OpenAI claims a Millennium Prize problem
10,000 AI agents prove forced Navier–Stokes blow-up; priority dispute erupts
Disputed
Importance: historic (5 of 5)The takeaway
On 8 Sep 2026 OpenAI released a 166-page paper and a Lean formalisation proving that the 3D incompressible Navier–Stokes equations with a smooth external force can develop a finite-time singularity from smooth initial data.
Status
- Claim
Disputed
- Our reporting
- Medium confidence
- Verification
- Lean-verified
- Importance
- Historic (5 of 5)
- Last verified
- 29 October 2026
Your AI and this story
- GPT-6 Astra131 days after its cutoff
- Claude Opus 5.570 days after its cutoff
- Gemini 4 Hydrogen161 days after its cutoff
- Grok 4.7100 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 70 days before it.
Key facts
- Autoformalisation critique (arXiv 2610.08144, 6 Oct; HN 334 points): Alexander Bastounis, Fabian Circelli and Anders C. Hansen argue that resolving ambiguity in natural-language mathematics for a ‘semantically faithful’ translation into Lean sits arbitrarily high in the SCI/arithmetical hierarchy, so Lean checks of AI autoformalisations ‘may offer no confidence in the original NL argument’; they give examples including this release and state that ‘the formalised Lean proof does not correspond to the NL proof of blow-up of solutions to the Navier-Stokes equations’. New Scientist covered it on 8 Oct (‘OpenAI mistranslated mathematics into code for its Navier-Stokes proof’). OpenAI had not responded publicly as of 8 Oct
- Scale: ~10,000 agents, 88 hours, ~2.7M messages (some reports ~5M), ~130B tokens; Lean formalisation in 17 more hours; led by Sébastien Bubeck
- Claim: a smooth fluid initially at rest, under smooth forcing, develops a singularity in finite time (velocity unbounded, energy bounded)
- Clay Institute (11 Sep): problem has ‘apparently been settled’ but its process is ‘deliberately unhurried’; no prize awarded
- Luis Silvestre: ‘The Clay problem is settled, but the main problem for the Navier-Stokes equations is not.’
- Charles Fefferman: ‘The heroes of the story… are Córdoba and Martínez-Zoroa’
- Buckmaster and Alpöge (with Matei Coiculescu) released forced blow-up results for IPM, 2D Boussinesq and 3D Euler on 7 Sep, obtained with Claude and Codex and Lean-verified on 22 Aug
- TechCrunch (8 Sep): Buckmaster says Bubeck pressed him to drop Alpöge’s credit (‘Why would you ruin your career?’); OpenAI says it ‘did not see any of their work through any means until released publicly’ but could not rule out that de-identified Codex usage data helped indirectly; the week-long OpenAI effort used ~300B output tokens (~$22.5M at Astra list rates)
Show 13 more
- Understanding AI (Kai Williams, 10 Sep) and MIT Technology Review (Grace Huckins, 8 Sep): OpenAI offered Buckmaster co-authorship only without Alpöge; he refused
- Córdoba and Martínez-Zoroa (guest post on Tao’s blog, 4 Oct): OpenAI’s announcement made ‘two separate claims’, forced Navier–Stokes blow-up and unforced Euler blow-up from smooth data; the first, ‘if verified under the stated hypotheses’, settles an allowed Millennium alternative but ‘is distinct from unforced Navier–Stokes blow-up’; they are ‘not in a position to say anything substantive’ about how it compares with their cascade method
- Valerio Capraro’s viral X post ‘OpenAI might have stolen another major proof’ (~9 Sep, HN front page) relayed Andreas Thom’s separate claim about the Astra soficity result (see OpenAI’s unreleased ‘Astra’ model claims ten advances in maths and theoretical CS, with Lean proofs)
- Buckmaster alleged OpenAI may have benefited from his Codex sessions; OpenAI’s statements shifted from ‘cannot rule out’ to denial (‘no user inputs past July 3rd’)
- The Verge (12 Sep): OpenAI spokesperson Laurance Fauconnet said ‘it is impossible for Dr. Buckmaster’s Codex prompts over the last two months to have influenced the system in any way, including training’; Buckmaster says Bubeck offered him near ‘unlimited compute’ and sole authorship of OpenAI’s paper if Alpöge (an Anthropic employee) was left out, which he called a ‘bribe’; Bubeck told the NYT he offered resources and asked ‘how can we have an internal OpenAI project with an Anthropic employee?’
