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math.DS — Dynamical Systems

Collision Systems, with Applications to Geodesic Networks and Point Billiards

Contributed by Alec Shelley, Henry Bosch

A collision system is a finite set of point particles with masses, moving freely with velocities given by the gradient of a convex energy function of the momentum, whose collisions conserve total momentum and total energy and are otherwise unconstrained. Stationary geodesic networks near a line of multiplicity $n$ , point billiards with momentum- and energy-conserving collisions, and relativistic point particles are collision systems. We prove that the number of collisions of a collision system is at most exponential in its total mass, with an explicit base, whenever the energy is close enough to the Newtonian one, and that it can be exponential for every energy on a line. For geodesic networks this replaces the previously reported bounds, of order $n^2$ from below and $n^{n^2}$ from above, by exponential bounds on both sides; for point billiards it gives explicit exponential bounds. The exponential growth rates remain open.

A mix of human-written and AI-generated textHuman understanding: some partscollision systemselastic collisionsgeodesic netsnumber of collisionspoint billiardsstationary varifolds

Oct 29, 2026PDF ↗View work →

hexagon:2610.00600v1

cs.CC — Computational Complexity

Deterministic Factoring in Time $N^{o(1)}$

Contributed by Isaac Rudich, Louis-Martin Rousseau

The current best upper bound for deterministic factoring is $N^{1/5+o(1)}$ . We provide a Lean certificate that there exists a deterministic Turing machine that computes the smallest prime factor of every $N$ in $N^{o(1)}$ steps: for every $k\ge 1$ it runs within $C_k\,2^{n/k}$ steps on $n$ -bit inputs. The algorithm reduces factoring to counting divisors with prescribed leading bits, and counts them by leveraging the counting engine used to refute the Exponential Time Hypothesis (ETH). The Lean statement that a reader must check to verify the proof is only 13 lines and only uses Mathlib. The full proof is about 35,000 lines. This note was written less than 24 hours after the ETH refutations were posted. No human-readable explanation of these results exists yet, and their Lean proofs are long: Anthropic's alone runs to about 152,000 lines. Our proof re-proves only the part of the counting engine it needs and imports nothing from those developments. We hope this application, with its short statement, serves as a stepping stone for those working to understand the engine; a follow-up paper will explain this proof in detail.

A mix of human-written and AI-generated textHuman understanding: some partsExponential Time HypothesisLeancounting algorithmsdeterministic factoringformal verificationinteger factorization

Oct 29, 2026PDF ↗View work →

hexagon:2610.00599v1

math.NT — Number Theory

A logarithmic error term for subset sums of zero-sum-free sets

Contributed by Chao Liu

For a positive integer $k$ , let $f(k)$ be the minimum number of distinct nonempty subset sums of a zero-sum-free $k$ -element set, where the minimum is taken over all finite abelian groups. We prove $f(k)\ge \frac{k^2}{2}-13k\log(k+1)-116k$ . Together with the classical upper bound, this gives $f(k)=k^2/2+O(k\log k)$ , with the optimal leading coefficient. As a consequence, the Olson constant of every finite abelian group $G$ satisfies $\mathrm{Ol}(G)\le\sqrt{2|G|}+O(\log|G|)$ . The argument combines a triangular lower bound for repeated supports, a count using blocks of two and three copies, and a greedy expansion argument. No restriction is imposed on the order of the group or on the stabilizer of the set of subset sums.

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

Oct 29, 2026PDF ↗View work →

hexagon:2610.00594v1

cs.CC — Computational Complexity

Subexponential Weighted Counting and Exact Optimisation

Contributed by Deep Bhattacharjee

The deterministic algorithm of \oai{} that counts the satisfying assignments of a $k$ -CNF in time $2^{o(n)}$ is reproduced in full and extended from counting to sums of local weights with values in any commutative $\Q$-algebra. The extension gives deterministic $2^{o(n)}$ algorithms for exact Max-Cut and Max- $k$ -SAT on sparse instances, for the independence polynomial, the maximum independent set and the knapsack problem with a conflict graph on every graph, for $0$ - $1$ integer programs whose rows involve boundedly many variables, for Vertex Cover in time $2^{o(k)}+\poly(n)$, for the density of states of Ising and Potts models on bounded-degree graphs, for the Tutte polynomial on the hyperbolas $(x-1)(y-1)=q$ , for the minimum distance and the maximum-likelihood decoding of codes with sparse parity checks, for Max-Cut and Max- $k$ -SAT at every average degree $o(\sqrt{\log n/\log\log n})$ , for Max-Cut on dense graphs close to blow-ups of graphs with $o(n/\log n)$ vertices, and for every problem in $\SNP$.

A mix of human-written and AI-generated textHuman understanding: some partsExponential Time HypothesisLDPC codes.Max-SATTutte polynomialVertex Coverindependence polynomialpartition functionsweighted counting

Oct 29, 2026PDF ↗View work →

hexagon:2610.00593v1

math.MP — Mathematical Physics

Stability and Convergence of a Regularized Immersed Boundary Method under Dynamic Cartesian Refinement. Part III: Fluid-Driven Rigid-Body Motion

Contributed by Sida He

We study a conforming regularized immersed boundary discretization of the two-dimensional Eulerian Stokes–Newton system for one freely moving rigid body. Regularized interior moments complete the surface constraints used for prescribed motion. Positive excess mass and inertia define a fixed fluid–body kinetic inner product. A skew inertial correction is derived from the fictitious rigid interior; it has zero power but is necessary for the physical translational momentum equation. The method advances the body configuration with the preceding computed rigid velocity and then solves a linear joint fluid–body step on a dynamically enriched Cartesian space. Exact orthogonal transfer gives an energy identity independent of the number of mesh changes. Interior and surface moment defects control the discrepancy between the discrete background energy and physical kinetic energy. A volume-preserving geometry comparison and a moving coarse recovery family yield conditional joint convergence of the fluid velocity, rigid velocity, and body configuration on an interval with positive wall clearance. The error bound distinguishes the energy norm after geometry alignment from the generally weaker estimate at the same Eulerian points. The analysis assumes exact integration and explicit strong-solution regularity; it does not cover arbitrary density ratios or collisions. Manufactured coupled computations, independent residual checks, and a Lean algebraic companion document the numerical and formal verification scope.

Primarily AI-generated textHuman understanding: no parts

Oct 29, 2026PDF ↗View work →

hexagon:2610.00592v1

math.MP — Mathematical Physics

Stability and Convergence of a Regularized Immersed Boundary Method under Dynamic Cartesian Refinement. Part II: General Prescribed Motion

Contributed by Sida He

We study a regularized immersed boundary discretization of unsteady Stokes flow around a rigid body with prescribed translation and rotation. The interface may change its position and orientation, while remaining uniformly separated from the outer wall. The fluid spaces are conforming, exactly divergence free, and constructed on fixed physical Cartesian grids with a retained coarse space and arbitrarily changing local enrichments. The method uses regularized surface-moment constraints, backward Euler time stepping, and current-space orthogonal projection. We construct a time-dependent coarse lifting that satisfies every discrete rigid-velocity constraint exactly. A moving collar approximation has a small spatial error in $H^1$ and a small Eulerian time-derivative error in $L^2$ , although its time derivative need not be accurate in $H^1$ . This distinction permits a quantitative convergence proof without a restriction coupling the time step to the number of mesh changes. A shifted-interface trace estimate accounts for the motion of the reaction surface during each time step and for physical slip between time nodes. Under stated piecewise regularity and an $L^2$ interface-reaction assumption, the energy error is bounded by the square roots of the retained coarse size, regularization width, surface-panel size, and maximum time step, together with a specified load-consistency error. The analysis assumes exact integration and does not prove adaptive optimality or cover fluid-driven motion. Manufactured moving-ellipse computations and a Lean algebraic companion test specified implementation and proof components.

Primarily AI-generated textHuman understanding: no parts

Oct 29, 2026PDF ↗View work →

hexagon:2610.00591v1

math.NT — Number Theory

Integrality of iterated averages and pointwise prime powers

Contributed by Darij Grinberg

Suppose that we repeatedly replace the values of an integer-valued function on $\Z$ by averages over $p$ consecutive arguments, where $p$ is prime. If the first $\ell$ rounds of averaging give integers, will this remain true when we begin with the $p$ th powers of the original values? We prove that the answer is yes for $\ell\le p$ . For $p=2$ and $p=3$ , the answer is yes for every $\ell$ , and therefore also when integrality is required at every stage. For larger primes, a known counterexample first fails at round $p+1$ . The proofs use finite differences and polynomial interpolation. We also explain connections with earlier work on divisibility of finite differences and Stirling numbers.

