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hexagon:2610.00501v1

math.CO — Combinatorics

A counterexample to a conjecture of Graffiti.pc on the forest number

Contributed by Deep Bhattacharjee

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.

A mix of human-written and AI-generated textHuman understanding: some partsGraffiti.pccomplementcounterexampledegree sequenceforest numberinduced forest
Oct 23, 2026PDF ↗View work →
hexagon:2610.00500v1

math.CO — Combinatorics

Counterexamples to five conjectures of Graffiti.pc on domination

Contributed by Deep Bhattacharjee

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.

A mix of human-written and AI-generated textHuman understanding: some partsGraffiti.pccaterpillarcounterexampleindependent dominationtotal dominationtreewell totally dominated graph
Oct 23, 2026PDF ↗View work →
hexagon:2610.00499v1

math.MG — Metric Geometry

Regular Morley tetrahedra from pp-secting planes

Contributed by Deep Bhattacharjee

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 pp-secting planes, with one parameter e,f,g>1e,f,g>1 for each pair of opposite edges; the resulting tetrahedron is denoted Me,f,g(ABCD)\mathcal M_{e,f,g}(ABCD). 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 Me,f,g(ABCD)\mathcal M_{e,f,g}(ABCD) is regular. We show that the conjecture is false. We place an isosceles tetrahedron in the box [−a,a]×[−b,b]×[−c,c][-a,a]\times[-b,b]\times[-c,c] and prove that Me,f,g(ABCD)\mathcal M_{e,f,g}(ABCD) is regular exactly when its vertex A′A' differs from the centre of the box and lies on one of the four lines through the centre in the directions (1,1,1)(1,1,1), (1,−1,−1)(1,-1,-1), (−1,1,−1)(-1,1,-1) and (−1,−1,1)(-1,-1,1). Every interior point of these lines other than the centre occurs for exactly one triple of parameters. When aa, bb and cc 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, e=f=g=4e=f=g=4, an isosceles tetrahedron has a regular Morley tetrahedron only if it is regular.

A mix of human-written and AI-generated textHuman understanding: some parts$p$-secting planeMorley tetrahedronMorley's theoremdihedral angledisphenoidisosceles tetrahedron
Oct 23, 2026PDF ↗View work →
hexagon:2610.00498v1

math.CO — Combinatorics

Upper kk-star-forming sets and kk-independence in bipartite graphs

Contributed by Deep Bhattacharjee

Sahbi conjectured that the kk-independence number βk\beta_k and the upper kk-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 k≥3k\ge3 we give a connected bipartite graph on 3k+33k+3 vertices with βk=2k+2\beta_k=2k+2 and $\SF_k\ge3k$. On chain graphs, however, $\beta_k=\SF_k$ for every kk. We also show, answering another of his questions, that deciding whether a bipartite graph has a minimal 22-star-forming set with at least tt vertices is NP-complete.

A mix of human-written and AI-generated textHuman understanding: some parts$k$-independence$k$-star-forming setNP-completenessbipartite graphchain graphformal verificationupper domination
Oct 23, 2026PDF ↗View work →
hexagon:2610.00497v1

math.RA — Rings and Algebras

Maximal sublattices of finite semidistributive lattices: the complement need not be an interval

Contributed by Deep Bhattacharjee

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.

A mix of human-written and AI-generated textHuman understanding: some partsbounded latticeformal verificationintervalmaximal sublatticesemidistributive lattice
Oct 23, 2026PDF ↗View work →
hexagon:2610.00496v1

math.AC — Commutative Algebra

Finite Field Hudzik-Landes-Dragomir-Kato-Saito-Tamura Inequality

Contributed by K. MAHESH KRISHNA

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.

Primarily human-written textHuman understanding: all partsFinite Field.Normed linear spaceSub-modulusSub-norm.Triangle inequality
Oct 23, 2026PDF ↗View work →
hexagon:2610.00495v1

math.RA — Rings and Algebras

Maximal sublattices of convex geometries: four counterexamples

Contributed by Deep Bhattacharjee

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.

A mix of human-written and AI-generated textHuman understanding: some partsconvex geometryformal verificationjoin-semidistributive latticemaximal sublatticemeet-irreducible element
Oct 23, 2026PDF ↗View work →
hexagon:2610.00494v2

math.CO — Combinatorics

Unbounded strong-product ratios for the Grundy domination number

Contributed by Deep Bhattacharjee

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 CC there are graphs whose strong product has Grundy domination number more than CC times the product of the two factors. The graphs are built on the five-element subsets of an mm-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 mm and (m2)\binom m2, while the diagonal of the product is an independent set of size (m5)\binom m5. We also compute the Grundy domination number of the graph on {0,…,N−1}\{0,\dots,N-1\} in which uu and vv are adjacent when ∣u−v∣≡2(mod3)|u-v|\equiv2\pmod3: it is ⌊(N+1)/3⌋+3\lfloor(N+1)/3\rfloor+3 for every N≥6N\ge6. Its strong product with the octahedron has a legal sequence of length NN, so the conjecture already fails for N=19N=19, on a graph with 114114 vertices, and for every N≥21N\ge21. Every statement is proved by hand.

A mix of human-written and AI-generated textHuman understanding: some partsGrundy domination numbercircular cliquedominating sequenceformal verificationstrong product
Oct 23, 2026PDF ↗View work →
hexagon:2610.00493v2

math-ph — Mathematical Physics

Explicit String Vacua and the Geometric Selection Conjecture

Contributed by Deep Bhattacharjee, Ushashi Bhattacharya

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.

A mix of human-written and AI-generated textHuman understanding: some partsCalabi-Yau threefoldGeometric Selection Conjecturegenerationsheterotic stringmirror symmetry
Oct 23, 2026PDF ↗View work →
hexagon:2610.00491v1

math.PR — Probability

Largest Gaps between Eigenvalues of the Gaussian Orthogonal Ensemble

Contributed by Patrick Lopatto

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.

Primarily AI-generated textHuman understanding: some partsrandom matrix theory
Oct 23, 2026PDF ↗View work →
hexagon:2610.00490v1

math.QA — Quantum Algebra

Vertex Operator Algebra Realizations of Twisted Haagerup–Izumi Doubles

Contributed by Andrew Riesen, David E. Evans

We construct vertex operator algebras realizing an infinite family of twisted, nonunitary Haagerup--Izumi doubles. For every odd integer N≥3N\geq 3, we construct a spherical fusion category FN\mathcal{F}_N with the Haagerup--Izumi fusion rules for Z/NZ\mathbb{Z}/N\mathbb{Z} and pointed associator of exact order NN, and realize its Drinfeld center as the ribbon category of modules of an explicit vertex operator algebra VN\mathscr{V}_N. 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 E7E_7 lattice vertex operator algebra removes the additional semion factor. The resulting VN\mathscr{V}_N is simple, regular, self-contragredient, and C2C_2-cofinite, with central charge −6N2−10-6N^2-10, effective central charge eight, and N2+3N^2+3 irreducible modules with explicit characters. Its weight-zero subspace has dimension 5656, so it is not of CFT type; we establish modularity and the ribbon equivalence without relying on the CFT-type hypothesis.

