All works

math.PR — Probability

Lower bounds for the Beardwood–Halton–Hammersley constant

Contributed by Tejasvi Singh Tomar

Let $\beta$ be the Beardwood–Halton–Hammersley constant: the shortest tour through $n$ independent uniform random points in the unit square has length asymptotic to $\beta\sqrt n$ almost surely. Dong and Cao proved $\beta\ge0.6421$ from an inequality $\beta\ge5/8+\sum_{k=3}^{K}C_k$, where $C_k$ is an integral over configurations of $k$ points in the plane that form a connected graph when points at distance at most $1$ are joined. For $C_3$, $C_4$ and $C_5$, sharper bounds come from a first-order expansion of the area of the union of the unit disks on boxes of configurations. For $C_6+\dots+C_{22}$, which Dong and Cao did not use, a lower bound comes from comparing each point only with the three points before it in the order of their first coordinates. We obtain $\beta\ge0.6496$, and $\beta\ge0.6508$ if the exponential function of the C library used to evaluate the bound for $C_6+\dots+C_{22}$ has a relative error below $10^{-9}$. The proofs are computer-assisted; the bound $0.6496$ is proved in Lean 4, with finite checks evaluated by compiled code, from the inequality of Dong and Cao and two statements about unions of disks that we prove but do not formalize.

Author comments: Published on Zenodo: https://doi.org/10.5281/zenodo.23716094

cs.IT — Information Theory

The automorphism group of an extremal [72,36,16] code is trivial

Contributed by Tejasvi Singh Tomar

Whether a binary doubly-even self-dual $[72,36,16]$ code exists is a question raised by Sloane in 1973. A series of papers, surveyed by Borello, shows that the automorphism group of such a code is trivial or isomorphic to $C_2$, $C_3$, $C_2\times C_2$ or $C_5$. We prove that such a code has no automorphism of order $2$, $3$ or $5$, and hence no nontrivial automorphism. Each such automorphism reduces the problem to a finite computer search. For order $2$ the search runs over symmetric bilinear forms on the $41$ self-dual $[36,18,8]$ codes. An automorphism of order $3$ leads to a search over labellings of the Hermitian self-dual $[24,12,8]$ codes over $\mathbb{F}_4$ by the extended Golay code. Order $5$ reduces to $7\times7$ unitary matrices over $\mathbb{F}_{16}$. Along the way we classify the Hermitian self-dual $[24,12,8]$ codes over $\mathbb{F}_4$: there are $26\,924\,931$ of them up to monomial equivalence.

Author comments: Published on Zenodo: https://doi.org/10.5281/zenodo.23789973

math.CO — Combinatorics

The Steiner ratio is at least 0.8564

Contributed by Tejasvi Singh Tomar

The Steiner ratio is the infimum, over finite sets of points in the Euclidean plane, of the length of a Steiner minimal tree divided by the length of a minimum spanning tree. Gilbert and Pollak conjectured in 1968 that it equals $\sqrt3/2\approx0.8660$, and the conjecture is open. We prove that it is at least $2141/2500=0.8564$, improving the bound $0.8559$ of Ke, Huang, Shu, He, Gai and Wang. Their method reduces the induction step of Du and Hwang's 1983 proof of the bound $0.8$ to finitely many inequalities on boxes in a space of local configurations of a Steiner minimal tree, which are checked by computer. These inequalities use estimates for the positions of terminals outside the local configuration. We prove all the geometric lemmas that our certificate uses, filling gaps in two proofs of Ke et al., and we add two such estimates that hold for more parameter values than the published ones. Within this method some new estimate is necessary: with the published estimates, no certificate of this kind reaches $0.856$, whatever the boxes. The computer-assisted part is also verified in Lean 4, by a proof whose checks all run in Lean's kernel; the rest of the argument enters the formal proof as hypotheses.

Author comments: Published on Zenodo: https://doi.org/10.5281/zenodo.23716009

math.DS — Dynamical Systems

The number of planar central configurations of four bodies

Contributed by Tejasvi Singh Tomar

For four positive masses in the plane we determine the number $N$ of central configurations, counted up to rotation, translation and scaling, in terms of one fold surface, whose certified enclosures decide $N$ for all masses outside a closed null set. Of these configurations, $12$ are collinear, by Moulton's theorem, and $6$ are convex, by the existence theorem of MacMillan and Bartky and the uniqueness proved in the author's preprint arXiv:2609.35632. Through Dziobek's relations, the concave configurations with inner body $i$ form the fibre of a proper mass map of degree $2$. Their number $c_i$, counted up to reflection, is $1$, $2$, $3$ or $4$, and we read it off from the position of the mass ratios relative to this surface. Altogether $N=18+2(c_1+c_2+c_3+c_4)$, an even number with $32\le N\le50$. Moreover $N=32$ exactly at Palmore's masses, and every even value in this range occurs. For four equal masses we recover Albouy's count $N=50$. The proof shows with interval certificates that the mass map is injective where its Jacobian is not positive, obtains its degree from the restricted problem, and analyses its folds, cusps and Palmore's umbilic. A formalization in Lean also proves these results.

Author comments: Published on Zenodo: https://doi.org/10.5281/zenodo.23256909

math.DS — Dynamical Systems

Smale's sixth problem for six bodies in the plane: finiteness of central configurations for generic masses, with a proof formalized in Lean

Contributed by Tejasvi Singh Tomar

Smale's sixth problem asks whether, for every choice of positive masses, the number of central configurations up to similarity is finite. We prove that it is finite for six bodies in the plane and all positive masses outside a proper algebraic subset of the mass space. This subset, which we describe only in part, contains every mass vector with three equal masses or with one mass equal to the sum of two others. Finiteness for all positive masses remains open. Infinitely many central configurations would give a curve of complex solutions; following Albouy and Kaloshin, we exclude such a curve using the valuations of the mutual distances at its points at infinity. The proof is computer-assisted and uses linear programs, Gröbner bases and interval arithmetic. The theorem is proved in the Lean proof assistant, whose kernel checks the computations as well.

Author comments: Published on Zenodo: https://doi.org/10.5281/zenodo.23720035

math.MG — Metric Geometry

The kissing numbers in dimensions five and seven are at most 43 and 132

Contributed by Tejasvi Singh Tomar

The kissing number $\tau_n$ is the largest number of non-overlapping unit balls in $\mathbb{R}^n$ that can touch a common unit ball. The best bounds so far were $40\le\tau_5\le44$ and $126\le\tau_7\le134$; both upper bounds come from high-precision floating-point computations of the three-point bound by Mittelmann and Vallentin. We prove $\tau_5\le 43$ and $\tau_7\le132$. Each proof is an exact rational solution of the second level of the Lasserre hierarchy for spherical codes at degrees $(d_1,d_2,\delta)=(14,20,20)$. We use the formulation of de Laat, Leijenhorst and de Muinck Keizer and check the solutions with their verification program. Each certificate is obtained by rounding a convex combination of two dual iterates of the interior-point method, one on each side of the target value. Exact certificates for the same hierarchy also give $A(4,\arccos\frac13)\le14$, $A(5,\arccos\frac13)\le22$ and $A(6,\arccos\frac13)\le34$ for spherical codes, improving the previous bounds $15$, $23$ and $35$; with a known code, $A(4,\arccos\frac13)=14$. The largest number of points at angle $\pi/3$ from a point of a kissing configuration in $\mathbb{R}^n$ equals $A(n-1,\arccos\frac13)$, so in $\mathbb{R}^5$ this number is $14$. We also prove the five bounds in the Lean proof assistant, assuming only the positivity of the zonal kernels given by the zonal polynomial files: once by checking the exact certificates with compiled code, and once, with every computation done by Lean's kernel, from the certificates rounded to $30$ decimal places.

