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math.FA — Functional Analysis
We derive a Riesz-Frechet representation for bounded linear functionals defined on the inner product spaces over sub-valued finite fields.
math.NT — Number Theory
Recent work on Artin's primitive root conjecture establishes a zero-free half-plane for finite-order Hecke -functions over cyclotomic fields containing . Following their strategy, we prove that if is any number field, then the Dedekind zeta function does not vanish in the half-plane . As a corollary, the -function of any finite-order Hecke character defined over also does not vanish in the half-plane , wherever it is holomorphic. These results were proved and formalized in Lean using AxiomProver. Compared to the work on Artin's primitive root conjecture, the hardest part was the formalization of the work of Kazhdan and Patterson on metaplectic forms in order to accommodate all number fields.
math.AC — Commutative Algebra
Let k be any infinite field. We prove that, for every fixed d ≥ 1 and all sufficiently large n, the quotient of k[x1, . . . , xn] by any number of general forms of degree d has the Hilbert function predicted by Fröberg’s conjecture through degree 2d. The threshold for n is independent of the field and the number of generators. For generating degrees at least three, a direct construction with two groups of variables gives a nonincreasing recurrence for the two defects at the critical generator counts. Divisibility of these bounded defects and polynomial equidistribution then force their eventual vanishing.
math.CO — Combinatorics
We construct strongly regular graphs with parameters (306,55,4,11), (400,171,74,72), (576,253,112,110), (784,351,158,156) and (1296,595,274,272), and hence also their complements. All five parameter sets were marked "?" (existence unknown) in Brouwer's online table on 10 and 11 October 2026. Three of the graphs were found by a computer search that extends the transitive coset fusion of our earlier papers to groups acting with two orbits: the search fuses the orbitals of a group G on G/H_1⊔ G/H_2, screens with tables of marks, solves the exact quadratic equations on the structure constants with a SAT-based solver, and rebuilds every solution from scratch. The groups are 2× M_12, 5^(1+2)_+:4A_5 and Sp(6,2). Also deposited on Zenodo with the full data archive (the citable record): https://doi.org/10.5281/zenodo.23296158 The graphs on 400, 576, 784 and 1296 vertices come from a short step beyond known constructions. A symmetric Bush-type Hadamard matrix of order n^2 whose n× n blocks have unit diagonal, for instance one of Kharaghani's matrices, defines a strongly regular graph Γ_0 with the parameters of the Latin square graph LS_(n/2+1)(n). In Γ_0 the block rows and the block columns form two orthogonal spreads of Delsarte cliques. Deleting one spread gives the known graphs with λ=μ of Bonato, Holzmann and Kharaghani. By an elementary spread lemma, deleting both gives a strongly regular graph with the parameters of LS_(n/2-1)(n). Kharaghani's matrices exist for every Hadamard order n, so such a graph exists for every n=4m≥8, for example srg(1600,741,344,342) and srg(1936,903,422,420). A second source, found by analysing the search hits, is the rule (a,b)∼(c,d) iff a≠ c, b≠ d and H_abH_bcH_cdH_da=1 for a symmetric Hadamard matrix H. Together the two sources give at least 5, 7, 5 and 6 pairwise non-isomorphic graphs on 400, 576, 784 and 1296 vertices. None of the graphs we tested on these four vertex numbers is a Latin square graph, so they say nothing about the orthogonal arrays OA(n,n/2-1) named in the table. The graph on 306 vertices has a short description on the 300 non-square points of the parabolic quadric Q(4,5) together with the six points of a totally singular line; this line is a Delsarte clique. The graph has the parameters of the collinearity graph of a generalized quadrangle GQ(5,10) but is not geometric, and the same rule over 𝔽_3 gives the collinearity graph of Q(4,3). Strong regularity of every graph is also checked by an exact, standalone, standard-library Python program.
math.PR — Probability
We study Langevin dynamics for the pure spherical -spin glass with , with covariance and generator , started from a uniform point independent of the disorder. We let the dimension tend to infinity on fixed time horizons and study the long-time behavior of the resulting hard-spherical Cugliandolo–Kurchan equations for the correlation and the response . At every positive temperature, their solution has a unique stationary limit at bounded time differences, which satisfies the fluctuation–dissipation theorem (FDT), and its energy converges. For , where is the dynamical transition, we show that the plateau of the stationary correlation is the larger solution of the marginality relation , and we compute the limiting energy. Correlations with every fixed earlier state tend to zero. The limits of large waiting time and large time difference therefore do not commute, which proves aging of the correlation, including at the transition point. We also prove that the response at a late time, as a function of the time difference, converges to its stationary profile in and uniformly over all earlier times; that FDT in the later time holds uniformly over all earlier times; and that the integral of the squared violation of FDT in the earlier time tends to zero, while for the positive part of this violation keeps a positive integrated mass. For , the stationary correlation approaches its plateau as a power law, with the exponent predicted by mode-coupling theory and a finite positive amplitude. For the cubic model and , including the transition point, we prove a lower bound on the total susceptibility and a product inequality for correlations at three times; for , we prove, for each fixed level below the plateau, an explicit asymptotic lower bound on the first passage time to that level multiplied by the entropy production during the passage. The aging proofs for the cubic and quartic models represent the correlation by vectors in a Hilbert space indexed by trees, and general pure interactions are treated with averages over finitely many independent futures and established algorithmic threshold bounds. The full fluctuation–dissipation relation of the aging regime and the time scale of aging remain open.
