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math.NT — Number Theory

Cyclic reductions of elliptic curves at a positive proportion of primes

Contributed by Manjul Bhargava

We prove that every elliptic curve over $\Q$ whose points of order two are not all rational has cyclic reduction at a positive lower density of primes. This is the only necessary restriction: full rational two-torsion prevents cyclic reduction at every odd prime of good reduction. More precisely, a common zero-free half-plane $\Re s>1-1/N$ for Dedekind zeta functions gives a lower proportion of at least $c_E/(N\log N)$ , with $c_E>0$ independent of $N$ ; the theorem of Cureton and Thorner supplies $N=32$ . The proof uses the varying-cofactor sieve of our earlier works on primitive roots and primitive points, in conjunction with the marked Type~II estimates of~OpenAI, to construct candidate primes. We then apply a smoothed explicit formula for Dedekind zeta functions, with nonnegative weights on prime ideals, to bound the number of candidates with noncyclic reduction. This gives a common argument for CM and non-CM curves.

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

Oct 31, 2026PDF ↗View work →

hexagon:2610.00698v1

math.OA — Operator Algebras

Noncommutative Lagrange-Fujiwara, Carmichael-Mason and Kuniyeda Bounds for Roots of Polynomials

Contributed by K. MAHESH KRISHNA

Two centuries ago, Lagrange derived an upper bound for every root of a complex polynomial using the maximum of powers of coefficients. In 1916, Fujiwara independently found the Lagrange bound. In 1914, Carmichael and Mason derived an upper bound using the sum of squares of the absolute values of the coefficients of a polynomial. This bound was extended by Kuniyeda in 1916. We derive noncommutative versions of the Lagrange-Fujiwara, Carmichael-Mason and Kuniyeda bounds.

Primarily human-written textHuman understanding: all partsBanach algebra.Carmichael-Mason boundKuniyeda boundLagrange-Fujiwara bound

Oct 31, 2026PDF ↗View work →

hexagon:2610.00697v1

math.NT — Number Theory

Deterministic Polynomial-Time Primality Testing with a Single Binomial Congruence

Contributed by Joseph Shunia

We give a deterministic polynomial-time primality test whose main verification is the congruence $(1+X)^n=1+X^n$ in $C_n[X]/(X^r-b(Y))$ , where $C_n=(\mathbb{Z}/n\mathbb{Z})[Y]/(\Phi_r(Y))$ . Two identities in the coefficient ring normalize $b$ and ensure compatibility with the main congruence. A finite-field root count and a lattice bound show that an accepted input is a prime power; elementary preprocessing excludes nontrivial perfect powers. We construct $b$ deterministically using Jacobi sums and the uniform Dedekind-zeta zero-free theorem of Cureton and Thorner, which extends OpenAI's Hecke zero-free theorem. The total bit complexity is $L^{32+o(1)}$ , where $L=\lceil\log_2 n\rceil$ , and verification after constructing $b$ costs $L^{6+o(1)}$ . More generally, for a fixed real algebraic $1/2\le\theta<1$ , a common zero-free boundary $\Re s>\theta$ for the relevant Dedekind zeta functions gives $L^{\max\{6,(1-\theta)^{-1}\}+o(1)}$ . This bound reaches $L^{6+o(1)}$ at $\theta=5/6$ , and hence under GRH. We state a fixed-base modification of Agrawal's conjecture for odd integers passing the base- $2$ Fermat test, using the modulus $X^r-2$ with $r$ the least positive integer not dividing $n(n-1)$ .

Primarily AI-generated textHuman understanding: all partsJacobi sumsKummer extensionsbinomial congruencescyclotomic ringsdeterministic algorithmspolynomial timeprimality testing

Oct 31, 2026PDF ↗View work →

hexagon:2610.00696v1

math.CV — Complex Variables

A contact and coefficient correction for Korenblum's constant

Contributed by Ricardo Pinho

Retaining an angular deficit in an annular coefficient estimate gives $C_{68}\ge0.44667$ for Korenblum's constant in the unweighted Bergman space. The flux input is David Niedbala Giraudin's. A contact-moment estimate and an even outer factor retain a correction involving the first two coefficients; a rational measure then satisfies the modified moment conditions. The argument builds on Wang's coefficient approach and Wikström's moment integration. This note is unrefereed, with no independent human review or formalization.

Primarily AI-generated textHuman understanding: some partsBergman spaceKorenblum's maximum principleannular Green functioncontact momentsinterval certificatesouter functions

Oct 31, 2026PDF ↗View work →

hexagon:2610.00695v1

math.NT — Number Theory

A trace upper-bound certificate with complex-root auxiliaries

Contributed by Ricardo Pinho

A rational piecewise-linear measure with 23 auxiliary polynomials gives $C_{86}\le1.80697883$ for the Schur--Siegel--Smyth trace constant. The proof uses Smith's arithmetic-measure criterion and converse theorem. Two degree-eight auxiliaries have nonreal roots; both polynomials already occur in Flammang (2009) and McKee (2011). Their use in this upper-bound certificate, rather than their discovery, is the change described here. This note is unrefereed, with no independent human review or formalization.

Primarily AI-generated textHuman understanding: some partsSchur-Siegel-Smyth trace constantauxiliary polynomialsinterval certificateslogarithmic potentialtotally positive algebraic integers

Oct 31, 2026PDF ↗View work →

hexagon:2610.00694v1

math.CO — Combinatorics

Fourier granularity and block sensitivity in the relevant-variable recurrence

Contributed by Ricardo Pinho

Rational Fourier lattice certificates, inserted into the existing local-degree recurrence, give $C_{44}\le4.1225323803$ for the number of relevant variables of a Boolean function. The finite packet contains 37 block-sensitivity exclusions through degree 40. We explain the certificate criterion and its passage to a bound valid in every degree, and identify the inherited analytic inputs. Neither the small-degree bounds $B_4\le9$ , $B_5\le14$ nor Fourier granularity is new. This note is unrefereed, with no independent human review or formalization.

Primarily AI-generated textHuman understanding: some partsBoolean functionsFourier granularityblock sensitivityrational certificatesrelevant variables

Oct 31, 2026PDF ↗View work →

hexagon:2610.00693v1

math.CO — Combinatorics

An exact computer-assisted certificate for R(5,6)≤84

Contributed by Dr. András Pongrácz

We give an exact computer-assisted proof of R(5,6)≤84. We derive a system of linear inequalities that must be satisfied by any graph on 84 vertices containing neither a five-vertex clique nor a six-vertex independent set. We provide explicit nonnegative rational weights for which adding these inequalities yields a contradiction. Published small Ramsey bounds and complete catalogues of smaller Ramsey graphs supply the external inputs. The accompanying verification programs check the derivation of the inequalities and their weighted sum using exact arithmetic. Two supplements give a worked local calculation and a detailed mathematical specification of the complete proof.

Primarily AI-generated textHuman understanding: some partsRamsey theory

Oct 31, 2026PDF ↗View work →

hexagon:2610.00692v1

cs.CC — Computational Complexity

The partition exponent in the graded pieces of the Claude 3-SAT algorithm is optimal for pieces of that shape

Contributed by Carlos Stein Naves de Brito

In Appendix B of "A $2^{o(n)}$ time algorithm for 3-SAT" (Claude, Anthropic, hexagon:2610.00577) the gain parameter $\rho = 1/\sqrt{2}$ comes from the Hardy--Ramanujan growth $\log p(E) \sim c\sqrt{E}$ of the partition numbers, and Appendix H observes that growth of order $E^\alpha$ would give $\rho = 2^{\alpha-1}$ by the same calculation. We point out that the rank floor stated in the same Appendix H already excludes $\alpha < 1/2$ for every graded family of the same shape: the pieces at weight $h$ have total rank at most $(4h+1)\dim V_{\le 4h}$ in one of the two directions, and the crossing they must add up to has rank $(\dim V_h)^2$ , so the construction needs $\log \dim V_{\le 4h} \ge 2\log\dim V_h - O(\log h)$ , that is $\alpha \ge 1/2$ . Restoring the rank with a larger weight costs $\rho \ge 2^{-\alpha}$ in the weight count, so the count never goes below $\max(2^{\alpha-1}, 2^{-\alpha}) \ge 1/\sqrt{2}$ , with equality only at $\alpha = 1/2$ ; when the identity holds on the whole weight space, as Lemma 9 of the paper gives it, the bound sharpens to $2^\alpha/(2(4^\alpha-1))$ , which exceeds 1 for $\alpha < 0.357$ . Theorem B of the paper is rank-tight up to a polynomial factor. The inequality is exact and we tabulate it: for the partition numbers it holds for every $h$ , for partitions into squares ( $\alpha = 1/3$ ) it fails for all $h \ge 5891$ . We also record that the Heisenberg Fock spaces on a sparse set of modes, the natural candidates for slower growth, do not carry Lemma 9, and that no choice of coefficients repairs it. The floor $\rho \ge 1/2$ remains open.