A mix of human-written and AI-generated textHuman understanding: some parts
Oct 23, 2026PDF ↗View work →
hexagon:2610.00489v1

math.OA — Operator Algebras

On the tightness of the Parthasarathy and Schmidt rank bounds for extremal tracial couplings

Contributed by Dipankar Dey, Somnath Jana, Soumyashant Nayak

Let 2≤a≤b2 \le a \le b, and let ρA,ρB\rho_A, \rho_B be density operators on $\C^a$ and $\C^b$, respectively. Let C(ρA,ρB)\mathcal{C}(\rho_A, \rho_B) 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 (a,b)(a, b) this bound is tight. We restrict attention to the tracial case ρA=Ia/a\rho_A = I_a/a, ρB=Ib/b\rho_B = I_b/b. For the Parthasarathy bound, we construct extreme couplings of rank rr from pairs of length-rr finite unit-norm tight frames (FUNTFs) in $\C^a$ and $\C^b$ satisfying a transversality condition. This yields an extreme coupling of rank bb for every 2≤a<b2 \le a < b, and hence tightness of $\KRP(a,b)$ whenever a<ba<b and b≥(a2−1)/2b\ge(a^2-1)/2; in particular, along the whole qubit-qudit line (2,b)(2,b), b≥3b\ge3. This recovers the two previously known tracial cases with a<ba<b, namely (2,3)(2,3) and (3,4)(3,4). 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 rr whenever b−(r−1)ab-(r-1)a divides aa, and show that for a<b≤2aa<b \le 2a, where this condition reads b−a=gcd⁡(a,b)b-a=\gcd(a,b), it is also necessary. For b>a(a−1)b>a(a-1), the divisibility condition is superfluous and the Schmidt bound is attained for every such bb. Thus, for each fixed aa, attainability of both bounds remains open for only finitely many b≥ab\ge a.

Primarily human-written textHuman understanding: all partsCompletely positive mapsExtremal tracial couplingFinite unit-norm tight frames (FUNTFs)Kraus operatorsQuantum marginal problem
Oct 23, 2026PDF ↗View work →
hexagon:2610.00487v1

math.CV — Complex Variables

Origin-Centered Möbius Circles and Dyadic Pairings of the Riemann Zeta Function on the Critical Line

Contributed by Ryan Hackbarth

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.

A mix of human-written and AI-generated textHuman understanding: all parts
Oct 23, 2026PDF ↗View work →
hexagon:2610.00486v1

math.PR — Probability

Explicit onsets, rates and the tail of the collision constant for heat-evolved Weyl polynomials

Contributed by Bingo Baggins

Let hkh_k be the orthonormal Hermite polynomials for the standard Gaussian weight, and let a(w)=∑j≥0ajwja(w)=\sum_{j\ge0}a_jw^j be a real power series with radius of convergence greater than 11, a(0)≠0a(0)\ne0, no zeros on the unit circle and only simple zeros in the unit disk. For all large nn the polynomial ∑j=0najhn−j\sum_{j=0}^{n}a_jh_{n-j} has exactly as many nonreal zeros as aa has in the disk, but the known proof gives the onset by compactness. We make it explicit: if aa has no zeros in {1−d≤∣w∣≤1}\{1-d\le|w|\le1\}, with dd below an explicit threshold, the count is exact as soon as nn exceeds a polynomial in 1/d1/d and in the inverses of ∣a(0)∣|a(0)|, min⁡∣w∣=1∣a∣\min_{|w|=1}|a|, min⁡∣w∣=1−d∣a∣\min_{|w|=1-d}|a| and min⁡[−1+d,1−d]max⁡(∣a∣,∣a′∣)\min_{[-1+d,1-d]}\max(|a|,|a'|); no separation of the nonreal zeros of aa is needed. By an exact identity this applies to the random Weyl polynomials GnG_n with independent standard Gaussian coefficients under the backward heat flow Gn,t=exp⁡(−t2n∂z2)GnG_{n,t}=\exp(-\frac t{2n}\partial_z^2)G_n, whose real-rooting time converges in distribution to T∗=R∗−2T_*=R_*^{-2}, where R∗R_* is the smallest modulus of a nonreal zero of the Gaussian power series q(w)=∑jηjwjq(w)=\sum_j\eta_jw^j. We show that the convergence holds with the rate n−αn^{-\alpha} on [1+ε,∞)[1+\varepsilon,\infty) for every α<350\alpha<\frac3{50}, an exponent that does not depend on ε\varepsilon; the dependence on ε\varepsilon is removed by a super-exponential tail for the number of zeros of qq in a disk. Under a coupling of the degrees that shares the top coefficients, the real-rooting time Λn\Lambda_n satisfies ∣Λn−T∗∣≤K/n|\Lambda_n-T_*|\le K/n for n≥n0n\ge n_0, with KK and n0n_0 depending on qq. We bound KK and n0n_0 by polynomials in condition parameters of qq, with absolute degrees and, when T∗≥1+εT_*\ge1+\varepsilon, coefficients polynomial in 1/ε1/\varepsilon. For the Gaussian power series this gives, on the event {T∗≥1+ε}\{T_*\ge1+\varepsilon\}, a power tail with exponent 1/9001/900 for the largest scaled displacement DD of the final collision of zeros, and without restriction the tail bound exp⁡(−c(log⁡x/log⁡log⁡x)1/2)\exp(-c(\log x/\log\log x)^{1/2}); the inputs include closed forms of the two-point correlation functions of the zeros of qq.

Primarily AI-generated textHuman understanding: no partsHermite polynomialsWeyl polynomialscollision of zeroscorrelation functions of zerosexplicit onsetheat flowlinear combinations of orthogonal polynomialsnonreal zerosrandom power seriesrate of convergence
Oct 23, 2026PDF ↗View work →
hexagon:2610.00485v1

math.NT — Number Theory

A 256-step certificate for Turán's pure power-sum constant

Contributed by Ricardo Pinho

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.

Primarily AI-generated textHuman understanding: some partsTurán power sumscoefficient cancellationcomputer-assisted proofinterval arithmeticrigorous numerics
Oct 23, 2026PDF ↗View work →
hexagon:2610.00484v1

math.MG — Metric Geometry

A counterexample to the table conjecture on the sphere

Contributed by Pablo Soberón

We disprove Matschke's table conjecture, which asserts that every continuous real-valued function on S2S^2 takes a common value on the vertices of some rotated copy of every prescribed inscribed square. The counterexample is the cubic polynomial H(x,y,z)=14z3−6z+y+xzH(x,y,z)=14z^3-6z+y+xz and a square contained in a plane at distance 1/71/\sqrt{7} from the origin. We give an elementary proof.

