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math.NT — Number Theory
We give complete solutions of thirty-eight Diophantine equations left open in Grechuk's 2024 monograph: fifteen affine equations, five generalized Fermat equations, eight plane quartics, and ten further homogeneous equations. All proofs were generated by AI and require substantial computer algebra computations.
quant-ph — Quantum Physics
We consider the rotated surface code of odd distance with its standard syndrome-extraction circuit, under circuit-level depolarizing noise in which every gate, idle step, preparation and measurement fails independently with probability , decoded by minimum-weight perfect matching with the log-likelihood weights of single faults. We prove that a memory experiment with rounds fails with probability at most , where , for every . Hence the logical error rate per round decays exponentially in for all . Earlier rigorous thresholds for matching on surface-code circuits are , due to Dennis, Kitaev, Landahl and Preskill, and , due to Fowler. We keep their reduction of a failure to a weighted self-avoiding path in the decoding graph, but bound these paths by a local criterion for non-backtracking walks, which gives the threshold in closed form. To reach we use the minimality of the correction along the path, for a canonical choice among the minimum-weight corrections: labellings of short stretches of the path into error and correction edges that contradict this minimality are excluded, and the remaining labelled walks are counted with a potential function that is checked in exact arithmetic. For a leaner schedule with fused measure-and-reset steps the two arguments give and , respectively. With error rates that depend on the type of operation, they certify exponential decay for error models built from the published component error rates of Quantinuum's Helios and H2 and of IBM's Heron r3 processor ibm_boston, mapped onto this circuit. All theorems are also proved in the proof assistant Lean 4, with all computations checked by its kernel. Any argument that bounds the failure probability by the probability of an increasing event, ours included, is limited by the smallest such event that contains all failures, and Monte Carlo estimates place the threshold of this event near for this circuit.
math.GR — Group Theory
Let and let be the unique subgroup of containing with . We prove that and have the same set of irreducible complex character degrees. Meanwhile, is centerless and is not isomorphic to for any finite abelian group . This provides a counterexample to Huppert's conjecture.
math.HO — History and Overview
Written numerically, the date October 24, 2026 corresponds to a Pythagorean triple---the last one of the twenty-first century! The associated ---- right triangle has area twice its perimeter; a classical factorization shows that exactly three integer-sided right triangles have this property and, more generally, counts those with any prescribed integer inradius. We also observe that is the smallest perimeter shared by two noncongruent integer-sided right triangles, one of which is the ---- triangle.
math.GM — General Mathematics
Let be the Taylor polynomials of a power series with nonzero rational coefficients. We study the proportions of degrees for which the Galois group over $\Q$ is symmetric, alternating, or neither. We prove that the set of degrees with symmetric group can be chosen arbitrarily, even for an entire function with positive rational coefficients whose Taylor polynomials have distinct negative real roots. More precisely, at each degree outside the chosen set, has exactly one rational root, and the group on the other roots is the full symmetric group. Thus the proportions need not converge; when they do, the proportion of other groups can be any number in . These conclusions answer negatively two questions of Katz and Rivin. Moreover, for fixed $s\in\Q\setminus\Z$, we also give a condition on the prime divisors of rational multipliers of the coefficients that forces symmetric groups in density one. For the Taylor polynomials of , we determine the groups in all sufficiently large degrees. The number of exceptions to the symmetric group up to degree is , with an explicit constant . We also determine the groups of the central-binomial Taylor polynomials in every degree and prove that every nonconstant Catalan Taylor polynomial has symmetric group. This proves Sun's Catalan group conjecture and his irreducible-reduction criterion for the monic reciprocals of the central-binomial polynomials.
math.DS — Dynamical Systems
We prove that a Newtonian central configuration of finitely many positive masses on a circle whose center is the center of mass must be a regular polygon with equal masses. The proof selects an extremal mass–gap pair and lifts the circle to a periodic configuration on the line. A positive integral representation of the circular kernel converts the force equations into sharp weighted quadrature bounds. A scalar convexity estimate forces equality at the selected pair, and the equality conditions determine every mass and gap. The argument applies to every finite number of bodies and includes all analytic and rational estimates needed for the conclusion.
math.NT — Number Theory
Let be a finite extension of and a monic prime of . Anglès asked whether the -adic Leopoldt defect of Taelman's module of units of the Carlitz module over always vanishes, and Lucas conjectured this, together with the equivalence " iff is injective", for Anderson modules. We show that both statements fail. For every odd prime , the extension of generated by a root of is real, has Galois group , and has positive -adic Leopoldt defect and vanishing -adic zeta value at for every . The obstruction comes from modular representation theory at the infinite place. We attach to every separable an integer , depending only on the Galois group of the Galois closure, its decomposition group at and the character by which this group acts on the Carlitz period, and prove that the -adic defect is at least for all . Using Bosser's -adic Baker-Brumer theorem, we prove that the defect vanishes for every for real abelian extensions and for abelian extensions with cyclic Sylow -subgroup. We also show that a -adic analogue of Papanikolas's algebraic independence theorem for Carlitz logarithms would imply that the defect equals in general.
