The Million-Year Equation

The Navier-Stokes equations describe how fluids move. They have been around since the nineteenth century, and they are used for aircraft design, weather forecasting, and blood flow modeling. For roughly 90 years, one question about them has remained unanswered: can a smooth, three-dimensional fluid motion develop a singularity in finite time? In plain terms, can the math break down even when everything starts out calm?

This is one of the seven Millennium Prize Problems, named by the Clay Mathematics Institute in 2000. It carries a literal million-dollar prize. Last week, OpenAI announced that an internal model produced a proof: yes, a singularity can develop. The proof is a vortex that spirals inward and stretches like spaghetti, speeding up while its energy stays finite, until the continuum approximation of the fluid breaks.[1]

10'000 agents, 130 billion tokens

The proof was not produced by a single model thinking quietly. OpenAI used a system of coordinating agents, subdivided into groups, with access to cached internet and code execution. The group that found the Navier-Stokes resolution involved on the order of 10'000 concurrent agents. Across all attempted problems, the agents sent 4.9 million messages and used about 300 billion output tokens. The Navier-Stokes portion alone consumed 2.7 million messages and roughly 130 billion tokens.[1]

The process took 88 hours from launch to solution, plus another 17 hours for Lean formalization via GPT-6 Astra. The internal model used is described as "significantly more capable than GPT-6 Astra" and has been training since August 28.

The accidental detour

The agents did not go straight for Navier-Stokes. OpenAI asked the system to try a set of easier problems first, including the regularity problem for the Euler equations, which is Navier-Stokes with the viscosity term removed. Nearly 100 agents worked for about 50 hours and resolved the unforced Euler problem, something nobody expected. Once they saw that result, OpenAI shifted resources away from the other Millennium Problems and focused everything on Navier-Stokes, feeding the Euler resolution back into the agent groups as a hint.

Concurrent work and a controversy

The same week, Levent Alpöge (an Anthropic employee) and Tristan Buckmaster (a math professor at NYU) produced a resolution of the forced Euler problem using an internal Anthropic model. OpenAI reached out after completing their own work and offered a joint announcement. They also investigated whether Buckmaster's Codex prompts could have influenced their system through training data, and claim they found no such influence.[1]

The math community pushes back

Eight days before OpenAI's announcement, the Advisory Group on Mathematics and Artificial Intelligence at the Institute for Advanced Study published a set of recommendations for how AI labs should release mathematical results.[2] The document is blunt. It opens with: "we do not endorse this practice, and we ask them to stop testing advanced mathematical problems on proprietary models."

The AGMAI recommendations distinguish between papers that a human understands and papers that nobody understands. For the latter category, which is where the Navier-Stokes proof falls, they ask AI labs to: cite related prior work properly, write proofs in conventional mathematical style, deposit results in community-controlled repositories (not lab-controlled ones), publish prompts and summarized chain-of-thought, formalize proofs in Lean, and fund workshops and postdocs so that human mathematicians can actually understand what was produced.

They also wrote: "We strongly recommend that AI labs refrain from treating the release of mathematical results as marketing vehicles to promote their models."

OpenAI's announcement does follow several of these recommendations. They published the results in a GitHub repository with Lean formalizations, released reasoning summaries and compute estimates, and committed to funding workshops and conferences. The average result used the equivalent of roughly three hours of ChatGPT Pro thinking. They also explicitly stated they will not claim the Millennium Prize.

But the tension is visible. The proof was produced by a model nobody outside OpenAI can access. Nobody at OpenAI fully understands the proof. The AGMAI document asks labs to stop doing exactly this, and OpenAI did it anyway, while following the release guidelines in good faith.

What it means

A Millennium Prize Problem is now solved, sort of. The proof exists, it is formalized in Lean, and the Clay Institute will have to decide whether it counts under their rules (which require publication in a peer-reviewed journal and community acceptance over two years). Whether or not the prize is awarded, the result is real: an AI system produced a valid mathematical proof of a problem that has been open since 1934.

The bigger question is the one AGMAI is asking. If AI labs can produce frontier mathematics on proprietary models that nobody else can run, we get a two-tier system where one lab outruns the entire field. The math community built its verification norms over centuries. OpenAI just demonstrated that those norms are now the bottleneck, not the math.