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math.PR — Probability

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

Contributed by Rajesh Dachiraju

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

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

Oct 13, 2026PDF ↗View work →

hexagon:2610.00213v1

math.QA — Quantum Algebra

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

Contributed by Tzu Chen Huang

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

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

Oct 13, 2026PDF ↗View work →

hexagon:2610.00212v1

math.AG — Algebraic Geometry

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

Contributed by Rafael Amichis Luengo

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

A mix of human-written and AI-generated textHuman understanding: some partsDegtyarev–Shimada conjectureFermat varietiesLean 4integral Hodge conjecturelinear subspacestorsion

Oct 13, 2026PDF ↗View work →

hexagon:2610.00211v1

math.AT — Algebraic Topology

Axiomatic degree theory on topological manifolds

Contributed by Stoyan Apostolov

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

Primarily human-written textHuman understanding: all partsdegree theory

Oct 12, 2026PDF ↗View work →

hexagon:2610.00210v1

math.MG — Metric Geometry

The truncated octahedral conjecture

Contributed by Thomas Hales, Lark Song

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

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

Oct 12, 2026PDF ↗View work →

hexagon:2610.00209v1

math.NT — Number Theory

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

Contributed by Anwesh Ray

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

Primarily human-written textHuman understanding: all parts

Oct 12, 2026PDF ↗View work →

hexagon:2610.00208v1

math.CO — Combinatorics

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

Contributed by Anton Kovaľ

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

Primarily AI-generated textHuman understanding: some parts

Oct 12, 2026PDF ↗View work →

hexagon:2610.00207v1

math.NT — Number Theory

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

Contributed by Pedro Caceres

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

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

Oct 12, 2026PDF ↗View work →

hexagon:2610.00206v1

math.CO — Combinatorics

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

Contributed by Jungang Chen

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

Primarily AI-generated textHuman understanding: no parts

Oct 12, 2026PDF ↗View work →

hexagon:2610.00205v1

math.CO — Combinatorics

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

Contributed by Rafael Amichis Luengo

Let $K (Q_{n})$ be the sandpile group (critical group, Jacobian) of the $n$ -dimensional hypercube graph $Q_{n}$ . Its odd part was determined by Bai in 2003; its $2$ -part has been open since then, and the latest work on it determines its $n - 1$ largest cyclic factors. We determine the whole 2-part of $K (Q_{n})$ for every $n$ , by an explicit rule. The group splits over the indecomposable tilting modules $T (\lambda)$ of $\mathrm{SL}_{2}$ in characteristic $2$ that occur in $V^{\otimes n}$ , with their multiplicities. The contribution of $T (\lambda)$ at the shift $i = (n - \lambda) / 2$ consists of explicit small cyclic factors and of $2^{| I |}$ large ones, where $I$ is the set of binary digits of $\lambda + 1$ below the leading one. The large ones are obtained by pooling a sequence of explicit integers — binomial valuations and carries — into a non-increasing sequence, in the manner of isotonic regression. The proof reduces each contribution, by elimination inside explicit $2$ -adic lattices, to the Smith form of an explicit $2^{| I |} \times 2^{| I |}$ integer matrix, and computes that Smith form from the valuations of its entries. It needs a unique cheapest matching for an upper bound and an inductive potential for a lower bound. As consequences we recover Bai's counts and the theorems of Gao, Marx-Kuo, McDonald and Yuen on the largest factors, and we prove their Conjectures 4.14 and 5.4 and the formula for the $(n + 1)$ -th factor stated in the 2019 poster of three of them. We also prove the fold law: the 2-part of the sandpile group of the folded cube $Q_{n} / \langle (1, \dots, 1) \rangle$ is $2 \cdot \operatorname{Syl}_{2} K (Q_{n})$ , the group obtained by lowering every exponent by one. With the results above this gives the whole $2$ -part of the sandpile group of the folded cube, and, with the known odd part, its whole sandpile group. The whole proof has been checked in the proof assistant Lean 4 with the Mathlib library, with nothing cited. This work has not been refereed by a human expert.

A mix of human-written and AI-generated textHuman understanding: some partsLeanSmith normal formcritical groupfolded cubehypercubesandpile grouptilting modules

Oct 12, 2026PDF ↗View work →

hexagon:2610.00201v1

math.QA — Quantum Algebra

Finite quantum theta functions for skew-symmetrizable seeds

Contributed by 琪越唐

For integral compatible skew-symmetrizable seeds, finite quantum theta functions form a bar-invariant, mutation-compatible basis of an algebra between the cluster and upper cluster algebras. We prove leading-coefficient positivity, valuative independence, unit Newton vertex coefficients, and independence under coefficientwise base change. We obtain upper theta bases for large cluster complexes, reddening classes, and Markov matrices and their one-way acyclic extensions. In mutation-acyclic classes, these are also bases of the cluster algebra.

Primarily AI-generated textHuman understanding: no partsquantum cluster algebra

Oct 12, 2026PDF ↗View work →

hexagon:2610.00168v1

math.PR — Probability

Mobility Edge for Sparse Complex Gaussian Matrices

Contributed by Patrick Lopatto

We consider the sparse Hermitian random matrix supported on the Erdős–Rényi graph $G(n,p/n)$ , whose nonzero entries are independent complex Gaussian variables of variance $1/p$ , together with its local limit, a weighted Poisson–Galton–Watson tree. We prove that for every fixed and sufficiently large $p$ , the limiting operator on the tree conditioned on survival has exactly one positive mobility edge $E_c(p)$ : almost surely, its spectrum is purely absolutely continuous on $(-E_c(p),0)\cup(0,E_c(p))$ and pure point outside $[-E_c(p),E_c(p)]$ . Moreover, $E_c(p)/\sqrt p$ converges to $\sqrt\pi/2$ as $p$ tends to infinity. Thus the mobility edge lies far in the tail of the spectrum, rather than near the edge $2$ of the limiting semicircle law, as had been predicted in the physics literature.

Primarily AI-generated textHuman understanding: some parts

Oct 12, 2026PDF ↗View work →

hexagon:2610.00177v1

math.PR — Probability

An $N^{-7/10}$ Bound for the Finite-Size Bias of the Sherrington-Kirkpatrick Free Energy

Contributed by P. M. Aronow, Patrick Lopatto

For every fixed inverse temperature $\beta>1$ , we prove that the expected free energy per spin of the zero-field Gaussian Sherrington-Kirkpatrick model differs from its Parisi limit by at most $C_\beta N^{-7/10}$ . This improves the exponent $2/3$ of our earlier work (arXiv:2609.09103). We also compare the bound with the finite-size correction of order $N^{-3/4}$ predicted by Franz, Parisi, and Ricci-Tersenghi, and with the strict improvement over $2/3$ that would follow from the ideas in OpenAI's recently released manuscripts.

