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hexagon:2610.00451v1

math.NT — Number Theory

Small algebraic integers and Frobenius prime sums

Contributed by Scott Kominers

The determinant argument in recent proofs of Landau–Siegel zero exclusion compares the size of a nonzero integer with its divisibility by inert primes. We observe that the same comparison applies to every Frobenius class in a finite Galois extension of Q\mathbb{Q}, bounding its weighted prime count in terms of the supply of small algebraic integers. For each fixed field, the resulting height bound implies a weighted Chebotarev theorem whose only analytic inputs are the elementary Chebyshev and Mertens estimates. In the quadratic case there are only two Frobenius classes, so the lower bound obtained by complementation remains uniform in the conductor; this recovers Landau–Siegel zero exclusion.

A mix of human-written and AI-generated textHuman understanding: all partsChebotarev density theoremFrobenius congruenceGalois extensionGeometry of numbersLandau–Siegel zeroWronskianinterpolation determinant
Oct 22, 2026PDF ↗View work →
hexagon:2610.00450v1

math.MP — Mathematical Physics

Superfluid Stiffness and Nonclassical Rotational Inertia of the Dilute Hard-Sphere Bose Gas

Contributed by P. M. Aronow, Patrick Lopatto

We study the three-dimensional Bose gas of hard spheres of diameter aa at density ρ\rho and temperature TT, in the thermodynamic limit with aa, ρ\rho, and TT fixed. When the gas parameter ρa3\rho a^3 is sufficiently small and 0<T<T∗0<T<T_*, we prove two equilibrium signatures of superfluidity. First, in a periodic box, the superfluid fraction, defined through the free-energy cost of twisted boundary conditions, has thermodynamic lower limit at least c (ρa3)3/8c\,(\rho a^3)^{3/8}, uniformly in TT in this interval. Second, in dilations of an arbitrary nonempty bounded open container with hard walls, the nonclassical rotational inertia fraction 1−I/Icl1-I/I_{\mathrm{cl}} has thermodynamic lower limit at least c′(ρa3)3/8c'(\rho a^3)^{3/8} for every fixed rotation axis, where II is the moment of inertia, IclI_{\mathrm{cl}} its classical value, and c′>0c'>0 depends only on the container and the axis. In both results the derivative of the free energy is taken at finite volume, before the thermodynamic limit. The temperature T∗T_* is of order (ρa3)1/3/log⁡(1/(ρa3))(\rho a^3)^{1/3}/\log(1/(\rho a^3)) times the critical temperature of the ideal Bose gas. We also bound the moment generating function of the winding vector of the Feynman–Kac paths by that of a centered Gaussian, at every density and positive temperature. Combining this upper bound with the first result, we determine the order of the winding fluctuations in the regime above. Further, for 0<ε≤10<\varepsilon\le1, ρa3≤(ε/100)3\rho a^3\le(\varepsilon/100)^3, and T≥(1+ε)4πρ2/3T\ge(1+\varepsilon)4\pi\rho^{2/3}, about 1.9(1+ε)1.9(1+\varepsilon) times the critical temperature of the ideal Bose gas, the superfluid fraction in a periodic box tends to zero exponentially in its side length. By identities of Pollock and Ceperley and of Sindzingre, Klein, and Ceperley, we can express the two responses as multiples of the variances of the winding and of the signed projected area of the paths. We bound these variances from below with a variance inequality of Hammersley–Chapman–Robbins type, applied to random rewirings of the paths in a short imaginary-time interval along closed lattice routes. The rewirings adapt route comparisons developed in OpenAI's preprints on Bose–Einstein condensation in dilute Bose gases, and we extend the thermal estimates of the positive-temperature hard-sphere preprint in that series to higher temperatures. We average the rewirings over randomly translated closed lattice walks whose independent copies rarely approach each other. This makes the changed law close to the original law in chi-square divergence. Using conditional independence between disjoint imaginary-time intervals, we obtain bounds uniform in temperature. We also transfer the thermal estimates to arbitrary containers.

Primarily AI-generated textHuman understanding: no partsbose gassuperfluid
Oct 22, 2026PDF ↗View work →
hexagon:2610.00449v1

math.NT — Number Theory

A quantitative refinement of Lu's elementary prime number theorem

Contributed by Rajat Bhattacharjee

We refine Lu's elementary method for the prime number theorem to obtain, for every fixed K>400K>400, \[ \psi(X)-X=O_K\!\left( X\exp\!\left\{-\frac{\sqrt{\log X}}{(\log\log X)^K}\right\}\right) \qquad(X\to\infty), \] where ψ\psi is Chebyshev's function. The main ingredient is a comparison estimate that remains uniform as the degree grows and the short-interval parameter shrinks. Combining this estimate with exponential weights in Balog's identity, a single finite energy step and positivity-based interpolation yields the stated bound, retaining the full degree tail and all arithmetic correction terms. The resulting remainder strengthens Lu's fixed-parameter estimates and Goz\'e's quantitative refinement within the real-variable framework; stronger remainder estimates are known by other methods.

Primarily AI-generated textHuman understanding: no parts
Oct 22, 2026PDF ↗View work →
hexagon:2610.00448v1

math.CO — Combinatorics

A polynomial algorithm for general d-position in trees

Contributed by Daniel Adam Pyle

A vertex set is in general dd-position if no three of its vertices lie on a shortest path of length at most dd. We give an exact algorithm for an nn-vertex tree in O(n(D+2)4)O(n(D+2)^4) time and O(n(D+2)2)O(n(D+2)^2) space, where D=min⁡{d,n−1}D=\min\{d,n-1\}. In particular, the problem is polynomial-time solvable when the threshold is part of the input. This answers the tree-complexity question posed by Klav\v zar, Rall, and Yero in 2021 and retained in the August 2026 version of the general position survey. The dynamic program uses two truncated distances at each rooted boundary: the nearest selected vertex and the nearest far endpoint of an ancestor-comparable selected pair. These distances characterize every forbidden triple created by joining child subtrees. The algorithm reconstructs an optimal set and extends to signed vertex weights and exact counting of optimal sets. It also computes the full cardinality generating polynomial in O(n2(D+2)4)O(n^2(D+2)^4) coefficient operations. Exhaustive comparisons with direct subset enumeration support the implementation independently of the proof.

Primarily AI-generated textHuman understanding: no parts
Oct 22, 2026PDF ↗View work →
hexagon:2610.00447v1

math.AG — Algebraic Geometry

Point Separation for Adjoint Line Bundles

Contributed by Brian Nugent

Let X be a smooth complex projective variety of dimension n and let L be an ample line bundle. Using the freeness and minimization inputs specified in the introduction, we prove that KX + mL separates distinct points for every integer m ≥ n + 2, the sharp bound predicted by Fujita’s very ampleness conjecture. Exponential-moment convergence produces point-centred minimizing valuations at both points in a common degree. Strict convexity then gives a divisor with discrepancy equality at one point and strict excess at the other. A rational perturbation on a resolution, followed by Kawamata–Viehweg vanishing and descent, gives the required sections

Primarily AI-generated textHuman understanding: some parts
Oct 22, 2026PDF ↗View work →
hexagon:2610.00446v1

math-ph — Mathematical Physics

Which String Vibrations Become Which Generations on the Dwork Quintic

Contributed by Deep Bhattacharjee, Ushashi Bhattacharya

We determine, for the heterotic string on the one-parameter family of quintic threefolds, which vibration modes become which families of charged particles. The hundred and one families are labelled by the characters of a symmetry group of order one hundred and twenty-five. Every Yukawa coupling that this symmetry allows is a power of the modulus, and every other coupling vanishes. The invariant deformation pairs each family with exactly one partner, so that fifty families are massless at the Fermat point and none anywhere else, and at the Fermat point the masses are given in closed form by the Gamma function. On the three other one-parameter families of the same type, in weighted projective spaces, the symmetry no longer tells the families apart and some families stay massless at every member. We also prove that the Geometric Selection Conjecture is equivalent to the existence of one viable vacuum fixed by mirror symmetry, and that a free quotient of a Calabi–Yau complete intersection of any of the thirteen one-parameter hypergeometric types never carries three families with the standard embedding; on seven of these types, among them the sextic and dectic pencils, no finite group acts freely at all.

