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math.NT — Number Theory
Let denote the Dirichlet beta function. For all sufficiently large even integers , we prove .
cs.CC — Computational Complexity
The Minimum Independent Dominating Set problem (MIDS) asks for a smallest maximal independent set of an undirected graph; unless , it cannot be approximated within for any . We present version 0.0.8 of \textsc{Siriaisa}, an ensemble of eighteen strategies that adapts the vertex cover algorithm \textsc{Salvador} and the independent set algorithm \textsc{Esperanza} to MIDS, with every exact subproblem a forest solved by a conditioned tree dynamic program. We prove that the output is an independent dominating set computed in expected time, that its ratio is at most , that it is optimal on forests, and that its ratio is at most on three classes of graphs. A degree-splitting strategy yields a dominating set of size at most for an explicit splitting excess , which vanishes on bounded-degree families for large and is largest on complete graphs, where the output is optimal. More generally, splitting at gives the bound for every . We state the ratio as a conjecture whose proof would imply . Through a reduction of Chleb{\'\i}k and Chleb{\'\i}kov{\'a}, every output decodes into a vertex cover, with conditional consequences for the Unique Games Conjecture. In experiments the output is optimal on of graphs with an exact integer-program optimum and on all bipartite graphs whose optimum was proved. As an application, we analyse the SAT solver \texttt{satypy} 0.0.4: it answers every unsatisfiable formula correctly, every SATISFIABLE answer carries a satisfying assignment, a wrong answer requires a ratio larger than for variables, and its construction gives a self-contained proof of Halld{\'o}rsson's inapproximability theorem.
math.GR — Group Theory
We prove that every pair of soluble subgroups of the symmetric group or the alternating group has trivially intersecting conjugates whenever . This answers the second question of Anagnostopoulou-Merkouri and Burness in Problem 21.3 of the Kourovka Notebook.
math-ph — Mathematical Physics
For a non-relativistic atom with protons and bound electrons, Lieb proved in 1984 that . Several improved bounds have been developed, but they all leave open the question of whether a bound helium dianion can exist. We prove that the answer is no. The ground-state energies of the four- and three-electron helium Hamiltonians coincide, , so a helium nucleus binds at most three electrons. A weighted form of Lieb's identity reduces the problem to the nonnegativity of an explicit two-electron Schrodinger operator , and Barta's inequality further reduces it to the pointwise nonnegativity of an explicit function of three variables. We prove the nonnegativity of this function by a combination of analytic estimates and a computer-assisted certification in interval arithmetic, and we publish the certificate with the paper.
math.NT — Number Theory
For a Casas-Alvero polynomial , the type of is one less than the minimal number of roots of needed to meet every Hasse derivative . We show that the type does not increase under reduction modulo a prime and under the descent from degree to degree in characteristic . Consequently, if no Casas-Alvero polynomial of degree and type at most exists over , then none of degree and type at most exists in characteristic , for every . For the hypothesis is equivalent to for an explicit integer , and we show by a finite computation that for and . In particular there is no counterexample of type at most in degree . This gives infinite families of degrees for which the conjecture is not known and every counterexample has type at least : the degrees , the degrees for primes that are bad for degree , the degrees , , and for , , and explicit primes that are bad for these degrees, and the degrees and for all primes outside a finite set (for , including ). For we obtain the first degrees in which every counterexample has type at least : the degrees for , the degrees , the degrees for and the degrees for primes .
math.CO — Combinatorics
We prove an exact formula for the diameter of the directed Cayley graph for the symmetric group with generators (left cyclic shift) and (exchange of the first two elements). For every , we show that the diameter equals for even and for odd . This resolves a longstanding question first proposed in the 1960's and refined in more recent work. The proof is AI-assisted and is based on a variety of combinatorial arguments. We also describe the agentic system we developed that led to this proof and looks highly promising for attacking other questions of this type. An accompanying Lean 4 formalization rigorously verifies the result.
math.FA — Functional Analysis
We study exact embedding obstructions for Hilbertian operator spaces in noncommutative -spaces. We prove a quantitative version of the column inequality, whose equality case forces the absolute values of the entries to be proportional. We deduce that an operator space embedded completely isometrically in a noncommutative -space is a row Hilbert space whenever its first-column norms at the two-by-two level are Hilbert-Schmidt norms. This gives an exact embedding criterion for sums of weighted row and column spaces, including coordinate-dependent weights which are ordered in the same direction. In particular, does not embed completely isometrically into any noncommutative -space when . For , whose complete isometric embeddability is established in a companion paper, we retain the common-support and commutative-target obstructions. We also show that distortion one is approached by matrix models but is attained by none of them, and that the matrix size must tend to infinity as the distortion tends to one.
