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hexagon:2610.00578v1

math.NT — Number Theory

Sun's supercongruence for an Apéry-like sequence

Contributed by Oleksiy Babanskyy

Let an=∑j=0n(nj)2(2jj)a_n=\sum_{j=0}^n\binom{n}{j}^2\binom{2j}{j}. We prove that, for every prime p>3p>3, the sum (−1)(p−1)/2∑n=0p−1(2nn)an/54n(-1)^{(p-1)/2}\sum_{n=0}^{p-1}\binom{2n}{n}a_n/54^n is congruent modulo p2p^2 to 4x2−2p4x^2-2p when p=x2+3y2≡1(mod3)p=x^2+3y^2\equiv1\pmod3, and to zero when p≡2(mod3)p\equiv2\pmod3. For the latter primes a quadratic differential invariant of the truncated generating polynomial lifts Sun's congruence modulo pp. 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 pp and both parity cases of the complex multiplication matrices are treated explicitly. A corresponding evaluation of the Franel sum with denominator 5050 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 ε>0\varepsilon>0, 33-SAT on nn variables with mm clauses can be decided in time Oε(2εn(n+m)O(1))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 TT with Laplacian L(T)L(T), Brualdi and Goldwasser (1984) introduced the Laplacian ratio π(T)=per⁡L(T)/∏vdeg⁡v\pi(T)=\operatorname{per} L(T)/\prod_v\deg v and asked which tree on nn vertices maximizes it. We answer the question for every nn. For every n≥4n\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≥492n\ge 492 the centre carries only arms, almost all with five cherries, and the exceptional arms are fixed by 6(n−1) mod 116(n-1)\bmod 11 through an explicit rule; for 4≤n≤4914\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, 64621ρn≤max⁡∣T∣=nπ(T)≤2ρn−1\tfrac{64}{621}\rho^{n}\le\max_{|T|=n}\pi(T)\le 2\rho^{n-1} with ρ=(621/64)1/11\rho=(621/64)^{1/11}, and prove the sharp inequality behind it: every pendant subtree bb has matching weight at most ρ∣b∣\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 Q(q)\mathbb Q(q), for every n≥4n\ge4, we calculate the full space of classical-weight-preserving solutions of the linearized braid equation at the standard type-DnD_n vector braiding. Its dimension is (n+1)(n+2)/2(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 nn; 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:2610.00577v1

cs.CC — Computational Complexity

A 2o(n)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 2o(n)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 XX, let D(X)D(X) be the hyperspace of subcontinua of XX equal to the closure of their interior, with the Hausdorff metric. Camargo, Ordo\~nez and Ram\'irez asked whether D(X)D(X) can be homeomorphic to the rationals or to the irrationals. We answer the latter question affirmatively. We construct a continuum X⊆ℓ2X\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 GδG_\delta subset of a Cantor cube. We then show that no point of D(X)D(X) has a compact neighborhood. The Alexandrov--Urysohn characterization yields D(X)≅NND(X)\cong\N^\N, and hence D(X)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)M=f(t), where MM is a monomial independent of tt with a variable occurring to the first power, and f∈Z[t]f\in\Z[t] splits completely over $\Q$. If MM involves at most two variables, ff 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 22-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 C5\mathbb C^5. We construct unital kk-positive maps from this fixed system into matrix algebras where distance from every ucp map is bounded below independently of kk. 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 22-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 U4U_4, the canonical raised Meissner pair, the canonical raise of U3U_3, the body B5B_5, and M4M_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 U4U_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:2610.00564v2

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 −2KB-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 2×22\times2 matrix in which the masses do not appear. We prove that this trace is less than 34\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 St\mathcal{S}_t underlying Deligne's Rep(St)\mathrm{Rep}(S_t) comes from varying tt. Here we remove the braided hypothesis: every pivotal deformation of St\mathcal{S}_t, t≠0t\neq0, over a local Artinian C\mathbb{C}-algebra comes from varying tt. The pivotal hypothesis turns out to be automatic, so the same holds for monoidal deformations. Since St\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 22- through 66-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 pp-Group Counterexamples to the Amit--Ashurst Conjecture

