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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.
math.OA — Operator Algebras
We develop the Poisson boundary theory for general von Neumann algebras with separable preduals. As applications, we generalize and give new proofs of the results on the weak Dixmier property for inclusions with expectations or operator-valued weights \cite{Pop99,Mar19,Iso25}. We also prove that every continuous factor admits irreducible embeddings into the hyperfinite factors of type , , and into some hyperfinite factor of type . Finally, using the Poisson boundary techniques developed above, we prove that for every inclusion with faithful normal conditional expectation , has the weak Dixmier property. By Marrakchi's characterization \cite[Theorem~D]{Ma25}, this gives the relative bicentralizer conjecture for von Neumann algebras with separable preduals.
math.QA — Quantum Algebra
We prove that every element of a strongly positive basis of a coefficient-free rank-two quantum cluster algebra has nonnegative Laurent-polynomial coefficients in the quantum greedy basis. This establishes the strongly positive basis assertion of Conjecture 16 of Lee, Li, Rupel, and Zelevinsky. We also give a counterexample to their Conjecture 13(b).
math.RA — Rings and Algebras
We prove that, over any field and with algebraically independent initial seed coordinates, two cluster -variables are algebraically dependent if and only if they are equal or reciprocal. In characteristic zero, this is equivalent to the vanishing of the wedge product of their differentials.
math.RA — Rings and Algebras
We give a counterexample to Chang and Zhu’s question of whether every explicit rooted cluster morphism is ideal.
math.AG — Algebraic Geometry
A systematic way to construct Poisson brackets on moduli stacks in algebraic geometry is through shifted Poisson structures on derived stacks. The standard construction, due to Calaque--Pantev--To{\"e}n--Vaqui{\'e}--Vezzosi, imports Gelfand--Kazhdan formal geometry to derived stacks in order to establish formal descent for polyvector fields. The result is a graded Lie algebra which we call the ``crystalline'' polyvector fields on a derived stack. Using instead the formal derived geometry machinery of Gaitsgory--Rozenblyum, we construct a graded Lie algebra of ``infinitesimal'' polyvector fields which has a direct geometric interpretation: it governs deformations of the formal shifted cotangent stack, together with its canonical exact symplectic form. Our main result is an equivalence between crystalline and infinitesimal polyvector fields on a derived stack as graded Lie algebra objects, providing a deformation-theoretic perspective on the graded Lie algebra constructed by Calaque--Pantev--To{\"e}n--Vaqui{\'e}--Vezzosi.
math.QA — Quantum Algebra
We prove that quantum theta and greedy bases coincide in coefficient-free rank two for all positive exchange parameters. Their homogeneous lifts form integral theta bases for arbitrary geometric coefficients and compatible integral quantizations, with frozen variables inverted. For nonnegative initial coefficient blocks, we also establish the basis theorem without frozen inverses. 2026.10.15:This version underwent AI-assisted correctness review using Rethlas-OSV with GPT-6-Astra at xhigh reasoning effort. The review comprised 24 completed segment checks and a completed supplementary full-paper review. No confirmed mathematical errors were reported within the reviewed scope. Rethlas-OSV: https://github.com/OSVerify/Rethlas-OSV
math.CO — Combinatorics
The \emph{arithmetic product} is a bilinear operation on the ring of symmetric functions over defined by \[ p_\lambda \ \boxdot\ p_\mu = \prod_{i,j}p_{\operatorname{lcm}(\lambda_i,\mu_j)}^{\gcd(\lambda_i,\mu_j)} \] on the power-sum basis . It is known that this operation preserves the subring of symmetric functions with integer coefficients, and is Schur-positive on pairs of Schur functions. Indeed, it corresponds to induction of representations from to . In this work, we consider the \emph{anti-arithmetic product} , which is defined by the same formula but with and interchanged. We show that it, too, preserves , even though it lacks the Schur positivity property. This was conjectured on MathOverflow in 2014. Our method is a more intricate variant of the representation-theoretical interpretation of . The anti-arithmetic product does not correspond to an operation on actual representations; but its adjoint can still be described as a map on characters, which (as we show) is a -ring morphism from the character ring of to that of . Now, a 1975 theorem of Boorman shows that the representation ring of is generated by the natural permutation representation as a -ring. Thus, the proof of integrality boils down to only computing the image of under this adjoint, which can be done using M\"obius inversion in the representation ring. This work is aimed at readers familiar with representation rings and the representation theory of symmetric groups. No prior knowledge of -rings is presumed. Three self-contained proofs of Boorman's theorem are given in the appendices, two of them lifting the theorem to the noncommutative symmetric functions (or Solomon's descent algebra), recovering a result of Schocker.
