Archive

Integer multiplication below n log n

OpenAI

• Mathematics
• Theoretical Computer Science

The manuscript reports deterministic multiplication in O(n (log n)^(1−κ)) time with κ = 2^−182 on a fixed finite-tape Turing machine.

An Upper Bound of 9/4 for the Matrix Multiplication Exponent

OpenAI

• Mathematics
• Theoretical Computer Science

The manuscript reports that the complex matrix-multiplication exponent is at most 9/4.

Every t-perfect graph is 50-colorable

Sang-il Oum

• Mathematics

Every finite t-perfect graph is 50-colorable. T-perfection means that its stable-set polytope is described by nonnegativity, edge and odd-cycle inequalities.

Proof of the Erdős-Szemerédi sunflower conjecture

Matija Bucić, Fankang He, Hyunwoo Lee

• Mathematics

For integers k≥3 and D,n≥1, a k-sunflower-free family in [D]^n has size at most ((2k)^(2/(k−1)) D^(1−1/(k−1)))^n. Here a vector sunflower has entries all equal or pairwise distinct in every coordinate. The authors’ reduction gives the set-family bound 3(2−1/(16k³))^n and resolves the Erdős–Szemerédi conjecture for every k≥3.

The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane Re(s) > 7/8

OpenAI

• Mathematics

OpenAI’s manuscript reports that all Dirichlet L-functions and finite-order Hecke L-functions over Q(√−3) have no zeros with real part greater than 7/8.

A gauge-invariant measure for lattice chiral fermions

Nathaniel Craig

• Theoretical Physics

For SU(5) overlap fermions in the representation 10⊕5̄ on finite periodic four-dimensional Euclidean lattices, the paper constructs a smooth exponentially local measure current satisfying the anomaly equation and full integrability on admissible gauge fields. It reconstructs a gauge-invariant measure on every nonempty component, up to a constant phase, and restricts it to the four stated global forms of the Standard Model gauge group.

Exact Storage-Repair-Bandwidth Tradeoffs for (6,5,5) and (7,6,6) Exact-Repair Regenerating Codes

Chao Tian

• Theoretical Computer Science

The manuscript gives converse bounds that completely characterize the storage/repair-bandwidth tradeoffs of exact-repair regenerating codes for (n,k,d)=(6,5,5) and (7,6,6), without assuming linear codes. Previously known layered codes with space sharing attain the bounds.

Pointwise convergence of ergodic averages for two commuting transformations

• Mathematics

For two commuting measure-preserving maps T and S of a probability space, the bilinear averages N^-1 ∑_{n<N} f(T^n x)g(S^n x) converge almost everywhere when f∈Lp, g∈Lq, p,q∈[1,∞] and 1/p+1/q≤1. No mixing or invertibility assumption is imposed.

Single-minus graviton tree amplitudes are nonzero

Alfredo Guevara, Alexandru Lupsasca, David Skinner, Andrew Strominger, Kevin Weil (on behalf of OpenAI)

• Theoretical Physics

The paper derives nonzero single-minus n-graviton tree amplitudes supported on half-collinear configurations in Klein signature or for complexified momenta. A Berends–Giele recursion has a sum-over-trees solution; in a restricted decay chamber it reduces to a product of soft factors. The Ward-identity reconstruction additionally assumes the stated analyticity conditions.

Global Hölder estimates for linearized Monge–Ampère equations in divergence form with applications to dual semigeostrophic equations and periodic homogenization

Guoqing Cui, Chong Gu, Nam Q. Le, Ling Wang, Bin Zhou

• Mathematics

For n≥3, Proposition 1.2 bounds the spatial L¹ norm of the gradient of the Dirichlet Green’s function for the linearized Monge–Ampère operator by C h^β, uniformly in its pole on a section of height h. The convex C³ potential has Hessian determinant between positive constants; the nested sections satisfy the paper’s compact-containment hypotheses.