Author comments: Published on Zenodo: https://doi.org/10.5281/zenodo.23992400

math.RA — Rings and Algebras

Local optimality of rank-47 schemes for 4×4 matrix multiplication over F2

Contributed by Tejasvi Singh Tomar

The tensor $\langle 4,4,4\rangle$ of $4\times4$ matrix multiplication has rank at most 47 over $\mathbb{F}_2$, and as it is not known whether 47 is optimal, we ask how far the known decompositions of rank 47 are from one of rank 46. A decomposition is locally optimal to depth $D$ if no $D$ or fewer of its rank-one terms can be replaced by fewer rank-one terms. We prove that five pairwise inequivalent decompositions of rank 47 over $\mathbb{F}_2$ are locally optimal to depth 18, as is every decomposition equivalent to one of them, so a decomposition of rank 46 over $\mathbb{F}_2$ shares at most 28 terms with any decomposition in these five classes. The five classes include the scheme of Fawzi et al. and every public decomposition of rank 47 over $\mathbb{F}_2$ known to us; one of them, found in our flip-graph searches, is new. The proof rests on a peeling lemma and, for four sets of terms that resist the other bounds, on a refinement of the substitution method, which we call substitution over subspaces with lifting. Local optimality to depth 18 is also proved in Lean 4.

Author comments: Published on Zenodo: https://doi.org/10.5281/zenodo.23721965

cs.DS — Data Structures and Algorithms

Exact sizes of networks that select the three or four largest of n elements

Contributed by Tejasvi Singh Tomar

We study $(t,n)$-selection networks, which move the $t$ largest of $n$ inputs to $t$ given lines in any order, and the least number $\hat U_t(n)$ of comparators in such a network. Alekseev showed that $\hat U_t(n)\ge(n-t)\lceil\lg(t+1)\rceil$. For $t=2^k\ge 4$ we raise this bound by $k-1$ for every $n>t$, and by $2k-5$ when $t\ge 32$; as far as we know, these are the first improvements at powers of two. The proofs follow the weights of the lines, and a pruning step of Yao extends the gain to every $t\ge 3$. For $t=3$, Yao's upper bound $2n-5+\lfloor\lg(n-3)\rfloor$ is exact for $4\le n\le 18$. For $t=4$ the new bound is $3n-11$ for all $n\ge 5$, and the value $3n-10$ that Knuth expected is exact for $6\le n\le 12$. Their lower bounds rest on exhaustive searches at a few sizes and on the fact that one more input costs at least two more comparators. Most of the results are also proved in Lean 4, some through compiled checks.

Author comments: Published on Zenodo: https://doi.org/10.5281/zenodo.23801189

cs.LO — Logic in Computer Science

Re-proving native_decide facts in Lean's kernel: a study of 16 public packages and a converter

Contributed by Tejasvi Singh Tomar

Lean's native_decide proves a decidable proposition by running compiled code and records the result as an axiom. Mathlib forbids such axioms, and lake check and Comparator in Lean 4.35 reject them, as do the independent proof checkers. NativeTrust re-proves the recorded facts in Lean's kernel, with only the standard axioms and without editing the projects that contain them. Where a definition blocks kernel evaluation and NativeTrust has a version proved equal to it (for example, a fuel-indexed form of well-founded recursion), the converter evaluates a copy that uses that version. It then proves the copy equal to the original, usually by congruence. With its default settings it converts 98.2% of 52,821 native facts in 16 public packages. The three largest packages hold 94% of these facts; outside them it converts 68.5%. Nearly every converted fact needed nothing beyond plain kernel evaluation. Two tools remove the native axioms that the kernel can prove within a budget: a single import that makes each call of the tactic try the kernel first, and a source rewriter.

Author comments: Published on Zenodo: https://doi.org/10.5281/zenodo.23818391

math.RA — Rings and Algebras

The rank of 4×4 matrix multiplication over F2 is at least 35

Contributed by Tejasvi Singh Tomar

We prove that the rank of the $4\times4$ matrix multiplication tensor over $\mathbb{F}_2$ is at least $35$; the previous lower bound was $34$. The bound follows by substitution from $\mathrm{R}_{\mathbb{F}_2}(\langle 3,3,4\rangle)\ge27$, which also gives $\mathrm{R}_{\mathbb{F}_2}(\langle 3,4,4\rangle)\ge31$. Each of the three bounds is one more than what follows from $\mathrm{R}_{\mathbb{F}_2}(\langle 3,3,3\rangle)=23$. A flattening argument reduces the bound for $\langle 3,3,4\rangle$ to lower bounds for two quotients of the tensor, by the span of a matrix of rank two and by the span of a matrix of rank three. These come from exhaustive searches over the possible patterns of a short decomposition, pruned by lower bounds for quotients by larger subspaces. The whole argument, including every step of the searches, is checked in the proof assistant Lean 4, with one compiled evaluation of a verified certificate checker.

Author comments: Published on Zenodo: https://doi.org/10.5281/zenodo.23749576

quant-ph — Quantum Physics

Certified lower bounds on the optimal-decoding threshold of the surface code

Contributed by Tejasvi Singh Tomar

Under independent bit flips, rigorous lower bounds on the threshold of the surface code from earlier work stay below $0.04$, while numerical estimates place the optimal threshold near $0.109$. By counting labelled failure paths with exact transfer-matrix certificates, we prove without sampling that every minimum-weight decoder has threshold at least $0.0465$ on the planar and rotated codes. We then give statistical certificates that, with confidence $1-10^{-9}$ each, the optimal threshold of the planar code is at least $0.103$ under bit flips and at least $0.02$ under phenomenological noise. Each statistical certificate bounds the failure probability of an auxiliary decoder that divides the code into blocks and corrects each block using a window around it; optimal decoding does at least as well, so the bound also holds for optimal decoding. A deterministic lemma shows that the auxiliary decoder fails only if bad blocks, in which the window corrections leave a defect or a long residual path, form a sparse hierarchy that joins the two rough boundaries. This gives an explicit bound once every block is bad with probability below a limit, about $4\times10^{-4}$ under bit flips, and a pre-registered Monte Carlo experiment bounds these probabilities. The confidence rests on a binomial model of the counts of bad blocks and on two tested properties of each decoding program. All these results are proved in the proof assistant Lean 4, the statistical ones with the binomial model and the two properties as hypotheses.

Author comments: Published on Zenodo: https://doi.org/10.5281/zenodo.24048739

quant-ph — Quantum Physics

Entanglement from the Diósi–Penrose Model: An Exact Criterion, and Why It Stays Small at Any Distance

Contributed by Marc Sperzel

Trillo and Navascués showed that the Diósi–Penrose model, in the Tilloy–Diósi form with the same smearing length $\sigma$ in force and noise, entangles two masses in superposition when they are closer than about $2.2\,\sigma$, and asked whether gravitationally induced entanglement can be observed beyond that distance. We answer for two rigid bodies whose branches are position eigenstates on finite sets, without kinetic term, for noise strength $C$. One kernel, the Coulomb energy of the smeared bodies, sets both the Newtonian coupling and the correlated noise, and the question becomes a single geometric comparison: a product state becomes entangled if and only if the largest canonical correlation between neutral charge patterns on the two branch sets exceeds $C/\sqrt{1+C^2}$. Below this threshold the evolution is a separable operation at all times. The published critical distances are two-branch cases of the criterion: for parallel two-branch superpositions the supremum is $2.21\,\sigma$ at $C=1$, and a straddling geometry reaches $2.74\,\sigma$. With many branches there is no critical distance. What limits the effect is the model's own dephasing, which sits in the same quadratic form. For bodies of effective size $\sigma_{\rm eff}$ ($\sigma$ for point masses) separated by a slab of width $d\gg\sigma$, the entangling rate at $C=1$ is at most $0.12\,\sigma_{\rm eff}/d$ times the single-body dephasing rate. The entanglement is therefore transient and, for well-separated branches, a fraction of order $\sigma_{\rm eff}/d$ of what unitary Newtonian gravity gives. Entanglement of the size unitary gravity predicts still excludes the model; the inference from any detected entanglement beyond a fixed multiple of $\sigma$ does not hold.