A mix of human-written and AI-generated textHuman understanding: some partsInteger SequencesInteger-valued polynomialsdivisibilityfinite differencessupercongruences

Oct 29, 2026PDF ↗View work →

hexagon:2610.00589v1

math.NT — Number Theory

Nonconstant congruence speeds of integer tetration: period spectra, prime bases, explicit bounds, and densities

Contributed by Marco Ripà

Building on earlier human research into the constant congruence speed of integer tetration, including Appendix B of Ripà and Di Pietro (2026), and on a lifting-the-exponent (LTE) criterion publicly presented by Alper Ferudun for OEIS A390535 and A390576, this model-generated note proposes an explicit account of nonconstant speeds in non-squarefree numeral systems. Its principal results are an eventual floor-difference representation, an exact spectrum of attainable least periods, and explicit arithmetic progressions containing infinitely many prime bases whose congruence speed never becomes constant for each non-squarefree radix. The latter supplies a proposed proof of the coprime-infinitude assertion left conjectural in Appendix B of the earlier human-authored article. We also derive bounds for the least such base and exact natural densities, both among coprime bases and among all bases not divisible by the radical of the radix, together with relative prime densities. The mathematical derivations and manuscript are primarily GPT-6-generated. Ripà supplied the theoretical setting, earlier results and specialist guidance, and subsequently examined selected deductions and historical test cases. The manuscript has not received comprehensive independent human proof verification.

Primarily AI-generated textHuman understanding: some partscongruence speedeventual periodicityinteger tetrationlifting-the-exponent lemmanatural densitynon-squarefree radicesp-adic valuationsprimes in arithmetic progressions

Oct 29, 2026PDF ↗View work →

hexagon:2610.00590v1

math.PR — Probability

The sharp iid Berry–Esseen inequality: Analytic reductions and exact certificate proof

Contributed by Kacper Rodziewicz, Bartosz Kołodziejek

Let $X_1, X_2, \dots$ be independent and identically distributed real random variables with $\mathbb{E}[X_1] = 0$ , $\mathbb{E}[X_1^2] = 1$ , and finite third absolute moment $\rho = \mathbb{E}|X_1|^3 < \infty$ . Write $F_n(x) = \mathbb{P}((X_1 + \cdots + X_n)/\sqrt{n} \le x)$ and let $\Phi$ denote the standard normal distribution function. Define \[ \Delta_n = \sup_{x \in \mathbb{R}} |F_n(x) - \Phi(x)|, \qquad C_n = \sup \frac{\sqrt{n}\,\Delta_n}{\rho}, \qquad C^* = \sup_{n \ge 1} C_n, \] and let \[ C_{\mathrm{E}} = \frac{3 + \sqrt{10}}{6\sqrt{2\pi}} \approx 0.4097321837 \] denote Esseen's 1956 two-point lower-bound constant. We present a computer-assisted proof architecture and complete verification suite establishing unconditionally that \[ C_n \le C_{\mathrm{E}} \quad \text{for every } n \ge 1, \qquad C^* = C_{\mathrm{E}}. \] The proof combines a regularized Prawitz smoothing inequality, a conserved zero-bias/characteristic-function reduction on the all- $n$ tail, and exact finite verification certificates across 13 roles covering all $n \ge 1$ (`full_pass: "ESTABLISHED"`). All certificates, native C++/Python checkers, and a single-command multi-core Docker verification setup (`git clone https://github.com/rodziewiczk/berry-esseen-certificates.git && cd berry-esseen-certificates && docker compose up --build --abort-on-container-exit`) are publicly archived at https://github.com/rodziewiczk/berry-esseen-certificates (tag `v1.0.0-full-pass`).

Primarily AI-generated textHuman understanding: some parts

Oct 29, 2026PDF ↗View work →

hexagon:2610.00595v1

math.FA — Functional Analysis

Metric, Schauder and Operator-Valued Frames

Contributed by K. MAHESH KRISHNA

Notion of frames and Bessel sequences for metric spaces have been introduced. This notion is related with the notion of Lipschitz free Banach spaces. It is proved that every separable metric space admits a metric $\mathcal{M}_d$ -frame. Through Lipschitz-free Banach spaces it is showed that there is a correspondence between frames for metric spaces and frames for subsets of Banach spaces. Several characterizations of metric frames are obtained. Stability results are also presented. Non linear multipliers are introduced and studied. This notion is connected with the notion of Lipschitz compact operators. Continuity properties of multipliers are discussed. For a subclass of approximated Schauder frames for Banach spaces, characterization result is derived using standard Schauder basis for standard sequence spaces. Duals of a subclass of approximate Schauder frames are completely described. Similarity of this class is characterized and interpolation result is derived using orthogonality. A dilation result is obtained. A new identity is derived for Banach spaces which admit a homogeneous semi-inner product. Some stability results are obtained for this class. A generalization of operator-valued frames for Hilbert spaces are introduced which unifies all the known generalizations of frames for Hilbert spaces. This notion has been studied in depth by imposing factorization property of the frame operator. Its duality, similarity and orthogonality are addressed. Connections between this notion and unitary representations of groups and group-like unitary systems are derived. Paley-Wiener theorem for this class are derived.

Primarily human-written textHuman understanding: all partsBessel sequenceFrameLipschitz functionRiesz basismetric space.multiplieroperator-valued frame

Oct 29, 2026PDF ↗View work →

hexagon:2610.00583v1

cs.LO — Logic in Computer Science

On the scope of computability and formal axiomatic systems

Contributed by Jon Bohlin

This survey explores the theoretical scope of effective computation and formal axiomatic systems, emphasizing a constructive perspective based on computable trees and their associated path spaces. No new technical results are presented; rather, the aim is to synthesize established results from computability theory and mathematical logic into a unified framework that clarifies what can be computed in principle, complementing the classical emphasis on undecidability results. It is argued that standard models of computation, such as Turing machines and equivalent formalisms, are computationally saturated: any effective extension of their instruction sets does not enlarge the class of effectively computable functions and algorithms. This saturation corresponds, on the logical side, to the representational completeness of Robinson arithmetic, which is sufficient to encode all computable functions despite being finitely axiomatized. The survey proceeds by showing how standard models of computation give rise to computable trees whose infinite paths correspond to complete and consistent, but typically non-computable, extensions. This construction links effectively closed classes in Cantor space with spaces of completions of formal theories and clarifies the boundary between effective computability and logical possibility. Since Cantor space is homeomorphic to the classical Cantor set, this framework provides a natural bridge between discrete computability-theoretic and logical structures on the one hand, and continuous topology on the other.

Primarily human-written textHuman understanding: all partsComputability

Oct 29, 2026PDF ↗View work →

hexagon:2610.00598v1

cs.AI — Artificial Intelligence

Do Self-Improving Coding Agents Beat Textbook Baselines? Evidence from a Darwin Gödel Machine Port and Two Published Systems

Contributed by Tejasvi Singh Tomar

Self-improving agent harnesses such as the Darwin Gödel Machine (DGM) let a coding agent edit its own scaffolding, and they keep the edits that score better and report large gains on coding benchmarks. In three pre-registered rounds, we compare what such loops find with standard engineering changes ("textbook moves"), one-shot redesigns by a language model, and placebo loops that make no edits; we also re-run released agents of the DGM and of ADAS with newer models. On hard LiveCodeBench problems with Claude Haiku 5.5, the starting agent often stops on an empty turn because its output-token limit is too small for the model's reasoning. A single textbook move, which raises this limit to 16,000 tokens and makes three other changes to the model loop, recovers about 90% of the gain of the best loop agent and costs less than half as much per task. Four of five one-shot redesigns by Claude Opus 5.5 also raise the limit, to 8,192 tokens, without evaluation feedback. The best loop agent still leads the textbook moves by 1–5 percentage points in pass rate. On held-out SWE-bench Verified tasks, run in the DGM's harness, the DGM's released best agent does not outperform its initial agent; three textbook moves on the initial agent outperform the best agent at a quarter of its cost, whether or not the patches of runs that ended early, most of them the best agent's, are kept. Claims of self-improvement should be measured against textbook moves, one-shot redesigns, placebo loops, held-out tasks and newer models.

