IT Home, October 17: In September, OpenAI released a 166-page proof related to the Navier–Stokes equations, claiming its internal AI system solved this Millennium Prize problem in approximately 88 hours.
However, as the mathematical community continues to review the work, researchers have begun to question the methods used in the proof, its relationship to existing research, and the extent to which large-scale AI agents contributed.
Mathematician Gonzalo Cao-Labora noted that OpenAI may have rushed to publish its paper without fully understanding it themselves.
He noted that the details of OpenAI's proof follow a different path from the method previously proposed by Spanish mathematicians Diego Córdoba and Luis Martínez-Zoroa, and any similarities between them may be more superficial than substantive.
Multiple groups of experts who reviewed the work told Scientific American that increasing evidence suggests OpenAI’s proof did not follow the “Spanish route,” meaning that OpenAI appeared to lack a full understanding of what its model actually did when it released the results.
IT Home note: The Navier-Stokes equations describe fluid motion and are one of the Millennium Prize Problems established by the Clay Mathematics Institute. On September 8, OpenAI released related research stating that its internal model completed a mathematical proof and simultaneously published the proof written in Lean, a formal verification language. OpenAI stated that its research team used approximately 10,000 AI agents over about 88 hours to obtain the result, followed by approximately 17 hours to complete formalization and verification.
Lean can verify whether logical deductions in formal proofs adhere to established rules, but formal verification does not guarantee that the problem being solved aligns with the original scientific question researchers intended to address, nor does it ensure that the proof’s reasoning is easily understandable to humans. The proof published by OpenAI is 166 pages long, and the mathematical community is currently reviewing its specific methods and conclusions.
One of the points of contention is how closely OpenAI’s research is related to the research paths previously explored by Córdova and Martínez-Zarauz.
Scientific American notes that OpenAI's proof exploited the external force term in the Navier-Stokes equations to construct a scenario where fluid velocity grows without bound in finite time.
Mathematicians hold differing views on the significance of this result: it aligns with one of the scenarios permitted by the Clay Mathematics Institute’s original formulation of the problem, but it does not resolve the more widely studied question of whether solutions to the three-dimensional Navier-Stokes equations remain smooth in the absence of external forces.
Regarding the attribution of research findings, New York University mathematician Tristan Buckmaster has questioned whether OpenAI’s research may have benefited from his related work conducted with Anthropic employee Levent Alpöge using publicly available models.
In response, OpenAI denied using Buckmaster’s data and, in an updated official statement on September 10, stated that after investigation, the prompts Buckmaster entered into Codex over the previous two months could not have influenced this research in any way, including through model training. However, as of now, the two parties still dispute the relationship between the research methodology and its outcomes.
Another unresolved question is how much these agents actually contributed to the outcome. OpenAI stated that its research involved approximately 10,000 agents running in parallel, collectively sending around 2.7 million messages and using roughly 130 billion output tokens. However, these figures only reflect the scale of computation and the process, not the direct contribution of the multi-agent architecture to the final result.
OpenAI researcher Noam Brown also noted that researchers currently do not know how much additional benefit such a large-scale agent cluster provides. Due to the high cost of conducting systematic controlled experiments with thousands of agents, researchers have not thoroughly compared the time required for a single agent to complete the same task, nor can they accurately measure the advantage of 10,000 agents over 1,000 agents. Brown estimates that the contribution of the multi-agent collaborative architecture to this success may be less than 10%.
In his view, because AI agent technology is novel and prone to widespread attention, it often receives recognition beyond its actual contributions; the truly critical factors may be the underlying model’s capabilities and its ability to sustain long-term task processing and concurrently explore multiple solutions.
In early October, OpenAI publicly released approximately 400 mathematical research findings. The mathematical community generally agrees that it would take years just to digest this content.
These results were collectively released in a relatively informal manner on GitHub, again causing dissatisfaction among some mathematical researchers. They are concerned that publishing a large volume of results without adequate explanation, organization, or academic discourse may reduce the readability of mathematical research and blur the distinction between rigorous scholarly validation and large-scale automated generation.
The mathematical community is currently evaluating OpenAI’s related research. The controversy extends beyond whether AI can generate formal proofs to include questions about whether the research methodology truly addresses the intended problem, how credit for the results should be assigned, and how to properly measure the practical contributions of multi-agent systems.