All works

math.AG — Algebraic Geometry

Terminal Symplectic Singularities for Parabolic Base Affine Spaces

Contributed by Tom Gannon

Let $G$ be a simply connected semisimple complex algebraic group, and let $U_P$ be the unipotent radical of a parabolic subgroup $P\subseteq G$. We prove that the affinization $\overline{T^*(G/U_P)}$ is factorial and has terminal symplectic singularities. To establish symplectic singularities, we apply a criterion of Fu--Liu after constructing a log Fano boundary on an associated projective bundle over the full flag variety. The boundary construction uses Frobenius splitting of flag varieties and reduction modulo primes to obtain log canonicity in characteristic zero. We then prove that every first-order Poisson deformation of $\overline{T^*(G/U_P)}$ is trivial. Namikawa's deformation theory implies that the singular locus has codimension at least four, yielding terminality. No conical grading is assumed.

math.NT — Number Theory

An improved upper bound on the number of representations of an integer as a binomial coefficient

Contributed by Ben Whitmore

Let $N(t)$ be the number of pairs of integers $(n,m)$ with $\binom nm=t$. Singmaster conjectured that $N(t)=O(1)$, and the best bounds in the literature are $N(t)=O(\log t\,\log\log\log t/(\log\log t)^3)$ unconditionally, and $N(t)=O((\log t)^{2/3+\varepsilon})$ conditional on Cramér's conjecture. We show that $N(t)=O((\log t)^{1/2+o(1)})$ unconditionally. Solutions with $m\ge\exp((\log n)^{2/3+\varepsilon})$ are handled by a theorem of Matomäki, Radziwiłł, Shao, Tao and Teräväinen. For smaller $m$ we show, using an interpolation determinant in the spirit of Bombieri and Pila, that the solutions with $m$ in a short interval lie on a single algebraic curve of low degree in $n$, and we bound the number of solutions on such a curve by a zero estimate for exponential polynomials.

math.PR — Probability

THE CRITICAL CAPACITY OF THE HOPFIELD MODEL

Contributed by Yuwen Zhou

Abstract. We prove that the critical capacity of the Hopfield model at zero temperature lies in [0.138198129, 0.13820106]. The best rigorous lower bound was 0.113, and our lower end exceeds the published one-step and two-step replica symmetry breaking values. The capacity is the supremum of the loads at which the free energy at fixed overlap with one pattern has a well. At every positive temperature, we give a variational formula for this free energy and show that binary patterns and Gaussian rows have the same wells, which characterizes the capacity. The proof builds on the Hamilton–Jacobi solution of the model by Ali, Boursier, Chen, Ho, Issa and Mourrat, and on a

math.CO — Combinatorics

Hadamard difference sets in groups with a cyclic odd quotient

Contributed by David Niedbala Giraudin

Let u > 1 be odd. We prove that the abelian group C_2 x C_{2u^2} contains no Hadamard difference set. Together with the recently announced proof of the circulant Hadamard conjecture, this shows that an abelian group of order 4u^2 (u > 1 odd) whose odd Sylow subgroups are all cyclic never contains a Hadamard difference set, so Lander's conjecture holds for Hadamard difference sets in these groups. More generally, let G be a group of order 4u^2 with a normal subgroup N such that G/N is isomorphic to E x C_{w^2}, where |E| = 4, w > 1, w | u and gcd(w, u/w) = 1. If every prime divisor of u/w is self-conjugate modulo rad(w) (modulo 4 rad(w) when E is cyclic), then G contains no Hadamard difference set. The proof adapts the alternating character products used for the cyclic case.

math.NT — Number Theory

The Hankel determinants of the Baum-Sweet sequence are 2-automatic

Contributed by David Niedbala Giraudin

Let b be the Baum-Sweet sequence and H_n = det(b_{i+j})_{0<=i,j<n} its Hankel determinants over F_2. We prove that the sequence chi_0(n) = [H_n != 0] is 2-automatic: chi_0(n+1) is computed by an explicit automaton with 8 states reading the binary expansion of n, and the minimal automaton of chi_0 reading least significant digits first has 74 states. The same holds for the shifted determinants det(b_{i+j+1}), with the same 8-state graph and 68 states. This contrasts with Mkaouar's theorem that the partial quotients of the Baum-Sweet series are not automatic. On the way we show that Theta = t f^2, a root of Z^3 + Z + t^3 in F_2((1/t)), has all its partial quotients in {t, t^3, t^3 + t}, given by an explicit coding of the Mills-Robbins expansion, and that the linear complexity profile of b satisfies -2 <= 2L(b,N) - N <= 3 for all N, all six values occurring. The proofs reduce to finite verifications, which an ancillary script performs.

math.DS — Dynamical Systems

Three Axes of Collatz Dynamics: 2-adic Measures, the Dimension Spectrum of Inverse Trees, and Critical Exponents of Density Propagation

Contributed by Zeng Chunping

This work presents a systematic measurement and critical diagnosis of three research axes on the Collatz (3n+1) problem, with the aim of determining which tools can advance the problem and which are structurally incapable of doing so. The three axes are: the 2-adic statistics of individual orbits, the combinatorial structure of the inverse tree, and the critical exponent of density propagation (the Terras-Krasikov-Lagarias line). On the first axis we establish a strict identity and a divergence criterion. For the odd-step map m -> m' = (3m+1)/2^a with a = v_2(3m+1), one has log m_{K+1} - log m_1 = K log 3 - (sum a_i) log 2 + sum log(1 + 1/(3m_i)), verified to 10^{-13}. Divergence is therefore equivalent to (average a) < log_2 3. Over 298 million odd steps spanning nine orders of magnitude we obtain average a = 1.996929, matching the i.i.d. geometric prediction E[a] = 2, with a safety margin of 0.412 (a 20.6% drop would be needed for divergence). The distribution of a is exactly geometric and the autocorrelation is approximately zero; a surrogate control (preserving the marginal, destroying the serial structure) shows no difference between real and shuffled sequences at long windows. On the second axis we derive a closed form for one-step propagation on the inverse tree. The two extensible residue classes give Lambda_1(beta) = (3/4)^beta/(1-4^{-beta}) and Lambda_2(beta) = (3/2)^beta/(1-4^{-beta}). Because preimages are uniformly distributed over the three residue classes, the extension matrix has rank one and rho(beta) = (Lambda_1 + Lambda_2)/3 = 3^{beta-1}/(2^beta - 1). At beta = 1 one has Lambda_1 = 1 (critical) and Lambda_2 = 2 (supercritical), the latter being exactly cancelled by the one-third loss into leaf nodes, so that the total spectral radius is exactly 1. This turns "density propagation stops at beta = 1" from an observation into a corollary of the closed form. On the third axis the depth-k reachable set is measured as a genuine fractal of dimension gamma_k (gamma_1 = 0.101 logarithmic, gamma_12 = 0.603, gamma_24 = 0.824, gamma_40 = 0.923, increasing monotonically to 1). A decisive test rejects the competing polylogarithmic hypothesis (at k = 24: measured growth x294, polylog prediction x1.7 x 10^7). We further show that the type-transfer spectral radius cannot produce any gamma_k: its zero is constantly the asymptotic ceiling 1; obtaining gamma_k requires the infinite-branch Ruelle operator on the spatial variable, which we construct and then falsify by a fit-free hard criterion (operator count 5398 exceeds the hard bound X/2 = 5000). Two popular frameworks are falsified: the local-entropy framework (its numerical verification is vacuous and its upper-bound theorem conflicts with the random model) and the conjecture that truncated propagation possesses a finite critical root (the truncated ratio is always below 1 and increases monotonically to 1). All numerics carry finite-size labels; the finite-size deviation is proved to be O(1/log X) rather than O(1/X). A methodological ledger records nine recurring errors and their countermeasures. The three axes are finally unified by the master identity rho(beta) = M(beta-1), where M is the moment generating function of the single-step log-increment: the inverse tree, the stopping-time large deviations, and the orbit drift are three readings of one function. Five corollaries follow, including rho'(1) = delta(3) (the spectral slope at criticality equals the drift) and beta_c = -log min rho (the far-tail rate is the negative logarithm of the spectral minimum). A three-way shootout against the Bessel(3)/meander hypothesis for the survival-profile shape falsifies that hypothesis (z = +/-50 to 69), isolating an open phenomenon: Collatz survival trajectories have a definite "initial plateau, mid-concentrated descent" shape that no model with the correct marginal reproduces.