Formally Verified Autoresearch for Analyzing the Expressivity of Multi-Head Attention

Karthik Viswanathan, Plamen Dimitrov, Stefano Rocca, Sarayu Sirikonda

• Theoretical Computer Science

For the paper’s one-layer, attention-only transformer on Boolean inputs with a dedicated query token and linear threshold readout, the autonomous core results characterize head complexity by sums of admissible linear-fractional functions, bound it below by threshold degree, and determine it exactly for symmetric Boolean functions by the number of sign changes. In particular, n-bit parity requires exactly n heads.

The normal order of the number of iterations of Euler’s function

Plasma AI

• Mathematics

Let f(n) count the iterations of Euler’s totient function needed to reach 1. The manuscript reports a constant α between 1/log 3 and 1/log 2 such that f(n)/log n tends to α on a set of natural density one. It also gives a quantitative mean-square bound, answering the first two parts of Erdős Problem 408.

At the Limits of Computability: Steps Towards Autonomous AI Discovery on the Busy Beaver Frontier

Marco Dos Santos, Vivek Veeriah, Alex Havrilla, Andrew Bolt, Goran Zuzic, Satinder Singh, Tom Zahavy

• Theoretical Computer Science

A Gemini agent pipeline discovered and formally proved closed systems of Collatz-like acceleration rules for eleven specified six-state, two-symbol Turing machines. Each system summarizes exact execution from its fixed blank-tape starting configuration; the machines’ eventual halting behavior remains unresolved.

Revisiting Separators in String Graphs, Region Intersection Graphs, and Induced-Minor-Free Graphs

James Davies, Paweł Rzążewski, Jack Spalding-Jamieson, Bartosz Walczak

• Mathematics
• Theoretical Computer Science

The AI-origin improvement in Appendix B gives non-expansive line embeddings with O(log h) average distortion for the stated vertex-weighted metrics of connected graphs excluding the one-subdivision of K_h as an induced minor. The paper derives balanced separators of size O(h log h √m) for the corresponding m-edge graphs.

Persistent failure of Euclidean quartic SOS exactness

• Mathematics
• Theoretical Computer Science

An explicit fixed rational cubic in 14 variables gives a quartically regularized polynomial whose exact minimum shift is nonnegative but not a sum of polynomial squares for every positive regularization parameter. A second explicit cubic has a unique global minimizer with positive-definite Hessian there and retains a strict SOS gap for every positive parameter.

Erdős Problem 111: bipartizing finite subgraphs of uncountably chromatic graphs

• Mathematics

If a graph is not countably colorable, h_G(n)/n tends to infinity, where h_G(n) is the largest minimum number of edge deletions needed to make an n-vertex subgraph bipartite. Equivalently, a uniform linear deletion bound on all finite subgraphs forces countable colorability. The limit holds for the whole sequence, for graphs of any uncountable chromatic cardinality.

Every two-coloring of the plane misses at most one triangle, which is equilateral

• Mathematics

In every arbitrary two-coloring of the Euclidean plane, each non-equilateral three-point set, including three distinct collinear points, has a monochromatic congruent copy. At most one equilateral side length can be missed, and an alternating-strip coloring shows that one exception can occur.

A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Characteristic Two

OpenAI

• Mathematics

There are a finite field K of characteristic two, a finitely presented group G containing odd-prime torsion, and scalar a,b in K[G] with ab=1 but ba≠1. The manuscript gives a terminating prescription for the witness; it does not assert K=F₂.

Ten Advances in Mathematics and Theoretical Computer Science

OpenAI

• Mathematics
• Theoretical Computer Science

For every finite two-player entangled game G with entangled value below one, the value of winning all n independent repetitions decays exponentially in n; the Chapter 6 bound depends on the gap and answer alphabet sizes.

