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323 works

cs.DS — Data Structures and Algorithms

Subset Sum in time O(2^{0.4877n})

Contributed by Annanay Kapila

We present a uniform randomized classical algorithm for Subset Sum with running time O(2^{0.4877n}) in the word-RAM model, for inputs whose integer bit lengths are polynomially bounded in the number n of input integers. The algorithm halts on every execution and answers correctly with probability at least 2/3. The construction builds on OpenAI’s “Subset Sum in Time O(2^{0.49n})” (4 October 2026), retaining its framework of isolation, a capped preliminary search, a modular checksum, and alias subtraction. Two components are replaced: an alias procedure that computes phase sums using a smaller modulus, and a family of 43 checksum filters selected according to the input pattern. The analysis combines an averaged small-layer lemma, grouping by reduced and unreduced subset sums, and rigorous numerical certificates. Verification scripts using Arb ball arithmetic check the alias constraints and 430 moment inequalities. The argument relies on results from the earlier paper, which are restated in an appendix; the numerical certificates verify specified inequalities rather than formally verifying the entire proof.

Primarily AI-generated textHuman understanding: some partsSubset Sumcomputational complexityexponential-time algorithmsrandomized algorithmsrigorous numerics

Oct 15, 2026PDF ↗View work →

hexagon:2610.00259v2

math.AG — Algebraic Geometry

The Hodge Conjecture for CM abelian varieties

Contributed by Pol van Hoften, Jack Sempliner

Following the recent work [Ope26], we give a proof of the Hodge conjecture for CM abelian varieties.

Primarily human-written textHuman understanding: all partsHodge theory

Oct 15, 2026PDF ↗View work →

hexagon:2610.00257v1

math.FA — Functional Analysis

Sequential Games

Contributed by Rajesh Dachiraju

We formulate a time-series game in which each player chooses a hidden Markov model (HMM) as a mixed strategy. The HMM generates a finite, variable-length sequence of probability vectors, and the payoff is assigned to the completed game as the sum of its stage rewards. We prove a conditional Nash-equilibrium existence result for games with a common finite bound on output lengths. The proof applies a fixed-point argument to the best-response correspondence when each player's admissible HMM-density set is nonempty, convex, and compact in $(L^1)$ , and the completed-game payoff is bounded and measurable. A common BV bound together with $(L^1)$ -closedness can ensure compactness; closure under suitable disjoint-state mixtures can ensure convexity. These properties do not follow from the HMM representation or BV regularity alone and must be verified for a specified model. Consistent estimation of HMM densities may support computation, but by itself does not establish equilibrium existence or replace the theorem's assumptions.

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

Oct 15, 2026PDF ↗View work →

hexagon:2610.00258v1

math.CO — Combinatorics

An Intrinsic Formula for the Ray–West Correction Term

Contributed by Donald Poindexter

For a nonempty permutation $\beta$ of length $m$ , let $g_2(\beta)$ denote the number of distinct permutations of length $m+2$ containing $\beta$ . Ray and West showed that $$ g_2(\beta)=\frac{m^4+2m^3+m^2+4m+4-2j(\beta)}{2}, \qquad 0\le j(\beta)\le m-1, $$ but their correction term was not expressed as a statistic of $\beta$ . This was posed explicitly as Problem 3.7 by Vatter. We give a polynomial-time formula defined entirely from $\beta$ : $j(\beta)=L(\beta)+R(\beta)$ , where $L$ counts licensed gaps and $R$ counts anchor-free two-deletion classes containing a collision whose canonical alignment support is a permutation interval. We also derive the Ray–West bound intrinsically and characterize its upper-bound equality cases as precisely the layered or reverse-layered permutations. The proof is combinatorial. Exact computation is used only for independent corroboration; the accompanying proof-and-replay archive contains the full proof supplement and bounded verification. This preliminary technical report was first deposited on Zenodo on August 9, 2026: https://doi.org/10.5281/zenodo.21857882.

Primarily AI-generated textHuman understanding: no partsRay–West correction termWilf equivalencedeletion classesenumerative combinatoricspermutation patternspermutation statisticsstrong intervals

Oct 15, 2026PDF ↗View work →

hexagon:2610.00253v1

math.QA — Quantum Algebra

$C_2$ -cofiniteness of logarithmic $W$ -algebras

Contributed by Andrew Riesen

Let $\mathfrak{g}$ be a finite-dimensional simple complex Lie algebra with root lattice $Q$ and lacing number $t$ , equipped with an invariant bilinear form normalized so that long roots have squared length two. We prove $C_2$ -cofiniteness of the Feigin--Tipunin logarithmic $W$ -algebra $\W_p(Q)$ for $p=tm\geq2$ and $m\geq h^\vee_{{}^L\g}-1$, where $h^\vee_{{}^L\g}$ is the dual Coxeter number of the Langlands dual. Drinfeld--Sokolov reduction of induced affine Weyl modules gives graded generators for the multiplicity spaces of the conformal group action. Their degree bounds show that the intrinsic $C_2$ -algebra is a finite module over the polynomial $C_2$ -algebra of its principal invariant subalgebra. Adjoint contractions produce a finite relation ideal in this polynomial ring. Heisenberg normal ordering and a constant-coefficient differential equation show that its common zero is the origin. The argument gives explicit sufficient bounds and strong fin

Primarily AI-generated textHuman understanding: no parts

Oct 15, 2026PDF ↗View work →

hexagon:2610.00252v2

math.NT — Number Theory

Cubic augmentation of a two-moment model: an exact finite absorption theorem and its limits

Contributed by Kunal Tyagi

We study adjoining a half-bandwidth signed cubic trace and a spectral count ladder to a bandwidth-one trace/Frobenius model motivated by zeta-zero certificates. In a finite circle model with mass N = 64, flat windows, and explicit budget centers, we prove exact absorption over the full atom-only class: the augmented and baseline optima agree for both the distinct-count and simple-fraction objectives. Three explicitly specified integer-marked grid columns attain the lower bounds. Their mixing weights are defined algebraically, and an Arb certificate proves all strict feasibility and inertia inequalities. The witness satisfies the ladder at every positive threshold, not just on a finite threshold grid. We also give the exact conjugate-pair spectrum, the isolated-block cubic identity, grid Parseval, and the sharp conditional spectral-capacity constant 2 sqrt(2). The result is a finite model theorem, not an unconditional statement about zeta zeros. Version 2 corrects an infeasible witness, a cubic-tolerance mismatch, and an incorrect rationality claim in version 1. Global absorption in the pair-containing or pinned classes remains unproved; equality on a searched column dictionary and heuristic pricing do not establish it.

Primarily AI-generated textHuman understanding: some parts

Oct 15, 2026PDF ↗View work →

hexagon:2610.00251v1

math.GT — Geometric Topology

Phase-Labeled Doubles and Disjoint Handleslide Cubes with Complete Markings

Contributed by Hao-Yu Sun

We prove two exact coherence results for marked handles in Walker's extended cobordism category. At fixed disjoint sites, a matching pair of forward and reverse phases contributes one local $S^1\times S^2$ summand, whereas a mismatch contributes an identity cylinder. Thus the resulting double depends only on the total Walker weight and the number of phase matches, not on the orders of attachment. For positive handleslides supported in pairwise-disjoint protected sectors, we construct the complete marked Boolean cube and prove that every two-face commutes with weight zero; consequently, every finite route has the same composite and the endpoint identities telescope. An optional odd-prime $\mathrm{SO}(3)$ -TQFT corollary evaluates the fixed-site double as $\kappa_p^N(D_p^{\mathrm{GM}})^b$ times the identity. Overlapping slides and their pentagon are treated by a separate companion theorem.

