OpenAI Proves Navier-Stokes Equation Faces Criticism for 'Jumping the Gun', First Time AI Company Solves Such Problem Instead of Human Mathematicians
OpenAI announced that its internal AI model proved that the three-dimensional Navier-Stokes equations would produce a "singularity" in a finite time, which is one of the "Millennium Prize Problems" set by the Clay Mathematics Institute (with a reward of $1 million). This is also the first time such a problem has been solved by an AI company rather than human mathematicians. However, this achievement was immediately challenged by NYU mathematician Tristan Buckmaster, who accused OpenAI of "jumping the gun", sparking a widespread controversy over the attribution of research results and the use of training data by AI companies.
Buckmaster had been privately collaborating with mathematician Levent Alpöge, currently at Anthropic, for the past year, extensively using AI tools like Claude and Codex to assist their research. They had resolved the Euler equations using the "forcing" method (previously proposed by mathematicians Diego Córdoba and Luis Martínez-Zoroa), which is seen as a key stepping stone towards the Navier-Stokes problem. The two also believe they have found a specific result for a variant of the Navier-Stokes equations, but it has not yet been formally published or peer-reviewed.
According to Buckmaster, after rumors of the "Millennium Prize Problem being solved" began circulating online in early September, he proactively contacted OpenAI on September 3 to clarify that this was a private collaboration unrelated to their institutions. OpenAI confirmed that its team only initiated the internal model challenge after hearing the rumors on September 1 and announced the proof results on September 8—on the same day Buckmaster publicly shared preliminary results. OpenAI CEO Sam Altman stated, "Now we can see their results, and the two methods seem different," confirming that the company would not claim the $1 million prize.
Buckmaster stated in a public statement that OpenAI researcher Sebastien Bubeck proposed two options: either Buckmaster's team would publish a paper first, followed by OpenAI the next day, or Buckmaster would publish a paper alone and remove Alpöge from the author list, citing Alpöge's employment at competitor Anthropic. He also claimed that after being rejected, the other party said things like, "Why would you ruin your career?" Bubeck responded that he had clearly told Alpöge that "the results should be credited to Alpöge and Buckmaster" and was willing to share all materials, including prompts. He admitted to saying the phrase about "ruining a career" but claimed it was a "slip of the tongue out of frustration" and was retracted on the spot, intending to help the mathematician retain authorship, which conveyed the opposite message. He also stated that the suggestion to remove Alpöge stemmed from a misunderstanding that he only learned about midway through the call, regarding the fact that the other party was solving the Euler equations rather than the complete Navier-Stokes problem, referring only to the hypothetical scenario if Buckmaster were to rewrite OpenAI's own results alone.
OpenAI invested about 10,000 parallel "agents", generating approximately 300 billion output tokens (valued at about $22.5 million based on current computing power pricing), and the proof process was completed with formal verification using the Lean programming language, which is considered "essentially ensuring correctness". Mathematician Diego Córdoba commented, "We are all a bit shocked," while another mathematician, Luis Silvestre, stated, "These past two days have been crazy; the entire academic community is discussing its impact."
The controversy was truly ignited online by a cautious response from OpenAI: the company stated, "While it is unlikely, we cannot rule out that user-generated, anonymized data from our products may have indirectly helped improve our models," while emphasizing that it generally does not use enterprise client data to train models, and individual ChatGPT users can opt out. OpenAI insisted that it had "not seen the other party's work in any way before the results were made public." However, this statement still raised concerns among many users that their unpublished research drafts, which they had long placed in paid Codex sessions, might have been indirectly used to train competing models—benefiting large AI labs that possess vast amounts of user session data and computing power far exceeding that of individual researchers, while putting pressure on independent scholars and employees of competing labs who rely on these tools for research but lack control over the flow of their draft data.
Source: Public Information
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This is not the first time an AI lab has sparked controversy for "jumping the gun on others' research directions" by leveraging scaled computing power—there have been multiple discussions in academia about large AI labs quickly mobilizing resources to replicate or even surpass progress in a research direction after observing signs of advancement. OpenAI publicly acknowledged that its team initiated this previously "seldom-trodden technical route" on September 1 after hearing rumors about a competitor, deploying tens of thousands of agents' worth of computing power within a week to catch up with a year's worth of research accumulated by two human mathematicians.
The core resource mobilized by OpenAI in this endeavor was computing power rather than human resources—exchanging approximately 300 billion output tokens (valued at about $22.5 million) in inference costs to achieve a historic result of "AI first solving" the highly publicized Millennium Prize Problem; the direct return on this investment is not the $1 million prize for the research itself (which OpenAI has made clear it will not claim), but the narrative that "AI can achieve human-level mathematical breakthroughs", which has a far greater impact on the company's valuation and technical credibility than the prize amount.
This aligns with the competitive logic among AI labs surrounding benchmark tests, competitions, and narratives of "being the first to achieve"—previously, several labs have also announced "surpassing human" achievements in standardized tasks like Olympiad mathematics and programming competitions. The difference this time is that the "race target" has shifted from publicly comparable standardized tests to the unpublished private research directions of two scholars, directly touching on the more sensitive area of academic priority. Currently, generative AI companies are at a critical juncture in transitioning from "research assistant" to "independent research competitor", and this incident marks the first large-scale public friction in this transition process.
This controversy is essentially a reconstruction of the industrial chain under the logic of technological substitution—when AI tools evolve from "research assistants" to potential "researchers" that may directly compete with human researchers, factors such as whose session data, whose computing power, and who can mobilize resources faster begin to determine who can secure the final attribution and honors more than who had the idea first; the mechanism at play is that AI companies simultaneously play the dual roles of "tool providers" and "potential competitors", and the data traces left by users while using their products theoretically have the potential to be used by the same company to train models that directly compete with the users, a conflict of interest that has been exposed in such a public and intense manner in academia for the first time as the gap in computing power has widened to an order of magnitude.
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