Correction, October 13, 2026: OpenAI reports 88 hours to its result, followed by 17 additional hours for Lean formalization and verification. Buckmaster said his early questions concerned both access to private Codex drafts and their possible use in training.
Update, October 9, 2026: OpenAI reported investigation findings on September 10. It said Buckmaster’s recent Codex prompts could not have influenced the result, including through training.
Buckmaster and Alpöge spent close to a year on fluid equations, using AI tools to help. Their August breakthrough concerned Euler equations with an external force. After rumors circulated, OpenAI says its agents produced a related Navier-Stokes result in 88 hours. Buckmaster wants to know exactly what OpenAI knew, and when.
What happened
Tristan Buckmaster is a mathematics professor at NYU’s Courant Institute. He and Levent Alpöge, a mathematician at Anthropic, studied related fluid equations. Their August 15 breakthrough concerned the Euler equations with an external force. This model omits viscosity, a fluid’s resistance to flow, which Navier-Stokes includes.
The Navier-Stokes existence and smoothness problem asks whether fluid-motion equations always produce well-behaved solutions or can develop a mathematical breakdown. It’s one of seven Millennium Prize Problems that have each carried a $1 million reward since May 24, 2000. Buckmaster and Alpöge used Claude (Anthropic’s AI assistant) and Codex (OpenAI’s AI coding agent) in their research.
Word of that breakthrough reached OpenAI. Buckmaster and Alpöge then learned OpenAI already had a team pursuing a closely related problem using a similar approach, and reached out to ask what OpenAI knew and when its own effort had actually started. In a statement he later published, Buckmaster described pressing OpenAI for a straight answer:
“I asked when the first prompt had been sent by them. This question was not answered directly by OpenAI for some time. Eventually it was agreed that it had been sent in the past few days, after information about our work had reached OpenAI.”
Buckmaster also asked whether the model had access to, or had been trained on, the drafts they submitted through Codex. He said OpenAI answered the access question but did not answer the training question at that time.
OpenAI’s own account of its result, published alongside its proof, says its agents produced a resolution in 88 hours, work that launched September 1 and reached its result September 5. Lean formalization and verification took another 17 hours. That effort alone used 2.7 million agent messages (individual steps in the agent’s automated back-and-forth work) and roughly 130 billion output tokens (the chunks of text the model generates as it works). OpenAI said it “did not see any of their work through any means.”
In a September 10 update, OpenAI said Buckmaster’s Codex prompts from the preceding two months could not have influenced the result, including through training.
OpenAI offered to publish jointly with Buckmaster, according to his account, but wanted the paper to exclude Alpöge as a co-author, citing his position at Anthropic, a competitor.
Who’s involved
Buckmaster is a professor at NYU’s Courant Institute, one of the country’s leading applied mathematics departments, and a winner of the Clay Research Award, a prize the same institute behind the Millennium Prize Problems gives to recognize major mathematical breakthroughs. Alpöge is a mathematician who works at Anthropic. Their accounts describe what happened to their research, not an official position from OpenAI or Anthropic.
What Tao added
Terence Tao, a UCLA mathematics professor and a Fields Medal winner, one of the highest honors in mathematics, posted a warning the same week about a broader version of the same dynamic:
“We have now seen that even the rumor of someone working on a problem can trigger a massive amount of AI-powered effort to flatten it before the original research project has time to reach its full potential. The incentives may now be pointing in the direction of no longer sharing any promising research directions with the broader community, which would reverse centuries of traditions of open science and do serious long-term damage to the future of the field.”
In the third post of that thread, Tao describes a similar concern without identifying Buckmaster: a private breakthrough, a rumor reaching a well-resourced AI lab, and an agent-driven effort producing a related result within days.
Timing and credit remain separate questions
OpenAI reports completing its agent effort in less than four days after hearing a rumor of a mathematical breakthrough. Its September 10 findings address the private-prompt concern, but timing and collaboration remain separate questions. Buckmaster’s statement records his concerns about when work began and how credit would be shared.
None of this proves OpenAI’s model saw or trained on Buckmaster and Alpöge’s private Codex sessions. Buckmaster’s own statement doesn’t claim that it did either. What it documents is that OpenAI wouldn’t give him a clear answer on the question for some time, and that the proposed authorship excluded his collaborator, according to his account. Their research was a personal collaboration, rather than an institutional Anthropic project.
Why it’s notable
The reported pace shows how much parallel work a well-resourced lab can attempt in days. It is not a direct efficiency comparison with Buckmaster and Alpöge’s work on different equations. Tao’s concern is about incentives: a rumor can prompt a competing AI-driven effort before the original researchers are ready to share.
What it means for builders
If you’re doing work you consider genuinely valuable and aren’t ready to share, a research result, a product design, a data pipeline, check what your AI coding tool’s data-usage policy actually says before you put that work into it. “De-identified” and “used to improve the model” aren’t the same thing as “walled off from everyone else.” Most providers, including OpenAI, publish a data usage policy describing what you can opt out of on your specific plan. Check the policy and your account’s data controls before sharing unpublished work.
The second lesson is about pace, not just privacy. In this case, OpenAI says a rumor prompted its research effort. The reported pace does not establish how quickly a competing team could reproduce a different project. That doesn’t mean stop collaborating or sharing direction with trusted partners. Decide which research directions you are ready to share, while recognizing that disclosure can prompt other teams to pursue them.
End of article