OpenAI Claims Navier-Stokes Proof; Credit Dispute and Verification Unresolved

OpenAI announced on September 8, 2026 that an internal model, described as significantly more capable than GPT-6 Astra, proved a finite-time singularity exists for the 3D incompressible Navier-Stokes equations. The system coordinated approximately 10,000 agents for 88 hours, producing 2.7 million messages and roughly 130 billion output tokens; GPT-6 Astra then completed Lean formalization in 17 additional hours, yielding a 166-page proof. OpenAI's full run across its Millennium Prize attempts used 4.9 million messages and 300 billion output tokens. Bubeck estimated the computational cost at several million dollars; at retail API prices the run would have cost approximately $22 million. OpenAI said it does not intend to claim the $1 million prize.

The Clay Mathematics Institute has not accepted the proof and still lists Navier-Stokes as an unsolved problem. Its rules require a two-year review window before any prize can be accepted. The Poincaré Conjecture remains the only Millennium Prize Problem previously resolved. Clay Mathematics Institute president Martin Bridson called OpenAI's announcement 'an exciting day' representing 'major advances in the human understanding of mathematics,' but no independent mathematical verification has been completed.

The model began training on August 28, 2026 and was still in training when it produced the proof. The research team initiated the project on September 1, 2026. Sam Altman said OpenAI pursued the problem because of 'rumors on the internet last week that Anthropic's models had solved a millennium problem.'

Tristan Buckmaster (NYU) and Levent Alpöge (Anthropic) had been working on the same mathematical program for nearly a year, using Claude and Codex. They achieved a Lean-verified proof for the 3D Euler equations by August 22. On September 8 they published preprints with Lean formalizations covering finite-time blowup for incompressible porous media, the Boussinesq system, and 3D incompressible Euler equations. Buckmaster stated: 'We believe we also have blowup for hypo-dissipative Navier-Stokes. We are not releasing that paper today: unlike the above, the Lean verification has not yet finished.' Buckmaster noted they were forced to publish early without time to produce readable proofs and apologized for the quality of the writeups. Buckmaster credited Diego Córdoba and Luis Martínez-Zoroa for originating the foundational mathematical program and stated his view that Martínez-Zoroa deserves a Fields Medal for this work.

Buckmaster alleged that when he asked OpenAI when it sent its first prompt, 'this question was not answered directly by OpenAI for some time' before OpenAI confirmed the effort had begun after information about their work had reached OpenAI. Buckmaster wrote that 'the circumstances of this claim strongly suggest that OpenAI somehow got wind of what Buckmaster and Alpöge were doing and decided to scoop them.' During September 6 calls, Buckmaster alleged that Bubeck 'twice asserted that he wanted Levent removed from authorship, and said it would all be simple if only it were not the case that, and it was so annoying that, Levent works at Anthropic.' Buckmaster said he was offered two proposals: post the Euler result first and let OpenAI post Navier-Stokes the next day, or write the Navier-Stokes result himself crediting the OpenAI model. Buckmaster said he was told 'Why would you ruin your career?' and 'If you don't want me to be nice, then I don't have to be nice.' Bubeck and OpenAI's Noam Brown denied wrongdoing and said OpenAI acted in good faith; Bubeck called Buckmaster's account 'false and inflammatory.' OpenAI acknowledged: 'While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models.'

Bubeck said OpenAI's model independently solved the Euler problem by different means than those employed by Buckmaster and Alpöge, but acknowledged the Navier-Stokes solution used a similar method to theirs. OpenAI conceded priority for the 3D Euler result to Buckmaster and Alpöge.

On September 11, OpenAI told the New York Times it had made 'substantial progress' on a second Millennium Prize problem without naming it. Andrew Curran said the second problem is rumored to be the Hodge Conjecture and the model may be called Aeon; OpenAI confirmed neither. Reporting indicated OpenAI has been using Millennium Prize Problems as performance benchmarks for frontier models.

Terence Tao warned that 'even the rumor of someone working on a problem can trigger a massive amount of AI-powered effort to flatten it before the original research project has time to reach its full potential,' and that current incentives may push researchers toward secrecy, which he said 'would reverse centuries of traditions of open science and do serious long-term damage to the future of the field.' Tao also cautioned that once an automated proof path is found, researchers stop exploring alternative approaches.