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math-ph — Mathematical Physics

Area law for three-dimensional U(1) lattice gauge theory with the Wilson action at every coupling

Contributed by Tejasvi Singh Tomar

We prove that three-dimensional U(1) lattice gauge theory with the Wilson action confines static charges at every coupling. More precisely, for each $\beta>0$ there are $\sigma(\beta)>0$ and $R_1(\beta)<\infty$ such that rectangular Wilson loops of sides $R,T\ge R_1(\beta)$ satisfy $\langle\cos\theta_C\rangle\le\exp(-\sigma(\beta)RT+c(R+T))$ , uniformly in the finite box, with an absolute constant $c$ and an explicit lower bound on $\sigma(\beta)$ . For the Villain action this is the theorem of G\"opfert and Mack. For the Wilson action it is cited as known but has remained open. The proof rests on a comparison. The Wilson plaquette weight is a mixture of Villain weights whose coupling has a very light upper tail. Hence, by Ginibre's inequality, the Wilson loop at coupling $\beta$ is at most an average, over a sparse Bernoulli set of frozen plaquettes, of the Villain loop at coupling $K\beta$ , and frozen plaquettes are insulating walls for the monopole gas. A variational regularization, built on the smallest minimizer of a cut functional, freezes more plaquettes, which only increases the loop. It produces slit domains on which the multiscale analysis of G\"opfert and Mack, with wall fields graded by their periods around the handles, runs with uniform constants. On these domains the loop decays like the exponential of the minimal spanning area with frozen plaquettes free. A Peierls estimate shows that this area is at least $(1-\eta)RT$ outside an event of probability at most $e^{-RT+O(R+T)}$ , with $\eta\to0$ as $K\to\infty$ . We carry over the steps cited from Brydges\textendash Federbush and G\"opfert\textendash Mack with explicit changes and prove them in full in Part S4 of the supplementary material, including a tree-graph counting step of Brydges whose printed deduction is incomplete. As corollaries, we obtain the area law at every coupling for all closed subgroups of U( $n$ ) containing the scalars, also with a Higgs field blind to the centre (as for U(2) with an adjoint Higgs field), and for the Georgi\textendash Glashow model at infinite Higgs coupling.

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

Oct 26, 2026PDF ↗View work →

hexagon:2610.00549v1

cs.DS — Data Structures and Algorithms

Sorting networks on 13 inputs need 45 comparators

Contributed by Tejasvi Singh Tomar

Let $S(N)$ be the least number of comparators in a sorting network on $N$ inputs (the Bose--Nelson problem). The values are known for $N\le12$ , and for $13$ inputs the smallest known sorting network, found by Juill\'e, has $45$ comparators. We prove $S(13)\ge45$ , so $S(13)=45$ . Deleting the comparators met by the largest and the second largest input leaves a sorting network on $N-2$ inputs, so $S(N)\ge S(N-2)+P(2,N)$ , where $P(2,N)$ is the least possible worst-case number of comparators met by these two values. Van Voorhis stated $P(2,N)\ge\lceil\lg F(N)\rceil$ for an explicit function $F$ , but Mericanii showed that his proof is invalid. We prove $P(2,N)=\lceil\lg F(N)\rceil$ for $3\le N\le32$ by a finite search, which rests on the structure of the tree of paths of the largest value, the comparators on these paths, and a forest of comparators that carries the second largest value. With $S(11)=35$ this bound gives only $S(13)\ge44$ . For $45$ we apply it to the suffix of a network: if a network with $L$ comparators sorts a set $X$ of $0$ - $1$ vectors, then for suitable pairs of inputs the two values meet at most $L-c$ comparators, where $c$ is the cost of sorting the residual set left when their paths are deleted. A breadth-first search over the sets produced by prefixes of a hypothetical sorting network with $44$ comparators discards each set for which no single network stays within all these budgets, and no set survives at depth $25$ . A separate verifier checks the certificate written by the search. Two proofs in Lean 4 check the same certificate, and in one of them Lean's kernel checks every computation, with only the standard axioms. Juill\'e's network is checked in the kernel as well. As a consequence, each of the twenty lower bounds on $S(N)$ for $13\le N\le32$ that rested on Van Voorhis's inequality is proved and raised by one.

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

Oct 26, 2026PDF ↗View work →

hexagon:2610.00546v1

math.GN — General Topology

ALL FINITE CARDINALITIES OF THE HYPERSPACE OF REGULAR SUBCONTINUA

Contributed by Lucas Almeida Almeida

For a metric continuum X, let D(X) be the family of subcontinua A of X with A = cl_X int_X A. Ordõnez asked which finite numbers occur as |D(X)|. Camargo, Ordõnez and Ramírez excluded 2, 4, 5, 8, 9 and asked whether every k ≥ 256 occurs. We prove that a positive integer k equals |D(X)| for some metric continuum X if and only if k not in {2, 4, 5, 8, 9, 16}. The finite cardinalities of D(X) are exactly the numbers of vertex sets inducing connected subgraphs of finite connected graphs; every finite connected graph is realized by copies of the Knaster continuum glued at points. We determine the values of this graph invariant without computer assistance. A short argument excludes 16, and an explicit recursive family of graphs realizes every k ≥ 17 with the least possible number of vertices. Consequently, for k ≥ 32 the least number of minimal regular subcontinua of a continuum X with |D(X)| = k is ⌊log2 k⌋ + 1. Programs check the explicit finite graph counts, and Theorem A is formally verified in Lean.