math.RT — Representation Theory
Let be a finite-dimensional algebra over a field. A finite-dimensional algebra over the same field is called reflexively equivalent to if their categories of reflexive modules are equivalent. We prove that there is a basic algebra , unique up to isomorphism, such that is reflexively equivalent to if and only if for some reflexive -module . Moreover, we compute explicitly. We extend these results to module-finite algebras over henselian local rings of dimension at most one under a dominant dimension condition at minimal primes. We also introduce reflexive modules over additive categories and prove that taking reflexive modules is idempotent: the reflexive modules over the category of reflexive modules form an equivalent category. As an application, we show that reflexive equivalence classes of algebras with finitely many indecomposable reflexive modules correspond bijectively to Morita equivalence classes of algebras whose reflexive modules are projective.
math.QA — Quantum Algebra
We prove that the rank-two quantum greedy basis is the least strongly positive basis when b | c or c | b, resolving Conjecture 14 of Lee–Li–Rupel–Zelevinsky. Every strongly positive basis has nonnegative expansion coefficients in the greedy basis.
math.QA — Quantum Algebra
We give a finite polyhedral construction of the quantum greedy basis for coefficient-free rank-two quantum cluster algebras. The construction specializes coefficientwise to the classical polyhedral construction at v = 1. Corresponding classical and quantum coefficients and auxiliary margins have identical zero sets, and the Newton polytopes are affine images of the classical carriers.
math.OC — Optimization and Control
It has long been known that the solutions to abstract linear control systems are continuous in time for controls in with . We prove that the same property remains valid for the endpoint case , giving a positive answer to Weiss' 1989 Problem 2.4. The proof relies on a direct semigroup argument based on Phillips' lemma, does not require the input map to have an integral representation (which is not always the case for ), and actually entails that such systems are all of the zero-class (which fails for ).
math.PR — Probability
We study the negative spherical perceptron, a mean-field model of jamming with a nonconvex solution space. We establish the Parisi formula for its free energy and identify the capacity for every constraint density . We also show that the jamming point is isostatic: optimal configurations have exactly contacts. For slightly above two, we show that replica symmetry is lost at the de Almeida–Thouless margin through a continuous transition to full replica symmetry breaking. Further, we show that Gardner's formula overestimates the capacity by order .
math.MG — Metric Geometry
We deduce a new asymptotic upper bound on the maximal lattice kissing number in dimension , using the explicit auxiliary functions constructed in OpenAI's "Ten Advances" preprint. The same bound holds for the average contact degree of a finite packing of congruent balls. The bound's base- exponential rate rounds to , matching the value extrapolated empirically by Afkhami-Jeddi, Cohn, Hartman, de Laat, and Tajdini in 2020.
math.NT — Number Theory
Communicated by Guy in 1994, the ternary pentagonal numbers conjecture of Blecksmith and Selfridge asserts that every integer larger than is the sum of three positive pentagonal numbers. Prior to this note, it was even unknown whether every sufficiently large integer is the sum of three positive pentagonal numbers. We resolve this in the affirmative, using the landmark work of Duke and Schulze-Pillot on ternary quadratic forms to handle all sufficiently large integers with , and present an explicit lift to handle the with .
math.NT — Number Theory
Let denote the least prime congruent to modulo , where . We give a computer-assisted proof that , improving the exponent of Xylouris. More precisely, for every sufficiently large , uniformly in . The main new ingredient is a graded near-density estimate for zeros of Dirichlet -functions, a weighted form of Heath-Brown's Lemma~12.1 of the kind he asked for in 1992. We combine it with the zero-location estimates and far-density method of Heath-Brown and Xylouris. The second novelty is the scale of the case analysis. Heath-Brown and Xylouris closed their final case analyses with and main cases, each by a chain of inequalities evaluated in floating point. In this paper, with the help of advanced AI models, we can push this much further. The middle range of the first zero is divided into root cases and terminal cases, each closed by its own linear program. On threshold boxes these programs give linear relaxations, and exact integer certificates for all of them bound the normalized zero sum strictly below~. Exceptional zeros and the remaining exterior range are treated separately. This case analysis is a kind of systematic brute force enabled by AI: an analysis of this size, with parameters tuned to each case, would be very laborious or nearly impossible to carry out by hand, and it lets the new estimate be applied separately in each case. The near-density lemma, certificate soundness, and a conditional passage from a certified case to a prime are formalized in Lean, assuming published analytic inputs and specified facts about zeros (PALOMAR-2026-10-01-000020 v1).
