AI Agent Solves Navier-Stokes, Sparking Priority Dispute

OpenAI says an internal artificial intelligence system organized roughly 10,000 coordinating agents to produce a proof of the Navier–Stokes existence and smoothness problem — one of the seven Clay Mathematics Institute Millennium Prize Problems — in about 88 hours. The announcement, published on 8 September, immediately ignited a dispute over credit and priority, with a New York University mathematician claiming that closely related advances by him and a collaborator preceded the AI’s breakthrough.

According to CNBC, OpenAI used a system of 10,000 “coordinating agents” powered by its internal AI model to tackle the problem, which relates to how fluids move and has resisted solution for some 90 years.

What the AI Actually Proved

The result is more nuanced than the headline suggests. Navier–Stokes is a system of partial differential equations describing the velocity and pressure of fluids. The Millennium problem asks whether smooth solutions always exist for incompressible three-dimensional flow. As the Wikipedia entry on the controversy explains, the problem is framed around four alternatives set out by mathematician Charles Fefferman.

OpenAI did not prove that solutions are always smooth. Instead, it pursued the negative “breakdown” direction — constructing a finite, smooth external force under which a smooth solution cannot be sustained for all time. In its release, the company described a system in which agents could read from a cached version of the internet and run code.

“The agents had access to tools such as the ability to read from a cached version of the internet and the ability to run code. Agents were subdivided into groups with the ability to communicate within the group,” OpenAI said. “The groups varied in size, and the group that produced the Navier–Stokes resolution involved on the order of 10,000 concurrent agents.”

“The agents arrived at their resolution on Saturday, September 5, about 88 hours after the first agents were launched,” the company added.

The Priority Dispute

OpenAI’s work drew immediate questions from Tristan Buckmaster, a professor of mathematics at the Courant Institute at New York University. In a statement on his website, Buckmaster said he had been working in a personal collaboration with mathematician Levent Alpöge, who works at OpenAI rival Anthropic, on problems including Navier–Stokes.

Buckmaster said Alpöge had received tips that information about the pair’s progress had been passed to OpenAI, and that the company’s route to the solution resembled their own work — which he said is “not the direction one arrives at in a few days by giving a model the problem statement.”

He also raised questions over whether OpenAI models had been trained on, or had access to, the mathematicians’ sessions in OpenAI’s Codex, noting they had used several large language models in their work. But he was careful to hedge: “I would like to be clear about what I am not claiming. I have not seen OpenAI’s proof. I do not know what their model did, or how. I do not know whether our data was used,” he said.

OpenAI said its effort began on 1 September after “hearing a rumor” about progress on the puzzle that it later learned related to Alpöge and Buckmaster. “We (the researchers and the agents) did not see any of their work through any means until they released it publicly — in particular, no specific user data was accessed in order to solve this problem,” the company said. “While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models.”

A Turning Point for Open Science

The most consequential reaction may have come from Fields Medalist Terence Tao, who warned that AI’s brute-force entry into mathematics is eroding centuries-old open-science traditions. In a thread on Mastodon, Tao argued that the collection of good, fruitful open problems is being mined “in a non-renewable fashion,” leading to a scenario in which those problems become scarce.

He compared the situation to a country surrounded by ocean yet short of drinking water: mathematical problems are abundant, but the scarce resource is knowing which questions are worth pursuing. AI, he wrote, acts without clear frontiers, flattening mathematics’ “difficulty landscape.”

“We have now seen that even the rumor of someone working on a problem can trigger a massive amount of AI-powered effort to flatten it before the original research project has time to reach its full potential,” Tao wrote. “The incentives may now be pointing in the direction of no longer sharing any promising research directions with the broader community, which would reverse centuries of traditions of open science and do serious long-term damage to the future of the field.”

His conclusion was blunt: “Good questions will become scarcer than good answers.”

What’s Next

Neither result has been peer-reviewed, and the Clay Mathematics Institute has not acknowledged either party. The Institute requires peer-reviewed publication before awarding any prize, and its president, Martin Bridson, has called the announcements “exciting” while stressing that standard. OpenAI has said it will not claim the $1 million prize.

The episode leaves several questions unresolved: whether the two efforts were genuinely independent, how credit should be apportioned if they were, and whether confidential user interactions with AI coding tools can influence model training in ways that reshape scientific priority. For now, the mathematical community must independently verify both the 166-page paper and the mathematicians’ claims — a process that will take far longer than the 88 hours it took to produce the result.

What the field watches next is not only whether the proof holds, but whether the norms that govern how discoveries are shared can survive the arrival of machines that solve problems faster than humans can discuss them.