Let Ta be the first hitting time of a nonzero point a by a
symmetric strictly α-stable L\'evy process, where 1<α<2.
We give a computer-assisted classification of its ordinary additive
infinite divisibility. There is a unique threshold α∗, with
1.6670659903<α∗<1.6670659915, such that Ta is infinitely
divisible if and only if 1<α≤α∗. The threshold is
characterized by a unique zero contact of the canonical inverse
logarithmic-derivative kernel. The proof combines explicit analytic
estimates, rigorous interval covers, Gaussian necessary conditions and
Rouch\'e's theorem. In particular, a fully analytic argument proves
failure of infinite divisibility for every 15/8≤α<2.
Scalar bounds at the two time ends reduce the compact interval
computations. The finite computations and their acceptance
conditions are supplied as ancillary material.
Primarily AI-generated textHuman understanding: no parts
Let V be a simple vertex algebra with a connected nonnegative
integer grading, and let G be a finite group of grading-preserving
automorphisms. We prove that C2-cofiniteness of V implies
C2-cofiniteness of VG. The proof bounds the length and height of monomials in strong generators
representing classes in a relative C2 quotient.
The length bound follows from a good filtration and finite-group
invariant theory. The height bound follows from differential equations
for products of fields, together with identities for invariants and vacuum
coefficients. The height bound also shows that finite strong generation
passes to VG, without assuming C2-cofiniteness.
For strongly rational vertex operator algebras, the cofiniteness result
and McRae's orbifold theorem imply strong rationality of every
finite-group fixed-point algebra.
A mix of human-written and AI-generated textHuman understanding: all parts
We construct a Boolean Lévy group with a free ergodic spatial nonsingular action on a standard probability space, answering a question of Glasner, Tsirelson and Weiss. The action admits no equivalent invariant σ-finite measure. The group is a quotient of a weighted summable Boolean group by a discrete subgroup. By the nonsingular fixed-point theorem of Avraham-Re'em and Peterzil, the resulting group is not locally Wasserstein.
A mix of human-written and AI-generated textHuman understanding: all parts
We prove that the lower limit of the sixth moment of the Riemann zeta
function, normalized by (a3/9!)T(logT)9, is greater than 41.007.
Our approach extends the two-piece amplifier construction used by Durkan and Page by allowing independent profiles, centered derivatives, and finite real linear combinations of two projected test families. An asymmetric twisted fourth-moment estimate and their amplified moment formula yield a common numerator kernel. Explicit balanced and unbalanced
kernels determine the projected Gram form. A rational certificate with
250 tests, followed by approximation by admissible profiles, gives the
stated bound.
A mix of human-written and AI-generated textHuman understanding: some partsRiemann Zeta Functionamplified momentssixth momenttwisted fourth moment
In every dimension d≥3, we construct nonspherical, origin-symmetric, strictly convex bodies of revolution with real-analytic boundary that float in equilibrium in every orientation. Their densities tend to the unit-ball cap fraction at height (d+4)−1/2, where we prove that a degree-four spherical resonance is isolated. More generally, our construction applies at isolated even-degree resonances whose waterline angle is an irrational multiple of π. Varying the density gives formal deformations. To obtain actual bodies, we use a finite-order diagonal reduction to establish inverse estimates away from small divisors. Newton iteration then corrects fixed-order truncations on closed parameter sets of asymptotically full measure.
A mix of human-written and AI-generated textHuman understanding: some partsUlam's floating body problembodies of revolutioncentral symmetryconvex geometryfloating bodiessmall divisors
We characterize weak amenability for arbitrary connected Lie groups and prove the product formula for the Cowling--Haagerup constant in terms of the simple factors of a Levi decomposition. Consequently, the constant is invariant under local isomorphism and decreases under quotients of connected locally compact groups. We also prove that a locally compact group G with amenable component group G/G0 has the Haagerup property if and only if it is weakly amenable with constant one. This proves Cowling's conjecture for connected Lie groups and almost connected locally compact groups. For groups with amenable component group, the Cowling--Haagerup constant is unchanged by every closed central quotient. We deduce applications to lattices and their central quotients. The proof combines Ozawa's invariant-state obstruction with averaging over central characters and the tensor identity for induced representations.
A mix of human-written and AI-generated textHuman understanding: some partsCowling--Haagerup constantHaagerup propertycentral extensionsconnected Lie groupsweak amenability
Contributed by Deep Bhattacharjee, Ushashi Bhattacharya, Sanjeevan Singha Roy
We prove that one finite set of points decides whether sensors cover a polygonal target under every failure rule in which an extra working sensor never hurts. The set is the same for every such rule, and on a face of dimension f the weakest point is fixed by f+1 sensors. The theorem holds in any dimension for functions that share a convex part and differ by affine terms. For sensing disks it gives the exact radius and power that keep the target covered under each rule, as rational numbers, the full trade-off between radius and budget, the smallest list of sensor groups whose loss breaks coverage, exact tests under independent placement errors, and protection plans proved optimal by a matching dual certificate.
A mix of human-written and AI-generated textHuman understanding: all partsmonotone failure rulesresilience certificatessensor coveragesupport theorem
Let A and B be real symmetric matrices with nonnegative entries. We prove that ρ(AB)k≤ρ(AkBk) for the spectral radius ρ(⋅) and every integer k≥1; for B=uuT this is the Blakley–Roy inequality. We prove it via the more general inequality ∥(AB)m∥k≤∥(AkBk)m∥ in the spectral norm, valid for all integers k,m≥1. The even exponents case follows from standard tools; for odd exponents k=2r+1 the claim follows via the more general product inequality (uTC1⋯Ctv)k≤(uTu)r(vTv)ruTC1k⋯Ctkv for nonnegative symmetric Ci and nonnegative u,v. Its proof is an entropy argument on paths that traverse each edge k times, back and forth.
A mix of human-written and AI-generated textHuman understanding: all partsBlakley-Roy inequalitymatrix productsnonnegative matricesspectral radius
Let A and B be positive semidefinite, and let 0≤t≤1. Lin (2017) conjectured that λj((1−t)A+tB)≥σj1/2(A2(1−t)B2t) for j=1,…,n. The case t=1/2 is Drury's theorem, which answered a long-standing question of Bhatia and Kittaneh. Lin remarked that Drury's method does not seem to reach other values of t. We prove his conjecture for every t∈[0,1]. The proof builds on Drury's argument, but replaces his block-matrix step (unique to the midpoint t=1/2) by an eigenvalue count based on an inertia identity and a one-variable inequality.