math.NT — Number Theory
Let be a normalized Hecke eigenform of level and weight , with eigenvalues . In 1981 Hatada asked whether whenever and . Rustom's congruence modulo answers this for . We show that the answer is affirmative for and negative for , and , the counterexamples coming from weight and . The proof combines Rustom's theorem with a certificate for modulo on period polynomials, a -adic Galois law (for instance for every ), and a reduction to a computation in a finite group. As a consequence, implies . At the prime we show that, on Hatada's lattice of period polynomials, , with refinements modulo . This gives an independent proof of Conjecture 9.1 of Rustom without any ramification hypothesis and of the level case of the Coleman--Stein conjecture, both of which also follow from Rustom's recent classification modulo . We also prove that for every , and a -adic Galois law modulo on the Galois group of , which gives congruences modulo and for suitable split primes and a bilinear law for primes . Beyond , for every prime we prove , where ; the exponent is attained for and fails for . Each statement is marked as proved, known, verified or open. The finite computations are certified by seven Python scripts, provided as ancillary files.
math.QA — Quantum Algebra
A Haagerup algebra is a positive strongly rational vertex operator algebra of central charge eight with $\Rep(U)$ braided equivalent to the Drinfeld center of the Haagerup fusion category. Such an algebra has been proposed by Jiao, with a computer-assisted proof, and announced by Riesen and by Huang. We show that has exactly three holomorphic extensions, each isomorphic to , and that the categories of modules over the corresponding Lagrangian algebras are the three Haagerup fusion categories of Grossman and Snyder. The extensions attached to and are the two lattice extensions of . Along a strategy outlined by Huang, we prove a recognition theorem for and among unitary fusion categories of rank six with three invertible objects, a noncommutative fusion ring and a pointed part with trivial associator. For regular unitary vertex operator algebras with Tener's bounded localized vertex operators, we prove that the fusion product of two unitary modules is unitarizable when one of them integrates to the conformal net. The explicit unitary algebra of our previous work, isomorphic to Jiao's algebra, is strongly local. Granting that Jiao's algebra is a Haagerup algebra, we deduce that the conformal net is completely rational with $\Rep(\cal A_S) \cong Z(\cal H_3)$, that the Haagerup subfactor arises from , and that solves a problem of Bischoff on generalized orbifolds. Under the same hypothesis, using a unitary structure on found by exact computer calculations, we prove that is completely unitary and that through the correspondence given by smeared vertex operators. The diagonal construction and the passage to the twin algebra also show that every Haagerup algebra is completely unitary and satisfies Condition I+.
math.NT — Number Theory
We prove uniform bounds for reciprocal sums and residue products along Collatz paths. Let be a finite nonrepeating shortcut path whose input states are greater than , let be the sum of the reciprocals of its odd inputs, and let be its finite-path residue. Then For the first-arrival trajectory of a terminating positive integer , these inequalities give where and are the division and odd-operation counts. More generally, when each odd value greater than is counted once, the corresponding reciprocal-sum and residue bounds hold for every forward orbit, without a termination assumption. The proof combines verified finite height-delay records and residue bounds, canonical-block reciprocal estimates, and an exact finite count organized by gap brackets. Gap brackets, closely related to parity vectors, record the divisions by two between successive odd inputs and organize the finite counting argument; together with a related coordinate representation, they also yield exact formulas for reconstructing finite trajectories. After taking base- logarithms, the bound leaves an uncertainty of less than in the relation between the even- and odd-operation counts. Since these counts are integers, either count is determined exactly by the other count and the path endpoints. We obtain nearest-integer restrictions on the changes in and between adjacent integers, apart from explicit small exceptions, and prove that adjacent terminating integers of equal height have identical operation counts. Thus every equal-height streak lies in a single operation-count class. We further introduce the charge and associated height and charge slice functions to study the persistence and eventual break of equal-height streaks. We distinguish class records, which are the least starting integers attaining specified values, from delay records, which attain new extrema relative to all smaller starting integers. The resulting record theory shows that -delay records form a subsequence of the odd-count delay records, which form a subsequence of the common division-count and height-delay list. On the class side, the nonpositive charge-class records lie among the odd-count class records, which lie among the division-count class records, which in turn lie among the height-class records. The corresponding hierarchies also hold for first-arrival paths to a common endpoint when or when is nonperiodic under the shortcut map.
math.NT — Number Theory
Allouche, Han and Niederreiter asked whether the sequence of Hankel determinants of a -automatic sequence must be -regular, or must become -automatic after reduction modulo an integer. We answer the first question in the negative, and the second one for the reduction modulo . Let be the root of in with , the quartic of Mills and Robbins, and let be its coefficient sequence, which is -automatic by Christol's theorem. The Hankel determinants of vanish in runs whose lengths are unbounded but of the order of the square root of their position, because the partial quotients of are unbounded while its irrationality exponent is ; and a lemma of Schlage-Puchta forbids runs in that regime in an automatic sequence. Hence these Hankel determinants are -automatic for no , and the integer Hankel determinants of any lift of are -regular for no .