Primarily AI-generated textHuman understanding: some partsWalker weightdisjoint coherence cubehandleslidemarked cobordismphase double

Oct 15, 2026PDF ↗View work →

hexagon:2610.00248v1

math.MG — Metric Geometry

Improved Kissing Numbers in Nineteen and Twenty-One Dimensions

Contributed by Kevin Li

We prove that the kissing numbers in 19 and 21 dimensions are at least 12460 and 31560, improving the best published bounds 11948 (Ho) and 29768 (Cohn and Li). Both constructions start from the odd-sign configurations of Cohn and Li, which leave room around the sign vectors they add. We use this room to reshape the added vectors so that codewords closer than n/4 can be used together. In 21 dimensions each added vector has entries ±2 on one line of the projective plane of order 4 formed by the weight-5 words of the punctured Golay code, and ±3 elsewhere; a linear labelling of the codewords by lines gives 3840 new points, and the proof is by hand. In 19 dimensions we cut the 20-dimensional configuration by a hyperplane and project the sign vectors of the 20-dimensional code into it. The projections have a 16-coordinate head and a three-dimensional tail, the tails separate words whose heads are close, and one point for each of the 1792 heads with a nonzero tail can be added, where Ho adds 1280.

Primarily AI-generated textHuman understanding: all partsKissing Number

Oct 14, 2026PDF ↗View work →

hexagon:2610.00249v1

math.NT — Number Theory

Explicit positive lower density for the Collatz Conjecture

Contributed by Kenny Lau

Lech Mazur proved that a positive proportion of integers $n$ reach $1$ under the Collatz map in at most $10.46\log n$ steps. We give a quantitative refinement of his construction. For any $X\ge 2^{2^{2^{2^{87640}}}}$ , at least $X/2^{2^{2^{87640}}}$ positive integers $n<X$ reach $1$ within $\floor{10.46\log n}$ ordinary steps, and $\floor{3.4881\log n}$ accelerated steps. We also obtain results for paths that include any prescribed integer $t$ not divisible by $3$ . AxiomProver was used to obtain and formalize this result in Lean.

A mix of human-written and AI-generated textHuman understanding: some partsCollatz conjecturearithmetic dynamics

Oct 14, 2026PDF ↗View work →

hexagon:2610.00241v1

math.CO — Combinatorics

A six-spin obstruction to antiferromagnetic biclique maximization

Contributed by Matěj Kripner, Václav Rozhoň, Robert Šámal, Martin Šindelář, Adrián Zámečník

Sah, Sawhney, Stoner and Zhao conjectured that for every spin model whose interaction matrix has exactly one positive eigenvalue, the partition function of any graph is at most a product of biclique partition functions determined by its edge degrees. We disprove this conjecture with a strictly positive six-spin model on a triangle with three pendant edges; the violation persists on an open set of nonsingular models. The mechanism is a negative mixed covariance trace at vertices of degrees two, three and four: the triangle contributes a favourable cubic term, whereas all biclique corrections start at fourth order.

Primarily AI-generated textHuman understanding: some partsantiferromagnetic modelsgraph homomorphismspartition functions

Oct 14, 2026PDF ↗View work →

hexagon:2610.00240v1

math.CO — Combinatorics

A cubic variance bound for matching deficiency

Contributed by Matěj Kripner, Václav Rozhoň, Robert Šámal, Martin Šindelář, Adrián Zámečník

Let $M$ be a uniformly random matching of a finite graph, let $D=\nu-\mathbb E|M|$ be its expected deficiency from a maximum matching, and let $V=\operatorname{Var}(|M|)$ . Kahn showed that $D$ and $V$ tend to infinity together, proved $D=O(V^4)$ when $V$ is bounded below, and conjectured $D=O(V^3)$ . We prove $D\le41V+104V^2+64V^3$ for every graph, confirming the conjecture, and show that the exponent is attained by connected bipartite graphs with perfect matchings. The proof combines alternating-path identities for monomer probabilities with the sharp estimate $|V(G)-V(G-v)|\le1/4$ .

Primarily AI-generated textHuman understanding: some partsmatching deficiencymatchingsvariance

Oct 14, 2026PDF ↗View work →

hexagon:2610.00238v1

math.CO — Combinatorics

A nonlinear connectivity code larger than every linear code

Contributed by Matěj Kripner, Václav Rozhoň, Robert Šámal, Martin Šindelář, Adrián Zámečník

A connectivity code on a graph $H$ is a family of edge sets of $H$ whose pairwise symmetric differences are connected spanning subgraphs. Alon asked whether a nonlinear connectivity code can ever be larger than every linear one on the same graph. We answer this affirmatively: we construct a $4$ -regular graph on fifteen vertices on which the largest connectivity code has nine members and the largest linear connectivity code has eight. The code is obtained by lifting a quaternary code of length six with pairwise agreement one through clique gadgets.

Primarily AI-generated textHuman understanding: some partsconnectivity codesgraph codesnonlinear codes

Oct 14, 2026PDF ↗View work →

hexagon:2610.00225v1

math.CO — Combinatorics

The Bannai–Munemasa–Venkov lattice method for strongly regular graphs: a uniform discriminant-form condition and new nonexistence results

Contributed by Anton Kovaľ

Bannai, Munemasa and Venkov, and later Nebe and Venkov, excluded tight spherical designs — among them strongly regular graphs of a special kind — by studying the lattice spanned by the design: they bound its discriminant group, pass to an even (over)lattice, and compare Gauss sums by the Milgram–Braun formula. We extend this method to arbitrary strongly regular graphs with integral eigenvalues. For an integral representation G=b(A-ρ I)+cJ that is positive semidefinite with even diagonal, an elementary identity for dual vectors ties every class of the discriminant form of a maximal even overlattice to the class of a rescaled sum of the vertex vectors. This gives a necessary condition on the parameters (v,k,λ,μ) alone: a finite quadratic module with a short list of properties must exist. Its simplest case applies the textbook congruence det≡(-1)^(n/2) (mod 4) for even lattices of odd determinant to the centred representation. A sweep of Brouwer's table (v≤1300) finds 70 complementary pairs of parameter sets listed as open that fail the condition, among them (96,35,10,14), the triangle-free set (176,25,0,4) and (169,42,5,12); the grade of evidence is given for each (hand proof, or exact computation by at least two independent programs). No existing parameter set of the table fails in any representation tested, real graphs satisfy every condition with their actual discriminant forms, and the method gives short independent second proofs of the known nonexistence of srg(75,32,10,16) and srg(289,54,1,12). The simplest case also yields an infinite family: if q is a power of a prime p≡1 (mod 4) and m is odd, there is no strongly regular graph with the parameters (q^2, m(q+1), -q+m^2+3m, m(m+1)) of the negative-Latin-square type NL_m(q). This family, the parity case and the nonexistence of srg(169,42,5,12) are formalized in Lean 4 with Mathlib. Also deposited on Zenodo with the full data archive (the citable record): https://doi.org/10.5281/zenodo.23281359

Primarily AI-generated textHuman understanding: some parts

Oct 14, 2026PDF ↗View work →

hexagon:2610.00245v1

math.NT — Number Theory

Finite Analytic Characters and Tannakian Source Reconstruction: A Finite Tensor–Quiver Framework for Beyond Endoscopy

Contributed by Shawn Schafer

This paper organizes finite tensor–quiver reconstruction for reductive source data. For a fixed reductive Zariski-closed subgroup H of a reductive complex group G and a faithful finite-dimensional representation, the standard mixed-tensor reconstruction theorem and Noetherianity imply that finitely many H-invariant tensors have common stabilizer H. Classical quiver invariant theory then implies that finitely many traces of oriented cycles separate closed orbits, including semisimple realizations. For GL₂/ℚ, we give a restricted application of Cheng’s symmetric-square limiting distribution. After a finite-level cuspidal block has been specified and a target constituent selected by an interpolating Hecke test function built from that block’s Hecke characters, ratios of trace-formula distributions recover the first two Satake power sums at auxiliary unramified places, provided Cheng’s hypotheses hold and the full selected coefficient is nonzero. This is post-selection recovery, not a method for identifying the target constituent, and applies only to the nonzero-coefficient visible subset.

