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57 works
hexagon:2610.00021v1

math.MG — Metric Geometry

A Fourier Consequence for Lattice Kissing Numbers and Average Contacts

Contributed by Scott Kominers

We deduce a new asymptotic upper bound Klat(n)≤(2e/π+o(1))nK_{\mathrm{lat}}(n)\leq\left(\sqrt{2e/\pi}+o(1)\right)^n on the maximal lattice kissing number Klat(n)K_{\mathrm{lat}}(n) in dimension nn, 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-22 exponential rate rounds to 0.39560.3956, matching the value extrapolated empirically by Afkhami-Jeddi, Cohn, Hartman, de Laat, and Tajdini in 2020.

A mix of human-written and AI-generated textHuman understanding: all partsContact numbersDiscrete geometryEuclidean latticesFourier linear programming boundsGeometry of numbersKissing numbersSphere packing
Oct 02, 2026PDF ↗View work →
hexagon:2610.00017v1

math.NT — Number Theory

On the ternary pentagonal numbers conjecture

Contributed by Glenn Bruda

Communicated by Guy in 1994, the ternary pentagonal numbers conjecture of Blecksmith and Selfridge asserts that every integer larger than 3306633066 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 nn with v3(24n+3)≤8v_3(24n+3)\leq8, and present an explicit lift to handle the nn with v3(24n+3)≥9v_3(24n+3)\geq9.

Primarily human-written textHuman understanding: all partsPentagonal numbersalmost universalityternary quadratic forms
Oct 02, 2026PDF ↗View work →
hexagon:2610.00018v1

math.NT — Number Theory

Linnik's constant is at most 3.993.99

Contributed by Eric Naslund

Let P(a,q)P(a,q) denote the least prime congruent to aa modulo qq, where (a,q)=1(a,q)=1. We give a computer-assisted proof that P(a,q)≪q3.99P(a,q)\ll q^{3.99}, improving the exponent 55 of Xylouris. More precisely, P(a,q)<q3.99P(a,q)<q^{3.99} for every sufficiently large qq, uniformly in aa. The main new ingredient is a graded near-density estimate for zeros of Dirichlet LL-functions, a weighted form of Heath-Brown's Lemma~12.1 of the kind he asked for in 1992. We combine it with the zero-location estimates and far-density method of Heath-Brown and Xylouris. The second novelty is the scale of the case analysis. Heath-Brown and Xylouris closed their final case analyses with 1414 and 2121 main cases, each by a chain of inequalities evaluated in floating point. 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 44534453 root cases and 47884788 terminal cases, each closed by its own linear program. On 4,196,8794{,}196{,}879 threshold boxes these programs give 29,397,33629{,}397{,}336 linear relaxations, and exact integer certificates for all of them bound the normalized zero sum strictly below~11. Exceptional zeros and the remaining exterior range are treated separately. This case analysis is a kind of systematic brute force enabled by AI: an analysis of this size, with parameters tuned to each case, would be very laborious or nearly impossible to carry out by hand, and it lets the new estimate be applied separately in each case. The near-density lemma, certificate soundness, and a conditional passage from a certified case to a prime are formalized in Lean, assuming published analytic inputs and specified facts about zeros (PALOMAR-2026-10-01-000020 v1).

Primarily AI-generated textHuman understanding: some parts
Oct 02, 2026PDF ↗View work →
hexagon:2610.00016v1

math.CO — Combinatorics

An Exponent of 1.04273 for the Unit Distance Problem

Contributed by Eric Naslund

Let u(U)u(U) count the unordered pairs at distance one in a finite planar set UU. We construct finite sets UjU_j with ∣Uj∣→∞|U_j|\to\infty and u(Uj)/∣Uj∣1.04273→∞u(U_j)/|U_j|^{1.04273}\to\infty. The largest exponent previously claimed, 1.03581.0358, is in the author's unpublished manuscript. The method is the number-field construction of OpenAI and Sawin: unit distances come from elements of relative norm one in quadratic extensions, and the fields come from an infinite pro-22 class field tower. The new ingredients are quadratic extensions of mixed signature, with an exact average over their norm-one units, and a tower over the real quadratic field $\Q(\sqrt{241})$, in which 22, 33 and 55 split. Its Golod--Shafarevich function contains two copies of the local conditions at these primes but only one constant term, and the extra room lets the tower be ramified only above 22, 33 and 55; the root discriminant of its fields is about 286286. The relative zeta value is bounded through the zeta function of the degree-512512 field generated over $\Q(\sqrt{241})$ by the square roots of its {2,3,5}\{2,3,5\}-units, which is the product of the Dedekind zeta function of $\Q(\sqrt{241})$ and 255255 quadratic Hecke LL-functions. Finite facts and numerical inequalities are certified by exact computation and interval arithmetic. A significant portion of this work was verified in Lean, reducing the result with exponent 1.04271.0427 to an explicit zeta function inequality (Palomar registry, PALOMAR-2026-10-01-000018, version 1).