Primarily human-written textHuman understanding: all partscounterexamplefixed pointsquare-peg
Oct 23, 2026PDF ↗View work →
hexagon:2610.00483v1

math.PR — Probability

The Real-Rooting Time Of Heat-Evolved Random Polynomials

Contributed by Bingo Baggins

Let Gn,t=exp⁡(−t2n∂z2)GnG_{n,t}=\exp(-\frac{t}{2n}\partial_z^2)G_n be a random Weyl polynomial of degree nn 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 t=1t=1. We show that the first time at which all zeros of Gn,tG_{n,t} are real, a finite-degree analogue of the de Bruijn--Newman constant, converges in distribution to T∗=R∗−2T_*=R_*^{-2}, where R∗R_* is the smallest modulus of a nonreal zero of the random power series ∑jηjwj\sum_j\eta_jw^j 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 [1,∞][1,\infty]. The proof rests on an exact count of the nonreal zeros of analytic combinations ∑j≤najhn−j\sum_{j\le n}a_jh_{n-j} of orthonormal Hermite polynomials: if the real power series aa has radius of convergence greater than 11, no zeros on the unit circle and simple zeros in the unit disk, then for all large nn the combination has exactly as many nonreal zeros as aa has in the disk. Its method expresses the polynomial and its derivative in the bulk through a 2×22\times2 matrix applied to (hn,hn−1)(h_n,h_{n-1}), built from only O(log⁡n)O(\log n) steps of the backward three-term recurrence, with the remaining terms controlled by a Christoffel--Darboux bound. For Gaussian coefficients the law of T∗T_* is a Fredholm determinant, P(T∗>t)∼43πt−3/2\mathbb{P}(T_*>t)\sim\frac4{3\pi}t^{-3/2} as t→∞t\to\infty, and (t−1)log⁡P(T∗≤t)→−π2/12(t-1)\log\mathbb{P}(T_*\le t)\to-\pi^2/12 as t↓1t\downarrow1, half of the constant for complex coefficients. At the scale 1/n1/n the last pair of nonreal zeros collides inside the real bulk, at relative position cos⁡θ∗\cos\theta_*, where R∗e±iθ∗R_*e^{\pm i\theta_*} are the nonreal zeros of minimal modulus, and the real-rooting time is within O(1/n)O(1/n) of T∗T_*. In a window of size 1/n1/n the polynomial is, up to a positive factor and O(1/n)O(1/n), the model ζcos⁡(ζ+c)+λsin⁡(ζ+c)\zeta\cos(\zeta+c)+\lambda\sin(\zeta+c), whose threshold for real-rootedness is given by Kepler's equation; the phase cc rotates with nn, so n(Λn−T∗)n(\Lambda_n-T_*) oscillates over an explicit interval and has an explicit limit law jointly with T∗T_*.

Primarily AI-generated textHuman understanding: no partsFredholm determinantsHermite polynomialsPfaffian point processesWeyl polynomialscollision of zerosde Bruijn-Newman constantheat flowlinear combinations of orthogonal polynomialsrandom power seriesreal zeros
Oct 22, 2026PDF ↗View work →
hexagon:2610.00481v1

math.AG — Algebraic Geometry

A counterexample to Liu's conjecture on Fano indices in dimension four

Contributed by Deep Bhattacharjee, Ushashi Bhattacharya

We prove that Liu's conjecture on Fano indices fails in dimension four: 13891389 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.

A mix of human-written and AI-generated textHuman understanding: some partsCalabi–Yau varietyFano indexindexlog canonical singularityweighted projective space
Oct 22, 2026PDF ↗View work →
hexagon:2610.00480v1

math.RA — Rings and Algebras

Nilpotent-free spaces of 3 x 3 matrices over the field with three elements

Contributed by Deep Bhattacharjee

Wigderson conjectured that for every finite field FF and every n≥1n\ge1 there is a space of n×nn\times n matrices over FF of dimension n2−(n2)n^2-\binom n2 whose only nilpotent element is zero. We show that every six-dimensional space of 3×33\times3 matrices over $\F_3$ contains a nonzero nilpotent matrix, so the conjecture fails for $F=\F_3$ and n=3n=3; the largest possible dimension is five.

A mix of human-written and AI-generated textHuman understanding: some partsGerstenhaber's theoremfinite fieldsformal verificationnilpotent matricesspaces of matrices
Oct 22, 2026PDF ↗View work →
hexagon:2610.00479v1

math.MG — Metric Geometry

Densest packings of two translates of a lattice in five dimensions

Contributed by Deep Bhattacharjee, Ushashi Bhattacharya

We prove that no packing of unit balls in R5\R^5 whose centres form two translates of a lattice is denser than the D5D_5 lattice packing. Up to congruence exactly three such packings attain this density: 2 D5\sqrt2\,D_5, a layered packing of Leech, and the packing with Szöllősi's kissing configuration.

A mix of human-written and AI-generated textHuman understanding: some parts$D_5$ latticeRyshkov polyhedronSphere packingVoronoi celldeep holeperiodic packing
Oct 22, 2026PDF ↗View work →
hexagon:2610.00478v1

math.NT — Number Theory

The Number of Terms in Greedy Fibonacci--Lucas Representations

Contributed by Deep Bhattacharjee, Ushashi Bhattacharya

We prove Kimberling's conjecture on greedy representations by Fibonacci and Lucas numbers: the ratio of the number of Zeckendorf summands of 1,…,n1,\dots,n to the number of terms of their greedy representations tends to a limit in (1.2,1.4)(1.2,1.4), and the limit is (5−5)/2(5-\sqrt5)/2. We also prove the conjectures of Greathouse and Kurkov on the number of terms of the greedy representation.

A mix of human-written and AI-generated textHuman understanding: some partsFibonacci numbersLucas numbersZeckendorf representationformal verificationgreedy algorithmsummatory function
Oct 22, 2026PDF ↗View work →
hexagon:2610.00477v1

math.AG — Algebraic Geometry

Weil classes on abelian sixfolds of Weil type for Q(i)\mathbb{Q}(i) with discriminant −3-3

Contributed by Benjamin Severn

Let K=Q(i)K=\mathbb{Q}(i). We prove that the Weil classes of every polarized complex abelian sixfold of Weil type for KK whose Hermitian discriminant is the non-split class [−3][-3] 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 W(E8)⋅μ4W(E_8)\cdot\mu_4, 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 1212 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

Primarily AI-generated textHuman understanding: some partsFermat varietiesHodge conjecturePrym-Tyurin varietiesWeil classesabelian coversabelian varieties of Weil type
Oct 22, 2026PDF ↗View work →
hexagon:2610.00475v1

math.GT — Geometric Topology

The bridge index and the nonorientable four-genus of knots with at most thirteen crossings

Contributed by Deep Bhattacharjee

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 33 or 44; we prove that \bclosed{} of them have bridge index and meridional rank 44, 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.

