OpenAI Says 10,000 Agents Solved Navier-Stokes Millennium Problem

The Navier–Stokes equations underpin models of fluid motion used in aircraft design, weather forecasting and blood-flow research. Mathematicians have spent roughly 90 years asking whether a smooth three-dimensional flow can develop a finite-time singularity, where velocity becomes unbounded. The Clay Mathematics Institute designated the question one of seven Millennium Prize Problems in 2000, carrying a $1 million award. A machine-generated, formally verified proof would mark a major advance for AI-assisted mathematics and automated reasoning.

OpenAI on Sept. 8, 2026, released a claimed solution produced by an internal model more capable than GPT-6 Astra and about 10,000 coordinating agents. The agents reached the result after 88 hours, followed by 17 hours of Lean formalization and verification. New York University mathematician Tristan Buckmaster and Anthropic researcher Levent Alpöge disputed the priority narrative and questioned whether Codex usage data influenced OpenAI’s model. OpenAI denied accessing specific user data, said the proofs and results differed, and does not plan to seek the $1 million prize. Independent review remains pending.

Generative AI is moving beyond computational assistance toward producing original mathematical arguments, raising the prospect of faster discovery across fields where progress can take decades. Fields Medal-winning Oxford professor James Maynard said the traditionally cautious mathematics community is rapidly adapting. The shift could reshape how researchers select problems, verify proofs and assign credit, while intensifying concerns that human intellectual contributions may be obscured.

OpenAI said on Aug. 1, 2026, that an internal version of Astra, its next major model, produced 10 results that resolved or substantially advanced long-standing problems. The work spans high-dimensional geometry, coding theory, group theory, quantum complexity and lattice cryptography. OpenAI estimated the tokens used to find the solutions would cost about $2,000 at Sol API rates. Humans prepared the arguments as manuscripts, after which Astra formalized each result as a machine-checkable Lean certificate.

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.