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hexagon:2610.00699v1

math.NT — Number Theory

Cyclic reductions of elliptic curves at a positive proportion of primes

Contributed by Manjul Bhargava

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

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

Oct 31, 2026PDF ↗View work →

hexagon:2610.00698v1

math.OA — Operator Algebras

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

Contributed by K. MAHESH KRISHNA

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

Primarily human-written textHuman understanding: all partsBanach algebra.Carmichael-Mason boundKuniyeda boundLagrange-Fujiwara bound

Oct 31, 2026PDF ↗View work →

hexagon:2610.00697v1

math.NT — Number Theory

Deterministic Polynomial-Time Primality Testing with a Single Binomial Congruence

Contributed by Joseph Shunia

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

Primarily AI-generated textHuman understanding: all partsJacobi sumsKummer extensionsbinomial congruencescyclotomic ringsdeterministic algorithmspolynomial timeprimality testing

Oct 31, 2026PDF ↗View work →

hexagon:2610.00696v1

math.CV — Complex Variables

A contact and coefficient correction for Korenblum's constant

Contributed by Ricardo Pinho

Retaining an angular deficit in an annular coefficient estimate gives $C_{68}\ge0.44667$ for Korenblum's constant in the unweighted Bergman space. The flux input is David Niedbala Giraudin's. A contact-moment estimate and an even outer factor retain a correction involving the first two coefficients; a rational measure then satisfies the modified moment conditions. The argument builds on Wang's coefficient approach and Wikström's moment integration. This note is unrefereed, with no independent human review or formalization.

Primarily AI-generated textHuman understanding: some partsBergman spaceKorenblum's maximum principleannular Green functioncontact momentsinterval certificatesouter functions

Oct 31, 2026PDF ↗View work →

hexagon:2610.00695v1

math.NT — Number Theory

A trace upper-bound certificate with complex-root auxiliaries

Contributed by Ricardo Pinho

A rational piecewise-linear measure with 23 auxiliary polynomials gives $C_{86}\le1.80697883$ for the Schur--Siegel--Smyth trace constant. The proof uses Smith's arithmetic-measure criterion and converse theorem. Two degree-eight auxiliaries have nonreal roots; both polynomials already occur in Flammang (2009) and McKee (2011). Their use in this upper-bound certificate, rather than their discovery, is the change described here. This note is unrefereed, with no independent human review or formalization.

Primarily AI-generated textHuman understanding: some partsSchur-Siegel-Smyth trace constantauxiliary polynomialsinterval certificateslogarithmic potentialtotally positive algebraic integers

Oct 31, 2026PDF ↗View work →

hexagon:2610.00694v1

math.CO — Combinatorics

Fourier granularity and block sensitivity in the relevant-variable recurrence

Contributed by Ricardo Pinho

Rational Fourier lattice certificates, inserted into the existing local-degree recurrence, give $C_{44}\le4.1225323803$ for the number of relevant variables of a Boolean function. The finite packet contains 37 block-sensitivity exclusions through degree 40. We explain the certificate criterion and its passage to a bound valid in every degree, and identify the inherited analytic inputs. Neither the small-degree bounds $B_4\le9$ , $B_5\le14$ nor Fourier granularity is new. This note is unrefereed, with no independent human review or formalization.

Primarily AI-generated textHuman understanding: some partsBoolean functionsFourier granularityblock sensitivityrational certificatesrelevant variables

Oct 31, 2026PDF ↗View work →

hexagon:2610.00693v1

math.CO — Combinatorics

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

Contributed by Dr. András Pongrácz

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

Primarily AI-generated textHuman understanding: some partsRamsey theory

Oct 31, 2026PDF ↗View work →

hexagon:2610.00692v1

cs.CC — Computational Complexity

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

Contributed by Carlos Stein Naves de Brito

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

Primarily AI-generated textHuman understanding: some parts3-SATExponential Time Hypothesisinteger partitionsrank bounds

Oct 31, 2026PDF ↗View work →

hexagon:2610.00690v1

math.DG — Differential Geometry

Isometric embeddings of parallelizable and stably parallelizable Riemannian manifolds

Contributed by Wenchang Yang, Yi Wang

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

A mix of human-written and AI-generated textHuman understanding: some partsHopf actionh-principleisometric embeddingmetric correctionparallelizable manifoldsecond fundamental form

Oct 31, 2026PDF ↗View work →

hexagon:2610.00688v1

math.NT — Number Theory

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

Contributed by Pedro Caceres

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

A mix of human-written and AI-generated textHuman understanding: all partsRiemann hypothesis; Cesaro; Mobius; Landau; Nyman; Dirichlet; Weil; Davenport

Oct 31, 2026PDF ↗View work →

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