Primarily AI-generated textHuman understanding: some parts3-SATExponential Time Hypothesisinteger partitionsrank bounds

Oct 31, 2026PDF ↗View work →

hexagon:2610.00690v1

math.DG — Differential Geometry

Isometric embeddings of parallelizable and stably parallelizable Riemannian manifolds

Contributed by Wenchang Yang, Yi Wang

Let $s_n=n(n+1)/2$ . We prove that every smooth parallelizable Riemannian $n$ -manifold, $n\ge 3$ , admits a smooth isometric embedding into $\mathbb{R}^{s_n+1}$ whose image lies in an arbitrarily small ball. For closed stably parallelizable manifolds, each of whose connected components has vanishing Euler characteristic, we obtain small-image isometric embeddings into $\mathbb{R}^{s_n+5}$ . The proof combines explicit partial isometries along commuting Hopf flows, an iterative scheme for exact metric correction using oscillatory perturbations constructed from the second fundamental form, and a transversality argument for metric-preserving perturbations.

A mix of human-written and AI-generated textHuman understanding: some partsHopf actionh-principleisometric embeddingmetric correctionparallelizable manifoldsecond fundamental form

Oct 31, 2026PDF ↗View work →

hexagon:2610.00688v1

math.NT — Number Theory

The Riemann hypothesis as a margin: a two-sided criterion verified to 2*10^29, the fourth-moment anatomy of Mobius cancellation, and one constant seen from every side

Contributed by Pedro Caceres

This paper collects, with uniform notation, a month of new results around one way of looking at the Riemann hypothesis (RH): as the statement that explicitly computable margins stay on one side of explicit constants for ever. Part I studies the Cesaro margin M(N) between a smooth curve and the weighted prime sum 2*sum_{q<N} Lambda(q) q^{-1/2} log(N/q): RH is equivalent to M(N) >= 0 for all large N (a theorem of Suzuki), and we verify 0 < M(N) < 0.18484 for every real 1 < N <= 2.15*10^29, with explicit unconditional bounds beyond; the margin is analyzed as a walk of the primes against a curve, as an energy, and as a one-sided mean: RH holds if and only if a weighted mean of the prime-counting error stays below -B = -(1 + gamma/2 - (1/2) log 4pi) for ever. Part II dissects the Mobius side of the same coin: an exact positive-definite variance identity for sum_{d<=D} mu(d)({x/d}-1/2), empirically Gaussian statistics, and an exact anatomy of the fourth moment — the conductor-1 orbit decouples, the cross term moves only at composite squarefree conductors, two of its four sectors are O(D log^4 D) unconditionally, and an empirical jump law, first refuted, is proved exactly in its sector. Part III shows that 2B is the Baez-Duarte constant of the Nyman-Beurling criterion and recomputes the Nyman-Beurling distances from scratch through Vasyunin's cotangent formula: d_N^2 log N reaches 0.0461 against 2B = 0.04619, with no zero of zeta used anywhere in the computation. Part IV distils the author's twenty-two-part study of the cross-energy of consecutive Dirichlet partial sums: the trichotomy that singles out the critical line as the trace of the pole, the energy law (the mean energy of the prime-counting error is >= 2B unconditionally, with equality if and only if RH holds with simple zeros), the energy slope as a truncated Weil functional exact in the primes, the functional reflection that reproduces Hardy's function from the integers, and the systematic Davenport-Heilbronn and Beurling controls. Part V states precisely what none of this can do: a no-go theorem shows that weights indistinguishable from the primes' to q^(delta-1/2) satisfy every averaged statement here and still carry an off-line zero, so the exact identity Lambda = mu * log must enter beyond its first moments. The paper opens with a list of its twenty-two original contributions and a matching statement of what is due to others. We do not prove the Riemann hypothesis, and no statement in this paper should be read as progress towards it beyond what it literally says.

A mix of human-written and AI-generated textHuman understanding: all partsRiemann hypothesis; Cesaro; Mobius; Landau; Nyman; Dirichlet; Weil; Davenport

Oct 31, 2026PDF ↗View work →

hexagon:2610.00686v1

math.NT — Number Theory

Uniformizability of elementary Anderson $t$ -motives of dimension $2$

Contributed by David Niedbala Giraudin

Let $M(A)$ , $A\in M_2(\mathbb{C}_\infty)$ , be the elementary Anderson $t$ -motive of dimension $2$ and rank $4$ defined by $Te=\theta e+A\tau e+\tau^2e$ . Using the affine equation of Ehbauer, Grishkov and Logachev, we give an explicit sufficient condition for $M(A)$ to be uniformizable, in terms of the valuations of the coefficients of that equation. It holds as soon as no cancellation occurs among the four summands of one of these coefficients; in particular $h^1(M(A))=4$ in the simple case where no cancellation occurs in any coefficient. This proves their Conjecture 3.7: $h^1(M(A))=4$ whenever $(\mathrm{ord}\,a_{11},\mathrm{ord}\,a_{12},\mathrm{ord}\,a_{21},\mathrm{ord}\,a_{22})$ lies outside an explicit union of nine hyperplanes.

Primarily AI-generated textHuman understanding: no partsAnderson t-motivesDrinfeld modulesfunction fieldsuniformizability

Oct 31, 2026PDF ↗View work →

hexagon:2610.00685v1

math.NT — Number Theory

Ultrametric Bourgain-Figiel-Milman, Enflo Type and Mendel-Naor Cotype Problems

Contributed by K. MAHESH KRISHNA

We ask for ultrametric version of following three: (1) Bourgain-Figiel-Milman Theorem, (2) Enflo Type, (3) Mendel-Naor Cotype.

Primarily human-written textHuman understanding: all partsBourgain-Figiel-Milman TheoremEnflo TypeMendel-Naor CotypeUltrametric space.

Oct 31, 2026PDF ↗View work →

hexagon:2610.00684v1

math.PR — Probability

Curvature-Induced Killing at Stochastic Algebraic Folds

Contributed by Zhenyu Sun

We study the lifetime of a regular algebraic branch under continuous forcing and infinite-activity tangent jumps. For a parabolic fold, a first-order tangent kick decreases the discriminant by an exact quadratic amount. Symmetric stable marks of index $\alpha\in(0,2)$ therefore generate a one-sided normal jump kernel of index $q=\alpha/2$ , with a coefficient determined by curvature. The normal diffusion and jump amplitudes depend on the evolving tangential state. We compute the full killed generator on powers of the discriminant and obtain occupation and terminal-jump bounds for this same joint process. Uniformly subcritical normal drift gives finite censored boundary lifetime and finite cumulative killing rate; continuous contact and jump exit both have positive probability. At and above the uniform critical threshold, a logarithmic nonlocal barrier excludes censored boundary contact, and the original branch terminates at a jump almost surely. A comparison after a stochastic time change yields an explicit stretched-exponential upper bound for survival and hence all positive lifetime moments. In the subcritical regime, an optimized upper bound for the probability of jump termination tends to zero as the curvature coefficient or the initial discriminant tends to zero. The analysis combines power estimates for the jump generator with a logarithmic barrier at the critical threshold. Numerical calculations illustrate the dependence of terminal causes on curvature.

A mix of human-written and AI-generated textHuman understanding: all partsLévy processesalgebraic foldsboundary accessibilitykilled jump diffusionsstochastic differential-algebraic equations

Oct 31, 2026PDF ↗View work →

hexagon:2610.00682v1

math.GM — General Mathematics

Arithmetic Expression Spaces: Expansion and Motion, Eikonal Equation, Horocycle Pullbacks, Commutativization, Contact Structure, and Tearing

Contributed by Mingli Yuan

An arithmetic expression is a word of operations; a motion is a flow on a manifold. For expressions whose evaluation has a single internal spine, the two are one structure: affine for addition and multiplication, projective once the second operand slot is divided, contact-geometric in the order defect. A regular arithmetic expression space (AES) is an oriented Riemannian surface $(M,g)$ with a smooth assignment $a\colon M\to\mathbb R$ satisfying the eikonal equation $|\nabla a|_g^2=\mu^2+\lambda^2a^2$ ; the four operations then act as motions translating and scaling $a$ . In the basic model $\mathfrak E_0$ , the upper half-plane with the left-invariant metric of $\mathrm{Aff}^+(1,\mathbb R)$ (curvature $-\lambda^2$ ), addition and positive multiplication are affine translations and dilations obeying a Baumslag--Solitar law, and $a$ is a Laplace--Beltrami eigenfunction of eigenvalue $2\lambda^2$ . Left-expanded words stay in the affine Borel cell fixing $\infty$ ; the right-expanded language, once second-slot division is admitted, generates all of $\mathrm{PGL}_2(\mathbb R)$ through the rank-one Bruhat decomposition, and pulling standard horocycles back along one operator word gives the dual reading, ripple geometry. Commutativizing the motions by total charge yields the accumulative commutative space, where the order defect of two histories is an exact weighted area, the tearing. The same defect is the constant $\mu\lambda$ in the bracket $[D_u,D_v]=\mu\lambda\partial_a$ of the horizontal fields of $\alpha=da-\mu\,du-\lambda a\,dv$ , a contact form whenever $\mu\lambda\ne0$ , and it is the curvature of that horizontal connection; for $\mathfrak E_0$ the basic model is proved to be the Reeb quotient of the contact model, and fixed-charge tearing and fixed-state Reeb drift are Legendre dual readings of the same curvature. The generators $\partial_a$ , $a\partial_a$ , and $a\log a\,\partial_a$ span an infinite-dimensional Lie algebra, so no finite-dimensional Lie group carries all three ranks, and no observable of the charge endpoint detects tearing. For every finite puncture count a conformally compactifiable punctured AES is constructed, with an explicit four-circle zero locus and exact reconstruction from three-ended pants blocks. All results are elementary, planar, or local to a three-dimensional contact model.