A mix of human-written and AI-generated textHuman understanding: some partsYukawa couplingsfree quotientsgenerationsheterotic stringquintic threefold
Oct 22, 2026PDF ↗View work →
hexagon:2610.00445v2

math.CT — Category Theory

Vanishing of Higher Magnitude Homology for Finite Prefix Models

Contributed by Ryoji Hirayama

Bradley and Vigneaux define a generalized distance on a finite prefix poset using next-token probabilities. They conjecture that its higher magnitude homology vanishes. Motivated by their expression of magnitude as a weighted sum of homological Euler characteristics, we verify this conjecture for their explicit complex, in which both endpoint faces are zero. For each finite length, the homology groups in degrees two and above are zero. The degree-one homology group is free abelian, with one generator for each positive-probability edge of that length. The proof constructs a contracting homotopy for each component whose endpoints have an interior prefix. This homotopy inserts the least interior prefix. Edge probabilities zero and one are both allowed.

A mix of human-written and AI-generated textHuman understanding: all partsenriched categorieslanguage modelsmagnitude homologyprefix posets
Oct 22, 2026PDF ↗View work →
hexagon:2610.00443v1

math.CO — Combinatorics

Distinct partial products in prime dihedral groups

Contributed by alexandre Thiery

Let D2pD_{2p} be the dihedral group of order 2p2p, where pp is prime. For every fixed 0<α<10<\alpha<1, we prove that every subset A⊆D2p∖{e}A\subseteq D_{2p}\setminus\{e\} with Cα≤∣A∣≤pαC_\alpha\le |A|\le p^\alpha admits an ordering whose nonempty partial products are pairwise distinct. Combined with the known small- and large-set theorems, this covers every subset size for all sufficiently large primes. Pairing reflections reduces the products to partial sums with fixed parity-labelled offsets. Two passes of block rotations remove distant collisions; bounded displacement allows the local condition to exclude the remaining collisions. The existence of suitable rotations follows from Pham--Sauermann's subset-sum anticoncentration theorem, an affine-rank estimate, and a count of clustered collision witnesses.

Primarily AI-generated textHuman understanding: some parts
Oct 22, 2026PDF ↗View work →
hexagon:2610.00442v1

math.GT — Geometric Topology

Inscribed Square Problem and an Embedding of M¨obius Band in a Family of Solid Toruses

Contributed by Faris ELmikdad

In this paper, we relate the space of line segments induced by a Jordan curve in the plane to the space of line segments induced by the plane by embedding the M¨obius band associated with the curve into the family of toruses associated with the plane; we further demonstrate how the geometric ”square condition” of the curve translates into a topological property through this embedding.

Primarily human-written textHuman understanding: all partsM¨obius bandRotationSquare.Torus
Oct 22, 2026PDF ↗View work →
hexagon:2610.00441v1

math.AC — Commutative Algebra

Zero-divisor methods for absolute concentration robustness

Contributed by Michael Mislan

We study the relation between absolute concentration robustness and zero divisors in steady-state coordinate rings. Explicit positive rational, two-sided bimolecular examples disprove both necessity and sufficiency of the proposed unrestricted criterion. A separate example shows why elimination order matters without contradicting lexicographic or product-order statements. We prove completeness of a block-compatible algebraic interface under explicit basis, leading-coverage, nonvacuity and zero-divisor hypotheses. The manuscript develops complementary criteria from regular positive geometry and quantitative algebraic certificates, with examples from loaded reactors, EnvZ and product-release networks. Selected counterexamples, the conditional completeness theorem and exact certificate identities are formalized in Lean 4. Geometric and nonlinear dynamical arguments supplied conventionally are identified separately.

Primarily AI-generated textHuman understanding: some partsLean formalizationabsolute concentration robustnesselimination ordersquantitative certificatessteady-state idealszero divisors
Oct 21, 2026PDF ↗View work →
hexagon:2610.00440v1

math.PR — Probability

Chemical heredity without genetic templates

Contributed by Michael Mislan

We prove two quantitative forms of chemical heredity that do not rely on copying a genetic sequence. In the first model, a population of four compartments carries three prescribed catalyst types that compete for a shared local fuel. Two types are present from the start; a third is supplied for a fixed seeding period, after which all catalyst input stops. We prove that, with failure probability at most 0.004517975690072193, a single finite-stock process acquires the third type and keeps all three present in both daughters of every division, delivers three million catalytic conversion credits on a fixed per-site schedule, and rebuilds the population's copy reserves eleven times, interleaved with ten divisions. The proof controls acquisition, catalytic service, recovery and resource consumption on one stochastic history; it does not assume that a compartment stays replenished between recovery and division. In the second model, an imperfect parent copies its majority conformation into an independent racemic blank, after which both outputs undergo cooperative restoration while spontaneous errors continue. At 4,000 molecules per compartment, 100 successive copying cycles succeed with probability at least 0.999, uniformly over a box of rate parameters. Complete conventional proofs and explicit finite certificates are supplied. The central finite-stock theorem, though not the restoration result, also has a machine-checked Lean formalization. A converse for random division explains why unlinked information units and linked carriers face different loss mechanisms. These results establish acquisition and retained function under prescribed chemistry and operations; chemically generated successive innovations and open-ended evolution require additional mechanisms.

Primarily AI-generated textHuman understanding: some partschemical heredityfinite resourcesorigin of lifestochastic reaction networkstemplate-free heredity
Oct 21, 2026PDF ↗View work →
hexagon:2610.00439v1

math.DS — Dynamical Systems

Unistationarity of Type II_l and Type V autocatalytic cores with nonnegative degradation

Contributed by Michael Mislan

We prove that reversible Type II_l autocatalytic cores with l at least three and Type V cores admit at most one positive stationary state when every degradation rate is nonnegative. The Type II result assumes source-minimality, while the Type V statement uses its literal normal form. Strictly positive reaction rate constants are retained throughout. A reduction to the fork skeleton, source-exhaustion identities and diagonal current criteria control both positive-loss and boundary cases, including internal chains. We also derive determinant estimates and local continuation results, with small-degradation existence and stability under the stated hypotheses. The main nonnegative-degradation uniqueness theorem is registered in Lean 4; the manuscript and supporting claim map distinguish the formal interface from additional conventional results.

Primarily AI-generated textHuman understanding: some partsLean formalizationType V networksautocatalytic coresmass-action kineticsnonnegative degradationunistationarity
Oct 21, 2026PDF ↗View work →
hexagon:2610.00438v1

math.DS — Dynamical Systems

Optimal affinity of autocatalytic networks beyond gross stoichiometry

Contributed by Michael Mislan

An explicit reversible autocatalytic network has a unique global maximum of controlled production whose affinity exponential is strictly below its gross stoichiometric seed ratio. This refutes the proposed general lower bound. We identify a response-profile capacity that retains the missing kinetic information, analyze a literal recycling family approaching the source boundary and give a zero-diagonal counterexample. Under the stated rooted accessibility and response hypotheses, a local mass-action realization theorem makes the capacity sharp; an explicit power family supplies global maxima, while interacting cores delimit broader claims. The registered Lean project verifies the original refutation and checks the explicit witness against the literature conventions. Broader historical formal evidence and conventional arguments are distinguished in the manuscript and supporting claim map.

Primarily AI-generated textHuman understanding: some partsLean formalizationautocatalytic networkscounterexamplemass-action kineticsoptimal affinityresponse-profile capacity
Oct 21, 2026PDF ↗View work →
hexagon:2610.00437v1

math.DS — Dynamical Systems

Unistationarity of Type II_l autocatalytic cores

Contributed by Michael Mislan

We prove unistationarity of reversible Type II_l autocatalytic cores with l at least three under strictly positive degradation. The source-minimal wiring forces source exhaustion and unit return tails, allowing comparison of positive stationary states through current ratios, chain reduction and a cyclic determinant argument. The proof includes separated stems and the exceptional coincident three-fork branch, with all internal chain species retained. Exact six- and seven-species reversible mass-action examples show that the minimality assumptions cannot be dropped: both admit two distinct positive stationary states with strictly positive rates and losses. The principal uniqueness theorem has a complete registered Lean formalization. The manuscript distinguishes that theorem, broader historical formal evidence and conventional exact calculations; the supplied claim map records the boundaries of each.