Primarily AI-generated textHuman understanding: some parts
Oct 02, 2026PDF ↗View work →
hexagon:2610.00015v1

math.CO — Combinatorics

Tilings of an equilateral triangle by at most five lattice trapezoids with 60° base angles: a complete structural classification

Contributed by Gonzalo Barria

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 for 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 reproduces the sequence for every n ≤ 127.

A mix of human-written and AI-generated textHuman understanding: all partsOEIS A391498lattice trapezoidsquasi-polynomialrational generating functiontilings of an equilateral triangle
Oct 01, 2026PDF ↗View work →
hexagon:2610.00014v1

math.PR — Probability

On the escape rate of favorite sites of planar random walks

Contributed by Heng Ma

For planar simple random walk, the favorite sites at time nn are the sites whose local time at time nn is maximal. We prove that, almost surely, for every γ>1/2\gamma>1/2 and every c>0c>0, all favorite sites lie outside the ball centered at the origin with radius cn/(log⁡n)γc\sqrt n/(\log n)^\gamma for all sufficiently large nn. At the critical exponent γ=1/2\gamma=1/2, almost surely, for every c>0c>0, the entire favorite sites lies within distance cn/log⁡nc\sqrt{n/\log n} of the origin infinitely often.

Primarily AI-generated textHuman understanding: no parts
Oct 01, 2026PDF ↗View work →
hexagon:2610.00011v1

math.MG — Metric Geometry

A generalized Gerver sofa for angled corridors

Contributed by Henrik Schou Guttesen

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…2.2195\dots, which was only recently proven to be optimal by J.~Back. Here, I consider the moving sofa problem in an angled corridor, CφC_{\varphi}, of unit width for turn angles 0<φ≤π20 < \varphi \le \frac{\pi}{2}. I first generalize Hammersley's arguments to devise simple analytical lower and upper area bounds for all 0<φ≤π20 < \varphi \le \frac{\pi}{2}. ChatGPT-6 Astra (Codex) is prompted to derive a generalized Gerver sofa parametrized by φ\varphi. The generalized Gerver sofa area is non-analytic at the unique root φ∗∈(0,π/2)\varphi_*\in(0,\pi/2) of exp⁡(φ∗cot⁡φ∗)=1+2cos⁡φ∗−cos⁡2φ∗\exp({\varphi_*\cot\varphi_*}) = 1+2\cos\varphi_*-\cos^2\varphi_*. For turn angles smaller than φ∗\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<φ≤π20 < \varphi \le \frac{\pi}{2}.

A mix of human-written and AI-generated textHuman understanding: some partsGeometryGerver-sofamoving-sofa-problemrecreational-math
Oct 01, 2026PDF ↗View work →
hexagon:2610.00010v1

math.DG — Differential Geometry

A contractible open four-manifold with a complete metric of uniformly positive scalar curvature

Contributed by Jiangcheng You, Heng Zhang

We construct a smooth contractible open four-manifold that is not homeomorphic to R4\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.

A mix of human-written and AI-generated textHuman understanding: all partsBaumslag–Solitar groupsGromov–Lawson surgerycontractible four-manifoldsgeometric topologyopen four-manifoldspositive scalar curvaturetopology at infinity
Oct 01, 2026PDF ↗View work →
hexagon:2610.00003v1

math.LO — Logic

Arrow and Gibbard Satterthwaite Theorems in Lean

Contributed by Arthur, David Barros Hulak, Ruy J. G. B. de Queiroz

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.

Primarily AI-generated textHuman understanding: some parts
Oct 01, 2026PDF ↗View work →
hexagon:2610.00004v1

math.LO — Logic

A Lean Formalization of the Shapley Value Characterization for Finite Cooperative Games

Contributed by Arthur, David Barros Hulak, Ruy J. G. B. de Queiroz

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.

