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math.NT — Number Theory

Optimal windowed test functions for bounding analytic ranks of elliptic curves

Contributed by Sabrina Hackfort

Assuming the Generalized Riemann Hypothesis, the explicit formula for the L-function of an elliptic curve E/Q bounds the analytic rank of E in terms of the traces a_p for p < e^(2L), for any admissible test function supported on a window [-L,L]. In rank computations this test function is usually fixed in advance, typically to a Fejer-type kernel; optimising it for an individual L-function goes back to Booker, and an earlier test on quadratic twists of rank about 7 found the gain to be small. We choose, for each individual curve, the test function that minimises the resulting bound within a finite-dimensional space supported on [-L,L], certify the result, and measure the gain systematically, up to the current record curves. The minimiser is the solution of a linear system with the Gram matrix of the Weil quadratic form of E, and the optimal bound equals the reciprocal of the corresponding reproducing kernel at the central point, which we call the radix Riemanniana of E. Because the Fejer-type kernel lies in this space, the optimised bound is never weaker. We certify every bound with ball arithmetic, check the prime-side evaluation against the zero side computed independently from the zeros of L(E,s), and benchmark against the fixed kernel on 1642 curves from Cremona's tables, 2500 curves from the Stein-Watkins database and a set of curves with conductors up to 3*10^16. Depending on the sample and on the stopping criterion, the number of primes required to pin down the analytic rank decreases by 8-23% in aggregate. For individual curves of large conductor the decrease ranges from 0% to 43%. Within each sample the gain is largest at rank 0 and decreases with the rank; it grows again, and by a much larger factor, in the regime of large rank and large conductor where the computation is most expensive. On the smallest-conductor curves of rank 28-31 in the ICARM leaderboard the optimised kernel needs 84-85% fewer primes to certify the rank. For the current record, the rank-32 curve #4239 of Elkies, Klagsbrun and Li, it certifies r_an<=32 under GRH with 2.53*10^9 primes - the first upper bound for this curve - and hence r_an=32 under GRH and BSD, while the fixed kernel, certified at L=12.40 with 2 481 819 193 primes, is still 0.39 above the target. On all 395 leaderboard curves of rank 24-31, with primes below 10^10, the optimised kernel certifies r_an<= r on 379 curves and the fixed kernel on 372; where both succeed the fixed kernel needs a median of 4.0 times as many primes. We validate the fixed-kernel baseline by reproducing every published value of Bober and of Klagsbrun, Sherman and Weigandt with an independent implementation.

Primarily human-written textHuman understanding: all parts

Oct 19, 2026PDF ↗View work →

hexagon:2610.00361v1

math.NT — Number Theory

Arithmetic structure in a complete atlas of power-residue runs below $2^{32}$ : Rabung profiles, Kummer densities, norm dependencies, and local codes

Contributed by Brice Pouly

We study the arithmetic behind long runs in a complete census of power-residue colourings below $2^{32}$ , covering every admissible order $2\le r\le9$ . The dataset contains $203\,280\,221$ primes and $677\,591\,249$ profiles, with exact quadratic four-window counts at every odd prime. A bounded study of its deep tail identifies algebraic windows with fewer independent character conditions than their lengths suggest. We certify their power-class groups and conjugate-root overlaps, obtaining exact Kummer densities for fixed templates, including eight additional order-four models. In a quadratic family, eleven constant positions force twenty-six neutral positions at odd character orders. In two further families, norm identities generate all remaining higher-root relations. A length- $53$ cubic realizes a three-wise independent binary code whose monochromatic probability is $4096$ times that of independent signs; its local run law is computed exactly. A separate stratified million-profile ternary sample is analysed with the same finite catalogue filter applied to the data and to an independent-colour comparator. Finite census statements, selected explanations and asymptotic template laws concern distinct populations. No universal law of global maxima or finite-range error estimate for the template densities is asserted.

Primarily AI-generated textHuman understanding: no partsKummer densitiesRabung profilesarithmetic dependenciescomplete finite atlaslocal codesmultiplicative charactersnorm identitiespower-residue runs

Oct 19, 2026PDF ↗View work →

hexagon:2610.00347v1

math.PR — Probability

Gaussian weights on fractals and Edelman's ghosts and shadows

Contributed by Djalil Chafaï

Can a Gaussian on a noninteger-dimensional fractal give a probabilistic realization of Edelman's ghosts and shadows? We distinguish a Gaussian-weighted Hausdorff measure, a radial Gaussian law, and a Gaussian random vector or field. For every real $\beta>1$ , we construct a closed cone of Hausdorff dimension $\beta$ whose Gaussian-weighted $\beta$ -dimensional Hausdorff measure has exactly the $\chi_\beta$ radial distribution. The construction relies on exact radial volume growth, not merely on a dimension calculation. We prove a metric obstruction below dimension one and explain why the cone does not supply ordinary Gaussian addition or a scalar algebra. A separate Gram-determinant argument rules out preserving all polynomially continued Euclidean Gaussian Gram identities in a positive model. We then give proofs of the angular beta-gamma identities, associativity of Bessel-Kingman convolution, and its Gaussian convolution semigroup for $\beta\geq1$ . Squared Bessel processes, tridiagonal and bidiagonal matrix models, Dunkl Gaussians, and Gaussian fields on fractals provide complementary constructions with explicitly different meanings of dimension. None of these statements asserts the existence of a full noninteger-dimensional scalar algebra.

Primarily AI-generated textHuman understanding: no partsRandom matrices; fractals; hypergroups

Oct 18, 2026PDF ↗View work →

hexagon:2610.00362v1

math.NT — Number Theory

Cubic-residue quadruplets: elliptic formulas and arithmetic fluctuations

Contributed by Brice Pouly

Let $A_p$ count the monochromatic consecutive four-windows of a nontrivial cubic character modulo a prime $p\equiv1\pmod3$ , with zero excluded and overlapping windows counted separately. We give an exact formula for $A_p$ in terms of fixed elliptic traces and a finite boundary correction. Its complex-multiplication terms are related cubic twists. This produces two arithmetic fluctuation regimes: the first four moments of $(27A_p-p)/\sqrt p$ tend to $(0,44,0,4224)$ when three is a cube and to $(0,17,0,738)$ otherwise. Over all admissible primes they tend to $(0,26,0,1900)$ . The proof combines a four-position identity following from Greene's hypergeometric formalism and Lennon's trace formula with symmetric-power automorphy, cyclic base change and a finite collection of Rankin-Selberg prime number theorems; it uses no GRH. A direct geometric proof exposes the phases and exceptional points, with the genus-two and isogeny constructions of Freeman-Satoh and Guillevic-Vergnaud explicitly credited. Point-order and Hasse certificates verify numerical examples without enumerating their billion-position character words. The conclusions concern fixed pattern length and four moments; no complete limiting distribution, constant-order centring of the raw count, or law for logarithmic-length maxima is proved.

Primarily AI-generated textHuman understanding: no partsElliptic Curvesarithmetic fluctuationsaveraged momentscomplex multiplicationcubic characterscubic twistsfinite fieldsfour-term patterns

Oct 18, 2026PDF ↗View work →

hexagon:2610.00365v1

math.NT — Number Theory

Adelic Galois representations are maximal for almost all Drinfeld modules

Contributed by Anwesh Ray

Let $r\geq 3$ be an integer, let $q$ be an odd prime power, and set $A=\mathbb{F}_q[T]$ and $F=\mathbb{F}_q(T)$ . We study the Galois representations associated with rank $r$ Drinfeld modules of the form $\phi_T=T+g_1\tau+\cdots+g_r\tau^r$ , where $g_i\in A$ and $g_r\neq 0$ . Ordering the coefficient tuples $\vec{g}=(g_1,\ldots,g_r)$ by their maximum degree, we prove that, for a density-one set of tuples, the $\lambda$ -adic Galois representation is surjective for every prime $\lambda$ of $A$ , and the adelic Galois image contains $\mathrm{SL}_r(\widehat A)$ . Moreover, for every tuple in this set, the index of the adelic image equals the index of the corresponding rank-one determinant representation and hence divides $q-1$ . These results extend Zywina's theorem in rank $2$ to arbitrary ranks $r\geq 3$ . As a further consequence, we establish the existence of rank $r$ Drinfeld modules with surjective adelic Galois representations for every $r\geq 3$ , extending previous existence results which were known for prime values of $r$ .

