On September 8, OpenAI announced it had purportedly solved the Navier-Stokes equations, a challenge recognized as one of the Millennium Prize Problems, for which the Clay Mathematics Institute has offered a $1 million reward.
We’re sharing a solution to the Navier-Stokes Millennium Prize Problem, one of the deepest problems at the frontier of mathematics.
The proof was produced by a group of agents, using an OpenAI next-generation model significantly more capable than GPT-6 Astra.
The problem… pic.twitter.com/8zol3BPTL4
— OpenAI (@OpenAI) September 8, 2026
The Navier-Stokes equations describe the behavior of fluids such as liquids and gases. A key inquiry is whether a smooth three-dimensional flow remains smooth over time.
Another approach to this problem involves demonstrating a scenario where a solution loses its smoothness and a singularity arises within a finite time frame.
OpenAI’s system constructed such an example involving a stationary fluid acted upon by a smooth external force, while keeping the system's energy finite.
“The solution represents a vortex—a swirling fluid flow that tightens and elongates like spaghetti,” OpenAI explained.
In this scenario, the central region compresses over time while the velocity increases indefinitely. According to the company, this provides solutions for options C and D of the official problem statement, which allow for the use of a smooth external force.
This means that the result does not imply that a similar singularity would naturally occur in ordinary water or air without external influence.
Local fluid motion in OpenAI's construction. Source: OpenAI.On August 28, OpenAI began training a new internal model, and on September 1, its team learned of potential breakthroughs by other researchers in hydrodynamic problems. They decided to test their system on remaining Millennium Problems and other complex mathematical challenges.
OpenAI initiated groups of interacting agents with access to a cached version of the internet, tools for executing code, and the ability to share intermediate results.
Initially, the system solved a simpler related problem about Euler's equations, which lack viscosity. Following this, the developers redirected computational resources towards the Navier-Stokes equations.
According to OpenAI, around 10,000 agents worked simultaneously on the main problem, achieving a result on September 5—approximately 88 hours after the experiment commenced. An additional 17 hours were spent formalizing and verifying the proof using GPT-6 Astra.
During the work on Navier-Stokes, the agents exchanged about 2.7 million messages and generated around 130 billion output tokens. Across the entire mathematical experiment, they accumulated 4.9 million messages and roughly 300 billion tokens.
As reported by Nature, OpenAI utilized an unprecedented amount of computational resources for a single mathematical investigation.
According to the rules set by the Clay Mathematics Institute, the proposed solution must first be published in a peer-reviewed scientific journal. Following publication, it must be recognized by the global mathematical community for at least two years before the institute can evaluate the result.
To date, only the Poincaré conjecture has been officially solved among the seven Millennium Problems.
Indication that the problem remains unsolved. Source: Clay Institute.OpenAI has stated it does not intend to claim the monetary reward:
“Our goal in publishing this result is to report significant progress made by our AI models. We do not intend to claim the prize for this result.”
Debate Surrounds Discovery's Priority
The OpenAI experiment coincided with the work of Professor Tristan Buckmaster from New York University and mathematician Levent Alpöge from Harvard University, who also collaborates with Anthropic.
On September 7, the researchers published results regarding the formation of singularities in finite time under smooth external forces for equations governing incompressible porous media, a two-dimensional Boussinesq system, and three-dimensional incompressible Euler equations. They utilized Claude and Codex in their preparation.
OpenAI acknowledged that it initially mistakenly believed its team had a solution directly for Navier-Stokes. After completing its proof, the company reached out to the researchers to coordinate their publications.
As reported by TechCrunch, Buckmaster claimed that information about the direction of their unpublished work reached OpenAI prior to the company concentrating significant computational resources on a similar approach. He also stated that OpenAI's head of mathematical research, Sebastian Bubek, suggested excluding Alpöge from authorship due to his work at Anthropic.
Buckmaster also acknowledged that OpenAI's internal system may theoretically have been influenced by data from his interactions with Codex.
We congratulate Levent Alpöge and Tristan Buckmaster on their remarkable mathematical work.
We (the researchers and the agents) did not see any of their work through any means until they released it publicly—in particular, no specific user data was accessed in order to solve… https://t.co/otJKRHnQgb
— OpenAI (@OpenAI) September 8, 2026
“We (the researchers and agents) did not see any of their work in any way until they published it publicly—in particular, no specific user data was accessible for solving this problem. Although it is unlikely, we cannot rule out that anonymized data obtained from their use of our products may have helped improve our models,” OpenAI acknowledged.
The company asserts that its proof is significantly distinct from the findings of Buckmaster and Alpöge.
ChatGPT Images 2.5
On the same day, OpenAI unveiled ChatGPT Images 2.5, a new version of its model for generating and editing images.
ChatGPT Images 2.5—faster, sharper, smarter, with better tools for creating whatever you can dream of.
- Faster image generation to keep your ideas flowing
- Improved fidelity for more natural, recognizable images
- Consistent details across multiple edits
- Comment-based… pic.twitter.com/SjHITKJUSV
— OpenAI (@OpenAI) September 8, 2026
The developers claim it retains people and other objects from reference images more accurately, performs local edits better, and maintains details more consistently during sequential edits. The generation delay has been reduced by up to 50% compared to Images 2.0.
ChatGPT now includes a Sketch feature, allowing users to draw a sketch directly in the interface and use it as a basis for the final image. The company has also added templates for common formats and the ability to share the original prompt with others alongside the generated image.
It’s worth noting that on August 25, OpenAI released the first measured results of its inference chip, Jalapeño. In tests, the accelerator demonstrated 1.5 to 1.9 times more computational work per watt and 1.7 to 3.6 times less latency than systems based on Nvidia's GB200 and GB300.