All works

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.

Author comments: Version 3 adds one citation (introduction).

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.

math.NT — Number Theory

Joint distribution of factorial units and valuations in translated intervals

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

For every fixed finite set of primes, the corresponding prime valuations of factorials, reduced modulo arbitrary fixed integers, and the prime-free parts, reduced modulo arbitrary fixed prime powers, have a joint uniform distribution. We give a proof with a power-saving error on every interval of consecutive nonnegative integers, uniform in the interval's starting point. A fixed congruence on the index may be imposed independently. The resulting statement for the part of a factorial coprime to an arbitrary fixed modulus gives the densities and relative density asked for in Problems 2 and 3 of Hajdu and Papp. The proof uses a finite digit-window identity, a strict local character contraction, and a carry estimate followed by finite Fourier analysis.

math.LO — Logic

Borel Ruziewicz problem

Contributed by Konstantin Slutsky

We prove that normalized surface measure is the unique finitely additive rotation-invariant probability on the Borel subsets of $S^n$, $n\geq2$. We also prove that Lebesgue measure is the unique finitely additive measure on bounded Borel subsets of $\R^d$, $d\geq3$, invariant under orientation-preserving rigid motions and normalized on the unit cube. The main step is the annihilation of Borel null sets. A spectral gap first gives bounded divergence certificates for compact null sets, chosen in a jointly Borel way. A shared random threshold then combines these certificates to cover arbitrary null sets. For Euclidean space, the construction is applied to families of spheres about two centers; spherical averaging completes the identification of the measure.

math.CV — Complex Variables

Smale's mean value constant (1981) exceeds 1.1559, and other miscellanea

Contributed by Dave Goel

We prove that Smale's mean value constant exceeds $1.1559837$, improving the bound $1.1018$ of Jatar and Ng; after the first version of this paper, the author proved that it equals $4$. Smale proved in 1981 that for every complex polynomial $P$ of degree at least $2$ and every point $z$ with $P'(z)\neq0$, some critical point $\zeta$ of $P$ satisfies $|P(z)-P(\zeta)|\le K\,|z-\zeta|\,|P'(z)|$ with $K=4$, and he conjectured that $K=1$ suffices; $K^*$ is the best constant, called Smale's mean value constant. That conjecture was disproved in 2026. Jatar and Ng showed that $K^*$ equals the supremum of $1/B(0)$ over the admissible pairs $(W,B)$: $W$ is a normalised univalent map of the unit disk, $B$ is a finite Blaschke product with $B(0)>0$, and the derivative of the product $WB$ has no zero in the disk. We construct such a pair explicitly, with $B$ of degree two, from a polynomial of degree $18$ with Gaussian rational coefficients, and we verify the required properties by a computation in exact arithmetic. Writing $\mathcal B_d$ for this supremum restricted to $\deg B\le d$, we prove, partly with certified computation, that $1.1363375<\mathcal B_1<1.177801$, $\mathcal B_2<100/61$, and $\mathcal B_d\le4-0.00597/(d\log d)$ for every $d\ge2$, which improves the bound of Jatar and Ng for $d\ge452$. If a pair attains $\mathcal B_d$ with $d\ge1$, then $W$ does not extend holomorphically to any neighbourhood of the closed disk and is not rational. If $1/B_n(0)\to4$ along a sequence of admissible pairs $(W_n,B_n)$ (such sequences exist, since $K^*=4$), then after rotation $W_n$ tends to the Koebe function, as Jatar and Ng observed, and the zeros $a$ of $B_n$, weighted by $\log(1/|a|)$, have an explicit limit distribution. Every admissible pair $(W,B)$ with $W$ meromorphic on a neighbourhood of the closed disk has $\deg B\le\operatorname{wt}(W)$, a weighted count of the points of the circle where $|W|$ is stationary and $\arg W$ decreases.

math.CV — Complex Variables

Smale's mean value constant (1981) equals 4

Contributed by Dave Goel

For a complex polynomial p of degree d ≥ 2 with p(0) = 0 and p′(0) ≠ 0, let S(p) be the minimum of |p(c)/(c p′(0))| over the critical points c of p, and let K(d) be the supremum of S(p) over all such p of degree d. In 1981 Smale proved that K(d) ≤ 4 for every d, and asked whether 4 can be replaced by 1. We prove that Smale's mean value constant sup_d K(d) equals 4 and that K(d) → 4 as d → ∞, combining a reduction due to Jatar and Ng with an explicit construction. Each K(d) is known to be below 4, so the supremum is not attained. The main result has also been formalized in the Lean 4 proof assistant.

