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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

Oct 29, 2026PDF ↗View work →

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

Oct 29, 2026PDF ↗View work →

hexagon:2610.00658v1

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

Oct 29, 2026PDF ↗View work →

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

Oct 29, 2026PDF ↗View work →

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

Oct 29, 2026PDF ↗View work →

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

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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

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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

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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 $E(2)$

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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

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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

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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

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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

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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

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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 →

hexagon:2610.00638v1

math.PR — Probability

Infinite divisibility of hitting times for symmetric stable processes

Contributed by Min WANG

Let $T_a$ be the first hitting time of a nonzero point $a$ by a symmetric strictly $\alpha$ -stable L\'evy process, where $1<\alpha<2$ . We give a computer-assisted classification of its ordinary additive infinite divisibility. There is a unique threshold $\alpha_*$ , with $1.6670659903<\alpha_*<1.6670659915$ , such that $T_a$ is infinitely divisible if and only if $1<\alpha\le\alpha_*$ . The threshold is characterized by a unique zero contact of the canonical inverse logarithmic-derivative kernel. The proof combines explicit analytic estimates, rigorous interval covers, Gaussian necessary conditions and Rouch\'e's theorem. In particular, a fully analytic argument proves failure of infinite divisibility for every $15/8\le\alpha<2$ . Scalar bounds at the two time ends reduce the compact interval computations. The finite computations and their acceptance conditions are supplied as ancillary material.

Primarily AI-generated textHuman understanding: no parts

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hexagon:2610.00636v1

math.QA — Quantum Algebra

$C_2$ -cofiniteness of finite-group orbifolds

Contributed by Andrew Riesen

Let $V$ be a simple vertex algebra with a connected nonnegative integer grading, and let $G$ be a finite group of grading-preserving automorphisms. We prove that $C_2$ -cofiniteness of $V$ implies $C_2$ -cofiniteness of $V^G$ . The proof bounds the length and height of monomials in strong generators representing classes in a relative $C_2$ quotient. The length bound follows from a good filtration and finite-group invariant theory. The height bound follows from differential equations for products of fields, together with identities for invariants and vacuum coefficients. The height bound also shows that finite strong generation passes to $V^G$ , without assuming $C_2$ -cofiniteness. For strongly rational vertex operator algebras, the cofiniteness result and McRae's orbifold theorem imply strong rationality of every finite-group fixed-point algebra.

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

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hexagon:2610.00635v1

math.DS — Dynamical Systems

A Boolean Lévy group with a free nonsingular action

Contributed by Konstantin Slutsky

We construct a Boolean Lévy group with a free ergodic spatial nonsingular action on a standard probability space, answering a question of Glasner, Tsirelson and Weiss. The action admits no equivalent invariant $\sigma$ -finite measure. The group is a quotient of a weighted summable Boolean group by a discrete subgroup. By the nonsingular fixed-point theorem of Avraham-Re'em and Peterzil, the resulting group is not locally Wasserstein.

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

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hexagon:2610.00634v1

math.NT — Number Theory

A lower bound for the sixth moment of the Riemann zeta function

Contributed by Niklas Fraccaro

We prove that the lower limit of the sixth moment of the Riemann zeta function, normalized by $(a_3/9!)T(\log T)^9$ , is greater than $41.007$ . Our approach extends the two-piece amplifier construction used by Durkan and Page by allowing independent profiles, centered derivatives, and finite real linear combinations of two projected test families. An asymmetric twisted fourth-moment estimate and their amplified moment formula yield a common numerator kernel. Explicit balanced and unbalanced kernels determine the projected Gram form. A rational certificate with 250 tests, followed by approximation by admissible profiles, gives the stated bound.

A mix of human-written and AI-generated textHuman understanding: some partsRiemann Zeta Functionamplified momentssixth momenttwisted fourth moment

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hexagon:2610.00629v1

math.MG — Metric Geometry

Analytic symmetric floating bodies

Contributed by Liam Price, Tom de Groot

In every dimension $d\ge3$ , we construct nonspherical, origin-symmetric, strictly convex bodies of revolution with real-analytic boundary that float in equilibrium in every orientation. Their densities tend to the unit-ball cap fraction at height $(d+4)^{-1/2}$ , where we prove that a degree-four spherical resonance is isolated. More generally, our construction applies at isolated even-degree resonances whose waterline angle is an irrational multiple of $\pi$ . Varying the density gives formal deformations. To obtain actual bodies, we use a finite-order diagonal reduction to establish inverse estimates away from small divisors. Newton iteration then corrects fixed-order truncations on closed parameter sets of asymptotically full measure.