Primarily AI-generated textHuman understanding: some parts
Oct 01, 2026PDF ↗View work →
hexagon:2610.00005v1

math.LO — Logic

Nash Bargaining Characterization in Lean

Contributed by Arthur, David Barros Hulak, Ruy J. G. B. de Queiroz

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.

Primarily AI-generated textHuman understanding: some parts
Oct 01, 2026PDF ↗View work →
hexagon:2610.00008v1

math.LO — Logic

Finite Nash Equilibria and Dependent Mixed Strategies in Lean

Contributed by Arthur, David Barros Hulak, Ruy J. G. B. de Queiroz

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.

Primarily AI-generated textHuman understanding: some partsBrouwer fixed pointLean 4MathlibNash equilibriumdependent action typesmixed strategiespotential games
Oct 01, 2026PDF ↗View work →
hexagon:2610.00002v1

math.OC — Optimization and Control

A Finite Coordinate Reduction for Approachability Loss in Lean

Contributed by Arthur, David Barros Hulak, Ruy J. G. B. de Queiroz

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.

Primarily AI-generated textHuman understanding: some parts
Oct 01, 2026PDF ↗View work →
hexagon:2610.00006v1

math.GR — Group Theory

An Executable Stallings Recognizer for Finite Generating Lists in Lean

Contributed by Arthur, David Barros Hulak, Ruy J. G. B. de Queiroz

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.

Primarily AI-generated textHuman understanding: some partsLean 4MathlibStallings foldingfinite inverse automatafree groupssubgroup membership
Oct 01, 2026PDF ↗View work →
hexagon:2610.00007v1

math.AC — Commutative Algebra

Prime-Generated Localization Descent for Unique Factorization in Lean

Contributed by Arthur, David Barros Hulak, Ruy J. G. B. de Queiroz, Anjolina Grisi de Oliveira

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.

Primarily AI-generated textHuman understanding: some partsLean 4MathlibNagata factorialitycommutative algebralocalizationunique factorization domains
Oct 01, 2026PDF ↗View work →
hexagon:2610.00001v1

cs.LO — Logic in Computer Science

A Lean Formalization of Myerson-Satterthwaite Impossibility for Continuous Bilateral Trade

Contributed by Arthur, David Barros Hulak, Ruy J. G. B. de Queiroz

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.

Primarily AI-generated textHuman understanding: some parts
Oct 01, 2026PDF ↗View work →
hexagon:2610.00009v1

cs.DS — Data Structures and Algorithms

A Lean Certificate for the Single Source Unsplittable Flow Cost Counterexample

Contributed by Arthur, David Barros Hulak, Ruy J. G. B. de Queiroz

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.

Primarily AI-generated textHuman understanding: some partsGoemans cost conjectureLean 4MathlibRybin counterexamplefinite counterexampleformal verificationsingle-source unsplittable flow
Oct 01, 2026PDF ↗View work →
hexagon:2609.00152v1

math.AT — Algebraic Topology

A Lean Formalization of the Classical Seifert van Kampen Groupoid Theorem

Contributed by Arthur, David Barros Hulak, Ruy J. G. B. de Queiroz

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.

Primarily AI-generated textHuman understanding: some partsLean 4MathlibSeifert van Kampen theoremcategorical pushoutfundamental groupoidopen cover
Sep 30, 2026PDF ↗View work →
hexagon:2609.00149v1

cs.DS — Data Structures and Algorithms

Deferred Acceptance in Lean for Finite Strict Preference Markets

Contributed by Arthur, David Barros Hulak, Ruy J. G. B. de Queiroz

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.

Primarily AI-generated textHuman understanding: some partsGale-ShapleyLean 4deferred acceptanceformalizationstable matching
Sep 30, 2026PDF ↗View work →
hexagon:2609.00151v1

math.OC — Optimization and Control

A Lean Formalization of Roberts Theorem on Unrestricted Valuations

Contributed by Arthur, David Barros Hulak, Ruy J. G. B. de Queiroz

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.

Primarily AI-generated textHuman understanding: some partsLean 4Roberts theoremaffine maximizersdominant strategiesformalizationmechanism design
Sep 30, 2026PDF ↗View work →
hexagon:2609.00150v1

math.OC — Optimization and Control

Finite VCG and Groves Payments in Lean

Contributed by Arthur, David Barros Hulak, Ruy J. G. B. de Queiroz

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.

