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math.MP — Mathematical Physics

Octonionic Heisenberg-Robertson-Schrodinger and Maccone-Pati Uncertainty Principles

Contributed by K. MAHESH KRISHNA

Let $\mathcal{O}$ be an octonionic Hilbert space. Let $A: \mathcal{D}(A)\subseteq \mathcal{O}\to \mathcal{O}$ and $B:\mathcal{D}(B)\subseteq \mathcal{O}\to \mathcal{O}$ be possibly unbounded self-adjoint linear operators. Then for all $h \in \mathcal{D}(AB)\cap \mathcal{D}(BA)$ with $\|h\|=1$ , we show that \begin{align*} (1) \quad \quad \quad \Delta _h(A)\Delta _h(B)\geq \frac{\sqrt{|\langle \{A,B\}h, h \rangle -2\langle Ah, h \rangle\langle Bh, h \rangle|^2+|\langle [A,B]h, h \rangle |^2}}{2} \end{align*} and \begin{align*} (2) \quad \quad \quad \Delta _h(A)^2+ \Delta _h(B)^2 \geq \frac{|\langle (A+B)h, y \rangle|^2+|\langle (A-B)h, y \rangle|^2}{2}, \quad \forall y \in \mathcal{O} \text{ satisfying } \|y\|\leq1 \text{ and } \langle h, y \rangle =0, \end{align*} where $\Delta _h(A):= \|Ah-\langle Ah, h \rangle h \|$ . We call Inequality (1) as octonionic Heisenberg-Robertson-Schrodinger uncertainty principle and Inequality (2) as octonionic Maccone-Pati uncertainty principle.

Primarily human-written textHuman understanding: all partsOctonionic Hilbert space.Uncertainty Principle

Oct 20, 2026PDF ↗View work →

hexagon:2610.00378v2

math.DG — Differential Geometry

Smooth Local Isometric Embeddings of Riemannian Manifolds in Optimal Dimension

Contributed by Wenchang Yang, Yi Wang

We prove that every smooth Riemannian manifold of dimension $n\geq5$ admits, near each point, a smooth isometric embedding into $\mathbb{R}^{n(n+1)/2}$ . This extends the local theorem of Janet and Cartan from real-analytic to arbitrary smooth metrics in these dimensions, and attains the optimal universal ambient dimension without any curvature or genericity assumption. The proof starts from a flat immersion whose second fundamental form has full normal rank. The linearized metric equation then reduces to a determined first-order system, whose characteristic variety is singular. We do not invert this system on its singular set. Instead, the metric is corrected by the quadratic interaction of characteristic waves: their phase-averaged quadratic terms generate the directions missing from the image of the reduced symbol, while the remaining oscillatory products and the residual linear errors are removed by parametrices on regular characteristic and elliptic patches. Exact Fourier localization of the increments yields an iteration that converges in $C^\infty$ on a fixed neighborhood.

A mix of human-written and AI-generated textHuman understanding: some partsIsometric embeddingsJanet–Cartan theoremcharacteristic wavesoptimal embedding dimensionsmooth Riemannian metrics

Oct 20, 2026PDF ↗View work →

hexagon:2610.00375v1

math.NT — Number Theory

Ramanujan's prime counting inequality holds for all x ≥ 38,358,837,683: ranges from the zeros, a spectral barrier, and trade-offs

Contributed by Sayed Barez

Ramanujan proved that $\pi(x)^2 < \frac{ex}{\log x}\pi(x/e)$ for all sufficiently large $x$ . Let $N_R$ be the least integer such that this holds for every real $x \ge N_R$ . Dudek and Platt showed that the Riemann hypothesis implies $N_R = 38{,}358{,}837{,}683$ . We prove that $N_R = 38{,}358{,}837{,}683$ . To our knowledge, this is the first determination of $N_R$ without assuming the Riemann hypothesis. The proof uses the zero-free half-plane $\mathrm{Re}s > 7/8$ of a recent OpenAI preprint. Its main statement for $\zeta$ is formalized in Lean 4 and has been rebuilt independently; its written proof has not been peer reviewed. The deduction is our Theorem B: if $\zeta(s) \ne 0$ for $\mathrm{Re}s > 0.945092$ , then $N_R = 38{,}358{,}837{,}683$ . The proof of Theorem B uses only refereed inputs, together with Bhattacharjee's correction of a theorem of Büthe. Relying only on the peer-reviewed literature, and on preprints where stated, we also prove the following by rigorous computer-assisted interval arithmetic. (A) The inequality holds for $38{,}358{,}837{,}683 \le x \le e^{753.98}$ . The previous range ended at $e^{103}$ . (C) It holds for $x \ge e^{3035.01}$ . This replaces Axler's threshold $e^{3158.442}$ , whose proof applies a lemma of Dudek and Platt in a case where that lemma's proof is defective. (B′, D) Either of the following also suffices: a zero-free region $\beta \le 1 - 1/(1.7373\log|\gamma|)$ above height $3\cdot10^{12}$ , or RH verified to height $4.1602\cdot10^{14}$ . We tabulate the trade-off between the two. (E) Arguments that use only non-negativity, analytic continuation and the poles up to height $T$ cannot go beyond $e^{754.40}$ when $T = 3\cdot10^{12}$ . So (A) is optimal among such arguments to within 0.41 in $\log x$ . Building on Bhattacharjee's discovery that the constant $E_2$ in Büthe's Theorem 1 lacks a factor $\log(3c)$ , we show that the affected rows of the Broadbent–Kadiri–Lumley–Ng–Wilk table inherit the error, by 1.3–2.2% after re-optimization. We also trace the consequences in the literature.

Primarily AI-generated textHuman understanding: all partsChebyshev functionRamanujan's inequalitycomputer-assisted proofexplicit estimatesinterval arithmeticprime-counting functionzero-density estimateszero-free half-planezero-free region

Oct 20, 2026PDF ↗View work →

hexagon:2610.00374v1

math.CO — Combinatorics

Antimatroid characterization of RAF interior operators

Contributed by Michael Mislan

Addressing the RAF-realizability question in Section 4 of Mike Steel's Interior Operators and Their Relationship to Autocatalytic Networks (arXiv:2305.16714v4, 2023), we characterize which interior operators on the subsets of a finite set E are maxRAF operators of catalytic reaction systems whose reaction set is exactly E. An interior operator is represented by its family of fixed sets, a union-closed family containing the empty set. A same-ground realization is a system with one reaction per element of E, finitely many molecule types, a food set and a catalysis relation; it realizes the empty set together with its RAFs, the nonempty reaction sets whose reactants arise from food by iterated closure under the selected reactions and whose reactions are each catalysed by a molecule of that closure. Such a family has a same-ground realization if and only if it is the intersection of an accessible union-closed family A and a predecessor-support family. Here A contains the empty set, is closed under unions, and every nonempty member of A has an element whose removal leaves a member of A; the members of A need not cover E. A set S is supported by a predecessor map P on E when every element e of S has some element of S in P(e), loops allowed. The family consisting of the empty set and all subsets of size at least three has a same-ground realization if and only if E has at most four elements. This gives the sharp threshold for the family considered in Proposition 5 and its subsequent remarks in Steel (2023). Both results are formalized in Lean 4 with Mathlib.

Primarily AI-generated textHuman understanding: some partsLean formalizationRAF setsantimatroidsfood generationinterior operatorsrealizability

Oct 20, 2026PDF ↗View work →

hexagon:2610.00373v1

math.CO — Combinatorics

Equivalence of the RAF abundance conjecture and Frankl's union-closed sets conjecture

Contributed by Michael Mislan

In a catalytic reaction system with finitely many reactions, a RAF is a nonempty set of reactions whose reactants are produced from the food set by iterated closure under the selected reactions and whose reactions are each catalysed by a molecule of that closure. Steel (arXiv:2305.16714v4, Section 4) asked whether some reaction always lies in at least half of the RAFs. Call a reaction food-ready if all its reactants are food, and abundant if it lies in at least half of the sets in the family formed by the RAFs and the empty set. If some RAF U consists entirely of food-ready reactions, then some reaction of U is abundant, with no condition on the other reactions. If every reaction of the system, except at most two, is food-ready and the system has a RAF, then some reaction is abundant. Finally, the statement that every catalytic reaction system with finitely many reactions and a RAF has an abundant reaction is equivalent to Frankl's union-closed sets conjecture in the form: every finite union-closed family containing the empty set and a nonempty set has an element lying in at least half of its members.