Primarily AI-generated textHuman understanding: some parts

Oct 12, 2026PDF ↗View work →

hexagon:2610.00198v1

math.NT — Number Theory

Infinitely Many Elliptic Curves of Exact Ranks Five, Six, and Seven

Contributed by P. M. Aronow, Patrick Lopatto

We prove that for each $r\in\{5,6,7\}$ , there exist infinitely many elliptic curves over $\mathbb{Q}$ , pairwise nonisomorphic over $\overline{\mathbb{Q}}$ , whose Mordell–Weil rank over $\mathbb{Q}$ is exactly $r$ . Zywina recently proved the analogous statement for $0\le r\le 4$ over every number field, and our proof follows his strategy. For each $r$ , we give an explicit pair of $2$ -isogenous elliptic curves whose discriminants are, up to constants, products of powers of twelve binary linear forms in two integer parameters. The families are obtained from Zywina's curves of rank four, or from a related pair of curves, by base change along a conic. Explicit rational points give lower bounds for the images of the two descent maps. We specialize at parameters in a fixed congruence class at which the twelve linear forms, divided by fixed constants, take prime values. There, necessary local conditions in the descent become linear equations over $\mathbb{F}_2$ , and their solution spaces have the same dimensions as the lower bounds. The coefficients of these equations are determined by the congruence class up to a single sign, and both values of this sign give the same bound. The Green–Tao–Ziegler theorem on linear equations in primes then gives infinitely many such specializations, with infinitely many $j$ -invariants.

Primarily AI-generated textHuman understanding: no partselliptic curverank

Oct 12, 2026PDF ↗View work →

hexagon:2610.00199v1

math.FA — Functional Analysis

Finite Field Riesz-Frechet Representation

Contributed by K. MAHESH KRISHNA

We derive a Riesz-Frechet representation for bounded linear functionals defined on the inner product spaces over sub-valued finite fields.

Primarily human-written textHuman understanding: all partsFinite Field.Hilbert SpaceRiesz-Frechet representation

Oct 12, 2026PDF ↗View work →

hexagon:2610.00200v1

math.NT — Number Theory

A quasi-Riemann hypothesis for Dedekind zeta functions

Contributed by Dean Cureton, Jesse Thorner

Recent work on Artin's primitive root conjecture establishes a zero-free half-plane $\Rea s>1-10^{-6}$ for finite-order Hecke $L$ -functions over cyclotomic fields containing $e^{2\pi i/12}$ . Following their strategy, we prove that if $K$ is any number field, then the Dedekind zeta function $\zeta_K(s)$ does not vanish in the half-plane $\Rea s>1-1/6000$. As a corollary, the $L$ -function of any finite-order Hecke character defined over $K$ also does not vanish in the half-plane $\Rea s > 1-1/6000$. The proofs were generated and verified in Lean using AxiomProver. Compared to the work on Artin's primitive root conjecture, the hardest part was the formalization of the work of Kazhdan and Patterson on metaplectic forms in order to accommodate all number fields.

A mix of human-written and AI-generated textHuman understanding: some partsDedekind zeta functionHecke characterL-functionnumber field

Oct 12, 2026PDF ↗View work →

hexagon:2610.00195v1

math.AC — Commutative Algebra

Fröberg's Conjecture through 2d in sufficiently many variables

Contributed by Vlado Vojdanovski

Let k be any infinite field. We prove that, for every fixed d ≥ 1 and all sufficiently large n, the quotient of k[x1, . . . , xn] by any number of general forms of degree d has the Hilbert function predicted by Fröberg’s conjecture through degree 2d. The threshold for n is independent of the field and the number of generators. For generating degrees at least three, a direct construction with two groups of variables gives a nonincreasing recurrence for the two defects at the critical generator counts. Divisibility of these bounded defects and polynomial equidistribution then force their eventual vanishing.

Primarily AI-generated textHuman understanding: some parts

Oct 12, 2026PDF ↗View work →

hexagon:2610.00194v2

math.CO — Combinatorics

Strongly regular graphs on 306, 400, 576, 784 and 1296 vertices from multi-orbit coset fusion and Bush-type Hadamard matrices

Contributed by Anton Kovaľ

We construct strongly regular graphs with parameters (306,55,4,11), (400,171,74,72), (576,253,112,110), (784,351,158,156) and (1296,595,274,272), and hence also their complements. All five parameter sets were marked "?" (existence unknown) in Brouwer's online table on 10 and 11 October 2026. Three of the graphs were found by a computer search that extends the transitive coset fusion of our earlier papers to groups acting with two orbits: the search fuses the orbitals of a group G on G/H_1⊔ G/H_2, screens with tables of marks, solves the exact quadratic equations on the structure constants with a SAT-based solver, and rebuilds every solution from scratch. The groups are 2× M_12, 5^(1+2)_+:4A_5 and Sp(6,2). Also deposited on Zenodo with the full data archive (the citable record): https://doi.org/10.5281/zenodo.23296158 The graphs on 400, 576, 784 and 1296 vertices come from a short step beyond known constructions. A symmetric Bush-type Hadamard matrix of order n^2 whose n× n blocks have unit diagonal, for instance one of Kharaghani's matrices, defines a strongly regular graph Γ_0 with the parameters of the Latin square graph LS_(n/2+1)(n). In Γ_0 the block rows and the block columns form two orthogonal spreads of Delsarte cliques. Deleting one spread gives the known graphs with λ=μ of Bonato, Holzmann and Kharaghani. By an elementary spread lemma, deleting both gives a strongly regular graph with the parameters of LS_(n/2-1)(n). Kharaghani's matrices exist for every Hadamard order n, so such a graph exists for every n=4m≥8, for example srg(1600,741,344,342) and srg(1936,903,422,420). A second source, found by analysing the search hits, is the rule (a,b)∼(c,d) iff a≠ c, b≠ d and H_abH_bcH_cdH_da=1 for a symmetric Hadamard matrix H. Together the two sources give at least 5, 7, 5 and 6 pairwise non-isomorphic graphs on 400, 576, 784 and 1296 vertices. None of the graphs we tested on these four vertex numbers is a Latin square graph, so they say nothing about the orthogonal arrays OA(n,n/2-1) named in the table. The graph on 306 vertices has a short description on the 300 non-square points of the parabolic quadric Q(4,5) together with the six points of a totally singular line; this line is a Delsarte clique. The graph has the parameters of the collinearity graph of a generalized quadrangle GQ(5,10) but is not geometric, and the same rule over 𝔽_3 gives the collinearity graph of Q(4,3). Strong regularity of every graph is also checked by an exact, standalone, standard-library Python program.

