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math.RT — Representation Theory

A note on counterexamples to homological conjectures

Contributed by Qiyue Tang

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. 2026.10.15:Version 3 of this manuscript has completed a full-paper review using Rethlas-OSV with GPT-6-Astra at xhigh reasoning effort, covering all 23 review scopes created by the system. A supplementary audit checked the manuscript’s use and attribution of the two cited OpenAI papers against locally supplied original PDFs and TeX sources. Neither review reported any confirmed mathematical errors within its scope. Rethlas-OSV: https://github.com/OSVerify/Rethlas-OSV OSVerify: https://github.com/OSVerify/OSVerify

A mix of human-written and AI-generated textHuman understanding: no partsRepresentation theory

Oct 07, 2026PDF ↗View work →

hexagon:2610.00140v6

math.RT — Representation Theory

A note of refinements to an algebra of infinite little finitistic dimension

Contributed by Qiyue Tang

Building on OpenAI’s An algebra of infinite little finitistic dimension, we simplify the exposition and extend the construction to every field. 2026/10/10：We have added formal verification：https://github.com/tqyqiqi1509/Note198-formalization

A mix of human-written and AI-generated textHuman understanding: no partsRepresentation theory

Oct 07, 2026PDF ↗View work →

hexagon:2610.00138v1

math.AT — Algebraic Topology

Weak chromatic splitting fails at n = p for every odd prime p

Contributed by Agnes Beaudry, Tomer Schlank

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 $p$ . This 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.

Primarily human-written textHuman understanding: all parts

Oct 07, 2026PDF ↗View work →

hexagon:2610.00137v1

math.NT — Number Theory

Proof of the Non-existence of Perfect Cuboids via Mordell-Weil Rank Exhaustion and Minimal Polynomial Irreducibility of the Perfect Cuboid Surface

Contributed by Jonathan 𝑓(n) Reed

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.

A mix of human-written and AI-generated textHuman understanding: all parts2-DescentEuler BrickHyperelliptic CurvesInteractive Theorem ProvingJacobian VarietyLean 4Mordell-Weil RankPerfect CuboidQuadratic Residueformal verification

Oct 07, 2026PDF ↗View work →

hexagon:2610.00136v2

math.NT — Number Theory

The generalized Fermat equation $x^3+y^3=z^n$

Contributed by Manvir Jaswal

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.

Primarily AI-generated textHuman understanding: no partsFrey representationscompatible systems of Galois representationsgeneralized Fermat equationhypergeometric motivesmodular method

Oct 07, 2026PDF ↗View work →

hexagon:2610.00135v1

math.NT — Number Theory

Improved sum-difference inequalities in abelian groups

Contributed by Logan Kleinwaks

For all finite subsets $X,Y$ of an abelian group, we prove \[ |X-Y|\le |X+Y|^{\lamI},\qquad \lamI=\frac{9451\e-3286}{5378\e+787}=1.454277448906\ldots, \] improving the classical exponent $3/2$ . We also prove that a universal two-set exponent $\lambda$ over the integers implies an upper bound $2-1/\lambda$ 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 $A,B\subset\Z$ . Consequently, \[ \theta^*\le\frac{13524\e-7359}{9451\e-3286}=1.312373302115\ldots, \] improving their bound $4/3$ . The first argument combines a coupling with distinct differences, entropy inequalities for independent sums, a finite certificate using non-Shannon inequalities, and an explicit limiting certificate with weights $\int_0^1t(1-t)^k\e^t\,dt$. The transfer uses localisation and the Pl\"unnecke--Ruzsa inequality over a sequence of scales. The proofs are formalised in Lean~4.

Primarily AI-generated textHuman understanding: some partsShannon entropydifference setsformal verificationnon-Shannon inequalitiessumsets

Oct 07, 2026PDF ↗View work →

hexagon:2610.00134v2

math.OA — Operator Algebras

Poisson boundaries of type III factors and the relative bicentralizer conjecture

Contributed by Shuoxing Zhou

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.

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

Oct 06, 2026PDF ↗View work →

hexagon:2610.00083v1

math.QA — Quantum Algebra

Strongly positive bases and quantum greedy expansions

Contributed by 琪越唐

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).

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

Oct 06, 2026PDF ↗View work →

hexagon:2610.00031v1

math.RA — Rings and Algebras

Algebraic dependence of cluster X-variables

Contributed by 琪越唐

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.

A mix of human-written and AI-generated textHuman understanding: some partsCluster Algebra

Oct 05, 2026PDF ↗View work →

hexagon:2610.00032v1

math.RA — Rings and Algebras

An explicit nonideal rooted cluster morphism

Contributed by 琪越唐

We give a counterexample to Chang and Zhu’s question of whether every explicit rooted cluster morphism is ideal.