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

Oct 28, 2026PDF ↗View work →

hexagon:2610.00597v1

math.AG — Algebraic Geometry

Serre’s conjecture on the Artin character for a regular local ring

Contributed by Mark Kisin

We prove Serre’s conjecture on the Artin character of a finite group G acting on a regular local ring A with isolated fixed points. The case of positive characteristic is due to Abe. In mixed characteristic we use an equivariant deformation to reduce to the case where A has an isolated singularity, in which case we can make use of the results of Beraldo and Pippi on the twisted conductor formula.

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

Oct 28, 2026PDF ↗View work →

hexagon:2610.00588v1

math.NT — Number Theory

Hankel determinants and the length of finite Pierce expansions

Contributed by Jason Davies

For integers $a>b>0$ , let $P(a,b)$ be the number of steps the iteration $x\mapsto a \bmod x$ takes to reach $0$ from $b$ ; it is the length of the Pierce expansion of $b/a$ . We prove $P(a,b)\ll_\varepsilon a^{\beta_*+\varepsilon}$ , where $\beta_*=(41301-3\sqrt{82849})/137480=0.2941336553\ldots$ , improving the exponent $1/3-2/177$ of Chase and Pandey. The tail of an orbit carries a sequence of positive integers, the Hankel determinants of a natural moment sequence, and these are small where the drops of the orbit are small. Grouping orbit states by such integers places the quotient differences of each group on curves of bounded degree, whose integer points we count with the uniform bounds of Castryck, Cluckers, Dittmann and Nguyen. The resulting exponent approaches $5/17$ when these curves have large degree relative to the logarithmic size of the box. We raise the degree by adjoining further integer invariants, by an effective Hilbert irreducibility sieve, and by an exact identity for the direction of the curvature vector, and come within $1.61\cdot10^{-5}$ of $5/17$ . We also give an elementary proof of the exponent $3/10+\varepsilon$ and a bound for orbits whose quotient increments rarely rise. All results are formalised in Lean 4 with Mathlib. Shallit's conjectured $O((\log a)^2)$ bound remains open.

Primarily AI-generated textHuman understanding: some partsHankel determinantsHilbert irreducibilityLean formalizationPierce expansionsinteger points on curvesiterated remainders

Oct 27, 2026PDF ↗View work →

hexagon:2610.00584v1

math.CA — Classical Analysis and ODEs

On some positivity properties of the Riemann $\Xi$ function

Contributed by Suvrit Sra

We prove strict log-concavity of the positive functions $\Theta$ and $\Lambda$ with one-sided and bilateral Laplace transforms $1/\xi(\tfrac12+\sqrt s)$ and $1/\Xi$ , respectively, where $\Xi(z)=\xi(\tfrac12+iz)$ . For either function, log-concavity is the Pólya frequency property of order two (PF $_2$ ). Without assuming RH, but using known zero verification through height $3\cdot10^{12}$ , we further prove that $\Lambda$ is a PF function of order ten (PF $_{10}$ ). We also prove local PF positivity for $\Theta$ : for each fixed order, the ordered minors of $\Theta(x-y)$ are positive when all differences $x_i-y_j$ are sufficiently small and positive. Additionally, we prove the generalized Laguerre inequalities: if RH holds up to height $T>1/2$ , then $L_n[\Xi](x)\ge0$ for every $n\ge0$ and $|x|\le T-1/2$ ; the case $n=1$ is due to Csordas, Ruttan and Varga (1991).

A mix of human-written and AI-generated textHuman understanding: all partsPolya frequencyRiemann Xicomplete monotonicitygeneralized Laguerrelog-concavity

Oct 27, 2026PDF ↗View work →

hexagon:2610.00587v1

math.QA — Quantum Algebra

Lefschetz coefficients of quantum theta functions

Contributed by Qiyue Tang

We prove that the coefficients of quantum theta functions in the skew-symmetric setting are characters of finite-dimensional $\mathfrak{sl}_2$ representations. This proves Conjectures 1.4, 2.17 and 3.13 of Davison–Mandel and the two-wall conjecture stated as Conjecture 3.12 in Davison’s ICM 2026 survey.

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

Oct 27, 2026PDF ↗View work →

hexagon:2610.00586v1

math.NT — Number Theory

Infinitely many elliptic curves with torsion $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/4\mathbb{Z}$ and rank at least six

Contributed by Manvir Jaswal

We show that there are infinitely many elliptic curves over $\mathbb{Q}$ with torsion subgroup $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/4\mathbb{Z}$ and rank at least $6$ , answering the case $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/4\mathbb{Z}$ of Problem 2.6 in Dujella's list of open problems. The largest rank previously known to occur infinitely often with this torsion was $5$ , due to Eroshkin. The curves are fibers of a one-parameter family of Dujella and Peral, which we show has rank exactly $3$ over the rational function field. The family is a base change of an elliptic K3 surface, and quadratic twists of that surface give it three further independent points over the function field of an elliptic curve of rank one. Congruences then carry the independence of the six points, and the torsion $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/4\mathbb{Z}$ , from one rational point of that curve to infinitely many. The result is unconditional. It contradicts the prediction of Park, Poonen, Voight and Wood that only finitely many curves with this torsion have rank greater than $5$ , although the curves form a sparse family. A second family, over an elliptic curve of rank two, gives another proof.

Primarily AI-generated textHuman understanding: no partsDiophantine triplesElliptic CurvesMordell-Weil Rankelliptic surfacesspecializationtorsion subgroup

Oct 27, 2026PDF ↗View work →

hexagon:2610.00571v1

math.NT — Number Theory

Thakur's Conjectures in Function Field Arithmetic: A Report On Queries to Frontier Agents

Contributed by Taylor Dupuy

This report was created on 23 October 2026 from six AI research reports generated with the frontier-dump skill on Dinesh Thakur's conjectures in function field arithmetic. Taylor Dupuy initiated those queries during his talk How to Use AI in Mathematical Research Effectively at the University of Rochester Number Theory Seminar on that date. The six reports were synthesized with harvest-dumps and developed into this manuscript with using-latexpowers. We survey the current status of several conjectures of Dinesh Thakur and his coauthors on the arithmetic of $\mathbb{F}_q[t]$ : the degree hypotheses for power sums, the characterisation of zeta-like multizeta values in genus zero and in higher genus, the relations among multizeta values, the distribution of zeros of Goss zeta functions, and the automata-theoretic transcendence questions for special values. For each conjecture we give the precise statement and its status in the literature. The six computational research dumps report several results and refutations. Among them, the recursion predicted by Thakur's second power-sum hypothesis fails for an explicit family of exponents at thirteen non-prime values of $q$ (a computer-assisted result); the shuffle relations, even with Frobenius and Carlitz's Eulerian evaluations, do not generate all relations among multizeta values; a closed formula for a $v$ -adic valuation difference conjectured by Thakur holds only after a sign change (and, for $d\ge4$ , halving), and the corrected formula is proved, with computer assistance, for degrees $d\le 11$ ; and the $v$ -adic Newton polygon of a Goss zeta function over $\mathbb{F}_2[t]$ at a prime of degree two has a slope of multiplicity one, which appears to contradict a result claimed in a 2023 preprint (this conflict needs human verification). The shuffle result and an assembly of published theorems describing all relations are elementary, and we claim no novelty for them. We also write out a proof of a theorem of Wen and Yao on $v$ -adic arithmetic gamma values. Computational evidence is labelled as such throughout, and the open problems are listed. This manuscript is AI-generated by Claude Code (model claude-opus-5-5) from AI-generated research dumps; its proofs received only AI review, and no human has checked them.

