OpenAI Claims Navier-Stokes Solution as Researchers Question How It Was Reached

OpenAI has announced what it describes as a solution to the Navier-Stokes existence and smoothness problem, one of mathematics' seven Millennium Prize Problems, using an internal artificial intelligence system.

The Navier-Stokes equations describe the motion of fluids and gases, including water and air. The long-standing mathematical question asks, in simplified terms, whether smooth three-dimensional fluid motion must always remain regular or whether a singularity can develop in finite time.

OpenAI said its system produced an analytical proof and a formalisation in Lean showing that an initially smooth fluid at rest can develop a singularity in finite time while subject to a smooth external force and while maintaining finite energy. The company presents this as a resolution of the Millennium Prize formulation, although it has said it does not intend to claim the associated prize.

The Clay Mathematics Institute selected the Navier-Stokes existence and smoothness problem as one of seven Millennium Prize Problems, with USD 1 million allocated to each problem solved in accordance with the institute's requirements.

According to OpenAI, the work was carried out using a new internal model that is significantly more capable than GPT-6 Astra. The company said training of the internal model began on August 28 and that the system had shown unprecedented performance on its internal benchmarks, including mathematics.

OpenAI said the effort used a coordinating multi-agent system and that the group which produced the Navier-Stokes result involved roughly 10,000 concurrent agents. The company said the agents had access to tools including a cached version of the internet and code execution.

The announcement has also prompted questions over the timing of the work and the relationship between OpenAI's result and concurrent research by Tristan Buckmaster, a mathematics professor at New York University, and Levent Alpöge, a researcher at Anthropic. QNA reported that Buckmaster had disclosed related research shortly before OpenAI's announcement and raised questions about whether work carried out through Codex could have influenced OpenAI's system.

OpenAI has rejected the suggestion that it accessed specific user data to solve the problem. In its official account, the company said neither its researchers nor its agents saw Buckmaster and Alpöge's work before it was made public and that no specific user data was accessed for the solution.

At the same time, OpenAI said it could not completely rule out the possibility that de-identified data derived from use of its products had contributed to improving its models. The company also said the concurrent result by Buckmaster and Alpöge concerned the forced Euler problem, while OpenAI's own work on the related Euler question and its Navier-Stokes proof differed from theirs.

The episode has therefore drawn attention not only to the mathematical claim itself but also to questions surrounding research priority, the use of AI tools in academic work and how data generated through those tools may contribute to future model development.

--

About Qatar Living:

Since 2005, Qatar Living has been the trusted destination for everything Qatar. As the country's largest online community and marketplace, Qatar Living connects people with opportunities through jobs, property, vehicles, services, classifieds, events, local insights, and breaking news. Trusted by residents, newcomers, visitors, and businesses alike, Qatar Living brings Qatar together in one place.

Instagram - @qatarliving

X - @qatarliving

Facebook - Qatar Living

YouTube - qatarlivingofficial