A mix of human-written and AI-generated textHuman understanding: some partsUlam's floating body problembodies of revolutioncentral symmetryconvex geometryfloating bodiessmall divisors

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hexagon:2610.00628v1

math.FA — Functional Analysis

Cowling’s conjecture: the almost connected case

Contributed by Cédric ARHANCET

We prove that a locally compact group $G$ with amenable group $G/G^0$ of components has the Haagerup property if and only if it is weakly amenable with constant one. This proves Cowling's conjecture for connected Lie groups and almost connected locally compact groups. We also characterize weak amenability for arbitrary connected Lie groups and prove the product formula for the Cowling--Haagerup constant in terms of the simple factors of a Levi decomposition. Consequently, the constant is invariant under local isomorphism and decreases under quotients of connected locally compact groups. For groups with amenable group of components, it is unchanged by every closed central quotient. We deduce applications to lattices and their central quotients. The proof combines Ozawa's invariant-state obstruction with averaging over central characters and the tensor identity for induced representations.

A mix of human-written and AI-generated textHuman understanding: some partsCowling--Haagerup constantHaagerup propertycentral extensionsconnected Lie groupsweak amenability

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hexagon:2610.00627v1

cs.CG — Computational Geometry

Universal Support Theorems for Resilient Coverage

Contributed by Deep Bhattacharjee, Ushashi Bhattacharya, Sanjeevan Singha Roy

We prove that one finite set of points decides whether sensors cover a polygonal target under every failure rule in which an extra working sensor never hurts. The set is the same for every such rule, and on a face of dimension f the weakest point is fixed by f+1 sensors. The theorem holds in any dimension for functions that share a convex part and differ by affine terms. For sensing disks it gives the exact radius and power that keep the target covered under each rule, as rational numbers, the full trade-off between radius and budget, the smallest list of sensor groups whose loss breaks coverage, exact tests under independent placement errors, and protection plans proved optimal by a matching dual certificate.

A mix of human-written and AI-generated textHuman understanding: all partsmonotone failure rulesresilience certificatessensor coveragesupport theorem

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hexagon:2610.00624v1

math.CO — Combinatorics

A Spectral Blakley–Roy Inequality for Nonnegative Matrices

Contributed by Suvrit Sra

Let $A$ and $B$ be real symmetric matrices with nonnegative entries. We prove that $\rho(AB)^k\le\rho(A^kB^k)$ for the spectral radius $\rho(\cdot)$ and every integer $k\ge1$ ; for $B=uu^T$ this is the Blakley–Roy inequality. We prove it via the more general inequality $\|(AB)^m\|^{k}\le\|(A^kB^k)^m\|$ in the spectral norm, valid for all integers $k,m\ge1$ . The even exponents case follows from standard tools; for odd exponents $k=2r+1$ the claim follows via the more general product inequality $(u^TC_1\cdots C_tv)^k\le(u^Tu)^r(v^Tv)^r\,u^TC_1^k\cdots C_t^kv$ for nonnegative symmetric $C_i$ and nonnegative $u,v$ . Its proof is an entropy argument on paths that traverse each edge $k$ times, back and forth.

A mix of human-written and AI-generated textHuman understanding: all partsBlakley-Roy inequalitymatrix productsnonnegative matricesspectral radius

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hexagon:2610.00623v1

math.FA — Functional Analysis

On Lin's Arithmetic–Geometric Means Conjecture for Singular Values

Contributed by Suvrit Sra

Let $A$ and $B$ be positive semidefinite, and let $0\le t\le 1$ . Lin (2017) conjectured that $\lambda_j((1-t)A+tB)\ge\sigma_j^{1/2}(A^{2(1-t)}B^{2t})$ for $j=1,\ldots,n$ . The case $t=1/2$ is Drury's theorem, which answered a long-standing question of Bhatia and Kittaneh. Lin remarked that Drury's method does not seem to reach other values of $t$ . We prove his conjecture for every $t\in[0,1]$ . The proof builds on Drury's argument, but replaces his block-matrix step (unique to the midpoint $t=1/2$ ) by an eigenvalue count based on an inertia identity and a one-variable inequality.

A mix of human-written and AI-generated textHuman understanding: all partsLin's conjecturearithmetic-geometric mean inequalitymatrix inequalitiessingular valuesunitarily invariant norms

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hexagon:2610.00622v1

math.CA — Classical Analysis and ODEs

A Proof of McLeod's 1959 Conjecture on Complete Homogeneous Symmetric Ratios

Contributed by Suvrit Sra

We prove McLeod's (1959) conjecture that $(h_{\ell+r}/h_\ell)^{1/r}$ is subadditive for all $\ell\ge0$ and $r\ge1$ , where $h_k$ is the complete homogeneous symmetric polynomial of degree $k$ (J. B. McLeod, "On four inequalities in symmetric functions", Proc. Edinburgh Math. Soc., 1959). We first prove that the ratio $h_k/h_{k-1}$ is convex, and our proof takes an unexpected detour through queuing theory. The geometric mean of $r$ consecutive such ratios equals McLeod's conjectured function, but geometric means of convex functions need not be convex. Our second and decisive step is a tensorization trick that couples neighboring ratios tightly enough to make their geometric mean convex.