Author comments: The expanded version of this paper is Hexagon 2611.00283, at https://hexagonmath.org/2611.00283 and on Zenodo at https://doi.org/10.5281/zenodo.24069916

math.CV — Complex Variables

Smale's mean value constant (1981) equals 4, and other miscellanea

Contributed by Dave Goel

For a complex polynomial p of degree d ≥ 2 with p(0) = 0 and p′(0) ≠ 0, let S(p) be the minimum of |p(c)/(c p′(0))| over the critical points c of p, and let K(d) be the supremum of S(p) over all such p of degree d. In 1981 Smale proved that K(d) ≤ 4 for every d, and asked whether 4 can be replaced by 1. We prove that Smale's mean value constant sup_d K(d) equals 4 and that K(d) → 4 as d → ∞, combining a reduction due to Jatar and Ng with an explicit construction. Each K(d) is known to be below 4, so the supremum is not attained. The main result has also been formalized in the Lean 4 proof assistant. We give three further proofs: by a variant of the main construction in which part of the boundary of an auxiliary domain is a level curve of an entire function; by a Riemann map onto a domain with thin hairs; and by a Riemann surface whose conformal radius at its base point is near 1. We also include a complete proof of the known fact that K(d) does not decrease.

Author comments: This is the expanded version of the paper Hexagon 2611.00282, "Smale's mean value constant (1981) equals 4", at https://hexagonmath.org/2611.00282 and on Zenodo at https://doi.org/10.5281/zenodo.24069884

math.QA — Quantum Algebra

Cayley graphs of abelian groups and their quantum symmetries

Contributed by Teo Banica

Building on the recent work of Marashdeh on the circulant graphs of prime order, we study the quantum symmetries of the Cayley graphs of the finite abelian groups $A$. Our main result states that if the coherent algebra is the orbital algebra of $A\rtimes E$, with $|E|=k$ and with the non-trivial elements of $E$ having at most $m$ fixed points, then the graph has no quantum symmetry when $|A|\geq k^2m$. We also settle the case of the circulant graphs with $p^n$ vertices.

math.DS — Dynamical Systems

Singular perpendicular orbits and complete periodicity in rational triangles

Contributed by Joshua Boorstein

Let $\Delta$ be a triangle whose angles are rational multiples of $\pi$. A billiard path leaving a side $\sigma$ at a right angle either meets a vertex first — its starting point is then singular — or meets a side at a right angle again: $\sigma$ itself, at a new point, or another side, where it retraces itself. The singular points cut $\sigma$ into finitely many intervals, and each interval retraces or is carried onto another by the return map, reversing orientation (a mirror pair) or preserving it (a translation pair). We count these intervals exactly. The singular points of the sides perpendicular to a direction class are in bijection with the slots — a vertex with a direction of the class pointing inward — whose path meets a side at a right angle before any vertex, so the number of intervals is $v + S - 2p - s'$: $S$ slots, $v$ perpendicular sides, $p$ pairs of mutually reverse vertex-to-vertex paths meeting no side at a right angle, and $s'$ slots whose path never reaches a vertex. This bounds the retracing bands and translation pairs, sharply on isosceles triangles, and makes $s' = 0$ equivalent to complete periodicity of the class. Our main theorem is that $p = 0$ forces complete periodicity: a minimal component needs a swapped pair. The converse fails. In every isosceles triangle $(a, a, b)$ with $N$ even and $a \geq 3$ the class of the base direction has $p \geq 1$, and it is completely periodic, by a companion paper (computer-assisted in part), whenever $b \geq 4$ and $N$ has at most one odd prime factor. At odd $N$, two infinite families of scalene triangles have $p \geq 1$ and are completely periodic in the checked range.