Primarily AI-generated textHuman understanding: some partsBeyond EndoscopyHecke operatorsSatake parametersTannakian reconstructionautomorphic representationsquiver invariant theorytensor invariantstrace formulas

Oct 14, 2026PDF ↗View work →

hexagon:2610.00243v1

math.NT — Number Theory

Steinhaus Piercing with Slack: The Exact Asymptotic Constant

Contributed by Anton Joha

Let s(d) be the largest order N for which the first n + d points of a sequence in [0,1) meet every uniform n-cell for all n ≤ N. We prove s(d) = d/(1 − log 2) + O(√d), determining the limit asked for by Anholcer et al. and disproving their conjectured coefficient. For strong piercing, which requires meeting every interval of length 1/n, we obtain an exact finite harmonic capacity and rational constructions. A Dirichlet covering lemma and a packing estimate for disjoint gaps transfer the capacity bound to ordinary grids. We also solve the finite largest-gap minimax problem, establish rigidity for infinite heap refinements, and extend the asymptotic estimate to affine arrival budgets. Ordinary near-extremizers satisfy universal laws for root sizes and first-split times, with a corresponding entropy correction. An explicit finite obstruction shows why root constraints alone do not characterize ordinary piercing.

A mix of human-written and AI-generated textHuman understanding: some partsSteinhaus problementropyharmonic capacityinterval refinementirregularities of distributionpiercing sequences

Oct 14, 2026PDF ↗View work →

hexagon:2610.00239v1

math.NT — Number Theory

Small witnesses for eventual Erdős--Straus binomial divisibility

Contributed by Matěj Kripner, Václav Rozhoň, Robert Šámal, Martin Šindelář, Adrián Zámečník

Erdős and Straus asked whether, for each fixed $n\ge2$ , every sufficiently large integer $m$ admits $1\le k\le m-n$ with $\binom{n+k}{n}\mid\binom{m+k}{k}$ . We prove that it does; previously this was known only for a set of $m$ of density one. The witness can be taken with $k=O_{n,\varepsilon}(m^\varepsilon)$ for every $\varepsilon>0$ : a factorial identity reduces the problem to finding a small value of a fixed polynomial coprime to $\binom mn$ , which an elementary sieve provides. Conversely, the least witness is at least logarithmic in $m$ along an infinite sequence.

Primarily AI-generated textHuman understanding: some partsErdős–Strausbinomial divisibilitysmall witnesses

Oct 14, 2026PDF ↗View work →

hexagon:2610.00236v1

math.NT — Number Theory

A counterexample to Haglund's monotonic-zeros conjecture for the Riemann $\Xi$ approximants

Contributed by Kunal Tyagi

Haglund introduced the partial sums $\Xi_N = \sum_{n \le N} \Phi_n$ of Riemann's expansion of the $\Xi$ -function in incomplete gamma functions and conjectured that for every $N$ the zeros of $\Xi_N$ in the closed first quadrant, listed by increasing real part, have nondecreasing imaginary parts. The conjecture implies the Riemann hypothesis; Haglund's computations supported a weak form of it for $N \le 10$ . We show that the conjecture is false for $N = 27$ : $\Xi_{27}$ has a real zero in $(3144.8946, 3144.8947)$ and a non-real zero $z_0 = 3143.2206824215\ldots + 0.3152587993\ldots\, i$ in the first quadrant, whose real part is smaller. The weak form fails as well. Both facts are certified twice, independently, in interval arithmetic (Arb ball arithmetic; outward-rounded intervals with proved truncation bounds), and $z_0$ is enclosed in a square of half-width $10^{-48}$ . The same violation occurs at $N = 24$ for the companion approximants $\xi_N = \Xi_N + c_N$ . The failure has a mechanism: on the critical line $\Xi_N < \Xi < \xi_N$ , so the real zeros of $\Xi_N$ lie in positive lobes of $\Xi$ that rise above a positive level, and a low lobe followed by a higher one near the height $4(N+1)^2$ , where the approximant departs from $\Xi$ , can leave a non-real zero below real zeros. The same structure shows that the number of positive real zeros of $\Xi_N$ , counted with multiplicity, is odd, and that $x^2\Xi_N(x)$ tends to a negative limit on the real axis for every $N$ .

Primarily AI-generated textHuman understanding: some parts

Oct 14, 2026PDF ↗View work →

hexagon:2610.00234v1

math.NT — Number Theory

An unavoidable set of divisors for the nearest integers to ξ(3/2)^n

Contributed by Ralf Stephan

We prove that for every real ξ ̸= 0 the sequence of nearest integers [ξ(3/2)^n + 1/2], n ∈ N, contains infinitely many elements divisible by 2 or by 5. In particular, for ξ > 0 infinitely many of these nearest integers are composite. This is the nearest-integer counterpart of the theorem of Forman and Shapiro on [ξ(3/2)^n], for which Dubickas and Novikas give the unavoidable set {2, 5, 7, 11}; for nearest integers two primes suffice. The proof follows the method of Dubickas and Novikas: if 2 and 5 divide only finitely many terms, the sequence of operations is forced to become (A^2B^2)^∞, which is impossible by their Lemma 2. All results are machine-verified in Lean 4.

Primarily AI-generated textHuman understanding: some partsuniform distribution

Oct 14, 2026PDF ↗View work →

hexagon:2610.00229v1

math.NT — Number Theory

Non-Archimedean Massera-Schaffer-Maligranda-Pecaric-Rajic Inequality

Contributed by K. MAHESH KRISHNA

Massera and Schaffer [\textit{Ann. Math. (2), 1958}] derived a breakthrough upper bound for the Clarkson angle between two nonzero vectors in a normed linear space, which was later improved by Maligranda [\textit{Am. Math. Mon., 2006}]. Pecaric and Rajic [\textit{Math. Inequal. Appl., 2007}] extended Maligranda's inequality to finitely many nonzero vectors. We derive a non-Archimedean version of Massera-Schaffer-Maligranda-Pecaric-Rajic inequality.

Primarily human-written textHuman understanding: all partsNon-Archimedean linear spaceNormed linear spaceTriangle inequalityUltra-norm.

Oct 14, 2026PDF ↗View work →

hexagon:2610.00223v1

math.NT — Number Theory

Long runs and rare patterns of a random completely multiplicative function

Contributed by Brice Pouly

Let $f$ be a random completely multiplicative function with independent symmetric signs at the primes. We study rare constant runs and prescribed words at logarithmic lengths, despite long-range multiplicative identities. A uniform weighted two-window square-relation estimate, with critical bound $O_\varepsilon(N^{5/3+\varepsilon})$ , separates exact rational relations from residual components. At critical intensity, this estimate gives Poisson comparisons retaining positions, exact excess lengths and signs. For dictionaries, uniform bounds under size and overlap conditions are complemented by typical bounds obtained by averaging the distance after fixing the dictionary. In a quantitatively controlled regime of diverging intensity, an exact one-word coupling by largest-odd-prime pivots compares the full signed run field under prescribed small-prime conditioning. For fixed admissible window parameters and almost every fixed realization of $f$ , run-start counts in prescribed near-macroscopic windows satisfy an empirical Poisson law along dyadic scales. In the exceptionally rare regime, a relative marked comparison separates the boundary event from bulk occurrences. Conditional on existence, the first location, sign and overshoot have a two-source lattice law with scale-dependent weights. The two overshoots use different clocks: prime rank at the boundary and integer distance in the bulk.

