All works

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.

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.

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.

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.

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.

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.

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.

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.

math.NT — Number Theory

The Riemann Hypothesis on Young's Lattice

Contributed by José Damián Espinosa

This work establishes a novel structural connection between the Riemann Hypothesis and Young's Lattice through partition-indexed harmonic numbers and the Generalized Assembly Theorem. We establish a new family of equivalences for the conjecture, define the partition-based harmonic numbers $H_n^{\boldsymbol{\lambda}}$, and reformulate the Riemann Hypothesis as the Young's Lattice equivalence: $\sum_{\substack{k \in \mathbb{N} \\ \boldsymbol{\lambda} \, \vdash \, k}}\frac{H_n^{\boldsymbol{\lambda}}}{\boldsymbol{\lambda}!} \, a_k > \sum_{d \mid n} d$ for every $n \in \mathbb{N}$, where $\boldsymbol{\lambda} \vdash k$ denotes a partition of $k$, $\boldsymbol{\lambda}!$ is the partition factorial, and $a_k = \frac{1}{k} + \dots + \frac{1}{k^2}$ or $a_k=\log(k+1)$. We then prove the Generalized Assembly Theorem, which shows that the assembly sum $\mathfrak{S}_n^m = \sum_{k=1}^{n} \sum_{\boldsymbol{\lambda} \in \wp_k^{k+m-1}} H_n^{\boldsymbol{\lambda}}$ equals $n$ for every excess parameter $m-1$, providing a systematic framework to analyze divisor bounds through the branch decomposition of the partition-indexed series. The resulting structure yields an exact, non-asymptotic realization of the divisors of $n$. Finally, we propose and explore a hierarchical expansion connecting the exponential of the Euler--Mascheroni constant, $e^\gamma$, to Multiple Zeta Values under Abel summability. $e^\gamma = 1 + \sum_{a\ge2}\frac{1}{a!}\sum_{m\ge0}(-1)^m\zeta^{\mathrm{A}}(a+m)+\sum_{a,b\ge2}\frac{1}{a!\,b!}\sum_{m,n\ge0}(-1)^{m+n}\zeta^{\mathrm{A}}(a+m,b+n)+\cdots$

math.NT — Number Theory

Finite prime gaps from weighted Selberg sieves and pair penalties

Contributed by Annanay Kapila

Let $H_m=\liminf_{n\to\infty}(p_{n+m}-p_n)$. We develop a pair-penalty refinement of a multidimensional Selberg sieve, together with explicit admissible tuples and interval-arithmetic evaluations. A mixed prime–divisor asymptotic is derived from distribution on triply densely divisible moduli. Combining it with the numerical certificates gives $H_2\le19{,}844$, $H_3\le1{,}357{,}780$, $H_4\le82{,}234{,}788$, and $H_5\le4{,}830{,}355{,}416$. The proof combines truncated marginal estimates, support-loss bounds for signed trial functions, a trace-preserving auxiliary sieve for prime pairs, and exact modular certificates. Exact finite data, interval enclosures, and reproducibility code accompany the manuscript.

Author comments: Updated manuscript dated 6 November 2026, with a pair-penalty refinement and improved bounds for H_2 through H_5.

math.CT — Category Theory

The third unit in the adjoint tower of group involutions

Contributed by Zhenyu Sun

The functor $G\mapsto\operatorname{Hom}(C_2,G)$ determines an adjoint tower through successive Eilenberg--Moore comparison functors. We prove that the unit of the third adjunction need not be an isomorphism, answering a question of Bergman. For the generalized dihedral group $H=(C_4\times C_4)\rtimes C_2$, with inversion action, we compute $F_3V_3H\cong H\times C_2$, with counit the first projection. The unit at $V_3H$ is injective, with underlying domain and codomain of cardinalities $20$ and $40$. For an arbitrary group $G$, let $E(G)=F_2V_2G$ and let $K_G$ be the kernel of the second counit. We prove that $K_G$ is central and that $F_3V_3G\cong E(G)/K_G[2]$, where $K_G[2]$ consists of the elements of $K_G$ whose square is one. The third unit at $V_3G$ is an isomorphism precisely when $K_G$ has no element of order four. A central extension of $H$ of order $128$ supplies such an element and yields the stated description of $F_3V_3H$.

math.NT — Number Theory

Superlinear complexity of the $(3/2)^n$ steering word

Contributed by Ralf Stephan

Write $(3/2)^n = m_n + \varepsilon_n$ with $m_n$ the nearest integer and $\varepsilon_n\in[-\tfrac12,\tfrac12)$, and let $T=(t_n)$, $t_n=2m_{n+1}-3m_n$, be the resulting \emph{steering word}: the step-by-step record of the map $x\mapsto\tfrac32 x$ on the orbit of 1, coded by nearest-integer rounding. We prove that the subword complexity $p_T(k)$ of $T$ is superlinear, $p_T(k)/k\to\infty$. A pigeonhole argument reduces this to exponential pair repulsion of the orbit modulo one, which we establish by two applications of the $p$-adic Subspace Theorem: once in three variables, which handles all gaps between the two orbit points at once, and once in two variables (Ridout's theorem) for the single degenerate family of constant gap. The argument is completely formalized in Lean~4, together with the Subspace Theorem it uses.

math.AG — Algebraic Geometry

Testing the non-vanishing of the Ceresa cycle of $X_0(N)$

Contributed by Leonardo Finzi

For a curve $C$ and a fixed degree-one divisor $e$ of $C$, we can embed $C$ into its Jacobian. This embedding induces an algebraic $1$-cycle on the Jacobian called the Ceresa cycle, which we denote by $\mathrm{Cer}(C,e)$. Although homologically trivial, Ceresa cycles of very general curves with genus $\geq 3$ are non-trivial under algebraic equivalence. It is generally unknown whether a very general curve admits a non-trivial Ceresa cycle modulo rational equivalence. In this paper, we explore the non-triviality under rational equivalence of the Ceresa cycle corresponding to the modular curve $X_0(N)$. By extending a method of Lupoian--Rawson involving computing the order of the shadow of the Atkin--Lehner involution $w_N$, we show $\mathrm{Cer}(X_0(kN),e)$ is non-trivial when $N = 57, 58, 77, 82, 85, 88, 91, 93, 111, 112, 115, 123, 133, 135, 141, 143, 145, 147, 155, 161$ for $k \geq 1$. Moreover, we detail an algorithm for computing the shadow of the Hecke operator $T_\ell$ for odd primes $\ell$ not dividing $N$, which can be used to determine the non-triviality of the Ceresa cycle of $X_0(N)$.