Author comments: Revised presentation: the idea is stated first, a contents page is added, and the numerical values are collected in an appendix. Results and title are unchanged.

math.NT — Number Theory

Euler's 1747 conjecture on sums of integers with restricted prime factors

Contributed by Will Howe, Grant Huffman, Jensen Kohlmeyer, Liam Kruer

We prove that every positive integer congruent to $2$ modulo $4$ is a sum of two positive integers whose prime factors are all congruent to $1$ modulo $4$, with $1$ admitted as a summand. Euler proposed this statement in a letter to Goldbach dated May 6, 1747. The proof counts representations by four squares and bounds those containing a common prime divisor congruent to $3$ modulo $4$ in either coordinate pair. An average over a nonsplit torus normalizer converts the quaternionic Hecke bound into an error independent of the sieving prime. Elementary estimates then give the explicit threshold $64^4$. An exact finite arithmetic certificate covers the remaining $4{,}194{,}304$ integers in the required residue class.

math-ph — Mathematical Physics

Osterwalder–Schrader axioms for scalar QED$_3$ in the Coulomb phase

Contributed by Mark Goldstein, Alfredo Guevara

King constructed the continuum, infinite-volume limit of three-dimensional scalar quantum electrodynamics, the $U(1)$ (abelian) Higgs model, along Neumann–Dirichlet cubes. For his state at fixed photon mass we prove the Osterwalder–Schrader axioms: regularity $\mathrm{E0}''$ with absolute constants, invariance under the full Euclidean group, reflection positivity about every plane, symmetry, and the measure with its ergodic decomposition, on whose components the cluster axiom holds in all its forms. Rotation invariance is proved by comparing King's cubes with periodic tori carrying a smooth mass wall on two lattices whose orientations differ by $\arctan(3/4)$. With real scalar sources, the joint field of the curvature and the renormalised Wick square satisfies, on a tube of small Wick amplitudes, the axioms: regularity, invariance under translations and the lattice point group, Euclidean invariance of the joint field outside an exceptional set of couplings, reflection positivity for Wick amplitudes below a threshold, symmetry, and the measure with the cluster axiom in its weak form; above the threshold the Euclidean clause is open. The curvature results hold on King's whole coupling range; the Wick results hold under a one-sided sign clause of the Wick leg, which we prove on explicit coupling regions. The limit of vanishing photon mass is treated in a subsequent paper.

Author comments: Companion paper: Osterwalder–Schrader axioms for scalar QED$_3$ in the Higgs phase (hexagon:2611.00321). Project: https://github.com/marikgoldstein/scalar-qed-existence

math.PR — Probability

Supersymmetric Analysis of a Random Band Matrix at Criticality, Volume I

Contributed by Patrick Lopatto

We study the $N \times N$ complex Hermitian Gaussian band matrix whose variance profile is the inverse of $I - W^2 \Delta$, where $\Delta$ is the Neumann Laplacian on $\{1, \dots, N\}$, in the critical regime $W^2 \asymp N$. We prove that, for energies $E$ in a neighborhood of zero, the unfolded eigenvalue process converges to a stationary point process, whose law depends only on the limit $X$ of $N/((4 - E^2) W^2)$ and is given by the partition functions of a supersymmetric sigma model on an interval. We also determine the macroscopic law of the mass of an eigenvector selected near a given energy. Its density is the normalized exponential of a Brownian motion with drift, under the law tilted by its total mass; the same law describes the eigenvectors of the critical one-dimensional random Schrödinger operator. Further, we compute the limits of the two-energy correlations of the local densities of states at all pairs of interior sites, of the mean eigenvector profile at the scale of single sites, and of $N$ times the mean inverse participation ratio, which is $2 + 2X/3$. As $X \to 0$ and $X \to \infty$, we recover the pair statistics of the GUE and of a Poisson process, and in the localized limit we obtain an explicit random shape for the eigenvectors. The number variance in a window of length $L$ grows like $\sqrt{XL}$ as $L \to \infty$ at every fixed $X$, and we describe the crossover to Poisson statistics in windows of length of order $X$. Several of these results confirm predictions that physicists obtained from the sigma model. The proof starts from the supersymmetric representation of averages of ratios of characteristic polynomials, which for this profile is a transfer operator on a one-dimensional chain. We show that the transfer chain converges to the heat semigroup of the sigma model, whose noncompact direction produces the Brownian motion in the eigenvector law.

math.PR — Probability

Supersymmetric Analysis of a Random Band Matrix at Criticality, Volume II

Contributed by Patrick Lopatto

We study the $N \times N$ complex Hermitian Gaussian band matrix whose variance profile is the inverse of $I - W^2 \Delta$, where $\Delta$ is the Neumann Laplacian on $\{1, \dots, N\}$, in the critical regime $W^2 \asymp N$. We prove that, for energies $E$ in a neighborhood of zero, the unfolded eigenvalue process converges to a stationary point process, whose law depends only on the limit $X$ of $N/((4 - E^2) W^2)$ and is given by the partition functions of a supersymmetric sigma model on an interval. We also determine the macroscopic law of the mass of an eigenvector selected near a given energy. Its density is the normalized exponential of a Brownian motion with drift, under the law tilted by its total mass; the same law describes the eigenvectors of the critical one-dimensional random Schrödinger operator. Further, we compute the limits of the two-energy correlations of the local densities of states at all pairs of interior sites, of the mean eigenvector profile at the scale of single sites, and of $N$ times the mean inverse participation ratio, which is $2 + 2X/3$. As $X \to 0$ and $X \to \infty$, we recover the pair statistics of the GUE and of a Poisson process, and in the localized limit we obtain an explicit random shape for the eigenvectors. The number variance in a window of length $L$ grows like $\sqrt{XL}$ as $L \to \infty$ at every fixed $X$, and we describe the crossover to Poisson statistics in windows of length of order $X$. Several of these results confirm predictions that physicists obtained from the sigma model. The proof starts from the supersymmetric representation of averages of ratios of characteristic polynomials, which for this profile is a transfer operator on a one-dimensional chain. We show that the transfer chain converges to the heat semigroup of the sigma model, whose noncompact direction produces the Brownian motion in the eigenvector law.

math.RT — Representation Theory

Lusztig's Fourier matrix conjecture for finite noncrystallographic Coxeter groups

Contributed by Qiyue Tang

We give a computer-assisted proof of Lusztig's Fourier matrix conjecture for all finite noncrystallographic Coxeter groups. For every two-sided cell, the normalized $S$-matrix of the Drinfeld center of its asymptotic Hecke category, equipped with the inherited pivotal structure, equals the corresponding Fourier matrix in the modular convention. In particular, we establish the exact $74\times74$ matrix identity for the middle cell of type $H_4$.

Author comments: 37 pages. Supplementary computational data and verification materials are available at https://github.com/tqyqiqi1509/lusztig-fourier-conjecture 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.NT — Number Theory

Divisors of Mersenne numbers under doubling

Contributed by Sunay Venkat

This paper studies a question of Erdős about how the total number of divisors of Mersenne numbers grows when the range is doubled. We prove that the doubling ratio is always at least 3, sharp, and nearly 4 whenever the relevant Fermat numbers are composite, as all known ones are. We also reduce the open question of whether the ratio tends to infinity to a statement about indices with few divisors. This leads to a new conjecture on the prime factors of Mersenne numbers next to near-Wagstaff numbers, which we study through implications, barrier results and a proven polynomial analogue.

math.NA — Numerical Analysis

A polynomial-time algorithm to compute Theta functions of Schottky groups

Contributed by Marc Masdeu

Abelian integrals, holomorphic differentials, period matrices and the Schottky--Klein prime function of a Riemann surface uniformized by a classical Schottky group are given by Poincar\'e theta series, which are sums over the elements of a free group. The algorithms that evaluate these series sum a truncated, possibly adaptively pruned, portion of the Cayley tree of the group, and the number of terms needed grows exponentially with the number of correct digits. We observe that the sum over all group elements starting with a fixed generator is a holomorphic function on the complement of a disc, and that these functions satisfy an exact linear functional equation coming from the self-similarity of the Cayley tree, given by a transfer operator built from the generators. Representing them by truncated Taylor expansions turns the evaluation of the series into a finite-dimensional linear problem, which we solve iteratively. The resulting algorithm computes the integrals $\Omega_n$, the period matrix and the prime function to $P$ bits of precision; for a fixed Schottky datum the number of arithmetic operations is cubic in $P$ and in the genus. We prove that it converges geometrically under a standard Schottky-type hypothesis on the group, which is closely related to the absolute convergence of the series. Along the way we give a complete proof of the cross-ratio formula for the period matrix, including a term that is missing from one version of it in the literature. We compare with the existing algorithms, and illustrate the method by reproducing a published value of the prime function and computing it to $30$ digits, and by computing period matrices of genus up to six to about $25$ digits.