A mix of human-written and AI-generated textHuman understanding: all partsMcLeod inequalityclosed queuing networkcomplete homogeneous symmetric polynomialconvexitydiscrete Poincare inequality

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hexagon:2610.00621v1

math.AG — Algebraic Geometry

Counterexamples to Hironaka's Characteristic- $p$ Resolution: A Report On Queries to Frontier Agents

Contributed by Taylor Dupuy

Created on 23 October 2026, this manuscript draws on a frontier-dump investigation of Hironaka's proposed positive-characteristic resolution method. Taylor Dupuy initiated the queries during his talk How to Use AI in Mathematical Research Effectively at the University of Rochester Number Theory Seminar on that date. The findings were synthesized with harvest-dumps and developed into this manuscript with using-latexpowers. We examine the proposed positive-characteristic resolution mechanism of Hironaka (2017). Its negative-degree pieces are the unit ideal at a closed singular point where, for a common multiple $m$ of the generator degrees, $\wp(E,m)$ contains an element of order $m$ . For the family $x^p+y^n$ , with $n>2p$ and $p\nmid n$ , the asserted transformation rule fails under the algebraic description of the characteristic algebra. With core focusing read by (44), the fixed local chain and the $\check E$ or $\hat E$ reading of the invariant, the length-one string agrees before and after the first blowup. With renewed chains and corresponding transported choices, the corresponding strings agree along the resolution sequence under either core-focusing reading. The two formulas for core focusing respectively retain the base-hiked exponent and, when its Inv-max stratum is smooth and closed, forget it. We give characteristic- $2$ limits on tail enclosure and a shade increase at a kangaroo point. These calculations do not supply a decreasing invariant or settle resolution. The mathematical draft of this manuscript was AI-generated by Claude Code from an AI-generated research record; its proofs received only AI review, and no human has checked them.

Primarily AI-generated textHuman understanding: no partsHironakacounterexamplespositive characteristicresolution of singularities

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hexagon:2610.00620v1

math.FA — Functional Analysis

Positive definite functions of noncommuting contractions, Hua-Bellman matrices, and a new distance metric

Contributed by Suvrit Sra

We study positive definiteness of Hua-Bellman matrices (i.e., matrices of the form $[\det(I-A_i^*A_j)^{-α}]_{ij}$ , where $A_i$ and $A_j$ are strict contractions and $α\in\mathbb{R}$ ). We start by revisiting a 1959 work of Bellman (Representation theorems and inequalities for Hermitian matrices; Duke Mathematical J., 26(3), 1959) that studies positive definiteness of Hua-Bellman matrices and claims a strengthening of Hua's representation-theoretic results (Inequalities involving determinants; Acta Mathematica Sinica, 5 (1955)). We uncover a critical error in Bellman's proof that has surprisingly escaped notice to date, and we show by an explicit example that his claim itself is false. We then provide conditions under which $\det(I-A^*B)^{-α}$ is a positive definite function. For complex contractions our condition is the known sharp one, with the integer exponents below Hua's range reached by an elementary argument. For real symmetric contractions we recover the part of Bellman's claims that survives. Building on our result, we introduce a new Poincaré-like distance metric on noncommuting strict contractions, and yet another `log-det' based distance in the appendix.

Primarily human-written textHuman understanding: all partsHua-Bellman matricesdistance metricnoncommuting contractionsoperator inequalitiespositive definite functions

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hexagon:2610.00631v1

cs.AI — Artificial Intelligence

Automated Conjecturing: Generation, Selection and Evaluation

Contributed by Ayush Agrawal, Vedant Shah, Navin Goyal, Ankit Anand, Aaron Courville

Deciding which conjectures are worth proving is a crucial part of mathematical discovery, and a system that learns mathematics from its axioms alone must make that decision without human guidance. Self-play provers such as Minimo learn a conjecturer alongside the prover and reward it for producing hard conjectures, but a conjecture that is harder to prove is not necessarily more useful for improving a prover: the learned conjecturer drifts toward long, artificial statements. We study conjecturing in a controlled, single-round setting in the Peano formal environment, across propositional logic, abelian groups, and natural-number multiplication, evaluating a set of conjectures by how much a fixed prover improves on held-out theorems after training on their proofs. We show that a far simpler generator does better: making small structural edits to statements the prover already knows yields conjectures that, at an equal training budget and under best-first search, improve the prover by 6 to 13 percentage points more than Minimo's learned conjectures. The gain holds even when conjectures are drawn at random from the generated pool. In contrast to the common practice of selecting the most difficult statements, we also propose Hard Neighbours (HN), a score that favors conjectures whose proof difficulty grows disproportionately relative to their structural distance from easier reference statements; HN gives the largest selection gains in propositional logic, but no selection method is best in every domain. In the current setting, how conjectures are generated matters more than how they are selected, and structural perturbation is the baseline that future conjecturers should be measured against. Apart from improvements in proving ability, we also study other attributes of generated proofs: the amount of search required and the length of the proofs.

