All works

math.NT — Number Theory

Irrationality of $\chi_2(1/q)$, $\mathrm{Ti}_2(1/q)$ and $\chi_2(2/q)$

Contributed by Shawn Krisman

Let $\chi_2(z)=\sum_{n\ge0}z^{2n+1}/(2n+1)^2$ be Legendre's chi function and $\mathrm{Ti}_2(z)=\sum_{n\ge0}(-1)^nz^{2n+1}/(2n+1)^2$ the inverse tangent integral. We prove three theorems: $\chi_2(1/q)$ is irrational for every integer $q\ge3$ with $q\ne4$; $\mathrm{Ti}_2(1/q)$ is irrational for every integer $q\ge5$; and $\chi_2(2/q)$ is irrational for every odd integer $q\ge175$. All three come from one construction: the Hankel determinants of a moment functional with half-integer poles and geometric weight $w$, read at $w=q^{-2}$, $w=-q^{-2}$ and $w=4q^{-2}$. Their denominators are bounded prime by prime, their size by a discrete log-gas argument against explicit dual certificates, and their nonvanishing by a reduction modulo an auxiliary prime. The three cases differ at the prime $2$, where each needs its own lemma, in the certificates, and in how the nonvanishing lemma is transferred. For large $q$ the three statements follow from the linear independence criteria of David, Hirata-Kohno and Kawashima; for the first two, a 1990 criterion of Hata, which we could not consult, also applies from a threshold unknown to us. Below these ranges the results are new as far as we know. All three theorems are formalised in Lean 4 with Mathlib and checked by the Lean kernel, using only Lean's three standard axioms.

math.NT — Number Theory

Regularity criteria and an effective bound for irregular extensions of Euler triples

Contributed by Sinisa Slijepcevic, Andrej Dujella

An Euler triple is a Diophantine triple $\{a,b,c\}$ with $a<b<c=a+b+2\sqrt{ab+1}$. It extends to the regular Diophantine quadruple $\{a,b,c,d_+\}$, and it is conjectured that it has no other extension $d>c$. We prove that an irregular extension forces $c<4\cdot10^{48}$, against $c<10^{98}$ in recent work of He and Togb\'e. The proof squeezes the index $m$ of the extension, which locates a common term of two Pell sequences, between a floor from congruences, $c<8m^4$, valid for every irregular extension, and ceilings from linear forms in logarithms. The same squeeze, with other bounds, gives three regularity criteria: by the size of $b/a$, which confines irregular extensions to $1+\frac1{1538}<b/a<3355^2$; by pair coordinates, through the hypergeometric method; and by coprime divisors of the index combinations, which settles every Euler triple with two elements in a ratio $j:k$ with $jk\le300$. As applications, the three-parameter family of Cipu, Fujita and Mignotte extends only regularly when $\nu A\ge2$ and $K\ge643$, and when $K=1$, $A\in\{2,3\}$ and $\nu\ge2$; the latter contains the Hoggatt--Bergum triples, for which this recovers Dujella's theorem for extensions $d>c$.

math.RT — Representation Theory

A proof of Lusztig's connectedness conjecture in finite type

Contributed by Qiyue Tang

We prove that every equal-parameter Kazhdan–Lusztig left cell in a finite Weyl group is connected by simple left multiplications.

Author comments: I am posting this manuscript to share the mathematical results themselves. In doing so, I do not seek to assert ownership of the results or claim priority. I welcome anyone interested in checking the arguments or preparing a finished account, and I believe the eventual scholarly credit should go to those who undertake that work.

math.RT — Representation Theory

A proof of Lusztig’s conjectures for extended affine Weyl groups

Contributed by Qiyue Tang

We prove Lusztig’s conjectures P1–P15 for extended affine Weyl groups with positive integer-valued weight functions.

Author comments: I am posting this manuscript to share the mathematical results themselves. In doing so, I do not seek to assert ownership of the results or claim priority. I welcome anyone interested in checking the arguments or preparing a finished account, and I believe the eventual scholarly credit should go to those who undertake that work.

math.CO — Combinatorics

An Upper Bound Exponent of 0.9999873 for Square-Difference-Free Sets

Contributed by Eric Naslund

Let $A\subseteq\{1,\ldots,N\}$ contain no two distinct elements whose difference is a nonzero square, and let $s(N)$ be the largest possible size of such a set. We prove \[ s(N)\le CN^{0.9999873} \] with an absolute constant $C$. A recent preprint of OpenAI proves $s(N)\le CN^{1-\eps}$ for some absolute $\eps>0$. Their proof utilizes a certain average of the Fourier transform over $(24!)^{161}$ points, and for the parameter choices displayed there we calculate that they obtained $\eps\approx \exp_{10}(-\exp_{10}(\exp_{10}(5782.77)))$, in the sense that $\log_{10}\log_{10}\log_{10}(1/\eps)\approx5782.77$. We follow the same overall strategy, with new work at almost every quantitative step. The largest gain is the use of $72$ slots (positions at which the Fourier transform is evaluated) in place of $(24!)^{161}$: the exponent is roughly a fixed constant divided by the number of slots, so this alone gains many orders of magnitude, but leaves far less room for loss elsewhere. Other savings come from the local construction at each prime. Small primes are handled jointly rather than one by one; and errors are measured in the actual half-distributions of the laws being compared, not bounded term by term against a product model. Where sums over the primes of a block of Weil-type error terms of size $O(p^{-1/2})$ would be too weak, we prove bounds of order $O(p^{-1})$ for the same error terms using \'etale cohomology over finite fields (Deligne's theorems and Katz--Sarnak monodromy for families of curves), the representation theory of $\mathrm{GL}_2(\mathbf F_p)$, exact counts at $p\equiv3\pmod4$, and a modular evaluation of Ahlgren and Ono. Dependence in our distributions is controlled by the correlation inequalities of Peyre and Kassabov. The remaining finitely many numerical inequalities are verified by exact computer arithmetic, listed in the appendices; all analytic steps are proved in the text. The companion repository \cite{NaslundUpperRepository2026} contains a partial Lean 4 formalization, totaling over 600,000 lines of lean. It establishes $s(N)\le C N^{0.99999}$, conditional on five named analytic inputs and explicit numerical hypotheses. This formalizes the most critical parts of the proof.

math.NT — Number Theory

The central charge of an arithmetic system: a seven-world framework

Contributed by Zeng Chunping

To an arithmetic system (prime elements, norms, weights, $L$-functions) we attach the central charge $C_\nu=\sum_{\mathfrak p}(\log N\mathfrak p)^\nu/(N\mathfrak p(N\mathfrak p^\nu-1))\cdot(\text{weight of }\mathrm{Frob}_{\mathfrak p})^{-1}$. Seven worlds share it: number fields, function fields, graphs, finite-type dynamics, knots, tropical geometry, and arithmetic dynamics. We record four universal laws with their exact status. (I) Index-set identity in all seven worlds (number fields to $10^{13}$, tropical skeletons to $10^{31}$). (II) Critical law $C_\nu\sim\Phi\cdot(\nu-1)!/\Delta^\nu$, with $\Delta$ the distance to the controlling singularity: a pole (Tate) or the natural boundary (cuspidal); $\Phi$ measures reachability ($\approx 0.7$ Tate, $\approx 0.07$ cusp). (III) Diagonal main-term theorem: $L$ entire iff cuspidal iff no diagonal main term iff vanishing connected/remainder ratio; item (v) (never-saturating spectral form factor) is at conjecture level, the bridge closed by a companion paper's recurrence-peak criterion. (IV) Law of forgetting in six formulations, unified: forgetting = increasing $\Delta$ = rising diagonal fraction = approaching the semicircle (free) = setting $\eta^{24}=1$. The core dichotomy (horizontal coordinate $\sigma=1+\kappa$ universal, amplitude fingerprinted) is checked on moments. The 24 network ($\Delta=q\eta^{24}$, Yau--Zaslow, $L(11a1,1)/\Omega=1/5$) and the Neron wild tables are included; everything carries a status label and a corrections register. We prove neither abc, Szpiro, RH, nor BSD.

