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math.FA — Functional Analysis

Column rigidity in noncommutative $\L^1$-spaces and completely isometric embeddings of Hilbertian operator spaces

Contributed by Cédric ARHANCET

We study exact embedding obstructions for Hilbertian operator spaces in noncommutative $\mathrm{L}^1$ -spaces. We prove a quantitative version of the column inequality, whose equality case forces the absolute values of the entries to be proportional. We deduce that an operator space embedded completely isometrically in a noncommutative $\mathrm{L}^1$ -space is a row Hilbert space whenever its first-column norms at the two-by-two level are Hilbert-Schmidt norms. This gives an exact embedding criterion for sums of weighted row and column spaces, including coordinate-dependent weights which are ordered in the same direction. In particular, $\mathrm{R}_n+\mathrm{C}_n$ does not embed completely isometrically into any noncommutative $\mathrm{L}^1$ -space when $n \geq 2$ . For $\mathrm{OH}_n$ , whose complete isometric embeddability is established in a companion paper, we retain the common-support and commutative-target obstructions. We also show that distortion one is approached by matrix models but is attained by none of them, and that the matrix size must tend to infinity as the distortion tends to one.

Primarily human-written textHuman understanding: all partsHaagerup Lp-spacesKhintchine inequalitiescomplete isometriesnoncommutative L1-spacesoperator Hilbert spacerow and column spaces

Nov 03, 2026PDF ↗View work →

hexagon:2611.00035v1

math.NT — Number Theory

Hecke eigenvalues in level 1 modulo powers of 2 and 3: Hatada's question (1981) and the Coleman-Stein conjecture

Contributed by David Niedbala Giraudin

Let $f$ be a normalized Hecke eigenform of level $1$ and weight $k$ , with eigenvalues $a_n(f)$ . In 1981 Hatada asked whether $a_p(f)\equiv 1+p \pmod{2^a}$ whenever $p\equiv\pm1\pmod{2^{a-1}}$ and $6\le a\le 13$ . Rustom's congruence modulo $128$ answers this for $a\le 7$ . We show that the answer is affirmative for $a\le 10$ and negative for $a=11$ , $12$ and $13$ , the counterexamples coming from weight $16$ and $p=12289$ . The proof combines Rustom's theorem with a certificate for $T_5$ modulo $512$ on period polynomials, a $2$ -adic Galois law (for instance $a_p\equiv 1+p^{k-1}\pmod{2^{10}}$ for every $p\equiv 1\pmod 8$ ), and a reduction to a computation in a finite group. As a consequence, $a_p(f)=0$ implies $p\equiv -1\pmod{2^{11}}$ . At the prime $3$ we show that, on Hatada's lattice of period polynomials, $T_2^2\equiv 9(k-1)^2\pmod{27}$ , with refinements modulo $81$ . This gives an independent proof of Conjecture 9.1 of Rustom without any ramification hypothesis and of the level $1$ case of the Coleman--Stein conjecture, both of which also follow from Rustom's recent classification modulo $81$ . We also prove that $a_n(f)\equiv\sigma_{k-1}(n)\pmod{81}$ for every $n\equiv 1\pmod 3$ , and a $3$ -adic Galois law modulo $243$ on the Galois group of $\mathbb{Q}(\sqrt{-3})$ , which gives congruences modulo $243$ and $729$ for suitable split primes and a bilinear law for primes $\ell\equiv 2\pmod 3$ . Beyond $729$ , for every prime $\ell\equiv 2\pmod 3$ we prove $a_\ell^2\equiv\sigma_{k-1}(\ell)^2\pmod{3^{\min(2j+2,10)}}$ , where $j=v_3(\ell+1)$ ; the exponent $2j+2$ is attained for $j\le 4$ and fails for $j=5$ . Each statement is marked as proved, known, verified or open. The finite computations are certified by seven Python scripts, provided as ancillary files.

