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cs.CC — Computational Complexity

A Polynomial Separation for Adaptive Fourier-Overlap Parity Decision Trees

Contributed by YURAN XIU

Consider the parity decision tree rule that, at every nonconstant restriction, queries a nonzero character maximizing the intersection of the current Fourier support with its translate. We construct an explicit infinite family for which every adaptive policy obeying this rule has polynomial worst-case depth, although an unrestricted parity decision tree has logarithmic depth. For every $\ell\geq1$ , let $R=2^\ell$ . Our function has $n=4\ell-1+R^2\ell+R$ input bits and exactly $k=R^6-R^5+R$ nonzero Fourier coefficients. Every history-dependent maximizing policy terminates and computes the function, but has a feasible path of length at least $R>k^{1/6}$ ; an explicit unrestricted tree has depth at most $5\ell\leq\log_2 k$ . The construction places one pointer behind each cell of a finite-field affine partition, with all pointers accessing a shared data table. After any sequence of equal-answer table comparisons, exact signed Fourier coefficients force every maximizing query to compare two remaining table blocks. This invariant accommodates all tie choices and all resulting table partitions. The complete separation, including the specified polynomial-field representation and the existence of every operational policy's finite run, has been checked in Lean. The result concerns this particular support-overlap rule; it is not a lower bound against arbitrary parity decision trees. HAL deposit: hal-05800236v1.

Primarily AI-generated textHuman understanding: no partsFourier analysisLean formalizationadaptive algorithmsparity decision trees

Oct 17, 2026PDF ↗View work →

hexagon:2610.00311v2

math.NT — Number Theory

On Landau--Siegel zeros

Contributed by Ben Green

This is a variant of the short OpenAI proof that there do not exist Landau-Siegel zeros. In v2, two parameters are decoupled to give a better constant in the main theorem.

Primarily human-written textHuman understanding: all parts

Oct 17, 2026PDF ↗View work →

hexagon:2610.00299v1

math.NT — Number Theory

Polylogarithmic Descent for Almost All Collatz Orbits in Natural Density

Contributed by Idris Ali Shaik

Korec proved that, for each fixed θ > log 3/log 4, almost every positive integer n has a Collatz iterate below n^θ in natural density. Tao proved descent below every prescribed diverging threshold in logarithmic density. We obtain polylogarithmic descent in ordinary natural density within logarithmically many steps, with bounds on the preceding orbit values. The shortcut map T sends even n to n/2 and odd n to (3n+1)/2. The unaccelerated map Col sends even n to n/2 and odd n to 3n+1. For any prescribed positive function f(n) → ∞, however slowly, we prove that almost every positive integer n, in natural density, has nonnegative integers k = k(n) and ℓ = ℓ(n) satisfying T^k(n) = Col^ℓ(n) < (log n)^A_FP f(n), where A_FP ≈ 9.9911 is an explicit exponent. The shortcut time is k = c_* log n + O_f(√(log n log log n)), where c_* = 2/log(4/3). The leading time c_* log n is predicted by the average logarithmic contraction of the equally weighted multipliers 1/2 and 3/2. The unaccelerated time is ℓ = (3/2)c_* log n + O_f(√(log n log log n)). Both orbits are bounded by nf(n) up to their respective times. For each fixed A > A_FP, put q = A/(2A_FP) − 1/2. For sufficiently large X, all but at most C_A X/(log X)^q integers n ≤ X satisfy the same conclusions with target (log n)^A and bound n(log n)^q on both orbits. The constant C_A and the time-error constants depend only on A. The proof counts finite parity sequences exactly and follows first passages through decreasing thresholds. We count later failures among the original starting values, without assuming that values reached after a descent are uniformly distributed. For starting values 2^M ≤ n < 2^(M+1), only O(√(M log M)) cumulative passage times need to be counted at each threshold. A finite random-walk survival estimate supplies the additional saving needed at the critical exponent. Density-one descent below a fixed multiple of (log n)^A_FP is not established, and the pointwise Collatz conjecture remains open. The theorem for arbitrary f is formalized in Lean 4 with Mathlib and registered as PALOMAR-2026-08-26-000005, version 4.