Primarily AI-generated textHuman understanding: some partsLean formalizationRAF setsabundant reactionselementary coresunion-closed sets conjecture

Oct 20, 2026PDF ↗View work →

hexagon:2610.00372v1

math.NT — Number Theory

χ₂(1/3) and χ₂(1/5) are irrational

Contributed by Shawn Krisman

Let $\chi_2(z)=\sum_{n\ge0}z^{2n+1}/(2n+1)^2$ be Legendre's chi function. We prove that $\chi_2(1/3)$ and $\chi_2(1/5)$ are irrational. Equivalently, $\mathrm{Li}_2(1/q)-\mathrm{Li}_2(-1/q)\notin\mathbb{Q}$ for $q=3,5$ . The proof uses the Hankel determinants $\Delta_h(\chi_2(1/q))$ of the moment functional $g\mapsto\sum_{t\ge1}q^{-2t}(g(u)\,u/D_h(u))'|_{u=t}$ , where $D_h(u)=\prod_{k<h}(u+k+1/2)$ . These determinants are values of polynomials $\Delta_h(X)\in\mathbb{Q}[X]$ of degree at most $h$ . We bound the denominators of $\Delta_h(X)$ prime by prime: a bound adapted to residue classes for the large primes, a Lagrange-basis bound for the small ones, and separate arguments for the primes $2$ , $3$ and $q$ . We bound the size of $\Delta_h(\chi_2(1/q))$ by a discrete log-gas argument against an explicit dual certificate, and prove that $\Delta_h(a/b)\neq0$ for every rational $a/b$ and infinitely many $h$ , through a reduction modulo an auxiliary prime $\ell=h/3$ . The margins are below $-0.054$ ( $q=3$ ) and $-0.256$ ( $q=5$ ) per $h^2$ . The whole proof, including the prime number theorem and both numerical certificates, has been formalised in Lean 4 with Mathlib and checked by the Lean kernel, using only Lean's three standard axioms.

Primarily AI-generated textHuman understanding: no partsHankel determinantsLeanLegendre chi functionPadé approximationdilogarithmformal verificationirrationalitylogarithmic potential theory

Oct 20, 2026PDF ↗View work →

hexagon:2610.00371v1

math.AG — Algebraic Geometry

Algebraicity of Weil classes on non-split abelian sixfolds of Weil type over $\mathbb{Q}(\sqrt{-3})$ with Hermitian determinant $-2$

Contributed by Antti Hippeläinen

Let $K=\mathbb{Q}(\sqrt{-3})$ . We give a proof that every Weil class is algebraic on every complex abelian sixfold of Weil type for $K$ whose Hermitian form has signature $(3,3)$ and determinant $-2$ modulo norms. These forms are not split, so neither Schoen's theorem on sixfolds nor the theorem on sixfolds claimed in a preprint of Markman, both of which concern split forms, applies. We model the sixfolds by the part of the Jacobian of a cover $C\to\mathbb{P}^1$ , branched at thirteen points, on which its group, the complex reflection group $G_{32}$ , acts through its reflection representation. A degeneration argument with a ball-arithmetic certificate shows that this family dominates the nine-dimensional Shimura variety. On every member, an integral relation among characters induced from linear characters of subgroups, realized by correspondences with coefficients in the group algebra, reduces the Weil classes to the determinants of eigenspaces of eleven cyclic subcovers. A theorem of Schoen makes these determinants algebraic for the three étale subcovers; for the eight ramified ones we prove it for all cyclic covers of curves of degree at least $3$ with at least $3$ branch points and fully paired branching data, through a cone over a Fermat variety and Shioda's results. Specialization and isogenies then reach every sixfold. The determinant of the Hermitian form and the relation among characters are exact computations.

Primarily AI-generated textHuman understanding: no partsFermat varietiesHodge conjectureHurwitz spacesWeil classesabelian varieties of Weil typecomplex reflection groupsperiod maps

Oct 20, 2026PDF ↗View work →

hexagon:2610.00370v1

math.CO — Combinatorics

Graph Puzzles IV.1-preview: Factorisations of hypohamiltonian and permutation snarks

Contributed by Nikolay Ulyanov

Every hypohamiltonian snark and every permutation snark has a unique multiset of cyclically 5-edge-connected factors of the same class under successive decompositions along cycle-separating 4-edge-cuts. We prove that hypohamiltonicity passes from a dot product to both of its snark factors. Together with the factorisation theorem for bicritical snarks, this answers a question of Goedgebeur and Zamfirescu. For permutation snarks, we extend the atom argument of Chladný and Škoviera using colourable cut multipoles and closure under their canonical completions.

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

Oct 20, 2026PDF ↗View work →

hexagon:2610.00369v1

math.DS — Dynamical Systems

Counterexamples to the D-unstable-core conjecture for parameter-rich and elementary mass-action networks

Contributed by Michael Mislan

Two exact four-species counterexamples refute necessity of a D-unstable core for instability of a positive equilibrium. The first has five reactions, parameter-rich kinetics, an explicit generalized-mass-action equilibrium, and a rational Jacobian with eigenvalue 1/500 + (51/500)i. The second has six reactions, at most two molecules on each side of every reaction, no catalysts, and classical mass-action kinetics with fixed integer kinetic orders. At a positive rational equilibrium its Jacobian has integer entries and an unstable nonreal eigenvalue with real part between 0 and 5, certified by an exact shifted-quartic intermediate-value argument. The networks share a reactant incidence relation and 24 nonempty child selections. Exact symmetrization, triangular-block and cubic-factorization certificates show that every child matrix is D-nonunstable in both networks. A permanent zero eigenvalue in an exceptional child distinguishes D-noninstability from strict D-stability. Both counterexample theorems have Lean 4 proofs with Mathlib. A separate asymptotic calculation explains the elementary example through circulation and a fast branching coordinate. The results establish strict linear instability, without asserting a Hopf bifurcation or periodic orbit.

Primarily AI-generated textHuman understanding: some partsD-unstable coresLean formalizationchemical reaction networksclassical mass actioncounterexampleelementary bimolecular reactionsgeneralized mass actionparameter-rich kinetics

Oct 20, 2026PDF ↗View work →

hexagon:2610.00367v1

math.DS — Dynamical Systems

A critical-point criterion for D-stability in every dimension

Contributed by Michael Mislan

A real square matrix A is D-stable if DA is Hurwitz for every positive diagonal matrix D. Applying the critical-point method to Johnson’s contact variety establishes that, in every dimension and for every positive choice of barrier weights, D-stability is equivalent to Hurwitz stability and absence of positive solutions of an explicit principal-minor Fritz–John system. In dimension five the system separates into Lagrange and tangency branches; both equivalences are registered in Lean. Further results treat regular-contact persistence, the boundary of the D-stable set, stability under bounded time-scale ratios and the minimal destabilizing ratio. The finite decision procedure also uses conventional finiteness and termination arguments and was executed exactly only in dimension three. For the five-dimensional Lagrange system the multihomogeneous Bézout number is 1200; 980 generic complex solutions were observed numerically. Worked examples include a Lotka–Volterra community, a published refinery-column gain matrix and an exactly recovered benchmark window. Formal statements, conventional arguments, exact certificates and numerical illustrations are distinguished throughout.

Primarily AI-generated textHuman understanding: some partsD-stabilityFritz-John conditionsLean formalizationcritical-point methoddiagonal scalingtime-scale robustness

Oct 20, 2026PDF ↗View work →

hexagon:2610.00366v1

math.CO — Combinatorics

Finite Multiset Codes and Affine Lattice Restrictions

Contributed by Daniel Adam Pyle

We study constant-weight nonnegative integer vectors under half-L1 distance and compare arbitrary codes with full intersections of affine additive-subgroup cosets with the discrete simplex. A heavy-coordinate decomposition gives the exact reduction A_q(n,n-s) = q + B_q(n,s) to codes whose coordinates are bounded by s. It determines the unrestricted optimum throughout 2n > (q+1)s. For every q ≥ 3 and s ≥ 2, at n = qs−1 and D = (q−1)s−1 the unrestricted and full-coset optima are respectively q+1 and q; we classify all q² unrestricted extremizers. We retain a complete classification of both optima for three coordinates and n < 2D. In the next ternary band, 2D ≤ n < 3D, eight words are attainable precisely when 3n ≥ 8D; equality yields an infinite eight-versus-seven full-coset separation. Exact finite certificates also give a six-versus-five gap at (q,n,D) = (4,5,3) and the quinary single-deletion values A_5(9,2) = 143 and A_5(10,2) = 201. The proofs use intersection bounds and explicit integer affine combinations. The unrestricted decomposition extends known methods, and bibliographic novelty remains unresolved, particularly against a closely related 2024 paper not compared in full.

Primarily AI-generated textHuman understanding: no partsaffine latticescoding theorydeletion codesdiscrete simplexexact certificatesmultiset codes

Oct 20, 2026PDF ↗View work →

hexagon:2610.00364v1

math.CO — Combinatorics

Partitioning $\{1,\dots,n\}$ into parts of size at most three whose sums are powers of three

Contributed by Deep Bhattacharjee, Ushashi Bhattacharya

We prove unconditionally the conjecture of Gurvich and Naumova that for every $n\ge1$ the set $\{1,\dots,n\}$ can be partitioned into parts of at most three elements whose sums are powers of $3$ . The partition is given by an explicit recursion.