A mix of human-written and AI-generated textHuman understanding: all partsBaumslag--Solitar relationLegendre dualityaffine cocyclesarithmetic expression geometrycontact structures and Reeb quotientseikonal equation on surfaceshyperbolic upper half-plane and horocyclesinfinite-dimensional Lie algebraspositive affine grouppunctured Riemann surfaces and pants decompositions

Oct 31, 2026PDF ↗View work →

hexagon:2610.00680v1

math.DG — Differential Geometry

Closed three-manifolds as hypersurfaces of Euclidean fourspace: compactness, the homogeneous geometries, and total absolute curvature

Contributed by Jay Thrussell

## Abstract Let $(\Sigma, g)$ be a closed Riemannian three-manifold, and let $\mathbb M = -G$ be its negative Einstein endomorphism. Where $\det\mathbb M > 0$ , the companion paper [MA] reduces realizability as a hypersurface of $E^4$ to one equation, $F(g) = 0$ , and studies it locally. Here we treat the global questions. - **Compactness.** A closed realization has $\det\mathbb M \geq R^{-6}$ at a point farthest from any centre, where $R$ is that distance, and so $\max\det\mathbb M \geq 125/(8\,\mathrm{diam}^6)$ by Jung's theorem. If $\det\mathbb M > 0$ everywhere, the realization is a strictly convex embedded sphere, unique up to rigid motion. So a closed connected metric with $\det\mathbb M > 0$ is realizable iff $\Sigma$ is simply connected and $F(g) = 0$ . No closed three-manifold other than $S^3$ carries such a realizable metric. - **The homogeneous geometries.** A realizable closed metric with $\det\mathbb M > 0$ and constant scalar curvature is round, and so is every compact locally homogeneous realization. Locally, constant $\mathbb M$ -eigenvalues with $\det\mathbb M > 0$ force constant curvature, a case of Tsukada's classification. This recovers Agaoka and Hashinaga's list of the left-invariant metrics that embed locally. The eight Thurston geometries are worked as checks. - **Total absolute curvature.** $J(g) = \int\sqrt{\det\mathbb M}\,dV$ equals $2\pi^2$ on every closed realizable metric with $\det\mathbb M > 0$ , so any other value excludes realizability. At the round sphere $J$ is stationary. Its second variation on transverse-traceless tensors of level $n$ is $-(\mu_n + 2)(\mu_n + 3)/16$ times the squared norm, $\mu_n = n^2 - 3$ , and its radical is exactly the conformal and Lie directions. On conformal rays $J$ is flat to third order. Along the conformal path of [MA], $J = 2\pi^2 - \pi^2t^4/12 + O(t^6)$ . - **Non-convex realizations.** Embedded hypersurfaces of revolution are non-convex exactly around their necks. For the radial graphs $r = 1 + \varepsilon\cos2\chi$ , convexity is lost at $\varepsilon = 1/5$ in a Whitney cusp of the Gauss map. With $\delta = \varepsilon - 1/5$ , the relative excess $J/2\pi^2 - 1$ then grows as $500\,\delta^{3/2}/(9\sqrt2\,\pi)$ . - **Connectivity, and the round sphere.** Modulo isometry, the closed realizable metrics with $\det\mathbb M > 0$ form a contractible family, contracted by an explicit equivariant path. The embedded hypersurfaces of revolution join the round sphere through the degenerate locus. We collect the characterisations of the round sphere, with their hypotheses. Every displayed result is reproduced by a script; Appendix V indexes them.

Primarily AI-generated textHuman understanding: no partsEinstein tensorGauss map singularitiesGauss–Codazzi equationsThurston geometrieshypersurfaces of Euclidean spaceisometric immersionsupport functiontotal absolute curvature

Oct 31, 2026PDF ↗View work →

hexagon:2610.00677v1

math.CO — Combinatorics

Local independent sets and colourings of triangle-free graphs

Contributed by Matthew Bowen, András Máthé, Oleg Pikhurko

This AI-assisted draft constructs, for every sufficiently large $D$ , a local rule of radius $O((\ln D)^2)$ that produces, on any triangle-free graph $G$ of maximum degree at most $D$ , a random independent set that includes each vertex with probability at least $(1-o_D(1))\ln D/D$ . Further properties of the rule yield, for every fixed $\varepsilon>0$ and all sufficiently large $D$ , a randomised LOCAL algorithm that colours any such finite graph $G$ with at most $(1+\varepsilon)D/\ln D$ colours, with high probability, in $O_\varepsilon(\ln D)+(\ln\ln |V(G)|)^{O(1)}$ communication rounds. Applying known transference principles for Borel graphs, we obtain the same asymptotic bound on the number of colours for measurable and Baire-measurable colourings, and for Borel colourings of graphs with finite asymptotic separation index.

Primarily AI-generated textHuman understanding: some partsChromatic numberLOCAL algorithmdescriptive combinatoricsindependent settriangle-free graphs

Oct 31, 2026PDF ↗View work →

hexagon:2610.00675v1

math.PR — Probability

Smallest Gaps between Eigenvalues of Wigner Matrices without Smoothness Assumptions

Contributed by Patrick Lopatto

We determine the asymptotic distribution of the smallest gaps between eigenvalues in the bulk of the spectrum of real symmetric Wigner matrices, including Bernoulli matrices. After multiplication by $N^{3/2}$ , the gaps and their locations converge to a Poisson point process with the same intensity as for the Gaussian orthogonal ensemble. Since changing a single entry can move eigenvalues by an amount comparable to the smallest gaps, standard perturbative comparison estimates do not directly resolve this scale for arbitrary entry distributions. Instead, we use that resampling a few random entries does not change the law of the matrix, and we show that the resampling moves each pair of nearby eigenvalues by a random walk that satisfies a local limit theorem; this requires excluding arithmetic structure in the eigenvectors. We also prove a law of large numbers for the number of gaps at all scales between $N^{-3/2}$ and $N^{-1}$ , and we sketch the proof of the analogous smallest-gap result for complex Hermitian Wigner matrices.

Primarily AI-generated textHuman understanding: some partsrandom matrix theory

Oct 31, 2026PDF ↗View work →

hexagon:2610.00674v1

math.IT — Information Theory

Finite Alphabet Communication with Noisy Forward Memory in a Normalized Loop

Contributed by Daniel Adam Pyle

We study one-use communication of a finite classical message through a stipulated normalized loop containing a backward binary symmetric channel and one noisy forward memory carrier. The carrier is either a classical bit or a qubit; no shared random seed, retained side register, or entanglement is supplied. For every finite message count, we characterize the qubit optimum by a constructive scalar feasibility test. For rational noise parameters and a rational target success, the test uses O(M²) constant-size rational arithmetic checks. The proof combines an exact decoder reduction, minimum-trace allocation for constant encoders, and a concave geometric cap with a matching polygon construction. For three messages, the qubit and classical-bit optima coincide throughout the allowed noise range; an explicit Bernstein certificate proves the remaining polynomial inequality. For four messages, an exact rational qubit protocol exceeds a bound valid for every classical-bit protocol. These are conditional mathematical results for the declared resource model, not a construction of a physical backward channel. Independent human review and originality assessment remain pending.

Primarily AI-generated textHuman understanding: no partsinformation theory quantum information

Oct 31, 2026PDF ↗View work →

hexagon:2610.00673v1

math.NT — Number Theory

Small-index subsequences in finite cyclic groups

Contributed by Chao Liu

For $r\ge1$ , let $I_r(n)$ be the least length forcing a nonempty zero-sum subsequence of index at most $r$ over the cyclic group $C_n$ . We prove $I_r(n)\le n^{1+o(1)}$ when $r\ge C[\omega(n)\log(2\omega(n))]^2$ , and $I_1(p^k)\ll(k+1)p^k\log(2p^k)$ for prime powers. We also construct sequences showing that $I_r(n)/n$ is unbounded for every fixed $r$ ; for $r=1$ , all terms may be generators.

