Archive

The Deligne-Drinfeld conjecture

OpenAI

• Mathematics

OpenAI reports that the rational Grothendieck–Teichmüller Lie algebra is free on one generator in each odd weight starting at three, with the corresponding weight-completed isomorphism.

The additive indecomposability of the primes

OpenAI

• Mathematics

OpenAI reports that no set differing from the positive primes at finitely many integers is a sumset A+B of two nonnegative-integer sets that each contain at least two elements.

The Weak Inhomogeneous Duffin–Schaeffer Conjecture

OpenAI

• Mathematics

OpenAI reports that divergence of the totient-weighted approximation sum gives infinitely many successful shifted approximations for almost every real number, for each fixed real shift and arbitrary finite-valued tolerances.

Primitive roots for every admissible integer base

OpenAI

• Mathematics

OpenAI reports at least c_a x/(log x)² primes in each sufficiently large interval (x,2x) for which a fixed integer a is a primitive root, whenever a is neither minus one nor a square.

Modularity of elliptic curves over imaginary quadratic fields

OpenAI

• Mathematics

OpenAI reports modularity of every elliptic curve over every imaginary quadratic number field, with an automorphic representation matching the local parameters at all places.

Fujita's freeness conjecture

OpenAI

• Mathematics

OpenAI reports that K_X+mL is globally generated on every smooth complex projective n-dimensional variety for every ample line bundle L and every integer m at least n+1.

Nagata's conjecture for plane curves

OpenAI

• Mathematics

OpenAI reports that, at r at least ten very general complex plane points, the total multiplicity of every nonzero effective degree-d curve is strictly less than d times the square root of r.

Brennan's conjecture and sharp inverse-square integral means

OpenAI

• Mathematics

OpenAI reports area-integrability of the s-th power of the derivative of every conformal bijection from a simply connected planar domain to the unit disk whenever 4/3<s<4, together with the stated inverse-square integral-means exponent.

The Falconer distance conjecture in all dimensions

OpenAI

• Mathematics

OpenAI reports that every compact subset of Euclidean d-space, d at least two, with Hausdorff dimension strictly greater than d/2 has a distance set of positive Lebesgue measure.

The Kakeya maximal conjecture in three dimensions

OpenAI

• Mathematics

OpenAI reports the three-dimensional Kakeya maximal L³ estimate with tube-radius loss at most C_ε δ^(-ε) for every positive ε, uniformly over unit tubes of radius δ.

The symmetric Mahler conjecture and its equality cases

OpenAI

• Mathematics

OpenAI reports that every origin-symmetric convex body in n-dimensional real space has volume product at least 4^n/n!, with equality precisely for invertible linear images of Hanner polytopes.

The Gaussian propeller bound in every dimension

OpenAI

• Mathematics

OpenAI reports that the sum of squared Gaussian first-moment lengths of the cells of every finite measurable Euclidean partition is at most 9/(8π), with the three-sector planar construction attaining the bound in the stated dimensions.

The Unique Games Theorem

OpenAI

• Mathematics
• Theoretical Computer Science

OpenAI reports a deterministic polynomial-time reduction from 3SAT to fixed-alphabet Unique Games with completeness at least 1−ε and soundness at most δ for each fixed ε,δ in (0,1/2).

Exact derandomization of logarithmic space: L = RL = BPL

OpenAI

• Mathematics
• Theoretical Computer Science

OpenAI reports a uniform deterministic logarithmic-space simulation of polynomial-time randomized logarithmic-space decision procedures with bounded error, yielding L=RL=BPL.

Squared-logarithmic randomized k-server on arbitrary metrics

OpenAI

• Mathematics
• Theoretical Computer Science

OpenAI reports randomized k-server competitive ratio O(log²(k+1)) on every metric against oblivious requests, matching the stated worst-case lower-bound order; distinct initial server positions need no additive term.

Deterministic Polynomial Factorization over Prime Fields

OpenAI

• Mathematics
• Theoretical Computer Science

OpenAI reports uniform deterministic complete factorization of a densely represented polynomial over a binary-specified prime field in bit complexity polynomial in (degree+1) log p, including irreducible factors and multiplicities.

Quasipolynomial Bounds for Arithmetic Progressions

OpenAI

• Mathematics

OpenAI reports that every positive-integer set with divergent reciprocal sum contains arithmetic progressions of every finite length, through fixed-length bounds r_k(N)≤C_k N exp(−c_k(log N)^ε_k).

A counterexample to Sidorenko's conjecture

OpenAI

• Mathematics

OpenAI reports a bipartite graph with 35 vertices and 66 edges whose homomorphism density in a finite simple host is below the predicted edge-density bound; it also reports failure of the associated forcing assertion.

