OpenAI Solved a $1M Millennium Prize Problem. Or Did It Just Steal the Answer?

OpenAI’s claim that an internal AI system solved a version of the Navier-Stokes Millennium Prize Problem has become entangled in a dispute over whether the lab’s breakthrough followed, or may have indirectly benefited from, unpublished work produced by two mathematicians using its own Codex platform.

The company said about 10,000 coordinating agents running a next-generation model reached a forced Navier-Stokes blow-up result after 88 hours of work.

However, the effort began after OpenAI heard rumors that mathematicians Tristan Buckmaster and Levent Alpöge had made major progress on closely related problems.

OpenAI says neither its researchers nor its agents saw the pair’s work before publication and that no specific user data was accessed to solve the problem.

It has, however, acknowledged a narrower possibility: “While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models.”

That qualification has become central to a dispute involving one of mathematics’ most prestigious open problems and the increasingly blurred boundary between researchers using frontier AI systems and the companies operating them.

A Year of Work Meets an 88-Hour Sprint

Buckmaster, a professor at NYU’s Courant Institute, and Alpöge, a mathematician employed by Anthropic, had spent most of the previous year extending a program developed by Diego Córdoba and Luis Martínez-Zoroa.

Their work sought finite-time blow-up results under smooth forcing, progressing through the incompressible porous-medium equation, the Boussinesq system and three-dimensional incompressible Euler.

Buckmaster said they obtained the Boussinesq and Euler results on August 15 and completed Lean verification on August 22.

They also believed they had a hypo-dissipative Navier-Stokes result, although it had not yet been fully written or formally verified.

The collaboration made extensive use of AI systems, including Anthropic’s Claude and OpenAI’s Codex. Buckmaster said the team fed drafts from the project into their Codex sessions throughout the year.

OpenAI’s own effort moved on a radically different timescale.

According to the company, its internal group used a model substantially more capable than GPT-6 Astra and deployed roughly 10,000 agents.

The system ultimately generated an analytical proof and Lean formalization showing a finite-time singularity under Navier-Stokes dynamics.

The lab described the result as a solution to the Millennium Prize Problem, although it said it would not claim the Clay prize.

Rumors Preceded OpenAI’s Push

The timing is one of the main points of contention.

Bubeck said OpenAI began working on Millennium problems after “viral Twitter rumors” suggested Anthropic had resolved two of them.

The aim, he said, was to test whether OpenAI’s new internal system could achieve something comparable.

Buckmaster said he contacted an OpenAI-affiliated mathematician on September 3 after hearing that information about his collaboration had reached the company.

His email stressed that the project was personal rather than an Anthropic initiative and that the results would be posted shortly.

OpenAI replied that further details could help it “avoid competing” and offered additional compute.

Three days later, Buckmaster said he was told during calls with OpenAI researchers that the company had an approximately 100-page proof of finite-time blow-up for forced Navier-Stokes equations in R3 and T3 with smooth forcing.

For Buckmaster, the use of smooth forcing was significant.

He said the route had been opened by Córdoba and Martínez-Zoroa and was the same relatively uncommon direction that he and Alpöge had been pursuing privately.

“When I heard ‘forced,’ it was a bright red flag,” he wrote.

Questions Over Codex

Buckmaster said OpenAI initially presented its result as having emerged after its model was essentially given the problem statement with very little human involvement.

His account says further discussion revealed a wider operation: a team had explored several problems, initially worked on the unforced version, tested easier equations including Euler, and used substantial compute.

He also said the prompt shown to him had itself been generated using Codex.

When Buckmaster asked when OpenAI’s first prompt had been issued, he said the eventual answer placed it within the preceding days, after information about his work had reached the company.

He then asked whether OpenAI’s model had been trained on, or otherwise had access to, the Codex sessions containing the pair’s drafts.

Buckmaster said he was told the model did not look up user data, but that his separate question about training went unanswered during the call.

OpenAI has since denied accessing the researchers’ unpublished work.

“We (the researchers and the agents) did not see any of their work through any means until they released it publicly,” the company said.

Its additional statement that de-identified product-usage data could nevertheless have contributed to model improvement raises a materially different question than whether employees or agents directly inspected the researchers’ private files.

No evidence presented by either side establishes that OpenAI staff or agents read Buckmaster and Alpöge’s Codex sessions.

The dispute therefore does not establish that OpenAI copied or stole a finished proof.

It instead centers on whether information derived from their use of OpenAI products could have influenced a model that was later directed toward the same narrow mathematical route after rumors of their progress reached the lab.

Authorship Dispute Deepens Rift

The disagreement also spilled into questions of credit.

Buckmaster said OpenAI offered two arrangements during the September 6 discussions. Under one, his team would publish its Euler result before OpenAI announced Navier-Stokes.

Under another, he said he would write up OpenAI’s Navier-Stokes result himself, acknowledging the internal model but without Alpöge as an author.

Buckmaster alleged that Bubeck repeatedly raised Alpöge’s Anthropic employment as the obstacle and later asked: “Why would you ruin your career?” when Buckmaster said he would publicly describe the episode.

Bubeck disputes that characterization.

He said he never asked for Alpöge to be removed from authorship of Alpöge’s own work.

Rather, he said one option discussed was for Buckmaster to lead a rewrite of OpenAI’s separate Navier-Stokes proof, and that including an Anthropic employee as an author of OpenAI work presented an institutional problem.

Bubeck acknowledged making the career remark but said it was intended as concern over Buckmaster making what he regarded as unfounded accusations.

He called it an “extremely poor choice of words” and said he retracted it immediately.

Alpöge, however, said he heard part of the discussion from a hallway and understood that a Millennium result was being offered on the condition that he was removed from the paper.

Different Proofs, Same Controversy

OpenAI stresses that the mathematical outputs are not identical.

Buckmaster and Alpöge published a forced Euler blow-up result, while OpenAI says its Euler proof concerns the unforced case.

Its Navier-Stokes result, meanwhile, concerns smooth forcing corresponding to the forced alternatives in the Clay problem.

OpenAI has congratulated the pair on their work and recognized their mathematical achievement.

Buckmaster has also stopped short of alleging direct theft.

“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 am not accusing anyone of anything,” he wrote.

The narrower controversy remains unresolved.

Two researchers spent roughly a year developing an obscure smooth-forcing program while feeding drafts through frontier AI systems.

OpenAI then learned rumors of their progress, deployed vastly greater compute against related Millennium problems and announced a forced Navier-Stokes result days later.

Whether those events amount only to aggressive competition, indirect model learning from customer usage, or something more consequential remains unclear from the material made public.

But the episode has exposed a problem that extends beyond Navier-Stokes: researchers may increasingly rely on proprietary AI systems to develop unpublished mathematics while the same companies can train more capable models, deploy them at vastly greater scale and compete on the resulting discoveries.

The argument over who solved what may eventually be settled by mathematicians examining the proofs.

The harder question, what an AI laboratory owes researchers whose unpublished work passes through its systems before its own models enter the same race, is only beginning.