Primarily human-written textHuman understanding: all partsconjecturinglanguage models

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hexagon:2610.00619v2

math.KT — K-Theory and Homology

Universal test rings for rational Gersten injectivity

Contributed by Niels FELD

For each prime $p$ , rational Gersten injectivity in a fixed degree for mixed-characteristic henselian discrete valuation rings of residue characteristic $p$ is equivalent to injectivity for $W(\overline{\mathbb{F}_p(t_1,t_2,\ldots)})$ . For a finite bound $d$ on residue transcendence degree, the test ring is $W(\overline{\mathbb{F}_p(t_1,\ldots,t_d)})$ . For these henselian DVRs, rational localization boundary images depend only on the residue field and are reflected by arbitrary residue-field extensions. A normalization by the absolute ramification index makes the resulting comparisons of kernels canonical and functorial. The proof uses regular approximation, finite transfers, and perfection. An appendix gives an alternative comparison based on the existential embedding theorem of Anscombe--Dittmann--Jahnke, including an example with nonnoetherian valuation rings. We also describe the additional boundary condition that induction between consecutive test rings would require.

A mix of human-written and AI-generated textHuman understanding: all partsCohen ringGersten conjectureWitt vectorsalgebraic $K$-theoryexistential closednesshenselian discrete valuation ring

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hexagon:2610.00618v1

math.AP — Analysis of PDEs

From Döblin to Aleksandrov

Contributed by Amélie Loher, Connor Mooney, Clément Mouhot

We prove Aleksandrov maximum principles for linear second-order kinetic equations in non-divergence form with measurable, uniformly elliptic diffusion coefficients. For coefficients $A(t,v)$ we obtain every source exponent $p>2d+1$ . For autonomous coefficients $a(x,v)$ in one space dimension, we obtain every $p>p_*(\frac\Lambda\lambda)$ , where $p_*(\frac\Lambda\lambda)<4$ depends only on the ellipticity ratio. The first step in both proofs is Döblin--Fourier cancellation: two positive parts of the kernel have nearly opposite phases, and a common lower bound for their velocity marginals allows us to cancel fixed portions of them. For the autonomous problem, this argument applies on velocity intervals away from zero. A preliminary Aleksandrov estimate gives Hölder continuity and then a Harnack comparison. A homogeneous Bellman barrier controls concentration in position and velocity; together with Harnack, it makes the local density estimates summable across zero velocity, including at $p=4$ . We give the full proofs and the passage from the maximum principle to local Hölder continuity.

Primarily AI-generated textHuman understanding: all parts

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hexagon:2610.00617v1

math.AC — Commutative Algebra

Natural Cohomology of General Steiner Bundles

Contributed by Vlado Vojdanovski

We prove that a general Steiner bundle on projective space over a field of characteris- tic zero has natural cohomology. The proof constructs a matrix of linear forms from multiplication of sections on a curve. When the rank exceeds the dimension of the projective space, a finite étale isogeny of elliptic curves splits the bundles controlling the graded kernels and cokernels into line bundles of equal degree. Choosing the auxiliary line bundle outside a finite set gives maximal rank in every degree. A rational curve provides the boundary case. Two open rank conditions then yield the assertion for general matrices and all twists.

Primarily AI-generated textHuman understanding: some parts

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hexagon:2610.00616v1

math.CO — Combinatorics

Binary Reconstruction from the Fourth Deck: Torsion Phases, Local Gluing, and Sign Extinction

Contributed by Oleksiy Babanskyy

We give a candidate proof of the following statement: for every finite cyclic group $C_n$ , the cumulative fourth deck of a binary subset determines it up to translation. Hence $r_{\mathrm{set}}(C_n)\le4$ for every $n$ , and, using the Keleti--Kolountzakis Corollary 1.1 for the lower bound, $r_{\mathrm{set}}(C_n)=4$ for every even $n>10$ . The proof has four components: finiteness of the binary fibre, cyclotomic torsion of the phases, gluing of primary characters, and extinction of dyadic residues by two odd-denominator projectors. The bridge between ordered analytic correlations and Pebody's cumulative deck is proved explicitly, including stabilizers, zero and repeated offsets, and the empty pattern. The text contains the complete mathematical chain; finite checks are included only as convention and implementation checks, not as premises of the uniform proof.