A mix of human-written and AI-generated textHuman understanding: some partsformal verificationpowers of threeset partitionzero-sum triples

Oct 20, 2026PDF ↗View work →

hexagon:2610.00363v1

math.AG — Algebraic Geometry

Durfee inequalities for quasihomogeneous complete intersections of codimension two

Contributed by Qiyue Tang

For positive-weight quasihomogeneous isolated complete intersections of minimal embedding codimension two, we prove $\mu\ge4p_g+1$ in dimension two and $\mu>C_{n,2}p_g$ in every dimension $n\ge3$ .

A mix of human-written and AI-generated textHuman understanding: some partsDurfee inequality

Oct 20, 2026PDF ↗View work →

hexagon:2610.00360v1

cs.MS — Mathematical Software

Certified rational-anchor filtering for exact GPU scans of power-residue runs

Contributed by Brice Pouly

We study exact longest monochromatic runs of power-residue colourings, including neutral prefixes and least canonical witnesses. A certified rational cover represents every relevant interval inside a short signed character table. A reflection-compatible schedule retains an anchor; conservative filters reject only known contradictions, bilateral completion recovers the full run, and tie-preserving reduction selects its least witness. We prove this exact-success contract and describe its CUDA realization with separate arithmetic domains below $2^{32}$ and $2^{33}$ . On one RTX 5090, two reserved cohorts give ratios $24.00$ and $10.95$ of sums of hot per-prime medians against the identified historical dense engine. A separate batch study gives throughput factors $1.23$ - $2.58$ against an already optimized reference. The experiments have different timing boundaries and their factors are reported separately. Short-table multiplicative extraction has close algorithmic antecedents in cryptanalysis; our contribution is its certified organization for exhaustive maxima and witnesses and its measured implementation. Square-root-sized cover parameters do not give a general $O(\sqrt p)$ running-time bound.

Primarily AI-generated textHuman understanding: no partsCUDANVIDIA RTX 5090SM120certified algorithmsexact GPU scansmultiplicative characterspower-residue runsrational-anchor filtering

Oct 20, 2026PDF ↗View work →

hexagon:2610.00359v1

math.CO — Combinatorics

Succinct Rabung certificates and recurrence-closed lower bounds for van der Waerden numbers

Contributed by Brice Pouly

Rabung's power-residue construction describes a large progression-free colouring by a prime, a colour count and a target length. We give six additional primitive certificates, checked by full ordinary CPU scans, which yield fourteen direct lower bounds. In colour-first notation, these include $W(3,17)>33\,166\,190\,897$ , $W(3,18)>100\,289\,348\,022$ , $W(3,19)>259\,446\,988\,783$ and $W(3,20)>326\,368\,775\,450$ . The larger ternary primes lie outside the complete census below $2^{32}$ . We prove the equivalence of the run and boundary tests with Rabung's criterion and separate pointwise verification from discovery. A finite closure of explicitly identified earlier constructions places the new bounds in context: direct ternary certificates win through length $23$ ; Berlekamp and binary recurrence inputs win thereafter. Exact-modulus cyclic witnesses also give six-colour product bounds. The previously published thirteen-claim evidence and its closed empirical analyses are retained separately. Comparisons refer to the stated source catalogue, without an exhaustive global priority or optimality claim.

Primarily AI-generated textHuman understanding: no partsRabung certificatesarithmetic progressionscomputer-assisted proofslower boundspower-residue colouringsrecurrence closurevan der Waerden numbers

Oct 20, 2026PDF ↗View work →

hexagon:2610.00358v1

math.CO — Combinatorics

Prescribed Eigenvalues in Power Bands of Finite Prime-Cover Graphs

Contributed by Idris Ali Shaik

Let G_X be the graph on the positive integers at most X, with an edge between n and np whenever p is prime and np <= X. Write lambda_1(X) >= lambda_2(X) >= ... > 0 for its positive adjacency eigenvalues, counted with multiplicity. We prove that, for every fixed 21/61 < theta < 1/2 and every integer X^theta/log X <= a <= X^theta, lambda_a(X)^2 = pi(X/a) + O_theta(a/log X), where pi is the prime-counting function. This identifies the eigenvalue at the prescribed rank a throughout a polynomially growing band of centres. For every 0 < theta < 1/2, ordered mean-square bounds give a weaker estimate outside o(X^theta) centres below X^theta. When 1/3 < theta < 1/2, the same O_theta(a/log X) error holds outside o(X^theta) centres using only global prime estimates. The results extend to deletion of any fixed finite set of prime directions. Edges labelled by primes exceeding sqrt(X) form disjoint stars. Retaining their first-exit interactions gives local comparison operators and a family of approximate eigenvectors. Bounds on their overlaps and residuals control eigenvalue counts across clusters, allowing ranks to be recovered where the global perturbation bound no longer separates neighbouring star levels. The Baker--Harman--Pintz lower bound and the Brun--Titchmarsh upper bound supply the short-interval prime counts used in the uniform every-centre theorem.

A mix of human-written and AI-generated textHuman understanding: all partsmatrix perturbationordered eigenvaluesprime number theoremprime-cover graphsshort intervalsspectral graph theory

Oct 20, 2026PDF ↗View work →

hexagon:2610.00357v1

math.NT — Number Theory

Arithmetic coding against unknown small-prime signs

Contributed by Brice Pouly

We encode binary messages in completely multiplicative sign functions and read them along the three sequences $n^2+1$ , $n^2+2$ and $n^2-2$ . Two observations with the same exact cofactor remove the unknown smaller-prime signs when multiplied. Allocating shared large primes before pairing embeds ordinary binary error-correcting codes in each trace. Under an explicit large-factor mass hypothesis, a computable schedule gives one infinite encoding with positive prefix rate and a linear errors-and-erasures allowance. A first-occurrence filter makes its raw readings depend causally on a linear message prefix, while encoding and decoding remain polynomial in the numerical observation length. Its supporting primes have arbitrarily small harmonic cost. A finite construction, independent of the analytic hypothesis, stores $273$ bits in $8736$ readings of each trace and corrects every pattern of $499$ reading errors. Complementary constructions give persistent sign prescriptions, uniform laws for the active coordinates of words, and compact multiplicative subgroups whose nontrivial quotients have positive quadratic entropy. The word-law and entropy conclusions are not asserted for the polynomially decoded encoding. A separate matrix construction gives another causal positive-rate map. Fixed-scale balancing provides an unconditional quarter-sign bound against every small-prime assignment. All coding statements concern deliberately constructed families, not recovery of an arbitrary multiplicative source. The sparse constructions at superlinear cutoffs keep their large-factor mass input explicit.

Primarily AI-generated textHuman understanding: no partsarithmetic codingcausal encodingerror-correcting codeserrors and erasuresmultiplicative functionsquadratic tracesunknown prime signs

Oct 20, 2026PDF ↗View work →

hexagon:2610.00356v1

math.NT — Number Theory

Correlation laws and identifiability of multiplicative functions on quadratic traces

Contributed by Brice Pouly

We study which completely multiplicative sign functions can be identified from the statistics of $h(n^2+1)$ , $h(n^2+2)$ and $h(n^2-2)$ . In a class of finite products of quadratic characters of odd prime conductor, completed without zeros and modified at primes with summable inverse square roots, diagonal correlations of orders two and four determine the function. The second-order data recover the conductor and prime corrections, leaving exactly the ramified local signs undetermined. Fourth-order detectors recover those signs, including at the small primes; the inverse also extends to quadratic components at two. No bounded horizon suffices uniformly: for every prescribed window, two finite-conductor sources can share its complete law and disagree with density arbitrarily close to one half on each trace. For an arbitrary prescribed source $h$ , we construct factors $fg=h$ such that, for each factor separately, the three traces jointly have the uniform word law at every horizon. The choice is continuous and computable relative to $h$ , with a finite bound uniform in $h$ for approximation of a canonical synchronous coupling. The same choice treats every horizon sequence $o(\log\log N)$ . The factors need not be independent. Their synchronous empirical law projects exactly to the product's empirical word law; when a limiting canonical coupling exists, it retains the product's word law, which need not identify $h$ as a function.

Primarily AI-generated textHuman understanding: no partscorrelation lawsfinite observation horizonsidentifiabilitymultiplicative functionsnormalityquadratic charactersquadratic traces

Oct 20, 2026PDF ↗View work →

hexagon:2610.00355v1

math.NT — Number Theory

Local freedom and capacity barriers in quadratic trace reconstruction

Contributed by Brice Pouly

A completely multiplicative sign function is determined by its values on $n^2+1$ , $n^2+2$ , and $n^2-2$ , but exact determination does not quantify how many readings detect a negative prime sign. We distinguish three losses of information: partial compatibility tests, geometric restrictions on certificates, and fractional relaxation of signed constraints. After fixing a prime head, we describe the exact fibre of a window and the parity constraints joining two windows. For any fixed head and any sequence $H_N=N^{o(1)}$ , one table with $o(N)$ negative entries passes every $H_N$ -window test and every two-reading test. A separate finite table passes these tests without any global source. We bound certificate capacity - fractional packing under unit reading loads - for narrow and binary certificates and for low-degree transport libraries whose links cancel odd prime valuations in pairs. A fixed algebraic curve of certificates for a target prime $p$ has at most logarithmic capacity, or capacity at most one when a reading is constant; in the infinite unanchored case its common poles have field $\mathbb{Q}(\sqrt p)$ . Conversely, selected values of $n^2+1$ can force linearly many errors while the capacity of all certificates stays bounded. Two internally disjoint pair families compose without loss on their existing readings, whereas sixteen actual values exhibit $\operatorname{Pack}=3<\operatorname{Cap}=4<\mu=5$ . The selected-sample constructions do not give an upper bound for the minimum on a complete prefix.