Primarily human-written textHuman understanding: all partsDrinfeld modulesGalois representations

Oct 18, 2026PDF ↗View work →

hexagon:2610.00332v1

math.NT — Number Theory

Rare events in multiplicative sequences: Poisson laws, clusters, and observation

Contributed by Brice Pouly

A multiplicative identity can persist at every scale while disappearing from a rare-event count. Another identity can survive as a cluster whose points lie far apart. Which distinction matters depends on what is observed. We explain this through one completely calculated model: a uniform germ can trigger two occurrences in disjoint spatial windows. Its two marginal counts become Poisson, but their sum need not do so. A coupling identifies the correct same-grid target and separates the mean number of points from the mean number of active germs. We then isolate the additional work required by multiplicative sources: private prime coordinates, weighted relations between windows, corrections to prime-site intensities, and relations visible at the target scale. A final passage from marked configurations to records explains the roles of ties, recognition and observation clocks. Selected results are stated with their hypotheses; a short concordance leads to their proofs. The elementary calculations are proved here, while the arithmetic and functional limit theorems are taken from the four cited research articles.

Primarily AI-generated textHuman understanding: no partsPoisson approximationarithmetic clustersexpository articlemarked point processesrandom multiplicative functionsrare events

Oct 18, 2026PDF ↗View work →

hexagon:2610.00336v1

math.NT — Number Theory

Sparse prime surgery in completely multiplicative sequences: Normality, rare-event realization and optimal repair

Contributed by Brice Pouly

We study how changing prime signs affects fixed-word frequencies, rare-event laws and extremes of completely multiplicative sequences. The cost is $d_\sigma(f,g)=\sum_{f(p)\ne g(p)}p^{-\sigma}$ . A known theorem of Frantzikinakis-Lemańczyk-de la Rue preserves normality at finite $d_1$ ; we give its elementary specialization. Using explicit arithmetic inputs from Long runs, we then establish three families of results. For each fixed $0<\sigma\le1$ and almost every base, reversible surgery realizes every stationary finite-intensity marked point-process law along selected scales, with signs, exact excess lengths and converging actual intensity, at arbitrarily small $d_\sigma$ cost and with a prescribed prefix unchanged. Almost surely over all integer scales, the minimum number of prime changes needed for a run of length $\lceil c\log_2N\rceil$ is asymptotic to $c\alpha_c\log_2N$ for $1<c<3/2$ , where $\alpha_c\in(0,1/2)$ satisfies $H_2(\alpha_c)=1-1/c$ and $H_2(u)=-u\log_2u-(1-u)\log_2(1-u)$ . At each scale, with probability tending to one, symbolic and private large-prime repair maxima coincide on the same random base for $k_N=o(\log N)$ ; unrestricted prime repair agrees when $k_N=o(\log\log N)$ . This comparison transfers the classical lattice laws of runs with errors to adaptive prime repair. Elementary programming from arbitrary bases and an autonomous finite-range-dependent process construction complement these results.

Primarily AI-generated textHuman understanding: no partscompletely multiplicative functionslong runsnormalityoptimal repairpoint-process realizationprime-sign perturbationsrare events

Oct 18, 2026PDF ↗View work →

hexagon:2610.00338v1

math.NT — Number Theory

A lean formalization of Roy's strong six exponentials theorem

Contributed by Shawn Krisman

We formalize in Lean 4 with Mathlib Roy's strong six exponentials theorem: if $x_1,x_2$ are linearly independent over $\overline{\mathbb{Q}}$ and $y_1,y_2,y_3$ are linearly independent over $\overline{\mathbb{Q}}$ , then one of the six products $x_iy_j$ lies outside the $\overline{\mathbb{Q}}$ -vector space spanned by $1$ and the logarithms of algebraic numbers. The proof closure uses only Lean's three standard axioms. Along the way the development proves Waldschmidt's transcendence theorem on $\mathbb{G}_a^{d_0}\times\mathbb{G}_m^{d_1}$ in the form used by Roy, and Philippon's multiplicity estimate with derivatives on $\mathbb{G}_a^{d_0}\times\mathbb{G}_m^{d_1}$ over every algebraically closed field of characteristic zero, following Roy's account in Waldschmidt's book. We reproduce the public Lean statements, summarize the mathematical route, and record how the artifact is verified. The code is available at https://github.com/telaviv/strong-six-exponentials.

Primarily AI-generated textHuman understanding: no partsLeanformal verificationinterpolation determinantslinear forms in logarithmsmultiplicity estimatesstrong six exponentials theorem

Oct 18, 2026PDF ↗View work →

hexagon:2610.00343v1

math.NT — Number Theory

An Euler-product mollifier for the Jensen moment of zeta and an energy criterion for the Riemann hypothesis

Contributed by Pedro Caceres

The Jensen moment of Balazard, Saias and Yor expresses the zeros of the Riemann zeta function outside the critical line through the mean value of the logarithm of its absolute value on that line. Jensen's inequality converts any mollifier into an upper bound for this contribution. For Dirichlet-polynomial mollifiers, the bound cannot fall below 0.0386, a barrier arising from the pole at s = 1. We instead use the reciprocal of a smoothed Euler factor, denoted E-star(X,s), constructed from prime powers up to a constant multiple of X together with an explicit pole term. The quotient of the zeta function by this factor is then expressed solely in terms of its nontrivial zeros. We prove that the resulting bound B(X,s0), for any center s0 with real part greater than one half, tends to zero under the Riemann hypothesis (RH). Consequently, RH is equivalent to the convergence of B(X,s0) to zero. If the supremum of the real parts of the zeros is attained and exceeds one half, the bound grows at least as fast as X to a positive power divided by log X. Under the same attainment assumption, RH is therefore equivalent to boundedness, as X tends to infinity, of a single weighted mean-square energy: the integral over the real line of the squared absolute value of zeta divided by E-star on the critical line, weighted by 1/(1+t squared). Numerically, the bound decreases to 0.0015 at X = 10,000 for the Balazard–Saias–Yor center s0 = 1. At s0 = 2, it decreases approximately as 0.037 divided by log X up to X = 1,000,000, close to a heuristic prediction involving the constant kappa = 1.828... . By contrast, the corresponding quantity grows for the Davenport–Heilbronn function. We describe the geometry of the factorization and use two control functions to identify the additional ingredients required for a proof of boundedness of the mean-square energy.

A mix of human-written and AI-generated textHuman understanding: all partsBeurling primesDavenport-Heilbronn functionEuler productJensen formulaPoisson-Jensen formulaRiemann Hypothesisexplicit formulamollifier

Oct 18, 2026PDF ↗View work →

hexagon:2610.00324v1

math.CO — Combinatorics

Three-Element Lucky Sets of Stirling Permutations

Contributed by YURAN XIU

We characterize the three-element sets of lucky car indices that occur when a Stirling permutation of order $n$ is used as a parking function. A set $\{1,x,y\}$ , with $1<x<y<2n$ , is admissible precisely when $x$ is odd, or $x$ is even and $y$ is odd with $x<n$ , or both are even with $x+y\leq2n$ . Explicit Stirling words realize every permitted case. The number of admissible sets is $(n-1)(n-2)+2nh-3h(h+1)/2$ , where $h=\lfloor(n-1)/2\rfloor$ . This answers the three-element characterization and enumeration question in Problem 46 of Colmenarejo et al. The complete classification, parking constructions, and enumeration have been checked in Lean. HAL deposit: hal-05800210v1.

Primarily AI-generated textHuman understanding: no partsLean formalizationStirling permutationslucky setsparking functions

Oct 18, 2026PDF ↗View work →

hexagon:2610.00325v1

math.CO — Combinatorics

A Nonunimodal Acyclic Polynomial with All Roots in the Left Half-Plane

Contributed by YURAN XIU

Let $A(G,x)$ count vertex subsets of a graph $G$ that induce forests, by their cardinalities. Barton, Brown, and Pike asked whether failure of coefficient unimodality forces $A(G,x)$ to have a root with positive real part. We give a negative answer. The join of $K_{46}$ with three disjoint copies of $P_3$ has a nonunimodal acyclic polynomial of degree nine, yet all its complex roots have strictly negative real part. We derive the polynomial by classifying induced forests and give its exact positive Hurwitz determinants. A Lean formalization verifies the graph-to-polynomial identity and the full counterexample, using an exact rational Lyapunov certificate for strict stability. HAL deposit: hal-05800220v1.