Primarily AI-generated textHuman understanding: some partsClarke pivot paymentsGroves characterizationLean 4VCG mechanismdominant strategiesformal verification
Sep 30, 2026PDF ↗View work →
hexagon:2609.00148v1

math.OC — Optimization and Control

A Lean Formalization of Finite Zero Sum Minimax via Sion's Theorem

Contributed by Arthur, David Barros Hulak, Ruy J. G. B. de Queiroz

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.

Primarily AI-generated textHuman understanding: some partsLeanSion's theoremformalizationminimax theoremzero-sum games
Sep 30, 2026PDF ↗View work →
hexagon:2609.00147v1

math.AG — Algebraic Geometry

Signed Mukai diagonals and derived equivalences of fivefolds

Contributed by Benjamin Antieau

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 h0,3h^{0,3} and h1,4h^{1,4} and reduce the unrestricted Hodge-number problem to two numerical parameters, one of which is the possible failure of invariance of h0,2h^{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.

Primarily AI-generated textHuman understanding: no partsDerived categoriesfivefolds
Sep 30, 2026PDF ↗View work →
hexagon:2609.00144v1

math.PR — Probability

On the Asymptotic W1W_1 Cost of Two-Dimensional Semi-Discrete Matching

Contributed by Heng Ma

Let X1,X2,…X_1,X_2,\ldots be independent uniform points in the unit square and let λ\lambda denote Lebesgue probability measure. We prove that the normalized expected W1W_1 distance between the empirical measure and Lebesgue measure converges to a positive constant: \[ \lim_{N\to\infty} \sqrt{\frac{N}{\log N}}\, \mathbb E W_1\!\left(\frac1N\sum_{i=1}^N\delta_{X_i},\lambda\right) =c_\star. \] The constant is characterized by the convex viscosity equation \[ \partial_tU=\frac1{4\pi}\sqrt{\det D^2U}, \qquad U(0,q)=|q|. \] Its solution UU is unique under certain growth and regularity conditions, and the limiting constant c⋆=U(1,0)/2c_\star=U(1,0)/\sqrt2.

Primarily AI-generated textHuman understanding: no parts
Sep 30, 2026PDF ↗View work →
hexagon:2609.00146v1

math.MG — Metric Geometry

FINITE HAUSDORFF HYPERSPACES AND GROMOV–HAUSDORFF GEOMETRY

Contributed by Yoshito Ishiki

For a nonempty finite metric space X, let Exp(X) be the space of its nonempty subsets equipped with the Hausdorff metric. We prove that the isometry type of Exp(X) determines X when X is strongly rigid, meaning that distinct unordered pairs of distinct points have dis- tinct distances. We also prove that the hyperspace operation preserves the ordinary Gromov–Hausdorff distance locally under explicit separa- tion conditions on the distance spectra. For finite ultrametric spaces, the hyperspace operation is an isometric embedding with respect to the Gromov–Hausdorff ultrametric. In contrast, we construct finite ultra- metric spaces for which the ordinary Gromov–Hausdorff distance strictly decreases under the hyperspace operation. Examples of equal cardinality exist in every cardinality at least six and realize every ratio in [1/2, 1). Their distances remain constant after the first hyperspace iteration. An explicit perturbation also yields strict contraction for finite strongly rigid metric spaces with all triangle inequalities strict.

Primarily AI-generated textHuman understanding: some partsGromov--Hausdorff spaceHyper space
Sep 30, 2026PDF ↗View work →
hexagon:2609.00141v1

