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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.
quant-ph — Quantum Physics
The finite-dimensional Pauli problem asks how many measurements in orthonormal bases determine every pure state of a -level quantum system up to a global phase. Four bases always suffice, and the answer is known to be three for and four for and . Dimension four remained open because the embedding argument used in other dimensions fails there: the pure-state space embeds in , the space of the nine independent probabilities of three bases. We show that three bases do not suffice in \(\C^4\), so exactly four are needed. The same argument shows that no ten vectors in \(\C^4\) do phase retrieval. Since eleven vectors are known to suffice, the smallest phase-retrieval frame and the smallest rank-one POVM distinguishing all pure states in \(\C^4\) both have eleven elements. All three lower bounds follow from one statement: every six-dimensional real space of traceless Hermitian matrices with a common isotropic vector, that is, a nonzero with for all in the space, contains a nonzero matrix of rank at most two. We prove it with complex -theory: otherwise, an odd unitary map on the five-sphere would have a -class that is nonzero by antipodal symmetry but vanishes because of the isotropic vector.
math.MG — Metric Geometry
We prove that the centroid Banach–Mazur diameter of planar convex bodies is strictly less than , (slightly) improving the prior upper bound due to Lassak. The proof combines a sharp covariance-ellipse sandwich, whose equality cases in any single direction occur only for triangles, with a compactness argument: an extremal pair at distance would consist of two triangles, which are linearly equivalent and hence must in fact be at distance . The resulting gap below is uniform but nonquantitative.
math.AC — Commutative Algebra
Let be a commutative ring, and let the symmetric group act on $P=\mathbf{k}\left[ x_{1} ,x_{2},\ldots,x_{n} \right] $ by permuting the variables. We prove four results that are known (at least in the case when is a field) but not easily found in the literature. First, the coinvariant algebra (the quotient of by the ideal generated by the symmetric polynomials with constant term ) is a free -module of rank , with the residue classes of the Artin monomials as a basis. Second, is a free module of rank over the ring of symmetric polynomials, again with the Artin monomials as a basis. Third, if is invertible in , the coinvariant algebra is the regular $\mathbf{k}\left[ \mathfrak{S}_{n} \right] $-module. Fourth, under the same hypothesis, is a free left -module of rank . The first result follows from an elementary normal-form lemma for monic polynomials with pairwise relatively prime leading monomials. The second is proved by lifting the Artin basis. For the third, we use orbit harmonics with a strongly discrete point orbit, and the fourth follows by equivariantly lifting a regular basis of the coinvariant algebra.
cs.CC — Computational Complexity
Four-row mancala games, including Malawian Bawo and the related East African Bao, have received limited attention in computational complexity theory despite their combinatorial richness. This paper introduces GENERALIZED-BAWO-WIN, a finite-state two-player game model inspired by Malawian Bawo. The model is parameterized by a half-board width n and a binary-encoded lap cap L, and incorporates relay sowing, marker-pit capture, a formalized capture-relay rule (abstracting kichwa), mandatory capture, a singleton-start prohibition, and an explicit lap-cap truncation rule that guarantees move termination. We prove that GENERALIZED-BAWO-WIN belongs to EXPTIME under these semantics. The proof proceeds by establishing that the game's state space has size at most 2poly(|I|), that legal successors can be generated in 2poly(|I|) time, and that a WIN/LOSS/DRAW retrograde attractor can be computed over the resulting finite game graph. We emphasize that this is an upper bound: no hardness result is established. We further emphasize that the theorem applies only to the formally defined generalized model and does not classify Bao as played in East Africa, whose rules differ materially.
math.CO — Combinatorics
We give a -vertex counterexample to a conjecture of Pawlowski asserting that the characteristic polynomial of a chromatic operator associated with a forest is independent of the ordering of its edges. The two operators already have different traces of fifth powers.
math.NT — Number Theory
This note answers a problem posed in Montgomery's CBMS book. The proof was found by ChatGPT 6, and written up by the author.
math.AC — Commutative Algebra
We show that each univariate polynomial over a commutative ring can be factored into linear factors over a suitable commutative ring extension of . The proof proceeds by universal construction: is defined as the tensor product , where is the polynomial ring $\ZZ[a_1, b_1, a_2, b_2, \ldots, a_m, b_m]$, and where is its subring generated by its ``homogenized elementary symmetric polynomials'' for all . The injectivity of the structure homomorphism is deduced from a combinatorial study of the diagonal subring of . In the process, a homogeneous variant of the Garsia--Stanton basis is constructed, and some classical properties of symmetric polynomials are recovered.
math.CO — Combinatorics
Lafay, Peltola and Roussillon conjectured that their realization of simple modules of the fused Hecke algebra by fused Specht polynomials extends from Young diagrams with two columns to Young diagrams of arbitrary shape. We give a counterexample for \[ n=8,\qquad \lambda=(3,2,2,1),\qquad \varsigma=(2,2,2,2). \] More precisely, we exhibit an explicit linear dependence among three fused Specht polynomials indexed by row-strict Young tableaux. The same example also yields an infinite family of counterexamples.
