kala, Man, Widmer and Yatsyna, PNAS 122(20), e2419414122 (2025), Question 3.4", "problem_url": "https://arxiv.org/abs/2409.11082v2", "status": "Answered negatively", "claim": "A totally positive definite quadratic lattice of rank n over any totally real algebraic field, with integral norms but not necessarily classical, represents at most n totally positive unit square classes, so n + 1 cannot occur." }, { "id": "readk-covariance", "title": "Logarithmic influence bounds and variance counterexamples for Boolean decision-tree covariance", "url": "https://dvgyl.github.io/math-and-theory/papers/logarithmic-influence-bounds-and-variance-counterexamples-for-boolean-decision-tree-covariance/", "pdf": "https://dvgyl.github.io/math-and-theory/papers/logarithmic-influence-bounds-and-variance-counterexamples-for-boolean-decision-tree-covariance/manuscript.pdf", "discipline": "Theoretical computer science", "area": "Boolean functions and decision trees", "subject": "Boolean functions and decision trees", "subject_url": "https://dvgyl.github.io/math-and-theory/subjects/boolean-functions-and-decision-trees/", "href": "/math-and-theory/papers/logarithmic-influence-bounds-and-variance-counterexamples-for-boolean-decision-tree-covariance/", "tone": "refuted", "queries": [ "Wan Wright Wu decision tree covariance conjecture", "Wan-Wright-Wu logarithmic variance bound counterexample", "read-k decision tree Fourier entropy influence covariance", "Shalev Fourier entropy influence conjecture extremal classes read-k", "decision tree covariance log k variance counterexample", "Boolean decision tree covariance influence bound log2 k", "Fourier entropy-influence conjecture read-k decision trees" ], "rank": 260, "result_type": "Finite covariance bounds and variance counterexamples", "formal_proof": "standalone", "standout": false, "year": "2026", "keywords": [ "decision trees", "Boolean functions", "covariance inequalities", "influences", "read-k formulas", "Wan–Wright–Wu log₂ k variance bound for read-k decision-tree covariance", "Wan Wright Wu decision tree covariance conjecture", "Wan-Wright-Wu logarithmic variance bound counterexample", "read-k decision tree Fourier entropy influence covariance", "Shalev Fourier entropy influence conjecture extremal classes read-k", "decision tree covariance log k variance counterexample", "Boolean decision tree covariance influence bound log2 k", "Fourier entropy-influence conjecture read-k decision trees" ], "abstract": "For a finite Boolean decision tree under uniform independent bits, the sum of absolute child covariances, weighted by node reach probabilities, is at most the sum of each coordinate influence times the base-two logarithm of its global query-node count. Counts zero and one receive weight zero. No coordinate repeats on a root-to-leaf path. A read-k tree therefore has absolute covariance at most I(f) log₂ k. The known dummy-prefix construction attains equality for powers of two. A four-variable witness and a nine-variable witness refute the coefficient-one logarithmic variance proposal. Every query in the latter witness is relevant. The formalization contains no Fourier-entropy theorem.", "scope": "For finite Boolean decision trees with no repeated coordinate on a root-to-leaf path, a logarithmic influence bound controls signed and absolute tree covariance. A known family attains equality at powers of two. Exact read-3 examples disprove the proposed coefficient-one variance replacement, including an example in which every query is relevant. No Fourier entropy theorem is formalized.", "problem": "Wan–Wright–Wu log₂ k variance bound for read-k decision-tree covariance", "posed_by": "Wan, Wright and Wu, ITCS 2014, Section 2.4; called a conjecture by Shalev, arXiv:1806.03646v2, Section 6.2", "problem_url": "https://doi.org/10.1145/2554797.2554806", "status": "Disproved", "claim": "Read-k decision trees with no repeated coordinate on a path have tree covariance at most I(f) log₂ k, sharp for k = 2^d, while exact read-3 trees refute the proposed bound Var(f) log₂ k." }, { "id": "diagonal-concavity", "title": "Diagonal concavity does not imply concavity for sums of elementary symmetric polynomials", "url": "https://dvgyl.github.io/math-and-theory/papers/diagonal-concavity-does-not-imply-concavity-for-sums-of-elementary-symmetric-polynomials/", "pdf": "https://dvgyl.github.io/math-and-theory/papers/diagonal-concavity-does-not-imply-concavity-for-sums-of-elementary-symmetric-polynomials/manuscript.pdf", "discipline": "Mathematics", "area": "Inequalities and real analysis", "subject": "Inequalities and real analysis", "subject_url": "https://dvgyl.github.io/math-and-theory/subjects/inequalities-and-real-analysis/", "href": "/math-and-theory/papers/diagonal-concavity-does-not-imply-concavity-for-sums-of-elementary-symmetric-polynomials/", "tone": "refuted", "queries": [ "Lachaume conjecture elementary symmetric polynomials concavity", "Lachaume diagonal concavity conjecture counterexample", "concavity of p-th root of sum of elementary symmetric polynomials", "fourth root elementary symmetric polynomials concave positive orthant", "diagonal criterion concavity symmetric functions", "arXiv 1712.10327 Conjecture 1", "concavity of hyperbolic polynomials elementary symmetric sums" ], "rank": 270, "result_type": "Counterexample and complete parameter classification", "formal_proof": "comparator", "standout": false, "year": "2026", "keywords": [ "elementary symmetric polynomials", "concavity", "Lachaume criterion", "hyperbolic polynomials", "fourth root concavity", "Lachaume's diagonal concavity conjecture for sums of elementary symmetric polynomials", "Lachaume conjecture elementary symmetric polynomials concavity", "Lachaume diagonal concavity conjecture counterexample", "concavity of p-th root of sum of elementary symmetric polynomials", "fourth root elementary symmetric polynomials concave positive orthant", "diagonal criterion concavity symmetric functions", "arXiv 1712.10327 Conjecture 1", "concavity of hyperbolic polynomials elementary symmetric sums" ], "abstract": "For positive M, let f=1+Mσ₁+σ₂+Mσ₃+σ₄ in four variables. Its fourth root is concave on the positive diagonal exactly when M≥1. It is concave on the full positive orthant exactly when M=1. Every M>1 therefore disproves Lachaume’s diagonal criterion with strictly positive coefficients.", "scope": "For positive M, let f=1+Mσ₁+σ₂+Mσ₃+σ₄ in four variables. Its fourth root is concave on the positive diagonal exactly when M≥1. It is concave on the full positive orthant exactly when M=1. Every M>1 therefore disproves Lachaume’s diagonal criterion with strictly positive coefficients.", "problem": "Lachaume's diagonal concavity conjecture for sums of elementary symmetric polynomials", "posed_by": "Lachaume, arXiv:1712.10327v1, Conjecture 1", "problem_url": "https://arxiv.org/abs/1712.10327v1", "status": "Disproved", "claim": "For f = 1 + Mσ₁ + σ₂ + Mσ₃ + σ₄ in four variables with M > 0, f^(1/4) is concave on the positive diagonal exactly when M ≥ 1 and on the whole positive orthant exactly when M = 1, so every M > 1 is a counterexample." }, { "id": "best-of-n", "title": "Counterexamples to a best-of-n win-rate bound at every budget", "url": "https://dvgyl.github.io/math-and-theory/papers/counterexamples-to-a-best-of-n-win-rate-bound-at-every-budget/", "pdf": "https://dvgyl.github.io/math-and-theory/papers/counterexamples-to-a-best-of-n-win-rate-bound-at-every-budget/manuscript.pdf", "discipline": "Theoretical computer science", "area": "Sampling algorithms and information theory", "subject": "Sampling algorithms and information theory", "subject_url": "https://dvgyl.github.io/math-and-theory/subjects/sampling-algorithms-and-information-theory/", "href": "/math-and-theory/papers/counterexamples-to-a-best-of-n-win-rate-bound-at-every-budget/", "tone": "refuted", "queries": [ "best-of-n win rate KL divergence conjecture", "Beirami best-of-n conjecture 7.2 counterexample", "theoretical guarantees on the best-of-n alignment policy", "best-of-n sampling KL divergence win rate tradeoff", "best of n alignment policy win rate bound", "BoN sampling KL bound log n - (n-1)/n", "best-of-n win rate curve counterexample" ], "rank": 280, "result_type": "Disproof for every budget n≥2 and every positive atom cap", "formal_proof": "comparator", "standout": false, "year": "2026", "keywords": [ "best-of-n sampling", "win rate", "KL divergence", "language model alignment", "counterexample", "Beirami et al.'s best-of-n win-rate versus KL divergence conjecture (Conjecture 7.2)", "best-of-n win rate KL divergence conjecture", "Beirami best-of-n conjecture 7.2 counterexample", "theoretical guarantees on the best-of-n alignment policy", "best-of-n sampling KL divergence win rate tradeoff", "best of n alignment policy win rate bound", "BoN sampling KL bound log n - (n-1)/n", "best-of-n win rate curve counterexample" ], "abstract": "For every n≥2 and δ>0, there is a finite positive reference distribution whose largest atom is at most δ and whose best-of-n policy violates the proposed