All works

math-ph — Mathematical Physics

Density functionals for the Pauli-Fierz Hamiltonian

Contributed by Mihaly A. Csirik, Andre Laestadius, Markus Penz, Erik I. Tellgren

We develop a density-functional theory for the Pauli-Fierz Hamiltonian of nonrelativistic quantum electrodynamics in the spirit of the convex-analytic formulation of Lieb. We consider finitely many electrons, subject to external scalar and vector potentials, coupled to the full continuum of massless transverse photons with an ultraviolet cutoff. The natural basic variables are the standard electron density and a combined current density, the sum of the paramagnetic current density and a field-induced current density generated by the quantized vector potential. We prove representability results for these density-current pairs and show that the mixed-state constrained-search functional attains its infimum and coincides with the Lieb functional obtained by convex conjugation. Since arbitrarily soft photons purify mixed states at a negligible energy cost without changing their density and combined current density, the pure-state Levy-Lieb functional equals the Lieb functional and is therefore convex, even though its infimum need not be attained. In addition, we establish the zero-coupling limit, which recovers the mixed-state Vignale-Rasolt functional of current-density-functional theory. Finally, we consider the long-wavelength (dipole) limit and show that the limiting functional can be written as a constrained search for the well-known length-gauge dipole Hamiltonian.

math.NT — Number Theory

Every real primitive Dirichlet L-function has de Bruijn–Newman constant at most 9/32: a kernel-checked proof in Lean 4

Contributed by Peter W. Murphy, MD

Let χ be a real primitive Dirichlet character of conductor N ≥ 3 and parity k ∈ {0,1}. Let Λ(s,χ) = (N/π)^{(s+k)/2} Γ((s+k)/2) L(s,χ) be its completed L-function, and let H^χ_t(z) = ∫_0^∞ e^{tu²} Φ_χ(u) cos(zu) du be the de Bruijn–Newman heat flow of Λ(·,χ), normalised so that H^χ_0(z) = (1/8) Λ(1/2 + iz/2, χ). We give a proof, checked by the Lean 4 kernel, that for every such χ and every t ≥ 9/32 all zeros of H^χ_t are real. Equivalently, the generalised de Bruijn–Newman constant satisfies Λ_χ ≤ 9/32. The mathematics is not new, and we say exactly whose it is. De Bruijn's strip-contraction theorem (1950) gives Λ_χ ≤ D²/2 whenever the zeros of H^χ_0 lie in |Im z| ≤ D. Freire, González-Riquelme, Muñoz, Poblete and Salinas observed (arXiv:2610.27343, Remarks 3.1 and 3.4) that the zero-free half-plane Re s > 7/8 for Dirichlet L-functions asserted by OpenAI would give D = 3/4, and hence Λ_χ ≤ 9/32. What this paper adds is a complete formal proof of that composition. For conductors up to 400, FGMPS's unconditional bounds are stronger for 136 of the 242 characters; what is new is that 9/32, for every conductor, is a kernel-checked theorem. OpenAI's half-plane enters as their own kernel-checked Lean theorem. Every other step is proved in Lean on top of Mathlib: de Bruijn's theorem for a general class of kernels, the identification of the theta integral with Mathlib's completed L-function, and Gauss's sign theorem for real primitive characters. The final theorem depends only on the axioms propext, Classical.choice and Quot.sound. In local, unsandboxed replays five checkers accept it: Lean's kernel (run twice), nanoda, and the con-leche/con-ron pair. All five reject a deliberately unsound control. A sandboxed Comparator run against a separately written challenge also accepts it. That run was made by another agent session, and the auditor reproduced its solution export byte for byte. The result rests on OpenAI's Lean proof, whose accompanying manuscript is unrefereed. It is not progress on the Generalised Riemann Hypothesis, which is the statement Λ_χ ≤ 0. A short, separable section records two companion results for ζ that use no OpenAI code, Λ ≤ 225/512 and ψ(x) = x + O_ε(x^{31/32+ε}), both from Cureton and Thorner's half-plane Re s > 31/32. Code, verification records and archive: https://github.com/DrMurphyIsIn/real-dirichlet-dbn-lean (release v1.0.0); DOI 10.5281/zenodo.23840082.

math.CO — Combinatorics

Claw-free perfect graphs are chromatic-choosable.

Contributed by Narayanan Narayanan, Suchismita Mishra, Umesh Shankar

We prove that every claw-free perfect graph $G$ satisfies $\ch(G)=\chi(G)=\omega(G)$. More strongly, if $S$ is a clique of simplicial vertices of $G$ and $k\ge\omega(G)$, every proper precoloring of $S$ extends to a proper coloring of $G$ from arbitrary lists of size $k$ on $V(G)\setminus S$. The proof has three coloring steps: a mixed-list theorem for cobipartite graphs, a precoloring-extension theorem for elementary graphs built on Galvin's kernel method and a homogeneous-pair exchange argument, and a polynomial method (Combinatorial Nullstellensatz) argument for graphs of interval type in which a Vandermonde factor is transferred across each incomplete pair. The three coloring steps are assembled along the structure theorems of Chv\'atal--Sbihi, Maffray--Reed and Chudnovsky--Seymour, using double clique cutsets for composition. As a consequence, every perfect line graph is chromatic-choosable, recovering the theorem of Peterson and Woodall. Our precoloring extension theorem also resolves a conjecture posed by Chudnovsky and Seymour.

math.NT — Number Theory

An Improved Subexponential Bound for Deterministic Integer Factorization

Contributed by Joseph Shunia

We prove that the complete prime factorization of an $n$-bit integer can be computed deterministically in time $2^{O(n/\sqrt{\log n})}$ and polynomial space. This establishes the bound conjectured by Rudich and Rousseau, sharpening their qualitative $N^{o(1)}$ result. The proof combines OpenAI's quantitative Boolean tensor-contraction theorem with the linear-size multiplication encoding of Akshima et al. The encoding permits exact counting of divisors with prescribed binary prefixes without loss in the exponent. The least prime factor is then recovered by successive prefix queries, and iteration yields the complete factorization within the same asymptotic bound.

math.PR — Probability

Upper bounds for the Chvátal–Sankoff constants $\gamma_k$, $k\ge3$

Contributed by David Niedbala Giraudin

Lueker (J. ACM 56(3), 2009) constructed a family of finite-state machines $\mathcal{M}_h$ whose associated transfer matrices yield upper bounds on the Chvátal–Sankoff constants $\gamma_k$, and proved (his Theorem 3.26) that these bounds converge to $\gamma_k$ for every alphabet size $k$. He published values only for $k=2$. For $k\ge3$ the best upper bounds we were able to find in the literature in September 2026 were still those of Dančík's 1994 thesis: subsequent work generalising Lueker has generalised his Section 2, the lower bound. We implement $\mathcal{M}_h$ for arbitrary $k$. One ingredient has no binary analogue — the generating function of a truncation class when both sides of a minimal match pair are truncated — and we derive it. We then give a reduction by relative forms under which the transition multiplicities of the transfer matrix are falling factorials in $k$; one alphabet-free construction consequently determines the exact matrix for every $k$. We obtain improved upper bounds for $k=3,\dots,15$, each supported by a Collatz–Wielandt certificate verified in exact rational arithmetic, among them $\gamma_5\le0.645033$. A note added on 31 October 2026 compares these values with those of a repository made public by L. Jack on 25 October 2026.

math.NT — Number Theory

A closed-form bound for linear forms in three $p$-adic logarithms of rational numbers

Contributed by David Niedbala Giraudin

We deduce from the $p$-adic kit for three logarithms of the companion note hexagon:2610.00706 a fully explicit bound in closed form. If $\gamma_1,\gamma_2,\gamma_3$ are multiplicatively independent rationals, congruent to $1$ modulo $p$ (modulo $4$ if $p=2$), of heights at most $a_j\log p$ with $a_j\ge1$, and if $|b_j|\le B$, then $v_p(\gamma_1^{b_1}\gamma_2^{b_2}\gamma_3^{b_3}-1)\le a_1a_2a_3\,\mathcal{B}^2\max\{27\,711,\ C_2(p)\max_j a_j\}$, where $\mathcal{B}=\max\{\log B/\log p,\,10\}$ and $C_2(p)=11\,107\,\frac{p}{p-1}\bigl(1+\frac{0.31}{\log p}\bigr)^2$, as soon as $\gamma_1^{b_1}\gamma_2^{b_2}\gamma_3^{b_3}\ne1$. By a change of basis in the multiplicative group, the result applies to arbitrary rationals prime to $p$. On the examples of that note, the bound is $42$ to $956$ times smaller than that of Yu's explicit theorem, and stays within a factor between $1.2$ and $2.5$ of the numerical optimisation of the parameters of the kit.