Primarily AI-generated textHuman understanding: some partsBrouwer tableBush-type Hadamard matrixHadamard matrixLatin square graphcoset fusiongeneralized quadranglespreadstrongly regular graph

Oct 11, 2026PDF ↗View work →

hexagon:2610.00190v1

math.PR — Probability

Aging in Spherical Spin Glass Dynamics

Contributed by P. M. Aronow, Patrick Lopatto

We study Langevin dynamics for the pure spherical $p$ -spin glass with $p\ge3$ , with covariance $Nq^p$ and generator $\Delta_S+\beta\nabla_SH_N\cdot\nabla_S$ , started from a uniform point independent of the disorder. We let the dimension tend to infinity on fixed time horizons and study the long-time behavior of the resulting hard-spherical Cugliandolo–Kurchan equations for the correlation $C$ and the response $R$ . At every positive temperature, their solution has a unique stationary limit at bounded time differences, which satisfies the fluctuation–dissipation theorem (FDT), and its energy converges. For $\beta\ge\beta_{\mathrm d}$ , where $\beta_{\mathrm d}$ is the dynamical transition, we show that the plateau $q$ of the stationary correlation is the larger solution of the marginality relation $\beta^2p(p-1)q^{p-2}(1-q)^2=1$ , and we compute the limiting energy. Correlations with every fixed earlier state tend to zero. The limits of large waiting time and large time difference therefore do not commute, which proves aging of the correlation, including at the transition point. We also prove that the response at a late time, as a function of the time difference, converges to its stationary profile in $L^2$ and uniformly over all earlier times; that FDT in the later time holds uniformly over all earlier times; and that the integral of the squared violation of FDT in the earlier time tends to zero, while for $\beta>\beta_{\mathrm d}$ the positive part $(\partial_sC-R)_+$ of this violation keeps a positive integrated mass. For $\beta\ge\beta_{\mathrm d}$ , the stationary correlation approaches its plateau as a power law, with the exponent predicted by mode-coupling theory and a finite positive amplitude. For the cubic model and $\beta\ge\beta_{\mathrm d}$ , including the transition point, we prove a lower bound on the total susceptibility and a product inequality for correlations at three times; for $\beta>\beta_{\mathrm d}$ , we prove, for each fixed level below the plateau, an explicit asymptotic lower bound on the first passage time to that level multiplied by the entropy production during the passage. The aging proofs for the cubic and quartic models represent the correlation by vectors in a Hilbert space indexed by trees, and general pure interactions are treated with averages over finitely many independent futures and established algorithmic threshold bounds. The full fluctuation–dissipation relation of the aging regime and the time scale of aging remain open.

Primarily AI-generated textHuman understanding: some partsagingdynamicsspin glass

Oct 11, 2026PDF ↗View work →

hexagon:2610.00192v1

cs.LO — Logic in Computer Science

A Machine-Checked Kernel for Federated Automated Adjudication: Credibility Lattice Completion, Certificate Lifecycle Invariants, Certified Numerical Anchors, and a Parameterized Rice Template

Contributed by Chepin Zhang

We present a formally specified and machine-checked kernel for a federated automated adjudication system, in which independent computational lines judge mathematical and operational claims and deposit their verdicts, evidence, and certificates in a shared ledger. The kernel comprises five components. (i) A three-rung, three-track credibility lattice obtained as a completion of a legacy five-element poset: the completion is justified by an exhaustive enumeration over all 11^{3}=1331 ordered triples and is fully re-formalized in Lean 4, where the equational lattice laws, the order axioms, and the extremality properties of join and meet are all proved by the decide procedure over the finite carrier. (ii) A five-state certificate lifecycle machine with nine legal transitions and machine-checked invariants, likewise re-formalized in Lean 4; the formalization caught a specification flaw in one invariant (the originally stated grant-gating condition is false), and we report the corrected, machine-proved form. (iii) Certified numerical anchors: uniqueness of the gauged fixed point of circulant entropic-optimal-transport (Sinkhorn) systems, established by a Krawczyk interval operator with outward rounding across a six-instance family (k in {6,10 }, varepsilon in {1,1/2,1/5 }, later extended to ten instances), together with a second anchor family of five auction-algorithm assignment instances certified by interval dual brackets. We state plainly that the certification of these numerical anchors is carried by the Python interval-arithmetic layer; the attempted port to the Lean 4 interval library leancert is blocked by structural capability boundaries of that library (absence of the logarithm from its supported expression grammar and infeasibility of its determinant computation at the required dimensions), and the Lean files are formal statements, not machine-verified certificates. (iv) A parameterized Lean 4 template for Rice-type undecidability, consisting of a bridge theorem, an extensionality lemma, an instantiation generator, and discharged instances of increasing difficulty. (v) A minimal trusted computing base result: the soundness of an interval certificate checker (accept implies correctness), itself machine-proved in Lean 4. All Lean artifacts were verified on cloud infrastructure (Axiom Axle, environment lean-4.28.0) under the strict verify_proof channel, with axiom audits restricted to propext, Classical.choice, and Quot.sound, and no sorryAx. Empirical fuzzing over three certificate families observed zero false accepts; we report rule-of-three 95% upper bounds on the false-accept rate and explicitly do not claim a guarantee.

Primarily AI-generated textHuman understanding: some partsLean 4Rice theoremfederated adjudicationformal verificationinterval arithmeticmachine-checked certificates

Oct 11, 2026PDF ↗View work →

hexagon:2610.00193v1

math.CO — Combinatorics

A characterization of cut-percolating bigraphs

Contributed by Václav Rozhoň

Conlon and Lee introduced reflection bigraphs and cut-percolation in their study of graph norms. We prove that, for connected bigraphs, these two notions coincide, confirming their prediction.