Primarily AI-generated textHuman understanding: all partsAI-assisted mathematicsContinuumLean 4enumeration of connected setshyperspacesindecomposable continuuminverse problemregular subcontinuu

Oct 26, 2026PDF ↗View work →

hexagon:2610.00545v1

math.NT — Number Theory

The missing Lin(ni)k

Contributed by Paul Pollack

We show how Selberg's symmetry formulas and the recent finding that there are no Landau--Siegel zeros connect to give a radically elementary proof of Linnik's theorem on the least prime in an arithmetic progression.

A mix of human-written and AI-generated textHuman understanding: all partsLandau--Siegel zerosLinnik's theoremSelberg's symmetry formulasieve methods

Oct 26, 2026PDF ↗View work →

hexagon:2610.00543v1

math.GN — General Topology

Recurrent tubes and connectedness-preserving bijections of Euclidean spaces

Contributed by Peter Lim

For every $n\ge 2$ we construct a connectedness-preserving bijection $F$ of $\mathbb{R}^n$ whose inverse does not preserve connectedness. The restriction of $F$ off the closed coordinate ray $E$ is a homeomorphism onto the complement of $L=K\setminus\{\infty\}$ , where $K\subset\mathbb{S}^n$ is a nondegenerate continuum with empty interior. The discontinuity sets of $F$ and $F^{-1}$ are exactly $E$ and $L$ , and both maps can be chosen to have exact Baire class $2$ . The preimage of a fixed straight segment can be any prescribed uncountable closed subset of $E$ with empty relative interior, including the middle-thirds Cantor set. The construction is uniform in the dimension. Homeomorphisms with compact support move finitely many collar points to a prescribed net, and narrowing the resulting collars produces a recurrent tube. Compact arcs on one retained boundary line supply dense connected tracks, while a gate lemma proves preservation of arbitrary connected sets. This answers W. Wong's question in the negative. We also prove that every component of the closure of the discontinuity set of a planar connectedness-preserving bijection is unbounded, and construct Borel counterexamples on manifolds with a standard spherical end.

Primarily AI-generated textHuman understanding: some parts

Oct 26, 2026PDF ↗View work →

hexagon:2610.00542v2

math.NT — Number Theory

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

Contributed by Steve Hubbard

Assuming the Riemann Hypothesis, we prove that for every real $x \ge 3$ the interval $(x, x + 4.45 \log^2 x)$ contains an even integer that is a sum of two odd primes. The proof combines a weighted mean square bound for the Chebyshev prime counting error in short intervals with a piecewise linear weight in a bilinear identity for short interval prime counts. The deduction is checked in Lean 4 conditionally on the explicit formula ExplicitA, the zero sum bound HadamardK, the zero count BPTN, the HSW zero counts, the bound $|\gamma| \ge 14$ , and the Goldbach verification to $4 \cdot 10^{18}$ , all stated as hypotheses.

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

Oct 26, 2026PDF ↗View work →

hexagon:2610.00541v1

math.CO — Combinatorics

Maximal Sublattices of Convex Geometries with Unbounded Complement Minima

Contributed by Y. T. Miah

For every integer $r\ge 3$ , we construct a finite atomistic convex geometry of convex dimension exactly three having a maximal proper sublattice whose complement has exactly $r$ minimal elements and exactly two maximal elements. The lattice has $(r+2)^2$ elements and the sublattice has $(r+1)(r+2)/2+3$ elements. At $r=3$ the construction is the 25-element example of Bhattacharjee. Thus the number of minimal elements in such a complement is unbounded even when the convex dimension is fixed at three, in contrast to the common-minimum theorem in convex dimension two.

A mix of human-written and AI-generated textHuman understanding: some partsconvex dimensionconvex geometriesjoin-semidistributive latticesmaximal sublattices

Oct 26, 2026PDF ↗View work →

hexagon:2610.00540v1

math.RT — Representation Theory

A note of the Gorenstein symmetry conjecture (II)

Contributed by Qiyue Tang

We prove the Gorenstein symmetry conjecture for every finite-dimensional split square-zero extension of an algebra derived equivalent, over the ground field, to a finite-dimensional hereditary algebra.