math.OA — Operator Algebras

Noncommutative von Neumann-Riesz and von Neumann-Riesz-Lorch Mean Ergodic Theorems

Contributed by K. MAHESH KRISHNA

We derive noncommutative von Neumann-Riesz and von Neumann-Riesz-Lorch mean ergodic theorems for adjointable contractions on a Hilbert C*-module. We crucially use a result obtained by Paschke [\textit{Trans. Amer. Math. Soc., 1973}]. We also note that the exact version of the noncommutative von Neumann–Riesz mean ergodic theorem fails.

math.NT — Number Theory

The Discriminant Structure of Frey Curve Isogeny Classes: Crater–Peak Duality, the Volcano Law, and a Mass Criterion

Contributed by Zeng Chunping

Let $a,b,c$ be coprime positive integers with $a+b=c$, and let $E: y^2=x(x-a)(x+b)$ be the associated Frey curve. We determine the structure of minimal discriminants across the $\mathbb{Q}$-isogeny class $\mathcal{C}(E)$ by closed formulas in $(a,b,c)$. The three $2$-torsion points of $E$ are rational, so $E$ is always the centre of its class; the first-level quotients via Velu's formula have monomial model discriminants $\{16(abc)^2,\ 256a^4bc,\ 256ab^4c,\ 256abc^4\}$, an exact collapse forced by the identity $u=4f(r)$ in Velu's algorithm. We prove the Crater--Peak Law: among the first-level vertices the minimal and maximal discriminants are $\min\{16(abc)^2,\,256\min(a,b)^4\max(a,b)c\}$ and $256abc^4$, the exponent $4$ landing on the smallest entry (crater) and the largest (peak); for star classes these are the class extrema, while on square or quartic triggers the extrema are the minimum and maximum over ten explicit candidates. The class size $|V|\in\{4,6,8\}$ is decided by the square and quartic character of $ac$ and $bc$ (107 samples, zero failures), matching the isogeny-torsion graph classification via the lcm of the isogeny matrix. When $ac$ or $bc$ is a square the class extends to depth $2$ with model discriminants $2^{10}ACu^8v^2$ and $-2^{10}ACu^2v^8$, and to depth $3$ on quartic triggers with discriminants $2^{11}pq\,u^{16}v^4e$ and $2^{11}pq\,u^4v^{16}e$. On the multiplicative bad-prime set $\pi$ we prove the Volcano Law: each vertex carries an integer height vector $k(V)\in\mathbb{Z}^{\pi}$ with $v_p(\Delta^{\min}_V)=2^{k_p(V)}v_p(\Delta^{\min}_{\mathrm{crater}})$, and along every $2$-edge $k\mapsto k+\sigma$ with $\sigma\in\{\pm1\}^{\pi}$ independently per prime (578 edge-prime samples, zero exceptions), via the Tate parameterization. For odd primes $p\mid\min(a,b)$ we have $v_p(j)=-2v_p(\min)$, so the entire law is read off the $j$-invariant. The peak curve is explicit ($y^2=x^3-2(b-a)x^2+c^2x$), and the $j$-denominator exponent is a theorem: $\log\,\mathrm{den}(j)/\log\,\mathrm{num}(j)=(\alpha+2)/3+O(1/\log c)$, with the exact discrepancy identity verified to machine precision on $2185$ samples --- the Pasten meeting point in its precise form; we also show the Pasten mechanism is information-free on the Frey family. As an application, for $A^x+B^y=C^z$ we give an explicit two-branch formula for the crater ratio $\rho_c$; and with $\alpha=\log\min(a,b)/\log c$ we prove $\sigma_{\max}\to(5+\alpha)q$, so Szpiro is equivalent on the Frey family to the mass bound $q\le 6/(5+\alpha)$, sharp at $\alpha=1$ where Szpiro and abc coincide, while the residual gap lives at $\alpha=0$, the Pillai/$S$-unit corner. A Sage-verified table of $17$ classes (Kraus minimization agreeing with Sage minimal_model() throughout) gives $\sigma_{\max}>6$ in 14 of 17 rows (non-refutations: Szpiro allows $O(1)/\log N$ slack). All absolute $\sigma$ values are certified against Sage; structural statements are proved or machine-verified symbolically. We prove no conjecture: abc, Szpiro, Beal and the height conjecture remain open.

math.NT — Number Theory

Derivatives of Hardy's function at close pairs of zeros and an exact unitary constant $\pi^2/15$

Contributed by Jaskirat Singh

Let $\gamma_n<\gamma_{n+1}$ be consecutive ordinates of zeros of $\zeta(\tfrac12+it)$ with normalized gap $s_n$. We study the derivatives $Z'(\gamma_n)$, $Z'(\gamma_{n+1})$ of Hardy's function at close pairs. For the circular unitary ensemble we prove that the conditional variance of $\log|\Lambda'(\theta_1)|-\log|\Lambda'(\theta_2)|$ at adjacent eigenangles with normalized gap $s$ satisfies $\operatorname{Var}=c_N s^2(1+o(1))$ as $s\to0$, with the exact value $c_N=\frac{\pi^2}{15}\bigl(1-\frac{4}{N^2}\bigr)$, so that $c_\infty=\pi^2/15$, which we call the Singh constant. The proof reduces the problem through a Cayley transform to the Hua ensemble and closes it with two loop equations. We state a conditional theorem transferring the constant to the zeta function under the Riemann Hypothesis, a local GUE hypothesis and a tail bound, and we formulate a conjecture on the size of $Z'$ at close pairs. Numerical evidence from Odlyzko's tables up to zero number $10^{22}$ agrees with the predictions. This paper is a revised and expanded version of the Zenodo record doi:10.5281/zenodo.23899228.

Author comments: v2: Section 1 now lists fourteen Singh Lemmas, each labeled proved (6, 8, 45), conditional (7, 27, 42, 44, 46, 48, 54, 67) or numerical (5, 51, 57). The unitary pi^2/15 constant and Hua moments are credited as known results, Lemma 20 is credited to Soundararajan and Milinovich, and method credits and an explicit transfer hypothesis for Lemma 67 are added.

math.NT — Number Theory

Nonconstant Congruence Speeds of Integer Tetration in Non-Squarefree Radices

Contributed by Marco Ripà

We further develop the theory of congruence speed for integer tetration in non-squarefree numeral systems, building on Appendix B of [M. Ripà and G. Di Pietro (2026), A compact notation for peculiar properties characterizing integer tetration, Notes on Number Theory and Discrete Mathematics, 32(3), 477–496]. For bases coprime to the radix, we establish an eventual floor-difference representation and determine the exact spectrum of attainable least periods. We construct explicit arithmetic progressions containing infinitely many prime bases with nonconstant congruence speed for every non-squarefree radix, thereby proving the coprime-infinitude conjecture posed in Appendix B. We also derive a uniform quadratic bound for the least coprime base with nonconstant congruence speed and establish exact natural densities of nonstabilizing bases, both among integers coprime to the radix and among those not divisible by its radical, together with the corresponding relative density among primes. Finally, we give necessary and sufficient conditions for the perfect-power congruence-speed identity (Equation (2.12) of the cited paper) to hold in arbitrary radices, determine the exact least eventual period when stabilization fails, and establish a separate criterion for the boundary case of the original parameter restriction.

math.NT — Number Theory

Fully ordinary subfamilies and the Fontaine--Mazur conjecture in the residually reducible mixed case