divergence–win-rate curve. A four-response best-of-two example gives exact probabilities and a strict logarithmic certificate. The proof uses finite interval bounds and gives a quantitative positive gap. It does not determine the sharp replacement curve.", "scope": "For every n≥2 and δ>0, there is a finite positive reference distribution whose largest atom is at most δ and whose best-of-n policy violates the proposed divergence–win-rate curve. A four-response best-of-two example gives exact probabilities and a strict logarithmic certificate. The proof uses finite interval bounds and gives a quantitative positive gap. It does not determine the sharp replacement curve.", "problem": "Beirami et al.'s best-of-n win-rate versus KL divergence conjecture (Conjecture 7.2)", "posed_by": "Beirami et al., ICML 2025 (PMLR 267), Conjecture 7.2 and Eq. (28)", "problem_url": "https://arxiv.org/abs/2401.01879v3", "status": "Disproved", "claim": "For every budget n ≥ 2 and cap δ > 0, a finite reference law with all probabilities ≤ δ has a best-of-n policy with D < ℓ(W), violating W ≤ ℓ⁻¹(D); for n = 2, p = (2/5,1/5,1/5,1/5) gives W = 163/250." }, { "id": "q-permanent", "title": "Real positive-definite counterexamples to extended q-permanent monotonicity", "url": "https://dvgyl.github.io/math-and-theory/papers/real-positive-definite-counterexamples-to-extended-q-permanent-monotonicity/", "pdf": "https://dvgyl.github.io/math-and-theory/papers/real-positive-definite-counterexamples-to-extended-q-permanent-monotonicity/manuscript.pdf", "discipline": "Mathematics", "area": "Matrix theory", "subject": "Matrix theory", "subject_url": "https://dvgyl.github.io/math-and-theory/subjects/matrix-theory/", "href": "/math-and-theory/papers/real-positive-definite-counterexamples-to-extended-q-permanent-monotonicity/", "tone": "refuted", "queries": [ "q-permanent monotonicity conjecture", "Da Fonseca q-permanent conjecture counterexample", "da Fonseca mu-permanent conjecture", "μ-permanent positive definite matrix monotone", "q-permanent positive definite matrix decreasing", "inversion-weighted permanent monotonicity", "Bapat-Lal q-permanent conjecture", "q-permanent correlation matrix counterexample" ], "rank": 290, "result_type": "Counterexample and all-parameter extension", "formal_proof": "comparator", "standout": false, "year": "2026", "keywords": [ "q-permanent", "permanent monotonicity", "positive definite matrices", "Da Fonseca conjecture", "correlation matrices", "matrix permanents", "Da Fonseca's q-permanent (μ-permanent) monotonicity conjecture", "q-permanent monotonicity conjecture", "Da Fonseca q-permanent conjecture counterexample", "da Fonseca mu-permanent conjecture", "μ-permanent positive definite matrix monotone", "q-permanent positive definite matrix decreasing", "inversion-weighted permanent monotonicity", "Bapat-Lal q-permanent conjecture", "q-permanent correlation matrix counterexample" ], "abstract": "For every real parameter q>1, a real symmetric positive-definite matrix has a negative q-permanent derivative at q. An explicit matrix of order five decreases from q=14 to q=15. Correlation matrices with no zero entries and integer-entry examples are also obtained. The dimension can depend on q. The real-symmetric question on [−1,1] remains outside the result.", "scope": "For every real parameter q>1, a real symmetric positive-definite matrix has a negative q-permanent derivative at q. An explicit matrix of order five decreases from q=14 to q=15. Correlation matrices with no zero entries and integer-entry examples are also obtained. The dimension can depend on q. The real-symmetric question on [−1,1] remains outside the result.", "problem": "Da Fonseca's q-permanent (μ-permanent) monotonicity conjecture", "posed_by": "Da Fonseca (2010), Conjecture 1; restated as Conjecture 2 in Da Fonseca (2018)", "problem_url": "https://doi.org/10.1016/j.laa.2009.10.036", "status": "Disproved", "claim": "A real symmetric positive-definite 5×5 matrix has q-permanent 8,968,822,249 at q = 14 and 7,441,960,000 at q = 15; for each real q₀ > 1 a positive-definite matrix (size depending on q₀) has negative derivative at q₀." }, { "id": "hofstadter-iteration", "title": "Counterexamples to a late-iterate ordering conjecture for generalized Hofstadter functions", "url": "https://dvgyl.github.io/math-and-theory/papers/counterexamples-to-a-late-iterate-ordering-conjecture-for-generalized-hofstadter-functions/", "pdf": "https://dvgyl.github.io/math-and-theory/papers/counterexamples-to-a-late-iterate-ordering-conjecture-for-generalized-hofstadter-functions/manuscript.pdf", "discipline": "Mathematics", "area": "Integer sequences and combinatorics", "subject": "Integer sequences and combinatorics", "subject_url": "https://dvgyl.github.io/math-and-theory/subjects/integer-sequences-and-combinatorics/", "href": "/math-and-theory/papers/counterexamples-to-a-late-iterate-ordering-conjecture-for-generalized-hofstadter-functions/", "tone": "refuted", "queries": [ "generalized Hofstadter functions pointwise order", "Hofstadter G H functions iterate conjecture counterexample", "Letouzey Li Steiner conjecture", "Letouzey-Li-Steiner Hofstadter conjecture Lemma 7.6", "pointwise order of generalized Hofstadter functions G, H, and beyond", "nested recursion F_k(n) = n - F_k^k(n-1)", "Hofstadter G function iterates ordering" ], "rank": 300, "result_type": "Counterexamples for every parameter k≥4", "formal_proof": "comparator", "standout": false, "year": "2026", "keywords": [ "Hofstadter functions", "nested recursions", "iterated functions", "counterexample", "integer sequences", "Letouzey–Li–Steiner late-iterate ordering conjecture for generalized Hofstadter functions", "generalized Hofstadter functions pointwise order", "Hofstadter G H functions iterate conjecture counterexample", "Letouzey Li Steiner conjecture", "Letouzey-Li-Steiner Hofstadter conjecture Lemma 7.6", "pointwise order of generalized Hofstadter functions G, H, and beyond", "nested recursion F_k(n) = n - F_k^k(n-1)", "Hofstadter G function iterates ordering" ], "abstract": "For every k≥4, some input n satisfies Fₖ^(2k+1)(n)>Fₖ₊₁^(2k+2)(n). The proof constructs inverse iterates whose exact gap is k−3. The finite witness F₄⁹(132)=8>7=F₅¹⁰(132) gives the first instance. The result concerns infinitely many parameters, not infinitely many violating inputs for each fixed parameter.", "scope": "For every k≥4, some input n satisfies Fₖ^(2k+1)(n)>Fₖ₊₁^(2k+2)(n). The proof constructs inverse iterates whose exact gap is k−3. The finite witness F₄⁹(132)=8>7=F₅¹⁰(132) gives the first instance. The result concerns infinitely many parameters, not infinitely many violating inputs for each fixed parameter.", "problem": "Letouzey–Li–Steiner late-iterate ordering conjecture for generalized Hofstadter functions", "posed_by": "Letouzey, Li and Steiner, Journal of Integer Sequences 29 (2026), Article 26.3.3, conjecture before Lemma 7.6", "problem_url": "https://arxiv.org/abs/2410.00529v2", "status": "Disproved", "claim": "For every k ≥ 4 some input n has Fₖ^(2k+1)(n) > Fₖ₊₁^(2k+2)(n), contradicting the conjectured order Fₖ^j ≤ Fₖ₊₁^(j+1) for j > 2k; an explicit witness is F₄⁹(132) = 8 > 7 = F₅¹⁰(132)." }, { "id": "admm-counterexample", "title": "A rational five-cycle for two-block ADMM with multiplier step below two", "url": "https://dvgyl.github.io/math-and-theory/papers/a-rational-five-cycle-for-two-block-admm-with-multiplier-step-below-two/", "pdf": "https://dvgyl.github.io/math-and-theory/papers/a-rational-five-cycle-for-two-block-admm-with-multiplier-step-below-two/manuscript.pdf", "discipline": "Mathematics", "area": "Optimization", "subject": "Optimization", "subject_url": "https://dvgyl.github.io/math-and-theory/subjects/optimization/", "href": "/math-and-theory/papers/a-rational-five-cycle-for-two-block-admm-with-multiplier-step-below-two/", "tone": "refuted", "queries": [ "ADMM dual step size golden ratio counterexample", "Glowinski open question ADMM step size", "ADMM multiplier step length (0,2) convergence", "two-block ADMM divergence counterexample", "ADMM periodic orbit nonconvergence", "ADMM dual step size larger than 1.618", "alternating direction method of multipliers tau 1.99 counterexample", "Tao Yuan Glowinski ADMM question" ], "rank": 310, "result_type": "Standalone theorem and reproducible proof package", "formal_proof": "standalone", "standout": false, "year": "2026", "keywords": [ "ADMM", "alternating direction method of multipliers", "ADMM divergence", "periodic orbit", "dual step size", "counterexample", "Convergence of two-block ADMM with multiplier (dual) step size τ in (0, 2)", "ADMM dual step size golden ratio counterexample", "Glowinski open question ADMM step size", "ADMM multiplier step length (0,2) convergence", "two-block ADMM divergence counterexample", "ADMM periodic orbit nonconvergence", "ADMM dual step size larger than 1.618", "alternating direction method of multipliers tau 1.99 counterexample", "Tao Yuan Glowinski ADMM question" ], "abstract": "An exact period-five orbit for multiplier-only two-block ADMM at