Primarily AI-generated textHuman understanding: some partsCoxeter groupsSidorenko's conjectureblock designscut-percolationgraph isomorphismgraph normsreflection groupsweakly norming graphs

Oct 11, 2026PDF ↗View work →

hexagon:2610.00191v1

math.NT — Number Theory

BRUDNO’S FAMILIES OF EQUAL SUMS OF SIXTH POWERS AND BREMNER’S K3 SURFACE

Contributed by Alon Amit

Brudno (1970, 1976) and Brudno–Kaplansky (1974) found an elliptic family and a degree-four polynomial family of solutions of x^6 + y^6 + z^6 = u^6 + v^6 + w^6. We explain both in terms of the K3 surface V ⊂P5, the complete intersection of three quadrics studied by Bremner (1981) and Kuwata (2007). The Brudno–Kaplansky curve is Cremona’s curve 480f4 and is the fibre at λ=∞ of the elliptic pencil on V that Bremner calls the Brudno pencil; the curve z^2 = x^4 + 38x^2y^2 + y^4 of 1970 is 480f1, and the classical recursions of Lebesgue and Desboves that Brudno iterated are the multiplication maps [2] and [−3] on it. Brudno’s 1976 quartic is the unique rational curve of degree four on V (Bremner’s Theorem 4.4) and a section of the Brudno pencil; its parameter is the fibre coordinate λ, and the involution Brudno noticed is a linear symmetry of V. Using the Mordell–Weil lattice of the pencil we prove a closed formula for the degree of a section, deg C_S = 2⟨S−γ/4, S−γ/4⟩, enumerate all sections of degree ≤12 (four of them new, of degrees 7,10,11,11), and enumerate, by a lattice computation, all smooth rational curves of degree ≤12 on V over Q(i) and over Q, reproducing Bremner’s theorems and testing his conjecture. A census of the 74 primitive solutions with all terms ≤ 1343 identifies which of them lie on V, on known curves, on sections, and on fibres of jumping rank, and records some phenomena off V. In the second part we compute equations and parametrisations of the 48 new curves that are not sections and show that seven of the eight census solutions on no known curve lie on them, leaving a single unexplained point; we determine the geometry over C of the sextic fourfold M6, of the threefold W= M6 ∩M2 (which has exactly 710 nodes and defect 95) and of V (transcendental lattice U(2) ⊕⟨12⟩, zeta function through the newform 320f2), and discuss in what sense the geometry drives the arithmetic; and we show that Bremner’s conjecture follows, with non-explicit finite sets, from the theorems of Sterk and Bright–Logan–van Luijk, describe how to make it explicit (Borcherds–Shimada), and run the first step of Bremner’s own cone procedure, which identifies the degree-8 pencil involutions that the proof requires.

Primarily AI-generated textHuman understanding: some partsDiophantine equationsElliptic CurvesSextic

Oct 11, 2026PDF ↗View work →

hexagon:2610.00187v2

math.NT — Number Theory

An Irrationality Measure for Catalan's Constant

Contributed by Sean Litsey

uilding on OpenAI's recent proof that Catalan's constant $G=\sum_{j\ge0}(-1)^j/(2j+1)^2$ is irrational, we prove that \[ \left|G-\frac ab\right|>b^{-8\,127} \] for all integers $a$ and all sufficiently large integers $b$ . Thus the irrationality exponent of $G$ is at most $8\,127$ ; without the prime number theorem in the final step the same argument gives $23\,168$ . The previous best bound, due to Jha, is $10^8$ . The new ingredient is a perturbation estimate for OpenAI's determinant, viewed as a polynomial $\Delta_N(X)$ in the constant: on the interval $|x-G|\le e^{-404.25N}$ the polynomial moves by no more than OpenAI's bound for $|\Delta_N(G)|$ . It is proved by a quadrature formula for the coefficient of the constant, an Andr\'eief identity with fixed columns, a generalized Vandermonde bound, and OpenAI's energy method rerun for configurations with fewer points. We also give a new certificate, with re-optimized trial sequences, for one of the two cases of OpenAI's logarithmic-energy dual; it raises the margin of the argument by a factor of almost eight. Every step, including the final theorem, is formalized in Lean~4 on top of OpenAI's formalization, without \texttt{sorry} and with only the standard axioms.

Primarily AI-generated textHuman understanding: no partsAndr\'eief identityLeanatalan's constantdeterminantsformal verificationirrationality exponentirrationality measure

Oct 11, 2026PDF ↗View work →

hexagon:2610.00178v1

math.NT — Number Theory

Unconditional implications of the Quasi-Riemann Hypothesis

Contributed by Shubhrajit Bhattacharya

This record identifies published conclusions formerly obtained under RH or GRH whose proofs require only a fixed zero-free half-plane for Dirichlet $L$ -functions, with further applications of finite-order Hecke $L$ -functions over $\mathbb Q(\sqrt{-3})$ . The input is the $7/8$ theorem stated in the recent OpenAI manuscript claiming to prove the quasi-Riemann Hypothesis. We give the uniform analytic deductions explicitly and distinguish them from previously established algebraic and combinatorial results used in the proofs. An accompanying audit records the source verification and rejected candidates. No theorem about all number-field or automorphic $L$ -functions is inferred from the Dirichlet statement; the pure cubic and cubic-nonresidue applications separately identify the precise fixed-field Hecke input.

Primarily AI-generated textHuman understanding: no partsRiemann Hypothesis

Oct 11, 2026PDF ↗View work →

hexagon:2610.00188v1

math.LO — Logic

A counterexample to Vaught's Conjecture for $L_{\omega_1,\omega}$

Contributed by Nathanael L. Ackerman, Cameron Freer, Robin W. Knight

We construct a sentence of $L_{\omega_1,\omega}$ having exactly $\aleph_1$ many countable models up to isomorphism, but no perfect set of non-isomorphic models. This disproves Vaught's conjecture for infinitary logic. The construction follows a framework devised by Robin Knight, and the proof has been formally verified in Lean.

Primarily human-written textHuman understanding: all partsKnight systemsVaught’s conjectureinfinitary logicnumber of models

Oct 11, 2026PDF ↗View work →

hexagon:2610.00189v1

math.GR — Group Theory

Bruck loops with abelian inner mapping groups: a short proof

Contributed by Bartosz Naskrecki

Phillips and Stanovský proved that a Bruck loop with abelian inner mapping group has central nilpotency class at most two and asked for a human-friendly proof. We give a short self-contained proof: a displacement cocycle produces a commutant-valued endomorphism, right alternativity for commutant elements forces its image to be fixed, and cocycle rigidity makes that image central. This yields the abelian center quotient.

Primarily AI-generated textHuman understanding: all partsBruck loopsFormalized mathematicsLoop theory

Oct 11, 2026PDF ↗View work →

hexagon:2610.00186v1

math.NT — Number Theory

Stark units for real quadratic fields and reciprocity laws

Contributed by Danylo Radchenko, Campbell Wheeler

We prove that every value of the arithmetic finite quantum dilogarithms constructed by Radchenko and Wheeler (arXiv:2609.21892) is an algebraic unit, and determine the Artin action for arbitrary quadratic pseudolattices. In particular, the associated Stark units belong to the predicted ray class fields and have the predicted Galois action. We use finite Fourier transforms and distribution relations to obtain Frobenius congruences. We also give conductor bounds for individual values and obtain the unitary conjugates needed for Zauner's conjecture.

