All works

math.NT — Number Theory

Joint distribution of factorial units and valuations in translated intervals

Contributed by Will Howe, Grant Huffman, Jensen Kohlmeyer, Liam Kruer

For every fixed finite set of primes, the corresponding prime valuations of factorials, reduced modulo arbitrary fixed integers, and the prime-free parts, reduced modulo arbitrary fixed prime powers, have a joint uniform distribution. We give a proof with a power-saving error on every interval of consecutive nonnegative integers, uniform in the interval's starting point. A fixed congruence on the index may be imposed independently. The resulting statement for the part of a factorial coprime to an arbitrary fixed modulus gives the densities and relative density asked for in Problems 2 and 3 of Hajdu and Papp. The proof uses a finite digit-window identity, a strict local character contraction, and a carry estimate followed by finite Fourier analysis.

math.LO — Logic

Borel Ruziewicz problem

Contributed by Konstantin Slutsky

We prove that normalized surface measure is the unique finitely additive rotation-invariant probability on the Borel subsets of $S^n$, $n\geq2$. We also prove that Lebesgue measure is the unique finitely additive measure on bounded Borel subsets of $\R^d$, $d\geq3$, invariant under orientation-preserving rigid motions and normalized on the unit cube. The main step is the annihilation of Borel null sets. A spectral gap first gives bounded divergence certificates for compact null sets, chosen in a jointly Borel way. A shared random threshold then combines these certificates to cover arbitrary null sets. For Euclidean space, the construction is applied to families of spheres about two centers; spherical averaging completes the identification of the measure.

math.CV — Complex Variables

Smale's mean value constant (1981) exceeds 1.1559, and other miscellanea

Contributed by Dave Goel

We prove that Smale's mean value constant exceeds $1.1559837$, improving the bound $1.1018$ of Jatar and Ng; after the first version of this paper, the author proved that it equals $4$. Smale proved in 1981 that for every complex polynomial $P$ of degree at least $2$ and every point $z$ with $P'(z)\neq0$, some critical point $\zeta$ of $P$ satisfies $|P(z)-P(\zeta)|\le K\,|z-\zeta|\,|P'(z)|$ with $K=4$, and he conjectured that $K=1$ suffices; $K^*$ is the best constant, called Smale's mean value constant. That conjecture was disproved in 2026. Jatar and Ng showed that $K^*$ equals the supremum of $1/B(0)$ over the admissible pairs $(W,B)$: $W$ is a normalised univalent map of the unit disk, $B$ is a finite Blaschke product with $B(0)>0$, and the derivative of the product $WB$ has no zero in the disk. We construct such a pair explicitly, with $B$ of degree two, from a polynomial of degree $18$ with Gaussian rational coefficients, and we verify the required properties by a computation in exact arithmetic. Writing $\mathcal B_d$ for this supremum restricted to $\deg B\le d$, we prove, partly with certified computation, that $1.1363375<\mathcal B_1<1.177801$, $\mathcal B_2<100/61$, and $\mathcal B_d\le4-0.00597/(d\log d)$ for every $d\ge2$, which improves the bound of Jatar and Ng for $d\ge452$. If a pair attains $\mathcal B_d$ with $d\ge1$, then $W$ does not extend holomorphically to any neighbourhood of the closed disk and is not rational. If $1/B_n(0)\to4$ along a sequence of admissible pairs $(W_n,B_n)$ (such sequences exist, since $K^*=4$), then after rotation $W_n$ tends to the Koebe function, as Jatar and Ng observed, and the zeros $a$ of $B_n$, weighted by $\log(1/|a|)$, have an explicit limit distribution. Every admissible pair $(W,B)$ with $W$ meromorphic on a neighbourhood of the closed disk has $\deg B\le\operatorname{wt}(W)$, a weighted count of the points of the circle where $|W|$ is stationary and $\arg W$ decreases.

math.CV — Complex Variables

Smale's mean value constant (1981) equals 4

Contributed by Dave Goel

For a complex polynomial p of degree d ≥ 2 with p(0) = 0 and p′(0) ≠ 0, let S(p) be the minimum of |p(c)/(c p′(0))| over the critical points c of p, and let K(d) be the supremum of S(p) over all such p of degree d. In 1981 Smale proved that K(d) ≤ 4 for every d, and asked whether 4 can be replaced by 1. We prove that Smale's mean value constant sup_d K(d) equals 4 and that K(d) → 4 as d → ∞, combining a reduction due to Jatar and Ng with an explicit construction. Each K(d) is known to be below 4, so the supremum is not attained. The main result has also been formalized in the Lean 4 proof assistant.

Author comments: The expanded version of this paper is Hexagon 2611.00283, at https://hexagonmath.org/2611.00283 and on Zenodo at https://doi.org/10.5281/zenodo.24069916

math.CV — Complex Variables

Smale's mean value constant (1981) equals 4, and other miscellanea

Contributed by Dave Goel

For a complex polynomial p of degree d ≥ 2 with p(0) = 0 and p′(0) ≠ 0, let S(p) be the minimum of |p(c)/(c p′(0))| over the critical points c of p, and let K(d) be the supremum of S(p) over all such p of degree d. In 1981 Smale proved that K(d) ≤ 4 for every d, and asked whether 4 can be replaced by 1. We prove that Smale's mean value constant sup_d K(d) equals 4 and that K(d) → 4 as d → ∞, combining a reduction due to Jatar and Ng with an explicit construction. Each K(d) is known to be below 4, so the supremum is not attained. The main result has also been formalized in the Lean 4 proof assistant. We give three further proofs: by a variant of the main construction in which part of the boundary of an auxiliary domain is a level curve of an entire function; by a Riemann map onto a domain with thin hairs; and by a Riemann surface whose conformal radius at its base point is near 1. We also include a complete proof of the known fact that K(d) does not decrease.