math.NT — Number Theory
Gallardo and Rahavandrainy studied an analogue of the Collatz map on , in which plays the role of and that of : an odd polynomial is sent to the odd part of . They proved that every orbit reaches , and conjectured that the orbit of the trinomial has length , where is the largest integer with , and that for it contains terms of the same degree. We prove both statements for every . The proof rests on the identity : it reduces the dynamics to a map from one plateau to the next, characterises the last plateau of any orbit, and shows that along the orbit of the plateaus start at the -th powers of the odd parts of the polynomials , the index following an explicit binary recursion. We also note that a later restatement of the conjecture, with a non-strict inequality, fails exactly when is a power of . A script checks all the statements numerically.
math.NT — Number Theory
A polynomial over is perfect if it equals the sum of all its divisors, and odd if it is coprime to . Canaday conjectured in 1941 that 1 is the only odd perfect polynomial. Known necessary conditions on a counterexample bound its number of distinct irreducible factors (, Gallardo and Rahavandrainy), its number of irreducible factors counted with multiplicity and its degree ( and , Cengiz, Pollack and Trevino), and, when all exponents equal 2, the degree of its smallest irreducible factor (at least 40, Gallardo). We bound a different invariant: every odd perfect polynomial has an irreducible factor of degree at least 20. The proof reduces the question, for each , to a finite system: if all irreducible factors of have degree at most , the exponents of belong to an explicit finite set, through the cyclotomic factorisation of and the multiplicative order of modulo the factors of . An iterated sieve on the resulting production relations empties for every . The computation was carried out twice, by two independent programs, with identical results; the second program is attached as an ancillary file. As a cross-check of the method, the same programs reproduce, by the orbit argument of Gallardo and Rahavandrainy, the exclusion of every prime of degree at most 30 as the smallest factor of a special perfect polynomial, in agreement with the published bounds.
math.CO — Combinatorics
For a finite graph , let be the number of nonempty vertex sets that induce a connected subgraph. We show that a positive integer equals for some finite connected graph if and only if . Every attainable value is attained on vertices, the least possible number, and needs six vertices. More precisely, for every each integer from to is the value of on a connected graph with exactly vertices. The proofs of these results are short and use no computer; computer calculations serve only as independent checks, and for two auxiliary results: the values on six to eight vertices, and the result on all connected subgraphs described below. The value falls into a gap above the path: a connected graph on vertices other than the path has at least more connected sets than . The values are realized by a recursive construction from cones, disjoint unions and a family of graphs that we call near-cones, starting from seven small graphs; it produces a graph with in time polynomial in . The question arose from a problem of Ordoñez in continuum theory, through a reduction by Camargo, Ordoñez and Ramírez, who found the values up to by a computer search. The proof is formalized in Lean 4. We also observe that a theorem of Czabarka, Székely and Wagner on subtrees reduces the corresponding problem for the number of all connected subgraphs to a finite computation, and we report its outcome. We did not find these results in the literature, but we make no claim of novelty or priority.
math.CO — Combinatorics
We prove Hilton’s conjecture on the number of cycles of each length in dense Hamiltonian graphs. This resolves the problem posed by A. J.W. Hilton and recorded by Sheehan in 1977.
math.NT — Number Theory
J.-L. Nicolas asked how often a power of belongs to the image of , where is the -adic valuation, and showed that both the number of exponents with in the image and the number of the other exponents tend to infinity. We prove the identity , the sum running over the positions of the binary digits of , so that lies in the image of if and only if is a sum of distinct terms of the superincreasing sequence . For every exponent there is at most one possible largest term , given by , and no such exists exactly when . The remaining exponents are decided by an explicit recursion on pairs of integers bounded by , in at most steps. We give the closed formula for , and , and infinite explicit families of exponents for which is not in the image, indexed by and ; the families given by Nicolas are the cases and . As a consequence, and . We prove that the states of the recursion form a forest, rooted at the states and at the failing states, in which every state has exactly children; that the sets of successful states are symmetric; that exactly half of them lie in the tree of the root ; that ; that for ; and that satisfies a recursion from the smallest elements upwards, with period for even . The symmetry gives further explicit families, among them , and the improved bounds and . For every prime , the function counts the factor in over ; we extend to it the reduction, the unique candidate, the recursions and the symmetry, give explicit families, and describe completely the exponents rejected at the first step, which occur only for odd . We also report experiments in exact integer arithmetic: every statement is checked by three independent methods, the values of reproduce the table of Nicolas, the sets are computed for , and lies between and for .