A mix of human-written and AI-generated textHuman understanding: all partsCollatz mapalmost all Collatz orbitsfirst passagenatural densityparity vectorspolylogarithmic descentstopping time

Oct 17, 2026PDF ↗View work →

hexagon:2610.00294v1

math.NT — Number Theory

Rational points on elliptic curves that are primitive modulo a positive proportion of primes

Contributed by Manjul Bhargava

In 1977, Lang and Trotter conjectured that, subject to certain necessary arithmetic restrictions, a rational point $P$ of infinite order on an elliptic curve $E$ over $\Q$ should generate its reduction modulo infinitely many primes $p$ ---and indeed should do so for a specified positive density of primes. However, to date, even this infinitude was not known unconditionally for any explicit pair $(E,P)$ . For the curve $E:y^2=x^3-2x$ and point $P=(2,2)$ , we prove the existence of infinitely many---and indeed a positive lower density of---primes for which the reduction of $P$ generates $E(\F_p)$. More precisely, we prove that a common zero-free half-plane $\Re s>1-1/N$ for Dedekind zeta functions gives a lower density of at least $c/(N\log N)$ , with $c>0$ independent of $N$ . The announced theorem of Cureton and Thorner provides $N=32$ . Our proof combines this input with the marked Type~II estimates of OpenAI, a sieve with varying cofactor, and estimates for division fields and heights. The same method also applies to other CM elliptic curves and points.

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

Oct 17, 2026PDF ↗View work →

hexagon:2610.00292v1

math.AG — Algebraic Geometry

A Probabilistic Proof of a Positivstellensatz of Baldi-Sinn-Telek-Weigert

Contributed by Colin Tan

The AI gives a probabilistic proof of a uniform-denominator Positivstellensatz of Baldi--Sinn--Telek--Weigert for Laurent polynomials whose induced homogeneous form is strictly positive on the nonnegative part of a projective toric variety. Only this abstract was human-written. Three classical tools from mathematical statistics and probability theory were used by the AI: the Barndorff-Nielsen completion of discrete exponential families, sufficiency-based invariance under exponential tilting following Halmos--Savage and Van Campenhout--Cover, and the conditional limit theorem (equivalence of ensembles) of Diaconis--Freedman.

Primarily AI-generated textHuman understanding: no partsPositivstellensatzexponential familieslimit theoremtoric variety

Oct 16, 2026PDF ↗View work →

hexagon:2610.00291v1

math.PR — Probability

A nontrivial critical window for autocatalytic emergence in the binary polymer model

Contributed by Michael Mislan

The limiting probability is determined for a reflexively autocatalytic and food-generated (RAF) set in two reversible binary-polymer models. The models have the same molecules and food but distinguish reaction identities differently: the split-position catalogue retains ordered factorizations, whereas the quotient catalogue merges swapped factors exactly when they commute. With independent molecule--reaction catalysis, one mark shared by both directions, and mean catalysis $f_n$ per molecule satisfying $f_n/n\to\lambda>0$ , the RAF probability converges in either model to $\Theta_\nu(\lambda)=S_\nu(1-e^{-\lambda}),\qquad \nu\in\{\mathrm{sp},\mathrm{qt}\},$ where $S_\nu$ is the survival probability of infinite ordinary reversible food closure in the corresponding independent open-reaction field. Each profile is continuous and nondecreasing, lies strictly between zero and one at every finite positive intensity, and tends to zero and one at its two endpoints. Thus the entire linear scale is a nontrivial critical window, replacing the finite-step conjecture of Hordijk and Steel in the stated model. The common mechanism is uniform growth from a fixed supplied seed, followed by same-parameter richness and an exact law for complete catalytic pruning histories. A shared proof is given, including the explicit quotient transfer: deterministic closure and RAF equivalence project independent split marks to heterogeneous OR marks, rather than to the homogeneous quotient ensemble. The profiles satisfy $\Theta_\mathrm{sp}(\lambda/2)\le\Theta_\mathrm{qt}(\lambda)\le\Theta_\mathrm{sp}(\lambda)$ ; productive food gateways give the conventional bounds $\Theta_\mathrm{sp}(\lambda)\le1-e^{-32\lambda}$ and $\Theta_\mathrm{qt}(\lambda)\le1-e^{-30\lambda}$ . The quotient gateway-conditioned bulk-factor limit is also identified. It concerns structural autocatalysis, without assertions about reaction kinetics or thermodynamic viability.

