All works

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

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.

math.CO — Combinatorics

The exact diameter of the symmetric group 𝑆𝑛 with LRX generators

Contributed by Sergei Uskov, Alexander Chervov, Fedor Levkovich-Maslyuk, Ivan Litvak, Andrei Smolensky, Alexander Soibelman, Dia Taha

We consider the undirected Cayley graph of the symmetric group 𝑆𝑛 with respect to the cyclic shift 𝐿, its inverse 𝑅, and the transposition 𝑋 of the first two positions. We prove that its diameter is given by 𝑛(𝑛 − 1)/2 for every 𝑛 ≥ 4, thus settling a conjecture open since at least 2011. This is done by separately establishing sharp upper and lower bounds, invoking a variety of combinatorial and group-theoretic arguments which leave a finite set of special cases that are verified in the accompanying C++ code. We also provide a Lean formalization of the results.

math.NT — Number Theory

Explicit bounds for odd perfect and quasiperfect numbers

Contributed by Frank Vega

Let $N$ be an odd perfect number or a quasiperfect number, and let $B(N) = \prod_{p \mid N} (1+1/p)$. We prove that $2 < N/\varphi(N) < 2.17372$ and that $105 \nmid N$, and we obtain explicit two-sided bounds for $B(N)$ according to the value of $\gcd(15,N)$. For an odd perfect number $N$, they give $B(N) > 1.90281$, $1.70584$, $1.83233$ and $1.67856$ when $\gcd(15,N) = 1$, $3$, $5$ and $15$, respectively, and $B(N) < 1.84348$, $1.93549$ and $1.84125$ when $\gcd(15,N) = 3$, $5$ and $15$. These follow from general bounds for $B(n)$ in terms of $n/\varphi(n)$, of the abundancy index $I(n) = \sigma(n)/n$, and of $\sum_{p \mid n} 1/p$. We also bound the abundancy indices of $N \operatorname{rad}(N)$ and $N/\operatorname{rad}(N)$ and the functions $\sigma^{\pm}$, $\varphi^{+}$ and $\psi^{+}$ of S\'andor and Atanassov at $N$, give an explicit lower bound for the number of distinct prime factors of $N$ in terms of its least prime factor, and show that the sum of the reciprocals of the prime factors of a quasiperfect number $N$ is less than $\log I(N)$. Finally, for the shifted abundancy indices $I^{+}(n) = 1 + \sum_{d \mid n, \, d > 1} 1/(d+1)$ and $I^{-}(n) = 1 + \sum_{d \mid n, \, d > 1} 1/(d-1)$, we determine the optimal constants in the comparisons of $I^{+}$, $I$ and $I^{-}$ on the odd integers and on the odd integers with at most one odd exponent, and we deduce $1.8471 < I^{+}(N) < 2 < I^{-}(N) < 2.2722$ and $I^{-}(N) - I^{+}(N) < 0.42493$ for every odd perfect number $N$.

math.NT — Number Theory

A proof that $\mathbb{Z}$ is Diophantine in $\mathbb{Q}$ through congruent-number twists

Contributed by Manvir Jaswal

We give a proof of Glazer and collaborators' theorem that $\mathbb{Z}$ is Diophantine in $\mathbb{Q}$, which builds on OpenAI's proof that Hilbert's tenth problem over $\mathbb{Q}$ has a negative answer. Integers are represented by multiples of a point on an elliptic curve of rank one over $\mathbb{Q}(\sqrt2)$. An existential condition on two congruent-number twists forbids poles of odd order at odd primes. A height bound at six rational points, from a quaternionic Shimura curve and Serre's conjecture (a theorem of Khare–Wintenberger and Kisin), and a prime pattern from the Green–Tao–Ziegler theorem turn congruences at chosen primes into identities between multiples, and the growth of a Lucas sequence forces integrality. Our contribution is the existence of the rational points that the condition on twists requires: for coprime squarefree integers $\beta>1$ and $\theta\ge1$ with $\beta\equiv\theta\equiv1\pmod8$, each a square modulo the primes of the other, there is an integer $e$ such that $y^2=x^3-e^2x$ and $y^2=x^3-e^2\theta^2x$ both have rank one and a rational point with abscissa in $\beta\mathbb{Q}^{\times2}$. Where Glazer and collaborators use a restricted $2$-converse theorem, this rests on published theorems: the genus criterion of Tian, Yuan and Zhang, with Gross–Zagier.