math.NT — Number Theory
We prove that a positive proportion of positive integers reach 1 within c log x ordinary Collatz steps for every fixed c > 3/log(4/3), with one positive lower natural-density bound independent of c. For each fixed positive target not divisible by three, the same range of times holds with a target-dependent bound, irrespective of its subsequent orbit. Targets divisible by three have predecessor basins of natural density zero. Diagonalization gives one positive-density set with a vanishing excess above the limiting time coefficient. The proof extends weighted inverse trajectories from a fixed finite family of roots and preserves positive weighted mass at every sufficiently large scale. We derive an explicit Fourier-decay estimate, at frequencies not divisible by three, for an auxiliary distribution on residue classes. Its proof uses paired geometric valuations and their exact binomial first-passage law. At order 6409, the coefficient C satisfies log_2 log_2 log_2 C < 176, compared with > 2^172 for the explicit coefficient in Mazur's work. Fractional-moment certificates, residue-capacity bounds, and an enlarged family of roots give explicit density estimates after harmonic and dyadic counting. Further refinements locate surviving mass within the root family, retain conditional residue information, and recover additional distinct sources with a bounded extra step cost. The primitive-decay theorem and the main qualitative density conclusions are formalized in Lean 4. The coefficient-size comparison and the complete optimized density bounds remain paper proofs.
math.MP — Mathematical Physics
Bayesian weak commutativity has been suggested as the most likely condition for reaching the Nagaoka–Hayashi (NH) bound in multiparameter estimation. We show that it is not sufficient. For a full-rank four-level ensemble obeying it exactly, the effects required for NH equality cannot sum to the identity. We find the optimal risk exactly: it lies strictly above the bound and needs seven outcomes, while projective readout on the four levels adds over seven times the NH gap. A robust normalization witness bounds the gap from below across a family including truncated oscillator ladders.
math.NT — Number Theory
We give a normalization-consistent coefficient formulation of the de Bruijn–Newman heat flow attached to the Riemann ξ-function. For the classical kernel Φ and the family Ht(z) = Z ∞ 0 e tu2 Φ(u) cos(zu) du, Ft(w) = Ht(i √ w) = X n≥0 qn(t)w n , we prove the exact evolution laws ∂tFt = 2F ′ t + 4wF′′ t , q(m) n (t) = (2n + 2m)! (2n)! qn+m(t). The coefficients have positive integral representations and the transformed entire function has order at most 1/2. We relate the zeros of Ft to the Pólya–frequency property of (qn(t)), derive the rectangular Jacobi–Trudi identity for the consecutive Toeplitz minors, and state carefully the strict consecutive-minor criterion needed to pass from these minors to total positivity. In particular, for this coefficient family the real-zero condition is equivalent to strict positivity of every consecutive minor Dt(n, r), not merely to nonnegativity of that selected family. This distinction avoids an invalid use of a weak version of Fekete’s criterion. We also correct the multiplicity bookkeeping for the squared-variable map and record a Chebyshev convolution identity without attributing to it an unjustified duplication of zeros. The paper proves no new unconditional upper bound for the de Bruijn–Newman constant and no proof of the Riemann hypothesis; its purpose is to provide an auditable set of exact
math.MP — Mathematical Physics
This note gives an affirmative answer to an open question posed in [Chelkak, Laslier, Russkikh'21]. Namely, it demonstrates the existence of perfect t-embeddings for each sufficiently non-degenerate planar bipartite graph with positive edge weights. The non-degeneracy condition on is formulated in terms of its perfect and almost perfect matchings and does not concern the choice of edge weights. The proof and the main part of the writing were produced by GPT-6 Astra in response to a series of prompts. We manually added Section 1.1 to put this result into a bigger context and slightly edited the introduction in order to improve the readability. We plan to provide a better human-digested version of this result later.
math.NT — Number Theory
Robin's criterion states that the Riemann Hypothesis holds if and only if for every . Let be the squarefree kernel of , its largest prime factor, and . We prove that Robin's inequality holds for every with , and for every whose prime factors are all below , and that both results remain valid with replaced by . Our main results concern superabundant numbers, among which there are infinitely many counterexamples to Robin's inequality if the Riemann Hypothesis is false. Using the classical error term in the prime number theorem, we show that every superabundant counterexample satisfies for absolute constants . Since the least counterexample satisfies , we deduce that the Riemann Hypothesis is true if there are constants and with , and a threshold , such that for every superabundant number with . We also give explicit versions of these bounds: every counterexample satisfies , and every superabundant counterexample satisfies . An enumeration of the superabundant numbers up to shows that only of the exceeding satisfy , namely those with , the largest of which is . Testing the hypothesis of the sufficient condition on the same range, we find that all superabundant numbers between and satisfy , and that for each the minimum of over them is positive, ranging from for to for , and is attained at the same superabundant number, whose largest prime factor is .
math.NT — Number Theory
We prove that at least one of the eight numbers is irrational. The previous best result of this kind not involving , a consequence of a theorem of Lai and Zhou, was that one of is irrational. We use Zudilin's hypergeometric construction: sums of fourth derivatives of rational functions with quintuple zeros give linear forms in with no or term. The improvement comes from spreading the poles over eighteen blocks with nearby offsets, a sharper -adic bound on the denominators, and a saddle-point analysis of a cotangent contour integral. The estimates rest on exact rational and interval-arithmetic certificates. The whole proof, including these certificates, is formalised in Lean 4 with Mathlib and checked by the Lean kernel using only the standard axioms.
math.MG — Metric Geometry
We optimize a single term in the cube–Euclidean-ball construction of V. Patankar to obtain C_70 >= exp(1/(2e)) > 1.20194336847. The strongest public proposal located and reproduced in our dated source audit is Patankar's 1.201927, whose specified witness evaluates near 1.2019271425725. A separate finite-dimensional witness certifies 1.20194336. Two separately written arithmetic implementations, using Arb and rational logarithm/Stirling bounds respectively, verify the numerical comparisons and reject corrupted or excessive claims. The geometric reduction and passage to the limit are proved here. This unrefereed preprint is not peer reviewed and not formally verified; exhaustive priority is not asserted.
