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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:2610.00560v3

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 →

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 $\beta>0$ there are $\sigma(\beta)>0$ and $R_1(\beta)<\infty$ such that rectangular Wilson loops of sides $R,T\ge R_1(\beta)$ satisfy $\langle\cos\theta_C\rangle\le\exp(-\sigma(\beta)RT+c(R+T))$ , uniformly in the finite box, with an absolute constant $c$ and an explicit lower bound on $\sigma(\beta)$ . 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 $\beta$ is at most an average, over a sparse Bernoulli set of frozen plaquettes, of the Villain loop at coupling $K\beta$ , 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 $(1-\eta)RT$ outside an event of probability at most $e^{-RT+O(R+T)}$ , with $\eta\to0$ as $K\to\infty$ . 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( $n$ ) 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

Oct 26, 2026PDF ↗View work →

hexagon:2610.00549v1

cs.DS — Data Structures and Algorithms

Sorting networks on 13 inputs need 45 comparators

Contributed by Tejasvi Singh Tomar

Let $S(N)$ be the least number of comparators in a sorting network on $N$ inputs (the Bose--Nelson problem). The values are known for $N\le12$ , and for $13$ inputs the smallest known sorting network, found by Juill\'e, has $45$ comparators. We prove $S(13)\ge45$ , so $S(13)=45$ . Deleting the comparators met by the largest and the second largest input leaves a sorting network on $N-2$ inputs, so $S(N)\ge S(N-2)+P(2,N)$ , where $P(2,N)$ is the least possible worst-case number of comparators met by these two values. Van Voorhis stated $P(2,N)\ge\lceil\lg F(N)\rceil$ for an explicit function $F$ , but Mericanii showed that his proof is invalid. We prove $P(2,N)=\lceil\lg F(N)\rceil$ for $3\le N\le32$ 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 $S(11)=35$ this bound gives only $S(13)\ge44$ . For $45$ we apply it to the suffix of a network: if a network with $L$ comparators sorts a set $X$ of $0$ - $1$ vectors, then for suitable pairs of inputs the two values meet at most $L-c$ comparators, where $c$ 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 $44$ comparators discards each set for which no single network stays within all these budgets, and no set survives at depth $25$ . 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 $S(N)$ for $13\le N\le32$ 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

Oct 26, 2026PDF ↗View work →

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

Oct 26, 2026PDF ↗View work →

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

Oct 26, 2026PDF ↗View work →

hexagon:2610.00543v1

math.GN — General Topology

Recurrent tubes and connectedness-preserving bijections of Euclidean spaces

Contributed by Peter Lim

For every $n\ge 2$ we construct a connectedness-preserving bijection $F$ of $\mathbb{R}^n$ whose inverse does not preserve connectedness. The restriction of $F$ off the closed coordinate ray $E$ is a homeomorphism onto the complement of $L=K\setminus\{\infty\}$ , where $K\subset\mathbb{S}^n$ is a nondegenerate continuum with empty interior. The discontinuity sets of $F$ and $F^{-1}$ are exactly $E$ and $L$ , and both maps can be chosen to have exact Baire class $2$ . The preimage of a fixed straight segment can be any prescribed uncountable closed subset of $E$ 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

Oct 26, 2026PDF ↗View work →

hexagon:2610.00542v3

math.NT — Number Theory

Goldbach numbers in intervals of length $4.45 \log^2 x$ under the Riemann Hypothesis

Contributed by Steve Hubbard

Assuming the Riemann Hypothesis, we prove that for every real $x \ge 3$ the interval $(x, x + 4.45 \log^2 x)$ 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 $|\gamma| \ge 14$ , and the Goldbach verification to $4 \cdot 10^{18}$ , all stated as hypotheses.

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

Oct 26, 2026PDF ↗View work →

hexagon:2610.00541v1

math.CO — Combinatorics

Maximal Sublattices of Convex Geometries with Unbounded Complement Minima

Contributed by Y. T. Miah

For every integer $r\ge 3$ , we construct a finite atomistic convex geometry of convex dimension exactly three having a maximal proper sublattice whose complement has exactly $r$ minimal elements and exactly two maximal elements. The lattice has $(r+2)^2$ elements and the sublattice has $(r+1)(r+2)/2+3$ elements. At $r=3$ 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

Oct 26, 2026PDF ↗View work →

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

Oct 26, 2026PDF ↗View work →

hexagon:2610.00539v1

math.RT — Representation Theory

A note of the Gorenstein symmetry conjecture (I)

Contributed by Qiyue Tang

Over any field, we reduce the Gorenstein symmetry conjecture to an explicit family of endomorphism algebras over a fixed five-dimensional commutative local symmetric algebra.