Primarily AI-generated textHuman understanding: no partsColeman-Stein conjectureGalois representationsHecke eigenvaluescongruences of modular formsperiod polynomials

Nov 03, 2026PDF ↗View work →

hexagon:2611.00034v1

math.QA — Quantum Algebra

Haagerup symmetries of the $E_8$ theory and conformal nets at central charge eight

Contributed by Cédric ARHANCET

A Haagerup algebra is a positive strongly rational vertex operator algebra $U$ of central charge eight with $\Rep(U)$ braided equivalent to the Drinfeld center $Z(\cal H_3)$ of the Haagerup fusion category. Such an algebra has been proposed by Jiao, with a computer-assisted proof, and announced by Riesen and by Huang. We show that $U$ has exactly three holomorphic extensions, each isomorphic to $V_{E_8}$ , and that the categories of modules over the corresponding Lagrangian algebras are the three Haagerup fusion categories of Grossman and Snyder. The extensions attached to $\cal H_2$ and $\cal H_3$ are the two lattice extensions of $V_{E_6 \oplus A_2}$ . Along a strategy outlined by Huang, we prove a recognition theorem for $\cal H_2$ and $\cal H_3$ among unitary fusion categories of rank six with three invertible objects, a noncommutative fusion ring and a pointed part with trivial associator. For regular unitary vertex operator algebras with Tener's bounded localized vertex operators, we prove that the fusion product of two unitary modules is unitarizable when one of them integrates to the conformal net. The explicit unitary algebra $S \subset V_{E_6 \oplus A_2}$ of our previous work, isomorphic to Jiao's algebra, is strongly local. Granting that Jiao's algebra is a Haagerup algebra, we deduce that the conformal net $\cal A_S$ is completely rational with $\Rep(\cal A_S) \cong Z(\cal H_3)$, that the Haagerup subfactor arises from $\cal A_S$ , and that $\cal A_S$ solves a problem of Bischoff on generalized orbifolds. Under the same hypothesis, using a unitary structure on $V_{E_8 \oplus E_8}$ found by exact computer calculations, we prove that $S$ is completely unitary and that $\mathrm{Rep}(\cal A_S) \cong \mathrm{Rep}(S)$ through the correspondence given by smeared vertex operators. The diagonal construction and the passage to the twin algebra also show that every Haagerup algebra is completely unitary and satisfies Condition I+.

A mix of human-written and AI-generated textHuman understanding: some partsDrinfeld centerHaagerup subfactorLagrangian algebraconformal netgeneralized orbifoldvertex operator algebra

Nov 03, 2026PDF ↗View work →

hexagon:2611.00033v1

math.NT — Number Theory

On Bounds for Reciprocal Sums and Residues in Collatz Orbits, and Their Consequences