Primarily AI-generated textHuman understanding: some partsLean formalizationautocatalytic coresmass-action kineticspositive degradationsource minimalityunistationarity
Oct 21, 2026PDF ↗View work →
hexagon:2610.00436v1

math.RT — Representation Theory

Counterexamples to the weakly Gorenstein symmetry conjecture

Contributed by Qiyue Tang

We construct a 29-dimensional algebra which is left weakly Gorenstein and is not right weakly Gorenstein. We also construct a 145-dimensional gendo-symmetric algebra with the same property. Both constructions work over every field containing an element of infinite multiplicative order. These examples disprove the weakly Gorenstein symmetry conjecture.

A mix of human-written and AI-generated textHuman understanding: some partsweakly Gorenstein symmetry conjecture
Oct 21, 2026PDF ↗View work →
hexagon:2610.00435v1

math.DG — Differential Geometry

Area Nondecreasing Drawstrings and Counterexamples to Euclidean Rigidity and Zero Mass Stability

Contributed by Jiangcheng You, Heng Zhang

Building on the drawstring construction of Kazaras--Xu, we develop a localized refinement that contracts prescribed curves while preserving positive scalar curvature and without decreasing two-dimensional area. As two applications, we disprove part~(b) of Gromov's Euclidean C0C^0 rigidity conjecture and Sormani's zero mass stability conjecture.

Primarily human-written textHuman understanding: all partsADM massAF manifoldsDrawstring constructionsGromov's Euclidean C^0 rigidity conjectureIntrinsic flat convergencePositive mass theoremScalar curvature rigidityZero mass stability
Oct 21, 2026PDF ↗View work →
hexagon:2610.00434v1

math.CO — Combinatorics

Counterexamples to the Two-Bases Conjecture for the Tutte Polynomial

Contributed by Jungang Chen

We disprove the two-bases conjecture of Chávez-Lomelí, Merino, Noble, and Ramírez-Ibañez (Conjecture 5.8 in *Some Inequalities for the Tutte Polynomial*), which asserts that max⁡{TM(2,0),TM(0,2)}≥TM(1,1)\max\{T_M(2,0),T_M(0,2)\}\ge T_M(1,1) whenever a matroid contains two disjoint bases or its ground set is the union of two bases. We construct a connected, self-dual, rationally representable matroid MM of rank 3232 on 6464 elements whose ground set is partitioned into two bases, but for which \[ T_M(2,0)=T_M(0,2)=263656235940864 <275621102802945=T_M(1,1). \] The construction extends to an explicit infinite family MmM_m of rank 2m2m on 4m4m elements such that \[ \frac{\max\{T_{M_m}(2,0),T_{M_m}(0,2)\}}{T_{M_m}(1,1)}\longrightarrow0. \] Consequently, the corresponding additive and multiplicative Merino–Welsh inequalities also fail under the two-bases hypothesis. We additionally construct a simple and cosimple matroid on 6060 elements whose ground set is partitioned into two bases and for which TM(2,0)<TM(1,1)T_M(2,0)<T_M(1,1). This disproves the simple-matroid, two-bases-covering formulation of Conjecture 4.2 of Merino, Ibañez, and Rodríguez; its dual disproves the corresponding cosimple-matroid, two-disjoint-bases formulation. The constructions and inequalities are established by explicit rank functions, exact Tutte-polynomial evaluations, and elementary coefficient estimates.

Primarily AI-generated textHuman understanding: some parts
Oct 21, 2026PDF ↗View work →
hexagon:2610.00433v1

math.CO — Combinatorics

Highly connected counterexamples to the permutation Tutte inequality

Contributed by Jungang Chen

We disprove Conjecture 7.3 of Csikvári, Around the Merino–Welsh conjecture: improving Jackson's inequality, which asserts that T~H(2,0)T~H(0,2)≥1\widetilde T_H(2,0)\widetilde T_H(0,2)\ge1 for every bipartite graph HH of minimum degree at least two. For each integer k≥2k\ge2, we construct an explicit sequence of finite simple balanced bipartite graphs whose minimum degree and vertex connectivity are both exactly kk. The graphs admit automorphisms interchanging their two vertex classes, and both T~H(2,0)\widetilde T_H(2,0) and T~H(0,2)\widetilde T_H(0,2) tend to zero along the sequence. Consequently, no fixed minimum-degree or connectivity threshold ensures the inequality, and even the corresponding maximum and additive inequalities fail in these classes. The construction consists of a complete bipartite core with additional vertices of degree kk attached to disjoint kk-element subsets on each side. A one-variable integral bound gives explicit exponential decay for each fixed kk and a sufficient size threshold. We also give a 1000-vertex, 2-connected example whose two axis values are each less than 7/107/10, certified by exact rational arithmetic. These are statements about the permutation Tutte polynomial; a failure of a local permutation inequality alone does not give a counterexample to a graph or matroid Merino–Welsh inequality.

Primarily AI-generated textHuman understanding: some parts
Oct 21, 2026PDF ↗View work →
hexagon:2610.00431v1

math.CO — Combinatorics

Counterexamples to Three Conjectures of Ok on Tutte-Polynomial Deletion Ratios

Contributed by Jungang Chen

We disprove Conjectures 7.2.1, 7.2.2, and 7.2.3 from Seongmin Ok's thesis *Aspects of the Tutte Polynomial*. Conjecture 7.2.1 asserts that deleting a non-bridge edge from a connected loopless graph cannot decrease the ratio of acyclic orientations to spanning trees; Conjecture 7.2.2 strengthens this to monotonicity of T_{G-e}(x,2-x)/T_G(x,2-x) on 0 ≤ x ≤ 2. Both fail for an explicit simple planar, 2-connected, chordal 2-tree on 16 vertices, for which deleting one edge multiplies the first ratio by 53/54. A two-parameter construction gives an exact criterion for infinitely many counterexamples. Conjecture 7.2.3, a differential inequality for Tutte-polynomial ratios of matroid minors, fails already for the cycle matroid of the five-edge diamond graph: its asserted nonnegative expression equals -4 at (0,2) and -25/16 at (1/4,7/4). We further construct counterexamples of arbitrarily high vertex connectivity and minimum degree, even when both graphs are chordal, and prove that the sharp supremum of the reverse ratio under one deletion is 2, already within simple planar series-parallel graphs. These counterexamples do not disprove the Merino–Welsh inequalities.

Primarily AI-generated textHuman understanding: some parts
Oct 21, 2026PDF ↗View work →
hexagon:2610.00430v1

math.CO — Combinatorics

Highly Connected Counterexamples to the Multiplicative Merino–Welsh Inequality

Contributed by Jungang Chen

For every integer k≥3k\ge 3, we construct kk-connected matroids representable over every infinite field, with girth kk, arbitrarily large cogirth, and Merino–Welsh quotient \[ \Phi(M)=\frac{T_M(1,1)^2}{T_M(2,0)T_M(0,2)} \] tending to infinity. In particular, this disproves **Conjecture 7.2 of Csikvári (2026)**, which asserts the multiplicative Merino–Welsh inequality for simple and cosimple matroids, and therefore also the stronger **Conjecture 7.3** therein, as observed by its author. Our examples are rank truncations of direct sums of kk-element circuits. We give a 3-connected counterexample on 1011 elements of rank 433, certified by exact integer arithmetic, and a 6048-element counterexample admitting a proof without computation. Taking the direct sum of a counterexample with its dual also gives a negative answer to **Problem 4.3 of Beke, Csáji, Csikvári, and Pituk (2024)**: the original maximum and additive Merino–Welsh inequalities can fail for simple and cosimple matroids. The same construction disproves **Conjecture 2 of Knauer, Martínez-Sandoval, and Merino (2026)**: for every fixed g≥3g\ge 3, Φ\Phi is unbounded even among matroids having both girth and cogirth exactly gg. The dual direct-sum examples need not be connected; these constructions do not disprove the graphical or binary versions.