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

Oct 26, 2026PDF ↗View work →

hexagon:2610.00539v1

math.RT — Representation Theory

A note of the Gorenstein symmetry conjecture (I)

Contributed by Qiyue Tang

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

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

Oct 26, 2026PDF ↗View work →

hexagon:2610.00538v1

math-ph — Mathematical Physics

Nonlinear Wave Limits of Reversible Finite-State Cellular Automata

Contributed by Zhenyu Sun

We construct reversible finite-state cellular automata whose integer lifts approximate smooth solutions of one-dimensional semilinear wave equations. The kick–drift update is bijective for every odd modulus. Separate rounding of spatial edge contributions and local nonlinear forces gives an exact one-cell support bound for the full local operator algebra. Under a strict Courant condition, a discrete energy estimate bounds the field and staggered-velocity errors by $C_T(h^2+\tau^2+\delta/\tau^2)$ on a fixed physical time interval, where $h$ , $\tau$ , and $\delta$ are the spatial, temporal, and field spacings. An explicit modulus bound prevents modular wrapping throughout that interval. The choices $\tau=\nu h$ , $0<\nu<1$ , and $\delta=O(h^4)$ yield second-order convergence. Coupled scalar fields have a common-speed, locally Lorentz-covariant classical limit. Equal bare masses and equal quartic self-couplings preserve a symmetric invariant manifold. Unequal quartic couplings introduce a cubic source into the antisymmetric field equation, and we derive the first variation of the antisymmetric field. Numerical experiments compare exact integer trajectories with a nonlinear traveling wave and an independent continuum solver, measure light and massive dispersion branches, and examine continuously interacting wave packets. The accompanying programs reproduce the reported calculations.

Primarily human-written textHuman understanding: all partscausalitycontinuum limitinteger integrationreversible cellular automatasemilinear wave equations

Oct 26, 2026PDF ↗View work →

hexagon:2610.00537v1

math.RT — Representation Theory

A note of the Cartan determinant conjecture

Contributed by Qiyue Tang

We prove the Cartan determinant conjecture for every corner of finite global dimension in an Artin algebra of global dimension at most two. Consequently, the conjecture holds for torsionless-finite Artin algebras.

A mix of human-written and AI-generated textHuman understanding: some partsCartan determinant conjecture

Oct 26, 2026PDF ↗View work →

hexagon:2610.00536v1

math.MG — Metric Geometry

Illumination of complex convex bodies

Contributed by אלון שכטר

For every complex $1$ -unconditional convex body $K\subset\mathbb{C}^n$ , we prove \[ \operatorname{ill}(K)\le (2^{m+1}-1)\binom nm+\sum_{k=m+1}^{n}\binom nk-(n-m), \qquad m=\lceil2n/3\rceil. \] This bound is asymptotic to $3^{n+1}/\sqrt{\pi n}$ . We further obtain $\operatorname{ill}^*(K)\le b_*^n$ and $\operatorname{ill}(K)=O(n\log(n+1)b_*^n)$ , where $b_*=2.4805534697\ldots$ is defined by an explicit scalar equation. For normalized bodies other than $\mathbb{D}^n$ that contain a full coordinate $(n-1)$ -polydisc, we prove the sharp bound $\operatorname{ill}(K)\le2^{n+1}-2$ . We also prove the sharp bound $14$ for every normalized complex $1$ -unconditional body in $\mathbb{C}^3$ other than $\mathbb{D}^3$ , and the sharp bound $30$ in $\mathbb{C}^4$ for every normalized non-polydisc. The four-dimensional proof includes one finite covering assertion verified by exact integer arithmetic; its complete verifier is included. Under full coordinate permutation symmetry, we improve the three-dimensional bound to $12$ by an explicit twelve-vector construction and an exact finite covering certificate. For complex Hanner bodies, we compute both illumination numbers exactly and determine the equality cases in their dimension bounds. If every extreme point has all coordinates nonzero, we prove $\operatorname{ill}(K)\le2^{n+1}-1$ and a quantitative strict fractional bound $\operatorname{ill}^*(K)<2^n$ for normalized non-polydiscs. For real $1$ -symmetric non-cubes we obtain bounds $12$ in dimension four, $22$ in dimension five, and $5\cdot2^{n-3}$ for every $n\ge6$ . We also determine the leading asymptotic maximum in the real $1$ -symmetric non-cube class: it is $(1+o(1))2^{n-1}$ . An explicit binary construction proves this without an external coding theorem. A connection to connected domination in cube graphs improves the upper estimate to $2^{n-1}(1+(1+o(1))/n)$ .