math.NT — Number Theory

Explicit Weak abc Inequalities and the Discrete Obstruction to Full abc in the DIC Framework

Contributed by Zeng Chunping

Independent audit and coefficient verification of C. Soulé's preprint "On the abc conjecture" (arXiv:2611.05217, 2026). This work neither proves nor disproves the abc conjecture; all No-Go statements are internal to the audited construction, and every statement that uses Soulé's Theorem A is explicitly tagged as conditional on it. We axiomatise the DIC framework and prove three stratified groups of results on Kodaira--Parshin arithmetic surface fibrations. (i) Unconditional geometric theorems: the adjunction identity $Z\cdot\omega_{\mathcal{X}/B}=-Z^{2}$, the Riemann--Hurwitz intersection numbers, the invariants of the general $(g_0,r,b)$ construction, and a Deligne--Riemann--Roch decomposition $c=\frac{2b^{2}+2b-1}{12}+(D_0-T)/\omega^{2}$ of the arithmetic coefficient. (ii) A meta-theorem: integrality of torsion lengths forces the coefficient of Theorem A onto the discrete set $t=12/N$, $N\equiv 1\pmod 3$ (at the literature point $(q,\lambda)=(5,3)$; for varying $q$ this is an arithmetic coincidence, not a structural barrier). A local hyperelliptic model gives $T=8k$, $t=12$ as the unique transverse output; second-order torsion gives $n_1=n_2$, hence $\Theta=T/k\in 2\mathbb{Z}$, leaving $(\Theta,q)=(12,27)$ as the sole lattice candidate. The unified criterion $E=12(q-1)/(qN\gamma)$ has the unique solution $(N,\Theta,q,\gamma)=(13,12,27,8/9)$; the height is the canonical-bundle height of a fixed genus-$2$ curve ($\gamma=2/5$, confirmed by $12/\gamma=30$ in the source), so $(12,27)$ requires genus $13$ and lies outside the theorem's hypotheses: the last cell is empty. (iii) Controlled weak abc: all unconditional bounds known for abc are of exponential type ($\log c\ll N^{1/3}(\log N)^{3}$; Stewart--Yu); the exponent below is of polynomial type, and its existence is what Theorem A supplies. We derive $\eta(q)=(q-1)/q$, unify the exponent as $E=t\eta/\gamma$, and obtain $E_{\max}=12\frac{q-1}{q}\bigl\lceil\frac{2q-2}{\ell}\bigr\rceil$. A discriminant stratification sharpens this to $c\le A_{\varepsilon}(N/M_5)^{24+\varepsilon}$; bootstrapping to exponent $1$ is impossible ($\inf\theta=0$); the discriminant-to-degree ratio is tower-monotone; $4/5$ is an Artin conductor identity; and the isometry step (32) of the source contributes only $O(d')$, derived unconditionally from a finite Čech complex, leaving $\frac32=m\cdot c$, $36$, $12$, $24$ and $E(c)=12/(5-18c)$ unchanged. A verbatim check of the source confirms twelve items, including the statement and scope of Theorem A; the constant $12$ decomposes as $\frac13\cdot\frac{m}{\rho-mc}$; and Route 1 is closed. Full abc is unattainable inside the audited construction for nine independent reasons, none depending on the truth of Theorem A: hypothesis exclusion, $\inf\theta=0$, tower monotonicity, bridge breaking, the $S$-smoothness loop, the double exclusion $T/k=47/3$, the $g_0=13$ collapse, index-ideal absorption with exact slope $2/(n-1)$, and the distributional vanishing of the discount (the multiplicity excess and $\theta$ have density $1-o(1)$, with explicit large-deviation rates).

Author comments: Audit of arXiv:2611.05217 (Soulé, 2026). Neither proves nor disclaims abc. Weak-abc statements are conditional on Soulé's Theorem A (tagged Th.A*); all nine No-Go statements are internal to the audited construction and independent of Theorem A's truth. This version incorporates nine audit rounds and three external review reports, including an independent re-derivation of the adelic metric layer (Appendix G), an unconditional O(d')-form of the source's step (32) from a finite Čech complex (Appendix I), a verbatim check of the source (Appendix J), a higher-genus recalibration closing the g0=13 route (Appendix K), a quantitative index-ideal absorption analysis (Appendix L), and a distributional analysis of the discount (Appendix M). Numerical cross-checks were performed in exact rational arithmetic (SymPy); the manuscript was drafted and typeset with AI assistance under the author's direction and line-by-line review.

cs.LO — Logic in Computer Science

Expressiveness, Verifiability, Tractability: Toward a Computational Thermodynamics

Contributed by Benjamin Kaminsky

An investigation into the existence of a conservation law governing computational behaviour, in the form of the Expressiveness, Verifiability, Tractability (EVT) Hypothesis, is undertaken. The hypothesis is motivated by a simple observation: mechanisms that increase what a system can directly express, powerful control, effects, reflection, meta-programming, and self-interpretation, tend to undermine what can be verified uniformly and mechanically, while regimes that maximize static guarantees typically do so by restricting expressiveness or relocating effort into compilers, proof artifacts, or human interpretation. We analyze canonical failure modes that arise when one attempts to push beyond this frontier, distinguishing collapse of correctness, collapse of compilability, and collapse of comprehensibility. We then develop an order-theoretic model for comparing languages and verification disciplines and define an “E/V boundary” as a Pareto frontier in a partially ordered design space. Finally, we sketch how analogous tradeoffs appear in other computational domains, suggesting that “thermodynamic” boundary behaviour may be a recurring feature of expressive formalisms that demand strong global guarantees and within physical methods of information processing.

math.DG — Differential Geometry

Gysin Sequences in Supergeometry

Contributed by Sebastian Denton

We study spectral sequences in supergeometry and construct the Gysin sequence within the supermanifold framework. First, the foundational definitions are established and utilized to develop differential forms and de Rham cohomology on supermanifolds. We then apply a standard de Rham-to-Cech cohomology isomorphism to assemble a spectral sequence that accounts for the ˇ Z2-graded nature of supermanifolds. Finally, we explore additional properties arising from this construction, leading directly into Gysin sequences.

Author comments: Comments and suggestions are welcome. Please direct any correspondence to the author.

math.CO — Combinatorics

The Cranston–Li–Su–Wang–Xu (2026) conjecture is true

Contributed by Dave Goel

Let A be a finite abelian group. A nowhere-zero A-flow of a graph can be changed along a cycle: going around the cycle, add a fixed non-zero element of A to the value of every edge that points forward and subtract it from the value of every edge that points backward. The Cranston–Li–Su–Wang–Xu (2026) conjecture on the reconfiguration of nowhere-zero flows states that if A has at least four elements, then in every graph in which all degrees are even, any two nowhere-zero A-flows are joined by a chain of such changes through nowhere-zero A-flows. It also states that, for every integer k ≥ 4, the same holds for integer flows whose values are between 1 and k − 1 in absolute value, where each change adds a non-zero integer. We prove this conjecture. The proof is by induction on the number of edges. If a flow enters a vertex along one edge and leaves it along another edge with the same value, the two edges can be replaced by a single edge; this gives a smaller graph. The main step is to change every flow into a flow with such a pair of edges. For A-flows we do this when all degrees are at most 4; repeatedly replacing a vertex of degree at least 6 by a triangle, following Cranston et al., reduces the general case to this one. For integer flows, we show that, in a graph in which all degrees are even, any two nowhere-zero flows with values in the cyclic group Z_k, k ≥ 4, are joined by changes that add only 1, −1, 2 or −2, and that such changes lift to integer flows. All theorems of this paper are also proved in the Lean 4 proof assistant.

Author comments: An expanded version of this paper is posted on Zenodo: https://doi.org/10.5281/zenodo.23605838

math.RT — Representation Theory

The Humphreys support conjecture

Contributed by Qiyue Tang

We prove the Humphreys support conjecture for connected reductive groups over algebraically closed fields of characteristic $p>h$.

