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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.
math.LO — Logic
We describe a Lean 4 formalization of the standard four-axiom characterization of the Shapley value for finite transferable-utility cooperative games. Games are arbitrary real-valued characteristic functions normalized at the empty coalition; the player type may itself be empty. The factorial coalition formula is proved efficient, symmetric for interchangeable players, null-player respecting, and additive. Uniqueness follows from the values of arbitrarily scaled unanimity games and an explicit Boolean-lattice Möbius decomposition. The proof requires neither a real-homogeneity axiom nor a continuity assumption on competing allocation rules. We explain the finite-sum arguments, the precise Lean statements, the separation between the registry’s challenge and implemented solution, and the historical verification evidence for the pinned source revision. The contribution is an auditable exposition of an existing formalization of classical mathematics, with no claim of new mathematics or formalization priority.
math.LO — Logic
We describe a Lean 4 formalization of the classical two-person Nash bargaining characterization. A bargaining problem consists of a compact convex subset of the real plane, a feasible disagreement point, and a feasible outcome strictly improving both utilities. The Nash product is maximized over the individually rational part of that set. The development proves existence and uniqueness of the maximizer, verifies Pareto optimality, symmetry, positive affine invariance, and independence of irrelevant alternatives, and proves that these four axioms characterize the maximizer among feasible selection rules. We explain the algebraic midpoint proof of uniqueness and the normalization argument, including an explicit small-step tangent bound and a compact symmetric enlargement. The exposition is tied to an immutable repository snapshot and four declarations registered in Palomar. Historical registry verification is distinguished from the source inspection used to prepare this manuscript. The contribution is a documented formalization of established mathematics; no new bargaining theorem or formalization-priority claim is made.
math.LO — Logic
We present two complementary Lean 4 developments of finite Nash equilibrium. The pure-game layer uses player-specific legal action sets, distinguishes one-way generalized ordinal potentials from bidirectional ordinal potentials, and proves potential-maximizer certificates, a local-maximum characterization, absence of strict-improvement cycles, and weak acyclicity. The mixed-game layers use either a common finite action type or a dependent finite action type for each player. They prove payoff equalization on equilibrium supports, Dirac payoff identities, continuity of the normalized payoff-excess map, and mixed equilibrium existence. We explain the finite-sum and sum-of-squares arguments and the reindexing bridge to an attributed, proved product-of-simplices Brouwer theorem. An additional legal-subtype interface is separated from the declarations selected by the registry comparator. The account is tied to two immutable source snapshots and dated verification records. It documents established mathematics and its formal interfaces, without claiming a new equilibrium theorem or formalization priority.
math.OC — Optimization and Control
We explain a Lean 4 development of a finite-coordinate loss-preserving construction motivated by Theorem 4 of Dann, Mansour, Mohri, Schneider, and Sivan. Nonempty finite constraint and action index sets give a joint probability simplex with a canonical action marginal. For each mixed constraint, a linear comparator adds its outer product with that marginal. The comparator has mass two and leaves the legal action set. A one-step pairing identity yields exact equality of finite-horizon approachability and comparator-regret objectives, with explicit anchored lifts and marginal decoders. The development also proves a row-wise decomposition with at most one rank-one term per constraint index. We describe the twelve declarations registered in Palomar and their historical verification provenance. The scope is deliberately narrow: the constraint simplex parametrizes coordinate functions, both causal strategy translations observe original loss histories, and the comparators have no fixed point in the joint simplex. Thus the checked statements establish finite algebraic loss preservation; they do not certify the full published reduction, its fixed-point-defined improper class, reduced-loss-only feedback, or asymptotic rate theory. No mathematical novelty or formalization priority is claimed.