Contributed by Shihao Wang, Zhiwei Zheng

For every prime pp, we construct a finite pp-group GG of order p498p^{498} and nilpotency class two for which the word w=∏j=112[x2j−1,x2j]w=\prod_{j=1}^{12}[x_{2j-1},x_{2j}] admits a value g≠1g\ne1 whose fibre has size strictly less than ∣G∣23|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 Fp\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 ff-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/λ)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≥exp⁡(exp⁡(13.375))N \ge \exp(\exp(13.375)) can be written as N=p+P2N = p + P_2, where pp is a prime and P2P_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 uu and α\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⋅10−34\cdot 10^{-3}, because the inequality π2′(N)≥π2(N)−N−1\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+P2N = p + P_2 as soon as π2(N)>N+1\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, ε(1500)<1/8349\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)(1,4,4,4,4) and (1,1,4,4,4)(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 KnK_n be the complete graph on n≥3n\ge3 vertices, and let SL(Kn)S_L(K_n) replace each edge with a path of LL edges, where L≥2L\ge2 is an integer. Let edim⁡m\operatorname{edim}_{\mathrm m} be the minimum number of vertices whose distance multisets distinguish all edges. For every fixed L≥3L\ge3, we prove edim⁡m(SL(Kn))=ΘL ⁣(n1+1/⌊L/2⌋)(n⟶∞). \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 Θ(n)\Theta(n) vertices. Without them, the cost is quadratic at length three, has order n3/2n^{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≥3n\ge3: no selection works at length two, whereas at length three at most (n2)\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
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hexagon:2610.00555v1

math.FA — Functional Analysis

The operator Hilbert space OH\mathrm{OH} embeds completely isometrically into a noncommutative L1\mathrm{L}^1-space

Contributed by Cédric ARHANCET

We prove that Pisier's operator Hilbert space OH\mathrm{OH} embeds completely isometrically into the predual of a QWEP\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 hk=Λ12ckΛ12h_k = \Lambda^{\frac{1}{2}}c_k\Lambda^{\frac{1}{2}} realized in the free group factor VN(F∞)\mathrm{VN}(\mathbb{F}_\infty), with the trace and circular family scaled according to the mass of the truncated measure t−53dtt^{-\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
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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 XnX_n over a number field with fixed bad-reduction set and gn→∞g_n \to \infty, we study the ratio of the stable Faltings height to the conductor of its Jacobian, κn:=hF(Jac(Xn))/log⁡cond(Jac(Xn))\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 log⁡p4 n\frac{\log p}{4}\,n and second-order intercept −6log⁡A-6\log A (AA the Glaisher-Kinkelin constant), yielding κFermat=18\kappa_{\mathrm{Fermat}} = \frac{1}{8}; the orbit stratification is strictly increasing and converges to 18\frac{1}{8} from below. (II) A one-third barrier κ≤13\kappa \leq \frac{1}{3} for GL2\mathrm{GL}_2-type towers with σ→1\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 X0(N)X_0(N) attains equality asymptotically, so the constant cannot be improved. (III) The barrier is strictly stronger than BMY: κ≤13⇒R≤4σ−1≤3\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 [112,13][\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 SS-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)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
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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 3×33\times 3 matrix multiplication over the field with two elements is at most 2323 by Laderman's algorithm, and Rudich and Rousseau recently proved that it is at least 2222. We prove that it equals 2323. Hence Laderman's algorithm uses the fewest multiplications among all bilinear algorithms over F2\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 111111 of Wang's 496496 classes of subspaces, constrain the ranks of the first factors through a 27×2727\times 27 flattening of the tensor, and list, up to symmetry, all sets of 2222 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
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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
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hexagon:2610.00552v1

math-ph — Mathematical Physics

Area law for three-dimensional U(1) lattice gauge theory with the Wilson action at every coupling

