OpenAI Says AI Solved Decades-Old Maths Problem

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OpenAI says an experimental AI system has solved a 90-year-old mathematics problem by proving that equations used to describe the movement of fluids can, under certain conditions, produce speeds that grow without limit.

The question, known as the Navier-Stokes existence and smoothness problem, is one of seven Millennium Prize Problems selected by the Clay Mathematics Institute in 2000. Each carries a $1 million prize for a solution that meets its rules.

The Navier-Stokes equations describe how fluids such as water and air move and are important in fields including aircraft design, weather forecasting and research into blood flow.

Mathematicians have long known that the equations work in many practical situations. What they have been unable to establish is whether a smooth three-dimensional flow must always remain well behaved or whether the equations can eventually produce a breakdown in which fluid speed becomes unlimited.

OpenAI says its AI found such a breakdown.

Its proof starts with a smooth fluid at rest and applies a smooth external force. The fluid develops an increasingly concentrated vortex until its speed becomes unbounded in a finite period, while its total energy remains finite. OpenAI says this satisfies two of the possible ways, known as statements C and D, that Clay’s official formulation allows the problem to be resolved.

That distinction is important. OpenAI has “not” shown that an unforced Navier-Stokes fluid will necessarily develop such a breakdown. Its result uses a carefully constructed smooth force.

But using a force does not put the result outside the Millennium Prize problem. Clay’s official formulation explicitly allows a solution that shows a breakdown with smooth initial conditions and a smooth external force.

OpenAI has released both a mathematical paper and a version of the proof written in Lean, software that can formally check each logical step. Its public repository says the proof covers both the whole-space and periodic versions corresponding to statements C and D in Clay’s formulation.

The formal check gives the claim unusual weight, but it does not make the Millennium problem officially solved.

Clay’s website continued Wednesday, September 9, to list Navier-Stokes among its unsolved problems. Its rules require a proposed solution to be published in a qualifying outlet, remain under scrutiny for at least two years and gain general acceptance among mathematicians before the institute will consider it.

OpenAI said it does not intend to seek the $1 million prize.

“Our goal in releasing this result is to report on the substantial progress of our AI models,” the company said.

How 10,000 AI Agents Attacked the Problem

OpenAI began the experiment on September 1 after hearing rumors that two Millennium Prize Problems had been resolved and while testing a new internal model that it says is significantly more capable than GPT-6 Astra.

The model is still being trained and has not been released publicly.

Rather than asking a single AI system to solve Navier-Stokes, OpenAI divided major mathematics problems among groups of coordinating agents and encouraged them to try different approaches.

It also gave the system several easier related problems.

One involved the Euler equations, which are closely related to Navier-Stokes but leave out viscosity. Nearly 100 agents worked for about 50 hours and produced a proof that the unforced Euler equations can develop a singularity, OpenAI said.

That result persuaded researchers to concentrate their resources on Navier-Stokes.

OpenAI shifted agents from the other Millennium problems, gave them the Euler result and used Codex to combine promising ideas generated by different groups.

Around 10,000 agents eventually worked concurrently on Navier-Stokes.

They reached the result on Saturday, September 5, about 88 hours after the experiment began. GPT-6 Astra then spent another 17 hours turning the argument into a formal Lean proof and checking it.

The Navier-Stokes agents exchanged about 2.7 million messages and generated roughly 130 billion output tokens. Across all the problems attempted, the system exchanged 4.9 million messages and produced about 300 billion tokens.

OpenAI research chief Mark Chen told Wired that the computing effort cost millions of dollars.

Mathematicians Assess the Proof

Independent reaction so far has been strikingly positive, though the work is still new.

Quanta reported that the Lean formalization has given mathematicians confidence that the proof is mathematically sound, while stressing that its significance and the circumstances surrounding its discovery will take time to assess.

Charles Fefferman of Princeton University, who wrote Clay’s official description of the Navier-Stokes problem, told Quanta he was “thrilled that the problem was solved.”

He also pointed to years of work by mathematicians Diego Córdoba and Luis Martínez-Zoroa as central to the route that eventually led to the breakthrough.

That route is now at the center of a second controversy.

Credit Dispute Follows the Breakthrough

New York University mathematician Tristan Buckmaster and Levent Alpöge, a mathematician who works at Anthropic, had separately been pursuing closely related fluid-dynamics problems.

Their work built on an approach developed by Córdoba and Martínez-Zoroa that seeks to produce singularities using smooth external forcing.

Buckmaster and Alpöge used several AI systems, including OpenAI’s Codex and Anthropic’s Claude, in their research. They produced results showing finite-time breakdown for the three-dimensional Euler equations and other related systems with smooth forcing.

OpenAI acknowledges that rumors about their work helped trigger its September 1 experiment.

But Buckmaster has questioned how OpenAI came to focus so quickly on the same unusual route after learning that he and Alpöge were making progress.

“The route to the Clay problem through a smooth force … is the route Luis and Diego opened and the one Levent and I had quietly chosen to attack,” Buckmaster wrote in a statement.

He has also raised the possibility that information from the pair’s extensive use of OpenAI products could have influenced OpenAI’s models.

OpenAI denies that its researchers or AI agents accessed Buckmaster and Alpöge’s private work while solving the problem.

“We did not see any of their work through any means until they released it publicly,” OpenAI said, adding that no specific user data was accessed to solve Navier-Stokes.

But the company added a qualification that has drawn attention.

“While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models,” it said.

OpenAI says its proofs differ substantially from Buckmaster and Alpöge’s work and has recognized their priority on their Euler result.

The dispute also extends to discussions over how the competing findings would be announced and who would receive credit.

The Financial Times reported Wednesday that Buckmaster said OpenAI researcher Sébastien Bubeck proposed arrangements under which the researchers’ work and OpenAI’s result could be coordinated for publication. Buckmaster objected to how Alpöge, who works for OpenAI rival Anthropic, would be treated under one proposed arrangement.

Bubeck has disputed Buckmaster’s characterization, saying he never proposed removing Alpöge from authorship of work that Alpöge had done. OpenAI says it initially sought a coordinated announcement because it believed Buckmaster and Alpöge had independently solved Navier-Stokes. It later learned that their completed result concerned the related Euler problem.

The disagreement does not by itself determine whether OpenAI’s Navier-Stokes proof is correct. But it has opened a broader question about how scientific credit should work when researchers use commercial AI systems on unpublished work and those same companies are developing models capable of competing with them.