New proofTheoretical computer science2026-10-07-matmul-exponent-2371045
Matrix multiplication exponent below 2.3710449, machine-checked over every commutative ring
GitHub: olliepro/LeanMMBound, the Lean development and paper
LFCR
Frontier
New proofs, counterexamples, bounds and formalizations reported by people and AI systems, newest first.
October 2026 40
New proofPDE and mathematical physicsopenai/math #362
Three-dimensional Vlasov–Maxwell regularity
Proves large-data global existence and uniqueness for the three-dimensional, one-species relativistic Vlasov–Maxwell system. Smooth admissible initial data may be arbitrary provided the particle density is compactly supported and the electromagnetic fields have finite energy and bounded derivatives of every order; the solution remains smooth on every finite time interval.
LFCR
New proofAlgebraopenai/math #287
Free group factor isomorphism
Resolves the free group factor isomorphism problem: , and hence all interpolated free group factors, including , are isomorphic. Their common factor has fundamental group .
LFCR
New proofCombinatoricsopenai/math #180
Barnette Hamiltonicity conjecture
Proves that every finite simple cubic bipartite planar 3-vertex-connected graph has a Hamiltonian cycle, resolving Barnette's conjecture. Equivalently, every three-edge path in a finite simple cubic 3-vertex-connected bipartite Pfaffian graph lies in a Hamiltonian cycle.
LFCR
New proofAnalysisopenai/math #143
Hilbert sixteenth limit-cycle bound
Resolves the uniform boundedness assertion in Hilbert's sixteenth problem: the number of isolated periodic orbits of a real planar polynomial vector field is bounded by a finite constant depending only on its degree. For classical quintic Liénard systems, the exact maximum is two limit cycles.
LFCR
New proofTheoretical computer scienceopenai/math #102
Unique Games conjecture
Proves Khot's Unique Games Conjecture. Independent direct reductions also establish NP-hardness, on unweighted graphs, of approximation beyond the Goemans–Williamson ratio for Max-Cut, below factor two for Vertex Cover, and within any fixed constant factor for Min-UnCut and directed feedback vertex set. These direct proofs use established PCP and Label Cover hardness results.
LFCR
New proofGeometry and topologyopenai/math #074
Kakeya maximal/set assertions
Resolves the Kakeya maximal conjecture in three dimensions and the Hausdorff-dimension conjecture in four. In three dimensions, the radius-δ tube maximal operator maps to with norm for every ε > 0. In four dimensions, every set containing a unit segment in every direction has Hausdorff dimension four.
LFCR
New proofGeometry and topologyopenai/math #032
Rational Hodge for CM abelian varieties
Proves the rational Hodge conjecture for every complex CM abelian variety, in every dimension and codimension. Through Milne's theorems, this also gives the Tate conjecture for all abelian varieties over finite fields and the Hodge standard conjecture for abelian varieties in every characteristic.
LFCR
New proofNumber theoryopenai/math #029
Artin primitive-root infinitude
Proves the infinitude assertion in Artin's primitive root conjecture for every integer a that is neither −1 nor a square. For each such base, at least primes in every sufficiently large interval have primitive root a, with .
LFCR
New proofNumber theoryopenai/math #006
Goldfeld twist-rank distribution
Proves Goldfeld's conjecture for quadratic twists of every elliptic curve over ℚ: analytic ranks zero and one each have density 1/2, and the mean analytic rank tends to 1/2. Both statements order signed squarefree twist parameters by absolute value.
LFCR
New proofNumber theoryopenai/math #005
Catalan constant irrationality
Proves that Catalan's constant is irrational.
LFCR
New proofNumber theoryopenai/math #004
Hilbert’s tenth problem over Q
Proves that no algorithm decides whether an integer-coefficient polynomial in an arbitrary number of variables has a rational zero, resolving Hilbert's tenth problem over ℚ negatively.
LFCR
New proofNumber theoryopenai/math #003
Quasi-Riemann hypothesis
Proves that every Dirichlet L-function, including , is zero-free in , resolving the quasi-Riemann hypothesis. The same half-plane is zero-free for every finite-order Hecke L-function over . A companion gives a different proof of the zero-free half-plane .
LFCR
New proofNumber theoryopenai/math #002
BSD in Selmer-corank 0/1 cases
Proves the full Birch–Swinnerton-Dyer leading-term formula for every elliptic curve over ℚ whose full q-power Selmer group has corank zero or one for some prime q, including finiteness of the Tate–Shafarevich group. With result 006, this gives full BSD for a density-one set of quadratic twists of every elliptic curve over ℚ.
LFCR
New proofOther2026-10-06-openai-math-collection
719 manuscripts in 372 result families from an unreleased OpenAI model
GitHub: openai/math, the 722 manuscripts, Lean library and reasoning summaries
LFCR
New proofPDE and mathematical physicsopenai/math #377
Interior regularity for infinity-harmonic functions
Proves uniform interior regularity for bounded infinity-harmonic functions in every dimension d ≥ 3, for a positive exponent depending only on dimension. The gradient's supremum norm and Hölder seminorm on the half unit ball are bounded by a dimension-dependent constant times the oscillation on the unit ball. The exponent is not explicit.