Author comments: First version. Companion papers: "The orphan theorem" (hexagon:2611.00191), "The necklace of a rational right triangle" (hexagon:2611.00213) and "No arrivals" (hexagon:2611.00250), whose results this paper imports. Code: https://doi.org/10.5281/zenodo.24093456

math.DS — Dynamical Systems

Non-perpendicular periodic orbits in mirror directions of right triangles

Contributed by Joshua Boorstein

Let $T$ be a right triangle with an acute angle $a$. A direction is a mirror direction of $T$ if it is the axis of a reflection in the group generated by the reflections in the sides of $T$ — for instance, any direction parallel or perpendicular to a side. A billiard path that meets a side at a right angle retraces itself, its directions are mirror directions, and it is periodic as soon as it meets a side at a right angle a second time. We ask whether every periodic billiard path in a mirror direction is of this kind. When $\cos 2a$ is transcendental it is: every such path meets a side at a right angle, and in the directions parallel to the hypotenuse there is no periodic path at all. The proof is a holonomy computation on Hooper's chain of rhombi, and it bounds the degree of $\cos 2a$ for any exception. When $\cos 2a$ is algebraic and $a/\pi$ irrational, exceptions can occur: we construct non-perpendicular periodic cylinders in mirror directions explicitly — among them an infinite family indexed by the solutions of $u^2 - 2v^2 = -1$ — and show that the triangles carrying them form a countable dense set in each of the four classes of mirror directions. Corollaries include the absence of odd-period orbits in right triangles and the existence of one-way periodic paths in every right triangle (answering two questions of Holt), a description of the odd orbits of isosceles triangles, and the statement that Ruijgrok's third conjecture, read for right triangles, fails on a countable dense set of triangles, all with $\cos 2a$ algebraic, and holds whenever $\cos 2a$ is transcendental.

Author comments: First version. Companion paper: "The necklace of a rational right triangle" (hexagon:2611.00213). Code: https://doi.org/10.5281/zenodo.24037744

math.CA — Classical Analysis and ODEs

Triangles in sets of positive upper density in the plane

Contributed by Polona Durcik

This note proves that every measurable subset of the plane with positive upper Banach density contains a translate of a rotation of every sufficiently large dilate of any fixed noncollinear triangle.

math.RT — Representation Theory

The Benson-Etingof-Ostrik conjecture

Contributed by Kevin Coulembier, Pavel Etingof, Johannes Flake, Joseph Newton, Victor Ostrik

We prove the conjecture of Benson--Etingof--Ostrik (2023): every symmetric tensor category of moderate growth over an algebraically closed field of positive characteristic admits a symmetric tensor functor to a higher Verlinde category, possibly of infinite height. Using maximal nilpotence and geometric reductivity, we reduce the problem to proving that finitely generated maximally nilpotent tensor categories are finite. The main ingredients are the construction of generic points using alternating norms and a Frobenius reduction of the fundamental coend to an ordinary Hopf subalgebra.

Author comments: The authors plan to turn this AI generated proof into a human paper.

math-ph — Mathematical Physics

Integral torus partition functions just above c=1 have no fixed spectral gap

Contributed by Tzu Chen Huang

We consider modular-invariant torus partition functions of central charge $c>1$ that have a unique vacuum, an expansion in non-degenerate Virasoro characters with positive integer multiplicities, integer spins, no nonvacuum state at the unitarity bound, and parity symmetry. We prove that for every $g>0$ there is $\epsilon(g)>0$ such that no such partition function with $1<c<1+\epsilon(g)$ has all nonvacuum primary dimensions at least $g$. Thus the spectral gap closes as $c\to 1^+$ in this class, as suggested by the numerical bootstrap study of Benjamin, Langhoff, Li and Thaler. The proof is by contradiction and compactness. Using the crystalline-measure reduction of Eberhardt and Levine, we show that every limit at $c=1$ is an equal-weight average of two compact-boson partition functions, at least one at a half-integer radius. Positive Gaussian test functions adapted to the two circle lattices then force every state of a nearby partition function, through a logarithmically growing energy window, to lie within a power of $c-1$ of a lattice point. On the other hand, the central-charge shift of the vacuum contributes a second-order term at zero momentum that no distribution supported on circle lattices and compatible with modular invariance can carry; a finite family of Gaussian tests detects this. Limits with two half-integer radii, where lattice points collide, require a separate analysis. The threshold $\epsilon(g)$ is not effective; we describe the additional estimates that would make it so.