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

Oct 26, 2026PDF ↗View work →

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 $C_T(h^2+\tau^2+\delta/\tau^2)$ on a fixed physical time interval, where $h$ , $\tau$ , and $\delta$ are the spatial, temporal, and field spacings. An explicit modulus bound prevents modular wrapping throughout that interval. The choices $\tau=\nu h$ , $0<\nu<1$ , and $\delta=O(h^4)$ 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

Oct 26, 2026PDF ↗View work →

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

Oct 26, 2026PDF ↗View work →

hexagon:2610.00536v1withdrawn

math.MG — Metric Geometry

Illumination of complex convex bodies

Contributed by אלון שכטר

For every complex $1$ -unconditional convex body $K\subset\mathbb{C}^n$ , 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 $3^{n+1}/\sqrt{\pi n}$ . We further obtain $\operatorname{ill}^*(K)\le b_*^n$ and $\operatorname{ill}(K)=O(n\log(n+1)b_*^n)$ , where $b_*=2.4805534697\ldots$ is defined by an explicit scalar equation. For normalized bodies other than $\mathbb{D}^n$ that contain a full coordinate $(n-1)$ -polydisc, we prove the sharp bound $\operatorname{ill}(K)\le2^{n+1}-2$ . We also prove the sharp bound $14$ for every normalized complex $1$ -unconditional body in $\mathbb{C}^3$ other than $\mathbb{D}^3$ , and the sharp bound $30$ in $\mathbb{C}^4$ 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 $12$ 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 $\operatorname{ill}(K)\le2^{n+1}-1$ and a quantitative strict fractional bound $\operatorname{ill}^*(K)<2^n$ for normalized non-polydiscs. For real $1$ -symmetric non-cubes we obtain bounds $12$ in dimension four, $22$ in dimension five, and $5\cdot2^{n-3}$ for every $n\ge6$ . We also determine the leading asymptotic maximum in the real $1$ -symmetric non-cube class: it is $(1+o(1))2^{n-1}$ . An explicit binary construction proves this without an external coding theorem. A connection to connected domination in cube graphs improves the upper estimate to $2^{n-1}(1+(1+o(1))/n)$ .

Primarily AI-generated textHuman understanding: some partsconvex geometry

Oct 26, 2026PDF ↗View work →

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

Oct 26, 2026PDF ↗View work →

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

Oct 26, 2026PDF ↗View work →

hexagon:2610.00533v1

math.CO — Combinatorics

Complete planar arcs over nonsquare binary fields

Contributed by Jose Voloch

Let $m'_2(q)$ be the largest size of a complete arc in $\operatorname{PG}(2,q)$ 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 $h\bmod3$ , and prove further refinements using a characteristic-two normal form. Exact finite computations give, for example, $m'_2(128)\le105$ , $m'_2(512)\le452$ , and $m'_2(2048)\le1875$ .

Primarily AI-generated textHuman understanding: some partsPlane arcs

Oct 26, 2026PDF ↗View work →

hexagon:2610.00532v2

math.NT — Number Theory

Forced $3$ -adic anomalies of $S_3$ cubic fields

Contributed by Shawn Krisman

Let $k$ be a totally real cubic field and $p$ a prime unramified in $k$ . The $p$ -adic regulator of $k$ is a $2\times2$ determinant of $p$ -adic logarithms of units and is always divisible by $p^2$ ; call $k$ anomalous at $p$ if it is divisible by $p^3$ . Among all 592421 totally real cubic fields of discriminant up to $10^7$ , the proportion of anomalous fields matches a random-matrix prediction at every prime $5\le p\le 31$ , but at $p=3$ it is $0.381$ instead of the predicted $0.407$ . We explain this. For cubic fields of fundamental discriminant $D$ prime to 3, we show that the 3-class group and the ray class group modulo 9 of the quadratic field $\mathbb{Q}(\sqrt{D})$ can force every cubic field of discriminant $D$ to be anomalous at 3. When the 3-class group of $\mathbb{Q}(\sqrt{D})$ is cyclic, we decide exactly when $k$ 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 $\mathbb{F}_3[S_3]$ and has no analogue for $p\ge 5$ . 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