math.LO — Logic

Ordinal-definable families in Cohen, random, and collapse extensions

Contributed by Elliot Glazer

We study ordinal-definable families of sets of arbitrary rank in Cohen, random, and collapse extensions. Over LL, countable OD families have OD enumerations after adding one Cohen or one random real. A single generalized Cohen subset gives the corresponding sharp theorem at every regular uncountable cardinal. When κ\kappa is singular strong limit and has uncountable cofinality, adding κ\kappa Cohen reals makes every member of a short-parameter definable family of size at most κ\kappa definable from a short parameter. Adding κ\kappa random reals gives a single short-parameter definable enumeration for every such family of size strictly below κ\kappa. The random bound is sharp. The proofs use coordinate amalgamation and small-index arguments. We also obtain arbitrary-rank descent for countable families after collapsing any infinite cardinal, with applications to choice in the relative Feferman--Levy model. In the full Solovay collapse extension of an arbitrary ZFC ground, every family definable from a real and ordinals that represents fewer than 2c2^{\mathfrak c} classes modulo null sets consists of measurable sets; the corresponding category assertion also holds. In the extensions of LL by ω1\omega_1 Cohen or random reals, no model of IΔ0I\Delta_0 of size at most ℵ1\aleph_1, definable from ordinals and a real, has full binary-coded standard system. Π11-CA0\Pi^1_1\text{-}\mathsf{CA}_0 proves a finite-fragment Borel obstruction to a full binary standard system for IΔ0I\Delta_0. For regular uncountable κ\kappa, the generalized Cohen extension by $\Add(\kappa,\Lambda)^L$, Λ>κ\Lambda>\kappa, has no ambiently κ+\kappa^+-saturated IΔ0I\Delta_0 model in HOD⁡Hκ+\operatorname{HOD}_{H_{\kappa^+}}, regardless of its size. Consequently every infinite regular κ\kappa of LL admits a cofinality-preserving GCH extension with no ODP(κ)\mathrm{OD}_{\mathcal P(\kappa)} saturated arithmetic presentation of size κ+\kappa^+. ZFC also proves that every singular strong limit κ\kappa admits an OD κ+\kappa^+-saturated elementary extension of N\mathbb N of size 2κ2^\kappa, hence of size κ+\kappa^+ under GCH.

Primarily AI-generated textHuman understanding: some partsBoolean algebrasSet theorydefinability
Sep 30, 2026PDF ↗View work →
hexagon:2609.00142v1

math.CO — Combinatorics

Collapsibility of Alexander duals for shade maps

Contributed by Darij Grinberg

Let EE be a nonempty finite set. A shade map on EE is a map T:P(E)→P(E)T:\mathcal{P}(E)\rightarrow\mathcal{P}(E) such that toggling an element u∉T(F)u\notin T(F) in the input FF does not change T(F)T(F). We prove that, if TT is an inclusion-reversing shade map and G⊆EG\subseteq E, then the simplicial complex \[ \{F\subseteq E\mid G\subseteq T(F)\} \] is collapsible. Equivalently, if SS is an inclusion-preserving shade map, then the Alexander dual of \[ \{F\subseteq E\mid G\not \subseteq S(F)\} \] is collapsible. This is an Alexander-dual companion to the shade-map collapsibility theorem in \emph{The Elser nuclei sum revisited}. The proof first matches all faces on which T(F)≠ET(F)\neq E by a toggle. The remaining faces, characterized by T(F)=ET(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.

A mix of human-written and AI-generated textHuman understanding: all partsantimatroidsconvex geometriesdiscrete Morse theorygraphssimplicial complexes
Sep 30, 2026PDF ↗View work →
hexagon:2609.00143v1

math.QA — Quantum Algebra

Strongly positive bases in rank-two quantum cluster algebras

Contributed by 琪越 唐

We prove Conjecture 12 of Lee, Li, Rupel, and Zelevinsky: for arbitrary positive integers b,cb,c, every strongly positive basis of the coefficient-free rank-two quantum cluster algebra Av(b,c)\mathcal A_v(b,c) contains a unique element pointed at each pair in Z2\mathbb Z^2. These pointed elements exhaust the basis.

A mix of human-written and AI-generated textHuman understanding: some partsquantum cluster algebra
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hexagon:2609.00140v1

math.CO — Combinatorics

Coefficientwise positivity for alternating sums of qq-factorials

Contributed by David Anderson

We prove that the alternating sum ∑i=0k(−1)i(ki)[m−i]!q\sum_{i=0}^{k}(-1)^i\binom ki[m-i]!_q has nonnegative coefficients whenever m≥2k−1m\ge 2k-1. For k≥3k\ge3, this range is exact. The case m=2km=2k proves a conjecture of Lewis and Morales arising from the enumeration of invertible matrices with prescribed zero entries. After removing a common qq-factorial, we express the sum in a Gaussian-binomial basis whose coefficients are independent of the ambient size. Their positivity follows from a coefficientwise domination argument using convexity of ordinary binomial coefficients.

Primarily AI-generated textHuman understanding: all parts
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hexagon:2609.00138v1

math.CO — Combinatorics

Signed seeds and G-gradings on cluster algebras

Contributed by Lauren Williams, Alan Yan

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 5.2] (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.

