OpenAI has produced a solution to the Navier-Stokes equations, one of the most enduring problems in mathematics and a designated Millennium Prize Problem carrying a $1 million award from the Clay Mathematics Institute. The result, achieved with artificial intelligence, indicates that the equations — which model fluid flows such as air over aircraft wings and blood through veins — can break down in certain situations, a phenomenon mathematicians call “blow-ups.”
The announcement has generated controversy, though not primarily over the mathematics itself. The work has not undergone traditional peer review, but it has been formalised, a process in which proofs are converted into computer code that machines can systematically check for logical flaws. That formalisation gives researchers reasonable confidence in the correctness of the finding. The dispute instead centres on timing and credit.
Shortly before OpenAI released its news, two researchers — Tristan Buckmaster at New York University and Levent Alpöge at the AI company Anthropic — published their own solution to a closely related problem involving the Euler equations. Experts suggested that work could open a path toward a full Navier-Stokes solution. Buckmaster and Alpöge, having heard rumours that OpenAI was preparing an announcement, contacted the company for clarification. Buckmaster said he received opaque responses and issued a statement noting that his own work was stored on OpenAI’s servers — as a customer, not as a research partner. OpenAI has denied any wrongdoing, stating that no person or AI from the company had seen the researchers’ work and that its solution was different in any case.
The full picture may become clearer once complete papers from all parties are published. For now, the episode underscores how artificial intelligence is reshaping mathematics. In recent months, AI has made rapid progress on longstanding problems, and the Navier-Stokes result marks a peak in that acceleration. The solution reportedly took only days of AI effort, with computing costs estimated at $15 million if charged to a customer — many times the prize money.
Mathematicians are divided on the implications. Some welcome the new capabilities, while others worry about their careers and the long-term health of the discipline. Terence Tao at the University of California, Los Angeles, often described as the world’s greatest living mathematician, has voiced strong concerns. He noted that competition among mathematicians to be first has always existed, but the difficulty of the work created a natural brake on the pace of discovery. AI has removed that brake. “Now there’s no speed limit, and suddenly things are breaking down,” he said.
Tao fears that findings will accumulate faster than the community can absorb them. Without time to fully understand results, integrate them into textbooks, and teach them to students, progress could begin to harm the field. He described the situation bluntly: “These companies are dumping carcasses of raw meat onto our communal village table and saying, ‘here you go, I solved your food problem’, and then they just leave. They’re expecting us to prepare the food and cook it and eat it. All that work is left to us, to clean up, and it’s demoralising.”
Whether anyone will claim the $1 million prize remains unclear. Martin Bridson, president of the Clay Mathematics Institute, told New Scientist that the evaluation process is “deliberately unhurried” and will be “absolutely rigorous.” If the prize is awarded, a question will arise over whether it should go to Buckmaster and Alpöge, as some have suggested, or to OpenAI. Beyond the mathematics, the episode raises broader questions about how results are credited when AI is involved and how the field will adapt to a new era of machine-assisted discovery.