Primarily AI-generated textHuman understanding: no partsGoss zeta functionsThakur conjecturesfunction field arithmeticmultizeta values

Oct 27, 2026PDF ↗View work →

hexagon:2610.00580v2

math.AG — Algebraic Geometry

The Generalized Vanishing Conjecture: The Two-Variable Theorem and the First Failing Dimension

Contributed by Roy van Rijn

We determine the exact dimensional range of the Generalized Vanishing Conjecture (GVC) for constant-coefficient differential operators over fields of characteristic zero. In two variables, we prove that $\Lambda^m(P^m)=0$ for every $m\geq 1$ implies $\Lambda^m(QP^m)=0$ for all sufficiently large $m$ and every fixed polynomial $Q$ . The proof combines Hall's marriage theorem, Newton polygon geometry, prime-dilation arguments, and the constant-term theorem of Duistermaat and van der Kallen. We then construct a homogeneous three-variable counterexample. Setting $\rho=t^2+xy$ , $A=\rho+x^2$ , $C=y\rho^2-2xt^2\rho-x^3t^2$ , and $\Delta=4\partial_x\partial_y+\partial_t^2$ , the pair $\Lambda=\Delta^6$ and $P=AC^2$ satisfies $\Lambda^m(P^m)=0$ for every $m\geq1$ , while $\Lambda^m(x^2P^m)\ne0$ for every $m\geq1$ . The construction is a homogeneous endpoint-contact lift of Long's three-dimensional Gaussian counterexample and extends to an explicit cusp-profile family with arbitrary winding, profile, and radial suspension. In particular, every $\Delta^k$ for $k\geq6$ admits such a failure. Consequently, unrestricted, and already homogeneous, GVC holds in dimension $n$ exactly when $n\leq2$ .

A mix of human-written and AI-generated textHuman understanding: some partsGeneralized Vanishing Conjecture

Oct 27, 2026PDF ↗View work →

hexagon:2610.00579v1

math.CO — Combinatorics

Analytic q-inequalities and a proof of Bergeron's conjecture

Contributed by Veronica Phan

For integers $1\le a\le b\le c\le d$ with $ad=bc$ , Bergeron asked whether the difference of Gaussian binomial coefficients $\binom{b+c}{b}_q-\binom{a+d}{a}_q$ has nonnegative coefficients. We prove that it does, and more generally that $\binom{b+c}{b}_q-\binom{a+d}{a}_q$ has nonnegative coefficients whenever $a\le b\le c$ and $ad\le bc$ , a strengthening proposed by Troyka. The proof is analytic rather than bijective. We develop a toolkit for coefficientwise inequalities, including a coefficientwise local limit theorem for Gaussian polynomials based on Irwin--Hall densities, and use it in a Pascal-type induction that proves the case $a\le 20$ . For general $a$ the problem reduces to a two-parameter inequality between linearly interpolated Gaussian polynomials. Near its base case this inequality reduces exactly to the monotonicity of rational $q$ -Catalan numbers. Using the circle method with explicit error terms, we prove the conjecture of Chen, Wang and Wang that the coefficients of MacMahon's $q$ -Catalan numbers $C_n(q)$ are unimodal for every $n\ge16$ , and an analogue for the rational $q$ -Catalan numbers $C_{x,N+1}(q)$ with $x\le N\le 2x$ . Beyond this range the difference splits into two nonnegative parts, which we compare degree by degree using local limit bounds for Gaussian polynomials that are uniform in both parameters. The proof is computer-assisted: it uses exact integer and rational arithmetic, ball arithmetic, and one floating-point computation with a proved error bound. The scripts and the logs of their runs are provided as ancillary files.

Primarily AI-generated textHuman understanding: no parts

Oct 27, 2026PDF ↗View work →

hexagon:2610.00578v1

math.NT — Number Theory

Sun's supercongruence for an Apéry-like sequence

Contributed by Oleksiy Babanskyy

Let $a_n=\sum_{j=0}^n\binom{n}{j}^2\binom{2j}{j}$ . We prove that, for every prime $p>3$ , the sum $(-1)^{(p-1)/2}\sum_{n=0}^{p-1}\binom{2n}{n}a_n/54^n$ is congruent modulo $p^2$ to $4x^2-2p$ when $p=x^2+3y^2\equiv1\pmod3$ , and to zero when $p\equiv2\pmod3$ . For the latter primes a quadratic differential invariant of the truncated generating polynomial lifts Sun's congruence modulo $p$ . For the former primes we certify two modular periods directly by identities of weight eight, construct a Hecke polynomial with an explicit degree bound, and specialize a finite polynomial identity at a complex multiplication point. The coefficient at degree $p$ and both parity cases of the complex multiplication matrices are treated explicitly. A corresponding evaluation of the Franel sum with denominator $50$ follows by the same arguments.

Primarily AI-generated textHuman understanding: some partsApéry-like sequencesFranel numberscomplex multiplicationmodular formssupercongruences

Oct 27, 2026PDF ↗View work →

hexagon:2610.00574v1

cs.CC — Computational Complexity

On counting satisfying assignments via polynomial separation

Contributed by Scott Kominers

We give a preliminary exposition of a qualitative form of the main result of OpenAI's preprint "The Exponential Time Hypothesis is false": for every fixed $\varepsilon>0$ , $3$ -SAT on $n$ variables with $m$ clauses can be decided in time $O_\varepsilon(2^{\varepsilon n}(n+m)^{O(1)})$ and polynomial space. We separate the general counting procedure from the construction that supplies its input. The procedure deforms the clause inputs so that, at designated points, the count factors into local sums, from which Hermite interpolation recovers the original count; the construction builds such deformations from coordinate swaps in a tensor basis, leveraging a parity interpolation in the spirit of Alman and Williams.

A mix of human-written and AI-generated textHuman understanding: some parts3-SATExponential Time HypothesisHermite interpolationpolynomial interpolationsparsification

Oct 27, 2026PDF ↗View work →

hexagon:2610.00573v1

math.DS — Dynamical Systems

Periodic Forcing, Echo Constraints and the Non-Periodicity of the Rule 30 Centre Column: Reductions, Structural Lemmas, Certified Computations and Barriers

Contributed by Isaac kaimfa

Wolfram's Rule 30 Prize Problem 1 asks whether the centre column of Rule 30, started from a single lit cell, is ever eventually periodic. We study the problem through a half-line system driven by a periodic boundary signal. Combining a known local identity with the standard reduction to finite seeds, Problem 1 follows from a single statement, the Echo Lemma: no finite half-line configuration driven by a non-constant periodic boundary can satisfy, at every lit boundary minute, the identity that Rule 30 forces on the neighbour of the centre. Around this formulation we prove a collection of exact structural results: a dependence-window lemma and a merging criterion for reconstructed columns, triangularity and monotonicity of the finite-support constraints, a forced-creation identity at boundary bounces, a description of the infinite survivors as a full shift for period two, and, for the two-sided period-two problem, uniqueness of the compatible left half together with exact forced identities near the centre, including a half-column that must vanish identically. For the linear relative Rule 150 we prove the analogue of the Echo Lemma for every odd period by a Frobenius self-similarity argument, and we identify exactly the two steps that fail for Rule 30. We report exhaustive and SAT-certified computations excluding counterexamples of bounded width, and we document a set of barriers: every exact mechanism found is confined to a bounded neighbourhood of the centre, no additive conserved quantity of small range exists, and the sideways reconstruction has a growing number of distinguishable states. The Echo Lemma, and with it Problem 1, remains open; the contribution of this article is a verified framework, labelled throughout as proved, certified or open.

A mix of human-written and AI-generated textHuman understanding: all partsRule 150Rule 30SAT certificatescellular automatacentre columneventual periodicityleft-permutive rules

Oct 27, 2026PDF ↗View work →

hexagon:2610.00572v1

cs.CC — Computational Complexity

Some exposition of “The Exponential Time Hypothesis Is False”

Contributed by Ryan O'Donnell

This is an exposition of the paper "The Exponential Time Hypothesis Is False" by OpenAI [Oct. 26, 2026], produced through discussion with an LLM. Among other things, it tries to highlight the approximating polynomial for parity, and writes the matrix version of this polynomial using (a light amount of) quantum language.

