OpenAI Claims Navier–Stokes Flows Can Form Finite-Time Singularities

Károly Zsolnai-Fehér presents OpenAI’s result as a likely solution to the Navier–Stokes existence and smoothness problem, one of the Millennium Prize Problems. OpenAI says an internal system produced a proof that fluid dynamics governed by the Navier–Stokes equations can develop a singularity in finite time, and has released both a written proof and a formalization in Lean.^†

The mathematical question is not whether the equations can describe familiar fluid motion in ordinary cases. It is whether a flow that begins smooth is guaranteed to remain mathematically well-behaved forever. In Zsolnai-Fehér’s account of the claimed proof, the answer is no: the equations can break down.

The construction, as he describes it, begins with fluid at rest and applies carefully designed external forces. The mechanism is an inward-spiraling vortex that stretches as it evolves. Its velocity grows without bound in a finite time even while total energy remains finite. That is the kind of finite-time singularity that defeats a global guarantee of smooth behavior.

This is a mathematical counterexample, not a prediction of a literal infinite velocity in a river or laboratory. Zsolnai-Fehér says such behavior is not known to occur in nature; if dynamics reached that extreme, molecular-scale physics would take over, and Navier–Stokes would no longer be the appropriate model.

For Zsolnai-Fehér, the striking feature of Navier–Stokes is that the visual complexity of fluids can be expressed through a small set of effects:

The terms shown on screen represent advection, pressure, and diffusion, alongside an external force. A separate incompressibility condition says that volume remains constant: fluid is neither created nor lost from nothing.

Advection transports material with the flow—an object dropped into a river follows the water—but fluid also advects itself, through the nonlinear directional-derivative term. Pressure pushes outward from crowded regions, which he compares to people pressing against one another on a packed bus. Diffusion averages differences over time, as a concentrated drop of ink spreads through water.

On a grid, he says, advection can be treated as moving a quantity to an appropriate neighboring location and diffusion as averaging. That makes approximate simulations practical, whether for visual effects, wind-tunnel-style work, or fluid-control research. But being able to compute such approximations does not resolve the deeper question at stake: whether a smooth solution must remain smooth indefinitely.

Zsolnai-Fehér says the claimed solution was preceded by meaningful work from Levent Alpöge and Tristan Buckmaster on a related problem. In a September 7 statement shown on screen, Terence Tao said the pair had found a variant of their method that extended to two other equations and had a “high likelihood” of extending to Navier–Stokes. That is a view about the method’s prospective reach, not evidence in the source that Alpöge and Buckmaster themselves had completed a Navier–Stokes proof.

After hearing rumors of that progress, Zsolnai-Fehér says, OpenAI launched a system more powerful than GPT-6 Astra and it arrived at a solution. A chart shown in the source labels the internal model’s pass rate as roughly 2.8 times Astra’s. He does not know what formal attribution Alpöge and Buckmaster will receive, but explicitly credits them for the earlier work.

A separate issue concerns the researchers’ use of proprietary language models. Zsolnai-Fehér says Alpöge and Buckmaster had used systems including ChatGPT and Claude extensively, and sought confirmation that this usage had not contributed to OpenAI’s systems. The displayed OpenAI response does not offer an absolute exclusion; it says de-identified data derived from their use of OpenAI products might, though unlikely, have helped improve its models. It also says the proofs differ significantly and that even the precise Euler results differ, describing one as forced and the other as unforced.

Zsolnai-Fehér treats that statement as a warning about proprietary AI systems: material entered into a chat interface may contribute to model improvement. He contrasts that with running free, open-weight systems locally, where prompts do not leave the user’s machine.

The reported speed is nearly as consequential, in Zsolnai-Fehér’s telling, as the claimed mathematical result. The agents reached their resolution about 88 hours—roughly three and a half days—after the first agents were launched.

His explanation for AI’s rapid progress in mathematics is verification. Prose quality generally requires a person to read and judge an output; he suggests a human might assess around a hundred examples per hour. Mathematics, by contrast, can often be checked automatically. In his framing, that permits something like a hundred million lessons per hour rather than a hundred.

That difference, he argues, is why mathematical capability can advance especially quickly, and why much larger gains are likely. It also underlies his concern that increasingly capable systems will require substantially more coordination on safety and alignment.

Zsolnai-Fehér says Demis Hassabis told him that AI could cure all disease in 10 years. Extending the verification argument himself, he suggests that medical progress might accelerate if disease could be made into a verifiable problem in something like the way mathematics is. He presents that as his own extrapolation rather than an established path to the outcome.