Primarily AI-generated textHuman understanding: some partsautocatalytic setsbinary polymer modelcritical windowreaction channelsreversible closure

Oct 16, 2026PDF ↗View work →

hexagon:2610.00262v1

math.CO — Combinatorics

A Formally Verified Certificate Proof of the Maximum Number of Stable Matchings for Six Men and Six Women

Contributed by YURAN XIU

We give a complete Lean 4 verification of the theorem that the maximum number of stable matchings for six men and six women with strict complete preferences is f(6)=48. Jo (2026) previously proved the unrestricted upper bound, and Benjamin, Converse, and Krieger (1995) supplied an attaining construction. Our contribution is an alternative, fully formalized proof using a finite certificate; the value 48 and the unrestricted theorem are already known. We reduce stable matchings of arbitrary profiles to order ideals of sequences of cyclic partner exchanges in which no man revisits a partner. A weighted frontier recurrence counts these ideals, and a sum of permanents bounds every future continuation. These bounds yield a finite certificate with 135,828 states. The complete argument, including the reduction for arbitrary profiles, certificate soundness, all concrete checks, and an attaining profile, has been formalized and checked in Lean 4. The final unconditional theorem depends only on the standard axioms of Lean and mathlib.

Primarily AI-generated textHuman understanding: no partsLeanextremal combinatoricsformal verificationstable marriagestable matchings

Oct 16, 2026PDF ↗View work →

hexagon:2610.00279v1

math.CO — Combinatorics

Pattern-avoiding binary arrays and a Stirling-type table: explicit formulas, exact recurrence orders, and non-D-finiteness

Contributed by Yunkun Song

Let $a_k(n)$ be the number of $n\times k$ binary arrays in which no row contains $001$ or $010$ and no column contains $001$ or $011$ as three consecutive entries (OEIS A207123). Splitting the rows by parity gives $a_k(n)=U_k(\lceil n/2\rceil)\,U_k(\lfloor n/2\rfloor)$ , where $U_k(m)$ counts sequences in $\{0,\dots,m\}^k$ in which every three consecutive entries $(a,b,c)$ satisfy $b=c$ or $a\ge\max(b,c)$ ; equivalently, $U_k(m)$ counts the multichains of length $m$ in a subposet $\Lambda_k$ of the Boolean lattice. We prove three main results. First, for each $m$ the minimal order of a linear recurrence in $k$ is exactly $3m+1$ , and a unique factorization into blocks gives a positive double-sum formula for $U_k(m)$ in terms of Stirling and $r$ -Stirling numbers. Second, the bivariate generating function is not D-finite, and every polynomial-coefficient recurrence that holds on a quadrant is a left multiple of the defining four-term recurrence; the same holds for an associated Stirling-type triangle $N(k,q)$ , and $U$ is not Stirling-like in the sense of Kauers. Third, the numerators $h_k(t)$ of $\sum_mU_k(m)t^m=h_k(t)/(1-t)^{k+1}$ , analogues of the Eulerian polynomials, have only simple real zeros; equivalently, the chain polynomials of the posets $\Lambda_k$ are real-rooted, and the rows of $N$ are log-concave. We also determine all binomial shapes with nonnegative parameters in which $U_k(m)$ is a single sum, and show that the near-diagonal entries $N(k,k-d)$ are polynomials in $k$ starting exactly at $k=2d+2$ . The reduction, the exact orders, the explicit formula, the non-D-finiteness, the description of the quadrant recurrences and the near-diagonal theorem are formalized in Lean 4 with Mathlib.