Primarily AI-generated textHuman understanding: some partsconvex geometry

Oct 26, 2026PDF ↗View work →

hexagon:2610.00535v1

math.NT — Number Theory

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

Contributed by Changxia Lin

Abstract We give a normalization-consistent coefficient formulation of the de Bruijn–Newman heat flow attached to the Riemann ξ-function. For the classical kernel Φ and the family Ht(z) = Z ∞ 0 e tu2 Φ(u) cos(zu) du, Ft(w) = Ht(i √ w) = X n≥0 qn(t)w n , we prove the exact evolution laws ∂tFt = 2F ′ t + 4wF′′ t , q(m) n (t) = (2n + 2m)! (2n)! qn+m(t). The coefficients have positive integral representations and the transformed entire function has order at most 1/2. We relate the zeros of Ft to the Pólya–frequency property of (qn(t)), derive the rectangular Jacobi–Trudi identity for the consecutive Toeplitz minors, and state carefully the strict consecutive-minor criterion needed to pass from these minors to total positivity. In particular, for this coefficient family the real-zero condition is equivalent to strict positivity of every consecutive minor Dt(n, r), not merely to nonnegativity of that selected family. This distinction avoids an invalid use of a weak version of Fekete’s criterion. We also correct the multiplicity bookkeeping for the squared-variable map and record a Chebyshev convolution identity without attributing to it an unjustified duplication of zeros. The paper proves no new unconditional upper bound for the de Bruijn–Newman constant and no proof of the Riemann hypothesis; its purpose is to provide an auditable set of exact identities and to isolate the remaining quantitative problem

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

Oct 26, 2026PDF ↗View work →

hexagon:2610.00534v1

math.CO — Combinatorics

The maximum cardinality of a planar seven-distance set

Contributed by Alon Amit

We give a computer-assisted proof that the maximum cardinality of a planar set determining at most seven distinct positive distances is sixteen. A diameter-graph argument reduces a hypothetical seventeen-point example to an eleven-point set with at most six distances, except for a thirteen-cycle obstruction. We exclude the cycle by an exact interval certificate. Using a published exhaustive count of sixty-two candidate distance patterns, we classify the eleven-point cores by exact Gröbner-basis calculations and prove that none can have the required seventeen- point extension. All extension calculations use rational or algebraic arithmetic. An explicit sixteen-point triangular-lattice set attains the bound.

Primarily AI-generated textHuman understanding: all partscombinatorial-geometry

Oct 26, 2026PDF ↗View work →

hexagon:2610.00533v1

math.CO — Combinatorics

Complete planar arcs over nonsquare binary fields

Contributed by Jose Voloch

Let $m'_2(q)$ be the largest size of a complete arc in $\operatorname{PG}(2,q)$ other than a hyperoval. We prove $m'_2(q) \le q-\floor{q^{2/3}-q^{1/3}}, q=2^h\ge512,\ h\text{ odd}$. The geometric statement underlying the proof is a degree obstruction for arbitrary geometrically irreducible Frobenius nonclassical plane curves; an origin from an arc is unnecessary. The argument uses the curve and a Frobenius twist of its dual, exact weighted point counts, and a local conductor inequality that allows arbitrary singularities. We give a sharper bound obtained by optimizing over powers of two, determine its leading constant according to $h\bmod3$ , and prove further refinements using a characteristic-two normal form. Exact finite computations give, for example, $m'_2(128)\le105$ , $m'_2(512)\le452$ , and $m'_2(2048)\le1875$ .

Primarily AI-generated textHuman understanding: some partsPlane arcs

Oct 26, 2026PDF ↗View work →

hexagon:2610.00532v2

math.NT — Number Theory

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

Contributed by Shawn Krisman

Let $k$ be a totally real cubic field and $p$ a prime unramified in $k$ . The $p$ -adic regulator of $k$ is a $2\times2$ determinant of $p$ -adic logarithms of units and is always divisible by $p^2$ ; call $k$ anomalous at $p$ if it is divisible by $p^3$ . Among all 592421 totally real cubic fields of discriminant up to $10^7$ , the proportion of anomalous fields matches a random-matrix prediction at every prime $5\le p\le 31$ , but at $p=3$ it is $0.381$ instead of the predicted $0.407$ . We explain this. For cubic fields of fundamental discriminant $D$ prime to 3, we show that the 3-class group and the ray class group modulo 9 of the quadratic field $\mathbb{Q}(\sqrt{D})$ can force every cubic field of discriminant $D$ to be anomalous at 3. When the 3-class group of $\mathbb{Q}(\sqrt{D})$ is cyclic, we decide exactly when $k$ is anomalous, in terms of one invariant for which we give an explicit formula by cubic residue and Hilbert symbols. The phenomenon comes from the failure of semisimplicity of $\mathbb{F}_3[S_3]$ and has no analogue for $p\ge 5$ . The results fix one or two 3-adic digits of the regulator and say nothing about Leopoldt's conjecture itself.

Primarily AI-generated textHuman understanding: no partsGalois cohomologyLeopoldt conjecturePoitou-Tate dualityS3 extensionscup productsp-adic regulatorp-rationalityray class groupsrestricted ramificationtotally real cubic fields

Oct 26, 2026PDF ↗View work →

hexagon:2610.00531v1

math.NT — Number Theory

Long solutions to small equations

Contributed by Bogdan Grechuk, Eugene Go, Avraham Eisenberg

We give complete solutions of thirty-eight Diophantine equations left open in Grechuk's 2024 monograph: fifteen affine equations, five generalized Fermat equations, eight plane quartics, and ten further homogeneous equations. All proofs were generated by AI and require substantial computer algebra computations.

