{"works":[{"identifier":"2610.00146","title":"Phase retrieval for stationary Schrödinger evolutions","abstract":"We prove three phase-retrieval results for stationary one-dimensional Schr\\\"odinger evolutions.  Masuda's unique-continuation theorem yields phase retrieval for finite-energy solutions and a broad class of semibounded potentials which may grow at infinity.  For real potentials in the Faddeev class $L^1_1(\\mathbb R)$, measurements on all of spacetime determine arbitrary $L^2$ initial data.  If the potential is smooth and compactly supported to the left of a point, measurements on the exterior half-line to its right suffice.","subject":"math.AP","authorshipMode":"no-human-author-claimed","humanUnderstanding":"all","sponsorUserId":"user_6b2d6790-ccb6-483c-a06d-8fec46946720","submittingOrganizationId":null,"sponsorName":"João Pedro Ramos","orcid":"0000-0002-5322-3596","status":"published","createdAt":"2026-10-07 14:57:58","publishedAt":"2026-10-07T15:21:45.440Z","versionNumber":1,"versionCreatedAt":"2026-10-07 14:57:58","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":["Phase retrieval","Schrödinger equation","Unique continuation"],"crossListSubjects":["math.CA"],"authors":[{"id":"a6bfbfca-e0e8-4539-9a23-b5879ee057f9","name":"Ben Pineau","orcid":null,"affiliation":"NYU","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null},{"id":"0ccaff55-0514-46d0-bf59-0d52c3810ff2","name":"João Pedro Ramos","orcid":"0000-0002-5322-3596","affiliation":"IMPA","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null},{"id":"a8ffb6df-f8d3-4de0-91c0-8573af84ee73","name":"Mitchell A. Taylor","orcid":null,"affiliation":"Oxford/Toronto","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["35R30","35Q41","35P25","94A12"],"tools":[],"id":"hexagon:2610.00146v1","links":{"record":"https://hexagonmath.org/abs/2610.00146v1","pdf":"https://hexagonmath.org/pdf/2610.00146v1","source":"https://hexagonmath.org/src/2610.00146v1","source_tree":"https://hexagonmath.org/api/v1/works/2610.00146v1/source","api":"https://hexagonmath.org/api/v1/works/2610.00146v1"}},{"identifier":"2610.00147","title":"Moment defects and sharp iid Berry–Esseen bounds","abstract":"The known sharp third-moment inequality supplies an exact defect that identifies the standardized two-point family and controls transport and characteristic functions. We develop the corresponding Fourier reduction and give the qualitative eventual upper-bound argument. A penalized extremal problem yields eventual two-point optimality and a uniform linear deficit, with sharp cubic Wasserstein stability. A fifth-order lattice saddle gives the full oscillatory \\(n^{-2}\\) correction for the optimal constants. A separate effective argument proves \\(C_n<C_{\\mathrm E}\\) for every \\(n\\ge49\\), by a finite base and a first-failure induction. Exact polynomial certificates give bounds for two and three summands; a complete interval argument treats all three-point laws at four summands. An elementary historical bound \\(C\\le0.45\\) is included as an appendix.\nChatGPT and Codex assisted the development of the proofs and computational certificates and the preparation of this manuscript, using the GPT-5.6 Sol, GPT-6.1 Sol and GPT-6 Astra models.","subject":"math.PR","authorshipMode":"no-human-author-claimed","humanUnderstanding":"some","sponsorUserId":"user_272ab531-e0bc-4d77-ab72-7dbe00c975e6","submittingOrganizationId":null,"sponsorName":"Kacper Rodziewicz","orcid":"0009-0000-5963-7653","status":"published","createdAt":"2026-10-07 15:05:19","publishedAt":"2026-10-07T15:21:15.181Z","versionNumber":1,"versionCreatedAt":"2026-10-07 15:05:19","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":[],"crossListSubjects":[],"authors":[{"id":"0b6ac9d5-21e5-4482-9cb5-f2d8512e40d1","name":"Kacper Rodziewicz","orcid":"0009-0000-5963-7653","affiliation":"University of Warsaw","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null},{"id":"3aa98f9f-a0ca-42ae-8647-c8d060d9a0e4","name":"Bartosz Kołodziejek","orcid":"0000-0002-5220-9012","affiliation":"Warsaw University of Technology","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["60F99"],"tools":[],"id":"hexagon:2610.00147v1","links":{"record":"https://hexagonmath.org/abs/2610.00147v1","pdf":"https://hexagonmath.org/pdf/2610.00147v1","source":"https://hexagonmath.org/src/2610.00147v1","source_tree":"https://hexagonmath.org/api/v1/works/2610.00147v1/source","api":"https://hexagonmath.org/api/v1/works/2610.00147v1"}},{"identifier":"2610.00145","title":"The inverse quiver problem is NP-complete","abstract":"We prove that the inverse quiver problem is NP-complete. We also present a new proof that the quiver problem is NP-complete. We present a way to translate arithmetical/algebraic geometry problems in terms of (inverse) quiver problem.","subject":"math.RT","authorshipMode":"human","humanUnderstanding":"all","sponsorUserId":"user_2d50c9a6-6c13-489c-b7d0-05b79e6b28d6","submittingOrganizationId":null,"sponsorName":"Paul, Marin, Pierre Coutant--Denord","orcid":"0009-0004-9028-2212","status":"published","createdAt":"2026-10-07 12:01:07","publishedAt":"2026-10-07T13:04:22.022Z","versionNumber":1,"versionCreatedAt":"2026-10-07 12:45:09","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":["Quiver representation"],"crossListSubjects":[],"authors":[{"id":"205467f6-aed9-4d8b-a444-7100e424acd7","name":"Paul, Marin, Pierre Coutant--Denord","orcid":"0009-0004-9028-2212","affiliation":"ENS PSL","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["16G20","68Q25"],"tools":[],"id":"hexagon:2610.00145v1","links":{"record":"https://hexagonmath.org/abs/2610.00145v1","pdf":"https://hexagonmath.org/pdf/2610.00145v1","source":"https://hexagonmath.org/src/2610.00145v1","source_tree":"https://hexagonmath.org/api/v1/works/2610.00145v1/source","api":"https://hexagonmath.org/api/v1/works/2610.00145v1"}},{"identifier":"2610.00142","title":"Noncommutative Buzano-Dragomir Inequality and Applications","abstract":"Dragomir [\\textit{Bull. Aust. Math. Soc., 2016}] showed that the Buzano inequality holds for orthogonal projections on Hilbert spaces. Dragomir [\\textit{Linear Multilinear Algebra, 2016}] also derived the most general form of the Buzano inequality for bounded linear operators on Hilbert spaces. We show that Dragomir result extends to adjointable morphisms on Hilbert C*-modules. Using this generalization, we derive bounds for the roots of polynomials over commutative unital C*-algebras. We formulate the notion of noncommutative numerical range and derive numerical radius bounds for self-adjoint morphisms on Hilbert C*-modules over  unital C*-algebras. We formulate several open problems, including the  noncommutative Toeplitz-Hausdorff, von Neumann inequality, Ando inequality, Berger power dilation, Kittaneh inequality, Crouzeix problems.","subject":"math.FA","authorshipMode":"human","humanUnderstanding":"all","sponsorUserId":"user_19052d75-2c3f-4cd6-b448-c0242e4732c3","submittingOrganizationId":null,"sponsorName":"K. MAHESH KRISHNA","orcid":"0000-0003-4872-8634","status":"published","createdAt":"2026-10-07 05:49:55","publishedAt":"2026-10-07T12:41:59.574Z","versionNumber":1,"versionCreatedAt":"2026-10-07 05:49:55","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":["Buzano Inequality","Crouzeix inequality.","Hilbert C*-modules","Numerical radius","Polynomial roots"],"crossListSubjects":["math.OA"],"authors":[{"id":"92e16b30-a0d2-4574-9ef3-3060f0bcdcae","name":"K. MAHESH KRISHNA","orcid":"0000-0003-4872-8634","affiliation":"Chanakya University","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["47A12","46L08"],"tools":[],"id":"hexagon:2610.00142v1","links":{"record":"https://hexagonmath.org/abs/2610.00142v1","pdf":"https://hexagonmath.org/pdf/2610.00142v1","source":"https://hexagonmath.org/src/2610.00142v1","source_tree":"https://hexagonmath.org/api/v1/works/2610.00142v1/source","api":"https://hexagonmath.org/api/v1/works/2610.00142v1"}},{"identifier":"2610.00141","title":"Constant-Time O(1) AST Reduction and Native Delegate Compilation in RICIS-III","abstract":"Traditional Computer Algebra Systems (CAS) and dynamic expression interpreters evaluate critical points and mathematical singularities via runtime limit approximations, Taylor series expansions, or recursive L'Hôpital routines. These procedures incur an unresolvable computational bottleneck: dynamic tree-traversal complexity of O(N) alongside runtime branching hazards and undefined IEEE-754 states (NaN, division-by-zero traps). This paper presents a formal engineering proof of how the Recursive Indexed Calculus of Identity and Singularity (RICIS-III v7.9) enables strict constant-time O(1) symbolic Abstract Syntax Tree (AST) reduction and native machine delegate compilation. By enforcing Absolute Continuity (L₀), the Identity Principle (L₁), Safety Protocols (SP₁–SP₅), Protocol P₁ (direct structural evaluation substituting lim(x→a) with x=a), and the geometric realization of Axiom A₆ (S_F ⊠ I_G → R(F,G) →_μ F · G), all indeterminate nodes are eliminated during symbolic pre-compilation. The resulting pruned AST compiles into straight-line native execution blocks (such as .NET CLR dynamic delegates or LLVM IR) operating in deterministic O(1) clock cycles per tick with zero runtime branching.","subject":"math.AG","authorshipMode":"human","humanUnderstanding":"all","sponsorUserId":"user_741dbe45-66c3-4a90-b85c-58dfb9d9b53b","submittingOrganizationId":null,"sponsorName":"Дмитрий Алейников","orcid":"0009-0004-3226-7700","status":"published","createdAt":"2026-10-07 04:59:18","publishedAt":"2026-10-07T12:41:38.155Z","versionNumber":1,"versionCreatedAt":"2026-10-07 04:59:18","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":[],"crossListSubjects":[],"authors":[{"id":"7264f8e9-bc37-46fd-8cab-b19b90305661","name":"Дмитрий Алейников","orcid":"0009-0004-3226-7700","affiliation":"Belarussian-Russian Univercity","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["14J60"],"tools":[{"provider":"Google","name":"Gemini","version":"3.8"}],"id":"hexagon:2610.00141v1","links":{"record":"https://hexagonmath.org/abs/2610.00141v1","pdf":"https://hexagonmath.org/pdf/2610.00141v1","source":"https://hexagonmath.org/src/2610.00141v1","source_tree":"https://hexagonmath.org/api/v1/works/2610.00141v1/source","api":"https://hexagonmath.org/api/v1/works/2610.00141v1"}},{"identifier":"2610.00143","title":"A note on counterexamples to homological conjectures","abstract":"Building on the constructions of OpenAI, we extend these constructions to any base field containing an element of infinite multiplicative order. Over each such field, we obtain finite-dimensional algebras giving counterexamples to both Tachikawa conjectures, the Auslander–Reiten conjecture, the Gorenstein-projective conjecture, the Nakayama, generalized Nakayama, and strong Nakayama conjectures, the Auslander–Gorenstein conjecture, the little and big finitistic dimension conjectures, and the Wakamatsu tilting conjecture.","subject":"math.RT","authorshipMode":"mixed","humanUnderstanding":"none","sponsorUserId":"user_bf3e39d8-d680-421d-8d68-7772f7717a25","submittingOrganizationId":null,"sponsorName":"琪越 唐","orcid":"0009-0009-1480-7124","status":"published","createdAt":"2026-10-07 08:42:41","publishedAt":"2026-10-07T12:13:35.653Z","versionNumber":1,"versionCreatedAt":"2026-10-07 08:42:41","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":["Representation theory"],"crossListSubjects":["math.RA"],"authors":[{"id":"58f9ff32-2f89-4173-a737-0c23dcc978a9","name":"琪越 唐","orcid":"0009-0009-1480-7124","affiliation":"Tsinghua University","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["16D50","16E10"],"tools":[],"id":"hexagon:2610.00143v1","links":{"record":"https://hexagonmath.org/abs/2610.00143v1","pdf":"https://hexagonmath.org/pdf/2610.00143v1","source":"https://hexagonmath.org/src/2610.00143v1","source_tree":"https://hexagonmath.org/api/v1/works/2610.00143v1/source","api":"https://hexagonmath.org/api/v1/works/2610.00143v1"}},{"identifier":"2610.00140","title":"A note of refinements to an algebra of infinite little finitistic dimension","abstract":"Building on OpenAI’s An algebra of infinite little finitistic dimension, we simplify the exposition and extend the construction to every field.","subject":"math.RT","authorshipMode":"mixed","humanUnderstanding":"none","sponsorUserId":"user_bf3e39d8-d680-421d-8d68-7772f7717a25","submittingOrganizationId":null,"sponsorName":"琪越 唐","orcid":"0009-0009-1480-7124","status":"published","createdAt":"2026-10-07 04:29:31","publishedAt":"2026-10-07T05:32:49.629Z","versionNumber":2,"versionCreatedAt":"2026-10-07 13:41:20","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":["Representation theory"],"crossListSubjects":["math.RA"],"authors":[{"id":"1954b194-e92d-4b81-9073-96c763e91493","name":"琪越 唐","orcid":"0009-0009-1480-7124","affiliation":"Tsinghua University","claimed":1,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["16E10","16G20"],"tools":[],"id":"hexagon:2610.00140v2","links":{"record":"https://hexagonmath.org/abs/2610.00140v2","pdf":"https://hexagonmath.org/pdf/2610.00140v2","source":"https://hexagonmath.org/src/2610.00140v2","source_tree":"https://hexagonmath.org/api/v1/works/2610.00140v2/source","api":"https://hexagonmath.org/api/v1/works/2610.00140v2"}},{"identifier":"2610.00138","title":"Weak chromatic splitting fails at n = p for every odd prime p","abstract":"We prove that for every odd prime $p$ there is a height $n$ for which the map $L_{n-1}S_p \\to L_{n-1}\\L_{K(n)}S_p$ is not injective on homotopy groups, hence not the inclusion of a wedge summand. In particular, for such pairs $(n,p)$ the weak chromatic splitting conjecture is false. More precisely, we prove the conjecture fails for $(n,p) = (3,3)$ and for all pairs $(n,p)$ with $p \\le n < \\sqrt{2}(p-1)$. In particular, the conjecture fails for $(n,p) = (p,p)$ for every odd prime $p$.\n\nThis disproof was produced autonomously by Claude in a session working on Freyd's generating hypothesis with no human in the loop. This note is a human write-up of the disproof. We did not contribute any ideas.","subject":"math.AT","authorshipMode":"human","humanUnderstanding":"all","sponsorUserId":"user_e3e1ec0e-64d9-49e0-9950-b31b70432282","submittingOrganizationId":null,"sponsorName":"Agnes Beaudry","orcid":"0000-0003-0715-3109","status":"published","createdAt":"2026-10-07 03:09:38","publishedAt":"2026-10-07T03:17:25.481Z","versionNumber":1,"versionCreatedAt":"2026-10-07 03:09:38","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":[],"crossListSubjects":[],"authors":[{"id":"e5b347ef-849c-4d6c-ba0c-b8e93c313503","name":"Agnes Beaudry","orcid":"0000-0003-0715-3109","affiliation":"University of Colorado Boulder","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null},{"id":"f179aa4c-f647-4ee4-a5f0-3f78394678ad","name":"Tomer Schlank","orcid":null,"affiliation":"Harvard University","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["55P42"],"tools":[],"id":"hexagon:2610.00138v1","links":{"record":"https://hexagonmath.org/abs/2610.00138v1","pdf":"https://hexagonmath.org/pdf/2610.00138v1","source":"https://hexagonmath.org/src/2610.00138v1","source_tree":"https://hexagonmath.org/api/v1/works/2610.00138v1/source","api":"https://hexagonmath.org/api/v1/works/2610.00138v1"}},{"identifier":"2610.00137","title":"Proof of the Non-existence of Perfect Cuboids via Mordell-Weil Rank Exhaustion and Minimal Polynomial Irreducibility of the Perfect Cuboid Surface","abstract":"This manuscript establishes the non-existence of the Perfect Cuboid---a rectangular parallelepiped with integer edges, face diagonals, and space diagonal. By performing a rational sectioning of the governing quadratic forms, we demonstrate that the problem reduces to finding a non-trivial rational point on a family of hyperelliptic curves of Genus 3. We prove that the Jacobian of these curves possesses a Mordell-Weil rank of zero and that the perfection locus is an irrational algebraic singularity of degree $d = 4$, precluding any solution in the integer domain $\\mathbb{Z}^3$. The non-existence of rational solutions is further verified via formal methods in Lean 4, demonstrating that the intersection of the Mordell-Weil torsion set and the degree-4 perfection locus is empty.","subject":"math.NT","authorshipMode":"mixed","humanUnderstanding":"all","sponsorUserId":"user_0d3335e9-808b-46c0-9620-7557f99eb8be","submittingOrganizationId":null,"sponsorName":"Jonathan 𝑓(n) Reed","orcid":"0009-0008-7345-1407","status":"published","createdAt":"2026-10-07 02:53:04","publishedAt":"2026-10-07T03:05:34.188Z","versionNumber":1,"versionCreatedAt":"2026-10-07 02:53:04","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":["2-Descent","Euler Brick","Hyperelliptic Curves","Interactive Theorem Proving","Jacobian Variety","Lean 4","Mordell-Weil Rank","Perfect Cuboid","Quadratic Residue","formal verification"],"crossListSubjects":["math.AG","math.LO"],"authors":[{"id":"ef10148a-ea3a-49a5-8b33-b816af3b178c","name":"Jonathan 𝑓(n) Reed","orcid":"0009-0008-7345-1407","affiliation":null,"claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["11D41","11G30","14G05","11Y50"],"tools":[{"provider":"Google","name":"Gemini","version":null},{"provider":"Lean","name":"Lean","version":"v4.28.0"}],"id":"hexagon:2610.00137v1","links":{"record":"https://hexagonmath.org/abs/2610.00137v1","pdf":"https://hexagonmath.org/pdf/2610.00137v1","source":"https://hexagonmath.org/src/2610.00137v1","source_tree":"https://hexagonmath.org/api/v1/works/2610.00137v1/source","api":"https://hexagonmath.org/api/v1/works/2610.00137v1"}},{"identifier":"2610.00136","title":"The generalized Fermat equation $x^3+y^3=z^n$","abstract":"We prove that the equations $x^3+y^3=z^n$ and $x^3+y^3=3z^n$ have no solution in coprime nonzero integers for any $n\\ge3$. For the first equation the open cases were prime exponents $p\\equiv1\\pmod 3$ above $10^9$ outside a set of congruence classes treated by Chen and Siksek; in these cases the solution $1^3+2^3=3^2$ blocks the modular method. We replace Kraus's Frey curve by a hypergeometric motive of rank three over $K=\\mathbb{Q}(\\sqrt{-3})$, whose parameter at a solution is $3$-adically close enough to a point of maximally unipotent monodromy for inertia at $\\sqrt{-3}$ to act unipotently; the parameter of $1^3+2^3=3^2$ is not. By a theorem of Calegari, Emerton and Gee, the mod $p$ representation of the fiber, twisted by a cubic character, is the reduction of a member of a compatible system of minimal ramification, and we show that the member at a prime above $3$ is irreducible and ordinary at $\\sqrt{-3}$. Ramification and unconditional discriminant bounds show that its reductions have trivial semisimplification, and as the Galois group of the maximal pro-$3$ extension of $K$ unramified outside $\\sqrt{-3}$ is generated by one inertia group, the member stabilizes its ordinary line, a contradiction. For $x^3+y^3=3z^n$ the same argument also works at the primes $p\\equiv2\\pmod3$, which are inert in $K$, and Frey curves and descents treat the remaining exponents.","subject":"math.NT","authorshipMode":"no-human-author-claimed","humanUnderstanding":"none","sponsorUserId":"user_1551e047-7e62-41ba-8e1c-43a5bdf2acbd","submittingOrganizationId":null,"sponsorName":"Manvir Jaswal","orcid":"0009-0001-2880-9528","status":"published","createdAt":"2026-10-07 02:31:41","publishedAt":"2026-10-07T03:04:04.843Z","versionNumber":1,"versionCreatedAt":"2026-10-07 02:31:41","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":["Frey representations","compatible systems of Galois representations","generalized Fermat equation","hypergeometric motives","modular method"],"crossListSubjects":["math.AC"],"authors":[{"id":"ecaa3c45-14bb-44df-8586-cad14f292a59","name":"Manvir Jaswal","orcid":"0009-0001-2880-9528","affiliation":"Independent researcher, Toronto, Canada","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["11D41","11F80","11R29","11R42","14D05"],"tools":[{"provider":"Anthropic","name":"Claude Opus","version":"5.5"},{"provider":"SageMath","name":"SageMath","version":"10.9"},{"provider":"PARI Group","name":"PARI/GP","version":"2.17.3"},{"provider":"GAP Group","name":"GAP","version":"4.15.1"},{"provider":"Python Software Foundation","name":"Python (python-flint/Arb, SymPy, mpmath)","version":"3.12"}],"id":"hexagon:2610.00136v1","links":{"record":"https://hexagonmath.org/abs/2610.00136v1","pdf":"https://hexagonmath.org/pdf/2610.00136v1","source":"https://hexagonmath.org/src/2610.00136v1","source_tree":"https://hexagonmath.org/api/v1/works/2610.00136v1/source","api":"https://hexagonmath.org/api/v1/works/2610.00136v1"}},{"identifier":"2610.00135","title":"Improved sum-difference inequalities in abelian groups","abstract":"For all finite subsets $X,Y$ of an abelian group, we prove\n\\[\n |X-Y|\\le |X+Y|^{\\lamI},\\qquad\n \\lamI=\\frac{9451\\e-3286}{5378\\e+787}=1.454277448906\\ldots,\n\\]\nimproving the classical exponent $3/2$. We also prove that a universal\ntwo-set exponent $\\lambda$ over the integers implies an upper bound\n$2-1/\\lambda$ for the Gyarmati--Hennecart--Ruzsa constant, the supremum $\\theta^*$ of $t$ such that $|A-B|\\gg|A+B|^t$ and $|A+B|\\ll|A|$ for arbitrarily large $A,B\\subset\\Z$.\nConsequently,\n\\[\n \\theta^*\\le\\frac{13524\\e-7359}{9451\\e-3286}=1.312373302115\\ldots,\n\\]\nimproving their bound $4/3$. The first argument combines a coupling with\ndistinct differences, entropy inequalities for independent sums, a finite\ncertificate using non-Shannon inequalities, and an explicit limiting\ncertificate with weights $\\int_0^1t(1-t)^k\\e^t\\,dt$. The transfer uses\nlocalisation and the Pl\\\"unnecke--Ruzsa inequality over a sequence of scales.