Primarily AI-generated textHuman understanding: some partsD-finite power seriesLean 4OEIS A207123Ore algebrasStirling numberschain polynomialsformal verificationpattern-avoiding arraysr-Stirling numbersreal-rootedness

Oct 16, 2026PDF ↗View work →

hexagon:2610.00286v1

math.CO — Combinatorics

Gallai's conjecture with two exceptional even vertices

Contributed by Idris Ali Shaik

Gallai conjectured that every connected finite simple graph on n vertices admits an edge partition into at most ⌈n/2⌉ simple paths. Botler and Sambinelli proved this when every even-degree vertex has at most three even neighbours. Fan, Hou and Zhou exempted one designated even vertex of positive degree and ensured that at least two paths ended there. Xie obtained the same endpoint conclusion for either of two adjacent designated even vertices, in separate partitions, when every other even vertex has at most three even neighbours. We prove that the same endpoint conclusion holds without requiring the two designated even vertices to be adjacent. Let F be the subgraph induced by the even-degree vertices of a connected finite simple graph G on n vertices, and let h and x be distinct vertices of F. Suppose every other vertex of F has degree at most three in F. Then for each of h and x there is an edge partition into at most ⌈n/2⌉ simple paths, at least two of which end at that vertex. Consequently, Gallai's bound holds whenever at most two vertices of F have degree greater than three. The two choices may use different partitions. We also obtain one partition with at least two paths ending at each member of the pair, using at most ⌊n/2⌋ + 1 paths. At odd order this is Gallai's bound. At even order the extra path can be necessary: K₄ minus an edge requires three paths when both degree-two vertices are prescribed. The ceiling and endpoint bounds are formalized in Lean 4 with Mathlib and registered as PALOMAR-2026-10-04-000006, version 1.

A mix of human-written and AI-generated textHuman understanding: all partsGallai's conjectureLean formalizationeven-degree verticesgraph theorypath decomposition

Oct 16, 2026PDF ↗View work →

hexagon:2610.00276v1

math.GT — Geometric Topology

Fixed-Level TQFT Transport of Marked Handleslide Coherence: Disjoint Cubes and a Three-Handle Pentagon

Contributed by Hao-Yu Sun

Fix an ordinary level $K\geq1$ and a fully specified family of pairwise disjoint positive handleslides on a compression body. The main question is how a literal geometric relation becomes a literal finite-level operator relation. We construct a finite reduced skein fibre and one normalized isomorphism $$ Q_{K,d}:\mathcal M_{K,d}\xrightarrow{\sim}\mathcal V^+_{K,d} $$ to the corresponding scalar-extended Walker--TQFT state space. Reusing this map at every vertex sends the supplied weight-zero slide cylinders to $E_K$ -linear automorphisms: geometric squares become operator squares, finite slide orders give the same operator, and forward/reverse maps telescope for all declared phases and integer weights. The construction is fibrewise and makes no comparison between levels. These operator relations are transported images of the geometric relations proved in Papers~1 and~2; they are not new geometric coherence theorems. We then treat a fully specified positive three-handle pentagon. Its vertex-zero compression body admits a compatible fixed-level comparison core: a rational disk-kernel cap, a normalized complementary handlebody, a reduced spine model, and one universal-construction isomorphism $Q_{K,d}^{\mathfrak q}:\mathcal M_{K,d}^{\mathfrak q}\xrightarrow{\sim}\mathcal V^+_{K,d}$ . The geometric weight-zero Walker pentagon consequently has a literal operator image over $E_K$ , with explicit inverse formulas and a fixed-basis matrix corollary. Different comparison cores are related by a typed comparison groupoid and give intertwined operator systems; no canonical comparison or literal equality between different coordinate choices is asserted.

A mix of human-written and AI-generated textHuman understanding: some partsanomaly normalizationfixed-level transporthandleslide coherenceskein moduletopological quantum field theory

Oct 16, 2026PDF ↗View work →

hexagon:2610.00287v1

math.GT — Geometric Topology

Integral tail signatures and the Hilbert--Smith conjecture

Contributed by Alexander Dabrowski

The obstruction to an effective $p$ -adic manifold action is an integer signature tail. On asymptotically controlled rational modules over one fixed free $C_p$ -space, finite-cover Mayer--Vietoris makes the difference between $p$ copies and quotient-permutation tensoring nilpotent. Every degree-four class therefore has eventually $p$ -divisible signatures. An effective action gives a class of signature $1$ : coordinate averaging provides degree-one sphere control, free sheets turn quotient representatives into a compatible homotopy idempotent, and its normalized Poincar\'e corner forgets to a scalar manifold germ. Successive localization boundaries recover one copy of $\CP^2$, up to sign. The resulting contradiction proves the Hilbert--Smith conjecture.

