OpenAI publishes a proof of a Millennium Prize problem, amid a credit dispute

What happened

On the morning of Tuesday 8 September, mathematicians at OpenAI announced that around 10,000 autonomous AI agents under their direction, running on an advanced model not available to the public, had found a “singularity” in the three-dimensional Navier–Stokes equations — resolving one of the Millennium Prize problems set by the Clay Mathematics Institute in 2000, as Quanta Magazine put it. According to The Guardian, OpenAI said the internal system was more powerful than its latest public model, GPT-6 Astra, and that it did not intend to claim the $1 million prize.

About 12 hours earlier, just before midnight on 7 September, NYU professor Tristan Buckmaster had published a statement announcing three results he and Anthropic mathematician Levent Alpöge had reached with help from several AI models. The second half of the statement described a clash with OpenAI.

Background: what is the problem?

The Navier–Stokes equations apply Newton’s second law to fluids, from ocean currents to air flows, and were written down in the mid-19th century. Engineers solve them approximately on computers every day, but one basic mathematical question has never been answered: can a solution “blow up”, so that at some moment an infinitely small part of the fluid moves infinitely fast?

According to Quanta, both teams built on the work of Madrid-based mathematicians Diego Córdoba and Luis Martínez-Zoroa, who had spent years on an unusual approach: using a carefully built external force to trigger a blow-up. By 2023 they had shown that the Euler equations (a version of the fluid equations without friction) blow up under a “messy” force. The remaining hurdle was to make the force smooth enough to meet the Millennium Prize criteria. Fefferman told Quanta he was “thrilled” the problem was solved, and that the heroes of the story are Córdoba and Martínez-Zoroa.

How OpenAI did it

According to OpenAI’s announcement, as reported by The Guardian, Quanta, TechCrunch and summarised on Wikipedia:

AI agents working at once

~10,000

Time to find the proof

88 hours

Time to formalise it in Lean

~17 hours

Output tokens over the week

~300 billion

Sources: Quanta, The Guardian and TechCrunch reporting on OpenAI's announcement, 8 Sep 2026

• The agents first assessed all the open Millennium Prize problems before focusing on Navier–Stokes (according to Wikipedia’s summary of OpenAI’s announcement).
• Separately, nearly 100 agents worked for about 50 hours on a related result for the Euler equations (OpenAI press release, quoted by Quanta).
• In total the agents sent almost 5 million messages to each other.
• Cost estimates differ. OpenAI researcher Sébastien Bubeck put the computing cost at several million dollars (Quanta). The Verge reported OpenAI as saying it used tens of millions of dollars of compute. TechCrunch calculated that the week’s 300 billion output tokens would cost $22.5 million at GPT-6 Astra’s published prices.

How the dispute unfolded

The two sides contest some details fiercely, but according to The Verge they broadly agree on the basic sequence of events:

1. 122 AugustBuckmaster and Alpöge's Euler blow-up result passes Lean verification.
2. 21 SeptemberOpenAI says rumours prompt it to start on Millennium problems.
3. 33–6 SeptemberBuckmaster contacts OpenAI; on the 6th he is told its model has a proof.
4. 47–8 SeptemberBuckmaster posts his statement; OpenAI publishes about 12 hours later.
5. 510 SeptemberOpenAI says its investigation found his Codex prompts had no influence.

Sources: Buckmaster's statement; TechCrunch, Quanta, The Verge, Wikipedia

Buckmaster’s account

Buckmaster wrote that his work with Alpöge was “a purely personal collaboration”, with no involvement from their employers. They used Anthropic’s Claude and OpenAI’s Codex, mainly with GPT-5.6 Sol and later GPT-6 Astra for write-ups and checking, and put all their drafts into Codex.

He says that on the 6 September calls he was shown a prompt and told the internal model “had simply been given the problem statement”, and Alpöge was told by Bubeck that “very little human input” had been used. But during the call, “it emerged that an entire team had been working on the problem” and “an insane amount of compute had been used”. “Eventually it was agreed that [the first prompt] had been sent in the past few days, after information about our work had reached OpenAI.”

He argued that the route through a smooth force was the one Córdoba and Martínez-Zoroa opened and that he and Alpöge had quietly chosen: “Almost nobody else I know of was working on it. It is not the direction one arrives at in a few days by giving a model the problem statement.”