Primarily AI-generated textHuman understanding: no partsHurwitz stabilityLean formalizationacyclic polynomialcounterexamplegraph polynomial

Oct 18, 2026PDF ↗View work →

hexagon:2610.00326v1

math.CO — Combinatorics

A Bipartite Graph with a Nonunimodal Acyclic Polynomial

Contributed by YURAN XIU

We construct a bipartite graph on 230 vertices whose acyclic polynomial, which enumerates vertex subsets inducing forests, is not unimodal. This answers Question 3.3 of Barton, Brown, and Pike negatively. The graph has a common set of 50 vertices and 15 branches, each with two sets of six vertices. A partition according to the number of selected common vertices gives a three-term formula for the complete induced-forest polynomial. Its coefficients have a strict descent from degree 62 to degree 63 and a later strict ascent from degree 66 to degree 67. The actual graph, the unrestricted subset-counting formula, and the exact inequalities have been checked in Lean. HAL deposit: hal-05800225v1.

Primarily AI-generated textHuman understanding: no partsLean formalizationacyclic polynomialbipartite graphscounterexampleunimodality

Oct 18, 2026PDF ↗View work →

hexagon:2610.00329v1

math.CO — Combinatorics

Counterexamples to the multicolor nonnested matching conjecture

Contributed by Pedro Machado Manhães de Castro

We show that excluding nested pairs can increase the number of vertices needed to force a matching of one color, disproving a conjecture of Barát, Gyárfás and Tóth. A matching consists of edges with no shared endpoint. In a fixed vertex order, two edges are nested if both endpoints of one lie between those of the other. For $r$ colors and a target of $k$ edges, the classical theorem guarantees a matching of one color on $(r-1)(k-1)+2k$ vertices when every pair is joined by an edge. For every $r,k\ge3$ , we construct a coloring at this very threshold in which every such matching contains a nested pair. Thus the nonnested threshold is larger by at least one vertex. Already with three colors, ten vertices force three disjoint edges of one color, while our coloring makes every such triple contain a nested pair. The self-contained proof combines endpoint counts with two edge recolorings that obstruct the pairings allowed by the vertex order.

Primarily AI-generated textHuman understanding: all partsRamsey numbernonnested matchingordered graph

Oct 18, 2026PDF ↗View work →

hexagon:2610.00339v1

math.GT — Geometric Topology

Marked Heegaard Boundary Modules: Filling, Joint Completion, and Integral Lattices

Contributed by Hao-Yu Sun

A marked handlebody state must retain its filling relations and coefficient control before complementary capping or root specialization. We construct boundary modules and regulated integral lattices that keep the disk and cyclotomic residues of one finite-stage representative together. Standard handle-attachment and recoupling results supply the geometric inputs; the comparison constructs endpoint lattices, separating transport, raw and saturated quotient towers, and exact cap domains. These objects supply the companion rootwise theory, while the two completed topologies remain distinct and root specialization remains restricted to its declared domain.

Primarily AI-generated textHuman understanding: some partsHeegaard splittingcapintegral latticejoint completionskein module

Oct 18, 2026PDF ↗View work →

hexagon:2610.00333v1

math.PR — Probability

Records, scale flows, and Gaussian limits from lattice Poisson fields

Contributed by Brice Pouly

How does a rare-event field determine the history of its extremes? Starting from the complete signed run-field comparison in Long runs and rare patterns, we develop records, censoring, occupation and Gaussian limits for the resulting lattice Poisson target. The arithmetic input retains every site, full exact length and actual sign, at growing intensity under a one-factor information-intensity budget. Strict integer records, their signed imbalance and logarithmic leadership time have a joint phase-uniform Brownian limit with explicit covariance, independent of the terminal critical field and its extremal paths. Right censoring gives scalar $M_1$ convergence for the visible maximum and $J_1$ convergence for the first-holder sign. Physical occupation instead has a nondegenerate phase-indexed limit. Canonical interpolation connects threshold thinning and a stationary scale flow without identifying their record rules or their Gaussian constants. The signed spatial theory includes weighted exact levels, fixed-window Ornstein-Uhlenbeck paths, two correlated threshold-compression noises, and rectangle fields with explicit grid centering. Direct partition and quadratic-reconstruction proofs provide alternative routes. For each specified observation, the target calculations identify its limit and the dependencies within the joint constructions.

Primarily AI-generated textHuman understanding: no partsGaussian limitsextremal processesfunctional limit theoremslattice Poisson fieldsoccupation timesrandom multiplicative functionsrecords

Oct 18, 2026PDF ↗View work →

hexagon:2610.00334v1

math.PR — Probability

Poisson laws and arithmetic clusters for rare multiplicative patterns

Contributed by Brice Pouly

We determine when rare observations of completely multiplicative functions have a Poisson law and when arithmetic relations force clusters. For independent prime values on the circle with uniformly bounded $L^2$ densities, we approximate the field of hits of site-dependent targets of Haar mass $O(N^{-1})$ on deterministic masks in $[2,CN]$ . The target retains the true prime-site intensities and uses Haar at composite sites, even when the total prime contribution diverges. The exponent $2$ is sharp in the uniform $L^q$ scale for this target and class of masks. Below this threshold, even correct prime intensities can fail at site resolution while the total count remains asymptotically Poisson. A fixed finite exceptional prime environment, possibly atomic and internally dependent, instead produces a conditional cluster law together with an exact environment-determined configuration. With Haar outside that environment, its integer relation lattice determines the compound-Poisson limit of the corrected prefix count and characterizes simple Poisson convergence. Further applications treat rational observations on sparse moving supports, permanent bounded prime biases, and - using a binary arithmetic relation profile - long Borel patterns in a finite abelian group times a torus. Classical marked Chen-Stein comparison provides a common interface. A separate deterministic-sequence interface gives correlation identities, transfer criteria and a resolution obstruction, without proving cancellation for Liouville.

Primarily AI-generated textHuman understanding: no partsPoisson approximationarithmetic clusterscompound Poisson lawsprime-value densitiesrandom multiplicative functionsrelation latticesshrinking targets

Oct 18, 2026PDF ↗View work →

hexagon:2610.00335v1

math.PR — Probability

Moving prime environments and arithmetic cluster limits

Contributed by Brice Pouly

We study shrinking-target occurrences of completely multiplicative functions with arbitrarily moving phases on a fixed finite set of exceptional primes and independent Haar values at the remaining primes. After exact removal of the environment-only points, effective integer-subgroup profiles classify all subsequential marked cluster limits, while target-scale pair overlaps characterize Poisson convergence. The quantitative input is a direct Haar comparison with the moving germ kernel, uniform in the exceptional phases and with stretched-exponential error. For deterministic profiles and interval targets, the complete spatial covariance measure reconstructs the unmarked law through primitive multiplicative rays and finite interval geometry. A positive forest formula makes this reconstruction valid even with unbounded cluster sizes. For a fixed known bounded Borel target, retaining positions and angular marks yields the corresponding identification from the complete marked covariance, using signed overlap channels. Restricting the observation to an interior spatial window imposes both angular and logarithmic-spatial constraints. The maximal number of descendants of one germ becomes a rectangle-occupancy number for a subgroup of $\mathbb{R}^2$ , finite exactly when that subgroup is discrete. Explicit rank-one laws illustrate what counts forget, while environmental mixtures show why deterministic identification does not extend by averaging covariances.

Primarily AI-generated textHuman understanding: no partsPoisson approximationarithmetic clusterscovariance reconstructionmoving environmentsrandom multiplicative functionsrelation latticesshrinking targets

Oct 18, 2026PDF ↗View work →

hexagon:2610.00345v1

math.MP — Mathematical Physics

A lower bound for two-interval quantum backflow

Contributed by Daniel Adam Pyle

An explicit compactly supported positive-momentum state yields a summed backflow greater than 0.06190 over two unit-duration intervals separated by a gap of 3/2, in units where the reduced Planck constant is 1 and the mass is 1/2. An executable numerical certificate encloses its value approximately between 0.0619551065 and 0.0619551266, conditional on stated arithmetic and library assumptions. The state is defined by stored binary-rational coefficients and is re-integrated with enclosed quadrature and rounding errors. This is a trial-state lower bound, not an optimality result. It exceeds a published conjectural numerical range for equal spacing, but does not establish superiority to the true equal-spacing supremum or publication priority.