math.CO — Combinatorics

The Sharp Exponential Constant for 6-Regular Circulant Matrices

Contributed by Yair Lavi

We prove that, for every integer $n\ge6$ and six-element set $S\subseteq\mathbb Z_n$, the binary circulant $C_n(S)=(1_{\{j-i\in S\}})_{i,j\in\mathbb Z_n}$ satisfies $\operatorname{per}C_n(S)\ge((3125/1296)e^{7/250000})^n$. For every integer $d\ge2$, we identify the optimal uniform base as $\inf_{n\ge d}\inf_{|S|=d}\operatorname{per}C_n(S)^{1/n}=\lim_{q\to\infty}\operatorname{per}C_{q^{d-1}}(\{0,1,q,\ldots,q^{d-2}\})^{1/q^{d-1}}$, where $S\subseteq\mathbb Z_n$ and $q$ runs through integers at least $2$. In degree six this base is at most $163812873663198^{1/35}$, while the lower bound strictly exceeds the sharp unrestricted base $3125/1296$. This resolves the surviving circulant lower-bound question in Minc's Problem 10; its proposed full upper bound fails even for circulants.

Author comments: The supplement cited in the text (supplement-v7.tar.gz) is in the Zenodo record https://doi.org/10.5281/zenodo.23044484.

math.CO — Combinatorics

A Conjecture on the Permanent of Positive Semidefinite Matrices

Contributed by Yair Lavi

We state the following conjecture of the author and prove it for $n\leq5$. Let $A=(a_{ij})$ be a complex positive semidefinite $n\times n$ matrix with $n\geq1$, and let $\chi_\lambda$ be an irreducible character of $S_n$ of degree $d_\lambda$. For $\sigma\in S_n$, put $a_\sigma=\prod_{i=1}^n a_{i,\sigma(i)}$. The conjecture asserts that $\sum_{\sigma,\tau\in S_n}\chi_\lambda(\sigma\tau)a_\sigma a_\tau \leq d_\lambda(\operatorname{per}A)^2.$ We show that Soules' permanent-on-top conjecture implies this inequality, and that it in turn implies both Lieb's immanant dominance and Chollet's permanent inequality. We also settle the previously unresolved order-four case of Soules' conjecture.

Author comments: The supplement cited in the text (exact certificates, order-four proof inputs and checking programs; ZIP, 176.5 MB) is in the Zenodo record https://doi.org/10.5281/zenodo.22976216.

math.NT — Number Theory

A √3 growth criterion for weighted reciprocal series

Contributed by twinheterodox

We prove that $\sum_{n\ge1}1/(a_n^2a_{n+1})$ is irrational for every strictly increasing sequence of positive integers satisfying $\limsup_{n\to\infty}\log a_n/(\sqrt{3})^n=\infty$. This lowers the sufficient growth base from $2$ to $\sqrt{3}$ in the criterion of Barreto, Kang, Kim, Kovač and Zhang (Bull. Lond. Math. Soc. 58 (2026), e70457, Theorem 3) for weights $(2,1)$. Irregular growth is allowed. The result also extends to nondecreasing sequences and positive integer numerators subject to polynomial size bounds.

Author comments: 6 pages. Includes a general criterion with positive integer numerators. Source and build instructions: https://github.com/twinheterodox/weighted-reciprocal-growth

math.AG — Algebraic Geometry

A proof of the Hausel–Rodriguez-Villegas conjecture

Contributed by Qiyue Tang

We prove the Hausel–Rodriguez-Villegas conjecture on the mixed Hodge polynomials of primitive-twisted $\mathrm{GL}_n$ character varieties.

Author comments: I am posting this manuscript to share the mathematical results themselves. In doing so, I do not seek to assert ownership of the results or claim priority. I welcome anyone interested in checking the arguments or preparing a finished account, and I believe the eventual scholarly credit should go to those who undertake that work.

math.GR — Group Theory

A proof of Brauer's $k(B)$-conjecture at the prime two

Contributed by Qiyue Tang

We prove that every $2$-block of a finite group has at most as many ordinary irreducible characters as the order of its defect groups.

Author comments: I am posting this manuscript to share the mathematical results themselves. In doing so, I do not seek to assert ownership of the results or claim priority. I welcome anyone interested in checking the arguments or preparing a finished account, and I believe the eventual scholarly credit should go to those who undertake that work.

math-ph — Mathematical Physics

Global twistor sources for analytic Einstein spacetimes

Contributed by Lior Isthmus

We globalize the local analytic projective twistor correspondence for the gravitational googly problem, retaining its convergent nonlinear treatment of both gravitational chiralities. For each fixed Einstein coefficient, the correspondence identifies independently defined atlases of two analytic Čech classes of weights $2$ and $-6$, structured operator sources on null monodromy presentations, and holomorphic oriented spin Einstein four-manifolds. Its inverse comparisons preserve the original source operations, Einstein scale, spin structure and complete point family, together with calibrated modules on parametrized incidence. For analytic Lorentz developments with a compact marked Cauchy surface, germs along the entire real development give the corresponding equivalence, and the maximal analytic Cauchy development determines a terminal source germ. A Nariai example admits no Hausdorff contact carrier containing the full complex skies over all points of the real development. In its product complexification and every allowed point restriction containing the entire real development, every real point sky has nonclosed saturation, so no ordinary locally finitely generated module has exactly that saturated support. These obstructions explain the use of null monodromy and parametrized incidence in the global correspondence, with compact point objects recovered by central specialization.

Author comments: v1.0. The same manuscript and the computational supplement are available at https://doi.org/10.5281/zenodo.24186390 and https://github.com/lioristhmus/global-twistor-sources/releases/tag/v1.0.

math.CO — Combinatorics

An order-eleven counterexample to triangular realization of sign matrix permanents

Contributed by Weiming Zhang

We construct an $11\times11$ sign matrix with permanent $P=2\,029\,568$. An exhaustive enumeration shows that \(\lvert\per B\rvert\ne P\) for every sign matrix $B$ of order eleven whose entries below the diagonal are all $+1$. This gives a negative answer to a question of Ingram and Razborov.

math.NT — Number Theory

Landau's problems through ratios of $L$-functions

Contributed by Haozhe Gou, Victor Wang, Max Wenqiang Xu

Very recently, Lee announced a proof of the Twin Prime Conjecture assuming the Ratios Conjecture for the Riemann zeta function. We obtain conditional asymptotic formulas with power-saving errors for each of Landau's four prime problems. Similar results may be possible for other problems in analytic number theory.

Author comments: 36 pages

math.RT — Representation Theory

A law of large numbers for infinitesimal characters of real reductive Lie algebras

Contributed by Andrew Salmon

We prove a law of large numbers for the infinitesimal characters of real reductive Lie algebras. Let $\pi$ be a unitary representation of a connected Lie group with reductive Lie algebra. For product states on the tensor powers of $\pi$ whose sites have uniformly bounded moments, for instance identical sites, the infinitesimal character divided by the number of sites concentrates at the spectrum of the limiting mean moment, with probabilities of deviation $O(n^{-m})$ for every $m$. The generators may be unbounded and the group non-compact. For compact groups this recovers the laws of large numbers of Keyl and Werner and of Botero, Christandl and Vrana, which enter the asymptotic analysis of linear programming bounds for binary codes; spherical designs require the non-compact case. We give concrete examples arising from Howe dual pairs on bosonic Fock space.

