On August 1, 2026, OpenAI researcher Noam Brown published the first results from Astra — OpenAI's next major model family — in a form that sidesteps hype entirely: ten mathematical proofs, written in Lean, covering problems that had been open for at least a decade each.
The ten problems span group theory, coding theory, quantum complexity, and high-dimensional geometry. Among the results:
- A construction proving that non-sofic groups exist, resolving a question Mikhail Gromov posed in 1999 and left untouched for 27 years
- A disproof of Connes's rigidity conjecture on von Neumann algebras
- A proof of Ehrhart's volume conjecture
- Resolutions of three problems from Paul Erdős's famous catalogue, including problem 183 on multicoloured Ramsey numbers
- The first improvement to the general upper bound on high-dimensional sphere-packing density since 1978
- A proof of a parallel repetition theorem for two-player quantum games
- New lower bounds on the circuit complexity of computing the permanent
The total compute cost for all ten solutions: roughly $2,000 at current API rates.
What makes this announcement harder to dismiss than the usual AI benchmark claim is the verification method. The proofs are delivered as Lean certificates — formal proofs written in a language that Lean 4's trusted kernel can check automatically. A Lean certificate gives a binary verdict: the proof compiles, or it does not. Unlike self-reported benchmark scores, these results can be independently verified by anyone with the toolchain installed.
That said, caveats apply. Astra is not yet publicly available. OpenAI published a 249-page manuscript, reasoning walkthroughs, and the Lean formalizations on GitHub, and independent mathematicians are currently reviewing the work. Lean verification confirms logical validity, not that the problems chosen are the hardest or most significant. Critics have also noted that the selected problems play to areas where AI systems naturally excel — formal symbolic reasoning over large search spaces.
Still, mathematicians had made no progress on any of these ten problems for at least a decade. A model that cracks them for $2,000 total — even with human researchers preparing the arguments — represents a meaningful shift in what AI-assisted mathematics can do.
The results are being examined now. When the peer review concludes, the conversation about what AI can contribute to mathematics will be different.