Primarily AI-generated textHuman understanding: no partsSAT

Oct 27, 2026PDF ↗View work →

hexagon:2610.00565v1

math.CO — Combinatorics

The maximum Laplacian ratio of a tree: a complete answer to a question of Brualdi and Goldwasser, kernel-checked in Lean

Contributed by Peter W. Murphy, MD

For a tree $T$ with Laplacian $L(T)$ , Brualdi and Goldwasser (1984) introduced the Laplacian ratio $\pi(T)=\operatorname{per} L(T)/\prod_v\deg v$ and asked which tree on $n$ vertices maximizes it. We answer the question for every $n$ . For every $n\ge 4$ the maximum is attained by a spider of cherry arms: a centre carrying pendant paths of length two (cherries) and arms, an arm being a vertex that carries cherries. For $n\ge 492$ the centre carries only arms, almost all with five cherries, and the exceptional arms are fixed by $6(n-1)\bmod 11$ through an explicit rule; for $4\le n\le 491$ the maximizer is given by an explicit table. Along the way we determine the exact exponential growth rate of the maximum, $\tfrac{64}{621}\rho^{n}\le\max_{|T|=n}\pi(T)\le 2\rho^{n-1}$ with $\rho=(621/64)^{1/11}$ , and prove the sharp inequality behind it: every pendant subtree $b$ has matching weight at most $\rho^{|b|}$ , with equality exactly for the five-cherry arm. The whole argument is formalized in Lean 4 with Mathlib and checked by the Lean kernel, depending only on Lean's three standard axioms. Large parts of it are certificates: exact data produced by untrusted generators and checked by the kernel. We explain the mathematics, show where each piece lives in the formal development and why it is organized that way, and discuss what the answer and the method suggest. Code, certificates and the Lean formalization: https://github.com/DrMurphyIsIn/brualdi-goldwasser (release v1.4.4; Zenodo doi:10.5281/zenodo.22983412).

A mix of human-written and AI-generated textHuman understanding: some partsLaplacian permanentLean 4Mathlibextremal graph theoryformal verificationtrees

Oct 27, 2026PDF ↗View work →

hexagon:2610.00582v1

math.MP — Mathematical Physics

Stability and Convergence of a Regularized Immersed Boundary Method under Dynamic Cartesian Refinement. Part I: Fixed Geometry

Contributed by Sida He

We analyze a conforming variational immersed boundary discretization of the two-dimensional unsteady Stokes equations with a fixed interface. Exactly divergence-free velocities are obtained from Cartesian bicubic Hermite streamfunctions; regularized surface moments impose the boundary condition, and backward Euler time stepping uses constrained orthogonal transfer. Arbitrary local fine-scale enrichments may be added or removed at every step while the complete coarse space is retained. For homogeneous interface data, a common coarse comparison family yields an exact energy identity, a physical no-slip defect of order (\sqrt{\varepsilon}+\sqrt{\ell}), and velocity convergence without a restriction coupling the time step to the number of mesh changes. We extend this analysis to a disk with prescribed time-dependent angular velocity. A coarse lifting reproduces rigid rotation exactly throughout every regularization support, independently of the refinement pattern and surface partition. Subtracting this lifting reduces the driven problem to the homogeneous constraint spaces with an explicitly controlled forcing perturbation. This gives driven stability, physical slip control, and a quantitative energy error estimate without assuming a uniform multiplier inf–sup condition. The method assumes exact integration and is not a standard staggered-grid AMR scheme. Manufactured unsteady stationary and rotating-disk calculations test the implementation and its quadrature sensitivity. General changes of interface geometry and fluid-driven rigid-body motion remain outside the analysis.

Primarily AI-generated textHuman understanding: no partsAdaptive Mesh RefinementComputational Fluid DynamicsImmersed Boundary Method

Oct 27, 2026PDF ↗View work →

hexagon:2610.00581v1

math.QA — Quantum Algebra

A Uniform Calculation of Weight-Preserving Infinitesimal Braid Deformations in Type D

Contributed by Hangang Wang

Over $\mathbb Q(q)$ , for every $n\ge4$ , we calculate the full space of classical-weight-preserving solutions of the linearized braid equation at the standard type- $D_n$ vector braiding. Its dimension is $(n+1)(n+2)/2$ . A coordinate slice for scalar and parameter variations, diagonal vector-basis changes, and alternating twists reduces the upper bound to a homogeneous vanishing problem. Outer-pair coefficients eliminate the zero-weight block through a recurrence closed by the two central weights. We also give a functorial formal local parametrization over local Artin algebras. The explicit integrable family belongs to earlier multiparameter constructions; an exact parameter dictionary is provided. No bibliographic priority claim is made for the full tangent-space exhaustion. The proof is uniform in $n$ ; finite-rank computations are diagnostics.

Primarily AI-generated textHuman understanding: some partsYang–Baxter equationexact computationformal deformationsinfinitesimal braid deformationsquantum groupstype D

Oct 27, 2026PDF ↗View work →

hexagon:2610.00576v1

math.OA — Operator Algebras

Improved Cauchy-Schwarz-Kaplansky Inequality

Contributed by K. MAHESH KRISHNA

We make an improvement and give a new proof for Cauchy-Schwarz-Kaplansky inequality derived by Kaplansky [\textit{Am. J. Math., 1953}].

Primarily human-written textHuman understanding: all partsC*-algebraCauchy-Schwarz-Kaplansky inequalityLagrange identity.

Oct 27, 2026PDF ↗View work →

hexagon:2610.00577v1

cs.CC — Computational Complexity

A $2^{o(n)}$ Time Algorithm for 3-SAT

Contributed by Anthropic

We refute the Exponential Time Hypothesis (ETH) by presenting a deterministic algorithm for 3-SAT on n variables that runs in $2^{o(n)}$ time. Our algorithm in fact solves the harder problem #3-SAT, providing a count of the number of satisfying assignments. The result was discovered by an internal research Claude model and has been formalized in Lean 4.

Primarily AI-generated textHuman understanding: some parts

Oct 27, 2026PDF ↗View work →

hexagon:2610.00570v1

math.GN — General Topology

A continuum whose hyperspace of regular subcontinua is homeomorphic to the irrationals

Contributed by Lucas Almeida Almeida

For a metric continuum $X$ , let $D(X)$ be the hyperspace of subcontinua of $X$ equal to the closure of their interior, with the Hausdorff metric. Camargo, Ordo\~nez and Ram\'irez asked whether $D(X)$ can be homeomorphic to the rationals or to the irrationals. We answer the latter question affirmatively. We construct a continuum $X\subseteq\ell^2$ as the closure of a countable tree of scaled copies of an indecomposable plane continuum. Two types of piece alternate: blocks with dense interior, and spines with empty interior carrying densely attached blocks. We characterize all regular subcontinua by nonempty convex sets of pieces satisfying a density condition, and prove that this coding is a homeomorphism onto a $G_\delta$ subset of a Cantor cube. We then show that no point of $D(X)$ has a compact neighborhood. The Alexandrov--Urysohn characterization yields $D(X)\cong\N^\N$ , and hence $D(X)$ is homeomorphic to the irrationals. We also explain why this construction does not extend to the rational case, which remains open. Github repository: https://github.com/LucasAlves8Sp/A-CONTINUUM-WHOSE-HYPERSPACE-OF-REGULAR-SUBCONTINUA-IS-HOMEOMORPHIC-TO-THE-IRRATIONALS

Primarily AI-generated textHuman understanding: all partsAI-assisted mathematicsBaire SpaceContinuumLean 4hyperspacesindecomposable continuumirrational numbersregular subcontinuum

Oct 27, 2026PDF ↗View work →

hexagon:2610.00569v1

math.NT — Number Theory

Polynomial parametrizations for two classes of Diophantine equations

Contributed by Bogdan Grechuk

We prove that the integer solution sets of two classes of Diophantine equations are finite unions of polynomial families with integer coefficients and unrestricted integer parameters. In the first class, each monomial, with at most one exception, contains a variable that occurs to the first power and in no other monomial. The second class consists of equations $M=f(t)$ , where $M$ is a monomial independent of $t$ with a variable occurring to the first power, and $f\in\Z[t]$ splits completely over $\Q$. If $M$ involves at most two variables, $f$ may instead be an arbitrary quadratic polynomial. We also give complete formulas for five equations listed as open in the monograph \emph{Polynomial Diophantine Equations: A Systematic Approach} (2024).

Primarily AI-generated textHuman understanding: some partsDiophantine equations; Polynomial parametrization; Integer solutions.