Primarily human-written textHuman understanding: all partsGrassmanniansamplituhedroncluster algebras
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hexagon:2609.00128v1

math.AG — Algebraic Geometry

On properties of Plancherel algebras

Contributed by Shurui Liu

We proved the predictions by Ben-Zvi, Sakellaridis, and Venkatesh that the Plancherel algebra PLX\mathrm{PL}_{X} is commutative and its loop-rotated version PLX,ℏ\mathrm{PL}_{X,\hbar} is flat over k[ℏ]k[\hbar].

Primarily AI-generated textHuman understanding: some partsPlancherel algebraRelative Langlandsgeometric Langlands
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hexagon:2609.00137v1

math.CO — Combinatorics

On a non-basis of the coinvariant algebra

Contributed by Darij Grinberg

A conjecture arising from a question of Procesi proposes a basis of the coinvariant algebra of SnS_n consisting of column-antisymmetrized monomials indexed by pairs of standard Young tableaux. We show that the proposed family need not even span an SnS_n-subrepresentation: for n=8n=8, a generator of degree 1515 is sent outside the span by the adjacent transposition (4,5)(4,5). The proof is an exact finite computation with an explicit separating functional, requiring neither a rank computation nor Gr\"obner reduction. We also retain an explicit relation for n=7n=7, where the basis assertion first fails, although the span is still invariant.

A mix of human-written and AI-generated textHuman understanding: all partsSpecht modulesYoung tableauxcoinvariant algebrahigher Specht polynomialsrepresentations of the symmetric group
Sep 29, 2026PDF ↗View work →
hexagon:2609.00132v1

math.NT — Number Theory

Primitive Integer Solutions of x^4 + 3y^3 + 2z^3 = 0

Contributed by Avraham Eisenberg

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.

Primarily AI-generated textHuman understanding: no parts
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hexagon:2609.00134v1

math.LO — Logic

Five equations for first-order logic, modulo the structure of existential graphs

Contributed by Anthony Hart

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.

Primarily AI-generated textHuman understanding: all partscylindric algebraexistential graphsfirst-order logic
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hexagon:2609.00133v1

math.RA — Rings and Algebras

A Universal Noncommutative Splitting Algebra for Polynomials

Contributed by Darij Grinberg

Let RR be a commutative ring. We show that every homogeneous polynomial f∈R[x1,…,xm]f\in R[x_{1},\ldots,x_{m}] of degree nn splits into a product of nn homogeneous linear forms over a suitable noncommutative ring extension SS of RR (that is, over a noncommutative RR-algebra SS whose structure morphism R→SR\rightarrow S is injective). The algebra SS is universal for such factorizations and is free as an RR-module. The proof uses Bergman's Diamond Lemma: we define SS by generators and relations, and the relations form a terminating reduction system that has no ambiguities and thus is confluent. An analogous result is also shown for inhomogeneous polynomials (with inhomogeneous factors). This easily follows from the homogeneous case by homogenizing and then setting the homogenizing variable equal to 11.

A mix of human-written and AI-generated textHuman understanding: all partsBergman's diamond lemmaGröbner basescombinatorial algebranoncommutative polynomials
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hexagon:2609.00135v1

math.CO — Combinatorics

A counterexample to the Burman--Kulishov conjecture on Lie elements

Contributed by Darij Grinberg

Burman and Kulishov defined Lie elements in the group algebra k[Sn]\mathbf k[S_n] by comparing, on every exterior power of the reflection representation VV, the usual action of k[Sn]\mathbf k[S_n] with the infinitesimal action induced by its action on VV. They conjectured that the Lie algebra Ln\mathcal L_n of all Lie elements is generated by the Kirchhoff differences 1−(ij)1-(ij). We disprove this conjecture for n=4n=4 by exhibiting an explicit counterexample arising from the (2,2)(2,2)-block of k[S4]\mathbf k[S_4]. More generally, we describe Ln\mathcal L_n in terms of the Artin--Wedderburn decomposition of k[Sn]\mathbf k[S_n]: its hook blocks are determined by the action on VV, whereas its non-hook blocks are arbitrary. Consequently, the primitive central idempotents of the non-hook blocks yield linearly independent obstructions to the conjecture. We also identify the Lie algebra generated by the Kirchhoff differences in terms of the derived algebra of the Lie algebra generated by transpositions. Along the way, we give an integral, and hence characteristic-free, proof that the exterior powers of VV are the hook-shaped Specht modules.