quant-ph — Quantum Physics

Genuinely multipartite bound entanglement of three qubits

Contributed by Antony Lewis

If a three-party state is positive under partial transposition (PPT) across each of its three bipartitions, no local protocol can distil any pure entanglement from it: neither singlets between any two parties nor GHZ states. All of its entanglement is bound. We show that such a state can nevertheless be genuinely multipartite entangled (GME), meaning that it cannot be prepared by mixing states in each of which some party is unentangled from the rest. This already happens for three qubits, the smallest system in which the question arises and a common experimental setting for multipartite entanglement; to our knowledge no three-qubit example was known. An explicit mixture of five orthogonal states (one GHZ-type, three $W$-type and one product state) is PPT across every cut and GME throughout an exact parameter interval, and its rank, five, is the minimum possible. Genuine entanglement is proved by a witness acting only on the five-dimensional support, whose nonnegativity on the biseparable states in that support reduces to a sum of squares on its curves of product vectors. Such states are invisible to the PPT-mixture test, the standard semidefinite relaxation used to certify GME, yet they form a set with nonempty interior among density matrices: an explicit full-rank example remains GME under $2.33\%$ white noise, while $2.336\%$ makes it biseparable. We also construct extreme examples with the smallest possible partial-transpose ranks $(5,5,5,5)$. All our explicit states, including the full-rank one, lie in the mixed $W$ class: they are mixtures of pure states with vanishing three-tangle, so their genuine entanglement needs no GHZ-type component. Full PPT therefore does not imply biseparability, even for three qubits.

math.AG — Algebraic Geometry

Laurent Phenomenon Algebras on Minuscule Grassmannians and their Mirrors

Contributed by Jiarui Fei

We construct Laurent phenomenon algebra structures on the homogeneous coordinate rings of the half-spin Grassmannians and the Freudenthal variety. Their frozen variables are the linear coordinates in a Coxeter orbit, and the resulting boundary divisors are anticanonical. Together with the construction of Daisey--Ducat for the Cayley plane, these examples admit explicit mirrors described by Newton polytopes. We identify the upper Laurent phenomenon algebras of the mirror seeds with the intrinsic mirror algebras of Keel--Yu at the specialization $q^\beta=1$, including the theta functions associated with the individual boundary components. The half-spin construction is uniform in the rank. In the exceptional cases, contact calculations and Laurent divisibility determine the local expansions; the Freudenthal comparison also uses exact calculations of divisorial valuations. We obtain integral comparisons and valuative independence of the intrinsic theta basis in every reachable seed.

Author comments: 57 pages, comments are welcome

math.RT — Representation Theory

Marked-point modules and central nilpotents at roots of unity

Contributed by Zhenyu Sun

We study extensions in the two-marked-point representation of the periodic Temperley--Lieb algebra at $q=e^{i\pi/p}$, $p\in\mathbb Z_{\geq3}$. For every positive even size $N$, we construct a submodule $Z_N$ extending $T_N=W_{N;2,\varepsilon}$ by $W_N=W_{N;0,\varepsilon q}$, where $\varepsilon=\pm1$. The dual trees of noncrossing matchings give an explicit injection $\Theta_N:T_N\to W_N$. Both central braid transfer elements have nilpotent part $g\Theta_N\pi_N$ on $Z_N$, where $\pi_N$ is the quotient map and $g=x+x^{-1}+\varepsilon(y+y^{-1})$ for the marked-point parameters $x,y\in\mathbb{C}^\times$. The extension splits exactly when $g=0$, and the formula determines the central Jordan forms at every even size. In the window $2\leq N<2p-2$, $Z_N$ is a joint central summand and, when $g\ne0$, has simple layers $T_N/S_N/T_N$. An explicit intertwiner and annular localization identify its two-layer quotient with a periodic XXZ sector. The associated basic image algebras have dimensions five and four; the quotient preserves first extensions between simples and kills a nonzero Yoneda product in degree two. An annular operator detects the extension with coefficient $2\varepsilon g$. Beyond the isolated window, we determine a five-simple image algebra and compute a $43$-dimensional joint central summand at $p=3,N=6$, on which the two central nilpotents generate a four-dimensional algebra. Exact matrix calculations and verification code accompany these finite computations.