math.GR — Group Theory
We explain a Lean 4 construction of a finite inverse automaton recognizing the subgroup of the rank-two free group generated by any finite list of signed words. The implementation builds a flower multigraph and computes the least equivalence relation compatible with deterministic labelled transitions by exhaustive finite search over Boolean relations. Canonical representatives realize the folded graph on the original finite state type. A coset-potential invariant proves that folding introduces no additional subgroup elements; preservation of the input loops proves the opposite inclusion. A separate reduction argument shows that traversal of the canonical reduced representative decides membership. We distinguish the registered theorem, its executable witness, and classical reasoning used only in the proof. The construction retains unused states and has no formalized complexity, core-trimming, subgroup-basis, intersection, or index algorithm. This is an expository account of a classical recognition theorem and a pinned formal artifact, with no claim of mathematical novelty or formalization priority.
math.AC — Commutative Algebra
We give a source-grounded exposition of a Lean 4 formalization of prime-generated Nagata factoriality descent. A commutative noetherian integral domain is a unique factorization domain if a localization is a unique factorization domain and every denominator is exactly a finite product of prime elements belonging to the denominator submonoid. The formal statement works with an arbitrary algebra carrying Mathlib’s IsLocalization structure. We explain the two cases for an irreducible element, the multiset cancellation and splitting arguments, and the corollary for submonoids generated by arbitrary sets of primes. A registered polynomial corollary illustrates reuse of the descent package; its Laurent-polynomial premise is proved using Mathlib’s existing polynomial UFD instance, a dependency made explicit here. The exposition is tied to the registered source and dependency pins, distinguishes historical mechanical verification from manuscript review, and situates the development relative to an earlier Lean preprint and an Isabelle/HOL formalization. No new algebraic theorem or priority claim is asserted.
cs.LO — Logic in Computer Science
We present a source-grounded account of a Lean 4 formalization of the Myerson-Satterthwaite impossibility theorem for continuous bilateral trade. Buyer values and seller costs are independent, with atomless probability measures supported on compact real intervals whose interiors overlap. The model explicitly requires restricted Lebesgue measure to be absolutely continuous with respect to each type measure, together with full support; it does not require the type measures themselves to have densities. Under allocation and transfer measurability and integrability conditions, the registered theorem excludes the simultaneous satisfaction of almost-sure ex post efficiency, Bayesian incentive compatibility at every admissible type and report, interim individual rationality, and weak ex ante budget balance. The proof derives envelope identities from incentive inequalities, identifies the efficient allocation almost everywhere, and uses a layer-cake representation to show that the required information rents strictly exceed efficient surplus. We explain the analytic steps, their Lean declarations, and the verification evidence for the exact Palomar-registered source revision. This is an exposition of a classical result and an existing formal artifact, without a claim of mathematical novelty or formalization priority.
cs.DS — Data Structures and Algorithms
We explain a standalone Lean 4 and Mathlib verification of the finite counterexample announced by Dmitry Rybin to the cost-preserving single-source unsplittable-flow conjecture. The instance has seven vertices, nine directed arcs, and three terminal demands. An exact rational path flow saturates every arc, has cost 58, and has largest demand 15. Exhaustive graph-path classification reduces every unsplittable routing to two choices per terminal. Three backbone-arc inequalities force any routing with load at most the fractional load plus 15 to use at least two paid routes, each costing 30. Thus every such routing costs at least 60, ruling out cost preservation even with a non-strict congestion bound. We present the numerical certificate, its graph realization, and the boundary between the pinned proved implementation and its statement-only verification interface. The contribution is an exposition of a source-based formal certificate; no new counterexample, first formalization, or general rounding algorithm is claimed.
math.AT — Algebraic Topology
We explain a Lean 4 formalization of the classical Seifert van Kampen theorem for the full fundamental groupoid. For an arbitrary topological space covered by two open subsets, the inclusion-induced square of ordinary continuous-path fundamental groupoids is a pushout in the category of categories. Every point is retained as an object; no basepoint, connectedness, or separation hypothesis is required. The development identifies an auxiliary directed path category for the indiscrete preorder with Mathlib's fundamental groupoid through strictly inverse functors and naturality equalities. It then assembles the pushout universal property from attributed path-subdivision and homotopy-grid helpers adapted from the directed-topology formalization of Basold, Bruin, and Lawson. We describe the descent construction, its transfer to ordinary fundamental groupoids, and the statement and verification boundaries of the exact artifact registered in Palomar. This is an expository account of a classical theorem and an existing formal artifact, with no claim of mathematical novelty or formalization priority.