math.CO — Combinatorics
Let count the unordered pairs at distance one in a finite planar set . We construct finite sets with and . The largest exponent previously claimed, apart from preliminary versions of the present work, is , in the author's unpublished manuscript. The method is the number-field construction of OpenAI and Sawin. Here unit distances come from elements of relative norm one in quadratic extensions in which has both real and complex places, and the fields come from an infinite pro- tower over $\Q(\sqrt{241})$, ramified only above , and , with Frobenius conditions at the two primes above and at one prime above . The relative zeta value is bounded through the dihedral quotients of the tower group: the tower group has quotients isomorphic to the dihedral group of order~; fifteen of them detect, by a census, the residue degree of the primes that split completely in the Kummer field up to norm , and seven of them generate a field of degree contained in every field of the tower. We write as a product of -functions of degrees , and~, evaluate the of degrees and rigorously at real points near~, and combine the values through a kernel inequality with coefficients of both signs. Finite facts and numerical inequalities are certified by exact computation and interval arithmetic. Partial formalization PALOMAR-2026-10-01-000018 v2
math.RA — Rings and Algebras
The augmentation of the Markov cluster algebra is explicit but not inducible as a rooted cluster morphism. This answers a question of Chang and Zhu negatively.
math.CO — Combinatorics
We study tilings of an equilateral triangle of side n in the triangular grid by k lattice trapezoids with base angles 60°, the objects behind the OEIS sequences A389392 (k = 4) and A391498 (k = 5). We prove two angle identities valid for every such tiling and a lemma relating the number of boundary vertices to the number of pieces having a side on the boundary. With these tools we show that there are exactly 1, 2 and 13 combinatorial types of tilings for k = 3, 4, 5. For k = 3 every tiling is a pinwheel. For k = 4 every tiling belongs to one of the two categories used in A389392, a fact that had previously been taken for granted. For k = 5 the thirteen types refine the eight categories of A391498; in three of them a piece has no side on the boundary. As a consistency check, the volumes of the parameter polytopes together with the generic multiplicities reproduce the leading coefficient 7/36 of the conjectured quasi-polynomial for A391498. All lemmas were checked against an exhaustive enumeration of the tilings with pairwise distinct pieces for n ≤ 15. Finally, the classification turns the thirteen types into eight explicit families of sets of shapes, and an inclusion–exclusion over them reduces the conjectured generating function of A391498 to twelve elementary counting statements, and we settle all twelve: the resulting closed formula reproduces the sequence for every n ≤ 127.
math.PR — Probability
For planar simple random walk, the favorite sites at time are the sites whose local time at time is maximal. We prove that, almost surely, for every and every , all favorite sites lie outside the ball centered at the origin with radius for all sufficiently large . At the critical exponent , almost surely, for every , the entire favorite sites lies within distance of the origin infinitely often.
math.RA — Rings and Algebras
We give a finite criterion for total sign skew symmetry of integer matrices of order three.
math.MG — Metric Geometry
The moving sofa problem asks for the maximal area of a two-dimensional object capable of being moved through a right-angled corridor of unit width. J. L. Gerver constructed a sofa of area , which was only recently proven to be optimal by J. Baek. Here, I consider the moving sofa problem in an angled corridor, , of unit width for turn angles . I first generalize Hammersley's arguments to devise simple analytical lower and upper area bounds for all . ChatGPT-6 Astra (Codex) is prompted to derive a generalized Gerver sofa parametrized by . The generalized Gerver sofa area is non-analytic at the unique root of . For turn angles smaller than a closed-form expression of the sofa area is obtained. The generalized Gerver sofa is conjectured to be the solution to the moving sofa problem for turn angles .
math.DG — Differential Geometry
We construct a smooth contractible open four-manifold that is not homeomorphic to and admits a complete Riemannian metric with scalar curvature at least one. This gives a negative answer to a question raised by Chang--Weinberger--Yu and Chodosh--M\'aximo--Mukherjee.
math.LO — Logic
We give a source-aligned account of two Lean 4 formalizations of classical finite social-choice impossibility theorems. For a finite nonempty electorate and a finite alternative set containing three distinct alternatives, Arrow’s theorem derives dictatorship from unanimity and independence of irrelevant alternatives. Its proof expands a weakly decisive coalition from one ordered pair to all pairs, then contracts decisive coalitions to a singleton. The Gibbard–Satterthwaite development proves that every onto strategy-proof choice function on the unrestricted strict-ranking domain is dictatorial by constructing a social welfare function and applying that Arrow implementation. We explain the essential construction in full: top-two profiles define pairwise comparisons; strategy-proof monotonicity and three-top profiles establish a strict total order; predecessor counts provide injective natural-number ranks. Exact hypotheses, vendored dependencies, pinned source identities, prior formalizations, and the distinction between historical registry verification and the present source audit are recorded. The contribution is an exposition and audit of reusable formalization artifacts, not a new social-choice theorem or a claim of first formalization.