Oct 26, 2026PDF ↗View work →

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

Oct 26, 2026PDF ↗View work →

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 $d$ with its standard syndrome-extraction circuit, under circuit-level depolarizing noise in which every gate, idle step, preparation and measurement fails independently with probability $p$ , decoded by minimum-weight perfect matching with the log-likelihood weights of single faults. We prove that a memory experiment with $T$ rounds fails with probability at most $0.061\,(T+1)(d+1)\,\lambda^{d}/(1-\lambda)$ , where $\lambda=0.999\,(p/p_{\ast\ast})^{9/25}$ , for every $p\le p_{\ast\ast}=2\times10^{-3}$ . Hence the logical error rate per round decays exponentially in $d$ for all $p\le p_{\ast\ast}$ . Earlier rigorous thresholds for matching on surface-code circuits are $1.7\times10^{-4}$ , due to Dennis, Kitaev, Landahl and Preskill, and $7.4\times10^{-4}$ , 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 $1.04\times10^{-3}$ in closed form. To reach $2\times10^{-3}$ 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 $1.53\times10^{-3}$ and $3\times10^{-3}$ , 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 $4\times10^{-3}$ for this circuit.

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

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

Primarily AI-generated textHuman understanding: all parts

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hexagon:2610.00528v1

math.HO — History and Overview

$10/24/26$ : 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 $10$ -- $24$ -- $26$ 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 $60$ is the smallest perimeter shared by two noncongruent integer-sided right triangles, one of which is the $10$ -- $24$ -- $26$ triangle.

A mix of human-written and AI-generated textHuman understanding: all partsPythagorean datesPythagorean triplesequable trianglesrecreational mathematics

Oct 26, 2026PDF ↗View work →

hexagon:2610.00527v3

math.GM — General Mathematics

Galois groups of Taylor polynomials: prescriptions and arithmetic families

Contributed by Khai-Hoan Nguyen-Dang

Let $f_n(x)=\sum_{j=0}^n a_jx^j$ be the Taylor polynomials of a power series with nonzero rational coefficients. We study the proportions of degrees for which the Galois group over $\Q$ is symmetric, alternating, or neither. We prove that the set of degrees $n\geq3$ with symmetric group can be chosen arbitrarily, even for an entire function with positive rational coefficients whose Taylor polynomials have distinct negative real roots. More precisely, at each degree $n\geq3$ outside the chosen set, $f_n$ has exactly one rational root, and the group on the other roots is the full symmetric group. Thus the proportions need not converge; when they do, the proportion of other groups can be any number in $[0,1]$ . These conclusions answer negatively two questions of Katz and Rivin. Moreover, for fixed $s\in\Q\setminus\Z$, we also give a condition on the prime divisors of rational multipliers of the coefficients $s/(s+j)$ that forces symmetric groups in density one. For the Taylor polynomials of $(1-x)^{-s}$ , we extend Wei's eventual group classification to fixed number fields. Over $\Q$, the number of exceptions to the symmetric group up to degree $X$ is $c_s\sqrt X+O_s(1)$ , with an explicit constant $c_s\geq0$ . We also determine the groups of the central-binomial Taylor polynomials in every degree and prove that every nonconstant Catalan Taylor polynomial has symmetric group. This proves Sun's Catalan group conjecture and his irreducible-reduction criterion for the monic reciprocals of the central-binomial polynomials.

A mix of human-written and AI-generated textHuman understanding: all partsGalois groupTaylor polynomial

Oct 26, 2026PDF ↗View work →

hexagon:2610.00526v1

math.DS — Dynamical Systems

Rigidity of Newtonian Centered Co-Circular Central Configurations

Contributed by Chih-Yao Wu

We prove that a Newtonian central configuration of finitely many positive masses on a circle whose center is the center of mass must be a regular polygon with equal masses. The proof selects an extremal mass–gap pair and lifts the circle to a periodic configuration on the line. A positive integral representation of the circular kernel converts the force equations into sharp weighted quadrature bounds. A scalar convexity estimate forces equality at the selected pair, and the equality conditions determine every mass and gap. The argument applies to every finite number of bodies and includes all analytic and rational estimates needed for the conclusion.

Primarily AI-generated textHuman understanding: no partsNewtonian n-body problemcentered co-circular configurationscentral configurationsregular polygonsweighted quadrature

Oct 26, 2026PDF ↗View work →

hexagon:2610.00525v1

math.NT — Number Theory

A modular obstruction to the $P$ -adic Leopoldt conjecture for the Carlitz module