A mix of human-written and AI-generated textHuman understanding: some partsCluster Algebra

Oct 05, 2026PDF ↗View work →

hexagon:2610.00030v1

math.AG — Algebraic Geometry

Crystalline and infinitesimal Poisson stacks

Contributed by Aleksandr Karapetyan

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.

Primarily human-written textHuman understanding: all partsDerived algebraic geometrydeformation theoryderived algebrainfinity-categoriesshifted Poisson structuresshifted symplectic structures

Oct 04, 2026PDF ↗View work →

hexagon:2610.00029v2

math.QA — Quantum Algebra

Theta Bases for Rank-Two Quantum Cluster Algebras of Geometric Type

Contributed by 琪越唐

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. 2026.10.15:This version underwent AI-assisted correctness review using Rethlas-OSV with GPT-6-Astra at xhigh reasoning effort. The review comprised 24 completed segment checks and a completed supplementary full-paper review. No confirmed mathematical errors were reported within the reviewed scope. Rethlas-OSV: https://github.com/OSVerify/Rethlas-OSV

A mix of human-written and AI-generated textHuman understanding: no partsquantum cluster algebra

Oct 04, 2026PDF ↗View work →

hexagon:2610.00026v1

math.CO — Combinatorics

The Anti-Arithmetic Product of Symmetric Functions

Contributed by Darij Grinberg

The \emph{arithmetic product} $\boxdot$ is a bilinear operation on the ring of symmetric functions over $\mathbb Q$ defined by \[ p_\lambda \ \boxdot\ p_\mu = \prod_{i,j}p_{\operatorname{lcm}(\lambda_i,\mu_j)}^{\gcd(\lambda_i,\mu_j)} \] on the power-sum basis $(p_\lambda)$ . It is known that this operation preserves the subring $\mathbf{Symm}_{\mathbb Z}$ of symmetric functions with integer coefficients, and is Schur-positive on pairs of Schur functions. Indeed, it corresponds to induction of representations from $S_n \times S_m$ to $S_{nm}$ . In this work, we consider the \emph{anti-arithmetic product} $\diamondsuit$ , which is defined by the same formula but with $\gcd$ and $\operatorname{lcm}$ interchanged. We show that it, too, preserves $\mathbf{Symm}_{\mathbb Z}$ , even though it lacks the Schur positivity property. This was conjectured on MathOverflow in 2014. Our method is a more intricate variant of the representation-theoretical interpretation of $\boxdot$ . The anti-arithmetic product does not correspond to an operation on actual representations; but its adjoint can still be described as a map on characters, which (as we show) is a $\lambda$ -ring morphism from the character ring of $S_{mn}$ to that of $S_m \times S_n$ . Now, a 1975 theorem of Boorman shows that the representation ring of $S_n$ is generated by the natural permutation representation $M_n = \mathbb Q^n$ as a $\lambda$ -ring. Thus, the proof of integrality boils down to only computing the image of $M_{mn}$ under this adjoint, which can be done using M\"obius inversion in the representation ring. This work is aimed at readers familiar with representation rings and the representation theory of symmetric groups. No prior knowledge of $\lambda$ -rings is presumed. Three self-contained proofs of Boorman's theorem are given in the appendices, two of them lifting the theorem to the noncommutative symmetric functions (or Solomon's descent algebra), recovering a result of Schocker.

A mix of human-written and AI-generated textHuman understanding: all partsSpecht modulesdescent algebralambda-ringsnoncommutative symmetric functionssymmetric functionssymmetric group representations

Oct 03, 2026PDF ↗View work →

hexagon:2610.00024v1

math.RT — Representation Theory

Reflexive equivalence and reflexive-minimal algebras

Contributed by Haruhisa Enomoto

Let $A$ be a finite-dimensional algebra over a field. A finite-dimensional algebra $B$ over the same field is called reflexively equivalent to $A$ if their categories of reflexive modules are equivalent. We prove that there is a basic algebra $A_{\min}$ , unique up to isomorphism, such that $B$ is reflexively equivalent to $A$ if and only if $B\cong\operatorname{End}_{A_{\min}}(A_{\min}\oplus X)$ for some reflexive $A_{\min}$ -module $X$ . Moreover, we compute $A_{\min}$ explicitly. We extend these results to module-finite algebras over henselian local rings of dimension at most one under a dominant dimension condition at minimal primes. We also introduce reflexive modules over additive categories and prove that taking reflexive modules is idempotent: the reflexive modules over the category of reflexive modules form an equivalent category. As an application, we show that reflexive equivalence classes of algebras with finitely many indecomposable reflexive modules correspond bijectively to Morita equivalence classes of algebras whose reflexive modules are projective.