Oct 27, 2026PDF ↗View work →

hexagon:2610.00575v1

math.GR — Group Theory

The Unbalanced Group Theorem

Contributed by Colin Defant, Ken Ono

The classification of finite simple groups (CFSG) identifies the basic building blocks of all finite groups. It grew from nineteenth-century work of Galois and Jordan into a collective effort involving many mathematicians, with Gorenstein playing a central organizing role. Its proof is a vast network of interdependent results such as the Feit--Thompson odd-order theorem, the signalizer functor theorem, and Aschbacher's classical involution theorem. The original papers make up the first-generation proof. The second-generation treatment, initiated by Gorenstein, Lyons, and Solomon, reorganizes and simplifies much of this work. A key structural result in the first-generation program is the Unbalanced Group Theorem (UGT). Under the inductive hypotheses of the classification, it says that a nontrivial subgroup of odd order normal in the centralizer of an involution forces the ambient simple group to be a group of Lie type in odd characteristic or one of the alternating groups of certain degrees. While formalizing the first-generation proof, we found counterexamples to a closure assertion used in a published route to UGT. This obstruction concerns that particular argument. UGT also has other proofs. Here we explain the obstruction and develop an alternative argument for the simple-group case using Solomon's 1976 theory of maximal $2$ -components, which provides a framework for comparing the simple pieces arising in different involution centralizers.

Primarily human-written textHuman understanding: all parts

Oct 27, 2026PDF ↗View work →

hexagon:2610.00568v1

math.NT — Number Theory

A proof Gauss might have found

Contributed by Roelof van Dijk

Ben Green’s historical variant of Tao’s recent tenth discriminant challenge [5] asks how early Gauss’s class-number-one conjecture could have been proved using mathematics then in print. Through discussions with GPT-6 Astra and Opus 5.5, we construct a proof of Gauss’s class-number-one conjecture [1] from ingredients published by 1808, specializing the Frobenius determinant argument [6]. Counting values of the principal form and summing factorial valuations reduce the proof to a residue sieve below 17000.

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

Oct 26, 2026PDF ↗View work →

hexagon:2610.00567v1

math.OA — Operator Algebras

The regular pentagonal system and the Smith--Ward problem

Contributed by Shengwei Qiu, Thomas Sinclair

The three-dimensional function system of the regular pentagon fails the lifting property, giving a counterexample to the operator-system formulation of the Smith--Ward problem with C $^*$ -envelope $\mathbb C^5$ . We construct unital $k$ -positive maps from this fixed system into matrix algebras where distance from every ucp map is bounded below independently of $k$ . We also prove that every regular polygon system with at least five vertices detects nuclearity, while its dual characterizes the weak expectation property. These tensor-product characterizations follow from a graph dichotomy: a graph operator system and its dual are C $^*$ -nuclear when the graph is $2$ -chordal; otherwise they detect nuclearity and WEP, respectively.

Primarily AI-generated textHuman understanding: all partsoperator systems; lifting property; nuclearity; weak expectation property

Oct 26, 2026PDF ↗View work →

hexagon:2610.00561v2

math.MG — Metric Geometry

Volume Computation and Comparison for Four-Dimensional Bodies of Constant Width

Contributed by Hangang Wang

We compute and compare the volumes of explicit four-dimensional bodies of constant width one. The original five-entry comparison consists of the simplex-symmetric projection body $U_4$ , the canonical raised Meissner pair, the canonical raise of $U_3$ , the body $B_5$ , and $M_4$ . We combine exact geometric formulas, numerical integration, interval enclosures and inscribed rational polytopes to obtain a certified strict ordering of their volumes. We also compare $U_4$ with the direct four-dimensional AMO peabody. The two raised Meissner bodies are proved noncongruent but exactly equal in volume. An introductory account explains why different constructions benefit from different computational methods, and distinguishes approximate volume estimates from inequalities certified by exact or outward-rounded arithmetic.

Primarily human-written textHuman understanding: all partsBodies of constant widthcertified comparisonfour-dimensional volumeinterval arithmetic

Oct 26, 2026PDF ↗View work →

hexagon:2610.00547v1

math.CO — Combinatorics

Functional van Lint-Seidel Relative and Gerzon Universal Bounds

Contributed by K. MAHESH KRISHNA

We introduce the notion of functional equiangular lines in finite rank modules over subrings of $\mathbb{R}$ . We show that van Lint-Seidel relative and Gerzon universal bounds hold for this most general lines.

Primarily human-written textHuman understanding: all partsEquiangular lines.

Oct 26, 2026PDF ↗View work →

hexagon:2610.00564v2

math-ph — Mathematical Physics

Proving Conjectures of Ashoke Sen: F-Theory Limits, Small Black Holes and Dyon Indices

Contributed by Deep Bhattacharjee

We examine thirty-six statements of Ashoke Sen on duality, dyon indices, tachyon condensation, F-theory and black hole entropy, and decide which are proved, which are supported but unproved, and where each stops. We prove that every degeneration of F-theory on an elliptic Calabi--Yau variety that is weakly coupled at the generic point of the base is, after base change and rescaling, Sen's family, and that one exists exactly when $-2K_B$ has a section. We prove that the entropy of a two-charge small black hole is fixed by one critical value for every curvature-corrected action, so that the Gauss--Bonnet truncation matches the microscopic count only in four and five dimensions and Prester's Lovelock completion is the unique one that matches it in every dimension. Exact product expansions reproduce the known negative dyon indices, and the tachyon-dust caustic concerns the pressureless equations, not the rolling solution.

A mix of human-written and AI-generated textHuman understanding: all partsS-dualitySiegel modular formsblack hole entropytachyon condensation

Oct 26, 2026PDF ↗View work →

hexagon:2610.00557v1

math-ph — Mathematical Physics

Gibbs Approximation from Coarse-Graining of Finite Systems

Contributed by Zhenyu Sun, Chen Chen

Thermodynamic integrability in finite systems admits concave probability-response potentials beyond the canonical ensemble. We investigate how replacing an unresolved block of energy levels by its effective free energy constrains this class and determines the Gibbs distribution, both exactly and quantitatively. For a response family defined across all finite spectra, exact single-block replacement implies conditional independence within each block and selects the Gibbs family with its canonical free-energy normalization. On a bounded energy window, conditional log-odds discrepancies determine a common reference inverse-temperature scale and yield a dimension-independent total-variation bound. A discrete calibration protocol incorporates grid rounding, interpolation under stated regularity bounds, and finite-sample concentration guarantees. We bound the conditional log-odds discrepancies in terms of coarse-grained block-probability residuals through finite block suppression, a bounded outer-control displacement, and a local internal curvature bound. The estimate separates auxiliary system size, measurement precision, and probability conditioning. Smooth local perturbations show that finitely many observations do not uniquely determine the global response. Applications to finite-capacity reservoirs and fixed-coupling mean-force models distinguish work-form closure from temperature-energy scaling. For two subsystems sharing small, bounded, symmetric log-temperature fluctuations, the leading mutual information is half the squared log-temperature variance times the product of their reference canonical heat capacities.

Primarily human-written textHuman understanding: all partsGibbs distributioncoarse grainingprobability boundsstatistical mechanicsthermodynamic integrability

Oct 26, 2026PDF ↗View work →

hexagon:2610.00551v1

math.DS — Dynamical Systems

Uniqueness of four-body convex central configurations for all masses

Contributed by Tejasvi Singh Tomar

We prove that for every choice of four positive masses and every cyclic ordering of the bodies there is exactly one strictly convex planar central configuration with that ordering, up to similarity. This answers Problem 10 in the list of Albouy, Cabral and Santos, which Santoprete calls the Simó--Yoccoz conjecture. The main step is a uniform lower bound for the Hessian at convex central configurations: it is at least one quarter of its radial part, which vanishes only on translations and rotations. Dziobek's relations turn the indefinite part of the Hessian into a negative multiple of a square, and a Cauchy--Schwarz argument bounds this term by the radial part times the trace of an explicit $2\times2$ matrix in which the masses do not appear. We prove that this trace is less than $\frac34$ by interval arithmetic on the three-dimensional set of normalized convex central configurations, in the coordinates of Corbera, Cors and Roberts. Uniqueness then follows by a covering argument from the case of four equal masses. The computation has been repeated with a second, independently written program. The Hessian bound and the uniqueness theorem, including the computation, have been formalized in the Lean proof assistant and checked by its kernel, using only the standard axioms. As consequences, every degenerate four-body central configuration is concave, the convex central configuration depends analytically on the masses, and the known symmetry theorems for kites, isosceles trapezoids and rhombi follow in a few lines. The analytic dependence and the symmetry theorems are also formalized.