Author comments: This is the expanded version of the paper Hexagon 2611.00282, "Smale's mean value constant (1981) equals 4", at https://hexagonmath.org/2611.00282 and on Zenodo at https://doi.org/10.5281/zenodo.24069884

math.QA — Quantum Algebra

Cayley graphs of abelian groups and their quantum symmetries

Contributed by Teo Banica

Building on the recent work of Marashdeh on the circulant graphs of prime order, we study the quantum symmetries of the Cayley graphs of the finite abelian groups $A$. Our main result states that if the coherent algebra is the orbital algebra of $A\rtimes E$, with $|E|=k$ and with the non-trivial elements of $E$ having at most $m$ fixed points, then the graph has no quantum symmetry when $|A|\geq k^2m$. We also settle the case of the circulant graphs with $p^n$ vertices.

math.DS — Dynamical Systems

Singular perpendicular orbits and complete periodicity in rational triangles

Contributed by Joshua Boorstein

Let $\Delta$ be a triangle whose angles are rational multiples of $\pi$. A billiard path leaving a side $\sigma$ at a right angle either meets a vertex first — its starting point is then singular — or meets a side at a right angle again: $\sigma$ itself, at a new point, or another side, where it retraces itself. The singular points cut $\sigma$ into finitely many intervals, and each interval retraces or is carried onto another by the return map, reversing orientation (a mirror pair) or preserving it (a translation pair). We count these intervals exactly. The singular points of the sides perpendicular to a direction class are in bijection with the slots — a vertex with a direction of the class pointing inward — whose path meets a side at a right angle before any vertex, so the number of intervals is $v + S - 2p - s'$: $S$ slots, $v$ perpendicular sides, $p$ pairs of mutually reverse vertex-to-vertex paths meeting no side at a right angle, and $s'$ slots whose path never reaches a vertex. This bounds the retracing bands and translation pairs, sharply on isosceles triangles, and makes $s' = 0$ equivalent to complete periodicity of the class. Our main theorem is that $p = 0$ forces complete periodicity: a minimal component needs a swapped pair. The converse fails. In every isosceles triangle $(a, a, b)$ with $N$ even and $a \geq 3$ the class of the base direction has $p \geq 1$, and it is completely periodic, by a companion paper (computer-assisted in part), whenever $b \geq 4$ and $N$ has at most one odd prime factor. At odd $N$, two infinite families of scalene triangles have $p \geq 1$ and are completely periodic in the checked range.

Author comments: First version. Companion papers: "The orphan theorem" (hexagon:2611.00191), "The necklace of a rational right triangle" (hexagon:2611.00213) and "No arrivals" (hexagon:2611.00250), whose results this paper imports. Code: https://doi.org/10.5281/zenodo.24093456

math.DS — Dynamical Systems

Non-perpendicular periodic orbits in mirror directions of right triangles

Contributed by Joshua Boorstein

Let $T$ be a right triangle with an acute angle $a$. A direction is a mirror direction of $T$ if it is the axis of a reflection in the group generated by the reflections in the sides of $T$ — for instance, any direction parallel or perpendicular to a side. A billiard path that meets a side at a right angle retraces itself, its directions are mirror directions, and it is periodic as soon as it meets a side at a right angle a second time. We ask whether every periodic billiard path in a mirror direction is of this kind. When $\cos 2a$ is transcendental it is: every such path meets a side at a right angle, and in the directions parallel to the hypotenuse there is no periodic path at all. The proof is a holonomy computation on Hooper's chain of rhombi, and it bounds the degree of $\cos 2a$ for any exception. When $\cos 2a$ is algebraic and $a/\pi$ irrational, exceptions can occur: we construct non-perpendicular periodic cylinders in mirror directions explicitly — among them an infinite family indexed by the solutions of $u^2 - 2v^2 = -1$ — and show that the triangles carrying them form a countable dense set in each of the four classes of mirror directions. Corollaries include the absence of odd-period orbits in right triangles and the existence of one-way periodic paths in every right triangle (answering two questions of Holt), a description of the odd orbits of isosceles triangles, and the statement that Ruijgrok's third conjecture, read for right triangles, fails on a countable dense set of triangles, all with $\cos 2a$ algebraic, and holds whenever $\cos 2a$ is transcendental.

Author comments: First version. Companion paper: "The necklace of a rational right triangle" (hexagon:2611.00213). Code: https://doi.org/10.5281/zenodo.24037744

math.CA — Classical Analysis and ODEs

Triangles in sets of positive upper density in the plane

Contributed by Polona Durcik

This note proves that every measurable subset of the plane with positive upper Banach density contains a translate of a rotation of every sufficiently large dilate of any fixed noncollinear triangle.

math.RT — Representation Theory

The Benson-Etingof-Ostrik conjecture

Contributed by Kevin Coulembier, Pavel Etingof, Johannes Flake, Joseph Newton, Victor Ostrik

We prove the conjecture of Benson--Etingof--Ostrik (2023): every symmetric tensor category of moderate growth over an algebraically closed field of positive characteristic admits a symmetric tensor functor to a higher Verlinde category, possibly of infinite height. Using maximal nilpotence and geometric reductivity, we reduce the problem to proving that finitely generated maximally nilpotent tensor categories are finite. The main ingredients are the construction of generic points using alternating norms and a Frobenius reduction of the fundamental coend to an ordinary Hopf subalgebra.

Author comments: The authors plan to turn this AI generated proof into a human paper.

math-ph — Mathematical Physics

Integral torus partition functions just above c=1 have no fixed spectral gap

Contributed by Tzu Chen Huang

We consider modular-invariant torus partition functions of central charge $c>1$ that have a unique vacuum, an expansion in non-degenerate Virasoro characters with positive integer multiplicities, integer spins, no nonvacuum state at the unitarity bound, and parity symmetry. We prove that for every $g>0$ there is $\epsilon(g)>0$ such that no such partition function with $1<c<1+\epsilon(g)$ has all nonvacuum primary dimensions at least $g$. Thus the spectral gap closes as $c\to 1^+$ in this class, as suggested by the numerical bootstrap study of Benjamin, Langhoff, Li and Thaler. The proof is by contradiction and compactness. Using the crystalline-measure reduction of Eberhardt and Levine, we show that every limit at $c=1$ is an equal-weight average of two compact-boson partition functions, at least one at a half-integer radius. Positive Gaussian test functions adapted to the two circle lattices then force every state of a nearby partition function, through a logarithmically growing energy window, to lie within a power of $c-1$ of a lattice point. On the other hand, the central-charge shift of the vacuum contributes a second-order term at zero momentum that no distribution supported on circle lattices and compatible with modular invariance can carry; a finite family of Gaussian tests detects this. Limits with two half-integer radii, where lattice points collide, require a separate analysis. The threshold $\epsilon(g)$ is not effective; we describe the additional estimates that would make it so.

math.FA — Functional Analysis

The Daugavet Property of Bidual Spaces

Contributed by Samir Hamad

We construct a separable Banach space whose bidual has the Daugavet property. The construction proceeds by successively extending finite-dimensional spaces so that prescribed vectors are approximated by averages of newly added unit vectors. A simultaneous extension argument produces finitely many fixed functionals whose norm-attaining faces approximate the unit ball of the bidual.

math.CO — Combinatorics

The Wang Transform Inequality

Contributed by Yair Lavi

We prove the Wang transform inequality in every order: if $n$ is a positive integer, $A$ is a nonnegative $n\times n$ matrix whose row and column sums equal one, and $J_n$ has every entry $1/n$, then $\operatorname{per}A\ge \operatorname{per}\!\left(\frac{nJ_n+A}{n+1}\right)$. Equality holds exactly when $A=J_n$. We also obtain an explicit positive quadratic gap for every $n\ge2$.