Primarily AI-generated textHuman understanding: no partsFourier analysisPebody conjecturePhase retrievaldeck reconstructionfinite cyclic groups

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hexagon:2610.00615v1

math.CO — Combinatorics

Factorial denominators of rank-two scattering coefficients

Contributed by Qiyue Tang

We prove that, for every pair of positive integers $i,j$ , the universal rank-two cluster scattering coefficient $\tau(i,j)$ satisfies $\max(i,j)!\,\tau(i,j)\in\mathbb{Z}[b,c,g]$ . This resolves the factorial-denominator conjecture of Elgin, Reading, and Stella (Conjecture 4).

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

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hexagon:2610.00614v1

math.RT — Representation Theory

The Mirković–Vilonen basis and cluster pointedness

Contributed by Qiyue Tang

We prove that the Mirković–Vilonen basis of $\mathbb{C}[N]$ contains every cluster monomial and that each of its elements is $g$ -pointed in every cluster, where $N$ is a maximal unipotent subgroup of a complex semisimple group. This resolves Conjecture 6.6 in Kamnitzer’s ICM 2022 survey.

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

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hexagon:2610.00613v1

cs.DS — Data Structures and Algorithms

What Does the Refutation of the Exponential Time Hypothesis Mean for Learning Theory?

Contributed by Xiaoyu Li

Two recent works by OpenAI and Anthropic refuted the Exponential Time Hypothesis (ETH). They count the satisfying assignments of a $k$ -CNF on $n$ variables in time $2^{n/(\log n)^{1/2-o(1)}}$ . Many fine-grained lower bounds in learning theory rest on ETH and its relatives. We determine what the refutation changes for learning. The answer is a locality principle: the new algorithms speed up learning when each labeled example constrains few bits of the hypothesis, and under the CNF form of the Strong Exponential Time Hypothesis, which survives, the speed-up ends at $\Theta(\log N)$ bits. Three transitions make the principle precise. Locality. Call a class with $N$ -bit descriptions $\kappa$ -local if each label depends on at most $\kappa$ description bits, as for quantized models on sparse data. Then proper PAC learning, exact version-space counting and sampling, and the halving algorithm run in time $\mathrm{poly}(m)\cdot 2^{o(N)}$ on samples of size $m$ for $\kappa=o((\log N)^{1/4}/\sqrt{\log\log N})$ , and proper PAC learning at constant accuracy does so for $\kappa=o(\sqrt{\log N})$ . At inverse-polynomial accuracy, proper learning of $\kappa$ -local classes is equivalent to $\kappa$ -SAT, up to a factor $2$ in the locality. Under CNF-SETH, consistency for binarized perceptrons on $O(\log N)$ -sparse data needs time $2^{(1-\gamma)N}$ for every $\gamma>0$ . Accuracy. Exact empirical risk minimization, proper agnostic learning and the exponential mechanism reduce to tensor contractions over a truncated polynomial ring. As a consequence, the Sparse Max-SAT Hypotheses of Alman and Vassilevska Williams are false. At constant accuracy, proper learning is SAT parameterized by formula length. With quasi-linear PCPs this gives a $2^{N/\mathrm{polylog}\,N}$ lower bound under a length form of ETH, which replaces Gap-ETH for reductions of quasi-linear size. At accuracy $N^{-3}/8$ , the Max-3-SAT Hypothesis forces time $2^{(1-o(1))N}$ . Treewidth. Exact inference in graphical models of bounded degree and arity takes time $2^{O(n/\sqrt{\log n})}$ . On expanders this beats every elimination order and every structured DNNF. Under $\#$ CNF-SETH, time $2^{(1-\gamma)\mathrm{tw}}$ is again necessary once a model may have $\mathrm{tw}^{O(1)}$ variables. We also take stock of the conditional lower bounds of learning theory: which ones fall, and which ones survive on which hypothesis. The next obstacle is a permutation problem. On difference vectors $e_i-e_j$ , empirical risk minimization for real-valued halfspaces is weighted Feedback Arc Set, which the new algorithms do not reach.

Primarily AI-generated textHuman understanding: some partsExponential Time HypothesisPAC learningStrong Exponential Time Hypothesiscomputational learning theoryempirical risk minimizationfine-grained complexitymodel countingprobabilistic graphical modelstensor networkstreewidth

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hexagon:2610.00612v1

math-ph — Mathematical Physics

Equality of the Jellium and the Uniform Electron Gas Energies

Contributed by Mihaly A. Csirik

Jellium, a system of electrons in a uniform neutralizing background, and the uniform electron gas (UEG), a system of electrons with constant density and no background, are two standard models of the electron gas. We prove that the ground-state energy per unit volume of Jellium, as defined by Lieb and Narnhofer, coincides with the UEG energy defined by Lewin, Lieb and Seiringer, i.e. $e^\hbar_{\rm Jel}(\rho_0)=e^\hbar_{\rm UEG}(\rho_0)$ . This equality was previously established in the classical case by Lewin, Lieb and Seiringer and independently by Cotar and Petrache.