- The Verge (12 Sep): Fauconnet also said ‘since the completion of Navier-Stokes we have made substantial progress on another Millennium Prize problem’; social media speculated about the Hodge conjecture (unconfirmed), and rumours said Anthropic was close to one too
- IEEE Spectrum (Oct 5, ‘OpenAI’s Navier–Stokes claim sparks a reckoning among mathematicians’), reporting from the Heidelberg Laureate Forum (from Sept 13): Peter Scholze: ‘They’re really just solving these difficult mathematical problems as benchmarks, as some kind of PR stunt’; Geordie Williamson: ‘I think that OpenAI behaved extremely poorly, and that we should acknowledge that in the community’; Michael Harris: reports ‘depict OpenAI as bullying and disrupting disciplinary norms’; Jacob Tsimerman: capabilities rose ‘even faster than many people, including myself, expected’. The claim was still described as pending verification
- King’s College London press release (27 Oct) on the Bastounis–Circelli–Hansen preprint: ‘the written proof and its computer-verified version make different mathematical claims’ (in one example the written proof claims more than the Lean version); the authors say this does not show the result is wrong, only that the written proof still needs independent checking. Bastounis: ‘verifying the correctness of the output of an AI remains exceptionally difficult’
- New Scientist (8 Oct) covered the Bastounis–Circelli–Hansen critique under the headline ‘OpenAI mistranslated mathematics into code for its Navier-Stokes proof’ (paywalled, not read). See also Lean kernel soundness bug #14576 for summer 2026 Lean kernel bugs that let false proofs pass
- Scientific American (Joseph Howlett, Oct 13): several groups of experts working through the paper told the magazine that the proof does not follow the ‘Spanish program’ of Diego Córdoba and Luis Martínez-Zoroa, which OpenAI officials had ‘clearly and repeatedly stated’ it did on Sept 8. Gonzalo Cao-Labora (EPFL): ‘When you go to the details, it is a different approach.’ Both use the same loophole in the Clay problem statement (an unphysical external force), but OpenAI’s proof builds the force ‘in a totally distinct way’, so its similarities to Buckmaster and Alpöge’s progress are ‘superficial’. Some experts say a smaller team of agents did follow the program for an intermediate result. Camillo De Lellis (IAS) on the dispute: ‘I honestly don’t see a factual resolution of this.’ The article gives the swarm as about 10,000 copies, 88 hours and 130 billion tokens
- The Decoder (Manuel Uth, Oct 17): relays Scientific American and adds OpenAI researcher Noam Brown on Dwarkesh Patel’s channel: nobody knows what the ~10,000-agent swarm actually contributed; Brown estimates less than 10% of the success came from the multi-agent setup, crediting mainly the base model’s ability to work long and in parallel
- Scott Armstrong (CNRS/NYU Courant) on X, 26 Oct, as relayed by 36Kr (28 Oct; the post itself not yet read): several groups of PDE experts have basically grasped the proof’s ideas; ‘It may go beyond the scope that humans could originally conceive, but it does not exceed human comprehension.’ (translated relay, unverified wording)
- Second wave (BBC, Pallab Ghosh, 28 Oct): ‘A bust-up between mathematicians is fuelling fears that AI could undermine science’ retells Buckmaster’s Codex question and OpenAI’s statements that nobody ‘saw the other teams’ work before it was published’ and that his prompts ‘could not have influenced the system in any way’. On 28 Oct a second mathematician, Jihao Liu, wrote that a draft he had ‘submitted to codex chat prior to July 2026’ may have fed OpenAI’s abundance proof (OpenAI claims the abundance conjecture; Jihao Liu raises priority)
What happened
OpenAI ran a massive swarm of agents on the forced Navier–Stokes blow-up problem, starting 1 Sep on a model in training since 28 Aug. It announced a complete proof with Lean code on 8 Sep. A day earlier, Buckmaster and Alpöge had released related forced-blow-up results for Euler-type equations using Claude and Codex, and Buckmaster accused OpenAI of rushing after learning of their work. Critics note that the official problem statement allows forcing (option C), but that experts regard the unforced question as the real open problem. Wikipedia now hosts a separate article on the priority controversy.
Why it matters
It is the first credible AI claim on a Clay Millennium Prize problem, even if only a technically permitted variant. It also crystallised disputes over credit, data provenance from AI products, and how AI labs announce results, which culminated in the Fields Medallists’ open letter three days later.
The authorship dispute (added 2026-09-29, verification pass)
Per Fortune’s timeline and Scientific American:
- 2026-08-15: Tristan Buckmaster and Levent Alpöge (Anthropic) privately proved that the Euler equations (Navier–Stokes without viscosity) can blow up. They built on the forcing methods of Diego Córdoba and Luis Martínez-Zoroa.
