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math.QA — Quantum Algebra

Low-rank Grothendieck rings: an AI-assisted classification project, constructions, and open problems

Contributed by Sebastien Palcoux

We present a public AI-generated research project on fusion rings and their categorifications over $\mathbb C$ , with data extending through rank twelve. The project reports unrestricted Grothendieck-ring classifications in ranks two, three, and four, a complete multiplicity-free classification through rank twelve, and classifications or partial classifications at higher multiplicities. The released census contains $1{,}434{,}042$ rings in ranks three through twelve; ten categorification questions remain open within its stated bounds. The resources include research manuscripts, categorical models, exclusion arguments, verification code, and an interactive atlas. One categorical model per ring suffices for our classification problem; classifying all realizing categories up to equivalence is a further task. We summarize the current scope, compare it with AnyonWiki, and link to the online constructions catalogue and unresolved cases. The project's original research content was generated by AI under human direction; independent human verification remains pending.

Primarily AI-generated textHuman understanding: some partsAI-assisted mathematicsFusion categoriesGrothendieck ringscategorificationfusion multiplicitylow ranknear-group categories

Oct 17, 2026PDF ↗View work →

hexagon:2610.00313v1

math.DS — Dynamical Systems

Attracting oscillations in sequential distributive multisite phosphorylation

Contributed by Michael Mislan

The sequential distributive $n$ -site phosphorylation cycle with one kinase and one phosphatase admits attracting positive periodic orbits under mass-action kinetics for every $n\ge3$ . An explicit three-site family has a fixed positive equilibrium, a transversal imaginary-eigenvalue crossing, a stable complementary spectrum and a negative first Lyapunov coefficient. A site-addition argument extends the supercritical Hopf bifurcation to every larger site count. Together with the one-site calculation and Vassena's two-site exclusion, this makes three the exact threshold for Hopf bifurcation in this family. Dynamic sequestration of the shared enzymes supplies the feedback: the equilibrium, productive fluxes and statically eliminated vector field remain fixed while stability changes. Further results compare enzyme clamps and selective reduction, quantify flux and amplitude laws, validate two finite-amplitude attracting orbits, and give a driven thermodynamic realization. Exact algebra and interval certificates support the quantitative claims; numerical continuation is distinguished from proof. A registered Lean theorem independently establishes an existential affine family with attracting periodic orbits for every $n\ge3$ . The manuscript's arbitrary-parent site-addition theorem and its quantitative extensions go beyond that registered statement.

Primarily AI-generated textHuman understanding: some partsHopf bifurcationattracting periodic orbitsenzyme sequestrationmultisite phosphorylation

Oct 17, 2026PDF ↗View work →

hexagon:2610.00302v1

math.CT — Category Theory

The Grothendieck construction between the telescope monad and contextads

Contributed by Kotaro Kamiya

A telescope is a list of stages in which each stage is a family indexed over the Grothendieck construction of the previous ones. The telescope monad stacks stages as the free-monoid 2-monad on spans, whereas a contextad collapses two stages into one by a dependent sum. This paper derives both from the span of single stages, and thereby explains why the strict algebras of the telescope monad are contextads. The link is the Grothendieck extension of a span. Because it is monoidal and preserves coproducts, it turns the telescopes into the free monad on the extension of the span of stages and the stage contextad into the Grothendieck monad, whose Kleisli arrows are the contextful arrows. The main theorem identifies the stage contextad with the pseudo-algebra of the telescope monad transported from the display contextad. Its structure map is the dependent sum, which the extension turns into the iterated multiplication of the Grothendieck monad.