Contributed by Tejasvi Singh Tomar

We prove that three-dimensional U(1) lattice gauge theory with the Wilson action confines static charges at every coupling. More precisely, for each β>0\beta>0 there are σ(β)>0\sigma(\beta)>0 and R1(β)<∞R_1(\beta)<\infty such that rectangular Wilson loops of sides R,T≥R1(β)R,T\ge R_1(\beta) satisfy ⟨cos⁡θC⟩≤exp⁡(−σ(β)RT+c(R+T))\langle\cos\theta_C\rangle\le\exp(-\sigma(\beta)RT+c(R+T)), uniformly in the finite box, with an absolute constant cc and an explicit lower bound on σ(β)\sigma(\beta). For the Villain action this is the theorem of G\"opfert and Mack. For the Wilson action it is cited as known but has remained open. The proof rests on a comparison. The Wilson plaquette weight is a mixture of Villain weights whose coupling has a very light upper tail. Hence, by Ginibre's inequality, the Wilson loop at coupling β\beta is at most an average, over a sparse Bernoulli set of frozen plaquettes, of the Villain loop at coupling KβK\beta, and frozen plaquettes are insulating walls for the monopole gas. A variational regularization, built on the smallest minimizer of a cut functional, freezes more plaquettes, which only increases the loop. It produces slit domains on which the multiscale analysis of G\"opfert and Mack, with wall fields graded by their periods around the handles, runs with uniform constants. On these domains the loop decays like the exponential of the minimal spanning area with frozen plaquettes free. A Peierls estimate shows that this area is at least (1−η)RT(1-\eta)RT outside an event of probability at most e−RT+O(R+T)e^{-RT+O(R+T)}, with η→0\eta\to0 as K→∞K\to\infty. We carry over the steps cited from Brydges\textendash Federbush and G\"opfert\textendash Mack with explicit changes and prove them in full in Part S4 of the supplementary material, including a tree-graph counting step of Brydges whose printed deduction is incomplete. As corollaries, we obtain the area law at every coupling for all closed subgroups of U(nn) containing the scalars, also with a Higgs field blind to the centre (as for U(2) with an adjoint Higgs field), and for the Georgi\textendash Glashow model at infinite Higgs coupling.

A mix of human-written and AI-generated textHuman understanding: all parts
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hexagon:2610.00549v1

cs.DS — Data Structures and Algorithms

Sorting networks on 13 inputs need 45 comparators

Contributed by Tejasvi Singh Tomar

Let S(N)S(N) be the least number of comparators in a sorting network on NN inputs (the Bose--Nelson problem). The values are known for N≤12N\le12, and for 1313 inputs the smallest known sorting network, found by Juill\'e, has 4545 comparators. We prove S(13)≥45S(13)\ge45, so S(13)=45S(13)=45. Deleting the comparators met by the largest and the second largest input leaves a sorting network on N−2N-2 inputs, so S(N)≥S(N−2)+P(2,N)S(N)\ge S(N-2)+P(2,N), where P(2,N)P(2,N) is the least possible worst-case number of comparators met by these two values. Van Voorhis stated P(2,N)≥⌈lg⁡F(N)⌉P(2,N)\ge\lceil\lg F(N)\rceil for an explicit function FF, but Mericanii showed that his proof is invalid. We prove P(2,N)=⌈lg⁡F(N)⌉P(2,N)=\lceil\lg F(N)\rceil for 3≤N≤323\le N\le32 by a finite search, which rests on the structure of the tree of paths of the largest value, the comparators on these paths, and a forest of comparators that carries the second largest value. With S(11)=35S(11)=35 this bound gives only S(13)≥44S(13)\ge44. For 4545 we apply it to the suffix of a network: if a network with LL comparators sorts a set XX of 00-11 vectors, then for suitable pairs of inputs the two values meet at most L−cL-c comparators, where cc is the cost of sorting the residual set left when their paths are deleted. A breadth-first search over the sets produced by prefixes of a hypothetical sorting network with 4444 comparators discards each set for which no single network stays within all these budgets, and no set survives at depth 2525. A separate verifier checks the certificate written by the search. Two proofs in Lean 4 check the same certificate, and in one of them Lean's kernel checks every computation, with only the standard axioms. Juill\'e's network is checked in the kernel as well. As a consequence, each of the twenty lower bounds on S(N)S(N) for 13≤N≤3213\le N\le32 that rested on Van Voorhis's inequality is proved and raised by one.