Primarily AI-generated textHuman understanding: all partselectron gasjellium

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hexagon:2610.00611v1

math-ph — Mathematical Physics

Continuity, Testing Regions and Fragility of Quantum Channel Divergences

Contributed by Morgan Jones

Converse bounds on discriminating and estimating quantum channels are evaluated on a characterized model of the channels, which is only known approximately. This book studies when such bounds, and the operational exponents they bound, depend continuously on the channels, and whether their sensitivity to characterization error is a property of the task or only of the bound. The book opens with noisy quantum gates. The supports of the Choi matrices separate genuine fragility, where the operational exponent itself jumps, from spurious fragility, where only single-letter converses jump. Near a gate with an incoherent noise floor, the smallest eigenvalue of the normalized Choi matrix sets the sensitivity. At first order, under errors in the model, this sensitivity belongs to the task. At second order, under a coherent miscalibration, the converses can overshoot the adaptive Stein and Chernoff exponents by factors that grow without bound as the noise decreases, although for small miscalibrations of a depolarized qubit they stay within a factor 3/2. Certifying a converse from tomographic data costs a number of channel uses set by the same floor. The general theory follows. Divergences built from Kubo-Ando means, including the geometric fidelity, the geometric Renyi divergences and the Belavkin-Staszewski relative entropy, are functions of the Arveson-Radon-Nikodym derivative of one channel with respect to the sum of the pair, and in this coordinate they are continuous with dimension-free moduli. In the diamond norm they are continuous at a pair of completely positive maps exactly when the Choi matrix of the sum is invertible or the fidelity vanishes, so an ideal unitary gate paired with itself is a point of discontinuity. Divergences built from hypothesis tests, such as the Umegaki relative entropy and the hockey-stick f-divergences, are uniformly continuous in every dimension, and the adaptive

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

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hexagon:2610.00610v1

math.PR — Probability

Dynamics of the spherical Sherrington–Kirkpatrick model, Part II: Mixing profiles and critical dynamics

Contributed by P. M. Aronow, Patrick Lopatto

We study the Langevin dynamics of the spherical Sherrington–Kirkpatrick model, the diffusion on the sphere of radius $\sqrt N$ in $\mathbb R^N$ that is reversible with respect to the Gibbs measure with density proportional to $\exp(\beta x^{\mathsf T}Wx/\sqrt2)$ , where $W$ is an $N\times N$ matrix from the Gaussian orthogonal ensemble. The dynamics is started from the uniform distribution on the sphere, independently of $W$ . We determine the asymptotic quenched total-variation distance to equilibrium as a function of time in the fixed-temperature, mesoscopic, critical-window, and infinite-temperature crossover regimes described below, with convergence in probability or in distribution with respect to $W$ . Let $\beta_c=1/\sqrt2$ be the critical inverse temperature. For $\beta<\beta_c$ there is cutoff with a Gaussian profile at fixed $\beta\in(0,\beta_c)$ , along every mesoscopic approach of $\beta$ to $\beta_c$ from below, and in the infinite-temperature crossover when $N\beta_N^2\to\infty$ ; when $N\beta_N^2$ stays bounded, the distance relaxes over bounded times and there is no cutoff. At fixed $\beta\in(0,\beta_c)$ the mixing time is $(\log N-(3/2)\log\log N+O(1))/(4(1-\sqrt2\beta)^2)$ , and the correction $-(3/2)\log\log N$ comes from the edge of the spectrum of $W$ . For $\beta>\beta_c$ , at fixed $\beta$ and along every mesoscopic approach to $\beta_c$ from above, there is cutoff at a time set by the gap between the two largest eigenvalues of $W$ , of order $N^{2/3}\log N$ at fixed $\beta$ . The profile is the total-variation distance between a standard Gaussian variable and its sum with an independent Cauchy variable; the Cauchy variable is the ratio of the first two eigencoordinates at an early stage of the evolution. In the critical window $\sqrt2\beta=1+\sigma N^{-1/3}$ , the distance at time $N^{2/3}s$ converges in distribution to a random profile determined by the Airy point process, and there is no cutoff. As $\sigma\to\pm\infty$ , this profile, suitably recentered and rescaled, converges to the profiles of the two mesoscopic regimes.

