All works

math.NT — Number Theory

Powers of a Prime in the Image of Summatory Valuation Functions

Contributed by Frank Vega

J.-L. Nicolas asked how often a power of $2$ belongs to the image of $c(n)=\sum_{k\le n}2^{v_2(k)}$, where $v_2$ is the $2$-adic valuation, and showed that the number $A(x)$ of exponents $a\le x$ with $2^a$ in the image and the number $B(x)$ of the other exponents both tend to infinity. We prove that $2^a$ lies in the image of $c$ if and only if $2^{a+3}$ is a sum of distinct terms of the superincreasing sequence $(j2^j)_{j\ge1}$. For every exponent there is at most one possible largest term $K$, given by $K+\lceil\log_2K\rceil=a+3$, and membership is then decided by an explicit recursion on pairs of integers at most $a+3$, in at most $a+1$ steps. The exponents without such a $K$ and the exponents decided in at most one step form explicit infinite families, described completely, which contain the three families of Nicolas. The states of the recursion form a forest and the set of successful states is symmetric; this yields further explicit families and the bounds $A(x)\ge(3.9655\ldots+o(1))\log_2x$ and $B(x)\ge(2.8178\ldots+o(1))\log_2x$, whereas the families of Nicolas give the constants $\frac12$ and $\frac32$. The reduction, the unique candidate, the recursion and the symmetry extend to $c_p(n)=\sum_{k\le n}p^{v_p(k)}$ for every prime $p$. Computations in exact integer arithmetic reproduce the table of Nicolas and give $A(x)$ for $x\le262000$, with $A(x)/x$ between $0.5132$ and $0.8750$ for $x=1000,2000,\dots,262000$.

Author comments: In this new version, we add new computer-based experiments.

math.CO — Combinatorics

Iterating the Gupta–Ndiaye–Norin–Wei method: a computer-verified bound $R(k)\le 3.7698^{k+o(k)}$

Contributed by Ella Puerto

Gupta, Ndiaye, Norin and Wei (GNNW) recast the Campos–Griffiths–Morris–Sahasrabudhe method as an inductive theorem that turns a known bound on off-diagonal Ramsey numbers into a better one. With two applications they obtained $R(k)\le (3.7992\ldots)^{k+o(k)}$. They also reported an unverified, AI-suggested third application giving $3.78233\ldots$. We make three observations about their framework. (i) The free function $M$ can be optimised pointwise. (ii) For a fixed input, the best output exponent is the least solution of an explicit first-order differential inequality. (iii) GNNW's Theorem 14 needs only $F\in C^1$ with locally Lipschitz derivative. We then construct an explicit chain of eight applications and verify every hypothesis by interval arithmetic on the whole range $0<\lambda\le 1$, with a hand-proved estimate covering $\lambda\to0$. The verification was done twice, with independent code, libraries and enclosure strategies. This proves $R(k)\le 3.76973^{k+o(k)}$, and in particular confirms GNNW's $3.78233$. Numerically, iterating the optimal step converges to about $3.7690$, which suggests this is close to the limit of the GNNW method in its present form.

math.NT — Number Theory

The invariant $c_M$ of Gallardo and Rahavandrainy at Mersenne prime exponents, and the case $p=7$

Contributed by David Niedbala Giraudin

A Mersenne prime polynomial is an irreducible polynomial over $\mathbb{F}_2$ of the form $x^a(x+1)^b+1$ with $a,b\ge1$. Gallardo and Rahavandrainy conjectured that, apart from two exceptions, $\sigma(M^{2h})$ always has an irreducible factor which is not a Mersenne prime. For a prime $p\ge5$ they proved it for every $M$ outside an exceptional set $\Sigma_p=\Sigma_p^1\cup\dots\cup\Sigma_p^4$, defined through an integer $c_M$ and an odd integer $m_M$ read off the leading coefficients of $\sigma(\sigma(M^{p-1}))$, and later they proved it for every Mersenne prime number $p\ge31$; their method does not reach $p=7$. We compute $c_M$ exactly when $p=2^d-1$ is a Mersenne prime number: $c_M=d+1$ if $\deg M\ne d$, and $c_M=d$ if $\deg M=d$. Consequently $\Sigma_p^1$ consists of the Mersenne primes of degree $d$ or $d+1$, $\Sigma_p^2$ is empty, and $\Sigma_p^3$, $\Sigma_p^4$ are described by $m_M=\deg M-d-1$ and $m_M=\deg M-d$ respectively. For $p=7$ this reduces the conjecture to the Mersenne primes $M$ with $m_M\in\{\deg M-4,\deg M-3\}$. A direct computation shows that $\sigma(M^6)$ has a non-Mersenne irreducible factor for every Mersenne prime $M$ of degree at most $500$, and that $m_M\le17$ in that range. Hence the conjecture holds whenever $7$ divides $2h+1$ and $\deg M\le500$, and a counterexample with $7\mid 2h+1$ would need $m_M\ge\deg M-4>496$. A script attached as an ancillary file checks these computations, and reproduces the table of exceptional sets of Gallardo and Rahavandrainy.

math.OA — Operator Algebras

Modular Dvoretzky, Type-Cotype, Khinchin-Kahane and Grothendieck Inequality Problems

Contributed by K. MAHESH KRISHNA

We ask for noncommutative versions of following four: (1) Dvoretzky Theorem, (2) Type-Cotype Definition, (3) Khinchin-Kahane Inequalities, (4) Grothendieck Inequality.

math.NT — Number Theory

PARAMETER MONOTONICITY FOR GENERALIZED k-MARKOV NUMBERS

Contributed by Qiyue Tang

We prove that the order reversal asked about in Question 35 of Banaian and Huang cannot occur for generalized k-Markov numbers, answering their question negatively.

math.SG — Symplectic Geometry

DEEP LOCI OF DOUBLE BOTT–SAMELSON VARIETIES

Contributed by Qiyue Tang

We prove that the deep locus of a complex double Bott–Samelson variety associated with a positive braid is its stabilizer locus. Consequently, the deep locus is empty if and only if any two components of the braid closure are unlinked. This resolves Conjecture 4.45 of Castronovo, Gorsky, Simental, and Speyer.

