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hexagon:2610.00474v1

math.CO — Combinatorics

Even-intersecting families of lattice paths and the Catalan numbers

Contributed by Deep Bhattacharjee

Shankar proved that a family of North--East lattice paths from $(0,0)$ to $(n,n)$ in which any two paths share an even number of edges has at most $2^n$ members, and conjectured that exactly $C_n$ families attain this bound, where $C_n$ is the $n$ th Catalan number. We prove the conjecture. The proof determines all equality cases of the inequality behind Shankar's bound: they correspond to noncrossing partial matchings of $\{1,\dots,m\}$ whose unmatched points lie under no arc, and there are $\binom{m}{\lfloor m/2\rfloor}$ of them.

A mix of human-written and AI-generated textHuman understanding: some partsCatalan numbersEventown theoremeven intersectionslattice pathsnoncrossing matchings

Oct 22, 202612 pages

hexagon:2610.00473v2

math.CO — Combinatorics

The minimum number of distinct abelian squares in a binary circular word

Contributed by Deep Bhattacharjee

We prove that every binary circular word of length $n$ contains at least $\fl{(n-1)/2}$ distinct abelian squares, and we determine all words attaining this bound. This settles a conjecture of Simpson: its odd case holds, and in its even case the list of extremal words misses $(ab)^{n/2}$ for $n\equiv 2\pmod 4$ , $n\ge 6$ . We then review the other problems of the survey: we conjecture, and verify to length $27$ , that for odd $n$ the least number of non-equivalent abelian squares in a circular word is also $(n-1)/2$ , and we show that the least number of occurrences of abelian squares in a binary word of length $n$ is asymptotically proportional to $n$ or grows faster, and is below $n^{3/2}$ .

A mix of human-written and AI-generated textHuman understanding: some partsParikh vectorabelian squarecircular wordcombinatorics on words

Oct 22, 202614 pages

hexagon:2610.00472v2

math.NT — Number Theory

The Welch function is not third-order sum-free

Contributed by Deep Bhattacharjee

Let $n=2m+1$ be odd and let $\F_{2^n}$ be the field with $2^n$ elements. A function on $\F_{2^n}$ is third-order sum-free when its values over any eight elements forming a three-dimensional affine subspace never add up to zero. The Welch function is the power map $x\mapsto x^{2^m+3}$ ; it is one of the classical almost perfect nonlinear functions, and Hou and Zhao conjectured that for $n\ge7$ it is not third-order sum-free. We prove this. The question comes down to a linearised polynomial $g(x,y)$ in $y$ whose roots form a vector space $Z(x)$ over $\F_2$ containing $1$ and $x$ , and the Welch function fails to be third-order sum-free exactly when $Z(x)$ has dimension three for some $x$ outside $\F_2$. We show that the dimension is always $2$ or $3$ , and that it is $3$ if and only if two explicit trace conditions hold: with $u=x^{2^m}$ , both $\Tr(1/(x+u))$ and $\Tr(xu/(x+u))$ equal $1$ . The proof applies the field automorphism $z\mapsto z^{2^{m+1}}$ , whose square is the Frobenius map, to turn $g$ into an additive polynomial of degree eight whose roots always lie in $\F_{2^n}$. Counting the elements that satisfy both trace conditions gives exactly $(2^n+1-3K_n)/4$ of them, where $K_n$ is the binary Kloosterman sum of $\F_{2^n}$, and Weil's bound $|K_n|\le 2^{n/2+1}$ makes this count positive for every $n\ge7$ . The Welch function sums to zero over exactly $(2^n+1-3K_n)(2^n-1)/168$ three-dimensional linear subspaces, and the count $2+(2^n+1-3K_n)/4$ agrees with the numbers that Hou and Zhao computed for $n\le15$ . Every statement is proved by hand.

A mix of human-written and AI-generated textHuman understanding: some partsKloosterman sumsWelch functionalmost perfect nonlinear functionslinearized polynomialssum-free functions

Oct 22, 20269 pages

hexagon:2610.00471v3

math-ph — Mathematical Physics

Residual-dimension obstructions for null-projected supersymmetric Yang--Mills theories

Contributed by Deep Bhattacharjee

We prove necessary conditions for null-projected supersymmetric Yang--Mills theories with a Majorana--Weyl gaugino in signature $(s,t)$ , under five explicit assumptions on the gaugino constraint, the transformation rules, the residual gauge field, the translation bracket and the absorption of the cubic gaugino terms. A real gaugino allows at most $\min(s,t)$ null vectors, the residual dimension must be two or ten, and $s-t$ must be $0$ or $\pm8$ . The main step is a converse of the cubic gamma-matrix identity: if symmetric gamma matrices acting on $N$ -component spinors in $d$ dimensions satisfy it, then $N=2(d-2)$ . Consequently Nishino's construction does not transfer to the signatures $(11+8n)+3$ proposed by Rios, Marrani and Chester. In $27+3$ a real null frame has at most three vectors against Nishino's ten, the translation bracket allows only two, and with two the residual spinors have $4096$ components where absorption needs $48$ . For unitary single-time theories with vanishing tensor charges on a massless multiplet, we also exclude every finite collection of scalar, spinor and vector polarizations when there are more than sixteen real supercharges. Theories outside the five assumptions are not decided.

A mix of human-written and AI-generated textHuman understanding: all partsClifford algebrasFierz identityMajorana-Weyl spinorsYang-Mills theorynull projectionspacetime signature

Oct 22, 202626 pages

hexagon:2610.00470v1

math.MP — Mathematical Physics

Gluing defects for projected coarse graining in finite spin-foam models

Contributed by Zhenyu Sun

Orthogonal truncation can discard states that contribute to a later boundary amplitude. For a self-adjoint contraction $T$ on a finite-dimensional Hilbert space and an isometric inclusion $s$ of retained states, we prove that the two-step residual $\Delta=s^*T^2s-(s^*Ts)^2$ controls every finite placement of intermediate projections in every Schatten norm. For blocks of lengths $n_1,\ldots,n_k$ , the error is at most $\sum_{i<j}n_in_j\|\Delta\|$ , and the constant is optimal even for positive contractions. The residual vanishes exactly when the retained subspace is invariant. Two-dimensional finite-group gauge sums realize an idempotent projected transfer operator with nonzero residual, as well as positive-semidefinite transfer operators with identical one-step retained data and different residuals. We derive their gluing and subdivision identities with explicit boundary measures and BF normalization. We also bound the accumulation of local errors under tensor contraction and the distance to compatible linear amplitude data. Colimits extend compatible amplitude families, and quotients by invariant subspaces preserve the comparison maps. A measurement map that is injective on the retained ansatz gives a unique normalized least-squares fit. Exact rational calculations reproduce the finite-group examples and the blocking comparisons.