Primarily AI-generated textHuman understanding: no partsLean 4Poisson approximationboundary effectscompletely multiplicative functionsformal verificationlong runsrandom multiplicative functionsrare patternssquare-product relations

Oct 14, 2026PDF ↗View work →

hexagon:2610.00246v2

math.AG — Algebraic Geometry

Algebraicity of Weil classes on abelian sixfolds of Weil type for $\mathbb{Q}(\sqrt{-3})$ with discriminant $-2$

Contributed by Benjamin Severn

Let $K=\mathbb{Q}(\sqrt{-3})$ . We prove that the Weil classes of every polarized abelian sixfold of Weil type for $K$ with Hermitian discriminant $-2$ are algebraic. This is a non-split discriminant, to which the theorems of Schoen and of Markman on split Weil classes do not apply. The proof uses Prym--Tyurin varieties of covers of $\mathbb{P}^1$ with monodromy the complex reflection group $G_{32}$ , and combines known mechanisms in a new configuration. First, on every such variety the Weil classes of the six-dimensional isotypic factor are algebraic. They are obtained, by a Lefschetz-class correspondence supplied by Milne's theorem, from Schoen's algebraic Weil classes on fifty-one cyclic intermediate quotients; the choice of quotients is governed by an integral identity among characters of a torus, verified by an exact computation. Second, the ten-dimensional family of these covers dominates the nine-dimensional moduli space: we exhibit an explicit cover at which the period map has rank nine, by a certified computation. Third, a Hilbert-scheme and Baire-category argument spreads algebraicity from an open set to the whole moduli space. The discriminant of the family is identified exactly. All computations are supplied as reproducible certificates. For the members whose special Mumford--Tate group is the full special unitary group, an invariant-theoretic lemma then gives the Hodge conjecture for every power. This version, with its computational certificates and review bundle, is archived on Zenodo: https://doi.org/10.5281/zenodo.23377274

Primarily AI-generated textHuman understanding: some partsHodge conjecturePrym-Tyurin varietiesWeil classesabelian varieties of Weil typecomplex reflection groupsperiod maps

Oct 14, 2026PDF ↗View work →

hexagon:2610.00244v1

cs.AI — Artificial Intelligence

First Proof Third Batch

Contributed by Mohammed Abouzaid, Nikhil Srivastava, Rachel Ward, Lauren Williams

To assess the ability of current AI systems to correctly solve research-level mathematics problems, we tested several AI systems on our ``third batch'' of ten problems in a broad range of mathematical fields; these problems arose naturally in the research process of the contributors. This document includes the problems, our methodology, and the results of our testing. We assessed the AI-generated solutions based on mathematical correctness, quality of exposition, and quality of attribution and citations. We provide links to supplementary documents including the human solutions, the AI-generated solutions, and the referee reports and logs for the AI-generated solutions.

Primarily human-written textHuman understanding: all partsAI systemsbenchmarklarge language models

Oct 14, 2026PDF ↗View work →

hexagon:2610.00227v1

math.GT — Geometric Topology

Marked Positive Handleslide Pentagons in Compression Bodies: Relative Coherence

Contributed by Hao-Yu Sun

Let $W:X\to Y_{\mathrm{geom}}$ be a completed oriented compression body with $N\geq 3$ ordered product $1$ -handles and a supplied protected three-handle sector. We realize one positive handleslide pentagon by five diffeomorphisms with compatible complete endpoint markings. An isotopy fixed on the entire protected boundary collar identifies the two-edge and three-edge route composites, so the comparison takes place at the level of marked maps rather than homology matrices. The construction fixes the marked disk systems at all five vertices before choosing any edge map, so incident edges use the same endpoint data. The cut-ball maps intertwine all fifteen disk pairings on common full-normal intervals and therefore descend smoothly through the paired faces. Extending these quotient maps by the identity gives global representatives. The same maps satisfy the finite compatibility conditions, including at $N=3$ , when the unused-handle domains are empty but remain part of the typed construction. The resulting boundary maps preserve the embedded meridian and product-framed dual-longitude rows, so their homological actions have blocks $M_e$ and $M_e^{-T}$ with no mixed shear. They also preserve the same rational disk-kernel Lagrangian. Walker's forward multiplication law lifts the relative isotopy to a literal weight-zero equality of mapping cylinders, including the seam and route weights.

A mix of human-written and AI-generated textHuman understanding: some partscompression bodymarked handlesliderelative isotopy

Oct 14, 2026PDF ↗View work →

hexagon:2610.00217v1

math.NA — Numerical Analysis

Low-discrepancy quasi-uniform digital nets in every dimension

Contributed by Leonhard Gruenschloss

Dick, Goda and Suzuki asked for low-discrepancy digital nets that are quasi-uniform, with minimum distance $\delta$ and covering radius both of the optimal order $N^{-1/s}$ , and stated the problem as open in dimension $s\ge3$ . We construct such nets in every dimension. For every $s\ge3$ , every base $q$ whose prime factors are at least $s-1$ , and every $k\ge1$ , we give explicit digital $(0,sk,s)$ -nets over $\Z/q\Z$ with $N=q^{sk}$ points, covering radius at most $\sqrt s\,N^{-1/s}$ , and $\delta N^{1/s}\ge1-a^*_s/q$ once $q\ge5a^*_s+4$ . The constant $a^*_s\le e^{s\log s+O(s)}$ is computed from $s$ by a deterministic polynomial-time lattice algorithm. In dimension three every $N=p^m$ , $m\ge3$ , is covered for each prime $p\ge37$ , and, with computer assistance, every $N=2^m$ and $N=3^m$ . In prime bases these families are well separated in the sense of Dick, Goda and Suzuki, and they stay so under every digital shift. One mechanism underlies all constructions. For $N=q^{sk}$ it reads as follows. The first $k$ base- $q$ digits of a point's coordinates select a cube of side $q^{-k}$ , and a $(0,sk,s)$ -net has exactly one point in each such cube. Digit $k+1$ of each coordinate is a linear function of the first $k$ digits of all coordinates, because the net is digital. We choose this linear function so that, between the planes where this digit wraps around modulo $q$ or where a carry occurs among the first $k$ digits, the points lie on a translate of one lattice; for the main theorem this lattice is a slightly distorted copy of $\Z^s$ . All later digits move a point by less than $1/q$ of the cube side. Part~II studies the constant $K$ in $\delta N^{1/s}=1-K/q+O(q^{-2})$ . We prove lower bounds on $K$ when the linear maps have bounded coefficients. At $N=q^3$ we find the optimal $K$ among nets whose points lie on lines parallel to the main diagonal. With the cube-tiling theorems of Haj\'os and Keller we show that idealized lattices with one point per cube, whose points sit independently inside their cubes, cannot exceed $\delta N^{1/s}=1$ . When the positions inside the cubes are coupled, the value $1$ can be exceeded: $\delta N^{1/3}\to\sqrt{281}/16$ at $N=p^3$ , $\delta N^{1/5}>1$ for $(0,5,5)$ -nets from $p=37$ on, and $\delta N^{1/s}\to\min(\sqrt s/2,\sqrt2)$ at $N=p^{sk}$ , which is the $E_8$ value at $s=8$ (limits as $p\to\infty$ ). The results are formalized in Lean~4, apart from some finite computations, which are checked by scripts.

Primarily AI-generated textHuman understanding: some partsLean 4 formalizationcovering radiusdigital netslatticeslow discrepancyminimum distancequasi-Monte Carloquasi-uniformity

Oct 14, 2026PDF ↗View work →

hexagon:2610.00224v1

math.DS — Dynamical Systems

Stable Algebraic Families for $\mathbb{P}^2$

Contributed by Jit Wu Yap

We classify one-dimensional families of endomorphisms on $\mathbb{P}^2$ that are algebraically stable, generalizing a theorem of McMullen. The non-isotrivial families are either homogeneous lifts of flexible Lattes maps on $\mathbb{P}^1$ , or a family of Lattes maps arising from abelian surfaces.

Primarily AI-generated textHuman understanding: some partsComplex dynamicsalgebraic dynamics

Oct 14, 2026PDF ↗View work →

hexagon:2610.00232v1

math.RA — Rings and Algebras

Another Counterexample to Little Finitistic Dimension Conjecture

Contributed by Andy Jiang

A characteristic-zero counterexample to the Little Finitistic Dimension Conjecture is given, arguably providing a simpler variant of OpenAI's construction.

Primarily AI-generated textHuman understanding: some partsLittle Finitistic Dimension Conjecture

Oct 14, 2026PDF ↗View work →

hexagon:2610.00237v1

math.QA — Quantum Algebra

Unipotent tensor categories and pronilpotent Lie quasibialgebras

Contributed by Pavel I Etingof

We construct a bijection between tensor equivalence classes of unipotent tensor categories over $\C$ and continuous Lie quasibialgebras with linearly compact pronilpotent underlying Lie algebra, modulo continuous isomorphisms and classical twists. The bijection is defined using a fixed universal quantization functor and solves the problem raised in Remark~4.6 of arXiv:0708.1487 by Etingof and Gelaki. The proof combines reconstruction with filtered dequantization modulo twists and convergence of universal formulas. A category admits a tensor fiber functor precisely when a classical twist removes its curvature.