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

Oct 31, 2026PDF ↗View work →

hexagon:2610.00672v1

cs.DS — Data Structures and Algorithms

Counting Hamiltonian cycles in sparse graphs in subexponential time, assuming the new #SAT algorithms

Contributed by Vaibhav Tulsyan

In October 2026 two manuscripts claimed refutations of the Exponential Time Hypothesis by deterministic algorithms for $\#k$ -SAT, each with a Lean formalisation of its top-level statement. Neither result applies directly to connectivity problems such as Hamiltonicity, because no linear-size CNF encoding of "this edge set is a single cycle" is known. This paper derives a corollary of the Anthropic manuscript: assuming three of its internal results (a network-to-clique reduction, an algorithm for cliques of all levels, and the existence of its "pieces"), the number of Hamiltonian cycles of a graph with $n$ vertices and at most $Dn$ edges can be computed exactly and deterministically in time $O(2^{\delta n})$ for every $\delta>0$ , and in time $2^{O(\max(m,n)/\sqrt{\log n})}$ for a graph with $m$ edges. A rank-encoded variant of the construction gives $2^{O(n\log\bar d/\sqrt{\log n})}$ for average degree $\bar d$ . Deterministic decision, search and Hamiltonian-path counting follow, and so do exact sparse TSP with polynomially bounded weights, exact counting of directed Hamiltonian cycles, and ATSP. The result is not a standalone breakthrough; it stands or falls with the claimed refutation. The ingredients are known: the folklore matrix-tree identity on $2$ -factors, the fermionic (Berezin) matrix-tree theorem of Caracciolo, Jacobsen, Saleur, Sokal and Sportiello, and the blob partitions of Cut&Count. What is new is the observation that connectivity can be encoded with $O(1)$ bits per edge through a Grassmann determinant whose only non-local part is a sign, and that this sign costs only $O(g^3)$ bits for $g$ blobs, so the whole construction is a network in the sense of the manuscript, whose value is $n\cdot\#\mathrm{HC}(G)$ . Every theorem we claim is formally verified in Lean, end to end, on top of the manuscript's own Lean development: the three counting theorems, the decision, search and path corollaries, sparse TSP, directed Hamiltonian cycles and ATSP, using only the standard axioms, with fresh kernel replays. The Lean sources are at https://github.com/xennygrimmato/eth-hamiltonicity. We also record weaker results that rest only on the machine-checked top-level theorems: the parity of $\#\mathrm{HC}(G)$ for every graph in time $2^{O((m+n)/\sqrt{\log(m+n)})}$ (machine-checked in Lean, with the hypotheses discharged by the labs' own Lean proofs), and one-sided randomized decision. We describe what is proved, what is checked numerically, what is formalised, what is assumed, and what the limitations are.

Primarily AI-generated textHuman understanding: some partsHamiltonian cycle; ETH; matrix-tree theorem; Grassmann algebra; Cut&Count; TSP

Oct 31, 2026PDF ↗View work →

hexagon:2610.00671v1

cs.DS — Data Structures and Algorithms

Improved basis-change transformations for $2 \times 2$ matrix multiplication

Contributed by Hugo Nielsen

To the best of our knowledge, we present a state-of-the-art change-of-basis algorithm for $2 \times 2$ matrix multiplication under the Karstadt--Schwartz-style basis change regime [1]. We do this by introducing basis-change matrices $B_U$ , $B_V$ , and $B_W$ that each require only a single addition and hence give rise to a matrix-multiplication algorithm performing $5n^{\log_2{7}} - 4n^2 + \frac{3}{4}n^2 \log_2{n},$ arithmetic operations when multiplying two $n \times n$ matrices, with $n$ a power of $2$ . This improves on the previous best bound, which had last term $\frac{3}{2}n^2\log_2{n}$ .

Primarily human-written textHuman understanding: all parts

Oct 31, 2026PDF ↗View work →

hexagon:2610.00670v1

math.CO — Combinatorics

Nonunimodal Laurent coefficients of cluster monomials

Contributed by Qiyue Tang

We prove that cluster monomials of coefficient-free type $A_5$ can have Laurent coefficient sequences that are neither unimodal nor log-concave. This disproves the log-concavity and unimodality assertions in Chen’s Conjecture 5.11 and gives a further family of counterexamples to Chen–Sun’s Conjecture 8.2, including its unimodality assertion.

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

Oct 31, 2026PDF ↗View work →

hexagon:2610.00669v1

math.CO — Combinatorics

The covering number C(13,7,4) is 30: A classification-assisted exact proof

Contributed by John Tripodo

We prove that $C(13,7,4)=30$ , using the classification of minimum $(12,6,3)$ coverings by Gordon, Patashnik, Petro and Taylor. A hypothetical $29$ -block cover has at least five points lying in exactly $15$ blocks. The link at one of these points leaves only $14$ blocks outside it. The other four points force complete families of outside blocks whose overlaps must agree. An exhaustive computation shows that every compatible choice requires $15$ outside blocks, giving a contradiction. The known $30$ -block construction is therefore optimal. We give reproducible exact computations and a public Lean formalization of a second argument for the finite obstruction. The Lean theorem states the published minimum bound and classification as explicit hypotheses; it does not reprove those results.

Primarily AI-generated textHuman understanding: no partsLeancovering designscovering numbersexact computation

Oct 31, 2026PDF ↗View work →

hexagon:2610.00668v1

math.CO — Combinatorics

Counterexamples to log-concavity of cluster monomials and theta structure constants

Contributed by Qiyue Tang

We disprove Conjectures 1.1 and 5.5 of Chen–Huang–Sun and the log-concavity assertions in Chen’s Conjecture 5.11 and Chen–Sun’s Conjecture 8.2 by constructing counterexamples for cluster monomials and theta structure constants in coefficient-free type $A_5$ .

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

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hexagon:2610.00667v1

math-ph — Mathematical Physics

First-order phase transition in Onsager's model of hard rods in $\mathbb R^3$

Contributed by Qidong He

We prove a first-order transition for hard rods in $\mathbb R^3$ whose orientations vary continuously over the sphere. For every sufficiently small positive ratio of diameter to length, the grand-canonical pressure has a nondifferentiable point, with a jump in number density at the intermediate Onsager scale. Following the phase-pressure strategy of Lebowitz, Mazel, and Presutti, we connect coexistence in the homogeneous Onsager functional to the microscopic pressure through an exact auxiliary-field representation, exponential decay of correlations in the rod gas at fixed auxiliary field, and estimates of the free-energy cost of interfaces between connected families of minimizing angular profiles.

Primarily AI-generated textHuman understanding: no parts

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hexagon:2610.00665v1

math.AG — Algebraic Geometry

A Linear In Rank Bound For Integral Arithmetic Progressions On Elliptic Curves

Contributed by Jacob Crim

Every known bound on the length N of an x-progression on an elliptic curve of rank r has the shape N ≤ C^{1+r}. We prove a bound that is linear in the rank for integral progressions, with an error depending only on the height of the model. Let y^2 = x^3 +a_2x^2 +a_4x+a_6 be an integral model and H_f = 1+|a2|+|a4|+|a6|. For every ε > 0, every integral x-progression of length N satisfies N ≤ (3/2 +ε)s+C_{ε} (1+log(H_f))^{1/2}, where s ≤ r is the rank of the span of the progression. In particular N ≤ (3/2 +o(1))s whenever log H_f = o(N^2). The proof has two halves. The local theory consists of exact point counts, the root-class lemma, pinned values, and a forbidden range of primes reaching N^2/49 by a shift-and-square argument. It forces log |b| ≫ N^2, where b is the common difference. The spectral cap is proved in an exact form. Splitting the Neron–Tate pairing into local kernels of positive type and summing the regularised diagonal terms over all places gives an identity in which the bad-prime corrections cancel. Combined with a uniform archimedean separation lemma, this yields λ_{max}(G) ≤ (3/4) logX + (1/4) ψ(N − 1) + (5/4) log(H_f )+ 2, where X =max|x_i|. Over function fields we prove an exact colouring formula for the Gram matrix of a progression of integral sections, which gives N ≤ min(6χ, (3/2)s). We also classify the 3-term Gram types on rational elliptic surfaces and show that all of them occur, and we show that all sign data are realised on elliptic K3 surfaces. The main bound is vacuous for every known example (all have N ≤14); its interest is that it is linear rather than exponential in the rank.

Primarily AI-generated textHuman understanding: some parts

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hexagon:2610.00664v1

math.NA — Numerical Analysis

Low-discrepancy quasi-uniform digital sequences in every dimension

Contributed by Leonhard Gruenschloss

A sequence of points in the unit cube is quasi-uniform if its first $N$ points are spread like a grid for every $N$ : no two of them are closer, and no empty ball between them is larger, than a constant times $N^{-1/s}$ in dimension $s$ . Dick, Goda and Suzuki asked for such sequences among the digital sequences of quasi-Monte Carlo integration, and Goda, Hofer and Suzuki asked whether any Halton-type sequence is quasi-uniform. We show that in dimension two and higher no Halton-type sequence, no Sobol' sequence, no classical Niederreiter sequence and no Faure sequence is quasi-uniform, and that linear scrambles of all coordinates but one do not help: a carry in the last digits of one coordinate leaves the leading digits of all others unchanged, giving explicit pairs of points that are far too close. We then show that one fixed linear scramble of a generalized Faure sequence, the same for all numbers of points, repairs this. In it, the digits of all coordinates at one scale are residues of one rational function, each taken at the coordinate's own point of the projective line, so the residue theorem ties the coordinates together and a carry in one coordinate is seen by the others. For every dimension $s$ and every prime base $p>2^s$ this gives explicit quasi-uniform digital $(0,s)$ -sequences, hence of low discrepancy, whose first $N=p^{sK-1}$ points are at least $(1-2p^{-1/s})N^{-1/s}$ apart, for every $K$ ; in dimensions two and three, for every prime $p\ge5$ , a variant keeps its first $N$ points at least $(1-2/p)N^{-1/s}$ apart whenever $N$ is a power of $p^{s^2}$ . For such scrambles, exact linear rules that place every point inside its cell as a function of the last few digits of the cell's index can hold at no more than about one in $s$ of the sizes $p^{sK}$ ; with allowed ranges instead of exact rules we certify all of these sizes, with points at least $(1-O(1/p))N^{-1/s}$ apart for large $p$ . Quasi-uniform digital sequences follow in every prime base $p\ge3$ in dimension three, and in every prime base in dimension two, base two included. Some steps are computer-assisted.