Contributed by Behzad Nikzad, Karl Frinkle, Fatemeh Bagherzadeh

We prove uniform bounds for reciprocal sums and residue products along Collatz paths. Let $\mathcal F$ be a finite nonrepeating shortcut path whose input states are greater than $1$ , let $\mathcal R_{\mathcal F}$ be the sum of the reciprocals of its odd inputs, and let $\rho_{\mathcal F}$ be its finite-path residue. Then $\mathcal R_{\mathcal F}<1, \qquad \rho_{\mathcal F}<\frac{4}{3}.$ For the first-arrival trajectory of a terminating positive integer $n$ , these inequalities give $\mathcal R(n)<1, \qquad \rho(n)=\frac{2^{E(n)}}{3^{O(n)}n}<\frac{4}{3},$ where $E(n)$ and $O(n)$ are the division and odd-operation counts. More generally, when each odd value greater than $1$ is counted once, the corresponding reciprocal-sum and residue bounds hold for every forward orbit, without a termination assumption. The proof combines verified finite height-delay records and residue bounds, canonical-block reciprocal estimates, and an exact finite count organized by gap brackets. Gap brackets, closely related to parity vectors, record the divisions by two between successive odd inputs and organize the finite counting argument; together with a related coordinate representation, they also yield exact formulas for reconstructing finite trajectories. After taking base- $2$ logarithms, the bound $\rho_{\mathcal F}<4/3$ leaves an uncertainty of less than $1/2$ in the relation between the even- and odd-operation counts. Since these counts are integers, either count is determined exactly by the other count and the path endpoints. We obtain nearest-integer restrictions on the changes in $E$ and $O$ between adjacent integers, apart from explicit small exceptions, and prove that adjacent terminating integers of equal height have identical operation counts. Thus every equal-height streak lies in a single operation-count class. We further introduce the charge $\lambda=E-2O$ and associated height and charge slice functions to study the persistence and eventual break of equal-height streaks. We distinguish class records, which are the least starting integers attaining specified values, from delay records, which attain new extrema relative to all smaller starting integers. The resulting record theory shows that $\lambda$ -delay records form a subsequence of the odd-count delay records, which form a subsequence of the common division-count and height-delay list. On the class side, the nonpositive charge-class records lie among the odd-count class records, which lie among the division-count class records, which in turn lie among the height-class records. The corresponding hierarchies also hold for first-arrival paths to a common endpoint $z$ when $z=1$ or when $z>2$ is nonperiodic under the shortcut map.

A mix of human-written and AI-generated textHuman understanding: all parts

Nov 03, 2026PDF ↗View work →

hexagon:2611.00032v1

math.NT — Number Theory

The Hankel determinants of an automatic sequence need not be automatic

Contributed by David Niedbala Giraudin

Allouche, Han and Niederreiter asked whether the sequence of Hankel determinants of a $d$ -automatic sequence must be $d$ -regular, or must become $d$ -automatic after reduction modulo an integer. We answer the first question in the negative, and the second one for the reduction modulo $3$ . Let $\alpha$ be the root of $f^4+f^2-Tf+1=0$ in $\mathbb{F}_3((1/T))$ with $|\alpha|<1$ , the quartic of Mills and Robbins, and let $S$ be its coefficient sequence, which is $3$ -automatic by Christol's theorem. The Hankel determinants of $S$ vanish in runs whose lengths are unbounded but of the order of the square root of their position, because the partial quotients of $\alpha$ are unbounded while its irrationality exponent is $2$ ; and a lemma of Schlage-Puchta forbids runs in that regime in an automatic sequence. Hence these Hankel determinants are $k$ -automatic for no $k$ , and the integer Hankel determinants of any lift of $S$ are $k$ -regular for no $k$ .

Primarily AI-generated textHuman understanding: no partsHankel determinantsautomatic sequencescontinued fractionspower series over finite fields

Nov 03, 2026PDF ↗View work →

hexagon:2611.00031v2

math.NT — Number Theory

The binary Collatz dynamics of the trinomials x^n+x+1: an independent proof of two conjectures of Gallardo and Rahavandrainy

Contributed by David Niedbala Giraudin

Gallardo and Rahavandrainy studied an analogue of the Collatz map on F_2[x], in which x(x+1) plays the role of 2 and x^2+x+1 that of 3: an odd polynomial A is sent to the odd part of 1+(x^2+x+1)A. They proved that every orbit reaches 1, and conjectured that the orbit of the trinomial x^n+x+1 has length 2^s+1, where s is the largest integer with 2^(s+1) < n, and that for 1 <= t <= s-1 it contains 2^t terms of the same degree. We prove both statements for every n >= 3. They were proved earlier, with a complete Lean 4 formalization, by W. Choi; the proof given here was obtained independently. The proof rests on the identity (x^2+x+1)+x(x+1)=1: it reduces the dynamics to a map from one plateau to the next, characterises the last plateau of any orbit, and shows that along the orbit of x^n+x+1 the plateaus start at the 2^j-th powers of the odd parts of the polynomials x^m(x^2+x+1)+1, the index m following an explicit binary recursion. We also note that a later restatement of the conjecture, with a non-strict inequality, fails exactly when n is a power of 2. A script checks all the statements numerically.