A mix of human-written and AI-generated textHuman understanding: all partsLin's conjecturearithmetic-geometric mean inequalitymatrix inequalitiessingular valuesunitarily invariant norms
We prove McLeod's (1959) conjecture that (hℓ+r/hℓ)1/r is subadditive for all ℓ≥0 and r≥1, where hk is the complete homogeneous symmetric polynomial of degree k (J. B. McLeod, "On four inequalities in symmetric functions", Proc. Edinburgh Math. Soc., 1959). We first prove that the ratio hk/hk−1 is convex, and our proof takes an unexpected detour through queuing theory. The geometric mean of r consecutive such ratios equals McLeod's conjectured function, but geometric means of convex functions need not be convex. Our second and decisive step is a tensorization trick that couples neighboring ratios tightly enough to make their geometric mean convex.
A mix of human-written and AI-generated textHuman understanding: all partsMcLeod inequalityclosed queuing networkcomplete homogeneous symmetric polynomialconvexitydiscrete Poincare inequality
Created on 23 October 2026, this manuscript draws on a frontier-dump investigation of Hironaka's proposed positive-characteristic resolution method. Taylor Dupuy initiated the queries during his talk How to Use AI in Mathematical Research Effectively at the University of Rochester Number Theory Seminar on that date. The findings were synthesized with harvest-dumps and developed into this manuscript with using-latexpowers. We examine the proposed positive-characteristic resolution mechanism of Hironaka (2017). Its negative-degree pieces are the unit ideal at a closed singular point where, for a common multiple m of the generator degrees, ℘(E,m) contains an element of order m. For the family xp+yn, with n>2p and p∤n, the asserted transformation rule fails under the algebraic description of the characteristic algebra. With core focusing read by (44), the fixed local chain and the Eˇ or E^ reading of the invariant, the length-one string agrees before and after the first blowup. With renewed chains and corresponding transported choices, the corresponding strings agree along the resolution sequence under either core-focusing reading. The two formulas for core focusing respectively retain the base-hiked exponent and, when its Inv-max stratum is smooth and closed, forget it. We give characteristic-2 limits on tail enclosure and a shade increase at a kangaroo point. These calculations do not supply a decreasing invariant or settle resolution. The mathematical draft of this manuscript was AI-generated by Claude Code from an AI-generated research record; its proofs received only AI review, and no human has checked them.
Primarily AI-generated textHuman understanding: no partsHironakacounterexamplespositive characteristicresolution of singularities
We study positive definiteness of Hua-Bellman matrices (i.e., matrices of the form [det(I−Ai∗Aj)−α]ij, where Ai and Aj are strict contractions and α∈R). We start by revisiting a 1959 work of Bellman (Representation theorems and inequalities for Hermitian matrices; Duke Mathematical J., 26(3), 1959) that studies positive definiteness of Hua-Bellman matrices and claims a strengthening of Hua's representation-theoretic results (Inequalities involving determinants; Acta Mathematica Sinica, 5 (1955)). We uncover a critical error in Bellman's proof that has surprisingly escaped notice to date, and we show by an explicit example that his claim itself is false. We then provide conditions under which det(I−A∗B)−α is a positive definite function. For complex contractions our condition is the known sharp one, with the integer exponents below Hua's range reached by an elementary argument. For real symmetric contractions we recover the part of Bellman's claims that survives. Building on our result, we introduce a new Poincaré-like distance metric on noncommuting strict contractions, and yet another `log-det' based distance in the appendix.
Deciding which conjectures are worth proving is a crucial part of mathematical discovery, and a system that learns mathematics from its axioms alone must make that decision without human guidance. Self-play provers such as Minimo learn a conjecturer alongside the prover and reward it for producing hard conjectures, but a conjecture that is harder to prove is not necessarily more useful for improving a prover: the learned conjecturer drifts toward long, artificial statements. We study conjecturing in a controlled, single-round setting in the Peano formal environment, across propositional logic, abelian groups, and natural-number multiplication, evaluating a set of conjectures by how much a fixed prover improves on held-out theorems after training on their proofs. We show that a far simpler generator does better: making small structural edits to statements the prover already knows yields conjectures that, at an equal training budget and under best-first search, improve the prover by 6 to 13 percentage points more than Minimo's learned conjectures. The gain holds even when conjectures are drawn at random from the generated pool. In contrast to the common practice of selecting the most difficult statements, we also propose Hard Neighbours (HN), a score that favors conjectures whose proof difficulty grows disproportionately relative to their structural distance from easier reference statements; HN gives the largest selection gains in propositional logic, but no selection method is best in every domain. In the current setting, how conjectures are generated matters more than how they are selected, and structural perturbation is the baseline that future conjecturers should be measured against. Apart from improvements in proving ability, we also study other attributes of generated proofs: the amount of search required and the length of the proofs.
Primarily human-written textHuman understanding: all partsconjecturinglanguage models
For each prime p, rational Gersten injectivity in a fixed degree for mixed-characteristic henselian discrete valuation rings of residue characteristic p is equivalent to injectivity for W(Fp(t1,t2,…)). For a finite bound d on residue transcendence degree, the test ring is W(Fp(t1,…,td)). For these henselian DVRs, rational localization boundary images depend only on the residue field and are reflected by arbitrary residue-field extensions. A normalization by the absolute ramification index makes the resulting comparisons of kernels canonical and functorial. The proof uses regular approximation, finite transfers, and perfection. An appendix gives an alternative comparison based on the existential embedding theorem of Anscombe--Dittmann--Jahnke, including an example with nonnoetherian valuation rings. We also describe the additional boundary condition that induction between consecutive test rings would require.
A mix of human-written and AI-generated textHuman understanding: all partsCohen ringGersten conjectureWitt vectorsalgebraic $K$-theoryexistential closednesshenselian discrete valuation ring
Contributed by Amélie Loher, Connor Mooney, Clément Mouhot
We prove Aleksandrov maximum principles for linear second-order kinetic equations in non-divergence form with measurable, uniformly elliptic diffusion coefficients. For coefficients A(t,v) we obtain every source exponent p>2d+1. For autonomous coefficients a(x,v) in one space dimension, we obtain every p>p∗(λΛ), where p∗(λΛ)<4 depends only on the ellipticity ratio. The first step in both proofs is Döblin--Fourier cancellation: two positive parts of the kernel have nearly opposite phases, and a common lower bound for their velocity marginals allows us to cancel fixed portions of them. For the autonomous problem, this argument applies on velocity intervals away from zero. A preliminary Aleksandrov estimate gives Hölder continuity and then a Harnack comparison. A homogeneous Bellman barrier controls concentration in position and velocity; together with Harnack, it makes the local density estimates summable across zero velocity, including at p=4. We give the full proofs and the passage from the maximum principle to local Hölder continuity.