A mix of human-written and AI-generated textHuman understanding: some partsHalton sequencesLean 4 formalizationcovering radiusdigital sequenceslow discrepancyminimum distancequasi-Monte Carloquasi-uniformityresidue theorem

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hexagon:2610.00663v1

math.CO — Combinatorics

Maximal Independent Sets in Long Graph Subdivisions

Contributed by Daniel Adam Pyle

We study maximal independent sets in finite simple graphs whose edges are replaced by paths with prescribed numbers of internal vertices. Using established finite-state transfer methods, we derive an exact boundary-state formula and an explicit leading asymptotic term that factors over the degrees of the original vertices. A graph-uniform relative error bound gives a sufficient condition for this approximation to remain valid when the base graph grows. Under the same condition, the joint membership indicators of the original vertices approach an explicit product of Bernoulli distributions in total variation. For fixed base graphs, we obtain the first adjacency-sensitive correction and a Gaussian limit for the size of a uniformly chosen maximal independent set. A degree-matched construction demonstrates that bounded aggregate error scale does not suffice for uniform degree-sequence universality. Reproducibility materials include exact enumeration, weighted and moment checks, and numerical spectral comparisons.

Primarily AI-generated textHuman understanding: no partsGraph subdivision; graph theory

Oct 31, 2026PDF ↗View work →

hexagon:2610.00661v1

math.NT — Number Theory

A two-sided criterion for the Riemann hypothesis from weighted prime sums, verified up to 2·10^29

Contributed by Pedro Caceres

For N > 1 let C(N) = 2 Σ_{n<N} Λ(n) n^{-1/2} log(N/n) and M(N) = R(N) − C(N), where R(N) = 8√N − 5.3721834 log N − 7.4013354 + O(N^{-5/2}) is explicit. By a theorem of Suzuki, the Riemann hypothesis (RH) holds if and only if M(N) ≥ 0 for all large N, and then 0 ≤ M(N) ≤ 4(log ξ)''(1/2) = 0.18484. We prove, with computer assistance, that this two-sided inequality holds for every real 1 < N ≤ 2.15·10^29: up to 10^10 from the primes with correctly rounded arithmetic, and beyond from the first 4520 zeros, certified with ball arithmetic, together with the verification of RH to 3·10^12 and an explicit zero-density estimate. The same ingredients give, for every N and without any hypothesis, −E(N) ≤ M(N) ≤ 0.18484 + E(N) with E(N) explicit and E(10^20) < 3·10^-5. We describe the local minima of M exactly and prove an unconditional energy identity for the slope α(x) = Σ_{n≤x} Λ(n) n^{-1/2} − 2√x + c0/2. We then show that positive weights within q^{δ−1/2} of Λ(q) q^{-1/2} satisfy the energy identity, the prime number theorem and Chebyshev’s identity, yet carry an off-line zero, so any proof must use Λ = μ∗log exactly. We do not prove the Riemann hypothesis.

A mix of human-written and AI-generated textHuman understanding: all partsRiemann hypothesis; explicit formula; Weil positivity; Riesz; ball arithmetic

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hexagon:2610.00660v1

math.NT — Number Theory

Computing Jacobsthal’s Function on Primorials: Exact Values and Constructive Bounds

Contributed by Alben Weeks

Let h(x) = j(Px), where j is Jacobsthal’s function and Px is the product of the first x primes. We describe a parallel residue-covering search. Exhaustive exclusion runs, together with independently checked covering witnesses, establish h(65) = 1134, h(66) = 1160, h(67) = 1192, h(68) = 1216, h(69) = 1248, h(70) = 1274, and h(71) = 1298. Method 5B gives explicit certificates for h(100) ≥ 2054, h(150) ≥ 3558, and h(200) ≥ 5172, using partial coverings with five reserve primes. Exhaustive searches give h(100) ≤ 2798, h(150) ≤ 6898, and h(200) ≤ 11998. The search combines parity compression, descending endpoint restrictions, canonical deduplication, a sliding-window bootstrap, and dynamic prime selection related to Ziller and Morack’s Greedy Permutation Algorithm. We state the conditions needed for capacity bounds and conditional propagation, and organize extensions of complete or partial coverings in a five-tier framework. A catalog of 62 coverings at index 65 supports a verified multi-certificate envelope. The solver source, run evidence, certificates, and arithmetic verifiers are publicly available in the accompanying GitHub release v1.0-paper. Upper-bound conclusions depend on exhaustive execution of the search; the record audits do not independently repeat those searches or supply standalone unsatisfiability certificates

A mix of human-written and AI-generated textHuman understanding: some partsJacobsthal functionbranch-and-boundparallel computationprimorialresidue covering

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hexagon:2610.00662v1

math.AG — Algebraic Geometry

Integral crystalline Tate classes of codimension two at supersingular points of Mumford's fourfolds

Contributed by Antti Hippeläinen

For a supersingular abelian variety $X$ over $\bar{\mathbb{F}}_p$ , let $T^4(X)$ be the lattice of integral crystalline Tate classes of codimension $2$ and $S(X)$ the span of products of divisor classes. Let $p\ge5$ and $\mathscr{G}_p$ be the $p$ -divisible group of the supersingular points of Mumford's families of abelian fourfolds at primes inert in their cubic field, for the hyperspecial levels given by tensor-induced lattices. For every abelian fourfold $X$ with $X[p^\infty]\cong\mathscr{G}_p$ we show that $T^4(X)/S(X)\cong(\mathbb{Z}/p)^4$ , and that the integral crystalline Tate conjecture holds in codimension $2$ on $X$ if statements in a preprint of Markman hold. At these primes, Kisin's crystalline tensors make the Dieudonné module a tensor product of three rank- $2$ modules permuted cyclically by Frobenius, and at supersingular points Frobenius maps a tensor basis to multiples of basis vectors. For such monomial modules all lattices involved are spanned by multiples of basis vectors, with exponents given by min-plus formulas independent of $p$ . Under the endomorphisms, $T^4/S$ is a non-split extension of $(\mathbb{Z}/p)^2$ by $(\mathbb{Z}/p)^2$ , generated by one Weil class of an inert imaginary quadratic action. Assuming these statements, an algebraic such class comes from specializing a Weil cycle on a fourfold with complex multiplication. A projector that is a polynomial in an endomorphism, and an intersection matrix with unit determinant, fix its $p$ -adic normalization without a comparison theorem. At unramified non-inert primes with such levels, $T^4=S$ at the supersingular points. Of the fourteen isomorphism classes of monomial supersingular Dieudonné modules of height $8$ , the conjecture holds unconditionally for nine, under the same assumption for that of $\mathscr{G}_p$ , and is open for four. The indices are computed exactly with integers.

Primarily AI-generated textHuman understanding: no partsDieudonne modulesMumford fourfoldsShimura varietiesWeil classescrystalline cohomologyintegral Tate conjecturesupersingular abelian varieties

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hexagon:2610.00625v1

math.LO — Logic

A true theorem Lean can state but can never verify

Contributed by William Banks

We give an expository account of a mathematical result that the Lean proof assistant can formally state but, by Gödel's second incompleteness theorem, can never verify, and which it also does not refute, namely, the consistency of Lean's own underlying type theory. Working in $\mathsf{ZFC}$ extended by the existence of infinitely many inaccessible cardinals, we sketch the construction of a set-theoretic model of Lean's Calculus of Inductive Constructions (with its universe hierarchy, quotient types, and choice), outline the proof of a soundness lemma, and conclude that no derivation of $\mathtt{False}$ exists. The argument is the model-theoretic consistency proof carried out in full by Carneiro, building on Werner's set-theoretic interpretation of the same calculus; all substantive results are due to these authors, and this paper claims no new mathematics. We explain why this theorem lies outside Lean's own reach, note that Gödel's second incompleteness theorem has itself been verified by Lean's kernel for first-order arithmetic, and discuss why every consistency proof ultimately rests on trust rather than proof.