cs.LO — Logic in Computer Science
We present a formally specified and machine-checked kernel for a federated automated adjudication system, in which independent computational lines judge mathematical and operational claims and deposit their verdicts, evidence, and certificates in a shared ledger. The kernel comprises five components. (i) A three-rung, three-track credibility lattice obtained as a completion of a legacy five-element poset: the completion is justified by an exhaustive enumeration over all 11^{3}=1331 ordered triples and is fully re-formalized in Lean 4, where the equational lattice laws, the order axioms, and the extremality properties of join and meet are all proved by the decide procedure over the finite carrier. (ii) A five-state certificate lifecycle machine with nine legal transitions and machine-checked invariants, likewise re-formalized in Lean 4; the formalization caught a specification flaw in one invariant (the originally stated grant-gating condition is false), and we report the corrected, machine-proved form. (iii) Certified numerical anchors: uniqueness of the gauged fixed point of circulant entropic-optimal-transport (Sinkhorn) systems, established by a Krawczyk interval operator with outward rounding across a six-instance family (k in {6,10 }, varepsilon in {1,1/2,1/5 }, later extended to ten instances), together with a second anchor family of five auction-algorithm assignment instances certified by interval dual brackets. We state plainly that the certification of these numerical anchors is carried by the Python interval-arithmetic layer; the attempted port to the Lean 4 interval library leancert is blocked by structural capability boundaries of that library (absence of the logarithm from its supported expression grammar and infeasibility of its determinant computation at the required dimensions), and the Lean files are formal statements, not machine-verified certificates. (iv) A parameterized Lean 4 template for Rice-type undecidability, consisting of a bridge theorem, an extensionality lemma, an instantiation generator, and discharged instances of increasing difficulty. (v) A minimal trusted computing base result: the soundness of an interval certificate checker (accept implies correctness), itself machine-proved in Lean 4. All Lean artifacts were verified on cloud infrastructure (Axiom Axle, environment lean-4.28.0) under the strict verify_proof channel, with axiom audits restricted to propext, Classical.choice, and Quot.sound, and no sorryAx. Empirical fuzzing over three certificate families observed zero false accepts; we report rule-of-three 95% upper bounds on the false-accept rate and explicitly do not claim a guarantee.
math.CO — Combinatorics
Conlon and Lee introduced reflection bigraphs and cut-percolation in their study of graph norms. We prove that, for connected bigraphs, these two notions coincide, confirming their prediction.
math.NT — Number Theory
Brudno (1970, 1976) and Brudno–Kaplansky (1974) found an elliptic family and a degree-four polynomial family of solutions of x^6 + y^6 + z^6 = u^6 + v^6 + w^6. We explain both in terms of the K3 surface V ⊂P5, the complete intersection of three quadrics studied by Bremner (1981) and Kuwata (2007). The Brudno–Kaplansky curve is Cremona’s curve 480f4 and is the fibre at λ=∞ of the elliptic pencil on V that Bremner calls the Brudno pencil; the curve z^2 = x^4 + 38x^2y^2 + y^4 of 1970 is 480f1, and the classical recursions of Lebesgue and Desboves that Brudno iterated are the multiplication maps [2] and [−3] on it. Brudno’s 1976 quartic is the unique rational curve of degree four on V (Bremner’s Theorem 4.4) and a section of the Brudno pencil; its parameter is the fibre coordinate λ, and the involution Brudno noticed is a linear symmetry of V. Using the Mordell–Weil lattice of the pencil we prove a closed formula for the degree of a section, deg C_S = 2⟨S−γ/4, S−γ/4⟩, enumerate all sections of degree ≤12 (four of them new, of degrees 7,10,11,11), and enumerate, by a lattice computation, all smooth rational curves of degree ≤12 on V over Q(i) and over Q, reproducing Bremner’s theorems and testing his conjecture. A census of the 74 primitive solutions with all terms ≤ 1343 identifies which of them lie on V, on known curves, on sections, and on fibres of jumping rank, and records some phenomena off V. In the second part we compute equations and parametrisations of the 48 new curves that are not sections and show that seven of the eight census solutions on no known curve lie on them, leaving a single unexplained point; we determine the geometry over C of the sextic fourfold M6, of the threefold W= M6 ∩M2 (which has exactly 710 nodes and defect 95) and of V (transcendental lattice U(2) ⊕⟨12⟩, zeta function through the newform 320f2), and discuss in what sense the geometry drives the arithmetic; and we show that Bremner’s conjecture follows, with non-explicit finite sets, from the theorems of Sterk and Bright–Logan–van Luijk, describe how to make it explicit (Borcherds–Shimada), and run the first step of Bremner’s own cone procedure, which identifies the degree-8 pencil involutions that the proof requires.