math.CO — Combinatorics
Gupta, Ndiaye, Norin and Wei (GNNW) recast the Campos–Griffiths–Morris–Sahasrabudhe method as an inductive theorem that turns a known bound on off-diagonal Ramsey numbers into a better one. With two applications they obtained . They also reported an unverified, AI-suggested third application giving . We make three observations about their framework. (i) The free function can be optimised pointwise. (ii) For a fixed input, the best output exponent is the least solution of an explicit first-order differential inequality. (iii) GNNW's Theorem 14 needs only with locally Lipschitz derivative. We then construct an explicit chain of eight applications and verify every hypothesis by interval arithmetic on the whole range , with a hand-proved estimate covering . The verification was done twice, with independent code, libraries and enclosure strategies. This proves , and in particular confirms GNNW's . Numerically, iterating the optimal step converges to about , which suggests this is close to the limit of the GNNW method in its present form.
math.NT — Number Theory
A Mersenne prime polynomial is an irreducible polynomial over of the form with . Gallardo and Rahavandrainy conjectured that, apart from two exceptions, always has an irreducible factor which is not a Mersenne prime. For a prime they proved it for every outside an exceptional set , defined through an integer and an odd integer read off the leading coefficients of , and later they proved it for every Mersenne prime number ; their method does not reach . We compute exactly when is a Mersenne prime number: if , and if . Consequently consists of the Mersenne primes of degree or , is empty, and , are described by and respectively. For this reduces the conjecture to the Mersenne primes with . A direct computation shows that has a non-Mersenne irreducible factor for every Mersenne prime of degree at most , and that in that range. Hence the conjecture holds whenever divides and , and a counterexample with would need . A script attached as an ancillary file checks these computations, and reproduces the table of exceptional sets of Gallardo and Rahavandrainy.
math.OA — Operator Algebras
We ask for noncommutative versions of following four: (1) Dvoretzky Theorem, (2) Type-Cotype Definition, (3) Khinchin-Kahane Inequalities, (4) Grothendieck Inequality.
math.NT — Number Theory
We prove that the order reversal asked about in Question 35 of Banaian and Huang cannot occur for generalized k-Markov numbers, answering their question negatively.
math.SG — Symplectic Geometry
We prove that the deep locus of a complex double Bott–Samelson variety associated with a positive braid is its stabilizer locus. Consequently, the deep locus is empty if and only if any two components of the braid closure are unlinked. This resolves Conjecture 4.45 of Castronovo, Gorsky, Simental, and Speyer.
math.GT — Geometric Topology
For the planar dodecahedron web D, the octahedral diagram of instanton homology reduces the possible values of the SO(3) instanton homology to 60 and 64. The remaining choice is determined by three coefficients of a secondary map. We give a proposed proof that all three vanish and that the dimension is 64. We express the coefficients as counts of instantons on specified foam cobordisms with one-parameter families of metrics. The index calculation requires these instantons to have negative action. An estimate uniform under neck stretching excludes them when the perturbations are sufficiently small. The construction includes local Floer complexes with zero differential, the marking calculation for excision, and a simultaneous choice of regular perturbations compatible with the families. Exact matrices and the supporting foam calculations are provided separately.
math-ph — Mathematical Physics
This work constructs a manifold consisting of twin helices to investigate the existence of computational geometric bordisms, as part of the broader program of the Cobordism Hypothesis. We use these structures to define a class of deterministic computational substrates, termed Geometrically Computable Manifolds, and study their geometric, algebraic, and computational properties. We show that, in a discrete regime, these manifolds encode arbitrary cyclic groups and support universal computation. This leads to a hypothesis describing the structural trade-offs in physical computation across distinct computational phases. We then analyze a regime in which abelian gauge computation is modeled with ordinary differential equations, linking the construction to continuous geometric dynamics. This signals a new research direction in physical mathematics at the intersection of geometry, gauge theory, and computability theory, termed Geometric Computability Theory, in which computation is treated as a property of geometric compositions. This perspective opens avenues for future investigation, motivating the development of a constructive computational gauge theory with applications to Yang–Mills theory. As of this writing, no complete unified compositional semantics yet exists for the full range of these manifolds; future work will develop such a formalism.
math.NT — Number Theory
For the Fermat tower , , we assemble the complete asymptotic machine for the stable Faltings height : the main-chain closed form ; an exact spectral identity with ; an arithmetic channel for the deviation functional through the Bernoulli-mass (, residual std ); a profile law proved by a carry-counting identity; and the layerwise height closure with , which explains the sign flip at and the explosion (, twenty-five times outside the corridor --). All identities are certified at the exact-arithmetic level on layers ( orbits and odd characters at layer , statistical verdicts at layers --). The irreducible content of the unified wall is reduced to a bounded per-type functional and an explicit five-parameter residual kernel ( on the orbit level).