Primarily AI-generated textHuman understanding: no partsarithmetic certificatescertificate capacityforced errorsfractional packinglocal compatibilitymultiplicative reconstructionquadratic traces

Oct 20, 2026PDF ↗View work →

hexagon:2610.00354v1

math.NT — Number Theory

Selective normality and algebraic rigidity of multiplicative traces

Contributed by Brice Pouly

A completely multiplicative sign function may be changed so that two quadratic traces remain identical at every index while a third becomes normal. We classify this freedom for the traces $n^2+1$ , $n^2+2$ and $n^2-2$ . After two traces and a finite prefix have been protected, a monic irreducible polynomial can be made normal from every prescribed source exactly when the field generated by one of its roots contains neither protected quadratic field. In the complementary case, every value of the polynomial is already fixed. A single decidable set of primes of relative density zero supports a continuous, relatively computable completion treating all admissible polynomials jointly, with uniform bounds for each fixed observation family. The proof combines weighted root counts with a sieve for moments whose order increases with the observation scale, followed by a selection using exact finite normalisers. We also give a sufficient condition on prime availability, explicit sparser reservoirs, and a divergent reservoir on which some sources have no completion with a convergent ordinary mean, despite convergence of all fixed word laws in probability. For larger protected families, the obstruction is governed by odd products of quadratic square classes. An explicit example distinguishes certificates for every numerical value from a rational norm identity in the variable.

Primarily AI-generated textHuman understanding: no partsChebotarev setsalgebraic rigiditymultiplicative functionsprime reservoirsquadratic tracesrelative computabilityselective normality

Oct 20, 2026PDF ↗View work →

hexagon:2610.00353v1

math.NT — Number Theory

Arithmetic reconstruction and error recovery from three quadratic traces

Contributed by Brice Pouly

We study completely multiplicative sign functions $h$ through their values at positive integers of the forms $n^2+1$ , $n^2+2$ and $n^2-2$ , with observed indices $1\le n\le N$ . First occurrences of prime factors give a canonical information set and a complete, prefix-compatible family of parity relations, each involving at most $\lfloor\log_2N\rfloor+3$ readings. These relations yield exact bounded-distance repair, including corrupted information coordinates, in time $N^{O(1)}(4+\lfloor\log_2N\rfloor)^t$ for $t$ errors; erasures enter through an explicit rank parameter. We then construct arithmetic redundancy without estimating a quadratic regulator. If $h(p)=-1$ and $K=\log N/\log\max(3,p)$ , the three traces contain at least $\exp(\exp(c\log K/\log\log K))$ negative readings for every fixed $0<c<\log2$ and sufficiently large $K$ , uniformly in the prime $p$ and all other prime signs. The same number of each sign occurs on one trace between $N^\alpha$ and $N$ , for each fixed $\alpha<1$ . The proof combines short reconstruction certificates, Lehmer divisibility and balanced divisors. Coprime blocks give compact access to these families without factoring large recurrence terms. Under independent reading noise with fixed erasure probability $0\le a<1$ and conditional flip probability $0\le b<1/2$ , every source prefix $P=\lfloor\exp((\log N)^\delta)\rfloor$ , for fixed $0<\delta<1$ , is recoverable from $O(P)$ readings with polynomially small failure probability in $P$ and preparation polynomial in the numerical $P$ and $\log N$ . A fixed positive erasure rate nevertheless leaves exponentially many assignments of observed large-prime signs. No positive-density inverse for unrestricted sources is asserted.

Primarily AI-generated textHuman understanding: no partsarithmetic reconstructioncompletely multiplicative functionserror recoveryinformation setsparity relationsquadratic tracesreading noise

Oct 20, 2026PDF ↗View work →

hexagon:2610.00352v1

math.NT — Number Theory

Prime perturbations of polynomial observations: instability and incidence criteria

Contributed by Brice Pouly

The cost of changing prime signs depends on how the resulting multiplicative function is observed. On a fixed prime support of finite harmonic sum, we prove that existence of a polynomial mean for every sign assignment is equivalent to uniform prime truncation and uniform convergence to local products. The criterion applies separately to ordinary and logarithmic means. For independent activations with finite expected harmonic cost, the almost-sure truncation defect is the averaged large-prime parity probability, not a sum of incidences. These criteria separate averaging methods and observation geometries. One common support is logarithmically stable for every fixed polynomial and every assignment, yet a fixed perturbation of $(-1)^{v_2(n)}$ on that support has divergent ordinary means on $n^2+1,n^2+2,n^2-2$ . Collective upper sieves give stability at critical counting exponents without critical summability, and harmonic stability on rectangles whose side lengths have asymptotically equal logarithms. For widths $X^\alpha$ , Gaussian prime estimates give the matching universal budget threshold when $119/500<\alpha<1$ ; the negative result concerns only the sum of two squares. Around every completely multiplicative sign function, fixed perturbations of arbitrarily small $\sum p^{-\sigma}$ cost can have divergent logarithmic means on $n^2+1$ for $125/164<\sigma\leq1$ . A positive oscillation bound is common to this range, although the starting scales may depend on $\sigma$ . Window laws further distinguish all origins from almost all origins.

Primarily AI-generated textHuman understanding: no partsharmonic costmultiplicative functionsordinary and logarithmic meansparity activationspolynomial observationsprime perturbationssieve methods

Oct 20, 2026PDF ↗View work →

hexagon:2610.00351v1

math.NT — Number Theory

Prime noise, polynomial observations and trace-preserving completion of multiplicative functions

Contributed by Brice Pouly

For every prescribed completely multiplicative sign function, every integer polynomial irreducible over $\mathbb{Q}$ of degree at least two and eventually positive, and every prescribed prefix, we construct a completion preserving the polynomial trace and the prefix exactly, with vanishing correlations along all fixed admissible affine families. The changed primes have relative density zero, although the changed integers have asymptotic density one half and the changed primes have infinite harmonic mass. One noise trajectory has these properties almost surely at all positive real times. The construction combines a classical root-free prime reservoir with decreasing clock rates. Its probabilistic input is affine regularisation against arbitrary bounded weights fixed before the noise, uniformly in time after a vanishing delay. Marks retain their possibly non-convergent empirical distribution. For non-selective noise, the local channel acts through irreducible factors rather than output coordinates; square identities determine its invariant characters, and rational constants supply a time-evolving random translation. The proofs separate arithmetic moments, temporal concentration and finite Fourier analysis. The constant-source existence case already follows from work of Klurman-Mangerel-Teräväinen. The conclusions concern the constructed function, not new cancellation for the unperturbed source.

Primarily AI-generated textHuman understanding: no partsaffine correlationscompletely multiplicative functionsfinite Fourier analysisnormalitypolynomial tracesprime noisetrace-preserving completion

Oct 20, 2026PDF ↗View work →

hexagon:2610.00350v1

math.NT — Number Theory

Reflected sign patterns of the Liouville function at every large total

Contributed by Brice Pouly

For signs $s,t$ , let $N_{s,t}(N)$ count the ordered representations $N=n+(N-n)$ with $\lambda(n)=s$ and $\lambda(N-n)=t$ . We prove that, for every fixed $\varepsilon>0$ , all four counts are at least $c_\varepsilon N^{1-\varepsilon}$ at every sufficiently large integer total. The constants are ineffective through Siegel's theorem. The same conclusion holds uniformly for completely multiplicative sign functions whose positive primes satisfy $\sum_{f(p)=1}p^{-\theta}\le M$ , with fixed $0<\theta<1$ and $M$ . Separately, at each sufficiently large prime $p$ , both patterns with product $(-1/p)$ have at least $p/(4(\log p)^{96})$ representations. All four satisfy this stronger bound in six reduced classes modulo $24$ and outside one very sparse family of primes. The proof combines finite reflection-to-dilation propagation, a short-product spectral test, reconstruction from the units, and signed counting of prime disagreements. For the uniform extension, candidates are united before a common exceptional population is removed. None of the conclusions gives a positive-density bound or cancellation of the reflected correlation at every total.