Contributed by Shawn Krisman

Let $L$ be a finite extension of $K=\mathbb{F}_q(\theta)$ and $P$ a monic prime of $\mathbb{F}_q[\theta]$ . Anglès asked whether the $P$ -adic Leopoldt defect of Taelman's module of units of the Carlitz module over $L$ always vanishes, and Lucas conjectured this, together with the equivalence " $L_P(E/\mathcal{O}_L)\neq 0$ iff $\exp_E$ is injective", for Anderson modules. We show that both statements fail. For every odd prime $p$ , the extension of $\mathbb{F}_p(\theta)$ generated by a root of $z^p-(-\theta)^{p-2}z-(-\theta)^{p-1}$ is real, has Galois group $\mathrm{AGL}(1,p)$ , and has positive $P$ -adic Leopoldt defect and vanishing $P$ -adic zeta value at $s=1$ for every $P$ . The obstruction comes from modular representation theory at the infinite place. We attach to every separable $L/K$ an integer $\delta_{\mathrm{gen}}(L)$ , depending only on the Galois group of the Galois closure, its decomposition group at $\infty$ and the character by which this group acts on the Carlitz period, and prove that the $P$ -adic defect is at least $\delta_{\mathrm{gen}}(L)$ for all $P$ . Using Bosser's $P$ -adic Baker-Brumer theorem, we prove that the defect vanishes for every $P$ for real abelian extensions and for abelian extensions with cyclic Sylow $p$ -subgroup. We also show that a $v$ -adic analogue of Papanikolas's algebraic independence theorem for Carlitz logarithms would imply that the defect equals $\delta_{\mathrm{gen}}(L)$ in general.

Primarily AI-generated textHuman understanding: no partsCarlitz logarithmsCarlitz moduleDrinfeld modulesGoss zeta functionLeopoldt conjectureP-adic L-seriesTaelman unitsmodular representation theory

Oct 26, 2026PDF ↗View work →

hexagon:2610.00523v1

math.NT — Number Theory

Positive density of Collatz convergence at every rate above 3/log(4/3)

Contributed by Idris Ali Shaik

We prove that a positive proportion of positive integers reach 1 within c log x ordinary Collatz steps for every fixed c > 3/log(4/3), with one positive lower natural-density bound independent of c. For each fixed positive target not divisible by three, the same range of times holds with a target-dependent bound, irrespective of its subsequent orbit. Targets divisible by three have predecessor basins of natural density zero. Diagonalization gives one positive-density set with a vanishing excess above the limiting time coefficient. The proof extends weighted inverse trajectories from a fixed finite family of roots and preserves positive weighted mass at every sufficiently large scale. We derive an explicit Fourier-decay estimate, at frequencies not divisible by three, for an auxiliary distribution on residue classes. Its proof uses paired geometric valuations and their exact binomial first-passage law. At order 6409, the coefficient C satisfies log_2 log_2 log_2 C < 176, compared with > 2^172 for the explicit coefficient in Mazur's work. Fractional-moment certificates, residue-capacity bounds, and an enlarged family of roots give explicit density estimates after harmonic and dyadic counting. Further refinements locate surviving mass within the root family, retain conditional residue information, and recover additional distinct sources with a bounded extra step cost. The primitive-decay theorem and the main qualitative density conclusions are formalized in Lean 4. The coefficient-size comparison and the complete optimized density bounds remain paper proofs.

A mix of human-written and AI-generated textHuman understanding: all partsCollatz problemFourier decayformal verificationhitting timesinverse trajectorieslower natural density

Oct 26, 2026PDF ↗View work →

hexagon:2610.00522v1

math.MP — Mathematical Physics

Bayesian Weak Commutativity Does Not Guarantee Attainability of the Nagaoka–Hayashi Bound

Contributed by Qihang Wang

Bayesian weak commutativity has been suggested as the most likely condition for reaching the Nagaoka–Hayashi (NH) bound in multiparameter estimation. We show that it is not sufficient. For a full-rank four-level ensemble obeying it exactly, the effects required for NH equality cannot sum to the identity. We find the optimal risk exactly: it lies strictly above the bound and needs seven outcomes, while projective readout on the four levels adds over seven times the NH gap. A robust normalization witness bounds the gap from below across a family including truncated oscillator ladders.

Primarily AI-generated textHuman understanding: some partsBayesian quantum estimationNagaoka–Hayashi boundmeasurement incompatibilitypositive operator-valued measuresquantum measurement theory

Oct 26, 2026PDF ↗View work →

hexagon:2610.00521v1

math.NT — Number Theory

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

Contributed by Changxia Lin, Pulin Shi

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

Primarily human-written textHuman understanding: all partsRiemann ξ-function; de Bruijn–Newman

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hexagon:2610.00520v1

math.MP — Mathematical Physics

Perfect t-embeddings: existence

Contributed by Mikhail Basok, Dmitry Chelkak

This note gives an affirmative answer to an open question posed in [Chelkak, Laslier, Russkikh'21]. Namely, it demonstrates the existence of perfect t-embeddings for each sufficiently non-degenerate planar bipartite graph $G$ with positive edge weights. The non-degeneracy condition on $G$ is formulated in terms of its perfect and almost perfect matchings and does not concern the choice of edge weights. The proof and the main part of the writing were produced by GPT-6 Astra in response to a series of prompts. We manually added Section 1.1 to put this result into a bigger context and slightly edited the introduction in order to improve the readability. We plan to provide a better human-digested version of this result later.