Primarily AI-generated textHuman understanding: no partscomputer-assisted proofinterval arithmeticmathematical physicsquantum backflowvalidated numerics

Oct 18, 2026PDF ↗View work →

hexagon:2610.00340v1

math.GT — Geometric Topology

Integral root fibers and closed factorizations: Pairing maps, Rees covariance, caps, and sewing

Contributed by Hao-Yu Sun

We construct rootwise and formal Rees boundary data from the marked modules and lattices of the companion boundary-module construction, keeping their distinct coefficient and completion domains. Using a Newton interpolation basis, we determine the Smith invariants and the integral image of the map induced by the Hopf pairing. Factorial/Rees covariance and compatible finite-stage representatives produce a strict rootwise/Rees module with its theta endpoint. At an admissible odd prime, no continuous homomorphism extends algebraic root specialization to the entire completed Rees source; further obstructions exclude unrestricted Newton padding and a state-natural lift of a contracted scalar identity. Established perturbative, Habiro, and stated-skein/Morita constructions give closed comparisons after the declared cap or sewing map, with their boundary state objects kept separate.

Primarily AI-generated textHuman understanding: some partsHabiro capHopf pairingMorita sewingRees completionroot of unity

Oct 18, 2026PDF ↗View work →

hexagon:2610.00341v1

math-ph — Mathematical Physics

Mass Gap of Lattice Yang--Mills Theory up to the Dobrushin Ceiling

Contributed by Deep Bhattacharjee, Ushashi Bhattacharya

We prove that the Hessian of the Wilson action on $d$ -dimensional lattice gauge fields is bounded by $4d$ , sharply, so lattice $\SU(N)$ Yang--Mills theory has a unique infinite-volume state with a positive mass gap for $|\beta|<1/(8d)$ . Every von Mises--Fisher law on the three-sphere moves by at most a quarter of its parameter in Kantorovich distance; consequently four-dimensional $\SU(2)$ theory has a mass gap for $\beta_W<2/9$ , the ceiling of all single-link Dobrushin arguments. A gauge-invariant finite-box criterion for decay and the area law is also given. The continuum mass gap remains open.

A mix of human-written and AI-generated textHuman understanding: some partsBakry-Emery curvatureDobrushin uniquenessKantorovich distancelattice gauge theorymass gapvon Mises-Fisher law

Oct 18, 2026PDF ↗View work →

hexagon:2610.00342v1

math-ph — Mathematical Physics

Ghosts in Poisson Brane Gases

Contributed by Deep Bhattacharjee, Ushashi Bhattacharya

A Poisson gas of branes with mass intensity $\nu$ dresses the graviton propagator by the random entire function $\prod_k(1+p^2/t_k)$ . We prove that the gas average of the propagator equals $p^{-2}\exp[-\int p^2\nu(\dd t)/(p^2+t)]$, so that the massive states carry total weight $e^{-\nu(0,\infty)}-1<0$ . For $\nu=\lambda(t)\,\dd t/t$ the averaged spectrum consists of ghosts alone exactly when $\lambda\le1$ , and every average with a finite four-dimensional tadpole has states of both signs. Frozen gases keep alternating residues.

A mix of human-written and AI-generated textHuman understanding: some partsKallen-Lehmann representationLaguerre-Polya classPoisson point processbrane gascomplete Bernstein functionghost

Oct 18, 2026PDF ↗View work →

hexagon:2610.00323v1

cs.CG — Computational Geometry

A sqrt(17/2) Lower Bound for Depth Reduction in Explainable Planar k-Center Clustering

Contributed by YURAN XIU

We prove that the factor- $2$ lower bound for depth reduction in explainable planar $k$ -center clustering is not tight. For every integer $k\ge2$ , an explicit set of $4k$ distinct integer points has both ordinary and explainable $k$ -center radius one, while every axis-parallel threshold tree with at most $k$ nonempty leaves and depth less than $k-1$ has radius at least $\sqrt{17/2}$ . Thus, for every fixed depth multiplier $H$ and every $C<\sqrt{17/2}$ , arbitrarily large instances admit no such tree of depth at most $H\lceil\log_2(k+1)\rceil$ and radius at most $C$ times the optimal explainable radius. The point set is a rescaled tail of the ladder of Deng, Gavva, Karthik, Patel, and Srinivasan. The strengthened analysis uses a four-color packing bound and the exact cluster budget to prove rigidity of every sufficiently small-diameter partition, without an explainability assumption. The complete infinite-family theorem and its unbounded-size corollary have been checked in Lean. The existence of a larger universal constant for logarithmic-depth explanations remains unresolved. HAL deposit: hal-05800242v1.

Primarily AI-generated textHuman understanding: no partsLean formalizationapproximation lower boundsdepth reductionexplainable clusteringk-center

Oct 18, 2026PDF ↗View work →

hexagon:2610.00327v1

cs.CC — Computational Complexity

A Three-Polynomial Counterexample to the Generalized Effective Hypercube Nullstellensatz

Contributed by YURAN XIU

We give a counterexample to the proposed extension of the effective Hypercube Nullstellensatz from two polynomials to an arbitrary fixed number of polynomials. For every $n\geq1$ , we construct three real polynomials in $2n+1$ variables, of degrees $2,2,1$ , respectively. Each is nonzero as a function on the Boolean cube, their product vanishes on the cube, and they have no common Boolean zero. Nevertheless, every polynomial certificate $\sum_{i=1}^3 h_i g_i=1$ on the cube satisfies $\deg\overline{h_1g_1}\geq n+1$ , where the bar denotes multilinearization. The construction applies a fresh selector variable to the hard pair of Kothari, Kovacs-Deak, Wang, and Yang. Thus its new ingredient is a short transfer of their existing degree obstruction to the product-zero setting. The full unbounded-degree counterexample has been formalized and checked in Lean 4. HAL deposit: hal-05800244v1.

Primarily AI-generated textHuman understanding: no partsBoolean cubeLean formalizationNullstellensatzdegree lower boundspolynomial certificates

Oct 18, 2026PDF ↗View work →

hexagon:2610.00328v1

cs.CC — Computational Complexity

A Counterexample to Affine-Subspace Union Stability under Small Deletions

Contributed by YURAN XIU

We construct a family of affine coordinate subspaces of $\mathbb{F}_2^N$ whose union is the entire space, but for which deleting a $1/N$ fraction from each member removes a fraction larger than $N^{-c}$ from the union, for every fixed $c>0$ and arbitrarily large $N$ . Each subspace has codimension $\log_2 N$ , and the family has at most $2N^2$ members. The construction uses all monochromatic coordinate conditions of size $k$ on $2k$ distinguished coordinates, where $N=2^k$ , and deletes their common intersection with the middle Hamming layer. The exact global loss is $\binom{2k}{k}/4^k$ . This disproves the affine-union conjecture of Alekseev and Gaevoy, already for their fixed parameters $q=2$ and $r=1$ . The all-exponent, arbitrarily-large-dimension counterexample has been formalized and checked in Lean 4. HAL deposit: hal-05800246v1.