math-ph — Mathematical Physics

Sharp Ionization Bounds for Helium and Lithium

Contributed by Alec Shelley, Charles Yang, Daniel Tartakovsky

How many electrons can a nucleus bind? In the non-relativistic Coulomb model, Lieb proved in 1984 that a nucleus of charge $Z$ binds fewer than $2Z+1$ electrons. While this bound has been improved, the sharp upper bound has remained unknown for any nucleus other than hydrogen. We solve this problem for $Z=2$ and $Z=3$ by proving that helium binds two or fewer electrons and lithium binds four or fewer. Both bounds are sharp. Our main tool is an adaptation of Lieb's argument with a quadratic weight, which reduces the nonbinding of doubly negative ions to finitely many inequalities that we verify in a computer-assisted proof.

quant-ph — Quantum Physics

Quantum-Information Probes of Collapse Dynamics in Causal Fermion Systems

Contributed by Benjamin Kaminsky

Finster, Kleiner, and Paganini derived from causal fermion systems (CFS) a nonrelativistic effective collapse model whose statistical operator obeys, to leading order, a Kossakowski--Lindblad equation. In the factorized regime of that derivation, the dissipator is built from purely spatial operators $N_{k,a}$ with positive rates fixed by microscopic time-nonlocal kernels. We translate this CFS-specific generator into quantities accessible to coherent quantum processors. The resulting CFS Dirichlet form obeys an exact purity-loss identity under arbitrary Hamiltonian control, and its null algebra is the commutant of the $*$-algebra generated by the projected $N_{k,a}$. For joint eigenbranches we obtain an exact distinguishability functional controlling exponential coherence loss and a lower bound on channel distance from ideal unitary evolution. In an $n$-qubit dephasing sector, the projected kernels generate a positive-semidefinite covariance matrix $C$, with GHZ visibility $\exp[-2T\bm{1}^{\mathsf T}C\bm{1}]$. We prove that uniformly summable finite-range correlations yield at most $O(n)$ GHZ decay, whereas $O(n^2)$ scaling requires a collective non-summable component. Single- and two-qubit cat coherences reconstruct $C$, giving a covariance-spectroscopy protocol, while Knill--Laflamme conditions characterize first-order correctability of the projected CFS error algebra. We also derive an explicit compatibility inequality between macroscopic state reduction and long coherent computation and show why absence of probe heating does not imply absence of coherence signatures. The results do not exclude CFS: its covariance strength, $\ellmin$, and hardware projection remain underdetermined. They instead identify the microscopic input required for a genuine no-go theorem and establish scalable quantum processors as direct probes of the CFS collapse sector.

math.RT — Representation Theory

The Quantum Kazhdan–Lusztig Conjecture for Quantum Loop Algebras

Contributed by Qiyue Tang

We prove the quantum Kazhdan–Lusztig conjecture for finite-dimensional type 1 representations of untwisted quantum loop algebras at every non-root-of-unity complex parameter.

Author comments: By posting this manuscript on this platform, I do not seek to assert ownership of the results or claim priority. I welcome anyone interested in preparing a finished account or checking the arguments; the eventual scholarly credit should go to those who undertake that work.