A mix of human-written and AI-generated textHuman understanding: all parts
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hexagon:2610.00546v1

math.GN — General Topology

ALL FINITE CARDINALITIES OF THE HYPERSPACE OF REGULAR SUBCONTINUA

Contributed by Lucas Almeida Almeida

For a metric continuum X, let D(X) be the family of subcontinua A of X with A = cl_X int_X A. Ordõnez asked which finite numbers occur as |D(X)|. Camargo, Ordõnez and Ramírez excluded 2, 4, 5, 8, 9 and asked whether every k ≥ 256 occurs. We prove that a positive integer k equals |D(X)| for some metric continuum X if and only if k not in {2, 4, 5, 8, 9, 16}. The finite cardinalities of D(X) are exactly the numbers of vertex sets inducing connected subgraphs of finite connected graphs; every finite connected graph is realized by copies of the Knaster continuum glued at points. We determine the values of this graph invariant without computer assistance. A short argument excludes 16, and an explicit recursive family of graphs realizes every k ≥ 17 with the least possible number of vertices. Consequently, for k ≥ 32 the least number of minimal regular subcontinua of a continuum X with |D(X)| = k is ⌊log2 k⌋ + 1. Programs check the explicit finite graph counts, and Theorem A is formally verified in Lean.

Primarily AI-generated textHuman understanding: all partsAI-assisted mathematicsContinuumLean 4enumeration of connected setshyperspacesindecomposable continuuminverse problemregular subcontinuu
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hexagon:2610.00545v1

math.NT — Number Theory

The missing Lin(ni)k

Contributed by Paul Pollack

We show how Selberg's symmetry formulas and the recent finding that there are no Landau--Siegel zeros connect to give a radically elementary proof of Linnik's theorem on the least prime in an arithmetic progression.

A mix of human-written and AI-generated textHuman understanding: all partsLandau--Siegel zerosLinnik's theoremSelberg's symmetry formulasieve methods
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hexagon:2610.00543v1

math.GN — General Topology

Recurrent tubes and connectedness-preserving bijections of Euclidean spaces

Contributed by Peter Lim

For every n≥2n\ge 2 we construct a connectedness-preserving bijection FF of Rn\mathbb{R}^n whose inverse does not preserve connectedness. The restriction of FF off the closed coordinate ray EE is a homeomorphism onto the complement of L=K∖{∞}L=K\setminus\{\infty\}, where K⊂SnK\subset\mathbb{S}^n is a nondegenerate continuum with empty interior. The discontinuity sets of FF and F−1F^{-1} are exactly EE and LL, and both maps can be chosen to have exact Baire class 22. The preimage of a fixed straight segment can be any prescribed uncountable closed subset of EE with empty relative interior, including the middle-thirds Cantor set. The construction is uniform in the dimension. Homeomorphisms with compact support move finitely many collar points to a prescribed net, and narrowing the resulting collars produces a recurrent tube. Compact arcs on one retained boundary line supply dense connected tracks, while a gate lemma proves preservation of arbitrary connected sets. This answers W. Wong's question in the negative. We also prove that every component of the closure of the discontinuity set of a planar connectedness-preserving bijection is unbounded, and construct Borel counterexamples on manifolds with a standard spherical end.

Primarily AI-generated textHuman understanding: some parts
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hexagon:2610.00542v2

math.NT — Number Theory

Goldbach numbers in intervals of length 4.45log⁡2x4.45 \log^2 x under the Riemann Hypothesis

Contributed by Steve Hubbard

Assuming the Riemann Hypothesis, we prove that for every real x≥3x \ge 3 the interval (x,x+4.45log⁡2x)(x, x + 4.45 \log^2 x) contains an even integer that is a sum of two odd primes. The proof combines a weighted mean square bound for the Chebyshev prime counting error in short intervals with a piecewise linear weight in a bilinear identity for short interval prime counts. The deduction is checked in Lean 4 conditionally on the explicit formula ExplicitA, the zero sum bound HadamardK, the zero count BPTN, the HSW zero counts, the bound ∣γ∣≥14|\gamma| \ge 14, and the Goldbach verification to 4⋅10184 \cdot 10^{18}, all stated as hypotheses.

