OpenAI has announced that its next major model, internally named Astra, has produced solutions to ten long-standing open problems in mathematics and theoretical computer science. These problems had been open for at least a decade (most much longer) and span high-dimensional geometry, coding theory, arithmetic circuit complexity, group theory, operator algebras, quantum complexity, lattice cryptography, and extremal combinatorics. The model generated the mathematical arguments; humans then prepared manuscripts and formalized each proof in Lean. The total token cost to produce these solutions was roughly $2,000 at current API rates, and the model’s reasoning process has been released alongside Lean certificates.

The results include: new upper bounds on high-dimensional sphere packing density down to the Cohn–Elkies threshold; exponentially improved bounds on binary and spherical codes at any prescribed minimum distance; a construction establishing the existence of non-sofic groups, addressing a central open question in group theory; a disproof of Connes’s rigidity conjecture that certain groups are uniquely determined by their von Neumann algebras; new lower bounds for arithmetic circuits computing the permanent, including an arithmetic-formula lower bound of order n^4 / log n; an exponential parallel repetition theorem for general two-player quantum games; polynomial-factor hardness of approximation for the closest vector problem in lattice cryptography, relevant to post-quantum security; determination of the maximum possible volume of a convex body whose centroid is its only interior lattice point in every dimension (Ehrhart’s volume conjecture); a superexponential lower bound for multicolor triangle Ramsey numbers, resolving Erdős problem 183; and results on the compactness and degeneracy conjectures in extremal graph theory, resolving Erdős problems 146 and 180.

OpenAI emphasizes the importance of proper attribution and transparency in AI-assisted research, noting that the mathematical arguments were generated by the system while humans prepared the manuscripts and formalized the proofs. They also announced ChatGPT for Academic Researchers, providing free access to 100,000 scientists and mathematicians, and hope the mathematical community will engage deeply with these results and bring the ideas to life through further research.