Primarily AI-generated textHuman understanding: some parts

Oct 03, 2026PDF ↗View work →

hexagon:2610.00025v2

math.QA — Quantum Algebra

Minimality of the Rank-Two Quantum Greedy Basis

Contributed by 琪越唐

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.

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

Oct 03, 2026PDF ↗View work →

hexagon:2610.00023v1

math.QA — Quantum Algebra

A Finite Polyhedral Construction of the Rank-Two Quantum Greedy Basis

Contributed by 琪越唐

We give a finite polyhedral construction of the quantum greedy basis for coefficient-free rank-two quantum cluster algebras. The construction specializes coefficientwise 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.

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

Oct 03, 2026PDF ↗View work →

hexagon:2610.00022v2

math.OC — Optimization and Control

Continuity of solutions to abstract linear control systems

Contributed by Frédéric Marbach

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. 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 $1 \le p < \infty$ ).

Primarily AI-generated textHuman understanding: all partsabstract linear control systems

Oct 03, 2026PDF ↗View work →

hexagon:2610.00019v1

math.PR — Probability

The Negative Spherical Perceptron: Capacity, Isostatic Jamming, and Full Replica Symmetry Breaking

Contributed by P. M. Aronow, Patrick Lopatto

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 $(\alpha-2)^3$ .

Primarily AI-generated textHuman understanding: some partsperceptronspin glass

Oct 02, 2026PDF ↗View work →

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 $K_{\mathrm{lat}}(n)\leq\left(\sqrt{2e/\pi}+o(1)\right)^n$ 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.

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 $33066$ 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 $v_3(24n+3)\geq9$ .

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 $3.99$

Contributed by Eric Naslund

Let $P(a,q)$ denote the least prime congruent to $a$ modulo $q$ , where $(a,q)=1$ . We give a computer-assisted proof that $P(a,q)\ll q^{3.99}$ , improving the exponent $5$ of Xylouris. More precisely, $P(a,q)<q^{3.99}$ for every sufficiently large $q$ , uniformly in $a$ . The main new ingredient is a graded near-density estimate for zeros of Dirichlet $L$ -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 $14$ and $21$ 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 $4453$ root cases and $4788$ terminal cases, each closed by its own linear program. On $4{,}196{,}879$ threshold boxes these programs give $29{,}397{,}336$ linear relaxations, and exact integer certificates for all of them bound the normalized zero sum strictly below~ $1$ . 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:2610.00016v2

math.CO — Combinatorics

An Exponent of 1.043 for the Unit Distance Problem

Contributed by Eric Naslund

Let $u(U)$ count the unordered pairs at distance one in a finite planar set $U$ . We construct finite sets $U_j$ with $|U_j|\to\infty$ and $u(U_j)/|U_j|^{1.043171}\to\infty$ . The largest exponent previously claimed, apart from preliminary versions of the present work, is $1.0358$ , in the author's unpublished manuscript. The method is the number-field construction of OpenAI and Sawin. Here unit distances come from elements of relative norm one in quadratic extensions $K/F$ in which $F$ has both real and complex places, and the fields come from an infinite pro- $2$ tower over $\Q(\sqrt{241})$, ramified only above $2$ , $3$ and $5$ , with Frobenius conditions at the two primes above $29$ and at one prime above $41$ . The relative zeta value is bounded through the dihedral quotients of the tower group: the tower group has $27$ quotients isomorphic to the dihedral group of order~ $8$ ; fifteen of them detect, by a census, the residue degree of the primes that split completely in the Kummer field up to norm $4\cdot10^{13}$ , and seven of them generate a field $E_{W''}$ of degree $8192$ contained in every field of the tower. We write $\zeta_{E_{W''}}$ as a product of $L$ -functions of degrees $2$ , $4$ and~ $8$ , evaluate the $576$ of degrees $4$ and $8$ rigorously at $21$ real points near~ $1$ , and combine the values through a kernel inequality with coefficients of both signs. Finite facts and numerical inequalities are certified by exact computation and interval arithmetic. Partial formalization PALOMAR-2026-10-01-000018 v2

Primarily AI-generated textHuman understanding: some parts

Oct 02, 2026PDF ↗View work →

hexagon:2610.00013v1

math.RA — Rings and Algebras

Explicit non-inducible rooted cluster morphisms

Contributed by 琪越唐

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.

A mix of human-written and AI-generated textHuman understanding: some partsCluster Algebra