math-ph — Mathematical Physics

Intrinsic Spectral Cosmology

Contributed by Rajesh Dachiraju

We propose an intrinsic spectral cosmology based on intrinsic geometric wave mechanics, spectral dynamics, localized resonances, HJ source flow, and Hamilton--Jacobi gravity for point masses. The intrinsic field is represented through its time-local spectrum $P(\mathbf k,t)$, whose second moment defines $M_2$, an intrinsic scale $R_I=M_2^{-1/2}$, and $H_I=\dot R_I/R_I=-\dot M_2/(2M_2)$. A canonical spectral Hamiltonian gives a conditional continuity equation and current $\mathbf J_k=\partial h/\partial(\nabla_k\theta)$; the actual Hamiltonian needed to determine that current is not fixed by the intrinsic space-time weak equation. We exhibit a minimal scale-drift Hamiltonian as a clearly labeled closure ansatz, not as a derived result. Gravity acts only between finite point masses through their Hamilton--Jacobi dynamics. Redshift, distance, structure formation, and the relation of localized resonances to masses remain hypotheses or open derivations. The framework makes no claim of empirical validation.

math-ph — Mathematical Physics

Beyond the Event Horizon: An Operator-First Theory of Persistent Geometry and Black Hole Dynamics

Contributed by Jeromie Beasley

A black hole carries one wall of each kind. Its horizon is a geometric fold: in the right coordinate it is smooth, and nothing breaks there. Its innermost stable circular orbit is a spectral wall, a Jordan block where steady circling turns into a plunge. This paper builds both kinds of wall exactly in a setting where every step can be proved and computed, and then compares them, structurally, with black holes. The setting is the shape space of four bodies. Its kinetic metric is the Bures metric of a real qutrit, and its one genuine wall, the coplanar configurations, is a fold at finite distance. Across it a tetrahedron meets its mirror image. The transport operator A(H) = TH + HT − 2Tr(TH)G is self-adjoint in the kinetic metric. Read in the Gram frame, its back-coupling vanishes like σ₃², and it forms a Jordan block whose square-root branch is exactly the orientation double cover. Lifted to the cover, it is analytic, and the wall point is a plain double point at the tip of a cone with slope 2|t₁−t₂|√(1−δ²). Both readings are exact. What a single eigenvector reports can fail at the wall; the eigenvalue pair and its trace-log ledger come through exact, and a loop through the mirror leaves a sign. New in edition 10 - Reflection and refraction on the projector metric. A wave meeting a step in the metric reflects through the metric along the step and the Clairaut constant C alone: Γ = (√(a₁−C) − √(a₂−C)) / (√(a₁−C) + √(a₂−C)). The transverse metric drops out at a sharp step. Total reflection begins exactly where a geodesic turns, so the optical threshold is a turning point. A change of spectral gap reflects with the peel coordinate tanh(½ ln Δ₂/Δ₁). - One Gram determinant does two jobs: it sets the wave admittance, and it caps the Berry curvature a path can feel (the angle safeguard of the intrinsic quantum geometric tensor). - Mirror and matched load. The polar stratum is a perfect mirror, |Γ| = 1. A Killing horizon, polar in the same sense, is a matched load, Γ = 0. - The photon ring, read by the same generator as the ISCO: each half orbit dims an image by e^−π = 0.0432, a value that belongs to the uncharged hole alone; charge lowers it. - The fermion record: its offset is a point on the same reflection chart, tanh(ħω/2k_BT_H). - The reflection tests were pre-registered before they ran, and each was pushed several ways: 400 random sharp steps, three smooth profiles, reciprocity, flux balance, the transverse metric in a thick layer, total reflection, and the gap form. Main results - The coplanar wall is a fold, and its distance is a Bures angle: arcsin√ε = arccos√F. - Quantum Fisher information on the shape space is eight times the shape kinetic energy wherever the rank is constant, and fidelity is blind to the mirror. - Two kinds of wall: spectral (a Jordan block in every frame) and geometric (a fold that a signed coordinate unfolds). The ISCO is spectral, at the point where gravity's effective power p = 2 + 6M/r reaches the inverse cube. The horizon is geometric. - Any distance potential is even across the wall, with stiffness 2∂V/∂w₃. A thin disk holds the shape at its own frequency ν. Gravity holds the square at ω = √(2(2√2−1)); its exact potential along the lifted square is given in closed form, and its stiffness turns over for a rhombus thinner than diagonal ratio k* = 0.144557. - Structural comparisons: Petrov types through Majorana stars (Kerr is type D), merging ringdown overtones, the horizon's surface resistance as a matched load (376.73 Ω), Bell violation by a Hawking pair, Ramanujan's mock theta functions, and the mirror rule for knots. The metric throughout is the intrinsic kinetic metric of the shape space, never a spacetime metric. Every statement about black holes is a structural comparison, and every claim carries its grade (derived, measured, cited, or structural) where it is made. Files - bh_paper_v10_1.pdf: the paper (56 pages). - bh_paper_v10_1.html: the web edition, with its cover; it runs no scripts. - bh_mixer_v10.html: the Event Horizon Mixer, edition 10. It runs offline in any browser and computes every curve live: the photon-ring orbits, the layered step, the reflection chart with the horizon and the fermion record, the sine guard, and the tip-and-tail matrices. Its Synchronize mode drives every eye from one wall dial, and its "Synchronize the universe" eye shows the connection energy draining as a ring of fields falls into step. - bh_v10_1_code_and_sources.zip: LaTeX source, figures, every check script with its saved output, the pre-registrations, the proof log and the ideas ledger, and a README. Each script states its kill condition before it runs (Python 3: numpy, scipy, sympy, mpmath, matplotlib). AI-assisted: the ideas and research direction are the author's; AI tools were used for literature search, computation, drafting and typesetting, and every number was checked by the scripts in the archive.

math.LO — Logic

Relative $P$-Ideals of Hypergraphs and Ultrafilter Spectra

Contributed by Zhenyu Sun

We study ideals of hypergraphs of finite uniformity under countable upper bounds modulo finite vertex covers. The canonical ideal generated by a free filter and finite weak colorings has this relative $P$-property exactly when the filter is the intersection of finitely many $P$-points. The classification is preserved by adjoining hypergraphs that are locally finite after a finite vertex deletion. The corresponding extension of a trace ideal induces capture ideals on increasing sequences. These are $\sigma$-ideals exactly for $P$-filters, and their uniformities are $\max\{\pi\chi(\mathcal{F}),\mathfrak{d}\}$. For an ultrafilter $\mathcal{U}$, the arity-two capture ideal equals the Louveau ideal exactly when $\mathcal{U}$ is a $P$-point. Successive capture ideals decrease with the arity, and equality characterizes $Q$-points. Every strict inclusion is witnessed by a closed set. Thus a non-$Q$ $P$-point yields a decreasing sequence of $\sigma$-ideals, distinguished by closed sets, with equal uniformities. We also establish invariance under locally finite witness enlargements and analyze Stone kernels and countable block products. Consistently, nested ray-positive $P$-sets have the same singleton endpoint spectrum and different ray transversal numbers.