A mix of human-written and AI-generated textHuman understanding: all partsLeanconsistencyinaccessible cardinalsincompletenessproof assistantstype theory

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hexagon:2610.00658v2

math.DG — Differential Geometry

Riemannian three-manifolds as hypersurfaces of Euclidean four-space: the realizability operator, its symbol, and its linearisation at the round sphere

Contributed by Jay Thrussell

For a Riemannian three-manifold $(\Sigma,g)$ with negative Einstein endomorphism $\mathbb{M}=-G$ , the Gauss equation of a hypersurface of $\mathbb{R}^4$ reads $\mathbb{M}=\mathrm{adj}\,K$ , and on $\{\det\mathbb{M}>0\}$ it is solved identically by $\tilde K=\sqrt{\det\mathbb{M}}\,\mathbb{M}^{-1}$ . Realizability with nondegenerate second fundamental form is therefore the single equation $F(g)=\mathrm{Cod}(\tilde K(g))=0$ . The reduction is classical (Beez, Killing, Thomas) and the zero set has been characterized intrinsically (Rivertz; Agaoka and Hashinaga); we study $F$ as a nonlinear operator on metrics. At every nondegenerate point its principal symbol has rank two, with kernel exactly the symbol of the linearised immersions, so $F$ is injective modulo immersions everywhere and nowhere elliptic modulo diffeomorphisms alone. The nine Codazzi components carry four identities, leaving five conditions at every nondegenerate point, and Cartan's test holds with equality, in exact arithmetic, off a proper real-algebraic set. On a closed manifold the zero set is nowhere dense in the nondegenerate domain. At the round $S^3$ the linearisation on transverse-traceless tensors is $\mathrm{Cod}\circ\frac{1}{2}(-\Delta+2)$ , with singular values $n(n^2-1)/\sqrt{2}$ and $\|DF\,h\|^2\ge 288\|h\|^2$ . For metrics foliated by round 2-spheres we solve the first-order problem in closed form and write down all hypersurfaces swept by round 2-spheres; the $\ell=1$ direction is realized exactly by tilting the spheres' 3-planes. We treat the degenerate locus by two polynomial forms, and give Gauss–Codazzi as the flatness of two $\mathfrak{su}(2)$ connections: on the nondegenerate domain, realizability is equivalent to a single parallel spinor. Every displayed result is checked by a deposited script.

Primarily AI-generated textHuman understanding: no partsCodazzi tensorsGauss–Codazzi equationshypersurfacesinvolutivityisometric embeddingprincipal symbolspinors

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hexagon:2610.00659v1

math.LO — Logic

The random graph ideal and fractional coloring

Contributed by Carlos López-Callejas

The random graph ideal is pathological, answering a question of López-Abad et al. We prove this by constructing a positive hereditarily pathological restriction using Kneser graphs. We also prove that every $F_\sigma$ ideal admitting a proper nonpathological $F_\sigma$ extension has a least such extension. Moreover, for an $F_\sigma$ ideal, being nonpathological, being an intersection of summable ideals, and being an intersection of matrix ideals are equivalent conditions. This answers both converses asked by Filipów and Tryba. The least nonpathological $F_\sigma$ extension of the random graph ideal is proper and is described by the fractional cochromatic numbers of its finite induced subgraphs.

A mix of human-written and AI-generated textHuman understanding: all partsF-sigma idealsKatětov orderfractional coloringleast nonpathological extensionsmatrix idealsnonpathological submeasurespathological idealsrandom graphsummable ideals

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hexagon:2610.00651v1

math.NT — Number Theory

Power-Kernel Fluctuations, the Lambert W Subpolynomial Source, and a Local CRT Conditional Law

Contributed by Zeng Chunping

For a positive integer $n$ we define the power kernel $d_n = n/\operatorname{rad}(n)$ and the fluctuation $X_n = \log d_n$ , where $\operatorname{rad}(n) = \prod_{p \mid n} p$ . This paper develops three mutually supporting lines around this ``relative fluctuation''. First, the microscopic theory. From the kernel decomposition $n = v_d s$ ( $s$ squarefree, $(s,d)=1$ , $v_d = d \operatorname{rad}(d)$ ) we obtain the exact discrete law $\mathbb{P}(X_n = \log d) = \frac{6}{\pi^2} \frac{w(d)}{d \operatorname{rad}(d)}$ , $w(d) := \prod_{p \mid d} \frac{p}{p+1}$ , together with a three-branch parity-conditioned refinement. Parity is a first-order effect: the measured gap $\mathbb{E}[X_n \mid n \text{ even}] - \mathbb{E}[X_n \mid n \text{ odd}] = 0.69008$ matches $\log 2 = 0.69315$ . Second, the conditional law. For fixed $c$ , the distribution of $X_{c-a}$ is completely determined by $\mathrm{mask}(a)$ through a three-case local CRT convolution: the local contribution vanishes identically when $p \mid c$ or $p \mid a$ , and otherwise follows a corrected law rather than the usual one, differing by a factor $\frac{p}{p-1}$ (a factor $2$ at $p=2$ ). The corrected law agrees with $220$ masks to $0.993$ , while the usual law underestimates the tail by $47\%$ -- $78\%$ . This replaces an earlier erroneous assertion of equidistribution. Third, the subpolynomial source. For radical-type Euler products $\prod_p \bigl(1 + \frac{\eta(p)}{p^\sigma - 1}\bigr)$ with $\eta(p) \to a$ , one has as $\sigma \to 0^+$ the asymptotic $\Phi(\sigma) = \frac{a}{\sigma W(a/\sigma)}\bigl[1 + \frac{\kappa_1}{W} + \frac{\kappa_2}{W^2} + \cdots\bigr]$ , where $W$ is the Lambert function. The main term comes not from small primes but from the prime range $[e^{v^*}, e^{1/\sigma}]$ , which explains why no finite prime table can reproduce it. The coefficients satisfy $\kappa_n = (n-1)! + \sum_j c_{n,j} \zeta(2j)$ , a sum over even zetas only, because the two-sided integral coefficients satisfy $\iota(r) = (-1)^{r+1} \rho(r)$ , forcing odd-zeta terms to cancel exactly at every order. This unifies $T(x) = \sum_{n \le x} 1/\operatorname{rad}(n)$ , $\Sigma(D)$ and $\Sigma_\perp(D;Q)$ , and explains at one stroke why all previously fitted ``constants'' are transient. Applications include: a quantitative formula for Mazur parameters in the open range $1 < s < 2$ ( $k_{\mathrm{eff}} = 1.91 \pm 0.17$ , excluding $k=1,3$ ); the ceiling theorem $q \le \frac{\log c}{\log 2 + \log \operatorname{rad}(c)}$ (equality iff $\{a,b\} = \{1,2^k\}$ ); the rad-- $P^+$ duality (principal versus negative Lambert branch); an exact $S$ -unit reformulation of $\rho(c) = \min_a \operatorname{rad}(a)\operatorname{rad}(c-a)$ ; and a mechanistic decomposition of the exponent of $\mathcal{N}_0(N)$ . We report honestly: the abc conjecture itself is not attackable within this framework. The three structural walls are: (i) the conditional law is provable for $D \le c^{1/2-\delta}$ while abc solutions live at $D \gtrsim c^{0.78}$ ; (ii) relaxing any one of the three radicals inflates the count by a factor $387$ ; (iii) the Fourier circle method is controllable iff $\alpha + \beta > 2$ , while Mazur parameters always have $\alpha, \beta < 1$ . Our weak abc bound is weaker than de Bruijn's 1962 trivial bound; this is recorded as well.

Primarily human-written textHuman understanding: all partspower kernel，radical of an integer，CRT conditional law

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hexagon:2610.00655v1

math.NT — Number Theory

An improved zero-free half-plane for Hecke L-functions over number fields and Quasi-Riemann Hypothesis

Contributed by Baiying Liu

In this paper, we obtain a zero-free half-plane Re(s) > B∗ for every finite-order Hecke L- function over every number field, where B∗ = 0.87446830769353045955... is an explicitly defined algebraic number. This improves the Quasi-Riemann Hypothesis over arbitrary num- ber fields. The proof refines the cubic-theta comparison. A sharper amplified Gauss-norm estimate permits a longer additional prime product, chosen separately for each dyadic sum- mation length. Combining the resulting moment saving with contour displacement on the same dyadic sum gives the principal improvement. The enlarged moment controls the critical summation length, where the contour contribution vanishes. Successive improvements of the character-counting estimate then reach the algebraic endpoint, while the numerator-moment parameter remains fixed. We prove optimality within the resulting comparison system. The argument uses the established theta, moment and continuation results of the earlier QRH papers and the general-field formulation of those results.