Primarily AI-generated textHuman understanding: no partsLiouville functionadditive representationscharacter sumsmultiplicative functionsquadratic charactersreflection-to-dilationsign patterns

Oct 20, 2026PDF ↗View work →

hexagon:2610.00349v1

math.CO — Combinatorics

Prime-Star Localization and Second-Order Spectra of Finite Divisibility Cover Graphs

Contributed by Idris Ali Shaik

Let G_X be the graph on the positive integers n <= X in which n is joined to np whenever p is prime and np <= X. Let A_X be its adjacency matrix, let lambda_1(A_X) be its spectral radius, and let pi(X) count the primes up to X. Then lambda_1(A_X)^2 = pi(X) + 4 sum_{p <= X} 1/(p - 1) + o(1). More generally, fix a finite set S of primes and retain only the positive integers divisible by none of the primes in S. Let a_1 < a_2 < ... list these integers, and let pi_S count the primes outside S. For every fixed j, the j-th largest positive eigenvalue satisfies lambda_j(A_{S,X})^2 = pi_S(X/a_j) + 4 sum_{p <= X, p not in S} 1/(p - 1) + c(a_j) + o_{S,j}(1), where c(a) = sum_{q divides a, q prime} (q - 5)/(q - 1). The proof splits the edges according to whether their prime label exceeds sqrt(X). The large-prime edges form disjoint stars, while the remaining adjacency has norm of order X^(1/4)/sqrt(log X) but zero first-order matrix element on each star eigenvector. An explicit first-exit resolvent calculation gives a degree-ratio correction with remainder O_{S,j}(1/log X) and localization of the extreme eigenvectors. The prime number theorem and Mertens estimates then evaluate that correction as 4 log log X plus an explicit constant, yielding the displayed o(1) formula. The same constant-layer formula and localization hold uniformly for retained centres a <= (log X)^(1/2-epsilon), with finite-sum error O_{S,epsilon}(a^2/log X). The paired positive and negative modes also yield uniform centre--shell adjacency evolution. The explicit finite correction determines its phase with uniform o(1) error whenever |t| = o(sqrt(X log X)/a^(5/2)). The uniform finite-prime-sum law, its largest-eigenvalue specialization, and the explicit-frequency centre--shell dynamics theorem are formalized in Lean 4 and registered in the Palomar Registry as PALOMAR-2026-08-29-000017, version 2.

A mix of human-written and AI-generated textHuman understanding: all partsFeshbach-Schur mapadjacency spectrumdivisibility cover grapheigenvector localizationprime-counting functionspectral graph theory

Oct 20, 2026PDF ↗View work →

hexagon:2610.00348v1

math.NT — Number Theory

Classical correspondences between per-prime Tate curves: a matching obstruction

Contributed by Kunal Tyagi

For the complex Tate curves E_p = C*/p^Z attached to primes, we compute the classical divisorial correspondence groups. For distinct primes, Hom(E_p,E_q) is nonzero exactly when log(p)log(q) belongs to 4 pi^2 Q. If log(p)log(q)/(4 pi^2) = u/v in lowest positive terms, this group has rank one and its generator has isogeny degree uv. The isogeny relation between distinct primes is a partial matching: unconditionally, each prime has at most one partner. These rigidity statements specialize results of Winkelmann; the explicit Hom generator and degree are included here. Outside the matching, the Neron-Severi group of E_p x E_q consists of the two fiber classes, which act trivially on H^1. On every diagonal surface End(E_p) = Z, by Gelfond-Schneider, and the intersection profile of multiplication graphs is independent of p. Consequently no connected classical correspondence system can involve three distinct prime fibers. Version 2 withdraws the broader universal no-go assertion of version 1 and states the exact cross-fiber support restriction required by the obstruction.

Primarily AI-generated textHuman understanding: some parts

Oct 20, 2026PDF ↗View work →

hexagon:2610.00321v1

math.DS — Dynamical Systems

Endotactic permanence and quantitative recovery with vanishing reverse rates

Contributed by Michael Mislan

For a finite ordinary endotactic reaction network, measurable reaction rates in a fixed positive compact box admit a compact convex invariant absorbing set in every positive stoichiometric class. Global positive Carathéodory solutions are unique, and entry times are uniform over compact initial sets and all admitted rate histories. The argument combines the finite affine construction of OpenAI with an endotactic witness estimate that retains the reaction projection lengths. We then quantify what permanence permits as reverse rates shrink. For autonomous weakly reversible networks with $k_e=\bar k_e\varepsilon^{a_e}$ , eventual reciprocal concentration bounds depend on the exponent spread $\Delta=a_+-a_-$ through $\varepsilon^{\Delta/(2H)}$ , where $H>0$ is a network-dependent geometric constant chosen before the individual rates. In an explicit three-species system with small positive rate parameters $p$ and $h$ , prepared concentrations have scales $ph^3$ , $ph^2$ , $ph$ , and the worst-case prepared recovery time lies between $(400/1313)(p^2h^7)^{-1}$ and $(31/10)(p^2h^7)^{-1}$ . A weighted inventory gives necessary restoration and depletion times from arbitrary positive seeds. Further results treat common clocks, reverse-exponent families and a conventional stochastic lower comparison. Two registered Lean developments support the permanence theorem, continuous dependence and selected deterministic quantitative results. Their precise coverage is separated from conventional extensions and numerical illustrations. Permanence at fixed positive rates supplies neither a uniform guarantee at zero reverse rate nor a global sharp-corridor recovery deadline.

Primarily AI-generated textHuman understanding: some partsLean formalizationendotactic reaction networksmass-action kineticsmeasurable ratespermanencerecovery timevanishing reverse rates

Oct 20, 2026PDF ↗View work →

hexagon:2610.00344v1

math.CA — Classical Analysis and ODEs

Image Reconstruction from Projections using Fourier Slice Theorem

Contributed by Rajesh Dachiraju

We study the problem of image reconstruction from projection data by combining the Fourier Slice Theorem with scattered-data approximation in the Fourier domain. The unknown image is assumed to belong to the space of functions of bounded variation (BV), allowing discontinuities that naturally arise in practical tomography. In contrast to conventional band-limited formulations, no assumption is imposed that the image itself is band-limited. Instead, the finite bandwidth is attributed to the image acquisition system through the detector response, finite sampling resolution, and associated measurement process. Consequently, the measured Radon projections contain only a finite range of spatial frequencies, and the reconstruction objective is the recovery of the instrument-limited image determined by the measurable Fourier bandwidth. To exploit the structure of the Fourier Slice Theorem, we introduce a symmetric embedding of the image on the torus together with the corresponding symmetrization of the Radon data. The resulting Fourier transform becomes real-valued and even, allowing the Fourier-transformed projection data to be interpreted as real-valued scattered samples of the two-dimensional Fourier transform on a fixed bounded frequency domain. This reformulates image reconstruction as a scattered-data interpolation problem in the Fourier domain. The interpolation problem is analyzed using the recently developed approximation theory for multivariate BV functions from scattered data. The resulting convergence error is expressed in terms of the total reconstruction error separating into an instrument error, determined solely by the finite bandwidth of the acquisition system, and a reconstruction error arising from interpolation of the measured Fourier data. The proposed framework provides a mathematically rigorous and physically realistic formulation of Fourier-slice tomography in which the limitations of the imaging hardware are distinguished from the accuracy of the reconstruction algorithm. Numerical experiments demonstrate the applicability of the method to image reconstruction from limited projection data and validate the predicted convergence behaviour.

Primarily human-written textHuman understanding: all parts

Oct 20, 2026PDF ↗View work →

hexagon:2610.00395v1

math.OC — Optimization and Control

Additive Rounding for LP-Tight Three-Item Bin Packing

Contributed by Ruixue Yuan, Da Teng, Zhenyu Sun

We consider bin-packing instances with $3m$ items of sizes in $(1/4,1/2)$ , total size $m$ , and configuration linear programming value $m$ . Let $d$ be the relative dimension of the polytope of zero-sum multiplicity patterns obtained by centering sizes at $1/3$ . We determine the maximum additive integrality gap for every $d\le10$ . It is zero for $d\le4$ and one for $5\le d\le10$ . More generally, a representation of the multiplicity vector by at most five zero-sum patterns admits an integral packing, and a representation by at most eleven patterns requires at most one additional bin. The integral-packing result has a structural proof; the additive-one result combines structural restrictions with exact finite enumeration. For arbitrary $d$ , we construct an LP-tight subinstance with at most $6d+3$ items and no smaller additive gap. An integer representation preserving the signs of triple sums then yields a finite rational family attaining the worst gap, which is computable in time $2^{O((d+1)^2)}$ . Finally, we construct a sharp family with simplex pattern polytopes and a linearly growing zero-sum matching deficit. Its eleven-dimensional member violates the zero-sum remainder condition used in the ten-dimensional proof. Thus the bound of eleven fractional patterns in that condition is sharp.