math.GT — Geometric Topology

The dodecahedron web and its instanton homology

Contributed by Peter Kronheimer, Tomasz Mrowka

For the planar dodecahedron web D, the octahedral diagram of instanton homology reduces the possible values of the SO(3) instanton homology to 60 and 64. The remaining choice is determined by three coefficients of a secondary map. We give a proposed proof that all three vanish and that the dimension is 64. We express the coefficients as counts of instantons on specified foam cobordisms with one-parameter families of metrics. The index calculation requires these instantons to have negative action. An estimate uniform under neck stretching excludes them when the perturbations are sufficiently small. The construction includes local Floer complexes with zero differential, the marking calculation for excision, and a simultaneous choice of regular perturbations compatible with the families. Exact matrices and the supporting foam calculations are provided separately.

math-ph — Mathematical Physics

Geometric Computation: Twin Helices and the Algebra of Flow

Contributed by Benjamin Kaminsky

This work constructs a manifold consisting of twin helices to investigate the existence of computational geometric bordisms, as part of the broader program of the Cobordism Hypothesis. We use these structures to define a class of deterministic computational substrates, termed Geometrically Computable Manifolds, and study their geometric, algebraic, and computational properties. We show that, in a discrete regime, these manifolds encode arbitrary cyclic groups and support universal computation. This leads to a hypothesis describing the structural trade-offs in physical computation across distinct computational phases. We then analyze a regime in which abelian gauge computation is modeled with ordinary differential equations, linking the construction to continuous geometric dynamics. This signals a new research direction in physical mathematics at the intersection of geometry, gauge theory, and computability theory, termed Geometric Computability Theory, in which computation is treated as a property of geometric compositions. This perspective opens avenues for future investigation, motivating the development of a constructive computational gauge theory with applications to Yang–Mills theory. As of this writing, no complete unified compositional semantics yet exists for the full range of these manifolds; future work will develop such a formalism.

Author comments: Updates: ---Specified the correct Lie algebra for conical helices in 1.1. ---Corrected "helicity" to "helices" in A.1.

math.NT — Number Theory

Asymptotic closed forms for the Faltings height of Fermat Jacobians and the spectrum of logarithmic derivatives of L-functions

Contributed by Zeng Chunping

For the Fermat tower $F_N\colon x^N+y^N=1$, $N=p^n$, we assemble the complete asymptotic machine for the stable Faltings height $h_F(J(F_N))$: the main-chain closed form $\hat h_n=\frac{\log p}{4}n-6\log A-\frac{p\log p}{4(p^2-1)}+\Theta\!\bigl(\frac{n\log p}{p^n}\bigr)$; an exact spectral identity $h(\Phi)=\frac{\mu(\Phi)}{2}-\sum_\chi m(\Phi,\chi)\rho_\chi$ with $\rho_\chi=R_\chi+\log f_\chi$; an arithmetic channel for the deviation functional through the Bernoulli-mass $M_1(\Phi)=\sum_\chi m(\Phi,\chi)\log|B_{1,\bar\chi}|$ ($R^2=0.78$, residual std $1.90$); a profile law $\mathbb{E}[M_{c'}]=3^{c'-2}\Lambda(n)$ proved by a carry-counting identity; and the layerwise height closure $\mathrm{mean}\,h^{\mathrm{Col}}(n)=\frac{3^{n-1}\log 3}{4}(n-n^*)$ with $n^*\to 5.69$, which explains the sign flip at $n=5$ and the $\Delta_9$ explosion ($\hat h^{\mathrm{Fal}}_9=+7.24$, twenty-five times outside the corridor $0.25$--$0.30$). All identities are certified at the exact-arithmetic level on layers $n\le 12$ ($2187$ orbits and $729$ odd characters at layer $7$, statistical verdicts at layers $8$--$12$). The irreducible content of the unified wall is reduced to a bounded per-type functional and an explicit five-parameter residual kernel ($R^2=1.0000$ on the orbit level).

cs.DS — Data Structures and Algorithms

Arithmetic Dynamics as the Seventh World: The Law of Forgetting for z^2+c

Contributed by Zeng Chunping

Among the seven worlds of the central-charge program (number fields, function fields, graphs, finite-type dynamics, knots, tropical geometry, and arithmetic dynamics), the seventh is the only one whose critical exponent is continuously tunable, and the only one whose "cusp layer" is visible point by point over the primes. For phi(z) = z^d + c reduced modulo p, the orbit structure splits into two layers: an algebraic-closure layer (universal, smooth, forgotten: N_m = #Fix(phi^m) = d^m + 1, independent of c) and an F_p-rational layer (arithmetic, fingerprinted, remembered: Per_p(phi), the Frobenius fixed-point count). We prove: the dynamic Lehmer constant L_dyn = h_{x^2+1}(0) = 0.203677261369740... (a complete minimization proof over Q); the three-axis separation theorem for the splitting family c = (1-m^2)/4 (amplitude = 2 is a free fingerprint carried entirely by the n = 1 layer; alpha = 0.5 is a theorem for all m >= 5; the Jensen gap J has no m-dependence and separates the monomial and Chebyshev cases that the scale exponent alpha confounds); the sqrt(2) anisotropy of escape near the cusp, A(phi) = |sin(phi/2)|/sqrt(6) (Joukowski coordinates, 15 directions at two scales digit-perfect); the Sha growth law lim_n log_2(Sha_n)/2^n = 2*alpha - 1 with the Stoll mod-square family classified (7/7 separation); the criterion alpha = 1 if and only if disguised group endomorphism, across polynomials and rational maps including both Lattes quotients; and the number-field PCF criterion h(0) = 0 if and only if c in M intersect O_K with all embeddings in M, verified against Baker-DeMarco with a six-field Lehmer table. The universal far-tail rate is identified with the random-mapping class, Phi = (4-pi)/4 = 0.2146, with a quantitative accounting of the PCC contamination that had drifted earlier measurements. All corrections found during the research (including two withdrawn headline predictions) are recorded in an appendix.

math.RT — Representation Theory

The Law of Forgetting in Moments of Sato-Tate Measures: Invariant Tensors, Maximal Tori, and the Rank Law