A mix of human-written and AI-generated textHuman understanding: some partsDonaldson diagonalisationGoeritz latticesKnotInfobridge indexmeridional ranknonorientable four-genuspinch movestransvections
Oct 22, 2026PDF ↗View work →
hexagon:2610.00474v1

math.CO — Combinatorics

Even-intersecting families of lattice paths and the Catalan numbers

Contributed by Deep Bhattacharjee

Shankar proved that a family of North--East lattice paths from (0,0)(0,0) to (n,n)(n,n) in which any two paths share an even number of edges has at most 2n2^n members, and conjectured that exactly CnC_n families attain this bound, where CnC_n is the nnth 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 {1,…,m}\{1,\dots,m\} whose unmatched points lie under no arc, and there are (m⌊m/2⌋)\binom{m}{\lfloor m/2\rfloor} of them.

A mix of human-written and AI-generated textHuman understanding: some partsCatalan numbersEventown theoremeven intersectionslattice pathsnoncrossing matchings
Oct 22, 2026PDF ↗View work →
hexagon:2610.00473v2

math.CO — Combinatorics

The minimum number of distinct abelian squares in a binary circular word

Contributed by Deep Bhattacharjee

We prove that every binary circular word of length nn 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 (ab)n/2(ab)^{n/2} for n≡2(mod4)n\equiv 2\pmod 4, n≥6n\ge 6. We then review the other problems of the survey: we conjecture, and verify to length 2727, that for odd nn the least number of non-equivalent abelian squares in a circular word is also (n−1)/2(n-1)/2, and we show that the least number of occurrences of abelian squares in a binary word of length nn is asymptotically proportional to nn or grows faster, and is below n3/2n^{3/2}.

A mix of human-written and AI-generated textHuman understanding: some partsParikh vectorabelian squarecircular wordcombinatorics on words
Oct 22, 2026PDF ↗View work →
hexagon:2610.00472v2

math.NT — Number Theory

The Welch function is not third-order sum-free

Contributed by Deep Bhattacharjee

Let n=2m+1n=2m+1 be odd and let $\F_{2^n}$ be the field with 2n2^n elements. A function on $\F_{2^n}$ is third-order sum-free when its values over any eight elements forming a three-dimensional affine subspace never add up to zero. The Welch function is the power map x↦x2m+3x\mapsto x^{2^m+3}; it is one of the classical almost perfect nonlinear functions, and Hou and Zhao conjectured that for n≥7n\ge7 it is not third-order sum-free. We prove this. The question comes down to a linearised polynomial g(x,y)g(x,y) in yy whose roots form a vector space Z(x)Z(x) over $\F_2$ containing 11 and xx, and the Welch function fails to be third-order sum-free exactly when Z(x)Z(x) has dimension three for some xx outside $\F_2$. We show that the dimension is always 22 or 33, and that it is 33 if and only if two explicit trace conditions hold: with u=x2mu=x^{2^m}, both $\Tr(1/(x+u))$ and $\Tr(xu/(x+u))$ equal 11. The proof applies the field automorphism z↦z2m+1z\mapsto z^{2^{m+1}}, whose square is the Frobenius map, to turn gg into an additive polynomial of degree eight whose roots always lie in $\F_{2^n}$. Counting the elements that satisfy both trace conditions gives exactly (2n+1−3Kn)/4(2^n+1-3K_n)/4 of them, where KnK_n is the binary Kloosterman sum of $\F_{2^n}$, and Weil's bound ∣Kn∣≤2n/2+1|K_n|\le 2^{n/2+1} makes this count positive for every n≥7n\ge7. The Welch function sums to zero over exactly (2n+1−3Kn)(2n−1)/168(2^n+1-3K_n)(2^n-1)/168 three-dimensional linear subspaces, and the count 2+(2n+1−3Kn)/42+(2^n+1-3K_n)/4 agrees with the numbers that Hou and Zhao computed for n≤15n\le15. Every statement is proved by hand.

A mix of human-written and AI-generated textHuman understanding: some partsKloosterman sumsWelch functionalmost perfect nonlinear functionslinearized polynomialssum-free functions
Oct 22, 2026PDF ↗View work →
hexagon:2610.00471v3

math-ph — Mathematical Physics

Residual-dimension obstructions for null-projected supersymmetric Yang--Mills theories

Contributed by Deep Bhattacharjee

We prove necessary conditions for null-projected supersymmetric Yang--Mills theories with a Majorana--Weyl gaugino in signature (s,t)(s,t), under five explicit assumptions on the gaugino constraint, the transformation rules, the residual gauge field, the translation bracket and the absorption of the cubic gaugino terms. A real gaugino allows at most min⁡(s,t)\min(s,t) null vectors, the residual dimension must be two or ten, and s−ts-t must be 00 or ±8\pm8. The main step is a converse of the cubic gamma-matrix identity: if symmetric gamma matrices acting on NN-component spinors in dd dimensions satisfy it, then N=2(d−2)N=2(d-2). Consequently Nishino's construction does not transfer to the signatures (11+8n)+3(11+8n)+3 proposed by Rios, Marrani and Chester. In 27+327+3 a real null frame has at most three vectors against Nishino's ten, the translation bracket allows only two, and with two the residual spinors have 40964096 components where absorption needs 4848. For unitary single-time theories with vanishing tensor charges on a massless multiplet, we also exclude every finite collection of scalar, spinor and vector polarizations when there are more than sixteen real supercharges. Theories outside the five assumptions are not decided.

A mix of human-written and AI-generated textHuman understanding: all partsClifford algebrasFierz identityMajorana-Weyl spinorsYang-Mills theorynull projectionspacetime signature
Oct 22, 2026PDF ↗View work →
hexagon:2610.00470v1

math.MP — Mathematical Physics

Gluing defects for projected coarse graining in finite spin-foam models

Contributed by Zhenyu Sun

Orthogonal truncation can discard states that contribute to a later boundary amplitude. For a self-adjoint contraction TT on a finite-dimensional Hilbert space and an isometric inclusion ss of retained states, we prove that the two-step residual Δ=s∗T2s−(s∗Ts)2\Delta=s^*T^2s-(s^*Ts)^2 controls every finite placement of intermediate projections in every Schatten norm. For blocks of lengths n1,…,nkn_1,\ldots,n_k, the error is at most ∑i<jninj∥Δ∥\sum_{i<j}n_in_j\|\Delta\|, and the constant is optimal even for positive contractions. The residual vanishes exactly when the retained subspace is invariant. Two-dimensional finite-group gauge sums realize an idempotent projected transfer operator with nonzero residual, as well as positive-semidefinite transfer operators with identical one-step retained data and different residuals. We derive their gluing and subdivision identities with explicit boundary measures and BF normalization. We also bound the accumulation of local errors under tensor contraction and the distance to compatible linear amplitude data. Colimits extend compatible amplitude families, and quotients by invariant subspaces preserve the comparison maps. A measurement map that is injective on the retained ansatz gives a unique normalized least-squares fit. Exact rational calculations reproduce the finite-group examples and the blocking comparisons.

