OpenAI Navier-Stokes Proof Faces Scrutiny Over Methods and Agent Contribution

However, as the mathematics community scrutinizes the work further, researchers have begun to question the methods used in the proof, its relationship to existing research, and the extent of the role played by large-scale AI agents.

Mathematician Gonzalo Cao-Labora noted that OpenAI may have rushed to publish without fully understanding their own results.

He stated that the details of OpenAI’s proof follow a different path from the methods previously proposed by Spanish mathematicians Diego Córdoba and Luis Martínez-Zoroa, suggesting that any similarities between the two are likely superficial.

Multiple expert groups reviewing the work told Scientific American that there is growing evidence that OpenAI’s proof did not follow the "Spanish route." This implies that OpenAI appeared to lack a complete understanding of what their model had actually accomplished when they released the result.

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 claiming that its internal models completed a mathematical proof, simultaneously publishing the proof written in the Lean formal verification language. OpenAI stated that its research team used approximately 10,000 AI agents for about 88 hours to obtain the results, followed by another ~17 hours to complete the formalization and verification.

Lean can check whether logical deductions in a formalized proof adhere to set rules, but passing formal verification does not guarantee that the problem solved aligns with the scientific questions originally intended by the researchers, nor does it ensure that the proof’s logic is easily understandable to humans. The proof published by OpenAI spans 166 pages, and the mathematics community is still examining its specific methods and conclusions.

One point of contention concerns the degree of connection between OpenAI’s research and the paths previously explored by Córdoba and Martínez-Zoroa.

Scientific American pointed out that OpenAI’s proof utilized the external force term in the Navier-Stokes equations to construct a scenario where fluid velocity grows unboundedly within finite time.

Mathematicians hold differing views on the significance of this result: while it fits one of the scenarios permitted by the original formulation of the Clay Mathematics Institute’s problem, it does not address the more prominent question of whether solutions to the three-dimensional Navier-Stokes equations remain smooth indefinitely when no external forces are applied.

Regarding attribution of the findings, Tristan Buckmaster, a mathematician at New York University, questioned whether OpenAI’s research might have benefited from his prior work with Levent Alpöge, an Anthropic employee, which involved using public models.

In response, OpenAI denied using data associated with Buckmaster. In an official update on September 10, they stated that after investigation, prompts entered by Buckmaster into Codex over the previous two months could not have influenced this research in any way, including through model training. However, disputes regarding the relationship between the methodologies and outcomes persist.

Another unresolved issue is the actual contribution of agent collaboration. OpenAI officially claimed that the research involved approximately 10,000 parallel-running agents, sending roughly 2.7 million messages and generating about 130 billion output tokens. Nevertheless, these figures only reflect computational scale and operational processes; they do not directly indicate the actual contribution of the multi-agent architecture to the final outcome.

OpenAI researcher Noam Brown also noted that researchers currently do not know how much additional benefit such a large-scale agent cluster provided. Due to the high cost of conducting systematic controlled experiments with thousands of agents, researchers have not sufficiently compared the time required for a single agent to complete the same task, making it difficult to accurately measure the advantage of 10,000 agents over 1,000. Brown estimated that the multi-agent collaborative architecture contributed less than 10% to this success.

In his view, because AI agent technology is novel and attracts significant attention, it tends to receive credit exceeding its actual contribution. The truly critical factors may be the inherent capabilities of the underlying models themselves, along with their ability to sustain long-duration task processing and explore multiple solution paths in parallel.

In early October, OpenAI publicly released approximately 400 mathematical research results. The mathematics community generally indicated that digesting this volume of content would take years.

These results were concentrated on GitHub in a relatively loose format, again sparking dissatisfaction among some mathematical researchers. They worry that releasing a large volume of results without adequate explanation, organization, or academic exchange reduces the readability of mathematical research and blurs the distinction between rigorous research verification and mass-generated automated results.

Currently, the mathematics community continues to evaluate OpenAI’s relevant research. The controversy involves not only whether AI can generate formal proofs, but also whether the research methods truly solve the target problems, how attribution should be handled, and how to measure the actual contribution of multi-agent systems.