math.CO — Combinatorics

A linear gap for the maximal cp-rank

Contributed by Yair Lavi

We prove that the maximal cp-rank $p_n$ satisfies $p_n\ge n(n-3)/2$ for odd $n\ge5$ and $p_n\ge n(n-3)/2-1$ for even $n\ge6$. Together with the known upper bound $p_n\le\binom{n+1}{2}-4$, this gives $p_n=n^2/2+O(n)$.

math.CO — Combinatorics

The Maximum Diagonal Product of a Stochastic Matrix of Bounded Rank

Contributed by Yair Lavi

Let $A=(a_{ij})$ be a nonnegative row-stochastic or column-stochastic $n\times n$ matrix of rank at most $k$, where $1\le k\le n$. Write $n=qk+s$, where $0\le s<k$. The author conjectured in 2018 that, for every permutation $σ$ of $\{1,\ldots,n\}$, $\prod_{i=1}^n a_{i,σ(i)}\le q^{-q(k-s)}(q+1)^{-(q+1)s}$. We prove this bound and classify all equality cases.

math-ph — Mathematical Physics

Notes on Wasserstein distance and the JT gravitational ensemble

Contributed by Ville Samuli Keränen

This note completes the study of the Boltzmann–Wasserstein (BW) distance, an optimal-transport metric on quantum theories, begun in a companion paper (arXiv:2606.01296). There the distance was evaluated between macroscopically different theories; here we take the opposite limit, two independent draws of the JT-gravity ensemble, which are identical at the classical level. The distance is computable and is controlled by the connected spectral form factor: the ramp, the double-trumpet wormhole, regulated by the plateau, the spectral discreteness. The mean is exponentially small, $\langle \mathcal{W}^2\rangle \sim S_0\, e^{-S_0}$, whereas uncorrelated (Poisson) levels would give an unsuppressed $O(1)$ distance: the self-distance is a sharp diagnostic of spectral chaos. As a metric, the BW distance renders the ensemble a point cloud: exponentially small, strongly anisotropic, its leading principal axis a coherent “breathing” deformation of the low-energy spectrum. Its effective dimension grows asymptotically as $D_{\mathrm{eff}} \simeq 0.42\, S_0^2$, so the distance self-averages, its relative width shrinking as $\mathrm{std}/\mathrm{mean} \sim 1/S_0$. At low temperatures it localizes on the Airy edge, where a single Tracy–Widom mode dominates and self-averaging fails. We test the picture with exact-diagonalization GUE and SYK simulations: in GUE the effective dimension grows logarithmically at every temperature and is predicted at finite $N$, with no free parameters, by the same covariance; unfolded SYK exhibits the predicted anisotropy and mode structure of the cloud. Together with the companion result, this fixes the resolution of the metric: the microstate rigidity computed here is the noise floor beneath a macroscopic difference between theories.

math.PR — Probability

The Ultimate Matching Theorem

Contributed by Michael Goldman, Francesco Grotto, Dario Trevisan

Let $X_1,\dots,X_N$ be independent uniform points in the unit square, matched either to an evenly spread deterministic configuration or to a second independent sample. For exponents $\alpha_1,\alpha_2>2$ with $1/\alpha_1+1/\alpha_2=1/2$, we prove that with probability tending to one a single permutation controls the $j$th coordinate displacements in the exponential Orlicz norm of exponent $\alpha_j$, at the scale $\sqrt{\log N/N}$, for both $j=1,2$. This settles Talagrand's Ultimate Matching Conjecture for all pairs of finite critical exponents. The proof constructs, on the flat torus, a current whose divergence is Gaussian white noise projected on a dyadic grid and whose coordinates have the critical anisotropic exponential Orlicz bounds, and transfers it to matching through a primal optimal-transport reduction. At the hard endpoints $(2,\infty)$ and $(\infty,2)$ the conjecture fails: a Brownian local-time construction shows that the normalized optimal matching cost diverges in probability.

math.IT — Information Theory

Energy complexity and dissipation of noisy consensus

Contributed by Zhenyu Sun

We characterize the signal-energy complexity of distributed agreement over independent Gaussian point-to-point channels. For $n$ processes deciding $L$ binary coordinates under coordinatewise unanimity validity and a joint failure probability of at most $\varepsilon$, every finite causal protocol requires a worst-input expected signal energy of $\Omega(N_0n[L+\ln(n/\varepsilon)])$. This lower bound combines two distinct mechanisms: an exact relative-entropy identity for locally interchanged reception histories yields the reliability term even in the presence of shared randomness and active noisy feedback, while a coordinatewise direct-sum argument establishes the payload term. In the fault-free setting, a deterministic fixed-leader protocol attains this optimal order for all $0<\varepsilon\leq1/8$. In the presence of $f\leq(1/3-\kappa)n$ static Byzantine faults, the same order is achieved in the polynomial-error regime $\varepsilon\geq n^{-a}$ via public committee sampling and robust joint decoding within explicit finite storage and computation budgets. Consequently, batching amortizes the reliability overhead, yielding an energy cost of $\Theta(N_0n[1+L^{-1}\ln(n/\varepsilon)])$ per coordinate. We then connect this communication optimum to physical dissipation through a stochastic-thermodynamic model of a thermal channel port. Under a non-recovering mode replacement interface, the communication converse translates directly into lower bounds on entropy production and, when the endpoint mean internal energies agree, cyclic work. When work recovery is permitted prior to mode reset, we derive the exact minimum reset entropy production within a bounded-displacement control class as a function of the information accessible to the recovery controller, thereby separating the fundamental communication cost from its control-dependent thermodynamic realization.

math.PR — Probability

Closing the identification step in the GFF limit: Uniqueness of the zero-slope six-vertex phase for −1 ≤ Δ < 1/2

Contributed by Yan Ru Pei

For the six-vertex model with $a=b=1$ and $c\in(1,2]$, that is $-1\le\Delta<\frac{1}{2}$, we prove that the plane measure, the limit of balanced torus measures, is the only translation-ergodic gradient Gibbs measure of slope zero, and hence the unique translation-invariant minimiser of the specific free energy. Both statements had been asserted by reference to Sheffield's uniqueness theory, which is proved for nearest-neighbour potentials; we show that for $c\ne1$ the six-vertex weights, which couple faces across the diagonals of each vertex, are not a nearest-neighbour potential in height coordinates. The proof extends Sheffield's cluster swapping to these diagonal interactions by means of residual energies on the diagonals, shows with the Lanford–Ruelle theorem for subshifts of finite type and a Burton–Keane argument that two distinct ergodic zero-slope phases are gradients of Gibbs measures on heights, and shows from the spin representation in the form given by Duminil-Copin, Kozlowski, Lammers and Manolescu that the plane measure is not. This supplies a step that we found missing in the chain of recent results on the Gaussian free field limit of the six-vertex height function. An unrefereed manuscript of OpenAI states the full-plane limit for every $c\in(0,2]$; for $\sqrt{2}\le c<2$ it identifies a state built from nearly balanced sectors with the plane measure through the characterisation above. Our result gives that identification, given two properties of the state stated in the manuscript. We also check that the manuscript and the work of Averous, Duminil-Copin, He, Lammers and Manolescu concern the same measure, so that their convergence in domains holds, for $c\in[1,2]$ and with constants depending on $\Delta$, wherever the manuscript's full-plane theorem holds.