math.CO — Combinatorics
We prove that the local-degree potential of a Boolean function of degree at most three is at most 5/4, using the published fact that such a function has at most ten relevant variables. Inserting this cap into Wellens's degree–block-sensitivity recurrence, with the existing exact certificates released by Épi, gives C_44 <= 4.3576549226. The reproduced preceding recurrence gives approximately 4.365467422574458; our exact degree-three cap lowers it by 1/128. Two separately written exact-arithmetic programs check the finite certificates, recurrence, infinite tail and finite cases of the added lemma. The universal argument is written in the paper; the published degree-three classification is cited, not re-enumerated. This preprint is not peer reviewed and not verified in a proof assistant; no optimality is claimed.
math.MP — Mathematical Physics
We consider the Wilson lattice gauge measure for on a periodic three-dimensional torus of fixed size, with bare coupling proportional to the lattice spacing , as dictated by the superrenormalizability of three-dimensional Yang–Mills theory. Observables are Wilson loops of the lattice configuration evolved by the Wilson flow, the lattice gradient flow of the Wilson action, at positive flow time. We prove that, at every fixed value of the coupling, the joint laws of finitely many flowed Wilson loops converge as the lattice spacing tends to zero along the sequences of lattice sides , for fixed odd and fixed positive divisible by four. For each such refinement class, the limits are realized by a unique probability measure on the space of gauge orbits of Yang–Mills heat-flow trajectories at positive times, a version of the state space introduced by Cao and Chatterjee. At each fixed positive time, the energies of the selected trajectories , with distributed under the lattice Wilson measures, are tight uniformly in the lattice spacing. The limiting measure is invariant under the lattice symmetries and the center symmetries. We also show that, at short distances, the rescaled deficits of the normalized fundamental Wilson loops of loops , flowed for time with fixed , converge to explicit quadratic functions of the heat-smoothed Gaussian free field. As the coupling tends to zero, the measure concentrates on flat connections, with Gaussian fluctuations of explicit covariance near regular commuting holonomies and fluctuations governed by the constant-mode quartic form near the central holonomies. The proof is a multiscale renormalization group analysis in the tradition of Bałaban, carried out on a small-field event whose complement we show to have small probability, uniformly in the lattice spacing, using reflection positivity and chessboard estimates. It is combined with stochastic estimates for the lattice heat flow and a deterministic analysis of the Wilson flow for rough initial data.
math.MP — Mathematical Physics
We consider the Wilson lattice gauge measure for on a periodic three-dimensional torus of fixed size, with bare coupling proportional to the lattice spacing , as dictated by the superrenormalizability of three-dimensional Yang–Mills theory. Observables are Wilson loops of the lattice configuration evolved by the Wilson flow, the lattice gradient flow of the Wilson action, at positive flow time. We prove that, at every fixed value of the coupling, the joint laws of finitely many flowed Wilson loops converge as the lattice spacing tends to zero along the sequences of lattice sides , for fixed odd and fixed positive divisible by four. For each such refinement class, the limits are realized by a unique probability measure on the space of gauge orbits of Yang–Mills heat-flow trajectories at positive times, a version of the state space introduced by Cao and Chatterjee. At each fixed positive time, the energies of the selected trajectories , with distributed under the lattice Wilson measures, are tight uniformly in the lattice spacing. The limiting measure is invariant under the lattice symmetries and the center symmetries. We also show that, at short distances, the rescaled deficits of the normalized fundamental Wilson loops of loops , flowed for time with fixed , converge to explicit quadratic functions of the heat-smoothed Gaussian free field. As the coupling tends to zero, the measure concentrates on flat connections, with Gaussian fluctuations of explicit covariance near regular commuting holonomies and fluctuations governed by the constant-mode quartic form near the central holonomies. The proof is a multiscale renormalization group analysis in the tradition of Bałaban, carried out on a small-field event whose complement we show to have small probability, uniformly in the lattice spacing, using reflection positivity and chessboard estimates. It is combined with stochastic estimates for the lattice heat flow and a deterministic analysis of the Wilson flow for rough initial data.
math.MP — Mathematical Physics
We consider the Wilson lattice gauge measure for on a periodic three-dimensional torus of fixed size, with bare coupling proportional to the lattice spacing , as dictated by the superrenormalizability of three-dimensional Yang–Mills theory. Observables are Wilson loops of the lattice configuration evolved by the Wilson flow, the lattice gradient flow of the Wilson action, at positive flow time. We prove that, at every fixed value of the coupling, the joint laws of finitely many flowed Wilson loops converge as the lattice spacing tends to zero along the sequences of lattice sides , for fixed odd and fixed positive divisible by four. For each such refinement class, the limits are realized by a unique probability measure on the space of gauge orbits of Yang–Mills heat-flow trajectories at positive times, a version of the state space introduced by Cao and Chatterjee. At each fixed positive time, the energies of the selected trajectories , with distributed under the lattice Wilson measures, are tight uniformly in the lattice spacing. The limiting measure is invariant under the lattice symmetries and the center symmetries. We also show that, at short distances, the rescaled deficits of the normalized fundamental Wilson loops of loops , flowed for time with fixed , converge to explicit quadratic functions of the heat-smoothed Gaussian free field. As the coupling tends to zero, the measure concentrates on flat connections, with Gaussian fluctuations of explicit covariance near regular commuting holonomies and fluctuations governed by the constant-mode quartic form near the central holonomies. The proof is a multiscale renormalization group analysis in the tradition of Bałaban, carried out on a small-field event whose complement we show to have small probability, uniformly in the lattice spacing, using reflection positivity and chessboard estimates. It is combined with stochastic estimates for the lattice heat flow and a deterministic analysis of the Wilson flow for rough initial data.