Primarily AI-generated textHuman understanding: no partsBoolean cubeLean formalizationaffine subspacescounterexampleunion stability

Oct 18, 2026PDF ↗View work →

hexagon:2610.00337v1

cs.CC — Computational Complexity

An Approximate Solution to the Minimum Vertex Cover Problem: The Salvador Algorithm

Contributed by Frank Vega

Minimum Vertex Cover has a classical linear-time ratio of $2$ , and under the Unique Games Conjecture no polynomial-time algorithm achieves a fixed ratio below $2$ unless $\mathrm{P}=\mathrm{NP}$ . We present version 0.1.4 of \textsc{Salvador}, a linear-time ensemble of seventeen strategies, numbered in the order in which they are computed: twelve base strategies (component amplification before a maximal matching, the maximal matching itself, degree and weight heuristics, reductions to auxiliary problems that are solved exactly, a decomposition into bipartite layers, a weighted subdivision reduction and a bounded-treewidth dynamic program), followed by five improvers (a union pass, local search, reduce-and-search, consensus forcing and a swap-maximize refinement). We prove that every strategy runs in $O((n+m)\,\alpha(n))$ time, and that the returned cover has at most $2\tau-r$ vertices, where $r$ is the number of components with an edge, so the ratio is at most $1.999999999$ whenever $\tau\le10^{9}r$ . We prove exactness or a bounded ratio on several classes of graphs, among them every bipartite graph on which the Hopcroft--Karp phases of the layer strategy reach a maximum matching, and we give a per-instance certificate: a cover with at most $2L-|M|$ vertices, for a lower bound $L\le\tau$ and the maximal matching $M$ , is within ratio $3/2$ on that instance. In a reproducible suite the returned cover satisfies $|C|\le2\tau-|M|$ on all $291$ tested graphs with a known optimum and is certified on the six without one, although single strategies violate it; on the $68$ DIMACS clique complements it is never worse than Hvala and optimal on $37$ instances against $18$ . Whether $|C|\le2\tau-|M|$ holds on every graph is stated as a conjecture: a proof would give a linear-time $3/2$ -approximation and would refute the Unique Games Conjecture unless $\mathrm{P}=\mathrm{NP}$ .

Primarily AI-generated textHuman understanding: all partsApproximation AlgorithmBipartite LayersEnsemble HeuristicsLinear TimeMaximal MatchingP versus NPSplit GraphUnique Games ConjectureVertex Cover

Oct 18, 2026PDF ↗View work →

hexagon:2610.00331v1

math.RA — Rings and Algebras

Values of the Huang–Ma invariant

Contributed by Qiyue Tang

We prove the binary-value conjecture of Huang and Ma for integer matrices admitting a fixed positive diagonal skew-symmetrizer. We determine the exact set of values in every size and construct matrices realizing each value.

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

Oct 18, 2026PDF ↗View work →

hexagon:2610.00330v1

math.NT — Number Theory

A construction of the sphere-packing magic functions in dimensions 8 and 24

Contributed by Thomas Hales

We develop an alternative route to the solutions of the sphere-packing problem in dimensions 8 and 24, through the auxiliary functions in the Cohn--Elkies Fourier bound. In each dimension we construct a radial Schwartz function with the prescribed lattice zeros by solving a countable system of linear equations in absolutely summable coefficients. We prove existence and uniqueness after normalization, without using the known magic functions or Fourier interpolation. Through a Cauchy boundary problem, solutions of the homogeneous equations are identified with global sections of a rank-six holomorphic vector bundle over the compactified modular curve of the full modular group, with its orbifold structure retained. Riemann--Roch shows that this bundle has Euler characteristic one, forcing a nonzero global section and hence a nontrivial solution of the equations. A separate argument proves uniqueness after normalization. An equivalent positive operator equation has a bounded inverse. The construction is motivated by the coefficient method for planar circle packing. A subsequent comparison identifies our solutions with the functions of Viazovska and of Cohn, Kumar, Miller, Radchenko, and Viazovska. The route to sphere packing is not yet independent in its entirety: we do not prove the required global sign inequalities from the coefficient equations. Those inequalities are inherited from the known functions after identification.

Primarily AI-generated textHuman understanding: some partsSphere packing

Oct 18, 2026PDF ↗View work →

hexagon:2610.00322v1

math.NT — Number Theory

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

Contributed by Changxia Lin

Let Ξ(t)=ξ(1/2+it) denote the Riemann ξ-function in the critical-line parametrisation, let Ξ(t)=Σ_{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 associated Toeplitz minors. By the Schoenberg--Edrei characterisation of Pólya frequency sequences, positivity of D(n,r) for all n≥0 and all r≥1 is equivalent to the Riemann hypothesis (RH); for each fixed r the inequalities D(n,r)≥0 supply a necessary condition for RH, and for r=2 they were proved unconditionally by Csordas, Norfolk and Varga, so that the second level of the resulting hierarchy is not discriminating. We show that this failure of discrimination persists far beyond level 2, and we determine the range over which it does so: every level r≤43 is non-discriminating. Parametrising the zeros of the generating function G(z)=Σ_{k≥0}c_k z^k by β_j=-1/z_j, the dual Jacobi--Trudi identity yields the exact identity D(n,r)=c_0^r s_{(r^n)}(β), where s_{(r^n)} denotes the Schur function of the rectangular partition (r^n) in the alphabet β. A zero of ζ off the critical line contributes to β a conjugate pair ρ e^{±iθ} whose argument θ coincides with the vertical angle of the corresponding violating pair, whereas zeros on the critical line contribute positive real variables. For an alphabet of cardinality two we prove the closed form s_{(a,b)}(ρ e^{iθ}, ρ e^{-iθ}) = ρ^{a+b} sin((a+1-b)θ)/sin θ = ρ^{a+b} U_{a-b}(cos θ), where U_k is the Chebyshev polynomial of the second kind; the sign is therefore governed by the criterion U_k(cos θ)>0 if and only if (k+1)θ<π. Through the coproduct expansion s_λ(X⊔Y)=Σ_μ s_μ(X)s_{λ/μ}(Y), the Littlewood--Richardson expansion s_{λ/μ}=Σ_ν c^λ_{μν}s_ν of each skew factor, and the positivity of Schur functions on positive real alphabets, the sign analysis reduces to two-variable constituents. Since c^λ_{μν}>0 forces ν⊆λ=(r^n), the index k=ν_1-ν_2 of each constituent is bounded by the number of columns r and is independent of the shift n. All constituents then have the same sign, so no cancellation occurs. The same bound persists for an alphabet containing several conjugate pairs: iterating the coproduct over the pairs and using the fact that the entry 1 occurs at most once in each column of a semistandard tableau bounds every constituent index by ν_1≤r, independently of the number of pairs. This yields the following result. Blind Zone Theorem. Let the alphabet contain any number of violating pairs, each subtending a vertical angle at most θ_max, and suppose θ_max<π/(r+1). Then every minor D(n,r') with r'≤r is strictly positive, notwithstanding the presence of non-real zeros. The critical strip together with γ_1=14.134725141... forces θ≤θ_max=2 arctan(1/(2γ_1))=0.07071826... radians, whence π/θ_max=44.424...; consequently every order r≤43 is non-discriminating: for all n≥0 and all r≤43 the minor D(n,r) is positive if RH fails and non-negative if RH holds, so no minor of order r≤43 can certify either the truth or the falsity of RH. The bound is on the order r and not on the shift n; the corresponding n-based assertion is false, and the mechanism responsible is exhibited in the remarks section. The threshold is established for an arbitrary number of violating pairs; the case of several pairs is settled in the coproduct section and requires no separation condition on the moduli and no bound on the number of pairs. No claim of a proof of the Riemann hypothesis is made.

Primarily human-written textHuman understanding: all parts1 Riemann zeta function 2 xi-function 3 Turan inequalities 4 Toeplitz minors

Oct 18, 2026PDF ↗View work →

hexagon:2610.00312v1

math.DS — Dynamical Systems

Sharp equilibrium and stable-state capacities of multisite phosphorylation

Contributed by Michael Mislan

For the sequential distributive $n$ -site phosphorylation cycle with one kinase and one phosphatase, the maximal number of positive equilibria is $2n-1$ and the maximal number of locally asymptotically stable equilibria is $n$ , for every $n\ge1$ . The stable-state ceiling includes nonhyperbolic attractors. Both bounds are attained simultaneously at common conserved totals by rational positive rate constants and rational hyperbolic equilibria: $n$ sinks alternate with $n-1$ saddles, each with one unstable direction. This configuration persists on a nonempty open subset of the full parameter space. An explicit polynomial recurrence and interlacing construct the equilibria. Stability follows by separating equilibrium geometry from kinetic time scales, retaining enzyme-bound substrate in the slow variables, and controlling a path matrix with positive rank-one feedback. Further results give a diagonal Lyapunov certificate, a quantitative eigenvalue splitting law, exact finite-size witnesses and a three-site robustness and recovery certificate. An accompanying registered Lean theorem establishes equilibrium attainment; the broader capacity results are proved here by conventional arguments.