Primarily AI-generated textHuman understanding: some partst-embeddings

Oct 26, 2026PDF ↗View work →

hexagon:2610.00519v1

math.NT — Number Theory

Robin's Inequality and Superabundant Counterexamples

Contributed by Frank Vega

Robin's criterion states that the Riemann Hypothesis holds if and only if $\sigma(n) < e^{\gamma} \cdot n \cdot \log\log n$ for every $n > 5040$ . Let $n'$ be the squarefree kernel of $n$ , $p_{k}$ its largest prime factor, and $d = e^{\gamma}/\zeta(2) \approx 1.08276$ . We prove that Robin's inequality holds for every $n > 5040$ with $d \cdot \log\log n' \leq \log\log n$ , and for every $n > 5040$ whose prime factors are all below $(\log n)^{1/d}$ , and that both results remain valid with $d$ replaced by $1.016$ . Our main results concern superabundant numbers, among which there are infinitely many counterexamples to Robin's inequality if the Riemann Hypothesis is false. Using the classical error term in the prime number theorem, we show that every superabundant counterexample satisfies $\log\log n - \log p_{k} < C \cdot \exp(-c \sqrt{\log p_{k}})$ for absolute constants $C, c > 0$ . Since the least counterexample satisfies $\log n > p_{k}$ , we deduce that the Riemann Hypothesis is true if there are constants $c > 0$ and $0 < \alpha < 2$ with $c \geq 0.0562/31.03^{2-\alpha}$ , and a threshold $N \leq 10^{10^{13.11485}}$ , such that $\log\log n - \log p_{k} \geq c/\log^{\alpha}(p_{k})$ for every superabundant number $n > N$ with $\log n > p_{k}$ . We also give explicit versions of these bounds: every counterexample satisfies $p_{k}^{1.000002} > \log n$ , and every superabundant counterexample satisfies $-0.010101/(\log\log n)^{3} < \log\log n - \log p_{k} < 0.056201/(\log\log n)^{2}$ . An enumeration of the superabundant numbers up to $10^{40}$ shows that only $3$ of the $244$ exceeding $5040$ satisfy $e^{0.0562} > 1 + (\log\log n - \log p_{k}) \cdot \log^{2}(p_{k})$ , namely those with $p_{k} > \log n$ , the largest of which is $10685862914126400$ . Testing the hypothesis of the sufficient condition on the same range, we find that all $168$ superabundant numbers between $10685862914126400$ and $10^{40}$ satisfy $\log\log n - \log p_{k} > 0$ , and that for each $\alpha \in \{1, 2, 3, 4, 5\}$ the minimum of $(\log\log n - \log p_{k}) \cdot \log^{\alpha}(p_{k})$ over them is positive, ranging from $0.0223$ for $\alpha = 1$ to $6.19$ for $\alpha = 5$ , and is attained at the same superabundant number, whose largest prime factor is $59$ .

Primarily AI-generated textHuman understanding: all partsChebyshev functionPrime NumbersRiemann HypothesisRobin's inequalitySquarefree kernelSum-of-divisors functionSuperabundant numbers

Oct 26, 2026PDF ↗View work →

hexagon:2610.00516v1

math.NT — Number Theory

One of $\zeta(7),\zeta(9),\dots,\zeta(21)$ is irrational

Contributed by Shawn Krisman

We prove that at least one of the eight numbers $\zeta(7),\zeta(9),\dots,\zeta(21)$ is irrational. The previous best result of this kind not involving $\zeta(5)$ , a consequence of a theorem of Lai and Zhou, was that one of $\zeta(7),\dots,\zeta(35)$ is irrational. We use Zudilin's hypergeometric construction: sums of fourth derivatives of rational functions with quintuple zeros give linear forms in $1,\zeta(7),\dots,\zeta(21)$ with no $\zeta(3)$ or $\zeta(5)$ term. The improvement comes from spreading the poles over eighteen blocks with nearby offsets, a sharper $p$ -adic bound on the denominators, and a saddle-point analysis of a cotangent contour integral. The estimates rest on exact rational and interval-arithmetic certificates. The whole proof, including these certificates, is formalised in Lean 4 with Mathlib and checked by the Lean kernel using only the standard axioms.