LFCR
New proofPDE and mathematical physicsopenai/math #376
Universal computation in forced Navier–Stokes flows
Constructs viscous incompressible flows starting from rest on a fixed flat three-dimensional domain that perform universal computation under smooth external forcing. A terminating compiler turns a Turing machine and input into a finite program for the force, so a designated particle reaches a fixed region exactly when the machine halts. The viscosity is fixed, positive and computable.
LFCR
New proofPDE and mathematical physicsopenai/math #375
De Giorgi's conjecture in dimension eight
Proves De Giorgi's conjecture at its sharp dimension-eight endpoint: every entire C2 solution of that is strictly increasing in one direction depends on only one linear coordinate. A stronger theorem classifies all stable entire solutions as constant wells or planar transitions, without an energy-growth assumption.
LFCR
New proofPDE and mathematical physicsopenai/math #373
Nonattainment of the three-marginal Coulomb Monge problem
An optimal three-particle Coulomb configuration need not be a deterministic function of the first particle, even for smooth identical spatial densities. The Monge and Kantorovich infima agree, but the Monge infimum is not attained. The same phenomenon occurs for every inverse-power Riesz exponent in each dimension at least two, with a suitable density in each case.
LFCR
New proofPDE and mathematical physicsopenai/math #372
Global uniqueness in smooth isotropic elasticity
Proves that full static boundary displacement-to-traction data determine both smooth real Lamé moduli on every bounded connected smooth domain in ℝ3, provided μ > 0 and on the closure. Neither analyticity, proximity to constant coefficients nor prior knowledge near the boundary is required.
LFCR
New proofPDE and mathematical physicsopenai/math #371
Stable blowup for the defocusing Schrödinger equation
For a sufficiently large odd nonlinearity power, constructs a nonempty open set of initial data in , with k > 8, whose solutions of the scalar defocusing nonlinear Schrödinger equation blow up in finite time. Thus finite-time blowup is stable under Sobolev perturbations in this supercritical regime.
LFCR
New proofPDE and mathematical physicsopenai/math #370
The Lane–Emden and Hénon–Lane–Emden conjectures
Resolves the subcritical Lane–Emden conjecture and its weighted Hénon extension. For n ≥ 2, and real A, B, the system , has no positive entire solution when , with solutions continuous at the origin and classical elsewhere. No symmetry or growth assumption is needed. Known radial existence gives the exact existence criterion for n ≥ 3 and .
LFCR
New proofPDE and mathematical physicsopenai/math #369
The hot spots conjecture for simply connected planar domains
Proves a strict form of Burdzy's simply connected hot spots conjecture. On every smooth bounded simply connected planar domain, each nonzero eigenfunction for the first positive Neumann eigenvalue has no interior critical point, so all global extrema lie on the boundary. Eigenvalue multiplicity is allowed.
LFCR
New proofPDE and mathematical physicsopenai/math #368
The three-dimensional Ball–Evans approximation problem
Resolves the three-dimensional Ball–Evans approximation problem: every homeomorphism between arbitrary bounded domains in ℝ3, for , is a strong limit of smooth diffeomorphisms onto the same target.
LFCR
New proofPDE and mathematical physicsopenai/math #367
The critical dimension for the one-phase Bernoulli problem
Establishes seven as the first dimension admitting a nonflat, one-homogeneous global minimizer of the one-phase Bernoulli energy. Consequently, minimizing free boundaries are smooth through dimension six, and their singular sets have dimension at most in higher dimensions.
LFCR
New proofPDE and mathematical physicsopenai/math #365
Joint metric and connection recovery from one boundary patch
Zero-frequency measurements on any nonempty open boundary patch determine a smooth metric and smooth unitary connection on a trivial Hermitian rank-two bundle over a compact connected manifold of dimension at least three, up to diffeomorphism and gauge fixed on that patch. Inputs and observations use the same patch. In contrast, distinct uniformly positive bounded measurable scalar conductivities on a three-dimensional ball can have identical full-boundary data.
LFCR
New proofPDE and mathematical physicsopenai/math #364
Kinetic limits and fluctuations over the Boltzmann lifespan
Derives the nonlinear Boltzmann equation from three-dimensional grand-canonical Newtonian gases throughout every regular kinetic interval with uniform Gaussian decay. Stable finite-range radial potentials may have attractive wells and a singular repulsive core; initial pair exclusion and spatially summable Gaussian density and gradient bounds are assumed. A companion gives finite-dimensional hard-sphere Gaussian fluctuations, centered at the exact microscopic expectation and governed by the linear fluctuating Boltzmann equation.