math.PR — Probability

Stochastic Connections on Surface Bundles

Contributed by Zhenyu Sun

We construct local stochastic connection dynamics on arbitrary principal bundles over closed oriented Riemannian surfaces. For each smooth reference connection, the bare additive spatial heat approximations of the Yang–Mills–DeTurck equation converge to a maximal local strong Markov family in a separable negative Hölder space. A finite Gaussian enhancement defines the singular products, and cancellation of the total logarithmic mass identifies the limiting equation. Rough comparison gauges and predictable rotations of the Wiener noise identify the local orbit laws for different references. The finite quadratic tensor anomaly vanishes after the two-dimensional Yang–Mills contraction, with the finite normalisation fixed by the spatial heat approximation. Continuing through bounded representatives gives a reference-independent maximal local strong Markov family on the full rough orbit space. Gaussian line primitives have every Hölder exponent below one on bounded regular curve families and define non-Abelian Young parallel transport. On a dense, gauge-invariant Borel full-heat trace domain, polygonal holonomies separate rough gauge orbits, admit Borel representative maps, and recover the bundle class. The orbit dynamics has continuous holonomy paths on this domain, and the regularised holonomies converge uniformly on bounded regular curve families and compact surviving positive-time intervals. We also prove regularisation-independent convergence of first-order Gaussian currents after time integration, including a harmonic example of infinite variation and zero quadratic variation. At fixed regularisation scale, the field-dependent covariant heat approximation has classical solutions and an explicit Itô correction in Coulomb coordinates. In every Abelian sector, the quotient diffusion is global, reference independent, and reversible with an explicitly identified unique invariant law.

math.CO — Combinatorics

Positivity for Weighted Linear Laurent Phenomenon Algebras

Contributed by Qiyue Tang

We prove Laurent positivity for ordinary Laurent phenomenon algebras whose initial exchange polynomials are nonconstant affine linear polynomials with positive constant terms and nonnegative numerical coefficients.

Author comments: By posting this manuscript on this platform, I do not seek to assert ownership of the results or claim priority. I welcome anyone interested in preparing a finished account or checking the arguments; the eventual scholarly credit should go to those who undertake that work.

math.RT — Representation Theory

Broué's Abelian Defect Group Conjecture for 2-Blocks

Contributed by Qiyue Tang

We prove that every 2-block of a finite group with an abelian defect group is splendidly Rickard equivalent to its Brauer correspondent.

Author comments: v2（2026/11/11）：Version 1 underwent several rounds of review and revision with GPT-6-Pro and subsequently passed the Rethlas-OSV review (GPT-6-Astra, high reasoning effort, four concurrent workers), which reported no mathematical issues. I am posting this manuscript to share the mathematical results themselves. In doing so, I do not seek to assert ownership of the results or claim priority. I welcome anyone interested in checking the arguments or preparing a finished account, and I believe the eventual scholarly credit should go to those who undertake that work.

math.CT — Category Theory

Higher Galois Recognition and Parameter Descent

Contributed by Zhenyu Sun

Let $n\geq0$ and let $\mathcal X$ be a bounded coherent $\infty$-topos. Suppose that finitely many noninitial coherent $n$-truncated objects cover its terminal object, and that every coherent $n$-truncated object in each of their slices has a finite covering family of sections. We identify $\mathcal X^{\mathrm{coh}}_{\leq n}$ with $\operatorname{Fun}(M,\mathcal S_{\pi,\leq n})$ for a $\pi$-finite $(n+1)$-type $M$, recovered by all finite models preserving finite limits, finite coproducts, and effective epimorphisms. Descent uses degrees through $n+2$ for objects and $n+1$ for maps; both bounds are sharp for finite homotopy actions. Every exact functor $F$ between recognized categories has a universal parameter $P_F=F_!(1)$ and a canonical factorization through its slice with fully faithful second functor. This parameter characterizes full faithfulness, conservativity, and all exact slice factorizations. For successive abelian Postnikov layers, we identify the space of marked exact lifts with a path space of Eilenberg--Mac Lane sections with local coefficients. The obstruction, classes of lifts, and relative automorphisms occur in degrees $r+1$, $r$, and $r-1$. We compute a twisted parity obstruction and, for odd primes, classify cyclic self-equivalence components by a $p$-adic valuation. A nonzero Postnikov class can persist in the identity component. For continuous finite families over a profinite $(n+1)$-type, we prove that a uniform finite-section atlas exists precisely when the complete model space is $\pi$-finite. Exact functors from finite model semantics, together with their lift spaces, satisfy finite-stage descent, while explicit profinite examples separate objectwise trivialization from uniform parameter elimination.

Author comments: 49 pages. Second paper in the series Model groupoids and finite homotopy semantics. Includes two verification scripts and their outputs.

math.LO — Logic

A Michael Space in the Laver model

Contributed by Morgan Jones

We show that a Michael space exists in the Laver model.

Author comments: I was stress testing Claude Fable 5.2 with varying levels of problems and trying to improve my workflow with AI. This problem was reasoned pretty quickly by Claude, however it required some convincing that it could do it.

math.NT — Number Theory

Watkins's Conjecture for Elliptic Curves with Rational Two-Torsion

Contributed by Qiyue Tang

We prove that the modular degree of every elliptic curve $E/\mathbb Q$ with $E(\mathbb Q)[2]\ne 0$ is divisible by $2^{\operatorname{rank}E(\mathbb Q)}$.

Author comments: By posting this manuscript on this platform, I do not seek to assert ownership of the results or claim priority. I welcome anyone interested in preparing a finished account or checking the arguments; the eventual scholarly credit should go to those who undertake that work.

math.CO — Combinatorics

Positive bases and polytopes for Laurent phenomenon algebras

Contributed by Qiyue Tang

We prove that the cluster monomials of every graph Laurent phenomenon algebra form an atomic positive basis. For every finite normalized Laurent phenomenon algebra over a unique factorization domain, we construct a simple lattice polytope with unimodular normal fan whose polar realizes the cluster complex. We also prove that cluster monomials form a basis of the upper algebra in the finite case. In particular, every linear Laurent phenomenon algebra admits such a polyhedral realization.