Author comments: Comments welcome

math.NT — Number Theory

Joint marked moments and optimized zero-free half-planes for unitary Hecke L-functions

Contributed by Baiying Liu

In this paper, we improve the uniform zero-free bound for the Quasi-Riemann Hypothesis and obtain the half-plane LK (s,ω) ̸= 0 (Re s>B∗ = 0.8733458561067185768...) for every continuous unitary Hecke character ω over every number field K, with the possible norm-character pole permitted. We prove joint moment estimates for inverse and ordinary character sums and apply them to zero detection and the averaged Mellin numerator. The proof uses the earlier number-field theta and moment theorems and the unitary estimates of Hu and Gao. The same half-plane holds for finite-dimensional continuous unitary Weil representations, with pole orders equal to the multiplicities of their norm-character constituents. We determine B∗ asanalgebraicnumberofdegreenineandastheinfimumofthespecifiedpositivecomparison, allowing unequal ordinary moments and variable comparison lengths.

math.LO — Logic

Bounded Presentations and Normalization in Arithmetic Reconstruction

Contributed by Zhenyu Sun

We study bounded presentations of selected constructions in absolute anabelian geometry and Frobenioid theory. In bounded Zermelo set theory with Choice, we prove uniform recursion for a common state space and a uniformly supplied transition rule. Under explicit index bounds, membership-table codes place the states and their histories at the same finite type level. We construct coherent comparisons for representative quotients, fiberwise full skeletons, isofibrational base changes, and countable towers, and verify the lifting properties for model Frobenioids and angular presentations. For nonzero numerical observables transforming by positive scalars, compatible normalization exists exactly when the associated character is trivial on every isotropy group. A self-equivalence of the realified arithmetic Frobenioid over $\mathbb{Q}$ preserves degree and every object isomorphism class but does not lift to the unit-trivialized perfection. The structural results give a relative uniform interpretation theorem: the weak theory constructs the bounded presentations and comparison maps, and specified arithmetic recognition theorems identify the resulting structures.

Author comments: 25 pages.

math.OA — Operator Algebras

Higher invariants of complex Hadamard matrices

Contributed by Teo Banica

Associated to any complex Hadamard matrix $H\in M_N(\mathbb T)$ is a quantum permutation group $G\subset_uS_N^+$, the Hopf image of the representation associated to the magic matrix formed by the rank 1 projections $P_{ij}=Proj(H_i/H_j)$, and its invariants $c_k=\dim(Fix(u^{\otimes k}))$ are the dimensions of the relative commutants of the subfactor associated to $H$. We compute these invariants well beyond the levels $k=2,3$ previously known, up to $k=9$ in the best cases, by using Jones--Wenzl and annular decompositions. For the Di\c t\u a deformations of the Fourier matrices we compute $G$ for all values of the parameters, and we study the order 8 families $F_{8,5}$, in connection with free wreath products. For the matrices of small order with $c_2=2$, such as the Karlsson ones at generic parameters, we find $c_k=Cat_k$ at all the levels computed. Finally, via an integrality principle and character sums, we prove that the Paley II matrices, and more generally those coming from symmetric conference matrices, have infinite depth, as well as the Paley I matrices with $11\leq q<10^5$.

math.DS — Dynamical Systems

Convergence Criteria for Measure-Valued Population Dynamics with Mutation and Horizontal Transfer

Contributed by Zhenyu Sun

We study a measure-valued population equation on a compact trait space with positive baseline growth and competition, an additional mutation birth gain, and frequency-dependent horizontal transfer with an antisymmetric net kernel. We give sufficient conditions for convergence to a unique nonzero equilibrium. Separating total mass from trait frequencies yields positive mass bounds. Without mutation, maximal time-averaged fitness on the initial support tends to zero, and the endpoint measure in each joint subsequential limit is supported on the zero set of the limiting averaged fitness. Potential transfer and recipient-independent competition reduce the frequency dynamics to a normalized positive semigroup. For general transfer, a common-reference balance gives an entropy criterion allowing arbitrarily strong circulation. An adjoint-weighted entropy identity includes singular population components and the normalization correction in transfer. Mutation dissipates deviations in weighted composition, while a quadratic-form condition controls the remaining mass deviation; in finite dimensions, mutation can also compensate for negative directions of that form. A separate contraction criterion compares arbitrary trajectories using only coefficient bounds. Invariant resident laws yield invasion indices for recurrent environments. Examples exhibit bistability under complete reset mutation and periodic populations whose time averages are not equilibria. For rank-one mutation, we classify stationary frequencies, prove a critical lower bound ruling out exponential convergence from an initial point mass at a maximizer, and determine the condensate when the initial population assigns positive mass to the maximizing set.

Author comments: 46 pages, 1 table.

math.NT — Number Theory

Every even integer greater than 2 is a prime plus a product of at most 18 primes

Contributed by Wren Aquila

We prove that every even integer $N \ge 4$ can be written as $N = p + \eta$ with $p$ prime and $\eta \ge 2$ a product of at most $18$ primes, counted with multiplicity. The best previously stated unconditional value we have located is due to Johnston and Starichkova: $395$ in the latest version of their paper, which revised the $369$ of its first two arXiv versions. Assuming that no Dirichlet $L$-function vanishes in the half-plane $\operatorname{Re} s > 7/8$, the same holds with at most $14$ primes, and under the generalised Riemann hypothesis for Dirichlet $L$-functions with at most $9$. The improvement comes from counting rather than bounding: an explicit lower bound for the number of primes $p$ with $N - p$ free of small primes is played against explicit Hardy--Ramanujan-type upper bounds for the integers with many prime factors, both in the range below the sieve threshold and inside the sieved sets. These upper bounds weight by Chebyshev's identity $\log n = \sum_{d \mid n} \Lambda(d)$, use the exact power-series coefficients of a majorant of a finite Euler product and the Mertens density of rough primes, and pre-sieve modulo $1155$, so that the integers to be excluded inherit the residue classes of the primes that produce them. All numerical steps are certified in interval or ball arithmetic, except some discrete convolutions with nonnegative terms, which are evaluated in binary64 under an explicit rounding-error bound and were also recomputed independently in ball arithmetic.

math-ph — Mathematical Physics

Algebra-Valued Yang-Mills-Type Theories: Polynomial Obstructions and Holonomy Reductions

Contributed by Yuri Ximenes Lecca, Willian Goulart Gomes Velasco

We study algebra-valued Yang-Mills-type ($A$-YMT) theories over a smooth $n$-manifold $M$, where connections take values in a general (not necessarily associative, commutative, or Lie) $\mathfrak{m}$-graded algebra $(A, *)$ and descend to an associated algebra bundle $E_A = P \times_\rho A \to M$ over a principal $G$-bundle $\pi : P \to M$. We first derive the variational field equations, infinitesimal gauge transformations, and generalized Bianchi identities, showing that affine Ehresmann compatibility forces the vertical image of $\mathfrak{g}$ into the Lie nucleus $\mathcal{N}_{\mathrm{Lie}}(A)$ of the commutator $[x,y]_* = x*y - y*x$, and proving that non-linear gauge covariance and the differential Bianchi identity are obstructed precisely by the Jacobiator $\mathrm{Jac}_{[\cdot,\cdot]_*}$ (leaving principal $G$-gauge transformations universally anomaly-free). We then establish polynomial and geometric obstructions in three settings. First, we prove a vanishing theorem for internal-product Chern-Weil and self-dual $A$-YMT functionals in weak $(2,s)$-solvable algebras, invariant under $(Q,S)$-splitting extensions. Second, in reductive Cartan-Yang-Mills theories valued in $A = \mathbb{R}^k \oplus \mathfrak{h}$, we compare quartic-block against recursive binary-tree parenthesization, proving a strict dimensional cutoff at $n \ge 8$ for quartic blocks and an exact Jacobsthal-number cutoff ($n - 4 \notin \bigcup_{r \ge 3}\{2 J_r, J_{r+1}\}$, plus coframe-rank vanishing for $n \ge k + 5$) for binary trees. We also catalog the admissible Berger holonomy reductions for $4 \le n \le 7$, establishing an on-shell vanishing (torsion-free) versus off-shell survival (torsionful) dichotomy both in 4D for $U(2)$ and $SU(2)$ connections---via $e \curlywedge_\rtimes (e \curlywedge_\rtimes \Omega_\omega) = \frac{1}{2}(e \curlywedge_\rtimes e) \curlywedge_\rtimes \Omega_\omega$ and the first Bianchi identity---and in 7D for $G_2$-connections. Third, we prove an associator-quotient collapse and Jacobiator vertex-dichotomy theorem for odd-dimensional internal $A$-Chern-Simons theories, with applications to $A$-$BF$ theories, octonionic and reductive Yang-Mills-Higgs Proca mass spectra, and reductive dualities.