Primarily AI-generated textHuman understanding: some parts
Oct 21, 2026PDF ↗View work →
hexagon:2610.00428v2

math.NT — Number Theory

Every algebraic number is a critical value of an integer polynomial

Contributed by RJ Acuna

We arrived at an independent proof of Petrov's theorem. For every irreducible f∈Z[x]f \in \mathbb{Z}[x] of positive degree there are g∈Z[x]g \in \mathbb{Z}[x] and an irreducible F∈Z[x]F \in \mathbb{Z}[x] of positive degree with F∣g′F \mid g' and F2∣f∘gF^{2} \mid f \circ g. Equivalently, every algebraic number is a critical value of an integer polynomial, so the set of critical values of Z[x]\mathbb{Z}[x] is exactly Q‾\overline{\mathbb{Q}}. The construction is explicit. We classify g∈Z[x]g \in \mathbb{Z}[x] with critical value α\alpha, and find a lower bound for deg⁡g\deg g. The proof was assisted by AI, and has been formalized in Lean.

A mix of human-written and AI-generated textHuman understanding: all parts
Oct 21, 2026PDF ↗View work →
hexagon:2610.00427v1

math.CO — Combinatorics

Total mutual visibility in Fibonacci and Lucas cubes

Contributed by Daniel Adam Pyle

A set of vertices is total mutual-visible if every pair of graph vertices has a shortest path whose internal vertices avoid the set. We prove that the total mutual-visibility numbers of the n-dimensional Fibonacci and Lucas cubes both have order ρⁿ/n, where ρ > 1 satisfies ρ³ = ρ² + 1. The eligible Fibonacci words are exactly those avoiding 11 for which zero padding avoids 0000. Their induced graph decomposes into fibres of recognizing subsets of maximal independent sets of a path, and each fibre is a Cartesian product of smaller Fibonacci cubes. Total visibility becomes open packing in these fibres. An elementary coloring and counting argument proves the order of growth. Typical degree and free-dimension estimates, together with a classical binary-code asymptotic, sharpen the normalized bounds to constants 4.032058… and 6.346255…. We give the cyclic analogue, an exact component inventory, and reproducible exact values through dimension 20 for Fibonacci cubes and 16 for Lucas cubes. Neither a leading asymptotic constant nor a closed formula in every dimension is established.

Primarily AI-generated textHuman understanding: no partsFibonacci cubeLucas cubeconstrained codeopen packingrecognizing subsettotal mutual visibility
Oct 21, 2026PDF ↗View work →
hexagon:2610.00426v1

math.NT — Number Theory

An explicit upper-bound certificate for the essential minimum of the Zhang–Zagier height

Contributed by Ricardo Pinho

We give an exact certificate proving C82<0.2534449C_{82}<0.2534449 for the essential minimum of h(α)+h(1−α)h(\alpha)+h(1-\alpha), where hh is the absolute logarithmic Weil height. We use Doche's limit-point method, building on Grinsztajn's factor library and, in particular, Épi's unequal-multiplicity construction. We retain Épi's denominator factors, add Doche's P7P_7, and add 25 numerator factors found by LP-dual-weighted evaluation lattices with LLL. Joint linear-program re-optimisation is followed by exact rational rounding. Two separately written ball-arithmetic checkers certify the bound; a further adversarial computational audit accepts the certificate.

Primarily AI-generated textHuman understanding: some partsDoche limit-point methodLLLWeil heightZhang-Zagier heightcomputer-assisted proofessential minimumlinear programming
Oct 21, 2026PDF ↗View work →
hexagon:2610.00425v1

math.NT — Number Theory

An explicit upper-bound certificate for the Schur–Siegel–Smyth trace constant

Contributed by Ricardo Pinho

We document an explicit certificate proving C86<1.8117251C_{86}<1.8117251 for the smallest limiting trace-to-degree ratio of totally positive algebraic integers. The construction uses the primal problem of Orloski and Talebizadeh Sardari, with their four-polynomial set enlarged to eleven. A discretised solution is rounded to a rational step density, then normalised to a probability measure. Exact rational calculations and rigorous ball-arithmetic inequalities verify Smith's sufficient criterion for an arithmetic measure. An independently written agent checker accepts the certificate and rejects four corrupted controls. This note records the proof, data provenance, reproduction results and limitations of the previously released certificate; it makes no optimality claim.

Primarily AI-generated textHuman understanding: some partsSchur-Siegel-Smyth trace problemball arithmeticcomputer-assisted prooflinear programmingtotally positive algebraic integers
Oct 21, 2026PDF ↗View work →
hexagon:2610.00421v1

math.CO — Combinatorics

Disjoint union activation of difference distance magic labelings

Contributed by Daniel Adam Pyle

A graph is difference distance magic orientable (DDMO) if its vertices can be labeled bijectively by consecutive positive integers and its edges oriented so that the incoming and outgoing neighbor-label sums agree at every vertex. We give an affirmative answer to the disjoint-union question posed by Aceska et al. in January 2026: two copies of the non-DDMO graph K₃,₃ have a DDMO disjoint union. More precisely, for k ≥ 1 and r ≥ 0, the graph consisting of k disjoint copies of K₃,₃ and r isolated vertices is difference distance magic orientable if and only if k is even or r ≥ 1. The proof reduces fixed-label bipartite orientations to signed zero sums and applies classical Skolem and hooked Skolem pairings. A compatible fixed labeling of K₃,₃ has exactly two orientations, related by reversal. We prove that twelve vertices is the minimum order of a DDMO graph whose connected components are all non-DDMO in isolation. A separate exact computer-assisted classification shows that two disjoint copies of K₃,₃ form the unique graph attaining that minimum.

Primarily AI-generated textHuman understanding: no partsSkolem sequencesdifference distance magicdisjoint unionexact computationgraph labeling
Oct 21, 2026PDF ↗View work →
hexagon:2610.00420v2

math.NT — Number Theory

The Riemann Hypothesis on Young's Lattice

Contributed by José Damián Espinosa

This work establishes a novel structural connection between the Riemann Hypothesis and Young's Lattice through partition-indexed harmonic numbers and the Generalized Assembly Theorem. We establish a new family of equivalences for the conjecture, define the partition-based harmonic numbers HnλH_n^{\boldsymbol{\lambda}}, and reformulate the Riemann Hypothesis as the Young's Lattice equivalence: $$\sum_{\substack{k \in \mathbb{N} \\ \boldsymbol{\lambda} \, \vdash \, k}}\frac{H_n^{\boldsymbol{\lambda}}}{\boldsymbol{\lambda}!} \, a_k > \sum_{d \mid n} d$$ for every n∈Nn \in \mathbb{N}, where λ⊢k\boldsymbol{\lambda} \vdash k denotes a partition of kk, λ!\boldsymbol{\lambda}! is the partition factorial, and ak=1k+⋯+1k2a_k = \frac{1}{k} + \dots + \frac{1}{k^2} or ak=log⁡(k+1)a_k=\log(k+1). We then prove the Generalized Assembly Theorem, which shows that the assembly sum Snm=∑k=1n∑λ∈℘kk+m−1Hnλ\mathfrak{S}_n^m = \sum_{k=1}^{n} \sum_{\boldsymbol{\lambda} \in \wp_k^{k+m-1}} H_n^{\boldsymbol{\lambda}} equals nn for every excess parameter m−1m-1, providing a systematic framework to analyze divisor bounds through the branch decomposition of the partition-indexed series. The resulting structure yields an exact, non-asymptotic realization of the divisors of nn. Finally, we propose and explore a hierarchical expansion connecting the exponential of the Euler--Mascheroni constant, eγe^\gamma, to Multiple Zeta Values under Abel summability. $$e^\gamma = 1 + \sum_{a\ge2}\frac1{a!}\sum_{m\ge0}(-1)^m\zeta^{\mathrm{A}}(a+m)+\sum_{a,b\ge2}\frac1{a!\,b!}\sum_{m,n\ge0}(-1)^{m+n}\zeta^{\mathrm{A}}(a+m,b+n)+\cdots$$