A mix of human-written and AI-generated textHuman understanding: all partsArtin reciprocityFrobenius congruencesReal quadratic fieldsStark unitsfinite quantum dilogarithmsray class fields

Oct 10, 2026PDF ↗View work →

hexagon:2610.00175v1

math.QA — Quantum Algebra

The Haagerup VOA at Central Charge Eight

Contributed by Andrew Riesen

We give an explicit construction of a vertex operator algebra $U$ of central charge eight inside $V_{E_6\oplus A_2}$ , and prove the ribbon equivalence $\Rep(U)\simeq Z(\HH_3)$, where $\HH_3$ is a Haagerup fusion category. The algebra is generated as a vertex algebra by six affine generators and three specified weight-two vectors. The argument combines exact lattice-vector calculations with an all-weight strong-generation induction, a finite presentation and semisimple descent for a full corner of Zhu's algebra, and the actual multiplication of two holomorphic $E_8$ extensions. The categorical identification uses splitting maps and a comparison with the Evans--Gannon reconstruction, not modular data alone.

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

Oct 10, 2026PDF ↗View work →

hexagon:2610.00182v1

math.RT — Representation Theory

A counterexample to Han’s conjecture over arbitrary fields

Contributed by 琪越唐

Based on the constructions of Kong–Liu–Shen and Liu–Shen, we give a counterexample to Han’s conjecture over every field.

Primarily AI-generated textHuman understanding: no partsHan’s conjecture

Oct 10, 2026PDF ↗View work →

hexagon:2610.00180v1

math.OA — Operator Algebras

A variational formula for affine semicircular determinants

Contributed by Sheng Yin

We express the Fuglede--Kadison determinant of an affine pencil in free semicircular operators as an infimum over positive definite matrices. The infimum is attained if and only if the pencil is boundedly invertible, in which case its minimizer is unique. At zero constant term, a scalar rescaling recovers the determinant-capacity identity of Mai and Speicher. For selfadjoint coefficients we compare the formula with a strictly constrained Hilbert--Schmidt variational principle and establish ordered min--max formulas under their respective positivity hypotheses. Strictly positive pencils admit a minimum over a linear matrix inequality. Applications to Brown measure include a uniform determinant lower bound, restrictions on atoms, radial distribution formulas, and a Hessian formula for the regularized density. The text of this manuscript was generated entirely by OpenAI's large language model Astra.

Primarily AI-generated textHuman understanding: no partsFree probabilityFuglede–Kadison determinantOperator-valued semicircular elements

Oct 10, 2026PDF ↗View work →

hexagon:2610.00185v1

math.CA — Classical Analysis and ODEs

Arithmetic progressions with prescribed lp gaps

Contributed by Vjekoslav Kovač

For every $n\ge3$ and $p\in[1,\infty)\setminus\{1,2,\ldots,n-1\}$ , we prove that every measurable subset of $\mathbf{R}^d$ of positive upper Banach density, with $d\ge2^{n-2}+1$ , contains an $n$ -term arithmetic progression with every sufficiently large prescribed $\ell^p$ gap.

Primarily AI-generated textHuman understanding: some partsEuclidean Ramsey theorydensity theoremmultilinear estimateoscillatory integralsingular integral

Oct 10, 2026PDF ↗View work →

hexagon:2610.00176v1

math.CA — Classical Analysis and ODEs

Superexponentially decaying sequences in the Erdos similarity problem

Contributed by Vjekoslav Kovač

We prove a non-universality criterion for the Erd\H{o}s similarity conjecture that applies beyond exponential (i.e., geometric) decay. In particular, it covers decreasing positive null sequences satisfying $\log(1/a_n)=o(n\log\log n)$ and $\log(a_n/a_{n+1})\ge e^{-n^\alpha}$ eventually for some $0<\alpha<1$ .

Primarily AI-generated textHuman understanding: some partsLebesgue measureconfigurationmeasure theory

Oct 10, 2026PDF ↗View work →

hexagon:2610.00179v1

math.RT — Representation Theory

Representation embeddings and the third Brauer--Thrall conjecture

Contributed by Haruhisa Enomoto

We show that every representation-infinite finite-dimensional algebra over an algebraically closed field has an indecomposable module of each infinite dimension. This settles, over algebraically closed fields, Simson's conjecture, also called the third Brauer--Thrall conjecture, that such an algebra has indecomposable modules of arbitrarily large cardinality. To construct these modules, we extend a representation embedding of Bongartz from finite-dimensional Kronecker modules without a simple injective summand to a representation embedding of such Kronecker modules of arbitrary dimension, which commutes with direct sums and filtered colimits and preserves the dimension of infinite-dimensional modules. Applying it to the indecomposable Kronecker modules of every infinite dimension constructed by G\"obel and Simson gives the required modules.

Primarily AI-generated textHuman understanding: some partsBrauer--Thrall conjectureGalois coveringKronecker modulelarge indecomposable modulerepresentation embedding

Oct 10, 2026PDF ↗View work →

hexagon:2610.00165v1

math.MG — Metric Geometry

Multi-coverings by lattice translates of convex bodies

Contributed by Scott Kominers

We show that for every convex body $K\subset\mathbb R^n$ ( $n\ge2$ ), every $0<\varepsilon<1$ , and every prescribed density $\rho\ge A_\varepsilon n\log n$ , there is an arrangement of translates of $K$ along a single lattice with mean multiplicity exactly $\rho$ and multiplicity at least $(1-\varepsilon)\rho$ everywhere. Consequently, lattice $k$ -fold coverings exist with density at most $\max\{(1+\varepsilon)k,C_\varepsilon n\log n\}$ , where $C_\varepsilon$ depends only on $\varepsilon$ . In sufficiently large dimension $n$ , a version with prescribed common covolume holds simultaneously for up to $\exp(n^{c_0})$ bodies whose uniform distributions have a common covariance matrix, with $c_0>0$ absolute. The proof builds on OpenAI's horizontal–vertical construction for single-lattice coverings and uses Gaussian layer averaging and Laplace bounds for covering counts.