math.MP — Mathematical Physics
We consider the Wilson lattice gauge measure for on a periodic three-dimensional torus of fixed size, with bare coupling proportional to the lattice spacing , as dictated by the superrenormalizability of three-dimensional Yang–Mills theory. Observables are Wilson loops of the lattice configuration evolved by the Wilson flow, the lattice gradient flow of the Wilson action, at positive flow time. We prove that, at every fixed value of the coupling, the joint laws of finitely many flowed Wilson loops converge as the lattice spacing tends to zero along the sequences of lattice sides , for fixed odd and fixed positive divisible by four. For each such refinement class, the limits are realized by a unique probability measure on the space of gauge orbits of Yang–Mills heat-flow trajectories at positive times, a version of the state space introduced by Cao and Chatterjee. At each fixed positive time, the energies of the selected trajectories , with distributed under the lattice Wilson measures, are tight uniformly in the lattice spacing. The limiting measure is invariant under the lattice symmetries and the center symmetries. We also show that, at short distances, the rescaled deficits of the normalized fundamental Wilson loops of loops , flowed for time with fixed , converge to explicit quadratic functions of the heat-smoothed Gaussian free field. As the coupling tends to zero, the measure concentrates on flat connections, with Gaussian fluctuations of explicit covariance near regular commuting holonomies and fluctuations governed by the constant-mode quartic form near the central holonomies. The proof is a multiscale renormalization group analysis in the tradition of Bałaban, carried out on a small-field event whose complement we show to have small probability, uniformly in the lattice spacing, using reflection positivity and chessboard estimates. It is combined with stochastic estimates for the lattice heat flow and a deterministic analysis of the Wilson flow for rough initial data.
math.QA — Quantum Algebra
For every odd integer , we construct an untwisted Haagerup--Izumi fusion category on the negative spherical branch and a simple, self-contragredient, regular, -cofinite vertex operator algebra of central charge , together with an equivalence of modular fusion categories: \[ \operatorname{Rep}(E_N)\simeq Z(\mathcal H_N). \] The VOA is rational, has exactly irreducible modules, and is lower bounded and integer graded, with least conformal degree . The proof begins with a CFT-type cyclic Virasoro extension of central charge , whose regularity we establish. Two categorical condensations recover , and a finite extension by shifted lattice modules removes the auxiliary pointed factor. We identify the complete ribbon modular data of with in the notation of Gannon--Schopieray--Yadav, for every odd .
math.MP — Mathematical Physics
We consider the Wilson lattice gauge measure for on a periodic three-dimensional torus of fixed size, with bare coupling proportional to the lattice spacing , as dictated by the superrenormalizability of three-dimensional Yang–Mills theory. Observables are Wilson loops of the lattice configuration evolved by the Wilson flow, the lattice gradient flow of the Wilson action, at positive flow time. We prove that, at every fixed value of the coupling, the joint laws of finitely many flowed Wilson loops converge as the lattice spacing tends to zero along the sequences of lattice sides , for fixed odd and fixed positive divisible by four. For each such refinement class, the limits are realized by a unique probability measure on the space of gauge orbits of Yang–Mills heat-flow trajectories at positive times, a version of the state space introduced by Cao and Chatterjee. At each fixed positive time, the energies of the selected trajectories , with distributed under the lattice Wilson measures, are tight uniformly in the lattice spacing. The limiting measure is invariant under the lattice symmetries and the center symmetries. We also show that, at short distances, the rescaled deficits of the normalized fundamental Wilson loops of loops , flowed for time with fixed , converge to explicit quadratic functions of the heat-smoothed Gaussian free field. As the coupling tends to zero, the measure concentrates on flat connections, with Gaussian fluctuations of explicit covariance near regular commuting holonomies and fluctuations governed by the constant-mode quartic form near the central holonomies. The proof is a multiscale renormalization group analysis in the tradition of Bałaban, carried out on a small-field event whose complement we show to have small probability, uniformly in the lattice spacing, using reflection positivity and chessboard estimates. It is combined with stochastic estimates for the lattice heat flow and a deterministic analysis of the Wilson flow for rough initial data.
math.RT — Representation Theory
Bautista, Gabriel, Roiter, and Salmer\'on proved that every finite-dimensional representation-finite algebra over an algebraically closed field admits a multiplicative basis, in which products of basis elements are zero or basis elements. Outside characteristic two, they also proved that such an algebra is standard: its category of indecomposable modules is presented by its Auslander--Reiten quiver and mesh relations. We give another proof of both theorems. For a basic algebra, grading morphisms between indecomposable modules by powers of the categorical radical and restricting to projectives yields its standard form, a standard algebra with the same Auslander--Reiten quiver. After constructing a multiplicative basis of the standard form, we recover the original algebra by filtered deformation. Positive-degree Hochschild cohomology embeds into cokernels measuring errors in mesh relations. In characteristic two, lifting derivations modulo four produces cocycles supported on zero basis products; the resulting polynomial family contains the original algebra with coefficients zero or one.