The Selmer converse for elliptic curves at every prime

OpenAI

• Mathematics

For every elliptic curve A/Q, every prime p and full p-power Selmer corank r in {0,1}, the manuscript claims analytic rank and Mordell–Weil rank r and finiteness of the whole Tate–Shafarevich group, without reduction, CM, residual-representation, torsion or isogeny restrictions.

Lean-checked candidate against the φ-mixing CLT

• Mathematics

GPT-6 Astra constructed a strictly stationary, centered φ-mixing sequence with finite second moments and divergent partial-sum variance. Along a subsequence, its variance-normalized sums converge to zero in probability rather than to a standard normal law.

On a conjecture on the Kasami APN function: reductions, structure theorems, a proof for k mod n ∈ {1, 2, n−2, n−1}, and exhaustive verification for n ≤ 13

Gábor P. Nagy, Attila Vajda

• Mathematics

For the Kasami function F(x)=x^(4^k−2^k+1) over F_(2^n), with gcd(k,n)=1, Nagy and Vajda prove the conjectured ordered triple count 2^(2n−3) for all distinct nonzero coefficients when k mod n is 1, 2, n−2 or n−1. They also check admissible n≤13 by exhaustive computation.

AI-found counterexample to Köthe’s matrix conjecture

• Mathematics

GPT-6 Astra constructed a nil algebra over the algebraic closure of F₂ with a nonnilpotent 2×2 matrix. In its unitization, the algebra is a nil two-sided ideal, refuting Krempa’s matrix form of Köthe’s conjecture.

AI-led proof gives a sharp global squared-law transport bound

• Mathematics
• Theoretical Computer Science

For any real law with X² in [A,B], A=3−√6 and B=3+√6, let δ=3(E[X²])²−E[X⁴] and let X₀ be the four-node Gaussian quadrature law identified by zero deficit. For every feasible 0≤δ≤30+12√6, W1(L(X²),L(X₀²))≤√(δ/3), using ordinary infimum-over-couplings Wasserstein distance. The bound is attained at every feasible deficit and does not require symmetry of X.

AI-led proof gives an optimal eighth-order MGF gap

• Mathematics
• Theoretical Computer Science

For every symmetric law with X² in [A,B], A=3−√6 and B=3+√6, set v=E[X²] and h=E[(X²−A)(B−X²)]. For every real t, exp(vt²/2)−E exp(tX)≥h t⁴/24+t⁸/25200. Both coefficients are optimal for this two-term bound. A positive coefficient on the intrinsic fourth deficit 3v²−E[X⁴] cannot be paired with the optimal eighth-order coefficient.

The exact reconstruction threshold for the three-state symmetric channel

OpenAI

• Mathematics

For the three-state symmetric channel Kλ(i,i)=(1+2λ)/3 and Kλ(i,j)=(1−λ)/3 for i≠j, with −1/2≤λ≤1, reconstruction from depth-ℓ spins and the observed tree on every regular b-ary tree (integer b≥2) and observed Poisson(d) tree (d>1) occurs exactly when dλ²>1 (d=b in the regular case). At admissible equality there is nonreconstruction. Poisson advantage averages over the tree without conditioning on survival.

Bi-Lipschitz Absorption of c0 Without a Linear Copy of c0

OpenAI

• Mathematics
• Theoretical Computer Science

A separable real Banach space Z has no closed linear copy of c₀ yet is globally bi-Lipschitz equivalent to Z×c₀. The construction also yields bi-Lipschitz universality for separable metric spaces.

An example of a very non-movable effective divisor

Jihao Liu

• Mathematics

Liu reports effective divisors D_n on one fixed smooth projective surface with D_n²=4 and h⁰(mD_n)=1 for every 1≤m<n²/4+1, for arbitrarily large odd n. Thus no uniform movable multiple depending only on that surface exists.

An Erdős–Kac law for palindromes and for reversed primes

• Mathematics
• Theoretical Computer Science

The archived preprint proves Erdős–Kac normal laws for the number of prime factors, with or without multiplicity, of base-b palindromes and reversed base-b primes for every b≥2. The reversed-prime theorem also allows a fixed leading digit.