A proof of Seymour’s second-neighborhood conjecture

OpenAI

• Mathematics

OpenAI reports that every nonempty finite oriented graph has a vertex with at least as many vertices at directed distance exactly two as at directed distance one.

A Torsion-Free Group Algebra with Zero Divisors

OpenAI

• Mathematics

OpenAI reports a finitely presented torsion-free group with a finite two-dimensional classifying space whose group algebra over the two-element field contains nonzero elements with product zero.

A Torsion-Free Group Algebra That Is Not Directly Finite

OpenAI

• Mathematics

OpenAI reports a finitely presented torsion-free group with a finite two-dimensional classifying complex and elements a,b in its characteristic-two group algebra satisfying ab=1 but ba≠1.

An algebra of infinite little finitistic dimension

OpenAI

• Mathematics

OpenAI reports one fixed finite-dimensional complex algebra with finite-dimensional modules of unbounded finite projective dimension, refuting universal finiteness of little finitistic dimension.

The low-temperature Sherrington–Kirkpatrick free-energy limiting law

OpenAI

• Mathematics

OpenAI reports a nondegenerate limiting law for centered and variance-normalized total free energy in the zero-field Gaussian Ising Sherrington–Kirkpatrick model at each fixed β>1, with variance asymptotic to a positive constant times n^(1/3).

Rigidity of the Turing degrees

OpenAI

• Mathematics
• Theoretical Computer Science

OpenAI reports that every order automorphism of the full Turing-degree partial order is the identity in ZFC, without a definability or regularity restriction on the automorphism.

The Partition Principle does not imply Choice

OpenAI

• Mathematics
• Theoretical Computer Science

OpenAI reports that consistency of ZF implies consistency of ZF plus the Partition Principle and Choice for ordinal-indexed families but without the full Axiom of Choice.

An infinite finitely presented residually finite 2-group

OpenAI

• Mathematics

OpenAI reports an infinite discrete group with an ordinary finite presentation that is residually finite and has finite two-power order for every element, with unbounded element orders.

Thompson's group F is nonamenable

OpenAI

• Mathematics

OpenAI reports that Thompson’s group F, viewed as a discrete group, is nonamenable.

Absolutely Continuous Spectrum for Weak-Disorder Anderson Models in Dimensions at Least Three

OpenAI

• Mathematics

OpenAI reports purely absolutely continuous spectrum of nonzero weight on an open energy interval for the lattice Anderson model with independent uniform site potentials at each fixed dimension d≥3 and sufficiently small positive disorder.

The maximum number of mutually unbiased bases in dimension six

OpenAI

• Mathematics

OpenAI reports that the maximum number of mutually unbiased orthonormal bases in complex dimension six is three, using an exhaustive computer-assisted exclusion of four bases under specified arithmetic and compiler conditions.

The periodic spin-one Haldane gap

OpenAI

• Mathematics

OpenAI reports a uniform positive spectral gap above the ground energy for the pure antiferromagnetic spin-one Heisenberg chain on even periodic rings.

Kadison's similarity theorem through uniform derivation estimates

OpenAI

• Mathematics

OpenAI reports that every bounded complex-linear unital homomorphism from a unital complex C*-algebra to operators on a Hilbert space is conjugate by a bounded invertible operator to a *-homomorphism.

The Singer conjecture in dimension four

OpenAI

• Mathematics

OpenAI reports vanishing of universal-cover L²-Betti numbers outside degree two for closed connected aspherical topological four-manifolds, and more generally finite aspherical integral Poincaré complexes of formal dimension four.

Generalized Cartan–Hadamard isoperimetry and Euclidean equality rigidity

OpenAI

• Mathematics

OpenAI reports the generalized Cartan–Hadamard isoperimetric inequality in every dimension for complete simply connected smooth manifolds with sectional curvature at most κ≤0, comparing finite-volume finite-perimeter sets with equal-volume model balls.

Uniformization of complete Kähler manifolds with positive bisectional curvature

OpenAI

• Mathematics

OpenAI reports that every complete connected noncompact Kähler manifold with strictly positive holomorphic bisectional curvature is biholomorphic to complex Euclidean space.

Strict hot spots and absence of interior critical points on smooth simply connected planar domains

OpenAI

• Mathematics

OpenAI reports that every nonzero first-positive Neumann eigenfunction on a smooth bounded simply connected planar domain has nonvanishing interior gradient, placing all global extrema on the boundary even for a multiple eigenvalue.