Primarily AI-generated textHuman understanding: some partsHilbert–Smith conjectureHilbert’s fifth problemLie groupsalgebraic L-theorycontrolled topologylocally compact groupsp-adic group actionssignaturestransformation groups

Oct 16, 2026PDF ↗View work →

hexagon:2610.00289v1

math.GM — General Mathematics

Linchpin Theorems

Contributed by Thomas Hales

A proof assistant checks a formal proposition. It does not check that the proposition expresses the mathematics that its author intended. This distinction becomes pressing when large bodies of mathematics are autoformalized. We propose the use of linchpin theorems: statements whose meanings can be audited with relatively little mathematical infrastructure, and whose proofs connect them to substantial results. Examples range from sphere packing and the classification of Lie algebras to regularity theory, the Weil conjectures, and computational complexity.

Primarily AI-generated textHuman understanding: some parts

Oct 16, 2026PDF ↗View work →

hexagon:2610.00282v1

math.OA — Operator Algebras

Noncommutative Maccone-Pati Uncertainty Principles

Contributed by K. MAHESH KRISHNA

Let $\mathcal{E}$ be a Hilbert C*-module over a unital C*-algebra $\mathcal{A}$ . Let $A: \mathcal{D}(A) \subseteq \mathcal{E} \to \mathcal{E}$ and $B: \mathcal{D}(B)\subseteq \mathcal{E}\to \mathcal{E}$ be possibly unbounded self-adjoint morphisms. Then for all $x \in \mathcal{D}(AB)\cap \mathcal{D}(BA)$ with $\langle x, x \rangle =1$ , we show that \begin{align*} (1) \quad\quad\quad d_x(A)^2+d_x(B)^2&\geq \mp \langle \{A,B\}x, x \rangle +\langle (A\pm B)x, y \rangle\langle y, (A\pm B)x \rangle\pm \{\langle Ax, x \rangle, \langle Bx, x \rangle\}, \\ &\quad \forall y \in \mathcal{E} \text{ satisfying } \|y\|\leq 1 \text{ and } \langle x,y \rangle =0 \end{align*} \begin{align*} (2) \quad \quad \quad d_x(A)^2+d_x(B)^2&\geq \mp i\langle [A,B]x, x \rangle +\langle (A\pm iB)x, y \rangle\langle y, (A\pm iB)x \rangle \mp i[\langle Ax, x \rangle, \langle Bx, x \rangle], \\ &\quad \forall y \in \mathcal{E} \text{ satisfying } \|y\|\leq 1 \text{ and } \langle x,y \rangle =0. \end{align*} where \begin{align*} & d_x(A):= \sqrt{\langle Ax, Ax \rangle -\langle Ax, x \rangle^2},\quad [A,B] := AB-BA, \\ & [\langle Ax, x \rangle,\langle Bx, x \rangle]:= \langle Ax, x \rangle\langle Bx, x \rangle -\langle Bx, x \rangle\langle Ax, x \rangle. \end{align*} We call Inequalities (1) and (2) as noncommutative Maccone-Pati uncertainty principles. They reduce to uncertainty principles derived by Maccone and Pati [\textit{Phys. Rev. Lett., 2014}] whenever $\mathcal{A}=\mathbb{C}$ .

Primarily human-written textHuman understanding: all partsHilbert C*-module.Uncertainty Principle

Oct 16, 2026PDF ↗View work →

hexagon:2610.00281v2

math.NT — Number Theory

Almost all Markoff numbers are unique.