Contributed by Zeng Chunping

For a compact group G with a finite-dimensional representation V, the moments of the character are, by a one-line argument, the dimensions of the invariant tensor spaces: <chi_V^m>_G = dim(V^{otimes m})^G. We call the passage from averaging over G to averaging over a maximal torus T the "law of forgetting": T-invariants only require weights to sum to zero, while G-invariants impose the additional invariance constraint. For the standard representation of SU(2) the two ladders differ by the exact factor F_k = binom(2k,k)/Cat(k) = k+1 (proved via the Temperley-Lieb non-crossing matching basis). The general forgetting factor admits a complete asymptotic F_k = c_V * k^s * (1 + gamma_V/k + O(1/k^2)), where s is the number of positive roots; both constants are given in closed form: c_V = |W| / (2^s <h_d>), a ratio of the Weyl denominator's lowest homogeneous term at the saddle to its Gaussian average, and gamma_V = (epsilon_T - epsilon_G)/2, whose inputs are the quartic term of -log|chi/dim V| and the next term of the Weyl denominator. The exponent s = (dim G - rank G)/2 is verified across six compact Lie groups (A_1, A_2, B_2, C_2, C_3, G_2) through the theorem F(1) = dim V (proved: weight multiplicities one, Schur's lemma), and the universal U(N) family gives the closed form F(N,k) ~ R_N k^{N choose 2} with R_N = (2/N)^{N choose 2} / G(N+1), G the Barnes function, identified with the reciprocal of a Mehta constant. On the arithmetic side, the master identity is verified on elliptic-curve Frobenius data; the two-copy test E x E versus E_1 x E_2 reads Schur's lemma directly from the second moment (a single number, 4 versus 2, separates isogenous from distinct copies); and the (lambda,lambda) pairing identity of Howe duality is identified with the Dirichlet coefficients of the Rankin-Selberg L-function, the N <-> k symmetry reading as convolution symmetry. A parallel shape fingerprint W = E[Y^2]/E[Y]^2 bypasses the heavy-tail divergence of the Jensen-gap functional J = E[Y]E[1/Y], with closed forms for twelve distribution families. All conclusions carry explicit status labels (proved / numerically anchored / identified), and every error found during the research is recorded with its correction in an appendix.

math.DS — Dynamical Systems

Existence of SRB measures for some Rössler systems

Contributed by Theophile Caby

We prove that for the Rössler systemẋ = −(y + z),ẏ = x + ay,ż = b + z(x − c) with a = b = 0.2, the set of parameters c within 6 · 10 −5 of the classical value c = 5.7 for which the ow has a Hénon-like strange attractor supporting a SinaiRuelleBowen (SRB) measure has positive Lebesgue measure. The key step, proved with computer assistance, is that at an explicit parameter within 5.5 · 10 −5 of 5.7 the period-one saddle of the Poincaré map has a quadratic homoclinic tangency that unfolds generically; the conclusion then follows from the theorems of MoraViana, BenedicksYoung and BenedicksViana. The same method gives two further such tangencies near 5.7 and one at the band-merging parameter c ≈ 4.507. The proof reduces the generic-tangency conditions to nitely many inequalities, through explicit C 2 graph-transform bounds for the local invariant manifolds of a Poincaré map whose area contraction is of order 10 −11 10 −14 , and a scalar tangency function. The inequalities are checked with interval arithmetic in CAPD and rechecked with an independent implementation based on the Kv library; the analytic tangency criterion is formalised in Lean 4. Whether an SRB measure exists at c = 5.7 itself remains open.

math.NT — Number Theory

Minimal recurrences for binomial power sums

Contributed by Khai-Hoan Nguyen-Dang

For every positive integer $r$, we prove that the least order of a polynomial-coefficient recurrence for $F_r(n)=\sum_{k=0}^n\binom nk^r$ is $\lfloor(r+1)/2\rfloor$. This establishes the unrestricted minimal-order assertion arising from Franel's work of 1894--1895. We also determine the least order $\lfloor r/2\rfloor$, for $r\geq2$, of the alternating sums $(-1)^n\sum_{k=0}^{2n}(-1)^k\binom{2n}{k}^r$. The lower bounds follow from real moment representations: every nonzero singularity of the representing measure forces a root of the leading characteristic polynomial of the recurrence. A common reflection argument gives telescoping recurrences attaining both bounds. For the ordinary sums, we combine the deformation of Straub and Zudilin with a shifted prime-power congruence to prove that the minimal shift operator is irreducible over $\C(n)$.

math.AG — Algebraic Geometry

Gersten injectivity and Morel's purity conjecture

Contributed by Niels FELD

Morel conjectured that the $\mathbb{A}^1$-homotopy sheaves of an $S^1$-spectrum built from suspension spectra of smooth projective schemes are pure; one condition in purity is injectivity of restriction to dense open subschemes. Over a regular base $S$, we show that this injectivity condition, for the homotopy sheaves of algebraic $K$-theory with coefficients in an arbitrary spectrum, is equivalent to Gersten injectivity at the henselizations of all local rings of smooth $S$-schemes; and that these $K$-theory spectra are pure in Morel's sense. Hence every henselian regular local ring at which Gersten injectivity fails, integrally, rationally, or with finite coefficients, gives a counterexample to Morel's conjecture. The author's finite-coefficient counterexample over $\mathbb{Z}_3[[x,y]]/(3+x^2-y^3)$ therefore disproves the conjecture, already over the regular scheme $\operatorname{Spec}\mathbb{Z}[1/2][x,y]/(3+x^2-y^3)$. Recently announced integral and rational Gersten counterexamples, including unpublished work of the author, would show, if confirmed and if they persist after henselization, that the conjecture fails already for the $K$-theory spectrum itself.

math.CO — Combinatorics

The ten minimum universal sets of MacMahon cubes

Contributed by Jason Davies

A collection of six-coloured MacMahon cubes is universal if eight of its cubes can be selected to build a $2\times2\times2$ model of any target variety, with matching exterior colours and no internal matching requirement. We prove that exactly ten labelled collections attain the known minimum size of twelve, even when repeated varieties are allowed, resolving the conjectures of Johnson and Roldán. Usable-cube inequalities force a symmetric multiplicity matrix with row and column sums two; corner coverage then forces two disjoint triangles in Conway's tableau. The written proof is elementary and requires no exhaustive search. The classification and its discrete geometric interpretation are formalised in Lean 4. Exhaustive computation provides a separate check and additional target-count distributions.

math.CO — Combinatorics

Positive polytopes and the full Fock–Goncharov conjecture

Contributed by Qiyue Tang

We prove the origin-interior form of the polytope criterion of Gross, Hacking, Keel and Kontsevich: over a field of characteristic zero, full Fock–Goncharov for a principal coefficient cluster variety is equivalent to the existence of a bounded full-dimensional positive polytope containing the origin in its interior. Replacing the interior condition by membership of the origin gives a false criterion.