math.FA — Functional Analysis

The Daugavet Property of Bidual Spaces

Contributed by Samir Hamad

We construct a separable Banach space whose bidual has the Daugavet property. The construction proceeds by successively extending finite-dimensional spaces so that prescribed vectors are approximated by averages of newly added unit vectors. A simultaneous extension argument produces finitely many fixed functionals whose norm-attaining faces approximate the unit ball of the bidual.

math.CO — Combinatorics

The Wang Transform Inequality

Contributed by Yair Lavi

We prove the Wang transform inequality in every order: if $n$ is a positive integer, $A$ is a nonnegative $n\times n$ matrix whose row and column sums equal one, and $J_n$ has every entry $1/n$, then $\operatorname{per}A\ge \operatorname{per}\!\left(\frac{nJ_n+A}{n+1}\right)$. Equality holds exactly when $A=J_n$. We also obtain an explicit positive quadratic gap for every $n\ge2$.

math.CO — Combinatorics

A linear gap for the maximal cp-rank

Contributed by Yair Lavi

We prove that the maximal cp-rank $p_n$ satisfies $p_n\ge n(n-3)/2$ for odd $n\ge5$ and $p_n\ge n(n-3)/2-1$ for even $n\ge6$. Together with the known upper bound $p_n\le\binom{n+1}{2}-4$, this gives $p_n=n^2/2+O(n)$.

math.CO — Combinatorics

The Maximum Diagonal Product of a Stochastic Matrix of Bounded Rank

Contributed by Yair Lavi

Let $A=(a_{ij})$ be a nonnegative row-stochastic or column-stochastic $n\times n$ matrix of rank at most $k$, where $1\le k\le n$. Write $n=qk+s$, where $0\le s<k$. The author conjectured in 2018 that, for every permutation $σ$ of $\{1,\ldots,n\}$, $\prod_{i=1}^n a_{i,σ(i)}\le q^{-q(k-s)}(q+1)^{-(q+1)s}$. We prove this bound and classify all equality cases.

math-ph — Mathematical Physics

Notes on Wasserstein distance and the JT gravitational ensemble

Contributed by Ville Samuli Keränen

This note completes the study of the Boltzmann–Wasserstein (BW) distance, an optimal-transport metric on quantum theories, begun in a companion paper (arXiv:2606.01296). There the distance was evaluated between macroscopically different theories; here we take the opposite limit, two independent draws of the JT-gravity ensemble, which are identical at the classical level. The distance is computable and is controlled by the connected spectral form factor: the ramp, the double-trumpet wormhole, regulated by the plateau, the spectral discreteness. The mean is exponentially small, $\langle \mathcal{W}^2\rangle \sim S_0\, e^{-S_0}$, whereas uncorrelated (Poisson) levels would give an unsuppressed $O(1)$ distance: the self-distance is a sharp diagnostic of spectral chaos. As a metric, the BW distance renders the ensemble a point cloud: exponentially small, strongly anisotropic, its leading principal axis a coherent “breathing” deformation of the low-energy spectrum. Its effective dimension grows asymptotically as $D_{\mathrm{eff}} \simeq 0.42\, S_0^2$, so the distance self-averages, its relative width shrinking as $\mathrm{std}/\mathrm{mean} \sim 1/S_0$. At low temperatures it localizes on the Airy edge, where a single Tracy–Widom mode dominates and self-averaging fails. We test the picture with exact-diagonalization GUE and SYK simulations: in GUE the effective dimension grows logarithmically at every temperature and is predicted at finite $N$, with no free parameters, by the same covariance; unfolded SYK exhibits the predicted anisotropy and mode structure of the cloud. Together with the companion result, this fixes the resolution of the metric: the microstate rigidity computed here is the noise floor beneath a macroscopic difference between theories.