Primarily AI-generated textHuman understanding: all parts
We prove that a general Steiner bundle on projective space over a field of characteris-
tic zero has natural cohomology. The proof constructs a matrix of linear forms from multiplication
of sections on a curve. When the rank exceeds the dimension of the projective space, a finite étale
isogeny of elliptic curves splits the bundles controlling the graded kernels and cokernels into line
bundles of equal degree. Choosing the auxiliary line bundle outside a finite set gives maximal
rank in every degree. A rational curve provides the boundary case. Two open rank conditions
then yield the assertion for general matrices and all twists.
Primarily AI-generated textHuman understanding: some parts
We give a candidate proof of the following statement: for every finite cyclic group Cn, the cumulative fourth deck of a binary subset determines it up to translation. Hence rset(Cn)≤4 for every n, and, using the Keleti--Kolountzakis Corollary 1.1 for the lower bound, rset(Cn)=4 for every even n>10. The proof has four components: finiteness of the binary fibre, cyclotomic torsion of the phases, gluing of primary characters, and extinction of dyadic residues by two odd-denominator projectors. The bridge between ordered analytic correlations and Pebody's cumulative deck is proved explicitly, including stabilizers, zero and repeated offsets, and the empty pattern. The text contains the complete mathematical chain; finite checks are included only as convention and implementation checks, not as premises of the uniform proof.
Primarily AI-generated textHuman understanding: no partsFourier analysisPebody conjecturePhase retrievaldeck reconstructionfinite cyclic groups
We prove that, for every pair of positive integers i,j, the universal rank-two cluster scattering coefficient τ(i,j) satisfies max(i,j)!τ(i,j)∈Z[b,c,g]. This resolves the factorial-denominator conjecture of Elgin, Reading, and Stella (Conjecture 4).
A mix of human-written and AI-generated textHuman understanding: some partsCluster Algebra
We prove that the Mirković–Vilonen basis of C[N] contains every cluster monomial and that each of its elements is g-pointed in every cluster, where N is a maximal unipotent subgroup of a complex semisimple group. This resolves Conjecture 6.6 in Kamnitzer’s ICM 2022 survey.
A mix of human-written and AI-generated textHuman understanding: some partsCluster Algebra
Two recent works by OpenAI and Anthropic refuted the Exponential Time Hypothesis (ETH). They count the satisfying assignments of a k-CNF on n variables in time 2n/(logn)1/2−o(1). Many fine-grained lower bounds in learning theory rest on ETH and its relatives. We determine what the refutation changes for learning.
The answer is a locality principle: the new algorithms speed up learning when each labeled example constrains few bits of the hypothesis, and under the CNF form of the Strong Exponential Time Hypothesis, which survives, the speed-up ends at Θ(logN) bits. Three transitions make the principle precise.
Locality. Call a class with N-bit descriptions κ-local if each label depends on at most κ description bits, as for quantized models on sparse data. Then proper PAC learning, exact version-space counting and sampling, and the halving algorithm run in time poly(m)⋅2o(N) on samples of size m for κ=o((logN)1/4/loglogN), and proper PAC learning at constant accuracy does so for κ=o(logN). At inverse-polynomial accuracy, proper learning of κ-local classes is equivalent to κ-SAT, up to a factor 2 in the locality. Under CNF-SETH, consistency for binarized perceptrons on O(logN)-sparse data needs time 2(1−γ)N for every γ>0.
Accuracy. Exact empirical risk minimization, proper agnostic learning and the exponential mechanism reduce to tensor contractions over a truncated polynomial ring. As a consequence, the Sparse Max-SAT Hypotheses of Alman and Vassilevska Williams are false. At constant accuracy, proper learning is SAT parameterized by formula length. With quasi-linear PCPs this gives a 2N/polylogN lower bound under a length form of ETH, which replaces Gap-ETH for reductions of quasi-linear size. At accuracy N−3/8, the Max-3-SAT Hypothesis forces time 2(1−o(1))N.
Treewidth. Exact inference in graphical models of bounded degree and arity takes time 2O(n/logn). On expanders this beats every elimination order and every structured DNNF. Under #CNF-SETH, time 2(1−γ)tw is again necessary once a model may have twO(1) variables.
We also take stock of the conditional lower bounds of learning theory: which ones fall, and which ones survive on which hypothesis. The next obstacle is a permutation problem. On difference vectors ei−ej, empirical risk minimization for real-valued halfspaces is weighted Feedback Arc Set, which the new algorithms do not reach.
Primarily AI-generated textHuman understanding: some partsExponential Time HypothesisPAC learningStrong Exponential Time Hypothesiscomputational learning theoryempirical risk minimizationfine-grained complexitymodel countingprobabilistic graphical modelstensor networkstreewidth
Jellium, a system of electrons in a uniform neutralizing background, and the uniform electron gas (UEG), a system of electrons with constant density and no background, are two standard models of the electron gas. We prove that the ground-state energy per unit volume of Jellium, as defined by Lieb and Narnhofer, coincides with the UEG energy defined by Lewin, Lieb and Seiringer, i.e. eJelℏ(ρ0)=eUEGℏ(ρ0). This equality was previously established in the classical case by Lewin, Lieb and Seiringer and independently by Cotar and Petrache.
Primarily AI-generated textHuman understanding: all partselectron gasjellium
Converse bounds on discriminating and estimating quantum channels are evaluated on a characterized model of the channels, which is only known approximately. This book studies when such bounds, and the operational exponents they bound, depend continuously on the channels, and whether their sensitivity to characterization error is a property of the task or only of the bound.
The book opens with noisy quantum gates. The supports of the Choi matrices separate genuine fragility, where the operational exponent itself jumps, from spurious fragility, where only single-letter converses jump. Near a gate with an incoherent noise floor, the smallest eigenvalue of the normalized Choi matrix sets the sensitivity. At first order, under errors in the model, this sensitivity belongs to the task. At second order, under a coherent miscalibration, the converses can overshoot the adaptive Stein and Chernoff exponents by factors that grow without bound as the noise decreases, although for small miscalibrations of a depolarized qubit they stay within a factor 3/2. Certifying a converse from tomographic data costs a number of channel uses set by the same floor.