math.NT — Number Theory

The exponent of $p$ in $x^{p-1}-1$: Yamada's constant 283 lowered to 69

Contributed by David Niedbala Giraudin

Yamada (J. Number Theory 130, 2010) deduced from the method of interpolation determinants of Bugeaud and Laurent the bound $v_p(x^{p-1}-1)\le 283\,(p-1)\log y\,\log(xy)/(\log p)^2+4$, for multiplicatively independent integers $x,y$ coprime to $p$. We replace 283 by 69. The new ingredient is an elementary bound, by rearrangement, on the spread of the exponents that occur in the Liouville inequality: it at least halves the height term of Yamada's criterion, involves no congruence class, and allows the parameters to be re-optimised. The proof is complete up to the lemmas of Bugeaud–Laurent (zero lemma, analytic upper bound) and of Laurent–Mignotte–Nesterenko (bound for the polynomial), which are used as they stand. A self-contained verifier checks the numerical inequalities.

math.NT — Number Theory

A kit for linear forms in three $p$-adic logarithms

Contributed by David Niedbala Giraudin

We transpose to the $p$-adic case the kit of Mignotte and Voutier for linear forms in three logarithms. The new tool is a lower bound for the $p$-adic valuation of interpolation determinants with no factorial loss: it is obtained by expanding the powers $\theta^{\ell z}$ as binomial series in $\theta-1$, and then transporting an order of vanishing that we compute optimally for generic points, by means of the initial forms of the space of exponential polynomials involved. It holds for every prime $p$. Combined with a rearrangement bound for the spread of the exponents and with Gouillon's zero lemma applied within a Teichmüller class, it gives a criterion of the form of criterion (2.8) of Mignotte–Voutier, in which $\log\rho$ is replaced by $\log p$ and the height term contains no analytic contribution. For two logarithms, in $\mathbb{Q}_p$ with $p$ odd, it improves Lemma 8 of Bugeaud–Laurent. A change of basis in the multiplicative group reduces to the case of principal units: the factor $g=p-1$ leaves the volumes and moves into the heights. On examples, with exponents up to $10^{10}$ or $10^{30}$, the complete bounds (degenerate cases included) are $154$ to $667$ times smaller than those of Yu's explicit theorem for $p\in\{5,13\}$, $608$ to $1980$ times for $p\in\{7,11\}$ (where the explicit form of Yu goes through $\mathbb{Q}(i)$), and $73$ to $105$ times for $p=101$.

math.MP — Mathematical Physics

Riordan lattice paths counted by area

Contributed by Ilmar Gahramanov

We study Riordan paths enumerated by the area between the path and the axis. We show that the generating function satisfies a $q$-Riccati equation and can be written in a closed form as a ratio of basic hypergeometric series. The auxiliary series admits an algebraic representation as the motivic generating series of a symmetric two-node quiver. We obtain explicit formulas for the coefficients, the exponential asymptotics of the number of paths of given area, and a linear recurrence whose coefficients are signed counts of partitions into distinct parts. For a random path of area $n$ we obtain a joint Gaussian limit law for the numbers of returns, horizontal steps and up steps. The height is of order $\sqrt{\log n}$, and its distribution concentrates on two adjacent integers determined explicitly by $n$.

math.NT — Number Theory

An optimal multiplicity estimate for interpolation determinants in three logarithms

Contributed by David Niedbala Giraudin

In the kit of Mignotte and Voutier for linear forms in three logarithms, the order of vanishing at the origin of the interpolation determinant is bounded below by Waldschmidt's general multiplicity lemma; the authors indicate that this estimate should be improvable. We give a lower bound for this order which is optimal for generic points. The functions involved span a space whose initial forms are described by a uniqueness property for a linear differential equation with constant coefficients. As a consequence, the left-hand side of their criterion (2.8) can be taken equal to $\frac{(2K+1)(L-1)}{3}\log\rho$, with main term $\frac{2KL}{3}\log\rho$ instead of $\frac{KL}{2}\log\rho$. On the two examples of their paper, with their public code modified in one line and the conditions of the zero lemma checked, the bounds given by the criterion go from $17.94\cdot10^6$ to $4.79\cdot10^6$ and from $28.85\cdot10^6$ to $7.00\cdot10^6$. The same estimate is the multiplicity input of our kit for three $p$-adic logarithms.

cs.CC — Computational Complexity

The Esperanza Algorithm for Maximum Independent Set

Contributed by Frank Vega

The maximum independent set problem asks for the largest set of pairwise non-adjacent vertices in an undirected graph. It is NP-hard to approximate within $n^{1-\varepsilon}$ for every $\varepsilon>0$, and under the Unique Games Conjecture it is hard to approximate within $O(\Delta/\log^2\Delta)$ on graphs of maximum degree $\Delta$; the Khot--Regev form of the conjecture, for which a proof has been reported but not yet peer reviewed, also rules out a $(2-\varepsilon)$-approximation of Minimum Vertex Cover. We present version 0.1.6 of \textsc{Esperanza}, which takes the better of a dynamic minimum-degree (Caro--Wei) greedy set and the complement of the \textsc{Salvador} vertex cover, and improves it by three budgeted local searches: repair-and-grow restarts at the cover vertices and at a maximum-degree vertex, chained local restarts realising $(k,k+1)$-improvements, and an iterated $(1,2)$-swap search. We prove that it runs in $O((n+m)\,\alpha_{\mathrm A}(n))$ expected time for fixed budgets, with $\alpha_{\mathrm A}$ the inverse Ackermann function. We prove that the output is a maximal independent set, that its complementary vertex cover has at most $2\tau-r$ vertices, where $\tau$ is the minimum cover size and $r$ the number of components with an edge, and that the ratio is at most $(\Delta+2)/3$ on graphs of maximum degree $\Delta$. We prove the ratio $2\sqrt n$ on every graph with $\Delta\le\sqrt n$ and on three further classes, and state it for all graphs as a conjecture, whose proof would imply $\mathrm{P}=\mathrm{NP}$. We also state the conjecture that the complementary cover has at most $2\tau-|M|$ vertices for the NetworkX minimum maximal matching $M$; a proof would give a $3/2$-approximation for Minimum Vertex Cover and refute the Unique Games Conjecture unless $\mathrm{P}=\mathrm{NP}$. In reproducible experiments the returned set is optimal on all $30\,000$ graphs of a suite certified by an exact integer program; on $40$ Khot--Regev YES instances with $640$ to $2\,560$ vertices the conjectured bound is certified on $38$ and open on $2$; and on $31$ bounded-degree graphs the certified ratio is at most $1.0769$ on the families with a planted independent set.

math.NT — Number Theory

Robin's Inequality at Colossally Abundant Numbers and Nicolas' Inequality at Their Largest Prime Factor

Contributed by Frank Vega

Robin's criterion states that the Riemann Hypothesis holds if and only if $\sigma(n) < e^{\gamma} \cdot n \cdot \log\log n$ for every $n > 5040$, and Nicolas' criterion states that it holds if and only if $c(q) = \log(q\#/\varphi(q\#)) - \gamma - \log\log\theta(q) > 0$ for every prime $q$, where $q\#$ is the primorial of $q$ and $\theta$ is the Chebyshev function. Let $N$ be a colossally abundant number, let $\xi(N)$ be the real number associated with $N$ by Nicolas, and let $q$ be the largest prime not exceeding $\xi(N)$. Writing $\Delta(N) = \log(\sigma(N)/N) - \gamma - \log\log\log N$, we prove without any hypothesis that, whenever $\log q \geq 31$, $1.353/(\sqrt{q} \log^{2} q) \leq \Delta(N) - c(q) + 2\sqrt{2}/(\sqrt{q} \log q) \leq 3.970/(\sqrt{q} \log^{2} q)$. The range $\log q \geq 31$ contains the prime $q$ of every colossally abundant counterexample, in accordance with the author's result that the largest prime factor of the least counterexample exceeds $e^{31.018}$. Thus the functions $\Delta(N)$ and $c(q)$ differ by $2\sqrt{2}/(\sqrt{q} \log q)$ up to an explicit error of order $1/(\sqrt{q} \log^{2} q)$. This comparison is two-sided, but it is not an equivalence between Robin's inequality at $N$ and Nicolas' inequality at $q$: it does not decide Robin's inequality at $N$ when $c(q) \sqrt{q} \log q$ lies between $2\sqrt{2}(1 - 1.4035/\log q)$ and $2\sqrt{2}(1 - 0.4787/\log q)$. As consequences, every colossally abundant counterexample to Robin's inequality yields a prime $q$ with $c(q) > 2.70/(\sqrt{q} \log q)$; conversely, every prime $q$ with $\log q \geq 31$ and $c(q) > 2\sqrt{2}(1 - 0.4787/\log q)/(\sqrt{q} \log q)$ yields a colossally abundant counterexample, which gives a derivation of Robin's known theorem on colossally abundant counterexamples from Nicolas' oscillation theorem; and under the Riemann Hypothesis every colossally abundant number with $\log q \geq 31$ satisfies $\Delta(N) < -0.654/(\sqrt{q} \log q)$, in accordance with the constant $2(\sqrt{2}-1)$ of Ramanujan's bound. For superabundant counterexamples, using the exponent bound of Nazardonyavi and Yakubovich, which gives $\log n \geq \psi(p_{k})$ for the largest prime factor $p_{k}$, we prove $c(p_{k}) > 0.9991/(\sqrt{p_{k}} \log p_{k})$, a bound that applies to the least counterexample, which is superabundant, and we show that the least colossally abundant number not smaller than the least counterexample $n$ is itself a counterexample $N$ with $\log N - \log\xi(N) \leq \log n$.

