A Fight Triggered Immediately Over Who Solved It First
An unreleased OpenAI model reportedly produced a machine-checked proof that three-dimensional fluid flow governed by the Navier–Stokes equations can break down in finite time, a question that has stood unresolved since the 1930s — but the announcement landed hours after two human mathematicians, one of them an Anthropic researcher, said they had independently solved a related version of the same problem.
On 8 September 2026, OpenAI published a claimed solution to the Navier–Stokes existence-and-smoothness problem, one of the seven Clay Mathematics Institute Millennium Prize Problems that have each carried a $1 million prize since 2000. Before this week, only one of the seven — the Poincaré conjecture, solved by Grigori Perelman in 2003 — had ever been resolved. If the OpenAI result withstands the scrutiny now converging on it, it would be the second, and the first ever produced primarily by an AI system rather than human mathematicians.
The Navier–Stokes equations describe how fluids — water, air, blood, weather systems — move under forces like viscosity and pressure. The specific open question, posed by the Clay Institute in 2000, asks whether smooth, well-behaved three-dimensional fluid flow can always be relied upon to stay smooth, or whether the equations can spontaneously develop a singularity — a point where some physical quantity, such as velocity, blows up to infinity in finite time. OpenAI’s model reportedly produced a configuration in which a vortex tightens and spins faster and faster while the fluid’s total energy remains bounded — a finite-time blowup, resolving the question by showing that smoothness is not guaranteed.
How the proof was reportedly produced
According to OpenAI’s own account, an internal model — described as “significantly more capable than GPT-6 Astra” and not yet publicly released — was set loose on the problem starting 1 September 2026, after the company heard rumours that two mathematicians outside OpenAI were closing in on a related result. The company says roughly 10,000 AI agents worked the problem concurrently over about 88 hours, exchanging 2.7 million messages and consuming on the order of 130 billion output tokens, at a computing cost reportedly running into the millions of dollars. A smaller group of about 100 agents is said to have first resolved an intermediate, unforced version of the Euler regularity problem as a stepping stone. The final proof — reported at 165 pages — was then formally verified using Lean, a proof-assistant language that checks each logical step of a mathematical argument by machine rather than relying solely on human peer review. OpenAI has said it does not intend to claim the $1 million prize itself.
The credit dispute
The announcement did not land cleanly. OpenAI has said that the rumour it acted on traced back to Tristan Buckmaster, a mathematics professor at New York University, and Levent Alpöge — who works at Anthropic. Alpöge and Buckmaster released their own paper on 7 September, one day before OpenAI’s announcement, describing a solution to a related fluid-equation problem that the two mathematicians say they had been developing for roughly a year using Anthropic’s models as a research aid. OpenAI has said it contacted the two researchers after completing and verifying its own proof on 6 September, intending to propose a joint announcement, only to learn the pair’s work was already essentially complete and about to be released independently. The overlapping timelines, and the fact that OpenAI’s own account places the origin of its urgency in a rumour about a rival lab’s researcher, have made the sequence of events itself a subject of scrutiny alongside the mathematics.
Martin Bridson, president of the Clay Mathematics Institute, called it “an exciting day” for mathematics while the claims are examined. Independent verification of a 165-page, largely machine-generated proof is not instantaneous: mathematicians outside both labs will need time to work through the argument and the Lean formalisation before the result can be considered established rather than claimed. Coverage in Nature‘s news pages on 8 September was notably measured, describing the announcement as significant but treating both the OpenAI and the Alpöge–Buckmaster results as claims requiring scrutiny rather than settled fact.
Why it matters
Independent of who is ultimately credited, the episode marks a genuine inflection point in how frontier mathematics gets done. AI systems have previously assisted with proof search, symbolic computation and formalisation, but a claimed resolution of a Millennium Prize Problem — even one requiring careful independent checking — is a different order of event: it suggests that large-scale, machine-verified mathematical reasoning at genuine research frontiers is now plausible rather than speculative. The credit dispute is also instructive in its own right, and not only as controversy: it surfaces a real and unresolved question for AI-assisted science generally, which is how research credit and safe collaboration norms should work when a researcher at one frontier AI lab is using a rival lab’s models to do unpublished work that the rival lab’s own systems may then independently converge on. For a discipline in which priority disputes are already common even between purely human teams, adding a fast, secretive, compute-intensive AI actor into the mix raises the stakes and the speed at which such disputes now unfold.
– Anil Atluri
Key facts
– OpenAI claims an internal, unreleased AI model produced a finite-time-blowup solution to the Navier–Stokes existence-and-smoothness problem, one of seven Clay Institute Millennium Prize Problems (only one, the Poincaré conjecture, had previously been solved)
– Proof: 165 pages, verified via the Lean formal-proof language; OpenAI says it will not claim the $1 million prize
– Process: ~10,000 AI agents over ~88 hours, begun 1 September 2026 after OpenAI heard rumours of a rival result; completed and internally verified 6 September
– A day earlier (7 September), NYU’s Tristan Buckmaster and Anthropic researcher Levent Alpöge released their own paper on a related fluid-equation result, developed over roughly a year using Anthropic’s models
– Neither claim has yet completed independent peer review; both are being treated by the mathematics community as significant but unverified as of 8–9 September 2026