Primarily AI-generated textHuman understanding: some partsGoldbach numbersLean 4Riemann Hypothesisexplicit boundsshort intervals
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hexagon:2610.00541v1

math.CO — Combinatorics

Maximal Sublattices of Convex Geometries with Unbounded Complement Minima

Contributed by Y. T. Miah

For every integer r≥3r\ge 3, we construct a finite atomistic convex geometry of convex dimension exactly three having a maximal proper sublattice whose complement has exactly rr minimal elements and exactly two maximal elements. The lattice has (r+2)2(r+2)^2 elements and the sublattice has (r+1)(r+2)/2+3(r+1)(r+2)/2+3 elements. At r=3r=3 the construction is the 25-element example of Bhattacharjee. Thus the number of minimal elements in such a complement is unbounded even when the convex dimension is fixed at three, in contrast to the common-minimum theorem in convex dimension two.

A mix of human-written and AI-generated textHuman understanding: some partsconvex dimensionconvex geometriesjoin-semidistributive latticesmaximal sublattices
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hexagon:2610.00540v1

math.RT — Representation Theory

A note of the Gorenstein symmetry conjecture (II)

Contributed by Qiyue Tang

We prove the Gorenstein symmetry conjecture for every finite-dimensional split square-zero extension of an algebra derived equivalent, over the ground field, to a finite-dimensional hereditary algebra.

A mix of human-written and AI-generated textHuman understanding: some partsGorenstein symmetry conjecture
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hexagon:2610.00538v1

math-ph — Mathematical Physics

Nonlinear Wave Limits of Reversible Finite-State Cellular Automata

Contributed by Zhenyu Sun

We construct reversible finite-state cellular automata whose integer lifts approximate smooth solutions of one-dimensional semilinear wave equations. The kick–drift update is bijective for every odd modulus. Separate rounding of spatial edge contributions and local nonlinear forces gives an exact one-cell support bound for the full local operator algebra. Under a strict Courant condition, a discrete energy estimate bounds the field and staggered-velocity errors by CT(h2+τ2+δ/τ2)C_T(h^2+\tau^2+\delta/\tau^2) on a fixed physical time interval, where hh, τ\tau, and δ\delta are the spatial, temporal, and field spacings. An explicit modulus bound prevents modular wrapping throughout that interval. The choices τ=νh\tau=\nu h, 0<ν<10<\nu<1, and δ=O(h4)\delta=O(h^4) yield second-order convergence. Coupled scalar fields have a common-speed, locally Lorentz-covariant classical limit. Equal bare masses and equal quartic self-couplings preserve a symmetric invariant manifold. Unequal quartic couplings introduce a cubic source into the antisymmetric field equation, and we derive the first variation of the antisymmetric field. Numerical experiments compare exact integer trajectories with a nonlinear traveling wave and an independent continuum solver, measure light and massive dispersion branches, and examine continuously interacting wave packets. The accompanying programs reproduce the reported calculations.

Primarily human-written textHuman understanding: all partscausalitycontinuum limitinteger integrationreversible cellular automatasemilinear wave equations
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hexagon:2610.00537v1

math.RT — Representation Theory

A note of the Cartan determinant conjecture

Contributed by Qiyue Tang

We prove the Cartan determinant conjecture for every corner of finite global dimension in an Artin algebra of global dimension at most two. Consequently, the conjecture holds for torsionless-finite Artin algebras.

