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Update: Paper Says OpenAI’s Lean Navier–Stokes Check Doesn’t Match Its Written Proof, as a New Group Urges Mathematicians to Stop Working With OpenAI

Updated Oct. 8: A preprint from researchers at Cambridge and King’s College London argues that the Lean formalisation behind OpenAI’s September Navier–Stokes blow-up claim does not correspond to the written proof, and that Lean acceptance does not show a translation is faithful. The authors say they make no claim about whether OpenAI’s written proof is correct. Separately, a new group calling itself the Association for Human Mathematics urged mathematicians to discontinue their work with OpenAI over the 722-manuscript release. Original Oct. 7 report: OpenAI published 722 manuscripts in 372 families from an unreleased model, 162 with a Lean-checked main result.

Times of AI Desk 10 min read San Francisco, CA View as Markdown
Engraved drafting table stacked with hundreds of math manuscripts bound in family folios, a Lean theorem seal stamped on a minority of covers, an unreleased model vault in the background, cobalt and graphite intaglio.

Updated October 8, 2026: A new preprint argues that OpenAI’s Lean formalisation of its September Navier–Stokes blow-up claim does not match the written proof, and a new group calling itself the Association for Human Mathematics has urged mathematicians to stop working with OpenAI. The new section directly below covers both; the original October 7 report follows, unchanged except for minor wording edits.

October 8: a challenge to the Lean checks, and a call to walk away#

OpenAI has offered Lean as a reason to trust its AI proofs: its September Navier–Stokes claim came with a Lean formalisation, and the October 6 catalogue counts 162 Lean-checked main results. A preprint posted to arXiv on October 6 argues that a Lean check is weaker evidence than it sounds. Alexander Bastounis (King’s College London) and Fabian Circelli and Anders C. Hansen (Cambridge) write that "the formalised Lean proof does not correspond to the NL [natural-language] proof of blow-up of solutions to the Navier–Stokes equations," the result OpenAI announced in September.

The paper does not say OpenAI’s proof is wrong. Its disclaimer reads: "We do not make claims about the correctness of OpenAI’s NL proof, we only make statements about mistranslations into Lean." The claim is narrower: a Lean file can compile and still prove something other than what the paper says.

What the authors point to#

  • A weaker estimate in Lean. In their Example 3.1, the written paper states an inverse estimate using m + 4 derivatives, while the cited Lean formalisation requires m + 5, which the authors call "strictly weaker." They say the m + 5 requirement recurs across related Lean results.
  • A mismatched pressure-flux bound. In Example 3.3, they argue that the paper’s pressure-flux bound and its proof are both mistranslated in the Lean code it cites.
  • A general limit. They argue that faithfully resolving the ambiguities of mathematical prose is, informally, harder than the Halting problem, so Lean acceptance of a translation "does not imply semantic preservation." On OpenAI’s companion Euler blow-up claim, they say, from preliminary examination, that it "will be another great case study for mistranslations" and that the likelihood of substantial mistranslations is "very high."

This is a preprint, not peer-reviewed, and OpenAI had not responded publicly to it in the sources we reviewed. It examines the September Navier–Stokes work, not the 722-manuscript catalogue below. But its argument bears on how to read that catalogue’s headline number: on the authors’ account, "162 papers with a formalized main result" shows that Lean accepted a formal proof for each, not, by itself, that each formalisation faithfully matches its manuscript.

The Association for Human Mathematics statement#

A group launched in September, calling itself the Association for Human Mathematics (AHM), per Scientific American, published a statement on OpenAI’s October 6 release. It says "Mathematicians did not ask for this work to be done," and that releasing "over 700 files at once is not a demonstration of scholarship, but a demonstration of power." It closes: "We urge mathematicians to discontinue their work with OpenAI and to return to a vision of science that centers human understanding." The statement is signed by the AHM’s Communications Working Group. India Today reported it as a call to boycott OpenAI.