A mix of human-written and AI-generated textHuman understanding: all partsGeometry of numbersconvex bodiesconvex geometrycovering densitycovering multiplicitylattice coveringmulti-covering

Oct 09, 2026PDF ↗View work →

hexagon:2610.00161v1

math.NT — Number Theory

THE CROSS-ENERGY OF DIRICHLET PARTIAL SUMS AND THE ZEROS OF THE RIEMANN ZETA FUNCTION

Contributed by Pedro Caceres

Let X_n(s) = Σ_{k≤n} k⁻ˢ and let C₂(n,s) = |X_n(s)|² − |X_{n−1}(s)|² − n⁻²σ be the cross-energy of consecutive partial sums. At every point of the critical strip the limit of C₂(n,s) is 1/(¼ + t²), 0 or +∞ according as σ = ½, σ > ½ or σ < ½; this trichotomy is the trace of the pole at s = 1. On the critical line the associated helix has a natural orientation whose signed radius is Hardy’s Z(t), and the zeros of the truncated functions spiral into the zeros of ζ at the rate n⁻ᵝ. We prove that the mean energy of the normalized error E(u) in the prime number theorem equals Σ_ρ lim_{n→∞} C₂(n,ρ) in [0,∞]; it is at least 2 + γ₀ − log(4π), with equality if and only if the Riemann hypothesis (RH) holds and all zeros are simple, and an unconditional local version expresses the energy in a Gaussian window as a double sum over zeros. We then study the pointwise inequality S(σ,t) = ∂σ log|ξ(σ + it)| > 0, which is equivalent to RH: an off-line pair lowers S exactly inside a disc, the failure set at abscissa σ up to height T has measure at most N(σ,T + 1), and S is Weil’s functional at a Poisson kernel, whose truncation is an exact finite expression in the primes up to eᴸ. Every criterion is tested against the Davenport–Heilbronn function, which has zeros off the critical line, and against its Euler-product twin L(s,χ) mod 5: the former violates each criterion in the predicted way, the latter behaves like ζ. For ζ, the primes up to 10⁸ give mean energy 0.0453 against 0.0462. On the circle of the Euler factor of a prime p the zeros lean toward the angle π, and toward π + argχ(p) for L(s,χ). We do not prove the Riemann hypothesis.

A mix of human-written and AI-generated textHuman understanding: all partsC2 FunctionDirichlet SumsRiemann Zeta Function

Oct 09, 2026PDF ↗View work →

hexagon:2610.00169v1

math.PR — Probability

Classical infinite divisibility, self-decomposability and bell-shape of free stable laws

Contributed by Min WANG

We determine the classical infinite divisibility, self-decomposability and extended Thorin regions of the strictly free stable family. Above index one, infinite divisibility and self-decomposability hold on two extremal skewness intervals, with distinct maximal indices $\alpha_{\mathrm{ID}}=1.5240739610\ldots$ and $\alpha_{\mathrm{SD}}=1.4283571142\ldots$ . Below one, every law is self-decomposable, whereas the Thorin region is a central interval; the optimal uniform bound is $4/5$ . We characterize the boundaries by global contact conditions and derive their different asymptotic scales at index one. A common phase representation also gives an exact bell-shape test, strictly bell-shaped examples outside the Thorin class, and an unbounded number of skewness components near one. The exceptional free one-stable family admits an explicit integer criterion. The proofs combine analytic estimates with finite interval certificates. The bell-shape results do not constitute a full parameter classification.

Primarily AI-generated textHuman understanding: no parts

Oct 09, 2026PDF ↗View work →

hexagon:2610.00173v1

math.MP — Mathematical Physics

Wave Mechanics

Contributed by Rajesh Dachiraju

We introduce a complex Hilbert space structure on $L^2(\mathbb{T}^m;\mathbb{C})$ in which the Hilbert transform on the torus is incorporated directly into the metric and inner product. The resulting geometry is defined via a bounded linear embedding that couples each function to its Hilbert transform, yielding an inner product with both symmetric and symplectic components. Within this framework, we redefine weak differentiation intrinsically using the complex Hilbert-space inner product rather than the ambient $L^2$ pairing. We show that this intrinsic weak derivative coincides with the classical weak derivative on its natural domain, while remaining geometrically well defined on the full space. The derivative operator is shown to be densely defined and skew-adjoint, with a purely imaginary, discrete spectrum. Its eigenfunctions are given by the Fourier modes on the torus, which induce a complete orthonormal basis adapted to the Hilbert-transform metric. This construction yields a spectral representation in which differentiation is diagonal and the Hilbert transform is absorbed into the notion of differentiability itself, providing a unified analytic and geometric framework. We formulate wave mechanics intrinsically in this complex Hilbert space, define wave function, scalar and vector potentials and associated energy functional, derive space time wave equation, derive its solution, prove existence and uniqueness and establish regularity of the solution.

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

Oct 09, 2026PDF ↗View work →

hexagon:2610.00167v1

math.GR — Group Theory

The Sharp Commutator Bound for the Polymath14 Inequality

Contributed by Jaeyeon Shim

We present a simplified proof of the Polymath14 inequality, along with a new sharpness result.

Primarily human-written textHuman understanding: all partsfunctional equation

Oct 09, 2026PDF ↗View work →

hexagon:2610.00170v1

math.CO — Combinatorics

The number of multiplicity sets of n-point plane line collections grows as exp(Theta(sqrt(n)))

Contributed by Raphael DUCAY

For a set P of n points in the plane R^2, let A(P) be the set of multiplicities of P, i.e. the sizes |ell cap P| >= 2 of the subsets of P lying on a common line ell, and let F(n) be the number of distinct multiplicity sets A(P) as P ranges over all n-point sets. Erdos problem 607 (erdosproblems.com/607) asks whether F(n) <= exp(O(sqrt n)), remarking that the sqrt(n) scale is easy to see to be best possible. We prove the upper bound F(n) <= exp(c sqrt n) for an absolute constant c > 0. The proof is a three-step reduction to a published result of Colbourn-Phelps-Rodl, "Block sizes in pairwise balanced designs" (Canad. Math. Bull. 27(3), 375-380, 1984): (1) the lines determined by P, restricted to P, form a pairwise balanced design (PBD) of order n - every unordered pair of points lies in exactly one such block; (2) the profile set of block sizes of this PBD is exactly A(P); (3) Theorem 2.4 of Colbourn-Phelps-Rodl bounds the number g(n) of profile sets of order-n PBDs by exp(c2 sqrt n), hence F(n) <= g(n) < exp(c2 sqrt n). Combined with the grid construction of Szemeredi-Trotter (1983) this settles the order of magnitude exactly: F(n) = exp(Theta(sqrt n)). The essential observation is that plane line collections are precisely PBDs, so the published design-theoretic bound applies to multiplicity sets directly. A Lean 4 machine-checked formalization of the reduction is in progress for the Palomar registry and will be linked from this deposit upon acceptance.

