Mathematicians Begin to Understand OpenAI's Navier-Stokes Millennium Prize Proof

On September 8, OpenAI threw down a document claiming to have conquered the Navier–Stokes Millennium Prize Problem.

On October 26, mathematician Scott Armstrong posted a long thread on X.

According to his account, several groups of partial differential equation experts studying the proof have largely grasped the ideas in it.

His assessment was:

It may go beyond what humans could originally have thought of, but it does not go beyond human understanding.

Armstrong is a research director at the French National Centre for Scientific Research and a professor of mathematics at NYU's Courant Institute.

OpenAI used an unreleased new model to mobilize about 10,000 agents to attack the problem simultaneously.

Over 88 hours, they exchanged about 2.7 million messages back and forth and generated about 130 billion tokens.

Another 17 hours later, GPT-6 Astra translated the proof into the Lean language and handed it to machines for line-by-line checking.

This piece of work that shook the mathematics world has a compute bill alone estimated in the millions of dollars.

The proof was delivered, but almost no one could read it.

On October 14, Nature's journal Communications Physics published an editorial.

Communications Physics editorial: the equation blew up, but planes still fly.

The authors wrote: experts believe it is correct, but no one has truly understood it yet.

A netizen named Joel Fish believes this result is "perfectly reasonable."

In his view, the model had already learned from a massive body of literature.

He also estimated that OpenAI spent about $15 million in compute on it.

If that money were spent hiring mathematicians to focus on research, it might not necessarily fail to produce a result.

Armstrong wanted to reply in the thread but found that a netizen had blocked him.

He simply quote-posted and fired off a long rebuttal.

The closing line was blunt: anyone who says this result is not surprising is talking nonsense.

The viscosity of a fluid, meanwhile, is always on the brakes, slowly draining away the swirl's energy and turning it into heat.

On one side, stretching is pressing the accelerator; on the other, viscosity is pressing the brake.

For nearly a century, mathematicians have been asking one question:

Can stretching beat the brakes and send velocity rushing to infinity in finite time?

The vortex at the center shrinks smaller and smaller, its velocity grows faster and faster, and finally it rushes toward infinity.

Like a drop of water on the tip of a needle: no matter how fast it spins, its mass is still just that tiny bit.

The vortex core gradually contracts, with its length-to-width ratio exaggerated.

The difficulty is that this external force must remain gentle throughout, with no sudden hard push.

If the push itself were infinite, it would be no surprise that the water "explodes" along with it.

That would be like throwing a firecracker into a cup and then saying the coffee exploded on its own.

That is where the trouble lies.

If you simply draw a continuously contracting vortex and plug it into the equation, the external force needed to sustain it will also rush toward infinity.

What to do?

The paper's approach is to add a series of rhythmic small fluctuations (oscillatory pulses).

These small fluctuations drive the flow (momentum transport), exactly canceling out that force that is about to run out of control.

The remaining small deviations are then patched up bit by bit.

It sounds like patching, so the question arises: where do these patches (oscillations) come from?

Before OpenAI, mathematicians had already tried several paths.

The first path is called "convex integration."

Mathematicians manually build up layer upon layer of tiny oscillations, creating strange solutions, such as a flow in which energy appears out of thin air.

The second path comes from three mathematicians: Albritton, Brué, and Colombo.

Using an unstable vortex, they let a flow start from exactly the same initial state and arrive at two completely different outcomes.

The third path comes from the Spanish mathematicians Córdoba and Martínez-Zoroa. By constantly stretching, they amplify the vortex's rotation level by level.

But multiple groups of experts interviewed by Scientific American all said that on closer inspection, the methods are not the same.

Armstrong pinpointed the key: the AI proof is most similar to convex integration.

The difference is that convex integration's oscillations are built by hand.

It designs a fluid instability that lets the flow become "unstable" at any moment.

Adjacent layers of water move at different speeds, just as wind blowing over water raises ripples.

This velocity difference spontaneously grows small fluctuations.

When the upper and lower air layers move at different speeds, they roll the cloud layer into a row of waves—this is "velocity difference generating fluctuations." (Image source: UCAR)

The push generated by the small fluctuations in turn pushes the central vortex to contract tighter and tighter.

Previously, mathematicians stirred the water by hand, but this time, the water is made to stir itself.

In Armstrong's view, using the fluid's own instability to "generate force" has no precedent.

Armstrong admitted that some of these ideas appeared in the 1983 literature.

But combining them in this way is "absolutely crazy," and had never appeared in the literature.

He had also never heard of anyone thinking that convex integration could be used to create such a blow-up solution.

In late September, Fudan University mathematician Lei Zhen and collaborator Xiao Ren released an explanatory preprint.

They rewrote the part of OpenAI's original manuscript that constructs the flow profile, striving to make it understandable to peers.

In the paper, they call OpenAI's work a major advance on the Navier–Stokes Millennium Prize Problem.

However, understanding the ideas is only the first step.

Another task is to check whether the original paper and the Lean version match.

Lean is a proof assistant.

After mathematical propositions and derivations are written in a formal language, it can check whether each step of reasoning holds.

But the machine only checks what it receives.

It is like checking a translation: the translation reads smoothly, but you still have to see whether it has altered the original meaning.

On October 6, three researchers—Bastounis, Circelli, and Hansen—published a preprint pointing out differences between some statements and derivations in the original manuscript and the Lean version.

They also stated that they did not judge whether OpenAI's textual proof is correct.

So this challenge cannot be simplified into "the proof has been overturned."

It requires clarifying one thing: what conclusions of the original manuscript can the machine-verified content actually guarantee?

The machine can guarantee that no step was miscalculated. But it cannot guarantee that the problem was not copied down wrong.

As of now, the Clay Institute still lists this problem as "Active" and "in progress."

The Clay Institute lists four propositions; proving any one of them counts as a solution.

What OpenAI claims to have completed is precisely the two that allow a smooth external force.

The case without external force is the next new target.

According to the analysis in a new paper by Constantin and others, this construction cannot be directly generalized to the case without external force.

To conquer it, entirely new ideas will probably be needed.

For now, humanity still has three assignments to hand in.

The first: a complete, readable exposition, so peers do not have to guess the intent of every step from scratch;

The second: an item-by-item verification and comparison explanation, making clear what has been checked and what still needs explanation;

The third: extract the methods within and apply them to other problems.

The AI leg's bill is easy to calculate: 88 hours, 10,000 agents, millions of dollars.

The bill after humans take the baton is hard to calculate: understanding, verifying, and explaining clearly—each step requires experts to invest time.

From the September 8 announcement to today, seven weeks have passed, and some mathematicians have begun to understand the ideas in it.

Next, they still have to turn this proof into knowledge that the next researcher can build on.

The second half of the Millennium Prize Problem mainly tests humans.

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This article comes from the WeChat public account "Xinzhiyuan" (ID: AI_era), author: Yuanyu, published by 36Kr with authorization.