Primarily AI-generated textHuman understanding: some partsDiophantine equations; Chabauty method; Brauer--Manin obstruction

Oct 26, 2026PDF ↗View work →

hexagon:2610.00530v1

quant-ph — Quantum Physics

A matching threshold of 2×10⁻³ for the rotated surface code under circuit-level noise

Contributed by Tejasvi Singh Tomar

We consider the rotated surface code of odd distance $d$ with its standard syndrome-extraction circuit, under circuit-level depolarizing noise in which every gate, idle step, preparation and measurement fails independently with probability $p$ , decoded by minimum-weight perfect matching with the log-likelihood weights of single faults. We prove that a memory experiment with $T$ rounds fails with probability at most $0.061\,(T+1)(d+1)\,\lambda^{d}/(1-\lambda)$ , where $\lambda=0.999\,(p/p_{\ast\ast})^{9/25}$ , for every $p\le p_{\ast\ast}=2\times10^{-3}$ . Hence the logical error rate per round decays exponentially in $d$ for all $p\le p_{\ast\ast}$ . Earlier rigorous thresholds for matching on surface-code circuits are $1.7\times10^{-4}$ , due to Dennis, Kitaev, Landahl and Preskill, and $7.4\times10^{-4}$ , due to Fowler. We keep their reduction of a failure to a weighted self-avoiding path in the decoding graph, but bound these paths by a local criterion for non-backtracking walks, which gives the threshold $1.04\times10^{-3}$ in closed form. To reach $2\times10^{-3}$ we use the minimality of the correction along the path, for a canonical choice among the minimum-weight corrections: labellings of short stretches of the path into error and correction edges that contradict this minimality are excluded, and the remaining labelled walks are counted with a potential function that is checked in exact arithmetic. For a leaner schedule with fused measure-and-reset steps the two arguments give $1.53\times10^{-3}$ and $3\times10^{-3}$ , respectively. With error rates that depend on the type of operation, they certify exponential decay for error models built from the published component error rates of Quantinuum's Helios and H2 and of IBM's Heron r3 processor ibm_boston, mapped onto this circuit. All theorems are also proved in the proof assistant Lean 4, with all computations checked by its kernel. Any argument that bounds the failure probability by the probability of an increasing event, ours included, is limited by the smallest such event that contains all failures, and Monte Carlo estimates place the threshold of this event near $4\times10^{-3}$ for this circuit.

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

Oct 26, 2026PDF ↗View work →

hexagon:2610.00529v1

math.GR — Group Theory

A Counterexample to Huppert's Conjecture

Contributed by Shihao Wang, Zhiwei Zheng

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

Primarily AI-generated textHuman understanding: all parts

Oct 26, 2026PDF ↗View work →

hexagon:2610.00528v1

math.HO — History and Overview

$10/24/26$ : Factoring a Pythagorean Date

Contributed by Scott Kominers

Written numerically, the date October 24, 2026 corresponds to a Pythagorean triple---the last one of the twenty-first century! The associated $10$ -- $24$ -- $26$ right triangle has area twice its perimeter; a classical factorization shows that exactly three integer-sided right triangles have this property and, more generally, counts those with any prescribed integer inradius. We also observe that $60$ is the smallest perimeter shared by two noncongruent integer-sided right triangles, one of which is the $10$ -- $24$ -- $26$ triangle.

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

Oct 26, 2026PDF ↗View work →

hexagon:2610.00527v3

math.GM — General Mathematics

Galois groups of Taylor polynomials: prescriptions and arithmetic families

Contributed by Khai-Hoan Nguyen-Dang

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

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

Oct 26, 2026PDF ↗View work →

hexagon:2610.00526v1

math.DS — Dynamical Systems

Rigidity of Newtonian Centered Co-Circular Central Configurations

Contributed by Chih-Yao Wu

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

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

Oct 26, 2026PDF ↗View work →

hexagon:2610.00525v1

math.NT — Number Theory

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

Contributed by Shawn Krisman

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

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

Oct 26, 2026PDF ↗View work →

hexagon:2610.00523v1

math.NT — Number Theory

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

Contributed by Idris Ali Shaik

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

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

Oct 26, 2026PDF ↗View work →

hexagon:2610.00522v1

math.MP — Mathematical Physics

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

Contributed by Qihang Wang

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

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

Oct 26, 2026PDF ↗View work →

hexagon:2610.00521v1

math.NT — Number Theory

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

Contributed by Changxia Lin, Pulin Shi

We give a normalization-consistent coefficient formulation of the de Bruijn–Newman heat flow attached to the Riemann ξ-function. For the classical kernel Φ and the family Ht(z) = Z ∞ 0 e tu2 Φ(u) cos(zu) du, Ft(w) = Ht(i √ w) = X n≥0 qn(t)w n , we prove the exact evolution laws ∂tFt = 2F ′ t + 4wF′′ t , q(m) n (t) = (2n + 2m)! (2n)! qn+m(t). The coefficients have positive integral representations and the transformed entire function has order at most 1/2. We relate the zeros of Ft to the Pólya–frequency property of (qn(t)), derive the rectangular Jacobi–Trudi identity for the consecutive Toeplitz minors, and state carefully the strict consecutive-minor criterion needed to pass from these minors to total positivity. In particular, for this coefficient family the real-zero condition is equivalent to strict positivity of every consecutive minor Dt(n, r), not merely to nonnegativity of that selected family. This distinction avoids an invalid use of a weak version of Fekete’s criterion. We also correct the multiplicity bookkeeping for the squared-variable map and record a Chebyshev convolution identity without attributing to it an unjustified duplication of zeros. The paper proves no new unconditional upper bound for the de Bruijn–Newman constant and no proof of the Riemann hypothesis; its purpose is to provide an auditable set of exact