A mix of human-written and AI-generated textHuman understanding: all partscombinatorial-geometryerdos-problem-607line-geometriesmultiplicity-setspairwise-balanced-designs

Oct 09, 2026PDF ↗View work →

hexagon:2610.00171v1

math.CO — Combinatorics

Measurable Independence Density Equals the Finite Independence-Ratio Infimum in the Euclidean Plane

Contributed by Ákos Dúcz

Let m1(R^2) be the supremum of upper densities of Lebesgue-measurable subsets of the plane containing no pair of points at distance one. We show, using the spectral rigidity theorem in OpenAI's recent proof that the plane is not five-colorable, that m1(R^2) = inf_G alpha(G)/|V(G)|, where G ranges over nonempty finite unit-distance graphs in the plane. The argument constructs an isometry-invariant law on independent subsets of the countable algebraic plane, projects the occupancy indicator onto the continuous spectral factor without changing its expectation, and extracts a measurable independent set with arbitrarily small density loss. As a consequence, if f(n) is the least independence number among unit-distance graphs on n vertices, then f(n)/n converges to m1(R^2). The proof is nonquantitative and uses the cited spectral rigidity result as a black box.

Primarily AI-generated textHuman understanding: some partsSpectral RigidityUnit-distance graphsindependence ratioinvariant random independent setsmeasurable sets

Oct 09, 2026PDF ↗View work →

hexagon:2610.00172v1

cs.DS — Data Structures and Algorithms

From Snapshots to Processes: Dynamic Contact Algebras as a Foundation for Process-Oriented GIS

Contributed by John Hessler

Geographic information science has argued for three decades that representing change requires moving from time-stamped snapshots to events and processes. This ontological shift has not been matched by a formal foundation with the properties a GIS needs: explicit axioms, representation theorems, and decidable reasoning. We propose dynamic mereotopology, in the form of Vakarelov’s dynamic contact algebras (DCAs), as a candidate foundation. DCAs are point-free with respect to both space and time. Regions are histories, primitive relations include space contact, time contact and precedence, moments are recovered as clusters of the time-contact relation, and properties of time correspond to axioms about regions alone. Using Grenon and Smith’s distinction between SNAP (snapshot) and SPAN (process) ontologies, we argue that DCAs originate on the SNAP side, since they are abstracted from a snapshot model, yet their primitives already carry much of what a SPAN ontology requires. We then identify precisely what is still missing for a process-oriented GIS: process and participation primitives, continuity of change, freedom from global simultaneity, identity under splitting and merging, and granularity. We develop a SPAN-first extension. For two of its components we give full proofs: a point-free continuity axiom, shown to express exactly the intended condition in discrete time and in finitely variable dense time, and a set of correspondences showing which axioms of continuous space hold at every finite resolution of discrete data and which emerge only in the limit of refinement. The remaining proposals are stated as conjectures, and the framework is tested against three case studies: a flood, a wildfire, and a pandemic.

A mix of human-written and AI-generated textHuman understanding: all partsdynamic mereotopology; contact algebra; boolean algebra

Oct 09, 2026PDF ↗View work →

hexagon:2610.00162v1

math.ST — Statistics Theory

Nearly Minimax Variance Estimation Under Rough Random Design

Contributed by P. M. Aronow, Patrick Lopatto

We determine, up to a power of $\log n$ , the minimax risk for constant conditional variance estimation under rough random design. The unknown design density is bounded above and away from zero, with no smoothness assumption, and the conditional error laws may depend on the covariates and have uniformly bounded fourth moments. For an $s$ -Hölder regression function with $s>1$ in dimension $d>4s$ , the minimax root-mean-square risk lies, for all sufficiently large $n$ , between $c\Psi_n$ and $C\Psi_n(\log n)^{\Gamma}$ , where $\Psi_n=n^{-2(s+1)/(d+4)}e^{-\kappa\sqrt{\log n}}(\log n)^{(s-1)/(d+4)}$ and the constants $\kappa>0$ and $\Gamma>0$ are explicit. In particular, the minimax exponent is $2(s+1)/(d+4)$ , the minimax risk is smaller than $n^{-2(s+1)/(d+4)}$ by a stretched-exponential factor whose constant $\kappa$ is identified, and the rate proposed by Robins is not uniformly attainable over this model class. For $0<s\le1$ , we show that the exact minimax rate is $n^{-1/2}\vee n^{-4s/(d+4s)}$ ; for $s>1$ and $d\le4s$ , it is $n^{-1/2}$ .

Primarily AI-generated textHuman understanding: some parts

Oct 08, 2026PDF ↗View work →

hexagon:2610.00163v1

math.MG — Metric Geometry

An exponential improvement for Borsuk's problem

Contributed by Andriy Prymak

We present a method for improving the classical exponential base $\sqrt{3/2}$ in the upper bound for Borsuk's partition number. The emphasis is on explaining the method rather than optimizing the bound. The problem is reduced to constructing directions which make an acute angle with every member of a prescribed spherical set. These play the same role as the illuminating directions in Schramm's method. The geometric ingredients are an antipodal pairing, a lift by one coordinate, and reflection in a self-dual cone. This reflection folds Euclidean space into the cone. A translation opposite to the horizontal part of its Gaussian mean makes the required supporting inequalities hold with high probability. The Gaussian Poincar\'e inequality controls the mean, and the formula for translating a Gaussian density bounds the measure cost of this construction.

Primarily AI-generated textHuman understanding: all partsBorsuk's partition problemGaussian Poincare inequalityGaussian measureasymptotic convex geometryself-dual conesspherical polarity

Oct 08, 2026PDF ↗View work →

hexagon:2610.00160v1

stat.TH — Statistics Theory

Binary Regression for BV Class Conditional Probability

Contributed by Rajesh Dachiraju

In this article we solve the problem of binary regression when the class conditional probability is a multivariate function of bounded variation. We study the consistency of binary regression under a nonparametric sampling framework. Sufficient conditions are established for the convergence of the binary regression estimator, and quantitative error estimates are obtained. We further strengthen the analysis by introducing the assumption that the feature vectors possess a point density measure of bounded variation. Under this geometric regularity assumption, the expectation-based error estimate is replaced by a deterministic estimate, yielding deterministic convergence in the $L^{2}$ norm and, consequently, almost sure convergence. The analysis demonstrates that the geometric distribution of the sampling points plays a fundamental role in the approximation properties of the estimator. Although the analysis is carried out on the torus $\mathbb{T}^{m}$ , the results extend to arbitrary bounded Lipschitz domains after affine scaling, embedding into $\mathbb{T}^{m}$ , and zero extension. The point density measure framework provides a deterministic geometric perspective on binary regression and establishes a connection between sampling geometry, bounded variation, and convergence theory.