math.NT — Number Theory
Let be Catalan's constant. We prove that for all integers and all sufficiently large integers . Hence the irrationality exponent of is at most . The proof refines OpenAI's recent proof that is irrational. That proof compares two bounds for a large determinant whose entries are linear forms in and . The lower bound comes from the denominators of the entries, and the upper bound from an integral representation. We improve both. On the arithmetic side we show that the matrix of rational parts has large corank modulo for every prime in an explicit range, which lowers the denominators. On the analytic side we bound a determinant that the energy method treats trivially, using Pick matrices, Heine's formula and an inequality for the logarithmic energy of signed measures. A quadrature formula extends the upper bound to an exponentially small neighbourhood of , as a measure requires. The proof is formalized in the Lean proof assistant.
math.NT — Number Theory
This record identifies published conclusions formerly obtained under RH or GRH whose proofs require only a fixed zero-free half-plane for Dirichlet -functions, with further applications of finite-order Hecke -functions over . The input is the theorem stated in the recent OpenAI manuscript claiming to prove the quasi-Riemann Hypothesis. We give the uniform analytic deductions explicitly and distinguish them from previously established algebraic and combinatorial results used in the proofs. An accompanying audit records the source verification and rejected candidates. No theorem about all number-field or automorphic -functions is inferred from the Dirichlet statement; the pure cubic and cubic-nonresidue applications separately identify the precise fixed-field Hecke input.
math.LO — Logic
We construct a sentence of having exactly many countable models up to isomorphism, but no perfect set of non-isomorphic models. This disproves Vaught's conjecture for infinitary logic. The construction follows a framework devised by Robin Knight, and the proof has been formally verified in Lean.
math.GR — Group Theory
Phillips and Stanovský proved that a Bruck loop with abelian inner mapping group has central nilpotency class at most two and asked for a human-friendly proof. We give a short self-contained proof: a displacement cocycle produces a commutant-valued endomorphism, right alternativity for commutant elements forces its image to be fixed, and cocycle rigidity makes that image central. This yields the abelian center quotient.
math.NT — Number Theory
We prove that every value of the arithmetic finite quantum dilogarithms constructed by Radchenko and Wheeler (arXiv:2609.21892) is an algebraic unit, and determine the Artin action for arbitrary quadratic pseudolattices. In particular, the associated Stark units belong to the predicted ray class fields and have the predicted Galois action. We use finite Fourier transforms and distribution relations to obtain Frobenius congruences. We also give conductor bounds for individual values and obtain the unitary conjugates needed for Zauner's conjecture.
math.QA — Quantum Algebra
We give an explicit construction of a vertex operator algebra of central charge eight inside , and prove the ribbon equivalence $\Rep(U)\simeq Z(\HH_3)$, where $\HH_3$ is a Haagerup fusion category. The algebra is generated as a vertex algebra by six affine generators and three specified weight-two vectors. The argument combines exact lattice-vector calculations with an all-weight strong-generation induction, a finite presentation and semisimple descent for a full corner of Zhu's algebra, and the actual multiplication of two holomorphic extensions. The categorical identification uses splitting maps and a comparison with the Evans--Gannon reconstruction, not modular data alone.
math.RT — Representation Theory
Based on the constructions of Kong–Liu–Shen and Liu–Shen, we give a counterexample to Han’s conjecture over every field.
math.OA — Operator Algebras
We express the Fuglede--Kadison determinant of an affine pencil in free semicircular operators as an infimum over positive definite matrices. The infimum is attained if and only if the pencil is boundedly invertible, in which case its minimizer is unique. At zero constant term, a scalar rescaling recovers the determinant-capacity identity of Mai and Speicher. For selfadjoint coefficients we compare the formula with a strictly constrained Hilbert--Schmidt variational principle and establish ordered min--max formulas under their respective positivity hypotheses. Strictly positive pencils admit a minimum over a linear matrix inequality. Applications to Brown measure include a uniform determinant lower bound, restrictions on atoms, radial distribution formulas, and a Hessian formula for the regularized density. The text of this manuscript was generated entirely by OpenAI's large language model Astra.
math.CA — Classical Analysis and ODEs
For every and , we prove that every measurable subset of of positive upper Banach density, with , contains an -term arithmetic progression with every sufficiently large prescribed gap.
math.CA — Classical Analysis and ODEs
We prove a non-universality criterion for the Erd\H{o}s similarity conjecture that applies beyond exponential (i.e., geometric) decay. In particular, it covers decreasing positive null sequences satisfying and eventually for some .
math.RT — Representation Theory
We show that every representation-infinite finite-dimensional algebra over an algebraically closed field has an indecomposable module of each infinite dimension. This settles, over algebraically closed fields, Simson's conjecture, also called the third Brauer--Thrall conjecture, that such an algebra has indecomposable modules of arbitrarily large cardinality. To construct these modules, we extend a representation embedding of Bongartz from finite-dimensional Kronecker modules without a simple injective summand to a representation embedding of such Kronecker modules of arbitrary dimension, which commutes with direct sums and filtered colimits and preserves the dimension of infinite-dimensional modules. Applying it to the indecomposable Kronecker modules of every infinite dimension constructed by G\"obel and Simson gives the required modules.