Contributed by Luis Dieulefait

Let $F$ be a totally real field, abelian over $\Q$, in which an odd prime $p$ splits completely, and let $\rho:G_F\to\GL_2(\overline\Q_p)$ be a continuous, absolutely irreducible, totally odd representation, unramified outside finitely many places and de Rham with distinct Hodge--Tate weights at the places above $p$, with reducible residual representation. We prove that $\rho$ is modular in the mixed case, in which $\rho$ is reducible at some places above $p$ and irreducible at the others, provided that the ratio of the two residual characters extends to $G_\Q$ and its restriction to $G_{\Q_p}$ is not $\omega^{\pm1}$. The case where this restriction is trivial is allowed. No assumption is made on the splitness of $\rho$ at the reducible places, on potential modularity, or on the local behaviour of $\rho$ away from $p$. A suitable irreducible component through the point of $\rho$ in a partially ordinary deformation space contains a fully ordinary subfamily of maximal dimension; when the restriction of the residual ratio to $G_{\Q_p}$ is trivial, this subfamily lives on a cover of the deformation space which records the ordinary characters, following Pan. A classical point of this subfamily is a regular point of the deformation space, by the vanishing of adjoint Selmer groups, and a dimension estimate for a partially ordinary Hecke algebra then identifies the component of $\rho$ with a component of the Hecke algebra.

math.CO — Combinatorics

A composite modulus and Frobenius digits in the Heilbronn triangle construction

Contributed by David Niedbala Giraudin

OpenAI proved that for every sufficiently large n there are n points in the unit square all of whose triangles have area at least c n^(-2+eta), with an explicit but extremely small eta (of order 10^(-235)). Subsequent public work lowered the cost of the construction, the latest claimed exponent being eta = 2/100866719, about 1.98*10^(-8). We introduce two changes. First, the digit base is taken to be a multiple of the label prime r and the main modulus is h = r B^(k-1) instead of B^k; this reduces the ratio h/tau between the modulus and the excluded determinant window from about B to about r, which allows the extension degree d = 3. Second, the norm polynomial is placed in the digits by a digit-wise convolution along the Frobenius expansion of the exponents, with a trace digit in a polynomial basis, giving k = 1440 positions. The result is Delta(n) >= c n^(-2+eta) with eta = 1/177022 > 5.64*10^(-6), for every sufficiently large n. The finite identities are checked by exact computation in the ancillary files.

quant-ph — Quantum Physics

Classical Extensions and Strongly Positive Realizations of Decoherent History Probabilities

Contributed by Zhenyu Sun

Probabilities on several decoherent families of histories may agree on shared events yet admit no classical extension to the Boolean algebra that the families generate. We characterize the strongly positive decoherence functionals that realize given local probabilities on a finite algebra. A covariance decomposition allows any signed extension to be prescribed at any attainable rank and gives the resulting increase in minimum trace. For two contexts, agreement on their intersection suffices for a strongly positive realization. On the bipartite graph of context atoms, the minimum negative mass of a signed extension is $(W_1-1)/2$, where $W_1$ is the transport cost for graph distance. A graph Laplacian formula gives the minimum trace. Its minimizers form an affine copy of a rectangular contraction ball whose extreme points have the smallest realization rank. For a pure-state implementation with one intermediate projective decomposition and a final postselection, the optimal success probability of a realization is the reciprocal of the least trace of a diagonal matrix dominating it. In a family of classical examples, this probability tends to zero for the unique trace minimizer even though another realization succeeds with probability one.

math.NT — Number Theory

REAL ZEROS OF L-FUNCTION DERIVATIVES: DENSITY-ONE ASYMPTOTICS IN ARITHMETIC FAMILIES

Contributed by Yuhyun Han

We study real zeros of the derivative of an uncompleted L-function on [1/2, 1]. The limiting stationary process has covariance sech2((t− s)/2) after the change of variables σ = 1/2+e^−t. Its nonzero-level Rice intensity gives the constant e^−1/2/(π√2) for quadratic Dirichlet characters, a specified family of quadratic Hecke characters over Q(i), and the unweighted level-one holomorphic family; the resulting density-one asymptotics are unconditional. For a fixed holomorphic modular form and its quadratic twists, and for non-Galois cubic Artin L- functions, we obtain the same constant under the respective family GRH. The real cyclic cubic augmentation instead has constant 1/(π√2). In these three GRH families a uniform quantile estimate controls all three counts: distinct real zeros, zeros with multiplicity, and strict sign changes. For an individual complex cubic character, the derivative has o(log log q) real zeros in density under cubic-character GRH. We also prove a uniform mean asymptotic and exponential concentration for the random Euler model.

math.CO — Combinatorics

Every $C_4$-free graph on $49$ vertices has a vertex of degree at most $6$: an independent verification

Contributed by David Niedbala Giraudin

ABSTRACT We give an independent proof that every graph on $49$ vertices containing no cycle of length four has a vertex of degree at most $6$; equivalently, $R(C_4,K_{1,42})\le 49$, hence $R(C_4,K_{1,42})=49$. In the notation of Erd\H{o}s, the least integer $f(n)$ such that every graph on $n$ vertices with minimum degree at least $f(n)$ contains a $C_4$ satisfies $f(49)=7$, while $f(48)=f(50)=8$, so $f$ is not non-decreasing. These values were first established by Walters, Claude Fable, GPT Sol, Astra and Claude Opus (12 October 2026), by a different case split whose computational part is checked by the formally verified checker cake_lpr and assembled in Lean; we were not aware of their work when this note was written. Our proof combines structural lemmas, valid for every odd $k$, on $C_4$-free graphs with $k^2$ vertices and minimum degree at least $k$, a symmetry reduction for $k=7$, and an exhaustive SAT search carried out by two separately written programs. Erd\H{o}s's question whether $f(n+1)\ge f(n)$ for all sufficiently large $n$ (Problem 85 of erdosproblems.com) remains open. COMMENTS 8 pages. Independent verification of f(49)=7 (Erdos Problem #85), first established by Walters, Claude Fable, GPT Sol, Astra and Claude Opus (Zenodo, doi:10.5281/zenodo.23325806, 12 October 2026). Ancillary folder anc/: pure-Python scripts, solver logs and SHA-256 hashes of all formulas. Technical assistance: Claude (Anthropic). COMMENTAIRE DE VERSION (cas B seulement) v2: credits the earlier proof of f(48)=8>f(49)=7 by R. Walters, Claude Fable, GPT Sol, Astra and Claude Opus (Zenodo, doi:10.5281/zenodo.23325806, 12 October 2026; v1.1 20 October 2026), of which the author was not aware when v1 was posted. Title, abstract, introduction (new paragraph "Related work") and Remark 3 updated; the 48-vertex graph is isomorphic to their witness. No mathematical change; ancillary files unchanged.

quant-ph — Quantum Physics

Exact amplitudes of sparse quantum circuits in subexponential time, assuming the new #SAT algorithms

Contributed by Vaibhav Tulsyan

Classical simulation lower bounds for quantum circuits have been derived from the Exponential Time Hypothesis (ETH). However, in October 2026 OpenAI and Anthropic claimed deterministic subexponential algorithms for #k-SAT, refuting ETH, and neither manuscript mentions quantum circuits. We record the corollary. Assuming three internal results of the Anthropic manuscript, one deterministic Turing machine computes the amplitude $\langle y|C|x\rangle$ of a Clifford+T circuit with $G$ two-qubit gates in arbitrary connectivity exactly, as an element of $\mathbb{Z}[\omega]/\sqrt2^{\,k}$ with $\omega=e^{i\pi/4}$, in time $2^{O(G/\sqrt{\log G})}$ whenever the input length $L$ satisfies $\log L\le G^{0.9}$. The proof reads the sum over paths, one bit per wire segment, as an Anthropic-type network, with a Kronecker map for $\mathbb{Z}[\omega]$ and a Galois bound. We machine-check this theorem in Lean 4 (amplitude_TM_rate', amplitude_TM_rate, amplitude_TM; axioms propext, Classical.choice, Quot.sound) on top of Anthropic's Lean development (sources at https://github.com/xennygrimmato/eth-hamiltonicity). On paper only, the method gives exact marginals and exact sampling, depth-$d$ circuits in $2^{O(dn/\sqrt{\log n})}$, and, through OpenAI's tensor-contraction theorem, polynomial space when the number of Hadamard-type gates is $O(G)$. Conversely, constant-depth circuits with $O(n)$ gates encode #3-SAT with $O(n)$ clauses, so the rate matches that of the refutations. We check the fine-grained supremacy conjectures we found against their sources; a counting form of the Calabro-Impagliazzo-Paturi width reduction refutes the gap conjecture of Morimae and Tamaki's depth-scaling paper. Dense statements (cubic phases, permanents, random circuits) survive. Nothing here concerns approximate sampling, the constants are galactic, and everything is conditional on the claimed refutations, which are under review.

