OpenAI on Saturday disclosed that an unreleased internal model it calls Astra produced formal proofs for ten problems in mathematics and theoretical computer science, each open for at least a decade, at a compute cost of roughly $2,000 at Sol API rates. The headline result is an explicit construction of a non-sofic group, resolving a question that has been open since Mikhail Gromov introduced soficity in 1999.

The evidence lives on GitHub. OpenAI published a 249-page manuscript alongside Lean 4 proof certificates under an Apache 2.0 licence, and the repository’s “sorry” count, Lean’s marker for unproven steps, stands at zero. That’s the analytically load-bearing detail. Machine-checkable proofs sidestep the usual argument about whether a language model “understands” what it output.

Other results include disproving Connes’s rigidity conjecture on von Neumann algebras, proving Ehrhart’s volume conjecture, and resolving three problems from Paul Erdős’s catalogue, among them problem 183 on multicolour Ramsey numbers. Thomas Bloom, the University of Manchester mathematician who maintains erdosproblems.com, called it “big news” on X, and said the batch outweighs May’s counterexample to the Erdős unit-distance conjecture. Noam Brown, an OpenAI researcher, conceded the limits on the same platform: “Sadly, no Millennium Prize Problems (yet).”

None of the ten results has been through peer review. That collides directly with the Leiden Declaration, endorsed by the International Mathematical Union in June, which warns that AI companies are “using published research without consent, bypassing peer review, and threatening the integrity of proof and attribution.”

OpenAI hasn’t said whether Astra ships as GPT-6 or a GPT-5 variant, and any launch is expected to run through the federal AI safety review process. For now the field is being asked to referee a repository, not a paper.

Sources