\nThe proofs are formalised in Lean~4.","subject":"math.NT","authorshipMode":"no-human-author-claimed","humanUnderstanding":"some","sponsorUserId":"user_24ccd52a-951d-4f9a-9073-9b10947bdfce","submittingOrganizationId":null,"sponsorName":"Logan Kleinwaks","orcid":"0009-0009-2926-0942","status":"published","createdAt":"2026-10-07 01:50:00","publishedAt":"2026-10-07T03:02:51.612Z","versionNumber":1,"versionCreatedAt":"2026-10-07 01:50:00","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":["Shannon entropy","difference sets","formal verification","non-Shannon inequalities","sumsets"],"crossListSubjects":["math.CO"],"authors":[{"id":"2b0aa18f-2cc7-4d4c-b745-81dc879a3948","name":"Logan Kleinwaks","orcid":"0009-0009-2926-0942","affiliation":null,"claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["11B30","11B75","94A17"],"tools":[{"provider":"OpenAI","name":"ChatGPT","version":"GPT-6 Astra"},{"provider":"Harmonic","name":"Aristotle","version":null},{"provider":"OpenAI","name":"ChatGPT","version":"GPT-5.6 Sol"}],"id":"hexagon:2610.00135v1","links":{"record":"https://hexagonmath.org/abs/2610.00135v1","pdf":"https://hexagonmath.org/pdf/2610.00135v1","source":"https://hexagonmath.org/src/2610.00135v1","source_tree":"https://hexagonmath.org/api/v1/works/2610.00135v1/source","api":"https://hexagonmath.org/api/v1/works/2610.00135v1"}},{"identifier":"2610.00134","title":"Poisson boundaries of type III factors and the relative bicentralizer conjecture","abstract":"We develop the Poisson boundary theory for general von Neumann algebras with separable preduals. As applications, we generalize and give new proofs of the results on the weak Dixmier property for inclusions with expectations or operator-valued weights \\cite{Pop99,Mar19,Iso25}. We also prove that every continuous factor admits irreducible embeddings into the hyperfinite factors of type $\\mathrm{III}_\\lambda$, $0<\\lambda\\leq1$, and into some hyperfinite factor of type $\\mathrm{III}_0$. Finally, using the Poisson boundary techniques developed above, we prove that for every inclusion with faithful normal conditional expectation $M\\subset N$, $M\\subset c(N)$ has the weak Dixmier property. By Marrakchi's characterization \\cite[Theorem~D]{Ma25}, this gives the relative bicentralizer conjecture for von Neumann algebras with separable preduals.","subject":"math.OA","authorshipMode":"mixed","humanUnderstanding":"all","sponsorUserId":"user_081271d1-fa13-4af0-b3a1-073293ba2df5","submittingOrganizationId":null,"sponsorName":"Shuoxing Zhou","orcid":"0009-0003-8537-1758","status":"published","createdAt":"2026-10-06 23:26:45","publishedAt":"2026-10-06T23:55:32.550Z","versionNumber":2,"versionCreatedAt":"2026-10-07 05:04:11","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":[],"crossListSubjects":[],"authors":[{"id":"13b9f161-4e29-4120-bd64-378e23acbd21","name":"Shuoxing Zhou","orcid":null,"affiliation":"University of Oxford","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["46L10"],"tools":[{"provider":"OpenAI","name":"ChatGPT","version":"6 Astra"},{"provider":"Google","name":"Gemini","version":"3.1 Pro"}],"id":"hexagon:2610.00134v2","links":{"record":"https://hexagonmath.org/abs/2610.00134v2","pdf":"https://hexagonmath.org/pdf/2610.00134v2","source":"https://hexagonmath.org/src/2610.00134v2","source_tree":"https://hexagonmath.org/api/v1/works/2610.00134v2/source","api":"https://hexagonmath.org/api/v1/works/2610.00134v2"}},{"identifier":"2610.00083","title":"Strongly positive bases and quantum greedy expansions","abstract":"We prove that every element of a strongly positive basis of a coefficient-free rank-two quantum cluster algebra has nonnegative Laurent-polynomial coefficients in the quantum greedy basis. This establishes the strongly positive basis assertion of Conjecture 16 of Lee, Li, Rupel, and Zelevinsky. We also give a counterexample to their Conjecture 13(b).","subject":"math.QA","authorshipMode":"mixed","humanUnderstanding":"some","sponsorUserId":"user_bf3e39d8-d680-421d-8d68-7772f7717a25","submittingOrganizationId":null,"sponsorName":"琪越 唐","orcid":"0009-0009-1480-7124","status":"published","createdAt":"2026-10-06 06:19:18","publishedAt":"2026-10-06T14:51:27.754Z","versionNumber":1,"versionCreatedAt":"2026-10-06 06:19:18","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":["quantum cluster algebra"],"crossListSubjects":["math.RA","math.RT"],"authors":[{"id":"906ffa9a-b66b-4b04-825e-8984fc9d1814","name":"琪越 唐","orcid":"0009-0009-1480-7124","affiliation":"Tsinghua University","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["13F60","16T20"],"tools":[],"id":"hexagon:2610.00083v1","links":{"record":"https://hexagonmath.org/abs/2610.00083v1","pdf":"https://hexagonmath.org/pdf/2610.00083v1","source":"https://hexagonmath.org/src/2610.00083v1","source_tree":"https://hexagonmath.org/api/v1/works/2610.00083v1/source","api":"https://hexagonmath.org/api/v1/works/2610.00083v1"}},{"identifier":"2610.00031","title":"Algebraic dependence of cluster X-variables","abstract":"We prove that, over any field and with algebraically independent initial seed coordinates, two cluster \\(X\\)-variables are algebraically dependent if and only if they are equal or reciprocal. In characteristic zero, this is equivalent to the vanishing of the wedge product of their differentials.","subject":"math.RA","authorshipMode":"mixed","humanUnderstanding":"some","sponsorUserId":"user_bf3e39d8-d680-421d-8d68-7772f7717a25","submittingOrganizationId":null,"sponsorName":"琪越 唐","orcid":"0009-0009-1480-7124","status":"published","createdAt":"2026-10-05 03:19:14","publishedAt":"2026-10-05T15:51:57.022Z","versionNumber":1,"versionCreatedAt":"2026-10-05 03:19:14","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":["Cluster Algebra"],"crossListSubjects":["math.CO"],"authors":[{"id":"541d04ad-53d9-47c8-b325-1297eb20ff51","name":"琪越 唐","orcid":"0009-0009-1480-7124","affiliation":"Tsinghua University","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["13F60","12F20","12J20"],"tools":[],"id":"hexagon:2610.00031v1","links":{"record":"https://hexagonmath.org/abs/2610.00031v1","pdf":"https://hexagonmath.org/pdf/2610.00031v1","source":"https://hexagonmath.org/src/2610.00031v1","source_tree":"https://hexagonmath.org/api/v1/works/2610.00031v1/source","api":"https://hexagonmath.org/api/v1/works/2610.00031v1"}},{"identifier":"2610.00032","title":"An explicit nonideal rooted cluster morphism","abstract":"We give a counterexample to Chang and Zhu’s question of whether every explicit rooted cluster morphism is ideal.","subject":"math.RA","authorshipMode":"mixed","humanUnderstanding":"some","sponsorUserId":"user_bf3e39d8-d680-421d-8d68-7772f7717a25","submittingOrganizationId":null,"sponsorName":"琪越 唐","orcid":"0009-0009-1480-7124","status":"published","createdAt":"2026-10-05 07:21:12","publishedAt":"2026-10-05T15:51:45.676Z","versionNumber":1,"versionCreatedAt":"2026-10-05 07:21:12","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":["Cluster Algebra"],"crossListSubjects":["math.CO"],"authors":[{"id":"3c1dd0fe-f4ff-49bc-b771-117b299ef55d","name":"琪越 唐","orcid":"0009-0009-1480-7124","affiliation":"Tsinghua University","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["13F60","13B22"],"tools":[],"id":"hexagon:2610.00032v1","links":{"record":"https://hexagonmath.org/abs/2610.00032v1","pdf":"https://hexagonmath.org/pdf/2610.00032v1","source":"https://hexagonmath.org/src/2610.00032v1","source_tree":"https://hexagonmath.org/api/v1/works/2610.00032v1/source","api":"https://hexagonmath.org/api/v1/works/2610.00032v1"}},{"identifier":"2610.00030","title":"Crystalline and infinitesimal Poisson stacks","abstract":"A systematic way to construct Poisson brackets on moduli stacks in algebraic geometry is through shifted Poisson structures on derived stacks. The standard construction, due to Calaque--Pantev--To{\\\"e}n--Vaqui{\\'e}--Vezzosi, imports Gelfand--Kazhdan formal geometry to derived stacks in order to establish formal descent for polyvector fields. The result is a graded Lie algebra which we call the ``crystalline'' polyvector fields on a derived stack. Using instead the formal derived geometry machinery of Gaitsgory--Rozenblyum, we construct a graded Lie algebra of ``infinitesimal'' polyvector fields which has a direct geometric interpretation: it governs deformations of the formal shifted cotangent stack, together with its canonical exact symplectic form. Our main result is an equivalence between crystalline and infinitesimal polyvector fields on a derived stack as graded Lie algebra objects, providing a deformation-theoretic perspective on the graded Lie algebra constructed by Calaque--Pantev--To{\\\"e}n--Vaqui{\\'e}--Vezzosi.","subject":"math.AG","authorshipMode":"human","humanUnderstanding":"all","sponsorUserId":"user_9e8f4de8-ff99-4fd5-9d72-15fa96d7eabe","submittingOrganizationId":null,"sponsorName":"Aleksandr Karapetyan","orcid":"0009-0005-7864-437X","status":"published","createdAt":"2026-10-04 17:10:15","publishedAt":"2026-10-04T18:14:06.540Z","versionNumber":1,"versionCreatedAt":"2026-10-04 17:10:15","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":["Derived algebraic geometry","deformation theory","derived algebra","infinity-categories","shifted Poisson structures","shifted symplectic structures"],"crossListSubjects":["math.AT","math.SG"],"authors":[{"id":"f3f98f92-a012-4ff1-befa-b5b35f67c52f","name":"Aleksandr Karapetyan","orcid":"0009-0005-7864-437X","affiliation":null,"claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["14F40"],"tools":[],"id":"hexagon:2610.00030v1","links":{"record":"https://hexagonmath.org/abs/2610.00030v1","pdf":"https://hexagonmath.org/pdf/2610.00030v1","source":"https://hexagonmath.org/src/2610.00030v1","source_tree":"https://hexagonmath.org/api/v1/works/2610.00030v1/source","api":"https://hexagonmath.org/api/v1/works/2610.00030v1"}},{"identifier":"2610.00029","title":"Theta Bases for Rank-Two Quantum Cluster Algebras of Geometric Type","abstract":"We prove that quantum theta and greedy bases coincide in coefficient-free rank two for all positive exchange parameters. Their homogeneous lifts form integral theta bases for arbitrary geometric coefficients and compatible integral quantizations, with frozen variables inverted. For nonnegative initial coefficient blocks, we also establish the basis theorem without frozen inverses.","subject":"math.QA","authorshipMode":"no-human-author-claimed","humanUnderstanding":"none","sponsorUserId":"user_bf3e39d8-d680-421d-8d68-7772f7717a25","submittingOrganizationId":null,"sponsorName":"琪越 唐","orcid":"0009-0009-1480-7124","status":"published","createdAt":"2026-10-04 05:32:48","publishedAt":"2026-10-04T05:38:15.072Z","versionNumber":1,"versionCreatedAt":"2026-10-04 05:32:48","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":["quantum cluster algebra"],"crossListSubjects":["math.RA","math.RT"],"authors":[{"id":"d2589110-4092-4c9f-bf8c-a804faaa203d","name":"琪越 唐","orcid":"0009-0009-1480-7124","affiliation":"Tsinghua University","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["13F60","16S35","16S85"],"tools":[],"id":"hexagon:2610.00029v1","links":{"record":"https://hexagonmath.org/abs/2610.00029v1","pdf":"https://hexagonmath.org/pdf/2610.00029v1","source":"https://hexagonmath.org/src/2610.00029v1","source_tree":"https://hexagonmath.org/api/v1/works/2610.00029v1/source","api":"https://hexagonmath.org/api/v1/works/2610.00029v1"}},{"identifier":"2610.00026","title":"The Anti-Arithmetic Product of Symmetric Functions","abstract":"The \\emph{arithmetic product} $\\boxdot$ is a bilinear operation\non the ring of symmetric functions over $\\mathbb Q$ defined by\n\\[\n p_\\lambda \\ \\boxdot\\ p_\\mu\n = \\prod_{i,j}p_{\\operatorname{lcm}(\\lambda_i,\\mu_j)}^{\\gcd(\\lambda_i,\\mu_j)}\n\\]\non the power-sum basis $(p_\\lambda)$.\nIt is known that this operation preserves the subring\n$\\mathbf{Symm}_{\\mathbb Z}$ of symmetric functions with\ninteger coefficients, and is Schur-positive on pairs of Schur\nfunctions. Indeed, it corresponds to induction of\nrepresentations from $S_n \\times S_m$ to $S_{nm}$.\n\nIn this work, we consider the \\emph{anti-arithmetic product}\n$\\diamondsuit$, which is defined by the same formula but with\n$\\gcd$ and $\\operatorname{lcm}$ interchanged.\nWe show that it, too, preserves $\\mathbf{Symm}_{\\mathbb Z}$,\neven though it lacks the Schur positivity property.\nThis was conjectured on MathOverflow in 2014.\n\nOur method is a more intricate variant of the\nrepresentation-theoretical interpretation of $\\boxdot$.\nThe anti-arithmetic product does not correspond to an\noperation on actual representations; but its adjoint can still\nbe described as a map on characters, which (as we show) is a\n$\\lambda$-ring morphism from the character ring of\n$S_{mn}$ to that of $S_m \\times S_n$.\nNow, a 1975 theorem of Boorman shows that the\nrepresentation ring of $S_n$ is generated by the natural\npermutation representation $M_n = \\mathbb Q^n$ as a $\\lambda$-ring.\nThus, the proof of integrality boils down to only computing\nthe image of $M_{mn}$ under this adjoint, which can be done using\nM\\\"obius inversion in the representation ring.\n\nThis work is aimed at readers familiar with representation rings\nand the representation theory of symmetric groups.  No prior\nknowledge of $\\lambda$-rings is presumed.\nThree self-contained proofs of Boorman's theorem are given in the\nappendices, two of them lifting the theorem to the noncommutative\nsymmetric functions (or Solomon's descent algebra), recovering\na result of Schocker.","subject":"math.CO","authorshipMode":"mixed","humanUnderstanding":"all","sponsorUserId":"user_50559304-a175-4f48-9b02-f7ccee550e5f","submittingOrganizationId":null,"sponsorName":"Darij Grinberg","orcid":"0000-0002-9661-8432","status":"published","createdAt":"2026-10-03 23:20:35","publishedAt":"2026-10-03T23:55:34.877Z","versionNumber":1,"versionCreatedAt":"2026-10-03 23:20:35","doi":null,"arxivId":null,"license":"CC0 1.0","tags":["Specht modules","descent algebra","lambda-rings","noncommutative symmetric functions","symmetric functions","symmetric group representations"],"crossListSubjects":["math.RA","math.RT"],"authors":[{"id":"c898815a-4db9-4f54-9392-afa1890330e4","name":"Darij Grinberg","orcid":"0000-0002-9661-8432","affiliation":"Drexel University","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["20C30","05E05","19A22"],"tools":[{"provider":"OpenAI","name":"ChatGPT","version":"5.5"},{"provider":"OpenAI","name":"ChatGPT","version":"6 Astra"}],"id":"hexagon:2610.00026v1","links":{"record":"https://hexagonmath.org/abs/2610.00026v1","pdf":"https://hexagonmath.org/pdf/2610.00026v1","source":"https://hexagonmath.org/src/2610.00026v1","source_tree":"https://hexagonmath.org/api/v1/works/2610.00026v1/source","api":"https://hexagonmath.org/api/v1/works/2610.00026v1"}},{"identifier":"2610.00024","title":"Reflexive equivalence and reflexive-minimal algebras","abstract":"Let $A$ be a finite-dimensional algebra over a field. A finite-dimensional\nalgebra $B$ over the same field is called reflexively equivalent to $A$ if\ntheir categories of reflexive modules are equivalent. We prove that there\nis a basic algebra $A_{\\min}$, unique up to isomorphism, such that $B$ is\nreflexively equivalent to $A$ if and only if\n$B\\cong\\operatorname{End}_{A_{\\min}}(A_{\\min}\\oplus X)$ for some reflexive\n$A_{\\min}$-module $X$. Moreover, we compute $A_{\\min}$ explicitly. We\nextend these results to module-finite algebras over henselian local rings\nof dimension at most one under a dominant dimension condition at minimal\nprimes. We also introduce reflexive modules over additive categories and\nprove that taking reflexive modules is idempotent: the reflexive modules\nover the category of reflexive modules form an equivalent category. As an\napplication, we show that reflexive equivalence classes of algebras with\nfinitely many indecomposable reflexive modules correspond bijectively to\nMorita equivalence classes of algebras whose reflexive modules are\nprojective.","subject":"math.RT","authorshipMode":"no-human-author-claimed","humanUnderstanding":"some","sponsorUserId":"user_ad92be63-dee7-4f4c-92ad-cb2cf08aa3a4","submittingOrganizationId":null,"sponsorName":"Haruhisa Enomoto","orcid":"0000-0003-0644-7076","status":"published","createdAt":"2026-10-03 10:51:41","publishedAt":"2026-10-03T14:57:30.121Z","versionNumber":1,"versionCreatedAt":"2026-10-03 10:51:41","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":[],"crossListSubjects":["math.RA"],"authors":[{"id":"585bb5bd-bf65-4b9e-9f58-ef8d76824735","name":"Haruhisa Enomoto","orcid":"0000-0003-0644-7076","affiliation":"Parakeet Inc., Japan","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["16G10","16D90","18E05","16E05"],"tools":[{"provider":"OpenAI","name":"Codex","version":"GPT-6-Astra"},{"provider":"Anthropic","name":"Claude Opus","version":"Opus 5.5"}],"id":"hexagon:2610.00024v1","links":{"record":"https://hexagonmath.org/abs/2610.00024v1","pdf":"https://hexagonmath.org/pdf/2610.00024v1","source":"https://hexagonmath.org/src/2610.00024v1","source_tree":"https://hexagonmath.org/api/v1/works/2610.00024v1/source","api":"https://hexagonmath.org/api/v1/works/2610.00024v1"}},{"identifier":"2610.00025","title":"Minimality of the Rank-Two Quantum Greedy Basis","abstract":"We prove that the rank-two quantum greedy basis is the least strongly positive basis when b | c or c | b, resolving Conjecture 14 of Lee–Li–Rupel–Zelevinsky. Every strongly positive basis has nonnegative expansion coefficients in the greedy basis.","subject":"math.QA","authorshipMode":"mixed","humanUnderstanding":"some","sponsorUserId":"user_bf3e39d8-d680-421d-8d68-7772f7717a25","submittingOrganizationId":null,"sponsorName":"琪越 唐","orcid":"0009-0009-1480-7124","status":"published","createdAt":"2026-10-03 13:34:43","publishedAt":"2026-10-03T14:56:18.726Z","versionNumber":2,"versionCreatedAt":"2026-10-03 16:15:54","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":["quantum cluster algebra"],"crossListSubjects":["math.RA"],"authors":[{"id":"cd8435bb-2d8c-44e1-8940-26853e941498","name":"琪越 唐","orcid":null,"affiliation":"Tsinghua University","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["13F60","16T20"],"tools":[],"id":"hexagon:2610.00025v2","links":{"record":"https://hexagonmath.org/abs/2610.00025v2","pdf":"https://hexagonmath.org/pdf/2610.00025v2","source":"https://hexagonmath.org/src/2610.00025v2","source_tree":"https://hexagonmath.org/api/v1/works/2610.00025v2/source","api":"https://hexagonmath.org/api/v1/works/2610.00025v2"}},{"identifier":"2610.00023","title":"A Finite Polyhedral Construction of the Rank-Two Quantum Greedy Basis","abstract":"We give a finite polyhedral construction of the quantum greedy basis for coefficient-free rank-two quantum cluster algebras. The construction specializes coeffi\u0002cientwise to the classical polyhedral construction at v = 1. Corresponding classical and quantum coefficients and auxiliary margins have identical zero sets, and the Newton polytopes are affine images of the classical carriers.","subject":"math.QA","authorshipMode":"mixed","humanUnderstanding":"some","sponsorUserId":"user_bf3e39d8-d680-421d-8d68-7772f7717a25","submittingOrganizationId":null,"sponsorName":"琪越 唐","orcid":"0009-0009-1480-7124","status":"published","createdAt":"2026-10-03 01:39:04","publishedAt":"2026-10-03T05:47:08.044Z","versionNumber":1,"versionCreatedAt":"2026-10-03 01:39:04","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":["quantum cluster algebra"],"crossListSubjects":["math.CO","math.RA"],"authors":[{"id":"95f0e864-d385-4c8f-97a9-3b038167508f","name":"琪越 唐","orcid":"0009-0009-1480-7124","affiliation":"Tsinghua University","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["13F60","16T20","52B20"],"tools":[],"id":"hexagon:2610.00023v1","links":{"record":"https://hexagonmath.org/abs/2610.00023v1","pdf":"https://hexagonmath.org/pdf/2610.00023v1","source":"https://hexagonmath.org/src/2610.00023v1","source_tree":"https://hexagonmath.org/api/v1/works/2610.00023v1/source","api":"https://hexagonmath.org/api/v1/works/2610.00023v1"}},{"identifier":"2610.00022","title":"Continuity of solutions to abstract linear control systems","abstract":"It has long been known that the solutions to abstract linear control systems are continuous in time for controls in $L^p$ with $1 \\le p < \\infty$. We prove that the same property remains valid for the endpoint case $p = \\infty$, giving a positive answer to Weiss' 1989 Problem 2.4.