math.MG — Metric Geometry
We show that for every convex body (), every , and every prescribed density , there is an arrangement of translates of along a single lattice with mean multiplicity exactly and multiplicity at least everywhere. Consequently, lattice -fold coverings exist with density at most , where depends only on . In sufficiently large dimension , a version with prescribed common covolume holds simultaneously for up to bodies whose uniform distributions have a common covariance matrix, with absolute. The proof builds on OpenAI's horizontal–vertical construction for single-lattice coverings and uses Gaussian layer averaging and Laplace bounds for covering counts.
math.NT — Number Theory
Let X_n(s) denote the sum of k^(-s) for k from 1 to n, and define the cross energy C_2(n,s) as |X_n(s)|^2 − |X_(n−1)(s)|^2 − n^(−2σ). At every point of the critical strip, the limit of C_2(n,s) is 1/(1/4 + t^2), zero, or positive infinity, according as σ = 1/2, σ > 1/2, or σ < 1/2. This trichotomy reflects the pole at s = 1. On the critical line, the associated helix has a natural orientation whose signed radius is Hardy’s Z(t), and the zeros of the truncated functions spiral toward the zeros of the Riemann zeta function at rate n^(−β). We prove that the mean energy of the normalized error E(u) in the prime number theorem equals the sum, over zeros ρ, of the limits of C_2(n,ρ), with values in [0,∞]. This energy is at least 2 + γ₀ − log(4π), with equality if and only if the Riemann hypothesis (RH) holds and all zeros are simple. An unconditional local formulation expresses the energy in a Gaussian window as a double sum over zeros. We also study the pointwise inequality S(σ,t) = ∂/∂σ log|ξ(σ + it)| > 0, which is equivalent to RH. An off-line pair reduces S precisely inside a disc; at abscissa σ, the failure set up to height T has measure at most N(σ,T + 1). Moreover, S is Weil’s functional evaluated at a Poisson kernel, and its truncation admits an exact finite expression involving primes up to e^L. We test each criterion against the Davenport–Heilbronn function, which has zeros off the critical line, and against its Euler-product counterpart L(s,χ) modulo 5. The former violates each criterion as predicted, whereas the latter behaves like the zeta function. For zeta, primes up to 10^8 yield a mean energy of 0.0453, compared with 0.0462. Assuming RH, simple zeros, and a spacing condition, we determine the next-order term in the energy: the integral from 1 to X of (ψ(x) − x)^2/x^2 dx equals C_B log X + K + Φ(log X) + o(1), where K = 2.5125… and Φ describes almost-periodic interference between pairs of zeros. Prime data up to 10^7 track this term with correlation 0.999; random models of primes and of the Möbius function μ carry 20 to 150 times more energy. On the circle associated with the Euler factor of a prime p, the zeros favor angles near π, and near π + arg χ(p) for L(s,χ). Restoring the endpoint term (1/2)N^(−s), we prove, with explicit constants, that zeros of the truncations converge to zeros ρ of zeta along helices with radius |ρ| N^(−β−1)/(12|ζ′(ρ)|). For the Davenport–Heilbronn function, the convergence rates for the two zeros in an off-line pair differ by 2(β − 1/2). Finally, Weil’s form, using two windows at the endpoints of its support, detects an isolated zero at β = 1/2 + δ and height γ using prime powers up to (C log γ)^(1/δ). The corresponding positivity holds for zeta up to 3 × 10^6 but fails for a random multiplicative model. We do not prove the Riemann hypothesis.
math.PR — Probability
We determine the classical infinite divisibility, self-decomposability and extended Thorin regions of the strictly free stable family. Above index one, infinite divisibility and self-decomposability hold on two extremal skewness intervals, with distinct maximal indices and . Below one, every law is self-decomposable, whereas the Thorin region is a central interval; the optimal uniform bound is . We characterize the boundaries by global contact conditions and derive their different asymptotic scales at index one. A common phase representation also gives an exact bell-shape test, strictly bell-shaped examples outside the Thorin class, and an unbounded number of skewness components near one. The exceptional free one-stable family admits an explicit integer criterion. The proofs combine analytic estimates with finite interval certificates. The bell-shape results do not constitute a full parameter classification.
math.MP — Mathematical Physics
We introduce a complex Hilbert space structure on in which the Hilbert transform on the torus is incorporated directly into the metric and inner product. The resulting geometry is defined via a bounded linear embedding that couples each function to its Hilbert transform, yielding an inner product with both symmetric and symplectic components. Within this framework, we redefine weak differentiation intrinsically using the complex Hilbert-space inner product rather than the ambient pairing. We show that this intrinsic weak derivative coincides with the classical weak derivative on its natural domain, while remaining geometrically well defined on the full space. The derivative operator is shown to be densely defined and skew-adjoint, with a purely imaginary, discrete spectrum. Its eigenfunctions are given by the Fourier modes on the torus, which induce a complete orthonormal basis adapted to the Hilbert-transform metric. This construction yields a spectral representation in which differentiation is diagonal and the Hilbert transform is absorbed into the notion of differentiability itself, providing a unified analytic and geometric framework. We formulate wave mechanics intrinsically in this complex Hilbert space, define wave function, scalar and vector potentials and associated energy functional, derive space time wave equation, derive its solution, prove existence and uniqueness and establish regularity of the solution.