- 2026-08-15 to 08-22: word of this unpublished work reached OpenAI. Sébastien Bubeck’s math team then produced the proof extending it to the forced Navier–Stokes equations.
- 2026-09-03 to 09-06: according to Buckmaster, OpenAI offered him sole authorship of a paper crediting OpenAI’s model, on condition that Alpöge be removed because of his Anthropic affiliation. Buckmaster says Bubeck asked “Why would you ruin your career?” Buckmaster also raised the possibility that OpenAI’s model had seen his Codex session drafts.
- 2026-09-08: Buckmaster posted his statement the night OpenAI announced its result. Bubeck replied “We did not use their prompt or models or proof” and called the account “false and inflammatory”. OpenAI pledged not to claim the Clay prize, and later recruited nine mathematicians to referee its math claims. This is the first public priority and misconduct dispute between frontier labs over a mathematical result.
Sources
42 sources from 28 sites. Numbers match the chips in the text.
42 sources: 4 primary, 23 press, 15 reactions
Primary
- OpenAI: Navier–Stokes solutionopenai.com, official
- London Mathematical Society statement on the Navier–Stokes developments (9 Sep)lms.ac.uk, official
- OpenAI on X: Navier-Stokes solution announcementx.com, official
- Bastounis, Circelli & Hansen: Navier–Stokes lost in translation (arXiv 2610.08144)arxiv.org, paper
Press
- IEEE Spectrum: OpenAI’s Navier–Stokes claim sparks a reckoning among mathematicians (Oct 5)spectrum.ieee.org, press
- TokenPost: OpenAI announces Navier-Stokes proof as Clay Prize review continuestokenpost.com, press
- OfficeChai: Bubeck says he tried to coordinate release with Buckmaster & Alpögeofficechai.com, press
- Quanta: AI has solved one of math’s $1 million Millennium Prize problemsquantamagazine.org, press
- Scientific American: Did OpenAI solve the wrong Navier–Stokes problem?scientificamerican.com, press
- Fortune: OpenAI says it cracked Navier–Stokes; Buckmaster accusationfortune.com, press
- CNBC: OpenAI claims to have solved 90-year-old Navier–Stokes problem in 88 hourscnbc.com, press
- Scientific American: OpenAI claims blockbuster math breakthrough amid swirl of controversyscientificamerican.com, press
- Startup Fortune: OpenAI recruits nine mathematicians to referee its AI math claimsstartupfortune.com, press
- TechCrunch: OpenAI fought dirty on career-making math problem, says NYU mathematiciantechcrunch.com, press
- MIT Technology Review: What OpenAI’s latest controversy tells us about the future of mathtechnologyreview.com, press
- Understanding AI: OpenAI spent millions to solve this famous math problem — mathematicians are furiousunderstandingai.org, press
Show all 23 sources
- The Verge (8 Sep): Drama swirls around OpenAI’s legendary mathematical milestonetheverge.com, press
- The Verge (9 Sep): OpenAI’s sly mathematical breakthrough sends a chill through academiatheverge.com, press
- The Verge (12 Sep): OpenAI just wants to win (Buckmaster and Thom interviews)theverge.com, press
- The Verge: All the drama around AI’s takeover of mathematics (storystream, updated Oct 5)theverge.com, press
- Quanta (Oct 5): Is AI the End of Math As We Know It?quantamagazine.org, press
- King’s College London: OpenAI’s Navier-Stokes proof called into question (27 Oct)kcl.ac.uk, press
- New Scientist: OpenAI mistranslated mathematics into code for its Navier-Stokes proof (8 Oct; paywalled, not read)newscientist.com, press
- Scientific American: OpenAI’s blockbuster Navier-Stokes proof isn’t what we thought it was (Joseph Howlett, Oct 13)scientificamerican.com, press
- The Decoder: Mathematicians say OpenAI doesn’t understand its own Navier-Stokes proof (Oct 17)the-decoder.com, press
- 36Kr: OpenAI’s 88-hour Navier–Stokes proof, mathematicians spend 7 weeks decoding it (28 Oct)eu.36kr.com, press
- BBC (28 Oct): A bust-up between mathematicians is fuelling fears that AI could undermine sciencebbc.co.uk, press
Reactions
- Sebastien Bubeck on X: allegations are “false and inflammatory”x.com, discussion
- Terence Tao: finite-time blowup with smooth forcing (Buckmaster–Alpöge–Coiculescu)terrytao.wordpress.com, discussion