The general theory follows. Divergences built from Kubo-Ando means, including the geometric fidelity, the geometric Renyi divergences and the Belavkin-Staszewski relative entropy, are functions of the Arveson-Radon-Nikodym derivative of one channel with respect to the sum of the pair, and in this coordinate they are continuous with dimension-free moduli. In the diamond norm they are continuous at a pair of completely positive maps exactly when the Choi matrix of the sum is invertible or the fidelity vanishes, so an ideal unitary gate paired with itself is a point of discontinuity. Divergences built from hypothesis tests, such as the Umegaki relative entropy and the hockey-stick f-divergences, are uniformly continuous in every dimension, and the adaptive
A mix of human-written and AI-generated textHuman understanding: some partsQuantum Information
We study the Langevin dynamics of the spherical Sherrington–Kirkpatrick model, the diffusion on the sphere of radius N in RN that is reversible with respect to the Gibbs measure with density proportional to exp(βxTWx/2), where W is an N×N matrix from the Gaussian orthogonal ensemble. The dynamics is started from the uniform distribution on the sphere, independently of W. We determine the asymptotic quenched total-variation distance to equilibrium as a function of time in the fixed-temperature, mesoscopic, critical-window, and infinite-temperature crossover regimes described below, with convergence in probability or in distribution with respect to W. Let βc=1/2 be the critical inverse temperature. For β<βc there is cutoff with a Gaussian profile at fixed β∈(0,βc), along every mesoscopic approach of β to βc from below, and in the infinite-temperature crossover when NβN2→∞; when NβN2 stays bounded, the distance relaxes over bounded times and there is no cutoff. At fixed β∈(0,βc) the mixing time is (logN−(3/2)loglogN+O(1))/(4(1−2β)2), and the correction −(3/2)loglogN comes from the edge of the spectrum of W. For β>βc, at fixed β and along every mesoscopic approach to βc from above, there is cutoff at a time set by the gap between the two largest eigenvalues of W, of order N2/3logN at fixed β. The profile is the total-variation distance between a standard Gaussian variable and its sum with an independent Cauchy variable; the Cauchy variable is the ratio of the first two eigencoordinates at an early stage of the evolution. In the critical window 2β=1+σN−1/3, the distance at time N2/3s converges in distribution to a random profile determined by the Airy point process, and there is no cutoff. As σ→±∞, this profile, suitably recentered and rescaled, converges to the profiles of the two mesoscopic regimes.
Primarily AI-generated textHuman understanding: some partsdynamicsspin glass
We study the Langevin dynamics of the spherical Sherrington–Kirkpatrick model, the diffusion on the sphere of radius N in RN that is reversible with respect to the Gibbs measure with density proportional to exp(βxTWx/2), where W is an N×N matrix from the Gaussian orthogonal ensemble. The dynamics is started from the uniform distribution on the sphere, independently of W. We determine the asymptotic quenched total-variation distance to equilibrium as a function of time in the fixed-temperature, mesoscopic, critical-window, and infinite-temperature crossover regimes described below, with convergence in probability or in distribution with respect to W. Let βc=1/2 be the critical inverse temperature. For β<βc there is cutoff with a Gaussian profile at fixed β∈(0,βc), along every mesoscopic approach of β to βc from below, and in the infinite-temperature crossover when NβN2→∞; when NβN2 stays bounded, the distance relaxes over bounded times and there is no cutoff. At fixed β∈(0,βc) the mixing time is (logN−(3/2)loglogN+O(1))/(4(1−2β)2), and the correction −(3/2)loglogN comes from the edge of the spectrum of W. For β>βc, at fixed β and along every mesoscopic approach to βc from above, there is cutoff at a time set by the gap between the two largest eigenvalues of W, of order N2/3logN at fixed β. The profile is the total-variation distance between a standard Gaussian variable and its sum with an independent Cauchy variable; the Cauchy variable is the ratio of the first two eigencoordinates at an early stage of the evolution. In the critical window 2β=1+σN−1/3, the distance at time N2/3s converges in distribution to a random profile determined by the Airy point process, and there is no cutoff. As σ→±∞, this profile, suitably recentered and rescaled, converges to the profiles of the two mesoscopic regimes.
Primarily AI-generated textHuman understanding: some partsdynamicsspin glass
For a fixed integer q≥3, we call a graph q-colorable if its vertices admit q colors with adjacent vertices colored differently. We prove that there exists dq>0 such that the random graph on n vertices with independent edge probability d/n is q-colorable with probability tending to one for fixed 0≤d<dq and to zero for fixed d>dq, as n→∞. The same holds for the uniform random graph with ⌊cn⌋ edges, for fixed c≥0, with threshold dq/2 in c. For fixed ϵ∈(0,1), the first loss of colorability under independent uniform additions of pairs of distinct vertices lies with probability at least 1−ϵ in a deterministic window of Oq,ϵ(n(logn)−3/2) steps, where log is natural. The proof uses interpolation to establish convergence and bounds the expected number of pairs needed to destroy colorability of the last colorable graph by identifying random vertices and controlling the growth of classes equal in every coloring.
Primarily AI-generated textHuman understanding: some parts
We prove the Erdős matching conjecture for four-uniform hypergraphs for every matching parameter and every admissible number of vertices. The proof combines the finite-board method of Hou, Hu and Liu with ordered local configurations, exact integer counts and certified inequalities. Small matching parameters are treated on full labelled configurations. Two fixed counting inequalities and two endpoint anchors propagate polynomially across the remaining range; 51 endpoint records and the two fixed inequalities suffice for the intermediate cardinality inputs. Dense triple supports admit direct slab and majority proofs. A classical shadow bound and a deletion argument complete the passage from these local bounds to the global theorem. The strict margins also yield clique stability with explicit constants for shifted families whose matching number is preserved on deleting the first vertex, and classifications of extremizers in specified ranges. The finite parts are supported by exact certificates and exhaustive computations with stated mathematical specifications.