Primarily human-written textHuman understanding: all partsbin packingconfiguration linear programminghypergraph matchinglattice polytopeszero-sum patterns

Oct 20, 2026PDF ↗View work →

hexagon:2610.00388v1

math.OC — Optimization and Control

Exact interfaces for modular autocatalytic networks

Contributed by Michael Mislan

We study which information about an autocatalytic reaction network suffices for exact composition with other modules. For the maximum amplification factor, threshold-indexed species-price cones compose by exact set operations, while scalar optima do not. Autocatalytic child-selection cores are reconstructed from complete, uniquely matched ordinary-core anchors using alternating exchange paths and certified rational semipositivity tests. For irreducible unary diluted source networks, positive-composition certificates characterize growth, criticality and extinction under arbitrary species degradation; a related certificate identifies partial-actuation stabilizability. Further manuscript results treat factor interactions, exact boundary traces, contextual indistinguishability and junction-tree composition under running intersection and scope extensionality. Separator width bounds message arity, without a runtime claim. Eleven foundation statements are Palomar registered in Lean 4; the broader historical formalization and conventional extensions retain their separately documented scope. Thermodynamic and passive-tail examples illustrate exact interfaces and information loss. Structural amplification and linear onset conclusions are distinguished from nonlinear kinetic dynamics.

Primarily AI-generated textHuman understanding: some partsLean formalizationautocatalytic networksboundary traceschild-selection coresdegradationmaximum amplification factormodular composition

Oct 20, 2026PDF ↗View work →

hexagon:2610.00389v1

math.PR — Probability

Linear-size autocatalytic sets are asymptotically absent in the sparse binary polymer model

Contributed by Michael Mislan

In the sparse binary polymer model with reversible ligation and cleavage, we prove that for every fixed food horizon, fixed nonnegative catalysis intensity and fixed linear reaction budget, the probability of a nonempty reflexively autocatalytic and food-generated set within that budget tends to zero, at rate O(1/n). The probability that a single molecule catalyses a self-constructing reaction set also vanishes, cleavage included. A productive-prefix argument excludes bounded catalyst counts, while an injective acyclic dependency encoding supplies a factorial saving in the count of productive reaction supports. Consequences include divergence of minimum RAF size relative to polymer length and of minimum catalyst count, together with conditional forms. Combining the lower bound with the published quadratic-witness result of Hordijk and Steel gives a superlinear-to-quadratic window. The principal estimates and consequences are formalized in Lean 4 against the literal split-position model; the published upper-witness input is cited separately.

Primarily AI-generated textHuman understanding: some partsLean formalizationRAF setsbinary polymer modelminimum RAF sizeprobabilistic combinatoricssparse catalysis

Oct 20, 2026PDF ↗View work →

hexagon:2610.00376v1

math.PR — Probability

Slow Equilibration and Persistent Stationary Correlations in Spherical Spin Glasses

Contributed by P. M. Aronow, Patrick Lopatto

We study the Langevin dynamics of spherical mixed $p$ -spin glasses at fixed temperature, on time scales exponential in the dimension $N$ . Below the static transition temperature, the dynamics started from the uniform distribution on the sphere stays at total variation distance close to one from the Gibbs measure up to a time exponential in $N$ , with probability tending to one. We prove this for pure models of degree at least four at every such temperature, and for mixtures of minimum degree at least four that satisfy an explicit condition on a function of one variable. It extends to initial distributions that are independent of the disorder or depend weakly on it. Between the dynamical and static transition temperatures, we show for general finite mixtures that, with probability exponentially close to one, the generator has exponentially many exponentially small eigenvalues; a fixed part of the centered spin lies in the corresponding slow modes; the stationary spin correlation stays above every level below $q_*^2$ up to an exponential time, where $q_*$ is an explicit root of the plateau equation of the physics literature; the dynamics started from most points of the Gibbs measure stays far from equilibrium; and the two endpoints of the stationary dynamics share mutual information at least linear in $N$ . With probability tending to one, the correlation stays above every level below $q_*$ . Near the dynamical transition, exponentially many eigenvalues stay below $e^{-cN}$ , where $c$ is proportional to an explicit power of the distance to the transition.

Primarily AI-generated textHuman understanding: some partsdynamicsspin glass

Oct 20, 2026PDF ↗View work →

hexagon:2610.00397v1

math.RT — Representation Theory

Reduction of the Gorenstein symmetry conjecture to finite prime fields

Contributed by Qiyue Tang

We prove that the Gorenstein symmetry conjecture for finite-dimensional algebras over arbitrary fields is equivalent to its restriction to the finite prime fields. A counterexample specializes to a finite field without changing its dimension or its finite self-injective dimension. In characteristic zero, a counterexample yields counterexamples of one fixed dimension over every prime field outside a finite set.

A mix of human-written and AI-generated textHuman understanding: some partsGorenstein symmetry conjecture

Oct 20, 2026PDF ↗View work →

hexagon:2610.00396v1

math.RT — Representation Theory

A reduction of the Gorenstein symmetry conjecture to gendo-symmetric algebras

Contributed by Qiyue Tang

Over any field, the Gorenstein symmetry conjecture for finite-dimensional algebras is equivalent to its restriction to gendo-symmetric algebras.

A mix of human-written and AI-generated textHuman understanding: some partsGorenstein symmetry conjecture

Oct 20, 2026PDF ↗View work →

hexagon:2610.00392v2

math.RT — Representation Theory

Cartan determinants in finite representation type over arbitrary fields

Contributed by Qiyue Tang

Let $A$ be a finite-dimensional representation-finite algebra over an arbitrary field, and let $M$ be a nonzero finite-dimensional right $A$ -module. We prove that $\operatorname{End}_A(M)$ has Cartan determinant one whenever it has finite global dimension. In particular, the Cartan determinant conjecture holds for representation-finite algebras over arbitrary fields.

A mix of human-written and AI-generated textHuman understanding: some partsCartan determinant

Oct 20, 2026PDF ↗View work →

hexagon:2610.00390v1

cs.CC — Computational Complexity

Unconstrained detection of productive autocatalytic cores is NP-complete

Contributed by Michael Mislan

We prove NP-completeness of deciding whether a finite reversible reaction network contains a productive autocatalytic core when neither a target species nor an allowed-food set is specified. The input is a binary encoding of literal nonnegative integer reactant and product matrices, with a signed real flow on each reaction identity. A nonsingular square-witness characterization gives polynomial certificates. A weighted incidence construction reduces prescribed two-path directed linkage to unrestricted core detection, and an explicit constant-size switch construction reduces SAT to the linkage instances. Actual machine-level NP membership and a total polynomial-time binary transducer complete the classification. The terminal theorem is verified in Lean 4. The result concerns structural possibility in this source model and imposes no elementary reaction arity or conservation constraint.

Primarily AI-generated textHuman understanding: some partsLean formalizationNP-completenessautocatalytic coresbinary encodingpolynomial-time reductionsreaction networks

Oct 20, 2026PDF ↗View work →

hexagon:2610.00368v1

cs.CC — Computational Complexity

Deciding D-stability is coNP-hard

Contributed by Michael Mislan

The computational difficulty of D-stability, raised by Chen, Fan and Yu (1995, doi:10.1016/0167-6911(94)00036-U), is addressed by an explicit integer-matrix reduction from PARTITION. For positive weights of total 2T, the construction produces a matrix of order m + 4 that is D-stable exactly when no subset has weight T. When an equal split exists, some positive diagonal scaling has an eigenvalue with strictly positive real part. A Pell recursion supplies the parameters, with a logarithmic search bound and an explicit bound on every matrix entry. These bounds give polynomial binary output length. Together with the classical NP-completeness of PARTITION, the reduction yields coNP-hardness of D-stability and D-semistability, and NP-hardness of strict D-instability, even for a four-dimensional core with attachments at one coordinate. The hardness is in the weak sense for binary encoding. Running-time analysis and the trivial-instance cases are treated conventionally; registered Lean statements verify the explicit arithmetic encoding, bounds and promise dichotomy.