Author comments: 98 pages. Computational code and data are available at https://github.com/tqyqiqi1509/humphreys-support-conjecture 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.AG — Algebraic Geometry

Bootstrapping GAGA Theorems

Contributed by Aniruddh Venkatesan

Let $X$ be a noetherian scheme with affine diagonal, and let $\lambda : \mathcal{X} \to X$ be a morphism of locally ringed spaces from a qcqs locally strongly noetherian analytic adic space. We prove that an equivalence on coherent sheaves, together with comparison of their cohomology, extends to every proper $X$-scheme. As an application, we obtain GAGA for proper schemes over the Fargues--Fontaine curve in equal and mixed characteristic.

math.NT — Number Theory

A unimodular continued fraction for cubic irrationals

Contributed by Will Howe, Grant Huffman, Jensen Kohlmeyer, Liam Kruer

We construct a single algorithm that assigns to every real number a recoverable sequence of positive integers which is eventually periodic exactly when the input is a cubic irrational. Its matrix digits are positive unimodular transformations on the two-dimensional simplex. Their products determine nested rational triangles whose diameters decrease by at least a factor of two at each step. An effectively specified partition combines Euclidean reduction with the dominant eigenvectors of cubic units. In both cubic signatures the matrix expansion eventually repeats a single matrix, whose Perron eigenvalue is a unit generating the input field.

math.NT — Number Theory

A congruence obstruction to Ibn al-Khawwām’s eleventh problem

Contributed by Will Howe, Grant Huffman, Jensen Kohlmeyer, Liam Kruer

We prove that there are no two rational squares whose sum is ten and such that adding its own square root to each gives another rational square. This is the existence question in Ibn al-Khawwām’s eleventh arithmetic problem. The proof uses only the residues of squares modulo three and a divisibility argument.

math.CO — Combinatorics

Optimal Packing of Eleven Unit Squares: The Formalized Voronoi-Cell Proof

Contributed by Manasseh Ahmed

We explain the computer-assisted proof that eleven independently rotated unit squares require a square container of side at least T = 3.8770835900228141773 . . ., the side attained by the known construction attributed to Walter Trump. The proof reduces arbitrary packings to 2,184 patterns of occupied closed Voronoi cells. Exact geometric certificates exclude 2,180 patterns, a square-symmetry argument reduces the remaining four to one, and a complete subdivision of that case places every possible smaller packing inside an explicitly certified isolation neighborhood. We give the geometric definitions, prove the continuous soundness rules underlying the finite calculations, and explain the common-cone local isolation argument used in the Lean development. The large certificate instances are identified by pinned source references; they are not replaced by numerical experiments. The reported formal verification uses Lean’s kernel together with its native compiler for specified numerical certificates.

math.CO — Combinatorics

The independence number of the generalized Johnson graph $J(13,5,3)$

Contributed by Mustafa Stanikzai

Let $J(13,5,3)$ be the graph on the 5-element subsets of a 13-element set, two subsets being adjacent when they share exactly 3 elements. We prove that $\alpha(J(13,5,3)) = 33$. The graph is an explicit unit-distance graph in $\mathbb{R}^{12}$, so $\chi(\mathbb{R}^{12}) \ge \lceil 1287/33 \rceil = 39$; earlier bounds from explicit finite graphs give at most 38. A recent preprint of Davies gives much larger bounds, none from an explicit finite graph. An explicit independent set gives $\alpha \ge 33$. For the upper bound, a symmetry reduction splits the problem into 200 clique subproblems, and a certifying solver refutes each one with a proof log that the checker VeriPB accepts. An uncertified computation reproduces the value $\alpha(J(13,6,4)) = 46$ reported by Exoo and Ismailescu, which gives $\chi(\mathbb{R}^{12}) \ge 38$.

Author comments: 16 pages. Programs, certificates and proof logs: https://doi.org/10.5281/zenodo.24145863 Changes from version 1: 28 cross-references to lemmas, propositions, claims, definitions and corollaries were printed with the word "Theorem" (for example "Theorem 2.4" for Lemma 2.4). They now carry the correct names. The reference numbers were already correct. Statements, proofs, results and bibliography are unchanged.

math.NT — Number Theory

A Fibonacci Analogue of the Fifteen Theorem

Contributed by Scott Kominers

We prove that a positive-definite quadratic form with integral Gram matrix represents every Fibonacci number if and only if it represents the first forty-two of them. The unique minimal criterion set consists of ten Fibonacci numbers, the largest of which is $F_{42}=267914296$.

math.GR — Group Theory

Finite-length permutation modules for\\free-amalgamation structures of arbitrary arity

Contributed by Dr. András Pongrácz

Let $\A$ be the countable homogeneous limit of a free-amalgamation class in a finite relational language, expanded by a generic linear order. We prove that its tuple permutation modules have finite composition length over every field, also after naming finitely many points. More precisely, each orbit of increasing $d$-tuples disjoint from the named points has a permutation module of length $2^d$. This extends the theorem of Yang--Boja\'nczyk--Klin from unary and binary relations to relations of arbitrary arity. The main step shows that every finitely supported array whose sums over each single coordinate vanish is a linear combination of alternating sums obtained by independently doubling its coordinates. Its proof combines a Ramsey separation theorem with a construction that moves finite configurations into prescribed order intervals. Over a finite field, we also obtain finitely many closed invariant linear subspaces of $\F^{A^k}$; each is generated, in the product topology, by functions definable using at most $k$ additional parameters.

math.CO — Combinatorics

The Maximum of $\operatorname{per}(I-A)$ in Odd Order

Contributed by Yair Lavi

Let $Ω_n$ denote the set of $n\times n$ doubly stochastic matrices. Kim and Roush conjectured in 1981 that, for $n=2k+1>1$, $\max_{A\inΩ_{2k+1}}\operatorname{per}(I-A)=3\cdot 2^{k-2}$. They proposed the block construction $A_\star=\frac12(J_3-I_3)\oplus P_2^{\oplus(k-1)}$, where $P_2=\begin{pmatrix}0&1\\1&0\end{pmatrix}$. Here $J_3$ is the $3\times3$ all-ones matrix. They did not claim uniqueness. We fully prove their conjecture and classify equality: the maximizers are exactly the simultaneous-permutation conjugates of $A_\star$.

math.CO — Combinatorics

A counterexample to the Foregger-Sinkhorn tie-point conjecture

Contributed by Yair Lavi

The Foregger-Sinkhorn tie-point conjecture asserts that if a nearly decomposable doubly stochastic matrix minimizes the permanent on a face and the permanental cofactor at a prescribed zero is larger than its permanent, then that zero is a tie point. We give a counterexample in dimension eight.

math.CO — Combinatorics

The Maximum Permanent of a Stochastic Matrix of Bounded Rank

Contributed by Yair Lavi

Let $A$ be a stochastic $n\times n$ matrix with $\operatorname{rank}A\leq k$, where $1\leq k\leq n$. Write $n=qk+s$, where $0\leq s<k$. The author conjectured in 2018 that $\operatorname{per}A\leq (q!/q^q)^{k-s}((q+1)!/(q+1)^{q+1})^s$, with equality if and only if $A=P(J_q^{\oplus(k-s)}\oplus J_{q+1}^{\oplus s})Q$, where $P,Q$ are permutation matrices and $J_t$ is the $t\times t$ matrix with every entry $1/t$. We prove this conjecture in full.

math.CO — Combinatorics

The Marcus-Minc Transform Inequality

Contributed by Yair Lavi

We prove the Marcus-Minc conjecture: let n be any integer at least 2, and let A be a nonnegative n by n matrix whose row and column sums are all one. Form a new matrix by subtracting A from the n by n matrix of ones and dividing by n minus one. Then the permanent of A is at least the permanent of the new matrix. We also determine all equality cases.

math.NT — Number Theory

Cross-Energy of Dirichlet Partial Sums. Part 38: The energy of the Chebyshev race: an explicit unconditional floor for the prime race between the lanes $6k\pm1$