A mix of human-written and AI-generated textHuman understanding: some partsGrothendieck constructioncomprehension categorycontextaddependent sumtelescope

Oct 17, 2026PDF ↗View work →

hexagon:2610.00297v1

math.CO — Combinatorics

Bounds for Erdős Problem #188: $7\le k\le 60$

Contributed by Carson Crocker

Let $\ell_k$ denote $k$ collinear points with consecutive distances $1$ . Erdős Problem #188 asks for the least $k$ such that the plane can be colored red and blue with no two red points at distance $1$ and no blue copy of $\ell_k$ . Previous work gives $6\le k\le 6330$ , with the lower bound due to Tsaturian and the upper bound due to Currier, Mody, Xie and Zhang. We prove $7\le k\le 60$ . For the lower bound we show that every red/blue coloring of the plane contains two red points at distance $1$ or a blue $\ell_6$ , following the skeleton of Arman and Tsaturian's proof in three dimensions; four distance lemmas are reduced to explicit unsatisfiable propositional formulas, certified by DRAT proofs. For the upper bound we give an explicit periodic coloring whose red set consists of $90$ closed regular $24$ -gons of $48$ different sizes, all of diameter less than $0.999$ , in each period, with rational centers and sizes. Any two cells are either far apart (all distances between their points exceed $1$ ) or close together (all distances are less than $1$ ), which is checked in exact arithmetic for all periodic translates and rules out red unit pairs. That every $\ell_{60}$ contains a red point is proved by a computer-assisted verification: for every unit direction $u$ , the sets $R-ju$ , $j=0,\dots,59$ , cover the period torus, with every inclusion checked in rigorous ball arithmetic. A second, separately written program, based on a reduction to one-dimensional covering problems solved in outward-rounded interval arithmetic, confirms this. The computations, the verification code and drafts of the text were produced with AI tools under the author's direction; all proof certificates and verification programs are public (doi:10.5281/zenodo.23405962).

Primarily AI-generated textHuman understanding: some partsDRATErdős problemsEuclidean Ramsey theorySATinterval arithmeticperiodic coloringunit distance

Oct 17, 2026PDF ↗View work →

hexagon:2610.00305v1

math.CO — Combinatorics

Three conjectures on the first two graph eigenvalues

Contributed by Idris Ali Shaik

Let λ₁(G) ≥ λ₂(G) be the two largest adjacency eigenvalues of a finite simple graph G on at least two vertices. We resolve the three conjectures appearing as 1.15–1.17 in Kannan, Kumar and Pragada. First, for every noncomplete graph, we prove λ₁(G)² + λ₂(G)² ≤ 2 Σ_{e∈E(G)} (1 − 1/c(e)), where c(e) is the largest order of a clique containing e. Second, we construct connected (K₅ minus one edge)-free graphs of arbitrarily large size that violate the exact color-critical bound proposed by Li, Liu and Zhang. Third, for every fixed F with χ(F) = r + 1 ≥ 3, we prove, uniformly over F-free graphs with m = e(G) → ∞, λ₁(G)² + λ₂(G)² ≤ 2(1 − 1/r)m − ((r − 2)/r)Δ(G) + o_F(m), where Δ(G) is the maximum degree. The coefficient of the degree term is best possible. The arguments use the half-plane complete-positivity theorem of Coutinho, Liu, Spier, Tang and Zhang, together with an explicit connected construction, weighted clique removal and spectral separation of vertices of high degree. Theorems 1.1–1.3 are formalized in Lean 4 and registered as PALOMAR-2026-10-15-000015 v1, PALOMAR-2026-10-16-000012 v1 and PALOMAR-2026-10-17-000005 v1, respectively.

A mix of human-written and AI-generated textHuman understanding: all partsadjacency eigenvaluesextremal graph theoryspectral graph theory

Oct 17, 2026PDF ↗View work →

hexagon:2610.00306v2

math.CO — Combinatorics

A Product Formula for Partition-Restricted Boundary Hypermatrix Counts

Contributed by YURAN XIU

We prove the Koprowski--Lewis product conjecture for nondegenerate $2\times(k+1)\times k$ hypermatrices over finite fields with two partition-shaped layers of prescribed zeros. The result holds for every positive $k$ , every finite field, and every admissible pair of partitions. After fixing the first slice, nondegeneracy is equivalent to simplicity of a partial linear map. Applying the simple-extension theorem of Arora, Ram, and Venkateswarlu inside the successive allowed row spaces gives the product directly. We include an elementary proof of that extension theorem and justify the equivalence with the defining projective critical locus of the hyperdeterminant. The complete counting statement and its semantic equivalences have been checked in Lean. HAL deposit: hal-05800194v1.