Primarily human-written textHuman understanding: all partscoarse grainingcylindrical consistencyfinite gauge theoryspin foamstensor gluingtruncation error

Oct 22, 202625 pages

hexagon:2610.00469v1

math.NT — Number Theory

A truncated theta-type series with triangular weights

Contributed by Manjil Saikia

Merca (2026) recently introduced the two-block odd partition function $a(n)$ , proved that its generating function factors as $A(q)=P(q)S(q)$ , where $P(q)$ is the generating function of partitions whose odd parts are distinct and whose even parts are unrestricted and $S(q)$ is a theta-type series with triangular-number weights, and conjectured that for $N\equiv1,2\pmod 4$ the truncation $P(q)S_N(q)$ of this product has nonnegative coefficients, positive from $q^{3}$ onward. We prove the conjecture in the stronger form $P(q)S_N(q)-A(q)\in\NN[[q]]$.

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

Oct 22, 202619 pages

hexagon:2610.00468v1

math.GM — General Mathematics

Arithmetic of Jacobi polynomials

Contributed by Khai Hoan Nguyen Dang

We study the arithmetic of Jacobi polynomials. As applications, we prove the Mahlburg--Ono conjecture on the irreducibility and Galois groups of the four Kaneko--Zagier families, and the Cullinan--Hajir conjecture on the Galois groups of Legendre polynomials. Other applications concern common zeros of Jacobi polynomials, harmonic functions on cones, and hyperelliptic Jacobians.

A mix of human-written and AI-generated textHuman understanding: some partsJacobi polynomialKaneko--Zagier polynomialLegendre polynomial

Oct 22, 202656 pages

hexagon:2610.00467v1

math.DS — Dynamical Systems

Finiteness of Planar Central Configurations for Five Positive Masses

Contributed by Chih-Yao Wu

We prove that, for any five positive masses, there are finitely many planar Newtonian central configurations up to translations, rotations, and positive scaling. Building on the singular-diagram framework of Albouy and Kaloshin, we treat the exceptional masses left open by their finiteness theorem. The proof retains a lower bound for the physical potential after passing to the complex algebraic equations. On each fixed potential level, energy estimates, torque identities, and local response calculations control the possible degenerations. A polynomial selection argument then yields finiteness. The exceptional local calculations and their algebraic certificates are given explicitly. A separate Lean development accompanies the proof.

Primarily AI-generated textHuman understanding: some partsLeanSmale sixth problemcelestial mechanicscentral configurationsfive-body problemformal verification

Oct 22, 202646 pages

hexagon:2610.00466v2

math.NT — Number Theory

Exceptional zeros of Hecke $L$ -functions

Contributed by Christopher Frei, Wataru Kai

We prove that Hecke $L$ -functions have no exceptional zeros. We do so by generalising a variant due to Kerr of the proof by OpenAI in the case of Dirichlet $L$ -functions. v2: Added a reference to independent concurrent work by Banks.

Primarily human-written textHuman understanding: all partsHecke L-functionsLandau--Siegel zerosexceptional zeros

Oct 22, 20269 pages

hexagon:2610.00465v1

math.AC — Commutative Algebra

Nonexistence of lim Cohen–Macaulay modules following [Che26] and [Ang26]

Contributed by Andy Jiang

Using the inequality stated in [ Che26, Main Claim 1.1], we prove nonexistence of lim Cohen–Macaulay sequences for the completed section rings of smooth projective surfaces over C with an ample globally generated divisor H satisfying KY ≡ 4H and 15H2/8 > χ(OY ). Such a pair is exhibited in [Ang26, Corollary 4.3], with H2 = 1080 and χ(OY ) = 1926. For a complete local domain (R, m) with ℓRHj m(R) < ∞ for 0 ≤ j < dim R, we prove that a lim Cohen–Macaulay sequence in the sense of Bhatt, Hochster, and Ma [ BHM26 ] exists if and only if a lim Γm-Cohen–Macaulay sequence exists (Definition 1.1). These may be different sequences. Chen’s geometric inequality [Che26, Main Claim 1.1] obstructs lim Γm-Cohen–Macaulay sequences over the completed section rings above. The comparison therefore rules out lim Cohen–Macaulay sequences over those rings.

Primarily AI-generated textHuman understanding: some parts

Oct 22, 202610 pages

hexagon:2610.00464v2

math.LO — Logic

Ramsey’s Theorem for Pairs in Pincus’s Model XI

Contributed by Frank Gilson

Howard and Rubin record Pincus’s ordered colored permutation model as N48, but do not determine whether Ramsey’s theorem for pairs holds there. We prove that N48 |= RT for pairs. The proof uses least finite supports and the homogeneity of Pincus’s ordered colored structure: after fixing all but one point of a support, the remaining point generates an infinite orbit whose unordered pairs form a single stabilizer orbit. Combining this with Pincus’s results on the dense linear ordering principle (DO), the Boolean Prime Ideal Theorem (BPI), and the failure of Part∞, together with Pincus’s transfer theorems, yields Con(ZF) ⇒ Con(ZF + RT_pairs + DO + BPI + ¬Part∞), where Part∞ states that every infinite set is the disjoint union of infinitely many infinite sets.

A mix of human-written and AI-generated textHuman understanding: some partsaxiom of choice; Ramsey's theorem

Oct 22, 20264 pages

hexagon:2610.00463v1

math.NT — Number Theory

Non-Archimedean Smale Mean Value and Dubinin–Sugawa Dual Mean Value Problems

Contributed by K. MAHESH KRISHNA

Motivated by the Smale mean value and the Dubinin–Sugawa dual mean value theorems, we formulate non-Archimedean Smale mean value and Dubinin–Sugawa dual mean value problems. We show that if a universal constant satisfying the non-Archimedean Smale mean value problem exists, it must be greater than or equal to one and if a universal constant satisfying the non-Archimedean Dubinin–Sugawa dual mean value problem exists, it must be less than or equal to one. We also derive other bounds whenever roots of unity exist in a non-Archimedean field.

Primarily human-written textHuman understanding: all partsDubinin-Sugawa dual mean value theoremNon-Archimedean field.Smale mean value theorem

Oct 22, 20267 pages

hexagon:2610.00462v1

math.CO — Combinatorics

Asymptotically optimal degree for the double Hall property

Contributed by John Tripodo

The minimum maximum degree for the double Hall property is asymptotic to sqrt(2 n). Deterministic balanced regular bipartite constructions attain this leading constant with the property on both shores and degree at most sqrt(2 n)+O(n^(21/80)) for every sufficiently large n. The construction repairs homology semibiplanes, pads by matched duplication, and completes to regularity. The all-order error term invokes the published Baker-Harman-Pintz prime-interval theorem; the formal transfer keeps that input as an explicit hypothesis.

Primarily AI-generated textHuman understanding: no parts

Oct 22, 20266 pages

hexagon:2610.00461v1

math.NT — Number Theory

An explicit elementary proof of no Landau--Siegel zeros

Contributed by Victor Z. Guo

We give an elementary proof of the absence of Landau--Siegel zeros with the explicit constant $c=1/100$ . For every primitive quadratic Dirichlet character $\chi$ of conductor $q\ge3$ , each real zero $\beta\in(0,1)$ of $L(s,\chi)$ satisfies $(1-\beta)\log q>c$ . The proof combines integer coefficient counting, polynomial congruences, a paired-tail hyperbola estimate, and a logarithmically weighted comparison of positive sums. All required constants are explicit, and one choice of parameters covers every conductor. The argument uses elementary arithmetic and real summation and integration, without separate zero computations for small conductors.