math.CV — Complex Variables

The Möbius pseudodistance of the annulus and an improved lower bound for Korenblum's constant

Contributed by David Niedbala Giraudin

We show that Korenblum's constant of the Bergman space satisfies $c_2\ge 0.446$. The best published lower bound was $c_2\ge 0.4263$ (Wikström, 2026). The new ingredient is a closed form, in Jacobi theta functions, of the Möbius pseudodistance of the annulus $\{c<|z|<1\}$ between two points of the same centred circle. It separates the variables and yields three monotonicity properties, one of which, a profile inequality, combined with Kirszbraun's theorem in the hyperbolic plane, improves the constant in Wang's annular estimate. Wikström's dual moment criterion is then verified at $c=223/500$ for an explicit measure with fourteen rational atoms, in rigorous arithmetic, by a self-contained verifier. The French version of this note is dated 21 October 2026.

math.QA — Quantum Algebra

Projected stated (SL_n)-skein algebras and the Huang–Wang conjectures

Contributed by Qiyue Tang

We prove Conjectures 1.5 and 1.6 of Min Huang and Zhihao Wang: the projected stated (SL_n)-skein algebra equals its quantum cluster and upper cluster algebras, and the specified exchangeable variables generate it over the frozen Laurent coefficient algebra. The results hold for triangulable unpunctured marked surfaces with at least two boundary marked points per connected component, over every coefficient domain and parameter in the conjectures.

math.GT — Geometric Topology

A certified upper bound for the distortion of the trefoil knot

Contributed by David Niedbala Giraudin

The distortion of a rectifiable closed curve $\gamma\subset\mathbb R^3$ is the supremum, over pairs of distinct points, of the ratio of the length of the shorter arc of $\gamma$ between them to their distance; the distortion $\delta(K)$ of a knot type $K$ is the infimum over all curves of type $K$. For the trefoil, the published lower bound is $\delta\ge5\pi/3$, valid for every non-trivial knot (a recent preprint claims $\delta>6.5321$), and the only upper bound we found in the literature is the value $7.16$ obtained by numerical simulation. We exhibit an explicit polygon with $192$ vertices, of knot type trefoil, whose distortion is at most $6.92$. The proof is computer-assisted: the knot type is certified by an explicit piecewise-linear path of embedded polygons starting from a polygon on a torus, and the distortion by exact rational arithmetic on every pair of edges. Two separately written verifiers are supplied.

math.MG — Metric Geometry

Improved and sharp bounds for illumination

Contributed by אלון שכטר

We prove improved ordinary illumination bounds and exact ordinary and fractional values for several classes of convex bodies. Every full-dimensional complex zonotope in $\C^2$ has illumination number five, except complex affine images of the bidisc, whose number is seven. For complex $1$-symmetric non-polydiscs in $\C^3$, an explicit construction gives twelve illuminating directions. For real $1$-symmetric non-cubes in $\R^n$, connected domination in the cube graph gives the upper bound $2^{n-1}+\gamma_c(Q_{n-1})+n-1 =2^{n-1}(1+(1+o(1))/n)$; an explicit family has illumination number $2^{n-1}$. We also give a product boundary measure proving fractional lower bounds $2^n$ for bodies between specified inner and outer bodies, and attain these bounds along Minkowski interpolations with a positive inner summand. Finally, the least size $U_n$ of a single family illuminating every normalized complex $1$-unconditional body satisfies $U_n\sim3^{n+1}/\sqrt{\pi n}$.

math.MG — Metric Geometry

Illumination of complex 1-unconditional convex bodies

Contributed by אלון שכטר

We study ordinary and fractional illumination of complex $1$-unconditional convex bodies. In every complex dimension $n$ we give an explicit ordinary bound asymptotic to $3^{n+1}/\sqrt{\pi n}$, and prove $\ill^*(K)\le b_*^n$ and $\ill(K)=O(n\log(n+1)b_*^n)$, where $b_*=2.4805534697\ldots$ is defined by an explicit scalar equation. For normalized non-polydiscs in dimensions two, three and four, the sharp ordinary bounds are $6$, $14$ and $30$. More generally, a full coordinate $(n-1)$-polydisc gives the sharp bound $2^{n+1}-2$ in every dimension $n\ge2$. In dimension two we prove the fractional bound $4$ and classify equality by the two maximal supporting coefficients at the coordinate axes. If every extreme point has all coordinates nonzero, we obtain the fractional bound $2^n$, with equality exactly for the polydisc, and a quantitative strict bound otherwise. Appendices give the exact sizes of simultaneous illuminating families in dimensions one through five, the exact finite verification used in dimension four, and a numerical comparison of the general estimates through dimension $100$.

math.NT — Number Theory

On Problem 5 of Vitorino and Vukusic: $\sup_{k\ge 0} B(2^k\alpha)\ge 12$

Contributed by David Niedbala Giraudin

For an irrational $\alpha$ let $B(\alpha)=\limsup_n a_n(\alpha)$ be the upper limit of its partial quotients. Vitorino and Vukusic proved $\sup_{k\ge 0}B(2^k\alpha)\ge 5$ for every irrational $\alpha$ and asked, as their Problem 5, for an improvement. We prove $\sup_{k\ge 0}B(2^k\alpha)\ge 12$. The proof is a finite computation organised around an invariant $N(C)$: the least number of doubling levels at which the recurrent kernel of a synchronous product of Raney transducers becomes a finite union of pure cycles. We show that $N(C)$ is intrinsic, that $N(C)<\infty$ for every $C$ would imply the $2$-adic Littlewood conjecture, and that $N(C)=\infty$ is equivalent to a rigidity failure; we compute $N(C)=1,1,3,5,5,8,15,21,29,30,31$ for $C=1,\dots,11$. In the language of Badziahin's set LMad this settles a level-bounded form of his Problem B for $C\le 11$. The cycles which survive are ideal classes of discriminant $D\equiv 1 \pmod 8$ whose whole $\mathfrak{p}_2$-orbit has partial quotients at most $C$; we list them, and check the lists by an independent arithmetic route, in both directions for $D\le 10^5$. The result concerns the upper limit $B$ only. On the side of the supremum $M$ it gives $\sup_k M(2^k\alpha)\ge 12$, which is weaker than what is known: Vitorino and Vukusic prove $15$, and Badziahin's bound $1/25$ yields $24$. Two self-contained Python files, without external dependency, reproduce the table and the lists of discriminants; the table up to $C=8$ takes about fourteen seconds.