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

Oct 26, 2026PDF ↗View work →

hexagon:2610.00559v1

math.MG — Metric Geometry

Finite Truncated Lattice-Length Data on Three-Dimensional Flat Tori

Contributed by Changxia Lin

We study what finite truncations of the lattice-length multiset can determine about a three-dimensional flat torus. For a full-rank lattice \Lambda \subset \mathbb{R}^3, let \lambda_1(\Lambda) \leq \lambda_2(\Lambda) \leq \lambda_3(\Lambda) be its successive minima, and let \mathcal{L}_T(\Lambda) record the lengths of all nonzero lattice vectors of norm at most T, counted with vector multiplicity. First, on the class defined by \lambda_1 = a and \lambda_2 > T, we prove an exact volume classification: the set of possible covolumes is (F(a,T), \infty), where F(a,T) = \frac{aT}{2}\sqrt{3T^2 - a^2}. The lower bound uses Oppenheim's classical determinant-minimization theorem for Minkowski-reduced ternary forms; an explicit perturbed basis proves every larger volume is attained. Second, we give a pair of positive-definite Gram matrices with equal diagonal entries and matching two-coordinate data but different determinants. This yields a quantitative obstruction to uniform finite identification and gives the lower bound \sqrt{3} for the worst-case identification-radius constant. Schiemann's finite theta-series determination theorem supplies the complementary upper bound \sqrt{7/2} after conversion from squared lengths to radii. We also record a cubic sign-product identity for the eight signed basis vectors and explain its limitations. All contributions are separated from classical inputs, and the possible optimality of the constant \sqrt{3} and the broader topological question are left open.

Primarily human-written textHuman understanding: all partslattice geometry; flat tori

Oct 26, 2026PDF ↗View work →

hexagon:2610.00566v1

math.QA — Quantum Algebra

Deformations of Deligne's S_t

Contributed by Noah Snyder

Snyder and Spencer showed that every *ribbon* infinitesimal deformation of the trivalent category $\mathcal{S}_t$ underlying Deligne's $\mathrm{Rep}(S_t)$ comes from varying $t$ . Here we remove the braided hypothesis: every pivotal deformation of $\mathcal{S}_t$ , $t\neq0$ , over a local Artinian $\mathbb{C}$ -algebra comes from varying $t$ . The pivotal hypothesis turns out to be automatic, so the same holds for monoidal deformations. Since $\mathcal{S}_t$ is not generated as a pivotal category by its trivalent vertex, the crossing must be added as a generator, and the Reidemeister moves only survive in a deformed form, with lower order terms in the $2$ - through $6$ -box spaces. An open condition on rotational eigenvalues lets us lift the crossing in a rotation invariant way. A handful of overlap computations, in the spirit of Liu's classification of Yang--Baxter relation planar algebras, then kill every lower order term up to a change of the chosen crossing.

Primarily AI-generated textHuman understanding: some partsDeligne interpolationTensor categoriesdeformations

Oct 26, 2026PDF ↗View work →

hexagon:2610.00563v1

math.GR — Group Theory

Explicit Finite $p$ -Group Counterexamples to the Amit--Ashurst Conjecture

Contributed by Shihao Wang, Zhiwei Zheng

For every prime $p$ , we construct a finite $p$ -group $G$ of order $p^{498}$ and nilpotency class two for which the word $w=\prod_{j=1}^{12}[x_{2j-1},x_{2j}]$ admits a value $g\ne1$ whose fibre has size strictly less than $|G|^{23}$ . This gives counterexamples to the Amit--Ashurst conjecture for every prime. We determine the exact fibre size by reducing the word equation to matrix equations over $\mathbb{F}_p$ and counting their solutions.

Primarily AI-generated textHuman understanding: all parts

Oct 26, 2026PDF ↗View work →

hexagon:2610.00556v1

math.OC — Optimization and Control

Risk Aversion and Information in Robust Contracting

Contributed by Ruixue Yuan, Zhenyu Sun

We study hidden-action contracting with a risk-averse agent and partial knowledge of actions in a compact convex envelope of distribution--cost pairs. With finitely many outcomes and bounded deterministic wages, the principal's worst-case payoff admits an exact dual representation with an endogenous $f$ -divergence governed by transfer utility and an incentive multiplier. Under strictly concave utility, finite-multiplier branch optima follow a truncated likelihood-ratio rule, including at zero-probability outcomes. To address nonattainment induced by adversarial tie-breaking, we characterize the infinite-multiplier limit as a restricted zero-sum game. For polyhedral envelopes, we establish two-sided $O(1/\lambda)$ convergence when inverse utility is Lipschitz. Positive tolerance for suboptimal agent responses restores existence and confines the multiplier search to an explicit compact interval. A finite optimal multiplier gives a linear upper bound on tolerance loss; an optimal infinite-multiplier endpoint in a polyhedral envelope gives a square-root upper bound. A two-action example attains the square-root rate asymptotically. Restricting contracts to a coarse performance statistic contracts the endogenous divergence. We characterize equality in data processing under wage bounds, bracket the value of finer contractible information using fine and coarse dual minimizers, and derive quantitative bounds for approximately sufficient statistics. Numerical experiments demonstrate strict gains over utility-affine contracts and distinguish technological knowledge from the richness of contractible performance signals.

Primarily human-written textHuman understanding: all partsapproximate best responsesconvex dualityrisk aversionrobust contractssufficient statisticsunknown actions

Oct 26, 2026PDF ↗View work →

hexagon:2610.00560v2

math.NT — Number Theory

A lower explicit threshold for Chen's theorem under GRH

Contributed by Steve Hubbard

Assume the Generalized Riemann Hypothesis (GRH). We show that every even integer $N \ge \exp(\exp(13.375))$ can be written as $N = p + P_2$ , where $p$ is a prime and $P_2$ is either a prime or a product of two primes. This result is conditional on GRH and on Theorems 5, 6, 7 and Lemma 9 of Bordignon and Starichkova (Ramanujan J. 64, 2024), together with their auxiliary lemmas, Table 1 and display (6) that their Section 5.4 uses, used at new parameter values inside their own Section 5.4. It is not a new method; it is a re-choice of the parameters $u$ and $\alpha$ in their argument, combined with the observation that Chen's theorem itself needs only that the Section 5.4 main-term combination be positive, not that it exceed $4\cdot 10^{-3}$ , because the inequality $\pi_2'(N) \ge \pi_2(N) - \sqrt{N} - 1$ from the proof of their Corollary 1 (Section 5.5, p. 1241), which does not involve the threshold, gives $N = p + P_2$ as soon as $\pi_2(N) > \sqrt{N} + 1$ . In Lean 4 the threshold statement itself is stated with GRH as an explicit hypothesis and proved conditional on the journal's Theorems 5, 6, 7 and the linear sieve Theorem 4 at the constants used; Lemma 9, the Section 5.5 step, $\varepsilon(1500) < 1/8349$ and the Section 5.4 bracket arithmetic are proved in Lean. Theorems 5, 6, 7 and Theorem 4 are not formalized.

Primarily AI-generated textHuman understanding: some partsChen's theoremGeneralized Riemann HypothesisLean 4explicit boundssieve methods

Oct 26, 2026PDF ↗View work →

hexagon:2610.00550v1

math.DS — Dynamical Systems

Smale's sixth problem for five bodies in the plane, with a proof formalized in Lean

Contributed by Tejasvi Singh Tomar

We prove that for every choice of five positive masses the planar five-body problem has finitely many central configurations up to similarity. This settles Smale's sixth problem for five bodies in the plane. Albouy and Kaloshin proved it for masses outside a closed algebraic subset of codimension two. We follow their method. Infinitely many central configurations would give an algebraic curve of complex solutions. At the places at infinity of the curve the valuations of the mutual distances must balance. We determine the possible valuation vectors at all positive masses, together with the mass conditions under which each can occur. We treat limits in which four bodies form a cluster of zero potential energy by expanding the potential, which is constant along the curve, at a place. This excludes, at every positive mass, one of the two diagrams in which such clusters occur, and part of the other. The same expansion excludes one further diagram at every positive mass. What remains is an exact analysis, by computer algebra, of the sets of mass conditions that can hold together at positive masses. Exact rational certificates show that the balance fails under each of them, except on a few one-parameter families of masses, which we treat separately. On the rays through $(1,4,4,4,4)$ and $(1,1,4,4,4)$ the certificates do not apply. On the first ray they cannot apply: there the complex equations have a continuum of solutions, as Albouy and Kaloshin observed. On these rays we count the real central configurations: there are 306 and 294 up to orientation-preserving similarity. The certificates, the programs that check them and a proof of the theorem in the Lean proof assistant are available.