A mix of human-written and AI-generated textHuman understanding: all partsFrobenius congruencesLandau--Siegel zeroselementary methodsexplicit estimatesquadratic characters

Oct 22, 202617 pages

hexagon:2610.00460v2

math.CO — Combinatorics

Two conjectures of Wormald on Hamilton cycles in tournaments

Contributed by Deep Bhattacharjee

A tournament is a complete graph in which every edge has been given a direction, and a Hamilton cycle in it is a directed cycle through all the vertices. Let $h(n)$ be the largest number of Hamilton cycles that a tournament on $n$ vertices can have, and let $g(n)=2^{-n}(n-1)!$ be the expected number of Hamilton cycles of a random tournament on $n$ vertices. Wormald obtained lower bounds for $h(n)$ by replacing every vertex of a fixed regular tournament $T$ on $d$ vertices by a random tournament on $n$ vertices. The expected number of Hamilton cycles of the result is governed by a number $H$ , the determinant of the Hessian of a function defined implicitly by $T$ ; the bound is $\sqrt{d/H}\,g(dn)$ , or $e^{1/d}\sqrt{d/H}\,g(dn)$ when the blocks are regular. Wormald computed $H$ for a few small tournaments, found that $d/H$ was the square of a rational number every time, and could not explain it. We prove that $H=d\det(d^2I+S^2)/\det(d^2I-S^2)$ , where $S$ is the skew adjacency matrix of $T$ , so that $H\neq0$ and $\sqrt{d/H}=\det(dI-S)/\det(dI-iS)$ is a ratio of two positive integers. This is his Conjecture~6. For the cyclic tournament on $d$ vertices the formula becomes $\sqrt{d/H}=\bigl((d+1)^d+(d-1)^d\bigr)/\bigl((d+i)^d+(d-i)^d\bigr)$ , which tends to $\cosh1/\cos1=2.8559578925\ldots$ as $d$ grows. This is his Conjecture~2, and it identifies the constant $2.855958$ that he estimated by fitting polynomials; it also explains why the terms in $1/d$ and $1/d^3$ of his fits were nearly zero. We add bounds for $\sqrt{d/H}$ over all regular tournaments on $d$ vertices, with equality in the lower bound exactly for doubly regular tournaments. Every statement is proved by hand.

A mix of human-written and AI-generated textHuman understanding: some partsHamilton cycleHessian determinantformal verificationrandom blow-upskew-symmetric matrixtournament

Oct 22, 202612 pages

hexagon:2610.00459v1

math.CO — Combinatorics

All $\tau_k$ -maximal graphs have the same size

Contributed by Deep Bhattacharjee

A graph is $\tau_k$ -maximal if none of its subgraphs has $k+1$ edge-disjoint spanning trees, but adding any missing edge creates such a subgraph. Wang and Tian conjectured that every $\tau_k$ -maximal graph on $n\ge 2k+2$ vertices has exactly $(k+1)(n-1)-1$ edges. We prove this conjecture.

A mix of human-written and AI-generated textHuman understanding: some partscount matroidedge-disjoint spanning treessaturated graphssparse graphs

Oct 22, 20265 pages

hexagon:2610.00458v1

math.NT — Number Theory

A 15/16 zero-free half-plane for angular Hecke L-functions over the Eisenstein field

Contributed by Troy Winarski

We describe the argument recorded in a frozen Lean development for nonvanishing of the unit-compatible angular Hecke L-functions over $\mathbb Q(\sqrt{-3})$ in the strict half-plane $\Re(s)>15/16$ , apart from the genuine principal pole. The family includes imprimitive finite parts, every integer angular type allowed by the unit relation, and real unitary norm twists. The proof route passes through a mean-square estimate for the actual sextic rows, sixth-power averaging, a smoothed Möbius estimate with a totient factor, and Mellin continuation. We reconstruct the angular reflection, quadratic energy estimate, finite descent, and the calculation producing the $15/16$ exponent, while identifying the technical source lemmas inherited from OAI-003 that remain to be independently reconstructed. The theorem is formally verified in Lean and the frozen source/evidence package has undergone statement comparison, axiom audit, recursive kernel replay, quotient postchecking, and separate AI-assisted review. This manuscript is an expository reconstruction rather than a claim of independent human certification. We describe the formally verified proof as a xenoproof, and the theorem provisionally as a xenotheorem, to distinguish formal mathematical correctness from human mathematical understanding.

Primarily AI-generated textHuman understanding: some partsEisenstein integersHecke L-functionsLeananalytic number theoryangular Hecke charactersformal verificationxenoproofxenotheoremzero-free regions

Oct 22, 202636 pages

hexagon:2610.00457v1

math.CO — Combinatorics

Projective-plane determinant bounds for sparse binary matrices

Contributed by John Tripodo

For every integer density r from 3 through 14, a binary square matrix with e ones satisfies an explicit affine-exponential determinant bound. Under e <= r n this gives the projective-plane determinant rate, with equality precisely for direct sums of projective-plane incidence matrices of order r-1. Individual row and column degrees are unrestricted. The written proof uses an exact finite scalar certificate with rigorous unbounded-degree tails; the companion contains a pinned 98-module Lean mathematics closure.

Primarily AI-generated textHuman understanding: no partscombinatorial matrix theorydeterminantsprojective planessparse matrices

Oct 22, 20268 pages

hexagon:2610.00456v1

math.CO — Combinatorics

Reflexive Descent to Minimal Seeds in the Kreuzer--Skarke List

Contributed by Deep Bhattacharjee, Ushashi Bhattacharya

Kreuzer and Skarke built the list of reflexive polytopes by growing it from small seed polytopes. We ask how far a polytope can be from its seeds when it is shrunk through reflexive polytopes only, one at a time and without skipping any. Such a descent always exists and always ends at a seed. In four dimensions no descent takes more than 348 steps, which proves the conjecture of a uniform transition depth, and in two and three dimensions the exact maxima are five and eleven. In four dimensions two polytopes with thirty-seven lattice points need exactly twenty-two steps, so the maximum lies between twenty-two and 348.

A mix of human-written and AI-generated textHuman understanding: some partsCalabi–Yau threefoldsKreuzer–Skarke listlattice polytopesreflexive polytopes

Oct 22, 202618 pages

hexagon:2610.00455v1

math.CO — Combinatorics

The book Ramsey number $R(B_{57},B_{58})=231$

Contributed by Deep Bhattacharjee

We construct a graph on $230$ vertices with no book $B_{57}$ whose complement contains no $B_{58}$ . With the upper bound of Rousseau and Sheehan this gives $R(B_{57},B_{58})=231$ , the smallest case of Wesley's conjecture $R(B_{n-1},B_n)=4n-1$ that was open. The graph is a two-block circulant graph over $\Z_{115}$ , and the proof reduces to $57$ inequalities.