Primarily AI-generated textHuman understanding: no partsErdős matching conjectureexact certificatesextremal set theoryhypergraph matchingsshifting
Abstract. A flat three-torus is determined by a Euclidean lattice up to orthogonal
changes of coordinates, or equivalently by a positive-definite Gram matrix up to integral
change of basis. A finite list of short lattice lengths is a much weaker datum. This
paper gives an explicit and fully elementary construction showing that the weakness
can persist over a large range of scales. For every real parameter A ≥ 2, we exhibit
two positive-definite Gram matrices G+(A) and G−(A) whose associated flat tori have
different volumes and hence are not isometric, while their length multisets agree below a
radius
RA =
p
2A2 − 2A−2 − 4A−4.
The shortest nonzero lattice length is exactly one in both examples, so RA is not made
large by a uniform rescaling; it tends to infinity with A. The proof isolates the coordinate
support on which the two quadratic forms differ and supplies a sharp enough lower
bound to exclude that support from every shorter vector. A second, isotropic example
gives a compact rational pair whose short length multisets agree below √
2 and differ at
that threshold. We then place the construction alongside the theory of ternary quadratic
forms and explain what it does—and does not—say about cosmic topology. In particular,
finite agreement of direct-lattice lengths is not equivalent to equality of CMB covariance
data, nor does it establish an observational degeneracy for any specified cosmological
model
Primarily human-written textHuman understanding: all partsGram matrixflat toruslattice
Reusing a stored Euclidean prefix, we show that every pair a≥b has a derivation of gcd(a,b) from a,b with O(loga/logloga) applications of the remainder function alone; Euclid's algorithm needs order loga in the worst case. These derivations bound the intrinsic complexity of gcd and coprimeness by the same order, refuting Moschovakis's conjectured logarithmic lower bounds for uniform processes; by his best-uniform-process theorem, a single uniform process attains this bound. The Main Conjecture for algorithms remains open. We also prove an Ω(logloga) lower bound on this complexity for almost all coprime pairs and give, for each ε>0, a nondeterministic register program computing gcd that, on every input a≥b≥1, has a successful computation using at most εloga+O(1) remainder calls. The upper bound also answers negatively an open problem of van den Dries and Moschovakis on the closure depth of the gcd. The main bounds and the refutations are formalised in Lean 4; the best-uniform-process theorem is cited.
Primarily AI-generated textHuman understanding: some partsEuclidean algorithmLean 4closure depthgreatest common divisorintrinsic complexityremainder functionuniform processes
We prove that the theta basis of the coordinate ring of a maximal unipotent subgroup of a complex semisimple group is biperfect. This answers affirmatively Question 6.5 in Kamnitzer's ICM 2022 survey.
A mix of human-written and AI-generated textHuman understanding: some partsTheta basis
We study symmetrized divergences induced by centroids of α-Tsallis relative entropies between density matrices σ and ρ; up to scaling, the induced divergence equals ∣2−∥ρα+σα∥1/α∣. Earlier work (S. Sra, "Metrics induced by Jensen–Shannon and related divergences on positive definite matrices," Linear Algebra Appl., 2021) studied such divergences and metrics induced by them. That paper allowed centroids to range over the positive definite cone, while we require that the centroid itself be a density matrix. This (more natural) choice makes the metricity in this paper considerably stronger than that of the aforementioned paper. Specifically, we prove that ∣2−∥ρα+σα∥1/α∣1/2 is a metric on density matrices for 1/2≤α≤2, α=1. For α<1, the proof uses a new four-point theorem for the S-divergence that might be of independent interest. For α>1, we first prove the endpoint α=2, and then transfer it to the other orders by a tensoring trick.
A mix of human-written and AI-generated textHuman understanding: all partsS-divergenceSchatten normSibson information radiusTsallis relative entropylog-determinant inequalitynegative typequantum Jensen-Shannon divergencequantum Renyi entropy
A collision system is a finite set of point particles with masses, moving freely with velocities given by the gradient of a convex energy function of the momentum, whose collisions conserve total momentum and total energy and are otherwise unconstrained. Stationary geodesic networks near a line of multiplicity n, point billiards with momentum- and energy-conserving collisions, and relativistic point particles are collision systems. We prove that the number of collisions of a collision system is at most exponential in its total mass, with an explicit base, whenever the energy is close enough to the Newtonian one, and that it can be exponential for every energy on a line. For geodesic networks this replaces the previously reported bounds, of order n2 from below and nn2 from above, by exponential bounds on both sides; for point billiards it gives explicit exponential bounds. The exponential growth rates remain open.
A mix of human-written and AI-generated textHuman understanding: some partscollision systemselastic collisionsgeodesic netsnumber of collisionspoint billiardsstationary varifolds
Contributed by Isaac Rudich, Louis-Martin Rousseau
The current best upper bound for deterministic factoring is N1/5+o(1). We provide a Lean certificate that there exists a deterministic Turing machine that computes the smallest prime factor of every N in No(1) steps: for every k≥1 it runs within Ck2n/k steps on n-bit inputs. The algorithm reduces factoring to counting divisors with prescribed leading bits, and counts them by leveraging the counting engine used to refute the Exponential Time Hypothesis (ETH). The Lean statement that a reader must check to verify the proof is only 13 lines and only uses Mathlib. The full proof is about 35,000 lines. This note was written less than 24 hours after the ETH refutations were posted. No human-readable explanation of these results exists yet, and their Lean proofs are long: Anthropic's alone runs to about 152,000 lines. Our proof re-proves only the part of the counting engine it needs and imports nothing from those developments. We hope this application, with its short statement, serves as a stepping stone for those working to understand the engine; a follow-up paper will explain this proof in detail.
A mix of human-written and AI-generated textHuman understanding: some partsExponential Time HypothesisLeancounting algorithmsdeterministic factoringformal verificationinteger factorization
For a positive integer k, let f(k) be the minimum number of distinct nonempty subset sums of a zero-sum-free k-element set, where the minimum is taken over all finite abelian groups. We prove f(k)≥2k2−13klog(k+1)−116k. Together with the classical upper bound, this gives f(k)=k2/2+O(klogk), with the optimal leading coefficient. As a consequence, the Olson constant of every finite abelian group G satisfies Ol(G)≤2∣G∣+O(log∣G∣). The argument combines a triangular lower bound for repeated supports, a count using blocks of two and three copies, and a greedy expansion argument. No restriction is imposed on the order of the group or on the stabilizer of the set of subset sums.