cs.DS — Data Structures and Algorithms
Among the seven worlds of the central-charge program (number fields, function fields, graphs, finite-type dynamics, knots, tropical geometry, and arithmetic dynamics), the seventh is the only one whose critical exponent is continuously tunable, and the only one whose "cusp layer" is visible point by point over the primes. For phi(z) = z^d + c reduced modulo p, the orbit structure splits into two layers: an algebraic-closure layer (universal, smooth, forgotten: N_m = #Fix(phi^m) = d^m + 1, independent of c) and an F_p-rational layer (arithmetic, fingerprinted, remembered: Per_p(phi), the Frobenius fixed-point count). We prove: the dynamic Lehmer constant L_dyn = h_{x^2+1}(0) = 0.203677261369740... (a complete minimization proof over Q); the three-axis separation theorem for the splitting family c = (1-m^2)/4 (amplitude = 2 is a free fingerprint carried entirely by the n = 1 layer; alpha = 0.5 is a theorem for all m >= 5; the Jensen gap J has no m-dependence and separates the monomial and Chebyshev cases that the scale exponent alpha confounds); the sqrt(2) anisotropy of escape near the cusp, A(phi) = |sin(phi/2)|/sqrt(6) (Joukowski coordinates, 15 directions at two scales digit-perfect); the Sha growth law lim_n log_2(Sha_n)/2^n = 2*alpha - 1 with the Stoll mod-square family classified (7/7 separation); the criterion alpha = 1 if and only if disguised group endomorphism, across polynomials and rational maps including both Lattes quotients; and the number-field PCF criterion h(0) = 0 if and only if c in M intersect O_K with all embeddings in M, verified against Baker-DeMarco with a six-field Lehmer table. The universal far-tail rate is identified with the random-mapping class, Phi = (4-pi)/4 = 0.2146, with a quantitative accounting of the PCC contamination that had drifted earlier measurements. All corrections found during the research (including two withdrawn headline predictions) are recorded in an appendix.
math.RT — Representation Theory
For a compact group G with a finite-dimensional representation V, the moments of the character are, by a one-line argument, the dimensions of the invariant tensor spaces: <chi_V^m>_G = dim(V^{otimes m})^G. We call the passage from averaging over G to averaging over a maximal torus T the "law of forgetting": T-invariants only require weights to sum to zero, while G-invariants impose the additional invariance constraint. For the standard representation of SU(2) the two ladders differ by the exact factor F_k = binom(2k,k)/Cat(k) = k+1 (proved via the Temperley-Lieb non-crossing matching basis). The general forgetting factor admits a complete asymptotic F_k = c_V * k^s * (1 + gamma_V/k + O(1/k^2)), where s is the number of positive roots; both constants are given in closed form: c_V = |W| / (2^s <h_d>), a ratio of the Weyl denominator's lowest homogeneous term at the saddle to its Gaussian average, and gamma_V = (epsilon_T - epsilon_G)/2, whose inputs are the quartic term of -log|chi/dim V| and the next term of the Weyl denominator. The exponent s = (dim G - rank G)/2 is verified across six compact Lie groups (A_1, A_2, B_2, C_2, C_3, G_2) through the theorem F(1) = dim V (proved: weight multiplicities one, Schur's lemma), and the universal U(N) family gives the closed form F(N,k) ~ R_N k^{N choose 2} with R_N = (2/N)^{N choose 2} / G(N+1), G the Barnes function, identified with the reciprocal of a Mehta constant. On the arithmetic side, the master identity is verified on elliptic-curve Frobenius data; the two-copy test E x E versus E_1 x E_2 reads Schur's lemma directly from the second moment (a single number, 4 versus 2, separates isogenous from distinct copies); and the (lambda,lambda) pairing identity of Howe duality is identified with the Dirichlet coefficients of the Rankin-Selberg L-function, the N <-> k symmetry reading as convolution symmetry. A parallel shape fingerprint W = E[Y^2]/E[Y]^2 bypasses the heavy-tail divergence of the Jensen-gap functional J = E[Y]E[1/Y], with closed forms for twelve distribution families. All conclusions carry explicit status labels (proved / numerically anchored / identified), and every error found during the research is recorded with its correction in an appendix.
math.DS — Dynamical Systems
We prove that for the Rössler systemẋ = −(y + z),ẏ = x + ay,ż = b + z(x − c) with a = b = 0.2, the set of parameters c within 6 · 10 −5 of the classical value c = 5.7 for which the ow has a Hénon-like strange attractor supporting a SinaiRuelleBowen (SRB) measure has positive Lebesgue measure. The key step, proved with computer assistance, is that at an explicit parameter within 5.5 · 10 −5 of 5.7 the period-one saddle of the Poincaré map has a quadratic homoclinic tangency that unfolds generically; the conclusion then follows from the theorems of MoraViana, BenedicksYoung and BenedicksViana. The same method gives two further such tangencies near 5.7 and one at the band-merging parameter c ≈ 4.507. The proof reduces the generic-tangency conditions to nitely many inequalities, through explicit C 2 graph-transform bounds for the local invariant manifolds of a Poincaré map whose area contraction is of order 10 −11 10 −14 , and a scalar tangency function. The inequalities are checked with interval arithmetic in CAPD and rechecked with an independent implementation based on the Kv library; the analytic tangency criterion is formalised in Lean 4. Whether an SRB measure exists at c = 5.7 itself remains open.
