OpenAI announced on Tuesday that an unreleased internal model, running roughly 10,000 coordinated AI agents, produced a proof concerning a forced formulation of the 3D incompressible Navier-Stokes equations. The Clay formulation includes Statements C and D concerning smooth applied forcing, so a forced result does not necessarily have to establish the unforced formulation first to qualify. The result, if it holds up, would not by itself resolve one of math's oldest open questions about how fluids move and whether smooth flow can suddenly break down. OpenAI paired the announcement with a machine-verified formalization of the proof in the Lean theorem prover.
According to CNBC, OpenAI launched its first agents about 88 hours before reaching the resolution, with the system arriving at its answer on September 5, 2026. The agents were organized into groups that could communicate within each group, and OpenAI has said its researchers and agents did not see any prior work by rival researchers before the proof's public release. The company also said no specific user data was accessed to solve the problem, though it acknowledged that de-identified data from product usage might have helped improve its underlying models.
A $15 Million Compute Bill and a 1,000-Fold Scale-Up
The scale of the effort was enormous. OpenAI researchers said during press briefings that the run cost millions of dollars in compute, first deploying 1,000 agents for 50 hours to tackle the related Euler equations before scaling up to 10,000 agents for an 11-hour final push on Navier-Stokes, according to XenoSpectrum. That represented a 1,000-fold scale-up over past math runs, with computational costs estimated at roughly $15 million in customer billing rates. The agents had access to a cached version of the internet and code-running tools while working the problem.
OpenAI researcher Sébastien Bubeck said publicly that the company kicked off its Millennium Prize initiative on September 1, 2026, specifically because viral social media rumors suggested rival lab Anthropic had resolved two Millennium Prize problems, as reported by VentureBeat. Bubeck posted that OpenAI wanted to test whether its internal models could pull off a comparable milestone. Those rumors, however, involved a misunderstanding: social media claims that week asserted that Anthropic's Claude had solved Navier-Stokes, when in fact they had confused an individual user's speculative forecast with Anthropic's actual, verified progress improving a bound on the Riemann hypothesis using 60 AI subagents, according to explainx.ai.
NYU Professor Raises Questions About Where the Data Came From
The announcement immediately collided with prior work by NYU mathematics professor Tristan Buckmaster and Anthropic researcher Levent Alpöge, who had achieved a breakthrough of their own on August 15, 2026, proving finite-time blowup with smooth forcing for the 3D Euler equations, the frictionless counterpart to Navier-Stokes. Per Unite.AI, the pair verified their own proof in Lean on August 22, 2026, using a forcing technique developed by Diego Córdoba and Luis Martínez-Zoroa over a year of research. Buckmaster has said he had not seen OpenAI's proof and did not know what its model did or whether his and Alpöge's data was used, but he has also said information about their progress had been passed to OpenAI, and he has publicly questioned whether OpenAI's models had been trained on or had accessed their Codex sessions.
A four-page statement published by Buckmaster on September 7, 2026, claimed that during negotiations on September 6, 2026, Bubeck offered a joint release of the findings but asked Buckmaster to remove Alpöge as a co-author, a request Buckmaster refused, according to AI Weekly. Bubeck has called allegations of improper pressure false and inflammatory. OpenAI has said its researchers and agents did not see Buckmaster and Alpöge's work before the public release, and the company denies inspecting any private user data, while conceding that de-identified platform usage could have helped train its models.
Fields Medalist Warns Against Turning Math Into a Production Game
Fields Medalist Warns Against Turning Math Into a Production Game The dispute has drawn sharp commentary from one of the field's most respected voices. Fields Medalist and UCLA mathematics professor Terence Tao publicly expressed concern about indiscriminate automated problem-solving and the loss of mathematical insight. Tao's concerns include the risk that black-box proofs could yield less mathematical insight.
Official Recognition Could Take Years, If It Comes at All
Even if OpenAI's proof withstands scrutiny, formal recognition is unlikely anytime soon. The Navier-Stokes problem is one of seven Millennium Prize Problems, each carrying a $1 million reward, established by the Clay Mathematics Institute, which is based in Cambridge, Massachusetts, in 2000. Official rules adopted by the institute in 2018 dictate that a proposed solution cannot be submitted directly to the Clay Mathematics Institute and must spend at least two years in a peer-reviewed journal while achieving widespread acceptance across the global mathematics community before any prize committee is even convened.
History suggests such recognition is rare. Since the seven problems were established, the only one officially verified and solved was the Poincaré conjecture, resolved in the 2000s by Russian mathematician Grigori Perelman, according to eNCA. Perelman famously turned down the $1 million reward in July 2010, partly because he felt the institute had ignored Richard Hamilton's foundational contributions to the work. The Clay Mathematics Institute has not commented on OpenAI's proposed solution.
OpenAI has explicitly said it does not intend to claim the $1 million prize payout even if its Navier-Stokes proof is ultimately verified by the mathematical community, framing the release instead as a benchmark demonstration of frontier AI capabilities rather than a financial pursuit. Buckmaster, for his part, has worked with Alpöge on math problems including Navier-Stokes and continues to press for clarity on how his unpublished work factored into a rival lab's headline-grabbing announcement.