Primarily AI-generated textHuman understanding: some partsmass-action kineticsmultisite phosphorylationmultistabilitysteady states

Oct 17, 2026PDF ↗View work →

hexagon:2610.00298v1

math.GT — Geometric Topology

Almost all simple lengths are unique up to isometry

Contributed by Nhat Minh Doan

For every fixed hyperbolic torus with one cusp or one geodesic boundary, we prove that, among simple curves of length at most $L$ , the number of isometry orbits and the number of distinct lengths differ by $o(L^2)$ . Thus almost every complete length level is a single isometry orbit of at most six curves, and multiplicity converges in mean to the generic orbit size. The modular specialization is exactly Chen's density theorem for Markoff uniqueness. We also prove the full Schmutz--Schaller bound for every punctured torus having a simple trace that is not an algebraic integer, including any transcendental trace, and for dense families with integral traces. More strongly, every rational length class on these surfaces contains at most two curves. Transcendental twists give multiplicity one, while symmetric families with geodesic boundary attain the sharp bounds four and six. The density proof extends Chen's method using the stable-norm estimates of Doan-Li-Nguyen.

Primarily AI-generated textHuman understanding: some parts

Oct 17, 2026PDF ↗View work →

hexagon:2610.00304v2

math.MG — Metric Geometry

An explicit finite cylinder cover of the regular tetrahedron saving $3.86\times10^{-4}$ below the half-area bound

Contributed by Ricardo Pinho

We give an explicit cover of a regular tetrahedron by $4096$ cylinders with compact triangular perpendicular bases. The sum of their base areas, divided by the tetrahedron's minimum orthogonal projection area, lies in $[0.499613830783714148,\,0.499613830783714149)$ . Thus the saving below $1/2$ exceeds $0.000386169216285851$ . The construction uses a finite tilt in the angular-sector framework of OpenAI's September 2026 preprint and rationally specified radial cutoffs. Coverage is certified by exact linear-programming dual witnesses, and independently verified by exhaustive rational polyhedron clipping. We describe the geometric reduction, the certificate obligations, and reproducible verification of the full closed tetrahedron and the perpendicular base areas. No optimality is asserted.

Primarily AI-generated textHuman understanding: some parts

Oct 17, 2026PDF ↗View work →

hexagon:2610.00295v1

math.MG — Metric Geometry

A converse to the discrete four-vertex theorem

Contributed by Kristoffer Josefsson

Tabachnikov asked whether every cyclic sequence of positive numbers with at least two local maxima and two local minima is the sequence of radii of the circles through consecutive vertex triples of a convex polygon whose circumcenters lie inside the corresponding vertex angles. We prove this for every n ≥ 4, with extrema counted as plateaus. This is the sharp form: by Dahlberg’s discrete four-vertex theorem the condition is also necessary for nonconstant sequences. Locally near every nonconstant admissible sequence the polygon can be chosen to depend continuously (indeed C1) on the sequence, but in general not globally: we give an explicit path of admissible hexagon data along which no continuous choice of realizing polygons exists. The existence and continuity results are formalized in Lean 4 with Mathlib. We also prove the corresponding characterization for circles tangent to three consecutive supporting lines, using a separate tangential-polygon construction and Wegner’s forward theorem.

A mix of human-written and AI-generated textHuman understanding: all partsfour-vertex theorempolygons

Oct 17, 2026PDF ↗View work →

hexagon:2610.00309v1

math.PR — Probability

Quantitative analysis of critical-window profiles for autocatalytic emergence

Contributed by Michael Mislan

The critical-window law for reflexively autocatalytic and food-generated (RAF) sets in the reversible binary-polymer model states that, when the mean number of reactions catalysed per molecule grows like $\lambda n$ , the RAF probability converges to $S_\nu(1-\mathrm{e}^{-\lambda})$ , where $S_\nu$ is the probability that ordinary reversible closure of the food set is unbounded in an infinite field of independently open reaction channels. The law is known for the ordered split-position catalogue ( $\nu=\mathrm{sp}$ ) and for the commuting-factor quotient catalogue ( $\nu=\mathrm{qt}$ ). It is an existence theorem: it names the limit but says nothing effective about its value, its shape near zero, or what a finite network actually does. This paper answers those three questions. For both profiles, a construction gives a terminating algorithm which, given rational openness $a$ and rational tolerance $\varepsilon$ , returns a rational interval of width below $\varepsilon$ containing $S_\nu(a)$ , with an explicit approximation error $1/m+(2^{L+1}-2)(1-a^{L+1})^r$ for a computable seed length $L$ and record index $r$ , uniform on every openness interval bounded away from zero. The key estimate is a finite-cap record contraction proved without conditioning on survival and without independence between repair attempts. Second, tracking total word length along productive closure histories replaces a coefficient of order $\binom{J}{r}$ over enormous channel caps by $C_r^\nu\le36^r2^{r(r-1)/2}$ , giving $\log S_\nu(a)\le-\log^2(1/a)/(2\log2)+O(\log(1/a))$ and exact certificates such as $S_\nu(10^{-20})<10^{-629}$ . Third, at fixed word length $n$ the probability that some singleton RAF exists is exactly $1-(1-p)^{248}$ (split) or $1-(1-p)^{234}$ (quotient), so the finite RAF probability is linear in $p$ at zero while the limiting profile is flatter than every power; this order-of-limits effect is quantified with rational two-sided predictions and two fully explicit examples at $n=16$ . Further results give strict monotonicity of the profile, an effective global modulus of continuity, terminating inverse-probability brackets, and non-analyticity at zero. The effective evaluator, its error contract with the finite catalytic model and productive-history extraction are checked in Lean~4; the printed numerical bounds are checked by the accompanying replay script; the sharper counting theorem, the singleton law and the regularity results have conventional proofs and are labelled as such.

Primarily AI-generated textHuman understanding: some partsautocatalytic setscomputable analysiscritical windowfinite-size effectslow-intensity asymptotics

Oct 17, 2026PDF ↗View work →

hexagon:2610.00300v1

math.QA — Quantum Algebra

Low-rank Grothendieck rings: an AI-assisted classification project, constructions, and open problems

Contributed by Sebastien Palcoux

We present a public research project on the Grothendieck rings of complex fusion categories of rank at most eight. The project reports unrestricted classification theorems in ranks two, three, and four, and classifications within specified fusion-multiplicity bounds in ranks five through eight, with two rank-six categorification questions unresolved. Its resources include research manuscripts, an exact fusion-ring census, verification code, and an interactive atlas. We describe seven constructions that are promising candidates for new tensor categories, subject to a separate assessment of their novelty up to tensor equivalence. We also state the two open problems explicitly. This announcement records the project's claims and their verification status; it invites independent mathematical review and does not provide a new proof of the complete classification.

Primarily AI-generated textHuman understanding: some partsAI-assisted mathematicsFusion categoriesGrothendieck ringscategorificationfusion multiplicitylow ranknear-group categories

Oct 17, 2026PDF ↗View work →

hexagon:2610.00313v1

math.DS — Dynamical Systems

Attracting oscillations in sequential distributive multisite phosphorylation

Contributed by Michael Mislan

The sequential distributive $n$ -site phosphorylation cycle with one kinase and one phosphatase admits attracting positive periodic orbits under mass-action kinetics for every $n\ge3$ . An explicit three-site family has a fixed positive equilibrium, a transversal imaginary-eigenvalue crossing, a stable complementary spectrum and a negative first Lyapunov coefficient. A site-addition argument extends the supercritical Hopf bifurcation to every larger site count. Together with the one-site calculation and Vassena's two-site exclusion, this makes three the exact threshold for Hopf bifurcation in this family. Dynamic sequestration of the shared enzymes supplies the feedback: the equilibrium, productive fluxes and statically eliminated vector field remain fixed while stability changes. Further results compare enzyme clamps and selective reduction, quantify flux and amplitude laws, validate two finite-amplitude attracting orbits, and give a driven thermodynamic realization. Exact algebra and interval certificates support the quantitative claims; numerical continuation is distinguished from proof. A registered Lean theorem independently establishes an existential affine family with attracting periodic orbits for every $n\ge3$ . The manuscript's arbitrary-parent site-addition theorem and its quantitative extensions go beyond that registered statement.

