OpenAI Cracked Navier-Stokes. Next Target: P vs NP

On the morning of September 8, 2026, OpenAI announced that an internal model had produced a proof — and a machine-checked Lean formalization — that the three-dimensional Navier–Stokes equations can develop a singularity in finite time. That resolves the Navier–Stokes existence and smoothness problem, one of the seven Millennium Prize Problems named by the Clay Mathematics Institute in 2000.

Within about eighteen hours, the AI research crowd on X had stopped talking about fluids entirely and started talking about what to point the machine at next. The most-quoted version of that impulse came from AI researcher Yam Peleg.

Aim that beast at P = NP let’s get it over with

— Yam Peleg (@Yampeleg) September 10, 2026

It is a joke with a serious premise underneath it. For the first time, "aim the model at a Millennium Prize Problem" is a sentence that describes something a lab has actually done.

What OpenAI actually proved

The Navier–Stokes equations apply Newton's second law to a fluid treated as a continuous medium rather than as individual molecules. They underpin aircraft design, weather forecasting, and models of blood flow. The open question, roughly since Jean Leray's 1934 work on generalized solutions, was whether a fluid that starts out perfectly smooth must stay smooth — or whether some vanishingly small parcel of it can be driven to infinite speed in finite time.

OpenAI's system produced the second answer. The solution is a vortex that spirals inward while stretching along its axis, shrinking and accelerating in a precisely balanced way so that total energy stays finite even as velocity runs away. The company says this establishes statements "C" and "D" in the official Clay formulation of the problem.

Two details make the claim harder to wave away than previous AI math announcements. The proof was formalized in Lean, so its correctness does not rest on a referee's patience. And OpenAI published both the writeup and the formalization repository rather than describing the result at a distance.

Quanta Magazine reported that roughly 10,000 autonomous agents were run under researcher direction on a model not available to the public. Charles Fefferman of Princeton, who wrote the Clay Institute's official description of the problem, told Quanta he was thrilled it was solved — and named Diego Córdoba and Luis Martínez-Zoroa, whose unconventional line of attack both AI-assisted teams leaned on, as the heroes of the story.

The number worth remembering

Here is the comparison we ran, and it is the one to carry out of this story.

Measured from Leray's 1934 paper to the announcement on September 8, 2026, the smoothness question stood open for about 92.2 years — roughly 808,000 hours. By OpenAI's own account, the agents reached the resolution 88 hours after launch, and Lean formalization and verification added 17 more, for 105 hours of wall-clock time.

That is about one hour of machine time for every 7,700 hours the problem had survived.

(Method: 92.2 years × 8,766 hours/year ÷ 105 hours = 7,695. The two runtimes are OpenAI's published figures; the 1934 start date is the conventional one and a reader who prefers the Clay Institute's 2000 formulation would get a ratio near 2,200 instead. Either way the order of magnitude is the point.)

Where the other six stand

Five of the seven Millennium problems remain open. Grigori Perelman resolved the Poincaré conjecture in 2003 and declined the prize; Navier–Stokes is now the second to fall, and the first to fall to a machine.

Years measured from the conventional first statement of each problem to September 2026, or to the year it was resolved. Sources: Clay Mathematics Institute problem descriptions; OpenAI announcement, September 8, 2026.

Note where P vs NP sits: at 55 years it is the youngest of the seven. Age is not difficulty, but it is a reminder that the problem Peleg wants aimed at has had the least time to accumulate failed attacks.

What the replies actually argued

The responses under Peleg's post were more interesting than the post. Three lines of argument surfaced quickly, and none of them was enthusiasm.

The first was that P vs NP is the wrong wish. "Room-temp superconductor or continuous learning would be more immediately useful," wrote @wyqtor, "since P =/= NP is probably the more likely and boring outcome." That is the consensus expectation compressed into one sentence: almost everyone believes P ≠ NP, so a proof would confirm the world we already assume we live in while a P = NP result would break modern cryptography overnight.

The second was closer to grief. "I think I'd rather never know," wrote @TheTarch. "We will miss the days of uncertainty, wish we could go back to a time when there were unsolved challenges we could comprehend." The complaint is not that machines will fail. It is that they will succeed, and that a solved problem stops being a place where a human can spend a career.

The third was pure noise, and worth one line: @ultima_centauri claimed to know the answer to P = NP "with 50% certainty," and it outperformed most of the serious replies. The crowd reacting to a Millennium Prize result is still, mostly, a crowd.

The credit fight nobody should skip

The result arrived attached to a dispute. Axios reported that NYU's Tristan Buckmaster and Anthropic's Levent Alpöge had announced closely related fluid-dynamics results twelve hours earlier, and that Buckmaster questioned whether OpenAI raced down a direction it learned about from their work. Sam Altman wrote on X that "now that we can see their work, the approaches appear to be different," and acknowledged the effort began after OpenAI heard rumors that a rival's model was close.

OpenAI says it did not access the researchers' user data, while conceding it cannot fully rule out that de-identified usage data improved its models. That caveat is the part working scientists should read twice. It is the first mainstream case of the question: what does the company supplying your research tools learn from you using them?

What comes next

Researchers at DrLambda, the parent company of ChatSlide, are collaborating with OpenAI to work on this final result. Rumor has it there is major progress.

Whether that lands on P vs NP or on one of the four other open problems, the shape of the last week has already changed what "aim the model at it" means. In May it was a metaphor. In September it is a budget line — OpenAI executives put the compute cost of the Navier–Stokes run in the millions of dollars, and said they do not intend to claim the $1 million Clay prize.

The economics of that are worth sitting with. A lab spent several times the prize money to demonstrate a capability, then declined the prize. The proof was never the product.

Cover image: von Kármán vortex street in the clouds downwind of the Cape Verde islands, captured by NASA's Terra satellite. Public domain, via Wikimedia Commons.