OpenAI faces dispute over AI-assisted Navier–Stokes proof effort

NYU mathematics professor Tristan Buckmaster says OpenAI moved to pursue a leading approach to the Navier-Stokes existence and smoothness problem after learning about work he was completing with Anthropic mathematician Levent Alpöge. The dispute emerged as Buckmaster and Alpöge announced preliminary results and OpenAI published what it described as a full proof of the same central problem.

The episode combines a potentially important mathematical development with a difficult question for AI research: when a model-assisted team learns that outside researchers have found a promising route, how much can it use that direction without improperly appropriating the researchers’ work? The available evidence consists primarily of Buckmaster’s account and OpenAI’s own statement. There is no independent confirmation in the source material that OpenAI’s proof is correct or that it used confidential user data.

A race around a $1 million mathematics problem

The Navier-Stokes existence and smoothness problem is one of the seven Millennium Prize problems established by the Clay Mathematics Institute. A correct solution carries a $1 million prize and would resolve a longstanding question about the mathematical behavior of fluid-flow equations.

Buckmaster announced three proofs with Alpöge, describing them as preliminary findings that moved toward the broader problem. Their work used several AI systems, primarily OpenAI’s Codex and Anthropic’s Claude, but Alpöge was acting independently rather than on behalf of Anthropic.

According to Buckmaster’s statement, the researchers learned while finalizing their results that information about their progress had reached OpenAI. He says OpenAI then told them it had already obtained a full proof. OpenAI’s own release said its latest effort began on September 1, after rumors circulated that two Millennium Prize problems had been solved.

That timing is central to the dispute. Buckmaster argues that the particular route his team pursued was not an obvious response to the problem statement and was being explored by relatively few researchers. He therefore found it suspicious that OpenAI’s team adopted what he described as the same direction soon after learning about the outside work.

What Buckmaster alleges, and what OpenAI says

Buckmaster says discussions with OpenAI became increasingly unclear when he asked when the company had begun its work and how much human guidance had been involved. He alleges that a large team was assigned to the problem and that an unusually large amount of computing power was used.

He also says OpenAI researcher Sébastien Bubeck asked him to remove Alpöge’s credit as part of a possible compromise. Buckmaster further alleges that Bubeck warned him against making the dispute public. Those statements are allegations from Buckmaster, not independently established facts in the available reporting.

OpenAI’s published account confirms that its effort began after September 1 and acknowledges ongoing conversations with Buckmaster and Alpöge. The company says its proof differs substantially from the researchers’ work, including a distinction between forced and unforced formulations in the Euler-related result.

The company also rejected the idea that its researchers or AI agents had directly viewed the team’s unpublished work before public release. OpenAI said that no specific user data was accessed to solve the problem. It nevertheless acknowledged that it could not rule out the possibility that de-identified data derived from product use had helped improve its models.

That qualification matters because Buckmaster used Codex extensively. OpenAI reserves the ability to train models on Codex interactions, although users can opt out. The source material does not establish that Buckmaster’s interactions were included in training, that any model reproduced his work, or that such data influenced OpenAI’s proof.

The proof claims remain unverified in this account

OpenAI says an unreleased next-generation model discovered the proof after a week-long effort involving 300 billion output tokens. TechCrunch reported a compute estimate of $22.5 million based on then-current Astra pricing. These are reported figures and company-linked claims, not independently audited measurements.

Likewise, the existence of a “full proof” should not be treated as equivalent to a solution accepted by the mathematics community. Millennium Prize solutions require detailed verification, and the source evidence does not include a peer-reviewed assessment, a Clay Mathematics Institute decision, or confirmation from independent mathematicians that OpenAI’s argument is sound.

Buckmaster and Alpöge’s results are also described as preliminary. Their significance may ultimately depend on whether the work can be completed, checked, and shown to resolve the exact conditions of the Navier-Stokes problem. In advanced mathematics, an AI-generated argument can be valuable without being correct, complete, or eligible for a formal prize.

The episode therefore has two separate claims that should not be conflated. One is mathematical: whether either team has produced a valid solution. The other is procedural: whether OpenAI used information obtained through conversations with the researchers or adopted their approach unfairly. Neither question is settled by the available statements alone.

Why the dispute matters for AI research teams

For AI builders and research organizations, the conflict highlights a growing governance problem around model-assisted discovery. A scientist using a hosted coding or reasoning system may generate novel work inside a platform whose provider retains some ability to use interaction data for model improvement. Even if no employee directly reads a user’s private work, questions can arise about training exposure, memorization, access controls, and the provenance of later outputs.

The case also puts pressure on disclosure practices. Teams using AI systems for original research may need clearer records of prompts, model versions, data-retention settings, human contributions, and the time at which key ideas emerged. Those records could become important when several groups report similar results or when a model produces a proof after being exposed to related research.

For enterprise buyers, the practical issue is not limited to mathematics. The same concerns apply when employees use AI tools to develop proprietary algorithms, product designs, legal strategies, or scientific findings. Organizations will want contractual clarity on training use, stronger isolation for sensitive sessions, and procedures for investigating possible overlap between customer work and vendor research.

The story also complicates competition among AI labs. Alpöge’s employment by Anthropic, despite his independent role in this project, appears to have added institutional tension. If researchers believe affiliation with a rival company can affect credit or access to collaboration, cross-company scientific work may become more difficult precisely when frontier AI research depends on it.

What to watch next

The first signal will be independent mathematical review of OpenAI’s claimed proof and of Buckmaster and Alpöge’s preliminary results. A formal submission, public technical details, or an assessment connected to the Clay Mathematics Institute would help distinguish a genuine solution from an impressive but incomplete model output.

Researchers should also watch for documentation about the chronology of the two efforts: when OpenAI first received relevant information, what its systems were shown, and whether any Codex interaction data was eligible for training. Clarification of the data-retention and opt-out settings used by Buckmaster would be relevant, although it still would not by itself prove model contamination.

Finally, the dispute may prompt AI labs and academic institutions to establish clearer norms for credit, confidentiality, and priority when models participate in research. The outcome could influence how future discoveries are recorded and how scientists choose between hosted AI tools and locally controlled systems.

Creati.ai perspective

The immediate temptation is to frame this as a contest over who solved a famous problem first. The more consequential issue is whether AI-assisted research has adequate provenance rules when a scientist’s work passes through a commercial model provider. OpenAI’s denial of direct access and its admission that de-identified influence cannot be ruled out leave a gap between technical possibility and evidentiary proof.

Until the mathematical claims and the research timeline are independently examined, the responsible conclusion is limited: OpenAI has reported a major proof effort, while Buckmaster has raised serious questions about priority, attribution, and data use. For builders and enterprises, the case is an argument for auditable AI workflows—not a reason to assume either that hosted models are copying users or that model-generated breakthroughs are already validated.