Author comments: By posting this manuscript on this platform, I do not seek to assert ownership of the results or claim priority. I welcome anyone interested in preparing a finished account or checking the arguments; the eventual scholarly credit should go to those who undertake that work.

math.LO — Logic

The 51-state beeper $M_{51}$: every battle ends, and what Peano Arithmetic can prove about it

Contributed by Annanay Kapila

We study an explicit 51-state, 2-symbol beeping Turing machine $M_{51}$ and its associated dispatch-level process $R$. An ordinal descent below $\varepsilon_0$ proves that every battle of $R$ terminates and hence that $M_{51}$ beeps infinitely often. A growth lemma shows that the battle counters and beep times eventually dominate every function provably computable in Peano Arithmetic. Together with a Rocq-checked simulation theorem, these results establish that quasihalting of $M_{51}$ is independent of Peano Arithmetic. The document includes the process definition, machine table, ordinal analysis, growth proof and an account of its external dependencies.

Author comments: Supporting materials: Rocq simulation development, https://github.com/a-kapila/maths-proofs/blob/2e2d689f198e7bd51d264cd5468d6b7606886d2e/beeping-busy-beaver-51/rocq51.tar.gz ; computational checks, https://github.com/a-kapila/maths-proofs/blob/2e2d689f198e7bd51d264cd5468d6b7606886d2e/beeping-busy-beaver-51/checks51.tar.gz . The Rocq materials support the simulation theorem; the complete independence argument is not claimed to be fully formalized.

math.FA — Functional Analysis

COUNTABLY INFINITE-DIMENSIONAL SPACES OF CRYSTALLINE MEASURES

Contributed by Jamie Martin

We construct a symmetric locally finite set Λ ⊂ R for which the space MΛ of crystalline measures with support and spectrum in Λ has countably infinite Hamel dimension; Meyer asked what the dimension of such spaces can be. More precisely, there is a sequence of crystalline measures such that every element of MΛ is a unique finite linear combination of them. Every layer of MΛ of fixed Hermite–Sobolev order is finite-dimensional, but the layer dimensions are unbounded; in one construction they grow linearly, and in general their growth can be prescribed up to a linear change of the order. (For every locally finite set the dimension is finite, countably infinite or equal to the continuum, and the countable case is exactly this behaviour of the layers.) The construction uses clusters of atoms that imitate derivatives of Dirac masses, which produce one-dimensional blocks of prescribed Hermite order. The main difficulty is to show that the union of infinitely many blocks supports no other crystalline measures. We prove this by placing the blocks at random scales, separated by rapidly growing gaps, and showing that every element of fixed Hermite order splits into finitely many block components. We also obtain disjoint sets A, B such that the crystalline measures with support in B and spectrum in A form a countably infinite-dimensional space, while MA = MB = {0}. The existence statement, without the symmetry of Λ, has been formalised and machine-checked in Lean 4; the formal development is recorded in Palomar

math.QA — Quantum Algebra

Another proof of quantum Laurent positivity, II

Contributed by Qiyue Tang

We give a further combinatorial proof of Laurent positivity for skew-symmetric quantum cluster algebras.

math.GR — Group Theory

Twisted conjugacy in spherical Artin groups and their crystallographic quotients

Contributed by Jason Davies

A group has property $R_\infty$ if every automorphism has infinitely many twisted conjugacy classes. Let $A$ be an irreducible spherical Artin group of Coxeter rank at least two, with pure subgroup $P$. We classify property $R_\infty$ for the crystallographic quotient $\Gamma=A/[P,P]$. If $A$ has Coxeter rank at least three, then $\Gamma$ has $R_\infty$. In type $I_2(m)$, this holds exactly when $4\mid m$, or when $m$ is odd and $3\mid m$. Together with the known dihedral Artin-group cases, the positive quotient results imply that every irreducible spherical Artin group of rank at least two has $R_\infty$. The positive proofs combine the rational representation theory of the holonomy action with integral lattice constraints and explicit fixed-vector witnesses. In the negative dihedral cases, we construct automorphisms compatible with the crystallographic extension. For type $I_2(5)$, we exhibit an automorphism with Reidemeister number $18$, showing concretely that the quotient construction does not preserve $R_\infty$. This example, including the identification with the presented Artin quotient and the exact Reidemeister count, is formalised in Lean. An alternative proof for seven Artin groups is formalised under explicitly stated external hypotheses.

math.GN — General Topology

Smooth functions with isolated critical values and systems of fibers

Contributed by Naok Kitazawa

In the 2020s, Marzantowicz and Michalak have introduced the notion of system of smooth hypersurfaces and discussed Morse functions compatible with them: the hypersurfaces are regular level sets of the functions. This is a kind of fundamental theory on Morse functions and related differential topology and geometric topology. We extend this to a certain case: we discuss systems of regular level sets and critical level sets (regular fibers and singular fibers) of smooth functions with isolated critical values.

math.NT — Number Theory

The Exact Second Order Term In Ramanujan's Bound For $\sigma(n)/n$

Contributed by Jacob Crim

Let $\sigma(n)$ be the sum of the divisors of $n$, and assume the Riemann hypothesis. Ramanujan showed that $\sigma(n)/n\le e^\gamma(\log\log n-2(\sqrt2-1)/\sqrt{\log n}+S_1(\log n)+O(1/(\sqrt{\log n}\log\log n)))$, where $S_1$ is a sum over the zeros of $\zeta$. Nicolas made this effective, placed the coefficient of $1/(\sqrt{\log n}\log\log n)$ in $[0.9567, 3.789]$ at colossally abundant numbers, and asked for its exact asymptotic value. We show that this coefficient is $\kappa+W_2(\log n)$ with $\kappa=\sqrt2(2+\log2)-2=1.80868\ldots$, where $W_2(x)=-\sum_\rho x^{i\Im\rho}/(1-\rho)^2$ oscillates in $[-\beta,\beta]$, $\beta=2+\gamma-\log4\pi$. The next coefficient is the constant $c_0=8-\sqrt2(8+4\log2+\frac34\log^2 2)=-7.7443\ldots$, up to an oscillation of amplitude at most $0.0066$. In particular, the rescaled residual studied numerically by Leshchiner, Musin and Thompson does not converge, and it exceeds $3/2$ at every colossally abundant number with parameter $\xi\ge4.4\cdot10^{15}$; this answers their Questions 3 and 4, the second in the negative. We also prove explicit two-sided bounds for the remainder at every colossally abundant number above Nicolas's threshold $N^{(0)}$, where $\log N^{(0)}\approx10^9$. This part is computer-assisted: rigorous interval arithmetic covers every real parameter $\xi\in[10^9,e^{300}]$, and an analytic argument covers $\xi>e^{300}$. As corollaries, $\sigma(n)/n\le e^\gamma(\log\log n-0.699/\sqrt{\log n})$ for all $n\ge N^{(0)}$, improving Nicolas's constant 0.582; Ramanujan's limsup is approached from above at every colossally abundant number with $\xi\ge10^9$; and an odd-integer analogue holds with an essentially sharp constant. The proof reads colossally abundant numbers as ground states of a free gas of prime modes, in which $\kappa$ splits into a prime-square term, a saturation-deficit term and a shell-energy term. All enclosures are checked against exact values computed from a sieve to $10^{10}$.