Primarily AI-generated textHuman understanding: some partsTensor categoriesquasi-Hopf algebras

Oct 14, 2026PDF ↗View work →

hexagon:2610.00231v1

math.AG — Algebraic Geometry

The Watermark and Double Ladder Theorems: the discriminant group of the Néron–Severi lattice of a Fermat surface of prime degree

Contributed by Rafael Amichis Luengo

Let $S_m\subset\mathbb{P}^3$ be the complex Fermat surface $x_0^m+x_1^m+x_2^m+x_3^m=0$ of prime degree $m\ge 5$ . Its Néron–Severi lattice $\mathrm{NS}(S_m)$ is generated by the classes of the $3m^2$ lines on $S_m$ (Schütt–Shioda–van Luijk, Degtyarev). We prove that $|\mathrm{disc}\,\mathrm{NS}(S_m)| = m^{3(m-3)^2}$ (the Watermark Theorem), which answers, for prime degree, the determinant question asked by Shioda in 1987, and we determine the discriminant group itself: $\mathrm{NS}(S_m)^*/\mathrm{NS}(S_m)\cong(\mathbb{Z}/m)^{3m^2-24m+59}\times(\mathbb{Z}/m^2)^{3m-16}$ for $m\ge 7$ , and $(\mathbb{Z}/5)^{10}\times\mathbb{Z}/25$ for $m=5$ (the Double Ladder Theorem). The transcendental lattice of $S_m$ has the same discriminant group. Both proofs work with the explicit integer intersection matrix $G$ of the lines and use only elementary lattice arithmetic: the action of $G$ on functions pulled back along the $m+1$ pencils of directions of $(\mathbb{Z}/m)^2$ , an explicit sublattice of $\ker G$ of index $m^{12}$ , and two index computations for the order; an integer identity $GYG=6m^2G$ for the exponent, and a factorization of $G$ modulo $m$ through twelve one-variable "shadows" for the number of cyclic factors of order $m^2$ . The lattice statements of both theorems are checked in the proof assistant Lean 4 with the Mathlib library, with no axiom beyond the three standard ones; the identification of $G$ with the intersection matrix of the lines is proved by hand and checked by computer. This work has not been refereed by a human expert.

Primarily AI-generated textHuman understanding: some partsFermat surfacesLean 4Néron–Severi latticediscriminant grouplatticeslines on surfaces

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hexagon:2610.00230v1

math.ST — Statistics Theory

A Positive Dispersion Optimum for the Rounded Sample Mean: Sufficiency and Necessity of Finite Variance

Contributed by Mark R. Diamond, Angela O'Brien-Malone

A continuous quantity, observable only after nearest-lattice rounding, is estimated by the mean of $n$ rounded readings of an imperfect instrument. Averaged over the position of the true value within a rounding cell, the mean-squared error of this estimator is a function of one dimensionless parameter: the pre-rounding standard deviation $s$ , measured in cell widths. A perfectly repeatable instrument ( $s=0$ ) returns the same rounded value on every reading, so averaging provides no additional information; its cell-averaged error is the rounding floor $1/12$ . We show that this floor is never optimal once more than one reading is taken. For \emph{any} centred pre-rounding law with finite, non-zero variance and any sample size $n>1$ , some strictly positive dispersion yields a smaller cell-averaged error than zero, so the global minimizer lies away from the no-scatter endpoint; and each of the three hypotheses--- $n>1$ , finite variance, non-degeneracy---is individually necessary, so existence is \emph{exactly} characterized. The mechanism is elementary: the autocorrelation of the rounding error has a \emph{corner} at zero lag, so a little scatter relieves the rounding bias at first order in $s$ while its variance contribution grows only at second order. We give the short corner-based proof and then derive the spectral form of the criterion, using it to compute the optimum for four laws. The Gaussian optimizer is unique and solves a Jacobi-theta equation. For the Laplace law the bias term has a hyperbolic closed form, and the optimizer is unique and characterized by the corresponding derivative equation. For the uniform and two-point laws the optimizer is explicit. The uniform law cancels every bias harmonic exactly at its first positive cancellation scale, which saturates the natural localization bound when $n=2$ ; and the two-point law, whose criterion has several local wells once $n\ge4$ , shows that finite variance ensures existence but not a single-well objective.

A mix of human-written and AI-generated textHuman understanding: all partsFourier seriescharacteristic functionsmean-squared errorquantization errorrounded sample meanrounding error

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hexagon:2610.00242v1

math.IT — Information Theory

Average-radius list decoding of random linear matrix codes at rank-metric capacity

Contributed by Matěj Kripner, Václav Rozhoň, Robert Šámal, Martin Šindelář, Adrián Zámečník

We show that for every fixed prime power $q$ and $0<p<1$ , a uniformly random $\mathbb F_q$ -linear rank-metric code in $\mathbb F_q^{n\times n}$ of rate $(1-p)^2-\varepsilon$ is, with high probability, average-radius list-decodable at radius $pn$ with list size $O_{p,q}(1/\varepsilon)$ . This answers a question of Guruswami, Li and Singhal; the case $q=2$ was known. The failure probability is explicit and remains small when $\varepsilon$ depends on $n$ with $\varepsilon n^2\to\infty$ . The proof combines a weighted affine extraction, span estimates for matrices from nearby rank shells, and Gaussian summation over rank profiles.

Primarily AI-generated textHuman understanding: some partsaverage-radius list decodingcapacityrandom linear codesrank metric

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hexagon:2610.00235v2

math.CO — Combinatorics

An Arbor Polytope with Nonreal Ehrhart Roots

Contributed by YURAN XIU

We construct a singleton-labeled arbor on 243 vertices whose Ehrhart polynomial has a nonreal root. This disproves the conjecture that the Ehrhart roots of every arbor polytope are real and lie in $[-1,0]$ . The polynomial factors as $(m+1)^{230}(7m+2)^8 f(m)/6144$ , where $f$ is an explicit quintic. We derive this identity by counting lattice points for every nonnegative integer dilation, using three moments of subtree deficits. A negative value of $f'^2-ff''$ at $-3/10$ gives an exact nonreal-root certificate. The lattice-count identity and the resulting counterexample have been formalized and checked in Lean. The source release and reproduction evidence are archived at https://doi.org/10.5281/zenodo.23435787.

Primarily AI-generated textHuman understanding: no partsEhrhart polynomialsLean formalizationarbor polytopescounterexamplereal-rootedness

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hexagon:2610.00218v2

math.KT — K-Theory and Homology

A Note on Laminar Presentations and Finite Chain Models for Ample Groupoid Homology

Contributed by Luciano Melodia

A finite free chain model for the homology of a Hausdorff ample groupoid $G$ is cut out of the Matui complex by a finite clopen partition of each nerve space. We ask what such a model must supply and isolate three requirements: a single model that is a quasi-isomorphism, a chain of models that is a complete system with ranks and transition maps read off the presenting data, and a bound on the number of atoms in terms of the presenting cover. An $n$ -Helly axiom on the presenting covers supplies none of them, being satisfied by every finite atomization for every $n$ , and a $1$ -Helly family of $N$ compact open sets can have $2^{N}-1$ atoms. We show that a laminar presentation, a laminar basis of compact open sets in each nerve degree, supplies the third, and the second as soon as it is countable: the atoms of a finite laminar family are exactly its nonempty nodes, so a subordinate model has rank at most the number of members presenting it, the subordinate models of finite height form a complete system containing, for a countable presentation, a chain that is itself complete, and the transition maps are the least-ancestor maps of the presenting subfamilies. Every second countable Hausdorff ample groupoid admits one, and the choice is made in each nerve degree separately, laminarity not being stable under products. The first requirement is a restriction on $G$ : the homology of a model is finitely generated over a Noetherian ring, whereas $H_0$ over $\mathbb Z$ is not for the unit groupoid on a Cantor set or for the CAR groupoid, and a truncated model carries free homology of arbitrary rank in arbitrarily high odd degree where the groupoid has none. The chain-level statements are accompanied by a Lean 4 formalization.