A mix of human-written and AI-generated textHuman understanding: some parts
The deterministic algorithm of \oai{} that counts the satisfying assignments of a k-CNF in time 2o(n) is reproduced in full and extended from counting to sums of local weights with values in any commutative $\Q$-algebra. The extension gives deterministic 2o(n) algorithms for exact Max-Cut and Max-k-SAT on sparse instances, for the independence polynomial, the maximum independent set and the knapsack problem with a conflict graph on every graph, for 0-1 integer programs whose rows involve boundedly many variables, for Vertex Cover in time $2^{o(k)}+\poly(n)$, for the density of states of Ising and Potts models on bounded-degree graphs, for the Tutte polynomial on the hyperbolas (x−1)(y−1)=q, for the minimum distance and the maximum-likelihood decoding of codes with sparse parity checks, for Max-Cut and Max-k-SAT at every average degree o(logn/loglogn), for Max-Cut on dense graphs close to blow-ups of graphs with o(n/logn) vertices, and for every problem in $\SNP$.
A mix of human-written and AI-generated textHuman understanding: some partsExponential Time HypothesisLDPC codes.Max-SATTutte polynomialVertex Coverindependence polynomialpartition functionsweighted counting
We study a conforming regularized immersed boundary discretization of the two-dimensional Eulerian Stokes–Newton system for one freely moving rigid body. Regularized interior moments complete the surface constraints used for prescribed motion. Positive excess mass and inertia define a fixed fluid–body kinetic inner product. A skew inertial correction is derived from the fictitious rigid interior; it has zero power but is necessary for the physical translational momentum equation. The method advances the body configuration with the preceding computed rigid velocity and then solves a linear joint fluid–body step on a dynamically enriched Cartesian space. Exact orthogonal transfer gives an energy identity independent of the number of mesh changes. Interior and surface moment defects control the discrepancy between the discrete background energy and physical kinetic energy. A volume-preserving geometry comparison and a moving coarse recovery family yield conditional joint convergence of the fluid velocity, rigid velocity, and body configuration on an interval with positive wall clearance. The error bound distinguishes the energy norm after geometry alignment from the generally weaker estimate at the same Eulerian points. The analysis assumes exact integration and explicit strong-solution regularity; it does not cover arbitrary density ratios or collisions. Manufactured coupled computations, independent residual checks, and a Lean algebraic companion document the numerical and formal verification scope.
Primarily AI-generated textHuman understanding: no parts
We study a regularized immersed boundary discretization of unsteady Stokes flow around a rigid body with prescribed translation and rotation. The interface may change its position and orientation, while remaining uniformly separated from the outer wall. The fluid spaces are conforming, exactly divergence free, and constructed on fixed physical Cartesian grids with a retained coarse space and arbitrarily changing local enrichments. The method uses regularized surface-moment constraints, backward Euler time stepping, and current-space orthogonal projection. We construct a time-dependent coarse lifting that satisfies every discrete rigid-velocity constraint exactly. A moving collar approximation has a small spatial error in H1 and a small Eulerian time-derivative error in L2, although its time derivative need not be accurate in H1. This distinction permits a quantitative convergence proof without a restriction coupling the time step to the number of mesh changes. A shifted-interface trace estimate accounts for the motion of the reaction surface during each time step and for physical slip between time nodes. Under stated piecewise regularity and an L2 interface-reaction assumption, the energy error is bounded by the square roots of the retained coarse size, regularization width, surface-panel size, and maximum time step, together with a specified load-consistency error. The analysis assumes exact integration and does not prove adaptive optimality or cover fluid-driven motion. Manufactured moving-ellipse computations and a Lean algebraic companion test specified implementation and proof components.
Primarily AI-generated textHuman understanding: no parts
Suppose that we repeatedly replace the values of an integer-valued
function on Z by averages over p consecutive arguments, where
p is prime. If the first ℓ rounds of averaging give integers,
will this remain true when we begin with the pth powers of the
original values? We prove that the answer is yes for ℓ≤p.
For p=2 and p=3, the answer is yes for every ℓ, and therefore
also when integrality is required at every stage. For larger primes,
a known counterexample first fails at round p+1. The proofs use
finite differences and polynomial interpolation. We also explain
connections with earlier work on divisibility of finite differences
and Stirling numbers.
A mix of human-written and AI-generated textHuman understanding: some partsInteger SequencesInteger-valued polynomialsdivisibilityfinite differencessupercongruences
Building on earlier human research into the constant congruence speed of integer tetration, including Appendix B of Ripà and Di Pietro (2026), and on a lifting-the-exponent (LTE) criterion publicly presented by Alper Ferudun for OEIS A390535 and A390576, this model-generated note proposes an explicit account of nonconstant speeds in non-squarefree numeral systems. Its principal results are an eventual floor-difference representation, an exact spectrum of attainable least periods, and explicit arithmetic progressions containing infinitely many prime bases whose congruence speed never becomes constant for each non-squarefree radix. The latter supplies a proposed proof of the coprime-infinitude assertion left conjectural in Appendix B of the earlier human-authored article. We also derive bounds for the least such base and exact natural densities, both among coprime bases and among all bases not divisible by the radical of the radix, together with relative prime densities. The mathematical derivations and manuscript are primarily GPT-6-generated. Ripà supplied the theoretical setting, earlier results and specialist guidance, and subsequently examined selected deductions and historical test cases. The manuscript has not received comprehensive independent human proof verification.
Primarily AI-generated textHuman understanding: some partscongruence speedeventual periodicityinteger tetrationlifting-the-exponent lemmanatural densitynon-squarefree radicesp-adic valuationsprimes in arithmetic progressions
Contributed by Kacper Rodziewicz, Bartosz Kołodziejek
Let X1,X2,… be independent and identically distributed real random variables with E[X1]=0, E[X12]=1, and finite third absolute moment ρ=E∣X1∣3<∞. Write Fn(x)=P((X1+⋯+Xn)/n≤x) and let Φ denote the standard normal distribution function. Define
\[
\Delta_n = \sup_{x \in \mathbb{R}} |F_n(x) - \Phi(x)|, \qquad C_n = \sup \frac{\sqrt{n}\,\Delta_n}{\rho}, \qquad C^* = \sup_{n \ge 1} C_n,
\]
and let
\[
C_{\mathrm{E}} = \frac{3 + \sqrt{10}}{6\sqrt{2\pi}} \approx 0.4097321837
\]
denote Esseen's 1956 two-point lower-bound constant.
We present a computer-assisted proof architecture and complete verification suite establishing unconditionally that
\[
C_n \le C_{\mathrm{E}} \quad \text{for every } n \ge 1, \qquad C^* = C_{\mathrm{E}}.