A mix of human-written and AI-generated textHuman understanding: some partsHecke L-functionsRiemann Zeta Functionquasi Riemann Hypothesis

Oct 29, 2026PDF ↗View work →

hexagon:2610.00630v1

math.NT — Number Theory

A lattice-theoretic approach to the abc conjecture: Weighted arithmetic derivatives, a master inequality, and the structure of the residual family

Contributed by Zeng Chunping

We develop a self-contained lattice-theoretic framework for the abc conjecture. For coprime positive integers $a+b=c$ with $k=\omega(ab)$ , the framework rests on a weighted arithmetic derivative $d^\psi$ whose additivity equation $d^\psi(a+b)=d^\psi(a)+d^\psi(b)$ is solvable modulo the modulus $g_c=\gcd_{p\parallel c}\bigl(\frac{c}{p}v_p(c)\bigr)$ , and on a Wronskian identity $W=a\,d^\psi(b)-b\,d^\psi(a)=M_{ab}\,H(u)$ , where $M_{ab}=ab/\operatorname{rad}(ab)$ and $H$ is an explicit linear form. A divisibility lemma $M_aM_bM_c\mid W$ and an elementary upper bound yield the master inequality $c\le\mu_{ab}\cdot\frac{\Omega(ab)}{2}\cdot\operatorname{rad}(abc)$ , in which the defect factor $\mu_{ab}$ is the shortest non-kernel weight of an explicit solution lattice $\Lambda$ of determinant $g_c/g$ . Writing $\eta=\log\mu_{ab}/\log c$ , geometry-of-numbers input (Hermite's bound and an orthogonal covering-radius sandwich; an earlier ``min--max barrier'' is falsified in Theorem 4.5) yields, conditionally on an explicit gap hypothesis $(\star')$ isolated in Section 7, the bound $\eta\le\frac1k+o(1)$ , whence $\limsup_{c\to\infty}q(a,b,c)\le\frac{k}{k-1}\le2$ , i.e.\ $c\lesssim\operatorname{rad}(abc)^{2+o(1)}$ . On the subclass $a+b=r^e$ the method closes unconditionally (Zsigmondy for $r\ge3$ ; cyclotomic/LTE control of valuations for $a=1$ ). The residual family is localized by a power--congruence bridge: $q^V\parallel b$ forces $e$ into a single residue class modulo $d\,q^{V-s}$ ( $d=\operatorname{ord}_q(r)$ , $s$ the Wieferich depth), of convergent total density; the unique unclosed layer reduces to growth-difference Pillai-type equations. We also prove a structural ceiling: the main term of $\eta$ is at most $\frac1k+o(1)$ and attains $\frac1k$ on the family $c=r^e$ , so the full abc conjecture ( $q\le1+\varepsilon$ ) is structurally unreachable inside this framework. A final section determines the reach and limits of the method elementarily: almost all $c$ have every coprime decomposition with $q\le1+o(1)$ (density theorem, by one-variable counting, exceptional set $\ll X^{1/2+o(1)}$ ), while any argument within the method class $\mathcal M$ cannot prove $\limsup q\le\alpha$ for any $\alpha<2$ (obstacle theorem), and the only remaining channel is equivalent to a sparse growth-difference Pillai core. All statements are proved in full or are explicitly labelled as working hypotheses; numerical cross-checks from two independent implementations are included, and no claim of a proof of the abc conjecture is made.

A mix of human-written and AI-generated textHuman understanding: all partsPillai equationWronskianabc conjecturedensity theoremerratalattice methodmaster inequalityobstacle theoremweighted arithmetic derivative

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hexagon:2610.00646v1

math.NT — Number Theory

An Integral-Structure Inflation in the Hull Operation of IUT Corollary 3.12

Contributed by Zeng Chunping

This paper identifies and rigorously quantifies a third-class gap in the passage from Theorem 3.11 to Corollary 3.12 in S. Mochizuki's inter-universal Teichmuller theory (IUT). It occupies the same position as, but is of a different type from, and neither implies nor is implied by, the Scholze-Stix gap (non-commutativity of the $j^2$ -scaling) and the compatibility gap (9-1) recorded in the LANA project. The holomorphic hull of IUT III, Remark 3.9.5, is the minimum over the constrained family $\{\lambda \cdot \mathcal{O}\}$ : each direct summand of $I^{\mathbb{Q}} \cong \bigoplus_{\alpha} F_{\alpha}$ is scaled independently. This operation "boxes" the image of the tensor-product integral structure and introduces an index $\rho$ . The main mathematical result of this paper is the exact closed form of this index, $\rho = |\mathfrak{d}_K|^{(n-1)d^{n-1}/2}$ , $d = [K:\mathbb{Q}_p]$ , proved as an identity of positive integers (not merely a $p$ -adic valuation), where $\mathfrak{d}_K$ is the discriminant of $K/\mathbb{Q}_p$ . Its $p$ -adic corollary is $v_p(\rho) = \frac{(n-1)d^{n-1}}{2}\, v_p(\mathfrak{d}_K)$ : the loss equals $\frac{n-1}{2}$ times (number of components) times (the $p$ -adic valuation of the local discriminant). The formula is uniform for tame and wild ramification, with no extra term; wild ramification only replaces the tame value $(d-1)$ by the actual $v_p(\mathfrak{d}_K)$ . Substituting IUT's own parameters ( $n = j+1$ , $\ell^* = (\ell-1)/2 \asymp h^{1/2}$ , $v \in V^{\mathrm{bad}}$ ), the inflation is doubly exponential: $\log\log\rho = \Theta(\sqrt{h})$ , i.e. $\rho = \exp(\exp(\Theta(\sqrt{h})))$ . Measured by the crossover scale $h^*(\kappa)$ , the conclusion is robust: even if the target bound were as generous as $h^{1000}$ , the $d = 2$ inflation overtakes it at $h \approx 4\times 10^{9}$ and the $d = 4$ inflation at $h \approx 9\times 10^{8}$ . This paper is not a verdict on IUT. What it proves is the narrower statement: even if (9-1) holds, layer-1 inflation alone already suffices to prevent Corollary 3.12 from yielding abc.

Primarily human-written textHuman understanding: all partsinter-universal Teichmuller theory，abc conjecture，holomorphic hull

Oct 29, 2026PDF ↗View work →

hexagon:2610.00654v1

math.DG — Differential Geometry

Collision Geometry and Holonomy of Finite Blaschke Model Spaces

Contributed by Zhenyu Sun

We investigate the rank- $n$ Hermitian vector bundle of finite Blaschke model spaces over the coefficient domain of monic polynomials whose zeros lie entirely in the open unit disk. By introducing a global anti-holomorphic rational frame, we show that both the bundle and its orthogonal-projection connection extend real-analytically across all internal root-collision strata. At the maximally coincident divisor, we compute the full curvature tensor explicitly; the successive differences of its matrix coefficients yield the standard matrix units, which implies that both the restricted and full holonomy groups coincide with the full unitary group $U(n)$ . Furthermore, the induced metric on the determinant line bundle is shown to agree with the diagonal restriction of the Szegő kernel of the symmetrized polydisc, and its power-sum Kähler potential endows the coefficient domain with a complete Kähler metric. By contrast, once the Clark phase circle is adjoined to the base, the Clark spectral lines together with the canonical model-space conjugation single out a unique compatible unitary connection. This real-spectral connection is flat, extends real-analytically across internal collisions, and exhibits a finite signed cyclic holonomy group. We also derive a weighted residue formula for the complex-bilinear pairing induced by the canonical conjugation, and analyze explicit low-dimensional subfamilies that illustrate the sharp geometric contrast between the curved projection transport of the moving model space and the flat parallel transport of its real spectral lines.

A mix of human-written and AI-generated textHuman understanding: all partsClark basesFinite Blaschke productsSzegő kernelholonomymodel spacesroot collisions

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hexagon:2610.00653v1

math.NT — Number Theory

Non-Archimedean John, Non-Archimedean Dvoretzky-Milman, Non-Archimedean Type-Cotype and Non-Archimedean Kwapien Problems

Contributed by K. MAHESH KRISHNA

We ask for non-Archimedean version of following four: (1) John Theorem, (2) Dvoretzky-Milman Theorem, (3) Type-Cotype of Banach space, (4) Kwapien Theorem.

Primarily human-written textHuman understanding: all partsDvoretzky TheoremJohn TheoremKwapien Theorem.Type-Cotype of Banach space

Oct 29, 2026PDF ↗View work →

hexagon:2610.00652v1

math.NT — Number Theory

Polynomial parametrization of quadratic Diophantine equations: decision and construction

Contributed by Bogdan Grechuk, Ashleigh Ratcliffe

We give an algorithm that, for any polynomial with integer coefficients of degree at most two, decides whether its integer zero set is a finite union of polynomial families with integer coefficients and constructs such families whenever they exist. The main step is a constructive proof that, for every nondegenerate integral quadratic form $Q$ isotropic over $\mathbb{Q}$ and every integer $m$ , the integer solutions of $Q(\mathbf{x})=m$ form a finite union of polynomial families. This proves the nondegenerate case of a result announced without proof by Vaserstein. For fixed $Q$ , the number of parameters in each family is bounded independently of $m$ . The new descent argument treats forms in at least five variables with rational Witt index one. We also settle the previously open cases $zt-x^2-y^2=1$ and $zt-x^2-y^2=-1$ by combining Larsen--Nguyen's group parametrization with a classical reduction of binary Hermitian forms, obtaining two and three polynomial families, respectively.

