OpenAI’s Astra solves 10 long-open math problems and publishes the proofs
OpenAI Group PBC revealed Saturday that an internal version of Astra, the model family it calls its next major release, produced new results for 10 problems in mathematics and theoretical computer science that had been open for at least a decade, and it published machine-checkable proofs alongside the claim.
The company posted a 249-page manuscript collection, model-written reasoning walkthroughs and Lean 4 certificates for all 10 results. The certificates sit on GitHub under an Apache 2.0 license, and the repository reports a “sorry” count of zero, meaning no step in any of the formalized proofs has been left unproven.
The headline result is an explicit construction of a non-sofic group, a question left open since Mikhail Gromov introduced soficity in 1999. Astra also disproved Connes’s rigidity conjecture, constructing infinitely many non-isomorphic groups with property (T) that share the same von Neumann algebra, and it proved Ehrhart’s volume conjecture. Three problems from Paul Erdős’s catalog fell as well, including problem 183 on multicolor Ramsey numbers.
Stripped of the terminology, a group is the mathematical description of a set of symmetries, and a sofic group is one whose structure can be approximated by shuffling a finite deck of cards. Every group anyone had examined turned out to be sofic, and no one could prove that all of them are. Astra built the exception. Connes’s conjecture, posed in 1980, held that for one rigid class of groups, a related algebraic object acts as a unique fingerprint, pinning down the group it came from. Astra produced infinitely many distinct groups sharing a single fingerprint.