math.NT — Number Theory
For every positive integer , we prove that the least order of a polynomial-coefficient recurrence for is . This establishes the unrestricted minimal-order assertion arising from Franel's work of 1894--1895. We also determine the least order , for , of the alternating sums . The lower bounds follow from real moment representations: every nonzero singularity of the representing measure forces a root of the leading characteristic polynomial of the recurrence. A common reflection argument gives telescoping recurrences attaining both bounds. For the ordinary sums, we combine the deformation of Straub and Zudilin with a shifted prime-power congruence to prove that the minimal shift operator is irreducible over $\C(n)$.
math.AG — Algebraic Geometry
Morel conjectured that the -homotopy sheaves of an -spectrum built from suspension spectra of smooth projective schemes are pure; one condition in purity is injectivity of restriction to dense open subschemes. Over a regular base , we show that this injectivity condition, for the homotopy sheaves of algebraic -theory with coefficients in an arbitrary spectrum, is equivalent to Gersten injectivity at the henselizations of all local rings of smooth -schemes; and that these -theory spectra are pure in Morel's sense. Hence every henselian regular local ring at which Gersten injectivity fails, integrally, rationally, or with finite coefficients, gives a counterexample to Morel's conjecture. The author's finite-coefficient counterexample over therefore disproves the conjecture, already over the regular scheme . Recently announced integral and rational Gersten counterexamples, including unpublished work of the author, would show, if confirmed and if they persist after henselization, that the conjecture fails already for the -theory spectrum itself.
math.CO — Combinatorics
A collection of six-coloured MacMahon cubes is universal if eight of its cubes can be selected to build a model of any target variety, with matching exterior colours and no internal matching requirement. We prove that exactly ten labelled collections attain the known minimum size of twelve, even when repeated varieties are allowed, resolving the conjectures of Johnson and Roldán. Usable-cube inequalities force a symmetric multiplicity matrix with row and column sums two; corner coverage then forces two disjoint triangles in Conway's tableau. The written proof is elementary and requires no exhaustive search. The classification and its discrete geometric interpretation are formalised in Lean 4. Exhaustive computation provides a separate check and additional target-count distributions.
math.CO — Combinatorics
We prove the origin-interior form of the polytope criterion of Gross, Hacking, Keel and Kontsevich: over a field of characteristic zero, full Fock–Goncharov for a principal coefficient cluster variety is equivalent to the existence of a bounded full-dimensional positive polytope containing the origin in its interior. Replacing the interior condition by membership of the origin gives a false criterion.
math.CV — Complex Variables
We show that Korenblum's constant of the Bergman space satisfies . The best published lower bound was (Wikström, 2026). The new ingredient is a closed form, in Jacobi theta functions, of the Möbius pseudodistance of the annulus between two points of the same centred circle. It separates the variables and yields three monotonicity properties, one of which, a profile inequality, combined with Kirszbraun's theorem in the hyperbolic plane, improves the constant in Wang's annular estimate. Wikström's dual moment criterion is then verified at for an explicit measure with fourteen rational atoms, in rigorous arithmetic, by a self-contained verifier. The French version of this note is dated 21 October 2026.
math.QA — Quantum Algebra
We prove Conjectures 1.5 and 1.6 of Min Huang and Zhihao Wang: the projected stated (SL_n)-skein algebra equals its quantum cluster and upper cluster algebras, and the specified exchangeable variables generate it over the frozen Laurent coefficient algebra. The results hold for triangulable unpunctured marked surfaces with at least two boundary marked points per connected component, over every coefficient domain and parameter in the conjectures.
math.GT — Geometric Topology
The distortion of a rectifiable closed curve is the supremum, over pairs of distinct points, of the ratio of the length of the shorter arc of between them to their distance; the distortion of a knot type is the infimum over all curves of type . For the trefoil, the published lower bound is , valid for every non-trivial knot (a recent preprint claims ), and the only upper bound we found in the literature is the value obtained by numerical simulation. We exhibit an explicit polygon with vertices, of knot type trefoil, whose distortion is at most . The proof is computer-assisted: the knot type is certified by an explicit piecewise-linear path of embedded polygons starting from a polygon on a torus, and the distortion by exact rational arithmetic on every pair of edges. Two separately written verifiers are supplied.