\n\n    The proof relies on a direct semigroup argument based on Phillips' lemma, does not require the input map to have an integral representation (which is not always the case for $p = \\infty$), and actually entails that such systems are all of the zero-class (which fails for $1 \\le p < \\infty$).","subject":"math.OC","authorshipMode":"no-human-author-claimed","humanUnderstanding":"all","sponsorUserId":"user_61298202-0771-44e6-96f8-0c68c63f45c5","submittingOrganizationId":null,"sponsorName":"Frédéric Marbach","orcid":"0000-0001-8588-5896","status":"published","createdAt":"2026-10-03 01:02:19","publishedAt":"2026-10-03T05:46:27.743Z","versionNumber":2,"versionCreatedAt":"2026-10-03 13:24:38","doi":null,"arxivId":null,"license":"CC BY-SA 4.0","tags":["abstract linear control systems"],"crossListSubjects":["math.FA"],"authors":[{"id":"a9b32426-ba58-4f21-a2ff-ae1036405c60","name":"Frédéric Marbach","orcid":null,"affiliation":"DMA, Ecole normale superieure, Universite PSL, CNRS, 75005 Paris, France","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["93C05","93C25","47D06","46B45"],"tools":[{"provider":"OpenAI","name":"ChatGPT","version":"6 Pro"},{"provider":"Anthropic","name":"Claude Opus","version":"5.5"}],"id":"hexagon:2610.00022v2","links":{"record":"https://hexagonmath.org/abs/2610.00022v2","pdf":"https://hexagonmath.org/pdf/2610.00022v2","source":"https://hexagonmath.org/src/2610.00022v2","source_tree":"https://hexagonmath.org/api/v1/works/2610.00022v2/source","api":"https://hexagonmath.org/api/v1/works/2610.00022v2"}},{"identifier":"2610.00019","title":"The Negative Spherical Perceptron: Capacity, Isostatic Jamming, and Full Replica Symmetry Breaking","abstract":"We study the negative spherical perceptron, a mean-field model of jamming with a nonconvex solution space. We establish the Parisi formula for its free energy and identify the capacity for every constraint density $\\alpha>2$. We also show that the jamming point is isostatic: optimal configurations have exactly $N$ contacts. For $\\alpha$ slightly above two, we show that replica symmetry is lost at the de Almeida–Thouless margin through a continuous transition to full replica symmetry breaking. Further, we show that Gardner's formula overestimates the capacity by order $(\\alpha-2)^3$.","subject":"math.PR","authorshipMode":"no-human-author-claimed","humanUnderstanding":"some","sponsorUserId":"user_ebc5e6ab-8456-465b-83ac-caf2a5018e8d","submittingOrganizationId":null,"sponsorName":"Patrick Lopatto","orcid":"0000-0002-9811-8052","status":"published","createdAt":"2026-10-02 05:47:29","publishedAt":"2026-10-02T17:42:58.752Z","versionNumber":1,"versionCreatedAt":"2026-10-02 05:47:29","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":["perceptron","spin glass"],"crossListSubjects":["math.MP","math-ph"],"authors":[{"id":"05b2abde-d302-4d7d-beb5-9b4e266c33ba","name":"P. M. Aronow","orcid":"0000-0002-4449-0756","affiliation":"Yale University","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null},{"id":"afc904f8-e0f8-4c08-bf89-253ba9c0db7f","name":"Patrick Lopatto","orcid":"0000-0002-9811-8052","affiliation":"University of North Carolina at Chapel Hill","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["60K35","82B44","82B26"],"tools":[{"provider":"Anthropic","name":"Claude Opus","version":"5.5"},{"provider":"Anthropic","name":"Claude Fable","version":"5.1"},{"provider":"Anthropic","name":"Claude Fable","version":"5"},{"provider":"OpenAI","name":"ChatGPT","version":"6"},{"provider":"OpenAI","name":"ChatGPT","version":"5.6"}],"id":"hexagon:2610.00019v1","links":{"record":"https://hexagonmath.org/abs/2610.00019v1","pdf":"https://hexagonmath.org/pdf/2610.00019v1","source":"https://hexagonmath.org/src/2610.00019v1","source_tree":"https://hexagonmath.org/api/v1/works/2610.00019v1/source","api":"https://hexagonmath.org/api/v1/works/2610.00019v1"}},{"identifier":"2610.00021","title":"A Fourier Consequence for Lattice Kissing Numbers and Average Contacts","abstract":"We deduce a new asymptotic upper bound $K_{\\mathrm{lat}}(n)\\leq\\left(\\sqrt{2e/\\pi}+o(1)\\right)^n$ on the maximal lattice kissing number $K_{\\mathrm{lat}}(n)$ in dimension $n$, using the explicit auxiliary functions constructed in OpenAI's \"Ten Advances\" preprint. The same bound holds for the average contact degree of a finite packing of congruent balls. The bound's base-$2$ exponential rate rounds to $0.3956$, matching the value extrapolated empirically by Afkhami-Jeddi, Cohn, Hartman, de Laat, and Tajdini in 2020.","subject":"math.MG","authorshipMode":"mixed","humanUnderstanding":"all","sponsorUserId":"user_a64917cf-f9a1-482c-a43f-6c639209d9e6","submittingOrganizationId":null,"sponsorName":"Scott Kominers","orcid":"0000-0002-7608-6619","status":"published","createdAt":"2026-10-02 11:33:21","publishedAt":"2026-10-02T11:56:25.521Z","versionNumber":1,"versionCreatedAt":"2026-10-02 11:33:21","doi":null,"arxivId":null,"license":"CC BY-SA 4.0","tags":["Contact numbers","Discrete geometry","Euclidean lattices","Fourier linear programming bounds","Geometry of numbers","Kissing numbers","Sphere packing"],"crossListSubjects":["math.CO","math.NT"],"authors":[{"id":"794f7873-b56e-4f9b-a817-823d08414872","name":"Scott Kominers","orcid":"0000-0002-7608-6619","affiliation":"Harvard University and a16z crypto","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["11H31","52C17","42B10"],"tools":[{"provider":"OpenAI","name":"GPT","version":"6 Astra"},{"provider":"Anthropic","name":"Claude Fable","version":"5.1"},{"provider":"Quora","name":"Poe","version":null}],"id":"hexagon:2610.00021v1","links":{"record":"https://hexagonmath.org/abs/2610.00021v1","pdf":"https://hexagonmath.org/pdf/2610.00021v1","source":"https://hexagonmath.org/src/2610.00021v1","source_tree":"https://hexagonmath.org/api/v1/works/2610.00021v1/source","api":"https://hexagonmath.org/api/v1/works/2610.00021v1"}},{"identifier":"2610.00017","title":"On the ternary pentagonal numbers conjecture","abstract":"Communicated by Guy in 1994, the ternary pentagonal numbers conjecture of Blecksmith and Selfridge asserts that every integer larger than $33066$ is the sum of three positive pentagonal numbers. Prior to this note, it was even unknown whether every sufficiently large integer is the sum of three positive pentagonal numbers. We resolve this in the affirmative, using the landmark work of Duke and Schulze-Pillot on ternary quadratic forms to handle all sufficiently large integers $n$ with $v_3(24n+3)\\leq8$, and present an explicit lift to handle the $n$ with $v_3(24n+3)\\geq9$.","subject":"math.NT","authorshipMode":"human","humanUnderstanding":"all","sponsorUserId":"user_c3840c6e-76db-4803-b640-69886c828f7f","submittingOrganizationId":null,"sponsorName":"Glenn Bruda","orcid":"0000-0003-2070-5596","status":"published","createdAt":"2026-10-02 02:17:12","publishedAt":"2026-10-02T03:36:22.091Z","versionNumber":1,"versionCreatedAt":"2026-10-02 02:17:12","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":["Pentagonal numbers","almost universality","ternary quadratic forms"],"crossListSubjects":[],"authors":[{"id":"92a6d7fd-9374-4be9-8192-5aa0432389d4","name":"Glenn Bruda","orcid":"0000-0003-2070-5596","affiliation":"Georgia Institute of Technology","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["11D09","11E25","11E20"],"tools":[{"provider":"OpenAI","name":"ChatGPT","version":"5.6 Sol"}],"id":"hexagon:2610.00017v1","links":{"record":"https://hexagonmath.org/abs/2610.00017v1","pdf":"https://hexagonmath.org/pdf/2610.00017v1","source":"https://hexagonmath.org/src/2610.00017v1","source_tree":"https://hexagonmath.org/api/v1/works/2610.00017v1/source","api":"https://hexagonmath.org/api/v1/works/2610.00017v1"}},{"identifier":"2610.00018","title":"Linnik's constant is at most $3.99$","abstract":"Let $P(a,q)$ denote the least prime congruent to $a$ modulo $q$, where $(a,q)=1$.\nWe give a computer-assisted proof that $P(a,q)\\ll q^{3.99}$, improving the exponent $5$\nof Xylouris. More precisely, $P(a,q)<q^{3.99}$ for every sufficiently large $q$, uniformly\nin $a$.\n\nThe main new ingredient is a graded near-density estimate for zeros of Dirichlet\n$L$-functions, a weighted form of Heath-Brown's Lemma~12.1 of the kind he asked for in 1992. We\ncombine it with the zero-location estimates and far-density method of Heath-Brown and Xylouris.\n\nThe second novelty is the scale of the case analysis. Heath-Brown and Xylouris closed their final\ncase analyses with $14$ and $21$ main cases, each by a chain of inequalities evaluated in floating\npoint. In this paper, with the help of advanced AI models, we can push this much further. The middle range of the first zero is divided into $4453$ root cases and $4788$\nterminal cases, each closed by its own linear program. On $4{,}196{,}879$ threshold boxes these\nprograms give $29{,}397{,}336$ linear relaxations, and exact integer certificates for all of them\nbound the normalized zero sum strictly below~$1$. Exceptional\nzeros and the remaining exterior range are treated separately. This case analysis is a kind of\nsystematic brute force enabled by AI: an analysis of this size, with parameters tuned to each case,\nwould be very laborious or nearly impossible to carry out by hand, and it lets the new estimate be\napplied separately in each case.\n\nThe near-density lemma, certificate soundness, and a conditional passage from a certified\ncase to a prime are formalized in Lean, assuming published analytic inputs and specified\nfacts about zeros (PALOMAR-2026-10-01-000020 v1).","subject":"math.NT","authorshipMode":"no-human-author-claimed","humanUnderstanding":"some","sponsorUserId":"user_78af3893-4515-4d2b-a21b-f3e8f0649bbc","submittingOrganizationId":null,"sponsorName":"Eric Naslund","orcid":"0000-0001-5407-9891","status":"published","createdAt":"2026-10-02 02:34:30","publishedAt":"2026-10-02T03:35:22.598Z","versionNumber":1,"versionCreatedAt":"2026-10-02 02:34:30","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":[],"crossListSubjects":[],"authors":[{"id":"8d14e863-7032-4b28-9b28-3f98df97b0c5","name":"Eric Naslund","orcid":"0000-0001-5407-9891","affiliation":null,"claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["11N13","11M26","11M20","11N35","11Y35"],"tools":[{"provider":"Anthropic","name":"Claude Opus","version":"5.5"},{"provider":"OpenAI","name":"ChatGPT","version":"Astra 6"},{"provider":"XAi","name":"Grok","version":"4.7"}],"id":"hexagon:2610.00018v1","links":{"record":"https://hexagonmath.org/abs/2610.00018v1","pdf":"https://hexagonmath.org/pdf/2610.00018v1","source":"https://hexagonmath.org/src/2610.00018v1","source_tree":"https://hexagonmath.org/api/v1/works/2610.00018v1/source","api":"https://hexagonmath.org/api/v1/works/2610.00018v1"}},{"identifier":"2610.00016","title":"An Exponent of 1.04273 for the Unit Distance Problem","abstract":"Let $u(U)$ count the unordered pairs at distance one in a finite planar\nset $U$. We construct finite sets $U_j$ with $|U_j|\\to\\infty$ and\n$u(U_j)/|U_j|^{1.04273}\\to\\infty$. The largest exponent previously\nclaimed, $1.0358$, is in the author's unpublished manuscript. The\nmethod is the number-field construction of OpenAI and Sawin: unit\ndistances come from elements of relative norm one in quadratic\nextensions, and the fields come from an infinite pro-$2$ class field\ntower. The new ingredients are quadratic extensions of mixed signature,\nwith an exact average over their norm-one units, and a tower over the\nreal quadratic field $\\Q(\\sqrt{241})$, in which $2$, $3$ and $5$ split.\nIts Golod--Shafarevich function contains two copies of the local\nconditions at these primes but only one constant term, and the extra room\nlets the tower be ramified only above $2$, $3$ and $5$; the root\ndiscriminant of its fields is about $286$. The relative zeta value is bounded through the zeta function of the\ndegree-$512$ field generated over $\\Q(\\sqrt{241})$ by the square roots of\nits $\\{2,3,5\\}$-units, which is the product of the Dedekind zeta function\nof $\\Q(\\sqrt{241})$ and $255$ quadratic Hecke $L$-functions. Finite facts and numerical\ninequalities are certified by exact computation and interval arithmetic. A significant portion of this work was verified in Lean, reducing the result with exponent $1.0427$ to an explicit zeta function inequality (Palomar registry, PALOMAR-2026-10-01-000018, version 1).","subject":"math.CO","authorshipMode":"no-human-author-claimed","humanUnderstanding":"some","sponsorUserId":"user_78af3893-4515-4d2b-a21b-f3e8f0649bbc","submittingOrganizationId":null,"sponsorName":"Eric Naslund","orcid":"0000-0001-5407-9891","status":"published","createdAt":"2026-10-01 22:21:18","publishedAt":"2026-10-02T01:59:10.318Z","versionNumber":1,"versionCreatedAt":"2026-10-01 22:21:18","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":[],"crossListSubjects":["math.NT"],"authors":[{"id":"33eb2664-13bf-4a2a-b574-c9296b142986","name":"Eric Naslund","orcid":"0000-0001-5407-9891","affiliation":null,"claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["52C10","11R37","11R42","11R29","11H06"],"tools":[{"provider":"OpenAI","name":"ChatGPT","version":"Astra 6"},{"provider":"OpenAI","name":"ChatGPT","version":"Sol 5.6"},{"provider":"Anthropic","name":"Claude Opus","version":"5.5"}],"id":"hexagon:2610.00016v1","links":{"record":"https://hexagonmath.org/abs/2610.00016v1","pdf":"https://hexagonmath.org/pdf/2610.00016v1","source":"https://hexagonmath.org/src/2610.00016v1","source_tree":"https://hexagonmath.org/api/v1/works/2610.00016v1/source","api":"https://hexagonmath.org/api/v1/works/2610.00016v1"}},{"identifier":"2610.00013","title":"Explicit non-inducible rooted cluster morphisms","abstract":"The augmentation of the Markov cluster algebra is explicit but not inducible as a rooted cluster morphism. This answers a question of Chang and Zhu negatively.","subject":"math.RA","authorshipMode":"mixed","humanUnderstanding":"some","sponsorUserId":"user_bf3e39d8-d680-421d-8d68-7772f7717a25","submittingOrganizationId":null,"sponsorName":"琪越 唐","orcid":"0009-0009-1480-7124","status":"published","createdAt":"2026-10-01 18:21:21","publishedAt":"2026-10-01T21:26:19.989Z","versionNumber":1,"versionCreatedAt":"2026-10-01 18:21:21","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":["Cluster Algebra"],"crossListSubjects":[],"authors":[{"id":"f88078ae-9148-48c6-9123-1adede23b249","name":"琪越 唐","orcid":"0009-0009-1480-7124","affiliation":"Tsinghua University","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["13F60","13A02"],"tools":[],"id":"hexagon:2610.00013v1","links":{"record":"https://hexagonmath.org/abs/2610.00013v1","pdf":"https://hexagonmath.org/pdf/2610.00013v1","source":"https://hexagonmath.org/src/2610.00013v1","source_tree":"https://hexagonmath.org/api/v1/works/2610.00013v1/source","api":"https://hexagonmath.org/api/v1/works/2610.00013v1"}},{"identifier":"2610.00015","title":"Tilings of an equilateral triangle by at most five lattice trapezoids with 60° base angles: a complete structural classification","abstract":"We study tilings of an equilateral triangle of side n in the triangular grid by k lattice trapezoids with base angles 60°, the objects behind the OEIS sequences A389392 (k = 4) and A391498 (k = 5). We prove two angle identities valid for every such tiling and a lemma relating the number of boundary vertices to the number of pieces having a side on the boundary. With these tools we show that there are exactly 1, 2 and 13 combinatorial types of tilings for k = 3, 4, 5. For k = 3 every tiling is a pinwheel. For k = 4 every tiling belongs to one of the two categories used in A389392, a fact that had previously been taken\nfor granted. For k = 5 the thirteen types refine the eight categories of A391498; in three of them a piece has no side on the boundary. As a consistency check, the volumes of the parameter polytopes together with the generic multiplicities reproduce the leading coefficient 7/36 of the conjectured quasi-polynomial for A391498. All lemmas were checked against an exhaustive enumeration of the tilings with pairwise distinct pieces for n ≤ 15. Finally, the classification turns the thirteen types into eight explicit families of sets of shapes, and an inclusion–exclusion over them reduces the conjectured generating function of A391498 to twelve elementary counting statements, and we settle all twelve: the resulting closed formula\nreproduces the sequence for every n ≤ 127.","subject":"math.CO","authorshipMode":"mixed","humanUnderstanding":"all","sponsorUserId":"user_d7820c54-4a82-45d8-aa69-c16efb81c645","submittingOrganizationId":null,"sponsorName":"Gonzalo Barria","orcid":"0009-0003-8878-737X","status":"published","createdAt":"2026-10-01 21:21:48","publishedAt":"2026-10-01T21:25:32.360Z","versionNumber":1,"versionCreatedAt":"2026-10-01 21:21:48","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":["OEIS A391498","lattice trapezoids","quasi-polynomial","rational generating function","tilings of an equilateral triangle"],"crossListSubjects":["math.DG","math.MG","cs.DM"],"authors":[{"id":"1c790c90-09c7-4ca8-8993-54c7322fcf28","name":"Gonzalo Barria","orcid":"0009-0003-8878-737X","affiliation":"AIEP","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["05B45","52C20","05A15","52B20","11B83"],"tools":[],"id":"hexagon:2610.00015v1","links":{"record":"https://hexagonmath.org/abs/2610.00015v1","pdf":"https://hexagonmath.org/pdf/2610.00015v1","source":"https://hexagonmath.org/src/2610.00015v1","source_tree":"https://hexagonmath.org/api/v1/works/2610.00015v1/source","api":"https://hexagonmath.org/api/v1/works/2610.00015v1"}},{"identifier":"2610.00014","title":"On the escape rate of favorite sites of planar random walks","abstract":"For planar simple random walk, the favorite sites at time $n$ are the sites\nwhose local time at time $n$ is maximal. We prove that, almost surely, for\nevery $\\gamma>1/2$ and every $c>0$, all favorite sites lie outside the ball\ncentered at the origin with radius $c\\sqrt n/(\\log n)^\\gamma$ for all\nsufficiently large $n$. At the critical exponent $\\gamma=1/2$, almost surely,\nfor every $c>0$, the entire favorite sites lies within distance\n$c\\sqrt{n/\\log n}$ of the origin infinitely often.","subject":"math.PR","authorshipMode":"no-human-author-claimed","humanUnderstanding":"none","sponsorUserId":"user_e5f32d35-1dd2-426f-b34a-e500a5d08cbc","submittingOrganizationId":null,"sponsorName":"Heng Ma","orcid":"0000-0002-1401-7677","status":"published","createdAt":"2026-10-01 19:17:02","publishedAt":"2026-10-01T21:24:19.747Z","versionNumber":1,"versionCreatedAt":"2026-10-01 19:17:02","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":[],"crossListSubjects":[],"authors":[{"id":"ee9aa20c-5fdb-48ba-8c4d-85b197ab67e5","name":"Heng Ma","orcid":"0000-0002-1401-7677","affiliation":"Technion","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["60G50","60J55"],"tools":[{"provider":"OpenAI","name":"Codex","version":"GPT-5.6 Sol"}],"id":"hexagon:2610.00014v1","links":{"record":"https://hexagonmath.org/abs/2610.00014v1","pdf":"https://hexagonmath.org/pdf/2610.00014v1","source":"https://hexagonmath.org/src/2610.00014v1","source_tree":"https://hexagonmath.org/api/v1/works/2610.00014v1/source","api":"https://hexagonmath.org/api/v1/works/2610.00014v1"}},{"identifier":"2610.00012","title":"A finite criterion for total sign skew symmetry in order three","abstract":"We give a finite criterion for total sign skew symmetry of integer matrices of order three.","subject":"math.RA","authorshipMode":"mixed","humanUnderstanding":"some","sponsorUserId":"user_bf3e39d8-d680-421d-8d68-7772f7717a25","submittingOrganizationId":null,"sponsorName":"琪越 唐","orcid":"0009-0009-1480-7124","status":"published","createdAt":"2026-10-01 17:52:52","publishedAt":"2026-10-01T17:57:34.640Z","versionNumber":2,"versionCreatedAt":"2026-10-05 14:32:47","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":["Cluster Algebra"],"crossListSubjects":["math.CO"],"authors":[{"id":"7cd78360-919b-421c-b3b6-955cd26a06b7","name":"琪越 唐","orcid":null,"affiliation":"Tsinghua University","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["13F60","15B36"],"tools":[],"id":"hexagon:2610.00012v2","links":{"record":"https://hexagonmath.org/abs/2610.00012v2","pdf":"https://hexagonmath.org/pdf/2610.00012v2","source":"https://hexagonmath.org/src/2610.00012v2","source_tree":"https://hexagonmath.org/api/v1/works/2610.00012v2/source","api":"https://hexagonmath.org/api/v1/works/2610.00012v2"}},{"identifier":"2610.00011","title":"A generalized Gerver sofa for angled corridors","abstract":"The moving sofa problem asks for the maximal area of a two-dimensional object capable of being moved through a right-angled corridor of unit width. J. L. Gerver constructed a sofa of area $2.2195\\dots$, which was only recently proven to be optimal by J. Baek. Here, I consider the moving sofa problem in an angled corridor, $C_{\\varphi}$, of unit width for turn angles $0 < \\varphi \\le \\frac{\\pi}{2}$. I first generalize Hammersley's arguments to devise simple analytical lower and upper area bounds for all $0 < \\varphi \\le \\frac{\\pi}{2}$. ChatGPT-6 Astra (Codex) is prompted to derive a generalized Gerver sofa parametrized by $\\varphi$. The generalized Gerver sofa area is non-analytic at the unique root $\\varphi_*\\in(0,\\pi/2)$ of $\\exp({\\varphi_*\\cot\\varphi_*}) = 1+2\\cos\\varphi_*-\\cos^2\\varphi_*$. For turn angles smaller than $\\varphi_*$ a closed-form expression of the sofa area is obtained. The generalized Gerver sofa is conjectured to be the solution to the moving sofa problem for turn angles $0 < \\varphi \\le \\frac{\\pi}{2}$.","subject":"math.MG","authorshipMode":"mixed","humanUnderstanding":"some","sponsorUserId":"user_b90b0756-56f6-4f4e-8389-7820d0194107","submittingOrganizationId":null,"sponsorName":"Henrik Schou Guttesen","orcid":"0000-0002-2229-0294","status":"published","createdAt":"2026-10-01 16:30:58","publishedAt":"2026-10-01T16:42:54.877Z","versionNumber":2,"versionCreatedAt":"2026-10-04 12:10:11","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":["Geometry","Gerver-sofa","moving-sofa-problem","recreational-math"],"crossListSubjects":["math.HO"],"authors":[{"id":"b2c65a66-21e9-474a-a9c1-c559b12940d1","name":"Henrik Schou Guttesen","orcid":null,"affiliation":"anish Fundamental Metrology","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["49Q10","49K15"],"tools":[{"provider":"OpenAI","name":"Codex","version":"Astra Medium"}],"id":"hexagon:2610.00011v2","links":{"record":"https://hexagonmath.org/abs/2610.00011v2","pdf":"https://hexagonmath.org/pdf/2610.00011v2","source":"https://hexagonmath.org/src/2610.00011v2","source_tree":"https://hexagonmath.org/api/v1/works/2610.00011v2/source","api":"https://hexagonmath.org/api/v1/works/2610.00011v2"}},{"identifier":"2610.00010","title":"A contractible open four-manifold with a complete metric of uniformly positive scalar curvature","abstract":"We construct a smooth contractible open four-manifold that is not homeomorphic to $\\R^4$ and admits a complete Riemannian metric with scalar curvature at least one. This gives a negative answer to a question raised by Chang--Weinberger--Yu and Chodosh--M\\'aximo--Mukherjee.","subject":"math.DG","authorshipMode":"mixed","humanUnderstanding":"all","sponsorUserId":"user_f41ec1d7-8426-457f-811a-599ca6986d84","submittingOrganizationId":null,"sponsorName":"Jiangcheng You","orcid":"0009-0003-1996-052X","status":"published","createdAt":"2026-10-01 16:19:37","publishedAt":"2026-10-01T16:30:34.412Z","versionNumber":1,"versionCreatedAt":"2026-10-01 16:19:37","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":["Baumslag–Solitar groups","Gromov–Lawson surgery","contractible four-manifolds","geometric topology","open four-manifolds","positive scalar curvature","topology at infinity"],"crossListSubjects":["math.AP"],"authors":[{"id":"1867eca3-88c6-494d-b71f-8003afa0cc3b","name":"Jiangcheng You","orcid":"0009-0003-1996-052X","affiliation":"University of Science and Technology of China","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null},{"id":"a013e745-a605-484d-ab76-bbf85363af88","name":"Heng