math.PR — Probability

The Ultimate Matching Theorem

Contributed by Michael Goldman, Francesco Grotto, Dario Trevisan

Let $X_1,\dots,X_N$ be independent uniform points in the unit square, matched either to an evenly spread deterministic configuration or to a second independent sample. For exponents $\alpha_1,\alpha_2>2$ with $1/\alpha_1+1/\alpha_2=1/2$, we prove that with probability tending to one a single permutation controls the $j$th coordinate displacements in the exponential Orlicz norm of exponent $\alpha_j$, at the scale $\sqrt{\log N/N}$, for both $j=1,2$. This settles Talagrand's Ultimate Matching Conjecture for all pairs of finite critical exponents. The proof constructs, on the flat torus, a current whose divergence is Gaussian white noise projected on a dyadic grid and whose coordinates have the critical anisotropic exponential Orlicz bounds, and transfers it to matching through a primal optimal-transport reduction. At the hard endpoints $(2,\infty)$ and $(\infty,2)$ the conjecture fails: a Brownian local-time construction shows that the normalized optimal matching cost diverges in probability.

math.IT — Information Theory

Energy complexity and dissipation of noisy consensus

Contributed by Zhenyu Sun

We characterize the signal-energy complexity of distributed agreement over independent Gaussian point-to-point channels. For $n$ processes deciding $L$ binary coordinates under coordinatewise unanimity validity and a joint failure probability of at most $\varepsilon$, every finite causal protocol requires a worst-input expected signal energy of $\Omega(N_0n[L+\ln(n/\varepsilon)])$. This lower bound combines two distinct mechanisms: an exact relative-entropy identity for locally interchanged reception histories yields the reliability term even in the presence of shared randomness and active noisy feedback, while a coordinatewise direct-sum argument establishes the payload term. In the fault-free setting, a deterministic fixed-leader protocol attains this optimal order for all $0<\varepsilon\leq1/8$. In the presence of $f\leq(1/3-\kappa)n$ static Byzantine faults, the same order is achieved in the polynomial-error regime $\varepsilon\geq n^{-a}$ via public committee sampling and robust joint decoding within explicit finite storage and computation budgets. Consequently, batching amortizes the reliability overhead, yielding an energy cost of $\Theta(N_0n[1+L^{-1}\ln(n/\varepsilon)])$ per coordinate. We then connect this communication optimum to physical dissipation through a stochastic-thermodynamic model of a thermal channel port. Under a non-recovering mode replacement interface, the communication converse translates directly into lower bounds on entropy production and, when the endpoint mean internal energies agree, cyclic work. When work recovery is permitted prior to mode reset, we derive the exact minimum reset entropy production within a bounded-displacement control class as a function of the information accessible to the recovery controller, thereby separating the fundamental communication cost from its control-dependent thermodynamic realization.

math.PR — Probability

Closing the identification step in the GFF limit: Uniqueness of the zero-slope six-vertex phase for −1 ≤ Δ < 1/2

Contributed by Yan Ru Pei

For the six-vertex model with $a=b=1$ and $c\in(1,2]$, that is $-1\le\Delta<\frac{1}{2}$, we prove that the plane measure, the limit of balanced torus measures, is the only translation-ergodic gradient Gibbs measure of slope zero, and hence the unique translation-invariant minimiser of the specific free energy. Both statements had been asserted by reference to Sheffield's uniqueness theory, which is proved for nearest-neighbour potentials; we show that for $c\ne1$ the six-vertex weights, which couple faces across the diagonals of each vertex, are not a nearest-neighbour potential in height coordinates. The proof extends Sheffield's cluster swapping to these diagonal interactions by means of residual energies on the diagonals, shows with the Lanford–Ruelle theorem for subshifts of finite type and a Burton–Keane argument that two distinct ergodic zero-slope phases are gradients of Gibbs measures on heights, and shows from the spin representation in the form given by Duminil-Copin, Kozlowski, Lammers and Manolescu that the plane measure is not. This supplies a step that we found missing in the chain of recent results on the Gaussian free field limit of the six-vertex height function. An unrefereed manuscript of OpenAI states the full-plane limit for every $c\in(0,2]$; for $\sqrt{2}\le c<2$ it identifies a state built from nearly balanced sectors with the plane measure through the characterisation above. Our result gives that identification, given two properties of the state stated in the manuscript. We also check that the manuscript and the work of Averous, Duminil-Copin, He, Lammers and Manolescu concern the same measure, so that their convergence in domains holds, for $c\in[1,2]$ and with constants depending on $\Delta$, wherever the manuscript's full-plane theorem holds.