math.RT — Representation Theory

Class-two Lie algebras and quadratic-dual current constructions

Contributed by Ammar Husain

For a reductive Lie algebra $\s$, a finite-dimensional $\s$-module $R$ and an $\s$-submodule $K\subseteq\bigwedge^2R$, the class-two radical $R\oplus(\bigwedge^2R/K)$ carries an $\s$-action. For semisimple $\s$ the semidirect products $\s\ltimes(R\oplus(\bigwedge^2R/K))$ are, up to isomorphism, all Lie algebras whose radical has class at most two. We study the isomorphisms that fix the acting algebra pointwise, including their central mixing and shear terms. On the locus where central mixing vanishes, the resulting groupoid admits a description by invariant Grassmannians and equivariant linear groups. We express the wild matrix-pair construction as a functor into class-two Lie algebras and give its equivariant square in $\s$-modules. We work out single-isotypic slices, signed multiplicity-free tensor squares, and explicit low-dimensional multiplicity-space examples. The acting algebra may be zero and its central action need not be semisimple. Finally, we study pairs of Lie brackets on one object whose mixed composites vanish. Tensoring them with algebras carrying two independent commutative associative products, for instance pointwise multiplication and convolution exchanged by Fourier transform, produces further Lie algebras. For these pairs, in representations of $\s$, we define the groupoid of objects that are isomorphic to their bracket exchange $\sigma X$; we give the structure of the switches $X\to\sigma X$, the obstruction to an involutive switch, and its indecomposable objects in small dimension for semisimple $\s$.

Author comments: The choices of structures are on the author. So them being maximally unhelpful is due to them. The structures can be too weak or too strong to be of any use and that is retained as the author's responsibility.

math.NT — Number Theory

Verification of the Markov Uniqueness Conjecture for the First One Trillion Markov Values: Meridians on the Markov Surface and Farey Addresses

Contributed by Elleard Felix Webster Heffern

The Markov unicity problem asks whether the largest entry of a positive integer solution determines the other two. We organize the triples into families called Meridians, with compact addresses and explicit evaluation rules. The construction starts with catenary profiles: a fixed host sets the curve, and two heights one unit apart solve the Markov equation with that host. Selected positions give integer triples. Reversing geometric descent gives an exact indexing of these triples. Remainder patterns, growth bounds, corridor paths, and Fibonacci-matrix and snake-graph arguments then restrict possible collisions. The Meridian description of starting scales and logarithmic spacing guided the organization of a large verification into magnitude bands. The implementation uses Farey addresses and certified matrix bounds to select each band; its correctness does not require the Meridian collision theorems. Production finished on 10 November 2026 at 03:54:13 UTC (9 November, 21:54:13 CST). The combined computation compared 1,101,388,071,945 configurations and certified uniqueness of the first one trillion Markov values in numerical order. The trillionth value has 1,021,609 decimal digits. All 12 repeated 64-bit residue groups were resolved by exact arithmetic, with no equal maxima. On eight workers, retained active band processing took 23 h 57 min 23.146 s, including the inherited 500-billion certificate; the maximum recorded temporary key-file footprint was 1,065,664,392 bytes (0.992 GiB). This elapsed processing time is distinct from summed CPU time and the calendar span. Both final replays passed. A separate computer-checked certificate covers all 5,067,077 Meridians of degree at most 10,000, at every integer phase. The general unicity problem remains open.

Author comments: Version 22 (12 November 2026), 44 pages: revised title and abstract, including the 1,021,609-digit trillionth value; added Arthur Baragar's 1996 benchmark (M < 10^140). The proofs, figures and verification evidence are unchanged. Chronology (distinct announcements and deposits): 21 October 2026, 15:18:25 UTC: X announcement of an earlier search exceeding one billion values: https://x.com/webystar/status/2112926487091282350 23 October 2026, 14:54:57.771 UTC: LinkedIn announcement of the earlier 9.3-billion search in 24 minutes: https://www.linkedin.com/posts/elleard-felix-webster-heffern-joseph-by-confirmation-2953a3104_markovnumbers-activity-7519411048115003392-4NaJ 2 November 2026: 50-billion verification and degree-10,000 computational supplements: https://zenodo.org/records/23874075 9 November 2026: selected full decimal expansions first deposited: https://zenodo.org/records/24093257 10 November 2026: v21 trillion-verification deposit (publication-date field: 9 November): https://zenodo.org/records/24096109 12 November 2026: corrected v22 manuscript; Zenodo DOI reserved (deposit pending): https://doi.org/10.5281/zenodo.24167894 12 November 2026: ResearchGate full-text mirroring began: https://www.researchgate.net/publication/417066856_Meridians_on_the_Markov_Surface_Growth_Structure_and_Verification_of_One_Trillion_Markov_Values_Using_Farey_Addresses The earlier X and LinkedIn announcements are not announcements of the trillion result. The source releases preserve the original verification timestamps and separate software licenses. The general unicity conjecture remains open.

math.NT — Number Theory

Cross-Energy of Dirichlet Partial Sums. Part 33: Optimised mollifiers and the bound $\Lambda\le0.174$ for the de Bruijn--Newman constant

Contributed by Pedro Caceres

The Polymath project reduced upper bounds for the de Bruijn--Newman constant $\Lambda$ to three verifiable hypotheses: the Riemann hypothesis up to height $X/2$, a zero-free region for $H_{t_0}$ to the right of $X$, and a barrier at $X$. With the verification of the Riemann hypothesis up to $3\cdot10^{12}$ by Platt and Trudgian, this gave $\Lambda\le0.2$. The bottleneck is the zero-free region, which Polymath established with Euler-product mollifiers. We replace the Euler product by a mollifier $\sum_{d\le D}\lambda_d d^{-s_*}$ whose weights are chosen by solving an $\ell^1$-regression problem, so that any choice of weights still yields a valid lower bound. At the same height $X\approx6\cdot10^{12}$ and $t_0=0.155$, $y_0=0.2$, the lower bound for the mollified sum improves from $-0.12$ (our Lemma~\ref{lem:moll} evaluated with the Euler mollifier on the primes up to $13$) to $+0.05$ with $D=120$; with $D=240$ it is still $+0.04$ at $t_0=0.154$. We cover all $x\ge X$ by a short chunking argument together with a tail estimate, verify the barrier with Polymath's Taylor scheme, and obtain \[ \Lambda\le 0.174 . \] The certificates are computer-assisted. They use Arb ball arithmetic for all transcendental functions and outward-rounded floating-point arithmetic for the large sums. The code, the weights and the logs are included, and the certificate was reproduced on a second platform. We also report that at this height the weights we found with $D=240$ do not reach $0.173$, and we compare with Polymath's table of conditional results. We do not prove the Riemann hypothesis.

Author comments: Revised 10 Nov 2026 after external examination (see CHANGES_v2.md). 20 pages, 3 figures. The ancillary scripts are those of the original part (ancillary zip in the part folder). Part 33 of the book-series Cross-Energy of Dirichlet Partial Sums (47 parts, Parts 0-46); formerly Part 13. Renumbered 2 Nov 2026; mathematics unchanged. The paper states once, in its discussion, that it does not prove the Riemann hypothesis.

math.AG — Algebraic Geometry

The Shapiro–Shapiro conjecture for ordered positive flags

Contributed by Qiyue Tang

We prove the Shapiro–Shapiro conjecture for ordered totally positive flags in Grassmannians.

