All works

math.OC — Optimization and Control

A Sixth-Order Hurwitz Interval Polynomial Family Without Degree-Five Weak Strictly Positive Real Numerator

Contributed by Tiancheng Xu, Zhengtai Xiong, Wensheng Yu

We construct an explicit sixth-order Hurwitz-stable interval polynomial family with independent closed coefficient intervals that admits a common strictly positive real (SPR) numerator of degree six, yet admits no common weak strictly positive real (WSPR) numerator of degree five. More strongly, no nonzero real polynomial of degree at most five makes the real part of the associated transfer functions nonnegative across the family at all real frequencies. This counterexample shows that, for sixth-order interval polynomial families, robust Hurwitz stability does not guarantee the existence of a strictly proper common SPR or WSPR numerator of degree five, and equal numerator degree six is necessary in general.

math.NT — Number Theory

Exact Intervals and Arithmetic Structure in Fuchs' Extremal Problem

Contributed by Zhenyu Sun

We determine Fuchs' weighted unit-circle extremum for monic integer polynomials for $0<\lambda\le3$. The proof combines a fractional difference energy with a sharp rearrangement inequality and an explicit gap for nonnegative integer sequences that are not interval indicators. We obtain the finite-degree minimum on $(1,2)$ at every degree and on $(2,3)$ at every degree at least five. Every minimizing sequence on these intervals eventually consists, up to monomial factors, of Dirichlet polynomials and products of two adjacent Dirichlet polynomials, respectively. At integer parameters, Parseval's identity reduces the problem to integer coefficient norms. The lower bound is attained exactly when a multiplicity-free ideal Prouhet--Tarry--Escott pair exists. Classical PTE constructions give exact values for the integers $1$ through $10$ and for $12$, and a two-value alternative at $11$. Subset counting gives a quadratic upper bound in general. The interval formulas and explicit constructions show that $2$, $3$ and $4$ are strict local maxima; the formula on $(2,3)$ has a unique minimum. A Pell construction gives an upper bound on $(4,5)$, and coefficient-norm gaps exclude Dirichlet products uniformly near certain integers. For each fixed degree bound, the extremum, divided by a common gamma factor, is locally the lower envelope of finitely many rational functions. Exact lattice enumeration identifies a nonproduct minimizer on an open interval containing $17/4$ for every degree bound from $61$ to $80$.

Author comments: 24 pages. Includes exact-arithmetic programs and data.

math.HO — History and Overview

Grothendieck Montpellier. Énoncés des lectures modernisées démontrés.

Contributed by Michel Hua

Working notes of Alexandre Grothendieck, put online by the University of Montpellier, have been transcribed and restated in current notation with a language model; the restatements may contain errors. This appendix proves statements taken from these restatements in seventy-five folders, in commutative algebra, algebraic geometry, plane affine geometry, general and tame topology, category and group theory, homotopical algebra, order theory, algebra and finite combinatorics. Each proof follows a proof checked in the Lean 4 proof assistant with its library mathlib. For each statement the text gives the context of the leaf, the proof, the hypotheses that are superfluous or necessary, and the corrections made to the restatement or to the page. It ends with the list of statements the sections report without a Lean check.

math.CO — Combinatorics

Finite Field Grothendieck Inequality, Finite Field Johnson-Lindenstrauss Flattening and Finite Field Bourgain-Tzafriri Restricted Invertibility Problems

Contributed by K. MAHESH KRISHNA

We ask for finite field versions of following three: (1) Grothendieck Inequality, (2) Johnson-Lindenstrauss Flattening Lemma, (3) Bourgain-Tzafriri Restricted Invertibility Theorem.

math-ph — Mathematical Physics

A projective twistor correspondence for local analytic Einstein germs

Contributed by Lior Isthmus

For each prescribed numerical cosmological constant, including zero, we establish a convergent projective twistor correspondence recovering every marked four-dimensional local analytic Einstein germ of any algebraic Weyl type. The inputs are two independent analytic Čech classes of weights $2$ and $-6$ on a fixed flat twistor germ $T_0$ along its whole marked zero line, encoding both complete chiral curvature records. The fixed source correction and calibration select a unit optical section, an intrinsic projective extension condition recognizes Einstein geometry, and the prescribed normalized contact, optical and affine maps and their inverses converge on common neighborhoods of the whole physical sky for bounded holomorphic input families. On independently admitted ordinary and operator domains we prove both inverse comparisons, with the two source operations acting in one projective analytic Heisenberg model. The same comparisons preserve the nonlinear source, compact point objects, optical evaluation and calibrated infinity forms, together with physical point and full spin-frame changes and coupled Lorentz reality. For fixed positive Schwarzschild mass, we obtain finite horizon-crossing presentations, a calibrated local primitive comparison and optical approximation bounds for the entire specified compact nonradial crossing family using finite local readouts and the exact benchmark cuts, coordinate comparisons and Sachs reference frames.