math.PR — Probability

Stochastic Connections on Surface Bundles

Contributed by Zhenyu Sun

We construct local stochastic connection dynamics on arbitrary principal bundles over closed oriented Riemannian surfaces. For each smooth reference connection, the bare additive spatial heat approximations of the Yang–Mills–DeTurck equation converge to a maximal local strong Markov family in a separable negative Hölder space. A finite Gaussian enhancement defines the singular products, and cancellation of the total logarithmic mass identifies the limiting equation. Rough comparison gauges and predictable rotations of the Wiener noise identify the local orbit laws for different references. The finite quadratic tensor anomaly vanishes after the two-dimensional Yang–Mills contraction, with the finite normalisation fixed by the spatial heat approximation. Continuing through bounded representatives gives a reference-independent maximal local strong Markov family on the full rough orbit space. Gaussian line primitives have every Hölder exponent below one on bounded regular curve families and define non-Abelian Young parallel transport. On a dense, gauge-invariant Borel full-heat trace domain, polygonal holonomies separate rough gauge orbits, admit Borel representative maps, and recover the bundle class. The orbit dynamics has continuous holonomy paths on this domain, and the regularised holonomies converge uniformly on bounded regular curve families and compact surviving positive-time intervals. We also prove regularisation-independent convergence of first-order Gaussian currents after time integration, including a harmonic example of infinite variation and zero quadratic variation. At fixed regularisation scale, the field-dependent covariant heat approximation has classical solutions and an explicit Itô correction in Coulomb coordinates. In every Abelian sector, the quotient diffusion is global, reference independent, and reversible with an explicitly identified unique invariant law.

math.CO — Combinatorics

Positivity for Weighted Linear Laurent Phenomenon Algebras

Contributed by Qiyue Tang

We prove Laurent positivity for ordinary Laurent phenomenon algebras whose initial exchange polynomials are nonconstant affine linear polynomials with positive constant terms and nonnegative numerical coefficients.

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.RT — Representation Theory

Broué's Abelian Defect Group Conjecture for 2-Blocks

Contributed by Qiyue Tang

We prove that every 2-block of a finite group with an abelian defect group is splendidly Rickard equivalent to its Brauer correspondent.

Author comments: v3（2026/11/14）：The mathematical content is unchanged. This revision adds a brief motivation statement and a link to the related research materials on GitHub.

math.CT — Category Theory

Higher Galois Recognition and Parameter Descent

Contributed by Zhenyu Sun

Let $n\geq0$ and let $\mathcal X$ be a bounded coherent $\infty$-topos. Suppose that finitely many noninitial coherent $n$-truncated objects cover its terminal object, and that every coherent $n$-truncated object in each of their slices has a finite covering family of sections. We identify $\mathcal X^{\mathrm{coh}}_{\leq n}$ with $\operatorname{Fun}(M,\mathcal S_{\pi,\leq n})$ for a $\pi$-finite $(n+1)$-type $M$, recovered by all finite models preserving finite limits, finite coproducts, and effective epimorphisms. Descent uses degrees through $n+2$ for objects and $n+1$ for maps; both bounds are sharp for finite homotopy actions. Every exact functor $F$ between recognized categories has a universal parameter $P_F=F_!(1)$ and a canonical factorization through its slice with fully faithful second functor. This parameter characterizes full faithfulness, conservativity, and all exact slice factorizations. For successive abelian Postnikov layers, we identify the space of marked exact lifts with a path space of Eilenberg--Mac Lane sections with local coefficients. The obstruction, classes of lifts, and relative automorphisms occur in degrees $r+1$, $r$, and $r-1$. We compute a twisted parity obstruction and, for odd primes, classify cyclic self-equivalence components by a $p$-adic valuation. A nonzero Postnikov class can persist in the identity component. For continuous finite families over a profinite $(n+1)$-type, we prove that a uniform finite-section atlas exists precisely when the complete model space is $\pi$-finite. Exact functors from finite model semantics, together with their lift spaces, satisfy finite-stage descent, while explicit profinite examples separate objectwise trivialization from uniform parameter elimination.