Author comments: I am posting this manuscript to share the mathematical results themselves. In doing so, I do not seek to assert ownership of the results or claim priority. I welcome anyone interested in checking the arguments or preparing a finished account, and I believe the eventual scholarly credit should go to those who undertake that work.

math.RT — Representation Theory

A proof of the scheme-theoretic Humphreys conjectures

Contributed by Qiyue Tang

We prove the relative and traditional scheme-theoretic Humphreys conjectures for connected reductive groups over algebraically closed fields of positive characteristic exceeding the Coxeter number of every simple factor. We also prove that the self-extension support scheme of every finite-dimensional tilting module is reduced.

Author comments: I am posting this manuscript to share the mathematical results themselves. In doing so, I do not seek to assert ownership of the results or claim priority. I welcome anyone interested in checking the arguments or preparing a finished account, and I believe the eventual scholarly credit should go to those who undertake that work.

math.PR — Probability

The Wasserstein cost of the Girsanov Theorem

Contributed by Simon Coste

This is an expository note on the asymptotics of the Wasserstein distance between the empirical measure of infinite-dimensional random paths and their common distribution. We present the classical proof for these asymptotics in the case of the Wiener measure, unifying quantization arguments from Dereich and coauthors or Boissard and Le Gouic, and we adapt it for It\^o diffusions sampled through the Girsanov theorem. More precisely, let $\mathbb{Q}$ be the law of the solution of a stochastic differential equation with a bounded Lipschitz drift. Let $W_1,\ldots,W_n$ be independent Brownian paths, and let $Z_i$ be their Girsanov weights for $\mathbb{Q}$. The reweighted empirical measure $$\widehat\mu_n=\frac{\sum_{i=1}^n Z_i\delta_{W_i}}{\sum_{i=1}^n Z_i}$$ approximates $\mathbb{Q}$ when $n$ is large. We prove that the Wasserstein cost $W_p^p(\widehat\mu_n,\mathbb{Q})$ is of order $(\log n)^{-p/2}$, in expectation and almost surely, for every fixed $p\geqslant1$. In the Brownian case where $Z_i=1$ these results show that $\sqrt{\log n}W_p(\widehat\mu_n,\mathbb{Q})$ is asymptotically sandwiched between $\sqrt{2\lambda_d}$ and $4\sqrt{2\lambda_d}$, where $\lambda_d$ is the first eigenvalue of the Laplacian operator in the unit ball. We provide numerical experiments to estimate the (conjectured) limiting constant in dimension $d=1$.

Author comments: 16 pages. Full Lean formalization publicly available at gitlab.com/SimonCoste/wasserstein

math.RT — Representation Theory

Domino insertion and cells in type B

Contributed by Qiyue Tang

We prove Conjectures A, A$^{+}$, and B of Bonnafé, Geck, Iancu, and Lam on Kazhdan–Lusztig cells in type $B$.

Author comments: I am posting this manuscript to share the mathematical results themselves. In doing so, I do not seek to assert ownership of the results or claim priority. I welcome anyone interested in checking the arguments or preparing a finished account, and I believe the eventual scholarly credit should go to those who undertake that work.

math.PR — Probability

Causal prediction of rare recovery responses and the optimal scale of observation time

Contributed by Man Huang

We determine the information and physical waiting needed to predict a rare intervention response with error smaller than its occurrence probability in a one-dimensional growth process with renewed Poisson interval catastrophes. Deletions of initial sources in macroscopic intervals define a mixed response, with all intervention arms sharing the same stationary past and future forcing. A fixed number of growing submacroscopic gaps determines its rare-event probability scale. We identify the response probability and amplitude, characterize when terminal extinction bits lose predictive information, and construct a gate that recovers this information from the recovery trajectory. For absolute and squared loss, a deterministic-age Bayes-risk expansion holds uniformly with additive error smaller than the rare-event probability. Our main waiting theorem covers every honest causal stopping rule, allowing the complete realized catastrophe history. Its average-time benchmark is the sum of each ordered gap length weighted by the extinction probabilities of preceding gaps. Prediction error smaller than the rare-event probability requires mean waiting divided by this benchmark to diverge; every prescribed diverging multiplier is achievable using actual recovery records. With gap scales $n^{2/5}$ and $n^{4/5}$ and integrated catastrophe tail $A_x\sim C/x$, deterministic observation requires age larger than the $n^{4/5}$ scale, whereas a stopping rule achieves mean waiting $O(n^{2/5}\log n)$. The proofs preserve the rare-event scale through weighted response approximation, risk transfer over arbitrary information fields, and posterior lower bounds on rare history branches.

Author comments: Complete English manuscript with Technical Supplement Appendices A–E. Full English–Chinese edition, sources and original MIT-licensed diagnostic code: https://doi.org/10.5281/zenodo.24138927. Mathematical content version 2026-11-11.2. Companion preprints: https://doi.org/10.5281/zenodo.23911961 and https://doi.org/10.5281/zenodo.23912425.

math.AG — Algebraic Geometry

Stability conditions on twisted derived categories I: Existence and full numerical support

Contributed by Zhirui Li

We prove that for every nonempty smooth projective variety $X$ over an arbitrary field and every Brauer class $\alpha$, the twisted derived category $\mathrm{D}^b(X,\alpha)$ admits a Bridgeland stability condition with full numerical support. Based on Chunyi Li's construction and its extension to arbitrary fields, we establish uniform phase control under tensoring by an Azumaya algebra and calibrate the resulting slicing to the induced central charge. We further show that the constructed stability condition is geometric and satisfies a mass-Hom bound, which persists throughout its connected component.

Author comments: 26 pages. Comments and suggestions are welcome.

math.NT — Number Theory

Automorphic realization of Langlands parameters for split groups over global function fields, III. Classical groups

Contributed by Boyang Hu

Let $G$ be a split semisimple group whose simple factors are of classical type, over a global function field $F$ of arbitrary positive characteristic. We prove that every semisimple elliptic Langlands parameter $\Gamma_F\to\widehat G(\overline{\mathbb Q}_\ell)$ is the parameter of a cusp form on $G$, in the sense of V. Lafforgue's excursion operators; we call such a form an automorphic realization of the parameter. Neither $F$ nor its field of constants is extended. We start from L. Lafforgue's theorem for general linear groups and use automorphic descent and the theta correspondence. The work is to prove that the descent does not vanish over $F$, to carry out the computations with root subgroups in small characteristic, in particular in characteristic two, and to identify the parameter itself and not only its unramified Hecke eigenvalues. This is the third of three papers. Together with the first, which contains the arithmetic foundations, and the second, which treats the exceptional types by a geometric method, it gives automorphic realizations of semisimple parameters on all split reductive groups, with a restriction on the characteristic for the exceptional types only.

Author comments: Part III of a series of three papers; Parts I (Arithmetic foundations and the main theorems) and II (The geometric construction) are deposited separately. The folder anc/ contains the Python programs described in the appendix on computer checks.

math.MG — Metric Geometry

Fixed points of the polar projection body operator and its square

Contributed by Sihan Chen

Let $TK=(\Pi K)^\circ$ be the polar projection body of a convex body $K$ in $\R^n$. In dimension $n\ge3$, we prove that $TK$ is a positive dilate of $K$ exactly when $K$ is an origin-centred ellipsoid, settling Petty's polar projection conjecture. The same classification holds under the weaker assumption that $T^2K$ is a positive dilate of $K$. The proof uses a distributional Laplacian identity to compare two integrals of central section volumes. Equality supplies contractive hyperplane projections with every kernel direction, and James's theorem gives ellipsoidal rigidity. The argument applies to arbitrary convex bodies and yields the translative constant volume conjecture of Horv\'ath and L\'angi.