cs.DS — Data Structures and Algorithms
We describe a Lean 4 development of men-proposing deferred acceptance for finite one-to-one markets with complete strict preferences and optional partners. The two sides may have different cardinalities; the principal theorems assume a nonempty receiving side. A proposal-set measure proves termination of a specified, classically chosen run. Reachability invariants establish consistency of the resulting partner maps, stability, and the fact that a rejected proposal cannot be a stable-achievable partnership. The public optimality theorem is conditional on a man receiving a partner; an internal lemma separately excludes stable-achievable partners for men unmatched at quiescence. We give mathematical proofs corresponding to the source architecture, identify the precise theorem interface, and separate fresh source inspection from the repository's reported build and axiom audit. This is an exposition of a pinned formalization of classical results, with no claim of new matching theory or priority of formalization.
math.OC — Optimization and Control
We describe a Lean 4 formalization of the finite unrestricted-domain version of Roberts theorem. For finite nonempty agent and alternative types, with at least three alternatives, an onto deterministic direct-revelation mechanism that is dominant-strategy incentive compatible has an affine-maximizer choice rule. Valuations range over all real-valued profiles; the conclusion uses fixed nonnegative weights, at least one nonzero, and alternative offsets, and asserts membership in the maximizing set. The development follows the modular proof route: taxation and weak monotonicity, perturbation-based tie-breaking to strong monotonicity, no-veto range analysis, affine prices for two agents, and induction through fixed-report slices. We explain the proved tie-set transport lemma, the construction of fresh payments for the tie-broken rule, and the transfer back to arbitrary ties. A version-pinned account distinguishes the proved library from statement-only comparator placeholders and separates source inspection from executable checks. This is an exposition of a classical result and its formal artifact, with no claim of mathematical novelty or formalization priority.
math.OC — Optimization and Control
We describe a Lean 4 development of finite direct mechanisms with quasi-linear utility and unrestricted real-valued valuations. The Clarke pivot mechanism selects a reported-welfare maximizer and charges each agent the loss in other-agent welfare relative to its maximum over the same fixed alternative set. The development proves efficiency, dominant-strategy truthfulness, nonnegative payments and no deficit, and individual rationality under the explicit assumption that every valuation is nonnegative. It also proves that every efficient, dominant-strategy incentive-compatible mechanism on the full valuation domain has Groves-form payments, with a term independent of the paying agent's report. We explain the finite-maximization interface, the report-update invariants, and the characterization argument using outcome-forcing reports and positive perturbations. The exposition is tied to a specific source revision and separates proved implementations from statement-only comparator placeholders. These are classical mechanism-design results; no new mathematical theorem or formalization-priority claim is made.
math.OC — Optimization and Control
We describe a Lean 4 formalization of the minimax theorem for finite nonempty two-player zero-sum games with arbitrary real payoffs. Mixed strategies are probability vectors, and the two game values are defined using real indexed suprema and infima. An explicit payoff bound supplies the boundedness obligations required by these conditional extrema. The reverse minimax inequality follows by applying Mathlib's proved Sion saddle-point theorem to compact convex coordinate simplices, with the minimizing player placed in its first argument. We explain the representation bridge, the proof obligations, and the passage from a saddle point to equality. The accompanying version-pinned artifact separates a reusable library from a statement-only challenge and its proved solution. This is an exposition of a finite-game specialization of existing formal analysis, with no claim of a new mathematical proof or formalization priority.