Primarily AI-generated textHuman understanding: no parts

Nov 03, 2026PDF ↗View work →

hexagon:2611.00030v1

math.NT — Number Theory

On the largest irreducible factor of an odd perfect polynomial over $\mathbb{F}_2$

Contributed by David Niedbala Giraudin

A polynomial $A$ over $\mathbb{F}_2$ is perfect if it equals the sum $\sigma(A)$ of all its divisors, and odd if it is coprime to $x^2+x$ . Canaday conjectured in 1941 that 1 is the only odd perfect polynomial. Known necessary conditions on a counterexample bound its number of distinct irreducible factors ( $\omega\ge 5$ , Gallardo and Rahavandrainy), its number of irreducible factors counted with multiplicity and its degree ( $\Omega\ge 12$ and $\deg A>200$ , Cengiz, Pollack and Trevino), and, when all exponents equal 2, the degree of its smallest irreducible factor (at least 40, Gallardo). We bound a different invariant: every odd perfect polynomial $A\ne 1$ has an irreducible factor of degree at least 20. The proof reduces the question, for each $D$ , to a finite system: if all irreducible factors of $A$ have degree at most $D$ , the exponents of $A$ belong to an explicit finite set, through the cyclotomic factorisation of $\sigma(P^h)$ and the multiplicative order of $P$ modulo the factors of $\sigma(P^h)$ . An iterated sieve on the resulting production relations empties for every $D\le 19$ . The computation was carried out twice, by two independent programs, with identical results; the second program is attached as an ancillary file. As a cross-check of the method, the same programs reproduce, by the orbit argument of Gallardo and Rahavandrainy, the exclusion of every prime of degree at most 30 as the smallest factor of a special perfect polynomial, in agreement with the published bounds.

Primarily AI-generated textHuman understanding: no partsCanaday conjecturefinite fieldsperfect polynomialssum of divisors

Nov 03, 2026PDF ↗View work →

hexagon:2611.00029v1

math.CO — Combinatorics

The inverse problem for the number of connected induced subgraphs

Contributed by Lucas Almeida Almeida

For a finite graph $G$ , let $c(G)$ be the number of nonempty vertex sets that induce a connected subgraph. We show that a positive integer $k$ equals $c(G)$ for some finite connected graph $G$ if and only if $k\notin\{2,4,5,8,9,16\}$ . Every attainable value $k\ne25$ is attained on $\lfloor\log_2 k\rfloor+1$ vertices, the least possible number, and $25$ needs six vertices. More precisely, for every $m\ge6$ each integer from $2^{m-1}$ to $2^m-1$ is the value of $c$ on a connected graph with exactly $m$ vertices. The proofs of these results are short and use no computer; computer calculations serve only as independent checks, and for two auxiliary results: the values on six to eight vertices, and the result on all connected subgraphs described below. The value $16$ falls into a gap above the path: a connected graph on $n\ge4$ vertices other than the path $P_n$ has at least $n-3$ more connected sets than $P_n$ . The values $k\ge17$ are realized by a recursive construction from cones, disjoint unions and a family of graphs that we call near-cones, starting from seven small graphs; it produces a graph with $c(G)=k$ in time polynomial in $\log k$ . The question arose from a problem of Ordoñez in continuum theory, through a reduction by Camargo, Ordoñez and Ramírez, who found the values up to $255$ by a computer search. The proof is formalized in Lean 4. We also observe that a theorem of Czabarka, Székely and Wagner on subtrees reduces the corresponding problem for the number of all connected subgraphs to a finite computation, and we report its outcome. We did not find these results in the literature, but we make no claim of novelty or priority.