math.LO — Logic

Inertia and Elementary Theories of Hermitian Nilpotent Operators

Contributed by Zhenyu Sun, Jilin Wang

Let $V$ be a finite-dimensional Hermitian space over an algebraically closed field with a nonidentity involution, and suppose its form $h$ is nondegenerate. For a self-adjoint nilpotent operator $N$, we recover the multiplicity of each signed Jordan block from finite differences of the ranks and signature differences of the forms $h(x,N^j y)$. On a fixed Hermitian space these inertias classify unitary conjugacy. Across scalar fields they classify elementary equivalence in the four-sorted language of scalars, vectors, endomorphisms, and projective points with $N$ named. At fixed dimension each admissible inertia sequence isolates a complete theory. These theories are model complete and decidable, and none has the independence property. A bivariate generating function counts them by ambient signature. For two self-adjoint square-zero operators with a common image of fixed positive dimension and positive definite quotient forms, simultaneous unitary conjugacy on a fixed Hermitian space is classified by the ordered generalized eigenvalues. The associated Hermitian pencil recovers these eigenvalues even when all operator words have constant ranks and inertias. At fixed signature and rank the family theory is model complete and decidable. Its completions correspond to the ordered-field types of the spectrum, and each completion has the Turing degree of its type. We also characterize the Witt index by square-zero ranks over arbitrary fields with involution. For positive definite spaces we prove that finite-measurement sentences at fixed dimensions and outcome count have the same truth value over $R(\sqrt{-1})$ as over $\mathbb C$ for every real closed field $R$. The least Naimark dilation dimension of a measurement is the sum of its effect ranks.

cs.DS — Data Structures and Algorithms

Counting Eulerian circuits in subexponential time, assuming the new #SAT algorithms

Contributed by Vaibhav Tulsyan

Counting the Eulerian circuits of an undirected graph is #P-complete, even for planar 4-regular graphs, while for directed graphs the BEST theorem counts them in polynomial time. However, we are unaware of any exact algorithm for undirected graphs of bounded degree that runs in time $2^{o(m)}$, where $m$ is the number of edges. We show that such an algorithm follows from the exact counting of Hamiltonian cycles that we proved earlier on top of Anthropic's claimed #SAT algorithm. The reduction is classical: by Kotzig's description, the Eulerian circuits of $G$ are in bijection with the Hamiltonian cycles of the line graph of the double subdivision of $G$, which has $3m$ vertices and at most $(\Delta+1)m$ edges. We prove in Lean 4 (theoremEuler, theoremEuler_noIso, theoremEuler_rate; axioms propext, Classical.choice, Quot.sound) that for every $\delta>0$ and $\Delta$ a finite-alphabet Mathlib Turing machine outputs the number of Eulerian circuits of every graph of maximum degree at most $\Delta$ within $\lfloor C(n+2)^C 2^{\delta m}\rfloor$ steps, and within $\lfloor C 2^{\delta m}\rfloor$ steps when there is no isolated vertex; and that one machine does so for all graphs within $\lfloor C(n+2)^C 2^{Cm\log_2(\Delta+2)/\sqrt{\log_2(m+2)}}\rfloor$ steps. The factor $(n+2)^C$ pays for reading the $n^2$-bit adjacency matrix and cannot be dropped. The bijection is checked by brute force on twelve small graphs, for example 132 circuits for $K_5$. The formal object is Kotzig's transition-system definition; its agreement with Mathlib's Eulerian walks is not formalised. The constants are astronomical, and the proof rests on Anthropic's Lean development of its claimed refutation of the Exponential Time Hypothesis, which is under review.

math.GM — General Mathematics

Uniform irreducibility and Galois groups of generalized Laguerre polynomials

Contributed by Khai-Hoan Nguyen-Dang

We prove that the generalized Laguerre polynomial $H_{n,s}$ is irreducible over $\mathbb Q$ and has Galois group containing $A_n$ for every positive integer $s$, once $n$ exceeds an effective absolute constant. Equivalently, every numerator and denominator in the exponential Pad\'e table has these properties once its own degree is sufficiently large. The exceptional positive integral pairs form an effectively computable finite set. The proof combines integral traces at degenerate fibers, Pad\'e resultants, and an energy inequality for Galois-invariant positive semidefinite forms on the roots.

cs.CC — Computational Complexity

Class Numbers of Imaginary Quadratic Orders in the Polynomial Hierarchy

Contributed by Frank Vega

For $d \geq 3$ with $-d \equiv 0, 1 \pmod 4$, let $h(-d)$ be the number of classes of primitive positive definite binary quadratic forms of discriminant $-d$. We prove that $d \mapsto h(-d)$ is computable in deterministic polynomial time with an oracle for a language in the second level $\Sigma_2^{\text{P}}$ of the polynomial hierarchy. The proof uses only reduction theory of binary quadratic forms, Lagrange's theorem and Hermite normal forms; it does not assume the generalized Riemann hypothesis, does not take a generating set of the class group as input, and uses no randomness. A $\Sigma_2^{\text{P}}$ oracle finds a generating set of $O(\log d)$ forms, and an NP oracle then determines the lattice of relations, whose index is $h(-d)$. Consequently, if $\text{P} = \text{NP}$, class numbers of imaginary quadratic fields are computable in polynomial time. As an application, the number $D_n$ of representations of $n$ by the ternary form $8x^2 + 2y^2 + 16z^2$ in Tunnell's congruent number criterion equals $2h(-4n)$ for square-free $n \equiv 2 \pmod 8$ and vanishes for square-free $n \equiv 6 \pmod 8$; hence $D_n$ lies in $\text{FP}^{\Sigma_2^{\text{P}}}$ on even square-free $n$, and an exact counting reduction from $\#\text{SAT}$ to $D_n$ would collapse the polynomial hierarchy to $\Delta_3^{\text{P}}$.

Author comments: In this new version, we have added a computer-based numerical verification.

math.NT — Number Theory

Selected character moments, zero-free half-planes 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 $\Rea s>B_2$ for every finite-order Hecke $L$-function over every number field, where \[ B_2=\frac{777-\sqrt{1081}}{852} =0.873381966613207519929\ldots. \] We adapt selected character-sum estimates and finite moment inductions to ideal-character sums over arbitrary number fields and restore ordinary sixth powers. Two finite moment inductions give an ordinary product estimate with length condition $n_1+n_2+2z\le M$ for sufficiently fine original prime factors. Applying this estimate to zero detection and the averaged numerator in the cubic-theta comparison gives the stated boundary. We determine the exact infimum of the resulting comparison system and verify its scalar inequalities by rational Bernstein bounds. The proof uses the earlier theta, moment and continuation theorems over number fields.

math.RT — Representation Theory

Nonpolynomial Hall numbers for tame and wild algebras

Contributed by Qiyue Tang

We give examples showing that fixed-triple Hall polynomials may fail to exist for tame nonhereditary algebras and for wild algebras, both hereditary and nonhereditary.

