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385 works

math-ph — Mathematical Physics

Cancellation of all ultraviolet divergences of quantum gravity by a scale-invariant gas of branes

Contributed by Deep Bhattacharjee, Ushashi Bhattacharya

A frozen Poisson gas of branes with no preferred size, each brane a layer of independent cells that shift the proper time of a crossing graviton, multiplies the graviton kinetic operator by an entire function without zeros, $\exp[\tfrac n2\Ein(\Box^2/M^4)]$ times a random factor that tends to a constant at high momentum, in almost every realization. We prove that for $n\ge4$ the theory of gravity built on it has no ultraviolet divergence at any order of perturbation theory: every graph with two or more loops converges, the one-loop poles are those of a local theory and are cancelled by three operators quartic in curvature with finite coefficients, Newton's constant and the cosmological constant do not run, the zero-point energy of the graviton is $3nM^4/(64\pi^2)$ for every regulator, and the physical $S$ matrix is unitary at every loop order. The propagator keeps the graviton pole alone, and the potential of a point mass is finite at the origin. A one-loop graph in closed form confirms the cutting rule and grows faster than any exponential of the energy, so the loop expansion is an expansion in a small parameter only below an energy between $1.4M$ and $1.9M$ . The free Euclidean field is a Gaussian measure with finite curvature that is not reflection positive, which states what a construction beyond perturbation theory would have to supply.

A mix of human-written and AI-generated textHuman understanding: some partsPoisson point processesbrane gasentire form factorsnonlocal quantum gravityperturbative unitarityultraviolet finiteness

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:2610.00306v2

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

Oct 17, 2026PDF ↗View work →

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

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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:2610.00267v2

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

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hexagon:2610.00315v2

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, that the real multiplications by $\sqrt6$ 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, and that the Hodge conjecture holds on every power and every Hilbert scheme of points of the K3 surfaces of the one-dimensional Fricke families of levels $5$ and $7$ , which carry complex multiplication by $\QQ(\sqrt{-5})$ and $\QQ(\sqrt{-7})$. 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; among the steps, every convolution of the Weil line bundles on $E_{0}^{8}$ whose shifts lie in seven consecutive integers fails the polarised criterion in eighteen of the twenty-eight arrangements of the shifts, an explicit curve of genus four makes the Prym map of cyclic triple covers \'etale onto the Weil domain of $\QQ(\sqrt{-3})$, and the conjecture for a class is equivalent to the vanishing of the mass gap of a multiple of it plus a complete intersection, which Koll\'ar's hypersurfaces show cannot be dropped for integral classes. Both statements remain open.

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

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hexagon:2610.00301v2

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

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hexagon:2610.00307v2

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

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hexagon:2610.00308v2

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

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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

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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

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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

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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

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hexagon:2610.00262v2

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

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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

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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

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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 →