math.NT — Number Theory
This paper presents a proof of Legendre’s Conjecture by analyzing the distribution of odd composite and their elimination through odd composite sequences. The odd integer sequence is expressed as 𝑂≥3 = 3 + 2𝑟𝑂, while odd composite numbers are represented as 𝑂𝑝 = 𝑝2 + 2𝑝𝑟𝑝, where 𝑝 is prime and 𝑟𝑝 ≥ 0. Each prime-square sequence introduces a systematic elimination of odd composite numbers, reducing the density of primes in the sequence. By sequentially applying elimination factors (𝑞 − 1)/𝑞 for primes 𝑞 ≥ 3, we derive a lower bound for prime density: d/(dr_O ) N_p≥2/p Considering the quadratic interval [𝑛2, (𝑛 + 1)2], the total change in 𝑟𝑂 is Δ𝑟𝑂 = (𝑛+1)2−𝑛2 = 𝑛 + ½ . Combining this with the density bound yields a lowest bound of prime count per interval: π_((n^2,(n+1)^2))^LB≥⌈2/p (n+1/2)⌉=2(p≈n+1) Thus, every quadratic interval [𝑛2, (𝑛 + 1)2] contains at least two primes, establishing Legendre’s Conjecture. This elimination-based framework provides a systematic proof based on the distribution of odd integers and odd composite sequences.
cs.CC — Computational Complexity
Circuit-width restrictions explain several polynomial cases of automatic test pattern generation. We give a different explanation for an arbitrary-size extension of the c432 priority controller, including a subfamily with linear treewidth and net cut-width in its native layout. For a specified first-stage bridge fault, we characterize exactly the reachable qualifier sets and construct a primary input realizing each one. On drive-one inputs where the first selector flag changes, the healthy set is the target singleton; the faulty sets are precisely the target-containing sets with one request bus and all nonempty sets with at least two buses. Drive zero is undetectable. This image reduces detection to comparing decoder values on sets. For supplied guarded decoders, built from zero, positive channel leaves, disjunction and conjunction of one computed child with negative channel literals, supporting paths and dual-Horn reasoning find a detecting set in O((n+q)G) time for n channels, q outputs and counted decoder size G, or O(G) for formula forests. Unsatisfiable instances of the complete guarded native CNF encoding have resolution refutations with O(N³) inferences, where N is the encoding size. Detection is P-complete, while minimum-set optimization and counting detecting sets are NP-hard and #P-complete, respectively. A specified final-qualifier stuck-at fault makes detection NP-complete on shared guarded decoders. The exact first-stage bridge-image statements are formalized in Lean and registered as PALOMAR-2026-10-19-000010, version 1.
math.NT — Number Theory
Let be a prime number and be an exact sequence of finite groups, where is a nontrivial -group. Fix a continuous surjection $\pi:\GQ\twoheadrightarrow T$ and finitely many -conjugation-invariant local conditions $\cD$. Let $N_{G,\pi,\cD}(X)$ count the continuous surjections $\rho:\GQ\twoheadrightarrow G$ satisfying and these local conditions, with , where $F_\rho=\overline{\Q}^{\ker\rho}$. If at least one such lift exists, we prove that, for every , \[ X^{p/(|G|(p-1))} \ll_{G,\pi,\cD} N_{G,\pi,\cD}(X) \ll_{G,\pi,\cD,\varepsilon} X^{p/(|G|(p-1))+\varepsilon} \] for all sufficiently large . This establishes the power of predicted by Alberts's conjecture for every finite -group kernel, without any solvability assumption on . If every element of order in lies in , then every nonempty family satisfies the asymptotic predicted by Alberts: \[ N_{G,\pi,\cD}(X) \sim C_{G,\pi,\cD} X^{p/(|G|(p-1))}(\log X)^{b_V-1}, \] with $C_{G,\pi,\cD}>0$. As an application, for odd and , we obtain discriminant asymptotics for surjective lifts of a fixed residual Galois representation to $\GL_d(\Z/p^m\Z)$, subject to finitely many local conditions, whenever such a lift exists.
math.CO — Combinatorics
Conjecture 66 of Graffiti.pc bounds the order of a largest induced forest of a connected graph from below by the even mode of the degrees of its complement divided by its average degree. We show that it is false, under every reading of its definitions and by an unbounded margin.
math.CO — Combinatorics
Written on the Wall II, the list of conjectures of the program Graffiti.pc, records Conjectures 319, 352, 358, 359 and 427 as open. We show that all five are false. The counterexamples are caterpillars for Conjectures 352, 358, 359 and 427, and graphs built from complete graphs for Conjecture 319; each comes in an infinite family, and for Conjectures 358, 359 and 427 the difference between the two sides is unbounded.
math.MG — Metric Geometry
Morley's theorem says that the adjacent angle trisectors of a triangle meet in the vertices of an equilateral triangle. Tran carried the construction to a tetrahedron by trisecting its six dihedral angles and, more generally, by -secting planes, with one parameter for each pair of opposite edges; the resulting tetrahedron is denoted . Tran showed that a regular Morley tetrahedron does not force the original tetrahedron to be regular, observed that in his examples some pair of opposite edges has four equal cross edges, and conjectured that this happens whenever is regular. We show that the conjecture is false. We place an isosceles tetrahedron in the box and prove that is regular exactly when its vertex differs from the centre of the box and lies on one of the four lines through the centre in the directions , , and . Every interior point of these lines other than the centre occurs for exactly one triple of parameters. When , and are pairwise different, no pair of opposite edges has four equal cross edges, so each of these tetrahedra is a counterexample. Two examples are given with the parameters in closed form. Finally, for quadrisecting planes, , an isosceles tetrahedron has a regular Morley tetrahedron only if it is regular.
math.CO — Combinatorics
Sahbi conjectured that the -independence number and the upper -star-forming number $\SF_k$ agree on bipartite graphs, and in particular on chain graphs, and asked whether $\beta_3=\SF_3$ on triangle-free graphs. For every we give a connected bipartite graph on vertices with and $\SF_k\ge3k$. On chain graphs, however, $\beta_k=\SF_k$ for every . We also show, answering another of his questions, that deciding whether a bipartite graph has a minimal -star-forming set with at least vertices is NP-complete.