Primarily AI-generated textHuman understanding: some partsD-stabilityLean formalizationPARTITIONPell equationcoNP-hardnessdiagonal scaling

Oct 20, 2026PDF ↗View work →

hexagon:2610.00346v1

math.NT — Number Theory

Optimal windowed test functions for bounding analytic ranks of elliptic curves

Contributed by Sabrina Hackfort

Assuming the Generalized Riemann Hypothesis, the explicit formula for the L-function of an elliptic curve E/Q bounds the analytic rank of E in terms of the traces a_p for p < e^(2L), for any admissible test function supported on a window [-L,L]. In rank computations this test function is usually fixed in advance, typically to a Fejer-type kernel; optimising it for an individual L-function goes back to Booker, and an earlier test on quadratic twists of rank about 7 found the gain to be small. We choose, for each individual curve, the test function that minimises the resulting bound within a finite-dimensional space supported on [-L,L], certify the result, and measure the gain systematically, up to the current record curves. The minimiser is the solution of a linear system with the Gram matrix of the Weil quadratic form of E, and the optimal bound equals the reciprocal of the corresponding reproducing kernel at the central point, which we call the radix Riemanniana of E. Because the Fejer-type kernel lies in this space, the optimised bound is never weaker. We certify every bound with ball arithmetic, check the prime-side evaluation against the zero side computed independently from the zeros of L(E,s), and benchmark against the fixed kernel on 1642 curves from Cremona's tables, 2500 curves from the Stein-Watkins database and a set of curves with conductors up to 3*10^16. Depending on the sample and on the stopping criterion, the number of primes required to pin down the analytic rank decreases by 8-23% in aggregate. For individual curves of large conductor the decrease ranges from 0% to 43%. Within each sample the gain is largest at rank 0 and decreases with the rank; it grows again, and by a much larger factor, in the regime of large rank and large conductor where the computation is most expensive. On the smallest-conductor curves of rank 28-31 in the ICARM leaderboard the optimised kernel needs 84-85% fewer primes to certify the rank. For the current record, the rank-32 curve #4239 of Elkies, Klagsbrun and Li, it certifies r_an<=32 under GRH with 2.53*10^9 primes - the first upper bound for this curve - and hence r_an=32 under GRH and BSD, while the fixed kernel, certified at L=12.40 with 2 481 819 193 primes, is still 0.39 above the target. On all 395 leaderboard curves of rank 24-31, with primes below 10^10, the optimised kernel certifies r_an<= r on 379 curves and the fixed kernel on 372; where both succeed the fixed kernel needs a median of 4.0 times as many primes. We validate the fixed-kernel baseline by reproducing every published value of Bober and of Klagsbrun, Sherman and Weigandt with an independent implementation.

Primarily human-written textHuman understanding: all parts

Oct 19, 2026PDF ↗View work →

hexagon:2610.00361v1

math.NT — Number Theory

Arithmetic structure in a complete atlas of power-residue runs below $2^{32}$ : Rabung profiles, Kummer densities, norm dependencies, and local codes

Contributed by Brice Pouly

We study the arithmetic behind long runs in a complete census of power-residue colourings below $2^{32}$ , covering every admissible order $2\le r\le9$ . The dataset contains $203\,280\,221$ primes and $677\,591\,249$ profiles, with exact quadratic four-window counts at every odd prime. A bounded study of its deep tail identifies algebraic windows with fewer independent character conditions than their lengths suggest. We certify their power-class groups and conjugate-root overlaps, obtaining exact Kummer densities for fixed templates, including eight additional order-four models. In a quadratic family, eleven constant positions force twenty-six neutral positions at odd character orders. In two further families, norm identities generate all remaining higher-root relations. A length- $53$ cubic realizes a three-wise independent binary code whose monochromatic probability is $4096$ times that of independent signs; its local run law is computed exactly. A separate stratified million-profile ternary sample is analysed with the same finite catalogue filter applied to the data and to an independent-colour comparator. Finite census statements, selected explanations and asymptotic template laws concern distinct populations. No universal law of global maxima or finite-range error estimate for the template densities is asserted.

Primarily AI-generated textHuman understanding: no partsKummer densitiesRabung profilesarithmetic dependenciescomplete finite atlaslocal codesmultiplicative charactersnorm identitiespower-residue runs

Oct 19, 2026PDF ↗View work →

hexagon:2610.00347v1

math.PR — Probability

Gaussian weights on fractals and Edelman's ghosts and shadows

Contributed by Djalil Chafaï

Can a Gaussian on a noninteger-dimensional fractal give a probabilistic realization of Edelman's ghosts and shadows? We distinguish a Gaussian-weighted Hausdorff measure, a radial Gaussian law, and a Gaussian random vector or field. For every real $\beta>1$ , we construct a closed cone of Hausdorff dimension $\beta$ whose Gaussian-weighted $\beta$ -dimensional Hausdorff measure has exactly the $\chi_\beta$ radial distribution. The construction relies on exact radial volume growth, not merely on a dimension calculation. We prove a metric obstruction below dimension one and explain why the cone does not supply ordinary Gaussian addition or a scalar algebra. A separate Gram-determinant argument rules out preserving all polynomially continued Euclidean Gaussian Gram identities in a positive model. We then give proofs of the angular beta-gamma identities, associativity of Bessel-Kingman convolution, and its Gaussian convolution semigroup for $\beta\geq1$ . Squared Bessel processes, tridiagonal and bidiagonal matrix models, Dunkl Gaussians, and Gaussian fields on fractals provide complementary constructions with explicitly different meanings of dimension. None of these statements asserts the existence of a full noninteger-dimensional scalar algebra.

Primarily AI-generated textHuman understanding: no partsRandom matrices; fractals; hypergroups

Oct 18, 2026PDF ↗View work →

hexagon:2610.00362v1

math.NT — Number Theory

Cubic-residue quadruplets: elliptic formulas and arithmetic fluctuations

Contributed by Brice Pouly

Let $A_p$ count the monochromatic consecutive four-windows of a nontrivial cubic character modulo a prime $p\equiv1\pmod3$ , with zero excluded and overlapping windows counted separately. We give an exact formula for $A_p$ in terms of fixed elliptic traces and a finite boundary correction. Its complex-multiplication terms are related cubic twists. This produces two arithmetic fluctuation regimes: the first four moments of $(27A_p-p)/\sqrt p$ tend to $(0,44,0,4224)$ when three is a cube and to $(0,17,0,738)$ otherwise. Over all admissible primes they tend to $(0,26,0,1900)$ . The proof combines a four-position identity following from Greene's hypergeometric formalism and Lennon's trace formula with symmetric-power automorphy, cyclic base change and a finite collection of Rankin-Selberg prime number theorems; it uses no GRH. A direct geometric proof exposes the phases and exceptional points, with the genus-two and isogeny constructions of Freeman-Satoh and Guillevic-Vergnaud explicitly credited. Point-order and Hasse certificates verify numerical examples without enumerating their billion-position character words. The conclusions concern fixed pattern length and four moments; no complete limiting distribution, constant-order centring of the raw count, or law for logarithmic-length maxima is proved.

Primarily AI-generated textHuman understanding: no partsElliptic Curvesarithmetic fluctuationsaveraged momentscomplex multiplicationcubic characterscubic twistsfinite fieldsfour-term patterns

Oct 18, 2026PDF ↗View work →

hexagon:2610.00365v1

math.NT — Number Theory

Adelic Galois representations are maximal for almost all Drinfeld modules

Contributed by Anwesh Ray

Let $r\geq 3$ be an integer, let $q$ be an odd prime power, and set $A=\mathbb{F}_q[T]$ and $F=\mathbb{F}_q(T)$ . We study the Galois representations associated with rank $r$ Drinfeld modules of the form $\phi_T=T+g_1\tau+\cdots+g_r\tau^r$ , where $g_i\in A$ and $g_r\neq 0$ . Ordering the coefficient tuples $\vec{g}=(g_1,\ldots,g_r)$ by their maximum degree, we prove that, for a density-one set of tuples, the $\lambda$ -adic Galois representation is surjective for every prime $\lambda$ of $A$ , and the adelic Galois image contains $\mathrm{SL}_r(\widehat A)$ . Moreover, for every tuple in this set, the index of the adelic image equals the index of the corresponding rank-one determinant representation and hence divides $q-1$ . These results extend Zywina's theorem in rank $2$ to arbitrary ranks $r\geq 3$ . As a further consequence, we establish the existence of rank $r$ Drinfeld modules with surjective adelic Galois representations for every $r\geq 3$ , extending previous existence results which were known for prime values of $r$ .

Primarily human-written textHuman understanding: all partsDrinfeld modulesGalois representations

Oct 18, 2026PDF ↗View work →

hexagon:2610.00332v1

math.NT — Number Theory

Rare events in multiplicative sequences: Poisson laws, clusters, and observation

Contributed by Brice Pouly

A multiplicative identity can persist at every scale while disappearing from a rare-event count. Another identity can survive as a cluster whose points lie far apart. Which distinction matters depends on what is observed. We explain this through one completely calculated model: a uniform germ can trigger two occurrences in disjoint spatial windows. Its two marginal counts become Poisson, but their sum need not do so. A coupling identifies the correct same-grid target and separates the mean number of points from the mean number of active germs. We then isolate the additional work required by multiplicative sources: private prime coordinates, weighted relations between windows, corrections to prime-site intensities, and relations visible at the target scale. A final passage from marked configurations to records explains the roles of ties, recognition and observation clocks. Selected results are stated with their hypotheses; a short concordance leads to their proofs. The elementary calculations are proved here, while the arithmetic and functional limit theorems are taken from the four cited research articles.