math.NT — Number Theory

A Frobenius determinant proof of the non-existence of Landau–Siegel zeros

Contributed by Roelof van Dijk

We give a variant of the OpenAI proof [1] of the non-existence of Landau–Siegel zeros, following the simplifications of Kerr [2], Green [3] and Barreto [4]. The Frobenius congruence proves both divisibility and nonvanishing of the determinant. Reduction modulo the largest inert prime below the cutoff reduces nonvanishing to a polynomial root count, replacing resultants and Wronskians. The argument was discovered in discussions with GPT-6 Astra and Opus 5.5 and the new arguments should be attributed to these models.

math.NT — Number Theory

On the irrationality of the $3$-adic zeta value $\zeta_3(5)$

Contributed by Antti Hippeläinen

We give a proof that the $3$-adic zeta value $\zeta_3(5)$ is irrational. The proof adapts to $p=3$ a Hankel-determinant method that a recent preprint applies to $\zeta_2(7)$. A Volkenborn integral of an explicit rational function defines a linear functional with values in $\mathbb{Q}+\mathbb{Q}\,\zeta_3(5)$, and its Hankel determinants are the values at $\zeta_3(5)$ of polynomials with rational coefficients. We bound the $3$-adic valuation of these values and the size and the denominators of the coefficients. The new ingredients at $p=3$ are a triangular basis of the polynomials that are integer valued on $3^{-1}\mathbb{Z}_3^\times$, a local estimate at primes of the order of $n$ in which a pole of order $3$ coming from harmonic sums cancels exactly against a zero of order $3$ of the rational function, and a non-vanishing theorem at an auxiliary prime, in which Wilson's theorem makes one block of a reduced Gram matrix invertible. Here $n$ is the parameter of the rational function. Two constants of the estimates are bounded by finite computations, and with these bounds the $3$-adic valuation exceeds the size and the denominators together by more than $4.5$ bits per $n^2$. An appendix gives a second proof, independent of the first, which adapts to $\zeta_3(5)$ and the modular curve $X_0(3)$ a determinant argument that a preprint of Yi gives for $\zeta_7(3)$, and which rests on published theorems on overconvergent modular forms.

math.OC — Optimization and Control

Classical regularity and attainment in portfolio choice with nonhomothetic utility

Contributed by Yuanhong Wei, Zhenyu Sun

We study classical regularity and attainment in terminal-utility portfolio choice with unspanned factors, nonnegative wealth, and exposures that are square-integrable in expectation. A joint estimate for logarithmic marginal utility controls relative risk aversion and factor sensitivity. In several dimensions, a risk-price component uniformly separated from the factor-hedging span gives classical solvability and a unique strong optimal portfolio on every finite horizon. A scalar comparison equation treats correlated single-factor markets. For state-independent $C^4$ utility, the preference assumptions reduce to positive lower and upper bounds on relative risk aversion. In an orthogonal single-factor market, a uniform finite-order condition allows the risk price to vanish and change sign. Subelliptic gradient estimates and a parabolic equation in the factor variable yield a classical value; verification and strict concavity give strong existence for its continuous feedback. In both settings, the optimal dual coefficient has a deterministic bound uniform over initial states and multipliers. Attainment has a separate integrability boundary. For deterministic opportunities with a finite static optimum, the prescribed strategy class attains the value exactly when the optimal payoff lies in $L^2$ under the physical probability. An explicit nonhomothetic utility gives a horizon and wealth classification with smooth finite values and nonattainment. Further results identify wealth-dependent purchasing-power hedges and a strict increase in the dual value under factor-information restrictions.

math.MP — Mathematical Physics

Quaternionic Heisenberg-Robertson-Schrodinger and Maccone-Pati Uncertainty Principles

Contributed by K. MAHESH KRISHNA

Let $\mathcal{Q}$ be a quaternionic Hilbert space. Let $A: \mathcal{D}(A)\subseteq \mathcal{Q}\to \mathcal{Q}$ and $B:\mathcal{D}(B)\subseteq \mathcal{Q}\to \mathcal{Q}$ be possibly unbounded self-adjoint linear operators. Then for all $h \in \mathcal{D}(AB)\cap \mathcal{D}(BA)$ with $\|h\|=1$, we show that \begin{align*} (1) \quad \quad \quad \Delta _h(A)\Delta _h(B)\geq \frac{\sqrt{|\langle \{A,B\}h, h \rangle -2\langle Ah, h \rangle\langle Bh, h \rangle|^2+|\langle [A,B]h, h \rangle |^2}}{2} \end{align*} and \begin{align*} (2) \quad \quad \quad \Delta _h(A)^2+ \Delta _h(B)^2 \geq \frac{|\langle (A+B)h, y \rangle|^2+|\langle (A-B)h, y \rangle|^2}{2}, \quad \forall y \in \mathcal{Q} \text{ satisfying } \|y\|\leq1 \text{ and } \langle h, y \rangle =0, \end{align*} where $\Delta _h(A)\coloneqq \|Ah-\langle Ah, h \rangle h \|$. We call Inequality (1) as quaternionic Heisenberg-Robertson-Schrodinger uncertainty principle and Inequality (2) as quaternionic Maccone-Pati uncertainty principle.

math.KT — K-Theory and Homology

Quillen's local parameter conjecture in dimension four

Contributed by Yunpeng Li

Let $R$ be a Noetherian regular local ring of mixed characteristic and dimension four, and let $r$ be a member of a regular system of parameters. We prove that every finitely generated projective module over $R[1/r]$ is free. We also prove that $\mathrm{Sp}_4(R[1/r])$ acts transitively on unimodular columns of length four and that every determinant orientation of a rank-two projective module is induced by a basis. The argument reduces freeness to two vanishing results: the degree-three Zariski cohomology of the naive degree-four Milnor--Witt sheaf and the alternating Witt group of $R[1/r]$. We establish these results using integral degree-two algebraic $K$-theory, supported hermitian $K$-theory, and local motivic cohomology. A finite-rank symplectic comparison relates the image of the completion obstruction to Milnor--Witt multiplication and yields the required symplectic frames.

math.NT — Number Theory

Optimal bounds for the length and number of unit fraction representations

Contributed by Thomas Bloom, Wouter van Doorn

This paper gives a human-written exposition and simplification of proofs that were recently posted by OpenAI. We prove that every $a/b\in (0, 1)$ can be written as the sum of $O(\log\log b)$ many distinct unit fractions. We also prove that the number of representations of $1$ as the sum of $k$ distinct unit fractions grows doubly exponentially in $k$.

math.RT — Representation Theory

Hall polynomials for representation-finite algebras

Contributed by Qiyue Tang

We prove the Hall polynomial existence conjecture for representation-finite algebras over finite fields.

math.PR — Probability

Lower and Upper Bounds for Glauber Dynamics from Typical States of the Low-Temperature Sherrington–Kirkpatrick Model