Contributed by William Y. Chen

A Markoff number is the largest coordinate of a positive integer solution of $a^2+b^2+c^2=3abc$ . Frobenius conjectured in 1913 that every Markoff number is the largest coordinate of exactly one such solution with $a\le b\le c$ . We prove that the conjecture holds for almost all Markoff numbers: the proportion of exceptions among the Markoff numbers up to $X$ tends to $0$ as $X\to\infty$ . Two such solutions with the same largest coordinate give simple closed geodesics of equal length on the modular torus, that is, primitive homology classes $v,w$ with equal values $N(v)=N(w)$ of the McShane--Rivin stable norm $N$ . Estimates for the boundary of the unit ball of $N$ reduce the problem to the equations $N(Gv)=N(v)$ with $G$ a rational linear map. For each $G$ that is not a symmetry of the torus, a zero-one law for the $3$ -adic analytic dynamics of the mapping class group on the Markoff surface shows that the solutions $v$ have density zero.

Primarily AI-generated textHuman understanding: some partsMarkoff triplesmapping class group dynamicspunctured torus

Oct 16, 2026PDF ↗View work →

hexagon:2610.00285v2

math.NT — Number Theory

One of β(4), β(6),..., β(36) is irrational

Contributed by Shawn Krisman

Let $\beta(s)=\sum_{m\ge 0}(-1)^m(2m+1)^{-s}$ be Dirichlet's beta function. We prove that at least one of the seventeen numbers $\beta(4),\beta(6),\dots,\beta(36)$ is irrational. All previous explicit results of this kind that we know of include $\beta(2)$ , Catalan's constant; our statement does not involve $\beta(2)$ at all. The proof uses the linear forms $L_n=\sum_{t\ge0}(-1)^tR_n''(t)$ attached to a very-well-poised rational function $R_n$ with triple integer zeros and half-integer poles. The second derivative, a $\beta$ -analogue of a device of Rivoal, removes $\beta(2)$ , and a symmetry removes the odd values. The zeros of $R_n$ run through the region of its poles, which makes every prime $p>n$ harmless. The common denominators are controlled by a $p$ -adic analysis and an arithmetic integral that we bound with an exact rational certificate. The size of $L_n$ is bounded by an elementary estimate on the real axis, and its nonvanishing for infinitely many $n$ follows from a saddle-point analysis along a certified contour. The whole proof, including every numerical certificate, has been formalised in Lean 4 with Mathlib and checked by the Lean kernel, using only Lean's three standard axioms.

Primarily AI-generated textHuman understanding: no partsDirichlet beta functionLeanformal verificationirrationalitylinear forms in zeta-type valuessaddle-point method

Oct 16, 2026PDF ↗View work →

hexagon:2610.00290v1

math.NT — Number Theory

Finite Valuation Words and Terminal Obstructions for the Accelerated Syracuse Map

Contributed by Tosho Karadzhov

We develop a finite-word obstruction framework for the accelerated Syracuse map on positive odd integers. A finite valuation word determines an exact iterate formula and a unique realization class modulo a power of two; compatible infinite prefixes define a natural two-adic inverse-limit object together with least positive representatives. The first part records the exact finite-word congruence machinery, finite-descent equivalence, terminal-surplus identity, terminal-ceiling break criterion, bad-prefix windows, activated density collapse, stabilization criterion, and strict-jump bookkeeping. The second part uses these tools to isolate terminal obstruction classes for a hypothetical counterexample: a bounded branch leading to a nontrivial accelerated cycle, and an unbounded eventual no-jump branch, successively reduced through a chain of geometric and combinatorial normal forms to a packet-surplus compensation target. The resulting reduction spine is lossless: Collatz follows from exclusion of nontrivial accelerated cycles together with a packet-surplus, or stronger record-ladder, compensation theorem for the unbounded branch. As a closed module, we include a closed class-exclusion theorem showing that no eventually periodic infinite low-surplus valuation chain is realized by a positive odd integer. On the bounded branch, we further record, as an explicit unproved roadmap step, that the terminal-ceiling strict-jump obstruction reduces to a single recurrent valuation-mismatch statement. This manuscript proves the architecture stated in its theorems, propositions, lemmas, and corollaries; it states the remaining terminal exclusions explicitly as problems, conjectures, or roadmap steps, and does not claim a completed proof of the full Collatz conjecture.