\]
The proof combines a regularized Prawitz smoothing inequality, a conserved zero-bias/characteristic-function reduction on the all-n tail, and exact finite verification certificates across 13 roles covering all n≥1 (`full_pass: "ESTABLISHED"`).
All certificates, native C++/Python checkers, and a single-command multi-core Docker verification setup (`git clone https://github.com/rodziewiczk/berry-esseen-certificates.git && cd berry-esseen-certificates && docker compose up --build --abort-on-container-exit`) are publicly archived at https://github.com/rodziewiczk/berry-esseen-certificates (tag `v1.0.0-full-pass`).
Primarily AI-generated textHuman understanding: some parts
Notion of frames and Bessel sequences for metric spaces have been introduced. This notion is related with the notion of Lipschitz free Banach spaces. It is proved that every separable metric space admits a metric Md-frame. Through Lipschitz-free Banach spaces it is showed that there is a correspondence between frames for metric spaces and frames for subsets of Banach spaces. Several characterizations of metric frames are obtained. Stability results are also presented. Non linear multipliers are introduced and studied. This notion is connected with the notion of Lipschitz compact operators. Continuity properties of multipliers are discussed.
For a subclass of approximated Schauder frames for Banach spaces, characterization result is derived using standard Schauder basis for standard sequence spaces. Duals of a subclass of approximate Schauder frames are completely described. Similarity of this class is characterized and interpolation result is derived using orthogonality. A dilation result is obtained. A new identity is derived for Banach spaces which admit a homogeneous semi-inner product. Some stability results are obtained for this class.
A generalization of operator-valued frames for Hilbert spaces are introduced which unifies all the known generalizations of frames for Hilbert spaces. This notion has been studied in depth by imposing factorization property of the frame operator. Its duality, similarity and orthogonality are addressed. Connections between this notion and unitary representations of groups and group-like unitary systems are derived. Paley-Wiener theorem for this class are derived.
This survey explores the theoretical scope of effective computation and formal axiomatic systems, emphasizing a constructive perspective based on computable trees and their associated path spaces. No new technical results are presented; rather, the aim is to synthesize established results from computability theory and mathematical logic into a unified framework that clarifies what can be computed in principle, complementing the classical emphasis on undecidability results.
It is argued that standard models of computation, such as Turing machines and equivalent formalisms, are computationally saturated: any effective extension of their instruction sets does not enlarge the class of effectively computable functions and algorithms. This saturation corresponds, on the logical side, to the representational completeness of Robinson arithmetic, which is sufficient to encode all computable functions despite being finitely axiomatized.
The survey proceeds by showing how standard models of computation give rise to computable trees whose infinite paths correspond to complete and consistent, but typically non-computable, extensions. This construction links effectively closed classes in Cantor space with spaces of completions of formal theories and clarifies the boundary between effective computability and logical possibility. Since Cantor space is homeomorphic to the classical Cantor set, this framework provides a natural bridge between discrete computability-theoretic and logical structures on the one hand, and continuous topology on the other.
Primarily human-written textHuman understanding: all partsComputability
Self-improving agent harnesses such as the Darwin Gödel Machine (DGM) let a coding agent edit its own scaffolding, and they keep the edits that score better and report large gains on coding benchmarks. In three pre-registered rounds, we compare what such loops find with standard engineering changes ("textbook moves"), one-shot redesigns by a language model, and placebo loops that make no edits; we also re-run released agents of the DGM and of ADAS with newer models. On hard LiveCodeBench problems with Claude Haiku 5.5, the starting agent often stops on an empty turn because its output-token limit is too small for the model's reasoning. A single textbook move, which raises this limit to 16,000 tokens and makes three other changes to the model loop, recovers about 90% of the gain of the best loop agent and costs less than half as much per task. Four of five one-shot redesigns by Claude Opus 5.5 also raise the limit, to 8,192 tokens, without evaluation feedback. The best loop agent still leads the textbook moves by 1–5 percentage points in pass rate. On held-out SWE-bench Verified tasks, run in the DGM's harness, the DGM's released best agent does not outperform its initial agent; three textbook moves on the initial agent outperform the best agent at a quarter of its cost, whether or not the patches of runs that ended early, most of them the best agent's, are kept. Claims of self-improvement should be measured against textbook moves, one-shot redesigns, placebo loops, held-out tasks and newer models.
A mix of human-written and AI-generated textHuman understanding: all parts
We prove Serre’s conjecture on the Artin character of a finite group G acting on a regular local ring A with isolated fixed points. The case of positive characteristic is due to Abe. In mixed characteristic we use an equivariant deformation to reduce to the case where A has an isolated singularity, in which case we can make use of the results of Beraldo and Pippi on the twisted conductor formula.
A mix of human-written and AI-generated textHuman understanding: all partsRegular rings
For integers a>b>0, let P(a,b) be the number of steps the iteration x↦amodx takes to reach 0 from b; it is the length of the Pierce expansion of b/a. We prove P(a,b)≪εaβ∗+ε, where β∗=(41301−382849)/137480=0.2941336553…, improving the exponent 1/3−2/177 of Chase and Pandey. The tail of an orbit carries a sequence of positive integers, the Hankel determinants of a natural moment sequence, and these are small where the drops of the orbit are small. Grouping orbit states by such integers places the quotient differences of each group on curves of bounded degree, whose integer points we count with the uniform bounds of Castryck, Cluckers, Dittmann and Nguyen. The resulting exponent approaches 5/17 when these curves have large degree relative to the logarithmic size of the box. We raise the degree by adjoining further integer invariants, by an effective Hilbert irreducibility sieve, and by an exact identity for the direction of the curvature vector, and come within 1.61⋅10−5 of 5/17. We also give an elementary proof of the exponent 3/10+ε and a bound for orbits whose quotient increments rarely rise. All results are formalised in Lean 4 with Mathlib. Shallit's conjectured O((loga)2) bound remains open.
Primarily AI-generated textHuman understanding: some partsHankel determinantsHilbert irreducibilityLean formalizationPierce expansionsinteger points on curvesiterated remainders
We prove strict log-concavity of the positive functions Θ and Λ with one-sided and bilateral Laplace transforms 1/ξ(21+s) and 1/Ξ, respectively, where Ξ(z)=ξ(21+iz). For either function, log-concavity is the Pólya frequency property of order two (PF2). Without assuming RH, but using known zero verification through height 3⋅1012, we further prove that Λ is a PF function of order ten (PF10). We also prove local PF positivity for Θ: for each fixed order, the ordered minors of Θ(x−y) are positive when all differences xi−yj are sufficiently small and positive. Additionally, we prove the generalized Laguerre inequalities: if RH holds up to height T>1/2, then Ln[Ξ](x)≥0 for every n≥0 and ∣x∣≤T−1/2; the case n=1 is due to Csordas, Ruttan and Varga (1991).