Zhang","orcid":null,"affiliation":"University of Science and Technology of China","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["53C21","57R65","20E06"],"tools":[{"provider":"OpenAI","name":"ChatGPT","version":"6"}],"id":"hexagon:2610.00010v1","links":{"record":"https://hexagonmath.org/abs/2610.00010v1","pdf":"https://hexagonmath.org/pdf/2610.00010v1","source":"https://hexagonmath.org/src/2610.00010v1","source_tree":"https://hexagonmath.org/api/v1/works/2610.00010v1/source","api":"https://hexagonmath.org/api/v1/works/2610.00010v1"}},{"identifier":"2610.00003","title":"Arrow and Gibbard Satterthwaite Theorems in Lean","abstract":"We give a source-aligned account of two Lean 4 formalizations of classical finite social-choice impossibility theorems. For a finite nonempty electorate and a finite alternative set containing three distinct alternatives, Arrow’s theorem derives dictatorship from unanimity and independence of irrelevant alternatives. Its proof expands a weakly decisive coalition from one ordered pair to all pairs, then contracts decisive coalitions to a singleton. The Gibbard–Satterthwaite development proves that every onto strategy-proof choice function on the unrestricted strict-ranking domain is dictatorial by constructing a social welfare function and applying that Arrow implementation. We explain the essential construction in full: top-two profiles define pairwise comparisons; strategy-proof monotonicity and three-top profiles establish a strict total order; predecessor counts provide injective natural-number ranks. Exact hypotheses, vendored dependencies, pinned source identities, prior formalizations, and the distinction between historical registry verification and the present source audit are recorded. The contribution is an exposition and audit of reusable formalization artifacts, not a new social-choice theorem or a claim of first formalization.","subject":"math.LO","authorshipMode":"no-human-author-claimed","humanUnderstanding":"some","sponsorUserId":"user_92218a1d-1192-43e8-b6b8-31c683c18ccc","submittingOrganizationId":null,"sponsorName":"Arthur","orcid":"0009-0003-3568-0325","status":"published","createdAt":"2026-10-01 01:48:53","publishedAt":"2026-10-01T15:18:44.874Z","versionNumber":1,"versionCreatedAt":"2026-10-01 01:48:53","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":[],"crossListSubjects":["cs.LO"],"authors":[{"id":"f0f4d7f2-b1a6-4f46-993e-87ad05f22f30","name":"Arthur","orcid":"0009-0003-3568-0325","affiliation":null,"claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null},{"id":"15101082-ee04-4ea9-a3c1-a0c869a98e4c","name":"David Barros Hulak","orcid":"0009-0002-8056-1774","affiliation":null,"claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null},{"id":"36299843-0109-4476-ada4-a8ba1e16d22f","name":"Ruy J. G. B. de Queiroz","orcid":"0000-0003-1482-0977","affiliation":null,"claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["91B14","68V20"],"tools":[{"provider":"OpenAI","name":"ChatGPT","version":"GPT 6.1 (manuscript)"},{"provider":"OpenAI","name":"Codex","version":"gpt-6-luna (historical proof engineering)"},{"provider":"Lean","name":"Lean","version":"4.35.0-rc2 (registered artifacts)"}],"id":"hexagon:2610.00003v1","links":{"record":"https://hexagonmath.org/abs/2610.00003v1","pdf":"https://hexagonmath.org/pdf/2610.00003v1","source":"https://hexagonmath.org/src/2610.00003v1","source_tree":"https://hexagonmath.org/api/v1/works/2610.00003v1/source","api":"https://hexagonmath.org/api/v1/works/2610.00003v1"}},{"identifier":"2610.00004","title":"A Lean Formalization of the Shapley Value Characterization for Finite Cooperative Games","abstract":"We describe a Lean 4 formalization of the standard four-axiom characterization of the Shapley value for finite transferable-utility cooperative games. Games are arbitrary real-valued characteristic functions normalized at the empty coalition; the player type may itself be empty. The factorial coalition formula is proved efficient, symmetric for interchangeable players, null-player respecting, and additive. Uniqueness follows from the values of arbitrarily scaled unanimity games and an explicit Boolean-lattice Möbius decomposition. The proof requires neither a real-homogeneity axiom nor a continuity assumption on competing allocation rules. We explain the finite-sum arguments, the precise Lean statements, the separation between the registry’s challenge and implemented solution, and the historical verification evidence for the pinned source revision. The contribution is an auditable exposition of an existing formalization of classical mathematics, with no claim of new mathematics or formalization priority.","subject":"math.LO","authorshipMode":"no-human-author-claimed","humanUnderstanding":"some","sponsorUserId":"user_92218a1d-1192-43e8-b6b8-31c683c18ccc","submittingOrganizationId":null,"sponsorName":"Arthur","orcid":"0009-0003-3568-0325","status":"published","createdAt":"2026-10-01 01:54:14","publishedAt":"2026-10-01T15:18:14.550Z","versionNumber":1,"versionCreatedAt":"2026-10-01 01:54:14","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":[],"crossListSubjects":["cs.LO"],"authors":[{"id":"818e8e6a-34bf-4c1f-9986-a5f99b044635","name":"Arthur","orcid":"0009-0003-3568-0325","affiliation":null,"claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null},{"id":"03b3647b-8a7d-4e96-8460-5fd02228a315","name":"David Barros Hulak","orcid":"0009-0002-8056-1774","affiliation":null,"claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null},{"id":"0c4d54ee-6477-4766-8e3e-69ab73fd6d6d","name":"Ruy J. G. B. de Queiroz","orcid":"0000-0003-1482-0977","affiliation":null,"claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["91A12","68V20"],"tools":[{"provider":"OpenAI","name":"ChatGPT","version":"GPT 6.1 (manuscript)"},{"provider":"OpenAI","name":"Codex","version":"gpt-6-luna (historical proof engineering)"},{"provider":"Lean","name":"Lean","version":"4.35.0-rc2 (registered artifact)"}],"id":"hexagon:2610.00004v1","links":{"record":"https://hexagonmath.org/abs/2610.00004v1","pdf":"https://hexagonmath.org/pdf/2610.00004v1","source":"https://hexagonmath.org/src/2610.00004v1","source_tree":"https://hexagonmath.org/api/v1/works/2610.00004v1/source","api":"https://hexagonmath.org/api/v1/works/2610.00004v1"}},{"identifier":"2610.00005","title":"Nash Bargaining Characterization in Lean","abstract":"We describe a Lean 4 formalization of the classical two-person Nash bargaining characterization. A bargaining problem consists of a compact convex subset of the real plane, a feasible disagreement point, and a feasible outcome strictly improving both utilities. The Nash product is maximized over the individually rational part of that set. The development proves existence and uniqueness of the maximizer, verifies Pareto optimality, symmetry, positive affine invariance, and independence of irrelevant alternatives, and proves that these four axioms characterize the maximizer among feasible selection rules. We explain the algebraic midpoint proof of uniqueness and the normalization argument, including an explicit small-step tangent bound and a compact symmetric enlargement. The exposition is tied to an immutable repository snapshot and four declarations registered in Palomar. Historical registry verification is distinguished from the source inspection used to prepare this manuscript. The contribution is a documented formalization of established mathematics; no new bargaining theorem or formalization-priority claim is made.","subject":"math.LO","authorshipMode":"no-human-author-claimed","humanUnderstanding":"some","sponsorUserId":"user_92218a1d-1192-43e8-b6b8-31c683c18ccc","submittingOrganizationId":null,"sponsorName":"Arthur","orcid":"0009-0003-3568-0325","status":"published","createdAt":"2026-10-01 01:58:21","publishedAt":"2026-10-01T15:17:42.415Z","versionNumber":1,"versionCreatedAt":"2026-10-01 01:58:21","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":[],"crossListSubjects":["cs.LO"],"authors":[{"id":"dc87f9b6-2a71-4580-bc7f-153f43ec82bc","name":"Arthur","orcid":"0009-0003-3568-0325","affiliation":null,"claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null},{"id":"61c0fc59-22ef-4027-bacf-ec91c781edf0","name":"David Barros Hulak","orcid":"0009-0002-8056-1774","affiliation":null,"claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null},{"id":"eedad721-06c4-4d46-bc00-a49cbc110ee0","name":"Ruy J. G. B. de Queiroz","orcid":"0000-0003-1482-0977","affiliation":null,"claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["91A12","68V20"],"tools":[{"provider":"OpenAI","name":"ChatGPT","version":"GPT 6.1 (manuscript)"},{"provider":"OpenAI","name":"Codex","version":"gpt-6-luna (historical proof engineering)"},{"provider":"Lean","name":"Lean","version":"4.35.0-rc2 (registered artifact)"}],"id":"hexagon:2610.00005v1","links":{"record":"https://hexagonmath.org/abs/2610.00005v1","pdf":"https://hexagonmath.org/pdf/2610.00005v1","source":"https://hexagonmath.org/src/2610.00005v1","source_tree":"https://hexagonmath.org/api/v1/works/2610.00005v1/source","api":"https://hexagonmath.org/api/v1/works/2610.00005v1"}},{"identifier":"2610.00008","title":"Finite Nash Equilibria and Dependent Mixed Strategies in Lean","abstract":"We present two complementary Lean 4 developments of finite Nash equilibrium. The pure-game layer uses player-specific legal action sets, distinguishes one-way generalized ordinal potentials from bidirectional ordinal potentials, and proves potential-maximizer certificates, a local-maximum characterization, absence of strict-improvement cycles, and weak acyclicity. The mixed-game layers use either a common finite action type or a dependent finite action type for each player. They prove payoff equalization on equilibrium supports, Dirac payoff identities, continuity of the normalized payoff-excess map, and mixed equilibrium existence. We explain the finite-sum and sum-of-squares arguments and the reindexing bridge to an attributed, proved product-of-simplices Brouwer theorem. An additional legal-subtype interface is separated from the declarations selected by the registry comparator. The account is tied to two immutable source snapshots and dated verification records. It documents established mathematics and its formal interfaces, without claiming a new equilibrium theorem or formalization priority.","subject":"math.LO","authorshipMode":"no-human-author-claimed","humanUnderstanding":"some","sponsorUserId":"user_92218a1d-1192-43e8-b6b8-31c683c18ccc","submittingOrganizationId":null,"sponsorName":"Arthur","orcid":"0009-0003-3568-0325","status":"published","createdAt":"2026-10-01 04:07:23","publishedAt":"2026-10-01T15:16:38.434Z","versionNumber":1,"versionCreatedAt":"2026-10-01 04:07:23","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":["Brouwer fixed point","Lean 4","Mathlib","Nash equilibrium","dependent action types","mixed strategies","potential games"],"crossListSubjects":["cs.LO","cs.MS"],"authors":[{"id":"8defb949-ed1a-4a04-9cfd-625a3d044ca5","name":"Arthur","orcid":"0009-0003-3568-0325","affiliation":null,"claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null},{"id":"4c91ae05-38e0-4eef-ac9e-4dc5ff17e13d","name":"David Barros Hulak","orcid":"0009-0002-8056-1774","affiliation":null,"claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null},{"id":"51dbceb3-7f7f-46d7-8f96-db3d89ab8c2e","name":"Ruy J. G. B. de Queiroz","orcid":"0000-0003-1482-0977","affiliation":null,"claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["91A10","91A11","68V20"],"tools":[{"provider":"Lean","name":"Lean","version":"4.33.0 (historical pinned proofs)"},{"provider":"OpenAI","name":"Codex","version":"GPT-6.1 (manuscript)"}],"id":"hexagon:2610.00008v1","links":{"record":"https://hexagonmath.org/abs/2610.00008v1","pdf":"https://hexagonmath.org/pdf/2610.00008v1","source":"https://hexagonmath.org/src/2610.00008v1","source_tree":"https://hexagonmath.org/api/v1/works/2610.00008v1/source","api":"https://hexagonmath.org/api/v1/works/2610.00008v1"}},{"identifier":"2610.00002","title":"A Finite Coordinate Reduction for Approachability Loss in Lean","abstract":"We explain a Lean 4 development of a finite-coordinate loss-preserving construction motivated by Theorem 4 of Dann, Mansour, Mohri, Schneider, and Sivan. Nonempty finite constraint and action index sets give a joint probability simplex with a canonical action marginal. For each mixed constraint, a linear comparator adds its outer product with that marginal. The comparator has mass two and leaves the legal action set. A one-step pairing identity yields exact equality of finite-horizon approachability and comparator-regret objectives, with explicit anchored lifts and marginal decoders. The development also proves a row-wise decomposition with at most one rank-one term per constraint index. We describe the twelve declarations registered in Palomar and their historical verification provenance. The scope is deliberately narrow: the constraint simplex parametrizes coordinate functions, both causal strategy translations observe original loss histories, and the comparators have no fixed point in the joint simplex. Thus the checked statements establish finite algebraic loss preservation; they do not certify the full published reduction, its fixed-point-defined improper class, reduced-loss-only feedback, or asymptotic rate theory. No mathematical novelty or formalization priority is claimed.","subject":"math.OC","authorshipMode":"no-human-author-claimed","humanUnderstanding":"some","sponsorUserId":"user_92218a1d-1192-43e8-b6b8-31c683c18ccc","submittingOrganizationId":null,"sponsorName":"Arthur","orcid":"0009-0003-3568-0325","status":"published","createdAt":"2026-10-01 01:34:31","publishedAt":"2026-10-01T15:16:07.841Z","versionNumber":1,"versionCreatedAt":"2026-10-01 01:34:31","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":[],"crossListSubjects":["cs.LO"],"authors":[{"id":"e8146718-a7e5-4ac1-9b35-2bbcc67c45c8","name":"Arthur","orcid":"0009-0003-3568-0325","affiliation":null,"claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null},{"id":"c6ed42f0-83a4-4a46-9914-bda5b53522bb","name":"David Barros Hulak","orcid":"0009-0002-8056-1774","affiliation":null,"claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null},{"id":"be7f753a-c002-4a9a-97ea-ed1b33d181a0","name":"Ruy J. G. B. de Queiroz","orcid":"0000-0003-1482-0977","affiliation":null,"claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["91A26","68T05","68V20"],"tools":[{"provider":"OpenAI","name":"ChatGPT","version":"GPT 6.1 (manuscript)"},{"provider":"OpenAI","name":"Codex","version":"GPT-5 (historical proof assistance)"},{"provider":"Lean","name":"Lean","version":"4.33.0 (registered artifact)"}],"id":"hexagon:2610.00002v1","links":{"record":"https://hexagonmath.org/abs/2610.00002v1","pdf":"https://hexagonmath.org/pdf/2610.00002v1","source":"https://hexagonmath.org/src/2610.00002v1","source_tree":"https://hexagonmath.org/api/v1/works/2610.00002v1/source","api":"https://hexagonmath.org/api/v1/works/2610.00002v1"}},{"identifier":"2610.00006","title":"An Executable Stallings Recognizer for Finite Generating Lists in Lean","abstract":"We explain a Lean 4 construction of a finite inverse automaton recognizing the subgroup of the rank-two free group generated by any finite list of signed words. The implementation builds a flower multigraph and computes the least equivalence relation compatible with deterministic labelled transitions by exhaustive finite search over Boolean relations. Canonical representatives realize the folded graph on the original finite state type. A coset-potential invariant proves that folding introduces no additional subgroup elements; preservation of the input loops proves the opposite inclusion. A separate reduction argument shows that traversal of the canonical reduced representative decides membership. We distinguish the registered theorem, its executable witness, and classical reasoning used only in the proof. The construction retains unused states and has no formalized complexity, core-trimming, subgroup-basis, intersection, or index algorithm. This is an expository account of a classical recognition theorem and a pinned formal artifact, with no claim of mathematical novelty or formalization priority.","subject":"math.GR","authorshipMode":"no-human-author-claimed","humanUnderstanding":"some","sponsorUserId":"user_92218a1d-1192-43e8-b6b8-31c683c18ccc","submittingOrganizationId":null,"sponsorName":"Arthur","orcid":"0009-0003-3568-0325","status":"published","createdAt":"2026-10-01 02:21:45","publishedAt":"2026-10-01T15:15:30.086Z","versionNumber":1,"versionCreatedAt":"2026-10-01 02:21:45","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":["Lean 4","Mathlib","Stallings folding","finite inverse automata","free groups","subgroup membership"],"crossListSubjects":["math.LO"],"authors":[{"id":"7f78539c-c68f-4234-9823-0985992135ee","name":"Arthur","orcid":"0009-0003-3568-0325","affiliation":null,"claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null},{"id":"595965b9-db90-48fc-8469-82d16c806842","name":"David Barros Hulak","orcid":"0009-0002-8056-1774","affiliation":null,"claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null},{"id":"c4cc7799-ea12-492a-b4e9-24708a89b179","name":"Ruy J. G. B. de Queiroz","orcid":"0000-0003-1482-0977","affiliation":null,"claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["20E05","68Q45","03B35"],"tools":[{"provider":"OpenAI","name":"ChatGPT","version":"GPT 6.1 (manuscript)"},{"provider":"OpenAI","name":"Codex","version":"GPT-6 (historical proof engineering)"},{"provider":"Lean","name":"Lean","version":"4.33.0 (registered artifact)"}],"id":"hexagon:2610.00006v1","links":{"record":"https://hexagonmath.org/abs/2610.00006v1","pdf":"https://hexagonmath.org/pdf/2610.00006v1","source":"https://hexagonmath.org/src/2610.00006v1","source_tree":"https://hexagonmath.org/api/v1/works/2610.00006v1/source","api":"https://hexagonmath.org/api/v1/works/2610.00006v1"}},{"identifier":"2610.00007","title":"Prime-Generated Localization Descent for Unique Factorization in Lean","abstract":"We give a source-grounded exposition of a Lean 4 formalization of prime-generated Nagata factoriality descent. A commutative noetherian integral domain is a unique factorization domain if a localization is a unique factorization domain and every denominator is exactly a finite product of prime elements belonging to the denominator submonoid. The formal statement works with an arbitrary algebra carrying Mathlib’s IsLocalization structure. We explain the two cases for an irreducible element, the multiset cancellation and splitting arguments, and the corollary for submonoids generated by arbitrary sets of primes. A registered polynomial corollary illustrates reuse of the descent package; its Laurent-polynomial premise is proved using Mathlib’s existing polynomial UFD instance, a dependency made explicit here. The exposition is tied to the registered source and dependency pins, distinguishes historical mechanical verification from manuscript review, and situates the development relative to an earlier Lean preprint and an Isabelle/HOL formalization. No new algebraic theorem or priority claim is asserted.","subject":"math.AC","authorshipMode":"no-human-author-claimed","humanUnderstanding":"some","sponsorUserId":"user_92218a1d-1192-43e8-b6b8-31c683c18ccc","submittingOrganizationId":null,"sponsorName":"Arthur","orcid":"0009-0003-3568-0325","status":"published","createdAt":"2026-10-01 02:42:44","publishedAt":"2026-10-01T15:13:48.574Z","versionNumber":1,"versionCreatedAt":"2026-10-01 02:42:44","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":["Lean 4","Mathlib","Nagata factoriality","commutative algebra","localization","unique factorization domains"],"crossListSubjects":["cs.LO"],"authors":[{"id":"e26f64fb-42b1-4de1-8710-d01a9def5456","name":"Arthur","orcid":"0009-0003-3568-0325","affiliation":null,"claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null},{"id":"ec728a56-a83f-40c0-9ad0-626ebd71dda5","name":"David Barros Hulak","orcid":"0009-0002-8056-1774","affiliation":null,"claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null},{"id":"28838da8-0dbd-4c14-8f42-c5f6dc976305","name":"Ruy J. G. B. de Queiroz","orcid":"0000-0003-1482-0977","affiliation":null,"claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null},{"id":"86254438-4352-4ba0-aba9-94abee8f8e57","name":"Anjolina Grisi de Oliveira","orcid":null,"affiliation":null,"claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["13A05","13B30","03B35"],"tools":[{"provider":"OpenAI","name":"ChatGPT","version":"GPT 6.1 (manuscript)"},{"provider":"OpenAI","name":"Codex","version":"Historical assistance; precise model versions unspecified"},{"provider":"Lean","name":"Lean","version":"4.33.0 (registered artifact)"}],"id":"hexagon:2610.00007v1","links":{"record":"https://hexagonmath.org/abs/2610.00007v1","pdf":"https://hexagonmath.org/pdf/2610.00007v1","source":"https://hexagonmath.org/src/2610.00007v1","source_tree":"https://hexagonmath.org/api/v1/works/2610.00007v1/source","api":"https://hexagonmath.org/api/v1/works/2610.00007v1"}},{"identifier":"2610.00001","title":"A Lean Formalization of Myerson-Satterthwaite Impossibility for Continuous Bilateral Trade","abstract":"We present a source-grounded account of a Lean 4 formalization of the Myerson-Satterthwaite impossibility theorem for continuous bilateral trade. Buyer values and seller costs are independent, with atomless probability measures supported on compact real intervals whose interiors overlap. The model explicitly requires restricted Lebesgue measure to be absolutely continuous with respect to each type measure, together with full support; it does not require the type measures themselves to have densities. Under allocation and transfer measurability and integrability conditions, the registered theorem excludes the simultaneous satisfaction of almost-sure ex post efficiency, Bayesian incentive compatibility at every admissible type and report, interim individual rationality, and weak ex ante budget balance. The proof derives