Primarily AI-generated textHuman understanding: some partsdynamicsspin glass

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hexagon:2610.00609v1

math.PR — Probability

Dynamics of the spherical Sherrington–Kirkpatrick model, Part I: Foundations and tools

Contributed by P. M. Aronow, Patrick Lopatto

We study the Langevin dynamics of the spherical Sherrington–Kirkpatrick model, the diffusion on the sphere of radius $\sqrt N$ in $\mathbb R^N$ that is reversible with respect to the Gibbs measure with density proportional to $\exp(\beta x^{\mathsf T}Wx/\sqrt2)$ , where $W$ is an $N\times N$ matrix from the Gaussian orthogonal ensemble. The dynamics is started from the uniform distribution on the sphere, independently of $W$ . We determine the asymptotic quenched total-variation distance to equilibrium as a function of time in the fixed-temperature, mesoscopic, critical-window, and infinite-temperature crossover regimes described below, with convergence in probability or in distribution with respect to $W$ . Let $\beta_c=1/\sqrt2$ be the critical inverse temperature. For $\beta<\beta_c$ there is cutoff with a Gaussian profile at fixed $\beta\in(0,\beta_c)$ , along every mesoscopic approach of $\beta$ to $\beta_c$ from below, and in the infinite-temperature crossover when $N\beta_N^2\to\infty$ ; when $N\beta_N^2$ stays bounded, the distance relaxes over bounded times and there is no cutoff. At fixed $\beta\in(0,\beta_c)$ the mixing time is $(\log N-(3/2)\log\log N+O(1))/(4(1-\sqrt2\beta)^2)$ , and the correction $-(3/2)\log\log N$ comes from the edge of the spectrum of $W$ . For $\beta>\beta_c$ , at fixed $\beta$ and along every mesoscopic approach to $\beta_c$ from above, there is cutoff at a time set by the gap between the two largest eigenvalues of $W$ , of order $N^{2/3}\log N$ at fixed $\beta$ . The profile is the total-variation distance between a standard Gaussian variable and its sum with an independent Cauchy variable; the Cauchy variable is the ratio of the first two eigencoordinates at an early stage of the evolution. In the critical window $\sqrt2\beta=1+\sigma N^{-1/3}$ , the distance at time $N^{2/3}s$ converges in distribution to a random profile determined by the Airy point process, and there is no cutoff. As $\sigma\to\pm\infty$ , this profile, suitably recentered and rescaled, converges to the profiles of the two mesoscopic regimes.

Primarily AI-generated textHuman understanding: some partsdynamicsspin glass

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hexagon:2610.00608v1

math.PR — Probability

Existence of the $q$ -Colorability Threshold for Sparse Random Graphs

Contributed by P. M. Aronow, Patrick Lopatto

For a fixed integer $q \ge 3$ , we call a graph $q$ -colorable if its vertices admit $q$ colors with adjacent vertices colored differently. We prove that there exists $d_q > 0$ such that the random graph on $n$ vertices with independent edge probability $d/n$ is $q$ -colorable with probability tending to one for fixed $0 \le d < d_q$ and to zero for fixed $d > d_q$ , as $n \to \infty$ . The same holds for the uniform random graph with $\lfloor cn \rfloor$ edges, for fixed $c \ge 0$ , with threshold $d_q/2$ in $c$ . For fixed $\epsilon \in (0, 1)$ , the first loss of colorability under independent uniform additions of pairs of distinct vertices lies with probability at least $1 - \epsilon$ in a deterministic window of $O_{q, \epsilon}(n(\log n)^{-3/2})$ steps, where $\log$ is natural. The proof uses interpolation to establish convergence and bounds the expected number of pairs needed to destroy colorability of the last colorable graph by identifying random vertices and controlling the growth of classes equal in every coloring.

Primarily AI-generated textHuman understanding: some parts

Oct 29, 2026PDF ↗View work →

hexagon:2610.00607v1

math.CO — Combinatorics

A computer-assisted proof of the four-uniform Erdős matching conjecture

Contributed by Oleksiy Babanskyy

We prove the Erdős matching conjecture for four-uniform hypergraphs for every matching parameter and every admissible number of vertices. The proof combines the finite-board method of Hou, Hu and Liu with ordered local configurations, exact integer counts and certified inequalities. Small matching parameters are treated on full labelled configurations. Two fixed counting inequalities and two endpoint anchors propagate polynomially across the remaining range; 51 endpoint records and the two fixed inequalities suffice for the intermediate cardinality inputs. Dense triple supports admit direct slab and majority proofs. A classical shadow bound and a deletion argument complete the passage from these local bounds to the global theorem. The strict margins also yield clique stability with explicit constants for shifted families whose matching number is preserved on deleting the first vertex, and classifications of extremizers in specified ranges. The finite parts are supported by exact certificates and exhaustive computations with stated mathematical specifications.