A mix of human-written and AI-generated textHuman understanding: some partsRamsey numberblock-circulant graphbook graphformal verificationlocal search

Oct 22, 202610 pages

hexagon:2610.00454v2

math.NT — Number Theory

A sixteen-prime-factor bound for Lehmer's totient problem and exact prime-completion criteria

Contributed by Deep Bhattacharjee

We study composite integers $n$ for which Euler's totient divides $n-1$ . The finite-search results give at least sixteen distinct prime factors, at least $18001$ when $(n-1)/\varphi(n)\ge3$ , and more than $10^9$ when $3\mid n$ . A solution with $k$ prime factors satisfies $n<2^{2^{k-7}}$ , and every solution exceeds $10^{35}$ . For the companion divisibility $\varphi(n)\mid n+1$ , the nine known solutions exhaust the cases with at most eight prime factors. For a fixed prime prefix $A=\prod p$ , $B=\prod(p-1)$ , the quotient is forced to be $M=\lceil A/B\rceil$ and a possible largest prime is uniquely $q=1+(A-1)/(MB-A)$ . We derive its exact divisibility, coprimality and valuation consequences, and prove that deleting the largest prime of a hypothetical counterexample cannot yield a smaller Lehmer counterexample. A two-prime factorisation identifies the additional integrality and primality conditions required by a replacement construction. A complete finite construction is proved for every prescribed factor count. Excluding its output at every factor count, or excluding the final prime for every prefix, remains unproved; the full Lehmer conjecture is not resolved.

A mix of human-written and AI-generated textHuman understanding: some partsCarmichael numbersEuler's totient functionFermat primesLehmer's totient problemlarge sieve

Oct 22, 202656 pages

hexagon:2610.00453v1

math-ph — Mathematical Physics

Faster-than-power-law decay of order parameter correlations in the two-dimensional RP2 model: a rigorous result by vibe-proving

Contributed by Masaki Oshikawa

We consider the $RP^2$ model on the two-dimensional square lattice with nearest-neighbor interactions. It is obtained by replacing the spin in the standard classical Heisenberg model with a director, which does not distinguish two opposite directions. Unlike the standard classical Heisenberg model or the O(3) nonlinear sigma model, the $RP^2$ (and its cousin $RP^3$ ) model allows point vortex defects classified by $\pi_1(RP^2)=Z_2$ in two dimensions. It is tempting to expect the $Z_2$ vortices to induce a phase transition analogous to the well-established Berezinskii-Kosterlitz-Thouless transition in the O(2) model (XY model) in two dimensions. Indeed, the possibilities of a critical phase or a critical point with power-law correlations of the order parameter in the $RP^2$ and $RP^3$ models have been discussed in the literature. However, the challenging nature of the problem has prevented a definitive conclusion on the phase diagram so far. In this paper, we report a rigorous mathematical proof by Artificial Intelligence (ChatGPT 6 Astra) that the absolute value of the two-point correlation function of order parameters at distance $r$ in the $RP^2$ and $RP^3$ models on the two-dimensional square lattice is bounded above by $C_\beta \exp\{-c_\beta[\log(2+r)]^2\}$ for any finite inverse temperature $\beta\ge0$ , with positive constants $C_\beta$ and $c_\beta$ depending only on $\beta$ . Namely, the correlation function decays faster than any power of $r$ . The proof is formalized and verified in Lean; the formalization is publicly available at \href{https://doi.org/10.5281/zenodo.23548847}{doi:10.5281/zenodo.23548847}. A heuristic argument based on a one-parameter effective theory ansatz at large distances combined with the rigorous correlation bound suggests (but does not prove) that the $Z_2$ vortices are always unbound at any nonzero temperature.

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

Oct 22, 202635 pages

hexagon:2610.00452v1

math.NT — Number Theory

Cross-Energy of Dirichlet Partial Sums. Part 9: Zeros of the prime zeta function in its half-plane of absolute convergence

Contributed by Pedro Caceres

The prime zeta function P(s) = ∑ₚ p⁻ˢ has zeros in its half-plane of absolute convergence. Belovas, Čepaitytė and Sabaliauskas proved that P(s) ≠ 0 for Re(s) > σₚ = 1.7795446535…, where σₚ solves P(σ) = 2¹⁻σ. We show that this bound is sharp: the closure of the real parts of the zeros in Re(s) > 1 is exactly [1, σₚ]. There are no zeros on Re(s) = σₚ, while zeros occur arbitrarily close to every line Re(s) = σ with 1 < σ < σₚ (Bohr’s method). By Jessen’s theorem, the density of real parts is J″(σ)/(2π), where J(σ) = −σ log 2 + 𝔼[log⁺(2^σ |Yσ|)], Yσ = ∑ₚ≥₃ p⁻σ eⁱθₚ, and the phases θₚ are independent and uniformly distributed. The function J is smooth, σₚ = inf{σ : J(σ) + σ log 2 = 0}, and the density vanishes at σₚ faster than any power. The argument principle, implemented with ball arithmetic, certifies exactly 68 zeros with 1.1 ≤ Re(s) ≤ 1.85 below height 2000 (70.0 predicted), and 12 zeros with Re(s) ≥ 1.5 below height 5000 (12.0 predicted). A heuristic large-deviation extrapolation predicts zeros with real part above 1.7 approximately once every 6 × 10⁶ units of height. In ½ < Re(s) < 1, P is single-valued on the half-plane with (½, 1] removed if and only if the Riemann hypothesis holds (a reformulation of a classical equivalence). Its zeros are canonical below height 3 × 10¹² and solve ζ(s) = exp(A(s)), A(s) = ∑ₘ≥₂ P(ms)/m, with |P(s)| < 2π. Below height 600, their counts for Re(s) ≥ 0.7 agree numerically with the random model. The zeros arise from cancellations between 2⁻ˢ and the contributions of the remaining primes, the prime-sum analogue of Euler-product interference behind the off-line zeros of the Davenport–Heilbronn function. We do not prove the Riemann hypothesis.

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

Oct 22, 202611 pages

hexagon:2610.00451v1

math.NT — Number Theory

Small algebraic integers and Frobenius prime sums

Contributed by Scott Kominers

The determinant argument in recent proofs of Landau–Siegel zero exclusion compares the size of a nonzero integer with its divisibility by inert primes. We observe that the same comparison applies to every Frobenius class in a finite Galois extension of $\mathbb{Q}$ , bounding its weighted prime count in terms of the supply of small algebraic integers. For each fixed field, the resulting height bound implies a weighted Chebotarev theorem whose only analytic inputs are the elementary Chebyshev and Mertens estimates. In the quadratic case there are only two Frobenius classes, so the lower bound obtained by complementation remains uniform in the conductor; this recovers Landau–Siegel zero exclusion.