Primarily AI-generated textHuman understanding: no parts3x+1 problemCollatz conjectureaccelerated Syracuse maplow-surplus chainsterminal obstructionsvaluation words

Oct 16, 2026PDF ↗View work →

hexagon:2610.00284v1

math.QA — Quantum Algebra

A fourteen-dimensional biperfect Lie bialgebra

Contributed by Pavel I Etingof

We construct a fourteen-dimensional Lie bialgebra for which both the underlying Lie algebra and its dual are perfect. The construction is defined over $\mathbb{Q}$ and is self-dual over $\mathbb{C}$ . It gives an affirmative answer to Question 4.1(3) of Etingof--Gelaki--Guralnick--Saxl, arXiv:9912068. Its twenty-eight-dimensional Drinfeld double is quasitriangular with injective cobracket, also answering Question 4.1(4).

Primarily AI-generated textHuman understanding: some partsLie bialgebrasPoisson-Lie groups

Oct 16, 2026PDF ↗View work →

hexagon:2610.00278v1

math.GR — Group Theory

A universal group with thirteen relators

Contributed by Qiuyu Ren

We construct a finitely presented group with 13 relators that contains an isomorphic copy of every finitely presented group. This improves Valiev's bound of 21 relators for universal groups. The proof combines Valiev's construction, subgroup calculations of Kegel--Li--Ren, and small-cancellation theory.

Primarily AI-generated textHuman understanding: some partsfinitely presented groupuniversal group

Oct 16, 2026PDF ↗View work →

hexagon:2610.00280v1

math.NT — Number Theory

An elliptic curve of rank $\geq 34$ and root number $-1$ over a quadratic field

Contributed by Dylan T. Costa, Alvaro Lozano-Robledo

In this article, we find a quadratic twist over $\mathbb{Q}$ of rank $\geq 2$ (and $\geq 3$ assuming parity) of an elliptic curve $E/\mathbb{Q}$ of rank $\geq 32$ , recently found by Elkies, Klagsbrun, and Li. We use a family of quadratic twists of rank $\geq 2$ over $\mathbb{Q}(t)$ constructed by Mestre to search for twists $E^D$ with root number $-1$ , computing all root numbers without factoring the (very large) twisting parameter $D$ . As a result, we exhibit an explicit integer $D$ , with $411$ digits (a square times an integer with $381$ digits), such that $E^D$ has root number $-1$ and two independent rational points. In particular, the rank of $E$ over $\mathbb{Q}(\sqrt{D})$ is at least $34$ , unconditionally, and at least $35$ if we assume the Parity Conjecture.

A mix of human-written and AI-generated textHuman understanding: all partselliptic curveparity conjecturequadratic fieldrank

Oct 16, 2026PDF ↗View work →

hexagon:2610.00265v1

math.NT — Number Theory

Rational Fourth Powers Summing to 82

Contributed by Natanael Wildner Fraga

We prove that the only primitive positive integer solutions of $x^4+y^4=82z^4$ are $(1,3,1)$ and $(3,1,1)$ . An arithmetic reduction lifts every other putative solution to an elliptic curve over $K=\mathbb{Q}(\sqrt{41},\sqrt{2})$ . An explicit 2-descent determines its rank as 3. A logarithmic functional then excludes all nonzero parameters in the relevant 2-adic disc, including both relative sign patterns at the two places above 2. The same argument determines every point of the auxiliary quartic whose parameter is rational: only the two points at infinity occur. Exact square identities also determine the quotient modulo 2, prove that the displayed subgroup has odd index, and give the vanishing of the 2-primary part of the Tate–Shafarevich group. Two reductions determine the torsion. The finite arithmetic certificates are supplied in two self-contained Python programs; no height cutoff or conjectural rank computation is used.

Primarily AI-generated textHuman understanding: no parts2-Descentbiquadratic fieldselliptic Chabautygeneralised Fermat equationrational fourth powers

Oct 16, 2026PDF ↗View work →

hexagon:2610.00256v1

math.NT — Number Theory

The P\'olya--Vinogradov inequality with $\log\log q$

Contributed by Deep Bhattacharjee, Ushashi Bhattacharya

We prove unconditionally that every primitive Dirichlet character $\chi$ modulo $q\ge3$ has character sums of size at most $C\sqrt q\log\log q$ . Montgomery and Vaughan proved this bound under the generalised Riemann hypothesis, and Paley showed that it is optimal. The proof uses the zero-free half-plane $\Re s>7/8$ , which also gives $|L'/L(1,\chi)|\ll\log\log q$ and two-sided bounds for the Euler--Kronecker constants of the cyclotomic fields $\mathbb Q(\zeta_q)$ .

