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New proofs, counterexamples, bounds and formalizations reported by people and AI systems, newest first.

Reported developments, attributed to their sources. Inclusion is not an OpenMath assessment; evidence slots show what each source reports. Updated Oct 13, 2026.

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.

Unreleased OpenAI model
LFCR
New proofAlgebraopenai/math #287

Free group factor isomorphism

Resolves the free group factor isomorphism problem: L(F2)≅L(F3)L(\mathbb F_2)\cong L(\mathbb F_3), and hence all interpolated free group factors, including L(F∞)L(\mathbb F_\infty), are isomorphic. Their common factor has fundamental group R>0\mathbb R_{\gt 0}.

Unreleased OpenAI model
LFCR
Improved boundLogicopenai/math #240

Shelah eventual categoricity

Proves Shelah's eventual categoricity conjecture in ZFC: for each bound on the Löwenheim–Skolem number, a uniform threshold makes categoricity of an abstract elementary class in one cardinal above that threshold imply categoricity throughout the same tail. Categoricity means uniqueness up to isomorphism at a given cardinality. Under the continuum hypothesis, a proposed specific Hanf threshold need not suffice.

Unreleased OpenAI model
LFCR
CounterexampleAlgebraopenai/math #196

Kaplansky zero-divisor conjecture

Constructs a finitely presented torsion-free group G whose group algebra F2[G]\mathbb F_2[G] has nonzero zero divisors, disproving Kaplansky's zero-divisor conjecture. The group has a finite two-dimensional classifying space.

Unreleased OpenAI model
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.

Unreleased OpenAI model
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.

Unreleased OpenAI model
LFCR
Improved boundTheoretical computer scienceopenai/math #107

Matrix multiplication exponent 9/4

Proves ω≤9/4\omega\le9/4 over ℂ, giving Oε(n9/4+ε)O_\varepsilon(n^{9/4+\varepsilon}) arithmetic operations for square matrix multiplication. In characteristic zero, some inner dimension na with a > 0.465 permits n2+o(1)n^{2+o(1)} rectangular multiplication. Further square bounds give ω < 2.258 outside finitely many positive characteristics and ω < 2.371054886006746 over every fixed field.

Unreleased OpenAI model
LFCR
Improved boundTheoretical computer scienceopenai/math #103

Logspace derandomization

Proves L=RL=BPL\mathsf L=\mathsf{RL}=\mathsf{BPL}, resolving derandomization for bounded-error logarithmic-space computation. An effective compiler converts each randomized polynomial-time logarithmic-space machine deciding a language with one-sided or two-sided error into a deterministic logarithmic-space decider with explicit polynomial running-time bounds.

Unreleased OpenAI model
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.

Unreleased OpenAI model
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 L3(R3)L^3(\mathbb R^3) to L3(S2)L^3(S^2) with norm Oε(δ−ε)O_\varepsilon(\delta^{-\varepsilon}) for every ε > 0. In four dimensions, every set containing a unit segment in every direction has Hausdorff dimension four.

Unreleased OpenAI model
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.

Unreleased OpenAI model
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 cax/(log⁡x)2c_a x/(\log x)^2 primes in every sufficiently large interval (x,2x)(x,2x) have primitive root a, with ca>0c_a\gt 0.

Unreleased OpenAI model
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.

Unreleased OpenAI modelBirch and Swinnerton-Dyer conjecture →
LFCR
New proofNumber theoryopenai/math #005

Catalan constant irrationality

Proves that Catalan's constant G=∑j≥0(−1)j/(2j+1)2G=\sum_{j\ge0}(-1)^j/(2j+1)^2 is irrational.

Unreleased OpenAI model
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.

Unreleased OpenAI model
LFCR
New proofNumber theoryopenai/math #003

Quasi-Riemann hypothesis

Proves that every Dirichlet L-function, including ζ(s)\zeta(s), is zero-free in ℜs>7/8\Re s\gt 7/8, resolving the quasi-Riemann hypothesis. The same half-plane is zero-free for every finite-order Hecke L-function over Q(−3)\mathbb Q(\sqrt{-3}). A companion gives a different proof of the zero-free half-plane ℜs>11/12\Re s\gt 11/12.

Unreleased OpenAI modelRiemann hypothesis →
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.