A mix of human-written and AI-generated textHuman understanding: all partsChebotarev density theoremFrobenius congruenceGalois extensionGeometry of numbersLandau–Siegel zeroWronskianinterpolation determinant

Oct 22, 20268 pages

hexagon:2610.00450v1

math.MP — Mathematical Physics

Superfluid Stiffness and Nonclassical Rotational Inertia of the Dilute Hard-Sphere Bose Gas

Contributed by P. M. Aronow, Patrick Lopatto

We study the three-dimensional Bose gas of hard spheres of diameter $a$ at density $\rho$ and temperature $T$ , in the thermodynamic limit with $a$ , $\rho$ , and $T$ fixed. When the gas parameter $\rho a^3$ is sufficiently small and $0<T<T_*$ , we prove two equilibrium signatures of superfluidity. First, in a periodic box, the superfluid fraction, defined through the free-energy cost of twisted boundary conditions, has thermodynamic lower limit at least $c\,(\rho a^3)^{3/8}$ , uniformly in $T$ in this interval. Second, in dilations of an arbitrary nonempty bounded open container with hard walls, the nonclassical rotational inertia fraction $1-I/I_{\mathrm{cl}}$ has thermodynamic lower limit at least $c'(\rho a^3)^{3/8}$ for every fixed rotation axis, where $I$ is the moment of inertia, $I_{\mathrm{cl}}$ its classical value, and $c'>0$ depends only on the container and the axis. In both results the derivative of the free energy is taken at finite volume, before the thermodynamic limit. The temperature $T_*$ is of order $(\rho a^3)^{1/3}/\log(1/(\rho a^3))$ times the critical temperature of the ideal Bose gas. We also bound the moment generating function of the winding vector of the Feynman–Kac paths by that of a centered Gaussian, at every density and positive temperature. Combining this upper bound with the first result, we determine the order of the winding fluctuations in the regime above. Further, for $0<\varepsilon\le1$ , $\rho a^3\le(\varepsilon/100)^3$ , and $T\ge(1+\varepsilon)4\pi\rho^{2/3}$ , about $1.9(1+\varepsilon)$ times the critical temperature of the ideal Bose gas, the superfluid fraction in a periodic box tends to zero exponentially in its side length. By identities of Pollock and Ceperley and of Sindzingre, Klein, and Ceperley, we can express the two responses as multiples of the variances of the winding and of the signed projected area of the paths. We bound these variances from below with a variance inequality of Hammersley–Chapman–Robbins type, applied to random rewirings of the paths in a short imaginary-time interval along closed lattice routes. The rewirings adapt route comparisons developed in OpenAI's preprints on Bose–Einstein condensation in dilute Bose gases, and we extend the thermal estimates of the positive-temperature hard-sphere preprint in that series to higher temperatures. We average the rewirings over randomly translated closed lattice walks whose independent copies rarely approach each other. This makes the changed law close to the original law in chi-square divergence. Using conditional independence between disjoint imaginary-time intervals, we obtain bounds uniform in temperature. We also transfer the thermal estimates to arbitrary containers.

Primarily AI-generated textHuman understanding: no partsbose gassuperfluid

Oct 22, 2026233 pages

hexagon:2610.00449v1

math.NT — Number Theory

A quantitative refinement of Lu's elementary prime number theorem

Contributed by Rajat Bhattacharjee

We refine Lu's elementary method for the prime number theorem to obtain, for every fixed $K>400$ , \[ \psi(X)-X=O_K\!\left( X\exp\!\left\{-\frac{\sqrt{\log X}}{(\log\log X)^K}\right\}\right) \qquad(X\to\infty), \] where $\psi$ is Chebyshev's function. The main ingredient is a comparison estimate that remains uniform as the degree grows and the short-interval parameter shrinks. Combining this estimate with exponential weights in Balog's identity, a single finite energy step and positivity-based interpolation yields the stated bound, retaining the full degree tail and all arithmetic correction terms. The resulting remainder strengthens Lu's fixed-parameter estimates and Goz\'e's quantitative refinement within the real-variable framework; stronger remainder estimates are known by other methods.

Primarily AI-generated textHuman understanding: no parts

Oct 22, 202629 pages

hexagon:2610.00448v1

math.CO — Combinatorics

A polynomial algorithm for general d-position in trees

Contributed by Daniel Adam Pyle

A vertex set is in general $d$ -position if no three of its vertices lie on a shortest path of length at most $d$ . We give an exact algorithm for an $n$ -vertex tree in $O(n(D+2)^4)$ time and $O(n(D+2)^2)$ space, where $D=\min\{d,n-1\}$ . In particular, the problem is polynomial-time solvable when the threshold is part of the input. This answers the tree-complexity question posed by Klav\v zar, Rall, and Yero in 2021 and retained in the August 2026 version of the general position survey. The dynamic program uses two truncated distances at each rooted boundary: the nearest selected vertex and the nearest far endpoint of an ancestor-comparable selected pair. These distances characterize every forbidden triple created by joining child subtrees. The algorithm reconstructs an optimal set and extends to signed vertex weights and exact counting of optimal sets. It also computes the full cardinality generating polynomial in $O(n^2(D+2)^4)$ coefficient operations. Exhaustive comparisons with direct subset enumeration support the implementation independently of the proof.

Primarily AI-generated textHuman understanding: no parts

Oct 22, 202610 pages

hexagon:2610.00447v1

math.AG — Algebraic Geometry

Point Separation for Adjoint Line Bundles

Contributed by Brian Nugent

Let X be a smooth complex projective variety of dimension n and let L be an ample line bundle. Using the freeness and minimization inputs specified in the introduction, we prove that KX + mL separates distinct points for every integer m ≥ n + 2, the sharp bound predicted by Fujita’s very ampleness conjecture. Exponential-moment convergence produces point-centred minimizing valuations at both points in a common degree. Strict convexity then gives a divisor with discrepancy equality at one point and strict excess at the other. A rational perturbation on a resolution, followed by Kawamata–Viehweg vanishing and descent, gives the required sections

Primarily AI-generated textHuman understanding: some parts

Oct 22, 202615 pages

hexagon:2610.00446v1

math-ph — Mathematical Physics

Which String Vibrations Become Which Generations on the Dwork Quintic

Contributed by Deep Bhattacharjee, Ushashi Bhattacharya

We determine, for the heterotic string on the one-parameter family of quintic threefolds, which vibration modes become which families of charged particles. The hundred and one families are labelled by the characters of a symmetry group of order one hundred and twenty-five. Every Yukawa coupling that this symmetry allows is a power of the modulus, and every other coupling vanishes. The invariant deformation pairs each family with exactly one partner, so that fifty families are massless at the Fermat point and none anywhere else, and at the Fermat point the masses are given in closed form by the Gamma function. On the three other one-parameter families of the same type, in weighted projective spaces, the symmetry no longer tells the families apart and some families stay massless at every member. We also prove that the Geometric Selection Conjecture is equivalent to the existence of one viable vacuum fixed by mirror symmetry, and that a free quotient of a Calabi–Yau complete intersection of any of the thirteen one-parameter hypergeometric types never carries three families with the standard embedding; on seven of these types, among them the sextic and dectic pencils, no finite group acts freely at all.

A mix of human-written and AI-generated textHuman understanding: some partsYukawa couplingsfree quotientsgenerationsheterotic stringquintic threefold

Oct 22, 202631 pages

hexagon:2610.00445v2

math.CT — Category Theory

Vanishing of Higher Magnitude Homology for Finite Prefix Models

Contributed by Ryoji Hirayama

Bradley and Vigneaux define a generalized distance on a finite prefix poset using next-token probabilities. They conjecture that its higher magnitude homology vanishes. Motivated by their expression of magnitude as a weighted sum of homological Euler characteristics, we verify this conjecture for their explicit complex, in which both endpoint faces are zero. For each finite length, the homology groups in degrees two and above are zero. The degree-one homology group is free abelian, with one generator for each positive-probability edge of that length. The proof constructs a contracting homotopy for each component whose endpoints have an interior prefix. This homotopy inserts the least interior prefix. Edge probabilities zero and one are both allowed.