Author comments: v1.0. The same manuscript is available at https://doi.org/10.5281/zenodo.23975732.

math.CT — Category Theory

Acyclic Networks: Causal Comparisons, Convex Minors, and Gauge Descent

Contributed by Zhenyu Sun

We study natural transformations, minors, and symmetric monoidal decorations of finite directed acyclic networks. For any weakly connected small source category, a natural transformation into a free acyclic path category is determined by one component. If more than one transformation exists, common prefixes and suffixes give a canonical factorization through two constant functors and identify the transformation set with a single set of directed paths. Consequences include boundary rigidity, a decision algorithm, and a fundamental-group obstruction to non-uniqueness. For fusion–embedding minors, right fillings exist exactly when the graph pushout is acyclic. A support invariant detects a nonexact minor under the full gauge congruence, while path-convex embeddings admit two-sided fillings. Inverse-image pullbacks give canonical partial-map normal forms, a scanning algorithm, and a coherent presentation of the cartesian quotient. For open networks, wire splicing gives units, and diagram groupoids satisfy descent by homotopy pullback over the interface groupoid, including through-wires. This descent is invariant under strong symmetric monoidal equivalence and compatible with admissible event contractions within each piece. Finite-field examples yield orbit classifications, stabilizer weights, and boundary convolution formulas. Roof-refinement spaces realize spheres whose higher homotopy is lost under groupoid completion.

math.CV — Complex Variables

Supremum estimates for $\bar\partial$ on bounded convex domains in $\mathbb{C}^2$

Contributed by Yun-Heng Du

For every bounded convex domain $\Omega\subset\mathbb{C}^2$, we give a bounded distributional solution of $\bar\partial u=f$ for every distributionally $\bar\partial$-closed form $f\in L^\infty_{(0,1)}(\Omega)$. The solution satisfies $\|u\|_\infty\le C_\Omega\|f\|_\infty$, where $C_\Omega>0$ depends only on $\Omega$. The estimate follows from holomorphic localization of Leray rows and boundary cancellation.

math.NT — Number Theory

Points of prime order on elliptic curves over number fields of degree $9$ to $22$

Contributed by Filip Najman

Let $S(d)$ be the set of primes $p$ for which there exists an elliptic curve over a number field of degree $d$ with a point of order $p$. The sets $S(d)$ are known for $d\leq 8$. We determine $S(d)$ for all $9\leq d\leq 22$. For example, the largest prime in $S(22)$ is $67$, and $S(16)$ consists of the primes up to $37$. To reduce the problem to finitely many primes we prove a version of Oesterlé's bound which uses reduction modulo $2$ instead of modulo $3$, and which is much smaller for $d\geq 13$. The remaining primes are dealt with using results of Derickx and Stoll and extensive computations with the modular curves $X_1(p)$ over $\mathbb{F}_2$ for $p\leq 139$.

Author comments: Code: https://github.com/F-Najman/Sd. Computational certificates: https://doi.org/10.7910/DVN/5HHEY4.

math.RA — Rings and Algebras

The L'vov–Kaplansky conjecture for polynomials of degree four

Contributed by Qiyue Tang

We prove the L'vov–Kaplansky conjecture for multilinear polynomials of degree four over algebraically closed fields of characteristic zero.

math.CO — Combinatorics

A new proof of van der Waerden's inequality, and beyond

Contributed by Suvrit Sra

We give a new, self-contained proof of van der Waerden's permanent inequality via a monotone-averaging argument. Specifically, we show that the minimum total correlation of a random permutation with prescribed doubly stochastic marginals never decreases when two rows or two columns of the marginal matrix are averaged. This quantity is largest at the uniform matrix, which yields van der Waerden's bound. The monotone-averaging rests on the quadratic Alexandrov–Fenchel inequality, which lets us extend the argument to many other problems. In this way we also prove the Friedland–Tverberg inequalities and Gurvits's coefficient bound for Lorentzian polynomials.