math.AG — Algebraic Geometry
For every Hochschild diagonal of a smooth projective complex variety, a Fourier–Mukai equivalence preserves both the ordinary Hodge-number sum and the sum signed by the parity of the antiholomorphic degree. The latter is the signature of a Hermitian form obtained from the generalized Mukai pairing. In dimension five these invariants determine and and reduce the unrestricted Hodge-number problem to two numerical parameters, one of which is the possible failure of invariance of . Albanese methods remove this structure-sheaf ambiguity when the Albanese image has dimension at least three. The general fivefold case remains open, while a theorem of Abuaf proves invariance of all Hodge numbers for fivefolds with trivial canonical bundle.
math.QA — Quantum Algebra
We prove Conjecture 11 of Lee, Li, Rupel, and Zelevinsky on the support of triangular basis elements in rank-two quantum cluster algebras.
math.PR — Probability
Let be independent uniform points in the unit square and let denote Lebesgue probability measure. We prove that the normalized expected distance between the empirical measure and Lebesgue measure converges to a positive constant: \[ \lim_{N\to\infty} \sqrt{\frac{N}{\log N}}\, \mathbb E W_1\!\left(\frac1N\sum_{i=1}^N\delta_{X_i},\lambda\right) =c_\star. \] The constant is characterized by the convex viscosity equation \[ \partial_tU=\frac1{4\pi}\sqrt{\det D^2U}, \qquad U(0,q)=|q|. \] Its solution is unique under certain growth and regularity conditions, and the limiting constant .
math.MG — Metric Geometry
For a nonempty finite metric space X, let Exp(X) be the space of its nonempty subsets equipped with the Hausdorff metric. We prove that the isometry type of Exp(X) determines X when X is strongly rigid, meaning that distinct unordered pairs of distinct points have dis- tinct distances. We also prove that the hyperspace operation preserves the ordinary Gromov–Hausdorff distance locally under explicit separa- tion conditions on the distance spectra. For finite ultrametric spaces, the hyperspace operation is an isometric embedding with respect to the Gromov–Hausdorff ultrametric. In contrast, we construct finite ultra- metric spaces for which the ordinary Gromov–Hausdorff distance strictly decreases under the hyperspace operation. Examples of equal cardinality exist in every cardinality at least six and realize every ratio in [1/2, 1). Their distances remain constant after the first hyperspace iteration. An explicit perturbation also yields strict contraction for finite strongly rigid metric spaces with all triangle inequalities strict.
math.LO — Logic
We study ordinal-definable families of sets of arbitrary rank in Cohen, random, and collapse extensions. Over , countable OD families have OD enumerations after adding one Cohen or one random real. A single generalized Cohen subset gives the corresponding sharp theorem at every regular uncountable cardinal. When is singular strong limit and has uncountable cofinality, adding Cohen reals makes every member of a short-parameter definable family of size at most definable from a short parameter. Adding random reals gives a single short-parameter definable enumeration for every such family of size strictly below . The random bound is sharp. The proofs use coordinate amalgamation and small-index arguments. We also obtain arbitrary-rank descent for countable families after collapsing any infinite cardinal, with applications to choice in the relative Feferman--Levy model. In the full Solovay collapse extension of an arbitrary ZFC ground, every family definable from a real and ordinals that represents fewer than classes modulo null sets consists of measurable sets; the corresponding category assertion also holds. In the extensions of by Cohen or random reals, no model of of size at most , definable from ordinals and a real, has full binary-coded standard system. proves a finite-fragment Borel obstruction to a full binary standard system for . For regular uncountable , the generalized Cohen extension by $\Add(\kappa,\Lambda)^L$, , has no ambiently -saturated model in , regardless of its size. Consequently every infinite regular of admits a cofinality-preserving GCH extension with no saturated arithmetic presentation of size . ZFC also proves that every singular strong limit admits an OD -saturated elementary extension of of size , hence of size under GCH.