step 199/100.", "scope": "Seven modules and eleven endpoint audits cover the orbit, minimizers and optimizer. Analytic regularity of the finite strongly convex extension retains a conventional proof.", "problem": "Convergence of two-block ADMM with multiplier (dual) step size τ in (0, 2)", "posed_by": "Question commonly associated with Glowinski; discussed by Tao and Yuan, J. Optim. Theory Appl. 179 (2018)", "problem_url": null, "status": "Counterexample", "claim": "For two-block ADMM with penalty one and multiplier-only step τ = 199/100, two explicit compact convex polytopes in ℝ⁴ give a nonconstant period-five orbit with squared primal residual at least 500000, so ADMM fails to converge." }, { "id": "alternating-motzkin", "title": "Sharp root log-concavity of alternating Motzkin sums", "url": "https://dvgyl.github.io/math-and-theory/papers/sharp-root-log-concavity-of-alternating-motzkin-sums/", "pdf": "https://dvgyl.github.io/math-and-theory/papers/sharp-root-log-concavity-of-alternating-motzkin-sums/manuscript.pdf", "discipline": "Mathematics", "area": "Integer sequences and combinatorics", "subject": "Integer sequences and combinatorics", "subject_url": "https://dvgyl.github.io/math-and-theory/subjects/integer-sequences-and-combinatorics/", "href": "/math-and-theory/papers/sharp-root-log-concavity-of-alternating-motzkin-sums/", "tone": "proved", "queries": [ "alternating sums of Motzkin numbers log-concave", "Zhao conjecture Motzkin root log-concavity", "log-balancedness alternating sums of Motzkin numbers", "Motzkin numbers alternating sum nth root log-concave", "OEIS A187306 log-concavity", "root log-concavity Motzkin conjecture proof", "Tn^(1/n) log-concave Motzkin" ], "rank": 320, "result_type": "Conjecture resolution with an optimal threshold", "formal_proof": "comparator", "standout": false, "year": "2026", "keywords": [ "Motzkin numbers", "alternating sums", "log-concavity", "root log-concavity", "Zhao conjecture", "Zhao's root log-concavity conjecture for alternating sums of Motzkin numbers", "alternating sums of Motzkin numbers log-concave", "Zhao conjecture Motzkin root log-concavity", "log-balancedness alternating sums of Motzkin numbers", "Motzkin numbers alternating sum nth root log-concave", "OEIS A187306 log-concavity", "root log-concavity Motzkin conjecture proof", "Tn^(1/n) log-concave Motzkin" ], "abstract": "Let Tₙ be the alternating sum of Motzkin numbers through index n. The sequence Tₙ^(1/n) is strictly log-concave on its tail starting at five. The inequality reverses at center five, so no earlier tail is log-concave. This strengthens Zhao’s proposed tail starting at nine. The source recurrence and finite sum are linked in Lean.", "scope": "Let Tₙ be the alternating sum of Motzkin numbers through index n. The sequence Tₙ^(1/n) is strictly log-concave on its tail starting at five. The inequality reverses at center five, so no earlier tail is log-concave. This strengthens Zhao’s proposed tail starting at nine. The source recurrence and finite sum are linked in Lean.", "problem": "Zhao's root log-concavity conjecture for alternating sums of Motzkin numbers", "posed_by": "Feng-Zhen Zhao, INTEGERS 26 (2026), A55, Conjecture 1", "problem_url": "https://doi.org/10.5281/zenodo.19949645", "status": "Proved", "claim": "With Tₙ the alternating sum of the Motzkin numbers M₀,…,Mₙ, the sequence Tₙ^(1/n) is strictly log-concave from n = 5 and the inequality reverses at center 5, so 5 is optimal; this contains Zhao's tail from n = 9." }, { "id": "motzkin-bounded-forcing", "title": "Sharp log-concavity for alternating Motzkin sums", "url": "https://dvgyl.github.io/math-and-theory/papers/sharp-log-concavity-for-alternating-motzkin-sums/", "pdf": "https://dvgyl.github.io/math-and-theory/papers/sharp-log-concavity-for-alternating-motzkin-sums/manuscript.pdf", "discipline": "Mathematics", "area": "Integer sequences", "subject": "Integer sequences and combinatorics", "subject_url": "https://dvgyl.github.io/math-and-theory/subjects/integer-sequences-and-combinatorics/", "href": "/math-and-theory/papers/sharp-log-concavity-for-alternating-motzkin-sums/", "tone": "proved", "queries": [ "log-concavity perturbed linear recurrence bounded error", "alternating Motzkin sums ratio log-concave", "Motzkin recurrence bounded forcing", "Riordan numbers root log-concavity threshold", "shifted Riordan numbers log-concave", "log-concavity inhomogeneous P-recursive sequence", "Zhao alternating Motzkin conjecture sharp threshold" ], "rank": 330, "result_type": "Standalone theorem and reproducible proof package", "formal_proof": "standalone", "standout": false, "year": "2026", "keywords": [ "Motzkin numbers", "log-concavity", "linear recurrences", "bounded forcing", "alternating Motzkin sums", "Bounded-forcing log-concavity for Motzkin-type recurrences and alternating Motzkin sums", "log-concavity perturbed linear recurrence bounded error", "alternating Motzkin sums ratio log-concave", "Motzkin recurrence bounded forcing", "Riordan numbers root log-concavity threshold", "shifted Riordan numbers log-concave", "log-concavity inhomogeneous P-recursive sequence", "Zhao alternating Motzkin conjecture sharp threshold" ], "abstract": "A robust bounded-forcing recurrence theorem and corrected scaled corollary, with applications to alternating Motzkin-type sums.", "scope": "The arbitrary-real recurrence theorem and scaled corollary, including zero forcing, are formalized. Concrete sequences, uniform-family applications, root transfer and threshold classification retain conventional proofs.", "problem": "Bounded-forcing log-concavity for Motzkin-type recurrences and alternating Motzkin sums", "posed_by": "Motivated by Zhao, INTEGERS 26 (2026), A55, Conjecture 1 (p. 8); the extensions answer no posed question", "problem_url": "https://doi.org/10.5281/zenodo.19949645", "status": "New theorem", "claim": "Real aₙ with (m+2)aₘ = m(2aₘ₋₁+3aₘ₋₂)+εₘ, |εₘ| ≤ 2 for m ≥ 14, a₁₂ > 1800 and L₁₂ < a₁₃/a₁₂ < U₁₂ satisfy aₙ > 8(n+3)² and a ratio log-concavity margin 800aₙ⁴/(n+3)³ for all n ≥ 12." }, { "id": "sidon-renyi", "title": "Sharp universal Rényi threshold for iid sums on finite Sidon sets", "url": "https://dvgyl.github.io/math-and-theory/papers/sharp-universal-renyi-threshold-for-iid-sums-on-finite-sidon-sets/", "pdf": "https://dvgyl.github.io/math-and-theory/papers/sharp-universal-renyi-threshold-for-iid-sums-on-finite-sidon-sets/manuscript.pdf", "discipline": "Mathematics", "area": "Additive combinatorics and entropy", "subject": "Discrete analysis, tilings, probability and entropy", "subject_url": "https://dvgyl.github.io/math-and-theory/subjects/discrete-analysis-tilings-probability-and-entropy/", "href": "/math-and-theory/papers/sharp-universal-renyi-threshold-for-iid-sums-on-finite-sidon-sets/", "tone": "proved", "queries": [ "Renyi entropy sum of independent random variables Sidon set", "Rényi entropy iid sum uniform distribution optimal", "Sidon set entropy maximization uniform input", "B2 set Renyi entropy sum of two iid draws", "2^(alpha-1) = 2 alpha - 1 Renyi threshold", "zero-error adder MAC Renyi entropy Sidon", "entropy of X+X' on a Sidon set" ], "rank": 340, "result_type": "Standalone theorem and reproducible proof package", "formal_proof": "standalone", "standout": false, "year": "2026", "keywords": [ "Sidon sets", "Rényi entropy", "sums of independent random variables", "additive combinatorics", "B2 sets", "entropy inequalities", "Rényi entropy of iid sums on Sidon sets: for which orders is uniform input optimal?", "Renyi entropy sum of independent random variables Sidon set", "Rényi entropy iid sum uniform distribution optimal", "Sidon set entropy maximization uniform input", "B2 set Renyi entropy sum of two iid draws", "2^(alpha-1) = 2 alpha - 1 Renyi threshold", "zero-error adder MAC Renyi entropy Sidon", "entropy of X+X' on a Sidon set" ], "abstract": "A sharp universal Renyi-order threshold for iid sums supported on finite strong-B2 sets.", "scope": "The finite-support iid theorem, zero probabilities and critical endpoint are formalized. Comparison with unverified source definitions is conditional.", "problem": "Rényi entropy of iid sums on Sidon sets: for which orders is uniform input optimal?", "posed_by": null, "problem_url": null, "status": "Determined exactly", "claim": "For Rényi order α > 1, the uniform law maximizes H_α(X+X') for iid X, X' on every nonempty finite strong Sidon (B₂) set exactly when 2^(α−1) ≤ 2α − 1, i.e. α ≤ α_c ≈ 3.65986." }, { "id": "kunisky", "title": "Unbounded coalesced-mass certificates for resolution trees", "url": "https://dvgyl.github.io/math-and-theory/papers/unbounded-coalesced-mass-certificates-for-resolution-trees/", "pdf": "https://dvgyl.github.io/math-and-theory/papers/unbounded-coalesced-mass-certificates-for-resolution-trees/manuscript.pdf", "discipline": "Theoretical computer science", "area": "Proof complexity", "subject": "Proof complexity", "subject_url": "https://dvgyl.github.io/math-and-theory/subjects/proof-complexity/", "href": "/math-and-theory/papers/unbounded-coalesced-mass-certificates-for-resolution-trees/", "tone": "proved", "queries": [ "Kunisky Komlós conjecture resolution", "Kunisky Komlos conjecture resolution trees", "discrepancy of unsatisfiable matrices Komlós constant", "Kunisky Conjecture 4.4 resolution certificate", "resolution refutation clause mass certificate", "Komlos conjecture lower bound resolution proofs", "unbounded certificate resolution trees 1+sqrt(2)" ], "rank": 350, "result_type": "Standalone theorem and reproducible proof package", "formal_proof": "standalone", "standout": false, "year": "2026", "keywords": [ "proof complexity", "resolution", "resolution trees", "clause mass", "Kunisky", "Coalesced clause-mass certificate for resolution trees (Kunisky's Komlós approach)", "Kunisky Komlós conjecture resolution", "Kunisky Komlos conjecture resolution trees", "discrepancy of unsatisfiable matrices Komlós constant", "Kunisky Conjecture 4.4 resolution certificate", "resolution refutation clause mass certificate", "Komlos conjecture lower bound resolution proofs", "unbounded certificate resolution trees 1+sqrt(2)" ], "abstract": "A coalesced dyadic clause-mass functional has exact supremum n+1 on the stated unrestricted resolution-tree class and is unbounded across dimensions.", "scope": "The internally defined resolution-tree functional and proof class are formalized. No unconditional identification with an external conjecture or a regular proof class is claimed.", "problem": "Coalesced clause-mass certificate for resolution trees (Kunisky's Komlós approach)", "posed_by": "Attributed to Kunisky, arXiv:2111.02974v1, Conjecture 4.4 and Eq. (41); literal correspondence unverified", "problem_url": "https://arxiv.org/abs/2111.02974", "status": "Determined exactly", "claim": "For every n ≥ 3, the coalesced clause-mass certificate over finite unrestricted resolution refutations of the formula of all 2ⁿ full signed clauses has exact supremum n + 1; a three-variable witness exceeds 61/25 > 1 + √2." }, { "id": "ordered-allocation", "title": "Ordered-instance allocation transfer with exact extra-copy preservation", "url": "https://dvgyl.github.io/math-and-theory/papers/ordered-instance-allocation-transfer-with-exact-extra-copy-preservation/", "pdf": "https://dvgyl.github.io/math-and-theory/papers/ordered-instance-allocation-transfer-with-exact-extra-copy-preservation/manuscript.pdf", "discipline": "Theoretical computer science", "area": "Fair division", "subject": "Fair division", "subject_url": "https://dvgyl.github.io/math-and-theory/subjects/fair-division/", "href": "/math-and-theory/papers/ordered-instance-allocation-transfer-with-exact-extra-copy-preservation/", "tone": "proved", "queries": [ "ordered instance reduction fair division", "ordered instances additive valuations indivisible goods", "maximin share allocation with copies of goods", "fair allocation extra copies indivisible goods", "MMS ordered instance reduction Hall's theorem", "Akrami fair division copies ordered instance", "reduction to ordered instances maximin share" ], "rank": 360, "result_type": "Standalone theorem and reproducible proof package", "formal_proof": "standalone", "standout": false, "year": "2026", "keywords": [ "fair division", "fair allocation", "indivisible goods", "ordered instances", "Hall's theorem", "additive valuations", "Ordered-instance reduction for allocating indivisible goods with extra copies", "ordered instance reduction fair division", "ordered instances additive valuations indivisible goods", "maximin share allocation with copies of goods", "fair allocation extra copies indivisible goods", "MMS ordered instance reduction Hall's theorem", "Akrami fair division copies ordered instance", "reduction to ordered instances maximin share" ], "abstract": "A set-valued ordered-instance transfer preserving utility improvement, the exact extra-copy count, per-good caps and coverage.", "scope": "The transfer theorem is formalized. Existence of a fair allocation before transfer is outside the result; external source comparison is provisional.", "problem": "Ordered-instance reduction for allocating indivisible goods with extra copies", "posed_by": "Attributed to Akrami et al., arXiv:2502.09377v3; source conventions unverified", "problem_url": "https://arxiv.org/abs/2502.09377v3", "status": "New theorem", "claim": "Any set-valued ordered allocation under nonnegative additive valuations transfers to named goods with every agent's utility weakly higher, the same total number of extra copies, the same per-good cap, and every good allocated." }, { "id": "leaf-deletion", "title": "Leaf deletion can move the unique mode of a tree domination polynomial by two", "url": "https://dvgyl.github.io/math-and-theory/papers/leaf-deletion-can-move-the-unique-mode-of-a-tree-domination-polynomial-by-two/", "pdf": "https://dvgyl.github.io/math-and-theory/papers/leaf-deletion-can-move-the-unique-mode-of-a-tree-domination-polynomial-by-two/manuscript.pdf", "discipline": "Mathematics", "area": "Graph polynomials", "subject": "Graph theory and graph polynomials", "subject_url": "https://dvgyl.github.io/math-and-theory/subjects/graph-theory-and-graph-polynomials/", "href": "/math-and-theory/papers/leaf-deletion-can-move-the-unique-mode-of-a-tree-domination-polynomial-by-two/", "tone": "refuted", "queries": [ "domination polynomial unimodal trees", "domination polynomial mode leaf deletion", "Burcroff O'Brien Question 8.1 domination polynomial", "Burcroff-O'Brien leaf deletion question", "unimodality and monotonic portions of certain domination polynomials", "Zhang conjecture domination polynomial modes", "mode of domination polynomial of a tree" ], "rank": 370, "result_type": "Standalone theorem and reproducible proof package", "formal_proof": "standalone", "standout": false, "year": "2026", "keywords": [ "domination polynomial", "unimodal sequences", "mode of a polynomial", "trees", "leaf deletion", "Burcroff–O'Brien leaf-deletion mode question for tree domination polynomials", "domination polynomial unimodal trees", "domination polynomial mode leaf deletion", "Burcroff O'Brien Question 8.1 domination polynomial", "Burcroff-O'Brien leaf deletion question", "unimodality and monotonic portions of certain domination polynomials", "Zhang conjecture domination polynomial modes", "mode of domination polynomial of a tree" ], "abstract": "Deleting a leaf can move the unique mode of an explicit tree domination polynomial by two.", "scope": "The fixed tree, deletion, actual domination counts and unique-mode movement are formalized. External question identity and historical priority remain provisional.", "problem": "Burcroff–O'Brien leaf-deletion mode question for tree domination polynomials", "posed_by": "Attributed to Burcroff and O'Brien, Discrete Math. 346 (2023), 113508, Question 8.1 (arXiv version), and Zhang, arXiv:2111.00641v2, Conjecture 4.1; numbering unverified", "problem_url": "https://arxiv.org/abs/2110.00709", "status": "Counterexample", "claim": "A tree on 7284 vertices has a unimodal domination polynomial with unique mode 3978, while deleting one specified leaf gives a 7283-vertex tree whose domination polynomial is unimodal with unique mode 3976." }, { "id": "unicyclic-skewness", "title": "A sharp universal skewness bound for unicyclic currents", "url": "https://dvgyl.github.io/math-and-theory/papers/a-sharp-universal-skewness-bound-for-unicyclic-currents/", "pdf": "https://dvgyl.github.io/math-and-theory/papers/a-sharp-universal-skewness-bound-for-unicyclic-currents/manuscript.pdf", "discipline": "Mathematical physics", "area": "Stochastic thermodynamics", "subject": "Stochastic thermodynamics", "subject_url": "https://dvgyl.github.io/math-and-theory/subjects/stochastic-thermodynamics/", "href": "/math-and-theory/papers/a-sharp-universal-skewness-bound-for-unicyclic-currents/", "tone": "proved", "queries": [ "Ptaszyński skewness bound unicyclic currents", "Ptaszynski conjecture C3/C1 -1/16", "bounds on skewness and kurtosis of steady-state currents proof", "third cumulant bound Markov ring current", "unicyclic network current fluctuations skewness bound", "normalized third cumulant winding current Markov chain", "thermodynamic uncertainty relation higher cumulants unicyclic", "stochastic thermodynamics cumulant ratio bound Lean proof" ], "rank": 380, "result_type": "Standalone theorem and reproducible proof package", "formal_proof": "standalone", "standout": false, "year": "2026", "keywords": [ "stochastic thermodynamics", "current fluctuations", "skewness", "Markov rings", "unicyclic networks", "cumulants", "Ptaszyński's conjectured interval −1/16 ≤ C₃/C₁ ≤ 1 for unicyclic Markov currents", "Ptaszyński skewness bound unicyclic currents", "Ptaszynski conjecture C3/C1 -1/16", "bounds on skewness and kurtosis of steady-state currents proof", "third cumulant bound Markov ring current", "unicyclic network current fluctuations skewness bound", "normalized third cumulant winding current Markov chain", "thermodynamic uncertainty relation higher cumulants unicyclic", "stochastic thermodynamics cumulant ratio bound Lean proof" ], "abstract": "The