math-ph — Mathematical Physics

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

Contributed by Mark Goldstein, Alfredo Guevara

We study scalar quantum electrodynamics in three dimensions (QED$_3$) in its Higgs phase: the three-dimensional lattice abelian Higgs model in unitary gauge, with the Villain action, on lattices whose spacing tends to zero with canonically scaled couplings. At each spacing we construct infinite-volume states, as limit points of the periodic states and as limits of states in finite boxes; they carry the lattice symmetries, reflection positivity and bounds uniform in the volume. For the smeared curvature and bounded functions of the density, the generating functionals converge along a subsequence of spacings to a translation-invariant limit functional with the continuity required for the Osterwalder–Schrader reconstruction. For the limits built on the box states, and on the periodic states along dyadic sequences of sides, the limit curvature field satisfies the axioms OS0–OS3, with invariance under the full Euclidean group. The cluster axiom OS4 is proved in a weak form, on almost every ergodic component. The tools are Gaussian domination of the Villain curvature, Ginibre inequalities and reflection positivity.

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

math.GM — General Mathematics

An Elementary Proof of Fundamental Theorem of Algebra over Rationals

Contributed by liang zhang, Xiaohua(Michael) Xuan

We, with the aid of AI, give an elementary proof of the Fundamental Theorem of Algebra over Rationals, i.e., for any nonconstant polynomial $f$ with rational complex number coefficients and every positive rational number $\eps$, there exists a rational complex number $z$ such that $Q(f(z))<\eps^2$. The proof is constructive and relies only on finite rational arithmetic, without using real numbers, infinite sets, or related concepts.

math.NT — Number Theory

An Explicit Lower Bound for the Proportion of Isolated Zeros of the Riemann Zeta Function

Contributed by Cheolmin Kim

We study isolation in the full sequence of nontrivial zeros of the Riemann zeta function with positive ordinate, counted with multiplicity. For every fixed $0<c\le10^{-5}$, we prove unconditionally that the lower limiting proportion of zeros whose two neighboring gaps both exceed $2\pi c/\log X$ is at least $0.672468$. Here $X$ is the common height cutoff and the neighbors are taken from the infinite sequence. This refines Lamzouri's small-fixed-gap bound $0.6552\ldots$. We retain his isolation argument and the Montgomery–Taylor kernel, while replacing the linear cost of horizontal displacement by a certified majorant with a positive threshold. Young's shifted mollified second moment bounds the corresponding truncated displacement moment. We supply the boundary and global-height arguments and obtain a gap loss $6\pi^2c^2$ uniform under the required smoothing. A fixed interval-arithmetic certificate completes the numerical estimate. The result is asymptotic and gives no effective first height.

Author comments: This submission corresponds to version v0.4 of the preprint available on Zenodo: https://doi.org/10.5281/zenodo.24126375 The author understands the research question, the historical context, and the meaning of the claimed result. However, the detailed analytic arguments and computer-assisted numerical certification have not been independently verified or fully understood by the author. The mathematical development, internal proof checking, and manuscript preparation relied substantially on AI assistance, primarily GPT and Claude, with supplementary input from Gemini. The Zenodo record includes the full manuscript, LaTeX source, numerical certificate, and reproducibility materials. The manuscript has not undergone independent human peer review.

math.PR — Probability

Finite-preparation response under heavy-tailed catastrophes: exact amplitudes and separation from Poisson coverage

Contributed by Man Huang

We determine the terminal response of a one-dimensional binary recovery process to a finite preparation history under independently renewed Poisson interval catastrophes. For regularly varying interval weights $\nu_r\sim r^{-p}L(r)$, $p>3$, and fixed small positive intensities, we prove an exact leading response when preparation and observation durations grow proportionally. The main result expresses the response as an explicit functional of the actual recovery-front speeds accumulated along each fixed finite-stage preparation history and of the selected target-phase open density. Its weak-intensity second variation is a strictly positive quadratic form in every nonzero fixed signed temporal modulation, including equal-dose redistributions. A separate four-update theorem identifies local cooperation through an explicit positive equal-dose contrast for every nonzero interval law with finite first moment. These positive contrasts separate the process from conditional Poisson Boolean coverage with fixed grain kernels, a fixed environment distribution and affine intensity control, whose contrast is nonpositive at every finite time. As an effective extension, relative power-law tail bounds yield a conservative computable sufficient threshold valid at every later integer time. The core proof combines nonstationary front transfer, uniform background forgetting and exact resetting at intensity switches.

