On September 8, 2026, OpenAI announced that one of its internal AI systems — roughly 10,000 agents working together — produced a proof that the Navier-Stokes equations, which describe the flow of every fluid from water in a pipe to air over a wing, can "blow up": reach infinite speed in finite time from smooth, reasonable starting conditions. The full argument was formalized and checked line-by-line in the Lean proof assistant.
This video builds the flow the proof is about from scratch: a spinning column of fluid that gets stretched thinner and thinner, spinning faster and faster — like a skater pulling in their arms — until (in the mathematical limit) its speed becomes infinite while its energy stays finite.
Why it matters: the 3D Navier-Stokes existence-and-smoothness problem is one of the seven Clay Mathematics Institute Millennium Prize Problems, open since 1934.
The visualization in the video is based on the original repo where OpenAI published their solution: https://github.com/openai/NavierStokesAndEuler. Based on a custom Three.js/WebGL renderer, with streamlines computed in Python from the exact flow solutions.
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Video Summary
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The video discusses the Navier-Stokes equations, which describe fluid dynamics. While used widely in engineering, it remained unproven whether they always provide sensible answers in 3D. A recent claim suggests a proof that fluid speed can become infinite in finite time, potentially solving a Millennium Prize problem and revealing a fundamental breakdown in these equations.
Air over a wing, water in a pipe, blood in an artery.
Every fluid can be described by the same equations.
Navier-Stokes, written down in the 1840s.
They work for most practical cases, engineers use them daily.
But works most of the time is not the same as guaranteed to work in all cases.
Nobody has proven that the equations always give a sensible answer.
A failure of the equations would look like this.
In a finite time, the speed somewhere becomes infinite and the equations break down.
Inside the equations, two effects fight each other.
One term has the fluid stretch and fold itself, which makes the motion sharper and faster.
The other is friction, which smooths it back out.
In two dimensions, this was settled long ago.
The stretching can never outpace the smoothing, so infinite speed cannot occur.
But the cases that matter are three-dimensional.
The problem for 3D constitutes a millennium problem,
for which the Clay Mathematics Institute has pledged to pay 1 million US dollars for the first correct solution.
On September 8th, 2026, OpenAI published a claimed proof that the stretching can win.
They construct a flow whose speed does become infinite in finite time,
with every step checked by a computer in the proof language Lean.
This is the flow that may earn OpenAI 1 million dollar.
A vortex spinning around a vertical line drawn from the profile in their proof.
Each strand is the path a tiny drop of fluid would flow.
The moving dots are those drops.
Start with the outer strands.
They spiral steadily inward, carrying fluid towards the center line.
The inward flow is what feeds everything else.
Fluid can't pile up there and it can't be squeezed smaller,
so it escapes the only way left, straight up and straight down.
That escape stretches the spinning column taller and thinner.
Exactly the stretching two dimensions could not do.
That's how a flow reaches infinite speed without infinite energy.
If this holds up, it answers a question open since 1934 and one of the seven Millennium Prize problems.
The equations behind weather forecasts and aircraft design can break down on their own.
