Hexagon is a repository for the preservation, dissemination, study, and evaluation of research-level works in the mathematical sciences and theoretical computer science, including both original and expository texts. Hexagon does not perform peer review. Every deposit is screened automatically and, during the beta, also reviewed by a human moderator before it appears; acceptance is not a judgment of correctness, quality, or novelty.

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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:2610.00016v1

math.CO — Combinatorics

An Exponent of 1.04273 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 $u(U_j)/|U_j|^{1.04273}\to\infty$ . The largest exponent previously claimed, $1.0358$ , 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- $2$ 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 $2$ , $3$ and $5$ 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 $2$ , $3$ and $5$ ; the root discriminant of its fields is about $286$ . The relative zeta value is bounded through the zeta function of the degree- $512$ field generated over $\Q(\sqrt{241})$ by the square roots of its $\{2,3,5\}$ -units, which is the product of the Dedekind zeta function of $\Q(\sqrt{241})$ and $255$ quadratic Hecke $L$ -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 $1.0427$ 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:2610.00013v1

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

Oct 01, 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 $n$ are the sites whose local time at time $n$ is maximal. We prove that, almost surely, for every $\gamma>1/2$ and every $c>0$ , all favorite sites lie outside the ball centered at the origin with radius $c\sqrt n/(\log n)^\gamma$ for all sufficiently large $n$ . At the critical exponent $\gamma=1/2$ , almost surely, for every $c>0$ , the entire favorite sites lies within distance $c\sqrt{n/\log n}$ of the origin infinitely often.

Primarily AI-generated textHuman understanding: no parts

Oct 01, 2026PDF ↗View work →

hexagon:2610.00012v1

math.RA — Rings and Algebras

A finite criterion for total sign skew symmetry in order three

Contributed by 琪越唐

We give a finite criterion for total sign skew symmetry of integer matrices of order three.

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

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

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 $\R^4$ 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 →

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