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Oct 21, 2026PDF ↗View work →
hexagon:2610.00419v1

math.NT — Number Theory

Improved bounds on HmH_m for 2≤m≤52\le m\le 5

Contributed by Annanay Kapila

Let Hm=lim inf⁡n→∞(pn+m−pn)H_m=\liminf_{n\to\infty}(p_{n+m}-p_n), where pnp_n denotes the nnth prime. We prove \[ H_2\le 20{,}370,\qquad H_3\le 1{,}380{,}156,\qquad H_4\le 82{,}959{,}170,\qquad H_5\le 5{,}040{,}082{,}836, \] improving the bounds 33,11833{,}118, 2,718,1082{,}718{,}108, 214,099,720214{,}099{,}720 and 14,541,349,28814{,}541{,}349{,}288 of Roeling (2026). We use the multidimensional Selberg sieve of Maynard and Tao in the form developed by Polymath. The arithmetic input is Stadlmann's equidistribution estimate for primes in arithmetic progressions, in the extension to triply densely divisible moduli given in OpenAI's 2026 preprint proving H1≤186H_1\le186. Together with a criterion from the same preprint for the triple dense divisibility of least common multiples, it allows the sieve weights to be supported, instead of on Polymath's truncated simplex, on a region in which single coordinates may be as large as about 0.380.38. On this region we use product trial functions multiplied by an optimised function of the sum of the coordinates, and we certify the resulting variational quotients in ball arithmetic, computing the relevant distribution functions by Fourier inversion.

Primarily AI-generated textHuman understanding: some parts
Oct 20, 2026PDF ↗View work →
hexagon:2610.00418v1

math.DS — Dynamical Systems

The Binder-DeMarco-Dupont Inequality

Contributed by Snir Ben Ovadia, Aaron Brown, Laura DeMarco

In the setting of endomorphisms of Pk(C)\mathbb{P}^k(\mathbb{C}), a dimension formula for the measure of maximal entropy was conjectured by Binder and DeMarco in 2003, asserting that $$\mathrm{dim}_\mathrm{Hausdorff}(\mu)=\log d\sum_{i=1}^k \frac{1}{\chi_i},$$ where dd is the degree of the (non-invertible) endomorphism and χi\chi_i's are the Lyapunov exponents with multiplicity. Indeed, this is well-defined as all Lyapunov exponents are known to be positive due to Briend and Duval. The most significant progress is due to Dupont, who in complex-dimension 2, proved the dimension lower-bound holds; And recently by Genes, who proved the equality for two-dimensional polynomial skew-product maps which satisfy a Lyapunov exponent condition. Recently, He and Bianchi were able to find a family of simple counter-examples. In Appendix A we present a single ``hands on'' description of such a counter example, which is given by a quadratic polynomial skew-product. In this manuscript, we provide the proof, which was achieved by lightly prompting ChatGPT, of extending Dupont’s inequality to all complex dimensions. We wish to share this manuscript with the community as we believe that this provides a full picture of the recently settled Binder-DeMarco conjecture. We do not claim ownership of the proof.

Primarily AI-generated textHuman understanding: some partsComplex dynamics
Oct 20, 2026PDF ↗View work →
hexagon:2610.00417v1

math.CT — Category Theory

The third unit in the adjoint tower of group involutions

Contributed by Zhenyu Sun

The functor G↦Hom⁡(C2,G)G\mapsto\operatorname{Hom}(C_2,G) determines an adjoint tower through successive Eilenberg--Moore comparison functors. We prove that the unit of the third adjunction need not be an isomorphism, answering a question of Bergman. For the generalized dihedral group H=(C4×C4)⋊C2H=(C_4\times C_4)\rtimes C_2, with inversion action, we compute F3V3H≅H×C2F_3V_3H\cong H\times C_2, with counit the first projection. The unit at V3HV_3H is injective, with underlying domain and codomain of cardinalities 2020 and 4040. For an arbitrary group GG, let E(G)=F2V2GE(G)=F_2V_2G and let KGK_G be the kernel of the second counit. We prove that KGK_G is central and that F3V3G≅E(G)/KG[2]F_3V_3G\cong E(G)/K_G[2], where KG[2]K_G[2] consists of the elements of KGK_G whose square is one. The third unit at V3GV_3G is an isomorphism precisely when KGK_G has no element of order four. A central extension of HH of order 128128 supplies such an element and yields the stated description of F3V3HF_3V_3H.

A mix of human-written and AI-generated textHuman understanding: all partsadjoint towercentral extensiongeneralized dihedral groupinvolutory quandlemonadic decomposition
Oct 20, 2026PDF ↗View work →
hexagon:2610.00414v1

math.NT — Number Theory

Superlinear complexity of the (3/2)n(3/2)^n steering word

Contributed by Ralf Stephan

Write (3/2)n=mn+εn(3/2)^n = m_n + \varepsilon_n with mnm_n the nearest integer and εn∈[−12,12)\varepsilon_n\in[-\tfrac12,\tfrac12), and let T=(tn)T=(t_n), tn=2mn+1−3mnt_n=2m_{n+1}-3m_n, be the resulting \emph{steering word}: the step-by-step record of the map x↦32xx\mapsto\tfrac32 x on the orbit of 1, coded by nearest-integer rounding. We prove that the subword complexity pT(k)p_T(k) of TT is superlinear, pT(k)/k→∞p_T(k)/k\to\infty. A pigeonhole argument reduces this to exponential pair repulsion of the orbit modulo one, which we establish by two applications of the pp-adic Subspace Theorem: once in three variables, which handles all gaps between the two orbit points at once, and once in two variables (Ridout's theorem) for the single degenerate family of constant gap. The argument is completely formalized in Lean~4, together with the Subspace Theorem it uses.

A mix of human-written and AI-generated textHuman understanding: all partsuniform distribution
Oct 20, 2026PDF ↗View work →
hexagon:2610.00413v2

math.AG — Algebraic Geometry

Testing the non-vanishing of the Ceresa cycle of X0(N)X_0(N)

Contributed by Leonardo Finzi

For a curve CC and a fixed degree-one divisor ee of CC, we can embed CC into its Jacobian. This embedding induces an algebraic 11-cycle on the Jacobian called the Ceresa cycle, which we denote by Cer(C,e)\mathrm{Cer}(C,e). Although homologically trivial, Ceresa cycles of very general curves with genus ≥3\geq 3 are non-trivial under algebraic equivalence. It is generally unknown whether a very general curve admits a non-trivial Ceresa cycle modulo rational equivalence. In this paper, we explore the non-triviality under rational equivalence of the Ceresa cycle corresponding to the modular curve X0(N)X_0(N). By extending a method of Lupoian--Rawson involving computing the order of the shadow of the Atkin--Lehner involution wNw_N, we show Cer(X0(kN),e)\mathrm{Cer}(X_0(kN),e) is non-trivial when N=57,58,77,82,85,88,91,93,111,112,115,123,133,135,141,143,145,147,155,161N = 57, 58, 77, 82, 85, 88, 91, 93, 111, 112, 115, 123, 133, 135, 141, 143, 145, 147, 155, 161 for k≥1k \geq 1. Moreover, we detail an algorithm for computing the shadow of the Hecke operator TℓT_\ell for odd primes ℓ\ell not dividing NN, which can be used to determine the non-triviality of the Ceresa cycle of X0(N)X_0(N).

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hexagon:2610.00412v1

math-ph — Mathematical Physics

Intrinsic Spectral Cosmology

Contributed by Rajesh Dachiraju

We propose an intrinsic spectral cosmology based on intrinsic geometric wave mechanics, spectral dynamics, localized resonances, HJ source flow, and Hamilton--Jacobi gravity for point masses. The intrinsic field is represented through its time-local spectrum P(k,t)P(\mathbf k,t), whose second moment defines M2M_2, an intrinsic scale RI=M2−1/2R_I=M_2^{-1/2}, and HI=R˙I/RI=−M˙2/(2M2)H_I=\dot R_I/R_I=-\dot M_2/(2M_2). A canonical spectral Hamiltonian gives a conditional continuity equation and current Jk=∂h/∂(∇kθ)\mathbf J_k=\partial h/\partial(\nabla_k\theta); the actual Hamiltonian needed to determine that current is not fixed by the intrinsic space-time weak equation. We exhibit a minimal scale-drift Hamiltonian as a clearly labeled closure ansatz, not as a derived result. Gravity acts only between finite point masses through their Hamilton--Jacobi dynamics. Redshift, distance, structure formation, and the relation of localized resonances to masses remain hypotheses or open derivations. The framework makes no claim of empirical validation.

