OpenAI's Claim to Have Solved Navier-Stokes Draws Scrutiny Over Rivals' Private Codex Sessions

OpenAI says a swarm of roughly 10,000 AI agents cracked one of mathematics' seven Millennium Prize Problems in 88 hours, but the announcement has been overshadowed by accusations that the work may have benefited from two rival mathematicians' private data. The company said an internal, unreleased model — described as more capable than its GPT-6 Astra system — coordinated the agent swarm starting September 1, 2026, exchanging roughly 2.7 million messages and generating about 130 billion output tokens to produce a proof that the three-dimensional Navier-Stokes equations can develop a singularity, or "blow up," in finite time, according to OpenAI's own writeup and reporting from CNBC. Formalizing the result as a machine-checkable proof in the Lean proof assistant took a further 17 hours. The Navier-Stokes problem, one of the Clay Mathematics Institute's Millennium Prize Problems, has resisted a rigorous existence-and-smoothness proof for nearly 90 years and carries a $1 million prize; a genuine resolution would rank among the most significant applications of AI to pure mathematics to date, extending a string of OpenAI and rival claims about AI systems tackling unsolved research problems. The credibility of the result has since been clouded by a dispute over how OpenAI came to focus on the problem in the first place. According to Fortune and VentureBeat, OpenAI researcher Sébastien Bubeck said the company mobilized its agent swarm on September 1 after hearing rumors that mathematicians Tristan Buckmaster of New York University and Levent Alpöge, who is affiliated with Anthropic, had already made progress on a version of the same problem using Anthropic's models and OpenAI's own Codex coding tool. Buckmaster has said he asked OpenAI directly whether its model had been trained on, or had access to, the private Codex sessions in which he and Alpöge had been drafting their unpublished work, and that he did not receive a clear answer. He has also said that the technical route OpenAI's system took mirrored an unusual approach he and Alpöge pursued privately, calling it "not the direction one arrives at in a few days by giving a model the problem statement." OpenAI's public position has shifted as the dispute has played out. Chief Research Officer Mark Chen initially said flatly that "no people or AI systems searched through user data to solve this problem," and Bubeck said the company "did not use their prompts or proofs to prompt our models or direct our agents" and that researchers did not see Buckmaster and Alpöge's work until it was made public. OpenAI has since qualified that stance, acknowledging it cannot rule out that de-identified data derived from users' product activity, including Codex usage, helped improve the underlying models over time — a distinction between deliberately searching a specific user's data and that data indirectly shaping model training through routine data pipelines. Buckmaster has additionally alleged that Bubeck pressured him over how to credit the work, proposing that Buckmaster publish and claim the prize himself only if he stated OpenAI's model had also solved the problem and dropped Alpöge's name from the paper because of his ties to Anthropic; he has quoted Bubeck as asking "Why would you ruin your career?" and warning that he did not have to "be nice." Bubeck has disputed that characterization as false and inflammatory and said he has since publicly credited Buckmaster and Alpöge's independent contribution. The episode has fed a broader debate among mathematicians and AI researchers about how much credit and verification AI-assisted proofs deserve, and about data-handling boundaries between competing AI labs that now double as users of each other's developer tools. Fields Medalist Terence Tao has used the controversy to warn that AI companies risk treating hard, decades-old open problems as marketing opportunities rather than as questions requiring the kind of exploratory, insight-building work mathematicians value, likening rushed, resource-heavy problem-solving to "strip-mining" a site whose greater value lies in the context and methods that get destroyed along the way. Developers and enterprise customers, meanwhile, have seized on the affair as a case study in why data-isolation guarantees between an individual's use of a coding assistant and a vendor's future model training matter in practice, not just in policy documents, particularly when the vendor is also racing to beat its own customers to the same result. OpenAI has not disclosed the name of the internal model used, and independent mathematicians have not yet completed a full verification of the Lean-formalized proof, leaving both the mathematical result and the provenance dispute unresolved for now.