A mix of human-written and AI-generated textHuman understanding: some partsCartan determinant conjecture
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hexagon:2610.00536v1

math.MG — Metric Geometry

Illumination of complex convex bodies

Contributed by אלון שכטר

For every complex 11-unconditional convex body K⊂CnK\subset\mathbb{C}^n, we prove \[ \operatorname{ill}(K)\le (2^{m+1}-1)\binom nm+\sum_{k=m+1}^{n}\binom nk-(n-m), \qquad m=\lceil2n/3\rceil. \] This bound is asymptotic to 3n+1/πn3^{n+1}/\sqrt{\pi n}. We further obtain ill⁡∗(K)≤b∗n\operatorname{ill}^*(K)\le b_*^n and ill⁡(K)=O(nlog⁡(n+1)b∗n)\operatorname{ill}(K)=O(n\log(n+1)b_*^n), where b∗=2.4805534697…b_*=2.4805534697\ldots is defined by an explicit scalar equation. For normalized bodies other than Dn\mathbb{D}^n that contain a full coordinate (n−1)(n-1)-polydisc, we prove the sharp bound ill⁡(K)≤2n+1−2\operatorname{ill}(K)\le2^{n+1}-2. We also prove the sharp bound 1414 for every normalized complex 11-unconditional body in C3\mathbb{C}^3 other than D3\mathbb{D}^3, and the sharp bound 3030 in C4\mathbb{C}^4 for every normalized non-polydisc. The four-dimensional proof includes one finite covering assertion verified by exact integer arithmetic; its complete verifier is included. Under full coordinate permutation symmetry, we improve the three-dimensional bound to 1212 by an explicit twelve-vector construction and an exact finite covering certificate. For complex Hanner bodies, we compute both illumination numbers exactly and determine the equality cases in their dimension bounds. If every extreme point has all coordinates nonzero, we prove ill⁡(K)≤2n+1−1\operatorname{ill}(K)\le2^{n+1}-1 and a quantitative strict fractional bound ill⁡∗(K)<2n\operatorname{ill}^*(K)<2^n for normalized non-polydiscs. For real 11-symmetric non-cubes we obtain bounds 1212 in dimension four, 2222 in dimension five, and 5⋅2n−35\cdot2^{n-3} for every n≥6n\ge6. We also determine the leading asymptotic maximum in the real 11-symmetric non-cube class: it is (1+o(1))2n−1(1+o(1))2^{n-1}. An explicit binary construction proves this without an external coding theorem. A connection to connected domination in cube graphs improves the upper estimate to 2n−1(1+(1+o(1))/n)2^{n-1}(1+(1+o(1))/n).

Primarily AI-generated textHuman understanding: some partsconvex geometry
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hexagon:2610.00535v1

math.NT — Number Theory

Coefficient Dynamics and Consecutive Toeplitz Minors for the de Bruijn–Newman Famil

Contributed by Changxia Lin

Abstract We give a normalization-consistent coefficient formulation of the de Bruijn–Newman heat flow attached to the Riemann ξ-function. For the classical kernel Φ and the family Ht(z) = Z ∞ 0 e tu2 Φ(u) cos(zu) du, Ft(w) = Ht(i √ w) = X n≥0 qn(t)w n , we prove the exact evolution laws ∂tFt = 2F ′ t + 4wF′′ t , q(m) n (t) = (2n + 2m)! (2n)! qn+m(t). The coefficients have positive integral representations and the transformed entire function has order at most 1/2. We relate the zeros of Ft to the Pólya–frequency property of (qn(t)), derive the rectangular Jacobi–Trudi identity for the consecutive Toeplitz minors, and state carefully the strict consecutive-minor criterion needed to pass from these minors to total positivity. In particular, for this coefficient family the real-zero condition is equivalent to strict positivity of every consecutive minor Dt(n, r), not merely to nonnegativity of that selected family. This distinction avoids an invalid use of a weak version of Fekete’s criterion. We also correct the multiplicity bookkeeping for the squared-variable map and record a Chebyshev convolution identity without attributing to it an unjustified duplication of zeros. The paper proves no new unconditional upper bound for the de Bruijn–Newman constant and no proof of the Riemann hypothesis; its purpose is to provide an auditable set of exact identities and to isolate the remaining quantitative problem