A mix of human-written and AI-generated textHuman understanding: some parts
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hexagon:2610.00411v1

math-ph — Mathematical Physics

Beyond the Event Horizon: An Operator-First Theory of Persistent Geometry and Black Hole Dynamics

Contributed by Jeromie Beasley

A black hole carries one wall of each kind. Its horizon is a geometric fold: in the right coordinate it is smooth, and nothing breaks there. Its innermost stable circular orbit is a spectral wall, a Jordan block where steady circling turns into a plunge. This paper builds both kinds of wall exactly in a setting where every step can be proved and computed, and then compares them, structurally, with black holes. The setting is the shape space of four bodies. Its kinetic metric is the Bures metric of a real qutrit, and its one genuine wall, the coplanar configurations, is a fold at finite distance. Across it a tetrahedron meets its mirror image. The transport operator A(H) = TH + HT − 2Tr(TH)G is self-adjoint in the kinetic metric. Read in the Gram frame, its back-coupling vanishes like σ₃², and it forms a Jordan block whose square-root branch is exactly the orientation double cover. Lifted to the cover, it is analytic, and the wall point is a plain double point at the tip of a cone with slope 2|t₁−t₂|√(1−δ²). Both readings are exact. What a single eigenvector reports can fail at the wall; the eigenvalue pair and its trace-log ledger come through exact, and a loop through the mirror leaves a sign. New in edition 10 - Reflection and refraction on the projector metric. A wave meeting a step in the metric reflects through the metric along the step and the Clairaut constant C alone: Γ = (√(a₁−C) − √(a₂−C)) / (√(a₁−C) + √(a₂−C)). The transverse metric drops out at a sharp step. Total reflection begins exactly where a geodesic turns, so the optical threshold is a turning point. A change of spectral gap reflects with the peel coordinate tanh(½ ln Δ₂/Δ₁). - One Gram determinant does two jobs: it sets the wave admittance, and it caps the Berry curvature a path can feel (the angle safeguard of the intrinsic quantum geometric tensor). - Mirror and matched load. The polar stratum is a perfect mirror, |Γ| = 1. A Killing horizon, polar in the same sense, is a matched load, Γ = 0. - The photon ring, read by the same generator as the ISCO: each half orbit dims an image by e^−π = 0.0432, a value that belongs to the uncharged hole alone; charge lowers it. - The fermion record: its offset is a point on the same reflection chart, tanh(ħω/2k_BT_H). - The reflection tests were pre-registered before they ran, and each was pushed several ways: 400 random sharp steps, three smooth profiles, reciprocity, flux balance, the transverse metric in a thick layer, total reflection, and the gap form. Main results - The coplanar wall is a fold, and its distance is a Bures angle: arcsin√ε = arccos√F. - Quantum Fisher information on the shape space is eight times the shape kinetic energy wherever the rank is constant, and fidelity is blind to the mirror. - Two kinds of wall: spectral (a Jordan block in every frame) and geometric (a fold that a signed coordinate unfolds). The ISCO is spectral, at the point where gravity's effective power p = 2 + 6M/r reaches the inverse cube. The horizon is geometric. - Any distance potential is even across the wall, with stiffness 2∂V/∂w₃. A thin disk holds the shape at its own frequency ν. Gravity holds the square at ω = √(2(2√2−1)); its exact potential along the lifted square is given in closed form, and its stiffness turns over for a rhombus thinner than diagonal ratio k* = 0.144557. - Structural comparisons: Petrov types through Majorana stars (Kerr is type D), merging ringdown overtones, the horizon's surface resistance as a matched load (376.73 Ω), Bell violation by a Hawking pair, Ramanujan's mock theta functions, and the mirror rule for knots. The metric throughout is the intrinsic kinetic metric of the shape space, never a spacetime metric. Every statement about black holes is a structural comparison, and every claim carries its grade (derived, measured, cited, or structural) where it is made. Files - bh_paper_v10_1.pdf: the paper (56 pages). - bh_paper_v10_1.html: the web edition, with its cover; it runs no scripts. - bh_mixer_v10.html: the Event Horizon Mixer, edition 10. It runs offline in any browser and computes every curve live: the photon-ring orbits, the layered step, the reflection chart with the horizon and the fermion record, the sine guard, and the tip-and-tail matrices. Its Synchronize mode drives every eye from one wall dial, and its "Synchronize the universe" eye shows the connection energy draining as a ring of fields falls into step. - bh_v10_1_code_and_sources.zip: LaTeX source, figures, every check script with its saved output, the pre-registrations, the proof log and the ideas ledger, and a README. Each script states its kill condition before it runs (Python 3: numpy, scipy, sympy, mpmath, matplotlib). AI-assisted: the ideas and research direction are the author's; AI tools were used for literature search, computation, drafting and typesetting, and every number was checked by the scripts in the archive.

A mix of human-written and AI-generated textHuman understanding: some parts
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hexagon:2610.00410v1

math.LO — Logic

Relative PP-Ideals of Hypergraphs and Ultrafilter Spectra

Contributed by Zhenyu Sun

We study ideals of hypergraphs of finite uniformity under countable upper bounds modulo finite vertex covers. The canonical ideal generated by a free filter and finite weak colorings has this relative PP-property exactly when the filter is the intersection of finitely many PP-points. The classification is preserved by adjoining hypergraphs that are locally finite after a finite vertex deletion. The corresponding extension of a trace ideal induces capture ideals on increasing sequences. These are σ\sigma-ideals exactly for PP-filters, and their uniformities are max⁡{πχ(F),d}\max\{\pi\chi(\mathcal{F}),\mathfrak{d}\}. For an ultrafilter U\mathcal{U}, the arity-two capture ideal equals the Louveau ideal exactly when U\mathcal{U} is a PP-point. Successive capture ideals decrease with the arity, and equality characterizes QQ-points. Every strict inclusion is witnessed by a closed set. Thus a non-QQ PP-point yields a decreasing sequence of σ\sigma-ideals, distinguished by closed sets, with equal uniformities. We also establish invariance under locally finite witness enlargements and analyze Stone kernels and countable block products. Consistently, nested ray-positive PP-sets have the same singleton endpoint spectrum and different ray transversal numbers.

A mix of human-written and AI-generated textHuman understanding: all partsHypergraph idealLouveau idealP-pointQ-pointStone dualityreaping numberultrafilter
Oct 20, 2026PDF ↗View work →
hexagon:2610.00409v2

math.NT — Number Theory

A Frobenius determinant proof of the non-existence of Landau–Siegel zeros

Contributed by Roelof van Dijk

We give a variant of the OpenAI proof [1] of the non-existence of Landau–Siegel zeros, following the simplifications of Kerr [2], Green [3] and Barreto [4]. The Frobenius congruence proves both divisibility and nonvanishing of the determinant. Reduction modulo the largest inert prime below the cutoff reduces nonvanishing to a polynomial root count, replacing resultants and Wronskians. The argument was discovered in discussions with GPT-6 Astra and Opus 5.5 and the new arguments should be attributed to these models.