Primarily human-written textHuman understanding: all partscoarse grainingcylindrical consistencyfinite gauge theoryspin foamstensor gluingtruncation error
Oct 22, 2026PDF ↗View work →
hexagon:2610.00469v1

math.NT — Number Theory

A truncated theta-type series with triangular weights

Contributed by Manjil Saikia

Merca (2026) recently introduced the two-block odd partition function a(n)a(n), proved that its generating function factors as A(q)=P(q)S(q)A(q)=P(q)S(q), where P(q)P(q) is the generating function of partitions whose odd parts are distinct and whose even parts are unrestricted and S(q)S(q) is a theta-type series with triangular-number weights, and conjectured that for N≡1,2(mod4)N\equiv1,2\pmod 4 the truncation P(q)SN(q)P(q)S_N(q) of this product has nonnegative coefficients, positive from q3q^{3} onward. We prove the conjecture in the stronger form $P(q)S_N(q)-A(q)\in\NN[[q]]$.

A mix of human-written and AI-generated textHuman understanding: all parts
Oct 22, 2026PDF ↗View work →
hexagon:2610.00468v1

math.GM — General Mathematics

Arithmetic of Jacobi polynomials

Contributed by Khai Hoan Nguyen Dang

We study the arithmetic of Jacobi polynomials. As applications, we prove the Mahlburg--Ono conjecture on the irreducibility and Galois groups of the four Kaneko--Zagier families, and the Cullinan--Hajir conjecture on the Galois groups of Legendre polynomials. Other applications concern common zeros of Jacobi polynomials, harmonic functions on cones, and hyperelliptic Jacobians.

A mix of human-written and AI-generated textHuman understanding: some partsJacobi polynomialKaneko--Zagier polynomialLegendre polynomial
Oct 22, 2026PDF ↗View work →
hexagon:2610.00467v1

math.DS — Dynamical Systems

Finiteness of Planar Central Configurations for Five Positive Masses

Contributed by Chih-Yao Wu

We prove that, for any five positive masses, there are finitely many planar Newtonian central configurations up to translations, rotations, and positive scaling. Building on the singular-diagram framework of Albouy and Kaloshin, we treat the exceptional masses left open by their finiteness theorem. The proof retains a lower bound for the physical potential after passing to the complex algebraic equations. On each fixed potential level, energy estimates, torque identities, and local response calculations control the possible degenerations. A polynomial selection argument then yields finiteness. The exceptional local calculations and their algebraic certificates are given explicitly. A separate Lean development accompanies the proof.

Primarily AI-generated textHuman understanding: some partsLeanSmale sixth problemcelestial mechanicscentral configurationsfive-body problemformal verification
Oct 22, 2026PDF ↗View work →
hexagon:2610.00466v2

math.NT — Number Theory

Exceptional zeros of Hecke LL-functions

Contributed by Christopher Frei, Wataru Kai

We prove that Hecke LL-functions have no exceptional zeros. We do so by generalising a variant due to Kerr of the proof by OpenAI in the case of Dirichlet LL-functions. v2: Added a reference to independent concurrent work by Banks.

Primarily human-written textHuman understanding: all partsHecke L-functionsLandau--Siegel zerosexceptional zeros
Oct 22, 2026PDF ↗View work →
hexagon:2610.00465v1

math.AC — Commutative Algebra

Nonexistence of lim Cohen–Macaulay modules following [Che26] and [Ang26]

Contributed by Andy Jiang

Using the inequality stated in [ Che26, Main Claim 1.1], we prove nonexistence of lim Cohen–Macaulay sequences for the completed section rings of smooth projective surfaces over C with an ample globally generated divisor H satisfying KY ≡ 4H and 15H2/8 > χ(OY ). Such a pair is exhibited in [Ang26, Corollary 4.3], with H2 = 1080 and χ(OY ) = 1926. For a complete local domain (R, m) with ℓRHj m(R) < ∞ for 0 ≤ j < dim R, we prove that a lim Cohen–Macaulay sequence in the sense of Bhatt, Hochster, and Ma [ BHM26 ] exists if and only if a lim Γm-Cohen–Macaulay sequence exists (Definition 1.1). These may be different sequences. Chen’s geometric inequality [Che26, Main Claim 1.1] obstructs lim Γm-Cohen–Macaulay sequences over the completed section rings above. The comparison therefore rules out lim Cohen–Macaulay sequences over those rings.

Primarily AI-generated textHuman understanding: some parts
Oct 22, 2026PDF ↗View work →
hexagon:2610.00464v2

math.LO — Logic

Ramsey’s Theorem for Pairs in Pincus’s Model XI

Contributed by Frank Gilson

Howard and Rubin record Pincus’s ordered colored permutation model as N48, but do not determine whether Ramsey’s theorem for pairs holds there. We prove that N48 |= RT for pairs. The proof uses least finite supports and the homogeneity of Pincus’s ordered colored structure: after fixing all but one point of a support, the remaining point generates an infinite orbit whose unordered pairs form a single stabilizer orbit. Combining this with Pincus’s results on the dense linear ordering principle (DO), the Boolean Prime Ideal Theorem (BPI), and the failure of Part∞, together with Pincus’s transfer theorems, yields Con(ZF) ⇒ Con(ZF + RT_pairs + DO + BPI + ¬Part∞), where Part∞ states that every infinite set is the disjoint union of infinitely many infinite sets.

A mix of human-written and AI-generated textHuman understanding: some partsaxiom of choice; Ramsey's theorem
Oct 22, 2026PDF ↗View work →
hexagon:2610.00463v1

math.NT — Number Theory

Non-Archimedean Smale Mean Value and Dubinin–Sugawa Dual Mean Value Problems

Contributed by K. MAHESH KRISHNA

Motivated by the Smale mean value and the Dubinin–Sugawa dual mean value theorems, we formulate non-Archimedean Smale mean value and Dubinin–Sugawa dual mean value problems. We show that if a universal constant satisfying the non-Archimedean Smale mean value problem exists, it must be greater than or equal to one and if a universal constant satisfying the non-Archimedean Dubinin–Sugawa dual mean value problem exists, it must be less than or equal to one. We also derive other bounds whenever roots of unity exist in a non-Archimedean field.

Primarily human-written textHuman understanding: all partsDubinin-Sugawa dual mean value theoremNon-Archimedean field.Smale mean value theorem
Oct 22, 2026PDF ↗View work →
hexagon:2610.00462v1

math.CO — Combinatorics

Asymptotically optimal degree for the double Hall property

Contributed by John Tripodo

The minimum maximum degree for the double Hall property is asymptotic to sqrt(2 n). Deterministic balanced regular bipartite constructions attain this leading constant with the property on both shores and degree at most sqrt(2 n)+O(n^(21/80)) for every sufficiently large n. The construction repairs homology semibiplanes, pads by matched duplication, and completes to regularity. The all-order error term invokes the published Baker-Harman-Pintz prime-interval theorem; the formal transfer keeps that input as an explicit hypothesis.

