OpenAI Says Astra Advances 10 Long-Standing Math Problems

OpenAI has spent years pairing large language models with formal proof systems, moving from Olympiad-level exercises toward research mathematics. Astra, described as the company’s next major long-horizon reasoning model, marks a more consequential test: whether AI can generate original results rather than retrieve or restate existing work. Machine-checkable proofs in Lean raise confidence by verifying each logical step, though researchers must still assess whether the formalized statements capture the intended problems and whether the results withstand peer review.

OpenAI said on August 1, 2026, that an internal version of Astra solved 10 problems that had remained open for at least a decade, spanning high-dimensional sphere packing, group theory, coding theory, quantum complexity and lattice cryptography. The company released a 249-page manuscript collection and a Lean certificate for each result. It estimated the successful solution runs would have cost roughly $2,000 in tokens at Sol API rates. Astra has not been publicly released, and independent scrutiny of the broader mathematical claims is still under way.

Frontier problems in mathematics and theoretical computer science underpin areas ranging from cryptography and algorithm design to high-dimensional data analysis, but many resist progress for decades. OpenAI in May disclosed an AI-generated disproof of the Erdős unit-distance conjecture, found while testing an unreleased model. Its latest collection broadens that effort across geometry, coding theory, group theory, quantum complexity and lattice cryptography, highlighting AI’s emerging role as a research collaborator rather than only a problem-solving assistant.

OpenAI said on Aug. 1, 2026, that an internal version of Astra, its next major model, produced results on 10 problems whose main questions had seen no progress for at least a decade. The advances include a construction of non-sofic groups, a disproof of Connes’s rigidity conjecture and polynomial-factor approximation hardness for the closest vector problem. OpenAI estimated the solution search cost at about $2,000 at Sol API rates; humans prepared manuscripts with the model, which then formalized each argument in Lean.