Contributed by Pedro Caceres

Let $R(x)=\pi(x;3,2)-\pi(x;3,1)$ be the Chebyshev race between the two prime lanes $6k\mp1$ (the prime $2$ counted, by the congruence $2\equiv-1$ mod $3$, in the $6k-1$ lane). We study the race as an \emph{energy} and establish three things (the energy law uses two estimates imported from the $\zeta$ case; see Proposition~\ref{prop:cramer}). First, an elementary ledger identity: the entire bias of the race is the bookkeeping of prime squares, which all land in the lane $6k+1$; subtracting the ledger $\tfrac12\pi(\sqrt x)$ turns a walk with \emph{zero} sign changes up to $10^8$ into one with $15{,}329$ ($21{,}329$ if every passage between $D>0$ and $D\le0$ at a prime is counted). Second, the mean energy of the normalized race signal $e(u)=e^{-u/2}\psi(e^u,\chi_3)$ admits an explicit unconditional lower bound for its upper limit, which we call the floor, \[ \limsup_{U\to\infty}\frac1U\int_0^U e(u)^2\,du\;\ge\;\kappa_3 :=\sum_{\rho}\frac{1}{\rho(1-\rho)} =\log\frac3\pi-\gamma_E+2\,\frac{L'}{L}(1,\chi_3) =0.113229969857\ldots, \] the sum running over the nontrivial zeros of $L(s,\chi_3)$ with multiplicity, and the limit exists and attains the floor if and only if the Riemann hypothesis holds for $L(s,\chi_3)$ with all zeros simple. The constant is known: under GRH it is the variance of the limiting distribution of Rubinstein and Sarnak for $(\log x/\sqrt x)R(x)$ (four times smaller than in their normalization $E(x;3,2)-E(x;3,1)$), and its closed form is that of Fiorilli and Martin (there under GRH); it is also the variance $V_\chi$ of Part~37, on the $6k\pm1$ race, where the prime-square mechanism of the bias is identified; what is added here is the unconditional floor with its equality case and exponential-growth law, the ledger correction of the counting race, and the exact measurement. The constant $\kappa_3$ is the exact analogue, for the prime race, of the B\'aez-Duarte constant $2+\gamma_E-\log4\pi$ of the Nyman--Beurling criterion. Third, we measure: from all prime powers up to $10^8$, by exact piecewise integration, the cumulative mean energy is $0.1201$ at $x=10^8$ and lies $4$ to $12\%$ above $\kappa_3$ for $10^4\le x\le10^8$. This is consistent with the theorem and does not test it: the floor is a statement about an upper limit, obtained from the dichotomy between GRH and its failure, and means over single decades fall below $\kappa_3$ ($0.102$ on $[10^4,10^5]$, $0.113$ on $[10^7,10^8]$). The second-order term is $(\text{mean}-\kappa_3)\cdot U\approx0.11$ (between $0.09$ and $0.13$), the race analogue of the second-order constant $K=2.5124797\ldots$ of the prime energy. The proofs of the energy law transplant the $C_2$ energy law for $\zeta$ to $L(s,\chi_3)$, where the absence of a pole simplifies every step.

Author comments: Revised 10 Nov 2026 after external examination (see CHANGES_v2.md). 12 pages, 1 figures. Ancillary scripts in ancillary/ reproduce the computations. Part 38 of the book-series Cross-Energy of Dirichlet Partial Sums (47 parts, Parts 0-46); formerly the standalone note 'The energy of the Chebyshev race' (revised 1 Nov 2026). Renumbered 2 Nov 2026; mathematics unchanged. The paper states once, in its discussion, that it does not prove the Riemann hypothesis.

math.NT — Number Theory

Cross-Energy of Dirichlet Partial Sums. Part 37: The $6k\pm1$ race: sieve structure, Chebyshev's bias and a GRH-equivalent inequality

Contributed by Pedro Caceres

Every prime $p>3$ is $6k+1$ or $6k-1$. We collect what can be proved about the two resulting sequences and add three results. The races are those modulo $3$ (the prime $2$ counts for the class $2$); the race modulo $6$ differs by that one prime. (i) In the weighting $n^{-1/2}\log(N/n)$ of the Ces\`aro margin, the race between the prime powers $\equiv2$ and $\equiv1\pmod3$ is an exact identity. The $6k-1$ side leads by $2\frac{L'}L(\frac12,\chi_{-3})\log N-\frac12\psi'(\frac34)=1.1320\log N-1.2709$ plus the margin $M_\chi(N)$ of $L(s,\chi_{-3})$. The generalized Riemann hypothesis (GRH) for $L(s,\chi_{-3})$ is equivalent to $M_\chi\ge0$ for all large $N$, and under GRH the weighted race is never lost after $N=2$. This is a criterion of the type of Hardy--Littlewood and Landau (1918) for Chebyshev's assertion, here with the Ces\`aro weight and with explicit constants. (ii) We verify $0.1024\le M_\chi(N)\le0.3549$ for $2\le N\le2^{27}$, inside the window $[0,0.4536]$ that GRH predicts. We also check the formula against the first $271$ zeros of $L(s,\chi_{-3})$. (iii) For the unweighted race, the bias is $1/\sqrt{V_\chi}=2.97$ standard deviations, with $V_\chi=\sum_\rho\frac1{\rho(1-\rho)}=0.113230$ (equal to $\sum_\rho|\rho|^{-2}$ under GRH); under GRH and linear independence, a recomputation with $271$ zeros gives the density $0.99906$, in agreement with the $0.9990$ of Rubinstein and Sarnak. Along the way we prove the sieve characterization of $6k\pm1$ and of twin primes, explain why Legendre-type counting cannot give the error term, compare twin primes and Goldbach representations modulo $6$ with the Hardy--Littlewood predictions, and prove the formula for $\gcd_p(p^{2n}-1)$ (the sequence $24,240,504,480,\dots$). Version~2 adds the generating function $\frac12[\zeta_6(s)^2\pm L(s,\chi_6)^2]$ of the three forms $6xy\pm x\pm y$, and a duality theorem: averaged over the zeros of $\zeta$, the cross-energy $\mathcal C_2(N,\frac12,\gamma)$ of Part~1 returns $\log N/N$; averaged over the zeros of $L(s,\chi_{-3})$ it returns $\frac1N\sum_{d\mid N}\chi_{-3}(d)\Lambda(d)$; and M\"obius inversion separates the primes of the two lanes. Version~3 makes this exact at finite height and without RH: with a Gaussian weight on the zeros, the cross-energy satisfies an explicit formula whose pole term is $\mathcal C_2$ itself at $s=0$, whose prime term is $\log N$ exactly plus a nonnegative remainder from the near coincidences $nj\approx N$, and which returns every prime power up to $60$ from the $1517$ zeros below $2000$. Nothing here proves RH, GRH, or the twin prime or Goldbach conjectures.

Author comments: Revised 10 Nov 2026 after external examination (see CHANGES_v2.md). 25 pages, 11 figures; version 3 (2 November 2026) adds Section 11.3, an exact explicit formula for the cross-energy at finite height (unconditional), with a term-by-term explanation of how C_2 hears the primes through the zeros; version 2 added Section 11, the cross-energy as a two-way detector. Ancillary scripts in ancillary/ reproduce the computations. Part 37 of the book-series Cross-Energy of Dirichlet Partial Sums (47 parts, Parts 0-46); formerly the standalone note 'The 6k±1 race'. Renumbered 2 Nov 2026; version 2 adds Section 11. The paper states once, in its discussion, that it does not prove the Riemann hypothesis.

math.NT — Number Theory

Cross-Energy of Dirichlet Partial Sums. Part 16: How many positive directions does the Archimedean place have? A counting law with a boundary phase