Simple F∞ overgroups of groups with decidable word problem

OpenAI

• Mathematics
• Theoretical Computer Science

Every finitely generated group with decidable word problem embeds injectively in a nontrivial simple group of type F∞, meaning it has a classifying CW complex with finitely many cells in each dimension.

A counterexample to the Foregger-Sinkhorn tie-point conjecture

Yair Lavi

• Mathematics
• Theoretical Computer Science

An explicit 8×8 nearly decomposable support has a unique global permanent minimizer on its full doubly stochastic face. At a specified zero, the permanental cofactor exceeds the permanent although that zero is not a tie point, refuting the Foregger–Sinkhorn implication.

The Proper Hat-Guessing Number of K₇−e

Matthew Protti

• Mathematics

The proper hat-guessing number of K7−e is 12. Either two disjoint S(5,6,12) completion designs or a sharply five-transitive orbit-map construction meets the twin-completion criterion; the general upper bound gives equality.

Positivity in classical enumerative geometry: a case study in synchronized AI-assisted mathematics

Gergely Bérczi, László M. Fehér

• Mathematics

Bérczi and Fehér report that, for every k, the elementary-basis coefficients of the kth Chern class of Sym^d(C^n) are universal rational polynomials in T_r(n,d)=binom(d+n−1,n+r−1), for 1≤r≤k.

The Proper Hat-Guessing Number of K₅−e

Matthew Protti

• Mathematics

The proper hat-guessing number of K5−e is 8. An F₂³ construction and Hall-completion argument give a winning eight-color strategy, matching the prior general upper bound.

On the largest sum-free subset of the lattice cube

Peter Keevash, Jeck Lim

• Mathematics

For every d≥1 and nonnegative r,s,t with r+s+t=dn, the uniform distributions on the lattice-cube slices L_d(r), L_d(s), and L_d(t) are jointly mixable. Keevash and Lim use this theorem to obtain the optimal limiting sum-free density of [n+1]^d and an O(n^(d−1)) upper error.

The Proper Hat-Guessing Number of K₆−e

Matthew Protti

• Mathematics

The proper hat-guessing number of K6−e is 10. Two order-sensitive twin rules from a sharply four-transitive action on 11 points satisfy the coordinate-line criterion; Hall completion gives a winning ten-color strategy, matching the general upper bound.

Short proofs in combinatorics, probability and number theory II

Boris Alexeev, Moe Putterman, Mehtaab Sawhney, Mark Sellke, Gregory Valiant

• Mathematics
• Theoretical Computer Science

For every m≥1, an explicit K₄-free graph on 20m+31 vertices is 4-chromatic and vertex-critical, while every cycle has at most ten chords. This refutes the proposed unbounded forced chord count under the stated local 3-colorability condition.

Almost-everywhere regularity of stationary integral varifolds

OpenAI

• Mathematics

For every stationary integral m-varifold in an open subset of R^(m+n), with m,n≥1, the m-dimensional Hausdorff measure is zero for points lacking a neighborhood where the varifold is a positive integer multiple of one smooth embedded minimal m-submanifold.

A Remark on the Odd Area of Unit Disks

Stefan Steinerberger

• Mathematics
• Theoretical Computer Science

AlphaEvolve found 51- and 151-unit-disk configurations whose odd-covered areas are estimated by Monte Carlo calculations at about 2.73 and 2.51, respectively.

Short proofs in combinatorics, probability and number theory II

Boris Alexeev, Moe Putterman, Mehtaab Sawhney, Mark Sellke, Gregory Valiant

• Mathematics

There is no universal Erdős–Turán angular-discrepancy bound of order √(ν(f) log M(f)) with polynomial degree replaced by the number ν(f) of nonzero terms. For each N, a polynomial with ν=N+2 and M<3 has a positive real zero of multiplicity N+1, giving discrepancy at least N+1/2 on the stated short arc.

