OpenAI has published what it says is a solution to the Navier-Stokes existence and smoothness problem, one of mathematics’ seven Millennium Prize Problems, shortly after NYU professor Tristan Buckmaster announced related AI-assisted results. The releases have also produced a dispute over when OpenAI began its effort, what information influenced it, and how credit should be assigned.
Buckmaster and Anthropic mathematician Levent Alpöge announced three results involving finite-time blowup with smooth forcing for incompressible porous media, Boussinesq and three-dimensional incompressible Euler. In his statement, Buckmaster said the pair used several AI models, primarily OpenAI Codex, alongside Claude, GPT-5.6 Sol and later Astra.
The researchers also believe they have a result for hypo-dissipative Navier-Stokes, although that work was not released because its Lean verification was unfinished. Their research developed earlier work by mathematicians Diego Córdoba and Luis Martínez-Zoroa on forced blowups.
OpenAI Says Its Model Solved the Millennium Problem
Hours after Buckmaster’s announcement, OpenAI published an official Navier-Stokes solution produced by an unreleased internal model that it says is significantly more capable than GPT-6 Astra. The company released both a written proof and a formal Lean verification.
OpenAI said its effort began on September 1 after hearing rumors that two Millennium Prize problems had been solved. Its system ultimately involved roughly 10,000 concurrent agents, with agents producing 4.9 million messages and around 300 billion output tokens across the mathematical problems attempted.
The Navier-Stokes effort alone used about 130 billion output tokens across 2.7 million messages. OpenAI said the agents reached the result on September 5, around 88 hours after work began, followed by another 17 hours for Lean formalization and verification.
The Navier-Stokes problem concerns whether smooth three-dimensional fluid solutions can remain smooth indefinitely or develop singularities. Each Millennium Prize Problem carries a $1 million award.
Buckmaster Questions the Timing and Attribution
Buckmaster said information about his collaboration reached OpenAI before the company began its concentrated effort. He argued that OpenAI subsequently pursued the same relatively uncommon route through smooth forcing that he and Alpöge had quietly been studying.
OpenAI confirmed that its project began after hearing rumors connected to Buckmaster and Alpöge, but said its researchers and agents did not access their work before publication. It added that it could not completely rule out the possibility that de-identified product usage had contributed indirectly to model improvements.
That distinction matters because Buckmaster said he had placed project drafts into Codex throughout the research. OpenAI’s data policy says content from individual services can be used to improve models unless users opt out.
Buckmaster also alleged that OpenAI researcher Sébastien Bubeck proposed removing Alpöge from authorship in a possible joint arrangement because Alpöge works at Anthropic. He further alleged that Bubeck responded to his plan to disclose the disagreement by asking, “Why would you ruin your career?” OpenAI’s published account does not address those specific allegations.
OpenAI said it does not intend to claim the Millennium Prize for its result.
Featured image credits: Wikimedia Commons
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