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

Oct 26, 2026PDF ↗View work →

hexagon:2610.00520v1

math.MP — Mathematical Physics

Perfect t-embeddings: existence

Contributed by Mikhail Basok, Dmitry Chelkak

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

Primarily AI-generated textHuman understanding: some partst-embeddings

Oct 26, 2026PDF ↗View work →

hexagon:2610.00519v1

math.NT — Number Theory

Robin's Inequality and Superabundant Counterexamples

Contributed by Frank Vega

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

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

Oct 26, 2026PDF ↗View work →

hexagon:2610.00516v1

math.NT — Number Theory

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

Contributed by Shawn Krisman

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

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

Oct 26, 2026PDF ↗View work →

hexagon:2610.00515v1

math.MG — Metric Geometry

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

Contributed by Ricardo Pinho

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

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

Oct 26, 2026PDF ↗View work →

hexagon:2610.00514v1

math.CO — Combinatorics

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

Contributed by Ricardo Pinho

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

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

Oct 26, 2026PDF ↗View work →

hexagon:2610.00512v1

math.MP — Mathematical Physics

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

Contributed by P. M. Aronow, Patrick Lopatto

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

Primarily AI-generated textHuman understanding: no partsquantum field theory

Oct 26, 2026PDF ↗View work →

hexagon:2610.00511v1

math.MP — Mathematical Physics

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

Contributed by P. M. Aronow, Patrick Lopatto

Primarily AI-generated textHuman understanding: no partsquantum field theory

Oct 26, 2026PDF ↗View work →

hexagon:2610.00510v1

math.MP — Mathematical Physics

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

Contributed by Patrick Lopatto, P. M. Aronow

Primarily AI-generated textHuman understanding: no partsquantum field theory

Oct 26, 2026PDF ↗View work →

hexagon:2610.00509v1

math.MP — Mathematical Physics

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

Contributed by Patrick Lopatto, P. M. Aronow

Primarily AI-generated textHuman understanding: no partsquantum field theory

Oct 26, 2026PDF ↗View work →

hexagon:2610.00508v1

math.QA — Quantum Algebra

Vertex operator algebra realizations of untwisted Haagerup--Izumi centers

Contributed by Andrew Riesen, David E. Evans

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

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

Oct 26, 2026PDF ↗View work →

hexagon:2610.00506v1

math.MP — Mathematical Physics

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

Contributed by P. M. Aronow, Patrick Lopatto

Primarily AI-generated textHuman understanding: no partsquantum field theory

Oct 26, 2026PDF ↗View work →

hexagon:2610.00505v1

math.RT — Representation Theory

Multiplicative bases and deformations of representation-finite algebras

Contributed by Haruhisa Enomoto

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

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

Oct 23, 2026PDF ↗View work →

hexagon:2610.00504v1

math.NT — Number Theory

Minimum Prime Bound in Square Intervals

Contributed by Min Min Oo Oo

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

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

Oct 23, 2026PDF ↗View work →

hexagon:2610.00503v1

cs.CC — Computational Complexity

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

Contributed by Idris Ali Shaik

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

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

Oct 23, 2026PDF ↗View work →

hexagon:2610.00502v1

math.NT — Number Theory

Galois embedding problems and number field counting

Contributed by Anwesh Ray

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

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

Oct 23, 2026PDF ↗View work →

hexagon:2610.00501v1

math.CO — Combinatorics

A counterexample to a conjecture of Graffiti.pc on the forest number

Contributed by Deep Bhattacharjee

Conjecture 66 of Graffiti.pc bounds the order of a largest induced forest of a connected graph from below by the even mode of the degrees of its complement divided by its average degree. We show that it is false, under every reading of its definitions and by an unbounded margin.

A mix of human-written and AI-generated textHuman understanding: some partsGraffiti.pccomplementcounterexampledegree sequenceforest numberinduced forest

Oct 23, 2026PDF ↗View work →

hexagon:2610.00500v1

math.CO — Combinatorics

Counterexamples to five conjectures of Graffiti.pc on domination

Contributed by Deep Bhattacharjee

Written on the Wall II, the list of conjectures of the program Graffiti.pc, records Conjectures 319, 352, 358, 359 and 427 as open. We show that all five are false. The counterexamples are caterpillars for Conjectures 352, 358, 359 and 427, and graphs built from complete graphs for Conjecture 319; each comes in an infinite family, and for Conjectures 358, 359 and 427 the difference between the two sides is unbounded.