math.MG — Metric Geometry
We prove improved ordinary illumination bounds and exact ordinary and fractional values for several classes of convex bodies. Every full-dimensional complex zonotope in $\C^2$ has illumination number five, except complex affine images of the bidisc, whose number is seven. For complex -symmetric non-polydiscs in $\C^3$, an explicit construction gives twelve illuminating directions. For real -symmetric non-cubes in , connected domination in the cube graph gives the upper bound $2^{n-1}+\gamma_c(Q_{n-1})+n-1 =2^{n-1}(1+(1+o(1))/n)$; an explicit family has illumination number . We also give a product boundary measure proving fractional lower bounds for bodies between specified inner and outer bodies, and attain these bounds along Minkowski interpolations with a positive inner summand. Finally, the least size of a single family illuminating every normalized complex -unconditional body satisfies .
math.MG — Metric Geometry
We study ordinary and fractional illumination of complex -unconditional convex bodies. In every complex dimension we give an explicit ordinary bound asymptotic to , and prove $\ill^*(K)\le b_*^n$ and $\ill(K)=O(n\log(n+1)b_*^n)$, where is defined by an explicit scalar equation. For normalized non-polydiscs in dimensions two, three and four, the sharp ordinary bounds are , and . More generally, a full coordinate -polydisc gives the sharp bound in every dimension . In dimension two we prove the fractional bound and classify equality by the two maximal supporting coefficients at the coordinate axes. If every extreme point has all coordinates nonzero, we obtain the fractional bound , with equality exactly for the polydisc, and a quantitative strict bound otherwise. Appendices give the exact sizes of simultaneous illuminating families in dimensions one through five, the exact finite verification used in dimension four, and a numerical comparison of the general estimates through dimension .
math.NT — Number Theory
For an irrational let be the upper limit of its partial quotients. Vitorino and Vukusic proved for every irrational and asked, as their Problem 5, for an improvement. We prove . The proof is a finite computation organised around an invariant : the least number of doubling levels at which the recurrent kernel of a synchronous product of Raney transducers becomes a finite union of pure cycles. We show that is intrinsic, that for every would imply the -adic Littlewood conjecture, and that is equivalent to a rigidity failure; we compute for . In the language of Badziahin's set LMad this settles a level-bounded form of his Problem B for . The cycles which survive are ideal classes of discriminant whose whole -orbit has partial quotients at most ; we list them, and check the lists by an independent arithmetic route, in both directions for . The result concerns the upper limit only. On the side of the supremum it gives , which is weaker than what is known: Vitorino and Vukusic prove , and Badziahin's bound yields . Two self-contained Python files, without external dependency, reproduce the table and the lists of discriminants; the table up to takes about fourteen seconds.
math.GT — Geometric Topology
The tight knot constant is the largest constant with for every knot , where is the ropelength (unit-radius rope) and the crossing number. We exhibit an explicit embedding of the torus knot , with , of thickness at least and length at most . Hence . The published bound is . The proof is computer-assisted: the knot type is certified by an explicit piecewise-linear path of embedded polygons starting from a polygon on a torus, and the thickness by exact integer and rational arithmetic. Two separately written verifiers are supplied, together with certificates for and .
math.MG — Metric Geometry
We determine ordinary and fractional illumination numbers for complex Hanner bodies and for bodies obtained as convex hulls of coordinate polydiscs indexed by the maximal independent sets of a graph. The formulas are sums over coordinate supports. For Hanner bodies we classify all equality cases in the ordinary and fractional dimension bounds, including the sharp ordinary bound for non-polydiscs. For graph bodies, fractional equality occurs exactly when the graph is a disjoint union of isolated vertices and edges; a vertex of degree at least two gives the strict bound . We also prove the fractional bound for bodies defined by nondecreasing submodular constraints on coordinate moduli. For finite systems of positive weighted inequalities satisfying an explicit condition on tight sets, we obtain the same fractional bound and an ordinary bound of , together with an explicit estimate in every dimension. This includes bounded systems on laminar families with arbitrary positive coefficients. We conclude with a conjectural classification of fractional equality for normalized complex -unconditional bodies.
math.RT — Representation Theory
We prove Kamnitzer's ICM 2022 conjecture that the Mirković–Vilonen and theta bases coincide.
math.CO — Combinatorics
Labbé, Lapointe and Steiner proved that is injective on Christoffel words, gave two families of identities showing that it is not injective on , and asked whether every coincidence comes from these two families. We show that it does not. The smallest counterexamples have length ; by an exhaustive computation, among words of length at most there are exactly classes of coincidences not explained by the two families, of length and of length . Each of them contains an instance of one of a few new identities, which we prove for a generic matrix in place of the matrix of a factor of the word. One of these identities belongs to an infinite family, indexed by , which we prove for every such .
