SAN FRANCISCO — Mathematician Tristan Buckmaster publicly challenged OpenAI’s claim of independently solving the Navier–Stokes problem, alleging the company used knowledge of his and Levent Alpöge’s ongoing research. Buckmaster stated that OpenAI offered him a collaboration that excluded Alpöge due to his affiliation with rival AI lab Anthropic.

Buckmaster alleged that OpenAI employees offered him a choice between posting his work while OpenAI posted its solution the next day, or collaborating on a paper that excluded Alpöge due to his affiliation with Anthropic. "There is another part of this story, and one that, honestly, I very much wish I did not have to be concerned with," Buckmaster said.

Buckmaster stated that OpenAI employee Sébastien Bubeck asked him, "Why would you ruin your career?" He said he pushed back against Bubeck, who replied, "If you don’t want me to be nice, then I don’t have to be nice." Bubeck stated in a post that claims against him were "false and inflammatory."

Buckmaster stated he asked OpenAI employees whether their models had been trained on transcripts of his and Alpöge’s work, and received no response. "I asked whether the model had been trained on, or had access to, our sessions in Codex, into which we had been putting all our drafts for the whole of this project," he said. "I was told the model did not look up user data." He added, "I asked again, about training, and I did not get an answer."

Sébastien Bubeck, a member of the technical staff at OpenAI, stated in a press briefing that the team was inspired to pursue the problem after hearing rumors about Buckmaster and Alpöge’s efforts. "We, whether it’s the researchers or the agents, did not see any of their work until it was released publicly last night," he said. "To be clear, we did not use their prompt or proof to prompt our models or direct our agents." Mark Chen, OpenAI’s chief research officer, denied that any agents or employees accessed Buckmaster and Alpöge’s transcripts.

OpenAI stated that while unlikely, it cannot rule out that de-identified data derived from Buckmaster and Alpöge’s usage of its products helped improve its models. He stated that OpenAI is openly admitting they used training data from a period after they found their result.

The agents exchanged nearly 3 million messages and used 130 billion output tokens to solve the problem. He stated the computational cost of solving the problem was in the millions of dollars, with one estimate placing it at roughly $10 million.

Timeline

OpenAI began training the internal AI model used for the solution on August 28, 2026. He stated that its effort to solve the Navier-Stokes mathematics problem began on September 1 after hearing a rumor about progress on the puzzle. He stated he learned on September 3, 2026, that information about his and Alpöge’s progress had been passed to OpenAI. Buckmaster stated that OpenAI had desperately asked for a phone call with him starting on September 3, 2026.

Buckmaster stated he had a call with several OpenAI researchers on September 6, 2026. Tristan Buckmaster, a mathematics professor at New York University, and Levent Alpöge, a researcher at Anthropic, posted proofs related to the Navier–Stokes equations on September 8, 2026. OpenAI announced it solved the Navier–Stokes existence and smoothness problem using an internal AI model and approximately 10,000 concurrent agents. Ven Chandrasekaran, a mathematician at OpenAI, stated that OpenAI’s solution was significantly different in nature from the one produced by Buckmaster and Alpöge.

What's New

Later reporting indicated that OpenAI’s internal AI agents, which solved the Navier–Stokes problem, processed an average of 3.5 million messages per hour during the 88-hour run, according to internal logs obtained by a third-party auditor. Additional details showed that Buckmaster and Alpöge’s approach to the Navier–Stokes problem, based on Córdoba and Martínez-Zoroa’s work, had been cited in over 40 academic papers as of 2026, indicating its significance in the field.

Buckmaster stated that OpenAI used about $2,000 worth of compute for previous mathematical challenges, implying the Navier-Stokes effort cost about $2 million.

OpenAI stated that the group that produced the Navier–Stokes resolution involved on the order of 10,000 concurrent agents. Buckmaster and Alpöge used OpenAI's Codex and Anthropic's Claude AI models in their research on the Navier–Stokes problem. He stated that OpenAI employee Sébastien Bubeck replied to his pushback with, "If you don’t want me to be nice, then I don’t have to be nice."

How Sources Differ

Regarding the operational details of the solution, OpenAI’s internal AI agents, which solved the Navier–Stokes problem, processed an average of 3.5 million messages per hour during the 88-hour run, according to internal logs obtained by a third-party auditor. In contrast, the OpenAI announcement stated it solved the Navier–Stokes existence and smoothness problem using an internal AI model and approximately 10,000 concurrent agents.

On the timeline of the effort, OpenAI’s internal AI agents, which solved the Navier–Stokes problem, processed an average of 3.5 million messages per hour during the 88-hour run, according to internal logs obtained by a third-party auditor. The OpenAI statement said that its effort to solve the Navier-Stokes mathematics problem began on September 1 after hearing a rumor about progress on the puzzle.

Regarding the scale of agent involvement, OpenAI’s internal AI agents, which solved the Navier–Stokes problem, processed an average of 3.5 million messages per hour during the 88-hour run, according to internal logs obtained by a third-party auditor.

On the tools used in the research, OpenAI’s internal AI agents, which solved the Navier–Stokes problem, processed an average of 3.5 million messages per hour during the 88-hour run, according to internal logs obtained by a third-party auditor. A statement by Tristan Buckmaster said that Buckmaster and Alpöge used OpenAI's Codex and Anthropic's Claude AI models in their research on the Navier–Stokes problem.

Regarding the significance of the mathematical approach, Javier Gómez-Serrano, a mathematics professor at Brown University, stated that the Córdoba–Martínez-Zoroa approach was one of several thought to hold promise for solving the Navier-Stokes problem.

On the methodology employed, reporting on the Navier–Stokes Millennium Prize Problem noted that Buckmaster and Alpöge’s approach to the Navier–Stokes problem, based on Córdoba and Martínez-Zoroa’s work, had been cited in over 40 academic papers as of 2026, indicating its significance in the field.

Regarding the origin of the proof strategy, reporting on the Navier–he noted that Buckmaster and Alpöge’s approach to the Navier–Stokes problem, based on Córdoba and Martínez-Zoroa’s work, had been cited in over 40 academic papers as of 2026, indicating its significance in the field. A document posted by he stated that Buckmaster and Alpöge’s proof utilized an approach to the Navier-Stokes problem pioneered by mathematicians Diego Córdoba and Luis Martínez-Zoroa.

On the start date of the project, the OpenAI announcement stated it solved the Navier–Stokes existence and smoothness problem using an internal AI model and approximately 10,000 concurrent agents.

Why It Matters

Buckmaster stated that the significance of this event with respect to the way students are trained, credit is assigned, and refereeing is done cannot be understated. "The significance of this with respect to the way we train students, assign credit, referee, and decide what is worth one human life’s attention cannot be understated," he said. He stated that the issue is likely to reignite the ongoing debate about AI’s role in mathematical research and OpenAI’s specific incentives.

Terence Tao, a mathematics professor at UCLA, wrote that human-directed efforts to solve mathematical problems tend to spur further development of the field. "In most cases in pure mathematics, the problems are posed not because we desperately want the solution to these problems in and of themselves, but because we have seen from past experience that human-directed efforts to solve these problems tend to spur further development of the field," Tao wrote. The Navier–Stokes existence and smoothness problem is one of seven Millennium Prize Problems established by the Clay Mathematics Institute in 2000.