This Tuesday, OpenAI claimed to have solved one of the greatest problems in the history of mathematics: the so-called “existence and smoothness problem” of Navier-Stokes, one of the seven Millennium Problems of the Clay Mathematics Institute, each awarded with one million dollars, and which has remained unsolved for almost 90 years. This had been one of the major goals of AI companies for years, and other companies, such as Google Deepmind, were in the same race. On paper, it is a milestone: only one Millennium Problem had been solved before (the Poincaré conjecture), and it would be the first time a machine solves a mathematical problem of this magnitude. No external scientist has yet independently verified OpenAI’s proof: and a review of this importance usually takes months.
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Navier-Stokes tries to answer an apparently simple question: Can a fluid (water, air, any liquid or gas) move smoothly, without infinite whirlpools or speeds that skyrocket, forever, no matter how much time passes? The Navier-Stokes equations are the mathematical formulas that describe how any fluid moves. They were written more than 250 years ago and work very well: they are used to design airplanes or predict the weather. The problem is that, mathematically, no one knows if these equations are safe in a very specific sense: if a fluid moves smoothly and orderly without any strange point, without any extreme turbulence from the start, can that fluid, simply with the passage of time and following those same equations, end up generating a point where the speed becomes infinite? This is called a “singularity,” and it is the question that has been unresolved for 90 years. Until, supposedly, now.
But the announcement has been overshadowed by a priority and ethics dispute that has shaken both the mathematical community and the artificial intelligence community.
It all started with a series of rumors: “It’s true that we got into this because last week rumors circulated on the internet that Anthropic’s models had solved a Millennium Problem, and we were curious to see if ours could do it too,” wrote Sam Altman, CEO of OpenAI, on X. OpenAI says that on September 1 it began training its new model to solve the Millennium Problems, after hearing that two researchers (Tristan Buckmaster, mathematician at New York University, and Levent Alpöge, Anthropic researcher working on the project personally) had already solved one.
That was not entirely true: Buckmaster and Alpöge, relying on a method devised by Spaniards Diego Córdoba and Luis Martínez-Zoroa, had on August 15 made a step towards Navier-Stokes, but not the complete solution. Terence Tao, Fields Medalist and one of the few people in the world capable of evaluating whether this holds, called that work a “remarkable achievement” and said he saw no obvious obstacles to extending the method to Navier-Stokes. It was precisely that direction that OpenAI used to complete its own proof. The company used an internal model far superior to the ChatGPt-6 Astra they just released. And they deployed nearly 10,000 autonomous AI agents over 88 hours.
I understand how, looking at today’s LLMs, people might think the only way we’d achieve NS is by using Levent’s/Tristan’s prompts. But I hope this plot conveys the model used is a huge step up from today’s LLMs. Nobody looked at Levent’s/Tristan’s prompts. That’d be insane. https://t.co/FHuPoLvcPk pic.twitter.com/hdAQbP486X
— Noam Brown (@polynoamial) September 8, 2026
The night before OpenAI’s announcement, ) with serious accusations: that OpenAI had learned of his private and unpublished research in the last week, that the company already had a proof of the resolution before contacting him, and that they offered him the choice between a joint publication or writing the result himself attributing it to an OpenAI model.
According to his account, OpenAI scientist Sébastien Bubeck pressured twice to exclude Alpöge from authorship due to his link with Anthropic. Bubeck has called Buckmaster’s version “false and incendiary” and has promised to provide more details. “One of the options we discussed was that Buckmaster would be the lead author of a rewrite of OpenAI’s Navier-Stokes proof. It was in that context that I said ‘it would be easier if Levent were not an Anthropic employee,’ because I thought it inappropriate for an Anthropic employee to sign an OpenAI paper,” says Bubeck, in one of the explanations he has given on X within a very controversial debate about who should get more credit in this solution: not only which humans, but also which machine.
unfortunately i contacted them on the night of wednesday, september 2nd making the following points
1. i had intel they had information about math work i was doing and a few days before they had spun up a group to try to compete.
2. i emphasized and reemphasized mine was a… https://t.co/MreJNmPBTQ— levent (@__alpoge__) September 9, 2026
Adding to this is another front: Buckmaster wonders if OpenAI could have accessed his notes and prompts in Codex, OpenAI’s programming tool to speed up research. That is, to plagiarize his work. This fact, if confirmed, would be a catastrophe for the public image of OpenAI tools like ChatGPT and Codex.
OpenAI maintains that its agents did not access the work of the two researchers by any means before it was made public, although it does not completely rule out access to “de-identified” data, meaning that OpenAI’s model may have used them because they remained in its system once entered. OpenAI explains this in its press release: “We (the researchers and agents) did not see their work by any means until they made it public; in particular, no specific user data was accessed to solve this problem. Although unlikely, we cannot rule out that de-identified data derived from their use of our products contributed to [improving our models].” This consequence would be a serious warning for anyone using these models with confidential or private data.
Beyond who said what, Tao has raised the question that truly worries the community: if a rumor can become the trigger for deploying thousands of agents with millions of dollars in computing by a better-resourced competitor, what incentive remains for anyone to share promising ideas before they are finalized? Tao has also criticized the hiding of failed attempts and the demonstration process, and has called for standards that value both the mathematical idea and the final result.
The involvement and pressure of AI companies in mathematics has already caused problems in the past: OpenAI claimed last summer that it had won a medal in the Mathematical Olympiad when in fact, according to this newspaper, it had not even participated.
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