envelope identities from incentive inequalities, identifies the efficient allocation almost everywhere, and uses a layer-cake representation to show that the required information rents strictly exceed efficient surplus. We explain the analytic steps, their Lean declarations, and the verification evidence for the exact Palomar-registered source revision. This is an exposition of a classical result and an existing formal artifact, without a claim of mathematical novelty or formalization priority.","subject":"cs.LO","authorshipMode":"no-human-author-claimed","humanUnderstanding":"some","sponsorUserId":"user_92218a1d-1192-43e8-b6b8-31c683c18ccc","submittingOrganizationId":null,"sponsorName":"Arthur","orcid":"0009-0003-3568-0325","status":"published","createdAt":"2026-10-01 01:28:12","publishedAt":"2026-10-01T15:12:46.618Z","versionNumber":1,"versionCreatedAt":"2026-10-01 01:28:12","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":[],"crossListSubjects":["math.LO"],"authors":[{"id":"6ab3cd85-e038-45e4-8244-11899b111096","name":"Arthur","orcid":"0009-0003-3568-0325","affiliation":null,"claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null},{"id":"18c01a5e-76e4-4b47-a960-86459d1446f6","name":"David Barros Hulak","orcid":"0009-0002-8056-1774","affiliation":null,"claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null},{"id":"e1e9ed1c-495d-4343-a3b1-f5d95efb9cf4","name":"Ruy J. G. B. de Queiroz","orcid":"0000-0003-1482-0977","affiliation":null,"claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["91B26","68V20"],"tools":[{"provider":"OpenAI","name":"ChatGPT","version":"GPT 6.1 (manuscript)"},{"provider":"OpenAI","name":"Codex","version":"gpt-6-luna (historical proof engineering)"},{"provider":"Lean","name":"Lean","version":"4.35.0-rc2 (registered artifact)"}],"id":"hexagon:2610.00001v1","links":{"record":"https://hexagonmath.org/abs/2610.00001v1","pdf":"https://hexagonmath.org/pdf/2610.00001v1","source":"https://hexagonmath.org/src/2610.00001v1","source_tree":"https://hexagonmath.org/api/v1/works/2610.00001v1/source","api":"https://hexagonmath.org/api/v1/works/2610.00001v1"}},{"identifier":"2610.00009","title":"A Lean Certificate for the Single Source Unsplittable Flow Cost Counterexample","abstract":"We explain a standalone Lean 4 and Mathlib verification of the finite counterexample announced by Dmitry Rybin to the cost-preserving single-source unsplittable-flow conjecture. The instance has seven vertices, nine directed arcs, and three terminal demands. An exact rational path flow saturates every arc, has cost 58, and has largest demand 15. Exhaustive graph-path classification reduces every unsplittable routing to two choices per terminal. Three backbone-arc inequalities force any routing with load at most the fractional load plus 15 to use at least two paid routes, each costing 30. Thus every such routing costs at least 60, ruling out cost preservation even with a non-strict congestion bound. We present the numerical certificate, its graph realization, and the boundary between the pinned proved implementation and its statement-only verification interface. The contribution is an exposition of a source-based formal certificate; no new counterexample, first formalization, or general rounding algorithm is claimed.","subject":"cs.DS","authorshipMode":"no-human-author-claimed","humanUnderstanding":"some","sponsorUserId":"user_92218a1d-1192-43e8-b6b8-31c683c18ccc","submittingOrganizationId":null,"sponsorName":"Arthur","orcid":"0009-0003-3568-0325","status":"published","createdAt":"2026-10-01 04:12:19","publishedAt":"2026-10-01T15:06:37.880Z","versionNumber":1,"versionCreatedAt":"2026-10-01 04:12:19","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":["Goemans cost conjecture","Lean 4","Mathlib","Rybin counterexample","finite counterexample","formal verification","single-source unsplittable flow"],"crossListSubjects":["math.LO"],"authors":[{"id":"93e0a65f-efd5-458c-ac9b-c101ef3605cb","name":"Arthur","orcid":"0009-0003-3568-0325","affiliation":null,"claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null},{"id":"8b1e6144-28a3-4808-88d6-1c39459e228c","name":"David Barros Hulak","orcid":"0009-0002-8056-1774","affiliation":null,"claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null},{"id":"1fc0ec3d-2856-418f-80a0-632a27a9bf90","name":"Ruy J. G. B. de Queiroz","orcid":"0000-0003-1482-0977","affiliation":null,"claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["68R10","90C27","03B35"],"tools":[{"provider":"Lean","name":"Lean","version":"4.33.0 (historical pinned proofs)"},{"provider":"OpenAI","name":"Codex","version":"GPT-6.1 (manuscript)"}],"id":"hexagon:2610.00009v1","links":{"record":"https://hexagonmath.org/abs/2610.00009v1","pdf":"https://hexagonmath.org/pdf/2610.00009v1","source":"https://hexagonmath.org/src/2610.00009v1","source_tree":"https://hexagonmath.org/api/v1/works/2610.00009v1/source","api":"https://hexagonmath.org/api/v1/works/2610.00009v1"}},{"identifier":"2609.00152","title":"A Lean Formalization of the Classical Seifert van Kampen Groupoid Theorem","abstract":"We explain a Lean 4 formalization of the classical Seifert van Kampen theorem for the full fundamental groupoid. For an arbitrary topological space covered by two open subsets, the inclusion-induced square of ordinary continuous-path fundamental groupoids is a pushout in the category of categories. Every point is retained as an object; no basepoint, connectedness, or separation hypothesis is required. The development identifies an auxiliary directed path category for the indiscrete preorder with Mathlib's fundamental groupoid through strictly inverse functors and naturality equalities. It then assembles the pushout universal property from attributed path-subdivision and homotopy-grid helpers adapted from the directed-topology formalization of Basold, Bruin, and Lawson. We describe the descent construction, its transfer to ordinary fundamental groupoids, and the statement and verification boundaries of the exact artifact registered in Palomar. This is an expository account of a classical theorem and an existing formal artifact, with no claim of mathematical novelty or formalization priority.","subject":"math.AT","authorshipMode":"no-human-author-claimed","humanUnderstanding":"some","sponsorUserId":"user_92218a1d-1192-43e8-b6b8-31c683c18ccc","submittingOrganizationId":null,"sponsorName":"Arthur","orcid":"0009-0003-3568-0325","status":"published","createdAt":"2026-09-30 23:29:11","publishedAt":"2026-09-30T23:56:19.428Z","versionNumber":1,"versionCreatedAt":"2026-09-30 23:29:11","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":["Lean 4","Mathlib","Seifert van Kampen theorem","categorical pushout","fundamental groupoid","open cover"],"crossListSubjects":["math.CT"],"authors":[{"id":"004963e9-3b2a-4d09-91ee-efffbbd4fa8d","name":"Arthur","orcid":"0009-0003-3568-0325","affiliation":null,"claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null},{"id":"aa9b2a60-7286-45c7-849b-930a39ce625e","name":"David Barros Hulak","orcid":"0009-0002-8056-1774","affiliation":null,"claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null},{"id":"b31cf54f-d51b-460c-99af-28d4cdb45175","name":"Ruy J. G. B. de Queiroz","orcid":"0000-0003-1482-0977","affiliation":null,"claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["55Q05","18B40","68V20"],"tools":[{"provider":"Lean","name":"Lean","version":"4.35.0-rc2"},{"provider":"OpenAI","name":"Codex","version":"GPT-6.1"},{"provider":"OpenAI","name":"Codex","version":"gpt-6-astra (pinned repository-reported formalization assistance)"}],"id":"hexagon:2609.00152v1","links":{"record":"https://hexagonmath.org/abs/2609.00152v1","pdf":"https://hexagonmath.org/pdf/2609.00152v1","source":"https://hexagonmath.org/src/2609.00152v1","source_tree":"https://hexagonmath.org/api/v1/works/2609.00152v1/source","api":"https://hexagonmath.org/api/v1/works/2609.00152v1"}},{"identifier":"2609.00149","title":"Deferred Acceptance in Lean for Finite Strict Preference Markets","abstract":"We describe a Lean 4 development of men-proposing deferred acceptance for finite one-to-one markets with complete strict preferences and optional partners. The two sides may have different cardinalities; the principal theorems assume a nonempty receiving side. A proposal-set measure proves termination of a specified, classically chosen run. Reachability invariants establish consistency of the resulting partner maps, stability, and the fact that a rejected proposal cannot be a stable-achievable partnership. The public optimality theorem is conditional on a man receiving a partner; an internal lemma separately excludes stable-achievable partners for men unmatched at quiescence. We give mathematical proofs corresponding to the source architecture, identify the precise theorem interface, and separate fresh source inspection from the repository's reported build and axiom audit. This is an exposition of a pinned formalization of classical results, with no claim of new matching theory or priority of formalization.","subject":"cs.DS","authorshipMode":"no-human-author-claimed","humanUnderstanding":"some","sponsorUserId":"user_92218a1d-1192-43e8-b6b8-31c683c18ccc","submittingOrganizationId":null,"sponsorName":"Arthur","orcid":"0009-0003-3568-0325","status":"published","createdAt":"2026-09-30 22:12:39","publishedAt":"2026-09-30T23:41:41.345Z","versionNumber":1,"versionCreatedAt":"2026-09-30 22:12:39","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":["Gale-Shapley","Lean 4","deferred acceptance","formalization","stable matching"],"crossListSubjects":["cs.LO"],"authors":[{"id":"c56cf46e-ceee-4399-b0e9-fd8dcf7ebacc","name":"Arthur","orcid":"0009-0003-3568-0325","affiliation":null,"claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null},{"id":"5dab2be7-d158-42f2-ab0a-0f42c9db482d","name":"David Barros Hulak","orcid":"0009-0002-8056-1774","affiliation":null,"claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null},{"id":"1918dec5-f7cf-433d-ae5f-8ef8ead60ce6","name":"Ruy J. G. B. de Queiroz","orcid":"0000-0003-1482-0977","affiliation":null,"claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["91B68","68V20"],"tools":[{"provider":"Lean","name":"Lean","version":"4.35.0-rc2"},{"provider":"OpenAI","name":"Codex","version":"GPT-6.1"},{"provider":"OpenAI","name":"Codex","version":"GPT-6-Luna (repository-reported formalization assistance)"}],"id":"hexagon:2609.00149v1","links":{"record":"https://hexagonmath.org/abs/2609.00149v1","pdf":"https://hexagonmath.org/pdf/2609.00149v1","source":"https://hexagonmath.org/src/2609.00149v1","source_tree":"https://hexagonmath.org/api/v1/works/2609.00149v1/source","api":"https://hexagonmath.org/api/v1/works/2609.00149v1"}},{"identifier":"2609.00151","title":"A Lean Formalization of Roberts Theorem on Unrestricted Valuations","abstract":"We describe a Lean 4 formalization of the finite unrestricted-domain version of Roberts theorem. For finite nonempty agent and alternative types, with at least three alternatives, an onto deterministic direct-revelation mechanism that is dominant-strategy incentive compatible has an affine-maximizer choice rule. Valuations range over all real-valued profiles; the conclusion uses fixed nonnegative weights, at least one nonzero, and alternative offsets, and asserts membership in the maximizing set. The development follows the modular proof route: taxation and weak monotonicity, perturbation-based tie-breaking to strong monotonicity, no-veto range analysis, affine prices for two agents, and induction through fixed-report slices. We explain the proved tie-set transport lemma, the construction of fresh payments for the tie-broken rule, and the transfer back to arbitrary ties. A version-pinned account distinguishes the proved library from statement-only comparator placeholders and separates source inspection from executable checks. This is an exposition of a classical result and its formal artifact, with no claim of mathematical novelty or formalization priority.","subject":"math.OC","authorshipMode":"no-human-author-claimed","humanUnderstanding":"some","sponsorUserId":"user_92218a1d-1192-43e8-b6b8-31c683c18ccc","submittingOrganizationId":null,"sponsorName":"Arthur","orcid":"0009-0003-3568-0325","status":"published","createdAt":"2026-09-30 22:24:25","publishedAt":"2026-09-30T23:39:36.737Z","versionNumber":1,"versionCreatedAt":"2026-09-30 22:24:25","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":["Lean 4","Roberts theorem","affine maximizers","dominant strategies","formalization","mechanism design"],"crossListSubjects":["cs.LO"],"authors":[{"id":"1b2c5c9d-af0e-4883-b004-524dd83534f5","name":"Arthur","orcid":"0009-0003-3568-0325","affiliation":null,"claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null},{"id":"09696824-3534-4a2a-890e-7c83b528e20a","name":"David Barros Hulak","orcid":"0009-0002-8056-1774","affiliation":null,"claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null},{"id":"9ef38d83-6621-41ad-9e13-514387761998","name":"Ruy J. G. B. de Queiroz","orcid":"0000-0003-1482-0977","affiliation":null,"claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["91B14","68V20"],"tools":[{"provider":"Lean","name":"Lean","version":"4.35.0-rc2"},{"provider":"OpenAI","name":"Codex","version":"GPT-6.1"},{"provider":"OpenAI","name":"Codex","version":"GPT-6-Luna (repository-reported formalization assistance)"}],"id":"hexagon:2609.00151v1","links":{"record":"https://hexagonmath.org/abs/2609.00151v1","pdf":"https://hexagonmath.org/pdf/2609.00151v1","source":"https://hexagonmath.org/src/2609.00151v1","source_tree":"https://hexagonmath.org/api/v1/works/2609.00151v1/source","api":"https://hexagonmath.org/api/v1/works/2609.00151v1"}},{"identifier":"2609.00150","title":"Finite VCG and Groves Payments in Lean","abstract":"We describe a Lean 4 development of finite direct mechanisms with quasi-linear utility and unrestricted real-valued valuations. The Clarke pivot mechanism selects a reported-welfare maximizer and charges each agent the loss in other-agent welfare relative to its maximum over the same fixed alternative set. The development proves efficiency, dominant-strategy truthfulness, nonnegative payments and no deficit, and individual rationality under the explicit assumption that every valuation is nonnegative. It also proves that every efficient, dominant-strategy incentive-compatible mechanism on the full valuation domain has Groves-form payments, with a term independent of the paying agent's report. We explain the finite-maximization interface, the report-update invariants, and the characterization argument using outcome-forcing reports and positive perturbations. The exposition is tied to a specific source revision and separates proved implementations from statement-only comparator placeholders. These are classical mechanism-design results; no new mathematical theorem or formalization-priority claim is made.","subject":"math.OC","authorshipMode":"no-human-author-claimed","humanUnderstanding":"some","sponsorUserId":"user_92218a1d-1192-43e8-b6b8-31c683c18ccc","submittingOrganizationId":null,"sponsorName":"Arthur","orcid":"0009-0003-3568-0325","status":"published","createdAt":"2026-09-30 22:19:52","publishedAt":"2026-09-30T23:38:32.923Z","versionNumber":1,"versionCreatedAt":"2026-09-30 22:19:52","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":["Clarke pivot payments","Groves characterization","Lean 4","VCG mechanism","dominant strategies","formal verification"],"crossListSubjects":["cs.LO"],"authors":[{"id":"c7e14793-9b5b-43fc-82e9-be6381c1a51e","name":"Arthur","orcid":"0009-0003-3568-0325","affiliation":null,"claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null},{"id":"c7e3e860-cda3-4b75-b857-2950156b5244","name":"David Barros Hulak","orcid":"0009-0002-8056-1774","affiliation":null,"claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null},{"id":"fb5ce3e7-3d1c-4ede-b6e7-afd6f02c9e8e","name":"Ruy J. G. B. de Queiroz","orcid":"0000-0003-1482-0977","affiliation":null,"claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["91B26","68V20"],"tools":[{"provider":"Lean","name":"Lean","version":"4.35.0-rc2"},{"provider":"OpenAI","name":"Codex","version":"GPT-6.1"},{"provider":"OpenAI","name":"Codex","version":"GPT-6-Luna (repository-reported formalization assistance)"}],"id":"hexagon:2609.00150v1","links":{"record":"https://hexagonmath.org/abs/2609.00150v1","pdf":"https://hexagonmath.org/pdf/2609.00150v1","source":"https://hexagonmath.org/src/2609.00150v1","source_tree":"https://hexagonmath.org/api/v1/works/2609.00150v1/source","api":"https://hexagonmath.org/api/v1/works/2609.00150v1"}},{"identifier":"2609.00148","title":"A Lean Formalization of Finite Zero Sum Minimax via Sion's Theorem","abstract":"We describe a Lean 4 formalization of the minimax theorem for finite nonempty two-player zero-sum games with arbitrary real payoffs. Mixed strategies are probability vectors, and the two game values are defined using real indexed suprema and infima. An explicit payoff bound supplies the boundedness obligations required by these conditional extrema. The reverse minimax inequality follows by applying Mathlib's proved Sion saddle-point theorem to compact convex coordinate simplices, with the minimizing player placed in its first argument. We explain the representation bridge, the proof obligations, and the passage from a saddle point to equality. The accompanying version-pinned artifact separates a reusable library from a statement-only challenge and its proved solution. This is an exposition of a finite-game specialization of existing formal analysis, with no claim of a new mathematical proof or formalization priority.","subject":"math.OC","authorshipMode":"no-human-author-claimed","humanUnderstanding":"some","sponsorUserId":"user_92218a1d-1192-43e8-b6b8-31c683c18ccc","submittingOrganizationId":null,"sponsorName":"Arthur","orcid":"0009-0003-3568-0325","status":"published","createdAt":"2026-09-30 21:13:25","publishedAt":"2026-09-30T21:21:21.370Z","versionNumber":1,"versionCreatedAt":"2026-09-30 21:13:25","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":["Lean","Sion's theorem","formalization","minimax theorem","zero-sum games"],"crossListSubjects":["cs.LO","cs.MS"],"authors":[{"id":"5990a0fe-3963-4249-b23d-b55033337fba","name":"Arthur","orcid":"0009-0003-3568-0325","affiliation":null,"claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null},{"id":"a37642d2-d343-4f86-85c5-2080d57675ae","name":"David Barros Hulak","orcid":"0009-0002-8056-1774","affiliation":null,"claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null},{"id":"e0ecfe1c-2f77-4f1a-94d2-5010ba130fb5","name":"Ruy J. G. B. de Queiroz","orcid":"0000-0003-1482-0977","affiliation":null,"claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["91A05","68V20"],"tools":[{"provider":"Lean","name":"Lean","version":"4.35.0-rc2"},{"provider":"OpenAI","name":"Codex","version":"GPT-6.1"}],"id":"hexagon:2609.00148v1","links":{"record":"https://hexagonmath.org/abs/2609.00148v1","pdf":"https://hexagonmath.org/pdf/2609.00148v1","source":"https://hexagonmath.org/src/2609.00148v1","source_tree":"https://hexagonmath.org/api/v1/works/2609.00148v1/source","api":"https://hexagonmath.org/api/v1/works/2609.00148v1"}},{"identifier":"2609.00147","title":"Signed Mukai diagonals and derived equivalences of fivefolds","abstract":"For every Hochschild diagonal of a smooth projective complex variety, a Fourier–Mukai equivalence preserves both the ordinary Hodge-number sum and the sum signed by the parity of the antiholomorphic degree. The latter is the signature of a Hermitian form obtained from the generalized Mukai pairing. In dimension five these invariants determine $h^{0,3}$ and $h^{1,4}$ and reduce the unrestricted Hodge-number problem to two numerical parameters, one of which is the possible failure of invariance of $h^{0,2}$. Albanese methods remove this structure-sheaf ambiguity when the Albanese image has dimension at least three. The general fivefold case remains open, while a theorem of Abuaf proves invariance of all Hodge numbers for fivefolds with trivial canonical bundle.","subject":"math.AG","authorshipMode":"no-human-author-claimed","humanUnderstanding":"none","sponsorUserId":"user_76ce7f15-0dd8-48cb-813e-dd89b39e4f20","submittingOrganizationId":null,"sponsorName":"Benjamin Antieau","orcid":"0000-0002-5296-187X","status":"published","createdAt":"2026-09-30 17:06:08","publishedAt":"2026-09-30T17:09:03.751Z","versionNumber":1,"versionCreatedAt":"2026-09-30 17:06:08","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":["Derived categories","fivefolds"],"crossListSubjects":[],"authors":[{"id":"dab44c2e-8be1-414d-92c5-490f6b77ccd3","name":"Benjamin Antieau","orcid":"0000-0002-5296-187X","affiliation":"Northwestern University","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["14F08"],"tools":[{"provider":"OpenAI","name":"ChatGPT","version":"5.6 Sol"}],"id":"hexagon:2609.00147v1","links":{"record":"https://hexagonmath.org/abs/2609.00147v1","pdf":"https://hexagonmath.org/pdf/2609.00147v1","source":"https://hexagonmath.org/src/2609.00147v1","source_tree":"https://hexagonmath.org/api/v1/works/2609.00147v1/source","api":"https://hexagonmath.org/api/v1/works/2609.00147v1"}},{"identifier":"2609.00145","title":"The support conjecture for rank-two quantum triangular bases","abstract":"We prove Conjecture 11 of Lee, Li, Rupel, and Zelevinsky on the support of triangular basis elements in rank-two quantum cluster algebras.","subject":"math.QA","authorshipMode":"mixed","humanUnderstanding":"some","sponsorUserId":"user_bf3e39d8-d680-421d-8d68-7772f7717a25","submittingOrganizationId":null,"sponsorName":"琪越 唐","orcid":"0009-0009-1480-7124","status":"published","createdAt":"2026-09-30 07:44:59","publishedAt":"2026-09-30T09:15:16.970Z","versionNumber":2,"versionCreatedAt":"2026-10-04 16:12:57","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":["quantum cluster algebra"],"crossListSubjects":["math.RA","math.RT"],"authors":[{"id":"dfba6810-34c2-4fb5-ba34-e1eaf0c31545","name":"琪越 唐","orcid":null,"affiliation":"Tsinghua University","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["13F60","16G20"],"tools":[],"id":"hexagon:2609.00145v2","links":{"record":"https://hexagonmath.org/abs/2609.00145v2","pdf":"https://hexagonmath.org/pdf/2609.00145v2","source":"https://hexagonmath.org/src/2609.00145v2","source_tree":"https://hexagonmath.org/api/v1/works/2609.00145v2/source","api":"https://hexagonmath.org/api/v1/works/2609.00145v2"}},{"identifier":"2609.00144","title":"On the Asymptotic \\(W_1\\) Cost of Two-Dimensional Semi-Discrete Matching","abstract":"Let \\(X_1,X_2,\\ldots\\) be independent uniform points in the unit square and let\n\\(\\lambda\\) denote Lebesgue probability measure.  