math.LO — Logic

Definability and Lifting in Model Subgroupoids

Contributed by Zhenyu Sun

Restriction to a wide subgroupoid of a model groupoid can introduce invariant relations, maps, and objects. For open topological groupoids, a map between restricted objects lifts exactly when its graph is invariant under the full groupoid, and a specified quotient presentation lifts exactly when its kernel relation is invariant. Under coherent restriction, these criteria recover definable maps and coherent quotient presentations. For finite continuous actions of a group with an open subgroup of finite index, coinduction identifies extensions with invariant subsets on which evaluation is bijective. For a finite group $G$ and a normal subgroup $N$, we classify extensions of finite $N$-sets by normalizer extensions of their orbit types. Multiplicities must be constant on conjugacy orbits of types; within each such block, the liftable multiplicities form the additive monoid generated by indices of subgroups over which the normalizer extension splits. The same subgroup data determine the groupoid of lifts and its isomorphism-class and mass generating functions. This classification also applies to finite continuous actions of profinite groups with an open normal subgroup. For a fixed action, a centralizer extension gives the splitting obstruction, with an $H^2$ obstruction and an $H^1$ classification when its kernel is abelian. The center of $\mathrm{SL}_2(\mathbb F_5)$ gives the monoid $\langle12,20\rangle$; Heisenberg groups over fields of odd prime order give unbounded lifting indices at fixed cohomological period. Finite torsor theories and infinite pure sets realize the definability criteria. For profinite extensions with finite kernel, one finite quotient detects all liftable multiplicities, though it need not carry every lift.

Author comments: 37 pages. Exact finite-group computations and their output are included as ancillary files.

math.NT — Number Theory

The Schur-Siegel-Smyth trace problem: the lower bound 1.8022246

Contributed by David Niedbala Giraudin

We prove lambda_SSS >= 9011123/5000000 = 1.8022246 for the smallest limiting trace-to-degree ratio of totally positive algebraic integers. The previous best bound is 1.8022225 (Tomar). The certificate consists of fourteen integer polynomials with rational multipliers and a continuous piecewise-linear density with 8678 rational hat weights; its dual inequality is verified on (0,infinity) in ball arithmetic. For the same family of polynomials the value of the underlying optimisation problem is 1.8022247 to within 10^-9 (numerical), so the bound is within 1.1*10^-7 of what this family can give.

Author comments: 3 pages. Ancillary files: the exact certificate (cert_SSS_1802224600.json), a rigorous checker in ball arithmetic (check_sss.py, python-flint), its run log, an independent numerical cross-check (pointcheck_mp.py) and SHA-256 sums.

math.AG — Algebraic Geometry

Ideal generation for general reduced projective points

Contributed by Vlado Vojdanovski

We prove the Ideal Generation Conjecture for general reduced points in projective space over an algebraically closed field of characteristic zero. Equivalently, if the ideal of the points has initial degree D, multiplication by linear forms from degree D to degree D + 1 has maximal rank. The proof combines hyperplane specialization with a construction using two transverse hyperplanes. In the latter construction, motion of their common trace gives the required rank. A uniform induction for all degrees D ≥ 4 reduces the theorem to explicit quadratic and cubic constructions.

math.PR — Probability

An exact moment criterion for survival under renewed spatial catastrophes

Contributed by Man Huang

We prove an exact moment criterion for survival in a binary growth process subject to independently renewed spatial catastrophes. Occupied sites spread through a finite set of allowed increments on the integer lattice, and Poisson catastrophes with unbounded cube sizes empty all occupied sites in each affected cube. Under finite one-site hitting intensity, a positive finite-seed survival phase exists if and only if the cube-size measure has a finite moment of order one more than the spatial dimension. The divergent endpoint yields extinction at every positive intensity. The proof centres on a one-dimensional positive-phase construction for connected interval errors under a finite second moment. Extremal-path resampling gives a conditional correlation inequality, which controls exact interval insertion counts and a finite-history potential. Coordinate restriction and envelope covering then establish the criterion in every dimension. We also prove the bulk-to-seed reversal identity and extend the criterion to uniformly fat shapes. For the one-sided one-dimensional rule, the stationary hole density has a quadratic dilute expansion under the same second-moment condition. Its excess over independently eroded Poisson events is strictly positive at every positive intensity, with an explicit quadratic coefficient in the dilute limit.

Author comments: Based on Zenodo v1 (3 November 2026), https://doi.org/10.5281/zenodo.23911961. Mathematical text and references unchanged; portable font and provenance updates only. Original source and finite diagnostics: same DOI.

math.PR — Probability

Quartic exponential matching in the unit square

Contributed by Will Howe, Grant Huffman, Jensen Kohlmeyer, Liam Kruer

Take two independent samples of $n$ uniform points in the unit square. We prove that, with probability at least one half, the points can be paired so that the average of the exponential of the fourth power of the normalized pair distance is at most two. The distance normalization is $L\sqrt{\log n/n}$, with $L$ an absolute constant. One pairing therefore satisfies both coordinate inequalities in the symmetric $(4,4)$ case of Talagrand's matching problem. The argument first controls a vector field transporting smoothed point densities, and then converts that control to a pairing. The main estimate comes from a Gaussian construction whose successive partitions depend only on noise already revealed; a correction using the classical bottleneck matching preserves the required density balance.

cs.FL — Formal Languages and Automata Theory

Sublinear-space recognition for Boolean grammars

Contributed by Will Howe, Grant Huffman, Jensen Kohlmeyer, Liam Kruer

For every fixed Boolean grammar in the standard language family considered by Okhotin, we give a deterministic membership algorithm using $O_G(n^{2/3}(\log(n+2))^{2/3})$ working space on an input of length $n$. In particular, the whole family is contained in $\operatorname{DSPACE}(n^{3/4})$, answering Okhotin's Problem 4 affirmatively. The construction partitions substring endpoints into blocks of size $b$. It represents the parsing recurrence by a constant-fan-in circuit with $O_G(b^2)$-bit values and dependency height $O(n/b+1)$. Splits in the endpoint blocks are resolved locally, and an ordering of the remaining splits controls the height even when a split is highly unbalanced. Applying the Cook–Mertz tree-evaluation algorithm to the implicitly unfolded circuit gives the space bound.