Primarily AI-generated textHuman understanding: no partsLean formalizationenumerationfinite fieldshypermatrixproduct formula

Oct 17, 2026PDF ↗View work →

hexagon:2610.00316v2

math.CO — Combinatorics

Tools for proving that graphs are not graceful

Contributed by Boon Suan Ho

A graph with $m$ edges is graceful if its vertices can be labeled with distinct integers in $[0,m]$ whose edge differences are exactly $1,\dots,m$ . Graceful labelings have been constructed for sixty years, but there are few general methods for showing that a graph is not graceful. We start from a single generating-function identity that restates gracefulness, and derive three tools from it: residues at roots of unity, extending the classical parity condition; positivity on the unit circle, whose semidefinite form amounts exactly to the Goemans–Williamson max-cut relaxation tested against the minimum of the Dirichlet kernel; and the label interval itself, which forces the labels of a dense subgraph to have a large span. The consequences include new families of ungraceful graphs: every edge-transitive graph that contains $K_6$ , the Paley graphs $P(q)$ with $q\ge37$ , the Kneser graphs $K(n,k)$ with $n\ge5.61k$ , almost every $d$ -regular graph with $d\ge84$ , and the polarity graphs $\mathrm{ER}_q$ with $q\ge23$ , which contain no $4$ -cycle. For the semidefinite form of positivity we also determine exactly what it can prove, and we show that a planar refinement is strictly stronger: it excludes the Kneser graph $K(11,2)$ with any seven edges deleted.

Primarily AI-generated textHuman understanding: some partsDirichlet kernelGolomb rulerdifference setsgraceful labelingmax-cutsemidefinite programming

Oct 17, 2026PDF ↗View work →

hexagon:2610.00317v1

math.CO — Combinatorics

No lattice tiling of $\mathbb{Z}^{35}$ or $\mathbb{Z}^{805}$ by Lee balls of radius three

Contributed by Deep Bhattacharjee, Ushashi Bhattacharya

We prove that there is no lattice tiling of $\mathbb{Z}^{35}$ or of $\mathbb{Z}^{805}$ by Lee balls of radius three. These were the only cases left open by Xiao and Zhou among the Lee balls of dimension at least three and radius at least two with fewer than $2^{30}$ points, so the lattice Golomb--Welch conjecture holds for all of them.

A mix of human-written and AI-generated textHuman understanding: some partsGolomb-Welch conjectureLee metricgroup ringlattice tilingperfect code

Oct 17, 2026PDF ↗View work →

hexagon:2610.00318v1

math.CO — Combinatorics

A graph with three card types and five vertex orbits

Contributed by Deep Bhattacharjee, Ushashi Bhattacharya

We prove that the answer to part~(b) of Problem~20.71 of the Kourovka Notebook is yes: there is a connected graph on $63$ vertices with exactly three isomorphism types of vertex-deleted subgraphs whose automorphism group has five orbits on the vertices. The graph is a Cayley graph of a Suzuki $2$ -group of order $64$ with one vertex removed.

A mix of human-written and AI-generated textHuman understanding: some partsCayley graphSuzuki 2-groupdivisible design graphpseudosimilar verticesvertex-deleted subgraph