Author comments: In the revised version we emphasize that Theorem 1.1 is unconditional. Nicolas’ estimates in [20] are proved under RH and, under RH, give a narrower interval than ours (Remark 3.5). Since a counterexample to Robin’s inequality exists only if RH is false, an estimate that assumes RH cannot be applied to one, whereas Theorem 1.1 applies to every colossally abundant number with log q ≥ 31. This is what Corollaries 4.2–4.4 use.

math.IT — Information Theory

The Shared Randomness Cost of Optimal One Use Loop Communication

Contributed by Daniel Adam Pyle

We determine the minimum support and Shannon entropy of a shared classical seed needed to attain optimal worst-message success in a specified one-use normalized-loop model. The loop contains a backward binary symmetric channel with crossover 0 < p < 1/2 and a fixed finite-dimensional forward quantum channel. Seed branches are mixed as unnormalized weights before one normalization for each message. For qubit output and interior forward trace-distance diameter, equality at the known unrestricted optimum forces every supported seed to activate exactly two message labels. The seed-cost problem becomes a fractional perfect matching problem: even message counts are optimally covered by disjoint pairs, while odd counts require a triangle and disjoint pairs. We give separate exact formulas at zero and unit diameter, prove a sufficient higher-dimensional extension of the interior result, and exhibit an erasure-channel counterexample to diameter-only seed costs in general dimension. Explicit near-optimal constructions show how the exact-attainment cost discontinuities can be avoided at finite tolerance. Compactness also gives a strict performance gap below the minimum seed support. A complete exact-certificate calculation determines the two-seed classical optimum for three messages at backward and forward crossover 1/10. These are resource-specific mathematical results, not a physical backward-communication protocol or an asymptotic capacity theorem.

math.IT — Information Theory

Shared Randomness and Normalization in One-Use Loop Communication

Contributed by Daniel Adam Pyle

We study a finite-message communication model defined by contracting one backward binary symmetric channel with one fixed, finite-dimensional forward quantum channel and then normalizing the resulting weights. The devices may share a finite classical seed that is independent of the message before normalization. The seed branches are mixed at the level of unnormalized weights, followed by a single normalization for each message. For M ≥ 2 messages, backward crossover probability 0 < p < 1/2, and forward channel Φ, we derive the exact worst-message success probability S_raw(Φ,p,M) = 2(1−p)/[M−(M−2)(1−2p)δ(Φ)], where δ(Φ) = max over input states ρ, σ of ½‖Φ(ρ)−Φ(σ)‖₁. The quantity δ is the established trace-distance contraction coefficient, not a new channel invariant. A cyclic construction with M seed values attains the bound. Crucially, normalization can make the posterior seed distribution message-dependent: the theorem does not describe an average of separately normalized branches or require success for every seed. We give an exact example separating these conventions. Matched depolarizing-qubit and classical binary symmetric forward channels consequently have equal optima in this resource model. The result is a restricted one-use optimization, not a physical backward-communication protocol or a general asymptotic capacity theorem.

math.NT — Number Theory

Cyclic reductions of elliptic curves at a positive proportion of primes

Contributed by Manjul Bhargava

We prove that every elliptic curve over $\Q$ whose points of order two are not all rational has cyclic reduction at a positive lower density of primes. This is the only necessary restriction: full rational two-torsion prevents cyclic reduction at every odd prime of good reduction. More precisely, a common zero-free half-plane $\Re s>1-1/N$ for Dedekind zeta functions gives a lower proportion of at least $c_E/(N\log N)$, with $c_E>0$ independent of $N$; the theorem of Cureton and Thorner supplies $N=32$. The proof uses the varying-cofactor sieve of our earlier works on primitive roots and primitive points, in conjunction with the marked Type~II estimates of~OpenAI, to construct candidate primes. We then apply a smoothed explicit formula for Dedekind zeta functions, with nonnegative weights on prime ideals, to bound the number of candidates with noncyclic reduction. This gives a common argument for CM and non-CM curves.

math.OA — Operator Algebras

Noncommutative Lagrange-Fujiwara, Carmichael-Mason and Kuniyeda Bounds for Roots of Polynomials

Contributed by K. MAHESH KRISHNA

Two centuries ago, Lagrange derived an upper bound for every root of a complex polynomial using the maximum of powers of coefficients. In 1916, Fujiwara independently found the Lagrange bound. In 1914, Carmichael and Mason derived an upper bound using the sum of squares of the absolute values of the coefficients of a polynomial. This bound was extended by Kuniyeda in 1916. We derive noncommutative versions of the Lagrange-Fujiwara, Carmichael-Mason and Kuniyeda bounds.

math.NT — Number Theory

Deterministic Polynomial-Time Primality Testing with a Single Binomial Congruence

Contributed by Joseph Shunia

We give a deterministic polynomial-time primality test whose main verification is the congruence $(1+X)^n=1+X^n$ in $C_n[X]/(X^r-b(Y))$, where $C_n=(\mathbb{Z}/n\mathbb{Z})[Y]/(\Phi_r(Y))$. Two identities in the coefficient ring normalize $b$ and ensure compatibility with the main congruence. A finite-field root count and a lattice bound show that an accepted input is a prime power; elementary preprocessing excludes nontrivial perfect powers. We construct $b$ deterministically using Jacobi sums and the uniform Dedekind-zeta zero-free theorem of Cureton and Thorner, which extends OpenAI's Hecke zero-free theorem. The total bit complexity is $L^{32+o(1)}$, where $L=\lceil\log_2 n\rceil$, and verification after constructing $b$ costs $L^{6+o(1)}$. More generally, for a fixed real algebraic $1/2\le\theta<1$, a common zero-free boundary $\Re s>\theta$ for the relevant Dedekind zeta functions gives $L^{\max\{6,(1-\theta)^{-1}\}+o(1)}$. This bound reaches $L^{6+o(1)}$ at $\theta=5/6$, and hence under GRH. We state a fixed-base modification of Agrawal's conjecture for odd integers passing the base-$2$ Fermat test, using the modulus $X^r-2$ with $r$ the least positive integer not dividing $n(n-1)$.

math.CV — Complex Variables

A contact and coefficient correction for Korenblum's constant

Contributed by Ricardo Pinho

We give an unrefereed computer-assisted proof that Korenblum's constant in unweighted Bergman space satisfies $C_{68}\ge 0.44685$. The new input retains the signed first contact harmonic in an annular coefficient estimate. An outer factor and square completion yield a correction of $19/200$ to the linear coefficient, replacing the $7/100$ correction of version 2. A rational measure verifies the resulting moment conditions. We include the analytic argument and describe the finite interval certificates and both infinite tails. The Green/flux input is V. Patankar's (PR 274); the coefficient and moment methods build on Wang, Pinho's earlier construction, and Wikström. There has been no independent human review or formalization.

math.NT — Number Theory

A trace upper-bound certificate with classical auxiliaries on an adaptive density

Contributed by Ricardo Pinho

An exactly specified rational piecewise-linear density gives $C_{86}\le1.80560940$ for the Schur--Siegel--Smyth trace constant. It uses 28 classical auxiliary polynomials, ten with nonreal roots, and 5,375 rational nodes. Their recorded list occurrences are: Flammang 2019 (all 28), Liang--Wu 2011 (26), Flammang 2009 (21), McKee 2011 (21), Tomar's public A22 pool (18), Aguirre--Peral 2007 (16), and Flammang 2024 (9). The density refines infrastructure originating in Tomar's adaptive-node certificate. We give the full deduction from finite potential and logarithmic-integral checks through Smith's arithmetic-measure criterion and converse theorem, and describe the rigorous continuous enclosures. The contribution is the explicit feasible measure and its smaller upper bound. This note is unrefereed, with no independent human review or formalization.

math.CO — Combinatorics

Derivative controls and Boolean completion in the relevant-variable recurrence

Contributed by Ricardo Pinho

We give the unrefereed bound $C_{44}<2.0417086280$ for the number of relevant variables of a Boolean function of real degree $d$, $r(f)\le C_{44}2^d$. The argument bounds the number of coordinates controlling a ternary derivative, exploits its dictator branches and Boolean completion, and propagates local-degree caps. The small-degree inputs are $R_4=22$, $R_5\le50$, and $R_6\le109$. The quartic proof uses an exact calculation on Rasoolzadeh's published cubic catalogue; the remaining finite arithmetic is rational. The lower bound remains $3/2$. There has been no independent human review or formalization.