Contributed by P. M. Aronow, Patrick Lopatto

We study the heat-bath (Glauber) dynamics of the Sherrington–Kirkpatrick model at zero external field and fixed inverse temperature $\beta>1$. We sample the disorder $J$ and an initial configuration $X_0$ from its Gibbs measure. Let $T^J(X_0)$ be the time at which the transition law, conditional on $J$ and the realized $X_0$, first comes within total-variation distance $1/4$ of equilibrium. In continuous time, with rate one per site, we prove that, for every fixed $\omega\in(0,1/4)$, with probability tending to one, \[ A_\beta n^{7/30}<\log T^J(X_0)\le n^{1-\omega}, \] where $A_\beta>0$ depends only on $\beta$. Corresponding bounds hold for the discrete-time chain. These results strengthen OpenAI’s typical-start lower and upper bounds while following their proof architectures. The lower-bound argument introduces a three-replica comparison with a cubic pressure deficit and uses Aronow and Lopatto’s quantitative mean-bias estimate. The upper-bound argument combines a pathwise information–energy cancellation, concentration estimates on a growing mesh, and optimized exponents. We also complete Lean formalizations of two known results: OpenAI’s fluctuation-scale theorem, giving variance $n^{1/3+o(1)}$ for the log partition function, and its mean-bias consequence of order at least $n^{-5/6-\varepsilon}$ for every fixed $\varepsilon>0$. The formal development proves the positive-Laplace-window input needed for the fluctuation theorem. Five specified result groups are formalized, including the continuous-time dynamics upper bound.

math.NT — Number Theory

Irrationality measures for $G\pm5\pi^2/48$ and $L(2,\chi_{-3})$

Contributed by Sean Litsey

We prove irrationality measures for several differences of values of the trigamma function $\psi'$ at rational points. Among them are \[ \mu\bigl(L(2,\chi_{-3})\bigr)\le\MuChi,\qquad \mu\Bigl(L(2,\chi_{-3})\pm\frac{\pi^2}9\Bigr)\le\MuThirdsU, \] \[ \mu\Bigl(G\pm\frac{5\pi^2}{48}\Bigr)\le\MuLamU, \] and $\mu(\psi'(\frac k5)-\pi^2/6)\le\MuFifthsU$ and $\mu(\psi'(\frac k7)-\pi^2/6)\le\MuSeventhsU$ for $k$ prime to the denominator. For $L(2,\chi_{-3})$ and $L(2,\chi_{-3})\pm\pi^2/9$ the best bound known before was $24781$, due to Calegari, Dimitrov and Tang. For $G\pm5\pi^2/48$, $\psi'(\frac k5)-\pi^2/6$ and $\psi'(\frac k7)-\pi^2/6$ we know of no earlier proof of irrationality. The approximations are determinants of rectangular L\"owner matrices built from the digamma function at integer nodes. We bound their denominators prime by prime, using the corank of the matrix modulo $p$ and cancellations at second and third order in its Smith normal form. Nonvanishing at rational points follows from Hermite interpolation in characteristic zero. The size of the determinants is bounded by Andr\'eief's identity and an energy estimate. The proof is formalized in Lean~4.

math.FA — Functional Analysis

Approximations in lp and Morrey sequence spaces.

Contributed by Nickolai Kudryavtsev

In this paper we introduce positive order differences for sequences and study approximations in lp and Morrey sequence spaces.

math.CV — Complex Variables

A covering-strip and moment certificate for Korenblum's constant

Contributed by Ricardo Pinho

We give an explicit certificate for c_2>=54/125=0.432, where c_2 is Korenblum's constant for the Bergman space. This is an unrefereed preprint. The construction combines an elementary two-zero Schwarz estimate on the covering strip of an annulus, the classical hyperbolic Jung inequality, and Wikström's moment-duality criterion. A rational twelve-atom measure with a Lebesgue tail satisfies the required inequalities for every moment index. Two independently implemented exact checkers, one using Arb and one using only integer interval arithmetic, verify the quantitative hypotheses. The best public certified lower bound located in the accompanying priority audit is Wikström's 0.4263. Neither optimality nor a new version of Jung's theorem is claimed.

math.NT — Number Theory

$\log_p 2 \cdot \log_p(3+\sqrt2)$ is not a $p$-adic linear form in logarithms

Contributed by Shawn Krisman

Let $p$ be a prime, $\mathbb{C}_p$ the completion of an algebraic closure of $\mathbb{Q}_p$, $\overline{\mathbb{Q}}$ the algebraic numbers in $\mathbb{C}_p$, $L$ the $\mathbb{Q}$-vector space of $p$-adic logarithms of algebraic numbers, and $\mathcal{L}=\overline{\mathbb{Q}}+\overline{\mathbb{Q}}L$ the space of $p$-adic linear forms in logarithms of algebraic numbers with algebraic coefficients. We prove, in Lean 4 with Mathlib and from the three standard axioms only, that $\log_p 2\cdot\log_p(3+\sqrt2)\notin\mathcal{L}$ for every prime $p\equiv\pm3 \pmod 8$, together with a family of similar statements for real quadratic fields, and the general principle behind them: if $\sigma$ is a continuous automorphism of $\mathbb{C}_p$ fixing a nonzero $t\in L$ and moving $u\in L$ to a conjugate $\sigma u\neq\pm u$ of finite orbit, then $tu\notin\mathcal{L}$. This is G. Diaz's use of complex conjugation with an element of $\mathrm{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$ in its place; the complex analogue $\log 2\cdot\log(3+\sqrt2)\notin\mathcal{L}$ is open. The deep input is Roy's form of Waldschmidt's linear-subgroup theorem for $\mathbb{G}_a^{d_0}\times\mathbb{G}_m^{d_1}$ over $\mathbb{C}_p$, with a derivative subspace, which we also prove in Lean, from Philippon's multiplicity estimate; the strong six exponentials theorem and the inhomogeneous Baker--Brumer theorem in two logarithms over $\mathbb{C}_p$ follow. The $p$-adic linear-subgroup theorem has been used since Laurent and Roy, but we were unable to locate a proof written for $\mathbb{C}_p$ of the version with derivatives. The Lean development is the proof of record; the paper explains the statements, the route, and how to check the certificate. The code is available at https://github.com/telaviv/p-adic-log-products.

cs.DM — Discrete Mathematics

An explicit counterexample to the six-neighbor square tiling conjecture

Contributed by Chunlei Liu, Aijin Lin

Luo, Sun and Wu conjectured that in a locally finite square tiling of the complex plane in which each square intersects exactly six others, all squares have the same size. We construct a counterexample by surrounding a ten-square core with alternating rectangular shells, each formed by six squares. The only point-only contacts occur at the three four-square points in the core; all subsequent contacts have positive length. A two-state induction shows that every new square has four contacts when inserted and acquires two more at the next shell. The nested rectangles exhaust the plane, so the construction is locally finite and gives a counterexample on the whole plane.