Primarily AI-generated textHuman understanding: some partsHopf bifurcationattracting periodic orbitsenzyme sequestrationmultisite phosphorylation

Oct 17, 2026PDF ↗View work →

hexagon:2610.00302v1

math.CT — Category Theory

The Grothendieck construction between the telescope monad and contextads

Contributed by Kotaro Kamiya

A telescope is a list of stages in which each stage is a family indexed over the Grothendieck construction of the previous ones. The telescope monad stacks stages as the free-monoid 2-monad on spans, whereas a contextad collapses two stages into one by a dependent sum. This paper derives both from the span of single stages, and thereby explains why the strict algebras of the telescope monad are contextads. The link is the Grothendieck extension of a span. Because it is monoidal and preserves coproducts, it turns the telescopes into the free monad on the extension of the span of stages and the stage contextad into the Grothendieck monad, whose Kleisli arrows are the contextful arrows. The main theorem identifies the stage contextad with the pseudo-algebra of the telescope monad transported from the display contextad. Its structure map is the dependent sum, which the extension turns into the iterated multiplication of the Grothendieck monad.

A mix of human-written and AI-generated textHuman understanding: some partsGrothendieck constructioncomprehension categorycontextaddependent sumtelescope

Oct 17, 2026PDF ↗View work →

hexagon:2610.00297v1

math.CO — Combinatorics

Bounds for Erdős Problem #188: $7\le k\le 60$

Contributed by Carson Crocker

Let $\ell_k$ denote $k$ collinear points with consecutive distances $1$ . Erdős Problem #188 asks for the least $k$ such that the plane can be colored red and blue with no two red points at distance $1$ and no blue copy of $\ell_k$ . Previous work gives $6\le k\le 6330$ , with the lower bound due to Tsaturian and the upper bound due to Currier, Mody, Xie and Zhang. We prove $7\le k\le 60$ . For the lower bound we show that every red/blue coloring of the plane contains two red points at distance $1$ or a blue $\ell_6$ , following the skeleton of Arman and Tsaturian's proof in three dimensions; four distance lemmas are reduced to explicit unsatisfiable propositional formulas, certified by DRAT proofs. For the upper bound we give an explicit periodic coloring whose red set consists of $90$ closed regular $24$ -gons of $48$ different sizes, all of diameter less than $0.999$ , in each period, with rational centers and sizes. Any two cells are either far apart (all distances between their points exceed $1$ ) or close together (all distances are less than $1$ ), which is checked in exact arithmetic for all periodic translates and rules out red unit pairs. That every $\ell_{60}$ contains a red point is proved by a computer-assisted verification: for every unit direction $u$ , the sets $R-ju$ , $j=0,\dots,59$ , cover the period torus, with every inclusion checked in rigorous ball arithmetic. A second, separately written program, based on a reduction to one-dimensional covering problems solved in outward-rounded interval arithmetic, confirms this. The computations, the verification code and drafts of the text were produced with AI tools under the author's direction; all proof certificates and verification programs are public (doi:10.5281/zenodo.23405962).

Primarily AI-generated textHuman understanding: some partsDRATErdős problemsEuclidean Ramsey theorySATinterval arithmeticperiodic coloringunit distance

Oct 17, 2026PDF ↗View work →

hexagon:2610.00305v1

math.CO — Combinatorics

Three conjectures on the first two graph eigenvalues

Contributed by Idris Ali Shaik

Let λ₁(G) ≥ λ₂(G) be the two largest adjacency eigenvalues of a finite simple graph G on at least two vertices. We resolve the three conjectures appearing as 1.15–1.17 in Kannan, Kumar and Pragada. First, for every noncomplete graph, we prove λ₁(G)² + λ₂(G)² ≤ 2 Σ_{e∈E(G)} (1 − 1/c(e)), where c(e) is the largest order of a clique containing e. Second, we construct connected (K₅ minus one edge)-free graphs of arbitrarily large size that violate the exact color-critical bound proposed by Li, Liu and Zhang. Third, for every fixed F with χ(F) = r + 1 ≥ 3, we prove, uniformly over F-free graphs with m = e(G) → ∞, λ₁(G)² + λ₂(G)² ≤ 2(1 − 1/r)m − ((r − 2)/r)Δ(G) + o_F(m), where Δ(G) is the maximum degree. The coefficient of the degree term is best possible. The arguments use the half-plane complete-positivity theorem of Coutinho, Liu, Spier, Tang and Zhang, together with an explicit connected construction, weighted clique removal and spectral separation of vertices of high degree. Theorems 1.1–1.3 are formalized in Lean 4 and registered as PALOMAR-2026-10-15-000015 v1, PALOMAR-2026-10-16-000012 v1 and PALOMAR-2026-10-17-000005 v1, respectively.

A mix of human-written and AI-generated textHuman understanding: all partsadjacency eigenvaluesextremal graph theoryspectral graph theory

Oct 17, 2026PDF ↗View work →

hexagon:2610.00306v1

math.CO — Combinatorics

A Product Formula for Partition-Restricted Boundary Hypermatrix Counts

Contributed by YURAN XIU

We prove the Koprowski--Lewis product conjecture for nondegenerate $2\times(k+1)\times k$ hypermatrices over finite fields with two partition-shaped layers of prescribed zeros. The result holds for every positive $k$ , every finite field, and every admissible pair of partitions. After fixing the first slice, nondegeneracy is equivalent to simplicity of a partial linear map. Applying the simple-extension theorem of Arora, Ram, and Venkateswarlu inside the successive allowed row spaces gives the product directly. We include an elementary proof of that extension theorem and justify the equivalence with the defining projective critical locus of the hyperdeterminant. The complete counting statement and its semantic equivalences have been checked in Lean. HAL deposit: hal-05800194v1.

Primarily AI-generated textHuman understanding: no partsLean formalizationenumerationfinite fieldshypermatrixproduct formula

Oct 17, 2026PDF ↗View work →

hexagon:2610.00316v2

math.CO — Combinatorics

Tools for proving that graphs are not graceful

Contributed by Boon Suan Ho

A graph with $m$ edges is graceful if its vertices can be labeled with distinct integers in $[0,m]$ whose edge differences are exactly $1,\dots,m$ . Graceful labelings have been constructed for sixty years, but there are few general methods for showing that a graph is not graceful. We start from a single generating-function identity that restates gracefulness, and derive three tools from it: residues at roots of unity, extending the classical parity condition; positivity on the unit circle, whose semidefinite form amounts exactly to the Goemans–Williamson max-cut relaxation tested against the minimum of the Dirichlet kernel; and the label interval itself, which forces the labels of a dense subgraph to have a large span. The consequences include new families of ungraceful graphs: every edge-transitive graph that contains $K_6$ , the Paley graphs $P(q)$ with $q\ge37$ , the Kneser graphs $K(n,k)$ with $n\ge5.61k$ , almost every $d$ -regular graph with $d\ge84$ , and the polarity graphs $\mathrm{ER}_q$ with $q\ge23$ , which contain no $4$ -cycle. For the semidefinite form of positivity we also determine exactly what it can prove, and we show that a planar refinement is strictly stronger: it excludes the Kneser graph $K(11,2)$ with any seven edges deleted.

Primarily AI-generated textHuman understanding: some partsDirichlet kernelGolomb rulerdifference setsgraceful labelingmax-cutsemidefinite programming

Oct 17, 2026PDF ↗View work →

hexagon:2610.00317v1

math.CO — Combinatorics

No lattice tiling of $\mathbb{Z}^{35}$ or $\mathbb{Z}^{805}$ by Lee balls of radius three

Contributed by Deep Bhattacharjee, Ushashi Bhattacharya

We prove that there is no lattice tiling of $\mathbb{Z}^{35}$ or of $\mathbb{Z}^{805}$ by Lee balls of radius three. These were the only cases left open by Xiao and Zhou among the Lee balls of dimension at least three and radius at least two with fewer than $2^{30}$ points, so the lattice Golomb--Welch conjecture holds for all of them.