A mix of human-written and AI-generated textHuman understanding: some partsGraffiti.pccaterpillarcounterexampleindependent dominationtotal dominationtreewell totally dominated graph

Oct 23, 2026PDF ↗View work →

hexagon:2610.00499v1

math.MG — Metric Geometry

Regular Morley tetrahedra from $p$ -secting planes

Contributed by Deep Bhattacharjee

Morley's theorem says that the adjacent angle trisectors of a triangle meet in the vertices of an equilateral triangle. Tran carried the construction to a tetrahedron by trisecting its six dihedral angles and, more generally, by $p$ -secting planes, with one parameter $e,f,g>1$ for each pair of opposite edges; the resulting tetrahedron is denoted $\mathcal M_{e,f,g}(ABCD)$ . Tran showed that a regular Morley tetrahedron does not force the original tetrahedron to be regular, observed that in his examples some pair of opposite edges has four equal cross edges, and conjectured that this happens whenever $\mathcal M_{e,f,g}(ABCD)$ is regular. We show that the conjecture is false. We place an isosceles tetrahedron in the box $[-a,a]\times[-b,b]\times[-c,c]$ and prove that $\mathcal M_{e,f,g}(ABCD)$ is regular exactly when its vertex $A'$ differs from the centre of the box and lies on one of the four lines through the centre in the directions $(1,1,1)$ , $(1,-1,-1)$ , $(-1,1,-1)$ and $(-1,-1,1)$ . Every interior point of these lines other than the centre occurs for exactly one triple of parameters. When $a$ , $b$ and $c$ are pairwise different, no pair of opposite edges has four equal cross edges, so each of these tetrahedra is a counterexample. Two examples are given with the parameters in closed form. Finally, for quadrisecting planes, $e=f=g=4$ , an isosceles tetrahedron has a regular Morley tetrahedron only if it is regular.

A mix of human-written and AI-generated textHuman understanding: some parts$p$-secting planeMorley tetrahedronMorley's theoremdihedral angledisphenoidisosceles tetrahedron

Oct 23, 2026PDF ↗View work →

hexagon:2610.00498v1

math.CO — Combinatorics

Upper $k$ -star-forming sets and $k$ -independence in bipartite graphs

Contributed by Deep Bhattacharjee

Sahbi conjectured that the $k$ -independence number $\beta_k$ and the upper $k$ -star-forming number $\SF_k$ agree on bipartite graphs, and in particular on chain graphs, and asked whether $\beta_3=\SF_3$ on triangle-free graphs. For every $k\ge3$ we give a connected bipartite graph on $3k+3$ vertices with $\beta_k=2k+2$ and $\SF_k\ge3k$. On chain graphs, however, $\beta_k=\SF_k$ for every $k$ . We also show, answering another of his questions, that deciding whether a bipartite graph has a minimal $2$ -star-forming set with at least $t$ vertices is NP-complete.

A mix of human-written and AI-generated textHuman understanding: some parts$k$-independence$k$-star-forming setNP-completenessbipartite graphchain graphformal verificationupper domination

Oct 23, 2026PDF ↗View work →

hexagon:2610.00497v1

math.RA — Rings and Algebras

Maximal sublattices of finite semidistributive lattices: the complement need not be an interval

Contributed by Deep Bhattacharjee

Adams, Freese, Nation and Schmid proved that the complement of a maximal sublattice of a finite bounded lattice is an interval, and asked whether the same holds for finite semidistributive lattices; Adaricheva, Mata, Silberger and Zamojska-Dzienio conjectured that it does. It does not. In a 26-element semidistributive sublattice of the square of a 14-element semidistributive lattice, the diagonal is a maximal sublattice whose complement has two minimal elements.

A mix of human-written and AI-generated textHuman understanding: some partsbounded latticeformal verificationintervalmaximal sublatticesemidistributive lattice

Oct 23, 2026PDF ↗View work →

hexagon:2610.00496v1

math.AC — Commutative Algebra

Finite Field Hudzik-Landes-Dragomir-Kato-Saito-Tamura Inequality

Contributed by K. MAHESH KRISHNA

Hudzik and Landes \textit{[Math. Ann., 1992]} derived a major generalization of the triangle inequality for two nonzero elements in normed linear spaces, which was extended to finitely many nonzero elements independently by Dragomir \textit{[Bull. Aust. Math. Soc., 2006]} and by Kato, Saito and Tamura \textit{[Math. Inequal. Appl., 2007]}. We derive a finite field version of Hudzik-Landes-Dragomir-Kato-Saito-Tamura inequality.

