In 2000, the Clay Mathematics Institute pledged $1 million to the first person to correctly solve one of seven complex math problems before the end of the century.
Among these Millennium Prize Problems, the Navier-Stokes Existence and Smoothness Problem has long been the subject of research by Tristan Buckmaster, a mathematics professor at New York University.
His work, which deals with the predictability of fluid motion, has since become the center of a fierce debate between academics and artificial intelligence companies. Buckmaster alleged that OpenAI spied on his private Codex sessions to develop its own proof of the same problem.
“There’s no proof that they did that but [the approach and the timeline in the OpenAI proof] is highly suggestive that that’s exactly what happened,” Buckmaster told The New York Times.
OpenAI has denied the allegations, instead claiming that the Navier-Stokes puzzle was solved by a coordinated swarm of 10,000 autonomous agents.
“A major goal of our work is to empower scientists to advance research and technology that benefits all of humanity. To solve the Navier–Stokes problem, we used an internal model that is significantly more capable than GPT‐6 Astra,” the company said in a statement.
Regardless of the integrity of OpenAI’s proof, the solution of the Navier-Stokes puzzle has already raised significant questions about the proof’s industrial ramifications.
In the aerospace industry, engineers at manufacturing companies such as Boeing and Airbus already use the Navier-Stokes equation to predict the aerodynamic performance of aircraft models. However, for these engineers, Buckmaster’s research and OpenAI’s proof are unlikely to immediately change workflows.
“OpenAI’s proof does provide an example in which the equations allow a fluid to hit an infinite velocity, but the scenario is so contrived that it’s almost certainly unlikely to have any direct relevance to any physical system,” Adam Kovac, a writer for Scientific American, wrote.
Instead, the immediate economic rewards of the Navier-Stokes proof are more likely to be felt by AI companies.
OpenAI, which has warned that its latest models are reaching “artificial general intelligence,” can use the Navier-Stokes solution as a successful proof of product in its rivalry with Anthropic.
With this proof of product in mind, the tech rivalry is made more potent by the fact that Buckmaster chose Levent Alpöge, an Anthropic employee, to be his research partner.
With initial public offerings on the Nasdaq in sight, both companies have since set new goals of solving the remaining Millennium Prize Problems.
Anthropic has also claimed progress on the long-unsolved Riemann hypothesis. For OpenAI’s investors, however, the current Navier-Stokes controversy offers mixed signals.
“The maths announcement allows OpenAI to cast its technology in a positive light after a period of alarm over how safely it is being developed,” Ian Sample and Dan Milmo wrote in a Guardian article.
On the other hand, concerns about data privacy could lead OpenAI to lose customers to closed, local models that offer better security. Even if Buckmaster’s allegations turn out to be false, the assumption that private conversations will be leaked for AI training could prevent clients from trusting the company with their most sensitive proprietary data.
For mathematicians who have long sought to separate their research from corporate rivalries, the Navier-Stokes controversy has created a new crisis.
If OpenAI’s latest models are indeed advanced enough to solve every Millennium Prize Problem, the role of mathematicians in guiding and developing scientific advancement becomes unclear.
Nonetheless, OpenAI has claimed that the purpose of its work is to empower scientists. Despite the $1 million prize, OpenAI has rejected the Clay Institute’s money, leaving the possibility open for Buckmaster to claim the award of the century.