A mix of human-written and AI-generated textHuman understanding: some partsGolomb-Welch conjectureLee metricgroup ringlattice tilingperfect code

Oct 17, 2026PDF ↗View work →

hexagon:2610.00318v1

math.CO — Combinatorics

A graph with three card types and five vertex orbits

Contributed by Deep Bhattacharjee, Ushashi Bhattacharya

We prove that the answer to part~(b) of Problem~20.71 of the Kourovka Notebook is yes: there is a connected graph on $63$ vertices with exactly three isomorphism types of vertex-deleted subgraphs whose automorphism group has five orbits on the vertices. The graph is a Cayley graph of a Suzuki $2$ -group of order $64$ with one vertex removed.

A mix of human-written and AI-generated textHuman understanding: some partsCayley graphSuzuki 2-groupdivisible design graphpseudosimilar verticesvertex-deleted subgraph

Oct 17, 2026PDF ↗View work →

hexagon:2610.00319v1

math.CO — Combinatorics

Every odd antihole is strongly chromatic-choosable

Contributed by Deep Bhattacharjee, Ushashi Bhattacharya

We prove the complete Fiujlaali--McDonald conjecture on odd antiholes unconditionally: for every $\ell\ge3$ the only assignments of $(\ell-1)$ -element lists from which the odd antihole $\cbar{2\ell-1}$ cannot be coloured are the constant ones. Consequently, an odd-hole-free $k$ -critical graph of maximum degree $\Delta$ containing an odd antihole of length $2(\Delta-k)+7$ is that antihole.

A mix of human-written and AI-generated textHuman understanding: some partschromatic-choosabilitycolouringcritical graphodd antiholeodd hole

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hexagon:2610.00320v1

math.CO — Combinatorics

The $\mathfrak{sl}(2)$ weight system at $c=-3/32$ extends to a graph 4-invariant

Contributed by Deep Bhattacharjee, Ushashi Bhattacharya

We prove unconditionally the conjecture of Fomichev, Karev, Pavutnitskiy and Usanov that the specialization of the $\mathfrak{sl}(2)$ weight system at $c=-3/32$ extends to a $4$ -invariant of graphs. The extension is $\xi(G)=(-32)^{-n}S_G(-2)$ , where $S_G(N)$ counts with signs the assignments of $2$ -element subsets of $\{1,\dots,N\}$ to the $n$ vertices of $G$ .

A mix of human-written and AI-generated textHuman understanding: some partsClifford algebrafour-term relationintersection graphoscillator representationweight system

Oct 17, 2026PDF ↗View work →

hexagon:2610.00314v1

math.AP — Analysis of PDEs

Non-unique vanishing-viscosity solutions of 2D Euler with vorticity in $L^p$

Contributed by Umberto Pappalettera, Massimo Sorella

For every finite $p>2$ , we construct one real compactly supported zero-circulation initial vorticity in $L^1(\mathbb R^2)\cap L^p(\mathbb R^2)$ whose unforced Navier--Stokes evolutions have two distinct strong vanishing-viscosity accumulation paths. Both limits are renormalized unforced Euler solutions with the same strong initial trace. The construction uses a fixed infinite concentric stack, a smooth unstable fourfold profile, actual positive scalar envelopes, and a finite nonlinear expansion retaining the interacting harmonic histories. Discounted source history is controlled through the complete prospective receiver stop, and exact-cap normalization yields a depth-uniform outgoing reset. Protected first crossings, phase degree and a sparse phase assignment then separate the two limits. The analytical preliminaries, finite-sector spectral scope, source-reference comparison and uniform parameter order are included in the proof.

Primarily AI-generated textHuman understanding: no parts

Oct 17, 2026PDF ↗View work →

hexagon:2610.00267v1

math.AG — Algebraic Geometry

The fifteenth cohomology of moduli spaces of stable curves

Contributed by zeming sun, guoxiong gao, shurui liu

We determine the fifteenth cohomology of the moduli stack of stable curves $\overline{\mathcal{M}}_{g,n}$ in every genus $g$ and number of markings $n$ , with its symmetric-group action, by providing explicit multiplicity formulas for its simple constituents. We also prove the generalized Hodge conjecture in this case. The main result of this paper was obtained using the Pharos system. The author has reviewed the arguments and developed an understanding of the proof. On the basis of this review, the author believes the paper to be correct.

Primarily AI-generated textHuman understanding: some partsHodge structurescohomologygeneralized Hodge conjecturemoduli spaces of stable curvesmotives

Oct 17, 2026PDF ↗View work →

hexagon:2610.00303v1

math.AG — Algebraic Geometry

A Counterexample to Beauville's Stability Conjecture

Contributed by Yi Gu, Lifan Guan

We construct a morphism $f: X\to Y$ from a genus-two curve to an elliptic curve such that $f_*\mathcal L$ is unstable for every $\mathcal L\in\Pic^3(X)$. This gives a counterexample to a conjecture of Beauville.

A mix of human-written and AI-generated textHuman understanding: all partsDirect imagesFrey–Kani construction.stability

Oct 17, 2026PDF ↗View work →

hexagon:2610.00315v1

math.AG — Algebraic Geometry

Dense Algebraic Loci of Hodge Classes

Contributed by Deep Bhattacharjee, Ushashi Bhattacharya

We prove unconditionally that the Hodge conjecture is equivalent to two statements about abelian varieties, that the locus on which either statement holds is dense in every family and is either the whole family or a meagre set, and that the real multiplications by $\sqrt5$ , $\sqrt6$ , $\sqrt7$ and by the cubic fields $\QQ(\zeta_{7})^{+}$ and $\QQ(\zeta_{9})^{+}$ are algebraic on explicit families of K3 surfaces, the cubic ones on the surfaces of Mumford type whose quaternion algebra is ramified at two real places. For each of the two statements we build the object it needs next, a complex on a product of two Weil fourfolds and a class on an abelian fourteenfold, prove what it does, and prove at which step it stops. Both statements remain open.

A mix of human-written and AI-generated textHuman understanding: some partsHodge conjectureK3 surfacesWeil classesabelian varieties

Oct 17, 2026PDF ↗View work →

hexagon:2610.00301v1

cs.CC — Computational Complexity

The Exact Junta Bound for Degree-Two Boolean Functions on q-ary Hamming Cubes

Contributed by YURAN XIU

We determine the maximum number of relevant variables of a Boolean function of degree at most two on a finite alphabet. For every integer $q\geq2$ , this maximum is exactly $\max\{4,q+1\}$ . Here degree means Hamming, or junta, degree: the function is a sum of real-valued functions, each depending on at most two coordinates. We also prove that any such function with more than four relevant variables has a selector coordinate: after that coordinate is fixed, the function depends on at most one other coordinate. The proof combines saturation on a binary four-dimensional restriction with a rigidity argument that applies to every symbol of the original alphabet. Two disjoint interactions force a four-junta; a function with more than four relevant variables has a star of interactions. The exact bound and selector theorem have been formalized in Lean 4, with transitive dependencies using only the standard axioms of Lean and mathlib.

Primarily AI-generated textHuman understanding: no partsBoolean functionsHamming graphsLean formalizationdegree twojunta theoremrelevant variables

Oct 17, 2026PDF ↗View work →

hexagon:2610.00307v1

cs.CC — Computational Complexity

Set Cover by Cyclic Rotations Is NP-Complete

Contributed by YURAN XIU

Given an explicitly listed binary pattern of positive length $n$ and a nonnegative integer $b$ in binary, can at most $b$ cyclic rotations of the pattern cover all $n$ positions? We prove that this decision problem is NP-complete. The reduction constructs a pattern in a cyclic group of polynomial order and outputs its entire indicator word. Its principal ingredient is a replicated-marker construction: a cover within the budget must choose one translate in every layer, and each failed difference constraint leaves a block of more holes than the remaining translates can cover. Combining this construction with the known embedding of graph cliques into circulant graphs using distinct three-term sums gives a polynomial-time reduction from $3$ SAT. The theorem allows every subset, including the empty pattern, and every binary budget. A Lean formalization checks the mathematical reduction, the full-word polynomial-time programs, NP membership, and the unconditional NP-hardness foundation. HAL deposit: hal-05800233v1.

Primarily AI-generated textHuman understanding: no partsLean formalizationNP-completenesscyclic rotationsset cover

Oct 17, 2026PDF ↗View work →