math.NT — Number Theory

Landau--Siegel zeros over varying number fields

Contributed by Baiying Liu

Let $K$ be a number field of degree $n$ and absolute discriminant $D_K$. We give a direct arithmetic proof that every real zero $\beta\in(0,1)$ of $\zeta_K$ satisfies \[ (1-\beta)\{\log D_K+n\log(2n)+6n+1\}\geq\frac1{1370}. \] The bound is explicit for arbitrary varying fields, including nonnormal fields, and gives $(1-\beta)\log D_K\gg1/\log(2n)$ for $K\ne\Q$. Exceptional-zero exclusion over each fixed base field was obtained by Banks and, independently, Frei--Kai. The proof here combines OpenAI's determinant method, Masser's zero estimate as used by Kerr, and Banks' selected-embedding calculation. The resulting bound is expressed directly in the degree and discriminant of $K$, without introducing those of its normal closure. Stronger nonvanishing conclusions follow from recent half-plane results and positivity. The contribution here is an arithmetic proof with fully displayed dependence on the varying field, using neither those half-plane results nor earlier zero-exclusion results for real Dirichlet $L$-functions or Hecke characters.

math.NT — Number Theory

GRAM OBSERVATIONS AND THETA-SERIES RECONSTRUCTION OF EUCLIDEAN LATTICES

Contributed by Changxia Lin

A finite collection of lattice-vector lengths probes a positive-definite Gram matrix through rank-one linear functionals. We use this observation map to give a local criterion for non-identifiability and to construct an exact three-dimensional collision at a closed squared-length cutoff of one: two non-isometric lattices have identical complete truncated vector-length data, although their Gram determinants are different. A signed-cycle construction gives a higherdimensional family whose complete rank-one length multisets agree at every cutoff below an explicit threshold, with an exact first-shell discrepancy. We then turn to multi-vector data. The full degree-n Gram theta data determines every rank-n Euclidean lattice, by a basis-witness and covolume-index argument. On each compact subset of lattice moduli space, the same argument yields a finite cutoff; a determinant-one cusp family shows that the basis-witness cutoff supplied by this method cannot be uniform on the full moduli space. Finally, a published rank-32 theta-series coincidence, together with a minimum-shell spanning identity, is stabilized by orthogonal direct sums to every rank n ≥ 32. If k(n) denotes the least degree of full Gram theta data that universally determines rank-n Euclidean lattices, we obtain 4 ≤ k(n) ≤ n for n ≥ 32. The paper separates proved statements, cited lattice-theoretic input, and computational checks; it does not claim a solution of the exact minimum-degree problem.

math.NT — Number Theory

A congruence modulo p^2 for the p-th coefficient of e(1-z)^((1-z)/z)

Contributed by David Niedbala Giraudin

Define E_n by sum_{n>=0} E_n z^n = exp(sum_{k>=1} z^k/(k(k+1))) = e(1-z)^((1-z)/z). Equivalently, e(1+1/x)^(-x) = sum_{n>=0} E_n (x+1)^(-n), the reciprocal of the expansion of e^(-1)(1+1/x)^x used by Yang in refinements of Carleman's inequality. We prove that for every odd prime p, E_p = -p/2 (mod p^2); in particular v_p(E_p) = 1. The proof is elementary. With Phi(u) = e(1+u)^(-1/u), it combines the two expressions of E_p through the coefficients of Phi that come from the identities e(1-z)^((1-z)/z) = Phi(-z)/(1-z) and e(1-z)^((1-z)/z) = Phi(z/(1-z)), and it reduces the congruence to the differential recurrence of the E_n read modulo p. A script checks the statement and the intermediate congruences for all odd primes below 200 and a few larger ones.

math.NT — Number Theory

The exact multiplicity of the Wilson supercongruence in F_q[t]

Contributed by David Niedbala Giraudin

Let q = p^s with p an odd prime, and let P be a monic irreducible polynomial of degree d >= 2 in F_q[t]. The Wilson analogue F_d = -1 mod P holds for the product F_d of all nonzero polynomials of degree less than d, and Thakur proved that P is a Wilson prime, i.e. F_d = -1 mod P^2, exactly when all exponents of P are congruent to 0 or 1 mod p, in which case the congruence holds modulo P^(p-1). He asked whether it can hold modulo a higher power, and what the exact multiplicities are. Let r be the largest integer such that all exponents of P are congruent to 0 or 1 modulo p^r. We prove that the exact multiplicity m = v_P(F_d + 1) is 1 if r = 0, p^r - 1 if 1 <= r <= s, and q + 1 if r > s. In particular m <= q + 1, and the multiplicity of a Wilson prime in the numerator of the sum over i < d of 1/(t^(q^i) - t) is exactly m - 1. Examples over F_3, F_5 and F_9 show that all regimes occur. A script checks the theorem on all primes of small degree and on these examples.

math.NT — Number Theory

The p-adic valuations of the coefficients in Yang's expansion of (1+1/x)^x

Contributed by David Niedbala Giraudin

Yang's coefficients b_n, defined by (1+1/x)^x = e(1 - sum_{n>=1} b_n (x+1)^{-n}), appear in refinements of Carleman's inequality, and the literature on them concerns their analytic properties. We study their p-adic valuations for an odd prime p. From the identity 1 - sum_n b_n u^n = exp(-sum_{k>=1} u^k/(k(k+1))) one gets v_p(b_n) >= -M(n), where M(n) is the optimum of a knapsack problem on the compositions of n. We compute M(n) in closed form and show that, for n >= 2, the optimum is attained by a unique composition exactly when n = (p-1)m with m in an explicit set U_p defined by a digit condition. For these n the bound is an equality: v_p(b_n) = -(m + v_p(m!)). The set U_p has density 1/(p-1), so the valuation of b_n is determined on a set N_p of indices of density 1/(p-1)^2. The set N_p contains the indices p^K - 1, at which b_{p^K-1} is the unique coefficient of minimal valuation among b_1, ..., b_{p^K-1}. The same statements hold for the coefficients of e(1-u)^{1/u}, and the minimality statement for those of e(1-u)^{1/u-1}. A script checks all statements numerically in finite ranges.

math.NT — Number Theory

Torsion of elliptic curves over number fields of degree $6$, $7$, $8$ and $9$

Contributed by Filip Najman

We determine all the groups that appear as the torsion group of an elliptic curve over a number field of degree $d$ for $d=6$, $7$, $8$ and $9$. Apart from the groups that occur infinitely often, exactly $4$ groups occur in degree $6$, $3$ in degree $7$, $8$ in degree $8$ and $9$ in degree $9$, and among them the groups $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/24\mathbb{Z}$ over sextic fields, $\mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/15\mathbb{Z}$ over octic fields and $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/26\mathbb{Z}$ over nonic fields were not known to occur. For $13$ of these $24$ pairs of a degree and a group we also determine all the elliptic curves that realize them. The proofs combine the global method of Derickx, Hecke sieves, formal immersion criteria at the prime $2$ with Hecke operators, and Riemann-Roch computations on modular curves and their quotients over finite fields.

Author comments: Fixed ten references to equations (6) and (10), which appeared as "??" in v1 (compiled with TeX Live 2025). No mathematical changes.

math.NT — Number Theory

A sharper Type I/II estimate for primes in arithmetic progressions to smooth moduli, and H_m << e^{3.8043 m}

Contributed by David Niedbala Giraudin

In Stadlmann's Type I/II equidistribution estimate for x^delta-smooth moduli, one of the three conditions, 64 varpi + 20 delta + 2 sigma < 1, can be replaced by 48 varpi + 18 delta + 2 sigma < 1. This weaker condition follows from the second one, 48 varpi + 16 delta + 4 sigma < 1, whenever delta <= sigma. The proof follows Stadlmann's argument and makes two changes: the length of a block in one Taylor expansion depends on the pair of shifts, and the bounds are summed over the pairs instead of being multiplied by their number. Combined with Stadlmann's general Harman-sieve construction, this gives a minorant for the primes with exponent of distribution 1/2 + 2 varpi to smooth moduli for every varpi < 7/544, instead of varpi = 0.01265. As a consequence, liminf (p_{n+m} - p_n) << e^{cm} for every c > 3.80428, and for every c > 3.80423 if a numerically computed value of the sieve deficit is used. This improves Stadlmann's exponent 3.8075.

