NYU mathematics professor Tristan Buckmaster said on Tuesday that he and Levent Alpöge had completed three proofs tied to the Navier-Stokes existence and smoothness problem, shortly before OpenAI published what it called a full proof of the central problem.. Buckmaster said the results were produced with Alpöge using OpenAI’s Codex and Anthropic’s Claude models, and said a separate OpenAI effort may have built on their approach before their work became public..
OpenAI said its full proof was discovered by an unreleased next-generation model that had been working on several unsolved problems over the past week. The company said the effort used 300 billion output tokens, which the article said would equal about $22.5 million at current Astra rates..
The Navier-Stokes existence and smoothness problem is one of the seven Millennium Prize problems. Each carries a $1 million award from the Clay Mathematics Institute for a valid solution.. Buckmaster said that while he and Alpöge were finalizing their results, they learned that information about their progress had been passed to OpenAI.
He said OpenAI later told them it had already obtained a full proof, but gave unclear answers when asked when work began and how much human input was involved.. “It emerged that an entire team had been working on the problem,” Buckmaster said, adding that “an insane amount of compute had been used.” He said it was later agreed that the first prompt had been sent in the past few days, after information about his team’s work had reached OpenAI..
OpenAI said its latest effort began on September 1 and was inspired by rumors that two Millennium Prize problems had been solved. The company also confirmed that it had been in contact with Buckmaster and Alpöge.. Buckmaster said he found it suspicious that OpenAI had pursued the same relatively uncommon route to the problem.
“Almost nobody else I know of was working on it,” he wrote. “It is not the direction one arrives at in a few days by giving a model the problem statement.”. Alpöge, an Anthropic mathematician, conducted the research independently and not on Anthropic’s behalf.
Buckmaster alleged that OpenAI researcher Sébastien Bubeck asked him to remove Alpöge’s credit as part of a proposed compromise.. Buckmaster also said Bubeck warned him against publicizing the dispute. According to Buckmaster, Bubeck replied, “Why would you ruin your career?” and then said, “If you don’t want me to be nice, then I don’t have to be nice.”.
Buckmaster said his heavy use of Codex raised concerns that his own interactions with the tool could have informed OpenAI’s work, because OpenAI reserves the right to train models on Codex interactions unless users opt out.. OpenAI rejected the claim that it had directly accessed Buckmaster’s work.
“We 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,” the company said, adding that it could not fully rule out whether de-identified usage data had helped improve its models.. Featured image credit.
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