Contributed by Pedro Caceres

Restrict the archimedean part of Weil's quadratic form, for $\zeta$ or for a primitive real Dirichlet $L$-function of conductor $q$ and parity $\kappa$, to test functions supported in an interval of length $a$, and let $n_+(a)$ be the number of its positive directions, the directions that the pole and the primes must absorb. A Szeg\H{o}-type argument gives $n_+(a)=t_0a/\pi+o(a)$, where $t_0$ is the zero of $\vartheta'$. We observe that the form is $\Lambda-\psi_\kappa(-\Delta)$ with $\psi_\kappa$ a complete Bernstein function (proved) and, using the boundary phase of Kaleta, Kwa\'snicki and Ma{\l}ecki, we state, as a conjecture, the asymptotic law \[ n_+(a)=\Big\lfloor\frac{t_0a}\pi+\frac{2\varpi_\kappa(t_0)}\pi\Big\rfloor , \] with an explicit phase $\varpi_\kappa$ that depends only on the parity, the conductor entering only through $t_0$. We test it in seven cases, $\zeta$ and $\chi_{-3},\chi_{-4},\chi_5,\chi_{-7},\chi_8,\chi_{-8}$. On the finest mesh the shifts of the jumps $m=2,\dots,12$ bracket the prediction $2\varpi_\kappa(t_0)/\pi$, which ranges from $0.18$ to $0.59$, to within $1$ to $3\cdot10^{-3}$ for the characters and $3\cdot10^{-2}$ for $\zeta$. A second, independent implementation extrapolates in the mesh on three to nine jumps per case; the extrapolated shifts lie above the prediction by $0.2$ to $2.3\cdot10^{-4}$ in every case, a one-sided offset of the size of the extrapolation error, which we have checked to be a bias in one case only. The first jump is an exception, and for $\zeta$ the shift reaches its limit only after about ten jumps. A smoothed count removes the integer part and leaves ``degree plus constant'' with the boundary phase as constant; on a circle the constant is carried by the end points and cancels between an arc and its complement; and treating $-\log q$ as a degree, $q\leftrightarrow1/q$ acts on the band edge as $t_0\mapsto4\pi^2/t_0$ to leading order, with $\zeta$ at the self-dual point $t_0\approx2\pi$. We record what the law implies for the signature of Weil's form and where it stands with respect to the Riemann--Roch formulas for $\overline{\operatorname{Spec}\mathbb Z}$. The law is a conjecture supported by these computations. The Bernstein structure is proved here; the first-order count is quoted from Part~14 and its proof is not repeated.

Author comments: Revised 10 Nov 2026 after external examination (see CHANGES_v2.md). 9 pages, 1 figure; Part 16 of the 47-part book-series Cross-Energy of Dirichlet Partial Sums (Parts 0-46), block III (The margin and the constant B-hat); formerly a standalone note (v1.0). Part 0 is the guide.

math.NT — Number Theory

Cross-Energy of Dirichlet Partial Sums. Part 10: The Caceres Prime Function. A square-root damped prime weight, its exact Prime Zeta structure, and a dual spectrum of the Riemann zeros

Contributed by Pedro Caceres

\sloppy We study the \emph{Caceres Prime Function} \[ \C(s)=\sum_{n\ge 2}\frac{a(n)}{n^{s}},\qquad a(n)=\sum_{p^k\,\|\,n}\frac{\log p}{\sqrt{p^{k}}}, \] the Dirichlet series of the additive function that weights each prime factor $p$ of $n$ by $\log p$ divided by the square root of the exact power of $p$ dividing $n$. \emph{Structure.} We prove that $\C=\zeta\K$, where the \emph{prime kernel}, $\K(s)=\sum_p\log p\,(1-p^{-s})/(p^{s+1/2}-1)$, is an exact telescoping series of shifted derivatives of the Prime Zeta function. So $\K$ is meromorphic in $\Re s>-\frac14$, and the line $\Re s=-\frac14$ is a natural boundary. Every zero $\frac12+i\gamma$ of $\zeta$ produces two singular points of $\K$, at $i\gamma$ and $i\gamma/2$, which are poles unless $2\gamma$ is again an ordinate with twice the multiplicity (respectively $\gamma/2$ an ordinate with half the multiplicity, for $i\gamma/2$); the exponent $\frac12$ is the normalization that puts both families on the imaginary axis. More precisely, $\K(s)=-\frac{\zeta'}\zeta(s+\frac12)+\frac{\zeta'}\zeta(2s+\frac12)-\frac1{12s}+H(s)$, with $H$ holomorphic in a neighbourhood of $\Re s\ge0$. \emph{RH criteria.} RH is equivalent to the holomorphy of $\K(s)-\frac1{s-1/2}+\frac1{2(s-1/4)}$ in $\Re s>0$. It is also equivalent to a bound $O(x^\varepsilon)$, or even a one-sided bound, for the remainder $R_b$ of the kernel's partial sums. Both are exact reformulations of RH, not proofs. \emph{Mean value and explicit formula.} Unconditionally, $\sum_{n\le x}a(n)=\K(1)x+2\zeta(\frac12)\sqrt x+O(\sqrt x\,e^{-c_\ast\sqrt{\log x}})$, with the explicit bound $5.51\sqrt x$ for the error after the first term when $x\ge10^6$. Under RH, the Riesz mean of $a$ has an explicit formula in which every zero contributes the pure phases $x^{i\gamma}$ and $x^{i\gamma/2}$ at constant amplitude. \emph{The Ces\`aro margin.} The kernel is Suzuki's one-variable margin $M$ seen at two scales. Unconditionally, $R_b(x)=\alpha(x)-\alpha(\sqrt x)+o(1)$, where $\alpha$ is the slope of the margin. Under RH, the kernel's Riesz margin is $M(N)-2M(\sqrt N)$. Its fluctuation has energy $2(\log\xi)''(\frac12)$, reduced by $2\gamma^{-2}$ for every pair of ordinates with $\gamma'=2\gamma$; the near pair $\gamma_{14}\approx2\gamma_4$ is visible in the data. \emph{Zeros of $\C$.} A computer-assisted proof shows that $\C$ has a zero near $1.379+8.639\,i$, in the half-plane of absolute convergence, and Bohr's theory gives $\gg T$ such zeros up to height $T$. So $\C$ is not an $L$-function. The theorems are proved here, except that Section~\ref{sec:margin} uses the formula for the margin and the almost periodicity of the slope from Part~11. Uncertified computations are labelled as numerical observations.

Author comments: Revised 10 Nov 2026 after external examination (see CHANGES_v2.md). 26 pages, 7 figures. Part 10 of a series of forty-seven on the cross-energy of Dirichlet partial sums (book edition of 2 November 2026, block II: Primes and Euler products; Part 0 is the guide, Part 1 the plan). Formerly a standalone note ("The Caceres Prime Function", October 2026); in the book it builds on Part 9 (the prime zeta function), Part 1 (the C-transform) and Part 11 (the Cesaro margin), and is used by Parts 11 and 37. All theorems are proved; uncertified computations are labelled as numerical observations. The ancillary archive (core.py, explicit.py, zeros.py, step*, rigor_meanvalue.py, rigor_rouche.py, ctransform.py, laplace.py, bridge_M*.py, variogram.py, slope_b.py) is the one of the standalone note, Caceres_Prime_Function_ancillary.zip.

math.NT — Number Theory

At least one of β(4), β(6), …, β(32) is irrational

Contributed by twinheterodox

For the Dirichlet beta function, we prove that at least one of $\beta(4),\beta(6),\ldots,\beta(32)$ is irrational. This reduces Krisman's seventeen-value set $\beta(4),\beta(6),\ldots,\beta(36)$ (Hexagon 2610.00285v2) to fifteen values. The improvement comes from a sharper denominator estimate for his construction.