The AHM also says AGMAI "opened their initial advisory statement by saying that frontier AI corporations should not test advanced mathematical problems on internal models." That matches AGMAI’s September 29 recommendations, which begin: "we do not endorse this practice, and we ask them to stop testing advanced mathematical problems on proprietary models." The same document warns that internal models risk "a two-tier system where labs outrun the rest of the field." OpenAI, per India Today, says it consulted the advisory group and used its advice and public recommendations in deciding how to release the results.

October 7: the original report#

OpenAI just moved AI mathematics from showcase proofs to industrial volume — and left the field to referee hundreds of claimed results from a model nobody outside the company can run.

On October 6, OpenAI published openai/math: 722 manuscripts grouped into 372 families of related results, produced by an internal model it has not released. The README says the model was posed about 4,000 problems during the evaluation, and that each result used on average the equivalent of about three hours of ChatGPT Pro thinking. Work on a zero-free region for the Riemann zeta function and a proof of the Hodge conjecture for CM abelian varieties did not follow that fixed procedure; the zeta write-up was human-edited for readability. OpenAI's accompanying post says it consulted the IAS-hosted Advisory Group on Mathematics and AI (AGMAI), will fund workshops on AI-produced results, and is "working to responsibly release the model."

What the repo actually contains#

  • 722 manuscripts / 372 families — OpenAI's framing is that results "resolve or make substantial progress" on open questions; AGMAI, via The Verge, characterises the release as solutions to "hundreds of open questions." Both characterisations are the sources’ own.
  • 162 papers with a formalized main result, counted from lean/formalization.yaml (catalog header: "papers with a formalized main result"). This is a point-in-time count; the repo says more formalizations will be added.
  • 10 abridged reasoning summaries, including the irrationality exponent of π, Kaplansky's direct-finiteness conjecture in characteristic two, and the Mézard–Parisi formula.
  • An explicit README warning: "some of the unformalized results could have issues."

This sits on top of earlier OpenAI math drops: a Lean-checked Navier–Stokes blowup proposal and the ten Astra-era advances. Those were showcase scale. This is a catalogue.

Claims vs checks#

Scientific American reports that claimed results include the four-dimensional Kakeya conjecture and progress toward the Riemann hypothesis. Those specifics are Scientific American's reading of the manuscripts; we did not re-inspect those PDFs. The Riemann work in the README is a zero-free region / progress, not a proof of the hypothesis.

An OpenAI spokesperson told Scientific American that almost every result came from a single prompt to a single agent, though some may have taken several attempts, and that OpenAI's own mathematicians do not yet understand many of them. That single-agent claim is a spokesperson statement with a caveat, not a README fact.

AGMAI recommended disclosing the model, exact prompt and compute behind each result. OpenAI is publishing average compute and some statistics, not prompts. The spokesperson said OpenAI is not bound by the recommendations.

Mathematicians quoted in Scientific American split: Andrew Sutherland (MIT) treats one-shot claims as unverified until the model is released; Daniel Litt (Toronto) called the release good for mathematics.

The trade#

A single unreleased model produced several hundred claimed results at a reported average cost of about three hours of Pro-level compute each. That is a capability signal about whatever OpenAI ships next — and a stress test of how a field absorbs artefacts faster than it can referee them. About a fifth of the manuscripts (162 of 722) have a machine-checked main result. The rest carry OpenAI's own warning. "Solved hundreds of open problems" is OpenAI's and AGMAI's characterisation, not an independent verdict.

Limits#

  • October 8: the Cambridge–KCL paper is an unreviewed preprint about the September Navier–Stokes formalisation; it does not assess the 722-manuscript catalogue or the correctness of OpenAI’s written proof.
  • October 8: the AHM statement is undated on its website; it refers to October 6 as "yesterday." Scientific American reported the group's launch in early September; we did not verify its size or membership.
  • Model, per-result prompts and per-result compute are withheld.
  • Formalization count 162 is a point-in-time recount of lean/formalization.yaml; the catalogue is designed to grow.
  • Kakeya / Riemann manuscript specifics = Scientific American's reading, not our PDF audit.
  • 372 families are related-result groups, not a count of solved problems.

Sources#

Prior Coverage

Earlier Times of AI reporting on this thread.

Related Coverage

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