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

Oct 26, 2026PDF ↗View work →

hexagon:2610.00562v1

math.CO — Combinatorics

Edge multiset dimension of subdivided complete graphs

Contributed by Pedro Machado Manhães de Castro

An edge of a graph can be identified by its distances to a suitable set of vertices. If the reports omit the names of these vertices, only a multiset of distances remains, and finding a suitable set becomes more difficult. We determine the asymptotic cost of this loss of information for subdivided complete graphs. Let $K_n$ be the complete graph on $n\ge3$ vertices, and let $S_L(K_n)$ replace each edge with a path of $L$ edges, where $L\ge2$ is an integer. Let $\operatorname{edim}_{\mathrm m}$ be the minimum number of vertices whose distance multisets distinguish all edges. For every fixed $L\ge3$ , we prove $\operatorname{edim}_{\mathrm m}(S_L(K_n))=\Theta_L\!\left(n^{1+1/\lfloor L/2\rfloor}\right) \qquad(n\longrightarrow\infty).$ Retaining the names requires only $\Theta(n)$ vertices. Without them, the cost is quadratic at length three, has order $n^{3/2}$ at lengths four and five, and has smaller exponents at longer fixed lengths. We also settle existence at the two shortest lengths for every $n\ge3$ : no selection works at length two, whereas at length three at most $\binom n2$ vertices suffice. The proofs combine a count of integer vectors with realizable vertex labels. Sidon sets distinguish endpoint pairs at odd lengths, and exact finite certificates complete the length-three construction.

Primarily AI-generated textHuman understanding: all partsSidon setsedge multiset dimensiongraph subdivisions

Oct 26, 2026PDF ↗View work →

hexagon:2610.00555v1

math.FA — Functional Analysis

The operator Hilbert space $\mathrm{OH}$ embeds completely isometrically into a noncommutative $\mathrm{L}^1$ -space

Contributed by Cédric ARHANCET

We prove that Pisier's operator Hilbert space $\mathrm{OH}$ embeds completely isometrically into the predual of a $\mathrm{QWEP}$ von Neumann algebra, answering a longstanding question introduced by Pisier and later by Haagerup and Musat. The embedding is obtained as an ultraproduct of free probabilistic models $h_k = \Lambda^{\frac{1}{2}}c_k\Lambda^{\frac{1}{2}}$ realized in the free group factor $\mathrm{VN}(\mathbb{F}_\infty)$ , with the trace and circular family scaled according to the mass of the truncated measure $t^{-\frac{5}{3}} \mathrm{d} t$ .

Primarily human-written textHuman understanding: all partscircular elementscomplete isometriesnoncommutative $\L^1$-spacesoperator Hilbert spaceoperator-valued free probabilitypower-law truncationsultraproducts

Oct 26, 2026PDF ↗View work →

hexagon:2610.00544v2

math.NT — Number Theory

The Height-Conductor Ratio on Curve Towers: Closed Forms for the Fermat Tower, the One-Third Barrier, and the Attainable Set

Contributed by Zeng Chunping

For a tower of curves $X_n$ over a number field with fixed bad-reduction set and $g_n \to \infty$ , we study the ratio of the stable Faltings height to the conductor of its Jacobian, $\kappa_n := h_F(\mathrm{Jac}(X_n))/\log \mathrm{cond}(\mathrm{Jac}(X_n))$ . We prove four groups of results. (I) Closed forms for the Fermat tower: slope $\frac{\log p}{4}\,n$ and second-order intercept $-6\log A$ ( $A$ the Glaisher-Kinkelin constant), yielding $\kappa_{\mathrm{Fermat}} = \frac{1}{8}$ ; the orbit stratification is strictly increasing and converges to $\frac{1}{8}$ from below. (II) A one-third barrier $\kappa \leq \frac{1}{3}$ for $\mathrm{GL}_2$ -type towers with $\sigma \to 1$ , conditional on arithmetic BMY in general (Problem 10.1) and unconditional on modular towers, where the BMY input is a theorem (Dolce-Mercuri, Mayer) and where $X_0(N)$ attains equality asymptotically, so the constant cannot be improved. (III) The barrier is strictly stronger than BMY: $\kappa \leq \frac{1}{3} \Rightarrow R \leq \frac{4}{\sigma} - 1 \leq 3$ . (IV) The closure of the attainable set in the abelian-variety tower world is $[\frac{1}{12}, \frac{1}{3}]$ : the lower end is unconditional (Zhang, Wilms) and attained by Heisenberg towers, the upper end is the barrier and is attained by modular towers, and the interior is densely (countably) filled by reindexed product convexity; the curve-tower interior filling remains open. We further record three structural negative results: the archimedean channel is absorbed into the discriminant channel by the product formula; totally rational branch-point hyperelliptic families are closed off by Evertse's theorem; and the Belyi/PU route is likewise closed by $S$ -unit equations. The discriminant channel is recast in the thickness language with an explicit upper bound (unconditional for automorphic towers, conditional on the analyticity of $L(A,s)$ in general), and a fully unconditional alternative based on the Brumer-Kramer conductor bound moves - but does not close - the remaining gap. All conditional dependencies are explicitly flagged in the body; all corrections are recorded in the revision history, Appendix B.

Primarily human-written textHuman understanding: all partsBrumer--Kramer conductor boundFaltings heightFermat towerHeisenberg towerSzpiro conjecturearithmetic Bogomolov--Miyaoka--Yauconductorcurve towers

Oct 26, 2026PDF ↗View work →

hexagon:2610.00548v1

math.RA — Rings and Algebras

The rank of 3×3 matrix multiplication over F2 is 23

Contributed by Tejasvi Singh Tomar

The rank of the tensor of $3\times 3$ matrix multiplication over the field with two elements is at most $23$ by Laderman's algorithm, and Rudich and Rousseau recently proved that it is at least $22$ . We prove that it equals $23$ . Hence Laderman's algorithm uses the fewest multiplications among all bilinear algorithms over $\mathbb{F}_2$ and among all bilinear algorithms with integer coefficients. The proof uses the substitution method in the form developed by D'Ambrosio, Wang and Yang et al., which bounds the number of first factors of a decomposition that lie in a given subspace of the space of first factors. We sharpen these bounds for $111$ of Wang's $496$ classes of subspaces, constrain the ranks of the first factors through a $27\times 27$ flattening of the tensor, and list, up to symmetry, all sets of $22$ first factors that satisfy these constraints. An exact completion search shows that none of them is the set of first factors of a decomposition. The computation emits certificates, which are checked in the Lean 4 proof assistant. In one formal proof the largest checks are evaluated as compiled code, and in a second one the kernel evaluates every check, so that its final theorem depends only on the three standard axioms of Lean. The Lean proofs build on the formalization of Rudich and Rousseau and import their table of bounds and their reduction for block algorithms.

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

Oct 26, 2026PDF ↗View work →

hexagon:2610.00558v1

math.AG — Algebraic Geometry

Uniform sheafy Tate rings that are not stably uniform

Contributed by Christopher Birkbeck, Alex Torzewski

Buzzard–Verberkmoes and Mihara show that a stably uniform Tate Huber ring is sheafy. We construct examples of uniform and strongly sheafy Tate rings which are not stably uniform, answering Question 7 of Kedlaya’s Nonarchimedean Scottish Book. In the first example a rational localisation is non-reduced, whilst in the second the rational localisation is an integral domain but is not uniform. The two main results are due to ChatGPT 5.6 Sol. Both constructions, the rational localisations witnessing failure of stable uniformity, and their strong sheafiness have been formalised in Lean 4

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

Oct 26, 2026PDF ↗View work →