math.AG — Algebraic Geometry

Algebraicity of the $\mathrm{G}_2$-invariant tensors on Yun's exceptional motives

Contributed by Benjamin Severn

Zhiwei Yun constructed, from a rigid Hecke eigensheaf for $\mathrm{G}_2$ on $\mathbb{P}^1$ minus three points, a family of rank-seven motives whose motivic Galois groups are dense in $\mathrm{G}_2$. We show that on this family the invariant cubic form is the class of an algebraic cycle: at every Hodge-generic parameter, every rational Hodge class on every tensor construction of the rank-seven piece is algebraic, and at every parameter the transported $\mathrm{G}_2$-invariant tensors are algebraic. The cycles are top components of Satake convolution fibres transported by Hecke correspondences. The projector that cuts out the rank-seven piece is made algebraic by a canonical weighted resolution of Yun's boundary divisor, built on the moduli stack of bundles from the stabiliser tori of its points, together with a characteristic-zero census of those points and a cycle-class comparison on varieties with quotient singularities. Yun announced that the cubic is motivated in Andr\'e's sense; the step from motivated to algebraic is new. This version, with its computational certificates and review bundle, is archived on Zenodo: https://doi.org/10.5281/zenodo.23478625

math.CO — Combinatorics

An intrinsic interval-deletion characterization of digraph support families

Contributed by Michael Mislan

For a finite directed graph with loops allowed, its supported subsets are those in which every vertex has an in-neighbour within the subset. We characterize exactly which set families arise this way on their original ground set. Starting from the power set, a legal step chooses an inclusion-maximal remaining nonmember, selects a previously unused root absent from every target-family member below it, and deletes the rooted interval. A family is a digraph support family if and only if some legal deletion sequence reaches it. The maximal residual set may follow any fixed valid policy; only roots require branching. Successful traces have at most one step per vertex and reconstruct a realizing digraph. Consequences include exact root sets and trace length, a finite recognizer, conventional NP membership for explicit lists, projection, deletion and product closure, and RAF realizability criteria using the published antimatroid characterization. The characterization and selected consequences and trace-checking interfaces are verified in Lean 4; conventional statements and computational examples are identified separately.

math.AG — Algebraic Geometry

Weil classes on every abelian sixfold of Weil type for $\mathbb{Q}(\sqrt{-3})$

Contributed by Benjamin Severn

We prove that the Weil classes of every polarized complex abelian sixfold of Weil type for $K=\mathbb{Q}(\sqrt{-3})$, of every Hermitian discriminant, are algebraic. The same holds for every abelian eightfold of Weil type for $K$ of signature $(4,4)$ with split discriminant, and every power of such a sixfold whose special Mumford--Tate group is $SU_H$ satisfies the Hodge conjecture. The proof applies the transfer theorem of the companion preprint to the Shephard--Todd group $G_{34}$ with twenty reflections as local monodromies, and passes from eightfolds to sixfolds by a compensation argument. Two general tools are proved on the way. If every local monodromy of a transfer datum is conjugate to its inverse, the lattice condition of the transfer theorem follows from Brauer's induction theorem. Dominance of the period map follows from exact data at a maximal degeneration, by the nilpotent orbit theorem. We also give an explicit higher-dimensional form of Schoen's compensation argument, which shows that one decided cell of $2n$-folds, $n\ge4$, decides every cell of sixfolds over the same imaginary quadratic field. No outside unrefereed manuscript is used. This version, with its computational certificates, is archived on Zenodo: https://doi.org/10.5281/zenodo.23422649

math.AG — Algebraic Geometry

A transfer theorem for Weil classes on Prym--Tyurin factors of Galois covers

Contributed by Benjamin Severn

Let $K$ be an imaginary quadratic field, $G$ a finite group, $\sigma$ an irreducible representation of $G$ with character field $K$, and $S$ a connected component of a Hurwitz space of $G$-covers of $\mathbb{P}^1$ with fixed local monodromy. When the $\sigma$-isotypic factors of the Jacobians are abelian $2n$-folds of Weil type for $K$, we prove that the Weil classes of every polarized abelian $2n$-fold of Weil type for $K$ with Hermitian discriminant $\mathfrak{d}$ are algebraic, provided three conditions hold: a lattice condition (L) on the characters of a group of multiplicative type attached to the cover, dominance (D) of the period map of $S$, and the equality of the Hermitian discriminant of the family over $S$ with $\mathfrak{d}$. None of the conditions refers to a particular group. The Hodge conjecture for every power of a member $A$ follows when the special Mumford--Tate group of $A$ itself is $SU_H$; a statement about the generic member of the family does not suffice, and members with smaller Hodge group are not treated here. Positive instances of the conditions are verified by finite input certificates, so a new example needs new certificates, whose validation may need its own arguments, and no new transfer proof. The theorem recovers the case $G=G_{32}$, $K=\mathbb{Q}(\sqrt{-3})$, $\mathfrak{d}=-2$ of the author's earlier preprint (hexagon:2610.00246), with every step of that proof an instance of a general lemma. A second source of cycles, balanced combinations of several characters of one subgroup, is available conditionally on a theorem of an unrefereed manuscript on abelian covers. With it the datum $(\mathbb{Q}(i),[-3])$ carried by $W(E_8)\cdot\mu_4$ satisfies every hypothesis, so, conditionally on that theorem, the Weil classes of every abelian sixfold of Weil type for $\mathbb{Q}(i)$ with discriminant $-3$ are algebraic. For this datum the Schoen-only condition (L) fails. A companion preprint applies the transfer theorem, with new tools, to prove that the Weil classes of every abelian sixfold of Weil type for $\mathbb{Q}(\sqrt{-3})$, of every Hermitian discriminant, are algebraic. We also state what the theorem cannot reach. This version, with its computational certificates, is archived on Zenodo: https://doi.org/10.5281/zenodo.23422651

math.NT — Number Theory

Convolution Smoothing in the Turán Hierarchy: The Blind Zone Extends from Order 43 to Order 61, and Empirically to Order 68