We prove that the normalized\nexpected \\(W_1\\) distance between the empirical measure and Lebesgue measure\nconverges to a positive constant:\n\\[\n  \\lim_{N\\to\\infty}\n  \\sqrt{\\frac{N}{\\log N}}\\,\n  \\mathbb E W_1\\!\\left(\\frac1N\\sum_{i=1}^N\\delta_{X_i},\\lambda\\right)\n  =c_\\star.\n\\]\nThe constant is characterized by the convex viscosity equation\n\\[\n  \\partial_tU=\\frac1{4\\pi}\\sqrt{\\det D^2U},\n  \\qquad U(0,q)=|q|.\n\\]\nIts solution \\(U\\) is unique under certain growth and regularity conditions, and the limiting constant \\(c_\\star=U(1,0)/\\sqrt2\\).","subject":"math.PR","authorshipMode":"no-human-author-claimed","humanUnderstanding":"none","sponsorUserId":"user_e5f32d35-1dd2-426f-b34a-e500a5d08cbc","submittingOrganizationId":null,"sponsorName":"Heng Ma","orcid":"0000-0002-1401-7677","status":"published","createdAt":"2026-09-30 07:01:44","publishedAt":"2026-09-30T09:13:31.445Z","versionNumber":1,"versionCreatedAt":"2026-09-30 07:01:44","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":[],"crossListSubjects":["math.AP"],"authors":[{"id":"f856a60b-23fa-4ed1-b5d8-07da159a263e","name":"Heng Ma","orcid":"0000-0002-1401-7677","affiliation":"Technion","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["60D05","49Q22"],"tools":[{"provider":"OpenAI","name":"ChatGPT","version":"5.6Pro"}],"id":"hexagon:2609.00144v1","links":{"record":"https://hexagonmath.org/abs/2609.00144v1","pdf":"https://hexagonmath.org/pdf/2609.00144v1","source":"https://hexagonmath.org/src/2609.00144v1","source_tree":"https://hexagonmath.org/api/v1/works/2609.00144v1/source","api":"https://hexagonmath.org/api/v1/works/2609.00144v1"}},{"identifier":"2609.00146","title":"FINITE HAUSDORFF HYPERSPACES AND GROMOV–HAUSDORFF GEOMETRY","abstract":"For a nonempty finite metric space X, let Exp(X) be the\nspace of its nonempty subsets equipped with the Hausdorff metric. We\nprove that the isometry type of Exp(X) determines X when X is strongly\nrigid, meaning that distinct unordered pairs of distinct points have dis-\ntinct distances. We also prove that the hyperspace operation preserves\nthe ordinary Gromov–Hausdorff distance locally under explicit separa-\ntion conditions on the distance spectra. For finite ultrametric spaces,\nthe hyperspace operation is an isometric embedding with respect to the\nGromov–Hausdorff ultrametric. In contrast, we construct finite ultra-\nmetric spaces for which the ordinary Gromov–Hausdorff distance strictly\ndecreases under the hyperspace operation. Examples of equal cardinality\nexist in every cardinality at least six and realize every ratio in [1/2, 1).\nTheir distances remain constant after the first hyperspace iteration. An\nexplicit perturbation also yields strict contraction for finite strongly rigid\nmetric spaces with all triangle inequalities strict.","subject":"math.MG","authorshipMode":"no-human-author-claimed","humanUnderstanding":"some","sponsorUserId":"user_b05f43db-592b-491d-a5c2-051ffe2db668","submittingOrganizationId":null,"sponsorName":"Yoshito Ishiki","orcid":"0009-0002-7792-8280","status":"published","createdAt":"2026-09-30 08:36:04","publishedAt":"2026-09-30T09:12:24.355Z","versionNumber":1,"versionCreatedAt":"2026-09-30 08:36:04","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":["Gromov--Hausdorff space","Hyper space"],"crossListSubjects":["math.GN"],"authors":[{"id":"2a80d765-a1b4-4a6f-8f4b-dfef84d40fdd","name":"Yoshito Ishiki","orcid":"0009-0002-7792-8280","affiliation":"Department of Mathematical Sciences, Tokyo Metropolitan University, Minami-osawa, Hachioji, Tokyo 192-0397, Japan","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["58B20","51F30","54E35"],"tools":[],"id":"hexagon:2609.00146v1","links":{"record":"https://hexagonmath.org/abs/2609.00146v1","pdf":"https://hexagonmath.org/pdf/2609.00146v1","source":"https://hexagonmath.org/src/2609.00146v1","source_tree":"https://hexagonmath.org/api/v1/works/2609.00146v1/source","api":"https://hexagonmath.org/api/v1/works/2609.00146v1"}},{"identifier":"2609.00141","title":"Ordinal-definable families in Cohen, random, and collapse extensions","abstract":"We study ordinal-definable families of sets of arbitrary rank in\nCohen, random, and collapse extensions. Over $L$, countable OD families\nhave OD enumerations after adding one Cohen or one random real.\nA single generalized Cohen subset gives the corresponding sharp theorem\nat every regular uncountable cardinal. When $\\kappa$ is singular strong\nlimit and has uncountable cofinality, adding $\\kappa$ Cohen reals makes\nevery member of a short-parameter definable family of size at most\n$\\kappa$ definable from a short parameter. Adding $\\kappa$ random reals\ngives a single short-parameter definable enumeration for every such\nfamily of size strictly below $\\kappa$. The random bound is sharp.\nThe proofs use coordinate amalgamation and small-index arguments. We also obtain arbitrary-rank descent for countable families\nafter collapsing any infinite cardinal, with applications to choice\nin the relative Feferman--Levy model.\nIn the full Solovay collapse extension of an arbitrary ZFC ground,\nevery family definable from a real and ordinals that represents fewer\nthan $2^{\\mathfrak c}$ classes modulo null sets consists of measurable\nsets; the corresponding category assertion also holds.\n\nIn the extensions of $L$ by $\\omega_1$ Cohen or random reals, no\nmodel of $I\\Delta_0$ of size at most $\\aleph_1$, definable from\nordinals and a real, has full binary-coded standard system.\n$\\Pi^1_1\\text{-}\\mathsf{CA}_0$ proves a finite-fragment Borel obstruction\nto a full binary standard system for $I\\Delta_0$. For\nregular uncountable $\\kappa$, the generalized Cohen extension by\n$\\Add(\\kappa,\\Lambda)^L$, $\\Lambda>\\kappa$, has no ambiently\n$\\kappa^+$-saturated $I\\Delta_0$ model in\n$\\operatorname{HOD}_{H_{\\kappa^+}}$, regardless of its size.\nConsequently every infinite regular $\\kappa$ of $L$ admits a\ncofinality-preserving GCH extension with no\n$\\mathrm{OD}_{\\mathcal P(\\kappa)}$\nsaturated arithmetic presentation of size $\\kappa^+$. ZFC also proves that every singular strong limit $\\kappa$ admits\nan OD $\\kappa^+$-saturated elementary extension of $\\mathbb N$ of\nsize $2^\\kappa$, hence of size $\\kappa^+$ under GCH.","subject":"math.LO","authorshipMode":"no-human-author-claimed","humanUnderstanding":"some","sponsorUserId":"user_8578724b-402e-4621-8bab-8f9b678f398a","submittingOrganizationId":null,"sponsorName":"Elliot Glazer","orcid":"0009-0001-1457-6540","status":"published","createdAt":"2026-09-30 03:03:01","publishedAt":"2026-09-30T09:11:19.820Z","versionNumber":1,"versionCreatedAt":"2026-09-30 03:03:01","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":["Boolean algebras","Set theory","definability"],"crossListSubjects":[],"authors":[{"id":"1c8ec98a-a6a2-4803-85f4-88722573fd22","name":"Elliot Glazer","orcid":"0009-0001-1457-6540","affiliation":"Principia Labs","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["03C62","03E99","03E25"],"tools":[{"provider":"OpenAI","name":"Codex","version":"Astra"}],"id":"hexagon:2609.00141v1","links":{"record":"https://hexagonmath.org/abs/2609.00141v1","pdf":"https://hexagonmath.org/pdf/2609.00141v1","source":"https://hexagonmath.org/src/2609.00141v1","source_tree":"https://hexagonmath.org/api/v1/works/2609.00141v1/source","api":"https://hexagonmath.org/api/v1/works/2609.00141v1"}},{"identifier":"2609.00142","title":"Collapsibility of Alexander duals for shade maps","abstract":"Let $E$ be a nonempty finite set. A shade map on $E$ is a map $T:\\mathcal{P}(E)\\rightarrow\\mathcal{P}(E)$ such that toggling an element $u\\notin T(F)$ in the input $F$ does not change $T(F)$. We prove that, if $T$ is an inclusion-reversing shade map and $G\\subseteq E$, then the simplicial complex\n\\[\n\\{F\\subseteq E\\mid G\\subseteq T(F)\\}\n\\]\nis collapsible. Equivalently, if $S$ is an inclusion-preserving shade map, then the Alexander dual of\n\\[\n\\{F\\subseteq E\\mid G\\not \\subseteq S(F)\\}\n\\]\nis collapsible. This is an Alexander-dual companion to the shade-map collapsibility theorem in \\emph{The Elser nuclei sum revisited}.\n\nThe proof first matches all faces on which $T(F)\\neq E$ by a toggle. The remaining faces, characterized by $T(F)=E$, form the free convex set complex of an associated antimatroidal quasi-closure operator. We give a self-contained recursive acyclic matching on this complex. For ordinary convex geometries, Korte--Lovász--Schrader prove the stronger fact that the free convex set system is non-evasive.","subject":"math.CO","authorshipMode":"mixed","humanUnderstanding":"all","sponsorUserId":"user_50559304-a175-4f48-9b02-f7ccee550e5f","submittingOrganizationId":null,"sponsorName":"Darij Grinberg","orcid":"0000-0002-9661-8432","status":"published","createdAt":"2026-09-30 04:02:00","publishedAt":"2026-09-30T05:21:27.732Z","versionNumber":1,"versionCreatedAt":"2026-09-30 04:02:00","doi":null,"arxivId":null,"license":"CC0 1.0","tags":["antimatroids","convex geometries","discrete Morse theory","graphs","simplicial complexes"],"crossListSubjects":["math.AT"],"authors":[{"id":"ad464169-eff6-41ad-b5a3-57ffc9e4d3c7","name":"Darij Grinberg","orcid":"0000-0002-9661-8432","affiliation":"Drexel University","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["57Q70","05B35"],"tools":[{"provider":"OpenAI","name":"ChatGPT","version":"5.6 Sol"}],"id":"hexagon:2609.00142v1","links":{"record":"https://hexagonmath.org/abs/2609.00142v1","pdf":"https://hexagonmath.org/pdf/2609.00142v1","source":"https://hexagonmath.org/src/2609.00142v1","source_tree":"https://hexagonmath.org/api/v1/works/2609.00142v1/source","api":"https://hexagonmath.org/api/v1/works/2609.00142v1"}},{"identifier":"2609.00143","title":"Strongly positive bases in rank-two quantum cluster algebras","abstract":"We prove Conjecture 12 of Lee, Li, Rupel, and Zelevinsky: for arbitrary positive integers \\(b,c\\), every strongly positive basis of the coefficient-free rank-two quantum cluster algebra \\(\\mathcal A_v(b,c)\\) contains a unique element pointed at each pair in \\(\\mathbb Z^2\\). These pointed elements exhaust the basis.","subject":"math.QA","authorshipMode":"mixed","humanUnderstanding":"some","sponsorUserId":"user_bf3e39d8-d680-421d-8d68-7772f7717a25","submittingOrganizationId":null,"sponsorName":"琪越 唐","orcid":"0009-0009-1480-7124","status":"published","createdAt":"2026-09-30 05:08:30","publishedAt":"2026-09-30T05:20:36.913Z","versionNumber":2,"versionCreatedAt":"2026-10-04 16:42:50","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":["quantum cluster algebra"],"crossListSubjects":["math.RA"],"authors":[{"id":"feb3374a-708f-47d5-9e52-9e722bd7d577","name":"琪越 唐","orcid":null,"affiliation":"Tsinghua University","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["13F60","16T20"],"tools":[],"id":"hexagon:2609.00143v2","links":{"record":"https://hexagonmath.org/abs/2609.00143v2","pdf":"https://hexagonmath.org/pdf/2609.00143v2","source":"https://hexagonmath.org/src/2609.00143v2","source_tree":"https://hexagonmath.org/api/v1/works/2609.00143v2/source","api":"https://hexagonmath.org/api/v1/works/2609.00143v2"}},{"identifier":"2609.00140","title":"Coefficientwise positivity for alternating sums of $q$-factorials","abstract":"We prove that the alternating sum\n$\\sum_{i=0}^{k}(-1)^i\\binom ki[m-i]!_q$ has nonnegative coefficients\nwhenever $m\\ge 2k-1$. For $k\\ge3$, this range is exact.\nThe case $m=2k$ proves a conjecture of Lewis and Morales arising from\nthe enumeration of invertible matrices with prescribed zero entries.\nAfter removing a common $q$-factorial, we express the sum in a\nGaussian-binomial basis whose coefficients are independent of the ambient\nsize. Their positivity follows from a coefficientwise domination argument\nusing convexity of ordinary binomial coefficients.","subject":"math.CO","authorshipMode":"no-human-author-claimed","humanUnderstanding":"all","sponsorUserId":"user_89a06eea-4fec-4c26-948c-f7c7bcb7dcd1","submittingOrganizationId":null,"sponsorName":"David Anderson","orcid":"0000-0002-3482-0440","status":"published","createdAt":"2026-09-30 02:33:36","publishedAt":"2026-09-30T03:44:57.564Z","versionNumber":1,"versionCreatedAt":"2026-09-30 02:33:36","doi":null,"arxivId":null,"license":"CC0 1.0","tags":[],"crossListSubjects":[],"authors":[{"id":"65f0b171-8e04-401f-92da-c3b9ac23866b","name":"David Anderson","orcid":"0000-0002-3482-0440","affiliation":"Ohio State University","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["05A10","05E10"],"tools":[{"provider":"Anthropic","name":"Claude Fable","version":"5.1"},{"provider":"OpenAI","name":"Codex","version":"6"},{"provider":"Lean","name":"Lean","version":"4"}],"id":"hexagon:2609.00140v1","links":{"record":"https://hexagonmath.org/abs/2609.00140v1","pdf":"https://hexagonmath.org/pdf/2609.00140v1","source":"https://hexagonmath.org/src/2609.00140v1","source_tree":"https://hexagonmath.org/api/v1/works/2609.00140v1/source","api":"https://hexagonmath.org/api/v1/works/2609.00140v1"}},{"identifier":"2609.00138","title":"Signed seeds and G-gradings on cluster algebras","abstract":"The theory of cluster algebras is closely connected to the theory of total positivity; indeed, the desire to better understand total positivity was one of the main motivations for Fomin and Zelevinsky’s introduction of cluster algebras [FZ02]. In particular, any cluster variety whose coordinate ring has a cluster structure has a natural notion of positive part: the subset of the variety where all cluster variables are positive. In this paper, we explain that there are other signed cells contained in cluster varieties that are equally natural from a cluster-theoretic point of view. These come from signed seeds, which can be thought of as a Z/2Z-grading on cluster variables, and which were introduced in [EZLP+ 23] in the context of the amplituhedron. More generally, given any abelian group G, we introduce the notion of a G-graded seed for a cluster algebra, which is a way of assigning elements of G to each cluster variable which is compatible with the cluster structure. When G is the multiplicative group {−1, 1}, this recovers the above notion of signed seed; when G = C∗ , this recovers the notion of cluster automorphism group [GSV10] or cluster dilation group [NS26]; and when G = Zd , this recovers the notion of graded cluster algebra studied by Grabowski-Launois [GL14], Grabowski [Gra15] and Gekhtman-Shapiro-Vainshtein [GSV10, Section\n5.2] (which had previously appeared in special cases in work of Fomin-Zelevinsky [FZ07]). The examples we study include the space of square matrices, symmetric matrices, skew-symmetric matrices, positroid varieties, and amplituhedron tiles. We also connect this notion to tropical mutation when G = R or Z.","subject":"math.CO","authorshipMode":"human","humanUnderstanding":"all","sponsorUserId":"user_2e415841-2e19-4119-8f00-4c9a8044b6c2","submittingOrganizationId":null,"sponsorName":"Lauren Williams","orcid":"0000-0003-3319-8220","status":"published","createdAt":"2026-09-30 00:21:34","publishedAt":"2026-09-30T00:51:24.262Z","versionNumber":1,"versionCreatedAt":"2026-09-30 00:30:14","doi":null,"arxivId":"2610.03612","license":"CC BY 4.0","tags":["Grassmannians","amplituhedron","cluster algebras"],"crossListSubjects":["math.RA"],"authors":[{"id":"25597de9-2a41-41db-9d5b-6cea0d773bd9","name":"Lauren Williams","orcid":"0000-0003-3319-8220","affiliation":"Harvard University","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null},{"id":"9f940ace-f452-4445-99e4-0f4af70667f2","name":"Alan Yan","orcid":"0009-0001-9392-8443","affiliation":"Harvard University","claimed":1,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["13F60","05E99"],"tools":[],"id":"hexagon:2609.00138v1","links":{"record":"https://hexagonmath.org/abs/2609.00138v1","pdf":"https://hexagonmath.org/pdf/2609.00138v1","source":"https://hexagonmath.org/src/2609.00138v1","source_tree":"https://hexagonmath.org/api/v1/works/2609.00138v1/source","api":"https://hexagonmath.org/api/v1/works/2609.00138v1"}},{"identifier":"2609.00128","title":"On properties of Plancherel algebras","abstract":"We proved the predictions by  Ben-Zvi, Sakellaridis, and Venkatesh that the Plancherel algebra $\\mathrm{PL}_{X}$ is commutative and its loop-rotated version $\\mathrm{PL}_{X,\\hbar}$ is flat over $k[\\hbar]$.","subject":"math.AG","authorshipMode":"no-human-author-claimed","humanUnderstanding":"some","sponsorUserId":"user_fc64d13d-8b65-4656-a45a-b375aaac54a5","submittingOrganizationId":null,"sponsorName":"Shurui Liu","orcid":"0009-0008-3424-435X","status":"published","createdAt":"2026-09-29 02:25:48","publishedAt":"2026-09-30T00:00:57.257Z","versionNumber":1,"versionCreatedAt":"2026-09-29 02:25:48","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":["Plancherel algebra","Relative Langlands","geometric Langlands"],"crossListSubjects":["math.NT","math.RT"],"authors":[{"id":"a7567986-abae-4a1d-9367-37e45847fe1f","name":"Shurui Liu","orcid":"0009-0008-3424-435X","affiliation":"Stanford University","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["14D24"],"tools":[{"provider":"OpenAI","name":"Codex","version":"GPT-6-Astra"},{"provider":"Rethlas","name":"https://github.com/frenzymath/Rethlas","version":"commit 887cc46427636bbdd235160a112f9a30ae81d040"},{"provider":"OSVerify","name":"OSVerify","version":null}],"id":"hexagon:2609.00128v1","links":{"record":"https://hexagonmath.org/abs/2609.00128v1","pdf":"https://hexagonmath.org/pdf/2609.00128v1","source":"https://hexagonmath.org/src/2609.00128v1","source_tree":"https://hexagonmath.org/api/v1/works/2609.00128v1/source","api":"https://hexagonmath.org/api/v1/works/2609.00128v1"}},{"identifier":"2609.00137","title":"On a non-basis of the coinvariant algebra","abstract":"A conjecture arising from a question of Procesi proposes a basis of the\ncoinvariant algebra of $S_n$ consisting of column-antisymmetrized monomials\nindexed by pairs of standard Young tableaux. We show that the proposed\nfamily need not even span an $S_n$-subrepresentation: for $n=8$, a generator\nof degree $15$ is sent outside the span by the adjacent transposition\n$(4,5)$. The proof is an exact finite computation with an explicit\nseparating functional, requiring neither a rank computation nor\nGr\\\"obner reduction. We also retain an explicit relation for $n=7$,\nwhere the basis assertion first fails, although the span is still invariant.","subject":"math.CO","authorshipMode":"mixed","humanUnderstanding":"all","sponsorUserId":"user_50559304-a175-4f48-9b02-f7ccee550e5f","submittingOrganizationId":null,"sponsorName":"Darij Grinberg","orcid":"0000-0002-9661-8432","status":"published","createdAt":"2026-09-29 22:29:30","publishedAt":"2026-09-29T22:45:38.255Z","versionNumber":1,"versionCreatedAt":"2026-09-29 22:29:30","doi":null,"arxivId":null,"license":"CC0 1.0","tags":["Specht modules","Young tableaux","coinvariant algebra","higher Specht polynomials","representations of the symmetric group"],"crossListSubjects":["math.AC"],"authors":[{"id":"624358ec-306c-4a11-81da-f0857ce66128","name":"Darij Grinberg","orcid":"0000-0002-9661-8432","affiliation":"Drexel University","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["20C30","05E40"],"tools":[{"provider":"OpenAI","name":"ChatGPT","version":"5.6 Sol"},{"provider":"OpenAI","name":"ChatGPT","version":"6 Astra"},{"provider":"SageMath","name":"SageMath","version":"10.10.beta9"}],"id":"hexagon:2609.00137v1","links":{"record":"https://hexagonmath.org/abs/2609.00137v1","pdf":"https://hexagonmath.org/pdf/2609.00137v1","source":"https://hexagonmath.org/src/2609.00137v1","source_tree":"https://hexagonmath.org/api/v1/works/2609.00137v1/source","api":"https://hexagonmath.org/api/v1/works/2609.00137v1"}},{"identifier":"2609.00132","title":"Primitive Integer Solutions of x^4 + 3y^3 + 2z^3 = 0","abstract":"We determine the primitive integer solutions of x^4 + 3y^3 + 2z^3 = 0. They are exactly (1,-1,1) and (-1,-1,1), where primitivity means gcd(x,y,z)=1. A fourth-power descent in Q(∛12) produces four plane quartics. A congruence modulo 4 excludes two of them for primitive input. We determine the rational points on the other two by elliptic descent, finite reduction sieves, and elliptic Chabauty. For the final quartic, an explicit nonzero Cassels–Tate pairing sharpens a rank-three descent bound to rank one. All finite certificates and their verification programs accompany the proof.","subject":"math.NT","authorshipMode":"no-human-author-claimed","humanUnderstanding":"none","sponsorUserId":"user_05fda04e-c83c-4e35-9d88-6f3577955885","submittingOrganizationId":null,"sponsorName":"Avraham Eisenberg","orcid":"0009-0005-1734-1914","status":"published","createdAt":"2026-09-29 16:19:56","publishedAt":"2026-09-29T20:31:40.983Z","versionNumber":1,"versionCreatedAt":"2026-09-29 16:19:56","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":[],"crossListSubjects":[],"authors":[{"id":"9f4c6a68-9e64-448a-9f2d-c083e4ca27ed","name":"Avraham Eisenberg","orcid":"0009-0005-1734-1914","affiliation":null,"claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["11D45"],"tools":[{"provider":"OpenAI","name":"ChatGPT","version":"Astra 6 Pro"}],"id":"hexagon:2609.00132v1","links":{"record":"https://hexagonmath.org/abs/2609.00132v1","pdf":"https://hexagonmath.org/pdf/2609.00132v1","source":"https://hexagonmath.org/src/2609.00132v1","source_tree":"https://hexagonmath.org/api/v1/works/2609.00132v1/source","api":"https://hexagonmath.org/api/v1/works/2609.00132v1"}},{"identifier":"2609.00134","title":"Five equations for first-order logic, modulo the structure of existential graphs","abstract":"Peirce’s existential graphs draw a first-order formula so that the order and grouping of conjuncts, the names and order of bound variables, the position of a quantifier relative to conjuncts that do not mention its variable, and how a network of equalities is written cannot be seen. We take these structural laws for granted. On top of them, five two-way laws are complete for ordinary first-order logic with equality over nonempty domains. That is, any two equivalent formulas are connected by applying the laws inside them, in either direction. The laws are double negation, deiteration X ∧ ¬(X ∧Y ) ⇔ X ∧ ¬Y , annihilation ⊥ ∧ X ⇔ ⊥, substitution of equals, and existential introduction X(a) ∧ ∃u.X(u) ⇔ X(a), and none of them follows from the other four. The first three were already known to be complete for the propositional part. We prove first-order completeness by reduction to Tarski’s representation theorem for locally finite cylindric algebras. A size-ordered search over first-order laws, run up to size 5, kept four of the five; deiteration has size 7 and was added by hand. We also show that a published basis for Spencer-Brown’s boundary algebra is incomplete.","subject":"math.LO","authorshipMode":"no-human-author-claimed","humanUnderstanding":"all","sponsorUserId":"user_c8070388-f277-4cd0-8a92-76ea18456345","submittingOrganizationId":null,"sponsorName":"Anthony Hart","orcid":"0009-0004-4116-5538","status":"published","createdAt":"2026-09-29 17:59:28","publishedAt":"2026-09-29T20:30:15.928Z","versionNumber":1,"versionCreatedAt":"2026-09-29 17:59:28","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":["cylindric algebra","existential graphs","first-order logic"],"crossListSubjects":[],"authors":[{"id":"52ab8f0c-0a6c-4baa-b1d7-4f6f4865c058","name":"Anthony Hart","orcid":"0009-0004-4116-5538","affiliation":null,"claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["03B10","03G15","08B05","03G05","68V15"],"tools":[{"provider":"Anthropic","name":"Claude Opus","version":"5.5"},{"provider":"University of New Mexico","name":"Mace4","version":null},{"provider":"University of New Mexico","name":"Prover9","version":null}],"id":"hexagon:2609.00134v1","links":{"record":"https://hexagonmath.org/abs/2609.00134v1","pdf":"https://hexagonmath.org/pdf/2609.00134v1","source":"https://hexagonmath.org/src/2609.00134v1","source_tree":"https://hexagonmath.org/api/v1/works/2609.00134v1/source","api":"https://hexagonmath.org/api/v1/works/2609.00134v1"}},{"identifier":"2609.00133","title":"A Universal Noncommutative Splitting Algebra for Polynomials","abstract":"Let $R$ be a commutative ring. We show that every\nhomogeneous polynomial $f\\in R[x_{1},\\ldots,x_{m}]$ of degree $n$ splits into\na product of $n$ homogeneous linear forms over a suitable noncommutative ring\nextension $S$ of $R$ (that is, over a noncommutative $R$-algebra $S$ whose\nstructure morphism $R\\rightarrow S$ is injective). The algebra $S$ is\nuniversal for such factorizations and is free as an $R$-module.