math.CT — Category Theory

Isomorphism Loci and Coherence in Equivalence Extension

Contributed by Zhenyu Sun

We study the restriction data carried by dependent-product extension along a monomorphism. In a locally cartesian closed category, the extended map is an isomorphism after a change of base if and only if the input map is an isomorphism on the restricted base. In a topos, its isomorphism locus is therefore the universal quantification of the input locus. A terminal characterization gives coherent comparisons for base change and successive extension, together with formulas for intersections and unions. Under explicit homotopical, cartesian, and universe hypotheses, this construction gives equivalence extension and preserves the boundary identifications used to prove fibrancy of a universe. Graph comparisons and strict fillers of a fixed universe boundary are represented by trivially fibrant objects. A single realignment over the comparison object induces a weak equivalence between these parameter objects, and both admit extensions of every finite cubical boundary. A finite-groupoid example shows that graph comparisons cannot be selected naturally under all isomorphisms of unstructured input.

math.NT — Number Theory

Automorphic realization of Langlands parameters for split groups over global function fields, II. The geometric construction

Contributed by Boyang Hu

Let $G$ be a split simply connected semisimple group over a global function field $F$ whose characteristic does not divide the order of the Weyl group of $G$. We prove that every semisimple elliptic Langlands parameter $\Gamma_F\to\widehat G(\overline{\mathbb Q}_\ell)$ is the parameter of a cusp form on $G$, in the sense of V. Lafforgue's excursion operators; we call such a form an automorphic realization of the parameter. The parameter may be ramified at finitely many places and may have finite monodromy, and neither $F$ nor its field of constants is extended. The starting point is the geometric Langlands correspondence in characteristic zero. It provides a sheaf, and the function which the trace of Frobenius attaches to a nonzero sheaf can vanish identically. The main point of the construction is to arrange that the trace in question is a determinant $\det(1-\phi)$ with no zero factor, and that it stays nonzero under specialization to the given field. This is the second of three papers. Together with the first, which contains the arithmetic foundations, and the third, which treats classical groups in every characteristic, it gives automorphic realizations of semisimple parameters on all split reductive groups, with a restriction on the characteristic for the exceptional types only.

Author comments: Part II of a series of three papers; Parts I (Arithmetic foundations and the main theorems) and III (Classical groups) are deposited separately. The folder anc/ contains the Python programs described in the appendix on computer checks.

math.NT — Number Theory

Automorphic realization of Langlands parameters for split groups over global function fields, I. Arithmetic foundations and the main theorems

Contributed by Boyang Hu

Let $G$ be a split connected reductive group over a global function field $F$. V. Lafforgue decomposed the space of cusp forms on $G$ according to semisimple Langlands parameters. This is the first of three papers which prove a converse: every semisimple elliptic parameter is the parameter of a cusp form on $G$ over $F$ itself, which we call an automorphic realization of the parameter. This holds in every positive characteristic for classical groups, in characteristic greater than $3$ for $G_2$ and $F_4$, greater than $5$ for $E_6$, and greater than $7$ for $E_7$ and $E_8$. A parameter that is not elliptic is the parameter of an automorphic form which is not cuspidal, under the bound for the smallest Levi subgroup through which the parameter factors. The form has the prescribed eigenvalue for every excursion operator, and we determine for which lattices in the centre it can be taken invariant. The present paper contains the arithmetic part of the proof. It defines excursion operators on functions that are not assumed cuspidal, proves their compatibility with constant terms, shows that a function on which they act through an elliptic parameter is a cusp form, and passes from simply connected groups to isogenous groups and from Levi subgroups to the group. This reduces the theorems to the existence of a cusp form with a given elliptic parameter on a simply connected simple group. That is proved in the second paper by a geometric method, when the characteristic does not divide the order of the Weyl group, and in the third paper for classical groups, by descent from general linear groups.

Author comments: Part I of a series of three papers; Parts II (The geometric construction) and III (Classical groups) are deposited separately. The theorems of this part depend on the two existence theorems proved in Parts II and III.

math.DS — Dynamical Systems

Limit Cycles under Concave Per-Capita Growth

Contributed by Zhenyu Sun

We study predator--prey systems with a fixed globally linear functional response and a strictly concave prey per-capita growth rate. For every $N\ge1$ and $K\ge2$, we construct a growth polynomial of degree $2N+2\lceil K/2\rceil+2$ with exactly $N$ positive periodic orbits, all hyperbolic. The construction retains a chosen center $K$-jet and the low-density growth rate and carrying capacity of its limiting profile, subject to an explicit relation between the center height and curvature. A limiting Lyapunov function and return-map estimates at three amplitude scales exclude additional cycles. A dissipative extension gives exact global counts with prescribed limiting center energies. We also prove that profiles with at least $N$ hyperbolic cycles are $C^2$-dense in the admissible class, while preserving the entire germ at the maximum and both ecological endpoint values. The first nonzero canonical odd term determines the sharp differentiability threshold. Further results give parity restrictions for fixed focus jets, degree lower bounds, quantitative polynomial approximation under concavity and interpolation constraints, and amplitude and multiplicity interpolation by Abelian integrals. Finally, a single smooth strictly concave profile has infinitely many nested hyperbolic cycles converging to coexistence.

Author comments: 45 pages, 3 figures. Symbolic and numerical verification scripts and data are included in anc/.

math.OC — Optimization and Control

Quadratic Strictly Positive Real Factors of Stable Transfer Functions

Contributed by Jinlin Wang, Zhenyu Sun

We characterize factorizations $G=G_0\circ H$ in which $G$ and $G_0$ are stable, minimum-phase, biproper real-rational functions with simple poles and zeros and positive gains at infinity, and $H$ is a quadratic strictly positive real function. Classical rational decomposition identifies the quadratic right-factor classes with real Möbius involutions fixing $G$. Within each class, we parameterize all normalized representatives and eliminate the scaling parameter and the frequency variable. The resulting sign conditions involve univariate polynomials of degree at most four. They determine all feasible denominator intervals and all factorizations in the class; for real-algebraic input, they give an exact decision procedure. We also give a scaling that maximizes the smaller of the inner positive-real margin and the outer spectral margin for a fixed denominator. For quadratic input, existence is equivalent to strict positive realness of $G$ and to an explicit coefficient inequality. Examples exhibit a disconnected feasible parameter set, separate algebraic and analytic obstructions, and a quartic input with exactly three admissible classes.

math.CA — Classical Analysis and ODEs

Certified lower bounds for the first and the signed third autocorrelation constants

Contributed by Tejasvi Singh Tomar

Let $C_1$ be the largest constant such that $\sup_{|t|\le 1/2}\int f(t-x)f(x)\,dx\ge C_1\bigl(\int_{-1/4}^{1/4}f\bigr)^2$ for every nonnegative function $f$ on the real line. We prove $C_1\ge 1.337$, improving the bound $1.2937$ of Kim and Pilanci, by a spatial branch-and-bound over the masses of $f$ on equal bins. Consequently, for every large $g$, every $g$-Sidon set in $\{1,\dots,N\}$ has fewer than $1.2231\sqrt{gN}$ elements once $N$ is large enough in terms of $g$. For real-valued $f$ supported in $[-1/4,1/4]$, let $C_3$ and $C_3'$ be the analogous constants for the essential suprema of $|f*f|$ and of $f*f$; no lower bound better than the trivial bound $1$ was recorded for them. We prove $C_3\ge C_3'\ge 4/3$ with the argument principle. Both proofs are computer-assisted, and both theorems are also proved in Lean 4.

Author comments: Published on Zenodo: https://doi.org/10.5281/zenodo.23715655

cs.LO — Logic in Computer Science

A formal proof in Lean of the finiteness of planar five-body central configurations for all positive masses

Contributed by Tejasvi Singh Tomar

Smale's sixth problem asks whether, for every choice of positive masses, there are finitely many central configurations up to similarity. For five bodies in the plane, Albouy and Kaloshin proved finiteness outside a set of masses of codimension two. In a companion paper the author proved finiteness for every choice of five positive masses by their method, adding steps that cannot be checked by hand: exact computer algebra, rational certificates and a branch and bound. We formalize this theorem in Lean 4 with Mathlib without assuming the results of Albouy and Kaloshin that the informal proof uses. Instead we prove what the proof needs of their rules on limit diagrams, of their classification of these diagrams and of their mass conditions. We do so at places of complex algebraic curves, taken as valuations of the function field rather than points of a projective model. Almost every computation becomes data for a checker that we write in Lean, prove sound with respect to Mathlib's real and complex numbers, and run in the kernel (proof by reflection). The main theorem depends on about $680{,}000$ theorems proved in this way, and Mathlib's tactics prove the few other computations. The proof uses only Lean's three standard axioms, and two independent Lean kernels, nanoda and lean4lean, accept it.