A mix of human-written and AI-generated textHuman understanding: all partsenriched categorieslanguage modelsmagnitude homologyprefix posets

Oct 22, 20263 pages

hexagon:2610.00443v1

math.CO — Combinatorics

Distinct partial products in prime dihedral groups

Contributed by alexandre Thiery

Let $D_{2p}$ be the dihedral group of order $2p$ , where $p$ is prime. For every fixed $0<\alpha<1$ , we prove that every subset $A\subseteq D_{2p}\setminus\{e\}$ with $C_\alpha\le |A|\le p^\alpha$ admits an ordering whose nonempty partial products are pairwise distinct. Combined with the known small- and large-set theorems, this covers every subset size for all sufficiently large primes. Pairing reflections reduces the products to partial sums with fixed parity-labelled offsets. Two passes of block rotations remove distant collisions; bounded displacement allows the local condition to exclude the remaining collisions. The existence of suitable rotations follows from Pham--Sauermann's subset-sum anticoncentration theorem, an affine-rank estimate, and a count of clustered collision witnesses.

Primarily AI-generated textHuman understanding: some parts

Oct 22, 202616 pages

hexagon:2610.00442v1

math.GT — Geometric Topology

Inscribed Square Problem and an Embedding of M¨obius Band in a Family of Solid Toruses

Contributed by Faris ELmikdad

In this paper, we relate the space of line segments induced by a Jordan curve in the plane to the space of line segments induced by the plane by embedding the M¨obius band associated with the curve into the family of toruses associated with the plane; we further demonstrate how the geometric ”square condition” of the curve translates into a topological property through this embedding.

Primarily human-written textHuman understanding: all partsM¨obius bandRotationSquare.Torus

Oct 22, 20269 pages

hexagon:2610.00441v1

math.AC — Commutative Algebra

Zero-divisor methods for absolute concentration robustness

Contributed by Michael Mislan

We study the relation between absolute concentration robustness and zero divisors in steady-state coordinate rings. Explicit positive rational, two-sided bimolecular examples disprove both necessity and sufficiency of the proposed unrestricted criterion. A separate example shows why elimination order matters without contradicting lexicographic or product-order statements. We prove completeness of a block-compatible algebraic interface under explicit basis, leading-coverage, nonvacuity and zero-divisor hypotheses. The manuscript develops complementary criteria from regular positive geometry and quantitative algebraic certificates, with examples from loaded reactors, EnvZ and product-release networks. Selected counterexamples, the conditional completeness theorem and exact certificate identities are formalized in Lean 4. Geometric and nonlinear dynamical arguments supplied conventionally are identified separately.

Primarily AI-generated textHuman understanding: some partsLean formalizationabsolute concentration robustnesselimination ordersquantitative certificatessteady-state idealszero divisors

Oct 21, 202624 pages

hexagon:2610.00440v1

math.PR — Probability

Chemical heredity without genetic templates

Contributed by Michael Mislan

We prove two quantitative forms of chemical heredity that do not rely on copying a genetic sequence. In the first model, a population of four compartments carries three prescribed catalyst types that compete for a shared local fuel. Two types are present from the start; a third is supplied for a fixed seeding period, after which all catalyst input stops. We prove that, with failure probability at most 0.004517975690072193, a single finite-stock process acquires the third type and keeps all three present in both daughters of every division, delivers three million catalytic conversion credits on a fixed per-site schedule, and rebuilds the population's copy reserves eleven times, interleaved with ten divisions. The proof controls acquisition, catalytic service, recovery and resource consumption on one stochastic history; it does not assume that a compartment stays replenished between recovery and division. In the second model, an imperfect parent copies its majority conformation into an independent racemic blank, after which both outputs undergo cooperative restoration while spontaneous errors continue. At 4,000 molecules per compartment, 100 successive copying cycles succeed with probability at least 0.999, uniformly over a box of rate parameters. Complete conventional proofs and explicit finite certificates are supplied. The central finite-stock theorem, though not the restoration result, also has a machine-checked Lean formalization. A converse for random division explains why unlinked information units and linked carriers face different loss mechanisms. These results establish acquisition and retained function under prescribed chemistry and operations; chemically generated successive innovations and open-ended evolution require additional mechanisms.

Primarily AI-generated textHuman understanding: some partschemical heredityfinite resourcesorigin of lifestochastic reaction networkstemplate-free heredity

Oct 21, 202634 pages

hexagon:2610.00439v1

math.DS — Dynamical Systems

Unistationarity of Type II_l and Type V autocatalytic cores with nonnegative degradation

Contributed by Michael Mislan

We prove that reversible Type II_l autocatalytic cores with l at least three and Type V cores admit at most one positive stationary state when every degradation rate is nonnegative. The Type II result assumes source-minimality, while the Type V statement uses its literal normal form. Strictly positive reaction rate constants are retained throughout. A reduction to the fork skeleton, source-exhaustion identities and diagonal current criteria control both positive-loss and boundary cases, including internal chains. We also derive determinant estimates and local continuation results, with small-degradation existence and stability under the stated hypotheses. The main nonnegative-degradation uniqueness theorem is registered in Lean 4; the manuscript and supporting claim map distinguish the formal interface from additional conventional results.

Primarily AI-generated textHuman understanding: some partsLean formalizationType V networksautocatalytic coresmass-action kineticsnonnegative degradationunistationarity

Oct 21, 202620 pages

hexagon:2610.00438v1

math.DS — Dynamical Systems

Optimal affinity of autocatalytic networks beyond gross stoichiometry

Contributed by Michael Mislan

An explicit reversible autocatalytic network has a unique global maximum of controlled production whose affinity exponential is strictly below its gross stoichiometric seed ratio. This refutes the proposed general lower bound. We identify a response-profile capacity that retains the missing kinetic information, analyze a literal recycling family approaching the source boundary and give a zero-diagonal counterexample. Under the stated rooted accessibility and response hypotheses, a local mass-action realization theorem makes the capacity sharp; an explicit power family supplies global maxima, while interacting cores delimit broader claims. The registered Lean project verifies the original refutation and checks the explicit witness against the literature conventions. Broader historical formal evidence and conventional arguments are distinguished in the manuscript and supporting claim map.

Primarily AI-generated textHuman understanding: some partsLean formalizationautocatalytic networkscounterexamplemass-action kineticsoptimal affinityresponse-profile capacity

Oct 21, 202625 pages

hexagon:2610.00437v1

math.DS — Dynamical Systems

Unistationarity of Type II_l autocatalytic cores

Contributed by Michael Mislan

We prove unistationarity of reversible Type II_l autocatalytic cores with l at least three under strictly positive degradation. The source-minimal wiring forces source exhaustion and unit return tails, allowing comparison of positive stationary states through current ratios, chain reduction and a cyclic determinant argument. The proof includes separated stems and the exceptional coincident three-fork branch, with all internal chain species retained. Exact six- and seven-species reversible mass-action examples show that the minimality assumptions cannot be dropped: both admit two distinct positive stationary states with strictly positive rates and losses. The principal uniqueness theorem has a complete registered Lean formalization. The manuscript distinguishes that theorem, broader historical formal evidence and conventional exact calculations; the supplied claim map records the boundaries of each.

