On August 1, OpenAI announced that an internal version of Astra, its "next major model family," produced solutions to ten problems that had been open for at least a decade each — spanning high-dimensional geometry, coding theory, group theory, operator algebras, quantum complexity, lattice cryptography and extremal combinatorics.
The results include a construction proving the existence of non-sofic groups, a central open question in group theory; a disproof of Connes's rigidity conjecture; new upper bounds on sphere-packing density down to the Cohn–Elkies threshold; an exponential parallel-repetition theorem for quantum games; and resolutions of multiple problems posed by Paul Erdős. They build on the AI-generated disproof of the Erdős unit-distance conjecture that OpenAI shared in May.
OpenAI published each argument as a Lean certificate — a machine-checkable formal proof — on GitHub, together with the model's narration of its own reasoning. The company estimates the compute needed would cost roughly $2,000 at its Sol API rates. Fields Medalist Timothy Gowers said he would recommend one of the proofs to the journal Annals of Mathematics "without hesitation"; other prominent mathematicians including Noga Alon, Arul Shankar and Jacob Tsimerman assessed the results.
OpenAI was explicit about attribution: the mathematical arguments were generated by the system, while humans prepared the manuscripts and formalized the proofs. The company said it takes responsibility for their correctness and hopes the community engages deeply with them. Analysts caution that the problems were well-suited to AI's systematic-search strengths — but note that verifiable proofs turn an extraordinary claim into something anyone can independently check.