A mix of human-written and AI-generated textHuman understanding: some partsEuler--Kronecker constantsP\'olya--Vinogradov inequalitySelberg sievecharacter sumszero-free regions

Oct 16, 2026PDF ↗View work →

hexagon:2610.00275v1

math.NT — Number Theory

The Willow Transform and the Blind Zone Theorem: A Sharp Information-Theoretic Obstruction in the Turán Hierarchy for the Riemann xi-Function

Contributed by Changxia Lin

Let Xi(t) = xi(1/2 + it) be the Riemann Xi-function, let Xi(t) = sum_{k>=0} (-1)^k c_k t^{2k} be its even Taylor expansion at the origin, and let D(n,r) = det[c_{n+i-j}] (i,j = 0..r-1) be the Toeplitz minors whose positivity for all n >= 0, r >= 1 is equivalent to the Riemann Hypothesis (RH). We prove a sharp and quantitative obstruction in the resulting hierarchy of criteria. Parametrising the zeros of G(z) = sum c_k z^k by beta_j = -1/z_j, we establish by the dual Jacobi-Trudi identity the exact identity D(n,r) = s_{(r^n)}(beta), the Schur function of the rectangular partition (r^n) in the variables beta. A zero of zeta off the critical line contributes to beta a conjugate pair rho e^{+-i theta} whose angle theta is exactly the vertical angle of the violating pair of zeros, while zeros on the critical line contribute positive real variables. For two variables we prove the closed form s_{(a,b)}(rho e^{i theta}, rho e^{-i theta}) = rho^{a+b} sin((a+1-b) theta) / sin theta = rho^{a+b} U_{a-b}(cos theta), U_k being the Chebyshev polynomial of the second kind, so that the sign is governed by the classical criterion U_k(cos theta) > 0 if and only if (k+1) theta < pi. Since a Schur function in two variables can be expanded through the coproduct s_lambda(X sqcup Y) = sum_mu s_mu(X) s_{lambda/mu}(Y), and since s_mu(X) > 0 for positive real X, positivity reduces to the two-variable factors, whose index k = a - b is bounded by the number of columns r, not by the shift n. This yields: Blind Zone Theorem. If a violating pair subtends a vertical angle theta < pi/(r+1), then every Toeplitz minor D(n,r') with r' <= r is strictly positive, notwithstanding the presence of non-real zeros. The critical strip together with gamma_1 = 14.134725141... forces theta <= theta_max = 2 arctan(1/(2 gamma_1)) = 0.07071826... rad, whence pi/theta_max = 44.424... and therefore every order r <= 43 is completely non-discriminating. We emphasise that the exponential bound is on the order r and not on the shift n; we exhibit the mechanism and show the contrary n-based bound is false. We also record the refutation of a "universal covering law" previously conjectured by the author, and a floating-point artefact that nearly produced a false refutation of the main theorem. No claim of a proof of RH is made.

A mix of human-written and AI-generated textHuman understanding: all partsChebyshev polynomialsRiemann Zeta FunctionSchur functionsToeplitz minorsTuran inequalitiesblind zonetotal positivityxi-function

Oct 16, 2026PDF ↗View work →

hexagon:2610.00270v1

math.CO — Combinatorics

Linear Trace-Deficit Decay for Permanents of Positive Semidefinite Contractions

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

Every Hermitian positive semidefinite contraction $A$ satisfies $\operatorname{per}(A)\leq\exp(-\operatorname{tr}(I-A)/68)$ . This proves the linear trace-deficit bound conjectured by Ebrahimnejad, Nagda, and Oveis Gharan, whose estimate had quadratic dependence on the normalized deficit. The proof combines a size-biased complex Gaussian representation, a block version of Hanson–Wright decoupling, and a rational Loewner majorant. The constant is uniform in the dimension and is not optimized.

Primarily AI-generated textHuman understanding: some partsGaussian representationmatrix inequalitiespositive semidefinite permanenttrace deficit