Primarily AI-generated textHuman understanding: no partsLeanRiemann Zeta Functionformal verificationhypergeometric linear formsirrationalityodd zeta valuessaddle-point method

Oct 26, 2026PDF ↗View work →

hexagon:2610.00515v1

math.MG — Metric Geometry

Optimizing a cube–ball lower bound for the reverse Brunn–Minkowski constant

Contributed by Ricardo Pinho

We optimize a single term in the cube–Euclidean-ball construction of V. Patankar to obtain C_70 >= exp(1/(2e)) > 1.20194336847. The strongest public proposal located and reproduced in our dated source audit is Patankar's 1.201927, whose specified witness evaluates near 1.2019271425725. A separate finite-dimensional witness certifies 1.20194336. Two separately written arithmetic implementations, using Arb and rational logarithm/Stirling bounds respectively, verify the numerical comparisons and reject corrupted or excessive claims. The geometric reduction and passage to the limit are proved here. This unrefereed preprint is not peer reviewed and not formally verified; exhaustive priority is not asserted.

Primarily AI-generated textHuman understanding: some partscentrally symmetric convex bodiescube and Euclidean ballexact certificatesintrinsic volumesreverse Brunn–Minkowski inequalityrigorous arithmetic

Oct 26, 2026PDF ↗View work →

hexagon:2610.00514v1

math.CO — Combinatorics

A degree-three local-weight cap and an improved relevant-variable bound

Contributed by Ricardo Pinho

We prove that the local-degree potential of a Boolean function of degree at most three is at most 5/4, using the published fact that such a function has at most ten relevant variables. Inserting this cap into Wellens's degree–block-sensitivity recurrence, with the existing exact certificates released by Épi, gives C_44 <= 4.3576549226. The reproduced preceding recurrence gives approximately 4.365467422574458; our exact degree-three cap lowers it by 1/128. Two separately written exact-arithmetic programs check the finite certificates, recurrence, infinite tail and finite cases of the added lemma. The universal argument is written in the paper; the published degree-three classification is cited, not re-enumerated. This preprint is not peer reviewed and not verified in a proof assistant; no optimality is claimed.

Primarily AI-generated textHuman understanding: some partsBoolean functionsblock sensitivitycomputer-assisted proofexact rational certificatespolynomial degreerelevant variables

Oct 26, 2026PDF ↗View work →

hexagon:2610.00512v1

math.MP — Mathematical Physics

The continuum limit of three-dimensional SU(2) lattice Yang–Mills theory on a torus V: Short-distance and weak-coupling asymptotics

Contributed by P. M. Aronow, Patrick Lopatto

We consider the Wilson lattice gauge measure for $\mathrm{SU}(2)$ on a periodic three-dimensional torus of fixed size, with bare coupling $g^2=g_3^2a$ proportional to the lattice spacing $a$ , as dictated by the superrenormalizability of three-dimensional Yang–Mills theory. Observables are Wilson loops of the lattice configuration evolved by the Wilson flow, the lattice gradient flow of the Wilson action, at positive flow time. We prove that, at every fixed value of the coupling, the joint laws of finitely many flowed Wilson loops converge as the lattice spacing tends to zero along the sequences of lattice sides $nM^k$ , for fixed odd $M\ge3$ and fixed positive $n$ divisible by four. For each such refinement class, the limits are realized by a unique probability measure on the space of gauge orbits of Yang–Mills heat-flow trajectories at positive times, a version of the state space introduced by Cao and Chatterjee. At each fixed positive time, the energies of the selected trajectories $\pi_k(U)$ , with $U$ distributed under the lattice Wilson measures, are tight uniformly in the lattice spacing. The limiting measure is invariant under the lattice symmetries and the center symmetries. We also show that, at short distances, the rescaled deficits $(1-W)/(g_3^2\varepsilon)$ of the normalized fundamental Wilson loops $W$ of loops $x+\varepsilon\ell$ , flowed for time $\sigma\varepsilon^2$ with fixed $\sigma>0$ , converge to explicit quadratic functions of the heat-smoothed Gaussian free field. As the coupling tends to zero, the measure concentrates on flat connections, with Gaussian fluctuations of explicit covariance near regular commuting holonomies and fluctuations governed by the constant-mode quartic form near the central holonomies. The proof is a multiscale renormalization group analysis in the tradition of Bałaban, carried out on a small-field event whose complement we show to have small probability, uniformly in the lattice spacing, using reflection positivity and chessboard estimates. It is combined with stochastic estimates for the lattice heat flow and a deterministic analysis of the Wilson flow for rough initial data.