Primarily AI-generated textHuman understanding: some partsLean formalizationautocatalytic coresmass-action kineticspositive degradationsource minimalityunistationarity

Oct 21, 202614 pages

hexagon:2610.00436v1

math.RT — Representation Theory

Counterexamples to the weakly Gorenstein symmetry conjecture

Contributed by Qiyue Tang

We construct a 29-dimensional algebra which is left weakly Gorenstein and is not right weakly Gorenstein. We also construct a 145-dimensional gendo-symmetric algebra with the same property. Both constructions work over every field containing an element of infinite multiplicative order. These examples disprove the weakly Gorenstein symmetry conjecture.

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

Oct 21, 202613 pages

hexagon:2610.00435v1

math.DG — Differential Geometry

Area Nondecreasing Drawstrings and Counterexamples to Euclidean Rigidity and Zero Mass Stability

Contributed by Jiangcheng You, Heng Zhang

Building on the drawstring construction of Kazaras--Xu, we develop a localized refinement that contracts prescribed curves while preserving positive scalar curvature and without decreasing two-dimensional area. As two applications, we disprove part~(b) of Gromov's Euclidean $C^0$ rigidity conjecture and Sormani's zero mass stability conjecture.

Primarily human-written textHuman understanding: all partsADM massAF manifoldsDrawstring constructionsGromov's Euclidean C^0 rigidity conjectureIntrinsic flat convergencePositive mass theoremScalar curvature rigidityZero mass stability

Oct 21, 202625 pages

hexagon:2610.00434v1

math.CO — Combinatorics

Counterexamples to the Two-Bases Conjecture for the Tutte Polynomial

Contributed by Jungang Chen

We disprove the two-bases conjecture of Chávez-Lomelí, Merino, Noble, and Ramírez-Ibañez (Conjecture 5.8 in *Some Inequalities for the Tutte Polynomial*), which asserts that $\max\{T_M(2,0),T_M(0,2)\}\ge T_M(1,1)$ whenever a matroid contains two disjoint bases or its ground set is the union of two bases. We construct a connected, self-dual, rationally representable matroid $M$ of rank $32$ on $64$ elements whose ground set is partitioned into two bases, but for which \[ T_M(2,0)=T_M(0,2)=263656235940864 <275621102802945=T_M(1,1). \] The construction extends to an explicit infinite family $M_m$ of rank $2m$ on $4m$ elements such that \[ \frac{\max\{T_{M_m}(2,0),T_{M_m}(0,2)\}}{T_{M_m}(1,1)}\longrightarrow0. \] Consequently, the corresponding additive and multiplicative Merino–Welsh inequalities also fail under the two-bases hypothesis. We additionally construct a simple and cosimple matroid on $60$ elements whose ground set is partitioned into two bases and for which $T_M(2,0)<T_M(1,1)$ . This disproves the simple-matroid, two-bases-covering formulation of Conjecture 4.2 of Merino, Ibañez, and Rodríguez; its dual disproves the corresponding cosimple-matroid, two-disjoint-bases formulation. The constructions and inequalities are established by explicit rank functions, exact Tutte-polynomial evaluations, and elementary coefficient estimates.

Primarily AI-generated textHuman understanding: some parts

Oct 21, 20266 pages

hexagon:2610.00433v1

math.CO — Combinatorics

Highly connected counterexamples to the permutation Tutte inequality

Contributed by Jungang Chen

We disprove Conjecture 7.3 of Csikvári, Around the Merino–Welsh conjecture: improving Jackson's inequality, which asserts that $\widetilde T_H(2,0)\widetilde T_H(0,2)\ge1$ for every bipartite graph $H$ of minimum degree at least two. For each integer $k\ge2$ , we construct an explicit sequence of finite simple balanced bipartite graphs whose minimum degree and vertex connectivity are both exactly $k$ . The graphs admit automorphisms interchanging their two vertex classes, and both $\widetilde T_H(2,0)$ and $\widetilde T_H(0,2)$ tend to zero along the sequence. Consequently, no fixed minimum-degree or connectivity threshold ensures the inequality, and even the corresponding maximum and additive inequalities fail in these classes. The construction consists of a complete bipartite core with additional vertices of degree $k$ attached to disjoint $k$ -element subsets on each side. A one-variable integral bound gives explicit exponential decay for each fixed $k$ and a sufficient size threshold. We also give a 1000-vertex, 2-connected example whose two axis values are each less than $7/10$ , certified by exact rational arithmetic. These are statements about the permutation Tutte polynomial; a failure of a local permutation inequality alone does not give a counterexample to a graph or matroid Merino–Welsh inequality.

Primarily AI-generated textHuman understanding: some parts

Oct 21, 20267 pages

hexagon:2610.00431v1

math.CO — Combinatorics

Counterexamples to Three Conjectures of Ok on Tutte-Polynomial Deletion Ratios

Contributed by Jungang Chen

We disprove Conjectures 7.2.1, 7.2.2, and 7.2.3 from Seongmin Ok's thesis *Aspects of the Tutte Polynomial*. Conjecture 7.2.1 asserts that deleting a non-bridge edge from a connected loopless graph cannot decrease the ratio of acyclic orientations to spanning trees; Conjecture 7.2.2 strengthens this to monotonicity of T_{G-e}(x,2-x)/T_G(x,2-x) on 0 ≤ x ≤ 2. Both fail for an explicit simple planar, 2-connected, chordal 2-tree on 16 vertices, for which deleting one edge multiplies the first ratio by 53/54. A two-parameter construction gives an exact criterion for infinitely many counterexamples. Conjecture 7.2.3, a differential inequality for Tutte-polynomial ratios of matroid minors, fails already for the cycle matroid of the five-edge diamond graph: its asserted nonnegative expression equals -4 at (0,2) and -25/16 at (1/4,7/4). We further construct counterexamples of arbitrarily high vertex connectivity and minimum degree, even when both graphs are chordal, and prove that the sharp supremum of the reverse ratio under one deletion is 2, already within simple planar series-parallel graphs. These counterexamples do not disprove the Merino–Welsh inequalities.

Primarily AI-generated textHuman understanding: some parts

Oct 21, 20266 pages

hexagon:2610.00430v1

math.CO — Combinatorics

Highly Connected Counterexamples to the Multiplicative Merino–Welsh Inequality

Contributed by Jungang Chen

For every integer $k\ge 3$ , we construct $k$ -connected matroids representable over every infinite field, with girth $k$ , arbitrarily large cogirth, and Merino–Welsh quotient \[ \Phi(M)=\frac{T_M(1,1)^2}{T_M(2,0)T_M(0,2)} \] tending to infinity. In particular, this disproves **Conjecture 7.2 of Csikvári (2026)**, which asserts the multiplicative Merino–Welsh inequality for simple and cosimple matroids, and therefore also the stronger **Conjecture 7.3** therein, as observed by its author. Our examples are rank truncations of direct sums of $k$ -element circuits. We give a 3-connected counterexample on 1011 elements of rank 433, certified by exact integer arithmetic, and a 6048-element counterexample admitting a proof without computation. Taking the direct sum of a counterexample with its dual also gives a negative answer to **Problem 4.3 of Beke, Csáji, Csikvári, and Pituk (2024)**: the original maximum and additive Merino–Welsh inequalities can fail for simple and cosimple matroids. The same construction disproves **Conjecture 2 of Knauer, Martínez-Sandoval, and Merino (2026)**: for every fixed $g\ge 3$ , $\Phi$ is unbounded even among matroids having both girth and cogirth exactly $g$ . The dual direct-sum examples need not be connected; these constructions do not disprove the graphical or binary versions.