math.GR — Group Theory
We present a simplified proof of the Polymath14 inequality, along with a new sharpness result.
math.CO — Combinatorics
For a set P of n points in the plane R^2, let A(P) be the set of multiplicities of P, i.e. the sizes |ell cap P| >= 2 of the subsets of P lying on a common line ell, and let F(n) be the number of distinct multiplicity sets A(P) as P ranges over all n-point sets. Erdos problem 607 (erdosproblems.com/607) asks whether F(n) <= exp(O(sqrt n)), remarking that the sqrt(n) scale is easy to see to be best possible. We prove the upper bound F(n) <= exp(c sqrt n) for an absolute constant c > 0. The proof is a three-step reduction to a published result of Colbourn-Phelps-Rodl, "Block sizes in pairwise balanced designs" (Canad. Math. Bull. 27(3), 375-380, 1984): (1) the lines determined by P, restricted to P, form a pairwise balanced design (PBD) of order n - every unordered pair of points lies in exactly one such block; (2) the profile set of block sizes of this PBD is exactly A(P); (3) Theorem 2.4 of Colbourn-Phelps-Rodl bounds the number g(n) of profile sets of order-n PBDs by exp(c2 sqrt n), hence F(n) <= g(n) < exp(c2 sqrt n). Combined with the grid construction of Szemeredi-Trotter (1983) this settles the order of magnitude exactly: F(n) = exp(Theta(sqrt n)). The essential observation is that plane line collections are precisely PBDs, so the published design-theoretic bound applies to multiplicity sets directly. A Lean 4 machine-checked formalization of the reduction is in progress for the Palomar registry and will be linked from this deposit upon acceptance.
math.CO — Combinatorics
Let m1(R^2) be the supremum of upper densities of Lebesgue-measurable subsets of the plane containing no pair of points at distance one. We show, using the spectral rigidity theorem in OpenAI's recent proof that the plane is not five-colorable, that m1(R^2) = inf_G alpha(G)/|V(G)|, where G ranges over nonempty finite unit-distance graphs in the plane. The argument constructs an isometry-invariant law on independent subsets of the countable algebraic plane, projects the occupancy indicator onto the continuous spectral factor without changing its expectation, and extracts a measurable independent set with arbitrarily small density loss. As a consequence, if f(n) is the least independence number among unit-distance graphs on n vertices, then f(n)/n converges to m1(R^2). The proof is nonquantitative and uses the cited spectral rigidity result as a black box.
cs.DS — Data Structures and Algorithms
Geographic information science has argued for three decades that representing change requires moving from time-stamped snapshots to events and processes. This ontological shift has not been matched by a formal foundation with the properties a GIS needs: explicit axioms, representation theorems, and decidable reasoning. We propose dynamic mereotopology, in the form of Vakarelov’s dynamic contact algebras (DCAs), as a candidate foundation. DCAs are point-free with respect to both space and time. Regions are histories, primitive relations include space contact, time contact and precedence, moments are recovered as clusters of the time-contact relation, and properties of time correspond to axioms about regions alone. Using Grenon and Smith’s distinction between SNAP (snapshot) and SPAN (process) ontologies, we argue that DCAs originate on the SNAP side, since they are abstracted from a snapshot model, yet their primitives already carry much of what a SPAN ontology requires. We then identify precisely what is still missing for a process-oriented GIS: process and participation primitives, continuity of change, freedom from global simultaneity, identity under splitting and merging, and granularity. We develop a SPAN-first extension. For two of its components we give full proofs: a point-free continuity axiom, shown to express exactly the intended condition in discrete time and in finitely variable dense time, and a set of correspondences showing which axioms of continuous space hold at every finite resolution of discrete data and which emerge only in the limit of refinement. The remaining proposals are stated as conjectures, and the framework is tested against three case studies: a flood, a wildfire, and a pandemic.
math.ST — Statistics Theory
We determine, up to a power of , the minimax risk for constant conditional variance estimation under rough random design. The unknown design density is bounded above and away from zero, with no smoothness assumption, and the conditional error laws may depend on the covariates and have uniformly bounded fourth moments. For an -Hölder regression function with in dimension , the minimax root-mean-square risk lies, for all sufficiently large , between and , where and the constants and are explicit. In particular, the minimax exponent is , the minimax risk is smaller than by a stretched-exponential factor whose constant is identified, and the rate proposed by Robins is not uniformly attainable over this model class. For , we show that the exact minimax rate is ; for and , it is .
math.MG — Metric Geometry
We present a method for improving the classical exponential base in the upper bound for Borsuk's partition number. The emphasis is on explaining the method rather than optimizing the bound. The problem is reduced to constructing directions which make an acute angle with every member of a prescribed spherical set. These play the same role as the illuminating directions in Schramm's method. The geometric ingredients are an antipodal pairing, a lift by one coordinate, and reflection in a self-dual cone. This reflection folds Euclidean space into the cone. A translation opposite to the horizontal part of its Gaussian mean makes the required supporting inequalities hold with high probability. The Gaussian Poincar\'e inequality controls the mean, and the formula for translating a Gaussian density bounds the measure cost of this construction.