math.LO — Logic
For the inner model HOD_{Ord^{\omega}}, we derive an analogue for Roguski's classical result that without nontrivial assumptions on V, HOD is an arbitrary model ZFC model. For any sentence \sigma, letting CM denote "class of measurables," the following theories are equiconsistent: (1) ZFC + CM + [HOD_{Ord^\omega} \models (\exists \kappa V_{\kappa} \models \sigma)]; (2) ZFC + GCH + CM + [HOD_{Ord^\omega} \models (SVC \wedge CM \wedge \exists \kappa V_{\kappa} \models \sigma)]; (3) ZF + DC + CM + \exists \kappa V_{\kappa} \models \sigma. Thus, under measurables we have that HOD_{Ord^{\omega}} is "locally arbitrary."
math.AT — Algebraic Topology
We establish rational Swan induction for ordinary perfect modules over the finite-local sphere . For a finite abelian ambient group, subgroups of -rank at most suffice, and this bound is sharp. The proof combines cyclic homotopy fixed points in telescopic spectra with the isotropy filtration of a chromatic quotient of finite genuine spectra. The result proves Conjecture~7.22 of Clausen--Mathew--Naumann--Noel for Morava -theory and gives a new proof of their chromatic upper bound for the algebraic -theory of -linear categories. The quotient argument also produces finite complexes realizing the induction relations. We formulate the problem of explicit realizations and give two geometric models at the prime two.
math.AG — Algebraic Geometry
For every odd prime , we construct a regular strictly henselian local ring whose generic fibre has \'etale cohomology classes in $H^2_{\et}(A[1/p],\mu_p^{\otimes2})$ which are not sums of cup products of degree-one classes. The ring is the strict henselization of a local ring on a finite-type arithmetic scheme. A unit on a principal divisor gives an element of order in , detected by its localization boundary on the special fibre. Adams--Riemann--Roch and the connectivity of the motivic filtration show that this element has nonzero image in $H^3_{\mot}(A[1/p],\Z(2))$. The integral coefficient sequence then gives the cohomological obstruction. Only the eigenvalues and of are needed.
math.AG — Algebraic Geometry
We construct finite free group schemes of rank four and exponent eight as stabilizers in a smooth two-dimensional affine group. We first compute the complete local deformation ring of the invariant space of constants and squares in characteristic two, using six Grassmannian coordinates. For every Artinian specialization, evaluation at the identity has a finite free rank-four stabilizer. Its fourth-power morphism is , and its eighth-power morphism is trivial. The fourth power is nonzero on the third-order neighbourhood of the universal deformation ring and on a specialization over . We give complete formulas for the latter group scheme. Right translation preserves the chosen quadratic space exactly when it contains the square of the scalar character. This identifies as the common obstruction to fourth-power vanishing, normality of the stabilizer, and two-sidedness in an equivalent cyclic-module construction.
math.NT — Number Theory
Let p be an odd prime and K/Qp a finite unramified extension of degree f > 1. Let Z(r) be the reduced special fiber of the Emerton-Gee stack of two-dimensional crystalline representations of Hodge type r of the absolute Galois group of K. We study the collection of stacks Z(r) as r varies over p-bounded Hodge types, as a set partially ordered under inclusion. We prove that aside from two degenerate cases, simple inclusions can be classified in terms of three operations on Hodge types, two of which have standard automorphic interpretations. We also prove, with one exception, that inclusions can be detected at the level of an inclusion of closed points (equivalently, semisimple mod p Galois representations).
math.AC — Commutative Algebra
We deduce from Bhatt's graded Cohen--Macaulay and vanishing results that an integer-graded absolute integral closure of a weighted polynomial ring over , modulo , is free with basis degrees smaller than the sum of the weights. A direct -adic expansion turns this degree bound into a uniform solution of the cusp relation modulo nilpotents. Consequently every ring admits a faithfully flat algebra with seminormal reduction, and every invertible -module becomes free after a faithfully flat extension of , answering a question of Drinfeld.
math.AG — Algebraic Geometry
A Fourier–Mukai equivalence between smooth proper complex varieties of common even dimension preserves the topological signature. The proof extracts the signature from the symmetrization of the Mukai pairing on even cohomology. Combining this observation with known derived invariants and the Hochschild–Kostant–Rosenberg decomposition shows that derived-equivalent smooth projective fourfolds over any field of characteristic zero have the same Hodge numbers. This argument and text was produced by ChatGPT 5.6 Sol.
math.AG — Algebraic Geometry
Chen and Larson study tautological classes on the strata of holomorphic abelian differentials, where they predict a vanishing result. Using known tautological relations on the moduli space of curves, they reduce this prediction to the non-vanishing of certain coefficients of a quotient of hypergeometric generating series, treating the residue classes and through two separate series; they verify the non-vanishing by computer for small genus. We prove it in general: in both cases the relevant coefficient is non-vanishing for every genus and every stratum, and in the first case it has, more strongly, a uniform strict sign. Along the way we correct an error in the Chen--Larson derivation of the series, where a constant of the underlying relation of Ionel had inadvertently been changed. The paper falls into two parts. Part~I gives the mathematical proofs; Part~II documents the Lean~4 and Mathlib formalization --- whose only non-standard trust assumption is the compiler invoked by \code{native\_decide} for the large finite computations --- together with the long-horizon, multi-model generative-AI process that produced the proofs, exposed false intermediate routes, and uncovered the error in the printed Proposition~5.2 relation noted above.