Primarily AI-generated textHuman understanding: no parts
Oct 22, 2026PDF ↗View work →
hexagon:2610.00461v1

math.NT — Number Theory

An explicit elementary proof of no Landau--Siegel zeros

Contributed by Victor Z. Guo

We give an elementary proof of the absence of Landau--Siegel zeros with the explicit constant c=1/100c=1/100. For every primitive quadratic Dirichlet character χ\chi of conductor q≥3q\ge3, each real zero β∈(0,1)\beta\in(0,1) of L(s,χ)L(s,\chi) satisfies (1−β)log⁡q>c(1-\beta)\log q>c. The proof combines integer coefficient counting, polynomial congruences, a paired-tail hyperbola estimate, and a logarithmically weighted comparison of positive sums. All required constants are explicit, and one choice of parameters covers every conductor. The argument uses elementary arithmetic and real summation and integration, without separate zero computations for small conductors.

A mix of human-written and AI-generated textHuman understanding: all partsFrobenius congruencesLandau--Siegel zeroselementary methodsexplicit estimatesquadratic characters
Oct 22, 2026PDF ↗View work →
hexagon:2610.00460v2

math.CO — Combinatorics

Two conjectures of Wormald on Hamilton cycles in tournaments

Contributed by Deep Bhattacharjee

A tournament is a complete graph in which every edge has been given a direction, and a Hamilton cycle in it is a directed cycle through all the vertices. Let h(n)h(n) be the largest number of Hamilton cycles that a tournament on nn vertices can have, and let g(n)=2−n(n−1)!g(n)=2^{-n}(n-1)! be the expected number of Hamilton cycles of a random tournament on nn vertices. Wormald obtained lower bounds for h(n)h(n) by replacing every vertex of a fixed regular tournament TT on dd vertices by a random tournament on nn vertices. The expected number of Hamilton cycles of the result is governed by a number HH, the determinant of the Hessian of a function defined implicitly by TT; the bound is d/H g(dn)\sqrt{d/H}\,g(dn), or e1/dd/H g(dn)e^{1/d}\sqrt{d/H}\,g(dn) when the blocks are regular. Wormald computed HH for a few small tournaments, found that d/Hd/H was the square of a rational number every time, and could not explain it. We prove that H=ddet⁡(d2I+S2)/det⁡(d2I−S2)H=d\det(d^2I+S^2)/\det(d^2I-S^2), where SS is the skew adjacency matrix of TT, so that H≠0H\neq0 and d/H=det⁡(dI−S)/det⁡(dI−iS)\sqrt{d/H}=\det(dI-S)/\det(dI-iS) is a ratio of two positive integers. This is his Conjecture~6. For the cyclic tournament on dd vertices the formula becomes d/H=((d+1)d+(d−1)d)/((d+i)d+(d−i)d)\sqrt{d/H}=\bigl((d+1)^d+(d-1)^d\bigr)/\bigl((d+i)^d+(d-i)^d\bigr), which tends to cosh⁡1/cos⁡1=2.8559578925…\cosh1/\cos1=2.8559578925\ldots as dd grows. This is his Conjecture~2, and it identifies the constant 2.8559582.855958 that he estimated by fitting polynomials; it also explains why the terms in 1/d1/d and 1/d31/d^3 of his fits were nearly zero. We add bounds for d/H\sqrt{d/H} over all regular tournaments on dd vertices, with equality in the lower bound exactly for doubly regular tournaments. Every statement is proved by hand.

A mix of human-written and AI-generated textHuman understanding: some partsHamilton cycleHessian determinantformal verificationrandom blow-upskew-symmetric matrixtournament
Oct 22, 2026PDF ↗View work →
hexagon:2610.00459v1

math.CO — Combinatorics

All τk\tau_k-maximal graphs have the same size

Contributed by Deep Bhattacharjee

A graph is τk\tau_k-maximal if none of its subgraphs has k+1k+1 edge-disjoint spanning trees, but adding any missing edge creates such a subgraph. Wang and Tian conjectured that every τk\tau_k-maximal graph on n≥2k+2n\ge 2k+2 vertices has exactly (k+1)(n−1)−1(k+1)(n-1)-1 edges. We prove this conjecture.

A mix of human-written and AI-generated textHuman understanding: some partscount matroidedge-disjoint spanning treessaturated graphssparse graphs
Oct 22, 2026PDF ↗View work →
hexagon:2610.00458v1

math.NT — Number Theory

A 15/16 zero-free half-plane for angular Hecke L-functions over the Eisenstein field

Contributed by Troy Winarski

We describe the argument recorded in a frozen Lean development for nonvanishing of the unit-compatible angular Hecke L-functions over Q(−3)\mathbb Q(\sqrt{-3}) in the strict half-plane ℜ(s)>15/16\Re(s)>15/16, apart from the genuine principal pole. The family includes imprimitive finite parts, every integer angular type allowed by the unit relation, and real unitary norm twists. The proof route passes through a mean-square estimate for the actual sextic rows, sixth-power averaging, a smoothed Möbius estimate with a totient factor, and Mellin continuation. We reconstruct the angular reflection, quadratic energy estimate, finite descent, and the calculation producing the 15/1615/16 exponent, while identifying the technical source lemmas inherited from OAI-003 that remain to be independently reconstructed. The theorem is formally verified in Lean and the frozen source/evidence package has undergone statement comparison, axiom audit, recursive kernel replay, quotient postchecking, and separate AI-assisted review. This manuscript is an expository reconstruction rather than a claim of independent human certification. We describe the formally verified proof as a xenoproof, and the theorem provisionally as a xenotheorem, to distinguish formal mathematical correctness from human mathematical understanding.

Primarily AI-generated textHuman understanding: some partsEisenstein integersHecke L-functionsLeananalytic number theoryangular Hecke charactersformal verificationxenoproofxenotheoremzero-free regions
Oct 22, 2026PDF ↗View work →
hexagon:2610.00457v1

math.CO — Combinatorics

Projective-plane determinant bounds for sparse binary matrices

Contributed by John Tripodo

For every integer density r from 3 through 14, a binary square matrix with e ones satisfies an explicit affine-exponential determinant bound. Under e <= r n this gives the projective-plane determinant rate, with equality precisely for direct sums of projective-plane incidence matrices of order r-1. Individual row and column degrees are unrestricted. The written proof uses an exact finite scalar certificate with rigorous unbounded-degree tails; the companion contains a pinned 98-module Lean mathematics closure.

Primarily AI-generated textHuman understanding: no partscombinatorial matrix theorydeterminantsprojective planessparse matrices
Oct 22, 2026PDF ↗View work →
hexagon:2610.00456v1

math.CO — Combinatorics

Reflexive Descent to Minimal Seeds in the Kreuzer--Skarke List

Contributed by Deep Bhattacharjee, Ushashi Bhattacharya

Kreuzer and Skarke built the list of reflexive polytopes by growing it from small seed polytopes. We ask how far a polytope can be from its seeds when it is shrunk through reflexive polytopes only, one at a time and without skipping any. Such a descent always exists and always ends at a seed. In four dimensions no descent takes more than 348 steps, which proves the conjecture of a uniform transition depth, and in two and three dimensions the exact maxima are five and eleven. In four dimensions two polytopes with thirty-seven lattice points need exactly twenty-two steps, so the maximum lies between twenty-two and 348.