math.DS — Dynamical Systems

Palindromic sections in two Kolakoski phase actions

Contributed by Will Howe, Grant Huffman, Jensen Kohlmeyer, Liam Kruer

We prove the two specified-phase orbit statements in Shen's Conjectures 3.4 and 3.7. The all-ones phase for the alphabet $\{1,n\}$, with $n$ positive and even, and both alternating phases for $\{-1,n\}$, with $n$ any even integer, have orbit length $2^{\lceil d/2\rceil}$ at depth $d$. The proof uses unreduced signed words and a palindrome class preserved by taking the section of the square at two phase coordinates. It also gives the all-ones statement for nonpositive even $n$ under Shen's displayed signed recursion. These are finite phase-orbit results; they do not imply the density conjecture for the classical Kolakoski sequence.

math.DS — Dynamical Systems

Two exact frequency extrema for smooth sequences over {1,3}

Contributed by Will Howe, Grant Huffman, Jensen Kohlmeyer, Liam Kruer

Using the frequency coding of Jamet, Marcovici, Poirier, and de la Rue, we determine the largest frequency of $1$ in a smooth sequence over $\{1,3\}$ and the largest such frequency below $1/2$. They are $1/2+1/(3+\sqrt{5})$ and $1/2-1/(13-\sqrt{5})$, respectively. The unique attaining type directives are $1^\infty$ and $0001^\infty$. The proof is an order comparison for two planar homographies. The first value corrects the numerical expression in the stated extremum question.

math.RA — Rings and Algebras

A Counterexample to the Subtraction-Free Conjecture for Bounded Ratios of Minors

Contributed by Qiyue Tang

We disprove the subtraction-free conjecture of Boocher and Froehle for bounded ratios of minors of totally positive matrices. We construct products (P,M,Q) of ordinary minors of a matrix of order (3207) such that [ 0<\frac{P}{MQ}\leq\frac1{64} ] throughout the totally positive locus, whereas (MQ-P) has a coefficient equal to (-1) in the standard bidiagonal parameters.

math.NT — Number Theory

Irrationality of χ₂ and Ti₂ at units of real quadratic fields

Contributed by Shawn Krisman

We prove that $\chi_2(\sqrt{m^2+1}-m)$, a value of Legendre's chi function at a unit of norm $-1$, is irrational for every integer $m\ge2$ except $m=3,5,7,9,15$. For $m=2$ this shows that $\pi^2-18\log^2\varphi$ is irrational, where $\varphi$ is the golden ratio. The method extends to all units of norm $-1$ in real quadratic fields, giving $\chi_2(\varepsilon)\notin\mathbb{Q}$ outside an explicit set of $47$ units, and, through a twist identity, to the inverse tangent integral $\mathrm{Ti}_2$ at units of norm $+1$, outside $57$ units. The proof combines Hankel determinants of a reflection-symmetric Padé machine, arithmetic bounds on their denominators, a discrete log-gas bound certified in interval arithmetic, and a nonvanishing argument modulo a prime. Below the range covered by the criteria of David, Hirata-Kohno and Kawashima, these results appear to be new. All three theorems are formalised in Lean 4 with Mathlib, using only the standard axioms.

math.AT — Algebraic Topology

Singularity Projection in Three-Dimensional Space and the Discretization of a Continuous Interval

Contributed by zoushanzhong zou

We study the projection geometry induced by a singular point on the boundary of a three-dimensional fan-shaped cylinder. The cylinder is defined in the first quadrant of the real coordinate plane, with x and y both between zero and one, and a singular point located at x=1 on the boundary. Through an iterative projection process involving the angle bisector of the quadrant, we prove that the continuous interval [0,1] is precisely discretized to an infinite set of boundary points.

math.CO — Combinatorics

The Cereceda (2007) conjecture holds for bipartite 2-degenerate graphs

Contributed by Dave Goel

For a graph $G$ and a positive integer $k$, the $k$-recolouring graph $\mathcal{R}_{k}(G)$ has the proper $k$-colourings of $G$ as vertices, two of them adjacent when they differ at exactly one vertex. A graph is $d$-degenerate if each of its non-empty subgraphs has a vertex of degree at most $d$. The Cereceda (2007) conjecture states that, for fixed $d$ and $k \ge d+2$, the diameter of $\mathcal{R}_{k}(G)$ is $O(n^2)$ for every $d$-degenerate graph $G$ on $n$ vertices. For $d = 2$, the conjecture was known to hold for $k \ge 5$ by a result of Cereceda; for $k = 4$, the best bound known for all $2$-degenerate graphs is $O(n^3)$, due to Bousquet and Heinrich. We prove that the diameter of $\mathcal{R}_{4}(G)$ is at most $2n(n+1)$ for every bipartite $2$-degenerate graph $G$ on $n$ vertices; hence the conjecture holds for these graphs. We also show that the order $n^2$ cannot be improved for this class, even for graphs of girth $6$. For the upper bound, we fix an ordering of the vertices in which each vertex has at most two later neighbours. In $n$ stages, we transform any proper $4$-colouring into the colouring in which every vertex has the least colour not used by its later neighbours, recolouring each vertex at most twice per stage. Both of our bounds have also been proved in the Lean proof assistant.

Author comments: An expanded version of this paper, with many other results, is posted on Zenodo: https://doi.org/10.5281/zenodo.23603237

math.HO — History and Overview

The Rising-Sea Theses: Twelve Theses on Mathematics and Machine Reasoning

Contributed by Vincent Liu

The Grothendieckian Programme is a set of seven works on machine reasoning about mathematics (34 volumes, 26,021 pages), released openly in October 2026 as first drafts that have not been peer-reviewed. This paper draws from them twelve theses on how mathematics and machine reasoning should be combined. Each thesis is stated in one sentence, argued, tied to the passages of the seven works on which it rests, related to existing work, and paired with an observation that would refute it. The theses concern the unit of mathematical knowledge for a reasoning system, the place of epistemic status, the latent space of mathematical reasoning, attention, failure, authority, and the method of discovery. The paper adds no mathematical result to the seven works; none of them claims a proof of the Riemann Hypothesis, the Birch and Swinnerton-Dyer conjecture, the Hodge conjecture, the abc conjecture or the Langlands correspondence.