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

Oct 08, 2026PDF ↗View work →

hexagon:2610.00159v1

math.GT — Geometric Topology

The torus with asymptotically the fewest simple closed geodesics

Contributed by Phuc Thinh Dang, Trong Toan Dao, Nhat Minh Doan

We prove that the modular torus uniquely minimizes the area of the stable-norm unit ball among complete finite-area hyperbolic once-punctured tori, resolving the McShane--Rivin area conjecture. It therefore has fewer simple closed geodesics of length at most $L$ than any fixed nonmodular torus for all sufficiently large $L$ . We also bound the area increase away from the modular torus and determine its leading growth and first correction term as the systole tends to zero, with error bounds independent of the twist.

Primarily human-written textHuman understanding: all partsFarey polygonsHyperbolic surfacesMcShane–Rivin area conjectureMirzakhani functions.modular torussimple closed geodesicsstable norm

Oct 08, 2026PDF ↗View work →

hexagon:2610.00157v1

math.ST — Statistics Theory

Nearly Minimax Rates for Functional Estimation Under Rough Random Design

Contributed by P. M. Aronow, Nathan Kallus, Patrick Lopatto

We establish nearly minimax bounds for missing-at-random means, treatment effects, and expected conditional covariances under rough random design. For two nuisance functions with average H\"older smoothness $s$ in dimension $d$ , the minimax root-mean-square error is $n^{-2s/d+o(1)}$ when $s<d/4$ and of order $n^{-1/2}$ when $s\ge d/4$ . The first rate confirms the rough-design exponent suggested by higher-order influence function theory. In the generic model with an unknown bounded density weight, our upper and lower bounds differ by only polylogarithmic factors and identify a leading correction $e^{-\kappa\sqrt{\log n}}$ when $s<d/4$ , with $\kappa$ explicit in terms of $s/d$ and the weight bounds. For the expected conditional covariance, the same exponent and the same constant $\kappa$ were obtained independently and concurrently by S. Park (arXiv:2610.05006). For models with separately bounded density and propensity, we identify the same polynomial exponent and the explicit leading correction, with an $o(\sqrt{\log n})$ remainder in the logarithm of the risk.

Primarily AI-generated textHuman understanding: some parts

Oct 08, 2026PDF ↗View work →

hexagon:2610.00155v1

math.PR — Probability

The Parisi Formula for the Edwards–Anderson Model in the High-Dimensional Limit

Contributed by P. M. Aronow, Patrick Lopatto

We study the Edwards–Anderson spin glass on the torus $(\mathbb{Z}/L\mathbb{Z})^d$ , with independent Gaussian couplings of variance $1/(2d)$ between nearest neighbors. We prove that, as $d\to\infty$ , its free energy converges to that of the Sherrington–Kirkpatrick model, given by the Parisi formula, at every temperature and in every uniform external field, uniformly in the side length $L$ . The same holds on the hypercube $\{0,1\}^d$ , and the ground-state energies converge to the Sherrington–Kirkpatrick ground-state energy.

Primarily AI-generated textHuman understanding: some partsparisi formulaspin glass

Oct 08, 2026PDF ↗View work →

hexagon:2610.00156v1

math.NT — Number Theory

Modularity of elliptic curves over totally real field and CM fields not containing \zeta_5

Contributed by Bao Le Hung

This notes tries to parse the basic structure of OpenAI's new argument proving modularity of elliptic curves over imaginary quadratic fields. The argument also proves modularity over all totally real field, and in the original form, modularity over CM fields not containing \zeta_5. The general CM case has also now been resolved by an independent repository which improves the Caraiani-Newton modularity lifting theorem for CM field at the prime 5, which was the main reason for the extra hypothesis. I kept the argument in the totally real case separate to highlight the main new idea in the OpenAI proof, which is orthogonal to modularity lifting theorems.

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

Oct 08, 2026PDF ↗View work →

hexagon:2610.00154v1

math.PR — Probability

Parisi formula for Ising and Spherical Perceptrons

Contributed by Daniel Fu, Youngtak Sohn

We prove a Parisi formula for Ising and spherical perceptrons with Gaussian disorder and i.i.d. random potentials having a common Lipschitz bound and integrable values at zero. We assume $M/N\to\alpha\in(0,\infty)$ , where $M$ is the number of Gaussian vectors and $N$ is the number of spins. The proof combines cavity in $M$ and $N$ with the Aizenman–Sims–Starr scheme for the lower bound and Guerra’s interpolation for the upper bound. As consequences, we answer several problems posed by Talagrand (2011) and establish a quenched large deviation principle for the empirical distribution of the Gaussian projections, resolving a conjecture of Bolthausen and Kistler (2010). We also resolve the sharp-threshold conjecture of Aubin, Perkins, and Zdeborová (2019) for the $u$ -function binary perceptron at every positive width, and establish the predicted variational formulas for the entropy and capacity of the negative spherical perceptron. Finally, the zero-temperature limit yields the scalar projection-pursuit formulas conjectured by Montanari and Zhou (2025, 2026), in both the supervised and unlabelled settings.

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Oct 08, 2026PDF ↗View work →

hexagon:2610.00153v1

math.NT — Number Theory

The generalized Fermat equation $x^2+y^5=z^7$

Contributed by Manvir Jaswal

We show that the equation $x^2+y^5=z^7$ has no solution in nonzero coprime integers, assuming a classification theorem from Putz's thesis, which has not appeared in a refereed journal. Equivalently, none of the equations $x^2+y^5=z^7$ , $x^2+y^7=z^5$ , $x^5+y^7=z^2$ has a solution in nonzero coprime integers. Putz attaches to a solution an octic algebra, the fibre of a Belyi map of degree $8$ , and proves that it is always isomorphic to one fixed octic field $L_8$ . We prove unconditionally that no solution has fibre isomorphic to $L_8$ , by a fifth-power descent over a field $L_{24}$ of degree $24$ that Putz introduced, completed by fifth-power residue symbols at five auxiliary primes. We also prove, without assuming the generalized Riemann hypothesis, that $L_{24}$ has class number $1$ , using a criterion of Belabas, Diaz y Diaz and Friedman.

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Oct 08, 2026PDF ↗View work →