Oct 17, 2026PDF ↗View work →

hexagon:2610.00319v2

math.CO — Combinatorics

Every odd antihole is strongly chromatic-choosable

Contributed by Deep Bhattacharjee, Ushashi Bhattacharya

A hole in a graph is an induced cycle of length at least four, and an antihole is the complement of a hole. Odd holes and odd antiholes are the graphs that, by the strong perfect graph theorem, force the chromatic number above the clique number. Fiujlaali and McDonald studied the odd antiholes inside $k$ -critical graphs of maximum degree $\Delta$ , proved that such an antihole has at most $2(\Delta-k)+7$ vertices, and conjectured that this bound is attained only when the graph is the antihole itself, provided the graph has no odd hole. They reduced this to a statement about list colouring: for every $\ell\ge3$ , the odd antihole on $2\ell-1$ vertices has chromatic number $\ell$ , and the only assignments of lists of $\ell-1$ colours from which it cannot be coloured are those in which all lists are equal. This is their Conjecture~9, and they proved it when all the lists lie in a common set of $\ell$ colours. We prove it for every $\ell$ with no restriction on the lists, and we deduce their Conjecture~3 for all $k$ and $\Delta$ . The proof works with the complements of disjoint unions of paths with an odd number of vertices. The vertices in even positions of the paths form a clique in the complement, so equal lists of one colour fewer than the number of these vertices cannot be completed; we show by induction, using Hall's theorem and the theorem of Erd\H{o}s, Rubin and Taylor on complete multipartite graphs with parts of size at most two, that this is the only obstruction, and a separate lemma on minimal tight sets settles the extremal case of a single path. The cycle is then closed by carrying a colour shared by two consecutive vertices around the odd cycle in steps of two. The theorem also gives an infinite family of strongly chromatic-choosable graphs that are not strongly critical in the sense of Stiebitz, Tuza and Voigt. Every statement is proved by hand.

A mix of human-written and AI-generated textHuman understanding: some partschromatic-choosabilitycolouringcritical graphodd antiholeodd hole

Oct 17, 2026PDF ↗View work →

hexagon:2610.00320v2

math.CO — Combinatorics

The $\mathfrak{sl}(2)$ weight system at $c=-3/32$ extends to a graph 4-invariant

Contributed by Deep Bhattacharjee, Ushashi Bhattacharya

A weight system is a function on chord diagrams that satisfies the four-term relations. The Lie algebra $\mathfrak{sl}(2)$ gives a weight system with values in the polynomial ring $\mathbb{Q}[c]$ , and its value on a chord diagram depends only on the intersection graph of the diagram. Fomichev, Karev, Pavutnitskiy and Usanov showed that this weight system does not come from a function on graphs, and that a specialisation at a value $c=c_0$ can come from one only for $c_0=0$ , $3/8$ , $1$ or $-3/32$ . The first three values were settled before. They conjectured that the last one, $c=-3/32$ , which is the eigenvalue of the Casimir element on the oscillator representation, also extends. We prove this. The extension is the function $\xi(G)=(-32)^{-n}S_G(-2)$ , where $G$ is a graph with $n$ vertices and $S_G(N)$ is a signed count of the maps that send each vertex of $G$ to a two-element subset of $\{1,\dots,N\}$ : a map counts with the sign $-1$ for every edge whose two endpoints are sent to subsets meeting in exactly one element. The count is a polynomial in $N$ , so its value at $N=-2$ makes sense. The four-term relation for $\xi$ is proved by an involution on these maps. The identity with the weight system is proved by letting the chord diagram act on Verma modules, then on the oscillator representation, where a supertrace on the Weyl algebra turns the value into a sum over perfect matchings, and finally by reading the same sum as a trace in a Clifford algebra on $N$ generators.

A mix of human-written and AI-generated textHuman understanding: some partsClifford algebrafour-term relationintersection graphoscillator representationweight system

Oct 17, 2026PDF ↗View work →

hexagon:2610.00314v1

math.AP — Analysis of PDEs

Non-unique vanishing-viscosity solutions of 2D Euler with vorticity in $L^p$

Contributed by Umberto Pappalettera, Massimo Sorella

For every finite $p>2$ , we construct one real compactly supported zero-circulation initial vorticity in $L^1(\mathbb R^2)\cap L^p(\mathbb R^2)$ whose unforced Navier--Stokes evolutions have two distinct strong vanishing-viscosity accumulation paths. Both limits are renormalized unforced Euler solutions with the same strong initial trace. The construction uses a fixed infinite concentric stack, a smooth unstable fourfold profile, actual positive scalar envelopes, and a finite nonlinear expansion retaining the interacting harmonic histories. Discounted source history is controlled through the complete prospective receiver stop, and exact-cap normalization yields a depth-uniform outgoing reset. Protected first crossings, phase degree and a sparse phase assignment then separate the two limits. The analytical preliminaries, finite-sector spectral scope, source-reference comparison and uniform parameter order are included in the proof.