math.RA — Rings and Algebras
Adams, Freese, Nation and Schmid proved that the complement of a maximal sublattice of a finite bounded lattice is an interval, and asked whether the same holds for finite semidistributive lattices; Adaricheva, Mata, Silberger and Zamojska-Dzienio conjectured that it does. It does not. In a 26-element semidistributive sublattice of the square of a 14-element semidistributive lattice, the diagonal is a maximal sublattice whose complement has two minimal elements.
math.AC — Commutative Algebra
Hudzik and Landes \textit{[Math. Ann., 1992]} derived a major generalization of the triangle inequality for two nonzero elements in normed linear spaces, which was extended to finitely many nonzero elements independently by Dragomir \textit{[Bull. Aust. Math. Soc., 2006]} and by Kato, Saito and Tamura \textit{[Math. Inequal. Appl., 2007]}. We derive a finite field version of Hudzik-Landes-Dragomir-Kato-Saito-Tamura inequality.
math.RA — Rings and Algebras
Adaricheva, Mata, Silberger and Zamojska-Dzienio conjectured that in a finite convex geometry, and more generally in a finite join-semidistributive lattice, the complement of a maximal sublattice is a union of intervals with a common least element, and proved this for convex dimension two. We give convex geometries of convex dimension four and three in which such a complement has two and three minimal elements, so their theorem does not extend. The same examples refute their Hypothesis 4 and one half of their Question 2; an eleven-element convex geometry refutes the other half, and a ten-element lattice their Hypothesis 3.
math.CO — Combinatorics
A sequence of vertices of a graph is legal when every vertex in it dominates some vertex that none of the earlier ones dominated, and the Grundy domination number of the graph is the length of its longest legal sequence. In the strong product of two graphs, two vertices are adjacent when each coordinate is equal or adjacent in its factor. It has been known since 2016 that the Grundy domination number of a strong product is at least the product of the Grundy domination numbers of the two factors, and it was conjectured, by Bre\v{s}ar, Bujt\'as, Gologranc, Klav\v{z}ar, Ko\v{s}mrlj, Patk\'os, Tuza and Vizer, that equality always holds. We show that the conjecture is false, and false by an unbounded factor: for every number there are graphs whose strong product has Grundy domination number more than times the product of the two factors. The graphs are built on the five-element subsets of an -element set, with adjacency decided by the size of the intersection; incidence matrices over the fields with two and three elements bound the two factors by and , while the diagonal of the product is an independent set of size . We also compute the Grundy domination number of the graph on in which and are adjacent when : it is for every . Its strong product with the octahedron has a legal sequence of length , so the conjecture already fails for , on a graph with vertices, and for every . Every statement is proved by hand.
math-ph — Mathematical Physics
We combine explicit spectral calculations for the Dwork and weighted Fermat pencils with an audit of the Geometric Selection Conjecture. The residue calculation determines the character labels, cubic couplings and invariant-direction mass maps, while an index argument excludes three standard-embedding families on free quotients of thirteen hypergeometric complete-intersection types. For a nonstandard construction we fix one tetraquadric equation, prove smoothness and a free degree-four action, and calculate equivariant bundle cohomology by an exact determinant certificate. Its projected charged spectrum contains three families and one Higgs doublet pair. At the specified split-bundle point, however, an additional Abelian gauge factor survives, the perturbative Yukawa matrices fail the required rank test, and the tree-level volume remains flat. We then show that the first of these failures is forced: on every Calabi--Yau threefold with at most four K\"ahler parameters, every polystable sum of five line bundles with vanishing first Chern class has a charge matrix of rank at most three and keeps at least one Abelian gauge boson massless at its split locus. On the tetraquadric the polystable K\"ahler classes of the fixed bundle form exactly the volume ray. For the second failure we decide the perturbative selection rules on eleven split data, the fixed bundle and ten further line-bundle sums with three families on the same manifold, by a flow criterion for the singlet insertions: six of the eleven lose a Yukawa matrix identically and five are not excluded. We then follow the extension branch: a single non-split extension never removes the forced gauge boson, while for the fixed bundle and five of the ten further sums a polystable sum of two rank-two extensions and a line bundle, with the same Chern character, exists on one side of a stability wall arbitrarily close to the split point and carries no massless Abelian gauge boson; for three further sums no such bundle exists at any K\"ahler class, and for one it does not exist on the general tetraquadric. We fix the acceptance windows for the stabilization, Yukawa and supersymmetry breaking conditions, so that the conjecture becomes a definite statement, and show that the failures found do not depend on them. No vacuum satisfying all nine conditions is constructed; the Geometric Selection Conjecture remains unproved.
math.PR — Probability
We determine the asymptotic distribution of the largest gaps between consecutive eigenvalues of the Gaussian orthogonal ensemble (GOE) in the bulk of the spectrum. After an explicit centering and scaling, the largest gaps in a fixed interval converge to a Poisson point process with an exponential intensity, so the largest gap has a shifted Gumbel limit. The analogous result for the Gaussian unitary ensemble was proved by Feng and Wei. Our proof expresses the squared probability that the GOE has no eigenvalues in a union of intervals as the corresponding probability for the Gaussian unitary ensemble times a determinant of fixed size, for every matrix dimension. We control this determinant on intervals whose length in units of the local mean spacing grows with the dimension, using an uncertainty principle of Nazarov and exact identities for Hermite functions.