stat.TH — Statistics Theory
In this article we solve the problem of binary regression when the class conditional probability is a multivariate function of bounded variation. We study the consistency of binary regression under a nonparametric sampling framework. Sufficient conditions are established for the convergence of the binary regression estimator, and quantitative error estimates are obtained. We further strengthen the analysis by introducing the assumption that the feature vectors possess a point density measure of bounded variation. Under this geometric regularity assumption, the expectation-based error estimate is replaced by a deterministic estimate, yielding deterministic convergence in the norm and, consequently, almost sure convergence. The analysis demonstrates that the geometric distribution of the sampling points plays a fundamental role in the approximation properties of the estimator. Although the analysis is carried out on the torus , the results extend to arbitrary bounded Lipschitz domains after affine scaling, embedding into , and zero extension. The point density measure framework provides a deterministic geometric perspective on binary regression and establishes a connection between sampling geometry, bounded variation, and convergence theory.
math.GT — Geometric Topology
We prove that the modular torus uniquely minimizes the area of the stable-norm unit ball among complete finite-area hyperbolic once-punctured tori, resolving the McShane--Rivin area conjecture. It therefore has fewer simple closed geodesics of length at most than any fixed nonmodular torus for all sufficiently large . We also bound the area increase away from the modular torus and determine its leading growth and first correction term as the systole tends to zero, with error bounds independent of the twist.
math.ST — Statistics Theory
We establish nearly minimax bounds for missing-at-random means, treatment effects, and expected conditional covariances under rough random design. For two nuisance functions with average H\"older smoothness in dimension , the minimax root-mean-square error is when and of order when . The first rate confirms the rough-design exponent suggested by higher-order influence function theory. In the generic model with an unknown bounded density weight, our upper and lower bounds differ by only polylogarithmic factors and identify a leading correction when , with explicit in terms of and the weight bounds. For the expected conditional covariance, the same exponent and the same constant were obtained independently and concurrently by S. Park (arXiv:2610.05006). For models with separately bounded density and propensity, we identify the same polynomial exponent and the explicit leading correction, with an remainder in the logarithm of the risk.
math.PR — Probability
We study the Edwards–Anderson spin glass on the torus , with independent Gaussian couplings of variance between nearest neighbors. We prove that, as , its free energy converges to that of the Sherrington–Kirkpatrick model, given by the Parisi formula, at every temperature and in every uniform external field, uniformly in the side length . The same holds on the hypercube , and the ground-state energies converge to the Sherrington–Kirkpatrick ground-state energy.
math.NT — Number Theory
This notes tries to parse the basic structure of OpenAI's new argument proving modularity of elliptic curves over imaginary quadratic fields. The argument also proves modularity over all totally real field, and in the original form, modularity over CM fields not containing \zeta_5. The general CM case has also now been resolved by an independent repository which improves the Caraiani-Newton modularity lifting theorem for CM field at the prime 5, which was the main reason for the extra hypothesis. I kept the argument in the totally real case separate to highlight the main new idea in the OpenAI proof, which is orthogonal to modularity lifting theorems.
math.PR — Probability
We prove a Parisi formula for Ising and spherical perceptrons with Gaussian disorder and i.i.d. random potentials having a common Lipschitz bound and integrable values at zero. We assume , where is the number of Gaussian vectors and is the number of spins. The proof combines cavity in and with the Aizenman–Sims–Starr scheme for the lower bound and Guerra’s interpolation for the upper bound. As consequences, we answer several problems posed by Talagrand (2011) and establish a quenched large deviation principle for the empirical distribution of the Gaussian projections, resolving a conjecture of Bolthausen and Kistler (2010). We also resolve the sharp-threshold conjecture of Aubin, Perkins, and Zdeborová (2019) for the -function binary perceptron at every positive width, and establish the predicted variational formulas for the entropy and capacity of the negative spherical perceptron. Finally, the zero-temperature limit yields the scalar projection-pursuit formulas conjectured by Montanari and Zhou (2025, 2026), in both the supervised and unlabelled settings.
math.NT — Number Theory
We show that the equation has no solution in nonzero coprime integers, assuming a classification theorem from Putz's thesis, which has not appeared in a refereed journal. Equivalently, none of the equations , , has a solution in nonzero coprime integers. Putz attaches to a solution an octic algebra, the fibre of a Belyi map of degree , and proves that it is always isomorphic to one fixed octic field . We prove unconditionally that no solution has fibre isomorphic to , by a fifth-power descent over a field of degree that Putz introduced, completed by fifth-power residue symbols at five auxiliary primes. We also prove, without assuming the generalized Riemann hypothesis, that has class number , using a criterion of Belabas, Diaz y Diaz and Friedman.