A mix of human-written and AI-generated textHuman understanding: all partsPolya frequencyRiemann Xicomplete monotonicitygeneralized Laguerrelog-concavity
We prove that the coefficients of quantum theta functions in the skew-symmetric setting are characters of finite-dimensional sl2 representations. This proves Conjectures 1.4, 2.17 and 3.13 of Davison–Mandel and the two-wall conjecture stated as Conjecture 3.12 in Davison’s ICM 2026 survey.
A mix of human-written and AI-generated textHuman understanding: some partsquantum cluster algebra
We show that there are infinitely many elliptic curves over Q with torsion subgroup Z/2Z×Z/4Z and rank at least 6, answering the case Z/2Z×Z/4Z of Problem 2.6 in Dujella's list of open problems. The largest rank previously known to occur infinitely often with this torsion was 5, due to Eroshkin. The curves are fibers of a one-parameter family of Dujella and Peral, which we show has rank exactly 3 over the rational function field. The family is a base change of an elliptic K3 surface, and quadratic twists of that surface give it three further independent points over the function field of an elliptic curve of rank one. Congruences then carry the independence of the six points, and the torsion Z/2Z×Z/4Z, from one rational point of that curve to infinitely many. The result is unconditional. It contradicts the prediction of Park, Poonen, Voight and Wood that only finitely many curves with this torsion have rank greater than 5, although the curves form a sparse family. A second family, over an elliptic curve of rank two, gives another proof.
This report was created on 23 October 2026 from six AI research reports generated with the frontier-dump skill on Dinesh Thakur's conjectures in function field arithmetic. Taylor Dupuy initiated those queries during his talk How to Use AI in Mathematical Research Effectively at the University of Rochester Number Theory Seminar on that date. The six reports were synthesized with harvest-dumps and developed into this manuscript with using-latexpowers.
We survey the current status of several conjectures of Dinesh Thakur and his coauthors on the arithmetic of Fq[t]: the degree hypotheses for power sums, the characterisation of zeta-like multizeta values in genus zero and in higher genus, the relations among multizeta values, the distribution of zeros of Goss zeta functions, and the automata-theoretic transcendence questions for special values. For each conjecture we give the precise statement and its status in the literature.
The six computational research dumps report several results and refutations. Among them, the recursion predicted by Thakur's second power-sum hypothesis fails for an explicit family of exponents at thirteen non-prime values of q (a computer-assisted result); the shuffle relations, even with Frobenius and Carlitz's Eulerian evaluations, do not generate all relations among multizeta values; a closed formula for a v-adic valuation difference conjectured by Thakur holds only after a sign change (and, for d≥4, halving), and the corrected formula is proved, with computer assistance, for degrees d≤11; and the v-adic Newton polygon of a Goss zeta function over F2[t] at a prime of degree two has a slope of multiplicity one, which appears to contradict a result claimed in a 2023 preprint (this conflict needs human verification). The shuffle result and an assembly of published theorems describing all relations are elementary, and we claim no novelty for them. We also write out a proof of a theorem of Wen and Yao on v-adic arithmetic gamma values. Computational evidence is labelled as such throughout, and the open problems are listed.
This manuscript is AI-generated by Claude Code (model claude-opus-5-5) from AI-generated research dumps; its proofs received only AI review, and no human has checked them.
Primarily AI-generated textHuman understanding: no partsGoss zeta functionsThakur conjecturesfunction field arithmeticmultizeta values
We determine the exact dimensional range of the Generalized Vanishing Conjecture (GVC) for constant-coefficient differential operators over fields of characteristic zero.
In two variables, we prove that
Λm(Pm)=0 for every m≥1 implies Λm(QPm)=0 for all sufficiently large m and every fixed polynomial Q. The proof combines Hall's marriage theorem, Newton polygon geometry, prime-dilation arguments, and the constant-term theorem of Duistermaat and van der Kallen.
We then construct a homogeneous three-variable counterexample. Setting ρ=t2+xy, A=ρ+x2, C=yρ2−2xt2ρ−x3t2, and Δ=4∂x∂y+∂t2, the pair Λ=Δ6 and P=AC2 satisfies Λm(Pm)=0 for every m≥1, while Λm(x2Pm)=0 for every m≥1.
The construction is a homogeneous endpoint-contact lift of Long's three-dimensional Gaussian counterexample and extends to an explicit cusp-profile family with arbitrary winding, profile, and radial suspension. In particular, every Δk for k≥6 admits such a failure.
Consequently, unrestricted, and already homogeneous, GVC holds in dimension n exactly when n≤2.
A mix of human-written and AI-generated textHuman understanding: some partsGeneralized Vanishing Conjecture
For integers 1≤a≤b≤c≤d with ad=bc, Bergeron asked whether the difference of Gaussian binomial coefficients (bb+c)q−(aa+d)q has nonnegative coefficients. We prove that it does, and more generally that (bb+c)q−(aa+d)q has nonnegative coefficients whenever a≤b≤c and ad≤bc, a strengthening proposed by Troyka. The proof is analytic rather than bijective. We develop a toolkit for coefficientwise inequalities, including a coefficientwise local limit theorem for Gaussian polynomials based on Irwin--Hall densities, and use it in a Pascal-type induction that proves the case a≤20. For general a the problem reduces to a two-parameter inequality between linearly interpolated Gaussian polynomials. Near its base case this inequality reduces exactly to the monotonicity of rational q-Catalan numbers. Using the circle method with explicit error terms, we prove the conjecture of Chen, Wang and Wang that the coefficients of MacMahon's q-Catalan numbers Cn(q) are unimodal for every n≥16, and an analogue for the rational q-Catalan numbers Cx,N+1(q) with x≤N≤2x. Beyond this range the difference splits into two nonnegative parts, which we compare degree by degree using local limit bounds for Gaussian polynomials that are uniform in both parameters. The proof is computer-assisted: it uses exact integer and rational arithmetic, ball arithmetic, and one floating-point computation with a proved error bound. The scripts and the logs of their runs are provided as ancillary files.
Primarily AI-generated textHuman understanding: no parts