math.CO — Combinatorics

An exact computer-assisted certificate for R(5,6)≤84

Contributed by Dr. András Pongrácz

We give an exact computer-assisted proof of R(5,6)≤84. We derive a system of linear inequalities that must be satisfied by any graph on 84 vertices containing neither a five-vertex clique nor a six-vertex independent set. We provide explicit nonnegative rational weights for which adding these inequalities yields a contradiction. Published small Ramsey bounds and complete catalogues of smaller Ramsey graphs supply the external inputs. The accompanying verification programs check the derivation of the inequalities and their weighted sum using exact arithmetic. Two supplements give a worked local calculation and a detailed mathematical specification of the complete proof.

cs.CC — Computational Complexity

The partition exponent in the graded pieces of the Claude 3-SAT algorithm is optimal for pieces of that shape

Contributed by Carlos Stein Naves de Brito

In Appendix B of "A $2^{o(n)}$ time algorithm for 3-SAT" (Claude, Anthropic, hexagon:2610.00577) the gain parameter $\rho = 1/\sqrt{2}$ comes from the Hardy--Ramanujan growth $\log p(E) \sim c\sqrt{E}$ of the partition numbers, and Appendix H observes that growth of order $E^\alpha$ would give $\rho = 2^{\alpha-1}$ by the same calculation. We point out that the rank floor stated in the same Appendix H already excludes $\alpha < 1/2$ for every graded family of the same shape: the pieces at weight $h$ have total rank at most $(4h+1)\dim V_{\le 4h}$ in one of the two directions, and the crossing they must add up to has rank $(\dim V_h)^2$, so the construction needs $\log \dim V_{\le 4h} \ge 2\log\dim V_h - O(\log h)$, that is $\alpha \ge 1/2$. Restoring the rank with a larger weight costs $\rho \ge 2^{-\alpha}$ in the weight count, so the count never goes below $\max(2^{\alpha-1}, 2^{-\alpha}) \ge 1/\sqrt{2}$, with equality only at $\alpha = 1/2$; when the identity holds on the whole weight space, as Lemma 9 of the paper gives it, the bound sharpens to $2^\alpha/(2(4^\alpha-1))$, which exceeds 1 for $\alpha < 0.357$. Theorem B of the paper is rank-tight up to a polynomial factor. The inequality is exact and we tabulate it: for the partition numbers it holds for every $h$, for partitions into squares ($\alpha = 1/3$) it fails for all $h \ge 5891$. We also record that the Heisenberg Fock spaces on a sparse set of modes, the natural candidates for slower growth, do not carry Lemma 9, and that no choice of coefficients repairs it. The floor $\rho \ge 1/2$ remains open.

math.DG — Differential Geometry

Isometric embeddings of parallelizable and stably parallelizable Riemannian manifolds

Contributed by Wenchang Yang, Yi Wang

Let $s_n=n(n+1)/2$. We prove that every smooth parallelizable Riemannian $n$-manifold, $n\ge 3$, admits a smooth isometric embedding into $\mathbb{R}^{s_n+1}$ whose image lies in an arbitrarily small ball. For closed stably parallelizable manifolds, each of whose connected components has vanishing Euler characteristic, we obtain small-image isometric embeddings into $\mathbb{R}^{s_n+5}$. The proof combines explicit partial isometries along commuting Hopf flows, an iterative scheme for exact metric correction using oscillatory perturbations constructed from the second fundamental form, and a transversality argument for metric-preserving perturbations.

math.NT — Number Theory

The Riemann hypothesis as a margin: a two-sided criterion verified to 2*10^29, the fourth-moment anatomy of Mobius cancellation, and one constant seen from every side

Contributed by Pedro Caceres

This paper collects, with uniform notation, a month of new results around one way of looking at the Riemann hypothesis (RH): as the statement that explicitly computable margins stay on one side of explicit constants for ever. Part I studies the Cesaro margin M(N) between a smooth curve and the weighted prime sum 2*sum_{q<N} Lambda(q) q^{-1/2} log(N/q): RH is equivalent to M(N) >= 0 for all large N (a theorem of Suzuki), and we verify 0 < M(N) < 0.18484 for every real 1 < N <= 2.15*10^29, with explicit unconditional bounds beyond; the margin is analyzed as a walk of the primes against a curve, as an energy, and as a one-sided mean: RH holds if and only if a weighted mean of the prime-counting error stays below -B = -(1 + gamma/2 - (1/2) log 4pi) for ever. Part II dissects the Mobius side of the same coin: an exact positive-definite variance identity for sum_{d<=D} mu(d)({x/d}-1/2), empirically Gaussian statistics, and an exact anatomy of the fourth moment — the conductor-1 orbit decouples, the cross term moves only at composite squarefree conductors, two of its four sectors are O(D log^4 D) unconditionally, and an empirical jump law, first refuted, is proved exactly in its sector. Part III shows that 2B is the Baez-Duarte constant of the Nyman-Beurling criterion and recomputes the Nyman-Beurling distances from scratch through Vasyunin's cotangent formula: d_N^2 log N reaches 0.0461 against 2B = 0.04619, with no zero of zeta used anywhere in the computation. Part IV distils the author's twenty-two-part study of the cross-energy of consecutive Dirichlet partial sums: the trichotomy that singles out the critical line as the trace of the pole, the energy law (the mean energy of the prime-counting error is >= 2B unconditionally, with equality if and only if RH holds with simple zeros), the energy slope as a truncated Weil functional exact in the primes, the functional reflection that reproduces Hardy's function from the integers, and the systematic Davenport-Heilbronn and Beurling controls. Part V states precisely what none of this can do: a no-go theorem shows that weights indistinguishable from the primes' to q^(delta-1/2) satisfy every averaged statement here and still carry an off-line zero, so the exact identity Lambda = mu * log must enter beyond its first moments. The paper opens with a list of its twenty-two original contributions and a matching statement of what is due to others. We do not prove the Riemann hypothesis, and no statement in this paper should be read as progress towards it beyond what it literally says.

math.NT — Number Theory

Uniformizability of elementary Anderson $t$-motives of dimension $2$

Contributed by David Niedbala Giraudin

Let $M(A)$, $A\in M_2(\mathbb{C}_\infty)$, be the elementary Anderson $t$-motive of dimension $2$ and rank $4$ defined by $Te=\theta e+A\tau e+\tau^2e$. Using the affine equation of Ehbauer, Grishkov and Logachev, we give an explicit sufficient condition for $M(A)$ to be uniformizable, in terms of the valuations of the coefficients of that equation. It holds as soon as no cancellation occurs among the four summands of one of these coefficients; in particular $h^1(M(A))=4$ in the simple case where no cancellation occurs in any coefficient. This proves their Conjecture 3.7: $h^1(M(A))=4$ whenever $(\mathrm{ord}\,a_{11},\mathrm{ord}\,a_{12},\mathrm{ord}\,a_{21},\mathrm{ord}\,a_{22})$ lies outside an explicit union of nine hyperplanes.

math.NT — Number Theory

Ultrametric Bourgain-Figiel-Milman, Enflo Type and Mendel-Naor Cotype Problems

Contributed by K. MAHESH KRISHNA

We ask for ultrametric version of following three: (1) Bourgain-Figiel-Milman Theorem, (2) Enflo Type, (3) Mendel-Naor Cotype.

math.PR — Probability

Curvature-Induced Killing at Stochastic Algebraic Folds

Contributed by Zhenyu Sun

We study the lifetime of a regular algebraic branch under continuous forcing and infinite-activity tangent jumps. For a parabolic fold, a first-order tangent kick decreases the discriminant by an exact quadratic amount. Symmetric stable marks of index $\alpha\in(0,2)$ therefore generate a one-sided normal jump kernel of index $q=\alpha/2$, with a coefficient determined by curvature. The normal diffusion and jump amplitudes depend on the evolving tangential state. We compute the full killed generator on powers of the discriminant and obtain occupation and terminal-jump bounds for this same joint process. Uniformly subcritical normal drift gives finite censored boundary lifetime and finite cumulative killing rate; continuous contact and jump exit both have positive probability. At and above the uniform critical threshold, a logarithmic nonlocal barrier excludes censored boundary contact, and the original branch terminates at a jump almost surely. A comparison after a stochastic time change yields an explicit stretched-exponential upper bound for survival and hence all positive lifetime moments. In the subcritical regime, an optimized upper bound for the probability of jump termination tends to zero as the curvature coefficient or the initial discriminant tends to zero. The analysis combines power estimates for the jump generator with a logarithmic barrier at the critical threshold. Numerical calculations illustrate the dependence of terminal causes on curvature.