Author comments: Published on Zenodo: https://doi.org/10.5281/zenodo.23723139

cs.LO — Logic in Computer Science

A Lean 4 formalization of the area law for three-dimensional U(1) lattice gauge theory with the Wilson action

Contributed by Tejasvi Singh Tomar

In three-dimensional $U(1)$ lattice gauge theory with the Wilson action, static charges are confined at every coupling: rectangular Wilson loops obey an area law, uniformly in the finite box. This was proved recently by the author in a separate paper; the proof runs to over three hundred pages with its supplement and adapts the Debye-screening analysis of Brydges and Federbush and the multiscale expansion of Göpfert and Mack. We describe a formalization of this proof in Lean 4 with Mathlib, which also covers the paper's corollaries and its appendix on comparing Wilson and Villain loops. The formal theorems take no hypotheses and use only the standard axioms of Lean. The proof became formalizable through explicit interfaces between its parts, written-out versions of the steps that the paper adapts from the literature or leaves out, and certified values for every numerical constant. The formalization also yielded 70 corrections and improvements to the paper, none of which changes its main results.

Author comments: Published on Zenodo: https://doi.org/10.5281/zenodo.23752572

cs.LO — Logic in Computer Science

Formal Verification of a Computer-Assisted Proof in Celestial Mechanics

Contributed by Tejasvi Singh Tomar

Albouy, Fu and Sun conjectured in 2008 that four positive masses admit, for each cyclic order of the bodies, exactly one convex central configuration up to similarity; the question is Problem 10 of a 2012 list of open problems on the $n$-body problem. We recently proved the conjecture by reducing it to an inequality for a trace at the real zeros of an equation in four variables; a branch and bound that starts from a grid of 136,448 boxes establishes the inequality. In this paper we describe a formalization of the whole proof in Lean 4 and Mathlib in which Lean's kernel checks the computation, using decide +kernel, so that no compiler has to be trusted. Our checker stores an interval as two natural numbers in a biased fixed-point format. The kernel can then hand all the arithmetic to GMP. Functions are enclosed over boxes by first-order slopes. An unverified search records the decisions for each box in one natural number, and the checker replays them. We write each formula once over a type class; evaluating it at a type that pairs a real number with an interval then proves the enclosure correct. Checking the 3418 certificate theorems takes 7.6 hours of processor time, and the independent type checker nanoda accepts the proof. A version rebuilt on one eighth of the boxes passed three type checkers in 2.9 hours on a public registry of formal proofs.

Author comments: Published on Zenodo: https://doi.org/10.5281/zenodo.23485952

cs.DS — Data Structures and Algorithms

Sorting networks on 13 inputs need at least 44 comparators

Contributed by Tejasvi Singh Tomar

Let $S(N)$ be the least number of comparators in a sorting network on $N$ inputs. We prove $S(13)\ge44$, one more than the previously best proved lower bound. We also prove $P(2,N)=\lceil \log_2 F(N)\rceil$ for $3\le N\le32$, where $P(2,N)$ is the least worst-case number of comparators met by the two largest inputs in a network that delivers them to fixed outputs and $F$ is an explicit function of Van Voorhis; the lower bound comes from a finite search based on the structure of the paths of these two values. Starting from Harder's $S(11)=35$ and $S(12)=39$, these values of $P(2,N)$, the two-value bound $S(N)\ge S(N-2)+P(2,N)$ and the one-value bound $S(N)\ge S(N-1)+\lceil \log_2 N\rceil$ give the twenty lower bounds on $S(N)$ for $13\le N\le32$ in Dobbelaere's table; these bounds rested on Van Voorhis's inequality $P(2,N)\ge\lceil \log_2 F(N)\rceil$, whose published proof is invalid. The bound $S(13)\ge44$ is also proved in Lean 4, with every step checked by Lean's kernel.

Author comments: Published on Zenodo: https://doi.org/10.5281/zenodo.23730883

math.MG — Metric Geometry

The 24-cell conjecture in the all-contact case and near D4

Contributed by Tejasvi Singh Tomar

The 24-cell conjecture states that every Voronoi cell of a packing of unit balls in $\mathbb{R}^4$ has volume at least $8$, the volume of the regular 24-cell. It implies that no packing of equal balls in $\mathbb{R}^4$ is denser than the lattice packing $D_4$. We prove the conjecture for two classes of cells; in each class, equality holds only for the $D_4$ root configuration. The first consists of the cells whose ball touches every ball centred at distance less than $2\sqrt2$ from its centre, and its proof uses a certificate at the second level of the Lasserre hierarchy. The second consists of the cells whose $24$ nearest centres lie in an explicit neighbourhood of the $D_4$ configuration, whatever the other centres are. For them the volume is at least $8$ plus a fixed multiple of the total outward displacement of the $24$ centres, which we prove by a quantitative rigidity estimate and a second-order expansion of the volume about $D_4$. Our proofs for both classes are computer-assisted and partly formalised in Lean.

Author comments: Published on Zenodo: https://doi.org/10.5281/zenodo.23720963

math.CO — Combinatorics

Subplanes of order 3 and maximal arcs in projective planes of order 12

Contributed by Tejasvi Singh Tomar

Whether a projective plane of order 12 exists is open. A subplane of order 3 of such a plane is a case of equality in Bruck's bound $n\ge m^2+m$; the 27 points on no line of the subplane then form a maximal arc of degree 3, and together with the subplane a maximal arc $K$ of degree 4. We show that every line of a subplane of order 3 contained in $K$ lies in at most two such subplanes, whereas in $\mathrm{PG}(3,3)$, which like $K$ has 40 points, a line lies in four planes; the proof uses Egan and Wanless's classification of triples of mutually orthogonal Latin squares of order 9. We also show that $\mathrm{AG}(3,3)$ is not the design of any maximal arc of degree 3 in a plane of order 12, and that a permutation of the points that maps a subplane of order 3 onto itself and each line meeting it onto a line cannot have prime order other than 3. The proofs are computer-assisted, and their SAT refutations are certified.

Author comments: Published on Zenodo: https://doi.org/10.5281/zenodo.23717956

cs.CC — Computational Complexity

Which Fine-Grained Lower Bounds Survive the Refutation of 3SUM and APSP?

Contributed by Tejasvi Singh Tomar

Alman and Vassilevska Williams have refuted the 3SUM and APSP hypotheses. Their algorithms rest on counting the triangles through prescribed edges of lopsided tripartite graphs, whose middle part has $n^{\varepsilon}$ vertices for a small $\varepsilon$. Majdoddin extended the method to every $\varepsilon<\ln4/(5\ln5)\approx0.172$. We determine which of the main conditional lower bounds derived from the refuted hypotheses survive. We also prove that the method extends to every $\varepsilon<0.17843$ when the prescribed edges are known in advance and there are at most $n^{2-\varepsilon/2}$ of them, which confirms Majdoddin's numerical estimate. Whether a bound survives depends mostly on the instances that its reduction produces. The reductions behind most bounds produce instances outside the range of the new algorithms: balanced graphs, or lopsided graphs with a larger middle part. Most of the set-disjointness, dynamic-graph and triangle bounds therefore follow from hypotheses that no known algorithm contradicts. These hypotheses, however, concern mainly such instances, and for several bounds the hypothesis is the bound itself, stated for these instances. Their evidence came largely from the refuted hypotheses. A few bounds are false, among them a lower bound of Goldstein, Lewenstein and Porat for Set Disjointness over small universes; the Strong SetDisjointness Conjecture also fails at small polynomial query times. Other bounds have lost their support, such as the Set Disjointness trade-offs of Kopelowitz, Pettie and Porat over small universes apart from constant query time. Finally, we collect the known obstacles to a larger closed-form range with a different base identity and use border-rank lower bounds to exclude several natural candidates.

