On September 8, 2026, OpenAI announced that an unreleased internal model had done something no human mathematician has managed in nearly a century: find a singularity in the three-dimensional Navier-Stokes equations, one of the Clay Mathematics Institute's seven Millennium Prize Problems, each worth $1 million. Within hours, the celebration turned into a fight over who actually deserves the credit.
What OpenAI Says It Proved
The Navier-Stokes equations describe how fluids move — everything from weather systems to airflow over a wing. The Millennium Prize question asks whether these equations always produce smooth, well-behaved solutions in three dimensions, or whether they can "blow up" into a singularity, a point where the math breaks down entirely. OpenAI says its system found that a singularity can occur.
The Numbers Behind the Claim
The scale of the effort is the headline in itself. According to OpenAI's own account:
- Roughly 10,000 agents worked in parallel on the Navier-Stokes problem for about 88 hours, after launching on September 1.
- Those agents exchanged roughly 2.7 million messages on the Navier-Stokes problem specifically, out of nearly 4.9 million messages across all the related problems the agents attempted.
- A separate, smaller run — about 100 agents over roughly 50 hours — worked on variants of the related Euler equations.
- An additional 17 hours were spent formalizing the result in Lean, a programming language used to check mathematical proofs step by step.
Reported compute costs vary by outlet, with estimates running from roughly $15 million to over $22 million based on the roughly 300 billion output tokens the full effort consumed — either way, a proof that was extraordinarily expensive to generate.
How the Proof Was Checked
The reason mathematicians are taking this seriously at all, despite the AI slop and hype that has flooded the field, is Lean. Formalizing the argument in Lean means the logical steps were checked by a computer rather than taken on faith, which is a much higher bar than an AI simply asserting it has solved something. That said, formal verification confirms the proof is logically valid within its stated assumptions — it doesn't by itself settle whether the mathematical community accepts the framing of the problem or the significance of the result.
Why It Isn't "Solved" Yet
Even a Lean-checked proof doesn't trigger a $1 million check. The Clay Mathematics Institute requires a proof to be published in a recognized journal and to withstand roughly two years of scrutiny by the mathematical community before a prize committee will even consider a recommendation. OpenAI's announcement is the opening move in that process, not the end of it.
The Credit Fight
The more contentious story is about timing. NYU mathematician Tristan Buckmaster and Levent Alpöge, who works at Anthropic, had been quietly pursuing the same route to a singularity result — one opened up by earlier analytic work from mathematicians Diego Córdoba and Luis Martínez-Zoroa. Buckmaster and Alpöge published three papers of their own, and by Quanta's account, did so about 12 hours before OpenAI's announcement landed.
What Buckmaster Alleges
Buckmaster says word of his and Alpöge's unpublished progress reached OpenAI beforehand, writing that the specific approach he and Alpöge had "quietly chosen to attack" was one "almost nobody else" was working on. He also alleges that OpenAI researcher Sébastien Bubeck pressured him to strip Alpöge's name from a paper because of Alpöge's Anthropic affiliation, allegedly telling him, "Why would you ruin your career?" and, when Buckmaster pushed back, "If you don't want me to be nice, then I don't have to be nice." Buckmaster has also expressed regret that some of the AI-assisted material rushed out in the process amounts to what he called "slop."
OpenAI's Response
OpenAI says its effort began on September 1, prompted by rumors that a Millennium Prize problem was close to falling, and that it reached out to Buckmaster and Alpöge to coordinate a concurrent release once its own proof was done. The company states it "did not see any of their work through any means until they released it publicly," and it concedes priority on the 3D Euler results to Buckmaster and Alpöge while maintaining its own priority claim on the Navier-Stokes singularity. Notably, OpenAI has also said it cannot fully rule out that de-identified data from Buckmaster's own use of OpenAI's Codex tool over the prior months may have shaped its models — an uncomfortable wrinkle given he was a paying user of the very product racing to beat him.
Credit Where It's Actually Due
Even OpenAI's staunchest critics agree the underlying mathematical breakthrough traces back to human insight, not machine originality. Princeton's Charles Fefferman singled out Córdoba and Martínez-Zoroa as the real "heroes of the story," and Buckmaster himself has said Martínez-Zoroa "deserves a Fields Medal" for the foundational technique both AI efforts ultimately built on.
What to Watch Next
Expect two parallel tracks over the coming months: mathematicians independently checking whether OpenAI's Lean-verified argument actually holds up to the Clay Institute's standards, and a messier reckoning over whether AI labs racing to claim "world first" math headlines are compressing the norms of credit and collaboration that academic research depends on. The equations may be settled long before the ethics are.
-EditorZ
Photo by Chris Stenger on Unsplash
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