Primarily AI-generated textHuman understanding: no parts

Oct 17, 2026PDF ↗View work →

hexagon:2610.00267v2

math.AG — Algebraic Geometry

The fifteenth cohomology of moduli spaces of stable curves

Contributed by zeming sun, guoxiong gao, shurui liu

We determine the fifteenth cohomology of the moduli stack of stable curves $\overline{\mathcal{M}}_{g,n}$ in every genus $g$ and number of markings $n$ , with its symmetric-group action, by providing explicit multiplicity formulas for its simple constituents. We also prove the generalized Hodge conjecture in this case. The main result of this paper was obtained using the Pharos system. The author has reviewed the arguments and developed an understanding of the proof. On the basis of this review, the author believes the paper to be correct.

Primarily AI-generated textHuman understanding: some partsHodge structurescohomologygeneralized Hodge conjecturemoduli spaces of stable curvesmotives

Oct 17, 2026PDF ↗View work →

hexagon:2610.00303v1

math.AG — Algebraic Geometry

A Counterexample to Beauville's Stability Conjecture

Contributed by Yi Gu, Lifan Guan

We construct a morphism $f: X\to Y$ from a genus-two curve to an elliptic curve such that $f_*\mathcal L$ is unstable for every $\mathcal L\in\Pic^3(X)$. This gives a counterexample to a conjecture of Beauville.

A mix of human-written and AI-generated textHuman understanding: all partsDirect imagesFrey–Kani construction.stability

Oct 17, 2026PDF ↗View work →

hexagon:2610.00315v2

math.AG — Algebraic Geometry

Dense Algebraic Loci of Hodge Classes

Contributed by Deep Bhattacharjee, Ushashi Bhattacharya

We prove unconditionally that the Hodge conjecture is equivalent to two statements about abelian varieties, that the locus on which either statement holds is dense in every family and is either the whole family or a meagre set, that the real multiplications by $\sqrt6$ and by the cubic fields $\QQ(\zeta_{7})^{+}$ and $\QQ(\zeta_{9})^{+}$ are algebraic on explicit families of K3 surfaces, the cubic ones on the surfaces of Mumford type whose quaternion algebra is ramified at two real places, and that the Hodge conjecture holds on every power and every Hilbert scheme of points of the K3 surfaces of the one-dimensional Fricke families of levels $5$ and $7$ , which carry complex multiplication by $\QQ(\sqrt{-5})$ and $\QQ(\sqrt{-7})$. For each of the two statements we build the object it needs next, a complex on a product of two Weil fourfolds and a class on an abelian fourteenfold, prove what it does, and prove at which step it stops; among the steps, every convolution of the Weil line bundles on $E_{0}^{8}$ whose shifts lie in seven consecutive integers fails the polarised criterion in eighteen of the twenty-eight arrangements of the shifts, an explicit curve of genus four makes the Prym map of cyclic triple covers \'etale onto the Weil domain of $\QQ(\sqrt{-3})$, and the conjecture for a class is equivalent to the vanishing of the mass gap of a multiple of it plus a complete intersection, which Koll\'ar's hypersurfaces show cannot be dropped for integral classes. Both statements remain open.

A mix of human-written and AI-generated textHuman understanding: some partsHodge conjectureK3 surfacesKuga-Satake constructionWeil classesabelian varieties

Oct 17, 2026PDF ↗View work →

hexagon:2610.00301v2

cs.CC — Computational Complexity

The Exact Junta Bound for Degree-Two Boolean Functions on q-ary Hamming Cubes

Contributed by YURAN XIU

We determine the maximum number of relevant variables of a Boolean function of degree at most two on a finite alphabet. For every integer $q\geq2$ , this maximum is exactly $\max\{4,q+1\}$ . Here degree means Hamming, or junta, degree: the function is a sum of real-valued functions, each depending on at most two coordinates. We also prove that any such function with more than four relevant variables has a selector coordinate: after that coordinate is fixed, the function depends on at most one other coordinate. The proof combines saturation on a binary four-dimensional restriction with a rigidity argument that applies to every symbol of the original alphabet. Two disjoint interactions force a four-junta; a function with more than four relevant variables has a star of interactions. The exact bound and selector theorem have been formalized in Lean 4, with transitive dependencies using only the standard axioms of Lean and mathlib.