Author comments: Published on Zenodo: https://doi.org/10.5281/zenodo.23716495

quant-ph — Quantum Physics

On the minimal dimension of range-criterion counterexamples to the PPT² conjecture

Contributed by Tejasvi Singh Tomar

The PPT$^2$ conjecture asserts that the composition of two completely positive and completely copositive (PPT) maps is entanglement breaking. A recently announced counterexample, with PPT maps $\mathrm{M}_{10}\to\mathrm{M}_6\to\mathrm{M}_{10}$, is certified by the range criterion: the range of the composite Choi matrix contains no nonzero product vector. We ask how small a counterexample certified in this way can be. For PPT maps $\Phi\colon\mathrm{M}_m\to\mathrm{M}_k$ and $\Psi\colon\mathrm{M}_k\to\mathrm{M}_n$ with $\Psi\circ\Phi\neq0$ and $\max(m,k,n)\le6$ we prove that the range of the Choi matrix of $\Psi\circ\Phi$ contains a nonzero product vector whenever $k\le5$ or $\min(m,n)\le5$. Hence the range criterion cannot certify a counterexample on $\mathrm{M}_4$ or $\mathrm{M}_5$, and among the types with all three dimensions at most six only $(6,6,6)$ remains open. Since the range is spanned by products of Kraus operators, the question is when such a span can avoid the matrices of rank one. The computer-assisted proof combines rank bounds for PPT maps with Chern class and excess intersection computations; its computed facts are checked in Lean 4, modulo classical, quoted or hand-proved inputs. The announced counterexample lifts through exterior powers a map on $\mathrm{M}_4$ built from a pencil of real matrices (a map of pencil type). When the map of pencil type loses rank transversally at finitely many points, its lifts through any Schur functor, exterior powers included, put product vectors into the kernel of the composite Choi matrix. When the composite is also nonzero, these product vectors form an unextendible product set only for exterior powers, and then one of the three dimensions is at least ten.

Author comments: Published on Zenodo: https://doi.org/10.5281/zenodo.23715651

quant-ph — Quantum Physics

The PPT² conjecture holds for spin-factor maps

Contributed by Tejasvi Singh Tomar

The PPT$^2$ conjecture asserts that the composition of two completely positive and completely copositive maps is entanglement breaking. A counterexample in dimension ten has recently been announced, and for maps on $\mathrm{M}_4$ the conjecture is open. Let $\mathcal{S}\subset\mathrm{M}_4$ be the real span of the identity $\gamma_0$ and five pairwise anticommuting Hermitian unitaries $\gamma_1,\dots,\gamma_5$. It is a spin factor whose positive cone is a Lorentz cone. We prove the conjecture for all compositions $\Psi\circ\Phi$ of completely positive maps $\Phi\colon\mathrm{M}_4\to\mathrm{M}_k$ and $\Psi\colon\mathrm{M}_k\to\mathrm{M}_4$, with $k$ arbitrary, such that the ranges of $\Psi$ and of the adjoint $\Phi^{*}$ lie in invertible congruences of the complex span of $\mathcal{S}$. Such maps are automatically completely copositive, and the class contains maps whose Choi matrices are entangled. A theorem of Hildebrand on separability for pairs of Lorentz cones reduces the claim to one trace inequality between the coefficient matrices of the two Choi matrices. We derive this inequality from the operator inequality $\sum_{j=1}^{5}F_jF_0^{-1}F_j\preceq F_0$, valid whenever $\sum_{\mu=0}^{5}F_\mu\otimes\gamma_\mu\succeq 0$ and $F_0\succ 0$, which in turn follows from the Kadison–Schwarz inequality. As a corollary, $\Psi\circ\Phi$ is entanglement breaking whenever $\Psi$ is such a map and $\Phi$ is a completely positive map with Kraus operators of rank at most two.

Author comments: Published on Zenodo: https://doi.org/10.5281/zenodo.23483252

math.PR — Probability

The L¹ Poincaré constant of the Hamming cube

Contributed by Tejasvi Singh Tomar

Let $C$ be the smallest constant such that $\mathbb{E}|f-\mathbb{E} f|\le C\,\mathbb{E}|\nabla f|$ for every real-valued function $f$ on every Hamming cube $\{-1,1\}^n$, where $|\nabla f|$ is the Euclidean length of $(D_1f,\dots,D_nf)$ and $D_jf(x)=(f(x)-f(x^{(j)}))/2$. Ben Efraim and Lust-Piquard proved $C\le\pi/2$, the Gaussian case gives $C\ge\sqrt{\pi/2}$, and Ivanisvili, Li, van Handel and Volberg asked whether $C=\sqrt{\pi/2}$. We prove that it is, with an explicit constant $c_n<\sqrt{\pi/2}$ in dimension $n$. The proof turns on an integral of first absolute moments of sums of independent biased random signs: we compute it exactly when the coefficients are nonnegative, and show, with a sampled Fejér kernel, that it can only grow when the coefficients are replaced by their absolute values. The equality $C=\sqrt{\pi/2}$ is also proved in Lean 4 with Mathlib.

Author comments: Published on Zenodo: https://doi.org/10.5281/zenodo.23715900

math.CO — Combinatorics

A thrackle on n vertices has fewer than 4n/3 edges

Contributed by Tejasvi Singh Tomar

A thrackle is a drawing of a graph in the plane in which every two edges meet exactly once, either at a common endpoint or at a proper crossing. Conway conjectured that a thrackle with $n$ vertices has at most $n$ edges. We prove that a thrackle with $n\ge4$ vertices has at most $\frac43(n-1)$ edges, improving the bound $\frac{18}{13}(n-1)$ obtained by Tóth and, independently, by Scoppetto. The proof combines Xu's discharging argument with two computer searches, by the method of Fulek and Pach, showing that two generalized dumbbells in the sense of Fulek and Pach are not thrackleable: $\mathrm{DB}(6,8,-1)$ is a $6$-cycle and an $8$-cycle sharing one edge; $\mathrm{DB}(6,10,-3)$ is a $6$-cycle and a $10$-cycle sharing a path of three edges. A formal proof in Lean 4 derives the bound from five properties of thrackles and checks the computational facts it uses; Section 6 states what this proof trusts.

Author comments: Published on Zenodo: https://doi.org/10.5281/zenodo.24065370

math.MG — Metric Geometry

A lower bound of 0.8366 for Lebesgue's universal covering problem

Contributed by Tejasvi Singh Tomar

A universal cover is a convex set in the plane that contains a congruent copy of every set of diameter at most $1$. Lebesgue asked in 1914 for a universal cover of least area; before this work, the infimum $\Lambda$ of the areas of universal covers was known to satisfy $0.83474\le\Lambda\le0.8441$. Lower bounds come from test sets: $\Lambda$ is bounded below by the least area of a convex hull of congruent copies of given sets of diameter at most $1$. The best earlier lower bounds used the disk, the Reuleaux triangle and the Reuleaux pentagon, but these three bodies are known to admit a placement with hull area below $0.83479$, so they cannot give a much better bound. Adding the regular Reuleaux heptagon and nonagon of width $1$, we improve the lower bound to $\Lambda\ge0.8366$. Without the nonagon, the other four bodies already give $\Lambda\ge0.836$. The proof is a computer-assisted branch and bound over the relative placements of the bodies. Its certificates are checked in interval arithmetic, and separate proofs of both bounds in Lean 4, with their own certificates, are checked entirely by Lean's kernel.

Author comments: Published on Zenodo: https://doi.org/10.5281/zenodo.23717911