Primarily AI-generated textHuman understanding: no partsquantum field theory

Oct 26, 2026PDF ↗View work →

hexagon:2610.00511v1

math.MP — Mathematical Physics

The continuum limit of three-dimensional SU(2) lattice Yang–Mills theory on a torus IV: Wilson flow and continuum measure

Contributed by P. M. Aronow, Patrick Lopatto

Primarily AI-generated textHuman understanding: no partsquantum field theory

Oct 26, 2026PDF ↗View work →

hexagon:2610.00510v1

math.MP — Mathematical Physics

The continuum limit of three-dimensional SU(2) lattice Yang–Mills theory on a torus III: Regional renormalization and large-field control

Contributed by Patrick Lopatto, P. M. Aronow

Primarily AI-generated textHuman understanding: no partsquantum field theory

Oct 26, 2026PDF ↗View work →

hexagon:2610.00509v1

math.MP — Mathematical Physics

The continuum limit of three-dimensional SU(2) lattice Yang–Mills theory on a torus II: Fluctuation integration and local reconstruction

Contributed by Patrick Lopatto, P. M. Aronow

Primarily AI-generated textHuman understanding: no partsquantum field theory

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hexagon:2610.00508v1

math.QA — Quantum Algebra

Vertex operator algebra realizations of untwisted Haagerup--Izumi centers

Contributed by Andrew Riesen, David E. Evans

For every odd integer $N\geq3$ , we construct an untwisted Haagerup--Izumi fusion category $\mathcal H_N$ on the negative spherical branch and a simple, self-contragredient, regular, $C_2$ -cofinite vertex operator algebra $E_N$ of central charge $-16N$ , together with an equivalence of modular fusion categories: \[ \operatorname{Rep}(E_N)\simeq Z(\mathcal H_N). \] The VOA $E_N$ is rational, has exactly $N^2+3$ irreducible modules, and is lower bounded and integer graded, with least conformal degree $-(N-1)/2$ . The proof begins with a CFT-type cyclic Virasoro extension of central charge $-5N$ , whose regularity we establish. Two categorical condensations recover $\mathcal H_N$ , and a finite extension by shifted lattice modules removes the auxiliary pointed factor. We identify the complete ribbon modular data of $Z(\mathcal H_N)$ with $\mathrm{MD}_{-}(\mathbb Z/N;\mathbb Z/(N^2+4),b\mapsto b^2/(N^2+4))$ in the notation of Gannon--Schopieray--Yadav, for every odd $N$ .

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

Oct 26, 2026PDF ↗View work →

hexagon:2610.00506v1

math.MP — Mathematical Physics

The continuum limit of three-dimensional SU(2) lattice Yang–Mills theory on a torus I: Foundations and classical fields

Contributed by P. M. Aronow, Patrick Lopatto

Primarily AI-generated textHuman understanding: no partsquantum field theory

Oct 26, 2026PDF ↗View work →

hexagon:2610.00505v1

math.RT — Representation Theory

Multiplicative bases and deformations of representation-finite algebras

Contributed by Haruhisa Enomoto

Bautista, Gabriel, Roiter, and Salmer\'on proved that every finite-dimensional representation-finite algebra over an algebraically closed field admits a multiplicative basis, in which products of basis elements are zero or basis elements. Outside characteristic two, they also proved that such an algebra is standard: its category of indecomposable modules is presented by its Auslander--Reiten quiver and mesh relations. We give another proof of both theorems. For a basic algebra, grading morphisms between indecomposable modules by powers of the categorical radical and restricting to projectives yields its standard form, a standard algebra with the same Auslander--Reiten quiver. After constructing a multiplicative basis of the standard form, we recover the original algebra by filtered deformation. Positive-degree Hochschild cohomology embeds into cokernels measuring errors in mesh relations. In characteristic two, lifting derivations modulo four produces cocycles supported on zero basis products; the resulting polynomial family contains the original algebra with coefficients zero or one.

Primarily AI-generated textHuman understanding: some partsAuslander--Reiten theorydeformationmesh categorymultiplicative basisrepresentation-finite algebra