cs.CC — Computational Complexity
We give a deterministic algorithm for binary constraint satisfaction whose exponent in the input size is sublinear in treewidth on a class of graphs with unbounded treewidth. There are a class G of graphs of bounded maximum degree, a constant c, and a function f such that one algorithm correctly decides every binary CSP instance and, on instances I with primal graph G ∈ G, runs in time f(G) · ||I||^(c · tw(G) / sqrt(log tw(G))) . This contradicts the literal formulation of Conjecture 1.4 of Marx (2010). The algorithm follows from a domain-uniform counting bound: for every fixed maximum degree ∆, complete-table in- stances with k variables and domain size D ≥ 2 can be counted in time LO∆ (1)DO∆ (k/√log k) for sufficiently large k. The constants are independent of D and of the constraint relations. We obtain this bound from the tensor-network recursion of Claude (Anthropic), and use attributed graph foundations from OpenAI to construct the required expander family. Support pruning connects the counting routine to sparse tuple-list inputs without expanding the original domain. The algo- rithm, its native finite-alphabet machine implementation, and the conjecture counterexample are formalized in Lean 4.
math.OA — Operator Algebras
In 1829, Cauchy derived an upper bound for every root of a complex polynomial using the maximum of the absolute values of the coefficients. In 1931, Montel derived an upper bound using the sum of the absolute values of the coefficients. We derive noncommutative versions of the Cauchy and Montel bounds.
math.NT — Number Theory
We construct an infinite reduced mutation sequence for which all three logarithmic Markov–Euclid ratios converge to 9/10. This disproves the irrationality conjecture of Zhichao Chen and Zelin Jia (Conjecture 6.12).
math.QA — Quantum Algebra
A well-known conjecture, central in quantum permutation group theory, and going back to around 2006, states that is maximal. We revisit here this conjecture, and related questions, 20 years after, with 2026 knowledge and tools.
math.NT — Number Theory
We give a family of counterexamples to Richard Guy's problem E27 with initial terms of length five adding onto the recent example of length eight. Additionally, we prove that for length one, one always has a ultimately periodic sequence. Lastly we pose a problem regarding the notion of a zth mex sequence.
math-ph — Mathematical Physics
We prove low-temperature density coexistence for identical hard disks or hard spheres with a smooth, nonnegative, finite-range attractive Kac pair kernel. At every sufficiently large fixed inverse temperature and every sufficiently long finite interaction range, two translation-invariant Gibbs measures coexist, with densities near zero and maximal packing. The pressure is therefore nondifferentiable in chemical potential. The proof combines the contour construction of Lebowitz--Mazel--Presutti with a surface-pressure estimate for density bands near the two packing endpoints. A packing bound controls the fine density profile; centering the interaction makes its residual boundary contribution quadratic in the band width. Compatible near-packed configurations and conditional entropy then control the hard-core boundary contribution. These estimates allow the dense restricted ensemble to retain positional order and long-range correlations.
math.QA — Quantum Algebra
Evans and Gannon showed that a positive vertex operator algebra of central charge eight whose representation category is the Drinfeld center of the Haagerup fusion category has one of two character vectors, with six or nineteen currents. Call such an algebra, if strongly rational, a Haagerup algebra. We prove that a Haagerup algebra has six currents, that embeds into in one of two explicit ways, with levels or , and we determine the lowest weight spaces of its modules and as -modules. For the levels , is one of four Galois conjugate subspaces, and the algebra it generates is unitary, -cofinite and isomorphic to the algebra constructed by Jiao; for the levels , contains the algebra announced by Riesen. The associated lattice of is that of the form , resp.\ , the two classes of the principal genus of discriminant . Exchanging them in the decomposition of over its Heisenberg commutant gives a Haagerup algebra of the other levels, and is a bijection between the isomorphism classes of the two cases. Consequently, if either explicit algebra of Jiao or Riesen is positive and strongly rational with representation category braided equivalent to , there are exactly two Haagerup algebras up to isomorphism, forming a single hyperbolic genus. The explicit algebras of Jiao and Riesen have rational forms. The proofs combine exact trace relations given by vector-valued modular forms, exact computations in a model of , and genera of vertex operator algebras.
math.FA — Functional Analysis
Gibilisco and Isola conjectured that the scalar curvature of the Wigner–Yanase–Dyson metric with parameter on the invertible density matrices increases under mixing for all in some interval ; the case is Petz's conjecture for the Kubo–Mori metric (resolved in a companion paper). We prove that for , with independent of the dimension, the scalar curvature is a concave function of the eigenvalues and hence increases under mixing. By Dittmann's formula it suffices to prove that one symmetric function of three positive variables is concave, which we do by showing that is a Stieltjes transform along every line in the positive cone, uniformly in near . The value of is not explicit.
math.CA — Classical Analysis and ODEs
We prove a sharp Brégman–Minc bound for permanents of nonnegative matrices that was conjectured by Samorodnitsky and Gurvits. Specifically, for , with , applied entrywise, we show that every nonnegative row-stochastic square matrix satisfies , which is the Brégman–Minc inequality when each row of is uniform on its support. The crux of our proof is a scalar inequality on the simplex, which we prove using a carefully chosen yet elementary integral representation. We note brief applications to the entropy of a random permutation and to sharp bounds for rectangular permanents and matchings.