math.CO — Combinatorics
Let be a nonempty finite set. A shade map on is a map such that toggling an element in the input does not change . We prove that, if is an inclusion-reversing shade map and , then the simplicial complex \[ \{F\subseteq E\mid G\subseteq T(F)\} \] is collapsible. Equivalently, if is an inclusion-preserving shade map, then the Alexander dual of \[ \{F\subseteq E\mid G\not \subseteq S(F)\} \] is collapsible. This is an Alexander-dual companion to the shade-map collapsibility theorem in \emph{The Elser nuclei sum revisited}. The proof first matches all faces on which by a toggle. The remaining faces, characterized by , form the free convex set complex of an associated antimatroidal quasi-closure operator. We give a self-contained recursive acyclic matching on this complex. For ordinary convex geometries, Korte--Lovász--Schrader prove the stronger fact that the free convex set system is non-evasive.
math.QA — Quantum Algebra
We prove Conjecture 12 of Lee, Li, Rupel, and Zelevinsky: for arbitrary positive integers , every strongly positive basis of the coefficient-free rank-two quantum cluster algebra contains a unique element pointed at each pair in . These pointed elements exhaust the basis.
math.CO — Combinatorics
We prove that the alternating sum has nonnegative coefficients whenever . For , this range is exact. The case proves a conjecture of Lewis and Morales arising from the enumeration of invertible matrices with prescribed zero entries. After removing a common -factorial, we express the sum in a Gaussian-binomial basis whose coefficients are independent of the ambient size. Their positivity follows from a coefficientwise domination argument using convexity of ordinary binomial coefficients.
math.CO — Combinatorics
The theory of cluster algebras is closely connected to the theory of total positivity; indeed, the desire to better understand total positivity was one of the main motivations for Fomin and Zelevinsky’s introduction of cluster algebras [FZ02]. In particular, any cluster variety whose coordinate ring has a cluster structure has a natural notion of positive part: the subset of the variety where all cluster variables are positive. In this paper, we explain that there are other signed cells contained in cluster varieties that are equally natural from a cluster-theoretic point of view. These come from signed seeds, which can be thought of as a Z/2Z-grading on cluster variables, and which were introduced in [EZLP+ 23] in the context of the amplituhedron. More generally, given any abelian group G, we introduce the notion of a G-graded seed for a cluster algebra, which is a way of assigning elements of G to each cluster variable which is compatible with the cluster structure. When G is the multiplicative group {−1, 1}, this recovers the above notion of signed seed; when G = C∗ , this recovers the notion of cluster automorphism group [GSV10] or cluster dilation group [NS26]; and when G = Zd , this recovers the notion of graded cluster algebra studied by Grabowski-Launois [GL14], Grabowski [Gra15] and Gekhtman-Shapiro-Vainshtein [GSV10, Section 5.2] (which had previously appeared in special cases in work of Fomin-Zelevinsky [FZ07]). The examples we study include the space of square matrices, symmetric matrices, skew-symmetric matrices, positroid varieties, and amplituhedron tiles. We also connect this notion to tropical mutation when G = R or Z.
math.AG — Algebraic Geometry
We proved the predictions by Ben-Zvi, Sakellaridis, and Venkatesh that the Plancherel algebra is commutative and its loop-rotated version is flat over .
math.CO — Combinatorics
A conjecture arising from a question of Procesi proposes a basis of the coinvariant algebra of consisting of column-antisymmetrized monomials indexed by pairs of standard Young tableaux. We show that the proposed family need not even span an -subrepresentation: for , a generator of degree is sent outside the span by the adjacent transposition . The proof is an exact finite computation with an explicit separating functional, requiring neither a rank computation nor Gr\"obner reduction. We also retain an explicit relation for , where the basis assertion first fails, although the span is still invariant.