\n\nThe proof uses Bergman's Diamond Lemma: we define $S$ by generators and\nrelations, and the relations form a terminating reduction system that has no\nambiguities and thus is confluent.\n\nAn analogous result is also shown for inhomogeneous polynomials (with\ninhomogeneous factors). This easily follows from the homogeneous case by\nhomogenizing and then setting the homogenizing variable equal to $1$.","subject":"math.RA","authorshipMode":"mixed","humanUnderstanding":"all","sponsorUserId":"user_50559304-a175-4f48-9b02-f7ccee550e5f","submittingOrganizationId":null,"sponsorName":"Darij Grinberg","orcid":"0000-0002-9661-8432","status":"published","createdAt":"2026-09-29 17:13:18","publishedAt":"2026-09-29T20:28:56.912Z","versionNumber":1,"versionCreatedAt":"2026-09-29 17:13:18","doi":null,"arxivId":null,"license":"CC0 1.0","tags":["Bergman's diamond lemma","Gröbner bases","combinatorial algebra","noncommutative polynomials"],"crossListSubjects":["math.AC","math.CO"],"authors":[{"id":"da22342e-a021-4b3d-93e9-7ff5e9150269","name":"Darij Grinberg","orcid":"0000-0002-9661-8432","affiliation":"Drexel University","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["13P10","16S15"],"tools":[{"provider":"OpenAI","name":"ChatGPT","version":"5.5"}],"id":"hexagon:2609.00133v1","links":{"record":"https://hexagonmath.org/abs/2609.00133v1","pdf":"https://hexagonmath.org/pdf/2609.00133v1","source":"https://hexagonmath.org/src/2609.00133v1","source_tree":"https://hexagonmath.org/api/v1/works/2609.00133v1/source","api":"https://hexagonmath.org/api/v1/works/2609.00133v1"}},{"identifier":"2609.00135","title":"A counterexample to the Burman--Kulishov conjecture on Lie elements","abstract":"Burman and Kulishov defined Lie elements in the group algebra\n$\\mathbf k[S_n]$ by comparing, on every exterior power of the reflection\nrepresentation $V$, the usual action of $\\mathbf k[S_n]$ with the\ninfinitesimal action induced by its action on $V$.  They conjectured\nthat the Lie algebra $\\mathcal L_n$ of all Lie elements is generated by the\nKirchhoff differences $1-(ij)$.  We disprove this conjecture for\n$n=4$ by exhibiting an explicit counterexample arising from the\n$(2,2)$-block of $\\mathbf k[S_4]$.  More generally, we describe $\\mathcal L_n$ in\nterms of the Artin--Wedderburn decomposition of $\\mathbf k[S_n]$: its hook\nblocks are determined by the action on $V$, whereas its non-hook blocks\nare arbitrary.  Consequently, the primitive central idempotents of the\nnon-hook blocks yield linearly independent obstructions to the\nconjecture.  We also identify the Lie algebra generated by the\nKirchhoff differences in terms of the derived algebra of the Lie\nalgebra generated by transpositions.  Along the way, we give an\nintegral, and hence characteristic-free, proof that the exterior powers\nof $V$ are the hook-shaped Specht modules.","subject":"math.CO","authorshipMode":"mixed","humanUnderstanding":"all","sponsorUserId":"user_50559304-a175-4f48-9b02-f7ccee550e5f","submittingOrganizationId":null,"sponsorName":"Darij Grinberg","orcid":"0000-0002-9661-8432","status":"published","createdAt":"2026-09-29 18:24:18","publishedAt":"2026-09-29T20:27:22.136Z","versionNumber":1,"versionCreatedAt":"2026-09-29 18:24:18","doi":null,"arxivId":null,"license":"CC0 1.0","tags":["Lie algebra","symmetric group algebra","symmetric group representations"],"crossListSubjects":["math.RT"],"authors":[{"id":"0cbcd707-fb7a-4899-bfb1-15c3809e1c9e","name":"Darij Grinberg","orcid":"0000-0002-9661-8432","affiliation":"Drexel University","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["20C30","05E99"],"tools":[{"provider":"OpenAI","name":"ChatGPT","version":"5.6 Sol"}],"id":"hexagon:2609.00135v1","links":{"record":"https://hexagonmath.org/abs/2609.00135v1","pdf":"https://hexagonmath.org/pdf/2609.00135v1","source":"https://hexagonmath.org/src/2609.00135v1","source_tree":"https://hexagonmath.org/api/v1/works/2609.00135v1/source","api":"https://hexagonmath.org/api/v1/works/2609.00135v1"}},{"identifier":"2609.00130","title":"Primitive Integer Solutions of x^4 + 4y^3 + z^3 = 0","abstract":"We determine all primitive integer solutions of the generalized Fermat equation\n\nx^4 + 4y^3 + z^3 = 0.\n\nHere primitive means gcd(x,y,z)=1. We prove that the only primitive solutions are\n\n(x,y,z) = (1,0,-1), (-1,0,-1).\n\nThe proof begins with a fourth-power descent in the pure cubic field Q(∛2), reducing the problem to the rational points on two explicit smooth plane quartics of genus 3. These quartics admit maps to elliptic curves over Q(∛2). Exact 2-descent determines the relevant Mordell-Weil ranks and finite-index subgroups. A complete reduction sieve at 127 leaves four residue disks, and an elliptic Chabauty calculation proves that each disk contains exactly one rational point. Reconstruction of the original variables then leaves precisely the two stated primitive solutions. The finite computations are exact and reproducible from the certificates and verifier source included in the appendices, and the argument applies at all heights.","subject":"math.NT","authorshipMode":"no-human-author-claimed","humanUnderstanding":"none","sponsorUserId":"user_05fda04e-c83c-4e35-9d88-6f3577955885","submittingOrganizationId":null,"sponsorName":"Avraham Eisenberg","orcid":"0009-0005-1734-1914","status":"published","createdAt":"2026-09-29 13:06:12","publishedAt":"2026-09-29T15:26:56.364Z","versionNumber":1,"versionCreatedAt":"2026-09-29 13:06:12","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":[],"crossListSubjects":[],"authors":[{"id":"4492c36d-5df3-4d75-9fb9-b4a39d78b7b4","name":"Avraham Eisenberg","orcid":"0009-0005-1734-1914","affiliation":null,"claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["11D45"],"tools":[{"provider":"OpenAI","name":"ChatGPT","version":"Astra 6 Pro"}],"id":"hexagon:2609.00130v1","links":{"record":"https://hexagonmath.org/abs/2609.00130v1","pdf":"https://hexagonmath.org/pdf/2609.00130v1","source":"https://hexagonmath.org/src/2609.00130v1","source_tree":"https://hexagonmath.org/api/v1/works/2609.00130v1/source","api":"https://hexagonmath.org/api/v1/works/2609.00130v1"}},{"identifier":"2609.00124","title":"Quantum Laurent Positivity for Mutation-Acyclic Cluster Algebras","abstract":"We prove quantum Laurent positivity for mutation-acyclic skew-symmetrizable cluster algebras of arbitrary finite rank.","subject":"math.QA","authorshipMode":"mixed","humanUnderstanding":"some","sponsorUserId":"user_bf3e39d8-d680-421d-8d68-7772f7717a25","submittingOrganizationId":null,"sponsorName":"琪越 唐","orcid":"0009-0009-1480-7124","status":"published","createdAt":"2026-09-29 00:11:19","publishedAt":"2026-09-29T15:21:14.138Z","versionNumber":2,"versionCreatedAt":"2026-09-29 16:13:16","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":["quantum cluster algebra"],"crossListSubjects":["math.RA","math.RT"],"authors":[{"id":"5a609239-518b-44b8-9d85-ea6a8aabca85","name":"琪越 唐","orcid":null,"affiliation":"Tsinghua University","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["13F60","16G20","17B37"],"tools":[],"id":"hexagon:2609.00124v2","links":{"record":"https://hexagonmath.org/abs/2609.00124v2","pdf":"https://hexagonmath.org/pdf/2609.00124v2","source":"https://hexagonmath.org/src/2609.00124v2","source_tree":"https://hexagonmath.org/api/v1/works/2609.00124v2/source","api":"https://hexagonmath.org/api/v1/works/2609.00124v2"}},{"identifier":"2609.00126","title":"Pauli problem in dimension 4","abstract":"The finite-dimensional Pauli problem asks how many measurements in orthonormal bases determine every pure state of a \\(d\\)-level quantum system up to a global phase.\nFour bases always suffice, and the answer is known to be three for \\(d=2\\) and four for \\(d=3\\) and \\(d\\geq5\\).\nDimension four remained open because the embedding argument used in other dimensions fails there: the pure-state space \\(\\mathbb{CP}^3\\) embeds in \\(\\R^9\\), the space of the nine independent probabilities of three bases.\nWe show that three bases do not suffice in \\(\\C^4\\), so exactly four are needed.\nThe same argument shows that no ten vectors in \\(\\C^4\\) do phase retrieval.\nSince eleven vectors are known to suffice, the smallest phase-retrieval frame and the smallest rank-one POVM distinguishing all pure states in \\(\\C^4\\) both have eleven elements.\nAll three lower bounds follow from one statement: every six-dimensional real space of traceless Hermitian \\(4\\times4\\) matrices with a common isotropic vector, that is, a nonzero \\(e\\) with \\(e^*Te=0\\) for all \\(T\\) in the space, contains a nonzero matrix of rank at most two.\nWe prove it with complex \\(K\\)-theory: otherwise, an odd unitary map on the five-sphere would have a \\(K^1\\)-class that is nonzero by antipodal symmetry but vanishes because of the isotropic vector.","subject":"quant-ph","authorshipMode":"no-human-author-claimed","humanUnderstanding":"some","sponsorUserId":"user_d95f42c0-af23-4257-a1f8-670299f105c8","submittingOrganizationId":null,"sponsorName":"Dmitry Grinko","orcid":"0000-0001-6438-1998","status":"published","createdAt":"2026-09-29 02:07:25","publishedAt":"2026-09-29T06:02:58.268Z","versionNumber":1,"versionCreatedAt":"2026-09-29 02:07:25","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":[],"crossListSubjects":["math.AT","math.KT"],"authors":[{"id":"c4fbde4f-58f3-47ee-841e-7c6e86521e4e","name":"Dmitry Grinko","orcid":"0000-0001-6438-1998","affiliation":"QuSoft, University of Amsterdam","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["81P15","81P18","15B57","19L64","94A12"],"tools":[],"id":"hexagon:2609.00126v1","links":{"record":"https://hexagonmath.org/abs/2609.00126v1","pdf":"https://hexagonmath.org/pdf/2609.00126v1","source":"https://hexagonmath.org/src/2609.00126v1","source_tree":"https://hexagonmath.org/api/v1/works/2609.00126v1/source","api":"https://hexagonmath.org/api/v1/works/2609.00126v1"}},{"identifier":"2609.00123","title":"Breaking the Fourth Wall: The Planar Centroid Banach–Mazur Diameter Is Strictly Less than $4$","abstract":"We prove that the centroid Banach–Mazur diameter of planar convex bodies is strictly less than $4$, (slightly) improving the prior upper bound $69/17\\approx 4.05882$ due to Lassak. The proof combines a sharp covariance-ellipse sandwich, whose equality cases in any single direction occur only for triangles, with a compactness argument: an extremal pair at distance $4$ would consist of two triangles, which are linearly equivalent and hence must in fact be at distance $1$. The resulting gap below $4$ is uniform but nonquantitative.","subject":"math.MG","authorshipMode":"mixed","humanUnderstanding":"all","sponsorUserId":"user_a64917cf-f9a1-482c-a43f-6c639209d9e6","submittingOrganizationId":null,"sponsorName":"Scott Kominers","orcid":"0000-0002-7608-6619","status":"published","createdAt":"2026-09-28 21:54:51","publishedAt":"2026-09-28T22:13:08.481Z","versionNumber":1,"versionCreatedAt":"2026-09-28 21:54:51","doi":null,"arxivId":null,"license":"CC BY-SA 4.0","tags":["Banach–Mazur diameter"],"crossListSubjects":[],"authors":[{"id":"a154dcf9-4ddd-40fc-aeef-1255e07626e4","name":"Scott Kominers","orcid":"0000-0002-7608-6619","affiliation":"Harvard University and a16z crypto","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["52A40","52A10","52A21","46B20"],"tools":[{"provider":"OpenAI","name":"GPT","version":"5.6 Sol"},{"provider":"Anthropic","name":"Claude Fable","version":"5"},{"provider":"Quora","name":"Poe","version":null}],"id":"hexagon:2609.00123v1","links":{"record":"https://hexagonmath.org/abs/2609.00123v1","pdf":"https://hexagonmath.org/pdf/2609.00123v1","source":"https://hexagonmath.org/src/2609.00123v1","source_tree":"https://hexagonmath.org/api/v1/works/2609.00123v1/source","api":"https://hexagonmath.org/api/v1/works/2609.00123v1"}},{"identifier":"2609.00121","title":"Polynomials over Symmetric Polynomials","abstract":"Let $\\mathbf{k}$ be a commutative ring, and let the\nsymmetric group $\\mathfrak{S}_{n}$ act on $P=\\mathbf{k}\\left[  x_{1}\n,x_{2},\\ldots,x_{n} \\right]  $ by permuting the variables. We prove four\nresults that are known (at least in the case when $\\mathbf{k}$ is\na field) but not easily found in the literature.\nFirst, the coinvariant algebra (the quotient of $P$ by the\nideal generated by the symmetric polynomials with constant term $0$) is a free\n$\\mathbf{k}$-module of rank $n!$, with the residue classes of the Artin\nmonomials as a basis.\nSecond, $P$ is a free module of rank $n!$ over the ring\n$P^{\\mathfrak{S}_{n}}$ of symmetric polynomials, again with the Artin\nmonomials as a basis.\nThird, if $n!$ is invertible in $\\mathbf{k}$, the\ncoinvariant algebra is the regular $\\mathbf{k}\\left[  \\mathfrak{S}_{n}\n\\right]  $-module.\nFourth, under the same hypothesis, $P$ is a free left\n$P^{\\mathfrak{S}_{n}}\\left[  \\mathfrak{S}_{n} \\right]  $-module of rank $1$.\nThe first result follows from an elementary normal-form lemma for monic\npolynomials with pairwise relatively prime leading monomials. The second is\nproved by lifting the Artin basis. For the third, we use orbit harmonics with\na strongly discrete point orbit, and the fourth follows by equivariantly\nlifting a regular basis of the coinvariant algebra.","subject":"math.AC","authorshipMode":"mixed","humanUnderstanding":"all","sponsorUserId":"user_50559304-a175-4f48-9b02-f7ccee550e5f","submittingOrganizationId":null,"sponsorName":"Darij Grinberg","orcid":"0000-0002-9661-8432","status":"published","createdAt":"2026-09-28 18:18:01","publishedAt":"2026-09-28T20:23:25.697Z","versionNumber":1,"versionCreatedAt":"2026-09-28 18:18:01","doi":null,"arxivId":null,"license":"CC0 1.0","tags":["Artin basis","Gröbner bases","coinvariant algebra","orbit harmonics","symmetric polynomials"],"crossListSubjects":["math.CO"],"authors":[{"id":"c748dcc4-3ca5-4337-a142-a65c66fcf56d","name":"Darij Grinberg","orcid":"0000-0002-9661-8432","affiliation":"Drexel University","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["05E40","20C30","13P10"],"tools":[{"provider":"OpenAI","name":"ChatGPT","version":"5.6 Sol"}],"id":"hexagon:2609.00121v1","links":{"record":"https://hexagonmath.org/abs/2609.00121v1","pdf":"https://hexagonmath.org/pdf/2609.00121v1","source":"https://hexagonmath.org/src/2609.00121v1","source_tree":"https://hexagonmath.org/api/v1/works/2609.00121v1/source","api":"https://hexagonmath.org/api/v1/works/2609.00121v1"}},{"identifier":"2609.00122","title":"Generalized Bawo: A Finite-State Model and an EXPTIME Upper Bound","abstract":"Four-row mancala games, including Malawian Bawo and the related East African Bao, have received\nlimited attention in computational complexity theory despite their combinatorial richness. This paper\nintroduces GENERALIZED-BAWO-WIN, a finite-state two-player game model inspired by\nMalawian Bawo. The model is parameterized by a half-board width n and a binary-encoded lap cap\nL, and incorporates relay sowing, marker-pit capture, a formalized capture-relay rule (abstracting\nkichwa), mandatory capture, a singleton-start prohibition, and an explicit lap-cap truncation rule that\nguarantees move termination. We prove that GENERALIZED-BAWO-WIN belongs to EXPTIME\nunder these semantics. The proof proceeds by establishing that the game's state space has size at\nmost 2poly(|I|), that legal successors can be generated in 2poly(|I|) time, and that a WIN/LOSS/DRAW\nretrograde attractor can be computed over the resulting finite game graph. We emphasize that this is\nan upper bound: no hardness result is established. We further emphasize that the theorem applies\nonly to the formally defined generalized model and does not classify Bao as played in East Africa,\nwhose rules differ materially.","subject":"cs.CC","authorshipMode":"mixed","humanUnderstanding":"all","sponsorUserId":"user_d68b5678-f0b2-4e88-98cd-9d78f4fcce1e","submittingOrganizationId":null,"sponsorName":"Isaac kaimfa","orcid":"0009-0006-3151-6714","status":"published","createdAt":"2026-09-28 19:51:38","publishedAt":"2026-09-28T20:19:37.748Z","versionNumber":1,"versionCreatedAt":"2026-09-28 19:51:38","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":["EXPTIME","combinatorial games","computational complexity","finite games","generalized Bawo","state-space complexity"],"crossListSubjects":["cs.DS"],"authors":[{"id":"1d7697cc-d9e9-4ffb-b21e-2110d9c44c86","name":"Isaac kaimfa","orcid":"0009-0006-3151-6714","affiliation":null,"claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["68Q15"],"tools":[],"id":"hexagon:2609.00122v1","links":{"record":"https://hexagonmath.org/abs/2609.00122v1","pdf":"https://hexagonmath.org/pdf/2609.00122v1","source":"https://hexagonmath.org/src/2609.00122v1","source_tree":"https://hexagonmath.org/api/v1/works/2609.00122v1/source","api":"https://hexagonmath.org/api/v1/works/2609.00122v1"}},{"identifier":"2609.00118","title":"A counterexample to ordering-independence for chromatic operators","abstract":"We give a $10$-vertex counterexample to a conjecture of Pawlowski asserting\nthat the characteristic polynomial of a chromatic operator associated with a\nforest is independent of the ordering of its edges.  The two operators already\nhave different traces of fifth powers.","subject":"math.CO","authorshipMode":"no-human-author-claimed","humanUnderstanding":"all","sponsorUserId":"user_50559304-a175-4f48-9b02-f7ccee550e5f","submittingOrganizationId":null,"sponsorName":"Darij Grinberg","orcid":"0000-0002-9661-8432","status":"published","createdAt":"2026-09-28 17:29:35","publishedAt":"2026-09-28T18:09:22.441Z","versionNumber":1,"versionCreatedAt":"2026-09-28 17:29:35","doi":null,"arxivId":null,"license":"CC0 1.0","tags":["Graph coloring","Specht modules","symmetric group algebra"],"crossListSubjects":[],"authors":[{"id":"c15cf171-9b52-43c4-a857-c08997dd42a2","name":"Darij Grinberg","orcid":"0000-0002-9661-8432","affiliation":"Drexel University","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["05E05","05C15"],"tools":[],"id":"hexagon:2609.00118v1","links":{"record":"https://hexagonmath.org/abs/2609.00118v1","pdf":"https://hexagonmath.org/pdf/2609.00118v1","source":"https://hexagonmath.org/src/2609.00118v1","source_tree":"https://hexagonmath.org/api/v1/works/2609.00118v1/source","api":"https://hexagonmath.org/api/v1/works/2609.00118v1"}},{"identifier":"2609.00119","title":"Hilbert-type inequality with primes","abstract":"This note answers a problem posed in Montgomery's CBMS book.  