cs.CC — Computational Complexity
While the Cook-Levin theorem is fundamental to computational complexity, formalizations in proof assistants often rely on high-level arguments rather than constructing the concrete SAT formula. Within the Lean community, a rigorous, constructive reduction from Turing machines to SAT is currently absent from \texttt{mathlib}. We present a complete, machine-checked formalization in Lean 4 that addresses this gap by providing a constructive reduction from a deterministic Turing Machine model to a CNF formula. By rigorously defining the interface between the high-level computation model and the low-level SAT encoding, we bridge the gap between theoretical complexity and practical SAT-based verification. Our work specifically handles the challenges of verified 3D-variable indexing, injective mapping proofs, and global soundness, providing a verified tool for both communities to interact.
math.CO — Combinatorics
Let be the largest cardinality of a subset of containing no two elements whose difference is a nonzero perfect square. We give a computer-assisted construction proving , improving the exponent of Krachun's ranked-block construction. The residues we use carry subintervals of that are ordered along every modular square difference. A finite stopping-word argument shows that if finitely many such alphabets, in pairwise coprime perfect-square moduli, have interval moments whose contributions sum to more than , then is an attainable exponent. Krachun's prime chains reproduce his exponent in this framework. The gain comes from an alphabet modulo a power of , built by a -state recursion, and from two composite alphabets, at the primes and at , in which the first differing digit is read separately at each prime. The numerical conclusion is computer-assisted. The theorem, including every finite certificate, is formalized in lean, PALOMAR-2026-09-19-000006 v2.
math.DS — Dynamical Systems
Let be an infinite finitely generated group whose proper subgroups are finite. We prove that a positive Kazhdan constant uniform over all finite generating sets forces every unit vector in every representation without invariant vectors to have a cofinite displacement gap. For ergodic probability-measure-preserving actions this gives \[ \liminf_{g\to\infty}\mu(gA\mathbin\triangle A) \geq \ku(G)^2\mu(A)(1-\mu(A)), \] and hence Schmidt's Property~. A tensor-square argument also gives a uniform gap from modulus one for diagonal matrix coefficients. Conditional on the graded-diagram package underlying the Osin--Sonkin construction, the theorem shows that their infinite finitely generated simple ICC group of bounded exponent has Property~. We include a corrected, constant-tracked proof of the Osin--Sonkin deduction from that package and conclude with questions about Ozawa's quasifinite Kazhdan quotients.
math.AP — Analysis of PDEs
We prove three phase-retrieval results for stationary one-dimensional Schr\"odinger evolutions. Masuda's unique-continuation theorem yields phase retrieval for finite-energy solutions and a broad class of semibounded potentials which may grow at infinity. For real potentials in the Faddeev class , measurements on all of spacetime determine arbitrary initial data. If the potential is smooth and compactly supported to the left of a point, measurements on the exterior half-line to its right suffice.
math.PR — Probability
Starting from Shevtsova’s sharp third-moment inequality, we develop an exact defect from the standardized two-point family and derive quantitative transport and characteristic-function bounds. A penalized extremal problem yields eventual two-point optimality, , together with a uniform linear deficit and sharp cubic Wasserstein stability. A fifth-order lattice saddle-point expansion gives the full oscillatory correction to the optimal constants. A separate effective argument proves for every , using finite-index certificates and a first-failure argument. Exact polynomial certificates give unrestricted bounds for two and three summands. For four summands, a common-measure Fourier comparison and rational certificates yield the unrestricted bound ; on laws supported on at most three points, we obtain the sharper bound . An opening analytic argument explains the passage from Rademacher comparison to the matching two-point law. An elementary historical proof of is included in an appendix. ChatGPT and Codex assisted the development of the proofs and computational certificates and the preparation of this manuscript, using the GPT-5.6 Sol, GPT-6.1 Sol and GPT-6 Astra models.
math.RT — Representation Theory
We prove that the inverse quiver problem is NP-complete. We also present a new proof that the quiver problem is NP-complete. We present a way to translate arithmetical/algebraic geometry problems in terms of (inverse) quiver problem.
math.FA — Functional Analysis
Dragomir [\textit{Bull. Aust. Math. Soc., 2016}] showed that the Buzano inequality holds for orthogonal projections on Hilbert spaces. Dragomir [\textit{Linear Multilinear Algebra, 2016}] also derived the most general form of the Buzano inequality for bounded linear operators on Hilbert spaces. We show that Dragomir result extends to adjointable morphisms on Hilbert C*-modules. Using this generalization, we derive bounds for the roots of polynomials over commutative unital C*-algebras. We formulate the notion of noncommutative numerical range and derive numerical radius bounds for self-adjoint morphisms on Hilbert C*-modules over unital C*-algebras. We formulate several open problems, including the noncommutative Toeplitz-Hausdorff, von Neumann inequality, Ando inequality, Berger power dilation, Kittaneh inequality, Crouzeix problems.
