OpenAI Claims Another Huge Mathematical Result Amid Fights Over Credit, Ethics, and Privacy

OpenAI has announced one of its unreleased artificial intelligence models has found an answer to one of the biggest open questions in mathematics, the Navier-Stokes Millennium Prize problem. The Millennium Prizes offer million-dollar rewards for solutions to seven notoriously difficult mathematical puzzles, and until now only one had been solved.

Mathematical problems at this level are extremely complex. A solution to the similarly difficult ABC conjecture ran to more than 500 pages, and it took mathematicians six years to understand it enough to spot potential flaws. Researchers can devote entire careers to these problems, in the hopes of getting close to a solution.

And this is what seems to have happened here. Building on the work of several human mathematicians, OpenAI unleashed a swarm of 10,000 AI agents running a new experimental model, which churned through millions of dollars’ worth of computing power in the space of a few days to complete what the American Mathematical Society called “the final steps” of the solution process.

So what is the Navier-Stokes problem, and what did OpenAI do? And why are a lot of mathematicians impressed with the result but unimpressed with the AI company’s behavior?

A Fluid Situation

OpenAI’s announcement concerns the Navier-Stokes equations, which model the behavior of fluids such as water and air. These equations underpin modern science and engineering, but our mathematical understanding of them is incomplete.

To explain the problem, imagine looking at the flow of water, before zooming in with a camera. According to the equations, both the zoomed and un-zoomed water should look exactly the same, except that things will look a little faster in the zoomed-in view.

We know this can’t be right in the real world. If we keep zooming in far enough, we will stop seeing a smooth fluid and start seeing a teeming crowd of molecules jostling against one another. So the Navier-Stokes equations must break down somewhere.

The biggest concern is whether fluid swirls can shift their energy into smaller, faster swirls, accelerating every time we zoom in. If this is possible, then the fluid may become impossibly fast, creating a “blow-up” in speed (also known as a “singularity”).

We know this can never happen in the real world, but the Millennium Prize was about finding out whether it could happen in the equations. OpenAI found that yes, the Navier-Stokes equations do allow a blow-up under certain conditions.

A Blow-Up in Finite Time

As OpenAI tells the tale, their researchers heard rumors mathematicians at rival company Anthropic were close to solving two Millennium problems on September 1. They deployed their latest in-development model in an effort to crack one first.

After launching a swarm of agents, the model produced a solution in just 88 hours, with verification taking another 17. The result is a coup for OpenAI, which is trying to demonstrate the capacity of its models against those of its leading competitor Anthropic, as both companies head towards planned share market listings.

One of the rival teams closing in on a Navier-Stokes solution featured Anthropic staffer Levent Alpöge, who was collaborating in a private capacity with mathematician Tristan Buckmaster from New York University.

As it turns out, OpenAI had contacted Buckmaster to discuss his work and theirs. In a statement published hours before OpenAI’s, Buckmaster said he and Alpöge had been using OpenAI’s publicly available models in their work for some time, and had been pursuing a line of thinking similar to what was used in OpenAI’s result.

He says he asked whether their data had been used by OpenAI’s model to produce its results, but received no answer to this question. Instead, he says OpenAI offered to collaborate with him if he removed Alpöge’s name from the work, because of Alpöge’s Anthropic affiliation. (OpenAI denies this claim.)