math.QA — Quantum Algebra
We construct vertex operator algebras realizing an infinite family of twisted, nonunitary Haagerup--Izumi doubles. For every odd integer , we construct a spherical fusion category with the Haagerup--Izumi fusion rules for and pointed associator of exact order , and realize its Drinfeld center as the ribbon category of modules of an explicit vertex operator algebra . The construction begins with a uniform family of strongly rational Virasoro extensions, whose irreducible modules and theta-function characters we determine. A simple-current extension involving a conformally shifted lattice vertex operator algebra removes the additional semion factor. The resulting is simple, regular, self-contragredient, and -cofinite, with central charge , effective central charge eight, and irreducible modules with explicit characters. Its weight-zero subspace has dimension , so it is not of CFT type; we establish modularity and the ribbon equivalence without relying on the CFT-type hypothesis.
math.OA — Operator Algebras
Let , and let be density operators on $\C^a$ and $\C^b$, respectively. Let denote the set of bipartite couplings on $\C^a \otimes \C^b$ with these marginals. Parthasarathy showed that every extreme coupling has rank at most $\KRP(a,b) := \lfloor\sqrt{a^2+b^2-1}\rfloor$. It is not known for which dimension pairs this bound is tight. We restrict attention to the tracial case , . For the Parthasarathy bound, we construct extreme couplings of rank from pairs of length- finite unit-norm tight frames (FUNTFs) in $\C^a$ and $\C^b$ satisfying a transversality condition. This yields an extreme coupling of rank for every , and hence tightness of $\KRP(a,b)$ whenever and ; in particular, along the whole qubit-qudit line , . This recovers the two previously known tracial cases with , namely and . The Schmidt decomposition and subadditivity of rank give the lower bound $\ES(a, b) := \lceil b/a\rceil$ for the rank of every tracial coupling, extreme or not. For the Schmidt bound, we use sparse Kraus operators built from weighted partial permutation matrices. Setting $r := \ES(a, b)$, we construct an extreme coupling of rank whenever divides , and show that for , where this condition reads , it is also necessary. For , the divisibility condition is superfluous and the Schmidt bound is attained for every such . Thus, for each fixed , attainability of both bounds remains open for only finitely many .
math.CV — Complex Variables
We investigate the complex geometry of the normalized Dirichlet eta terms tn(s) = 1 1−2 1−s (−1)n−1 ns (s = a + bi). By analyzing the Möbius geometry of each term, we demonstrate that the base-2 resonant partial sum S2(s) = t1(s) + t2(s) forms an exact circle. Remarkably, this base circle is centered strictly at the origin (0, 0) if and only if Re(s) = 1 2 . We further establish a dyadic pairing theorem showing that every oddeven pair Pm(s) = tm(s) + t2m(s) inherits this origin-centered circular symmetry on the critical line. Analyzing the phase distribution on these concentric circles reveals four geometric mechanisms that distinguish non-trivial Riemann zeta zeros b = bn from generic values of b, including right-half-plane slow-zone clustering of P1 and opposinghemisphere alignment of the leading inner circles P3, P5, P7. Finally, we prove that removing all odd-even dyadic pairs leaves the multiples-of-4 subseries P∞ k=1 t4k(s) = t4(s)ζ(s), whose partial sums spiral around a target center Ctarget(b) that coincides with the origin (0, 0) if and only if b is a Riemann zeta zero.
math.PR — Probability
Let be the orthonormal Hermite polynomials for the standard Gaussian weight, and let be a real power series with radius of convergence greater than , , no zeros on the unit circle and only simple zeros in the unit disk. For all large the polynomial has exactly as many nonreal zeros as has in the disk, but the known proof gives the onset by compactness. We make it explicit: if has no zeros in , with below an explicit threshold, the count is exact as soon as exceeds a polynomial in and in the inverses of , , and ; no separation of the nonreal zeros of is needed. By an exact identity this applies to the random Weyl polynomials with independent standard Gaussian coefficients under the backward heat flow , whose real-rooting time converges in distribution to , where is the smallest modulus of a nonreal zero of the Gaussian power series . We show that the convergence holds with the rate on for every , an exponent that does not depend on ; the dependence on is removed by a super-exponential tail for the number of zeros of in a disk. Under a coupling of the degrees that shares the top coefficients, the real-rooting time satisfies for , with and depending on . We bound and by polynomials in condition parameters of , with absolute degrees and, when , coefficients polynomial in . For the Gaussian power series this gives, on the event , a power tail with exponent for the largest scaled displacement of the final collision of zeros, and without restriction the tail bound ; the inputs include closed forms of the two-point correlation functions of the zeros of .
math.NT — Number Theory
We give an exact-input certificate for the upper bound C_42 <= 0.6888863712 in the pure power-sum problem. The construction refines the step profiles of Griego, Röhrig and Plata to 256 middle blocks, without changing their coefficient-cancellation mechanism. It improves the strongest public proposal located in our 22 October 2026 audit, 0.688970, by 0.0000836288. An unmodified public interval checker and a separately written Arb checker verify the strict finite inequalities. The asymptotic implication is proved in the note; no explicit finite threshold, optimality, human peer review or formal verification is claimed.
math.MG — Metric Geometry
We disprove Matschke's table conjecture, which asserts that every continuous real-valued function on takes a common value on the vertices of some rotated copy of every prescribed inscribed square. The counterexample is the cubic polynomial and a square contained in a plane at distance from the origin. We give an elementary proof.