The proof was found by ChatGPT 6, and written up by the author.","subject":"math.NT","authorshipMode":"human","humanUnderstanding":"all","sponsorUserId":"user_a9ebc462-7731-49b8-ac0f-76516aa8a5ee","submittingOrganizationId":null,"sponsorName":"Kannan Soundararajan","orcid":"0000-0002-7595-0700","status":"published","createdAt":"2026-09-28 17:44:08","publishedAt":"2026-09-28T18:01:17.662Z","versionNumber":1,"versionCreatedAt":"2026-09-28 17:44:08","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":["Collar-Stein Lemma","Hilbert's inequality","Sieves"],"crossListSubjects":["math.CA"],"authors":[{"id":"3a27dc6b-1e4a-4abe-a99f-31b2475123bc","name":"Kannan Soundararajan","orcid":"0000-0002-7595-0700","affiliation":"Stanford University","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["11N05"],"tools":[],"id":"hexagon:2609.00119v1","links":{"record":"https://hexagonmath.org/abs/2609.00119v1","pdf":"https://hexagonmath.org/pdf/2609.00119v1","source":"https://hexagonmath.org/src/2609.00119v1","source_tree":"https://hexagonmath.org/api/v1/works/2609.00119v1/source","api":"https://hexagonmath.org/api/v1/works/2609.00119v1"}},{"identifier":"2609.00117","title":"Splitting a Polynomial into Linear Factors after an Injective Ring Extension","abstract":"We show that each univariate polynomial $P = p_0 + p_1 X + \\cdots + p_m X^m \\in R[X]$ over a commutative ring $R$ can be factored into linear factors over a suitable commutative ring extension $S$ of $R$. The proof proceeds by universal construction: $S$ is defined as the tensor product $R \\otimes_{C_m} B_m$, where $B_m$ is the polynomial ring $\\ZZ[a_1, b_1, a_2, b_2, \\ldots, a_m, b_m]$, and where $C_m$ is its subring generated by its ``homogenized elementary symmetric polynomials'' $E_r=\\sum_{\\substack{I\\subseteq [m];\\\\ |I|=r}} \\prod_{i\\in I}a_i\\prod_{i\\notin I}b_i$ for all $0 \\leq r \\leq m$. The injectivity of the structure homomorphism $R \\to S$ is deduced from a combinatorial study of the diagonal subring of $B_m$. In the process, a homogeneous variant of the Garsia--Stanton basis is constructed, and some classical properties of symmetric polynomials are recovered.","subject":"math.AC","authorshipMode":"mixed","humanUnderstanding":"all","sponsorUserId":"user_50559304-a175-4f48-9b02-f7ccee550e5f","submittingOrganizationId":null,"sponsorName":"Darij Grinberg","orcid":"0000-0002-9661-8432","status":"published","createdAt":"2026-09-28 17:16:53","publishedAt":"2026-09-28T17:27:04.790Z","versionNumber":1,"versionCreatedAt":"2026-09-28 17:16:53","doi":null,"arxivId":null,"license":"CC0 1.0","tags":[],"crossListSubjects":["math.CO","math.RA"],"authors":[{"id":"9ac9ae16-a1f0-4eec-9610-293b904846c0","name":"Darij Grinberg","orcid":"0000-0002-9661-8432","affiliation":"Drexel University","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["13F20","05E05","13P10"],"tools":[{"provider":"OpenAI","name":"ChatGPT","version":"5.5"}],"id":"hexagon:2609.00117v1","links":{"record":"https://hexagonmath.org/abs/2609.00117v1","pdf":"https://hexagonmath.org/pdf/2609.00117v1","source":"https://hexagonmath.org/src/2609.00117v1","source_tree":"https://hexagonmath.org/api/v1/works/2609.00117v1/source","api":"https://hexagonmath.org/api/v1/works/2609.00117v1"}},{"identifier":"2609.00116","title":"A Counterexample to a Conjecture on Fused Specht Polynomials","abstract":"Lafay, Peltola and Roussillon conjectured that their realization of simple\nmodules of the fused Hecke algebra by fused Specht polynomials extends from\nYoung diagrams with two columns to Young diagrams of arbitrary shape.  We\ngive a counterexample for\n\\[\n n=8,\\qquad \\lambda=(3,2,2,1),\\qquad\n \\varsigma=(2,2,2,2).\n\\]\nMore precisely, we exhibit an explicit linear dependence among three fused\nSpecht polynomials indexed by row-strict Young tableaux.  The same example\nalso yields an infinite family of counterexamples.","subject":"math.CO","authorshipMode":"no-human-author-claimed","humanUnderstanding":"some","sponsorUserId":"user_50559304-a175-4f48-9b02-f7ccee550e5f","submittingOrganizationId":null,"sponsorName":"Darij Grinberg","orcid":"0000-0002-9661-8432","status":"published","createdAt":"2026-09-28 16:22:13","publishedAt":"2026-09-28T17:11:38.538Z","versionNumber":1,"versionCreatedAt":"2026-09-28 16:22:13","doi":null,"arxivId":null,"license":"CC0 1.0","tags":["Specht modules","Specht polynomials","Young tableaux"],"crossListSubjects":[],"authors":[{"id":"b19f6d2d-c65d-4414-ad42-606d7bf23d86","name":"Darij Grinberg","orcid":"0000-0002-9661-8432","affiliation":null,"claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["05E99"],"tools":[],"id":"hexagon:2609.00116v1","links":{"record":"https://hexagonmath.org/abs/2609.00116v1","pdf":"https://hexagonmath.org/pdf/2609.00116v1","source":"https://hexagonmath.org/src/2609.00116v1","source_tree":"https://hexagonmath.org/api/v1/works/2609.00116v1/source","api":"https://hexagonmath.org/api/v1/works/2609.00116v1"}},{"identifier":"2609.00114","title":"Under Measurables, HOD_{Ord^{\\omega}} is locally arbitrary","abstract":"For the inner model HOD_{Ord^{\\omega}}, we derive an analogue for Roguski's classical result that without nontrivial assumptions on V, HOD is an arbitrary model ZFC model. For any sentence \\sigma, letting CM denote \"class of measurables,\" the following theories are equiconsistent:\n\n(1) ZFC + CM + [HOD_{Ord^\\omega} \\models (\\exists \\kappa V_{\\kappa} \\models \\sigma)];\n(2) ZFC + GCH + CM + [HOD_{Ord^\\omega} \\models (SVC \\wedge CM \\wedge \\exists \\kappa V_{\\kappa} \\models \\sigma)];\n(3) ZF + DC + CM + \\exists \\kappa V_{\\kappa} \\models \\sigma.\n\nThus, under measurables we have that HOD_{Ord^{\\omega}} is \"locally arbitrary.\"","subject":"math.LO","authorshipMode":"no-human-author-claimed","humanUnderstanding":"some","sponsorUserId":"user_8578724b-402e-4621-8bab-8f9b678f398a","submittingOrganizationId":null,"sponsorName":"Elliot Glazer","orcid":"0009-0001-1457-6540","status":"published","createdAt":"2026-09-28 05:07:15","publishedAt":"2026-09-28T15:38:03.520Z","versionNumber":4,"versionCreatedAt":"2026-09-30 23:32:40","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":["Set theory","axiom of choice"],"crossListSubjects":[],"authors":[{"id":"0e8e0bfe-8a5d-43a8-a2be-8285fd896811","name":"Elliot Glazer","orcid":null,"affiliation":"Principia Labs","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["03E99"],"tools":[{"provider":"OpenAI","name":"Codex","version":null}],"id":"hexagon:2609.00114v4","links":{"record":"https://hexagonmath.org/abs/2609.00114v4","pdf":"https://hexagonmath.org/pdf/2609.00114v4","source":"https://hexagonmath.org/src/2609.00114v4","source_tree":"https://hexagonmath.org/api/v1/works/2609.00114v4/source","api":"https://hexagonmath.org/api/v1/works/2609.00114v4"}},{"identifier":"2609.00112","title":"Swan induction for the finite-local sphere","abstract":"We establish rational Swan induction for ordinary perfect modules over\nthe finite-local sphere $L_n^{p,f}\\mathbb S$. For a finite abelian\nambient group, subgroups of $p$-rank at most $n+1$ suffice, and this\nbound is sharp. The proof combines cyclic homotopy fixed points in\ntelescopic spectra with the isotropy filtration of a chromatic quotient\nof finite genuine spectra. The result proves Conjecture~7.22 of\nClausen--Mathew--Naumann--Noel for Morava $E$-theory and gives a new\nproof of their chromatic upper bound for the algebraic $K$-theory of\n$L_n^{p,f}\\mathbb S$-linear categories. The quotient argument also\nproduces finite complexes realizing the induction relations. We\nformulate the problem of explicit realizations and give two geometric\nmodels at the prime two.","subject":"math.AT","authorshipMode":"no-human-author-claimed","humanUnderstanding":"some","sponsorUserId":"user_12fd56c8-1454-4444-9120-366f4739638f","submittingOrganizationId":null,"sponsorName":"Akhil Mathew","orcid":"0000-0002-3899-2872","status":"published","createdAt":"2026-09-27 17:47:14","publishedAt":"2026-09-27T18:21:54.575Z","versionNumber":1,"versionCreatedAt":"2026-09-27 17:47:14","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":["Burnside rings","Swan induction","chromatic homotopy theory"],"crossListSubjects":["math.KT"],"authors":[{"id":"2f8319bf-ccbc-4d37-9a1a-88b3e0f72032","name":"Akhil Mathew","orcid":"0000-0002-3899-2872","affiliation":null,"claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["55P42","19A22","55P91","55P60","19D10"],"tools":[],"id":"hexagon:2609.00112v1","links":{"record":"https://hexagonmath.org/abs/2609.00112v1","pdf":"https://hexagonmath.org/pdf/2609.00112v1","source":"https://hexagonmath.org/src/2609.00112v1","source_tree":"https://hexagonmath.org/api/v1/works/2609.00112v1/source","api":"https://hexagonmath.org/api/v1/works/2609.00112v1"}},{"identifier":"2609.00111","title":"Torsion in $K_1$ and $p$-adic vanishing cycles","abstract":"For every odd prime $p$, we construct a regular strictly henselian\nlocal ring $A$ whose generic fibre has \\'etale cohomology classes in\n$H^2_{\\et}(A[1/p],\\mu_p^{\\otimes2})$ which are not sums of cup products\nof degree-one classes. The ring is the strict henselization of a local\nring on a finite-type arithmetic scheme. A unit on a principal divisor gives\nan element of order $p$ in $SK_1(A[1/p])$, detected by its\nlocalization boundary on the special fibre. Adams--Riemann--Roch and the\nconnectivity of the motivic filtration show that this element has nonzero\nimage in $H^3_{\\mot}(A[1/p],\\Z(2))$. The integral coefficient sequence\nthen gives the cohomological obstruction. Only the eigenvalues $4$ and\n$8$ of $\\psi^2$ are needed.","subject":"math.AG","authorshipMode":"no-human-author-claimed","humanUnderstanding":"all","sponsorUserId":"user_12fd56c8-1454-4444-9120-366f4739638f","submittingOrganizationId":null,"sponsorName":"Akhil Mathew","orcid":"0000-0002-3899-2872","status":"published","createdAt":"2026-09-27 17:37:03","publishedAt":"2026-09-27T18:20:31.693Z","versionNumber":1,"versionCreatedAt":"2026-09-27 17:37:03","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":["Adams operations","Milnor K-theory","p-adic vanishing cycles"],"crossListSubjects":["math.KT"],"authors":[{"id":"b46029c3-3d8b-403e-88c8-7025633a38e4","name":"Akhil Mathew","orcid":"0000-0002-3899-2872","affiliation":"University of Chicago","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["19D45","14F20","14F42","19E20"],"tools":[],"id":"hexagon:2609.00111v1","links":{"record":"https://hexagonmath.org/abs/2609.00111v1","pdf":"https://hexagonmath.org/pdf/2609.00111v1","source":"https://hexagonmath.org/src/2609.00111v1","source_tree":"https://hexagonmath.org/api/v1/works/2609.00111v1/source","api":"https://hexagonmath.org/api/v1/works/2609.00111v1"}},{"identifier":"2609.00110","title":"Frobenius orbit maps and rank-four group schemes","abstract":"We construct finite free group schemes of rank four and exponent eight as stabilizers in a smooth two-dimensional affine group. We first compute the complete local deformation ring of the invariant space of constants and squares in characteristic two, using six Grassmannian coordinates. For every Artinian specialization, evaluation at the identity has a finite free rank-four stabilizer. Its fourth-power morphism is $(x,y)\\mapsto(2bxy,2axy)$, and its eighth-power morphism is trivial. The fourth power is nonzero on the third-order neighbourhood of the universal deformation ring and on a specialization over $\\Z[a,b]/(a^2b-2,a^3,b^3)$. We give complete formulas for the latter group scheme. Right translation preserves the chosen quadratic space exactly when it contains the square of the scalar character. This identifies $(2a,2b)$ as the common obstruction to fourth-power vanishing, normality of the stabilizer, and two-sidedness in an equivalent cyclic-module construction.","subject":"math.AG","authorshipMode":"no-human-author-claimed","humanUnderstanding":"some","sponsorUserId":"user_12fd56c8-1454-4444-9120-366f4739638f","submittingOrganizationId":null,"sponsorName":"Akhil Mathew","orcid":"0000-0002-3899-2872","status":"published","createdAt":"2026-09-27 01:43:14","publishedAt":"2026-09-27T16:03:11.731Z","versionNumber":1,"versionCreatedAt":"2026-09-27 01:43:14","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":["Hopf algebras","group schemes"],"crossListSubjects":[],"authors":[{"id":"68220f8b-cd2f-4537-b9b3-e169e8b8bdd5","name":"Akhil Mathew","orcid":"0000-0002-3899-2872","affiliation":"University of Chicago","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["14L15","14L17"],"tools":[],"id":"hexagon:2609.00110v1","links":{"record":"https://hexagonmath.org/abs/2609.00110v1","pdf":"https://hexagonmath.org/pdf/2609.00110v1","source":"https://hexagonmath.org/src/2609.00110v1","source_tree":"https://hexagonmath.org/api/v1/works/2609.00110v1/source","api":"https://hexagonmath.org/api/v1/works/2609.00110v1"}},{"identifier":"2609.00102","title":"Inclusions between p-bounded crystalline loci in dimension two","abstract":"Let p be an odd prime and K/Qp a finite unramified extension of degree f > 1. Let Z(r) be the reduced special fiber of the Emerton-Gee stack of two-dimensional crystalline representations of Hodge type r of the absolute Galois group of K. We study the collection of stacks Z(r) as r varies over p-bounded Hodge types, as a set partially ordered under inclusion. We prove that aside from two degenerate cases, simple inclusions can be classified in terms of three operations on Hodge types, two of which have standard automorphic interpretations. We also prove, with one exception, that inclusions can be detected at the level of an inclusion of closed points (equivalently, semisimple mod p Galois representations).","subject":"math.NT","authorshipMode":"human","humanUnderstanding":"all","sponsorUserId":"reviewer_69ccf5b20910abddf6faa05d4520486b","submittingOrganizationId":null,"sponsorName":"David Savitt","orcid":"0000-0001-5761-1177","status":"published","createdAt":"2026-09-25 01:26:01","publishedAt":"2026-09-26T21:14:40.721Z","versionNumber":1,"versionCreatedAt":"2026-09-26 16:59:54","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":[],"crossListSubjects":[],"authors":[{"id":"fed61e41-4d77-4cf6-99ed-95e7861b8614","name":"Kalyani Kansal","orcid":"0000-0002-2279-8805","affiliation":"Imperial College London","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null},{"id":"2615c60c-88a2-427b-860a-24b54bfe422d","name":"Brandon Levin","orcid":"0009-0002-1369-4390","affiliation":"Rice University","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null},{"id":"2e1b8e12-bc30-437a-8107-e32eb229fa09","name":"David Savitt","orcid":"0000-0001-5761-1177","affiliation":"Johns Hopkins University","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["11F80"],"tools":[],"id":"hexagon:2609.00102v1","links":{"record":"https://hexagonmath.org/abs/2609.00102v1","pdf":"https://hexagonmath.org/pdf/2609.00102v1","source":"https://hexagonmath.org/src/2609.00102v1","source_tree":"https://hexagonmath.org/api/v1/works/2609.00102v1/source","api":"https://hexagonmath.org/api/v1/works/2609.00102v1"}},{"identifier":"2609.00105","title":"Graded absolute integral closures and Picard groups","abstract":"We deduce from Bhatt's graded Cohen--Macaulay and vanishing results that\nan integer-graded absolute integral closure of a weighted polynomial ring\nover $\\Z_p$, modulo $p$, is free with basis degrees smaller than the sum\nof the weights. A direct $p$-adic expansion turns this degree bound into\na uniform solution of the cusp relation modulo nilpotents. Consequently\nevery ring $R$ admits a faithfully flat algebra with seminormal reduction,\nand every invertible $R[T]$-module becomes free after a faithfully flat\nextension of $R$, answering a question of Drinfeld.","subject":"math.AC","authorshipMode":"no-human-author-claimed","humanUnderstanding":"all","sponsorUserId":"user_12fd56c8-1454-4444-9120-366f4739638f","submittingOrganizationId":null,"sponsorName":"Akhil Mathew","orcid":"0000-0002-3899-2872","status":"published","createdAt":"2026-09-26 19:10:04","publishedAt":"2026-09-26T21:13:06.954Z","versionNumber":1,"versionCreatedAt":"2026-09-26 19:10:04","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":["Picard groups","absolute integral closure","seminormality"],"crossListSubjects":["math.AG"],"authors":[{"id":"cfa98cf9-267d-4890-8889-1379bfb1ed22","name":"Akhil Mathew","orcid":"0000-0002-3899-2872","affiliation":"University of Chicago","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["13B22","13A02","13F45","13C10"],"tools":[],"id":"hexagon:2609.00105v1","links":{"record":"https://hexagonmath.org/abs/2609.00105v1","pdf":"https://hexagonmath.org/pdf/2609.00105v1","source":"https://hexagonmath.org/src/2609.00105v1","source_tree":"https://hexagonmath.org/api/v1/works/2609.00105v1/source","api":"https://hexagonmath.org/api/v1/works/2609.00105v1"}},{"identifier":"2609.00104","title":"Derived invariance of the signature and Hodge numbers of fourfolds","abstract":"A Fourier–Mukai equivalence between smooth proper complex varieties of common even dimension preserves the topological signature. The proof extracts the signature from the symmetrization of the Mukai pairing on even cohomology. Combining this observation with known derived invariants and the Hochschild–Kostant–Rosenberg decomposition shows that derived-equivalent smooth projective fourfolds over any field of characteristic zero have the same Hodge numbers. This argument and text was produced by ChatGPT 5.6 Sol.","subject":"math.AG","authorshipMode":"no-human-author-claimed","humanUnderstanding":"all","sponsorUserId":"user_76ce7f15-0dd8-48cb-813e-dd89b39e4f20","submittingOrganizationId":null,"sponsorName":"Benjamin Antieau","orcid":"0000-0002-5296-187X","status":"published","createdAt":"2026-09-26 15:41:14","publishedAt":"2026-09-26T16:31:39.731Z","versionNumber":1,"versionCreatedAt":"2026-09-26 16:20:09","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":["Derived categories","Fourier–Mukai equivalence","fourfolds"],"crossListSubjects":[],"authors":[{"id":"479ff4cd-db45-4d27-8257-d9ce4de9292c","name":"Benjamin Antieau","orcid":"0000-0002-5296-187X","affiliation":"Northwestern University","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null},{"id":"ec80e546-5efd-4f66-8969-19894dc7a85c","name":"Andrei Căldăraru","orcid":null,"affiliation":"University of Wisconsin Madison","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null},{"id":"f6f82820-17b1-4d9b-a649-a50aa2b422ac","name":"Ruoxi Li","orcid":null,"affiliation":"University of California Berkeley","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null},{"id":"325a43f3-fb74-4c80-be4e-74da48856f73","name":"Akhil Mathew","orcid":null,"affiliation":"University of Chicago","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null},{"id":"56380d28-6a74-41eb-a70a-d8556ec26d25","name":"Joshua Mundinger","orcid":null,"affiliation":"University of California Berkeley","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null},{"id":"51e99fa2-c524-46b1-b009-66c210e4ed9b","name":"Noah Olander","orcid":null,"affiliation":"University of California Berkeley","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null},{"id":"ce97cb3e-78d5-4f2b-a793-1173857e67a7","name":"Martin Olsson","orcid":null,"affiliation":"University of California Berkeley","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":["14F08"],"tools":[{"provider":"OpenAI","name":"ChatGPT 5.6 Sol","version":null}],"id":"hexagon:2609.00104v1","links":{"record":"https://hexagonmath.org/abs/2609.00104v1","pdf":"https://hexagonmath.org/pdf/2609.00104v1","source":"https://hexagonmath.org/src/2609.00104v1","source_tree":"https://hexagonmath.org/api/v1/works/2609.00104v1/source","api":"https://hexagonmath.org/api/v1/works/2609.00104v1"}},{"identifier":"2609.00001","title":"Strict negativity and non-vanishing of Chen--Larson hypergeometric coefficients","abstract":"Chen and Larson study tautological classes on the strata of holomorphic\nabelian differentials, where they predict a vanishing result.  Using known\ntautological relations on the moduli space of curves, they reduce this\nprediction to the non-vanishing of certain coefficients of a quotient of\nhypergeometric generating series, treating the residue classes\n$g\\equiv0,2\\pmod3$ and $g\\equiv1\\pmod3$ through two separate series; they verify\nthe non-vanishing by computer for small genus.\n\nWe prove it in general: in both cases the relevant coefficient is non-vanishing\nfor every genus and every stratum, and in the first case it has, more strongly,\na uniform strict sign.  Along the way we correct an error in the Chen--Larson\nderivation of the $g\\equiv1\\pmod3$ series, where a constant of the underlying\nrelation of Ionel had inadvertently been changed.\n\nThe paper falls into two parts.  Part~I gives the mathematical proofs; Part~II\ndocuments the Lean~4 and Mathlib formalization --- whose only non-standard trust\nassumption is the compiler invoked by \\code{native\\_decide} for the large finite\ncomputations --- together with the long-horizon, multi-model generative-AI\nprocess that produced the proofs, exposed false intermediate routes, and\nuncovered the error in the printed Proposition~5.2 relation noted above.","subject":"math.AG","authorshipMode":"no-human-author-claimed","humanUnderstanding":null,"sponsorUserId":"user_2665dece-6156-420c-91c6-33107bd20f11","submittingOrganizationId":null,"sponsorName":"Johannes Schmitt","orcid":"0000-0001-5774-3508","status":"published","createdAt":"2026-09-01 12:46:39","publishedAt":"2026-09-01T13:22:34.635Z","versionNumber":1,"versionCreatedAt":"2026-09-01 13:18:39","doi":null,"arxivId":null,"license":"CC BY 4.0","tags":["power series","tautological classes"],"crossListSubjects":["cs.AI","math.AC"],"authors":[{"id":"f045ae6b-6e94-48fe-b79e-5d56fc5646cf","name":"Johannes Schmitt","orcid":"0000-0001-5774-3508","affiliation":"ETH Zurich","claimed":0,"contributorKind":"person","organizationId":null,"organizationSlug":null}],"mscCodes":[],"tools":[{"provider":"OpenAI","name":"GPT-5.5 Pro","version":null},{"provider":"OpenAI","name":"codex CLI","version":null},{"provider":"Anthropic","name":"Claude Opus 4.8","version":null},{"provider":"Anthropic","name":"Claude Fable 5","version":null}],"id":"hexagon:2609.00001v1","links":{"record":"https://hexagonmath.org/abs/2609.00001v1","pdf":"https://hexagonmath.org/pdf/2609.00001v1","source":"https://hexagonmath.org/src/2609.00001v1","source_tree":"https://hexagonmath.org/api/v1/works/2609.00001v1/source","api":"https://hexagonmath.org/api/v1/works/2609.00001v1"}}],"page":{"limit":100,"next_cursor":null,"has_more":false,"total":80}}