math.NT — Number Theory

Serre modularity over real quadratic fields

Contributed by Lassina Dembélé, Luis V. Dieulefait, Nuno Freitas, Ariel Pacetti

We prove Serre's modularity conjecture, in every odd residual characteristic and with arbitrary conductor, over twenty real quadratic fields $K=\Q(\sqrt d)$. The proofs for $d=2,5,13,17,41$ use classical discriminant bounds. For $d=33$ the proof uses the zero-free region $\Re s>31/32$ of Cureton and Thorner, and for $d=6$ the region $\Re s>0.9586$, which we derive from specified analytic inputs. For the remaining thirteen fields we assume GRH for Dedekind zeta functions. The main tools are conductor reductions at an odd split prime and at two, a weight-reduction step by congruences, a scalar dyadic modularity criterion, and integral trace orders. We also use a modularity theorem for the mixed ordinary--nonordinary case and, for the seven fields in which two splits, dyadic propagation results.

math.CO — Combinatorics

Three-colour cuboids from four-colour cubes

Contributed by Jason Davies

We determine the least number of four-colour cubes that guarantees a cuboid with monochromatic faces using exactly three colours. Each source uses all four colours of a fixed palette; only proper rotations are allowed, and interior faces need not match. For integer side lengths $2\le a\le b\le c$, the sharp threshold for the corner-and-edge frame is $\max\{16a-3,4(a+b+c)-16,\lfloor(24c-1)/5\rfloor\}$, and the complete-cuboid threshold is $\max\{abc,16a-3,\lfloor(24c-1)/5\rfloor\}$. The three frame bounds arise from colour-bucket capacity, the number of positions and balanced checkerboards. The upper bound combines corner containment, a weighted graph selecting edge directions, and Hall's theorem. For cubic frames the threshold is $16n-3$, with exactly two failing collections of size $16n-4$. For $n\ge3$ we classify every failure of size at least $14n$ and count these collections and the inclusion-maximal ones. The case $n=2$ answers the threshold question in Problem 2 of Berkove, Eickemeyer and Schulman (2018). All threshold, stability and counting results are formalised in Lean.

math.NT — Number Theory

Turán's pure power sum problem: certified bounds for finite $n$ and $\limsup R_n\le 0.68637$

Contributed by Vasileios Bakeas, Ricardo Pinho

For $n\ge1$ let $R_n=\inf\max_{1\le k\le n}|z_1^k+\dots+z_n^k|$, the infimum being over all complex $z_1=1,z_2,\dots,z_n$. This is Turán's pure power sum problem (Erdős problem #519). The best published upper bound for the constant $\limsup R_n$ is $0.69368$, from a computation of Harcos reported by Biró in 2000. We prove $\limsup_{n\to\infty}R_n\le0.68637$. The construction is the Biró–Harcos linearization, with the first half of the power sums prescribed as $s_k=Te^{i\varphi(k/n)}$ for a piecewise-linear phase $\varphi$ that varies from the start. We prove a limit theorem: the quantity that controls the second half converges to an explicit functional of the solution of a linear Volterra equation whose associated operator is a contraction. The resulting inequality is certified in ball arithmetic by an a posteriori error bound, with two separately written programs that follow the same mathematical argument. We also give rigorous upper bounds for $R_n$ at 13 values of $n$ between $300$ and $10^6$, from explicit exact certificates, each checked by one or two programs written independently of the generator. In an appendix, R. Pinho applies the limit theorem to a finer profile, with sharper error bounds, and certifies $\limsup R_n\le0.6863607950$. The work was done with substantial help from AI models; it has not been refereed.

math.NT — Number Theory

Arcs of Π(1+z^p): prime races, the Gaussian core, Dirichlet L-functions and a certified solver for Subset Sum

Contributed by Antonio Mennillo

The subsets of $\{1\} \cup \{p \le N\}$ reach every integer between 0 and their total, by Bertrand's postulate, so deciding whether a target is a subset sum of the primes is trivial. Counting the solutions is not: the count carries the arithmetic of the primes. We study the generating function $F_N(z) = \Pi(1 + z^w)$ on the unit circle and follow one object, the arc at a rational $a/q$, through three settings. For the primes, the centre of the arc $a/q$ rises above the sea of the minor arcs exactly when $q = 2^k (k \ge 2)$ or $q = 2p^j$, with excess $\Lambda(q/2)/\varphi(q)$, and the neighbourhoods of all the other arcs are bounded uniformly in $q$; Theorem E gives the expansion arc by arc. The height of an arc is a race of the primes modulo $q$ with weights $\log |\cos (\pi ra/q)|$. By Landau's theorem a one-sided bound on it is equivalent to GRH for the even characters it sees, unconditionally at $q = 8$, 10 and 14, where the zero-free segments it needs are certified; at $q = 14$ the race changes sign infinitely often. The principal arc carries $\zeta$ in the centre and width of its Gaussian. For arbitrary weights, the arcs become the decision rule of a Subset Sum solver that reads them on the original weights and ends with a meet-in-the-middle filtered by a random residue class. Every answer it returns is verified. The dummy weights of HURS explain why the reading is exact: they blur the counts by a flat window, and the blur is invertible on counts, though not on reachable sets. On random weights of $b$ bits the method has rigorous guarantees, in time $2^{b/2+o(b)}$ and, with the filter, $2^{b/3+o(b)}$, resting on a local limit theorem uniform in the size of the weights and on a concentration bound for signed subset sums. Finally, the solver is read inside a general theory of constraint propagation, in which every network factors as $T = T_{\mathrm{comp}} \circ T_{\mathrm{struct}}$.

math.NT — Number Theory

On the Fontaine--Mazur conjecture for $2$-adic representations

Contributed by Luis Dieulefait

Let $F$ be a totally real field, abelian over $\Q$, in which $2$ splits completely, and let $\rho:G_F\to\GL_2(\overline\Q_2)$ be an odd irreducible representation, de Rham with distinct Hodge--Tate weights at the places above $2$, whose residual representation is reducible with distinct characters at the places above $2$. We prove that $\rho$ is modular in the following cases: if $\rho$ is non-split at every place above $2$ at which it is reducible, which includes the case where $\rho$ is irreducible at all places above $2$, and, without any condition on the splitness, if $\rho$ is of Steinberg type at some finite place not dividing $2$, using an analogue at $2$, which we prove, of Pan's finiteness theorem for fully ordinary deformation rings. Both results include the mixed case, in which $\rho$ is reducible at some places above $2$ and irreducible at others. The pro-modularity input is obtained by extending to these fields the method of \cite{OAI}, which treats representations of $G_\Q$; this is where we use that the residual characters are distinct at $2$. For $F=\Q(\sqrt q)$ with $q\equiv1\pmod8$ prime, we remove the condition at $2$ from the first result in most cases; for $F=\Q(\sqrt{17})$ this includes every case where the residual semisimplification is scalar. The first result also holds when the residual representation is irreducible with solvable image, where residual modularity follows from an odd Artin lift. For the second result we combine the method of \cite{Die26b} with Thorne's potential modularity theorem for ordinary representations of weight two and the local indecomposability theorem of Hida and Zhao.

math.CO — Combinatorics

Off-Diagonal Commonality of a Shorter Even Cycle and a Longer Odd Cycle

Contributed by Taeyoung Kim

Let $n\ge4$ be even and let $\ell>n$ be odd. We determine the complete off-diagonal commonality region of the ordered pair $(C_n,C_\ell)$: it is the half-open interval $(0,a_{n,\ell}^{\star}]$ cut out by the unique root $a_{n,\ell}^{\star}\in(1/2,1)$ of $\frac{a^{n-1}\bigl(n+(\ell-n)a\bigr)}{\ell}=2^{1-n}.$ The upper endpoint is witnessed by the balanced disjoint union of two cliques. For the lower bound, cycle densities are expressed as traces of graphon operators. Rank-one eigenvalue majorisation separates the Perron eigenvalue from the rest of the spectrum, while the Hilbert--Schmidt norm leaves room for at most one negative eigenvalue to contribute with the wrong sign. That eigenvalue is paid for by the Perron eigenvalue, leaving the elementary inequality $x^{n}+(1-x)^{n}\ge2^{1-n}$.