Primarily AI-generated textHuman understanding: no partsBoolean functionsHamming graphsLean formalizationdegree twojunta theoremrelevant variables

Oct 17, 2026PDF ↗View work →

hexagon:2610.00307v2

cs.CC — Computational Complexity

Set Cover by Cyclic Rotations Is NP-Complete

Contributed by YURAN XIU

Given an explicitly listed binary pattern of positive length $n$ and a nonnegative integer $b$ in binary, can at most $b$ cyclic rotations of the pattern cover all $n$ positions? We prove that this decision problem is NP-complete. The reduction constructs a pattern in a cyclic group of polynomial order and outputs its entire indicator word. Its principal ingredient is a replicated-marker construction: a cover within the budget must choose one translate in every layer, and each failed difference constraint leaves a block of more holes than the remaining translates can cover. Combining this construction with the known embedding of graph cliques into circulant graphs using distinct three-term sums gives a polynomial-time reduction from $3$ SAT. The theorem allows every subset, including the empty pattern, and every binary budget. A Lean formalization checks the mathematical reduction, the full-word polynomial-time programs, NP membership, and the unconditional NP-hardness foundation. HAL deposit: hal-05800233v1.

Primarily AI-generated textHuman understanding: no partsLean formalizationNP-completenesscyclic rotationsset cover

Oct 17, 2026PDF ↗View work →

hexagon:2610.00308v2

cs.CC — Computational Complexity

A Polynomial Separation for Adaptive Fourier-Overlap Parity Decision Trees

Contributed by YURAN XIU

Consider the parity decision tree rule that, at every nonconstant restriction, queries a nonzero character maximizing the intersection of the current Fourier support with its translate. We construct an explicit infinite family for which every adaptive policy obeying this rule has polynomial worst-case depth, although an unrestricted parity decision tree has logarithmic depth. For every $\ell\geq1$ , let $R=2^\ell$ . Our function has $n=4\ell-1+R^2\ell+R$ input bits and exactly $k=R^6-R^5+R$ nonzero Fourier coefficients. Every history-dependent maximizing policy terminates and computes the function, but has a feasible path of length at least $R>k^{1/6}$ ; an explicit unrestricted tree has depth at most $5\ell\leq\log_2 k$ . The construction places one pointer behind each cell of a finite-field affine partition, with all pointers accessing a shared data table. After any sequence of equal-answer table comparisons, exact signed Fourier coefficients force every maximizing query to compare two remaining table blocks. This invariant accommodates all tie choices and all resulting table partitions. The complete separation, including the specified polynomial-field representation and the existence of every operational policy's finite run, has been checked in Lean. The result concerns this particular support-overlap rule; it is not a lower bound against arbitrary parity decision trees. HAL deposit: hal-05800236v1.

Primarily AI-generated textHuman understanding: no partsFourier analysisLean formalizationadaptive algorithmsparity decision trees

Oct 17, 2026PDF ↗View work →

hexagon:2610.00311v2

math.NT — Number Theory

On Landau--Siegel zeros

Contributed by Ben Green

This is a variant of the short OpenAI proof that there do not exist Landau-Siegel zeros. In v2, two parameters are decoupled to give a better constant in the main theorem.