Contributed by Changxia Lin

Let Xi(t)=xi(1/2+it) be the Riemann xi-function on the critical line, Xi(t)=sum_{k>=0}(-1)^k c_k t^{2k} its even Taylor expansion, and D(n,r)=det[c_{n+i-j}]_{i,j=0}^{r-1} the associated Toeplitz minors. Writing G(z)=sum_{k>=0}c_k z^k and beta_j=-1/z_j for the zeros of G, the dual Jacobi-Trudi identity gives D(n,r)=c_0^r s_{(r^n)}(beta), the Schur function of the rectangular partition (r^n) in the alphabet beta. Positivity of D(n,r) for all n>=0 and all r>=1 is equivalent to the Riemann hypothesis (RH) by the Schoenberg-Edrei characterisation of Polya frequency sequences; for each fixed r, the inequalities D(n,r)>=0 are a necessary condition, and for r=2 they hold unconditionally. A previous paper of the author proved that every level r<=43 is blind: the minors D(n,r) stay positive even in the presence of non-real zeros. The mechanism there was the two-variable Chebyshev criterion U_k(cos theta)>0 iff (k+1)theta<pi, together with the uniform bound theta<=theta_max=2 arctan(1/(2 gamma_1)) =0.07071826... The present paper shows that the bound 43 is not the truth but an artefact of ignoring two structural features of the true alphabet. First, the evenness Xi(-t)=Xi(t) forces every zero of G to occur with multiplicity two in beta, so the relevant constituent is not U_r but the self-convolution W_r = sum_{i=0}^{r} U_i(cos theta) U_{r-i}(cos theta), with generating function sum_{r>=0} W_r z^r = 1/(1-2z cos theta + z^2)^2. We prove the identity W_r = (1/2) U'_{r+1}(cos theta) and the pole-free sign criterion of Lemma (sign), and we show that W_r>0 for all r<=61 while W_62<0: the convolution of the Chebyshev sequence against itself smooths away the sign change that occurs at U_44, pushing the threshold from 43 to 61. Secondly, the zeros on the critical line contribute a positive real sub-alphabet whose complete symmetric functions h_i(X)>0 convolve with the violating part, h_r(X union Y)=sum_{i+j=r} h_i(X) h_j(Y), pushing the threshold further. Main theorem (proved). For every r<=61 and every n=1, and regardless of whether RH is true, D(1,r)>0. Consequently the first 61 levels of the Turan hierarchy are blind on the shift-1 row. High-precision computation of the full rectangular Schur functions s_{(r^n)}(beta) on the true alphabet (truncated at N critical-line zeros) shows that the first discriminating order is r*=69 at shift n=1 and n=3, so that all orders r<=68 are blind; the value is stable under truncation (N=45,60,80 all give 69). A scaling law r* ~ 1.5 pi/theta is observed across two and a half decades of the violating offset. At shift n=2 no negative minor was found for any r<=400, suggesting a qualitatively stronger phenomenon. The number 61 is proved; the number 68 is numerical, and the Status section separates the two without ambiguity. No claim of a proof of RH is made.

math.AC — Commutative Algebra

A counterexample to the Eisenbud–Mazur conjecture

Contributed by Chung Pang Mok, Yiqi Xu

We construct a height-five prime ideal $P$ in the six-variable formal power series ring $R$ over $\mathbb{Q}$ such that $P^{(2)}\not\subseteq\mathfrak{m}P$, thus providing a prime-ideal counterexample to the Eisenbud--Mazur conjecture (with $\mathbb{Q}$ being the coefficient field). The ideal is the kernel of an explicit quadratic parametrization over a number field of degree $28$ over $\mathbb{Q}$. A witness has order three, with leading term the determinant of a symmetric $3\times3$ matrix. We construct the witness by eliminating critical values to all orders and then straightening the resulting formal curve. Interpolation at the $28$ nodes of an arrangement of eight lines supplies the required lifting maps. A rational linear functional on third and fourth jets certifies nonmembership in $\mathfrak{m}P$. The construction also yields an unmixed radical ideal with $28$ smooth branches over $\mathbb{C}$, and provides a counterexample to the more general form of the Eisenbud--Mazur conjecture stated by More (with $\mathbb{C}$ being the coefficient field). Finally, the prime ideal counterexample descends to the localized polynomial ring $\mathbb{Q}[x_0,\ldots,x_5]_{(x_0,\ldots,x_5)}$, where a polynomial witness with the same fourth jet exists. All finite identities used in the proof admit exact verification.

math.AG — Algebraic Geometry

Non-flat toroidal compactifications of the universal abelian variety in genus five and above

Contributed by David Holmes

We prove that two mixed Delaunay--Voronoi models of the universal abelian variety are not flat in genus five: the classical model over the second-Voronoi compactification and the model over the finite cover that records a symmetric theta divisor. Both counterexamples arise from the same monoid-theoretic obstruction: two inequivalent minimal elements in one fibre-degree slice acquire a common positive translate. The ordinary Delaunay cells and cones are certified by exact all-lattice-point computations; after this geometric input, the obstruction is verified by the integer calculations displayed below. Combining the genus-five example in the fixed-theta model with a theorem of Nakamura--Sugawara gives the sharp threshold: the morphism is flat exactly in genera at most four.

cs.AI — Artificial Intelligence

BLLM: Beyond the LLM with Deep Human-Mind-AI

Contributed by Deep Bhattacharjee, Ushashi Bhattacharya

A system that improves itself keeps every change that passes a check, so a check that makes errors lets mistakes pile up. We compute the least checking that keeps such a system sound. It is an entropy of the costs of the changes, and an endless chain of improvements stays sound at finite cost exactly when that entropy is finite. A stack of checked layers costs more than its layers would cost alone. We also compute how many guesses a proposer needs before a right one appears: human authors come close to this least number, and language models need 1.5 to 4.3 times as many. Anaxia, an engine built on these laws, rebuilt the published 2016 counterexample to Steinberg's conjecture from the rules of a colouring problem alone. The engine is a work in progress, and its rules and search are being revised and extended for the open problems it has not yet settled.

math.DG — Differential Geometry

A proof of the second pinching theorem in dimension four

Contributed by JINXI ZHANG

Let M⁴ be a closed connected manifold minimally immersed in the unit sphere S⁵(1), and let h be its second fundamental form. We prove that constant S = |h|² > 4 implies S > 8. No constancy of tr(h³) is assumed. The proof combines integral curvature identities with a joint estimate for two contractions of the symmetrized Hessian. The remaining algebraic inequality is verified in Mathematica using exact rational arithmetic. The code and data are included in the supplementary material.

math.NT — Number Theory

Note on variable role-reversals

Contributed by Runbo Li

In [1], the author uses variable role-reversals only when $\theta < \frac{45}{89} - \varepsilon$. In this note, we shall discuss how to use variable role-reversals when $\frac{45}{89} - \varepsilon \leqslant \theta < \frac{11}{20} - \varepsilon$.

math.MP — Mathematical Physics

Octonionic Heisenberg-Robertson-Schrodinger and Maccone-Pati Uncertainty Principles

