OpenAI’s Navier-Stokes Breakthrough Has a Trust Problem

The Timeline That Sparked the Controversy

OpenAI says the project began on September 1 after rumors reached the company that two Millennium Prize Problems might have been resolved. Researchers sent groups of AI agents after several of the remaining problems and tried different formulations. After nearly 100 agents produced an unforced Euler blowup result, OpenAI dramatically scaled up its attack on Navier-Stokes. At its peak, the company says around 10,000 agents were working concurrently, with the Navier-Stokes effort producing roughly 130 billion output tokens. By September 5, the agents had their result, followed by another 17 hours spent formalizing and verifying it in Lean.

Those rumors were eventually traced to NYU mathematician Tristan Buckmaster and Levent Alpöge, a mathematician who works at Anthropic. For much of the previous year, the two had been working on fluid-dynamics problems closely related to Navier-Stokes, building on earlier research by mathematicians Diego Córdoba and Luis Martínez-Zoroa. According to Buckmaster’s account, he and Alpöge made a major breakthrough on August 15, showing that several related fluid equations could develop singularities in finite time even with smooth external forcing. One of those results involved the three-dimensional incompressible Euler equations, which are closely related to Navier-Stokes, and they formally verified that proof in Lean a week later.

This does not mean Buckmaster and Alpöge had already solved the Navier-Stokes Millennium Prize Problem. Their results concerned related equations, and the broader approach they were following was not secret. Buckmaster openly credits Córdoba and Martínez-Zoroa with developing the research program they built on, and OpenAI cites that earlier work in its own paper. What raised Buckmaster’s suspicions was the sequence of events: OpenAI learned that his team had made significant progress and then, within days, produced a Navier-Stokes result that also relied on smooth external forcing. OpenAI says its actual mathematical construction is substantially different, but Buckmaster questioned how the company had arrived at that particular direction so quickly. Adding to his concern, he and Alpöge had been using OpenAI’s Codex throughout their research, and Buckmaster says they had uploaded drafts from the project into their Codex sessions.

Could Private Codex Data Have Influenced the Result?

Buckmaster contacted OpenAI on September 3 after learning that word of their progress had reached the company. During conversations a few days later, he asked whether OpenAI’s internal model had been trained on, or had access to, their Codex sessions. Buckmaster says he was told that the model had not looked up user data. When he pressed specifically on training, he says he did not receive an answer at the time. He has stopped short of claiming he can prove OpenAI used their private research, instead raising the question of whether information from their Codex activity could have found its way into the development of a model that later produced a breakthrough in closely related mathematics.

OpenAI says its researchers and agents did not see Buckmaster and Alpöge’s unpublished work before it became public and that “no specific user data was accessed” to solve Navier-Stokes. But the company’s response includes an important qualification: while it considers the possibility unlikely, OpenAI says it “cannot rule out that de-identified data derived from their usage of our products helped improve our models.”

That is not an admission that OpenAI trained on their drafts. The company has not said that any such data contained their mathematics, entered the model responsible for the proof, or influenced the Navier-Stokes result. No public evidence currently traces an unpublished Buckmaster-Alpöge argument from a Codex session into OpenAI’s solution. Still, this distinction shows why proving independence in cases like this may be difficult. Saying that researchers or agents did not access someone’s chats does not necessarily establish that nothing derived from those interactions ever contributed to model development. WIRED reported that OpenAI denied inspecting the pair’s Codex prompts or using their proof to direct its agents, while the company acknowledged that it could not completely exclude indirect influence through de-identified product data.

Even whether their Codex material was eligible for model improvement remains unclear publicly. OpenAI’s treatment of user content depends on the product and data controls involved. Individual users can have content used for model improvement depending on their settings, while Business, Enterprise, Edu, and API data are not used for training by default. We do not know which settings applied throughout Buckmaster and Alpöge’s work, so simply putting research into Codex does not establish that OpenAI trained on it.

The dispute also became personal. Buckmaster says OpenAI discussed an arrangement in which he would help present its Navier-Stokes result without Alpöge as an author, with Alpöge’s employment at Anthropic raised as an issue. He also alleges that after saying he would make the situation public, he was asked why he would “ruin” his career. OpenAI disputes his characterization of the authorship discussion, saying the concern was whether an Anthropic employee should be added to a separate paper presenting work produced inside OpenAI, not whether Alpöge should lose credit for his own research.

The rivalry between OpenAI and Anthropic is difficult to ignore in that context. OpenAI acknowledges that rumors of another team’s progress helped kick off its experiment, after which thousands of AI agents were directed toward some of mathematics’ hardest problems. None of that proves OpenAI saw private research, but it does offer a glimpse of what competition between frontier AI labs can look like when it spills beyond model benchmarks and into scientific discovery. Former OpenAI and Anthropic researcher Jacob Coxon added to that wider conversation when he resigned from Anthropic this week and criticized the pace at which frontier labs are pursuing increasingly capable AI. His resignation has no known connection to Navier-Stokes, but his broader concern about companies racing because they fear a competitor will get there first has an obvious parallel with an experiment OpenAI says was itself prompted by rumors of another team’s breakthrough.

When the AI Helping Researchers Becomes Their Competitor

Whether OpenAI did anything improper here remains unresolved, but the more lasting issue may be the relationship the controversy has exposed. Scientists are no longer using commercial AI only to polish finished work. They are using it while discoveries are taking shape, asking models to challenge arguments, find errors, work through proofs, write code, and examine drafts containing ideas nobody else has seen. Meanwhile, the companies running those tools are developing AI systems capable of pursuing original scientific results themselves. The AI helping someone develop a discovery can now belong to a company capable of competing for that same discovery.

Academic publishers already recognize part of the risk. Elsevier’s generative AI policy, for example, warns peer reviewers against uploading unpublished manuscripts to generative AI tools when doing so could compromise confidentiality. Researchers voluntarily using AI throughout their own work presents a harder problem. If an unpublished idea develops inside a commercial AI system and its maker later arrives at something similar, proving that the discoveries were independent may be difficult for both sides.

For now, there is no public evidence that OpenAI copied Buckmaster and Alpöge’s unpublished work or that their Codex sessions caused its Navier-Stokes breakthrough. There is also plenty of mathematical work left before OpenAI’s proposed solution can be treated as the answer to a problem that has resisted mathematicians for decades. But even if the proof holds up and OpenAI’s result was entirely independent, this episode has exposed a problem science is likely to face again. We are getting better at checking whether AI-generated mathematics is correct, but we may also need better ways of establishing where AI-generated discoveries came from. Lean can check the proof, but it cannot check the provenance.

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