Truly Subquadratic 3SUM and Truly Subcubic APSP via Triangles in Sparse Lopsided Graphs

Josh Alman, Virginia Vassilevska Williams

• Mathematics
• Theoretical Computer Science

Claude’s original sparse-output thin-matrix multiplication algorithm, as presented in Section 2 and related to the certified 3SUM/APSP theorems. Human-optimized numerical parameters, new reductions, data structures and Hinted OMv extensions retain human credit.

An abelian envelope without the quotient property

Johannes Flake, Jonathan Gruber, Thorsten Heidersdorf

• Mathematics

Flake, Gruber and Heidersdorf report that the universal rigid monoidal category on one object has an abelian envelope without the quotient property.

Medvedev logic is undecidable

Rodrigo Nicolau Almeida, Søren Brinck Knudstorp

• Mathematics
• Theoretical Computer Science

Almeida and Knudstorp report that Medvedev’s logic of finite problems is undecidable, by reducing periodic plane tiling to non-theoremhood.

The k-server conjecture is true

Christian Coester, Elias Koutsoupias, Marek Zbysiński

• Mathematics
• Theoretical Computer Science

Coester, Koutsoupias and Zbysiński report that the work function algorithm is k-competitive on every metric space.

Scoped AI contribution: optimal packing proofs for one to seven squares

• Mathematics
• Theoretical Computer Science

The model-origin optimality arguments for five through seven freely rotated unit squares in a disk, separating known small cases and later human commentary.

AI-assisted six-task pinwheel scheduling counterexample

• Mathematics
• Theoretical Computer Science

The artifact reports (3,4,5,20,22,36) schedulable but (3,4,5,20,22,32) unschedulable, refuting the proposed six-task kernel cap.

An exponential lower bound for the bit pigeonhole principle in resolution over parities

Kamil Braun

• Mathematics
• Theoretical Computer Science

The model-extracted generic polynomial-calculus/Res(parity) sufficient criterion in Theorem 9.1, rather than attributing the complete bit-PHP lower-bound architecture to AI.

Existence of the Core in Approval-Based Committee Elections

Patrick Becker, Matthias Greger, Dominik Peters

• Mathematics
• Theoretical Computer Science

Becker, Greger and Peters report that every approval-based multiwinner election has a core committee and that one can be found in polynomial time.

Closing the Oracle-Complexity Gap in Derivative-Free Convex Optimization: A Near-Quadratic Lower Bound from Exact Function Values

Phillip Kerger

• Mathematics
• Theoretical Computer Science

Kerger reports Ω(d²/log(d+1)) exact-function-value queries at accuracy Θ(d^−1/2) for deterministic convex Lipschitz minimization over a d-dimensional ball.

Remarks on the disproof of the unit distance conjecture

Noga Alon, Thomas F. Bloom, W. T. Gowers, Daniel Litt, Will Sawin, Arul Shankar, Jacob Tsimerman, Victor Wang, Melanie Matchett Wood

• Mathematics

The human-verified writeup gives a fixed ε>0 and growing planar point sets with at least n^(1+ε) unit distances.

Improving the matrix multiplication exponent with modern optimization and AlphaEvolve

Emilien Dupont, Marvin Eisenberger, Borislav Kozlovskii, Abbas Mehrabian, Francisco J. R. Ruiz, Abigail See, Renfei Zhou, Josh Alman, Virginia Vassilevska Williams, Matej Balog

• Mathematics
• Theoretical Computer Science

Dupont and collaborators report ω<2.371177 by improving the combination-loss optimization and refining it with AlphaEvolve.

Global classical solutions of the three-dimensional relativistic Vlasov–Maxwell system

OpenAI

• Mathematics
• Theoretical Physics

OpenAI claims globally smooth unique evolution for arbitrary admissible large initial data in the three-dimensional one-species relativistic Vlasov–Maxwell model, under its support and field assumptions.

Smooth isometric immersions of closed surfaces into Euclidean four-space

OpenAI

• Mathematics

OpenAI claims that every compact boundaryless smooth Riemannian surface has a smooth length-preserving immersion into R⁴, including nonorientable surfaces.

A direct proof of the complete Crouzeix inequality

OpenAI

• Mathematics

OpenAI reports the sharp constant-two numerical-range inequality for matrix-valued polynomials, uniformly over all finite base and coefficient dimensions.

The Hilbert–Smith conjecture in every finite dimension

OpenAI

• Mathematics

OpenAI claims Lie-group structure for any locally compact second-countable Hausdorff group with a faithful jointly continuous action on a connected finite-dimensional manifold under the stated topological conventions.