math.GT — Geometric Topology

Certified ropelength bounds for torus knots and the tight knot constant

Contributed by David Niedbala Giraudin

The tight knot constant $\alpha_0$ is the largest constant with $L(K)\ge\alpha_0\,\mathrm{Cr}(K)^{3/4}$ for every knot $K$, where $L$ is the ropelength (unit-radius rope) and $\mathrm{Cr}$ the crossing number. We exhibit an explicit $C^{1,1}$ embedding of the torus knot $T(19,24)$, with $\mathrm{Cr}=432$, of thickness at least $1$ and length at most $939.984$. Hence $\alpha_0\le 9.9200$. The published bound is $10.76$. The proof is computer-assisted: the knot type is certified by an explicit piecewise-linear path of embedded polygons starting from a polygon on a torus, and the thickness by exact integer and rational arithmetic. Two separately written verifiers are supplied, together with certificates for $T(14,17)$ and $T(11,14)$.

math.MG — Metric Geometry

Illumination of several families of complex convex bodies

Contributed by אלון שכטר

We determine ordinary and fractional illumination numbers for complex Hanner bodies and for bodies obtained as convex hulls of coordinate polydiscs indexed by the maximal independent sets of a graph. The formulas are sums over coordinate supports. For Hanner bodies we classify all equality cases in the ordinary and fractional dimension bounds, including the sharp ordinary bound for non-polydiscs. For graph bodies, fractional equality $2^n$ occurs exactly when the graph is a disjoint union of isolated vertices and edges; a vertex of degree at least two gives the strict bound $3\cdot2^{n-2}$. We also prove the fractional bound $2^n$ for bodies defined by nondecreasing submodular constraints on coordinate moduli. For finite systems of positive weighted inequalities satisfying an explicit condition on tight sets, we obtain the same fractional bound and an ordinary bound of $(2+o(1))n\log n\,2^n$, together with an explicit estimate in every dimension. This includes bounded systems on laminar families with arbitrary positive coefficients. We conclude with a conjectural classification of fractional equality for normalized complex $1$-unconditional bodies.

math.RT — Representation Theory

Equality of the Mirković–Vilonen and theta bases

Contributed by Qiyue Tang

We prove Kamnitzer's ICM 2022 conjecture that the Mirković–Vilonen and theta bases coincide.

math.CO — Combinatorics

Coincidences of $q$-analogues of Markoff numbers outside the known families: an answer to a question of Labbé, Lapointe and Steiner

Contributed by David Niedbala Giraudin

Labbé, Lapointe and Steiner proved that $w\mapsto\mu_q(w)_{12}$ is injective on Christoffel words, gave two families of identities showing that it is not injective on $\{\mathtt{0},\mathtt{1}\}^*$, and asked whether every coincidence $\mu_q(x)_{12}=\mu_q(y)_{12}$ comes from these two families. We show that it does not. The smallest counterexamples have length $24$; by an exhaustive computation, among words of length at most $32$ there are exactly $50$ classes of coincidences not explained by the two families, $10$ of length $24$ and $40$ of length $30$. Each of them contains an instance of one of a few new identities, which we prove for a generic $2\times2$ matrix in place of the matrix of a factor of the word. One of these identities belongs to an infinite family, indexed by $k\equiv1\pmod 4$, which we prove for every such $k$.

cs.CC — Computational Complexity

A Sublinear Treewidth Exponent for Binary Constraint Satisfaction

Contributed by Troy Tri Nguyen

We give a deterministic algorithm for binary constraint satisfaction whose exponent in the input size is sublinear in treewidth on a class of graphs with unbounded treewidth. There are a class G of graphs of bounded maximum degree, a constant c, and a function f such that one algorithm correctly decides every binary CSP instance and, on instances I with primal graph G ∈ G, runs in time f(G) · ||I||^(c · tw(G) / sqrt(log tw(G))) . This contradicts the literal formulation of Conjecture 1.4 of Marx (2010). The algorithm follows from a domain-uniform counting bound: for every fixed maximum degree ∆, complete-table in- stances with k variables and domain size D ≥ 2 can be counted in time LO∆ (1)DO∆ (k/√log k) for sufficiently large k. The constants are independent of D and of the constraint relations. We obtain this bound from the tensor-network recursion of Claude (Anthropic), and use attributed graph foundations from OpenAI to construct the required expander family. Support pruning connects the counting routine to sparse tuple-list inputs without expanding the original domain. The algo- rithm, its native finite-alphabet machine implementation, and the conjecture counterexample are formalized in Lean 4.

math.OA — Operator Algebras

Noncommutative Cauchy Bound and Noncommutative Montel Bound for Roots of Polynomials

Contributed by K. MAHESH KRISHNA

In 1829, Cauchy derived an upper bound for every root of a complex polynomial using the maximum of the absolute values of the coefficients. In 1931, Montel derived an upper bound using the sum of the absolute values of the coefficients. We derive noncommutative versions of the Cauchy and Montel bounds.

math.NT — Number Theory

Rational limits in Markov–Euclid asymptotics

Contributed by Qiyue Tang

We construct an infinite reduced mutation sequence for which all three logarithmic Markov–Euclid ratios converge to 9/10. This disproves the irrationality conjecture of Zhichao Chen and Zelin Jia (Conjecture 6.12).

math.QA — Quantum Algebra

On the maximality conjecture for quantum permutation groups

Contributed by Teo Banica

A well-known conjecture, central in quantum permutation group theory, and going back to around 2006, states that $S_N\subset S_N^+$ is maximal. We revisit here this conjecture, and related questions, 20 years after, with 2026 knowledge and tools.

math.NT — Number Theory

On a Family of Counterexamples to Richard Guy's Mex Sequence Problem

Contributed by Shuta Iwai

We give a family of counterexamples to Richard Guy's problem E27 with initial terms of length five adding onto the recent example of length eight. Additionally, we prove that for length one, one always has a ultimately periodic sequence. Lastly we pose a problem regarding the notion of a zth mex sequence.

math-ph — Mathematical Physics

Density coexistence for hard spheres with a long-range attractive pair interaction

