Did an OpenAI Model Really Solve the Navier-Stokes Problem? Here’s What’s Actually True

September 8, 2026 · 5 min read

Quick answer: No — not the famous $1 million Millennium Prize version. What’s verified is a narrower result from NYU mathematician Tristan Buckmaster and Anthropic researcher Levent Alpöge, built with AI assistance over the past year. OpenAI separately claims an internal model produced a proof of a related, easier variant — but that proof hasn’t been published or independently checked. Buckmaster has also publicly accused OpenAI of racing to catch up after learning of his unpublished work, and of pressuring him over credit. OpenAI’s Sébastien Bubeck calls the allegations false.

Key Takeaways

Question	What’s actually known
Is the Millennium Prize problem solved?	No. The full, unforced 3D Navier-Stokes existence-and-smoothness question remains open.
What’s confirmed and published?	Buckmaster and Alpöge’s AI-assisted, Lean-verified blow-up proofs for forced fluid equations.
What did OpenAI claim?	An internal model produced a ~100-page proof of blow-up for forced Navier-Stokes — unpublished, unverified.
Who is Tristan Buckmaster?	NYU Courant Institute mathematician specializing in fluid dynamics and Navier-Stokes.
Who is Levent Alpöge?	Mathematician and Anthropic researcher; Buckmaster’s collaborator.
Who is Sébastien Bubeck?	OpenAI scientist, co-author of the 2023 “Sparks of AGI” paper; central to the dispute.
What’s disputed?	Whether OpenAI learned of Buckmaster’s work early and raced to match it, and whether it pressured him to drop Alpöge as co-author.
What’s unresolved?	Whether OpenAI had access to Buckmaster and Alpöge’s private draft sessions. Buckmaster says he doesn’t know.
Official response	Bubeck calls the allegations “false and inflammatory” and says a fuller account is coming.

What Is the Navier-Stokes Problem?

Did an OpenAI Model Really Solve the Navier-Stokes Problem

The Navier-Stokes equations describe how fluids like water and air move, and engineers have used them for roughly two centuries. The open mathematical question is narrower: in 3D incompressible flow, does a smooth starting condition always stay smooth, or can it collapse into a singularity in finite time?

The Verified Part: Buckmaster and Alpöge’s Result

For nearly a year, Buckmaster worked with Alpöge on a technical route toward blow-up results, building on the Córdoba–Martínez-Zoroa program and applying it to forced versions of the equations. They used AI tools, including OpenAI’s Codex, throughout, and formally verified their proofs using the Lean proof assistant. Fields Medalist Terence Tao reportedly praised the work.

This is the one part of the story that’s independently confirmed: a real, published, AI-assisted result — just not the full Millennium Prize problem. Separately, Tao published essays on September 3 speculating about AI’s future role in tackling problems like Navier-Stokes, which some social media posts appear to have conflated with an actual solved proof once this story broke.

What OpenAI Reportedly Claimed

According to Buckmaster’s account, he learned OpenAI had turned an internal model onto the same forced Navier-Stokes route he and Alpöge were pursuing. He says the first prompt on the problem went in only days before his conversations with the company — after his and Alpöge’s unpublished progress had reportedly reached OpenAI. The model was reportedly tested first on simpler problems, including a version of the Euler equations.

On September 6, 2026, Buckmaster had two calls with OpenAI staff, including Bubeck. He says OpenAI told him a model produced a roughly 100-page proof of finite-time blow-up for forced Navier-Stokes. He also says the story shifted under questioning — an initial claim of minimal human input reportedly gave way to an account involving a full team and heavy compute over time.

Important caveat: OpenAI hasn’t released or demonstrated this proof publicly, and no outside mathematician has verified it. It’s a claim relayed second-hand by Buckmaster, not a confirmed result.

The Credit Dispute

Buckmaster says OpenAI offered two options: a coordinated joint announcement, with his and Alpöge’s result published first and OpenAI’s following the next day; or an arrangement where Buckmaster alone would write up the Navier-Stokes result, crediting an OpenAI model, without Alpöge as co-author.

He alleges Bubeck argued twice that Alpöge should be excluded specifically because he works at Anthropic, a competitor. Buckmaster says he rejected both proposals, and that when he said he’d go public, someone asked him a pointed question about risking his career — which he took as an implicit threat.

He also raised an unresolved question: whether OpenAI’s model had access to the private Codex sessions where he and Alpöge stored their unpublished drafts. Buckmaster is careful to note the limits of his own knowledge here — he hasn’t seen OpenAI’s alleged proof and doesn’t know what data went into it. That question remains open.

OpenAI’s Response

Bubeck responded on X the same day, calling the circulating claims false and inflammatory, saying he followed academic norms, and promising a fuller response. As of this writing, that fuller account hadn’t been published. Other OpenAI and Anthropic-affiliated researchers exchanged pointed remarks on social media, but neither company has issued an official corporate statement.

Why the Story Blew Up

Several separate things got compressed into one headline: a real, published result from Buckmaster and Alpöge; an unverified OpenAI claim about a tougher (but still not Millennium-level) variant; and a personal dispute over credit. Some posts went further, suggesting OpenAI had solved the full Millennium Prize problem — a claim beyond anything either side has actually stated. The original, unforced 3D Navier-Stokes question remains open.

Frequently Asked Questions

Q. Has the Navier-Stokes Millennium Prize Problem been solved?

A. No. It remains unsolved as of September 2026.

Q. What was actually proven?

A. Buckmaster and Alpöge produced AI-assisted, Lean-verified proofs of blow-up for forced fluid equations — narrower than the Millennium case.

Q. Did OpenAI prove a version of Navier-Stokes?

A. It reportedly told Buckmaster an internal model produced a proof of forced Navier-Stokes blow-up, but this hasn’t been released or verified.

Q. Did OpenAI copy Buckmaster’s work?

A. That’s his allegation. He says he doesn’t know if his private data was used and isn’t accusing anyone of a proven act of misconduct. Bubeck disputes the broader claims.

Sources

Disclaimer: OpenAI’s reported proof remains unverified, and the $1 million Navier-Stokes Millennium Prize problem is still unsolved.

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Reviewed by editorial staff before publication.
Fact-checking and source verification applied.
Updated regularly for accuracy and clarity.
Aligned with newsroom ethics and publishing standards.

About The Author

Jordan Drew Senior Editor

Jordan Drew is a Technology and Economy Correspondent at New York Editor, covering technology, business, finance, markets, digital innovation, and major economic developments. Jordan focuses on explaining how emerging technologies and changing economic trends are influencing businesses, consumers, and the wider world. With an interest in both innovation and the forces shaping modern economies, Jordan produces clear, informative, and well-researched stories that make complex topics easier to understand. Their coverage includes artificial intelligence, consumer technology, startups, digital markets, business trends, economic policies, and the growing relationship between technology and the global economy. Jordan believes good technology and economy journalism should go beyond the headlines by giving readers useful context and helping them understand what developments mean in real life. "Technology is changing more than the way we work and communicate. My goal is to help readers understand the ideas, innovations, and economic trends shaping the world around them."

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