Primarily human-written textHuman understanding: all partsRiemann ξ-function; de Bruijn–Newman constant; heat flow
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hexagon:2610.00534v1

math.CO — Combinatorics

The maximum cardinality of a planar seven-distance set

Contributed by Alon Amit

We give a computer-assisted proof that the maximum cardinality of a planar set determining at most seven distinct positive distances is sixteen. A diameter-graph argument reduces a hypothetical seventeen-point example to an eleven-point set with at most six distances, except for a thirteen-cycle obstruction. We exclude the cycle by an exact interval certificate. Using a published exhaustive count of sixty-two candidate distance patterns, we classify the eleven-point cores by exact Gröbner-basis calculations and prove that none can have the required seventeen- point extension. All extension calculations use rational or algebraic arithmetic. An explicit sixteen-point triangular-lattice set attains the bound.

Primarily AI-generated textHuman understanding: all partscombinatorial-geometry
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hexagon:2610.00533v1

math.CO — Combinatorics

Complete planar arcs over nonsquare binary fields

Contributed by Jose Voloch

Let m2′(q)m'_2(q) be the largest size of a complete arc in PG⁡(2,q)\operatorname{PG}(2,q) other than a hyperoval. We prove $m'_2(q) \le q-\floor{q^{2/3}-q^{1/3}}, q=2^h\ge512,\ h\text{ odd}$. The geometric statement underlying the proof is a degree obstruction for arbitrary geometrically irreducible Frobenius nonclassical plane curves; an origin from an arc is unnecessary. The argument uses the curve and a Frobenius twist of its dual, exact weighted point counts, and a local conductor inequality that allows arbitrary singularities. We give a sharper bound obtained by optimizing over powers of two, determine its leading constant according to h mod 3h\bmod3, and prove further refinements using a characteristic-two normal form. Exact finite computations give, for example, m2′(128)≤105m'_2(128)\le105, m2′(512)≤452m'_2(512)\le452, and m2′(2048)≤1875m'_2(2048)\le1875.

Primarily AI-generated textHuman understanding: some partsPlane arcs
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hexagon:2610.00532v2

math.NT — Number Theory

Forced 33-adic anomalies of S3S_3 cubic fields

Contributed by Shawn Krisman

Let kk be a totally real cubic field and pp a prime unramified in kk. The pp-adic regulator of kk is a 2×22\times2 determinant of pp-adic logarithms of units and is always divisible by p2p^2; call kk anomalous at pp if it is divisible by p3p^3. Among all 592421 totally real cubic fields of discriminant up to 10710^7, the proportion of anomalous fields matches a random-matrix prediction at every prime 5≤p≤315\le p\le 31, but at p=3p=3 it is 0.3810.381 instead of the predicted 0.4070.407. We explain this. For cubic fields of fundamental discriminant DD prime to 3, we show that the 3-class group and the ray class group modulo 9 of the quadratic field Q(D)\mathbb{Q}(\sqrt{D}) can force every cubic field of discriminant DD to be anomalous at 3. When the 3-class group of Q(D)\mathbb{Q}(\sqrt{D}) is cyclic, we decide exactly when kk is anomalous, in terms of one invariant for which we give an explicit formula by cubic residue and Hilbert symbols. The phenomenon comes from the failure of semisimplicity of F3[S3]\mathbb{F}_3[S_3] and has no analogue for p≥5p\ge 5. The results fix one or two 3-adic digits of the regulator and say nothing about Leopoldt's conjecture itself.

Primarily AI-generated textHuman understanding: no partsGalois cohomologyLeopoldt conjecturePoitou-Tate dualityS3 extensionscup productsp-adic regulatorp-rationalityray class groupsrestricted ramificationtotally real cubic fields
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hexagon:2610.00531v1

math.NT — Number Theory

Long solutions to small equations

Contributed by Bogdan Grechuk, Eugene Go, Avraham Eisenberg

We give complete solutions of thirty-eight Diophantine equations left open in Grechuk's 2024 monograph: fifteen affine equations, five generalized Fermat equations, eight plane quartics, and ten further homogeneous equations. All proofs were generated by AI and require substantial computer algebra computations.