net marked-edge current on every finite bidirectional Markov ring with at least three states, positive rates and unequal forward and reverse rate products satisfies −1/16 ≤ C₃/C₁ ≤ 1. The two-channel two-state case has the stronger interval [1/4,1]. This ratio is not standardized skewness.", "scope": "The interval for actual tilted-matrix derivatives and a local branch is formalized. Identification with long-time stochastic cumulants and sharpness have conventional proofs. Lean does not construct a continuous-time sample-path probability space or a separate optimizing-sequence theorem.", "problem": "Ptaszyński's conjectured interval −1/16 ≤ C₃/C₁ ≤ 1 for unicyclic Markov currents", "posed_by": "K. Ptaszyński, Phys. Rev. E 106, 024119 (2022), Eq. (21)", "problem_url": "https://doi.org/10.1103/PhysRevE.106.024119", "status": "Proved", "claim": "For the net crossing count of a marked edge on every finite bidirectional Markov ring with n ≥ 3 states, positive rates and unequal forward and reverse rate products, −1/16 ≤ C₃/C₁ ≤ 1, and both constants are sharp." }, { "id": "quantum-exact-contraction", "title": "Exact conditional contraction for Pauli channels and depolarization in every dimension", "url": "https://dvgyl.github.io/math-and-theory/papers/exact-conditional-contraction-for-pauli-channels-and-depolarization-in-every-dimension/", "pdf": "https://dvgyl.github.io/math-and-theory/papers/exact-conditional-contraction-for-pauli-channels-and-depolarization-in-every-dimension/manuscript.pdf", "discipline": "Mathematical physics", "area": "Quantum information", "subject": "Quantum information", "subject_url": "https://dvgyl.github.io/math-and-theory/subjects/quantum-information/", "href": "/math-and-theory/papers/exact-conditional-contraction-for-pauli-channels-and-depolarization-in-every-dimension/", "tone": "proved", "queries": [ "conditional contraction coefficient quantum channel", "hockey-stick divergence contraction coefficient", "Pauli channel trace distance contraction coefficient exact", "depolarizing channel contraction coefficient quantum side information", "quantum Doeblin coefficient depolarizing channel", "strong data processing inequality quantum reference system", "conditional contraction coefficients Hirche George Nuradha Wilde" ], "rank": 390, "result_type": "Standalone theorem and reproducible proof package", "formal_proof": "standalone", "standout": false, "year": "2026", "keywords": [ "contraction coefficients", "hockey-stick divergence", "Pauli channels", "depolarizing channel", "strong data processing", "quantum channels", "Exact conditional hockey-stick contraction coefficients of Pauli and depolarizing channels", "conditional contraction coefficient quantum channel", "hockey-stick divergence contraction coefficient", "Pauli channel trace distance contraction coefficient exact", "depolarizing channel contraction coefficient quantum side information", "quantum Doeblin coefficient depolarizing channel", "strong data processing inequality quantum reference system", "conditional contraction coefficients Hirche George Nuradha Wilde" ], "abstract": "The conditional hockey-stick contraction coefficient is determined exactly for every stochastic Pauli qubit channel and for depolarization throughout its physical parameter interval in every dimension. The formulas include all boundary cases and every hockey-stick parameter γ ≥ 1, with arbitrary finite quantum references.", "scope": "The exact conditional hockey-stick contraction coefficients are formalized for actual finite-reference density states, every stochastic Pauli qubit channel, the full physical signed depolarizing family in every dimension d ≥ 2, and every γ ≥ 1. Channel constructors, unrestricted upper bounds and matching supremum witnesses are included. Unital-qubit normal-form existence is cited; erasure-simulation and ensemble-guessing consequences retain conventional proofs.", "problem": "Exact conditional hockey-stick contraction coefficients of Pauli and depolarizing channels", "posed_by": null, "problem_url": null, "status": "Determined exactly", "claim": "For every γ ≥ 1 the conditional hockey-stick contraction coefficient is 1 − 8ab/(a+b) for a Pauli channel with smallest weights a ≤ b, and s[1+(d²−1)s]/[1−(d²−3)s] for depolarization λ = −s, 0 ≤ s ≤ 1/(d²−1), in dimension d." }, { "id": "steering-bounds", "title": "Exact three-setting correlation bounds at fixed steering centre", "url": "https://dvgyl.github.io/math-and-theory/papers/exact-three-setting-correlation-bounds-at-fixed-steering-centre/", "pdf": "https://dvgyl.github.io/math-and-theory/papers/exact-three-setting-correlation-bounds-at-fixed-steering-centre/manuscript.pdf", "discipline": "Mathematical physics", "area": "Quantum information", "subject": "Quantum information", "subject_url": "https://dvgyl.github.io/math-and-theory/subjects/quantum-information/", "href": "/math-and-theory/papers/exact-three-setting-correlation-bounds-at-fixed-steering-centre/", "tone": "proved", "queries": [ "CJWR steering inequality two qubits maximum violation", "steering ellipsoid centre length bound", "three-setting linear steering inequality bound", "quantum steering ellipsoid maximally obese states", "two-qubit correlation matrix Frobenius norm steering bound", "fully entangled fraction steering ellipsoid centre", "Costa-Angelo steering criterion bound", "two-qubit steering threshold c < 2/3" ], "rank": 400, "result_type": "Standalone theorem and reproducible proof package", "formal_proof": "standalone", "standout": false, "year": "2026", "keywords": [ "quantum steering", "steering ellipsoids", "two-qubit states", "CJWR steering inequality", "fully entangled fraction", "Pauli correlations", "Maximum CJWR three-setting steering value at fixed steering-ellipsoid centre (two qubits)", "CJWR steering inequality two qubits maximum violation", "steering ellipsoid centre length bound", "three-setting linear steering inequality bound", "quantum steering ellipsoid maximally obese states", "two-qubit correlation matrix Frobenius norm steering bound", "fully entangled fraction steering ellipsoid centre", "Costa-Angelo steering criterion bound", "two-qubit steering threshold c < 2/3" ], "abstract": "The exact least upper bound of the optimized three-setting squared correlation value is determined for every two-qubit steering-centre length. The threshold for a value greater than one is c < 2/3. Attainment, nonattainment and the singular marginal boundary are included.", "scope": "Actual two-qubit density matrices, canonical filters, Pauli correlations, the exact least upper bound, attainment and nonattainment, and fully entangled fraction are formalized. The steering-ellipsoid and optimized CJWR measurement interpretations use cited external correspondences; general steerability is outside the result.", "problem": "Maximum CJWR three-setting steering value at fixed steering-ellipsoid centre (two qubits)", "posed_by": null, "problem_url": null, "status": "Determined exactly", "claim": "For two-qubit states with steering-ellipsoid centre length c, the exact supremum of F₃² = ‖T‖²_F is 1 + (2−3c)²/(2−2c−c²) for c ≤ 2/3 and 1 for c > 2/3, so F₃ > 1 is possible exactly when c < 2/3." }, { "id": "power-rank", "title": "A positive constant-recursive sequence with nonquasipolynomial power ranks", "url": "https://dvgyl.github.io/math-and-theory/papers/a-positive-constant-recursive-sequence-with-nonquasipolynomial-power-ranks/", "pdf": "https://dvgyl.github.io/math-and-theory/papers/a-positive-constant-recursive-sequence-with-nonquasipolynomial-power-ranks/manuscript.pdf", "discipline": "Mathematics", "area": "Integer sequences", "subject": "Integer sequences and combinatorics", "subject_url": "https://dvgyl.github.io/math-and-theory/subjects/integer-sequences-and-combinatorics/", "href": "/math-and-theory/papers/a-positive-constant-recursive-sequence-with-nonquasipolynomial-power-ranks/", "tone": "refuted", "queries": [ "power ranks constant-recursive sequence", "rank of powers of C-finite sequence", "Rowland Sistos Barron conjecture", "Rowland-Sistos Barron finiteness conjecture counterexample", "complexity of powers of a constant-recursive sequence", "minimal recurrence order of termwise powers of linear recurrence", "C-finite sequence powers recurrence order quasipolynomial" ], "rank": 410, "result_type": "Standalone theorem and reproducible proof package", "formal_proof": "standalone", "standout": false, "year": "2026", "keywords": [ "C-finite sequences", "constant-recursive sequences", "linear recurrences", "power ranks", "quasipolynomial", "recurrence order of powers", "Power ranks of constant-recursive sequences: Rowland–Sistos Barrón finiteness conjecture", "power ranks constant-recursive sequence", "rank of powers of C-finite sequence", "Rowland Sistos Barron conjecture", "Rowland-Sistos Barron finiteness conjecture counterexample", "complexity of powers of a constant-recursive sequence", "minimal recurrence order of termwise powers of linear recurrence", "C-finite sequence powers recurrence order