Contributed by K. MAHESH KRISHNA

Let $\mathcal{O}$ be an octonionic Hilbert space. Let $A: \mathcal{D}(A)\subseteq \mathcal{O}\to \mathcal{O}$ and $B:\mathcal{D}(B)\subseteq \mathcal{O}\to \mathcal{O}$ be possibly unbounded self-adjoint linear operators. Then for all $h \in \mathcal{D}(AB)\cap \mathcal{D}(BA)$ with $\|h\|=1$, we show that \begin{align*} (1) \quad \quad \quad \Delta _h(A)\Delta _h(B)\geq \frac{\sqrt{|\langle \{A,B\}h, h \rangle -2\langle Ah, h \rangle\langle Bh, h \rangle|^2+|\langle [A,B]h, h \rangle |^2}}{2} \end{align*} and \begin{align*} (2) \quad \quad \quad \Delta _h(A)^2+ \Delta _h(B)^2 \geq \frac{|\langle (A+B)h, y \rangle|^2+|\langle (A-B)h, y \rangle|^2}{2}, \quad \forall y \in \mathcal{O} \text{ satisfying } \|y\|\leq1 \text{ and } \langle h, y \rangle =0, \end{align*} where $\Delta _h(A):= \|Ah-\langle Ah, h \rangle h \|$. We call Inequality (1) as octonionic Heisenberg-Robertson-Schrodinger uncertainty principle and Inequality (2) as octonionic Maccone-Pati uncertainty principle.

math.DG — Differential Geometry

Smooth Local Isometric and Conformal Embeddings in Optimal Dimension

Contributed by Wenchang Yang, Yi Wang

Let $s_n=n(n+1)/2$. For every $n\geq5$, the smallest Euclidean dimension in which every smooth $n$-dimensional Riemannian metric admits a local isometric embedding is $s_n$. The corresponding dimension for smooth local conformal embedding is $s_n-1$. We construct two flat reference embeddings with the same reduced principal symbol. In the conformal reference, the metric line takes the place of one normal direction. The proof uses quadratic products of characteristic waves to correct the metric and gives a sequence converging in every derivative order on a fixed neighborhood. Polynomial metric germs give both lower bounds.

math.NT — Number Theory

Ramanujan's prime counting inequality holds for all x ≥ 38,358,837,683: ranges from the zeros, a spectral barrier, and trade-offs

Contributed by Sayed Barez

Ramanujan proved that $\pi(x)^2 < \frac{ex}{\log x}\pi(x/e)$ for all sufficiently large $x$. Let $N_R$ be the least integer such that this holds for every real $x \ge N_R$. Dudek and Platt showed that the Riemann hypothesis implies $N_R = 38{,}358{,}837{,}683$. We prove that $N_R = 38{,}358{,}837{,}683$. To our knowledge, this is the first determination of $N_R$ without assuming the Riemann hypothesis. The proof uses the zero-free half-plane $\mathrm{Re}s > 7/8$ of a recent OpenAI preprint. Its main statement for $\zeta$ is formalized in Lean 4 and has been rebuilt independently; its written proof has not been peer reviewed. The deduction is our Theorem B: if $\zeta(s) \ne 0$ for $\mathrm{Re}s > 0.945092$, then $N_R = 38{,}358{,}837{,}683$. The proof of Theorem B uses only refereed inputs, together with Bhattacharjee's correction of a theorem of Büthe. Relying only on the peer-reviewed literature, and on preprints where stated, we also prove the following by rigorous computer-assisted interval arithmetic. (A) The inequality holds for $38{,}358{,}837{,}683 \le x \le e^{753.98}$. The previous range ended at $e^{103}$. (C) It holds for $x \ge e^{3035.01}$. This replaces Axler's threshold $e^{3158.442}$, whose proof applies a lemma of Dudek and Platt in a case where that lemma's proof is defective. (B′, D) Either of the following also suffices: a zero-free region $\beta \le 1 - 1/(1.7373\log|\gamma|)$ above height $3\cdot10^{12}$, or RH verified to height $4.1602\cdot10^{14}$. We tabulate the trade-off between the two. (E) Arguments that use only non-negativity, analytic continuation and the poles up to height $T$ cannot go beyond $e^{754.40}$ when $T = 3\cdot10^{12}$. So (A) is optimal among such arguments to within 0.41 in $\log x$. Building on Bhattacharjee's discovery that the constant $E_2$ in Büthe's Theorem 1 lacks a factor $\log(3c)$, we show that the affected rows of the Broadbent–Kadiri–Lumley–Ng–Wilk table inherit the error, by 1.3–2.2% after re-optimization. We also trace the consequences in the literature.

math.CO — Combinatorics

Antimatroid characterization of RAF interior operators

Contributed by Michael Mislan

Addressing the RAF-realizability question in Section 4 of Mike Steel's Interior Operators and Their Relationship to Autocatalytic Networks (arXiv:2305.16714v4, 2023), we characterize which interior operators on the subsets of a finite set E are maxRAF operators of catalytic reaction systems whose reaction set is exactly E. An interior operator is represented by its family of fixed sets, a union-closed family containing the empty set. A same-ground realization is a system with one reaction per element of E, finitely many molecule types, a food set and a catalysis relation; it realizes the empty set together with its RAFs, the nonempty reaction sets whose reactants arise from food by iterated closure under the selected reactions and whose reactions are each catalysed by a molecule of that closure. Such a family has a same-ground realization if and only if it is the intersection of an accessible union-closed family A and a predecessor-support family. Here A contains the empty set, is closed under unions, and every nonempty member of A has an element whose removal leaves a member of A; the members of A need not cover E. A set S is supported by a predecessor map P on E when every element e of S has some element of S in P(e), loops allowed. The family consisting of the empty set and all subsets of size at least three has a same-ground realization if and only if E has at most four elements. This gives the sharp threshold for the family considered in Proposition 5 and its subsequent remarks in Steel (2023). Both results are formalized in Lean 4 with Mathlib.

math.CO — Combinatorics

Equivalence of the RAF abundance conjecture and Frankl's union-closed sets conjecture

Contributed by Michael Mislan

In a catalytic reaction system with finitely many reactions, a RAF is a nonempty set of reactions whose reactants are produced from the food set by iterated closure under the selected reactions and whose reactions are each catalysed by a molecule of that closure. Steel (arXiv:2305.16714v4, Section 4) asked whether some reaction always lies in at least half of the RAFs. Call a reaction food-ready if all its reactants are food, and abundant if it lies in at least half of the sets in the family formed by the RAFs and the empty set. If some RAF U consists entirely of food-ready reactions, then some reaction of U is abundant, with no condition on the other reactions. If every reaction of the system, except at most two, is food-ready and the system has a RAF, then some reaction is abundant. Finally, the statement that every catalytic reaction system with finitely many reactions and a RAF has an abundant reaction is equivalent to Frankl's union-closed sets conjecture in the form: every finite union-closed family containing the empty set and a nonempty set has an element lying in at least half of its members.