math.PR — Probability
Let be a random Weyl polynomial of degree under the backward heat flow, with independent coefficients of a law that is not a point mass and has finite variance. Hall, Ho, Jalowy and Kabluchko showed that the empirical zero distribution collapses onto the real line at . We show that the first time at which all zeros of are real, a finite-degree analogue of the de Bruijn--Newman constant, converges in distribution to , where is the smallest modulus of a nonreal zero of the random power series with coefficients of the same law; the convergence is almost sure under a coupling of the degrees that shares the top coefficients, whereas under the standard coupling the set of limit points of the real-rooting time is almost surely . The proof rests on an exact count of the nonreal zeros of analytic combinations of orthonormal Hermite polynomials: if the real power series has radius of convergence greater than , no zeros on the unit circle and simple zeros in the unit disk, then for all large the combination has exactly as many nonreal zeros as has in the disk. Its method expresses the polynomial and its derivative in the bulk through a matrix applied to , built from only steps of the backward three-term recurrence, with the remaining terms controlled by a Christoffel--Darboux bound. For Gaussian coefficients the law of is a Fredholm determinant, as , and as , half of the constant for complex coefficients. At the scale the last pair of nonreal zeros collides inside the real bulk, at relative position , where are the nonreal zeros of minimal modulus, and the real-rooting time is within of . In a window of size the polynomial is, up to a positive factor and , the model , whose threshold for real-rootedness is given by Kepler's equation; the phase rotates with , so oscillates over an explicit interval and has an explicit limit law jointly with .
math.AG — Algebraic Geometry
We prove that Liu's conjecture on Fano indices fails in dimension four: is the index of a terminal Calabi--Yau fourfold and of a four-dimensional log canonical singularity, but it is not the Fano index of any canonical weighted projective space of dimension four.
math.RA — Rings and Algebras
Wigderson conjectured that for every finite field and every there is a space of matrices over of dimension whose only nilpotent element is zero. We show that every six-dimensional space of matrices over $\F_3$ contains a nonzero nilpotent matrix, so the conjecture fails for $F=\F_3$ and ; the largest possible dimension is five.
math.MG — Metric Geometry
We prove that no packing of unit balls in whose centres form two translates of a lattice is denser than the lattice packing. Up to congruence exactly three such packings attain this density: , a layered packing of Leech, and the packing with Szöllősi's kissing configuration.
math.NT — Number Theory
We prove Kimberling's conjecture on greedy representations by Fibonacci and Lucas numbers: the ratio of the number of Zeckendorf summands of to the number of terms of their greedy representations tends to a limit in , and the limit is . We also prove the conjectures of Greathouse and Kurkov on the number of terms of the greedy representation.
math.AG — Algebraic Geometry
Let . We prove that the Weil classes of every polarized complex abelian sixfold of Weil type for whose Hermitian discriminant is the non-split class are algebraic, and that every Hodge class on every power of such a sixfold is algebraic when its Hodge group is the full special unitary group. The proof starts from Galois covers of the projective line with monodromy group , branched over twelve points: the multiplicity space of the natural representation in the cohomology of such a cover is the first cohomology of an abelian sixfold of Weil type, and a certified rank computation at one explicit cover shows that these sixfolds fill an open set of the nine-dimensional moduli space. Algebraic cycles reach these sixfolds through Milne's theorem on Lefschetz classes, from two sources: Schoen's cycles on cyclic covers, which a computer search shows do not suffice here, and products of determinant lines of characters on abelian covers of curves. For the second source we prove a general theorem (Theorem C): if such a product is of Hodge type at every Galois conjugate, and each of the cyclic covers involved has at least two branch points and order dividing after removing the common factor of its branch exponents, then the product spans a space of algebraic classes. Its proof runs Schoen's symmetric-product construction on each factor, localises at a special fibre, transfers to a product of Fermat hypersurfaces, and applies Shioda's theorem on products of Fermat varieties. This version is archived on Zenodo: https://doi.org/10.5281/zenodo.23566607; its computational certificates: https://doi.org/10.5281/zenodo.23549514
math.GT — Geometric Topology
We determine two invariants for knots that the KnotInfo database leaves open. KnotInfo lists the bridge index of \btotal{} prime knots with thirteen crossings as either or ; we prove that \bclosed{} of them have bridge index and meridional rank , so the meridional rank conjecture holds for them. The lower bound comes from representations of the knot group in $\GL_4(\F_q)$ that send a meridian to a transvection or a reflection and fix no nonzero vector. KnotInfo leaves the smooth nonorientable four-genus open for \ntotal{} prime knots with eleven to thirteen crossings; we determine it for \nsettled{} of them and narrow the interval for \nnarrowed{} more. Upper bounds come from pinch moves, lower bounds from the Goeritz-lattice obstructions of Jabuka and Kelly evaluated by an exhaustive search for lattice embeddings. Every bound is a certificate checked by a program independent of the search that found it.
math.CO — Combinatorics
Shankar proved that a family of North--East lattice paths from to in which any two paths share an even number of edges has at most members, and conjectured that exactly families attain this bound, where is the th Catalan number. We prove the conjecture. The proof determines all equality cases of the inequality behind Shankar's bound: they correspond to noncrossing partial matchings of whose unmatched points lie under no arc, and there are of them.
math.CO — Combinatorics
We prove that every binary circular word of length contains at least $\fl{(n-1)/2}$ distinct abelian squares, and we determine all words attaining this bound. This settles a conjecture of Simpson: its odd case holds, and in its even case the list of extremal words misses for , . We then review the other problems of the survey: we conjecture, and verify to length , that for odd the least number of non-equivalent abelian squares in a circular word is also , and we show that the least number of occurrences of abelian squares in a binary word of length is asymptotically proportional to or grows faster, and is below .