A mix of human-written and AI-generated textHuman understanding: all partsLean 4connected induced subgraphsconnected setsformal verificationgraph invariantsinverse problem

Nov 03, 2026PDF ↗View work →

hexagon:2611.00028v1

math.CO — Combinatorics

Hilton’s conjecture on cycle multiplicities

Contributed by Qiyue Tang

We prove Hilton’s conjecture on the number of cycles of each length in dense Hamiltonian graphs. This resolves the problem posed by A. J.W. Hilton and recorded by Sheehan in 1977.

A mix of human-written and AI-generated textHuman understanding: some partsHilton’s conjecture

Nov 03, 2026PDF ↗View work →

hexagon:2611.00027v1

math.NT — Number Theory

Powers of a Prime in the Image of Summatory Valuation Functions

Contributed by Frank Vega

J.-L. Nicolas asked how often a power of $2$ belongs to the image of $c(n)=\sum_{k\le n}2^{v_2(k)}$ , where $v_2$ is the $2$ -adic valuation, and showed that both the number $A(x)$ of exponents $a\le x$ with $2^a$ in the image and the number $B(x)$ of the other exponents tend to infinity. We prove the identity $c(n)=n+\frac12\sum_{i}i\,2^{i}$ , the sum running over the positions of the binary digits $1$ of $n$ , so that $2^a$ lies in the image of $c$ if and only if $2^{a+3}$ is a sum of distinct terms of the superincreasing sequence $(j2^j)_{j\ge1}$ . For every exponent there is at most one possible largest term $K$ , given by $K+\lceil\log_2K\rceil=a+3$ , and no such $K$ exists exactly when $a=2^r+r-2$ . The remaining exponents are decided by an explicit recursion on pairs of integers bounded by $a+3$ , in at most $a+1$ steps. We give the closed formula $c(2^{K-2}+2^{K-s-2})=2^{K+h-3}$ for $s=2^k$ , $h\equiv k\pmod{2^{k+1}}$ and $K=(2^{h+s}+s)/(2^s+1)$ , and infinite explicit families of exponents $a$ for which $2^a$ is not in the image, indexed by $s=3$ and $s=2^i$ ; the families given by Nicolas are the cases $s=1$ and $s=2$ . As a consequence, $A(x)\ge(2+o(1))\log_2x$ and $B(x)\ge(\frac54+\frac12\sum_{i\ge1}\frac{1}{2^i-1}+o(1))\log_2x=(2.0533\ldots+o(1))\log_2x$ . We prove that the states of the recursion form a forest, rooted at the states $(J,0)$ and at the failing states, in which every state $(J',w')$ has exactly $v_2(J'+1+w')$ children; that the sets $S_J$ of successful states are symmetric; that exactly half of them lie in the tree of the root $(1,0)$ ; that $N(2k+1)=N(2k)$ ; that $4\mid N(J)$ for $J\ge4$ ; and that $S_J$ satisfies a recursion from the smallest elements upwards, with period $J/2$ for even $J$ . The symmetry gives further explicit families, among them $c(2^{2^r}-1)=2^{2^r+r-1}$ , and the improved bounds $A(x)\ge(3.9655\ldots+o(1))\log_2x$ and $B(x)\ge(2.8178\ldots+o(1))\log_2x$ . For every prime $p$ , the function $c_p(n)=\sum_{k\le n}p^{v_p(k)}$ counts the factor $1-X$ in $(1-X)\cdots(1-X^n)$ over $\mathbb{F}_p$ ; we extend to it the reduction, the unique candidate, the recursions and the symmetry, give explicit families, and describe completely the exponents rejected at the first step, which occur only for odd $p$ . We also report experiments in exact integer arithmetic: every statement is checked by three independent methods, the values of $A(x)$ reproduce the table of Nicolas, the sets $S_J$ are computed for $J\le131072$ , and $A(x)/x$ lies between $0.5132$ and $0.8750$ for $x=1000,2000,\dots,131000$ .

