A new paper says OpenAI's Navier-Stokes proof and its Lean code don't match
OpenAI said its AI agents cracked one of math's seven Millennium Prize Problems in 88 hours. A paper posted October 6 says the computer-verified proof it published doesn't actually match the English-language argument it was supposed to confirm.
On September 8, OpenAI announced that an internal system, roughly 10,000 AI agents working in parallel for 88 hours, had produced a solution to the Navier-Stokes existence and smoothness problem, one of the seven Millennium Prize Problems the Clay Mathematics Institute has backed with a $1 million award since 2000. OpenAI published two versions: a natural-language proof written the way a mathematician would write one, and a separate version translated into Lean, the formal verification software mathematicians use to machine-check that a proof's logic actually holds.
That second part is where the trouble starts. A paper titled "Navier-Stokes lost in translation," posted to arXiv on October 6 by Alexander Bastounis, Fabian Circelli, and Anders C. Hansen, argues that the Lean-verified version doesn't correspond to the natural-language proof it was meant to formalize. The AI mistranslated the math when it converted the English argument into code, according to the authors. Lean confirmed the logic of a proof, just not the one OpenAI said it was confirming.
Hansen was careful about what the paper does and doesn't claim. "We are not saying that the natural-language proof is wrong," he said, according to a New Scientist report covering the findings. "Nor do we say that it is correct." The point isn't that OpenAI's math is false. It's that the Lean check, the part meant to remove human doubt from the equation, didn't actually do that job here.
Finding the gap wasn't fast, either. Reporting on the paper noted it took the authors roughly two weeks of work, first asking ChatGPT to flag possible discrepancies and then checking them by hand, to identify a genuine divergence between what the English proof asserted and what the Lean code actually verified. OpenAI's agents reached the original result in 88 hours, then spent another 17 hours on Lean formalization. Catching the mismatch took a human team more than three times as long as that combined run.
Mathematician Steven Miller accused OpenAI of reusing his 2016 sphere-packing argument without credit, while a separate arXiv audit led by Sienicki found no surviving errors across 18 chapter reviews of Astra's ten proofs. - independent mathematicians verify OpenAI math proof claims - how Lean certificates validate mathematical research results
OpenAI told New Scientist it's aware of the gap between the two versions and said that doesn't mean either proof is wrong. The company said it would fix errors in the natural-language proof as they're identified. That's a reasonable response to a bug report. It's a much weaker one to the underlying question raised by the paper: if Lean verification doesn't guarantee the formal proof says what the English proof says, what exactly was verified?
The timing made this worse from the start. OpenAI's announcement landed one day after NYU mathematician Tristan Buckmaster and Anthropic researcher Levent Alpöge posted their own Lean-verified blow-up proof for a related fluid equation. Buckmaster has said he emailed OpenAI on September 3 to flag that he and Alpöge had related work nearing publication, after rumors about their progress started circulating. According to Buckmaster, OpenAI offered him sole authorship credit on its Navier-Stokes result, on the condition that Alpöge's name be dropped.
OpenAI researcher Sébastien Bubeck disputed that account on social media. Buckmaster has also alleged that when he threatened to make the exchange public, Bubeck asked him, "Why would you ruin your career?" Sam Altman backed Bubeck's version of events in a post on X less than half an hour after Bubeck published it. OpenAI's own statement said its researchers and agents never saw Buckmaster and Alpöge's unpublished work, while acknowledging the company "cannot rule out" that de-identified usage data from their Codex sessions helped improve its models.
There's a narrower technical question sitting underneath the credit dispute, too. OpenAI's proof covers the "forced" version of the Navier-Stokes equations, where an external force acts on the fluid. OpenAI says that establishes statements C and D in the Clay Institute's official formulation, and the Clay Institute said on September 11 that the problem has "apparently been settled," while stressing that its prize process is deliberately unhurried.
None of this proves OpenAI's underlying math is wrong. The Bastounis-Circelli-Hansen paper goes further than a single bug report, though: it argues that guaranteeing a semantically faithful translation between natural language and Lean sits beyond what's theoretically achievable, placing the problem outside the range computer science can solve even in principle, in the same territory as the halting problem. If that holds up, "we verified it in Lean" stops being the trump card AI labs have been treating it as, for this proof and potentially for others built the same way.
Five weeks ago, OpenAI had what looked like a historic math result and a credibility fight over who deserved credit for it. Now it has a second question stacked on the first: whether the one part of the announcement that was supposed to be beyond dispute, the machine-checked proof, actually checked what OpenAI said it checked.
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OpenAI's October 6 GitHub dump spans 372 research families and roughly 4,000 posed problems, each surviving result costing about three hours of ChatGPT Pro compute, following a Navier-Stokes claim that triggered a 25-signature Fields Medalist rebuke over credit and review. - openai releases 722 mathematical proofs via github - mathematicians criticize openai ai generated math proofs
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