math.GM — General Mathematics

Arithmetic Expression Spaces: Expansion and Motion, Eikonal Equation, Horocycle Pullbacks, Commutativization, Contact Structure, and Tearing

Contributed by Mingli Yuan

An arithmetic expression is a word of operations; a motion is a flow on a manifold. For expressions whose evaluation has a single internal spine, the two are one structure: affine for addition and multiplication, projective once the second operand slot is divided, contact-geometric in the order defect. A regular arithmetic expression space (AES) is an oriented Riemannian surface $(M,g)$ with a smooth assignment $a\colon M\to\mathbb R$ satisfying the eikonal equation $|\nabla a|_g^2=\mu^2+\lambda^2a^2$; the four operations then act as motions translating and scaling $a$. In the basic model $\mathfrak E_0$, the upper half-plane with the left-invariant metric of $\mathrm{Aff}^+(1,\mathbb R)$ (curvature $-\lambda^2$), addition and positive multiplication are affine translations and dilations obeying a Baumslag--Solitar law, and $a$ is a Laplace--Beltrami eigenfunction of eigenvalue $2\lambda^2$. Left-expanded words stay in the affine Borel cell fixing $\infty$; the right-expanded language, once second-slot division is admitted, generates all of $\mathrm{PGL}_2(\mathbb R)$ through the rank-one Bruhat decomposition, and pulling standard horocycles back along one operator word gives the dual reading, ripple geometry. Commutativizing the motions by total charge yields the accumulative commutative space, where the order defect of two histories is an exact weighted area, the tearing. The same defect is the constant $\mu\lambda$ in the bracket $[D_u,D_v]=\mu\lambda\partial_a$ of the horizontal fields of $\alpha=da-\mu\,du-\lambda a\,dv$, a contact form whenever $\mu\lambda\ne0$, and it is the curvature of that horizontal connection; for $\mathfrak E_0$ the basic model is proved to be the Reeb quotient of the contact model, and fixed-charge tearing and fixed-state Reeb drift are Legendre dual readings of the same curvature. The generators $\partial_a$, $a\partial_a$, and $a\log a\,\partial_a$ span an infinite-dimensional Lie algebra, so no finite-dimensional Lie group carries all three ranks, and no observable of the charge endpoint detects tearing. For every finite puncture count a conformally compactifiable punctured AES is constructed, with an explicit four-circle zero locus and exact reconstruction from three-ended pants blocks. All results are elementary, planar, or local to a three-dimensional contact model.

math.DG — Differential Geometry

Closed three-manifolds as hypersurfaces of Euclidean fourspace: compactness, the homogeneous geometries, and total absolute curvature

Contributed by Jay Thrussell

## Abstract Let $(\Sigma, g)$ be a closed Riemannian three-manifold, and let $\mathbb M = -G$ be its negative Einstein endomorphism. Where $\det\mathbb M > 0$, the companion paper [MA] reduces realizability as a hypersurface of $E^4$ to one equation, $F(g) = 0$, and studies it locally. Here we treat the global questions. - **Compactness.** A closed realization has $\det\mathbb M \geq R^{-6}$ at a point farthest from any centre, where $R$ is that distance, and so $\max\det\mathbb M \geq 125/(8\,\mathrm{diam}^6)$ by Jung's theorem. If $\det\mathbb M > 0$ everywhere, the realization is a strictly convex embedded sphere, unique up to rigid motion. So a closed connected metric with $\det\mathbb M > 0$ is realizable iff $\Sigma$ is simply connected and $F(g) = 0$. No closed three-manifold other than $S^3$ carries such a realizable metric. - **The homogeneous geometries.** A realizable closed metric with $\det\mathbb M > 0$ and constant scalar curvature is round, and so is every compact locally homogeneous realization. Locally, constant $\mathbb M$-eigenvalues with $\det\mathbb M > 0$ force constant curvature, a case of Tsukada's classification. This recovers Agaoka and Hashinaga's list of the left-invariant metrics that embed locally. The eight Thurston geometries are worked as checks. - **Total absolute curvature.** $J(g) = \int\sqrt{\det\mathbb M}\,dV$ equals $2\pi^2$ on every closed realizable metric with $\det\mathbb M > 0$, so any other value excludes realizability. At the round sphere $J$ is stationary. Its second variation on transverse-traceless tensors of level $n$ is $-(\mu_n + 2)(\mu_n + 3)/16$ times the squared norm, $\mu_n = n^2 - 3$, and its radical is exactly the conformal and Lie directions. On conformal rays $J$ is flat to third order. Along the conformal path of [MA], $J = 2\pi^2 - \pi^2t^4/12 + O(t^6)$. - **Non-convex realizations.** Embedded hypersurfaces of revolution are non-convex exactly around their necks. For the radial graphs $r = 1 + \varepsilon\cos2\chi$, convexity is lost at $\varepsilon = 1/5$ in a Whitney cusp of the Gauss map. With $\delta = \varepsilon - 1/5$, the relative excess $J/2\pi^2 - 1$ then grows as $500\,\delta^{3/2}/(9\sqrt2\,\pi)$. - **Connectivity, and the round sphere.** Modulo isometry, the closed realizable metrics with $\det\mathbb M > 0$ form a contractible family, contracted by an explicit equivariant path. The embedded hypersurfaces of revolution join the round sphere through the degenerate locus. We collect the characterisations of the round sphere, with their hypotheses. Every displayed result is reproduced by a script; Appendix V indexes them.

math.CO — Combinatorics

Local independent sets and colourings of triangle-free graphs

Contributed by Matthew Bowen, András Máthé, Oleg Pikhurko

This AI-assisted draft constructs, for every sufficiently large $D$, a local rule of radius $O((\ln D)^2)$ that produces, on any triangle-free graph $G$ of maximum degree at most $D$, a random independent set that includes each vertex with probability at least $(1-o_D(1))\ln D/D$. Further properties of the rule yield, for every fixed $\varepsilon>0$ and all sufficiently large $D$, a randomised LOCAL algorithm that colours any such finite graph $G$ with at most $(1+\varepsilon)D/\ln D$ colours, with high probability, in $O_\varepsilon(\ln D)+(\ln\ln |V(G)|)^{O(1)}$ communication rounds. Applying known transference principles for Borel graphs, we obtain the same asymptotic bound on the number of colours for measurable and Baire-measurable colourings, and for Borel colourings of graphs with finite asymptotic separation index.

math.PR — Probability

Smallest Gaps between Eigenvalues of Wigner Matrices without Smoothness Assumptions

Contributed by Patrick Lopatto

We determine the asymptotic distribution of the smallest gaps between eigenvalues in the bulk of the spectrum of real symmetric Wigner matrices, including Bernoulli matrices. After multiplication by $N^{3/2}$, the gaps and their locations converge to a Poisson point process with the same intensity as for the Gaussian orthogonal ensemble. Since changing a single entry can move eigenvalues by an amount comparable to the smallest gaps, standard perturbative comparison estimates do not directly resolve this scale for arbitrary entry distributions. Instead, we use that resampling a few random entries does not change the law of the matrix, and we show that the resampling moves each pair of nearby eigenvalues by a random walk that satisfies a local limit theorem; this requires excluding arithmetic structure in the eigenvectors. We also prove a law of large numbers for the number of gaps at all scales between $N^{-3/2}$ and $N^{-1}$, and we sketch the proof of the analogous smallest-gap result for complex Hermitian Wigner matrices.

math.IT — Information Theory

Finite Alphabet Communication with Noisy Forward Memory in a Normalized Loop

Contributed by Daniel Adam Pyle

We study one-use communication of a finite classical message through a stipulated normalized loop containing a backward binary symmetric channel and one noisy forward memory carrier. The carrier is either a classical bit or a qubit; no shared random seed, retained side register, or entanglement is supplied. For every finite message count, we characterize the qubit optimum by a constructive scalar feasibility test. For rational noise parameters and a rational target success, the test uses O(M²) constant-size rational arithmetic checks. The proof combines an exact decoder reduction, minimum-trace allocation for constant encoders, and a concave geometric cap with a matching polygon construction. For three messages, the qubit and classical-bit optima coincide throughout the allowed noise range; an explicit Bernstein certificate proves the remaining polynomial inequality. For four messages, an exact rational qubit protocol exceeds a bound valid for every classical-bit protocol. These are conditional mathematical results for the declared resource model, not a construction of a physical backward channel. Independent human review and originality assessment remain pending.

math.NT — Number Theory

Small-index subsequences in finite cyclic groups

Contributed by Chao Liu

For $r\ge1$, let $I_r(n)$ be the least length forcing a nonempty zero-sum subsequence of index at most $r$ over the cyclic group $C_n$. We prove $I_r(n)\le n^{1+o(1)}$ when $r\ge C[\omega(n)\log(2\omega(n))]^2$, and $I_1(p^k)\ll(k+1)p^k\log(2p^k)$ for prime powers. We also construct sequences showing that $I_r(n)/n$ is unbounded for every fixed $r$; for $r=1$, all terms may be generators.