Author comments: 49 pages. Second paper in the series Model groupoids and finite homotopy semantics. Includes two verification scripts and their outputs.

math.LO — Logic

A Michael Space in the Laver model

Contributed by Morgan Jones

We show that a Michael space exists in the Laver model.

Author comments: I was stress testing Claude Fable 5.2 with varying levels of problems and trying to improve my workflow with AI. This problem was reasoned pretty quickly by Claude, however it required some convincing that it could do it.

math.NT — Number Theory

Watkins's Conjecture for Elliptic Curves with Rational Two-Torsion

Contributed by Qiyue Tang

We prove that the modular degree of every elliptic curve $E/\mathbb Q$ with $E(\mathbb Q)[2]\ne 0$ is divisible by $2^{\operatorname{rank}E(\mathbb Q)}$.

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.CO — Combinatorics

Positive bases and polytopes for Laurent phenomenon algebras

Contributed by Qiyue Tang

We prove that the cluster monomials of every graph Laurent phenomenon algebra form an atomic positive basis. For every finite normalized Laurent phenomenon algebra over a unique factorization domain, we construct a simple lattice polytope with unimodular normal fan whose polar realizes the cluster complex. We also prove that cluster monomials form a basis of the upper algebra in the finite case. In particular, every linear Laurent phenomenon algebra admits such a polyhedral realization.

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.LO — Logic

The 51-state beeper $M_{51}$: every battle ends, and what Peano Arithmetic can prove about it

Contributed by Annanay Kapila

We study an explicit 51-state, 2-symbol beeping Turing machine $M_{51}$ and its associated dispatch-level process $R$. An ordinal descent below $\varepsilon_0$ proves that every battle of $R$ terminates and hence that $M_{51}$ beeps infinitely often. A growth lemma shows that the battle counters and beep times eventually dominate every function provably computable in Peano Arithmetic. Together with a Rocq-checked simulation theorem, these results establish that quasihalting of $M_{51}$ is independent of Peano Arithmetic. The document includes the process definition, machine table, ordinal analysis, growth proof and an account of its external dependencies.

Author comments: Supporting materials: Rocq simulation development, https://github.com/a-kapila/maths-proofs/blob/2e2d689f198e7bd51d264cd5468d6b7606886d2e/beeping-busy-beaver-51/rocq51.tar.gz ; computational checks, https://github.com/a-kapila/maths-proofs/blob/2e2d689f198e7bd51d264cd5468d6b7606886d2e/beeping-busy-beaver-51/checks51.tar.gz . The Rocq materials support the simulation theorem; the complete independence argument is not claimed to be fully formalized.

math.FA — Functional Analysis

COUNTABLY INFINITE-DIMENSIONAL SPACES OF CRYSTALLINE MEASURES

Contributed by Jamie Martin

We construct a symmetric locally finite set Λ ⊂ R for which the space MΛ of crystalline measures with support and spectrum in Λ has countably infinite Hamel dimension; Meyer asked what the dimension of such spaces can be. More precisely, there is a sequence of crystalline measures such that every element of MΛ is a unique finite linear combination of them. Every layer of MΛ of fixed Hermite–Sobolev order is finite-dimensional, but the layer dimensions are unbounded; in one construction they grow linearly, and in general their growth can be prescribed up to a linear change of the order. (For every locally finite set the dimension is finite, countably infinite or equal to the continuum, and the countable case is exactly this behaviour of the layers.) The construction uses clusters of atoms that imitate derivatives of Dirac masses, which produce one-dimensional blocks of prescribed Hermite order. The main difficulty is to show that the union of infinitely many blocks supports no other crystalline measures. We prove this by placing the blocks at random scales, separated by rapidly growing gaps, and showing that every element of fixed Hermite order splits into finitely many block components. We also obtain disjoint sets A, B such that the crystalline measures with support in B and spectrum in A form a countably infinite-dimensional space, while MA = MB = {0}. The existence statement, without the symmetry of Λ, has been formalised and machine-checked in Lean 4; the formal development is recorded in Palomar

math.QA — Quantum Algebra

Another proof of quantum Laurent positivity, II

Contributed by Qiyue Tang

We give a further combinatorial proof of Laurent positivity for skew-symmetric quantum cluster algebras.