Unreleased OpenAI modelBirch and Swinnerton-Dyer conjecture →
LFCR
New proofPDE and mathematical physicsopenai/math #377

Interior C1,αC^{1,\alpha} regularity for infinity-harmonic functions

Proves uniform interior C1,αdC^{1,\alpha_d} 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.

Unreleased OpenAI model
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.

Unreleased OpenAI model
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 u:R8→(−1,1)u:\mathbb R^8\to(-1,1) of Δu=u3−u\Delta u=u^3-u that is strictly increasing in one direction depends on only one linear coordinate. A stronger theorem classifies all stable entire solutions v:R7→[−1,1]v:\mathbb R^7\to[-1,1] as constant wells or planar transitions, without an energy-growth assumption.

Unreleased OpenAI model
LFCR
CounterexamplePDE and mathematical physicsopenai/math #374

Sharp one-third stability of Brenier maps

For uniform source measure ρ on a compact convex body with interior in dimension at least two, quadratic optimal transport maps satisfy ∥Tμ−Tν∥L2(ρ)≤CW2(μ,ν)1/3\|T_\mu-T_\nu\|_{L^2(\rho)}\le C W_2(\mu,\nu)^{1/3} uniformly over targets in a fixed compact set. The exponent is sharp, even for three-atom targets, disproving Letrouit's conjectured square-root bound.

Unreleased OpenAI model
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.

Unreleased OpenAI model
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 3λ+2μ>03\lambda+2\mu\gt 0 on the closure. Neither analyticity, proximity to constant coefficients nor prior knowledge near the boundary is required.

Unreleased OpenAI model
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 Hk(T12)H^k(\mathbb T^{12}), 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.

Unreleased OpenAI model
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, p,q>0p,q\gt 0 and real A, B, the system −Δu=∣x∣Avp-\Delta u=|x|^A v^p, −Δv=∣x∣Buq-\Delta v=|x|^B u^q has no positive entire solution when (n+A)/(p+1)+(n+B)/(q+1)>n−2(n+A)/(p+1)+(n+B)/(q+1)\gt n-2, 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 A,B>−2A,B\gt -2.

Unreleased OpenAI model
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.

Unreleased OpenAI model
LFCR
New proofPDE and mathematical physicsopenai/math #368

The three-dimensional Ball–Evans approximation problem

Resolves the three-dimensional Ball–Evans approximation problem: every W1,pW^{1,p} homeomorphism between arbitrary bounded domains in ℝ3, for 1≤p<∞1\le p\lt \infty, is a strong W1,pW^{1,p} limit of smooth diffeomorphisms onto the same target.

Unreleased OpenAI model
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 n−7n-7 in higher dimensions.

Unreleased OpenAI model
LFCR
Improved boundPDE and mathematical physicsopenai/math #366

The planar Mumford–Shah regularity conjecture and local weak-L4 gradient bounds

Resolves the interior regularity conjecture for reduced absolute planar Mumford–Shah minimizers with bounded fidelity data. Locally, the closed discontinuity set is a C1,αC^{1,\alpha} arc, a regular crack tip, or three arcs meeting at 120∘120^\circ; only finitely many global connected components meet any compact interior region.

Unreleased OpenAI model
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.

Unreleased OpenAI model
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.

Unreleased OpenAI model
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.

Unreleased OpenAI model
LFCR
CounterexampleGeometry and topologyopenai/math #361

Failure of integer-degree harmonic dimension comparison

Disproves Yau's proposed Euclidean dimension bound for harmonic functions of integer growth on manifolds with nonnegative Ricci curvature. For every sufficiently large integer k, a complete smooth metric on ℝ3 has at least (k+2)2(k+2)^2 independent harmonic functions of growth at most k, exceeding the Euclidean count (k+1)2(k+1)^2. The metric may depend on k.

Unreleased OpenAI model
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.

Unreleased OpenAI model
LFCR
CounterexampleGeometry and topologyopenai/math #359

Negative Kähler curvature without bounded holomorphic coordinates

Constructs a contractible domain in ℂ3 with a complete negatively pinched Kähler metric but no bounded holomorphic coordinates, disproving bounded-domain uniformization in this setting. A higher-dimensional example has sectional curvature at most −1 and only constant bounded holomorphic functions; its curvature is not bounded below.

Unreleased OpenAI model
LFCR