quasipolynomial" ], "abstract": "Exact minimal recurrences for every power of a rank-thirteen sequence, nonquasipolynomial ranks, and infinitely many power-rank profiles at each rank at least three.", "scope": "The main and arbitrary-rank companion theorems are formalized. Generating-function and polynomial corollaries retain conventional proofs.", "problem": "Power ranks of constant-recursive sequences: Rowland–Sistos Barrón finiteness conjecture", "posed_by": "Rowland and Sistos Barrón, 'Complexity of powers of a constant-recursive sequence' (arXiv:2501.14643), unnumbered fixed-rank finiteness conjecture on page 2 of the author-hosted preprint", "problem_url": "https://arxiv.org/abs/2501.14643", "status": "Counterexample", "claim": "s(n) = 1 + 2^n + 3^n(1 − 2^n)² + n9^n(1 + 2^n)³ has rank 13 and power ranks P(M) − 2Q(M), not eventually quasipolynomial; a companion family gives infinitely many power-rank sequences at each rank r ≥ 3." }, { "id": "higuchi-sudbery", "title": "A sum-of-squares proof of the four-qubit Higuchi–Sudbery entropy bound", "url": "https://dvgyl.github.io/math-and-theory/papers/a-sum-of-squares-proof-of-the-four-qubit-higuchisudbery-entropy-bound/", "pdf": "https://dvgyl.github.io/math-and-theory/papers/a-sum-of-squares-proof-of-the-four-qubit-higuchisudbery-entropy-bound/manuscript.pdf", "discipline": "Mathematical physics", "area": "Quantum information", "subject": "Quantum information", "subject_url": "https://dvgyl.github.io/math-and-theory/subjects/quantum-information/", "href": "/math-and-theory/papers/a-sum-of-squares-proof-of-the-four-qubit-higuchisudbery-entropy-bound/", "tone": "proved", "queries": [ "Higuchi–Sudbery state", "Higuchi-Sudbery state maximally entangled", "Higuchi-Sudbery conjecture proof", "maximally entangled four qubits average entropy", "four-qubit state maximal two-qubit marginal entropy", "how entangled can two couples get", "four qubit marginal entropy log 2 + log 3/2", "sum of squares proof four-qubit entanglement bound" ], "rank": 420, "result_type": "Four-qubit entropy bound with attainment", "formal_proof": "standalone", "standout": false, "year": "2026", "keywords": [ "Higuchi–Sudbery", "Higuchi-Sudbery state", "four-qubit entanglement", "marginal entropy", "maximally entangled four qubits", "sum of squares", "Higuchi–Sudbery problem: maximal average two-qubit marginal entropy of four qubits", "Higuchi–Sudbery state", "Higuchi-Sudbery state maximally entangled", "Higuchi-Sudbery conjecture proof", "maximally entangled four qubits average entropy", "four-qubit state maximal two-qubit marginal entropy", "how entangled can two couples get", "four qubit marginal entropy log 2 + log 3/2", "sum of squares proof four-qubit entanglement bound" ], "abstract": "The average von Neumann entropy of the three two-qubit marginals of a pure four-qubit state is at most log(2) + log(3)/2. The Higuchi–Sudbery state attains this bound. The proof uses all sixteen complex amplitudes, actual reduced-matrix eigenvalues, an exact sum-of-squares identity, and an explicit maximizing state.", "scope": "The proof uses all sixteen complex amplitudes, actual reduced-matrix eigenvalues, an exact sum-of-squares identity, and an explicit maximizing state.", "problem": "Higuchi–Sudbery problem: maximal average two-qubit marginal entropy of four qubits", "posed_by": "Higuchi and Sudbery, 'How entangled can two couples get?', Physics Letters A 273 (2000), 213–217, which proposed the candidate maximizing state", "problem_url": "https://arxiv.org/abs/quant-ph/0005013", "status": "Proved", "claim": "For every pure four-qubit state the average von Neumann entropy of the three two-qubit marginals ρ_AB, ρ_AC, ρ_AD is at most log 2 + (1/2) log 3 (natural logarithms), and the Higuchi–Sudbery state attains it." }, { "id": "fidelity-three-state", "title": "The three-state root-fidelity entropy bound", "url": "https://dvgyl.github.io/math-and-theory/papers/the-three-state-root-fidelity-entropy-bound/", "pdf": "https://dvgyl.github.io/math-and-theory/papers/the-three-state-root-fidelity-entropy-bound/manuscript.pdf", "discipline": "Mathematical physics", "area": "Quantum information", "subject": "Quantum information", "subject_url": "https://dvgyl.github.io/math-and-theory/subjects/quantum-information/", "href": "/math-and-theory/papers/the-three-state-root-fidelity-entropy-bound/", "tone": "proved", "queries": [ "Fannes de Melo Roga Zyczkowski conjecture", "Fannes-de Melo-Roga-Zyczkowski conjecture", "matrices of fidelities Holevo quantity conjecture", "Holevo quantity bound fidelity matrix entropy", "root fidelity matrix entropy Holevo bound three states", "Holevo information upper bound Gram matrix of fidelities", "Roga Zyczkowski fidelity matrix conjecture proof" ], "rank": 430, "result_type": "Standalone theorem and reproducible proof package", "formal_proof": "standalone", "standout": false, "year": "2026", "keywords": [ "root fidelity", "Holevo quantity", "quantum ensembles", "von Neumann entropy", "fidelity matrix", "Fannes–de Melo–Roga–Życzkowski conjecture: root-fidelity matrix entropy bounds Holevo χ", "Fannes de Melo Roga Zyczkowski conjecture", "Fannes-de Melo-Roga-Zyczkowski conjecture", "matrices of fidelities Holevo quantity conjecture", "Holevo quantity bound fidelity matrix entropy", "root fidelity matrix entropy Holevo bound three states", "Holevo information upper bound Gram matrix of fidelities", "Roga Zyczkowski fidelity matrix conjecture proof" ], "abstract": "The entropy of the weighted three-by-three root-fidelity matrix bounds the Holevo quantity of every three-state ensemble in arbitrary finite dimension. The result includes mixed and singular states and zero probabilities.", "scope": "The unrestricted three-state ensemble theorem is formalized for actual positive semidefinite complex matrices of trace one, nonnegative weights summing to one, spectral entropy and nuclear-norm root fidelity. Credited QICLean foundations and prior-work adapters retain their attribution.", "problem": "Fannes–de Melo–Roga–Życzkowski conjecture: root-fidelity matrix entropy bounds Holevo χ", "posed_by": "Fannes, de Melo, Roga and Życzkowski, 'Matrices of fidelities for ensembles of quantum states and the Holevo quantity', Quantum Inf. Comput. 12 (2012), 472–489, Conjecture 1, eq. (18)", "problem_url": "https://doi.org/10.26421/QIC12.5-6-8", "status": "Proved", "claim": "For any three density matrices ρᵢ of finite dimension and probabilities pᵢ, the Holevo quantity S(Σ pᵢρᵢ) − Σ pᵢS(ρᵢ) is at most the entropy of the 3 × 3 matrix C_ij = √(pᵢpⱼ) ‖√ρᵢ √ρⱼ‖₁." }, { "id": "binary-spectral-tile", "title": "A 64-point spectral tile in F₂⁹ with no periodic complement", "url": "https://dvgyl.github.io/math-and-theory/papers/a-64-point-spectral-tile-in-f29-with-no-periodic-complement/", "pdf": "https://dvgyl.github.io/math-and-theory/papers/a-64-point-spectral-tile-in-f29-with-no-periodic-complement/manuscript.pdf", "discipline": "Mathematics", "area": "Spectral sets and tilings", "subject": "Discrete analysis, tilings, probability and entropy", "subject_url": "https://dvgyl.github.io/math-and-theory/subjects/discrete-analysis-tilings-probability-and-entropy/", "href": "/math-and-theory/papers/a-64-point-spectral-tile-in-f29-with-no-periodic-complement/", "tone": "refuted", "queries": [ "Fuglede conjecture F_2^9 spectral set", "Ferguson Iosevich Question 8.1 counterexample", "Ferguson-Iosevich Fuglede conjecture eight dimensions field with two elements", "spectral set without subspace tiling complement binary vector space", "periodic tiling property finite abelian groups F2^9", "Fan Zhang periodic tiling property counterexample", "Fuglede conjecture finite abelian groups binary dimension nine", "aperiodic spectral tile Z_2^9" ], "rank": 440, "result_type": "Explicit counterexample, classification of complements and reproducible proof package", "formal_proof": "standalone", "standout": false, "year": "2026", "keywords": [ "Fuglede conjecture", "spectral sets", "translational tiling", "periodic tiling property", "finite abelian groups", "binary vector spaces", "subspace tiling complement", "Ferguson–Iosevich", "Ferguson–Iosevich Question 8.1: subspace tiling complements of spectral sets in F₂⁹", "Fuglede conjecture F_2^9 spectral set", "Ferguson Iosevich Question 8.1 counterexample", "Ferguson-Iosevich Fuglede conjecture eight dimensions field with two elements", "spectral set without subspace tiling complement binary vector space", "periodic tiling property finite abelian groups F2^9", "Fan Zhang periodic tiling property counterexample", "Fuglede conjecture finite abelian groups binary dimension nine", "aperiodic spectral tile Z_2^9" ], "abstract": "An explicit 64-point set E ⊂ F₂⁹ is spectral and tiles by translation, but has no periodic tiling complement: its complements are exactly the 512 translates of one aperiodic eight-point set. In particular no linear subspace complements E, and F₂⁹ fails the periodic tiling property of Fan and Zhang. The construction places eight three-dimensional subspaces in parallel six-dimensional fibres; the