AI Claims to Solve the Navier-Stokes Equations — A Millennium Prize Problem Breakthrough

One of the most famous unsolved problems in mathematics may have just fallen — not to a human, but to an AI model. OpenAI has announced that its internal model found a proof related to the Navier-Stokes equations, a set of 19th-century formulas that describe the motion of water, air, and fluids and gases in general.

What the Navier-Stokes Equations Are and Why They Matter

The central question of the problem is deceptively simple: if you start with a perfectly normal flow, will the equations always produce a well-behaved solution? Or can the velocity mathematically “blow up” to infinity within a finite amount of time?

For more than 200 years, nobody knew.

OpenAI’s Claim: A Proof of Blowup in Three Dimensions

Today, OpenAI announced that its internal model has found a proof of exactly such a “blowup” in three dimensions. If confirmed, this would be only the second Millennium Prize Problem — each carrying a $1 million award — ever solved, after Grigori Perelman’s proof of the Poincaré Conjecture roughly 20 years ago.

The Forcing Caveat: Loophole or Legitimate Solution?

There is a catch, however. The proof relies on what mathematicians call “forcing” — an external force applied to the fluid. In the popular framing of the problem, the question is usually about a fluid left entirely to itself. But the official Clay Institute formulation does include variants where forcing is permitted.

So this is simultaneously a “loophole” and a completely legitimate solution to the problem exactly as it was defined when the million-dollar prize was announced. More details are still to come.

The Scandal Behind the Navier-Stokes Breakthrough

As seems increasingly common with AI, the announcement has not been without controversy. The key idea behind the proof was proposed by mathematicians Diego Córdoba and Luis Martínez-Soroa. Tristan Buckmaster and Levent Alpöge — an Anthropic employee, the same researcher who used AI to disprove the Jacobi conjecture during a World Cup final — then used Anthropic’s models to develop it into a proof for the Euler equations.

According to Buckmaster, word of the method reached OpenAI, and over a weekend its internal model completed the full problem. He was then allegedly offered sole authorship on the condition that Alpöge be excluded. When he threatened to go to the press, the response was reportedly along the lines of “why would you want to destroy your own career.”

OpenAI has not commented. Also still open is the question of whether the model was trained on the researchers’ own prompts.

A Moment Compared to Deep Blue Beating Kasparov

Everyone is already drawing comparisons to Deep Blue’s defeat of Kasparov in chess. Whether this holds up as a true milestone depends on what peer review reveals — but the momentum is unmistakable.

For now: we wait for details, watch the IPO race, and set a reminder for December 25th.