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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 $S_n$ consisting of column-antisymmetrized monomials indexed by pairs of standard Young tableaux. We show that the proposed family need not even span an $S_n$ -subrepresentation: for $n=8$ , a generator of degree $15$ is sent outside the span by the adjacent transposition $(4,5)$ . 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 $n=7$ , 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

Sep 29, 2026PDF ↗View work →

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

Sep 29, 2026PDF ↗View work →

hexagon:2609.00133v1

math.RA — Rings and Algebras

A Universal Noncommutative Splitting Algebra for Polynomials

Contributed by Darij Grinberg

Let $R$ be a commutative ring. We show that every homogeneous polynomial $f\in R[x_{1},\ldots,x_{m}]$ of degree $n$ splits into a product of $n$ homogeneous linear forms over a suitable noncommutative ring extension $S$ of $R$ (that is, over a noncommutative $R$ -algebra $S$ whose structure morphism $R\rightarrow S$ is injective). The algebra $S$ is universal for such factorizations and is free as an $R$ -module. The proof uses Bergman's Diamond Lemma: we define $S$ 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 $1$ .

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

Sep 29, 2026PDF ↗View work →

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 $\mathbf k[S_n]$ by comparing, on every exterior power of the reflection representation $V$ , the usual action of $\mathbf k[S_n]$ with the infinitesimal action induced by its action on $V$ . They conjectured that the Lie algebra $\mathcal L_n$ of all Lie elements is generated by the Kirchhoff differences $1-(ij)$ . We disprove this conjecture for $n=4$ by exhibiting an explicit counterexample arising from the $(2,2)$ -block of $\mathbf k[S_4]$ . More generally, we describe $\mathcal L_n$ in terms of the Artin--Wedderburn decomposition of $\mathbf k[S_n]$ : its hook blocks are determined by the action on $V$ , 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 $V$ are the hook-shaped Specht modules.

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

Sep 29, 2026PDF ↗View work →

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

Sep 29, 2026PDF ↗View work →

hexagon:2609.00124v3

math.QA — Quantum Algebra

Quantum Laurent Positivity for Mutation-Acyclic Cluster Algebras

Contributed by 琪越唐

We prove quantum Laurent positivity for mutation-acyclic skew-symmetrizable cluster algebras of arbitrary finite rank.

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

Sep 29, 2026PDF ↗View work →

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 $d$ -level quantum system up to a global phase. Four bases always suffice, and the answer is known to be three for $d=2$ and four for $d=3$ and $d\geq5$ . Dimension 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. 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 $4\times4$ 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. We 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.

Primarily AI-generated textHuman understanding: some parts

Sep 29, 2026PDF ↗View work →

hexagon:2609.00123v1

math.MG — Metric Geometry

Breaking the Fourth Wall: The Planar Centroid Banach–Mazur Diameter Is Strictly Less than $4$

Contributed by Scott Kominers

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.

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 $\mathbf{k}$ be a commutative ring, and let the symmetric group $\mathfrak{S}_{n}$ 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 $\mathbf{k}$ is a field) but not easily found in the literature. First, the coinvariant algebra (the quotient of $P$ by the ideal generated by the symmetric polynomials with constant term $0$ ) is a free $\mathbf{k}$ -module of rank $n!$ , with the residue classes of the Artin monomials as a basis. Second, $P$ is a free module of rank $n!$ over the ring $P^{\mathfrak{S}_{n}}$ of symmetric polynomials, again with the Artin monomials as a basis. Third, if $n!$ is invertible in $\mathbf{k}$ , the coinvariant algebra is the regular $\mathbf{k}\left[ \mathfrak{S}_{n} \right] $-module. Fourth, under the same hypothesis, $P$ is a free left $P^{\mathfrak{S}_{n}}\left[ \mathfrak{S}_{n} \right]$ -module of rank $1$ . 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

Sep 28, 2026PDF ↗View work →

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

Sep 28, 2026PDF ↗View work →

hexagon:2609.00118v1

math.CO — Combinatorics

A counterexample to ordering-independence for chromatic operators

Contributed by Darij Grinberg

We give a $10$ -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

Sep 28, 2026PDF ↗View work →

hexagon:2609.00119v1

math.NT — Number Theory

Hilbert-type inequality with primes

Contributed by Kannan Soundararajan

This note answers a problem posed in Montgomery's CBMS book. The proof was found by ChatGPT 6, and written up by the author.