Contributed by Qidong He

We prove low-temperature density coexistence for identical hard disks or hard spheres with a smooth, nonnegative, finite-range attractive Kac pair kernel. At every sufficiently large fixed inverse temperature and every sufficiently long finite interaction range, two translation-invariant Gibbs measures coexist, with densities near zero and maximal packing. The pressure is therefore nondifferentiable in chemical potential. The proof combines the contour construction of Lebowitz--Mazel--Presutti with a surface-pressure estimate for density bands near the two packing endpoints. A packing bound controls the fine density profile; centering the interaction makes its residual boundary contribution quadratic in the band width. Compatible near-packed configurations and conditional entropy then control the hard-core boundary contribution. These estimates allow the dense restricted ensemble to retain positional order and long-range correlations.

math.QA — Quantum Algebra

Haagerup vertex operator algebras at central charge eight

Contributed by Cédric ARHANCET

Evans and Gannon showed that a positive vertex operator algebra of central charge eight whose representation category is the Drinfeld center $Z(\cal H_3)$ of the Haagerup fusion category has one of two character vectors, with six or nineteen currents. Call such an algebra, if strongly rational, a Haagerup algebra. We prove that a Haagerup algebra $U$ has six currents, that $U_1 \cong \mathfrak{sl}_2 \oplus \mathfrak{sl}_2$ embeds into $V_{E_6 \oplus A_2}$ in one of two explicit ways, with levels $(1,39)$ or $(3,13)$, and we determine the lowest weight spaces of its modules and $U_2$ as $U_1$-modules. For the levels $(3,13)$, $U_2$ is one of four Galois conjugate subspaces. Together with $U_1$ it generates the unitary $C_2$-cofinite algebra $S$, which is isomorphic to the algebra constructed by Jiao. For the levels $(1,39)$, $U$ contains the explicit algebra of Riesen. The associated lattice of $U$ is that of the form $x^2+xy+10y^2$, resp.\ $3x^2+3xy+4y^2$, the two classes of the principal genus of discriminant $-39$. Exchanging them in the decomposition of $U$ over its Heisenberg commutant gives a Haagerup algebra $U^\natural$ of the other levels, and $U \mapsto U^\natural$ is a bijection between the isomorphism classes of the two cases. The classification gives exactly two isomorphism classes, forming a single hyperbolic genus, as soon as either explicit candidate is known to be a Haagerup algebra. This property is proved for $S$ in [ArhIII], where the twin $S^\natural$ is also identified, with the algebra of Huang and Riesen, using the results of Dong, Jiao and Xu [DJX26]. The explicit algebras of Jiao and Riesen have rational forms. The proofs combine exact trace relations given by vector-valued modular forms, exact computations in a model of $V_{E_6 \oplus A_2}$, and genera of vertex operator algebras.

Author comments: 75 pages

math.FA — Functional Analysis

Wigner–Yanase–Dyson Metrics Have Monotone Curvature

Contributed by Suvrit Sra

Gibilisco and Isola conjectured that the scalar curvature of the Wigner–Yanase–Dyson metric with parameter $p$ on the invertible density matrices increases under mixing for all $p$ in some interval $(1,1+\varepsilon)$; the case $p\downarrow1$ is Petz's conjecture for the Kubo–Mori metric (resolved in a companion paper). We prove that for $p\in(1-\varepsilon,1+\varepsilon)$, with $\varepsilon>0$ independent of the dimension, the scalar curvature is a concave function of the eigenvalues and hence increases under mixing. By Dittmann's formula it suffices to prove that one symmetric function $H_p$ of three positive variables is concave, which we do by showing that $-1/H_p$ is a Stieltjes transform along every line in the positive cone, uniformly in $p$ near $1$. The value of $\varepsilon$ is not explicit.

math.CA — Classical Analysis and ODEs

A sharp Brégman–Minc bound for permanents

Contributed by Suvrit Sra

We prove a sharp Brégman–Minc bound for permanents of nonnegative matrices that was conjectured by Samorodnitsky and Gurvits. Specifically, for $\phi(x)=\Gamma(1+1/x)^{-x}$, with $\phi(0)=0$, applied entrywise, we show that every nonnegative row-stochastic square matrix $A$ satisfies $\operatorname{per}\phi[A]\le1$, which is the Brégman–Minc inequality when each row of $A$ is uniform on its support. The crux of our proof is a scalar inequality on the simplex, which we prove using a carefully chosen yet elementary integral representation. We note brief applications to the entropy of a random permutation and to sharp bounds for rectangular permanents and matchings.

math.PR — Probability

The Macdonald $(q,t)$-permanent II: Probability laws and their regions of validity

Contributed by Suvrit Sra

The Macdonald $(q,t)$-permanent replaces the constant cycle weight of the Vere-Jones $\alpha$-permanent by $(1-t^k)/(1-q^k)$ for a cycle of length $k$. We determine which of the probability laws built on the $\alpha$-permanent survive this deformation, and where. The scalar gamma-type law survives on the whole parameter square, but the matrix law, a compound Wishart matrix, is more fragile. We prove that it exists in every dimension exactly when $t\in\{0,q,q^2,\ldots\}$, and in dimension two exactly when $t\le q$. We conjecture that in each fixed dimension the matrix law exists exactly on the region that we show to be necessary, a $q$-analogue of Gindikin's theorem.

math.CO — Combinatorics

The Macdonald $(q,t)$-permanent I: Algebra and positivity

Contributed by Suvrit Sra

We introduce the Macdonald $(q,t)$-permanent, a two-parameter cycle-weighted permanent that contains the usual permanent and determinant, as well as the $\alpha$-permanent of Vere-Jones, as degenerate cases. We prove a MacMahon-type master theorem for its generating function, and determine when it induces a valid kernel on tuples of vectors. Nonnegativity of the $(q,t)$-permanent itself on semidefinite matrices turns out to be subtler: for $t=q^\alpha$ nonnegativity holds exactly at the positive Wallach exponents; for a fixed $q$ we determine the region in rank two and for higher ranks under an additional condition, while for small $q$ positivity can even reach beyond the kernel region. Whether positivity forces $t\in\{0,q,q^2,\ldots\}$ remains an intriguing open problem.

math.FA — Functional Analysis

Weak positivity of Schur forms under decomposable curvature

Contributed by Suvrit Sra

Griffiths's 1969 conjecture predicts positivity of Schur forms for Griffiths-positive vector bundles. Finski reduced its weak form to a sign question for the double mixed discriminant $\Phi_n(H)$ of a positive linear map. Du's rank-nine counterexample shows that $\Phi_n(H)\ge0$ fails in general. We prove $\Phi_n(H)\ge0$ for every decomposable map $H$, a sum of a completely positive and a completely copositive map, using a quaternionic Gaussian determinant representation. Combined with Finski's incidence formula, this inequality gives weak nonnegativity of every Schur form under decomposable curvature, without splitting assumptions. Beyond the decomposable cone, nonnegativity of $\Phi_k(H_{V,U})$ on every square compression is equivalent to $\Phi_n(H+S)\ge0$ for every entanglement-breaking map $S$. This class contains explicit indecomposable maps, but is nonconvex in every size $n\ge9$. We prove the nonconvexity by expressing Du's map as an average of four members, and the obstruction persists when all compression inequalities are strict.