A Limiting Satisfiability Threshold for Every Fixed Clause Size

OpenAI

• Mathematics
• Theoretical Computer Science

For every fixed k≥3, random k-SAT with independent uniformly signed proper clauses sampled with replacement has a positive finite limiting density αₖ: satisfiability probability tends to one below αₖ and zero above it. The claim makes no assertion at αₖ.

Hyperbolicity Cones Without Semidefinite Lifts

OpenAI

• Mathematics
• Theoretical Computer Science

An OpenAI manuscript claims a real homogeneous hyperbolic polynomial whose closed cone has no exact finite affine semidefinite lift, even with arbitrary real coefficients and any finite number of auxiliary variables.

Erdős-Sós for digraphs

Dhruv Mubayi, Jacques Verstraete

• Mathematics
• Theoretical Computer Science

Every finite loopless Eulerian digraph with no repeated ordered arc, including digraphs with opposite arcs, contains every orientation of every t-edge tree when it has n vertices and more than (t−1)n arcs. Complete bidirected graph unions show the bound is sharp.

Exact mass asymptotics for the two-dimensional O(4) lattice model

OpenAI

• Mathematics
• Theoretical Physics

For the nearest-neighbor square-lattice O(4) model, the full Osterwalder–Schrader transfer gap, including rotation-invariant local-observable sectors and measured per original lattice time step, is asymptotic to 32 exp(π/4−1/2) √β exp(−πβ) as β tends to infinity.

Aperiodic maximisers of Lyapunov dimension: A counterexample to the unrestricted form of Eden's conjecture

Matthew Colbrook

• Mathematics

For each c>4, Colbrook’s polynomial ODE on ℝ⁵ has uniformly negative divergence and a compact global attractor. Its finite-time local Lyapunov and asymptotic Kaplan–Yorke dimension maxima are 4+4/c on an irrational-flow invariant two-torus, while equilibrium and periodic-point maxima are 3.

Prescribed large-volume charges on threefolds with trivial canonical bundle

OpenAI

• Mathematics

For each smooth projective complex threefold X, ample integral H and rational B₀, the manuscript gives k≥0 and R*>0 depending only on (X,H,B₀) such that every finite-slope, zero-tilt-slope semistable E satisfies ch₃^(B₀+bH)(E)≤(a²/6+k)H²ch₁^(B₀+bH)(E) for all a>0 and real b. The k correction vanishes uniformly in b for a>R*.

A Counterexample to Talagrand's Operator Cotype Problem

Xinglong Wu

• Mathematics

For sufficiently large n=2ᵏ, Wu’s Walsh–Hadamard norm and identity map Uₙ:Xₙ→ℓ∞ⁿ make C₂ʳ(Uₙ)/max(C₂ᵍ(Uₙ),‖Uₙ‖(2,1)) grow at least as c√log(n+1). Thus the proposed universal operator-cotype bound fails already at q=2.

Uniqueness of four-body convex central configurations for all masses

Tejasvi Singh Tomar

• Mathematics

For four positive masses in the planar Newtonian problem, each cyclic order has exactly one strictly convex central configuration up to similarity. The v2 theorem also gives Q≥K/4 at every convex configuration, nondegenerate local minima and analytic dependence.

The a = 3 slice of two permutational Ramsey conjectures: R_dih(P3alt, K_b) = R_cyc(P3alt, K_b) = 2b − 1 for all b

• Mathematics

For every integer b≥1, the dihedral and cyclic Ramsey numbers of the alternating 3-path versus K_b both equal 2b−1.

A Smooth Counterexample to the Trautman Conjecture

Sean N. Curry

• Mathematics

Curry constructs a nonnegative smooth, arbitrarily C²-small perturbation of a strongly pseudoconvex CR 3-structure with a nowhere-zero closed canonical-bundle section, but every local C¹ CR function has zero differential at the origin. It is therefore not locally embeddable.