Author comments: 26 pages, 5 figures. Version 1.0 of 28 October 2026; the same text is deposited at Zenodo, DOI 10.5281/zenodo.23734424 (all versions: 10.5281/zenodo.23734423). This copy was typeset with pdfLaTeX for Hexagon: its text is identical to version 1.0 at Zenodo, whose sealed PDF was typeset with LuaLaTeX, except that a broken cross-reference on page 3 ("igref{works}") is corrected to "Figure 1"; a few line and page breaks differ. The twelve theses are also set out at https://github.com/AlienMuse/grothendieckian-problems/blob/main/THESES.md; the seven works on which they rest, each with its DOI, are listed at https://github.com/AlienMuse/grothendieckian-programme. Corrections recorded since version 1.0 to the seven works and to the papers: https://github.com/AlienMuse/grothendieckian-problems/blob/main/ERRATA.md. Companion papers: The Grothendieckian Programme: An Overview (DOI 10.5281/zenodo.23734421) and The 57 Grothendieckian Problems (DOI 10.5281/zenodo.23734426).

math.CV — Complex Variables

Smale's mean value constant (1981) exceeds 1.1559

Contributed by Dave Goel

We prove that Smale's mean value constant exceeds $1.1559837$, improving the bound $1.1018$ of Jatar and Ng. Smale proved in 1981 that for every complex polynomial $P$ of degree at least $2$ and every point $z$ with $P'(z)\neq0$, some critical point $\zeta$ of $P$ satisfies $|P(z)-P(\zeta)|\le K\,|z-\zeta|\,|P'(z)|$ with $K=4$, and he conjectured that $K=1$ suffices. The best constant $K^*$ is called Smale's mean value constant. Smale's conjecture is false: a proof that $K^*>1$ appeared in 2026. Jatar and Ng showed that every pair $(W,B)$ of a normalized univalent map $W$ of the unit disk and a finite Blaschke product $B$ with $B(0)>0$, such that $(WB)'$ has no zero in the disk, gives $K^*\ge1/B(0)$. We construct such a pair explicitly, with $B$ of degree two, from a polynomial of degree $18$ with Gaussian rational coefficients, and we verify the required properties by a computation in exact arithmetic. Since the best constant $K(d)$ for degree $d$ is nondecreasing in $d$ (Crane), $K(d)$ also exceeds $1.1559837$ for all large $d$. All theorems, lemmas and corollaries of this paper, and the results of Jatar and Ng that it uses, are also proved in the Lean 4 proof assistant and checked with the Lean comparator.

math.NT — Number Theory

Spectral synthesis for the Báez–Duarte step space

Contributed by Will Howe, Grant Huffman, Jensen Kohlmeyer, Liam Kruer

We prove that the closed Báez–Duarte subspace of the Mellin step space is exactly the common kernel of the derivative evaluations prescribed by the zeros of the Riemann zeta function in $\operatorname{Re}s>1/2$, with their multiplicities. No finiteness or sparsity assumption on this zero set is required. The proof combines a uniform fractional regularity estimate for Mellin transforms of step functions with a comparison obtained by reflecting zeros in a shifted vertical line. These estimates produce a holomorphic Hardy-space family whose membership in the Báez–Duarte subspace continues from large positive shifts to every sufficiently small positive shift. A uniformly bounded weak limit then gives the synthesis theorem.

math.NT — Number Theory

Period relations as period terms: linear independence at CM and Beta periods

Contributed by Shawn Krisman

We prove linear independence results for periods of CM abelian varieties and Beta values by using a known relation between periods as the period term in the algebraic subgroup theorem. For example, $1$, $B(\frac27,\frac27)$, $B(\frac37,\frac37)$ and $B(\frac37,\frac37)/B(\frac17,\frac17)$ are linearly independent over $\overline{\mathbb{Q}}$, and families of such relations attached to the CM types of $\mathbb{Q}(\zeta_N)$ give the linear independence of $1$, a common ratio and $\kappa\ge2$ periods. The same method proves Waldschmidt's Conjecture 42 at six classes of CM ratios (for instance, $\wp(\log 2/\varpi)$ and $\wp(\log 3/\varpi)$ are not both algebraic on the lemniscatic curve) and several four exponentials statements at CM periods. All results are explicit consequences of theorems of Waldschmidt and of Roy and Waldschmidt.

math.QA — Quantum Algebra

Prime factors of fusion-category dimensions: weak group-theoreticity, rank and decomposition

Contributed by Sebastian Burciu, Jingcheng Dong, Sebastien Palcoux

Let $D$ be the Frobenius–Perron dimension of a weakly integral fusion category and $Q(D)$ its square-full part. For a divisor-minimal braided obstruction to weak group-theoreticity, we bound the conductors of character fields by formal codegrees and relate residue blocks to primary twists through local Weyl operators. For primes $p<q<r<s$ these restrictions give weak group-theoreticity whenever $Q(D)$ divides $p^5q^ar^b$ or $p^3q^3r^as^b$, with $a,b$ arbitrary. Fixing the cofactor of a prime power gives a finite test even when its exponent is unrestricted. A braided category is weakly group-theoretical for $Q(D)<29\,160\,000$ and solvable for odd $Q(D)<6\,087\,156\,075$. The corresponding bounds without a braiding are $D<5400$ and odd $D<121\,275$. Combining the dimension families with an exhaustive degree search gives rank at least $32$ for a non-pointed simple integral modular category, at least $33$ in nonsquare dimension, $57$ in odd dimension and $75$ in odd nonsquare dimension. We also obtain rank bounds without simplicity and prime-divisor criteria for a braided product decomposition.

Author comments: Strengthened the general rank bound to 32, incorporating the optimized degree search with dimension-family pruning. Unified verification code and certificates through rank 31, refreshed the manuscript–GitHub links, and preserved the 252-page length.

math.NT — Number Theory

The rational points of the non-split Cartan modular curve of level 121

Contributed by Manvir Jaswal

We prove that the modular curve $X_{\mathrm{ns}}^{+}(121)$ of the normalizer of a non-split Cartan subgroup of level 121, of genus 511, has exactly seven rational points, all of them CM. This is the level 121 case of a conjecture of Rouse, Sutherland and Zureick-Brown, and with their theorem it determines the 11-adic images of Galois of non-CM elliptic curves over $\mathbb{Q}$. The Jacobian has rank equal to the genus. We work at the primes 17 and 43, at each of which one Hecke block of the Mordell–Weil group has logarithm divisible by $p^2$ while its $p$-adic height has valuation exactly one. A comparison of valuations in the quadratic Chabauty identity then shows that the residue disc of each CM point holds no other rational point, with no computation of the local height at $p$, no basis of the Mordell–Weil group and no model of the curve. The inputs come from expansions of differentials at CM points and from a formula we prove for intersection numbers of CM divisors under Hecke correspondences on $X_{\mathrm{ns}}^{+}(\ell^e)$, at every prime $v \ne \ell$, and at $v = \ell$ when $\ell \ge 7$, $\ell \equiv 3 \bmod 4$ and $e \le 2$.