A mix of human-written and AI-generated textHuman understanding: all partsLie algebrasymmetric group algebrasymmetric group representations
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hexagon:2609.00130v1

math.NT — Number Theory

Primitive Integer Solutions of x^4 + 4y^3 + z^3 = 0

Contributed by Avraham Eisenberg

We determine all primitive integer solutions of the generalized Fermat equation x^4 + 4y^3 + z^3 = 0. Here primitive means gcd(x,y,z)=1. We prove that the only primitive solutions are (x,y,z) = (1,0,-1), (-1,0,-1). The 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.

Primarily AI-generated textHuman understanding: no parts
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hexagon:2609.00126v1

quant-ph — Quantum Physics

Pauli problem in dimension 4

Contributed by Dmitry Grinko

The finite-dimensional Pauli problem asks how many measurements in orthonormal bases determine every pure state of a dd-level quantum system up to a global phase. Four bases always suffice, and the answer is known to be three for d=2d=2 and four for d=3d=3 and d≥5d\geq5. Dimension four remained open because the embedding argument used in other dimensions fails there: the pure-state space CP3\mathbb{CP}^3 embeds in R9\R^9, the space of the nine independent probabilities of three bases. We show that three bases do not suffice in \(\C^4\), so exactly four are needed. The same argument shows that no ten vectors in \(\C^4\) do phase retrieval. Since 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. All three lower bounds follow from one statement: every six-dimensional real space of traceless Hermitian 4×44\times4 matrices with a common isotropic vector, that is, a nonzero ee with e∗Te=0e^*Te=0 for all TT in the space, contains a nonzero matrix of rank at most two. We prove it with complex KK-theory: otherwise, an odd unitary map on the five-sphere would have a K1K^1-class that is nonzero by antipodal symmetry but vanishes because of the isotropic vector.

Primarily AI-generated textHuman understanding: some parts
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hexagon:2609.00123v1

math.MG — Metric Geometry

Breaking the Fourth Wall: The Planar Centroid Banach–Mazur Diameter Is Strictly Less than 44

Contributed by Scott Kominers

We prove that the centroid Banach–Mazur diameter of planar convex bodies is strictly less than 44, (slightly) improving the prior upper bound 69/17≈4.0588269/17\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 44 would consist of two triangles, which are linearly equivalent and hence must in fact be at distance 11. The resulting gap below 44 is uniform but nonquantitative.

A mix of human-written and AI-generated textHuman understanding: all partsBanach–Mazur diameter
Sep 28, 2026PDF ↗View work →
hexagon:2609.00121v1

math.AC — Commutative Algebra

Polynomials over Symmetric Polynomials

Contributed by Darij Grinberg

Let k\mathbf{k} be a commutative ring, and let the symmetric group Sn\mathfrak{S}_{n} act on $P=\mathbf{k}\left[ x_{1} ,x_{2},\ldots,x_{n} \right] $ by permuting the variables. We prove four results that are known (at least in the case when k\mathbf{k} is a field) but not easily found in the literature. First, the coinvariant algebra (the quotient of PP by the ideal generated by the symmetric polynomials with constant term 00) is a free k\mathbf{k}-module of rank n!n!, with the residue classes of the Artin monomials as a basis. Second, PP is a free module of rank n!n! over the ring PSnP^{\mathfrak{S}_{n}} of symmetric polynomials, again with the Artin monomials as a basis. Third, if n!n! is invertible in k\mathbf{k}, the coinvariant algebra is the regular $\mathbf{k}\left[ \mathfrak{S}_{n} \right] $-module. Fourth, under the same hypothesis, PP is a free left PSn[Sn]P^{\mathfrak{S}_{n}}\left[ \mathfrak{S}_{n} \right] -module of rank 11. The first result follows from an elementary normal-form lemma for monic polynomials with pairwise relatively prime leading monomials. The second is proved by lifting the Artin basis. For the third, we use orbit harmonics with a strongly discrete point orbit, and the fourth follows by equivariantly lifting a regular basis of the coinvariant algebra.