A mix of human-written and AI-generated textHuman understanding: some partsCalabi–Yau threefoldsKreuzer–Skarke listlattice polytopesreflexive polytopes
Oct 22, 2026PDF ↗View work →
hexagon:2610.00455v1

math.CO — Combinatorics

The book Ramsey number R(B57,B58)=231R(B_{57},B_{58})=231

Contributed by Deep Bhattacharjee

We construct a graph on 230230 vertices with no book B57B_{57} whose complement contains no B58B_{58}. With the upper bound of Rousseau and Sheehan this gives R(B57,B58)=231R(B_{57},B_{58})=231, the smallest case of Wesley's conjecture R(Bn−1,Bn)=4n−1R(B_{n-1},B_n)=4n-1 that was open. The graph is a two-block circulant graph over Z115\Z_{115}, and the proof reduces to 5757 inequalities.

A mix of human-written and AI-generated textHuman understanding: some partsRamsey numberblock-circulant graphbook graphformal verificationlocal search
Oct 22, 2026PDF ↗View work →
hexagon:2610.00454v2

math.NT — Number Theory

A sixteen-prime-factor bound for Lehmer's totient problem and exact prime-completion criteria

Contributed by Deep Bhattacharjee

We study composite integers nn for which Euler's totient divides n−1n-1. The finite-search results give at least sixteen distinct prime factors, at least 1800118001 when (n−1)/φ(n)≥3(n-1)/\varphi(n)\ge3, and more than 10910^9 when 3∣n3\mid n. A solution with kk prime factors satisfies n<22k−7n<2^{2^{k-7}}, and every solution exceeds 103510^{35}. For the companion divisibility φ(n)∣n+1\varphi(n)\mid n+1, the nine known solutions exhaust the cases with at most eight prime factors. For a fixed prime prefix A=∏pA=\prod p, B=∏(p−1)B=\prod(p-1), the quotient is forced to be M=⌈A/B⌉M=\lceil A/B\rceil and a possible largest prime is uniquely q=1+(A−1)/(MB−A)q=1+(A-1)/(MB-A). We derive its exact divisibility, coprimality and valuation consequences, and prove that deleting the largest prime of a hypothetical counterexample cannot yield a smaller Lehmer counterexample. A two-prime factorisation identifies the additional integrality and primality conditions required by a replacement construction. A complete finite construction is proved for every prescribed factor count. Excluding its output at every factor count, or excluding the final prime for every prefix, remains unproved; the full Lehmer conjecture is not resolved.

A mix of human-written and AI-generated textHuman understanding: some partsCarmichael numbersEuler's totient functionFermat primesLehmer's totient problemlarge sieve
Oct 22, 2026PDF ↗View work →
hexagon:2610.00453v1

math-ph — Mathematical Physics

Faster-than-power-law decay of order parameter correlations in the two-dimensional RP2 model: a rigorous result by vibe-proving

Contributed by Masaki Oshikawa

We consider the RP2RP^2 model on the two-dimensional square lattice with nearest-neighbor interactions. It is obtained by replacing the spin in the standard classical Heisenberg model with a director, which does not distinguish two opposite directions. Unlike the standard classical Heisenberg model or the O(3) nonlinear sigma model, the RP2RP^2 (and its cousin RP3RP^3) model allows point vortex defects classified by π1(RP2)=Z2\pi_1(RP^2)=Z_2 in two dimensions. It is tempting to expect the Z2Z_2 vortices to induce a phase transition analogous to the well-established Berezinskii-Kosterlitz-Thouless transition in the O(2) model (XY model) in two dimensions. Indeed, the possibilities of a critical phase or a critical point with power-law correlations of the order parameter in the RP2RP^2 and RP3RP^3 models have been discussed in the literature. However, the challenging nature of the problem has prevented a definitive conclusion on the phase diagram so far. In this paper, we report a rigorous mathematical proof by Artificial Intelligence (ChatGPT 6 Astra) that the absolute value of the two-point correlation function of order parameters at distance rr in the RP2RP^2 and RP3RP^3 models on the two-dimensional square lattice is bounded above by Cβexp⁡{−cβ[log⁡(2+r)]2}C_\beta \exp\{-c_\beta[\log(2+r)]^2\} for any finite inverse temperature β≥0\beta\ge0, with positive constants CβC_\beta and cβc_\beta depending only on β\beta. Namely, the correlation function decays faster than any power of rr. The proof is formalized and verified in Lean; the formalization is publicly available at \href{https://doi.org/10.5281/zenodo.23548847}{doi:10.5281/zenodo.23548847}. A heuristic argument based on a one-parameter effective theory ansatz at large distances combined with the rigorous correlation bound suggests (but does not prove) that the Z2Z_2 vortices are always unbound at any nonzero temperature.

A mix of human-written and AI-generated textHuman understanding: some parts
Oct 22, 2026PDF ↗View work →
hexagon:2610.00452v1

math.NT — Number Theory

Cross-Energy of Dirichlet Partial Sums. Part 9: Zeros of the prime zeta function in its half-plane of absolute convergence

Contributed by Pedro Caceres

The prime zeta function P(s) = ∑ₚ p⁻ˢ has zeros in its half-plane of absolute convergence. Belovas, Čepaitytė and Sabaliauskas proved that P(s) ≠ 0 for Re(s) > σₚ = 1.7795446535…, where σₚ solves P(σ) = 2¹⁻σ. We show that this bound is sharp: the closure of the real parts of the zeros in Re(s) > 1 is exactly [1, σₚ]. There are no zeros on Re(s) = σₚ, while zeros occur arbitrarily close to every line Re(s) = σ with 1 < σ < σₚ (Bohr’s method). By Jessen’s theorem, the density of real parts is J″(σ)/(2π), where J(σ) = −σ log 2 + 𝔼[log⁺(2^σ |Yσ|)], Yσ = ∑ₚ≥₃ p⁻σ eⁱθₚ, and the phases θₚ are independent and uniformly distributed. The function J is smooth, σₚ = inf{σ : J(σ) + σ log 2 = 0}, and the density vanishes at σₚ faster than any power. The argument principle, implemented with ball arithmetic, certifies exactly 68 zeros with 1.1 ≤ Re(s) ≤ 1.85 below height 2000 (70.0 predicted), and 12 zeros with Re(s) ≥ 1.5 below height 5000 (12.0 predicted). A heuristic large-deviation extrapolation predicts zeros with real part above 1.7 approximately once every 6 × 10⁶ units of height. In ½ < Re(s) < 1, P is single-valued on the half-plane with (½, 1] removed if and only if the Riemann hypothesis holds (a reformulation of a classical equivalence). Its zeros are canonical below height 3 × 10¹² and solve ζ(s) = exp(A(s)), A(s) = ∑ₘ≥₂ P(ms)/m, with |P(s)| < 2π. Below height 600, their counts for Re(s) ≥ 0.7 agree numerically with the random model. The zeros arise from cancellations between 2⁻ˢ and the contributions of the remaining primes, the prime-sum analogue of Euler-product interference behind the off-line zeros of the Davenport–Heilbronn function. We do not prove the Riemann hypothesis.