Primarily human-written textHuman understanding: all partsFinite Field.Normed linear spaceSub-modulusSub-norm.Triangle inequality

Oct 23, 2026PDF ↗View work →

hexagon:2610.00495v1

math.RA — Rings and Algebras

Maximal sublattices of convex geometries: four counterexamples

Contributed by Deep Bhattacharjee

Adaricheva, Mata, Silberger and Zamojska-Dzienio conjectured that in a finite convex geometry, and more generally in a finite join-semidistributive lattice, the complement of a maximal sublattice is a union of intervals with a common least element, and proved this for convex dimension two. We give convex geometries of convex dimension four and three in which such a complement has two and three minimal elements, so their theorem does not extend. The same examples refute their Hypothesis 4 and one half of their Question 2; an eleven-element convex geometry refutes the other half, and a ten-element lattice their Hypothesis 3.

A mix of human-written and AI-generated textHuman understanding: some partsconvex geometryformal verificationjoin-semidistributive latticemaximal sublatticemeet-irreducible element

Oct 23, 2026PDF ↗View work →

hexagon:2610.00494v2

math.CO — Combinatorics

Unbounded strong-product ratios for the Grundy domination number

Contributed by Deep Bhattacharjee

A sequence of vertices of a graph is legal when every vertex in it dominates some vertex that none of the earlier ones dominated, and the Grundy domination number of the graph is the length of its longest legal sequence. In the strong product of two graphs, two vertices are adjacent when each coordinate is equal or adjacent in its factor. It has been known since 2016 that the Grundy domination number of a strong product is at least the product of the Grundy domination numbers of the two factors, and it was conjectured, by Bre\v{s}ar, Bujt\'as, Gologranc, Klav\v{z}ar, Ko\v{s}mrlj, Patk\'os, Tuza and Vizer, that equality always holds. We show that the conjecture is false, and false by an unbounded factor: for every number $C$ there are graphs whose strong product has Grundy domination number more than $C$ times the product of the two factors. The graphs are built on the five-element subsets of an $m$ -element set, with adjacency decided by the size of the intersection; incidence matrices over the fields with two and three elements bound the two factors by $m$ and $\binom m2$ , while the diagonal of the product is an independent set of size $\binom m5$ . We also compute the Grundy domination number of the graph on $\{0,\dots,N-1\}$ in which $u$ and $v$ are adjacent when $|u-v|\equiv2\pmod3$ : it is $\lfloor(N+1)/3\rfloor+3$ for every $N\ge6$ . Its strong product with the octahedron has a legal sequence of length $N$ , so the conjecture already fails for $N=19$ , on a graph with $114$ vertices, and for every $N\ge21$ . Every statement is proved by hand.

A mix of human-written and AI-generated textHuman understanding: some partsGrundy domination numbercircular cliquedominating sequenceformal verificationstrong product

Oct 23, 2026PDF ↗View work →

hexagon:2610.00493v2

math-ph — Mathematical Physics

Explicit String Vacua and the Geometric Selection Conjecture

Contributed by Deep Bhattacharjee, Ushashi Bhattacharya

We combine explicit spectral calculations for the Dwork and weighted Fermat pencils with an audit of the Geometric Selection Conjecture. The residue calculation determines the character labels, cubic couplings and invariant-direction mass maps, while an index argument excludes three standard-embedding families on free quotients of thirteen hypergeometric complete-intersection types. For a nonstandard construction we fix one tetraquadric equation, prove smoothness and a free degree-four action, and calculate equivariant bundle cohomology by an exact determinant certificate. Its projected charged spectrum contains three families and one Higgs doublet pair. At the specified split-bundle point, however, an additional Abelian gauge factor survives, the perturbative Yukawa matrices fail the required rank test, and the tree-level volume remains flat. We then show that the first of these failures is forced: on every Calabi--Yau threefold with at most four K\"ahler parameters, every polystable sum of five line bundles with vanishing first Chern class has a charge matrix of rank at most three and keeps at least one Abelian gauge boson massless at its split locus. On the tetraquadric the polystable K\"ahler classes of the fixed bundle form exactly the volume ray. For the second failure we decide the perturbative selection rules on eleven split data, the fixed bundle and ten further line-bundle sums with three families on the same manifold, by a flow criterion for the singlet insertions: six of the eleven lose a Yukawa matrix identically and five are not excluded. We then follow the extension branch: a single non-split extension never removes the forced gauge boson, while for the fixed bundle and five of the ten further sums a polystable sum of two rank-two extensions and a line bundle, with the same Chern character, exists on one side of a stability wall arbitrarily close to the split point and carries no massless Abelian gauge boson; for three further sums no such bundle exists at any K\"ahler class, and for one it does not exist on the general tetraquadric. We fix the acceptance windows for the stabilization, Yukawa and supersymmetry breaking conditions, so that the conjecture becomes a definite statement, and show that the failures found do not depend on them. No vacuum satisfying all nine conditions is constructed; the Geometric Selection Conjecture remains unproved.

A mix of human-written and AI-generated textHuman understanding: some partsCalabi-Yau threefoldGeometric Selection Conjecturegenerationsheterotic stringmirror symmetry

Oct 23, 2026PDF ↗View work →