AI and Mathematics: Conflict Over Data and Credit at OpenAI

OpenAI claims to have solved a Millennium Prize problem related to the Navier–Stokes equations, while acknowledging that it cannot rule out the possibility that de-identified user data may have improved its models. This assertion comes as two researchers, who have used LLMs in their own related work, describe contentious exchanges regarding authorship and the coordination of announcements.

OpenAI trains by default on inputs and does not exclude de-identified contributions

OpenAI's terms of service stipulate that its models are trained on data submitted by users unless explicitly opted out. The company states that users can withdraw via its privacy portal by selecting "do not train on my content." In March, it explained that sharing content makes its models more accurate, better on specific problems, and enhances their overall capabilities and security. In this case, OpenAI asserts that no specific user data was consulted to solve Navier–Stokes, but it cannot rule out that de-identified data from the use of its products may have contributed to improving its models. The company adds that none of its researchers or agents saw the duo's work before its publication, "by any means," while congratulating the authors. It further claims that its proofs differ significantly, particularly regarding the Euler case (forced vs. unforced). Tristan Buckmaster's statement specifies that he personally paid for the tools, without indicating the product used, the type of account, or any potential opt-out choice.

Prior work, methods, and limitations claimed by academic authors

Tristan Buckmaster credits Diego Cordoba and Luis Martinez-Zoroa for the foundational approach, stemming from a 2024 study. He then notes that his work with Levent Alpöge employed language models to make progress, particularly relying on Claude and Codex. The two mathematicians claim to have identified situations where current tools do not accurately describe smooth air and water flows, without claiming to have solved the question that would qualify for the Clay Mathematics Institute's million-dollar prize. Terence Tao described these results as "remarkable." On Monday, the duo published new papers on these topics.

Tristan Buckmaster claims he was informed of OpenAI's advancement after its announcement

According to Tristan Buckmaster, after notifying a mathematician at OpenAI of their intention to publish, members of OpenAI – including Sebastian Bubeck – informed him that their internal model had produced a proof for Navier–Stokes and suggested publication modalities. Buckmaster claims he was offered that Levent Alpöge be removed from the authorship list due to his employment at Anthropic. He writes that he has not seen OpenAI's proof, does not know if the duo's data was used, and does not accuse anyone, while stating he is reporting what was said and proposed to him. Sebastian Bubeck contested elements of this account in a post on X, denying having requested Alpöge's removal, stating he wanted to coordinate announcements, and asserting that OpenAI was motivated by viral rumors on Twitter mentioning the resolution of two Millennium problems by Anthropic. He had previously denounced false and inflammatory allegations against him.

Communities in turmoil and absence of formal reactions from the parties involved

The dispute has animated online communities in mathematics and AI. Sholto Douglas from Anthropic called the situation "extremely sad." Other actors have echoed the most striking allegations in press releases or offered summaries of the conflict. OpenAI, Anthropic, Buckmaster, and Alpöge have not responded to requests for comment from a business media outlet.

OpenAI's claim of a solution and the claimed scope

On Tuesday, OpenAI announced that it had found a solution to the Navier–Stokes problem in the context of the Millennium Prizes. The company asserts that no specific user data was consulted to achieve this and states that no member of its team or its AI agents had access to the duo's work before its public online release. OpenAI clarifies that its proofs differ significantly and that proven precise results do not coincide on the Euler case, distinguishing between forced and unforced frameworks.

Scientific credit and user data, two debates now linked

The unfolding of this matter raises questions about the sharing of credit between researchers and AI labs, especially since models can be trained on prior breakthroughs. It also brings to the forefront the handling of data collected from clients and the potential for competitive use. Tristan Buckmaster indicates that OpenAI was not clear with him about a potential training of its models on the duo's work, while the provider's policy allows training on user data. The final reminder emphasizes that AI companies continuously improve their models and that, unless opted out or accounts are set up to exclude it, this includes the input content. OpenAI, for its part, states it cannot rule out that the authors' use of its products may have contributed in a de-identified manner to the overall improvement of its models.