cs.DS — Data Structures and Algorithms

Counting Hamiltonian cycles in sparse graphs in subexponential time, assuming the new #SAT algorithms

Contributed by Vaibhav Tulsyan

In October 2026 two manuscripts claimed refutations of the Exponential Time Hypothesis by deterministic algorithms for $\#k$-SAT, each with a Lean formalisation of its top-level statement. Neither result applies directly to connectivity problems such as Hamiltonicity, because no linear-size CNF encoding of "this edge set is a single cycle" is known. This paper derives a corollary of the Anthropic manuscript: assuming three of its internal results (a network-to-clique reduction, an algorithm for cliques of all levels, and the existence of its "pieces"), the number of Hamiltonian cycles of a graph with $n$ vertices and at most $Dn$ edges can be computed exactly and deterministically in time $O(2^{\delta n})$ for every $\delta>0$, and in time $2^{O(\max(m,n)/\sqrt{\log n})}$ for a graph with $m$ edges. A rank-encoded variant of the construction gives $2^{O(n\log\bar d/\sqrt{\log n})}$ for average degree $\bar d$. Deterministic decision, search and Hamiltonian-path counting follow, and so do exact sparse TSP with polynomially bounded weights, exact counting of directed Hamiltonian cycles, and ATSP. The result is not a standalone breakthrough; it stands or falls with the claimed refutation. The ingredients are known: the folklore matrix-tree identity on $2$-factors, the fermionic (Berezin) matrix-tree theorem of Caracciolo, Jacobsen, Saleur, Sokal and Sportiello, and the blob partitions of Cut&Count. What is new is the observation that connectivity can be encoded with $O(1)$ bits per edge through a Grassmann determinant whose only non-local part is a sign, and that this sign costs only $O(g^3)$ bits for $g$ blobs, so the whole construction is a network in the sense of the manuscript, whose value is $n\cdot\#\mathrm{HC}(G)$. Every theorem we claim is formally verified in Lean, end to end, on top of the manuscript's own Lean development: the three counting theorems, the decision, search and path corollaries, sparse TSP, directed Hamiltonian cycles and ATSP, using only the standard axioms, with fresh kernel replays. The Lean sources are at https://github.com/xennygrimmato/eth-hamiltonicity. We also record weaker results that rest only on the machine-checked top-level theorems: the parity of $\#\mathrm{HC}(G)$ for every graph in time $2^{O((m+n)/\sqrt{\log(m+n)})}$ (machine-checked in Lean, with the hypotheses discharged by the labs' own Lean proofs), and one-sided randomized decision. We describe what is proved, what is checked numerically, what is formalised, what is assumed, and what the limitations are.

cs.DS — Data Structures and Algorithms

Improved basis-change transformations for $2 \times 2$ matrix multiplication

Contributed by Hugo Nielsen

To the best of our knowledge, we present a state-of-the-art change-of-basis algorithm for $2 \times 2$ matrix multiplication under the Karstadt--Schwartz-style basis change regime [1]. We do this by introducing basis-change matrices $B_U$, $B_V$, and $B_W$ that each require only a single addition and hence give rise to a matrix-multiplication algorithm performing $5n^{\log_2{7}} - 4n^2 + \frac{3}{4}n^2 \log_2{n},$ arithmetic operations when multiplying two $n \times n$ matrices, with $n$ a power of $2$. This improves on the previous best bound, which had last term $\frac{3}{2}n^2\log_2{n}$.

math.CO — Combinatorics

Nonunimodal Laurent coefficients of cluster monomials

Contributed by Qiyue Tang

We prove that cluster monomials of coefficient-free type $A_5$ can have Laurent coefficient sequences that are neither unimodal nor log-concave. This disproves the log-concavity and unimodality assertions in Chen’s Conjecture 5.11 and gives a further family of counterexamples to Chen–Sun’s Conjecture 8.2, including its unimodality assertion.

math.CO — Combinatorics

The covering number C(13,7,4) is 30: A classification-assisted exact proof

Contributed by John Tripodo

We prove that $C(13,7,4)=30$, using the classification of minimum $(12,6,3)$ coverings by Gordon, Patashnik, Petro and Taylor. A hypothetical $29$-block cover has at least five points lying in exactly $15$ blocks. The link at one of these points leaves only $14$ blocks outside it. The other four points force complete families of outside blocks whose overlaps must agree. An exhaustive computation shows that every compatible choice requires $15$ outside blocks, giving a contradiction. The known $30$-block construction is therefore optimal. We give reproducible exact computations and a public Lean formalization of a second argument for the finite obstruction. The Lean theorem states the published minimum bound and classification as explicit hypotheses; it does not reprove those results.

math.CO — Combinatorics

Counterexamples to log-concavity of cluster monomials and theta structure constants

Contributed by Qiyue Tang

We disprove Conjectures 1.1 and 5.5 of Chen–Huang–Sun and the log-concavity assertions in Chen’s Conjecture 5.11 and Chen–Sun’s Conjecture 8.2 by constructing counterexamples for cluster monomials and theta structure constants in coefficient-free type $A_5$.

math-ph — Mathematical Physics

First-order phase transition in Onsager's model of hard rods in $\mathbb R^3$

Contributed by Qidong He

We prove a first-order transition for hard rods in $\mathbb R^3$ whose orientations vary continuously over the sphere. For every sufficiently small positive ratio of diameter to length, the grand-canonical pressure has a nondifferentiable point, with a jump in number density at the intermediate Onsager scale. Following the phase-pressure strategy of Lebowitz, Mazel, and Presutti, we connect coexistence in the homogeneous Onsager functional to the microscopic pressure through an exact auxiliary-field representation, exponential decay of correlations in the rod gas at fixed auxiliary field, and estimates of the free-energy cost of interfaces between connected families of minimizing angular profiles.

math.AG — Algebraic Geometry

A Linear In Rank Bound For Integral Arithmetic Progressions On Elliptic Curves

Contributed by Jacob Crim

Every known bound on the length N of an x-progression on an elliptic curve of rank r has the shape N ≤ C^{1+r}. We prove a bound that is linear in the rank for integral progressions, with an error depending only on the height of the model. Let y^2 = x^3 +a_2x^2 +a_4x+a_6 be an integral model and H_f = 1+|a2|+|a4|+|a6|. For every ε > 0, every integral x-progression of length N satisfies N ≤ (3/2 +ε)s+C_{ε} (1+log(H_f))^{1/2}, where s ≤ r is the rank of the span of the progression. In particular N ≤ (3/2 +o(1))s whenever log H_f = o(N^2). The proof has two halves. The local theory consists of exact point counts, the root-class lemma, pinned values, and a forbidden range of primes reaching N^2/49 by a shift-and-square argument. It forces log |b| ≫ N^2, where b is the common difference. The spectral cap is proved in an exact form. Splitting the Neron–Tate pairing into local kernels of positive type and summing the regularised diagonal terms over all places gives an identity in which the bad-prime corrections cancel. Combined with a uniform archimedean separation lemma, this yields λ_{max}(G) ≤ (3/4) logX + (1/4) ψ(N − 1) + (5/4) log(H_f )+ 2, where X =max|x_i|. Over function fields we prove an exact colouring formula for the Gram matrix of a progression of integral sections, which gives N ≤ min(6χ, (3/2)s). We also classify the 3-term Gram types on rational elliptic surfaces and show that all of them occur, and we show that all sign data are realised on elliptic K3 surfaces. The main bound is vacuous for every known example (all have N ≤14); its interest is that it is linear rather than exponential in the rank.

math.NA — Numerical Analysis

Low-discrepancy quasi-uniform digital sequences in every dimension

Contributed by Leonhard Gruenschloss

A sequence of points in the unit cube is quasi-uniform if its first $N$ points are spread like a grid for every $N$: no two of them are closer, and no empty ball between them is larger, than a constant times $N^{-1/s}$ in dimension $s$. Dick, Goda and Suzuki asked for such sequences among the digital sequences of quasi-Monte Carlo integration, and Goda, Hofer and Suzuki asked whether any Halton-type sequence is quasi-uniform. We show that in dimension two and higher no Halton-type sequence, no Sobol' sequence, no classical Niederreiter sequence and no Faure sequence is quasi-uniform, and that linear scrambles of all coordinates but one do not help: a carry in the last digits of one coordinate leaves the leading digits of all others unchanged, giving explicit pairs of points that are far too close. We then show that one fixed linear scramble of a generalized Faure sequence, the same for all numbers of points, repairs this. In it, the digits of all coordinates at one scale are residues of one rational function, each taken at the coordinate's own point of the projective line, so the residue theorem ties the coordinates together and a carry in one coordinate is seen by the others. For every dimension $s$ and every prime base $p>2^s$ this gives explicit quasi-uniform digital $(0,s)$-sequences, hence of low discrepancy, whose first $N=p^{sK-1}$ points are at least $(1-2p^{-1/s})N^{-1/s}$ apart, for every $K$; in dimensions two and three, for every prime $p\ge5$, a variant keeps its first $N$ points at least $(1-2/p)N^{-1/s}$ apart whenever $N$ is a power of $p^{s^2}$. For such scrambles, exact linear rules that place every point inside its cell as a function of the last few digits of the cell's index can hold at no more than about one in $s$ of the sizes $p^{sK}$; with allowed ranges instead of exact rules we certify all of these sizes, with points at least $(1-O(1/p))N^{-1/s}$ apart for large $p$. Quasi-uniform digital sequences follow in every prime base $p\ge3$ in dimension three, and in every prime base in dimension two, base two included. Some steps are computer-assisted.