Author comments: 15 pages. An isolated-pole denominator refinement reduces Krisman's seventeen candidate even beta values to fifteen. Includes exact certificates, reproduction instructions and an independent mathematical audit with its verification scope.

math.CA — Classical Analysis and ODEs

Some positivity properties of Riemann $\Xi$ under smoothing

Contributed by Suvrit Sra

We show that smoothing the zeros of the Riemann $\Xi$-function makes it easy to establish positivity properties that are otherwise hard (as they are equivalent to the Riemann hypothesis). Specifically, we smooth each zero with a small density centered at it, and then test the impact of this smoothing on complete monotonicity (CM) of the heat sum $H(t)=\sum_\rho e^{-tz_\rho^2}$ over the zeros $\tfrac12+iz_\rho$. A sufficiently wide density quickly implies CM of the smoothed heat sum. We bound the width (i.e., half-variance) the density needs using increasingly precise facts about the zeros. If we only use that they lie in the critical strip, half-variance $1/8$ suffices; the primes lower this width to $\pi^2/288$; and zeros verified through height $T$ further lower it to $\pi^2/(8\log^2T)$ asymptotically. Ultimately, however, smoothing seems to be of limited utility: despite the CM result, the smoothed reciprocal laws of $\Xi$ are not even log-concave, unlike the unsmoothed ones.

quant-ph — Quantum Physics

Exact projector-rank thresholds for state-independent contextuality

Contributed by Sunay Venkat

For every integer r ≥ 1 we construct a finite exclusivity graph Fr with the following two properties. First, Fr has a projective representation by pairwise distinct projectors of rank r that exhibits state-independent contextuality. Second, no projective representation of Fr by projectors of any common rank s < r, in any finite dimension, exhibits state-independent contextuality; in every such representation the maximally mixed state is noncontextual. This establishes, for every r, the “rank efficient” scenarios sought by Hohn, Xu and Kleinmann, for which only numerical evidence at r = 2, 3 was available. The graph is the disjunctive product of an odd cycle C2r+1 with the orthogonality structure of a rank-one contextuality set whose fractional chromatic number exceeds its dimension by an arbitrarily small amount. Such sets are obtained from the infinite family of Kochen–Specker parity proofs in R^4 of Elford and Lisonek, together with a coning construction that raises the dimension by one.

math.GN — General Topology

On the structure of countable hyperspaces of regular subcontinua

Contributed by Lucas Almeida Almeida

Let $X$ be a metric continuum and let $\mathcal D(X)$ be its hyperspace of regular subcontinua, that is, of the subcontinua of $X$ which are the closure of their interior. We prove that if $\mathcal D(X)$ is countable, then the interiors of its minimal elements are dense in $X$, and every regular subcontinuum is determined by the minimal elements it contains. We further establish that $\mathcal D(X)$ is a $G_\delta$ subset of the hyperspace $C(X)$ of subcontinua whenever $X$ admits a family of regular subcontinua whose interiors have dense union and which satisfies an absorption property. Consequently, every countable $\mathcal D(X)$ is completely metrizable and scattered. In particular, $\mathcal D(X)$ is never homeomorphic to the rational numbers, which answers a question of Ramírez Angarita and of Camargo, Ordoñez and Ramírez. The theorems of the paper are formalized in Lean 4.

math.NT — Number Theory

An Unconditional Artin-Matthews Theorem For Simultaneous Primitive Roots

Contributed by Vlado Vojdanovski

Let S be a finite set of integers different from 0 and ±1. We prove that the primes for which every member of S is a primitive root have positive lower relative density precisely when neither of Matthews’ two obstructions occurs. The proof uses the marked estimates for prime predecessors and uniform splitting estimates of OpenAI. Its principal step is a refinement of the large-prime-factor law for prime predecessors by a fixed cubic Frobenius condition. We obtain this refinement from a shifted bilinear estimate for cubic theta series, a divisor-weighted positive sieve estimate, and a prime-extraction argument that retains the natural Hecke coefficients. The resulting lower bound holds in every sufficiently large dyadic interval.

math.MG — Metric Geometry

On the local geometry of the tight 39-point code in the octonionic projective plane

Contributed by Robert R. MacGregor

Cohn, Kumar and Minton constructed a four-parameter family of tight $39$-point codes in the octonionic projective plane. We show that near each member outside a closed nowhere-dense set of resonant parameters, the labeled tight codes modulo $F_4$ form exactly four ten-dimensional families, any two meeting in the four-parameter family, although the first-order deformations of the member form a $28$-dimensional space modulo $F_4$. The originally constructed code is resonant, and we do not determine the tight codes near it; but every neighborhood of it in the space of unlabeled tight $39$-point codes modulo $F_4$ contains a nonempty open subset homeomorphic to an open subset of $\mathbb R^{10}$, so any neighborhood of it that is a manifold has dimension ten.

Author comments: 55 pages, 5 figures. Ancillary files: anc/native-data.json holds the exact coefficients of Appendix C, and anc/finite-identities.py replays the finite identities of Proposition 6.5 from it in exact arithmetic. Lean source: https://github.com/macgregor-sudo/tight-39-code at commit 46f01c1.

math.NT — Number Theory

An effective André–Oort theorem for Shimura varieties

Contributed by Manvir Jaswal

We give effective bounds for the special subvarieties $V$ that are maximal in a subvariety $Z$ of a Shimura variety, in terms of the degree of $Z$: for the degree of $V$, with no hypothesis; and for the discriminant of a number field $L_V$ attached to $V$ and the Klingler–Yafaev invariant $\alpha_V$ of the level, if for some $\delta>0$ the Dedekind zeta function of $L_V$ has no zero with real part greater than $1-\delta$. Here $Z$ is geometrically irreducible and defined over a number field, the level is neat and of product form, and the bounds are explicit in the degree of $Z$ and, for $L_V$ and $\alpha_V$, also in the degree of the field and $\delta$, with constants that are effectively computable from the Shimura datum, the level and representatives of the connected components of the adjoint Shimura variety. A preprint of Cureton and Thorner, whose proof has been formalized in Lean, establishes such a half-plane with $\delta=1/32$ for every number field; granting it, all these bounds hold with no further hypothesis. For abelian type, an algorithm lists these subvarieties from the bounds, given a finite étale cover by a Shimura variety of Hodge type inside one of Mumford's moduli spaces and equations for the preimage of $Z$.

math.NT — Number Theory

The classic enumeration of genera of even definite lattices

Contributed by Mathieu Dutour Sikiric

We revisit the enumeration of a genus of even positive definite lattices by Kneser neighbours, certified by the mass. We work at primes dividing the determinant, where the projective spaces are smallest, schedule the exploration of the neighbour graph by a random walk and by orders based on a Hecke bookkeeping, and compute canonical forms through the Weyl chamber of the root system. Genera of rank up to 19 with up to 40,617 classes are enumerated in minutes.

Author comments: 22 pages. The algorithms are implemented in the program GENUS_Enumerate of polyhedral_common, https://github.com/MathieuDutSik/polyhedral_common

math.NT — Number Theory

Irrational even beta values from powered hypergeometric kernels

Contributed by twinheterodox

For the Dirichlet beta function, we prove that at least one of $\beta(6),\beta(8),\ldots,\beta(74)$ is irrational, and that at least one of $\beta(8),\beta(10),\ldots,\beta(110)$ is irrational. These shorten the previously stated intervals with the same starting indices, whose upper endpoints are $92$ and $128$, respectively (twinheterodox, Hexagon 2611.00160v2, Section 8.2). The new bounds come from a different construction, based on powers of Krisman's hypergeometric kernel (Hexagon 2610.00285v2).

Author comments: 13 pages. Powers of a hypergeometric kernel give smaller shifted intervals of even Dirichlet beta values containing an irrational member. Includes exact seed and arithmetic certificates, source records, and an independent AI audit. Sources and checks accompany the submission.

math.MG — Metric Geometry

Classification of fixed points of the centroid body operator

Contributed by Sihan Chen

We classify the fixed points, up to dilation, of the classical volume-normalized centroid body operator in every dimension. We prove that ΓK = cK holds precisely for ellipsoids centered at the origin. The proof combines a signed spherical moment-gap argument in low dimensions with boundary integral operators, compact-group structure, and quantitative regression estimates in higher dimensions. Affine eigenfunctions force a large rotation subspace; curvature and log-concave regression bounds then exclude every proper such subspace. The remaining finite comparisons are certified by exact rational inequalities. The manuscript supplies proofs of the auxiliary analytic estimates and explicitly identifies the external results used.

math.NT — Number Theory

Unitary character moments and a zero-free half-plane for unitary Weil L-functions

Contributed by Zhongpai Gao

We extend Liu's moment estimate for ordinary ideal-character sums to two independent fixed unitary twists, preserving the full length range of the finite-order theorem. The proof keeps the two character coefficients separate through the Gauss transformations and controls exceptional terms by the poles of their Mellin transforms. The case of two equal twists strengthens the fourth moment used in zero detection. Using the analytic results of Hu and Liu specified in the paper, we obtain the zero-free half-plane $\operatorname{Re}s>(777-\sqrt{1081})/852=0.8733819666\ldots$ for every unitary Hecke $L$-function over every number field, including characters of infinite order. This extends the known finite-order bound to the full unitary family. Unitary induction gives the same half-plane for finite-dimensional continuous unitary Weil representations, with the norm-character poles counted exactly.