Ten advances in mathematics and theoretical computer science

OpenAI announced Friday it has produced new results on ten longstanding open problems in mathematics and theoretical computer science using an internal version of its next major model, Astra. The problems span high-dimensional geometry, coding theory, arithmetic circuit complexity, group theory, operator algebras, quantum complexity, lattice cryptography and extremal combinatorics. Each problem had seen no progress on the main result for at least a decade. The results include new upper bounds on sphere-packing density, exponentially improved bounds on binary and spherical codes, and a construction establishing the existence of non-sofic groups. The system also disproved Connes's rigidity conjecture, set new lower bounds for computing the permanent using arithmetic circuits, and proved an exponential parallel repetition theorem for general two-player quantum games. Additional results cover polynomial-factor hardness of approximation for the closest vector problem, a determination of maximum volumes for Ehrhart's volume conjecture, a superexponential lower bound for multicolor triangle Ramsey numbers, and resolutions of two extremal number conjectures. OpenAI said the total tokens needed to find solutions would cost roughly $2,000 at Sol API rates. The arguments were prepared into manuscripts by humans with the same model and then formalized in Lean certificates. The company is also releasing a model narration of the thinking process for each solution. OpenAI said it takes responsibility for the correctness of the proofs while the mathematical arguments were generated by its system.