Author comments: Mathematical content and references follow Zenodo v1 (3 November 2026), https://doi.org/10.5281/zenodo.23912425. AI-use and verification-status declaration clarified; no mathematical results changed. Supplement and reproducibility archive: same DOI.

math.NT — Number Theory

A boundary limit for Suzuki's arithmetic canonical system

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

For the shifted Riemann xi-function, we prove the large-radius boundary limit of Suzuki's arithmetic canonical system. The proof applies to every positive shift for which the reflected quotient is pole-free in the upper half-plane and the two entire functions have no common zero. It therefore applies unconditionally to shifts at least one half, including Suzuki's original range greater than one. We give the weak-kernel construction needed at the smaller shifts, two integral identities for its positive determinant ratio, and decay estimates faster than every fixed exponential in logarithmic radius. The hypotheses on the shift are retained throughout.

cs.FL — Formal Languages and Automata Theory

A superlogarithmic lower bound for deterministic word separation

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

Let $S(n)$ be the maximum, over distinct binary words of length $n$, of the least number of states in a complete deterministic finite automaton that accepts exactly one of them. We prove $S(n)\geq(1-o(1))\log n\log\log n$. The main step is a common word of length $\exp(O(\sqrt{k}\log k))$ that, after suitable powering of the generators, reaches the sink components of every transformation pair on $k$ states and attains minimum rank on each sink. This transfers the balanced positive permutation identities of Xu and Regan to the full transformation monoid without changing their leading logarithmic length. The conclusion concerns deterministic automata; it makes no assertion about nondeterministic separators.

math.DS — Dynamical Systems

An ergodic classification of smooth type subshifts over odd two-letter alphabets

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

Fix distinct positive odd integers $\alpha<\beta$. We classify the invariant probability measures of the smooth subshifts obtained by prescribing the successive run types. Every such subshift is uniquely ergodic if $\alpha\ge3$ or $\{\alpha,\beta\}=\{1,3\},\{1,5\}$. For $\alpha=1$ and $\beta\ge7$, unique ergodicity holds exactly for directives containing infinitely many zeros; every other directive gives precisely two ergodic measures. The proof combines a two-dimensional frequency recursion with recognized run decompositions and explicit control of the density of decomposition boundaries.

math.CO — Combinatorics

Periodic counterexamples to three conjectures on Kolakoski transforms

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

The simple Kolakoski transform need not preserve the limiting mean of a periodic word on $\{1,2\}$. We give a counterexample with input period $24$ and output period $12$, and a balanced counterexample whose input has mean $3/2$ while its output has mean $13/9$. Both examples remain unchanged under every realization of a nonconstant summable perturbation of the weights. Their discrepancies from the input means are linear. Consequently they contradict Conjectures (1), (2), and (4) of Cloitre. These statements concern auxiliary transforms of periodic inputs; the classical Kolakoski frequency problem remains open.

cs.FL — Formal Languages and Automata Theory

Nonstabilization of the Predictive Boolean Lookahead Hierarchy

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

We study the hierarchy of languages generated by predictive Boolean $\mathrm{LL}(k)$ grammars. The model uses a deterministic recursive-descent table, strongly non-left-recursive rules, and at least one positive conjunct per rule. For every integer $r\geq1$, we construct a language over a fixed eight-letter alphabet that has an ordinary $\mathrm{LL}(r+r2^r+2)$ grammar and establish that it has no predictive Boolean $\mathrm{LL}(r)$ grammar. The construction asks for a specified token in a uniquely decodable word. The lower bound represents parsing on a long unary prefix by finitely many forward-moving endpoint streams and a finite Boolean recurrence. Random binary blocks followed by homogeneous blocks separate streams with a persistent timing error from streams having an exact finite-state potential. A bounded-lookahead argument hides a token-count change from each remaining stream. An encoding of the resulting independent queries is then shorter than their arbitrary binary payload. This gives nonstabilization under the stated predictive convention. The conclusion is for the stated parser model, including positive-conjunct and strong non-left-recursion requirements. Path-dependent tables satisfying these requirements give the same language families.