OpenAI’s Navier‑Stokes Claim Triggers Credit and Data Controversy

Earlier this week, OpenAI claimed to have solved the Navier-Stokes problem, one of mathematics's hardest problems, which dates back to the 1850s. This claim has sparked controversy, involving a math professor from NYU, a mathematician from Anthropic, and OpenAI's Sam Altman, all offering differing accounts of events.

The Navier-Stokes Equations

The Navier-Stokes equations, developed by Claude-Louis Navier and George Stokes, describe the movement of fluids and gases. They are essentially an application of "force equals mass times acceleration" to water. Engineers have relied on these equations for nearly 200 years for critical applications such as hurricane forecasting, airplane wing design, and simulating blood flow.

However, mathematicians require proof, and a fundamental question about these equations remained unanswered: "Can they ever break?" This question was deemed so difficult that in 1999, the Clay Mathematics Institute designated it one of their seven Millennium Prize Problems, offering a million-dollar reward for its solution. To claim the prize, one must either prove the equations never break or demonstrate a single instance where they do.

Tristan Buckmaster's Work

Tristan Buckmaster, an NYU math professor and recipient of the 2019 Clay Research Award for his work on these equations, teamed up with Levent Alpagay, a mathematician at Anthropic. They were making steady progress on the problem. In mid-August, they began to heavily utilize Claude Code and Codex, which significantly accelerated their research.

Their approach was novel, and on August 15th, they successfully caused a simpler version of Navier-Stokes, known as the Euler equations, to "fall apart." While not the million-dollar problem, it was the closest anyone had come to a solution.

OpenAI's Claim and the Controversy

Shortly after Buckmaster and Alpagay's breakthrough, OpenAI became intensely interested in the problem. In late August, while training a new model, they reportedly fed it all unsolved Millennium problems. After investing 10,000 agents and $20 million in computing power, OpenAI claimed to have solved Navier-Stokes.

Crucially, the process OpenAI used was reportedly the exact novel approach that Buckmaster and Alpagay had been developing and feeding into Codex over the past year.

On September 3rd, Buckmaster emailed OpenAI for clarification. The following Sunday, he had a phone call with Sebastian Bubck, the OpenAI researcher leading the project. Buckmaster was initially told that OpenAI's model was given only the problem statement with minimal input. However, as the call progressed, it became apparent that an entire team had worked on it, the prompt was written by Codex, and they had only started a few days prior after hearing rumors about Buckmaster's work.

Buckmaster then inquired if the model had been trained on his Codex sessions. He was told that user data was not looked up, but when he pressed specifically on training, he received no answer.

According to Buckmaster, OpenAI then presented two options: 1. Buckmaster and Alpagay could publish their Euler results first, after which OpenAI would publish its Navier-Stokes proof, crediting Buckmaster for getting closest to the solution. 2. Buckmaster could write the Navier-Stokes paper himself, but without including Alpagay, due to Alpagay's affiliation with Anthropic.

Buckmaster refused both options and threatened to go public. He claims Bubck then questioned why he would "ruin his career" and stated that if Buckmaster didn't want Bubck to be nice, he didn't have to be.

OpenAI, however, presented a different account, denying that they ever asked for Alpagay to be removed from the work. Sam Altman stated his belief that everyone on OpenAI's side acted with integrity and that Buckmaster was the only one making threats.

Public Disclosure and Aftermath

On Tuesday morning, Buckmaster and Alpagay published their papers along with a four-page statement detailing their version of events. That afternoon, OpenAI released their own solution and a blog post asserting that no user data was accessed and that the two proofs were "significantly different."

The renowned mathematician Terence Tao offered a "reasonable take" on the situation, congratulating Buckmaster and Alpagay. He warned that if the mere rumor of research can trigger a "swarm of agents racing to front-run it," mathematicians might cease sharing ideas, which would undermine centuries of open science.

Rumors also began circulating that OpenAI is close to verifying another Millennium Problem.