math.NT — Number Theory

χ₂(1/3) and χ₂(1/5) are irrational

Contributed by Shawn Krisman

Let $\chi_2(z)=\sum_{n\ge0}z^{2n+1}/(2n+1)^2$ be Legendre's chi function. We prove that $\chi_2(1/3)$ and $\chi_2(1/5)$ are irrational. Equivalently, $\mathrm{Li}_2(1/q)-\mathrm{Li}_2(-1/q)\notin\mathbb{Q}$ for $q=3,5$. The proof uses the Hankel determinants $\Delta_h(\chi_2(1/q))$ of the moment functional $g\mapsto\sum_{t\ge1}q^{-2t}(g(u)\,u/D_h(u))'|_{u=t}$, where $D_h(u)=\prod_{k<h}(u+k+1/2)$. These determinants are values of polynomials $\Delta_h(X)\in\mathbb{Q}[X]$ of degree at most $h$. We bound the denominators of $\Delta_h(X)$ prime by prime: a bound adapted to residue classes for the large primes, a Lagrange-basis bound for the small ones, and separate arguments for the primes $2$, $3$ and $q$. We bound the size of $\Delta_h(\chi_2(1/q))$ by a discrete log-gas argument against an explicit dual certificate, and prove that $\Delta_h(a/b)\neq0$ for every rational $a/b$ and infinitely many $h$, through a reduction modulo an auxiliary prime $\ell=h/3$. The margins are below $-0.054$ ($q=3$) and $-0.256$ ($q=5$) per $h^2$. The whole proof, including the prime number theorem and both numerical certificates, has been formalised in Lean 4 with Mathlib and checked by the Lean kernel, using only Lean's three standard axioms.

math.AG — Algebraic Geometry

Algebraicity of Weil classes on non-split abelian sixfolds of Weil type over $\mathbb{Q}(\sqrt{-3})$ with Hermitian determinant $-2$

Contributed by Antti Hippeläinen

Let $K=\mathbb{Q}(\sqrt{-3})$. We give a proof that every Weil class is algebraic on every complex abelian sixfold of Weil type for $K$ whose Hermitian form has signature $(3,3)$ and determinant $-2$ modulo norms. These forms are not split, so neither Schoen's theorem on sixfolds nor the theorem on sixfolds claimed in a preprint of Markman, both of which concern split forms, applies. We model the sixfolds by the part of the Jacobian of a cover $C\to\mathbb{P}^1$, branched at thirteen points, on which its group, the complex reflection group $G_{32}$, acts through its reflection representation. A degeneration argument with a ball-arithmetic certificate shows that this family dominates the nine-dimensional Shimura variety. On every member, an integral relation among characters induced from linear characters of subgroups, realized by correspondences with coefficients in the group algebra, reduces the Weil classes to the determinants of eigenspaces of eleven cyclic subcovers. A theorem of Schoen makes these determinants algebraic for the three étale subcovers; for the eight ramified ones we prove it for all cyclic covers of curves of degree at least $3$ with at least $3$ branch points and fully paired branching data, through a cone over a Fermat variety and Shioda's results. Specialization and isogenies then reach every sixfold. The determinant of the Hermitian form and the relation among characters are exact computations.

math.CO — Combinatorics

Graph Puzzles IV.1-preview: Factorisations of hypohamiltonian and permutation snarks

Contributed by Nikolay Ulyanov

Every hypohamiltonian snark and every permutation snark has a unique multiset of cyclically 5-edge-connected factors of the same class under successive decompositions along cycle-separating 4-edge-cuts. We prove that hypohamiltonicity passes from a dot product to both of its snark factors. Together with the factorisation theorem for bicritical snarks, this answers a question of Goedgebeur and Zamfirescu. For permutation snarks, we extend the atom argument of Chladný and Škoviera using colourable cut multipoles and closure under their canonical completions.

math.DS — Dynamical Systems

Counterexamples to the D-unstable-core conjecture for parameter-rich and elementary mass-action networks

Contributed by Michael Mislan

Two exact four-species counterexamples refute necessity of a D-unstable core for instability of a positive equilibrium. The first has five reactions, parameter-rich kinetics, an explicit generalized-mass-action equilibrium, and a rational Jacobian with eigenvalue 1/500 + (51/500)i. The second has six reactions, at most two molecules on each side of every reaction, no catalysts, and classical mass-action kinetics with fixed integer kinetic orders. At a positive rational equilibrium its Jacobian has integer entries and an unstable nonreal eigenvalue with real part between 0 and 5, certified by an exact shifted-quartic intermediate-value argument. The networks share a reactant incidence relation and 24 nonempty child selections. Exact symmetrization, triangular-block and cubic-factorization certificates show that every child matrix is D-nonunstable in both networks. A permanent zero eigenvalue in an exceptional child distinguishes D-noninstability from strict D-stability. Both counterexample theorems have Lean 4 proofs with Mathlib. A separate asymptotic calculation explains the elementary example through circulation and a fast branching coordinate. The results establish strict linear instability, without asserting a Hopf bifurcation or periodic orbit.

math.DS — Dynamical Systems

A critical-point criterion for D-stability in every dimension

Contributed by Michael Mislan

A real square matrix A is D-stable if DA is Hurwitz for every positive diagonal matrix D. Applying the critical-point method to Johnson’s contact variety establishes that, in every dimension and for every positive choice of barrier weights, D-stability is equivalent to Hurwitz stability and absence of positive solutions of an explicit principal-minor Fritz–John system. In dimension five the system separates into Lagrange and tangency branches; both equivalences are registered in Lean. Further results treat regular-contact persistence, the boundary of the D-stable set, stability under bounded time-scale ratios and the minimal destabilizing ratio. The finite decision procedure also uses conventional finiteness and termination arguments and was executed exactly only in dimension three. For the five-dimensional Lagrange system the multihomogeneous Bézout number is 1200; 980 generic complex solutions were observed numerically. Worked examples include a Lotka–Volterra community, a published refinery-column gain matrix and an exactly recovered benchmark window. Formal statements, conventional arguments, exact certificates and numerical illustrations are distinguished throughout.

math.CO — Combinatorics

Finite Multiset Codes and Affine Lattice Restrictions

Contributed by Daniel Adam Pyle

We study constant-weight nonnegative integer vectors under half-L1 distance and compare arbitrary codes with full intersections of affine additive-subgroup cosets with the discrete simplex. A heavy-coordinate decomposition gives the exact reduction A_q(n,n-s) = q + B_q(n,s) to codes whose coordinates are bounded by s. It determines the unrestricted optimum throughout 2n > (q+1)s. For every q ≥ 3 and s ≥ 2, at n = qs−1 and D = (q−1)s−1 the unrestricted and full-coset optima are respectively q+1 and q; we classify all q² unrestricted extremizers. We retain a complete classification of both optima for three coordinates and n < 2D. In the next ternary band, 2D ≤ n < 3D, eight words are attainable precisely when 3n ≥ 8D; equality yields an infinite eight-versus-seven full-coset separation. Exact finite certificates also give a six-versus-five gap at (q,n,D) = (4,5,3) and the quinary single-deletion values A_5(9,2) = 143 and A_5(10,2) = 201. The proofs use intersection bounds and explicit integer affine combinations. The unrestricted decomposition extends known methods, and bibliographic novelty remains unresolved, particularly against a closely related 2024 paper not compared in full.