Primarily human-written textHuman understanding: all partsAmple GroupoidsHK-Conjecture

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hexagon:2610.00228v1

math.NT — Number Theory

Every nonsquare integer not equal to $-1$ is a primitive root modulo a positive lower density of primes

Contributed by Manjul Bhargava

For every integer $a$ that is neither a square nor $-1$ , we prove that a positive lower density of primes have $a$ as a primitive root. More precisely, a zero-free half-plane $\Re s>1-1/N$ for all Dedekind zeta functions gives a lower proportion of at least $c_a/(N\log N)$ , with $c_a>0$ independent of $N\ge2$ . The theorem of Cureton and Thorner supplies $N=6000$ , so the positive-proportion conclusion is unconditional. The proof uses the marked Type II estimates of OpenAI to count primes $p$ such that $p-1$ is a small cofactor times one large prime. Allowing the cofactor to vary supplies an additional factor of $\log x$ in the count. We give the counting and sieve deductions in detail and explain the dependence on the zero-free width. The argument gives a lower bound rather than the exact Artin density.

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

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hexagon:2610.00226v1

math.NT — Number Theory

Milne's rationality conjecture

Contributed by Keerthi Madapusi

We prove Milne's rationality conjecture for reductions of Hodge classes on abelian varieties: The cohomological realizations of such classes pair rationally with Lefschetz classes (that is, classes in the algebra generated by divisor classes) in complementary codimension. The proof proceeds by establishing a dichotomy for reductions of Weil classes (an expected consequence of the rationality conjecture): Either they are orthogonal to all Lefschetz classes or are themselves Lefschetz classes. The key new input are lifting results of Kisin--Zhou for Hodge-type Shimura varieties. This is a cleaned up version of a preprint produced and released by OpenAI on Oct 6, 2026.

Primarily AI-generated textHuman understanding: all partsHodge cyclesShimura varietiesabelian varietiesalgebraic cycles

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hexagon:2610.00221v1

cs.DS — Data Structures and Algorithms

Deterministic Integer 3SUM in O(n^1.999017) Time via a Sharper Binomial-Tail Analysis

Contributed by 宇轩徐

We refine the analysis of the deterministic integer 3SUM algorithm of Alman and Vassilevska Williams (arXiv:2610.06783v1). Within their framework of thin matrix products and reductions through Exact Triangle, we obtain a worst-case running time of O(n^1.999017) for polynomially bounded integer inputs on a word RAM with O(log n)-bit words. The main refinement retains the binomial tail in the preprocessing cost instead of replacing it immediately by a uniform geometric bound. An entropy estimate yields a strictly larger preprocessing saving exponent, while preserving the query algorithm and its complexity. With explicit rational parameters c = 21, theta = 0.11202, a = 0.05665, and s = 0.06945, the resulting 3SUM exponent is 1.999016414375 + o(1), giving the stated fixed exponent with a positive margin. Separately, retuning the existing analysis without the tail refinement gives O(n^1.999073). We account for encoding, queries, prime selection, instance construction, witness scanning, and parameter rounding. The numerical inequalities are certified using exact rational interval arithmetic with explicit logarithm remainder bounds. Independent computations provide additional consistency checks. The contribution is a refinement of an existing algorithmic analysis, rather than a new matrix multiplication construction or reduction. No complete Lean formalization of the improved running-time bounds is claimed, and no global optimality or priority claim is made.

Primarily AI-generated textHuman understanding: no parts3SUMExact Trianglebinomial tailscomputer-assisted analysisdeterministic algorithmsfine-grained complexitythin matrix products

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hexagon:2610.00216v1

math.NT — Number Theory

A seven-eighths zero-free half-plane for unitary Weil L-functions

Contributed by Boyang Hu

We prove that for every number field $F$ and every finite-dimensional continuous unitary representation $r$ of the global Weil group $W_F$ , the $L$ -function $L_F(s,r)$ has no zeros in $\operatorname{Re} s>7/8$ , and that its only poles in this half-plane are at $s=1+it$ , with order equal to the multiplicity of $\mathcal N_F^{it}$ in $r$ . The analytic core is the rank-one case: $L_F(s,\psi)\ne0$ for $\operatorname{Re} s>7/8$ for every unitary Hecke character $\psi$ of every number field, including characters of infinite order. This extends the cubic-theta method of the OpenAI quasi-Riemann preprint, which treats finite-order characters over $\mathbb Q(\sqrt{-3})$ , by transferring its estimates to number fields containing $\mu_{12}$ and to families of fixed infinity type, followed by an abelian descent. Brauer induction and Frobenius reciprocity then give the result in every dimension. Consequences include the same half-plane for all Artin $L$ -functions and their unitary Hecke twists, an effective Chebotarev theorem, cuspidal weight-one newforms, and CM motives after the appropriate weight shift.

Primarily AI-generated textHuman understanding: some partsArtin L-functionsChebotarev density theoremHecke L-functionsWeil groupcubic theta functionzero-free regions

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hexagon:2610.00220v1

math.NT — Number Theory

A zero-free strip for the $L$ -functions of the elliptic curves $y^2=x^3+d$

Contributed by Michael Rubinstein, Mateo Restrepo Borrero

For a nonzero integer $d$ let $E_d$ be the elliptic curve $y^2=x^3+d$ over $\mathbb Q$ . Part I adapts OpenAI's $11/12$ argument to prove that $L(E_d,s)\ne0$ for $\Re s>17/12$ . Part II adapts OpenAI's $7/8$ argument to prove $L(s,\overline\alpha\eta)\ne0$ for $\Re s>7/8$ , where $\eta$ is any finite-order Hecke character of $\mathbb Q(\sqrt{-3})$ and $\alpha(n)=n/|n|$ on primary generators. Consequently $L(E_d,s)\ne0$ for $\Re s>11/8$ . The analytic normalization is $L(E_d,s+\frac12)$ , so the two boundaries are $11/12$ and $7/8$ within $0<\Re s<1$ . The curve characters have infinite order, and require new transformation and moment arguments. These are proved here; steps identical to the original arguments are cited. Part II also uses the ideas from OpenAI's finite-order $7/8$ theorem. Higher-weight and symmetric-power extensions are discussed; more general CM forms are the subject of a second paper.

Primarily AI-generated textHuman understanding: some partsCMelliptic curvequasi Riemann Hypothesis

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hexagon:2610.00215v1

math.NT — Number Theory

Non-Archimedean Hudzik-Landes-Dragomir-Kato-Saito-Tamura Inequality

Contributed by K. MAHESH KRISHNA

In 1992, Hudzik and Landes \textit{[Math. Ann.]} derived a breakthrough generalization of the triangle inequality for two nonzero elements in normed linear spaces, which was generalized to finitely many nonzero elements independently in 2006 by Dragomir \textit{[Bull. Aust. Math. Soc.]} and in 2007 by Kato, Saito and Tamura \textit{[Math. Inequal. Appl]}. We derive a non-Archimedean version of Hudzik-Landes-Dragomir-Kato-Saito-Tamura inequality.

Primarily human-written textHuman understanding: all partsNon-Archimedean linear spaceNormed linear spaceTriangle inequalityUltra-norm.

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hexagon:2610.00219v1

math.NT — Number Theory

Gonality and quintic points on intermediate modular curves

Contributed by Marin Varivoda

For $\{\pm 1\}\subseteq\Delta\subseteq(\mathbb{Z}/N\mathbb{Z})^\times$ let $X_\Delta(N)$ be the intermediate modular curve between $X_1(N)$ and $X_0(N)$ . We determine the $\mathbb{Q}$ -gonality of every strict intermediate curve $X_\Delta(N)$ with $N\le 50$ , and of $X_1(N)$ for $41\le N\le 50$ , where it equals the upper bound of Derickx and van Hoeij at every level. We also settle the thirty levels left open by Derickx, Hwang, Jeon and Orli\'c, completing the list of curves $X_0(N)$ with infinitely many points of degree $5$ , and we show that exactly $43$ strict intermediate curves have infinitely many points of degree $5$ . As a consequence, the strict intermediate curves of density degree $5$ are exactly the four $\mathbb{Q}$ -pentagonal curves found by Orli\'c. The proofs use published classifications, the Castelnuovo--Severi inequality, and three computational methods; all computations can be rerun from the accompanying code.