shifts disjoint from E are the 28 pair sums of an affine eight-circuit, which form the line graph of K₈ and contain no three-dimensional subspace. This answers negatively the stronger subspace-complement part of Question 8.1 of Ferguson and Iosevich and its equivalent Question 8.2. Whether every spectral 64-point subset of F₂⁹ tiles remains open.", "scope": "Spectrality with an actual basis of all complex functions on E, the tiling E ⊕ T = F₂⁹, aperiodicity of E, the exclusion of every periodic and every subspace complement, the classification of all complements as the 512 translates of T, and the failure of the periodic tiling property for F₂⁹ are formalized in 16 Lean modules. Minimality of dimension nine depends on an external dimension-eight theorem and is not formalized.", "problem": "Ferguson–Iosevich Question 8.1: subspace tiling complements of spectral sets in F₂⁹", "posed_by": "Ferguson and Iosevich, arXiv:2610.36417, Question 8.1 (subspace-complement part) and its equivalent Question 8.2", "problem_url": "https://arxiv.org/abs/2610.36417", "status": "Counterexample", "claim": "An explicit 64-point set E ⊂ F₂⁹ is spectral and tiles F₂⁹, but its tiling complements are exactly the 512 translates of one aperiodic 8-point set, so no subspace complements E and F₂⁹ fails the periodic tiling property." }, { "id": "graph-roots", "title": "Sharp escape of complex domination roots in square flowers", "url": "https://dvgyl.github.io/math-and-theory/papers/sharp-escape-of-complex-domination-roots-in-square-flowers/", "pdf": "https://dvgyl.github.io/math-and-theory/papers/sharp-escape-of-complex-domination-roots-in-square-flowers/manuscript.pdf", "discipline": "Mathematics", "area": "Graph polynomials", "subject": "Graph theory and graph polynomials", "subject_url": "https://dvgyl.github.io/math-and-theory/subjects/graph-theory-and-graph-polynomials/", "href": "/math-and-theory/papers/sharp-escape-of-complex-domination-roots-in-square-flowers/", "tone": "proved", "queries": [ "domination polynomial roots", "complex domination roots unbounded modulus", "domination roots friendship graph square flower", "domination polynomial book graph real roots", "largest root of domination polynomial Lambert W", "domination polynomial generalized friendship graphs roots", "zeros of domination polynomial asymptotics" ], "rank": 450, "result_type": "Standalone theorem and reproducible proof package", "formal_proof": "standalone", "standout": false, "year": "2026", "keywords": [ "domination roots", "domination polynomial", "complex roots of graph polynomials", "square flowers", "book graphs", "Largest complex domination roots of square flowers (n four-cycles sharing one vertex)", "domination polynomial roots", "complex domination roots unbounded modulus", "domination roots friendship graph square flower", "domination polynomial book graph real roots", "largest root of domination polynomial Lambert W", "domination polynomial generalized friendship graphs roots", "zeros of domination polynomial asymptotics" ], "abstract": "Sharp complex-root escape for square flowers, with complete flower and book real-root classifications and multiplicities.", "scope": "All numbered manuscript conclusions are formalized in seventy modules with 258 endpoint audits. Historical novelty and exact external-source correspondence remain provisional.", "problem": "Largest complex domination roots of square flowers (n four-cycles sharing one vertex)", "posed_by": null, "problem_url": null, "status": "New theorem", "claim": "For the square flower F_{4,n} of n four-cycles sharing one vertex, the largest domination-root modulus R_n satisfies R_n < 2 + √(4n/W(4n)) for all n ≥ 1 and 4n/R_n² = log(4n) − log log(4n) + o(1)." }, { "id": "thermal-kurtosis-counterexample", "title": "A three-state thermal counterexample to a heat-current kurtosis conjecture", "url": "https://dvgyl.github.io/math-and-theory/papers/a-three-state-thermal-counterexample-to-a-heat-current-kurtosis-conjecture/", "pdf": "https://dvgyl.github.io/math-and-theory/papers/a-three-state-thermal-counterexample-to-a-heat-current-kurtosis-conjecture/manuscript.pdf", "discipline": "Mathematical physics", "area": "Stochastic thermodynamics", "subject": "Stochastic thermodynamics", "subject_url": "https://dvgyl.github.io/math-and-theory/subjects/stochastic-thermodynamics/", "href": "/math-and-theory/papers/a-three-state-thermal-counterexample-to-a-heat-current-kurtosis-conjecture/", "tone": "refuted", "queries": [ "Ptaszyński kurtosis bound counterexample", "Ptaszynski bounds on skewness and kurtosis of steady-state currents", "heat current kurtosis conjecture thermal Markov", "negative fourth cumulant heat current local detailed balance", "heat current full counting statistics fourth cumulant sign", "three-level thermal model negative kurtosis heat current", "tilted generator fourth cumulant counterexample" ], "rank": 460, "result_type": "Exact finite spectral counterexample", "formal_proof": "standalone", "standout": false, "year": "2026", "keywords": [ "stochastic thermodynamics", "heat current", "full counting statistics", "kurtosis", "Markov jump processes", "counterexample", "Ptaszyński's heat-current kurtosis bound C₄/C₂ ≥ 0 for thermal Markov systems", "Ptaszyński kurtosis bound counterexample", "Ptaszynski bounds on skewness and kurtosis of steady-state currents", "heat current kurtosis conjecture thermal Markov", "negative fourth cumulant heat current local detailed balance", "heat current full counting statistics fourth cumulant sign", "three-level thermal model negative kurtosis heat current", "tilted generator fourth cumulant counterexample" ], "abstract": "A three-state model with two positive-temperature baths satisfies local detailed balance and has positive total off-diagonal transition rates. Its dominant tilted-generator eigenvalue has fourth-to-second derivative ratio −3470947982622836/176627429595846691. This refutes the thermal nonnegativity proposal in its finite spectral formulation. The proof constructs a smooth local eigenvalue branch, proves strict dominance over distinct complex eigenvalues, and calculates the exact derivatives.", "scope": "A three-state two-temperature thermal rate model satisfies local detailed balance and has an exact negative fourth-to-second cold-heat spectral cumulant ratio. Lean checks the finite matrices, complex spectrum, a smooth locally dominant eigenvalue branch, and the exact derivative ratio. It does not formalize the stochastic long-time identification theorem.", "problem": "Ptaszyński's heat-current kurtosis bound C₄/C₂ ≥ 0 for thermal Markov systems", "posed_by": "Ptaszyński, Physical Review E 106, 024119 (2022), Eq. (19)", "problem_url": "https://doi.org/10.1103/PhysRevE.106.024119", "status": "Counterexample", "claim": "An explicit three-state, two-bath thermal rate model with local detailed balance has spectral heat-cumulant ratio C₄/C₂ = −3470947982622836/176627429595846691 < 0, contradicting Eq. (19) in its tilted-generator form." }, { "id": "domirank-entropy", "title": "An entropy increase for a structurally disassortative DomiRank graph", "url": "https://dvgyl.github.io/math-and-theory/papers/an-entropy-increase-for-a-structurally-disassortative-domirank-graph/", "pdf": "https://dvgyl.github.io/math-and-theory/papers/an-entropy-increase-for-a-structurally-disassortative-domirank-graph/manuscript.pdf", "discipline": "Mathematics", "area": "Graph theory and matrix analysis", "subject": "Graph theory and graph polynomials", "subject_url": "https://dvgyl.github.io/math-and-theory/subjects/graph-theory-and-graph-polynomials/", "href": "/math-and-theory/papers/an-entropy-increase-for-a-structurally-disassortative-domirank-graph/", "tone": "refuted", "queries": [ "DomiRank centrality entropy conjecture", "DomiRank entropy monotonicity disassortative graph", "Zhang Zhao DomiRank Conjecture 4.9 counterexample", "Zhang-Zhao DomiRank conjecture", "structural properties of DomiRank centrality", "DomiRank resolvent centrality entropy", "degree log-degree covariance disassortative entropy" ], "rank": 470, "result_type": "Counterexample to an interval monotonicity conjecture", "formal_proof": "comparator", "standout": false, "year": "2026", "keywords": [ "DomiRank", "network centrality", "graph entropy", "disassortative graphs", "resolvent centrality", "Zhang–Zhao DomiRank entropy monotonicity conjecture for disassortative graphs", "DomiRank centrality entropy conjecture", "DomiRank entropy monotonicity disassortative graph", "Zhang Zhao DomiRank Conjecture 4.9 counterexample", "Zhang-Zhao DomiRank conjecture", "structural properties of DomiRank centrality", "DomiRank resolvent centrality entropy", "degree log-degree covariance disassortative entropy" ], "abstract": "A connected graph with 14 vertices and 32 edges has negative degree–log-degree edge covariance. All resolvent scores are positive on [251/1000,63/250] inside the stable domain. Its entropy nevertheless increases by more than 1/10000 in natural-logarithm units. This refutes the source’s assertion for