OpenAI announced on Sept. 8 that researchers using an unreleased internal model had solved a historic math problem. But the claimed breakthrough comes amid allegations from two mathematicians who say the company may have built on their work without proper credit.
The Navier-Stokes equations are considered one of the hardest in math, classified in 2000 by the Clay Mathematics Institute as one of seven "Millennium Prize Problems." The institute offers $1 million to whoever can solve it, and until now, only one of the seven problems had been solved.
New York University professor Tristan Buckmaster said in a Sept. 8 statement that he and Anthropic researcher Levent Alpöge have spent much of the past year working on the Navier-Stokes problem as independent
researchers. After OpenAI learned of their progress, Buckmaster alleged the company raced to solve the problem before them using a similar route.
OpenAI called the researchers' claims "categorically false" in an email to USA TODAY. The company's public statement adds that OpenAI researchers "did not see any of their work through any means until they released it publicly — in particular, no specific user data was accessed in order to solve this problem."
USA TODAY reached out to Buckmaster for comment. He had not responded at the time of publication.
What the mathematicians claim
Mathematicians have used AI to help solve historic math problems for several years. In February, an AI model helped formalize one called the sphere-packing problem, and in July, Alpöge made headlines for using an Anthropic model to disprove a well-known conjecture.
The Navier-Stokes equations describe how fluids such as water and air flow. Solving key questions about those equations could help explain and predict breezes and turbulence.
According to a definition from UCLA mathematician Terrence Tao, proving that the equations don't always behave as expected implies that the laws of physics can break down under certain conditions – that, for example, water could spontaneously explode, as mathematician.
"The significance of this with respect to the way we train students, assign credit, referee, and decide what is worth one human life’s attention cannot be understated," Buckmaster wrote in the statement.
Using models from both OpenAI and Anthropic, the pair went about proving the problem by building on research done by mathematicians Diego Cordoba and Luis Martinez-Zoroa, the statement continued. Buckmaster said they do not have a proof for the million-dollar prize, but do claim to have a proof for a similar problem — the Euler equation — that could help pave the way to solving the Millennium Prize Problem.
According to his statement, Buckmaster emailed a contact at OpenAI after rumors circulated that Anthropic solved the problem. He said the company requested a meeting and that, during the call, OpenAI researcher Sebastian Bubeck told him the lab was also working on the Navier-Stokes problem. Buckmaster said the approach was very similar to the one he and Alpöge were pursuing.
"I asked whether the model had been trained on, or had access to, our sessions in Codex, into which we had been putting all our drafts for the whole of this project," Buckmaster wrote in his statement. "I was told the model did not look up user data. I asked again, about training, and I did not get an answer."
Buckmaster added that Bubeck proposed releasing the results of their findings together, but with a caveat — leaving Alpöge off the authorship.
Bubeck denied asking to remove Alpöge, saying in a post on X that he wanted to "offer access to our internal model so that they could try to finish their proof and bridge the gap between Euler and NS. Again I did not know how to navigate giving access to internal OpenAI IP to an Anthropic employee."
At the end of his statement, Buckmaster wrote: "I would like to be clear about what I am not claiming. I have not seen OpenAI’s proof. I do not know what their model did, or how. I do not know whether our data was used. I'm not accusing anyone of anything. I am stating what I was told, when, and what was proposed to me. I am stating it because the alternative is to let a sequence of announcements say something I know to be false. If indeed an OpenAI model did close the gap to Navier-Stokes, that is."
OpenAI's response
OpenAI denied that the researchers' work was looked at by the company to solve the problem.
"We began working on the Millennium problems due to viral Twitter rumors that Anthropic had resolved 2 Millennium problems," Bubeck said in a post. "Our aim was to see whether our system was also capable of this impressive feat."
In a statement, the company admits that user feedback and data — like an individual's chats with ChatGPT or its coding agent, Codex — are used in its chatbot's training.
"While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models," OpenAI said in a statement.
CEO Sam Altman added in a social media post that "the approaches appear to be different. It is also worth noting that our latest model can solve many, many other math problems."
Sholto Douglas, Anthropic researcher, backed Altman and his competitor, writing in a post that "there is no way OAI would pull user transcripts for this, or knowingly train on it in a way that would've influenced this."
This article originally appeared on USA TODAY: OpenAI touts math breakthrough. Mathematicians dispute credit