A mix of human-written and AI-generated textHuman understanding: no parts
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hexagon:2610.00408v1

math.NT — Number Theory

On the irrationality of the 33-adic zeta value ζ3(5)\zeta_3(5)

Contributed by Antti Hippeläinen

We give a proof that the 33-adic zeta value ζ3(5)\zeta_3(5) is irrational. The proof adapts to p=3p=3 a Hankel-determinant method that a recent preprint applies to ζ2(7)\zeta_2(7). A Volkenborn integral of an explicit rational function defines a linear functional with values in Q+Q ζ3(5)\mathbb{Q}+\mathbb{Q}\,\zeta_3(5), and its Hankel determinants are the values at ζ3(5)\zeta_3(5) of polynomials with rational coefficients. We bound the 33-adic valuation of these values and the size and the denominators of the coefficients. The new ingredients at p=3p=3 are a triangular basis of the polynomials that are integer valued on 3−1Z3×3^{-1}\mathbb{Z}_3^\times, a local estimate at primes of the order of nn in which a pole of order 33 coming from harmonic sums cancels exactly against a zero of order 33 of the rational function, and a non-vanishing theorem at an auxiliary prime, in which Wilson's theorem makes one block of a reduced Gram matrix invertible. Here nn is the parameter of the rational function. Two constants of the estimates are bounded by finite computations, and with these bounds the 33-adic valuation exceeds the size and the denominators together by more than 4.54.5 bits per n2n^2. An appendix gives a second proof, independent of the first, which adapts to ζ3(5)\zeta_3(5) and the modular curve X0(3)X_0(3) a determinant argument that a preprint of Yi gives for ζ7(3)\zeta_7(3), and which rests on published theorems on overconvergent modular forms.

Primarily AI-generated textHuman understanding: no partsHankel determinantsKubota-Leopoldt L-functionVolkenborn integralirrationalityoverconvergent modular formsp-adic zeta values
Oct 20, 2026PDF ↗View work →
hexagon:2610.00406v1

math.OC — Optimization and Control

Classical regularity and attainment in portfolio choice with nonhomothetic utility

Contributed by Yuanhong Wei, Zhenyu Sun

We study classical regularity and attainment in terminal-utility portfolio choice with unspanned factors, nonnegative wealth, and exposures that are square-integrable in expectation. A joint estimate for logarithmic marginal utility controls relative risk aversion and factor sensitivity. In several dimensions, a risk-price component uniformly separated from the factor-hedging span gives classical solvability and a unique strong optimal portfolio on every finite horizon. A scalar comparison equation treats correlated single-factor markets. For state-independent C4C^4 utility, the preference assumptions reduce to positive lower and upper bounds on relative risk aversion. In an orthogonal single-factor market, a uniform finite-order condition allows the risk price to vanish and change sign. Subelliptic gradient estimates and a parabolic equation in the factor variable yield a classical value; verification and strict concavity give strong existence for its continuous feedback. In both settings, the optimal dual coefficient has a deterministic bound uniform over initial states and multipliers. Attainment has a separate integrability boundary. For deterministic opportunities with a finite static optimum, the prescribed strategy class attains the value exactly when the optimal payoff lies in L2L^2 under the physical probability. An explicit nonhomothetic utility gives a horizon and wealth classification with smooth finite values and nonattainment. Further results identify wealth-dependent purchasing-power hedges and a strict increase in the dual value under factor-information restrictions.

Primarily human-written textHuman understanding: all partsNonhomothetic utilityclassical regularityconvex dualitydegenerate HJB equationssquare-integrable strategiesstochastic factors
Oct 20, 2026PDF ↗View work →
hexagon:2610.00405v1

math.MP — Mathematical Physics

Quaternionic Heisenberg-Robertson-Schrodinger and Maccone-Pati Uncertainty Principles

Contributed by K. MAHESH KRISHNA

Let Q\mathcal{Q} be a quaternionic Hilbert space. Let A:D(A)⊆Q→QA: \mathcal{D}(A)\subseteq \mathcal{Q}\to \mathcal{Q} and B:D(B)⊆Q→QB:\mathcal{D}(B)\subseteq \mathcal{Q}\to \mathcal{Q} be possibly unbounded self-adjoint linear operators. Then for all h∈D(AB)∩D(BA)h \in \mathcal{D}(AB)\cap \mathcal{D}(BA) with ∥h∥=1\|h\|=1, we show that \begin{align*} (1) \quad \quad \quad \Delta _h(A)\Delta _h(B)\geq \frac{\sqrt{|\langle \{A,B\}h, h \rangle -2\langle Ah, h \rangle\langle Bh, h \rangle|^2+|\langle [A,B]h, h \rangle |^2}}{2} \end{align*} and \begin{align*} (2) \quad \quad \quad \Delta _h(A)^2+ \Delta _h(B)^2 \geq \frac{|\langle (A+B)h, y \rangle|^2+|\langle (A-B)h, y \rangle|^2}{2}, \quad \forall y \in \mathcal{Q} \text{ satisfying } \|y\|\leq1 \text{ and } \langle h, y \rangle =0, \end{align*} where Δh(A)≔∥Ah−⟨Ah,h⟩h∥ \Delta _h(A)\coloneqq \|Ah-\langle Ah, h \rangle h \|. We call Inequality (1) as quaternionic Heisenberg-Robertson-Schrodinger uncertainty principle and Inequality (2) as quaternionic Maccone-Pati uncertainty principle.

Primarily human-written textHuman understanding: all partsQuaternionic Hilbert space.Uncertainty Principle
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hexagon:2610.00404v1

math.KT — K-Theory and Homology

Quillen's local parameter conjecture in dimension four

Contributed by Yunpeng Li

Let RR be a Noetherian regular local ring of mixed characteristic and dimension four, and let rr be a member of a regular system of parameters. We prove that every finitely generated projective module over R[1/r]R[1/r] is free. We also prove that Sp4(R[1/r])\mathrm{Sp}_4(R[1/r]) acts transitively on unimodular columns of length four and that every determinant orientation of a rank-two projective module is induced by a basis. The argument reduces freeness to two vanishing results: the degree-three Zariski cohomology of the naive degree-four Milnor--Witt sheaf and the alternating Witt group of R[1/r]R[1/r]. We establish these results using integral degree-two algebraic KK-theory, supported hermitian KK-theory, and local motivic cohomology. A finite-rank symplectic comparison relates the image of the completion obstruction to Milnor--Witt multiplication and yields the required symplectic frames.

Primarily AI-generated textHuman understanding: no parts
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hexagon:2610.00407v1

math.NT — Number Theory

Optimal bounds for the length and number of unit fraction representations

Contributed by Thomas Bloom, Wouter van Doorn

This paper gives a human-written exposition and simplification of proofs that were recently posted by OpenAI. We prove that every a/b∈(0,1)a/b\in (0, 1) can be written as the sum of O(log⁡log⁡b)O(\log\log b) many distinct unit fractions. We also prove that the number of representations of 11 as the sum of kk distinct unit fractions grows doubly exponentially in kk.

Primarily human-written textHuman understanding: all partsegyptian fractionsunit fractions
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hexagon:2610.00402v1

math.PR — Probability

Lower and Upper Bounds for Glauber Dynamics from Typical States of the Low-Temperature Sherrington–Kirkpatrick Model

Contributed by P. M. Aronow, Patrick Lopatto

We study the heat-bath (Glauber) dynamics of the Sherrington–Kirkpatrick model at zero external field and fixed inverse temperature β>1\beta>1. We sample the disorder JJ and an initial configuration X0X_0 from its Gibbs measure. Let TJ(X0)T^J(X_0) be the time at which the transition law, conditional on JJ and the realized X0X_0, first comes within total-variation distance 1/41/4 of equilibrium. In continuous time, with rate one per site, we prove that, for every fixed ω∈(0,1/4)\omega\in(0,1/4), with probability tending to one, \[ A_\beta n^{7/30}<\log T^J(X_0)\le n^{1-\omega}, \] where Aβ>0A_\beta>0 depends only on β\beta. Corresponding bounds hold for the discrete-time chain. These results strengthen OpenAI’s typical-start lower and upper bounds while following their proof architectures. The lower-bound argument introduces a three-replica comparison with a cubic pressure deficit and uses Aronow and Lopatto’s quantitative mean-bias estimate. The upper-bound argument combines a pathwise information–energy cancellation, concentration estimates on a growing mesh, and optimized exponents. We also complete Lean formalizations of two known results: OpenAI’s fluctuation-scale theorem, giving variance n1/3+o(1)n^{1/3+o(1)} for the log partition function, and its mean-bias consequence of order at least n−5/6−εn^{-5/6-\varepsilon} for every fixed ε>0\varepsilon>0. The formal development proves the positive-Laplace-window input needed for the fluctuation theorem. Five specified result groups are formalized, including the continuous-time dynamics upper bound.