math.CO — Combinatorics

Maximal Independent Sets in Long Graph Subdivisions

Contributed by Daniel Adam Pyle

We study maximal independent sets in finite simple graphs whose edges are replaced by paths with prescribed numbers of internal vertices. Using established finite-state transfer methods, we derive an exact boundary-state formula and an explicit leading asymptotic term that factors over the degrees of the original vertices. A graph-uniform relative error bound gives a sufficient condition for this approximation to remain valid when the base graph grows. Under the same condition, the joint membership indicators of the original vertices approach an explicit product of Bernoulli distributions in total variation. For fixed base graphs, we obtain the first adjacency-sensitive correction and a Gaussian limit for the size of a uniformly chosen maximal independent set. A degree-matched construction demonstrates that bounded aggregate error scale does not suffice for uniform degree-sequence universality. Reproducibility materials include exact enumeration, weighted and moment checks, and numerical spectral comparisons.

math.NT — Number Theory

Cross-Energy of Dirichlet Partial Sums. Part 11: A two-sided criterion for the Riemann hypothesis from weighted prime sums, verified up to $2\cdot10^{29}$

Contributed by Pedro Caceres

For $N>1$ let $C(N)=2\sum_{n<N}\Lambda(n)n^{-1/2}\log(N/n)$ and let $M(N)=R(N)-C(N)$, where $R(N)=8\sqrt N-5.3721834\log N-7.4013354+O(N^{-5/2})$ is explicit. By a theorem of Suzuki, the Riemann hypothesis (RH) holds if and only if $M(N)\ge0$ for all large $N$, and then $0\le M(N)\le4(\log\xi)''(\frac12)=0.18484$. We prove, with computer assistance, that this two-sided inequality holds for every real $1<N\le2.15\cdot10^{29}$: up to $10^{10}$ from the primes with correctly rounded arithmetic, and beyond from the first $4520$ zeros, certified with ball arithmetic, together with the verification of RH to $3\cdot10^{12}$ and an explicit zero-density estimate. The same ingredients give, for every $N$ and without any hypothesis, $-E(N)\le M(N)\le0.18484+E(N)$ with $E(N)$ explicit and $E(10^{20})<3\cdot10^{-5}$. We describe the local minima of $M$ exactly, through the slope $\alpha(x)=\sum_{n\le x}\Lambda(n)n^{-1/2}-2\sqrt x+\frac{c_0}2$. Under RH the margin is the variogram of $\alpha$, and RH is equivalent to $\int_1^X\alpha^2\,dx/x=X^{o(1)}$. Without any hypothesis we prove an energy identity, which expresses the energy of the slope as a C-transform of the margin against the prime measure, and a lower bound for the mean energy that is attained exactly under RH with simple zeros. We then show what such arguments cannot do. Positive weights within $q^{\delta-1/2}$ of $\Lambda(q)q^{-1/2}$ satisfy the energy identity, the prime number theorem and Chebyshev's identity up to $x^{1/2+\delta}$, yet carry an off-line zero. So the structural inputs listed do not suffice, and a proof along these lines has to use $\Lambda=\mu*\log$ exactly. Carrying that identity through the slope, $\alpha$ is numerically, to within $1\%$ in the range tested, a M\"obius-weighted sawtooth, and the exact argument reduces to a type~I sum of $\mu$ against the fractional parts of $x/d$, which is RH again.

Author comments: Revised 10 Nov 2026 after external examination (see CHANGES_v2.md). 36 pages, 10 figures; ancillary scripts reproduce the computations; Part 11 of the 47-part book-series Cross-Energy of Dirichlet Partial Sums (Parts 0-46), block III (The margin and the constant B-hat); formerly a standalone note; a public version is the Hexagon preprint hexagon:2610.00661v1. Part 0 is the guide. The paper states the Riemann-hypothesis caveat once (2 Nov 2026: removed from the abstract and from a remark).

math.NT — Number Theory

Computing Jacobsthal’s Function on Primorials: Exact Values and Constructive Bounds

Contributed by Alben Weeks

Let h(x) = j(Px), where j is Jacobsthal’s function and Px is the product of the first x primes. We describe a parallel residue-covering search. Exhaustive exclusion runs, together with independently checked covering witnesses, establish h(65) = 1134, h(66) = 1160, h(67) = 1192, h(68) = 1216, h(69) = 1248, h(70) = 1274, and h(71) = 1298. Method 5B gives explicit certificates for h(100) ≥ 2054, h(150) ≥ 3558, and h(200) ≥ 5172, using partial coverings with five reserve primes. Exhaustive searches give h(100) ≤ 2798, h(150) ≤ 6898, and h(200) ≤ 11998. The search combines parity compression, descending endpoint restrictions, canonical deduplication, a sliding-window bootstrap, and dynamic prime selection related to Ziller and Morack’s Greedy Permutation Algorithm. We state the conditions needed for capacity bounds and conditional propagation, and organize extensions of complete or partial coverings in a five-tier framework. A catalog of 62 coverings at index 65 supports a verified multi-certificate envelope. The solver source, run evidence, certificates, and arithmetic verifiers are publicly available in the accompanying GitHub release v1.0-paper. Upper-bound conclusions depend on exhaustive execution of the search; the record audits do not independently repeat those searches or supply standalone unsatisfiability certificates

math.AG — Algebraic Geometry

Integral crystalline Tate classes of codimension two at supersingular points of Mumford's fourfolds

Contributed by Antti Hippeläinen

For a supersingular abelian variety $X$ over $\bar{\mathbb{F}}_p$, let $T^4(X)$ be the lattice of integral crystalline Tate classes of codimension $2$ and $S(X)$ the span of products of divisor classes. Let $p\ge5$ and $\mathscr{G}_p$ be the $p$-divisible group of the supersingular points of Mumford's families of abelian fourfolds at primes inert in their cubic field, for the hyperspecial levels given by tensor-induced lattices. For every abelian fourfold $X$ with $X[p^\infty]\cong\mathscr{G}_p$ we show that $T^4(X)/S(X)\cong(\mathbb{Z}/p)^4$, and that the integral crystalline Tate conjecture holds in codimension $2$ on $X$ if statements in a preprint of Markman hold. At these primes, Kisin's crystalline tensors make the Dieudonné module a tensor product of three rank-$2$ modules permuted cyclically by Frobenius, and at supersingular points Frobenius maps a tensor basis to multiples of basis vectors. For such monomial modules all lattices involved are spanned by multiples of basis vectors, with exponents given by min-plus formulas independent of $p$. Under the endomorphisms, $T^4/S$ is a non-split extension of $(\mathbb{Z}/p)^2$ by $(\mathbb{Z}/p)^2$, generated by one Weil class of an inert imaginary quadratic action. Assuming these statements, an algebraic such class comes from specializing a Weil cycle on a fourfold with complex multiplication. A projector that is a polynomial in an endomorphism, and an intersection matrix with unit determinant, fix its $p$-adic normalization without a comparison theorem. At unramified non-inert primes with such levels, $T^4=S$ at the supersingular points. Of the fourteen isomorphism classes of monomial supersingular Dieudonné modules of height $8$, the conjecture holds unconditionally for nine, under the same assumption for that of $\mathscr{G}_p$, and is open for four. The indices are computed exactly with integers.

math.LO — Logic

A true theorem Lean can state but can never verify

Contributed by William Banks

We give an expository account of a mathematical result that the Lean proof assistant can formally state but, by Gödel's second incompleteness theorem, can never verify, and which it also does not refute, namely, the consistency of Lean's own underlying type theory. Working in $\mathsf{ZFC}$ extended by the existence of infinitely many inaccessible cardinals, we sketch the construction of a set-theoretic model of Lean's Calculus of Inductive Constructions (with its universe hierarchy, quotient types, and choice), outline the proof of a soundness lemma, and conclude that no derivation of $\mathtt{False}$ exists. The argument is the model-theoretic consistency proof carried out in full by Carneiro, building on Werner's set-theoretic interpretation of the same calculus; all substantive results are due to these authors, and this paper claims no new mathematics. We explain why this theorem lies outside Lean's own reach, note that Gödel's second incompleteness theorem has itself been verified by Lean's kernel for first-order arithmetic, and discuss why every consistency proof ultimately rests on trust rather than proof.