math.PR — Probability

Computer-assisted bounds 0.556921 <= p_c <= 0.6463 for site percolation on the square lattice

Contributed by David Niedbala Giraudin

Let p_c be the critical probability of Bernoulli site percolation on Z^2. We prove 0.556921 <= p_c <= 0.6463. The previous best rigorous bounds known to us are p_c >= 0.5567 (Ashton, 2026, extending van den Berg and Ermakov, 1996) and p_c <= 0.666894 (Wierman and Oberly, 2024). The lower bound uses the growth-process comparison of van den Berg and Ermakov with the exact posterior on the exploration tree. We reduce this posterior to a scalar interval recursion and explore the tree generation by generation, with a prescribed order among siblings. This passes the value 0.556788... that the same scalar computation gives when the order is not used, and the value 0.556788090 reported by Ashton as the limit of his relaxation. The upper bound uses the substitution method of Wierman, in two stages with 4x4 regions, from the line graph of Z^2 (site threshold 1/2) to Z^2. Both bounds are certified by short pure-Python programs in exact rational and integer arithmetic.

math.LO — Logic

Intuitionistic Compression and Optimal Verification Trees

Contributed by Zhenyu Sun, Jilin Wang

In strict finitistic generation semantics, a formula is assertible at a generation if some stage in its verification tree forces it. We ask when assertibility agrees with intuitionistic truth on the poset of generations. The leaves of the verification trees, ordered by generation extension, form an intuitionistic Kripke model that preserves truth at every leaf, even without atomic prevalence. For a finite bundle, compression to the generations is exact if and only if each fiber of leaves has a greatest intuitionistic theory and the map selecting these theories has the back property. Compression remains exact under every restriction of the generation poset if and only if atomic prevalence holds. Conversely, consider coherent tree representations of a finite Kripke model whose atomic valuation reflects its order. Among representations with atomic prevalence, the minimum numbers of leaves and nodes over $u$ are $h(u)$ and $\delta_C+\sum_{x\preceq u}h(x)$, where $h(x)$ counts saturated paths from minimal worlds to $x$, and $\delta_C$ is $1$ if the comparability component of $u$ has no least element and $0$ otherwise. Saturated-path trees attain both minima simultaneously. Among representations attaining the node minimum at every world, the common root labels in components without a least element determine the representation up to isomorphism over the outer frame. Allowing rooted posets in place of trees can reduce the number of nodes exponentially.

math.NT — Number Theory

A certified lower bound for the essential minimum of the Zhang-Zagier height

Contributed by David Niedbala Giraudin

Let h be the absolute logarithmic Weil height. We prove that the essential minimum of the Zhang-Zagier height h(a)+h(1-a) is at least 0.2497815: for every d > 0, only finitely many algebraic numbers a satisfy h(a)+h(1-a) < 0.2497815 - d. The previous best lower bound was 0.24874 (Flammang, 2018). The proof combines Smyth's auxiliary-function method, with the polynomials of Doche and of Flammang, and the condition that the discriminant of the minimal polynomial is a nonzero integer, in the potential-theoretic form used by Smith and by Orloski, Talebizadeh Sardari and Smith for the trace problem. The infimum of the resulting auxiliary function is bounded below rigorously by monotone interval bounds and adaptive bisection, from a stored certificate. The statement concerns the essential minimum and is weaker in nature than the pointwise bounds of Doche and Flammang.

math.RA — Rings and Algebras

Cartan determinants for almost laura, quasi-laura, semilinear clannish and piecewise Bäckström algebras

Contributed by Qiyue Tang

We prove the Cartan determinant conjecture for corners of finite global dimension in almost laura, quasi-laura and (m)-quasi-laura Artin algebras, finite-dimensional piecewise Bäckström algebras, and semilinear clannish Artin algebras of finite dimension over their defining division ring.

math.CO — Combinatorics

Kernel Dimension of the Z/2 Connection Laplacian on a Finite Simple Graph (A Formalisation in Lean 4 / Mathlib v4.11.0)

Contributed by Jesus Vilela Jato

We present a Lean 4/Mathlib v4.11.0 formalization of the kernel-dimension theory of the $\mathbb{Z}/2$ connection Laplacian on finite simple graphs equipped with designated wrap edges. Each edge carries either the identity or the coordinate-swap matrix. A Hadamard–Kronecker change of basis decomposes the resulting rank-two connection Laplacian into the ordinary and signed graph Laplacians. An analogous splitting holds for the scalar Laplacian of the associated two-sheeted cover. A connected component is balanced precisely when its covering preimage has two connected components, equivalently when its sign holonomy is trivial along every closed walk. Consequently, the nullity of the signed Laplacian equals the number of balanced connected components, while the nullity of the full connection Laplacian equals the number of connected components plus the number of balanced components. For the flat connection, the nullity is twice the number of connected components. We also formalize exact characteristic polynomials and eigenvalue multisets, including multiplicities, for explicit ordinary and one-wrap signed cycle matrices for $n \geq 3$. Transfer to general connection-graph encodings remains conditional on specified matrix-reindexing identities. The contribution is a machine-checked formalization, exposition, and documented verification of results rooted in classical signed-graph spectral theory, rather than a claim of priority for those identities. The accompanying repository preserves proof artifacts, computational cross-checks, historical verification evidence, and documented corrections.

cs.DS — Data Structures and Algorithms

Three-machine unit-job scheduling in time O((L+2)^276)

Contributed by David Niedbala Giraudin

OpenAI recently gave a deterministic polynomial-time algorithm for P3|prec,p_j=1|C_max (problem OPEN8 of Garey and Johnson), with running time O((L+2)^150020) on a multitape Turing machine, L being the input length; the same exponent appears in the accompanying Lean statement. The exponent comes from enumerating all Boolean formulas with at most 10000 leaves, each leaf choosing its own triple of jobs. We show that every description actually used by the structural proof involves at most 21 distinct job triples and at most 3 distinct rank thresholds. Restricting the enumerated family accordingly, and using the length bound on accepting paths already proved there to limit the number of reachability passes, the same algorithm runs in time O((L+2)^276).

math.NT — Number Theory

Two improved lower bounds for the 2-adic valuation of the constant of the Prouhet-Tarry-Escott problem

Contributed by David Niedbala Giraudin

For an ideal solution of the Prouhet--Tarry--Escott problem of size $n$ the associated constant $C$ is divisible by a large power of $2$. Restricting to symmetric solutions, we prove two new lower bounds for $v_2(C)$: $v_2(C)\ge14$ at $n=15$, improving $12$, and $v_2(C)\ge19$ at $n=20$, improving $18$; both earlier bounds are those of the companion article hexagon:2611.00057. The first rests on two identities satisfied by every symmetric solution of odd size, one relating $v_2(C)$ to a difference of two products and one turning the index relation into an equality, together with an arithmetic criterion for a block of entries of equal valuation. The second rests on an identity internal to each of the two sides, from which we deduce that the vanishing of the cross invariant $R$ forces $k_0\equiv0\pmod4$ when $k_0\ge4$. Two self-contained verifiers accompany the note.

math.NT — Number Theory

The 2-adic valuation of the constant for symmetric ideal Prouhet-Tarry-Escott solutions