Primarily AI-generated textHuman understanding: no partsErdős matching conjectureexact certificatesextremal set theoryhypergraph matchingsshifting

Oct 29, 2026PDF ↗View work →

hexagon:2610.00606v1

math.NT — Number Theory

FINITE LENGTH DATA DO NOT UNIVERSALLY DETERMINE FLAT THREE-TORI EXPLICIT CONSTRUCTIONS, PROOFS, AND A CAUTION FOR COSMIC TOPOLOGY

Contributed by Changxia Lin

Abstract. A flat three-torus is determined by a Euclidean lattice up to orthogonal changes of coordinates, or equivalently by a positive-definite Gram matrix up to integral change of basis. A finite list of short lattice lengths is a much weaker datum. This paper gives an explicit and fully elementary construction showing that the weakness can persist over a large range of scales. For every real parameter A ≥ 2, we exhibit two positive-definite Gram matrices G+(A) and G−(A) whose associated flat tori have different volumes and hence are not isometric, while their length multisets agree below a radius RA = p 2A2 − 2A−2 − 4A−4. The shortest nonzero lattice length is exactly one in both examples, so RA is not made large by a uniform rescaling; it tends to infinity with A. The proof isolates the coordinate support on which the two quadratic forms differ and supplies a sharp enough lower bound to exclude that support from every shorter vector. A second, isotropic example gives a compact rational pair whose short length multisets agree below √ 2 and differ at that threshold. We then place the construction alongside the theory of ternary quadratic forms and explain what it does—and does not—say about cosmic topology. In particular, finite agreement of direct-lattice lengths is not equivalent to equality of CMB covariance data, nor does it establish an observational degeneracy for any specified cosmological model

Primarily human-written textHuman understanding: all partsGram matrixflat toruslattice

Oct 29, 2026PDF ↗View work →

hexagon:2610.00604v1

math.LO — Logic

The Euclidean algorithm is not optimal among uniform processes

Contributed by Jason Davies

Reusing a stored Euclidean prefix, we show that every pair $a\geq b$ has a derivation of $\gcd(a,b)$ from $a,b$ with $O(\log a/\log\log a)$ applications of the remainder function alone; Euclid's algorithm needs order $\log a$ in the worst case. These derivations bound the intrinsic complexity of gcd and coprimeness by the same order, refuting Moschovakis's conjectured logarithmic lower bounds for uniform processes; by his best-uniform-process theorem, a single uniform process attains this bound. The Main Conjecture for algorithms remains open. We also prove an $\Omega(\log\log a)$ lower bound on this complexity for almost all coprime pairs and give, for each $\varepsilon>0$ , a nondeterministic register program computing gcd that, on every input $a\geq b\geq1$ , has a successful computation using at most $\varepsilon\log a+O(1)$ remainder calls. The upper bound also answers negatively an open problem of van den Dries and Moschovakis on the closure depth of the gcd. The main bounds and the refutations are formalised in Lean 4; the best-uniform-process theorem is cited.

Primarily AI-generated textHuman understanding: some partsEuclidean algorithmLean 4closure depthgreatest common divisorintrinsic complexityremainder functionuniform processes

Oct 29, 2026PDF ↗View work →

hexagon:2610.00603v1

math.RT — Representation Theory

Biperfectness of the theta basis of a maximal unipotent subgroup

Contributed by Qiyue Tang

We prove that the theta basis of the coordinate ring of a maximal unipotent subgroup of a complex semisimple group is biperfect. This answers affirmatively Question 6.5 in Kamnitzer's ICM 2022 survey.

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

Oct 29, 2026PDF ↗View work →

hexagon:2610.00602v1

math.FA — Functional Analysis

Tsallis Metrics on Density Matrices via the S-Divergence

Contributed by Suvrit Sra

We study symmetrized divergences induced by centroids of $\alpha$ -Tsallis relative entropies between density matrices $\sigma$ and $\rho$ ; up to scaling, the induced divergence equals $|2-\|\rho^\alpha+\sigma^\alpha\|_{1/\alpha}|$ . Earlier work (S. Sra, "Metrics induced by Jensen–Shannon and related divergences on positive definite matrices," Linear Algebra Appl., 2021) studied such divergences and metrics induced by them. That paper allowed centroids to range over the positive definite cone, while we require that the centroid itself be a density matrix. This (more natural) choice makes the metricity in this paper considerably stronger than that of the aforementioned paper. Specifically, we prove that $|2-\|\rho^\alpha+\sigma^\alpha\|_{1/\alpha}|^{1/2}$ is a metric on density matrices for $1/2\le\alpha\le2$ , $\alpha\ne1$ . For $\alpha<1$ , the proof uses a new four-point theorem for the S-divergence that might be of independent interest. For $\alpha>1$ , we first prove the endpoint $\alpha=2$ , and then transfer it to the other orders by a tensoring trick.

A mix of human-written and AI-generated textHuman understanding: all partsS-divergenceSchatten normSibson information radiusTsallis relative entropylog-determinant inequalitynegative typequantum Jensen-Shannon divergencequantum Renyi entropy

Oct 29, 2026PDF ↗View work →