math.OC — Optimization and Control

The generalized Lax conjecture holds for mixed determinants

Contributed by Suvrit Sra

We prove that the mixed determinant $\eta(\cdot)$ of Borcea and Brändén has a spectrahedral hyperbolicity cone. More precisely, for Hermitian matrices $A_1,\ldots,A_m$ of order $d$, the hyperbolicity cone of $\eta(A_1,\ldots,A_m)$ is cut out by $\sum_j\Phi_j(A_j)\succeq0$, where $\Phi_1,\ldots,\Phi_m$ are explicit unital completely positive maps. We argue by induction on $d$, adding one row and one column to every argument and expressing the logarithmic derivatives of $\eta$ as extremal eigenvalues. By compressing along polynomial boundary kernels, we reduce the order of the pencil $\sum_j\Phi_j(A_j)$ to $\exp(O(md^3\log m))$; for one fixed positive semidefinite $n\times n$ matrix $A$ a separate pencil yields the order $\exp(O(n^2\log n))$. We also answer four problems of Borcea and Brändén negatively.

math.NT — Number Theory

The parity of the prime-counting function in time $x^{o(1)}$

Contributed by Vaibhav Tulsyan

Computing the prime-counting function $\pi(x)$ exactly takes $x^{1/2+o(1)}$ steps by the analytic method of Lagarias and Odlyzko, and to our knowledge its parity is known faster only on short intervals: Tao, Croot and Helfgott decide whether an interval of length at most $x^{1/2+c}$ near $x$ contains an odd number of primes in time $x^{1/2-c+o(1)}$. However, their identity needs the Mobius function $\mu(d)$ for all $d\le\sqrt x$, and we know of no bound of the form $x^{o(1)}$ for the parity, even conditionally. We present deterministic algorithms that output $\pi(x) \bmod 2$ and the parity of the number of primes in any interval $[A,B]$ in time $2^{O(n/\sqrt{\log n})}=x^{o(1)}$, where $n$ is the input bit length. Our approach replaces $\mu(d)$ by its residue modulo 2, which is the squarefree indicator again: the number of chains $d_1^2\mid m$, $d_2^2\mid d_1,\dots$ of depth $L$ is odd exactly when $m$ is squarefree, for $m<2^{2^L}$. The parity of $\pi$ on $[A,B]$ is then the parity of one count of tuples satisfying $O(\log n)$ multiplications, which Anthropic's counting engine evaluates through a network whose variables have geometrically shrinking widths. We prove both theorems in Lean 4 (piParity_TM_rate, piIntervalParity_TM_rate; axioms propext, Classical.choice, Quot.sound). The result does not give deterministic prime finding or the exact value of $\pi(x)$: the parity can be even on every interval one queries, and exact counts seem to need iterated products, for whose product-tree verifiers we prove a lower bound in Lean. The constants are astronomical, and the result rests on Anthropic's Lean development of its claimed refutation of the Exponential Time Hypothesis, which is under review.

math.NT — Number Theory

Class numbers, square roots and quadratic non-residues in time $2^{O(n/\sqrt{\log n})}$, assuming the new #SAT algorithms

