OpenAI Claims Navier-Stokes Proof, Credit Fight Erupts
Two mathematicians got there first on August 15; OpenAI's agents finished on September 5 after 88 hours and 130 billion output tokens.
Symbolic image: a technician walks past humming compute racks while a wall display shows an abstract swirling flow pattern.
OpenAI says its own agents produced a complete proof of Navier-Stokes, but two mathematicians finished three weeks earlier and are now asking how closely the company tracked their work.
At a glance
- Navier-Stokes is one of seven Millennium Prize Problems; the $1,000,000 award has stood since May 24, 2000.
- Buckmaster and Alpoge reached their result on August 15, 2026; OpenAI's agents reported theirs on September 5, 2026.
- OpenAI's run: 88 agent hours, 2.7 million messages and 130 billion output tokens for Navier-Stokes alone.
- Formalizing the proof in Lean took another 17 hours, according to OpenAI.
- Across all problems attempted, OpenAI cites 4.9 million messages, 300 billion tokens and roughly $15 million in cost.
OpenAI says an unreleased internal model has settled existence and smoothness for the Navier-Stokes equations, one of the seven Millennium Prize Problems and a $1,000,000 award. The work was done by agents running on a model the company calls GPT-6 Astra. Two human mathematicians reached the same summit three weeks earlier, using Claude alongside Codex on GPT-5.6 Sol. What followed was less a celebration than an argument about timestamps.
The compute bill is the story
OpenAI reports 2.7 million messages and 130 billion output tokens for Navier-Stokes on its own. Across every problem the team attempted, the totals rise to 4.9 million messages and 300 billion output tokens, or roughly $15 million at public GPT-6 Astra pricing. The agents finished on September 5, 2026, 88 hours after the run began, and a Lean formalization was checked over the following 17 hours.
Who got there first
Tristan Buckmaster, a mathematics professor at New York University, and Levent Alpoge, a mathematician who works at Anthropic, had their result on August 15, 2026, after close to a year of work. OpenAI says it first heard rumors of solved problems on September 1, 2026. That leaves a narrow window between hearing the rumor and shipping a proof of its own.
The question OpenAI has not answered
Buckmaster asked something simple: when was the first prompt sent? He reports that no direct answer came for some time. He has also asked whether the model could reach his Codex sessions, which held drafts of the project.
OpenAI's position is that neither its researchers nor its agents saw the outside work before it was public, and that no specific user data was used to solve the problem. It stops short of ruling out that de-identified data derived from product usage helped improve its models. For researchers who built a proof inside a vendor's tools, that residual gap is the whole complaint.
OpenAI also offered Buckmaster a choice: wait for his publication, or co-author a paper on OpenAI's own result. Alpoge was excluded from that offer because of his employer.
What is not verified
Every figure above comes from OpenAI's own account, quoted and analyzed in Simon Willison's write-up. We could not retrieve OpenAI's announcement page for this article, so none of it is independently confirmed here, and no peer review of either proof has been reported. OpenAI itself notes that the statements proved in the Euler case differ from those of the competing group.
FAQ
Has the Navier-Stokes problem actually been solved?
Two groups claim it independently. Neither claim has been reported as peer reviewed, and OpenAI concedes that the results differ in the Euler case.
Which model did OpenAI use for the proof?
An unreleased internal model it calls GPT-6 Astra. Buckmaster and Alpoge worked with Claude and with Codex on GPT-5.6 Sol.
Why are Buckmaster and Alpoge upset with OpenAI?
Buckmaster wants to know when OpenAI sent its first prompt and whether the model could reach his Codex sessions holding project drafts. No direct answer came for some time.
