OpenAI Cracks Half of Navier Stokes Problem in 88 Hours
The Bot Swarm Against a Nine-Decade-Old Proof
Eighty-eight hours. That is how long, by OpenAI's account, it took a group of about 10,000 AI agents to crack a piece of a problem that has stood unproven for 90 years. The problem is the Navier-Stokes existence and smoothness problem, a notoriously difficult mathematical challenge arising from the equations that describe how fluids move. OpenAI announced the result on Tuesday, called it a "milestone," and offered it as evidence that AI tools are improving quickly. An AI agent is not an ordinary piece of software; it is a bot that undertakes its task with some autonomy. The scale of what happened deserves attention before the mathematics does. OpenAI says the agents exchanged nearly 3 million messages with one another during the work. The output they generated came to 130 billion tokens, where a token is the individual unit of text and code that an AI model produces in its answers. At OpenAI's own pricing for output from its most advanced models, an effort of this size would have cost about $10m, or roughly 7.3m pounds, if someone had paid for it as a commercial service. The computation at the heart of this announced result was therefore expensive by any standard except one: the formal prize attached to the problem is only $1m. The machines were working on a question that has frustrated mathematicians for nine decades, but the underlying capability is fresh. OpenAI says it only began training the new model at the end of August. The model quickly showed that it was unusually good at mathematics, according to the company. OpenAI describes the model as "significantly more capable" than the one it has most recently released, and for now the model remains an internal tool rather than a public product. What turned that tool toward Navier-Stokes was a piece of news. OpenAI says that on 1 September it heard rumors that two problems tied to the Millennium Prize had already been resolved. The company then put thousands of AI bots trained on the new internal model to work on some of the remaining open problems. By 5 September, roughly 88 hours after the bots were set to work, the Navier-Stokes problem had produced what the company calls a solution. OpenAI is careful about what it claims. It says it does not intend to claim the Millennium Prize for this result, and it describes its goal as reporting on the substantial progress of its AI models. The result itself resolves part of the problem, not the whole of it.
A Prize That Demands Four Statements
[Pic1] The mathematical target needs explaining because its name is technical but its structure is simple. The Navier-Stokes equations describe how fluids move, and the existence and smoothness problem asks whether certain mathematical statements about those equations can be proven. For 90 years, important aspects of the problem have lacked a proof. In mathematics, a proof is the argument that underlies a correct mathematical equation, and without that argument a result is not considered established. The problem is not a purely abstract puzzle; it sits at the heart of turbulence, a physical phenomenon that is still not well understood. The formal framework for this kind of challenge is the Millennium Prize, which is run by the Clay Mathematics Institute, a mathematics organization based in the United States. The Institute offers $1m to whoever first solves one of the nominated mathematical conundrums. [2] For the Navier-Stokes problem, the prize rules demand four statements in the proof. [2] OpenAI says its solution resolves two of those four statements. In other words, the announced result is not a complete solution of the Millennium Prize problem, but a substantial claim about half of it. The missing half is the part that independent mathematicians must still examine. The solution has yet to be verified independently, and it has yet to be publicly accepted by the Clay Mathematics Institute. Those two steps are not formalities; they are what separate an interesting claim from an accepted proof. A company can announce a result from a private laboratory, but a prize problem is governed by an institution with its own procedures. Until the Institute speaks, the result exists in a kind of mathematical limbo: announced, detailed, and unreviewed. OpenAI's own statements reflect an awareness of that gap. The company says it has no intention of claiming the $1m prize for this result. Its stated purpose is to report on what its models can now do. That distinction is important for anyone trying to read the news accurately. OpenAI is presenting a capability claim, not a finished mathematical victory. The distinction also matters because the company says the work was completed without external review and without the blessing of the institution that owns the prize.
The Dispute That Followed the Announcement
The announcement would have been remarkable even without controversy. The controversy arrived anyway, and it involves assumptions about who knew what, and when. Tristan Buckmaster, a mathematics professor at New York University, says he and Levent Alpöge, a mathematician working for OpenAI's rival Anthropic, had been working toward solutions of the same problem. [3] The two researchers had been using Codex, a tool built by OpenAI, in their work. That detail gives the dispute its sharp edge: competing mathematicians were using the winning company's own product. Buckmaster says that on 3 September he found out that information about his group's progress had been passed to OpenAI. He claims OpenAI did not begin working on the Navier-Stokes equations until after information about his team's work had reached the company. His statement became public on the same day as OpenAI's announcement, but hours before it. Buckmaster also says he had not yet read OpenAI's full proof when he spoke. He felt compelled, he explained, to make public what he was told, when he was told it, and what was proposed to him. His reason was direct: the alternative was to let a sequence of announcements say something he knows to be false. [Pic2] Buckmaster's account includes emails. He says he exchanged emails with OpenAI about the work and about his questions regarding the company's timing and methods, and he included the text of those emails in his statement. He did not claim to have evaluated the mathematics of OpenAI's result. His complaint was about process and sequence, not about equations. A dispute of this kind is common in mathematics, where priority is a form of currency. What makes this case unusual is that one of the parties is a company with an internal model and a commercial interest in demonstrating its power. OpenAI's response has been measured but firm. The company congratulated the work of Buckmaster and Alpöge, called it "remarkable," and described it as "concurrent work." OpenAI says it had not seen any of their work through any means until it was released publicly, and that no user data was accessed during its own Navier-Stokes research. Then comes the sentence that researchers will likely examine with the most care. OpenAI said it cannot rule out that de-identified data derived from the pair's usage of its products helped improve its models, even though it considers such an outcome unlikely. The company also defended the independence of its result, insisting that the proofs differ significantly and that even the precise results proved are different. If OpenAI is right about that last point, the two research efforts are not duplicates, and both could turn out to be mathematically valid. If the company is wrong, the discrepancy will surface where all mathematical claims are eventually tested: through independent verification. Success, if it comes, will look unspectacular. Independent mathematicians will work through the two claimed statements, the Clay Mathematics Institute will decide whether to accept them, and the record will show precisely what the machines contributed to a 90-year-old problem. OpenAI says it designed the exercise not to claim a prize but to demonstrate progress. The verification process, not the announcement, will determine whether that progress deserves to be called mathematics.