Contributed by Vaibhav Tulsyan

Basic problems of computational number theory have proven deterministic bounds that are fixed powers of the input. The class number $h(D)$ of an imaginary quadratic discriminant takes $|D|^{1/2+o(1)}$ steps unconditionally and $|D|^{1/5+o(1)}$ under the generalised Riemann hypothesis (GRH); the least quadratic non-residue modulo a prime $p$ takes $p^{1/(4\sqrt e)+o(1)}$ by Burgess's 1957 bound, polynomial only under GRH; exact divisor sums $\sum_{m\le x}\tau(m)$ take $x^{1/3+o(1)}$. However, none of these bounds is of the form $X^{o(1)}$, even conditionally. We present deterministic algorithms in time $2^{O(n/\sqrt{\log n})}$, where $n$ is the input bit length, for six such problems: the number of reduced forms of discriminant $-N$, for fundamental discriminants $h(-N)$; the least square root of $a$ modulo any $m$; the least quadratic non-residue modulo a prime; and $\sum_{m\le x}\tau(m)$, $\sum_{m\le x}\sigma(m)$ and the Gauss circle count. Our approach writes each answer as a count of solutions of equations with one or two products and evaluates it with Anthropic's counting engine through one network library: products by a two-dimensional Schönhage-Strassen network, sums by signed carry chains, inequalities by comparison chains, with $O(n)$ shared bits. We prove the seven resulting theorems in Lean 4 (for example hForm_TM_rate; axioms propext, Classical.choice, Quot.sound): a finite-alphabet Mathlib Turing machine outputs the answer within $\lfloor C\cdot2^{Cn/\sqrt{\log_2(n+2)}}\rfloor$ steps on every input. Lean proves the count of Gauss's reduced forms; its equality with the ideal class number of $\mathbb{Q}(\sqrt{-N})$ is classical and not formalised. The constants are astronomical, the machines use exponential space, and every result rests on Anthropic's Lean development of its claimed refutation of the Exponential Time Hypothesis, which is under review.

math.OA — Operator Algebras

Universal Modules, Natural Pairings, and Weighted Isometries for Noncommutative $L^p$-Spaces

Contributed by Zhenyu Sun, Jilin Wang

For $1\le p<\infty$, we consider the functor $L^p$ on finite tracial von Neumann algebras with trace-preserving embeddings, including nonunital ones. For a functor of unital contractive left modules with compatible distinguished vectors $\xi_M$, finite direct sums of scalars determine the best constant in $\|x\xi_M\|\le K\|x\|_p$. The pointed $L^p$ functor is initial among the families satisfying this estimate with $K=1$. Retaining only the bound on scalar points gives an atomic norm. On $\mathbb C^n$ with unit weights the norm of the identity from $\ell^p_n$ to this atomic space is asymptotic to $p^{-1}(\log n)^{1-1/p}$ for $p>1$. For $1\le p,q<\infty$, every objectwise bounded natural bilinear family on $L^p\times L^q$ has the form $c\,\tau(xy)+d\,\tau(x)\tau(y)$ for $x,y\in M$, with $c,d$ independent of the object and $c=0$ when $1/p+1/q>1$. A uniformly bounded natural inner product on every $L^p(M,\tau)$ exists exactly at $p=2$. For finite-dimensional unital embeddings preserving a faithful positive functional, we determine the nearest modularly compatible target density both in Hilbert--Schmidt norm and in the Hilbert--Schmidt distance between positive square roots. The minimum squared distance between square roots lies between a weighted sum of the losses in squared $L^2$ norm on matrix units and twice that sum. Both constants are sharp.

math.GR — Group Theory

Three-Relator Universal Group, Four-Relator Adian–Rabin Family, and Unrecognizability of $\#^2(S^2\times S^2)$

Contributed by Qiuyu Ren

We construct a universal group with 2 generators and 3 relators, and an Adian–Rabin family with 2 generators and 4 relators whose nontrivial groups are universal. The resulting presentations show that the standard smooth manifold $\#^n(S^2\times S^2)$ is unrecognizable up to homeomorphism or diffeomorphism for every fixed n ≥ 2.

cs.LO — Logic in Computer Science

Voblint: End-to-End Soundness of a Context-Sensitive Goblint-Style Analyzer in Isabelle/HOL

Contributed by Manuel Lerchner

Goblint-style analyzers solve side-effecting equations, where evaluating one unknown can contribute to other unknowns. An existing Isabelle/HOL proof establishes solver partial correctness without connecting the equations to execution. We mechanize this connection for a scalar imperative language with recursive procedures but no pointers or heap memory. We adapt the local traces of Schwarz et al. to record procedure activations and the callers resumed by returns. A context policy relates concrete calls to one or more analysis contexts. A cover assigns sets of stores to program points and contexts as candidate overapproximations of reachability. We prove that five local obligations suffice for a cover to contain every valid trace’s final store in every admitted context, on any control-flow graph. A post-solution assigns abstract values that bound each equation’s result and side effects. When the abstract operations are sound and the routing agrees with the context policy, every post-solution induces a cover. The verified solver computes a post-solution when it returns. For every accepted configuration, whenever the Isabelle-defined analyzer returns a report, it overapproximates the stores reached by finite source executions from the prescribed initial states. Whenever execution reaches a check, PROVED guarantees that its condition holds, and REFUTED guarantees that it fails. Neither verdict establishes reachability. This source-level guarantee applies to the Isabelle-defined analyzer; solver termination, parsing, and correctness of generated code are not established.