Primarily AI-generated textHuman understanding: some partsgonalityintermediate modular curveslow degree pointsmodular curves

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hexagon:2610.00214v2

math.AC — Commutative Algebra

Hilbert functions of chopped ideals of general plane points

Contributed by Vlado Vojdanovski

Let Z be a finite set of general points in the projective plane over a field of character- istic zero. We determine the Hilbert function of the ideal generated by the degree-d equations of Z, for every d. This proves the plane case of the Expected Syzygy Conjecture. The proof realizes the quotient of the point ideal by its degree-d equations as the cokernel of a linear block in a Hilbert–Burch matrix, and applies natural cohomology for general plane Steiner bundles.

Primarily AI-generated textHuman understanding: all parts

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hexagon:2610.00203v1

math.PR — Probability

Point Density Measures: A Measure-Theoretic Framework with Applications to Geometric Probability

Contributed by Rajesh Dachiraju

The notion of point density is widely used in geometry, probability, sampling theory, computational science, and spatial statistics to describe the distribution of discrete point sets. Although commonly interpreted as the number of points per unit volume, the concept is often used without a unified measure-theoretic formulation, leading to ambiguities between counting measures, empirical measures, sampling measures, intensity measures, and density functions. This paper develops a unified measure-theoretic framework centered on the concept of a \emph{point density measure}, regarded as the intrinsic mathematical object describing the spatial distribution of point density over a measurable space. Counting measures and empirical measures are considered as fundamental constructions associated with discrete point sets, while point density functions arise only when the corresponding point density measure is absolutely continuous with respect to a prescribed reference measure, in which case they are obtained through the Radon--Nikodym theorem. The framework clarifies the relationships among these measure-theoretic objects and provides a consistent mathematical language for deterministic point configurations and probabilistic sampling procedures. As an application, Bertrand's paradox is reinterpreted within this framework by viewing the classical sampling procedures as inducing different point density measures on a common measurable parameter space. The resulting probability measures are obtained by normalization of the corresponding point density measures, illustrating that the differing probabilities arise from different choices of underlying measures rather than from an inconsistency in probability theory. The proposed framework provides a unified perspective on point distributions with potential applications in geometric probability, stochastic geometry, spatial statistics, computational geometry, sampling theory, image analysis, and point cloud processing.

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

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hexagon:2610.00213v2

math.QA — Quantum Algebra

An order-five Haagerup--Izumi vertex operator algebra at central charge eight

Contributed by Tzu Chen Huang

We construct an explicit unitary strongly rational vertex operator algebra $U$ of central charge eight inside $V_{A_4\oplus A_4}$ and identify $Rep(U)$ with the Drinfeld center of the positive order-five Haagerup--Izumi category. The construction adjoins one algebraic weight-two field to a diagonal affine $\mathfrak{sl}_2$ at level forty and the ambient conformal vector. Exact finite lattice identities imply strong generation and $C_2$ -cofiniteness. A finite integral order in the actual Zhu algebra descends semisimplicity from a $1590$ -dimensional upper presentation over $\mathbb F_{10069}$ . To determine the modules, we combine the good-prime block bound with lattice character growth and a Wronskian inequality. The resulting full branching has an irreducible congruence complement of dimension at most fifteen; its exact classification forces all fourteen denominator- $29$ sectors. Finally, the actual holomorphic $E_8$ extension determines the boundary fusion category and its associator. The proof is computer-assisted, with exact arithmetic certificates separated from exploratory numerical evidence.

A mix of human-written and AI-generated textHuman understanding: some partsFusion categoriesvertex operator algebra

Oct 13, 2026PDF ↗View work →

hexagon:2610.00212v1

math.AG — Algebraic Geometry

The Chaise Longue Theorem: Conjecture 1.2 of Degtyarev–Shimada for the Fermat varieties of every degree in every even dimension, and the integral Hodge conjecture for those whose degree is prime, equal to 4, or prime to (n+1)!

Contributed by Rafael Amichis Luengo

Let $X \subset \mathbb{P}^{n+1}$ be the complex Fermat variety of degree $m \ge 3$ and even dimension $n = 2k$ , and let $L(X) \subseteq H_n(X; \mathbb{Z})$ be the subgroup generated by the classes of its standard $k$ -dimensional linear subspaces. Degtyarev and Shimada conjectured that $H_n(X; \mathbb{Z})/L(X)$ is torsion free [DS, Conjecture 1.2]. We prove this conjecture for every degree $m \ge 3$ in every even dimension, modulo two topological results used by [DS]: Pham's theorem on the homology of the affine Fermat variety, and their intersection numbers. Its algebraic form — the quotient $\mathbb{Z}[G]/(\psi_J)$ of the group ring of [DS] is free abelian of rank $m^{2k+1} - Q_k(m)$ — is proved with no topology, and it has been verified for every degree by a complete proof in the proof assistant Lean 4 with the Mathlib library, with no gap and no axiom beyond the three standard ones. The proof reduces the conjecture, prime by prime, to dimension counts in truncated polynomial rings, proved by peeling off one variable at a time. Consequences: the integral Hodge conjecture for $X$ whenever $m$ is prime, or $m = 4$ , or $m$ is prime to $(n+1)!$ , with the integral Hodge classes of middle degree generated by the standard linear subspaces — in particular for the Fermat quartic of every even dimension; and the primitivity of the sublattices spanned by the linear subspaces that [DS] exhibit on the partial Fermat varieties, which [DS] proved conditionally on the conjecture. This work has not been refereed by a human expert.

Primarily AI-generated textHuman understanding: some partsDegtyarev–Shimada conjectureFermat varietiesLean 4integral Hodge conjecturelinear subspacestorsion

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hexagon:2610.00211v1

math.AT — Algebraic Topology

Axiomatic degree theory on topological manifolds

Contributed by Stoyan Apostolov

In this note we study extension of standard Euclidean degree in nonlinear analysis to the setting of general oriented topological manifolds. The axiomatization of Euclidean degree is adapted to this setting. Existence and uniqueness are proven. The existence does not rely on smooth approximations as in the Euclidean case -- this is irrelevant for manifolds which do not admit a smooth structure and could be ambiguous for manifolds admitting many smooth structures. Rather, we rely on a cohomological approach using only topological information. When compared to the Euclidean case, we think this approach makes the proofs simpler and more transparent. The standard notion of a degree of maps between closed oriented manifolds studied in algebraic topology is recovered as a special case. A generalization of the notion of winding number is developed and related to the notion of degree. The familiar composition formula of the Euclidean degree is also obtained in this context.

Primarily human-written textHuman understanding: all partsdegree theory

Oct 12, 2026PDF ↗View work →

hexagon:2610.00210v1

math.MG — Metric Geometry

The truncated octahedral conjecture

Contributed by Thomas Hales, Lark Song

Among three-dimensional parallelohedra of fixed volume, the Archimedean truncated octahedron uniquely minimizes surface area, affirming a conjecture of K. Bezdek from 2006. One inequality in the proof is verified by an exact computer-assisted certificate. The simplex, frame, and graphical-zonotope inequalities used in the proof hold in arbitrary dimension.

A mix of human-written and AI-generated textHuman understanding: all partscomputer-assisted proofisoperimetric inequalitytranslational tiling

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hexagon:2610.00209v1

math.NT — Number Theory

An analogue of the Erdos-Kac theorem for signed Iwasawa invariants of modular forms

Contributed by Anwesh Ray

Erdos and Kac proved that the number of distinct prime factors of an integer has a Gaussian limiting distribution. We prove an analogue for the signed Iwasawa invariants of non-ordinary weight-two modular forms. Under the hypotheses of this paper, we construct infinitely many forms congruent to a given form $g$ that realize every common nonnegative increase of its two signed $\lambda$ -invariants. We count the levels realizing each increase. In the resulting family, the common increase is asymptotically Gaussian, with mean and variance $\delta_\Pi\log\log X+O_g(1)$ for a positive constant $\delta_\Pi$ .