Oct 23, 2026PDF ↗View work →

hexagon:2610.00504v1

math.NT — Number Theory

Minimum Prime Bound in Square Intervals

Contributed by Min Min Oo Oo

This paper presents a proof of Legendre’s Conjecture by analyzing the distribution of odd composite and their elimination through odd composite sequences. The odd integer sequence is expressed as 𝑂≥3 = 3 + 2𝑟𝑂, while odd composite numbers are represented as 𝑂𝑝 = 𝑝2 + 2𝑝𝑟𝑝, where 𝑝 is prime and 𝑟𝑝 ≥ 0. Each prime-square sequence introduces a systematic elimination of odd composite numbers, reducing the density of primes in the sequence. By sequentially applying elimination factors (𝑞 − 1)/𝑞 for primes 𝑞 ≥ 3, we derive a lower bound for prime density: d/(dr_O ) N_p≥2/p Considering the quadratic interval [𝑛2, (𝑛 + 1)2], the total change in 𝑟𝑂 is Δ𝑟𝑂 = (𝑛+1)2−𝑛2 = 𝑛 + ½ . Combining this with the density bound yields a lowest bound of prime count per interval: π_((n^2,(n+1)^2))^LB≥⌈2/p (n+1/2)⌉=2(p≈n+1) Thus, every quadratic interval [𝑛2, (𝑛 + 1)2] contains at least two primes, establishing Legendre’s Conjecture. This elimination-based framework provides a systematic proof based on the distribution of odd integers and odd composite sequences.

Primarily human-written textHuman understanding: all partsLegendre's ConjecturePrime Numbers

Oct 23, 2026PDF ↗View work →

hexagon:2610.00503v1

cs.CC — Computational Complexity

Polynomial Fault Detection Without Bounded Width: Exact Reachable Sets in Priority Controllers

Contributed by Idris Ali Shaik

Circuit-width restrictions explain several polynomial cases of automatic test pattern generation. We give a different explanation for an arbitrary-size extension of the c432 priority controller, including a subfamily with linear treewidth and net cut-width in its native layout. For a specified first-stage bridge fault, we characterize exactly the reachable qualifier sets and construct a primary input realizing each one. On drive-one inputs where the first selector flag changes, the healthy set is the target singleton; the faulty sets are precisely the target-containing sets with one request bus and all nonempty sets with at least two buses. Drive zero is undetectable. This image reduces detection to comparing decoder values on sets. For supplied guarded decoders, built from zero, positive channel leaves, disjunction and conjunction of one computed child with negative channel literals, supporting paths and dual-Horn reasoning find a detecting set in O((n+q)G) time for n channels, q outputs and counted decoder size G, or O(G) for formula forests. Unsatisfiable instances of the complete guarded native CNF encoding have resolution refutations with O(N³) inferences, where N is the encoding size. Detection is P-complete, while minimum-set optimization and counting detecting sets are NP-hard and #P-complete, respectively. A specified final-qualifier stuck-at fault makes detection NP-complete on shared guarded decoders. The exact first-stage bridge-image statements are formalized in Lean and registered as PALOMAR-2026-10-19-000010, version 1.

A mix of human-written and AI-generated textHuman understanding: all partsBoolean satisfiabilityautomatic test pattern generationfault detectionpriority controllersproof complexity

Oct 23, 2026PDF ↗View work →

hexagon:2610.00502v1

math.NT — Number Theory

Galois embedding problems and number field counting

Contributed by Anwesh Ray

Let $p$ be a prime number and $1\to H\to G\xrightarrow{q}T\to1$ be an exact sequence of finite groups, where $H$ is a nontrivial $p$ -group. Fix a continuous surjection $\pi:\GQ\twoheadrightarrow T$ and finitely many $H$ -conjugation-invariant local conditions $\cD$. Let $N_{G,\pi,\cD}(X)$ count the continuous surjections $\rho:\GQ\twoheadrightarrow G$ satisfying $q\circ\rho=\pi$ and these local conditions, with $|\Delta_{F_\rho}|\leq X$ , where $F_\rho=\overline{\Q}^{\ker\rho}$. If at least one such lift exists, we prove that, for every $\varepsilon>0$ , \[ X^{p/(|G|(p-1))} \ll_{G,\pi,\cD} N_{G,\pi,\cD}(X) \ll_{G,\pi,\cD,\varepsilon} X^{p/(|G|(p-1))+\varepsilon} \] for all sufficiently large $X$ . This establishes the power of $X$ predicted by Alberts's conjecture for every finite $p$ -group kernel, without any solvability assumption on $T$ . If every element of order $p$ in $H$ lies in $Z(H)$ , then every nonempty family satisfies the asymptotic predicted by Alberts: \[ N_{G,\pi,\cD}(X) \sim C_{G,\pi,\cD} X^{p/(|G|(p-1))}(\log X)^{b_V-1}, \] with $C_{G,\pi,\cD}>0$. As an application, for odd $p$ and $m\geq2$ , we obtain discriminant asymptotics for surjective lifts of a fixed residual Galois representation to $\GL_d(\Z/p^m\Z)$, subject to finitely many local conditions, whenever such a lift exists.

Primarily human-written textHuman understanding: all partsMalle's conjecturedeformation theoryembedding problems

Oct 23, 2026PDF ↗View work →