A mix of human-written and AI-generated textHuman understanding: all partsC2 FunctionDirichlet SumsRiemann Zeta Function
Oct 22, 2026PDF ↗View work →
hexagon:2610.00451v1

math.NT — Number Theory

Small algebraic integers and Frobenius prime sums

Contributed by Scott Kominers

The determinant argument in recent proofs of Landau–Siegel zero exclusion compares the size of a nonzero integer with its divisibility by inert primes. We observe that the same comparison applies to every Frobenius class in a finite Galois extension of Q\mathbb{Q}, bounding its weighted prime count in terms of the supply of small algebraic integers. For each fixed field, the resulting height bound implies a weighted Chebotarev theorem whose only analytic inputs are the elementary Chebyshev and Mertens estimates. In the quadratic case there are only two Frobenius classes, so the lower bound obtained by complementation remains uniform in the conductor; this recovers Landau–Siegel zero exclusion.

A mix of human-written and AI-generated textHuman understanding: all partsChebotarev density theoremFrobenius congruenceGalois extensionGeometry of numbersLandau–Siegel zeroWronskianinterpolation determinant
Oct 22, 2026PDF ↗View work →
hexagon:2610.00450v1

math.MP — Mathematical Physics

Superfluid Stiffness and Nonclassical Rotational Inertia of the Dilute Hard-Sphere Bose Gas

Contributed by P. M. Aronow, Patrick Lopatto

We study the three-dimensional Bose gas of hard spheres of diameter aa at density ρ\rho and temperature TT, in the thermodynamic limit with aa, ρ\rho, and TT fixed. When the gas parameter ρa3\rho a^3 is sufficiently small and 0<T<T∗0<T<T_*, we prove two equilibrium signatures of superfluidity. First, in a periodic box, the superfluid fraction, defined through the free-energy cost of twisted boundary conditions, has thermodynamic lower limit at least c (ρa3)3/8c\,(\rho a^3)^{3/8}, uniformly in TT in this interval. Second, in dilations of an arbitrary nonempty bounded open container with hard walls, the nonclassical rotational inertia fraction 1−I/Icl1-I/I_{\mathrm{cl}} has thermodynamic lower limit at least c′(ρa3)3/8c'(\rho a^3)^{3/8} for every fixed rotation axis, where II is the moment of inertia, IclI_{\mathrm{cl}} its classical value, and c′>0c'>0 depends only on the container and the axis. In both results the derivative of the free energy is taken at finite volume, before the thermodynamic limit. The temperature T∗T_* is of order (ρa3)1/3/log⁡(1/(ρa3))(\rho a^3)^{1/3}/\log(1/(\rho a^3)) times the critical temperature of the ideal Bose gas. We also bound the moment generating function of the winding vector of the Feynman–Kac paths by that of a centered Gaussian, at every density and positive temperature. Combining this upper bound with the first result, we determine the order of the winding fluctuations in the regime above. Further, for 0<ε≤10<\varepsilon\le1, ρa3≤(ε/100)3\rho a^3\le(\varepsilon/100)^3, and T≥(1+ε)4πρ2/3T\ge(1+\varepsilon)4\pi\rho^{2/3}, about 1.9(1+ε)1.9(1+\varepsilon) times the critical temperature of the ideal Bose gas, the superfluid fraction in a periodic box tends to zero exponentially in its side length. By identities of Pollock and Ceperley and of Sindzingre, Klein, and Ceperley, we can express the two responses as multiples of the variances of the winding and of the signed projected area of the paths. We bound these variances from below with a variance inequality of Hammersley–Chapman–Robbins type, applied to random rewirings of the paths in a short imaginary-time interval along closed lattice routes. The rewirings adapt route comparisons developed in OpenAI's preprints on Bose–Einstein condensation in dilute Bose gases, and we extend the thermal estimates of the positive-temperature hard-sphere preprint in that series to higher temperatures. We average the rewirings over randomly translated closed lattice walks whose independent copies rarely approach each other. This makes the changed law close to the original law in chi-square divergence. Using conditional independence between disjoint imaginary-time intervals, we obtain bounds uniform in temperature. We also transfer the thermal estimates to arbitrary containers.

Primarily AI-generated textHuman understanding: no partsbose gassuperfluid
Oct 22, 2026PDF ↗View work →
hexagon:2610.00449v1

math.NT — Number Theory

A quantitative refinement of Lu's elementary prime number theorem

Contributed by Rajat Bhattacharjee

We refine Lu's elementary method for the prime number theorem to obtain, for every fixed K>400K>400, \[ \psi(X)-X=O_K\!\left( X\exp\!\left\{-\frac{\sqrt{\log X}}{(\log\log X)^K}\right\}\right) \qquad(X\to\infty), \] where ψ\psi is Chebyshev's function. The main ingredient is a comparison estimate that remains uniform as the degree grows and the short-interval parameter shrinks. Combining this estimate with exponential weights in Balog's identity, a single finite energy step and positivity-based interpolation yields the stated bound, retaining the full degree tail and all arithmetic correction terms. The resulting remainder strengthens Lu's fixed-parameter estimates and Goz\'e's quantitative refinement within the real-variable framework; stronger remainder estimates are known by other methods.

Primarily AI-generated textHuman understanding: no parts
Oct 22, 2026PDF ↗View work →
hexagon:2610.00448v1

math.CO — Combinatorics

A polynomial algorithm for general d-position in trees

Contributed by Daniel Adam Pyle

A vertex set is in general dd-position if no three of its vertices lie on a shortest path of length at most dd. We give an exact algorithm for an nn-vertex tree in O(n(D+2)4)O(n(D+2)^4) time and O(n(D+2)2)O(n(D+2)^2) space, where D=min⁡{d,n−1}D=\min\{d,n-1\}. In particular, the problem is polynomial-time solvable when the threshold is part of the input. This answers the tree-complexity question posed by Klav\v zar, Rall, and Yero in 2021 and retained in the August 2026 version of the general position survey. The dynamic program uses two truncated distances at each rooted boundary: the nearest selected vertex and the nearest far endpoint of an ancestor-comparable selected pair. These distances characterize every forbidden triple created by joining child subtrees. The algorithm reconstructs an optimal set and extends to signed vertex weights and exact counting of optimal sets. It also computes the full cardinality generating polynomial in O(n2(D+2)4)O(n^2(D+2)^4) coefficient operations. Exhaustive comparisons with direct subset enumeration support the implementation independently of the proof.

Primarily AI-generated textHuman understanding: no parts
Oct 22, 2026PDF ↗View work →