every positive-score interval.", "scope": "A connected graph with 14 vertices and 32 edges has negative degree–log-degree edge covariance. All resolvent scores are positive on [251/1000,63/250] inside the stable domain. Its entropy nevertheless increases by more than 1/10000 in natural-logarithm units. This refutes the source’s assertion for every positive-score interval.", "problem": "Zhang–Zhao DomiRank entropy monotonicity conjecture for disassortative graphs", "posed_by": "Zhang and Zhao, arXiv:2610.00107v4, Conjecture 4.9", "problem_url": "https://arxiv.org/abs/2610.00107v4", "status": "Disproved", "claim": "A connected graph with 14 vertices and 32 edges has negative degree–log-degree edge covariance and positive DomiRank scores on [251/1000, 63/250], yet its entropy increases by more than 1/10000 nats across that interval." }, { "id": "projection-amgm", "title": "An unbounded obstruction to projection AM–GM in Loewner order", "url": "https://dvgyl.github.io/math-and-theory/papers/an-unbounded-obstruction-to-projection-amgm-in-loewner-order/", "pdf": "https://dvgyl.github.io/math-and-theory/papers/an-unbounded-obstruction-to-projection-amgm-in-loewner-order/manuscript.pdf", "discipline": "Mathematics", "area": "Matrix theory", "subject": "Matrix theory", "subject_url": "https://dvgyl.github.io/math-and-theory/subjects/matrix-theory/", "href": "/math-and-theory/papers/an-unbounded-obstruction-to-projection-amgm-in-loewner-order/", "tone": "refuted", "queries": [ "matrix AM-GM inequality orthogonal projections", "noncommutative arithmetic-geometric mean inequality Loewner order", "projection AM-GM conjecture counterexample", "symmetrized product of projections Loewner order", "Sun Luo Ye random permutation coordinate descent AM-GM", "Recht-Re noncommutative AM-GM conjecture projections", "matrix AM–GM Loewner order three projections" ], "rank": 480, "result_type": "Focused research note and reproducible proof package", "formal_proof": "standalone", "standout": false, "year": "2026", "keywords": [ "matrix AM–GM inequality", "orthogonal projections", "Loewner order", "symmetrized products", "operator inequalities", "Projection AM–GM inequality in Loewner order for symmetrized products of projections", "matrix AM-GM inequality orthogonal projections", "noncommutative arithmetic-geometric mean inequality Loewner order", "projection AM-GM conjecture counterexample", "symmetrized product of projections Loewner order", "Sun Luo Ye random permutation coordinate descent AM-GM", "Recht-Re noncommutative AM-GM conjecture projections", "matrix AM–GM Loewner order three projections" ], "abstract": "Three rank-one orthogonal projections in dimension three violate the symmetrized-product Loewner AM–GM bound. No finite real multiplier repairs it, even with a positive-definite mean. The examples do not satisfy the additional nonsingular block structure of the motivating coordinate-descent application.", "scope": "Lean verifies a rational three-projection matrix obstruction and a parameterized obstruction for every real multiplier in dimension three; rank-one, positive-definite mean, rational all-multiplier choices, and the standard minimal-factor observation have complete conventional proofs.", "problem": "Projection AM–GM inequality in Loewner order for symmetrized products of projections", "posed_by": "Motivated by Sun, Luo and Ye, Math. Oper. Res. 45(1) (2020); exact match with their published conjecture is conditional", "problem_url": "https://doi.org/10.1287/moor.2019.0990", "status": "Counterexample", "claim": "Three rational rank-one orthogonal projections in ℝ³ have symmetrized product G ⋠ M³, M their mean (quadratic form −8/243), and for every real C some such triple with positive-definite mean has G ⋠ C·M³." }, { "id": "pure-ecqc", "title": "A pure-state matched-basis information inequality violation in dimension five", "url": "https://dvgyl.github.io/math-and-theory/papers/a-pure-state-matched-basis-information-inequality-violation-in-dimension-five/", "pdf": "https://dvgyl.github.io/math-and-theory/papers/a-pure-state-matched-basis-information-inequality-violation-in-dimension-five/manuscript.pdf", "discipline": "Mathematical physics", "area": "Quantum information", "subject": "Quantum information", "subject_url": "https://dvgyl.github.io/math-and-theory/subjects/quantum-information/", "href": "/math-and-theory/papers/a-pure-state-matched-basis-information-inequality-violation-in-dimension-five/", "tone": "refuted", "queries": [ "ECQC conjecture pure state counterexample", "complementary quantum correlations conjecture", "CQC conjecture mutually unbiased bases mutual information", "Iqbal CQC conjecture extension", "classical mutual information mutually unbiased bases quantum mutual information bound", "complementary measurements mutual information inequality pure states", "ECQC dimension five MUB counterexample" ], "rank": 490, "result_type": "Standalone theorem and reproducible proof package", "formal_proof": "standalone", "standout": false, "year": "2026", "keywords": [ "ECQC", "mutual information", "mutually unbiased bases", "pure states", "quantum information inequality", "counterexample", "Pure-state ECQC: matched-basis mutual-information inequality with mutually unbiased bases", "ECQC conjecture pure state counterexample", "complementary quantum correlations conjecture", "CQC conjecture mutually unbiased bases mutual information", "Iqbal CQC conjecture extension", "classical mutual information mutually unbiased bases quantum mutual information bound", "complementary measurements mutual information inequality pure states", "ECQC dimension five MUB counterexample" ], "abstract": "A dimension-five pure-state matched-basis information inequality violation, with a conventional all-prime companion.", "scope": "The dimension-five identical-local-basis theorem is formalized. The prime-family extension is conventional; arbitrary or independently optimized basis choices are outside the result.", "problem": "Pure-state ECQC: matched-basis mutual-information inequality with mutually unbiased bases", "posed_by": "Attributed to Iqbal, Quantum Information Processing 25 (2026), Article 250, Conjecture 3.1; correspondence unverified", "problem_url": "https://doi.org/10.1007/s11128-026-05258-2", "status": "Counterexample", "claim": "A normalized pure state in local dimension 5 with six fixed mutually unbiased bases, used identically by both parties, has five-setting classical information sum 5 log 5 versus quantum mutual information 2 log 5 (ratio 5/2)." }, { "id": "kovacec", "title": "A polygon enclosure fails for sums of normal matrices", "url": "https://dvgyl.github.io/math-and-theory/papers/a-polygon-enclosure-fails-for-sums-of-normal-matrices/", "pdf": "https://dvgyl.github.io/math-and-theory/papers/a-polygon-enclosure-fails-for-sums-of-normal-matrices/manuscript.pdf", "discipline": "Mathematics", "area": "Matrix theory", "subject": "Matrix theory", "subject_url": "https://dvgyl.github.io/math-and-theory/subjects/matrix-theory/", "href": "/math-and-theory/papers/a-polygon-enclosure-fails-for-sums-of-normal-matrices/", "tone": "refuted", "queries": [ "Kovačec conjecture normal matrices", "Kovacec conjecture determinant sum of normal matrices", "Marcus–de Oliveira conjecture", "Marcus-de Oliveira conjecture determinant normal matrices", "a conjecture more precise and stronger than the one by Marcus and de Oliveira", "determinant of sum of normal matrices permutation values", "Kovacec polygon enclosure counterexample" ], "rank": 500, "result_type": "Standalone theorem and reproducible proof package", "formal_proof": "standalone", "standout": false, "year": "2026", "keywords": [ "normal matrices", "determinants of sums of normal matrices", "Kovačec", "Marcus–de Oliveira conjecture", "permutation values", "polygon enclosure", "Kovačec's polygon enclosure conjecture for determinants of sums of normal matrices", "Kovačec conjecture normal matrices", "Kovacec conjecture determinant sum of normal matrices", "Marcus-de Oliveira conjecture determinant normal matrices", "a conjecture more precise and stronger than the one by Marcus and de Oliveira", "determinant of sum of normal matrices permutation values", "Kovacec polygon enclosure counterexample" ], "abstract": "A polygon enclosure fails for a specified pair of normal matrices and its complete transposition graph.", "scope": "Actual normal matrices, determinants, straight-segment exterior and component exclusion are formalized. Correspondence with an external source conjecture is conditional.", "problem": "Kovačec's polygon enclosure conjecture for determinants of sums of normal matrices", "posed_by": "Kovačec, Universidade de Coimbra preprint 26-11 (April 2026), Conjecture 7; literal statement unverified", "problem_url": null, "status": "Counterexample", "claim": "Normal 3×3 matrices with spectra (0, 1, i) and (−1−i, 1, −1+i) give det(A+B) = 2/3 + 2i/9, outside every bounded component closure of ℂ∖Γ, Γ the exterior boundary of the transposition graph, yet inside the permutation-value hull." } ] }