Primarily AI-generated textHuman understanding: no partsPoisson approximationarithmetic clustersexpository articlemarked point processesrandom multiplicative functionsrare events

Oct 18, 2026PDF ↗View work →

hexagon:2610.00336v1

math.NT — Number Theory

Sparse prime surgery in completely multiplicative sequences: Normality, rare-event realization and optimal repair

Contributed by Brice Pouly

We study how changing prime signs affects fixed-word frequencies, rare-event laws and extremes of completely multiplicative sequences. The cost is $d_\sigma(f,g)=\sum_{f(p)\ne g(p)}p^{-\sigma}$ . A known theorem of Frantzikinakis-Lemańczyk-de la Rue preserves normality at finite $d_1$ ; we give its elementary specialization. Using explicit arithmetic inputs from Long runs, we then establish three families of results. For each fixed $0<\sigma\le1$ and almost every base, reversible surgery realizes every stationary finite-intensity marked point-process law along selected scales, with signs, exact excess lengths and converging actual intensity, at arbitrarily small $d_\sigma$ cost and with a prescribed prefix unchanged. Almost surely over all integer scales, the minimum number of prime changes needed for a run of length $\lceil c\log_2N\rceil$ is asymptotic to $c\alpha_c\log_2N$ for $1<c<3/2$ , where $\alpha_c\in(0,1/2)$ satisfies $H_2(\alpha_c)=1-1/c$ and $H_2(u)=-u\log_2u-(1-u)\log_2(1-u)$ . At each scale, with probability tending to one, symbolic and private large-prime repair maxima coincide on the same random base for $k_N=o(\log N)$ ; unrestricted prime repair agrees when $k_N=o(\log\log N)$ . This comparison transfers the classical lattice laws of runs with errors to adaptive prime repair. Elementary programming from arbitrary bases and an autonomous finite-range-dependent process construction complement these results.

Primarily AI-generated textHuman understanding: no partscompletely multiplicative functionslong runsnormalityoptimal repairpoint-process realizationprime-sign perturbationsrare events

Oct 18, 2026PDF ↗View work →

hexagon:2610.00338v1

math.NT — Number Theory

A lean formalization of Roy's strong six exponentials theorem

Contributed by Shawn Krisman

We formalize in Lean 4 with Mathlib Roy's strong six exponentials theorem: if $x_1,x_2$ are linearly independent over $\overline{\mathbb{Q}}$ and $y_1,y_2,y_3$ are linearly independent over $\overline{\mathbb{Q}}$ , then one of the six products $x_iy_j$ lies outside the $\overline{\mathbb{Q}}$ -vector space spanned by $1$ and the logarithms of algebraic numbers. The proof closure uses only Lean's three standard axioms. Along the way the development proves Waldschmidt's transcendence theorem on $\mathbb{G}_a^{d_0}\times\mathbb{G}_m^{d_1}$ in the form used by Roy, and Philippon's multiplicity estimate with derivatives on $\mathbb{G}_a^{d_0}\times\mathbb{G}_m^{d_1}$ over every algebraically closed field of characteristic zero, following Roy's account in Waldschmidt's book. We reproduce the public Lean statements, summarize the mathematical route, and record how the artifact is verified. The code is available at https://github.com/telaviv/strong-six-exponentials.

Primarily AI-generated textHuman understanding: no partsLeanformal verificationinterpolation determinantslinear forms in logarithmsmultiplicity estimatesstrong six exponentials theorem

Oct 18, 2026PDF ↗View work →

hexagon:2610.00343v1

math.NT — Number Theory

An Euler-product mollifier for the Jensen moment of zeta and an energy criterion for the Riemann hypothesis

Contributed by Pedro Caceres

The Jensen moment of Balazard, Saias and Yor expresses the zeros of the Riemann zeta function outside the critical line through the mean value of the logarithm of its absolute value on that line. Jensen's inequality converts any mollifier into an upper bound for this contribution. For Dirichlet-polynomial mollifiers, the bound cannot fall below 0.0386, a barrier arising from the pole at s = 1. We instead use the reciprocal of a smoothed Euler factor, denoted E-star(X,s), constructed from prime powers up to a constant multiple of X together with an explicit pole term. The quotient of the zeta function by this factor is then expressed solely in terms of its nontrivial zeros. We prove that the resulting bound B(X,s0), for any center s0 with real part greater than one half, tends to zero under the Riemann hypothesis (RH). Consequently, RH is equivalent to the convergence of B(X,s0) to zero. If the supremum of the real parts of the zeros is attained and exceeds one half, the bound grows at least as fast as X to a positive power divided by log X. Under the same attainment assumption, RH is therefore equivalent to boundedness, as X tends to infinity, of a single weighted mean-square energy: the integral over the real line of the squared absolute value of zeta divided by E-star on the critical line, weighted by 1/(1+t squared). Numerically, the bound decreases to 0.0015 at X = 10,000 for the Balazard–Saias–Yor center s0 = 1. At s0 = 2, it decreases approximately as 0.037 divided by log X up to X = 1,000,000, close to a heuristic prediction involving the constant kappa = 1.828... . By contrast, the corresponding quantity grows for the Davenport–Heilbronn function. We describe the geometry of the factorization and use two control functions to identify the additional ingredients required for a proof of boundedness of the mean-square energy.

A mix of human-written and AI-generated textHuman understanding: all partsBeurling primesDavenport-Heilbronn functionEuler productJensen formulaPoisson-Jensen formulaRiemann Hypothesisexplicit formulamollifier

Oct 18, 2026PDF ↗View work →

hexagon:2610.00324v1

math.CO — Combinatorics

Three-Element Lucky Sets of Stirling Permutations

Contributed by YURAN XIU

We characterize the three-element sets of lucky car indices that occur when a Stirling permutation of order $n$ is used as a parking function. A set $\{1,x,y\}$ , with $1<x<y<2n$ , is admissible precisely when $x$ is odd, or $x$ is even and $y$ is odd with $x<n$ , or both are even with $x+y\leq2n$ . Explicit Stirling words realize every permitted case. The number of admissible sets is $(n-1)(n-2)+2nh-3h(h+1)/2$ , where $h=\lfloor(n-1)/2\rfloor$ . This answers the three-element characterization and enumeration question in Problem 46 of Colmenarejo et al. The complete classification, parking constructions, and enumeration have been checked in Lean. HAL deposit: hal-05800210v1.

Primarily AI-generated textHuman understanding: no partsLean formalizationStirling permutationslucky setsparking functions

Oct 18, 2026PDF ↗View work →

hexagon:2610.00325v1

math.CO — Combinatorics

A Nonunimodal Acyclic Polynomial with All Roots in the Left Half-Plane

Contributed by YURAN XIU

Let $A(G,x)$ count vertex subsets of a graph $G$ that induce forests, by their cardinalities. Barton, Brown, and Pike asked whether failure of coefficient unimodality forces $A(G,x)$ to have a root with positive real part. We give a negative answer. The join of $K_{46}$ with three disjoint copies of $P_3$ has a nonunimodal acyclic polynomial of degree nine, yet all its complex roots have strictly negative real part. We derive the polynomial by classifying induced forests and give its exact positive Hurwitz determinants. A Lean formalization verifies the graph-to-polynomial identity and the full counterexample, using an exact rational Lyapunov certificate for strict stability. HAL deposit: hal-05800220v1.

Primarily AI-generated textHuman understanding: no partsHurwitz stabilityLean formalizationacyclic polynomialcounterexamplegraph polynomial

Oct 18, 2026PDF ↗View work →

hexagon:2610.00326v1

math.CO — Combinatorics

A Bipartite Graph with a Nonunimodal Acyclic Polynomial

Contributed by YURAN XIU

We construct a bipartite graph on 230 vertices whose acyclic polynomial, which enumerates vertex subsets inducing forests, is not unimodal. This answers Question 3.3 of Barton, Brown, and Pike negatively. The graph has a common set of 50 vertices and 15 branches, each with two sets of six vertices. A partition according to the number of selected common vertices gives a three-term formula for the complete induced-forest polynomial. Its coefficients have a strict descent from degree 62 to degree 63 and a later strict ascent from degree 66 to degree 67. The actual graph, the unrestricted subset-counting formula, and the exact inequalities have been checked in Lean. HAL deposit: hal-05800225v1.

Primarily AI-generated textHuman understanding: no partsLean formalizationacyclic polynomialbipartite graphscounterexampleunimodality

Oct 18, 2026PDF ↗View work →

hexagon:2610.00329v1

math.CO — Combinatorics

Counterexamples to the multicolor nonnested matching conjecture

Contributed by Pedro Machado Manhães de Castro

We show that excluding nested pairs can increase the number of vertices needed to force a matching of one color, disproving a conjecture of Barát, Gyárfás and Tóth. A matching consists of edges with no shared endpoint. In a fixed vertex order, two edges are nested if both endpoints of one lie between those of the other. For $r$ colors and a target of $k$ edges, the classical theorem guarantees a matching of one color on $(r-1)(k-1)+2k$ vertices when every pair is joined by an edge. For every $r,k\ge3$ , we construct a coloring at this very threshold in which every such matching contains a nested pair. Thus the nonnested threshold is larger by at least one vertex. Already with three colors, ten vertices force three disjoint edges of one color, while our coloring makes every such triple contain a nested pair. The self-contained proof combines endpoint counts with two edge recolorings that obstruct the pairings allowed by the vertex order.

Primarily AI-generated textHuman understanding: all partsRamsey numbernonnested matchingordered graph

Oct 18, 2026PDF ↗View work →