Primarily AI-generated textHuman understanding: some parts

Oct 21, 20267 pages

hexagon:2610.00428v2

math.NT — Number Theory

Every algebraic number is a critical value of an integer polynomial

Contributed by RJ Acuna

We arrived at an independent proof of Petrov's theorem. For every irreducible $f \in \mathbb{Z}[x]$ of positive degree there are $g \in \mathbb{Z}[x]$ and an irreducible $F \in \mathbb{Z}[x]$ of positive degree with $F \mid g'$ and $F^{2} \mid f \circ g$ . Equivalently, every algebraic number is a critical value of an integer polynomial, so the set of critical values of $\mathbb{Z}[x]$ is exactly $\overline{\mathbb{Q}}$ . The construction is explicit. We classify $g \in \mathbb{Z}[x]$ with critical value $\alpha$ , and find a lower bound for $\deg g$ . The proof was assisted by AI, and has been formalized in Lean.

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

Oct 21, 20264 pages

hexagon:2610.00427v1

math.CO — Combinatorics

Total mutual visibility in Fibonacci and Lucas cubes

Contributed by Daniel Adam Pyle

A set of vertices is total mutual-visible if every pair of graph vertices has a shortest path whose internal vertices avoid the set. We prove that the total mutual-visibility numbers of the n-dimensional Fibonacci and Lucas cubes both have order ρⁿ/n, where ρ > 1 satisfies ρ³ = ρ² + 1. The eligible Fibonacci words are exactly those avoiding 11 for which zero padding avoids 0000. Their induced graph decomposes into fibres of recognizing subsets of maximal independent sets of a path, and each fibre is a Cartesian product of smaller Fibonacci cubes. Total visibility becomes open packing in these fibres. An elementary coloring and counting argument proves the order of growth. Typical degree and free-dimension estimates, together with a classical binary-code asymptotic, sharpen the normalized bounds to constants 4.032058… and 6.346255…. We give the cyclic analogue, an exact component inventory, and reproducible exact values through dimension 20 for Fibonacci cubes and 16 for Lucas cubes. Neither a leading asymptotic constant nor a closed formula in every dimension is established.

Primarily AI-generated textHuman understanding: no partsFibonacci cubeLucas cubeconstrained codeopen packingrecognizing subsettotal mutual visibility

Oct 21, 202613 pages

hexagon:2610.00426v1

math.NT — Number Theory

An explicit upper-bound certificate for the essential minimum of the Zhang–Zagier height

Contributed by Ricardo Pinho

We give an exact certificate proving $C_{82}<0.2534449$ for the essential minimum of $h(\alpha)+h(1-\alpha)$ , where $h$ is the absolute logarithmic Weil height. We use Doche's limit-point method, building on Grinsztajn's factor library and, in particular, Épi's unequal-multiplicity construction. We retain Épi's denominator factors, add Doche's $P_7$ , and add 25 numerator factors found by LP-dual-weighted evaluation lattices with LLL. Joint linear-program re-optimisation is followed by exact rational rounding. Two separately written ball-arithmetic checkers certify the bound; a further adversarial computational audit accepts the certificate.

Primarily AI-generated textHuman understanding: some partsDoche limit-point methodLLLWeil heightZhang-Zagier heightcomputer-assisted proofessential minimumlinear programming

Oct 21, 20265 pages

hexagon:2610.00425v1

math.NT — Number Theory

An explicit upper-bound certificate for the Schur–Siegel–Smyth trace constant

Contributed by Ricardo Pinho

We document an explicit certificate proving $C_{86}<1.8117251$ for the smallest limiting trace-to-degree ratio of totally positive algebraic integers. The construction uses the primal problem of Orloski and Talebizadeh Sardari, with their four-polynomial set enlarged to eleven. A discretised solution is rounded to a rational step density, then normalised to a probability measure. Exact rational calculations and rigorous ball-arithmetic inequalities verify Smith's sufficient criterion for an arithmetic measure. An independently written agent checker accepts the certificate and rejects four corrupted controls. This note records the proof, data provenance, reproduction results and limitations of the previously released certificate; it makes no optimality claim.

Primarily AI-generated textHuman understanding: some partsSchur-Siegel-Smyth trace problemball arithmeticcomputer-assisted prooflinear programmingtotally positive algebraic integers

Oct 21, 20266 pages

hexagon:2610.00423v1

math.NT — Number Theory

Waring--Goldbach Problem: One square, four fifth powers and twenty-two seventh powers

Contributed by Runbo Li

We prove that for every sufficiently large odd integer $n$ , the equation $$ n = p_1^2 + p_2^5 + p_3^5 + p_4^5 + p_5^5 + p_6^7 + \cdots + p_{27}^7 $$ has a solution with all $p_j$ prime.

Primarily human-written textHuman understanding: all partsWaring--Goldbach problemcircle method

Oct 21, 20267 pages

hexagon:2610.00422v1

math.NT — Number Theory

Waring--Goldbach Problem: One square, four fifth powers and fifteen sixth powers

Contributed by Runbo Li

We prove that for every sufficiently large even integer $n$ , the equation $$ n = p_1^2 + p_2^5 + p_3^5 + p_4^5 + p_5^5 + p_6^6 + \cdots + p_{20}^6 $$ has a solution with all $p_j$ prime.

Primarily human-written textHuman understanding: all partsWaring--Goldbach problemcircle method

Oct 21, 20267 pages

hexagon:2610.00421v1

math.CO — Combinatorics

Disjoint union activation of difference distance magic labelings

Contributed by Daniel Adam Pyle

A graph is difference distance magic orientable (DDMO) if its vertices can be labeled bijectively by consecutive positive integers and its edges oriented so that the incoming and outgoing neighbor-label sums agree at every vertex. We give an affirmative answer to the disjoint-union question posed by Aceska et al. in January 2026: two copies of the non-DDMO graph K₃,₃ have a DDMO disjoint union. More precisely, for k ≥ 1 and r ≥ 0, the graph consisting of k disjoint copies of K₃,₃ and r isolated vertices is difference distance magic orientable if and only if k is even or r ≥ 1. The proof reduces fixed-label bipartite orientations to signed zero sums and applies classical Skolem and hooked Skolem pairings. A compatible fixed labeling of K₃,₃ has exactly two orientations, related by reversal. We prove that twelve vertices is the minimum order of a DDMO graph whose connected components are all non-DDMO in isolation. A separate exact computer-assisted classification shows that two disjoint copies of K₃,₃ form the unique graph attaining that minimum.

Primarily AI-generated textHuman understanding: no partsSkolem sequencesdifference distance magicdisjoint unionexact computationgraph labeling

Oct 21, 20269 pages