math.DG — Differential Geometry

Riemannian three-manifolds as hypersurfaces of Euclidean four-space: the realizability operator, its symbol, and its linearisation at the round sphere

Contributed by Jay Thrussell

For a Riemannian three-manifold $(\Sigma,g)$ with negative Einstein endomorphism $\mathbb{M}=-G$, the Gauss equation of a hypersurface of $\mathbb{R}^4$ reads $\mathbb{M}=\mathrm{adj}\,K$, and on $\{\det\mathbb{M}>0\}$ it is solved identically by $\tilde K=\sqrt{\det\mathbb{M}}\,\mathbb{M}^{-1}$. Realizability with nondegenerate second fundamental form is therefore the single equation $F(g)=\mathrm{Cod}(\tilde K(g))=0$. The reduction is classical (Beez, Killing, Thomas) and the zero set has been characterized intrinsically (Rivertz; Agaoka and Hashinaga); we study $F$ as a nonlinear operator on metrics. At every nondegenerate point its principal symbol has rank two, with kernel exactly the symbol of the linearised immersions, so $F$ is injective modulo immersions everywhere and nowhere elliptic modulo diffeomorphisms alone. The nine Codazzi components carry four identities, leaving five conditions at every nondegenerate point, and Cartan's test holds with equality, in exact arithmetic, off a proper real-algebraic set. On a closed manifold the zero set is nowhere dense in the nondegenerate domain. At the round $S^3$ the linearisation on transverse-traceless tensors is $\mathrm{Cod}\circ\frac{1}{2}(-\Delta+2)$, with singular values $n(n^2-1)/\sqrt{2}$ and $\|DF\,h\|^2\ge 288\|h\|^2$. For metrics foliated by round 2-spheres we solve the first-order problem in closed form and write down all hypersurfaces swept by round 2-spheres; the $\ell=1$ direction is realized exactly by tilting the spheres' 3-planes. We treat the degenerate locus by two polynomial forms, and give Gauss–Codazzi as the flatness of two $\mathfrak{su}(2)$ connections: on the nondegenerate domain, realizability is equivalent to a single parallel spinor. Every displayed result is checked by a deposited script.

math.LO — Logic

Nonpathological extensions of $F_\sigma$ ideals and fractional coloring

Contributed by Carlos López-Callejas

The random graph ideal is pathological, answering a question of López-Abad et al. We prove this by constructing a positive hereditarily pathological restriction using Kneser graphs. For an $F_\sigma$ ideal $\mathcal I$ and any lower semicontinuous submeasure $\varphi$ with $\operatorname{FIN}(\varphi)=\mathcal I$, we show that the nonpathological envelope $\widehat\varphi$ generates an ideal that depends only on $\mathcal I$ and, when proper, is the least nonpathological $F_\sigma$ extension of $\mathcal I$. Thus, $\mathcal I$ is nonpathological if and only if $\operatorname{FIN}(\varphi)=\operatorname{FIN}(\widehat\varphi)$. We also characterize nonpathological $F_\sigma$ ideals both as intersections of summable ideals and as intersections of matrix ideals, answering both converses asked by Filipów and Tryba. The least nonpathological $F_\sigma$ extension of the random graph ideal is proper and is described by the fractional cochromatic numbers of its finite induced subgraphs.

math.NT — Number Theory

Power-Kernel Fluctuations, the Lambert W Subpolynomial Source, and a Local CRT Conditional Law

Contributed by Zeng Chunping

For a positive integer $n$ we define the power kernel $d_n = n/\operatorname{rad}(n)$ and the fluctuation $X_n = \log d_n$, where $\operatorname{rad}(n) = \prod_{p \mid n} p$. This paper develops three mutually supporting lines around this ``relative fluctuation''. First, the microscopic theory. From the kernel decomposition $n = v_d s$ ($s$ squarefree, $(s,d)=1$, $v_d = d \operatorname{rad}(d)$) we obtain the exact discrete law $\mathbb{P}(X_n = \log d) = \frac{6}{\pi^2} \frac{w(d)}{d \operatorname{rad}(d)}$, $w(d) := \prod_{p \mid d} \frac{p}{p+1}$, together with a three-branch parity-conditioned refinement. Parity is a first-order effect: the measured gap $\mathbb{E}[X_n \mid n \text{ even}] - \mathbb{E}[X_n \mid n \text{ odd}] = 0.69008$ matches $\log 2 = 0.69315$. Second, the conditional law. For fixed $c$, the distribution of $X_{c-a}$ is completely determined by $\mathrm{mask}(a)$ through a three-case local CRT convolution: the local contribution vanishes identically when $p \mid c$ or $p \mid a$, and otherwise follows a corrected law rather than the usual one, differing by a factor $\frac{p}{p-1}$ (a factor $2$ at $p=2$). The corrected law agrees with $220$ masks to $0.993$, while the usual law underestimates the tail by $47\%$--$78\%$. This replaces an earlier erroneous assertion of equidistribution. Third, the subpolynomial source. For radical-type Euler products $\prod_p \bigl(1 + \frac{\eta(p)}{p^\sigma - 1}\bigr)$ with $\eta(p) \to a$, one has as $\sigma \to 0^+$ the asymptotic $\Phi(\sigma) = \frac{a}{\sigma W(a/\sigma)}\bigl[1 + \frac{\kappa_1}{W} + \frac{\kappa_2}{W^2} + \cdots\bigr]$, where $W$ is the Lambert function. The main term comes not from small primes but from the prime range $[e^{v^*}, e^{1/\sigma}]$, which explains why no finite prime table can reproduce it. The coefficients satisfy $\kappa_n = (n-1)! + \sum_j c_{n,j} \zeta(2j)$, a sum over even zetas only, because the two-sided integral coefficients satisfy $\iota(r) = (-1)^{r+1} \rho(r)$, forcing odd-zeta terms to cancel exactly at every order. This unifies $T(x) = \sum_{n \le x} 1/\operatorname{rad}(n)$, $\Sigma(D)$ and $\Sigma_\perp(D;Q)$, and explains at one stroke why all previously fitted ``constants'' are transient. Applications include: a quantitative formula for Mazur parameters in the open range $1 < s < 2$ ($k_{\mathrm{eff}} = 1.91 \pm 0.17$, excluding $k=1,3$); the ceiling theorem $q \le \frac{\log c}{\log 2 + \log \operatorname{rad}(c)}$ (equality iff $\{a,b\} = \{1,2^k\}$); the rad--$P^+$ duality (principal versus negative Lambert branch); an exact $S$-unit reformulation of $\rho(c) = \min_a \operatorname{rad}(a)\operatorname{rad}(c-a)$; and a mechanistic decomposition of the exponent of $\mathcal{N}_0(N)$. We report honestly: the abc conjecture itself is not attackable within this framework. The three structural walls are: (i) the conditional law is provable for $D \le c^{1/2-\delta}$ while abc solutions live at $D \gtrsim c^{0.78}$; (ii) relaxing any one of the three radicals inflates the count by a factor $387$; (iii) the Fourier circle method is controllable iff $\alpha + \beta > 2$, while Mazur parameters always have $\alpha, \beta < 1$. Our weak abc bound is weaker than de Bruijn's 1962 trivial bound; this is recorded as well.

math.NT — Number Theory

An improved zero-free half-plane for Hecke L-functions over number fields and Quasi-Riemann Hypothesis

Contributed by Baiying Liu

In this paper, we obtain a zero-free half-plane Re(s) > B_ex for every finite-order Hecke L-function over every number field, where B_ex = 0.87407284766682790777... is an explicitly defined algebraic number of degree four. The proof refines the cubic-theta comparison in two parts. In Part I, we require sufficient prime-product length for length restoration and fix the short side of the equal-product comparison rectangle at 1/4. Direct and complementary Poisson estimates give an amplified Gauss norm with an additional prime product. Its separate applications to the numerator and the counting detector, combined with contour displacement and successive moment bounds, give the intermediate boundary B′ = 0.87422125049290208328.... In Part II, complete prime removal is permitted even when a positive length excess remains. The no-prime moment estimate then controls the resulting ordinary sums, and the comparison rectangle’s short side may vary. These changes remove the coupled condition on the remaining prime length and enlarge the allowed norm widths. The improved detector count and numerator estimate give successive proved bounds converging to B_ex. We prove optimality for this specified direct, detector and fourth-moment comparison system, retaining its geometric length inequality while allowing the sum of the length exponents to vary. Both parts use the established theta, moment and continuation results of the earlier QRH papers and their general-field formulation.