Primarily AI-generated textHuman understanding: all parts$2$-adic valuationArithmetic functionsBinary expansionOpen questions of NicolasPolynomials over $\mathbb{F}_2$Superincreasing sequences

Nov 03, 2026PDF ↗View work →

hexagon:2611.00026v1

math.CO — Combinatorics

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

Contributed by Ella Puerto

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

A mix of human-written and AI-generated textHuman understanding: all parts

Nov 03, 2026PDF ↗View work →

hexagon:2611.00025v1

math.NT — Number Theory

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

Contributed by David Niedbala Giraudin

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

Primarily AI-generated textHuman understanding: no partsCyclotomic PolynomialsMersenne prime polynomialsfinite fieldsperfect polynomialssum of divisors

Nov 03, 2026PDF ↗View work →

hexagon:2611.00024v1

math.OA — Operator Algebras

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

Contributed by K. MAHESH KRISHNA

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

Primarily human-written textHuman understanding: all partsC*-algebraDvoretzky TheoremGrothendieck InequalityHilbert C*-module.Khinchin-Kahane InequalityType-Cotype

Nov 03, 2026PDF ↗View work →

hexagon:2611.00023v1

math.NT — Number Theory

PARAMETER MONOTONICITY FOR GENERALIZED k-MARKOV NUMBERS

Contributed by Qiyue Tang

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

A mix of human-written and AI-generated textHuman understanding: some partsMarkov equation

Nov 03, 2026PDF ↗View work →

hexagon:2611.00022v1

math.SG — Symplectic Geometry

DEEP LOCI OF DOUBLE BOTT–SAMELSON VARIETIES

Contributed by Qiyue Tang

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

A mix of human-written and AI-generated textHuman understanding: some partsDeep locus

Nov 03, 2026PDF ↗View work →

hexagon:2611.00021v2

math.GT — Geometric Topology

The dodecahedron web and its instanton homology

Contributed by Peter Kronheimer, Tomasz Mrowka

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

Primarily AI-generated textHuman understanding: some partsInstanton homologyTait colorings

Nov 03, 2026PDF ↗View work →

hexagon:2610.00678v1

math-ph — Mathematical Physics

Geometric Computation: Twin Helices and the Algebra of Flow

Contributed by Benjamin Kaminsky

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

A mix of human-written and AI-generated textHuman understanding: all partsYang-Mills theorycobordism hypothesisgeometric computationheliceshigher gauge theoryprequantum geometrytopological quantum field theory

Nov 03, 2026PDF ↗View work →

hexagon:2610.00712v1

math.NT — Number Theory

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

Contributed by Zeng Chunping

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

Primarily human-written textHuman understanding: all partsColmez formulaFaltings heightFermat curvesL-functionscomplex multiplicationcyclotomic towersspecial values

Nov 03, 2026PDF ↗View work →

hexagon:2610.00691v1

cs.DS — Data Structures and Algorithms

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

Contributed by Zeng Chunping

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

Primarily human-written textHuman understanding: all partsarithmetic dynamics，z^2+c，arboreal Galois representation

Nov 03, 2026PDF ↗View work →

hexagon:2610.00687v1

math.RT — Representation Theory

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

Contributed by Zeng Chunping

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

Primarily human-written textHuman understanding: all partsSato-Tate conjecture，invariant tensors，maximal torus，forgetting factor

Nov 03, 2026PDF ↗View work →

hexagon:2610.00676v1

math.DS — Dynamical Systems

Existence of SRB measures for some Rössler systems

Contributed by Theophile Caby

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

Primarily AI-generated textHuman understanding: all partsRossler system

Nov 03, 2026PDF ↗View work →

hexagon:2611.00019v1

math.NT — Number Theory

Minimal recurrences for binomial power sums

Contributed by Khai-Hoan Nguyen-Dang

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

A mix of human-written and AI-generated textHuman understanding: some partsAp\'ery limitsFranel numbersFranel’s recurrence problembinomial power sums