- Fluid Breakdown: OpenAI claims modeled fluid speeds can grow without limit in finite time under a finite external push without abrupt changes.
- Research Credit: Mathematician Tristan Buckmaster disputes OpenAI’s handling of his work with Levent Alpöge and proposed authorship.
- Data Question: OpenAI denies accessing their private work directly but leaves possible contributions to model training unresolved.
- Proof Review: OpenAI reports computer-checked proofs; outside mathematical acceptance and recognition by the prize-awarding Clay Mathematics Institute remain separate.
OpenAI claims its AI agents have solved the difficult Navier-Stokes equations and proved that equations describing fluid motion can produce speeds growing without limit in finite time. The Navier–Stokes equations mathematically express momentum balance for Newtonian fluids.
The proposed result to the previously unresolved Navier-Stokes problem, released September 8, would settle a longstanding mathematical question about whether these equations always retain well-behaved solutions under specified conditions. It also arrives amid a dispute with two mathematicians over research credit and unpublished work entered into OpenAI’s tools.
Tristan Buckmaster, a mathematics professor at New York University, and his collaborator Levent Alpöge had announced related results hours earlier. Their work and OpenAI’s draw on an existing academic approach to fluid singularities. Buckmaster questions how OpenAI’s effort developed and how the company proposed assigning authorship; OpenAI disputes his characterization of their discussions.
How a Smooth Push Can Produce a Singularity
The Navier-Stokes equations describe fluids with viscosity, the internal friction that smooths differences in motion. A singularity is a breakdown of the smooth mathematical solution. OpenAI’s proposed construction starts at rest and works for every positive viscosity: speed becomes unbounded in finite time while total kinetic energy stays bounded.
Its vortex spins faster as its core shrinks. Both its width and height decrease, concentrating the highest speeds into a vanishingly small volume. That allows rising local speed without infinite total energy.
Keeping the external force smooth is the difficult part. The vortex alone would require a force that also becomes singular at its edge. OpenAI’s argument adds oscillating pulses that draw energy from differences in the surrounding flow and transport momentum, supplying the missing balance internally. For example, outward motion carrying extra rotational velocity and inward motion carrying a deficit both transport angular momentum outward.
The official Clay problem formulation explicitly allows it in two breakdown alternatives: one for fluid filling three-dimensional space, the other for a spatially repeating flow. OpenAI claims both. These alternatives differ from showing breakdown without an external force.
That choice of approach helps explain the credit dispute. Mathematicians Diego Córdoba and Luis Martínez-Zoroa developed a program that combines increasingly fine layers of flow to create singularities. Quanta magazine explains how keeping the combined force smooth was a remaining hurdle. Charles Fefferman, the Princeton mathematician who wrote Clay’s formulation, credited the pair as the intellectual heroes of the story.
OpenAI’s Navier-Stokes manuscript and Buckmaster and Alpöge’s Euler paper acknowledge those predecessors. Buckmaster says the smooth-forcing approach was the unusual direction he and Alpöge had quietly pursued, which is why hearing that OpenAI had taken it alarmed him.
The Euler Results Address Different Equations
Euler equations omit viscosity. Buckmaster and Alpöge’s Euler manuscript claims a singularity with smooth external forcing and a smooth, initially swirling flow. The quantities becoming unbounded concern spatial changes in the flow and vorticity, a measure of local rotation.
OpenAI’s separate Euler manuscript uses no external force. It starts from a smooth velocity confined to a bounded region and claims that the velocity gradient, which measures how motion changes across space, becomes unbounded. Its Navier-Stokes result instead includes viscosity and smooth forcing and claims unbounded speed itself. Euler itself is outside Clay’s list of prize problems.
Buckmaster also reported a possible result for a modified Navier-Stokes equation with weaker dissipation. He released a statement saying computer checking remained unfinished and the paper was not ready for release.
Human Direction Behind the Agent Effort
OpenAI says it launched its research effort on September 1 after hearing rumors that two Millennium Prize problems had been solved. The company says groups of agents explored different problem variants, communicated, read a cached internet and ran code.
Nearly 100 agents worked for about 50 hours on the Euler result. Humans then redirected resources toward Navier-Stokes, supplied that result to the agents and used Codex, OpenAI’s coding assistant, to consolidate insights from different groups. The successful group involved roughly 10,000 concurrent agents. OpenAI reports reaching its Navier-Stokes result on September 5, about 88 hours after launch, followed by 17 hours to translate the proof into the Lean computer-verification system and check it.
The company attributes approximately 130 billion output tokens, units of generated text, and 2.7 million messages to Navier-Stokes. Its roughly 300 billion tokens and 4.9 million messages cover all the problems attempted.
Buckmaster says he and Alpöge used AI tools including Claude and Codex throughout their project. Progress was slow for most of the preceding year as they worked through the literature and preliminary results. He dates their Euler and related breakthroughs to August 15 and Lean checking to August 22, followed by sustained work to understand the arguments and make them readable.
Although Alpöge works at Anthropic, Buckmaster says their collaboration was personal, without official employer involvement, and that his research funds paid for their tools.
What the Authorship Dispute Concerns
Buckmaster says two September 6 calls with OpenAI researcher Sébastien Bubeck and another company mathematician included a proposal that he write up OpenAI’s Navier-Stokes result without Alpöge. He attributes that exclusion to Bubeck’s objection to Alpöge’s Anthropic employment.
By September 10, Bubeck had denied seeking to remove Alpöge from authorship of Alpöge’s own work. He said the proposal concerned a rewrite of OpenAI’s proof, and that he considered an Anthropic employee authoring that work inappropriate.
When Buckmaster said he would go public if OpenAI released its result as proposed, he says Bubeck asked, “Why would you ruin your career?” Bubeck subsequently apologized for his wording, said he had retracted it immediately and described his concern as risking a career over accusations he considered unfounded.
Buckmaster says they put their unpublished drafts into Codex throughout the project. He says a denial that the model looked up user data did not answer his separate question about whether the drafts helped train it. “I do not know whether our data was used,” he wrote.
OpenAI denies that its researchers and agents saw the pair’s work before public release and says no specific user data was accessed to solve the problem. Separately, it says it cannot rule out de-identified data from their product usage helping improve its models, although it considers that unlikely. Whether these particular drafts contributed to training or the result remains unresolved.
Researchers need clarity about what happens to unpublished ideas entered into AI tools, particularly when the provider also conducts research.
What Formal Checking Establishes
OpenAI has released Lean formalizations alongside its manuscripts on GitHub. Lean is a proof assistant that checks logical steps in software. The reported successful check concerns the statements encoded in that system. As Quanta explains, humans still need to establish that those statements match the mathematical problem intended.
Before considering a proposed solution, the Clay Mathematics Institute requires publication in a qualifying outlet, at least two years since publication and general acceptance by the global mathematics community. OpenAI says it does not intend to claim the prize.