Primarily human-written textHuman understanding: all partsCollar-Stein LemmaHilbert's inequalitySieves

Sep 28, 2026PDF ↗View work →

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 $P = p_0 + p_1 X + \cdots + p_m X^m \in R[X]$ 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.

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

Sep 28, 2026PDF ↗View work →

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:2609.00114v5

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 of $\ZFC$. For any sentence $\sigma$ , letting $\CM$ denote ``class of measurables,'' the following theories are equiconsistent: \begin{enumerate} \item $\ZFC+\CM+ [\HOD_{\Ord^{\omega}}\models (\exists\kappa\,V_{\kappa}\models\sigma)]$; \item $\ZFC+\GCH+\CM+ [\HOD_{\Ord^{\omega}}\models (\SVC\wedge\CM\wedge\exists\kappa\,V_{\kappa}\models\sigma)]$; \item $\ZF+\DC+\CM+ \exists\kappa\,V_{\kappa}\models\sigma$. \end{enumerate} Thus, under measurables the only local constraint on $\HOD_{\Ord^{\omega}}$ is the requisite choice principle. We also derive an analogue for Woodins. Letting $\CW$ denote ``class of Woodins,'' the following are equiconsistent: \begin{enumerate} \item $\ZFC+\CW+ [\HOD_{\Ord^{\omega}}\models (\exists\kappa\,V_{\kappa}\models\sigma)]$; \item $\ZFC+\GCH+\CW+ [\HOD_{\Ord^{\omega}}\models (\exists\kappa\,V_{\kappa}\models\sigma)]+$ $\HOD_{\Ord^{\omega}}$ is a ground of $V$ ; \item $\ZFC+\CW+$ some symmetric extension $V(G) \models \DC+\exists\kappa\,V_{\kappa}\models\sigma$. \end{enumerate}

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

Sep 28, 2026PDF ↗View work →

hexagon:2609.00112v1

math.AT — Algebraic Topology

Swan induction for the finite-local sphere

Contributed by Akhil Mathew

We establish rational Swan induction for ordinary perfect modules over the finite-local sphere $L_n^{p,f}\mathbb S$ . For a finite abelian ambient group, subgroups of $p$ -rank at most $n+1$ suffice, and this bound is sharp. The proof combines cyclic homotopy fixed points in telescopic spectra with the isotropy filtration of a chromatic quotient of finite genuine spectra. The result proves Conjecture~7.22 of Clausen--Mathew--Naumann--Noel for Morava $E$ -theory and gives a new proof of their chromatic upper bound for the algebraic $K$ -theory of $L_n^{p,f}\mathbb S$ -linear categories. The quotient argument also produces finite complexes realizing the induction relations. We formulate the problem of explicit realizations and give two geometric models at the prime two.

Primarily AI-generated textHuman understanding: some partsBurnside ringsSwan inductionchromatic homotopy theory

Sep 27, 2026PDF ↗View work →

hexagon:2609.00111v1

math.AG — Algebraic Geometry

Torsion in $K_1$ and $p$ -adic vanishing cycles

Contributed by Akhil Mathew

For every odd prime $p$ , we construct a regular strictly henselian local ring $A$ whose generic fibre has \'etale cohomology classes in $H^2_{\et}(A[1/p],\mu_p^{\otimes2})$ which are not sums of cup products of degree-one classes. The ring is the strict henselization of a local ring on a finite-type arithmetic scheme. A unit on a principal divisor gives an element of order $p$ in $SK_1(A[1/p])$ , detected by its localization boundary on the special fibre. Adams--Riemann--Roch and the connectivity of the motivic filtration show that this element has nonzero image in $H^3_{\mot}(A[1/p],\Z(2))$. The integral coefficient sequence then gives the cohomological obstruction. Only the eigenvalues $4$ and $8$ of $\psi^2$ are needed.

