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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 , $\HOD$ is an arbitrary model of $\ZFC$. For any sentence , letting $\CM$ denote ``class of measurables,'' the following theories are equiconsistent: \begin{enumerate} \item $\ZFC+\CM+ [\HOD_{\Ord^{\omega}}\models (\exists\kappa\,V_{\kappa}\models\sigma)]$; \item $\ZFC+\GCH+\CM+ [\HOD_{\Ord^{\omega}}\models (\SVC\wedge\CM\wedge\exists\kappa\,V_{\kappa}\models\sigma)]$; \item $\ZF+\DC+\CM+ \exists\kappa\,V_{\kappa}\models\sigma$. \end{enumerate} Thus, under measurables the only local constraint on $\HOD_{\Ord^{\omega}}$ is the requisite choice principle. We also derive an analogue for Woodins. Letting $\CW$ denote ``class of Woodins,'' the following are equiconsistent: \begin{enumerate} \item $\ZFC+\CW+ [\HOD_{\Ord^{\omega}}\models (\exists\kappa\,V_{\kappa}\models\sigma)]$; \item $\ZFC+\GCH+\CW+ [\HOD_{\Ord^{\omega}}\models (\exists\kappa\,V_{\kappa}\models\sigma)]+$ $\HOD_{\Ord^{\omega}}$ is a ground of ; \item $\ZFC+\CW+$ some symmetric extension $V(G) \models \DC+\exists\kappa\,V_{\kappa}\models\sigma$. \end{enumerate}
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.