Primarily AI-generated textHuman understanding: some partsarithmetic groupsintegral pointspolynomial parametrizationquadratic fieldsquadratic forms

Oct 29, 2026PDF ↗View work →

hexagon:2610.00650v1

math.MG — Metric Geometry

Symmetric Volume Descent from U4 and a Four-Dimensional Lower Bound

Contributed by Hangang Wang

We explain a symmetric deformation that decreases the volume of the four-dimensional constant-width body $U_4$ . The emphasis is on why the deformation works, rather than the size of the decrease. Oddness preserves width; convexity requires control of the curvature radii. A positive cap on one support branch is coupled to a negative correction on the adjacent branch. Their first-order volume contributions have different orders in the localization parameter, so the favorable contribution dominates. A small motion towards the ball makes the direction feasible for positive time. In a separate part we give a spectral lower bound for all four-dimensional bodies of constant width one, including an explicit positive refinement. A short projection argument using a three-dimensional theorem of Jiang gives the stronger lower bound $0.17872203$ . A substantial gap between the lower and upper bounds remains.

A mix of human-written and AI-generated textHuman understanding: all partsconstant widthspectral lower boundsvolume descent

Oct 29, 2026PDF ↗View work →

hexagon:2610.00649v1

math.FA — Functional Analysis

A bridge between analysis on local fields and quantum groups: the noncompact case

Contributed by Cédric ARHANCET

Let $K$ be a non-Archimedean local field with residue cardinality $Q$ and let $d \geq 1$ . We establish a radial correspondence between analysis on $K^d$ and Woronowicz's quantum group $\mathrm{E}_r(2)$ , with $r^2 = Q^{-d}$ . For every $\alpha > 0$ , the radial Taibleson operator of order $\alpha$ is unitarily equivalent, through a unitary commuting with dilations, to a bilateral birth–death operator with Hahn–Exton $q$ -Bessel eigenfunctions. Their Dirichlet forms are equivalent on the same weighted sequence space, with optimal comparison constants. For $\alpha = d$ , the bilateral operator is the radial part of a self-adjoint Casimir on the quantum plane $\mathrm{E}_r(2)/\mathbb{T}$ , whose heat semigroup is covariant and completely positive. We further obtain these quantum-plane dynamics by contraction of the compact Casimir dynamics on the Podleś sphere, with Mosco convergence of the forms and pointwise norm convergence of the compressed heat maps.

A mix of human-written and AI-generated textHuman understanding: some parts$q$-Hankel transformDirichlet formsHahn--Exton $q$-Bessel functionsPoisson kernelsTaibleson operatorlocal fieldsquantum Markov semigroupsquantum group $E(2)$

Oct 29, 2026PDF ↗View work →

hexagon:2610.00648v1

math.FA — Functional Analysis

A bridge between analysis on local fields and quantum groups: the compact case

Contributed by Cédric ARHANCET

Let $K$ be a non-Archimedean local field with residue cardinality $Q$ and let $d \geq 1$ . We establish equivalence of Dirichlet forms between radial regional Taibleson operators on the unit ball of $K^d$ and little $q$ -Jacobi operators. Under the parameter relation $r^2 = Q^{-d}$ , the zonal Casimir $L$ of the standard Podleś sphere yields equivalent forms through its powers $L^{\alpha/d}$ precisely for $0 < \alpha < 3d$ . Covariant quantum Markov realizations up to equivalence of forms exist precisely for $0 < \alpha \leq d$ . We also relate this correspondence to boundary energies: harmonic extensions on the tree of balls recover a Taibleson energy, while the energy per level of quantum Poisson integrals converges to a multiple of the Casimir energy. Finally, we extend the correspondence and the sharp covariance threshold to quantum projective spaces, identifying the Taibleson orders associated with their zonal Casimir operators.

A mix of human-written and AI-generated textHuman understanding: some partsDirichlet formsPodle\'s spherePoisson boundariesTaibleson operatorlocal fieldsnoncommutative $\L^p$-spacesquantum Markov semigroupsquantum groupsquantum projective spacesrandom walks on trees

Oct 29, 2026PDF ↗View work →

hexagon:2610.00645v1

math.CO — Combinatorics

Algebraicity of rank-two scattering wall functions

Contributed by Qiyue Tang

We prove that every wall function in the rank-two scattering factorization determined by two polynomial initial walls over a field of characteristic zero is algebraic. This resolves the algebraicity question posed by Kontsevich and recorded by Gross and Pandharipande, and its cluster-scattering formulation in the work of Burcroff, Lee, Mou, Musiker, and Reineke.

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

Oct 29, 2026PDF ↗View work →

hexagon:2610.00644v1

math.NT — Number Theory

Three cubic trinomial series for $1/\pi$

Contributed by Khai-Hoan Nguyen-Dang

We prove the three identities in Sun's Conjecture VI, involving cubes of generalized central trinomial coefficients. A common double integral represents their generating functions. On two rational curves, this period reduces to quadratic expressions in Gauss hypergeometric functions associated with elliptic integrals. A first-order identity recovers the derivative transverse to each curve and supplies the linear weights in the three series. The constants are determined by an endpoint calculation and Bauer's identity. Classical modular equations of degrees three, five, and seven then complete the evaluations. We give explicit rational differential certificates for the period reductions and the transverse derivative identities.

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

Oct 29, 2026PDF ↗View work →

hexagon:2610.00643v1

math.NT — Number Theory

Davenport constants with growing binary rank

Contributed by Chao Liu

We study the correction to the standard lower bound $D^*$ for Davenport constants of finite abelian groups with a growing elementary two-component. For every fixed nontrivial abelian $p$ -group $H$ of odd order, we prove $D(H\oplus C_2^r)=D^*(H\oplus C_2^r)+(D(H)-1)\log_2 r+O_H(1)$ , with a uniform version for varying $H$ . For $C_3\oplus C_2^r$ , we determine exact values on explicit infinite intervals of ranks. More generally, the correction is logarithmic or smaller precisely when the fixed group has at most one odd prime divisor. We show that the two-primary part affects the logarithmic coefficient: for $C_{24}\oplus C_2^r$ , the ratio $(D-D^*)/\log_2 r$ has limsup at least $2.4933231\ldots>2$ , whereas for $C_3\oplus C_2^r$ the limit is $2$ . These upper limiting coefficients are unbounded even with the odd prime fixed. The proofs use binary codes and finite-field Fourier analysis.

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

Oct 29, 2026PDF ↗View work →

hexagon:2610.00642v1

math-ph — Mathematical Physics

Near-Null Limits and Contact Distributions for Maxwell Composite Operators

Contributed by Zhenyu Sun

We investigate the simultaneous near-null, longitudinal-continuum, and ultraviolet limits of four-point correlation functions of quadratic composite operators in Maxwell theory. Taking the external momenta to be collinear and spacelike, we weight each loop integral by a smooth test function of the routed longitudinal momentum. All three connected Wick contraction cycles are generated from a common Gaussian source functional and regulated by a cutoff on the total proper time. For the resulting integrals restricted to a fixed ultraviolet proper-time interval, we establish the uniform bound $\left|\mathcal I_{\delta,M}^{\epsilon}[\phi]-\mathcal I_{0,\infty}^{\epsilon}[\phi]\right|\le C\|\phi\|_{C_b^8}\left(\delta\epsilon^{-3}+M^{-2}\delta^{-3}\epsilon^{-1}\right)$ , where $\delta$ denotes the dimensionless near-null tilt parameter and $M$ specifies the longitudinal lattice resolution. This estimate determines the joint scaling regimes under which a universal counterterm subtraction yields an identical renormalized finite part. We further show that the longitudinal logarithmic divergence of the alternating chiral cycle is governed by the measure $8\mathbf 1_{(0,1)}\mathrm d\xi+3\delta_0+3\delta_1$ , whereas each of the other two cycles contributes $8\mathbf 1_{(0,2)}\mathrm d\xi$ . These distributional coefficients emerge from the complete covariant numerators, including the instantaneous longitudinal terms. Finally, for the alternating cycle and any smooth compactly supported test function coinciding with $\xi^2$ in a neighborhood of $[0,1]$ , the critical trajectory $\delta=c\epsilon^3$ with $M^2\delta^3\epsilon\to\infty$ leaves a finite discrepancy of $30c$ , demonstrating the sharpness of the tilt scaling. The proof combines exact Gaussian moment reductions, whole-line quadrature error bounds, and an explicit evaluation of boundary jets.

Primarily human-written textHuman understanding: all partsMaxwell theorycontact distributionslight-front quantizationnear-null limitsproper-time regularization

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hexagon:2610.00640v1

math.CO — Combinatorics

Laurent and g-vector counterexamples for quivers with homotopies

Contributed by Qiyue Tang

We give negative answers to the general Laurent-phenomenon question and the g-vector sign-coherence question for quivers with homotopies posed by Li, Liu, Mou and Pan.

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

Oct 29, 2026PDF ↗View work →

hexagon:2610.00639v1

math.CO — Combinatorics

Asymptotic growth of rank-two scattering coefficients in the Badlands

Contributed by Qiyue Tang

We prove that the normalized logarithmic growth rate of ordinary wall-function coefficients in every wild rank-two cluster scattering diagram exists along all integer approaches to an interior Badlands slope and is continuous in that slope. This resolves Conjecture 9.2 of Burcroff, Lee, Mou, Musiker, and Reineke.

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

Oct 29, 2026PDF ↗View work →