math.AG — Algebraic Geometry
Traditional Computer Algebra Systems (CAS) and dynamic expression interpreters evaluate critical points and mathematical singularities via runtime limit approximations, Taylor series expansions, or recursive L'Hôpital routines. These procedures incur an unresolvable computational bottleneck: dynamic tree-traversal complexity of O(N) alongside runtime branching hazards and undefined IEEE-754 states (NaN, division-by-zero traps). This paper presents a formal engineering proof of how the Recursive Indexed Calculus of Identity and Singularity (RICIS-III v7.9) enables strict constant-time O(1) symbolic Abstract Syntax Tree (AST) reduction and native machine delegate compilation. By enforcing Absolute Continuity (L₀), the Identity Principle (L₁), Safety Protocols (SP₁–SP₅), Protocol P₁ (direct structural evaluation substituting lim(x→a) with x=a), and the geometric realization of Axiom A₆ (S_F ⊠ I_G → R(F,G) →_μ F · G), all indeterminate nodes are eliminated during symbolic pre-compilation. The resulting pruned AST compiles into straight-line native execution blocks (such as .NET CLR dynamic delegates or LLVM IR) operating in deterministic O(1) clock cycles per tick with zero runtime branching.
math.RT — Representation Theory
Building on the constructions of OpenAI, we extend these constructions to any base field containing an element of infinite multiplicative order. Over each such field, we obtain finite-dimensional algebras giving counterexamples to both Tachikawa conjectures, the Auslander–Reiten conjecture, the Gorenstein-projective conjecture, the Nakayama, generalized Nakayama, and strong Nakayama conjectures, the Auslander–Gorenstein conjecture, the little and big finitistic dimension conjectures, and the Wakamatsu tilting conjecture. 2026.10.15:Version 3 of this manuscript has completed a full-paper review using Rethlas-OSV with GPT-6-Astra at xhigh reasoning effort, covering all 23 review scopes created by the system. A supplementary audit checked the manuscript’s use and attribution of the two cited OpenAI papers against locally supplied original PDFs and TeX sources. Neither review reported any confirmed mathematical errors within its scope. Rethlas-OSV: https://github.com/OSVerify/Rethlas-OSV OSVerify: https://github.com/OSVerify/OSVerify
math.RT — Representation Theory
Building on OpenAI’s An algebra of infinite little finitistic dimension, we simplify the exposition and extend the construction to every field. 2026/10/10：We have added formal verification：https://github.com/tqyqiqi1509/Note198-formalization
math.AT — Algebraic Topology
We prove that for every odd prime there is a height for which the map $L_{n-1}S_p \to L_{n-1}\L_{K(n)}S_p$ is not injective on homotopy groups, hence not the inclusion of a wedge summand. In particular, for such pairs the weak chromatic splitting conjecture is false. More precisely, we prove the conjecture fails for and for all pairs with . In particular, the conjecture fails for for every odd prime . This disproof was produced autonomously by Claude in a session working on Freyd's generating hypothesis with no human in the loop. This note is a human write-up of the disproof. We did not contribute any ideas.
math.NT — Number Theory
This manuscript establishes the non-existence of the Perfect Cuboid---a rectangular parallelepiped with integer edges, face diagonals, and space diagonal. By performing a rational sectioning of the governing quadratic forms, we demonstrate that the problem reduces to finding a non-trivial rational point on a family of hyperelliptic curves of Genus 3. We prove that the Jacobian of these curves possesses a Mordell-Weil rank of zero and that the perfection locus is an irrational algebraic singularity of degree , precluding any solution in the integer domain . The non-existence of rational solutions is further verified via formal methods in Lean 4, demonstrating that the intersection of the Mordell-Weil torsion set and the degree-4 perfection locus is empty.
math.NT — Number Theory
We prove that the equations and have no solution in coprime nonzero integers for any . For the first equation the open cases were prime exponents above outside a set of congruence classes treated by Chen and Siksek; in these cases the solution blocks the modular method. We replace Kraus's Frey curve by a hypergeometric motive of rank three over , whose parameter at a solution is -adically close enough to a point of maximally unipotent monodromy for inertia at to act unipotently; the parameter of is not. By a theorem of Calegari, Emerton and Gee, the mod representation of the fiber, twisted by a cubic character, is the reduction of a member of a compatible system of minimal ramification, and we show that the member at a prime above is irreducible and ordinary at . Ramification and unconditional discriminant bounds show that its reductions have trivial semisimplification, and as the Galois group of the maximal pro- extension of unramified outside is generated by one inertia group, the member stabilizes its ordinary line, a contradiction. For the same argument also works at the primes , which are inert in , and Frey curves and descents treat the remaining exponents.
math.NT — Number Theory
For all finite subsets of an abelian group, we prove \[ |X-Y|\le |X+Y|^{\lamI},\qquad \lamI=\frac{9451\e-3286}{5378\e+787}=1.454277448906\ldots, \] improving the classical exponent . We also prove that a universal two-set exponent over the integers implies an upper bound for the Gyarmati--Hennecart--Ruzsa constant, the supremum of such that and for arbitrarily large . Consequently, \[ \theta^*\le\frac{13524\e-7359}{9451\e-3286}=1.312373302115\ldots, \] improving their bound . The first argument combines a coupling with distinct differences, entropy inequalities for independent sums, a finite certificate using non-Shannon inequalities, and an explicit limiting certificate with weights $\int_0^1t(1-t)^k\e^t\,dt$. The transfer uses localisation and the Pl\"unnecke--Ruzsa inequality over a sequence of scales. The proofs are formalised in Lean~4.