math.GR — Group Theory

Twisted conjugacy in spherical Artin groups and their crystallographic quotients

Contributed by Jason Davies

A group has property $R_\infty$ if every automorphism has infinitely many twisted conjugacy classes. Let $A$ be an irreducible spherical Artin group of Coxeter rank at least two, with pure subgroup $P$. We classify property $R_\infty$ for the crystallographic quotient $\Gamma=A/[P,P]$. If $A$ has Coxeter rank at least three, then $\Gamma$ has $R_\infty$. In type $I_2(m)$, this holds exactly when $4\mid m$, or when $m$ is odd and $3\mid m$. Together with the known dihedral Artin-group cases, the positive quotient results imply that every irreducible spherical Artin group of rank at least two has $R_\infty$. The positive proofs combine the rational representation theory of the holonomy action with integral lattice constraints and explicit fixed-vector witnesses. In the negative dihedral cases, we construct automorphisms compatible with the crystallographic extension. For type $I_2(5)$, we exhibit an automorphism with Reidemeister number $18$, showing concretely that the quotient construction does not preserve $R_\infty$. This example, including the identification with the presented Artin quotient and the exact Reidemeister count, is formalised in Lean. An alternative proof for seven Artin groups is formalised under explicitly stated external hypotheses.

math.GN — General Topology

Smooth functions with isolated critical values and systems of fibers

Contributed by Naok Kitazawa

In the 2020s, Marzantowicz and Michalak have introduced the notion of system of smooth hypersurfaces and discussed Morse functions compatible with them: the hypersurfaces are regular level sets of the functions. This is a kind of fundamental theory on Morse functions and related differential topology and geometric topology. We extend this to a certain case: we discuss systems of regular level sets and critical level sets (regular fibers and singular fibers) of smooth functions with isolated critical values.

math.NT — Number Theory

The Exact Second Order Term In Ramanujan's Bound For $\sigma(n)/n$

Contributed by Jacob Crim

Let $\sigma(n)$ be the sum of the divisors of $n$, and assume the Riemann hypothesis. Ramanujan showed that $\sigma(n)/n\le e^\gamma(\log\log n-2(\sqrt2-1)/\sqrt{\log n}+S_1(\log n)+O(1/(\sqrt{\log n}\log\log n)))$, where $S_1$ is a sum over the zeros of $\zeta$. Nicolas made this effective, placed the coefficient of $1/(\sqrt{\log n}\log\log n)$ in $[0.9567, 3.789]$ at colossally abundant numbers, and asked for its exact asymptotic value. We show that this coefficient is $\kappa+W_2(\log n)$ with $\kappa=\sqrt2(2+\log2)-2=1.80868\ldots$, where $W_2(x)=-\sum_\rho x^{i\Im\rho}/(1-\rho)^2$ oscillates in $[-\beta,\beta]$, $\beta=2+\gamma-\log4\pi$. The next coefficient is the constant $c_0=8-\sqrt2(8+4\log2+\frac34\log^2 2)=-7.7443\ldots$, up to an oscillation of amplitude at most $0.0066$. In particular, the rescaled residual studied numerically by Leshchiner, Musin and Thompson does not converge, and it exceeds $3/2$ at every colossally abundant number with parameter $\xi\ge4.4\cdot10^{15}$; this answers their Questions 3 and 4, the second in the negative. We also prove explicit two-sided bounds for the remainder at every colossally abundant number above Nicolas's threshold $N^{(0)}$, where $\log N^{(0)}\approx10^9$. This part is computer-assisted: rigorous interval arithmetic covers every real parameter $\xi\in[10^9,e^{300}]$, and an analytic argument covers $\xi>e^{300}$. As corollaries, $\sigma(n)/n\le e^\gamma(\log\log n-0.699/\sqrt{\log n})$ for all $n\ge N^{(0)}$, improving Nicolas's constant 0.582; Ramanujan's limsup is approached from above at every colossally abundant number with $\xi\ge10^9$; and an odd-integer analogue holds with an essentially sharp constant. The proof reads colossally abundant numbers as ground states of a free gas of prime modes, in which $\kappa$ splits into a prime-square term, a saturation-deficit term and a shell-energy term. All enclosures are checked against exact values computed from a sieve to $10^{10}$.