Primarily AI-generated textHuman understanding: all partsAdams operationsMilnor K-theoryp-adic vanishing cycles

Sep 27, 2026PDF ↗View work →

hexagon:2609.00110v1

math.AG — Algebraic Geometry

Frobenius orbit maps and rank-four group schemes

Contributed by Akhil Mathew

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.

Primarily AI-generated textHuman understanding: some partsHopf algebrasgroup schemes

Sep 27, 2026PDF ↗View work →

hexagon:2609.00102v1

math.NT — Number Theory

Inclusions between p-bounded crystalline loci in dimension two

Contributed by Kalyani Kansal, Brandon Levin, David Savitt

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).

Primarily human-written textHuman understanding: all parts

Sep 26, 2026PDF ↗View work →

hexagon:2609.00105v1

math.AC — Commutative Algebra

Graded absolute integral closures and Picard groups

Contributed by Akhil Mathew

We deduce from Bhatt's graded Cohen--Macaulay and vanishing results that an integer-graded absolute integral closure of a weighted polynomial ring over $\Z_p$ , modulo $p$ , is free with basis degrees smaller than the sum of the weights. A direct $p$ -adic expansion turns this degree bound into a uniform solution of the cusp relation modulo nilpotents. Consequently every ring $R$ admits a faithfully flat algebra with seminormal reduction, and every invertible $R[T]$ -module becomes free after a faithfully flat extension of $R$ , answering a question of Drinfeld.

Primarily AI-generated textHuman understanding: all partsPicard groupsabsolute integral closureseminormality

Sep 26, 2026PDF ↗View work →

hexagon:2609.00104v1

math.AG — Algebraic Geometry

Derived invariance of the signature and Hodge numbers of fourfolds

Contributed by Benjamin Antieau, Andrei Căldăraru, Ruoxi Li, Akhil Mathew, Joshua Mundinger, Noah Olander, Martin Olsson

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.

Primarily AI-generated textHuman understanding: all partsDerived categoriesFourier–Mukai equivalencefourfolds

Sep 26, 2026PDF ↗View work →

hexagon:2609.00001v1

math.AG — Algebraic Geometry

Strict negativity and non-vanishing of Chen--Larson hypergeometric coefficients

Contributed by Johannes Schmitt

Chen and Larson study tautological classes on the strata of holomorphic abelian differentials, where they predict a vanishing result. Using known tautological relations on the moduli space of curves, they reduce this prediction to the non-vanishing of certain coefficients of a quotient of hypergeometric generating series, treating the residue classes $g\equiv0,2\pmod3$ and $g\equiv1\pmod3$ through two separate series; they verify the non-vanishing by computer for small genus. We prove it in general: in both cases the relevant coefficient is non-vanishing for every genus and every stratum, and in the first case it has, more strongly, a uniform strict sign. Along the way we correct an error in the Chen--Larson derivation of the $g\equiv1\pmod3$ series, where a constant of the underlying relation of Ionel had inadvertently been changed. The paper falls into two parts. Part~I gives the mathematical proofs; Part~II documents the Lean~4 and Mathlib formalization --- whose only non-standard trust assumption is the compiler invoked by \code{native\_decide} for the large finite computations --- together with the long-horizon, multi-model generative-AI process that produced the proofs, exposed false intermediate routes, and uncovered the error in the printed Proposition~5.2 relation noted above.

Primarily AI-generated textHuman understanding: not declaredpower seriestautological classes

Sep 01, 2026PDF ↗View work →