Primarily AI-generated textHuman understanding: some partsDiophantine equations; Chabauty method; Brauer--Manin obstruction
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hexagon:2610.00530v1

quant-ph — Quantum Physics

A matching threshold of 2×10⁻³ for the rotated surface code under circuit-level noise

Contributed by Tejasvi Singh Tomar

We consider the rotated surface code of odd distance dd with its standard syndrome-extraction circuit, under circuit-level depolarizing noise in which every gate, idle step, preparation and measurement fails independently with probability pp, decoded by minimum-weight perfect matching with the log-likelihood weights of single faults. We prove that a memory experiment with TT rounds fails with probability at most 0.061 (T+1)(d+1) λd/(1−λ)0.061\,(T+1)(d+1)\,\lambda^{d}/(1-\lambda), where λ=0.999 (p/p∗∗)9/25\lambda=0.999\,(p/p_{\ast\ast})^{9/25}, for every p≤p∗∗=2×10−3p\le p_{\ast\ast}=2\times10^{-3}. Hence the logical error rate per round decays exponentially in dd for all p≤p∗∗p\le p_{\ast\ast}. Earlier rigorous thresholds for matching on surface-code circuits are 1.7×10−41.7\times10^{-4}, due to Dennis, Kitaev, Landahl and Preskill, and 7.4×10−47.4\times10^{-4}, due to Fowler. We keep their reduction of a failure to a weighted self-avoiding path in the decoding graph, but bound these paths by a local criterion for non-backtracking walks, which gives the threshold 1.04×10−31.04\times10^{-3} in closed form. To reach 2×10−32\times10^{-3} we use the minimality of the correction along the path, for a canonical choice among the minimum-weight corrections: labellings of short stretches of the path into error and correction edges that contradict this minimality are excluded, and the remaining labelled walks are counted with a potential function that is checked in exact arithmetic. For a leaner schedule with fused measure-and-reset steps the two arguments give 1.53×10−31.53\times10^{-3} and 3×10−33\times10^{-3}, respectively. With error rates that depend on the type of operation, they certify exponential decay for error models built from the published component error rates of Quantinuum's Helios and H2 and of IBM's Heron r3 processor ibm_boston, mapped onto this circuit. All theorems are also proved in the proof assistant Lean 4, with all computations checked by its kernel. Any argument that bounds the failure probability by the probability of an increasing event, ours included, is limited by the smallest such event that contains all failures, and Monte Carlo estimates place the threshold of this event near 4×10−34\times10^{-3} for this circuit.

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Oct 26, 2026PDF ↗View work →
hexagon:2610.00529v1

math.GR — Group Theory

A Counterexample to Huppert's Conjecture

Contributed by Shihao Wang, Zhiwei Zheng

Let S=PSL4(5)S = \mathrm{PSL}_4(5) and let GG be the unique subgroup of PGL4(5)\mathrm{PGL}_4(5) containing SS with [G:S]=2[G:S] = 2. We prove that GG and SS have the same set of irreducible complex character degrees. Meanwhile, GG is centerless and is not isomorphic to S×AS \times A for any finite abelian group AA. This provides a counterexample to Huppert's conjecture.

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hexagon:2610.00528v1

math.HO — History and Overview

10/24/2610/24/26: Factoring a Pythagorean Date

Contributed by Scott Kominers

Written numerically, the date October 24, 2026 corresponds to a Pythagorean triple---the last one of the twenty-first century! The associated 1010--2424--2626 right triangle has area twice its perimeter; a classical factorization shows that exactly three integer-sided right triangles have this property and, more generally, counts those with any prescribed integer inradius. We also observe that 6060 is the smallest perimeter shared by two noncongruent integer-sided right triangles, one of which is the 1010--2424--2626 triangle.

A mix of human-written and AI-generated textHuman understanding: all partsPythagorean datesPythagorean triplesequable trianglesrecreational mathematics
Oct 26, 2026PDF ↗View work →