- Wikipedia: Navier–Stokes priority controversyen.wikipedia.org, discussion
- Alexander Gamburd: The Siren Call of Silicon Leviathan (arXiv 2609.28591, reflective essay)arxiv.org, discussion
- Córdoba & Martínez-Zoroa (guest post on Tao’s blog): On classical solutions and singularity formation in incompressible fluidsterrytao.wordpress.com, discussion
- Anima Anandkumar: Stable singularity of the Euler equations on R³ (concurrent unforced result)terrytao.wordpress.com, discussion
- Techmeme cluster, 2026-09-08techmeme.com, discussion
- Noam Brown on X: OpenAI mathematicians’ ‘Lee Sedol moment’x.com, discussion
- Tristan Buckmaster on Mastodon: three blow-up results and statementmastodon.social, discussion
- Terence Tao on Mathstodon: Alpöge–Buckmaster, a remarkable achievementmathstodon.xyz, discussion
- Terence Tao on Mathstodon: open problems as a non-renewable resource (thread)mathstodon.xyz, discussion
- Valerio Capraro on X: ‘OpenAI might have stolen another major proof’ (Thom/soficity)x.com, discussion
Show all 15 sources
- Hacker News: OpenAI might have stolen another major proofnews.ycombinator.com, discussion
- HN discussion: Navier–Stokes Lost in Translation (334 pts)news.ycombinator.com, discussion
- Dwarkesh Patel: OpenAI researcher on agent swarms & recursive self-improvement (Noam Brown; cited at 5:35)youtu.be, video
Changes
- Added BBC’s 28 Oct second-wave piece and the parallel Codex claim by Jihao Liu
- Armstrong’s 26 Oct X comment via 36Kr (wording unverified; science agent)
- KCL press release (27 Oct) on the Bastounis–Circelli–Hansen critique (news agent)
- Fixed ai_system (an unquoted comma in ‘10,000’ split it into two items)
- Added video(s) rest-is-science-mathematicians-canary-ai-coal-mine, kanal-zero-dragan-openai-navier-stokes-skandal, samuel-alexander-allegations-openai-stole-navier-stokes
- Added videos primetime-the-navier-stokes-situation-is-insane, quanta-why-navier-stokes-pushes-math-and-physics-to-the-edge, alex-hyman-navier-stonks-lost-in-translation-informal-vs-lean
Show all 23 changes
- Video added (dwarkesh-noam-brown-agent-swarms-recursive-self-improvement)
- Added The Decoder second wave (Oct 17) and Noam Brown’s <10% swarm-contribution estimate (Dwarkesh Patel video)
- Added Scientific American’s report that experts find the proof does not follow the Córdoba–Martínez-Zoroa program (Google News)
- Added New Scientist’s headline on the autoformalisation critique and a pointer to the Lean kernel soundness-bug entry
- Added the Bastounis–Circelli–Hansen autoformalisation critique (arXiv 2610.08144) claiming the Lean proof does not match the natural-language proof
- Science block: verification says what the Lean proof checks and what is disputed (the forced variant, not the proof)
- Added IEEE Spectrum’s Heidelberg Laureate Forum reactions (Scholze, Williamson, Harris, Tsimerman)
- Linked the videos “OpenAI’s biggest math breakthrough is getting ugly...”, “The Controversy Behind AI Solving A Million Dollar Math Problem”, “The world’s best mathematicians already overtaken by AI”, “Has a Millennium Prize Problem been solved by AI?”
- Added The Verge’s Sept 8/9/12 articles (found via its Oct 5 storystream; the 10 Sep original is linked from OpenAI’s unreleased ‘Astra’ model claims ten advances in maths and theoretical CS, with Lean proofs), OpenAI’s ‘substantial progress on another Millennium Prize problem’ statement, and Quanta’s Oct 5 essay
- Added Córdoba & Martínez-Zoroa’s 4 Oct guest post on Tao’s blog (their survey of the cascade method and their non-committal statement on OpenAI’s two claims)
- Added TechCrunch (300B tokens/~$22.5M, Bubeck quote), MIT TR, Understanding AI and Capraro/HN links; The Verge (10 Sep) and Economist (11 Sep) originals not found
- Added Bubeck’s X reply and OfficeChai coverage of his fuller response
- Added authorship/misconduct dispute and SciAm/Fortune sources (verification pass)
- Added post link(s)
- Added post link(s) (3) from Google/DeepMind + math posts pass
- Added Gamburd essay (arXiv 2609.28591: 616k lines of Lean per its abstract), LMS statement, the concurrent Anandkumar Euler result, and related Royal Society/ICIAM entries
- Filed