Primarily human-written textHuman understanding: all parts

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hexagon:2610.00208v1

math.CO — Combinatorics

Strongly regular graphs on 160, 232, 261 and 280 vertices from fusions of orbitals of PΓL(2,8) and 2^4:(S3×S3)

Contributed by Anton Kovaľ

We construct strongly regular graphs with parameters (160,54,18,18), (232,63,14,18), (261,64,14,16) and (280,63,14,14), and hence also their complements. All four parameter sets were marked "?" (existence unknown) in Brouwer's online table on 11 and 12 October 2026; they are the parameters of the hypothetical partial geometries pg(9,5,3), pg(7,8,2), pg(8,7,2) and pg(9,6,2). We give eight pairwise non-isomorphic graphs: three on 160 vertices, one on 232, two on 261 and two on 280. None of them is geometric. As far as we can determine, they are the first graphs known with these parameters. Five of the graphs have a short description over the projective line PG(1,8) and its quadratic extension PG(1,64). The vertices are pairs of points of PG(1,8) or conjugate pairs of points of PG(1,64) ∖ PG(1,8), each carrying a label in 𝔽_8*, together with a small orbit of points, pairs, or conjugate pairs. Adjacency is decided by cross ratios: for two labelled vertices on disjoint pairs, the quotient (or the product) of the two labels must equal the cross ratio of the two pairs or its inverse; for pairs sharing a point it must equal 1, and a labelled pair and a conjugate pair are adjacent when their cross ratio is a primitive cube root of unity. The "quotient" graphs on 232, 261 and 280 vertices have the automorphism group (SL(2,8) × C_7) ⋊ C_6 ≅ GL(2,8) ⋊ C_6 of order 21168; the two "product" graphs have C_2 × PΓL(2,8), of order 3024. One graph on 160 vertices lives on the 16 cells of a 4 × 4 grid and three copies of the 48 flags of its even transversals; its automorphism group is C_3 × 2^4:(S_3 × S_3), of order 1728. The other two graphs on 160 vertices have automorphism groups of orders 1728 and 576. The graphs were found by a computer search over fusions of orbitals of groups acting with two or three orbits, using the GAP library of tables of marks; the search reported five graphs, and enumerating all fusions of the same group actions gave three more. We do not classify the graphs with these parameters. The same constructions over PG(1,q) give, for q=4, the classical skew-lines graph of PG(3,2) and a second known srg(35,16,6,8), and no strongly regular graph for q=16 or q=32 under the groups searched (complete solver verdicts): within this model the graphs for q=8 appear to be sporadic. Strong regularity of every graph is checked by an exact, standalone, standard-library Python program, the constructions are re-implemented from the text of this paper, and all computer results were reproduced by an independent audit with separately written code and checked again by a referee with its own code. Also deposited on Zenodo with the full data archive (the citable record): https://doi.org/10.5281/zenodo.23313399

Primarily AI-generated textHuman understanding: some parts

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hexagon:2610.00207v1

math.NT — Number Theory

How long can an off-line zero survive the de Bruijn-Newman heat flow? An explicit unconditional lower bound for Lambda

Contributed by Pedro Caceres

Let $\Lambda$ be the de Bruijn-Newman constant. The Riemann hypothesis (RH) is equivalent to $\Lambda\le 0$ , and Rodgers and Tao proved $\Lambda\ge 0$ . We ask how small $\Lambda$ can be if $\zeta$ has a zero $1/2+\delta+i\gamma$ with $\delta>0$ . We use de Bruijn's strip, the zero dynamics of the Polymath project, and an explicit local zero count for the heat-flow deformations $H_\tau$ that we prove here. With these we show unconditionally that $\Lambda \ge \delta^2/(70\log\gamma\log\log\gamma)$ . Equivalently, every zero $\beta+i\gamma$ of $\zeta$ with $\gamma\ge 3\cdot 10^{12}$ satisfies $|\beta-1/2|\le (70\,\Lambda\log\gamma\log\log\gamma)^{1/2}$ ; below that height RH has been verified. We also give the bound as an explicit formula at each height, which improves slowly with $\gamma$ . The proof follows a weighted sum $\sum\psi(\mathrm{Re}\,z)(\mathrm{Im}\,z)^p$ over all zeros of $H_\tau$ in the upper half-plane, which is insensitive to collisions and to other off-line zeros. For a single pair, the imaginary part obeys an exact equation whose interaction term at $\tau=0$ is the energy slope $\partial_\sigma\log|\xi|$ of the other zeros. Under local isolation and regular spacing we prove a sharper bound, linear in $\delta$ . Simulations with the zeros up to height $10^4$ agree with both bounds. We do not prove RH.

Primarily human-written textHuman understanding: all partsC2 FunctionDirichlet SumsRiemann Zeta Function

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hexagon:2610.00206v1

math.CO — Combinatorics

A 23-Point Counterexample to a Modified Half-Frequency Conjecture for Union–Intersection Systems

Contributed by Jungang Chen

We consider a variant of Vince's half-frequency conjecture for union-intersection systems in which an entire system must be nontrivial, while a system is declared trivial only when all its blocks are singletons or when its only block is the whole ground set; every such trivial system is stipulated to be irreducible. For nontrivial systems, irreducibility retains Vince's two-part partition criterion, and complete irreducibility is tested on the system and all subsystems induced by actual blocks. We give a system on 23 points, generated by 17 three-element sets, whose union-intersection closure consists of 111 distinct blocks. Every point belongs to at most 55 blocks, yet the system is nontrivial and completely irreducible under the stipulated convention. A normal-form count, a membership-preserving quotient, and explicit certificates for the induced subsystems make the argument self-contained. This is a counterexample to the explicitly modified conjecture, not an assertion that the original wording of Vince's Conjecture 5 has a unique settled interpretation.

Primarily AI-generated textHuman understanding: no parts

Oct 12, 2026PDF ↗View work →

hexagon:2610.00205v2

math.CO — Combinatorics

The Dancing Sand Theorem: the 2-part of the sandpile group of the hypercube, for every n

Contributed by Rafael Amichis Luengo

Let $K (Q_{n})$ be the sandpile group (critical group, Jacobian) of the $n$ -dimensional hypercube graph $Q_{n}$ . Its odd part was determined by Bai in 2003; its $2$ -part has been open since then. Gao, Marx-Kuo, McDonald and Yuen determined its $n - 1$ largest cyclic factors, and Betzer the $n$ -th, through a remark of theirs stated without proof. We determine the whole 2-part of $K (Q_{n})$ for every $n$ , by an explicit rule. The group splits over the indecomposable tilting modules $T (\lambda)$ of $\mathrm{SL}_{2}$ in characteristic $2$ that occur in $V^{\otimes n}$ , with their multiplicities. The contribution of $T (\lambda)$ at the shift $i = (n - \lambda) / 2$ consists of explicit small cyclic factors and of $2^{| I |}$ large ones, where $I$ is the set of binary digits of $\lambda + 1$ below the leading one. The large ones are obtained by pooling a sequence of explicit integers — binomial valuations and carries — into a non-increasing sequence, in the manner of isotonic regression. The proof reduces each contribution, by elimination inside explicit $2$ -adic lattices, to the Smith form of an explicit $2^{| I |} \times 2^{| I |}$ integer matrix, and computes that Smith form from the valuations of its entries. It needs a unique cheapest matching for an upper bound and an inductive potential for a lower bound. As consequences we recover Bai's counts and the theorems of Gao, Marx-Kuo, McDonald and Yuen on the largest factors, we prove their Conjecture 4.14 without that remark, and we prove their Conjecture 5.4 and the formula for the $(n + 1)$ -th factor stated in the 2019 poster of three of them. We also prove the fold law: the 2-part of the sandpile group of the folded cube $Q_{n} / \langle (1, \dots, 1) \rangle$ is $2 \cdot \operatorname{Syl}_{2} K (Q_{n})$ , the group obtained by lowering every exponent by one. With the results above this gives the whole $2$ -part of the sandpile group of the folded cube, and, with the known odd part, its whole sandpile group. The whole proof has been checked in the proof assistant Lean 4 with the Mathlib library, with nothing cited. This work has not been refereed by a human expert.

Primarily AI-generated textHuman understanding: some partsLeanSmith normal formcritical groupfolded cubehypercubesandpile grouptilting modules

Oct 12, 2026PDF ↗View work →