math.NT — Number Theory
We determine the primitive integer solutions of x^4 + 3y^3 + 2z^3 = 0. They are exactly (1,-1,1) and (-1,-1,1), where primitivity means gcd(x,y,z)=1. A fourth-power descent in Q(∛12) produces four plane quartics. A congruence modulo 4 excludes two of them for primitive input. We determine the rational points on the other two by elliptic descent, finite reduction sieves, and elliptic Chabauty. For the final quartic, an explicit nonzero Cassels–Tate pairing sharpens a rank-three descent bound to rank one. All finite certificates and their verification programs accompany the proof.
math.LO — Logic
Peirce’s existential graphs draw a first-order formula so that the order and grouping of conjuncts, the names and order of bound variables, the position of a quantifier relative to conjuncts that do not mention its variable, and how a network of equalities is written cannot be seen. We take these structural laws for granted. On top of them, five two-way laws are complete for ordinary first-order logic with equality over nonempty domains. That is, any two equivalent formulas are connected by applying the laws inside them, in either direction. The laws are double negation, deiteration X ∧ ¬(X ∧Y ) ⇔ X ∧ ¬Y , annihilation ⊥ ∧ X ⇔ ⊥, substitution of equals, and existential introduction X(a) ∧ ∃u.X(u) ⇔ X(a), and none of them follows from the other four. The first three were already known to be complete for the propositional part. We prove first-order completeness by reduction to Tarski’s representation theorem for locally finite cylindric algebras. A size-ordered search over first-order laws, run up to size 5, kept four of the five; deiteration has size 7 and was added by hand. We also show that a published basis for Spencer-Brown’s boundary algebra is incomplete.
math.RA — Rings and Algebras
Let be a commutative ring. We show that every homogeneous polynomial of degree splits into a product of homogeneous linear forms over a suitable noncommutative ring extension of (that is, over a noncommutative -algebra whose structure morphism is injective). The algebra is universal for such factorizations and is free as an -module. The proof uses Bergman's Diamond Lemma: we define by generators and relations, and the relations form a terminating reduction system that has no ambiguities and thus is confluent. An analogous result is also shown for inhomogeneous polynomials (with inhomogeneous factors). This easily follows from the homogeneous case by homogenizing and then setting the homogenizing variable equal to .
math.CO — Combinatorics
Burman and Kulishov defined Lie elements in the group algebra by comparing, on every exterior power of the reflection representation , the usual action of with the infinitesimal action induced by its action on . They conjectured that the Lie algebra of all Lie elements is generated by the Kirchhoff differences . We disprove this conjecture for by exhibiting an explicit counterexample arising from the -block of . More generally, we describe in terms of the Artin--Wedderburn decomposition of : its hook blocks are determined by the action on , whereas its non-hook blocks are arbitrary. Consequently, the primitive central idempotents of the non-hook blocks yield linearly independent obstructions to the conjecture. We also identify the Lie algebra generated by the Kirchhoff differences in terms of the derived algebra of the Lie algebra generated by transpositions. Along the way, we give an integral, and hence characteristic-free, proof that the exterior powers of are the hook-shaped Specht modules.
math.NT — Number Theory
We determine all primitive integer solutions of the generalized Fermat equation x^4 + 4y^3 + z^3 = 0. Here primitive means gcd(x,y,z)=1. We prove that the only primitive solutions are (x,y,z) = (1,0,-1), (-1,0,-1). The proof begins with a fourth-power descent in the pure cubic field Q(∛2), reducing the problem to the rational points on two explicit smooth plane quartics of genus 3. These quartics admit maps to elliptic curves over Q(∛2). Exact 2-descent determines the relevant Mordell-Weil ranks and finite-index subgroups. A complete reduction sieve at 127 leaves four residue disks, and an elliptic Chabauty calculation proves that each disk contains exactly one rational point. Reconstruction of the original variables then leaves precisely the two stated primitive solutions. The finite computations are exact and reproducible from the certificates and verifier source included in the appendices, and the argument applies at all heights.
math.QA — Quantum Algebra
We prove quantum Laurent positivity for mutation-acyclic skew-symmetrizable cluster algebras of arbitrary finite rank.