LFCR
New proofPDE and mathematical physicsopenai/math #363
Nonuniqueness with local conservation for the hard-sphere Boltzmann equation
Constructs two distinct global entropy solutions of the three-dimensional periodic hard-sphere Boltzmann equation from the same nonnegative initial density, with bounded velocity support and finite mass, energy and absolute entropy. Both are strongly continuous in L1 and satisfy exact local conservation of mass, momentum and kinetic energy.
LFCR
New proofGeometry and topologyopenai/math #360
Weak MTW curvature gives convexity and regular optimal transport
On every closed connected Riemannian manifold of dimension at least two satisfying weak Ma–Trudinger–Wang curvature, all tangent injectivity domains are convex, resolving Villani’s conjecture in this setting. For squared-distance transport between measurable probability densities bounded above and away from zero, the optimal map and its inverse are Hölder continuous.
LFCR
New proofGeometry and topologyopenai/math #358
A three-manifold without conjugate points or nonpositive curvature
Constructs a closed connected orientable smooth three-manifold that admits a metric without conjugate points but no metric of nonpositive sectional curvature. This answers negatively, already in dimension three, whether the first metric-existence property implies the second.
LFCR
New proofGeometry and topologyopenai/math #357
Bi-Lipschitz coordinates at every regular RCD point
Proves that every regular point of a noncollapsed space, for K ∈ ℝ and integer n ≥ 2, has an open neighborhood bi-Lipschitz to an open subset of ℝn. Regularity requires all pointed tangents to be Euclidean, the reference measure is exactly , and the chart compares ambient distances with a point-dependent finite constant.
LFCR
New proofGeometry and topologyopenai/math #356
Gigli’s characterization of Alexandrov curvature
Proves Gigli's conjecture: in every integer dimension n ≥ 2, Alexandrov curvature at least κ is characterized by the full-support condition with reference measure and distributional sectional curvature at least κ in the original global test classes. The RCD condition is unreduced.
LFCR
New proofGeometry and topologyopenai/math #355
Unique tangent flows at the first surface singularity
Proves the first-singular-time case of tangent-flow uniqueness for smooth compact connected embedded surfaces without boundary in ℝ3. At every singular point, all fixed-center backward tangent flows agree as area measures at every negative time in the original ambient coordinates, without mean-convexity or a prescribed tangent model.
LFCR
New proofGeometry and topologyopenai/math #354
The isoperimetric profile of the cubic three-torus
Determines the isoperimetric profile of the unit cubic flat three-torus and classifies every finite-perimeter minimizer: balls, circular tubes around shortest closed geodesics, coordinate slabs, and their complements. The transition volumes are and , with exactly the adjacent two types minimizing at each transition.
LFCR
New proofGeometry and topologyopenai/math #349
The Solomon–Yau least-volume conjecture
Proves the Solomon–Yau least-volume conjecture for minimal hypersurfaces of round spheres. For every m ≥ 2, a closed connected minimal immersion into the unit sphere with non-totally-geodesic image has volume at least that of the smallest minimal Clifford product, counting covering multiplicity.
LFCR
New proofGeometry and topologyopenai/math #348
Nonnegative-curvature Einstein classification and an L2 topological gap
Classifies closed connected Einstein four-manifolds with positive Einstein constant and nonnegative sectional curvature: up to scaling, their universal Riemannian covers are the round S4, Fubini–Study , or a product of equal round two-spheres. A closed simply connected nonnegatively curved four-manifold is diffeomorphic to one of these whenever the scale-invariant L2 norm of its trace-free Ricci curvature lies below a universal positive constant.
LFCR
New proofGeometry and topologyopenai/math #345
Infinitely many closed geodesics on Riemannian spheres and closed three-manifolds
Proves that every smooth Riemannian metric on Sn, n ≥ 2, has infinitely many prime closed geodesics with pairwise distinct images. The same conclusion holds on every closed manifold admitting a finite smooth spherical cover and on every closed three-manifold, without orientability or nondegeneracy restrictions.
LFCR
New proofGeometry and topologyopenai/math #344
The metric Blaschke conjecture
Proves the metric Blaschke conjecture: every closed connected Riemannian manifold of positive dimension whose injectivity radius equals its diameter is, up to scaling, a standard compact rank-one symmetric space.
LFCR
New proofGeometry and topologyopenai/math #343
Symplectic ball packing in higher dimensions
Resolves the Siegel–Yao conjecture for arbitrary capacities in every dimension . Finitely many closed symplectic balls of capacities embed disjointly into an open ball of capacity R exactly when and for every distinct pair i, j.
LFCR
New proofGeometry and topologyopenai/math #342
Donaldson's tamed-to-compatible conjecture
Proves Donaldson's tamed-to-compatible conjecture: every smooth almost complex structure on a closed four-manifold that is tamed by a symplectic form admits a compatible symplectic form. The almost complex structure stays fixed; the form's cohomology class may change.
LFCR