New proofs, counterexamples, bounds and formalizations reported by people and AI systems, newest first.
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.
Resolves the free group factor isomorphism problem: , and hence all interpolated free group factors, including , are isomorphic. Their common factor has fundamental group .
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.
Constructs a finitely presented torsion-free group G whose group algebra has nonzero zero divisors, disproving Kaplansky's zero-divisor conjecture. The group has a finite two-dimensional classifying space.
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.
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.
Proves over ℂ, giving arithmetic operations for square matrix multiplication. In characteristic zero, some inner dimension na with a > 0.465 permits rectangular multiplication. Further square bounds give ω < 2.258 outside finitely many positive characteristics and ω < 2.371054886006746 over every fixed field.
Proves , 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.
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.
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.
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.
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 .
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.
Proves that Catalan's constant is irrational.
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.
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 .
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.
Optimal packings and lower-bound proofs are reported for these finite cases.
Two October preprints report dimension-independent bounds; their arguments and version histories are distinct.
A six-task counterexample is reported; exact periods and the reduction rule must be taken from the original witness before publishing.
An unforced C¹ blowup construction is reported alongside the Navier–Stokes work.
OpenAI reports examples for every ν>0 in R³ and on the three-torus, under its specified forcing and initial-data conditions.
A new AI-assisted end-to-end Lean development is reported in September, formalizing an already proved theorem.
A paper proves an upper bound of 212. The Lean certificate verifies a deduction from stated distribution estimates and other analytic inputs.
A lower bound about 67.250% for simple critical-line zeros; a related distinct-zero bound about 83.625% is also reported.
Sendov is reported for all degrees; the stricter distance conclusion and exceptional equality cases are addressed by the same argument.
Noncircular counterexamples are reported, including a real-analytic Jordan domain from computer-assisted existence estimates and a separate bifurcation construction.
A MAX CUT reduction establishes NP-completeness of the decision version and hardness of the optimization problem.
A sharp finite-n bound, and bounds for general δ, are reported; the δ≥1 conclusion settles the stated Feige assertion if the proof is valid.
A seven-vertex, nine-arc witness: fractional cost 58; every allowed unsplittable routing costs at least 60, with demands 15,10,15.
For the specified one-cluster automata with cycle length m and positive level ℓ, a reset-word bound (m-1)(n-1)+mℓ≤(n-1)² is reported.
A proof of the modulo-four assertion is reported.
A proof is reported in a July preprint.