Primarily human-written textHuman understanding: all parts

Oct 17, 2026PDF ↗View work →

hexagon:2610.00299v1

math.NT — Number Theory

Polylogarithmic Descent for Almost All Collatz Orbits in Natural Density

Contributed by Idris Ali Shaik

Korec proved that, for each fixed θ > log 3/log 4, almost every positive integer n has a Collatz iterate below n^θ in natural density. Tao proved descent below every prescribed diverging threshold in logarithmic density. We obtain polylogarithmic descent in ordinary natural density within logarithmically many steps, with bounds on the preceding orbit values. The shortcut map T sends even n to n/2 and odd n to (3n+1)/2. The unaccelerated map Col sends even n to n/2 and odd n to 3n+1. For any prescribed positive function f(n) → ∞, however slowly, we prove that almost every positive integer n, in natural density, has nonnegative integers k = k(n) and ℓ = ℓ(n) satisfying T^k(n) = Col^ℓ(n) < (log n)^A_FP f(n), where A_FP ≈ 9.9911 is an explicit exponent. The shortcut time is k = c_* log n + O_f(√(log n log log n)), where c_* = 2/log(4/3). The leading time c_* log n is predicted by the average logarithmic contraction of the equally weighted multipliers 1/2 and 3/2. The unaccelerated time is ℓ = (3/2)c_* log n + O_f(√(log n log log n)). Both orbits are bounded by nf(n) up to their respective times. For each fixed A > A_FP, put q = A/(2A_FP) − 1/2. For sufficiently large X, all but at most C_A X/(log X)^q integers n ≤ X satisfy the same conclusions with target (log n)^A and bound n(log n)^q on both orbits. The constant C_A and the time-error constants depend only on A. The proof counts finite parity sequences exactly and follows first passages through decreasing thresholds. We count later failures among the original starting values, without assuming that values reached after a descent are uniformly distributed. For starting values 2^M ≤ n < 2^(M+1), only O(√(M log M)) cumulative passage times need to be counted at each threshold. A finite random-walk survival estimate supplies the additional saving needed at the critical exponent. Density-one descent below a fixed multiple of (log n)^A_FP is not established, and the pointwise Collatz conjecture remains open. The theorem for arbitrary f is formalized in Lean 4 with Mathlib and registered as PALOMAR-2026-08-26-000005, version 4.

A mix of human-written and AI-generated textHuman understanding: all partsCollatz mapalmost all Collatz orbitsfirst passagenatural densityparity vectorspolylogarithmic descentstopping time

Oct 17, 2026PDF ↗View work →

hexagon:2610.00294v1

math.NT — Number Theory

Rational points on elliptic curves that are primitive modulo a positive proportion of primes

Contributed by Manjul Bhargava

In 1977, Lang and Trotter conjectured that, subject to certain necessary arithmetic restrictions, a rational point $P$ of infinite order on an elliptic curve $E$ over $\Q$ should generate its reduction modulo infinitely many primes $p$ ---and indeed should do so for a specified positive density of primes. However, to date, even this infinitude was not known unconditionally for any explicit pair $(E,P)$ . For the curve $E:y^2=x^3-2x$ and point $P=(2,2)$ , we prove the existence of infinitely many---and indeed a positive lower density of---primes for which the reduction of $P$ generates $E(\F_p)$. More precisely, we prove that a common zero-free half-plane $\Re s>1-1/N$ for Dedekind zeta functions gives a lower density of at least $c/(N\log N)$ , with $c>0$ independent of $N$ . The announced theorem of Cureton and Thorner provides $N=32$ . Our proof combines this input with the marked Type~II estimates of OpenAI, a sieve with varying cofactor, and estimates for division fields and heights. The same method also applies to other CM elliptic curves and points.

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

Oct 17, 2026PDF ↗View work →

hexagon:2610.00292v1

math.AG — Algebraic Geometry

A Probabilistic Proof of a Positivstellensatz of Baldi-Sinn-Telek-Weigert

Contributed by Colin Tan

The AI gives a probabilistic proof of a uniform-denominator Positivstellensatz of Baldi--Sinn--Telek--Weigert for Laurent polynomials whose induced homogeneous form is strictly positive on the nonnegative part of a projective toric variety. Only this abstract was human-written. Three classical tools from mathematical statistics and probability theory were used by the AI: the Barndorff-Nielsen completion of discrete exponential families, sufficiency-based invariance under exponential tilting following Halmos--Savage and Van Campenhout--Cover, and the conditional limit theorem (equivalence of ensembles) of Diaconis--Freedman.

Primarily AI-generated textHuman understanding: no partsPositivstellensatzexponential familieslimit theoremtoric variety