Primarily AI-generated textHuman understanding: some parts
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hexagon:2610.00401v1

math.NT — Number Theory

Irrationality measures for G±5π2/48G\pm5\pi^2/48 and L(2,χ−3)L(2,\chi_{-3})

Contributed by Sean Litsey

Building on OpenAI's determinant method for the irrationality of Catalan's constant G=∑j≥0(−1)j/(2j+1)2G=\sum_{j\ge0}(-1)^j/(2j+1)^2, we prove \[ \begin{gathered} \mu\Bigl(G-\frac{5\pi^2}{48}\Bigr)\le 38.35,\qquad \mu\Bigl(G+\frac{5\pi^2}{48}\Bigr)\le 38.35,\\ \mu\bigl(L(2,\chi_{-3})\bigr)\le 23.82 , \end{gathered} \] where μ\mu is the irrationality exponent. We know of no earlier irrationality measure for G±5π2/48G\pm5\pi^2/48. For L(2,χ−3)L(2,\chi_{-3}), which Calegari, Dimitrov and Tang proved irrational, the previous best bound we know of is their effective bound μ≤24781\mu\le24781. The approximations come from determinants of \emph{rectangular} L\"owner matrices of digamma differences at integer nodes. We bound their denominators with pp-adic assignment problems and rank deficits modulo pp. We prove nonvanishing modulo a prime by reducing it to bordered Krylov determinants, which we evaluate through Jacobi matrices and Darboux transformations. We bound their size with Andr\'eief's identity and a certified lower bound for a discrete logarithmic energy. Every result is formalized in Lean~4, with no unproved steps and only the standard axioms.

Primarily AI-generated textHuman understanding: no parts$L(2Andr\'eief identityCatalan's constantCauchy determinantsDarboux transformationL\"owner matricesLean\chi_{-3})$formal verificationirrationality exponent
Oct 20, 2026PDF ↗View work →
hexagon:2610.00399v1

math.CV — Complex Variables

A covering-strip and moment certificate for Korenblum's constant

Contributed by Ricardo Pinho

We give an explicit certificate for c_2>=54/125=0.432, where c_2 is Korenblum's constant for the Bergman space. This is an unrefereed preprint. The construction combines an elementary two-zero Schwarz estimate on the covering strip of an annulus, the classical hyperbolic Jung inequality, and Wikström's moment-duality criterion. A rational twelve-atom measure with a Lebesgue tail satisfies the required inequalities for every moment index. Two independently implemented exact checkers, one using Arb and one using only integer interval arithmetic, verify the quantitative hypotheses. The best public certified lower bound located in the accompanying priority audit is Wikström's 0.4263. Neither optimality nor a new version of Jung's theorem is claimed.

Primarily AI-generated textHuman understanding: some partsBergman spaceKorenblum's constantcomputer-assisted proofhyperbolic Jung inequalityinterval arithmeticmoment duality
Oct 20, 2026PDF ↗View work →
hexagon:2610.00398v2

math.NT — Number Theory

log⁡p2⋅log⁡p(3+2)\log_p 2 \cdot \log_p(3+\sqrt2) is not a pp-adic linear form in logarithms

Contributed by Shawn Krisman

Let pp be a prime, Cp\mathbb{C}_p the completion of an algebraic closure of Qp\mathbb{Q}_p, Q‾\overline{\mathbb{Q}} the algebraic numbers in Cp\mathbb{C}_p, LL the Q\mathbb{Q}-vector space of pp-adic logarithms of algebraic numbers, and L=Q‾+Q‾L\mathcal{L}=\overline{\mathbb{Q}}+\overline{\mathbb{Q}}L the space of pp-adic linear forms in logarithms of algebraic numbers with algebraic coefficients. We prove, in Lean 4 with Mathlib and from the three standard axioms only, that log⁡p2⋅log⁡p(3+2)∉L\log_p 2\cdot\log_p(3+\sqrt2)\notin\mathcal{L} for every prime p≡±3(mod8)p\equiv\pm3 \pmod 8, together with a family of similar statements for real quadratic fields, and the general principle behind them: if σ\sigma is a continuous automorphism of Cp\mathbb{C}_p fixing a nonzero t∈Lt\in L and moving u∈Lu\in L to a conjugate σu≠±u\sigma u\neq\pm u of finite orbit, then tu∉Ltu\notin\mathcal{L}. This is G. Diaz's use of complex conjugation with an element of Gal(Q‾p/Qp)\mathrm{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p) in its place; the complex analogue log⁡2⋅log⁡(3+2)∉L\log 2\cdot\log(3+\sqrt2)\notin\mathcal{L} is open. The deep input is Roy's form of Waldschmidt's linear-subgroup theorem for Gad0×Gmd1\mathbb{G}_a^{d_0}\times\mathbb{G}_m^{d_1} over Cp\mathbb{C}_p, with a derivative subspace, which we also prove in Lean, from Philippon's multiplicity estimate; the strong six exponentials theorem and the inhomogeneous Baker--Brumer theorem in two logarithms over Cp\mathbb{C}_p follow. The pp-adic linear-subgroup theorem has been used since Laurent and Roy, but we were unable to locate a proof written for Cp\mathbb{C}_p of the version with derivatives. The Lean development is the proof of record; the paper explains the statements, the route, and how to check the certificate. The code is available at https://github.com/telaviv/p-adic-log-products.

Primarily AI-generated textHuman understanding: no partsLean 4Mathlibformal verificationinterpolation determinantslinear forms in logarithmslinear-subgroup theoremmultiplicity estimatesp-adic logarithmp-adic transcendencestrong six exponentials theorem
Oct 20, 2026PDF ↗View work →
hexagon:2610.00394v1

cs.DM — Discrete Mathematics

An explicit counterexample to the six-neighbor square tiling conjecture

Contributed by Chunlei Liu, Aijin Lin

Luo, Sun and Wu conjectured that in a locally finite square tiling of the complex plane in which each square intersects exactly six others, all squares have the same size. We construct a counterexample by surrounding a ten-square core with alternating rectangular shells, each formed by six squares. The only point-only contacts occur at the three four-square points in the core; all subsequent contacts have positive length. A two-state induction shows that every new square has four contacts when inserted and acquires two more at the next shell. The nested rectangles exhaust the plane, so the construction is locally finite and gives a counterexample on the whole plane.

Primarily human-written textHuman understanding: all partsSquare Tiling Conjecture
Oct 20, 2026PDF ↗View work →
hexagon:2610.00393v3

math.AG — Algebraic Geometry

Algebraicity of the G2\mathrm{G}_2-invariant tensors on Yun's exceptional motives

Contributed by Benjamin Severn

Zhiwei Yun constructed, from a rigid Hecke eigensheaf for G2\mathrm{G}_2 on P1\mathbb{P}^1 minus three points, a family of rank-seven motives whose motivic Galois groups are dense in G2\mathrm{G}_2. We show that on this family the invariant cubic form is the class of an algebraic cycle: at every Hodge-generic parameter, every rational Hodge class on every tensor construction of the rank-seven piece is algebraic, and at every parameter the transported G2\mathrm{G}_2-invariant tensors are algebraic. The cycles are top components of Satake convolution fibres transported by Hecke correspondences. The projector that cuts out the rank-seven piece is made algebraic by a canonical weighted resolution of Yun's boundary divisor, built on the moduli stack of bundles from the stabiliser tori of its points, together with a characteristic-zero census of those points and a cycle-class comparison on varieties with quotient singularities. Yun announced that the cubic is motivated in Andr\'e's sense; the step from motivated to algebraic is new. This version, with its computational certificates and review bundle, is archived on Zenodo: https://doi.org/10.5281/zenodo.23478625

Primarily AI-generated textHuman understanding: some partsHecke eigensheavesHodge conjectureexceptional motivesgeometric Satakemotivated cyclesweighted blow-ups
Oct 20, 2026PDF ↗View work →