He says OpenAI offered two options: that he and Alpöge post their Euler result and OpenAI post Navier–Stokes the next day, or that he alone write a paper presenting OpenAI’s result. He says Bubeck twice said he wanted Alpöge removed from authorship because Alpöge works at Anthropic. Buckmaster declined both and said he would go public if OpenAI released its result as proposed. He says the reply was: “Why would you ruin your career?” and later, “If you don’t want me to be nice, then I don’t have to be nice.”

He asked whether OpenAI’s model had been trained on his Codex sessions and says he got no answer on training. He also stressed: “I have not seen OpenAI’s proof… I do not know whether our data was used. I am not accusing anyone of anything.”

OpenAI’s account

• Timeline. OpenAI’s post says the effort began on 1 September, prompted by rumours that two Millennium Prize problems had been solved, and confirms the conversations with Buckmaster and Alpöge (TechCrunch). According to Wikipedia, Bubeck also wrote that day that OpenAI began work on Navier–Stokes partly because of vague tweets by Alpöge, and that he had tried to coordinate publication of “concurrent discoveries”.
• Data. The post says: “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.” It adds: “While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models. However, our proofs differ significantly and even the precise results proved are different in the Euler case (forced vs unforced).” On 10 September OpenAI said an investigation had confirmed that Buckmaster’s Codex prompts over the previous two months “could not have influenced the system in any way, including through training”.
• The calls. According to The Verge, citing The New York Times, Bubeck rejected Buckmaster’s characterisation of the conversations. He acknowledged offering OpenAI’s resources to help Buckmaster finish his own proof or to have him write the company’s paper, and said OpenAI had made similar arrangements with other mathematicians. He was frank that Alpöge’s employer was a sticking point: “From our perspective, how can we have an internal OpenAI project with an Anthropic employee?” The Guardian reported that Bubeck denied at a press briefing that OpenAI had used the pair’s work.
• Priority. According to Quanta, OpenAI cedes priority for the 3D Euler result to Buckmaster and Alpöge while claiming the Navier–Stokes result.

Responding to OpenAI’s 10 September statement, Buckmaster said: “Given their behavior up until this point, one should take such statements with great skepticism.” What was said on the private calls rests on each side’s account and cannot be checked independently.

What mathematicians say

• About the result. “The Clay problem is settled, but the main problem for the Navier-Stokes equations is not,” University of Chicago mathematician Luis Silvestre told Scientific American. On 17 September, Peter Constantin, Mihaela Ignatova and Vlad Vicol posted a preprint arguing that OpenAI’s method cannot be extended to the case without an external force. If it turns out blow-up only happens with a force, mathematician Gonzalo Cao-Labora said, people might conclude “we shouldn’t have put the external force in the statement”.
• The Clay Institute said on 11 September that the problem “has apparently been settled”, but that its process “is deliberately unhurried”. Under its rules, two years must pass after publication, and the result must gain general acceptance among mathematicians, before a prize is considered.
• Worries about the field. Many of the more than a dozen mathematicians The Verge spoke to worried that AI companies would push researchers to keep unfinished work secret. Fields medallist Shing-Tung Yau said competing with companies whose resources far exceed academia’s could make PhD students and junior researchers even more reluctant to tackle ambitious problems. On 11 September, 28 Fields medallists published an open letter criticising AI companies (see 28 Fields Medallists warn AI companies over mathematics).

Things to keep in mind

• The proof has not yet been independently reviewed. According to Wikipedia, as of 18 September it had not been independently verified. The Lean check adds confidence, but people still need to confirm the statement is the right one.
• It solves the version with a force. That counts under the Clay Institute’s original wording, but several experts say the question most researchers wanted answered, without a force, is still open.
• The key facts of the dispute can’t be verified. The private calls are known only from each side’s account, and OpenAI’s model and full process are not public.
• Cost figures are estimates. Several million, tens of millions and $22.5 million come from different people using different methods.

What it means for ordinary users

Barely a year earlier, AI was just reaching the gold-medal line at the International Mathematical Olympiad (see AI reaches gold-medal level at the Maths Olympiad). Now, with enormous computing power, an AI system has produced a result at the research frontier that mathematicians take seriously. But it was an unreleased internal model; ordinary users don’t have access to anything like it.

The dispute also raises a practical point: what you type into AI tools may be used to train models. According to TechCrunch, OpenAI reserves the right to train models on Codex interactions, although users can opt out. If you use AI with unpublished research, code or business material, check your data settings first.