Contributed by David Niedbala Giraudin

For an ideal solution of the Prouhet--Tarry--Escott problem of size $n$, let $C$ denote the associated constant, and let $\overline{C_n}$ be the greatest common divisor of these constants. Filaseta and Markovich obtained $v_2(\overline{C_8})\ge6$ and determined $v_2(\overline{C_9})=9$, and asked for improved lower bounds in the range $10\le n\le12$; Bera and Nair gave bounds for $n=10$ and $n=12$. We treat the symmetric case at every size, even and odd, by $2$-adic arguments throughout. A single identity, the row law, governs both parities. For $n$ even, over $\mathbb{Z}$, it splits into two regular systems and yields $v_2(C)\ge6,8,11,12,12,16,18,18$ for $n=6,8,10,\dots,20$. The values at $n=6,8,10,12$ are attained and hence exact for the symmetric constant: in the symmetric case this closes both brackets of Bera--Nair, and gives $v_2(C'_8)=8$, for which the published information was $v_2(\overline{C_8})\ge6$. We also record that the two lower bounds of Bera--Nair, $9$ at $n=10$ and $11$ at $n=12$, are attained by $2$-adic solutions symmetric about $-1/2$, so that no argument using $2$-adic information alone can raise them. For $n$ odd, over $\mathbb{Z}_2$, it combines with a finite-difference lemma refining a congruence of Filaseta--Markovich to give $v_2(C)\ge11,12,12$ for $n=11,13,15$, against $8$, $10$ and $11$ from Kleiman's bound $(n-1)!\mid C$; the chain also reproduces the known values $4,6,9$ at $n=5,7,9$. At $n=11$ the bound is sharp: the two quadratic near-solutions of Coppersmith, Mossinghoff, Scheinerman and VanderKam are symmetric ideal solutions over $\mathbb{Z}_2$ with $v_2(C)=11$, since in both of the real quadratic fields involved the prime $2$ splits. This answers, for $n=11$ and $p=2$, two questions of Filaseta--Markovich, and exhibits for those sizes the case $p\le n$ of the question closing Rees--Smyth, who construct $p$-adic ideal solutions only for $p>n$: at the prime $2$ there is no local obstruction at these sizes. At $n=13$ we exhibit a symmetric ideal solution over $\mathbb{Z}_2$, obtained from a second object of the same authors, although its constant has valuation $17$.

math.CO — Combinatorics

Log-concavity in the Newton basis and support of rank-two scattering coefficients

Contributed by Qiyue Tang

We prove that polynomials with nonnegative, interval-supported log-concave Newton coefficients are closed under multiplication and, for integer Newton coefficients, under $D(m)\mapsto\binom{D(m)}k$ for integers $k\ge0$. Coordinatewise Newton multiplication preserves two-variable Schur positivity. We also determine the exact positive Newton support of symmetric rank-two cluster scattering coefficients.

cs.DS — Data Structures and Algorithms

Ulam Metric $k$-Center in Parameterized Linear Time

Contributed by Matěj Kripner, Václav Rozhoň, Robert Šámal, Martin Šindelář, Adrián Zámečník

Continuous Ulam Metric $k$-Center asks for at most $k$ permutations whose radius-$d$ balls cover the input rankings. For $m$ rankings of $n$ symbols, a deterministic algorithm takes $2^{O(kd\log(k+d+2))}mn$ word operations. Permanent reference rankings, short insertion lists, and small obstructions obtained by patience sorting control the search. For one center, this answers a question of Bai, Fomin, Golovach, More, and Wietheger with linear input dependence. An arbitrary-alphabet Hamming embedding with a nonexpansive retraction also gives the matching $2^{o(d\log d)}$ exclusion under the Exponential Time Hypothesis.

cs.DS — Data Structures and Algorithms

The Exact One-Half Threshold for Arbitrary-Capacity Bottleneck Multiple Subset Sum

Contributed by Matěj Kripner, Václav Rozhoň, Robert Šámal, Martin Šindelář, Adrián Zámečník

In bottleneck multiple subset sum, items with weights are packed into knapsacks with capacities, possibly leaving items unpacked, so as to maximize the minimum load. For identical capacities the best possible approximation ratio is $2/3$, and Chen et al.\ asked whether $2/3$ is also achievable for arbitrary capacities. We show that it is not: for arbitrary capacities we give a $1/2$-approximation using $O(n\log(n+1)+m\log(m+1))$ arithmetic and comparison operations for $n$ items and $m$ knapsacks, and prove that no fixed ratio above $1/2$ is achievable unless $\mathsf P=\mathsf{NP}$. We also give a short proof of the classical $1/2$ guarantee of Friesen and Langston for unit profits.

cs.CC — Computational Complexity

Universal-Clique Amplification and Hardness of Assignment-Optimal Edge Clique Covers

Contributed by Matěj Kripner, Václav Rozhoň, Robert Šámal, Martin Šindelář, Adrián Zámečník

An edge clique cover has assignment cost equal to the sum of its clique sizes. Unless $P=NP$, this cost admits no polynomial-time $O(N^{1/2-\eta})$-approximation for any fixed $0<\eta<1/2$, even on connected $N$-vertex graphs. A universal-clique join transfers coloring hardness without the additive obstruction in earlier reductions. Its exact residual formula also exhibits a change of optimal cover when a universal twin is added. Explicit counterexamples refute two proposed twin-extension rules, while weighted compression of true-twin classes preserves the optimum.

math.PR — Probability

Stationary diffusion on nonnegatively curved graphs

Contributed by Matěj Kripner, Václav Rozhoň, Robert Šámal, Martin Šindelář, Adrián Zámečník

Lazy random walk on a bounded-degree graph of nonnegative Ollivier–Ricci curvature has diffusive second moment and Gaussian upper tails when started in stationarity. If the maximum degree is $D$, then $\mathbb E d(X_0,X_n)^2\le4Dn$, with optimal order of dependence on $D$. The results also hold for stationary random rooted infinite graphs. The proof combines known heat-kernel total-variation decay with a reversible energy identity for the conditional mean displacement. Its exponential form controls variation between starting vertices; martingale concentration then controls the walk itself.

math.IT — Information Theory

Least-Atom Extremizers for Resampling Noise

Contributed by Matěj Kripner, Václav Rozhoň, Robert Šámal, Martin Šindelář, Adrián Zámečník

A single least-mass atom determines the extremal behavior of resampling noise on a finite probability space. A fixed-moment comparison gives the sharp generalized Samorodnitsky inequality for every $1<q<2$ through one scalar curve, and identifies all nonconstant scalar extremizers. The same principle determines every $L^p$-to-$L^q$ resampling norm for $1<p<q<\infty$, resolving a conjecture of Cao, Fan, Han, Qiu, and Wang. The sharp hypercontractive parameter increases strictly with the least atom. The uniform-binary phase diagram is explicit, and product equality cases are classified for every nonuniform reference law and for the uniform binary law.

math.NT — Number Theory

Nested Squares and Distinct-Prime Goldbach Partitions

Contributed by Frank Vega

The distinct-prime Goldbach conjecture asserts that for every $N\geq 4$ there is $M\in A_N=[1,N-3]$ with $N-M$ and $N+M$ both prime; geometrically, nested squares of side lengths $N$ and $M$ have an L-shaped difference of semiprime area $(N-M)(N+M)$. With $C_N$ and $D_N$ the sets of $M\in A_N$ for which $N-M$, respectively $N+M$, is prime, the conjecture for $2N$ is $C_N\cap D_N\neq\varnothing$. We embed this centered condition in a straddling one: $C_N\cap D_N$ lies in the set $E_N$ of half-differences $(Q-P)/2\in A_N$ of primes $2<P<N<Q<2N$. From Mikawa's theorem on prime pairs with almost all differences we deduce $|A_N\setminus E_N|\ll_D N/(\log N)^D$ for every $D>0$, and with explicit constants we prove $|E_N|>N/760$ for every $N\geq 3275$. Consequently, for all large $N$ and all but $O_D(N/(\log N)^D)$ primes $P<N$, the number $2N-P$ becomes prime after adding the difference $P'-P$ of two primes $P'<N$; Goldbach's partition is the case $P'=P$. In contrast, $|C_N\cap D_N|<(N-3)-\log^2(2N)$ for every $N\geq 34$, and covering all but $\log^2(2N)$ values of $A_N$ for all large $N$ would imply the Cram\'er-type bound $p'-p<2\log^2(2p+2)+2$. The method also shows that $2N$ is a sum of two distinct primes for more than $X/9600$ integers $N\leq X$, for every $X\geq 10^6$. Computations up to $N=10^6$ compare the number of distinct-prime partitions with the Hardy--Littlewood prediction and measure the missing values of $E_N$ forced by prime gaps. The conjecture itself is not proved here.