math.RA — Rings and Algebras

A Koteljanskii inequality for permanents

Contributed by Suvrit Sra

We prove a permanental analogue of Koteljanskii's inequality. If $A$ is an inverse $M$-matrix that becomes symmetric after a positive diagonal similarity, then $\mathrm{per}(A_{S\cup T})\,\mathrm{per}(A_{S\cap T})\ge\mathrm{per}(A_S)\,\mathrm{per}(A_T)$ for all $S,T\subseteq[n]$, where $A_S$ is the principal submatrix indexed by $S$. The proof expresses permanents as moments of a complex Gaussian vector and uses Ginibre's correlation inequality. Finally, an explicit counterexample shows that symmetry cannot be dropped.

math.FA — Functional Analysis

The $p$-Bures–Wasserstein distance via $3\times3$ block matrices (with application to $p$-Bures–Wasserstein barycenters)

Contributed by Suvrit Sra

We study the $p$-Bures–Wasserstein distance $D_p(A,B)=(\mathrm{tr}\,A^p+\mathrm{tr}\,B^p-2\|A^{1/2}B^{1/2}\|_p^p)^{1/2}$ on positive semidefinite matrices; it equals the Bures–Wasserstein distance for $p=1$, and the Hilbert–Schmidt distance for $p=2$. We prove that $D_p$ is a metric for every $1\le p\le2$, by proving a stronger 2014 conjecture of Lin and van den Driessche on $3\times3$ block matrices. Our main tool is a partition inequality for Schatten norms, proved by complex interpolation. We also characterize Fréchet means for $D_p$ by a fixed-point equation and prove geometric convergence of the associated iteration.

math.CO — Combinatorics

On majorization inequalities for normalized Jack and Macdonald polynomials

Contributed by Suvrit Sra

We study the inequality $P_\lambda(x;q,t)/P_\lambda(1^n;q,t)\geq P_\mu(x;q,t)/P_\mu(1^n;q,t)$ for all $x\in\mathbb{R}_{>0}^n$ whenever the partition $\lambda$ majorizes $\mu$; Sra suggested it in 2015, and Chen, Khare and Sahi stated it as a conjecture in 2025. We prove the Jack case for every $n$ from Sun's positive Gelfand–Tsetlin integral and the Dixon–Anderson recursion. For Macdonald polynomials we prove the case $n=2$, the covers $(d)\succeq(d-1,1)$ for every $n$ when $t\leq q$, and, by exact certificates, the case $n=3$ with $|\lambda|\leq 8$. The HCIZ mechanism behind the Jack proof has no direct finite-$q$ analogue: at two explicit pairs $(q,t)$, no positive representation of the normalized ratio has a kernel multiplicative in $\lambda$, and normalizing at the principal specialization $t^\delta$ instead fails the inequality.

math.CO — Combinatorics

The $q$-Foulkes problem III: On Foulkes' conjecture with a graded inner factor

Contributed by Suvrit Sra

We show that surjectivity of the canonical map $\mathrm{Sym}^b(\mathrm{Sym}^aV)\to\mathrm{Sym}^a(\mathrm{Sym}^bV)$ survives grading of the inner factor. Consequently $h_b[H_a]-H_a[h_b]$ is Schur positive whenever the canonical map is surjective, where $H_a$ is the (modified) Hall--Littlewood polynomial; the same holds for (modified) Macdonald polynomials. We prove this surjectivity for $b\ge(a-1)^2$, improving the known bound $a(a-1)$. For Bergeron's $q$-analog of Foulkes' conjecture that lets both factors be graded, we prove nonnegativity of every Schur coefficient for certain special cases. We also study the full conjecture under an expansion hypothesis on an explicit space of maps.

math.CO — Combinatorics

The $q$-Foulkes problem II: Schur positivity with high probability

Contributed by Suvrit Sra

Bergeron's $q$-analog of Foulkes' conjecture asserts that $(H_b[H_a]-H_a[H_b])/(1-q)$ is Schur positive for $a\le b$, where $H_n$ is the (modified) Hall--Littlewood polynomial. We view this conjecture probabilistically: on each irreducible representation of $S_{ab}$, the two plethysms induce degree distributions, and positivity means that the first is stochastically smaller than the second. We prove that, for a random (Plancherel) irreducible representation and a degree sampled from its graded multiplicity, the corresponding coefficient is negative with probability at most $C\exp(-c(ab)^{1/3})$, uniformly in $a\le b$. Consequently, the $q$-Foulkes conjecture holds after deleting a fraction at most $C\exp(-c(ab)^{1/3})$ of each of the two graded modules. Additionally, we show that exact conditional means and variances yield complete positivity for explicit width and height families.

math.CO — Combinatorics

The $q$-Foulkes problem I: A fusion inequality for Macdonald plethysms

Contributed by Suvrit Sra

We prove that the plethysm $H_m[\widetilde H_\mu]$ of a Macdonald polynomial by the Hall–Littlewood polynomial $H_m$ lies above the Macdonald polynomial $\widetilde H_{m\mu}$ obtained by fusing $m$ copies of $\widetilde H_\mu$. More specifically, the quotient $(H_m[\widetilde H_\mu]-\widetilde H_{m\mu})/(1-q^m)$ is Schur positive with coefficients in $\mathbb N[q,t]$, and it is the character of an explicit module that we call the fusion defect. For one-row shapes the inequality places both $q$-Foulkes plethysms $H_b[H_a]$ and $H_a[H_b]$ above the same polynomial $H_{ab}$, and comparing their two defects we obtain a distorted version of Bergeron's $q$-analog of Foulkes' conjecture, in which the subtracted plethysm is shifted by $q^{a(a-1)\lfloor b/2\rfloor}$.