math.NT — Number Theory

TRANSCENDENCE OF THE 3x+ 1 CONJUGACY MAP ON STURMIAN WORDS

Contributed by Elias De Jesús

Let T be the 3x + 1 map on the 2-adic integers and let Φ : ℤ₂ → ℤ₂ be the Bernstein–Lagarias conjugacy map, which sends a parity vector to the unique 2-adic integer realising it. We prove that Φ(cγ), Φ(1cγ), Φ(0cγ) and Φ of every shift of these words are transcendental for every irrational slope γ < γ* = (3 + √5)/(4 log₂ 3) = 0.8258977…, with no hypothesis whatever on the partial quotients of γ; in particular at the resonance slope γ = log₃ 2. The mechanism is a bridge: by the Bernstein–Lagarias isometry, the 2-adic approximation depth of a periodic shadow equals a longest common prefix, so the Diophantine behaviour of Φ is governed by the repetition structure of the parity word, at a height cost A(γ) = max(1, γ log₂ 3) that we control at every level. Quantitatively ω₁⁽²⁾(Φ(cγ)) ≥ ice(cγ)/A(γ) − 1, where ice is the initial critical exponent of Berthé–Holton–Zamboni; their floor ice(cγ) ≥ 1 + φ then beats the Ridout threshold ω₁⁽²⁾ ≤ 1 by a margin bounded away from zero. The result is not a corollary of the digit-based transcendence criteria: the 2-adic Hensel digits of Φ(c_{log₃ 2}) are measured to have full complexity 2ⁿ, against the parity word’s n + 1, and the shadow denominators 2^ℓ − 3^k are not S-units for any fixed finite S. We also record the Liouville alternative for unbounded partial quotients and a finite effective irrationality measure for bounded ones at slopes γ ≤ log₃ 2.

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.

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 neither $1$ nor $\omega^{\pm1}$. 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. 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 $\Delta_3^{\text{P}}$.

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 $K\ne\Q$. The proof 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 explicit field dependence, using neither those half-plane results nor earlier zero-exclusion results for real Dirichlet $L$-functions.

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 Luis V. Dieulefait, Nuno Freitas, Ariel Pacetti

Assuming GRH for Dedekind zeta functions, we prove Serre’s conjecture over the real quadratic fields $\mathbf Q(\sqrt d)$ for \[ d\in\{2,3,5,6,7,10,11,13,17,21,33,37,41,57,65,73,89\}. \] More precisely, every continuous, absolutely irreducible, totally odd two-dimensional Galois representation over these fields in odd residual characteristic is modular. The result for $d=17$ is unconditional; for $d=33$, GRH can be replaced by the assertion that Dedekind zeta functions have no zeros with real part greater than $31/32$. The proof combines conductor reduction with local ramification estimates and global discriminant bounds. Its principal new ingredients are a modularity criterion for $2$-adic representations with scalar reduction and integral trace arguments that force quadratic descent or bound coefficient fields. Uniform local lifting constructions allow these methods to treat arbitrary conductor.

math.CO — Combinatorics

Three-colour solutions from four-colour cubes

Contributed by Jason Davies

For every $n\ge2$, any $16n-3$ cubes, each using all four colours of a fixed palette, contain an $n\times n\times n$ frame whose exterior faces are monochromatic in exactly three colours; this bound is sharp. For a fixed palette, exactly two collections of $16n-4$ cubes fail, and reflection exchanges them. For $n\ge3$, every failure of size at least $14n$ lies in one ordinary orientation class with four bounded colour buckets. This gives $2\binom{d+15}{15}$ failures at size $16n-4-d$, for $0\le d\le2n-4$, of which $2\binom{d+11}{11}$ are inclusion-maximal. The threshold for a complete cube is $\max\{n^3,16n-3\}$; in particular, any $n^3$ source cubes suffice when $n\ge4$. The case $n=2$ answers the threshold question in Problem 2 of Berkove, Eickemeyer and Schulman (2018). The proof combines their corner-containment reduction with orientation constraints and double counting.

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 $T = T_{\mathrm{comp}} \circ T_{\mathrm{struct}}$.