A mix of human-written and AI-generated textHuman understanding: all partsArtin basisGröbner basescoinvariant algebraorbit harmonicssymmetric polynomials
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hexagon:2609.00122v1

cs.CC — Computational Complexity

Generalized Bawo: A Finite-State Model and an EXPTIME Upper Bound

Contributed by Isaac kaimfa

Four-row mancala games, including Malawian Bawo and the related East African Bao, have received limited attention in computational complexity theory despite their combinatorial richness. This paper introduces GENERALIZED-BAWO-WIN, a finite-state two-player game model inspired by Malawian Bawo. The model is parameterized by a half-board width n and a binary-encoded lap cap L, and incorporates relay sowing, marker-pit capture, a formalized capture-relay rule (abstracting kichwa), mandatory capture, a singleton-start prohibition, and an explicit lap-cap truncation rule that guarantees move termination. We prove that GENERALIZED-BAWO-WIN belongs to EXPTIME under these semantics. The proof proceeds by establishing that the game's state space has size at most 2poly(|I|), that legal successors can be generated in 2poly(|I|) time, and that a WIN/LOSS/DRAW retrograde attractor can be computed over the resulting finite game graph. We emphasize that this is an upper bound: no hardness result is established. We further emphasize that the theorem applies only to the formally defined generalized model and does not classify Bao as played in East Africa, whose rules differ materially.

A mix of human-written and AI-generated textHuman understanding: all partsEXPTIMEcombinatorial gamescomputational complexityfinite gamesgeneralized Bawostate-space complexity
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hexagon:2609.00118v1

math.CO — Combinatorics

A counterexample to ordering-independence for chromatic operators

Contributed by Darij Grinberg

We give a 1010-vertex counterexample to a conjecture of Pawlowski asserting that the characteristic polynomial of a chromatic operator associated with a forest is independent of the ordering of its edges. The two operators already have different traces of fifth powers.

Primarily AI-generated textHuman understanding: all partsGraph coloringSpecht modulessymmetric group algebra
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hexagon:2609.00117v1

math.AC — Commutative Algebra

Splitting a Polynomial into Linear Factors after an Injective Ring Extension

Contributed by Darij Grinberg

We show that each univariate polynomial P=p0+p1X+⋯+pmXm∈R[X]P = p_0 + p_1 X + \cdots + p_m X^m \in R[X] over a commutative ring RR can be factored into linear factors over a suitable commutative ring extension SS of RR. The proof proceeds by universal construction: SS is defined as the tensor product R⊗CmBmR \otimes_{C_m} B_m, where BmB_m is the polynomial ring $\ZZ[a_1, b_1, a_2, b_2, \ldots, a_m, b_m]$, and where CmC_m is its subring generated by its ``homogenized elementary symmetric polynomials'' Er=∑I⊆[m];∣I∣=r∏i∈Iai∏i∉IbiE_r=\sum_{\substack{I\subseteq [m];\\ |I|=r}} \prod_{i\in I}a_i\prod_{i\notin I}b_i for all 0≤r≤m0 \leq r \leq m. The injectivity of the structure homomorphism R→SR \to S is deduced from a combinatorial study of the diagonal subring of BmB_m. In the process, a homogeneous variant of the Garsia--Stanton basis is constructed, and some classical properties of symmetric polynomials are recovered.

A mix of human-written and AI-generated textHuman understanding: all parts
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hexagon:2609.00116v1

math.CO — Combinatorics

A Counterexample to a Conjecture on Fused Specht Polynomials

Contributed by Darij Grinberg

Lafay, Peltola and Roussillon conjectured that their realization of simple modules of the fused Hecke algebra by fused Specht polynomials extends from Young diagrams with two columns to Young diagrams of arbitrary shape. We give a counterexample for \[ n=8,\qquad \lambda=(3,2,2,1),\qquad \varsigma=(2,2,2,2). \] More precisely, we exhibit an explicit linear dependence among three fused Specht polynomials indexed by row-strict Young tableaux. The same example also yields an infinite family of counterexamples.

Primarily AI-generated textHuman understanding: some partsSpecht modulesSpecht polynomialsYoung tableaux
Sep 28, 2026PDF ↗View work →
hexagon:2609.00114v4

math.LO — Logic

Under Measurables, HOD_{Ord^{\omega}} is locally arbitrary

Contributed by Elliot Glazer

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: (1) ZFC + CM + [HOD_{Ord^\omega} \models (\exists \kappa V_{\kappa} \models \sigma)]; (2) ZFC + GCH + CM + [HOD_{Ord^\omega} \models (SVC \wedge CM \wedge \exists \kappa V_{\kappa} \models \sigma)]; (3) ZF + DC + CM + \exists \kappa V_{\kappa} \models \sigma. Thus, under measurables we have that HOD_{Ord^{\omega}} is "locally arbitrary."

Primarily AI-generated textHuman understanding: some partsSet theoryaxiom of choice
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