math.CA — Classical Analysis and ODEs

Sharp order-four sign regularity of a Jacobi theta kernel

Contributed by Suvrit Sra

We prove that the Hankel kernel $\Phi(\sqrt{x+y})$, where $\Phi$ is the Jacobi theta kernel whose cosine transform is the Riemann $\Xi$-function, is strictly reverse-regular through order four. The order is sharp: an interval enclosure of the fifth Hankel–Wronskian at $t=10^{-2}$ shows failure at order five. The third-order condition is Csordas's conjecture that $s'^2-ss''$ is strictly log-concave for $s(t)=\Phi(\sqrt t)$. The proof reduces the kernel minors to Wronskian signs and establishes the signs by exact polynomial estimates for $t\ge1$ and interval arithmetic on $[0,1]$. The fourth-order sign also yields three positive iterates of $s\mapsto s'^2-ss''$, the triple Turán inequalities for the normalized Taylor coefficients of $\Xi$, and quartic integral inequalities for $\Xi$.

math.CA — Classical Analysis and ODEs

Monotonicity of Turán-ratios of symmetric polynomials

Contributed by Suvrit Sra

We study the Turán-ratio $S_{\lambda,i}(x)^2/(S_{\lambda,i-1}(x)S_{\lambda,i+1}(x))$, where $S_{\lambda,i}(x):=s_\lambda(1^{[n-i]},x^{[i]})$, and $s_\lambda$ is the Schur polynomial in $n$ variables (indexed by partition $\lambda$) with $i$ arguments equal to $x$ and $n-i$ of them equal to $1$. Specifically, we prove that this Turán-ratio is strictly increasing for $x>1$ and $1\le i<n$ (unless all $n$ padded parts of $\lambda$ are equal, then it is identically one). For one-row shapes, our result answers a question of Ait-Haddou about complete homogeneous symmetric polynomials. For this case, we also offer a second proof through the zeros of Meixner polynomials that gives an explicit positive formula for the logarithmic derivative. We conclude with a Turán-ratio conjecture for Macdonald polynomials.

math.OC — Optimization and Control

Lower bounds for the quantum permanent

Contributed by Suvrit Sra

Griffiths's 1969 conjecture on the positivity of Schur forms reduces to the positivity of the quantum permanent $\Phi_n$ on positive bistochastic maps. This conjecture was refuted by Du with a map for which $\Phi_9<0$. Therefore, we ask: how negative can $\Phi_n$ be? We prove a lower bound $\Phi_n\ge-3n/(3n^2+20n+103)$ (for $n\ge 8$), and an asymptotically stronger bound $\Phi_n\ge-1/n+(3e^2-7-o(1))\log n/n^2$. The key to our bounds is the orbital polynomial of the map, which turns $\Phi_n$ into a linear function of a distribution on partitions and positivity into linear constraints. We also construct explicit maps for $n=8, 9$ that improve on Du's example. Finally, if $d_*\in\{4,\ldots,8\}$ is the first dimension for which $\Phi_n<0$, the minimum $\mu_n$ of $\Phi_n$ over positive bistochastic maps satisfies $-\mu_n=\Omega(n^{-d_*})$; we conjecture the matching order $-\mu_n=O(n^{-d_*})$.

math.PR — Probability

Beyond Small Balls: A Determinant Inequality for Functions of Matrix Gaussian Series

Contributed by Suvrit Sra

We prove a functional extension of Gaussian operator-norm small-ball inequalities. Let $X=\sum_{i=1}^d g_iB_i$, where $g\sim N(0,I_d)$, the matrices $B_i\in\mathbb{R}^{d\times d}$ are symmetric, and $\sum_iB_i^2\preceq I_d$. For every even log-concave $f\colon\mathbb{R}\to[0,\infty)$, we obtain $\mathbb{E}\det f(X)\ge(\mathbb{E} f(\kappa G))^d$ at dilation $\kappa=13/5$, where $G$ is standard Gaussian. The proof reduces each function to one small-ball estimate, obtained from matrix-weighted Poincaré inequalities, and controls the rest by averages over the spectrum. We give an equivalent entropy formulation. An explicit counterexample shows that no universal dilation below $\sqrt{\pi/2}$ can work; we conjecture that this value is optimal.

math.FA — Functional Analysis

A Proof of Petz's Conjecture on the Scalar Curvature of the Kubo–Mori Metric

Contributed by Suvrit Sra

Petz (1994) conjectured that the scalar curvature of the Kubo–Mori metric on invertible density matrices increases as the state becomes more mixed. Dittmann (2000) reduced the conjecture to the concavity of an explicit symmetric function $h$ on $(0,\infty)^3$. We prove that $h$ is strictly concave, and hence Petz's conjecture. The proof writes $h=-1/(8M)$ with $M$ positive and homogeneous of degree one, and its central idea is to show that $t\mapsto 1/M(X+tY)$ is a Stieltjes function for all $X,Y>0$; Cauchy–Schwarz and homogeneity then give concavity of $h$ on $(0,\infty)^3$.

math.NT — Number Theory

A 7/8 zero-free half-plane for angular Hecke L-functions over the Eisenstein field

Contributed by Troy Winarski

We reconstruct the argument recorded in a frozen Lean development establishing nonvanishing of the unit-compatible angular Hecke $L$-functions over $\mathbb{Q}(\sqrt{-3})$ in the strict half-plane $\Re(s)>7/8$, apart from the genuine principal pole. The family includes every admissible integer angular order, all nonzero finite ideal moduli, primitive and imprimitive presentations, and real unitary norm twists. This strengthens the earlier $15/16$ result for the same angular family. The proof compares two estimates for a single normalized physical probe: a low bound with exponent $3/16+\omega$ and an approximation by a reciprocal Mellin signal with error exponent $\beta_n-11/16-\sigma$, where $\beta_n$ is the supremum of the real parts of zeros within a finite angular cap. Crucially, the positive saving $\sigma$ is chosen before the target character. Mellin continuation converts these estimates into a strict improvement of the zero supremum. Induction on angular order, together with conjugation, establishes nonvanishing for the full family. The repaired public proof closure passed a clean compilation of 8,213 Lean modules and fresh native Lean kernel replay. An independent AI-assisted audit recorded five ACCEPT verdicts for the repaired package, following the identification and correction of a reproducibility defect in the original archive. We document the mathematical argument, formal verification evidence, and remaining limitations. This manuscript is an expository reconstruction, not independent human mathematical certification. We describe the formally verified proof as a *xenoproof* and the result provisionally as a *xenotheorem*, distinguishing machine-checkable correctness from established human mathematical understanding.