OpenAI's internal Astra model produced Lean 4-verified proofs for ten long-standing open mathematics problems

OpenAI published Lean 4-verified proofs for ten long-standing open problems in mathematics and theoretical computer science on August 1, 2026, attributed to an internal model called Astra. Each result is posted to a public GitHub repository (openai/ten-proofs) as a machine-checkable Lean 4 certificate. Problems addressed include a counterexample to Connes's 1980 rigidity conjecture, the first known construction of a non-sofic group (a question open since Gromov introduced soficity in 1999), a superexponential lower bound for multicolor Ramsey numbers resolving Erdős problem 183, improved sphere-packing bounds reaching the Cohn-Elkies threshold, proof of Ehrhart's volume conjecture, and stronger bounds for binary and spherical codes. The workflow involved Astra generating arguments, humans preparing manuscripts in collaboration with the model, and Astra then formalizing each proof in Lean 4. The estimated token cost for all ten results was approximately $2,000 at OpenAI's Sol API rates. As of August 1, no specialist proof-by-proof review had been publicly reported, and no gaps had been reported either. Astra is not publicly available and has no announced pricing or release date; OpenAI has not confirmed whether it will carry GPT-6 branding.