The Week Navier-Stokes Cracked, and What It Says About How Science Works Now

2026-09-08 - Adam Murphy

The Week Navier-Stokes Cracked, and What It Says About How Science Works Now

Two teams announced finite-time singularities around Navier-Stokes this week. It is a huge week. It is not solved. The way they got there is the story.

For a hundred and some years the Navier-Stokes equations have described every fluid you have ever touched. Water in a pipe, air over a wing, blood in an artery. Engineers use them daily. And yet nobody could prove whether a smooth fluid, left to its own physics in three dimensions, always stays smooth or can tear itself into a singularity in finite time. The Clay Mathematics Institute put a million dollars on the question in 2000. This week, two teams announced they had answered a large piece of it. The way they got there, and what it could mean, is what I want to look at today.

What was actually shown

Headlines are going to say one thing. I want to look at the bigger story, and try to see it from a factual standpoint as much as we can, because this is a historic time.

On September 7, Tristan Buckmaster of NYU and Levent Alpöge of Anthropic posted three papers, building on earlier work by Diego Córdoba and Luis Martínez-Zoroa. They constructed finite-time singularities for the incompressible porous medium equation, the two-dimensional Boussinesq equations, and the three-dimensional Euler equations, each driven by a smooth external force. Euler is Navier-Stokes with the viscosity removed. All three results are formalized in Lean, the proof-checking language, so the logic has been verified by machine.

On September 8, OpenAI announced that an unreleased internal system had produced a roughly 100-page proof, also formalized in Lean, that the full three-dimensional Navier-Stokes equations with a smooth force can blow up in finite time. The described mechanism is a vortex that spirals inward and stretches, the core shrinking as it speeds up, so that velocity becomes unbounded while total energy stays finite. Several terms in the equations grow enormous and cancel each other almost exactly. That claim addresses statements C and D of the Clay problem, the "singularity exists" direction. OpenAI also says the same system settled the unforced Euler case. The company now says it is sharing a writeup and a Lean formalization, and that it will not seek the prize. Independent reading of that argument is still in its first hours.

What remains open is the unforced Navier-Stokes problem, the version most people mean when they say "Navier-Stokes." Terence Tao has noted that viscous energy is a real obstacle to removing the forcing. The prize requires publication and a two-year waiting period. Nobody has claimed it. Clay still lists the problem as unsolved.

So: two teams, one public and machine-verified on Euler, one newly posted on forced Navier-Stokes, the hardest version still standing. That is a huge week. It is not "solved."

Two ways to get there

Here is the part I find most interesting, and it is the reason I am writing this instead of just sharing the link.

Buckmaster and Alpöge spent close to a year making slow progress. Then from mid-August they accelerated, working with Claude and Codex. By their own account the models identified key elements of prior proofs, reproduced arguments, and handled the bookkeeping of inductive orders and constants. The first solution arrived on August 15. The first writeup, in words Tao relayed on his blog, was "the worst writeup we had ever seen in the history of mathematics." The two humans then spent weeks simplifying it by hand until it was something a mathematician could actually read. Five weeks from first solution to posting.

OpenAI's approach, by its own description, was to launch about 10,000 concurrent agents on September 1, split into communicating groups, seeded with the preliminary Euler results. About 88 hours, 2.7 million messages, and 130 billion output tokens later, they had a Navier-Stokes proof. A separate model checked the Lean formalization in another 17 hours. Fortune reports the compute bill at roughly two million dollars.

I spent 20+ years writing software and running IT organizations, and I have watched similar patterns play out with code over the last three years. AI can now write the program, and increasingly it can write the whole program. But sometimes the build drifts. An architecture or design decision gets out of step in a way the AI has decided will work, maybe in the code structure, maybe in the database design, maybe in something that is just not in any training set. Or the bug lives in the gap between what the spec says and what the customer meant. A person who understands the problem space finds the path through, while a swarm of agents will circle for days, because the swarm is not actually looking at the problem the way we are from the outside.

That is what the Buckmaster and Alpöge result looks like to me. Humans chose the problem, chose the strategy of adding high-frequency corrections to a background solution with an exploitable instability, and did the messy final work of turning machine output into mathematics. OpenAI's result looks like the other thing: brute force at a scale no university or researcher could buy, aimed at the same strategy. Both worked, and there are lessons in both. Only one of them we can read as a paper a mathematician was willing to put their name on.

Today, the brute force is expensive and the human-steered path is more legible. I do not expect that balance to hold. But I also do not think the human role disappears. It moves upstream, into choosing what is worth attacking, and downstream, into deciding what the answer means. In between, at some point, the software or the math just runs.

The messy spots are not only mathematical

The story got uglier over the weekend, and I will keep this brief and neutral because I only know what has been reported.

Buckmaster says that in a September 6 meeting OpenAI's Sébastien Bubeck told him about the internal Navier-Stokes proof, proposed arrangements for how the two results would be announced and credited, and that when Buckmaster declined he was asked "Why would you ruin your career?" Buckmaster also asked whether his team's Codex sessions had been used to train or steer the model and says he did not get a clear answer. He is careful to say he has not seen OpenAI's proof and is making no accusation.

Bubeck has called the allegations "false and inflammatory" and stated, "We did not use their prompts or proofs to prompt our models or direct our agents." OpenAI's own announcement credits Buckmaster and Alpöge with priority on the forced Euler result and says that while unlikely, it "cannot rule out" that de-identified usage data helped improve its models.

I do not know who is right. I do know that credit, provenance, and trust are exactly the questions the world we are moving into is going to keep asking.

Understanding is the product

Tao wrote, "the actual solving of these problems is only a proxy goal for the primary goal of developing mathematical understanding." In Fortune he went further, warning that AI companies "strip-mining" famous open problems for marketing could damage the ecosystem that produces new mathematical techniques in the first place. Someone put it well in a post on X that I did not bookmark and have since lost: a proof is not only there to show a result is correct, it is there to show that we understood each step along the way. That is not the way the journey went in this case, at least not yet.

When a proof can be generated in a week and the writeup is unreadable, the constraint shifts from the proof to the reading, the understanding, the comprehension. It is the verification, the simplification, the checking of whether the thing that was proved is the thing that was claimed. Lean helps enormously with logic. It does not tell you whether the theorem matters, whether the forcing term is physically meaningful, or whether a 100-page argument contains an insight or just a certificate that it is true.

That is the problem we built TheoryOfEverything.ai to work on. It is not uncommon for me to learn physics and math by asking questions of the research submitted by independent researchers. Today, unlike any other time in history, we have access to serious review and we can ask questions about it. If you go to any paper or framework on the site and read the summary, the AI chat bubble at the bottom of the page can answer any question you ask about the research. That lets us increase our understanding, because I will be honest, a lot of the time things are way over my head. The expertise and knowledge being submitted is amazing. And what we are now looking at is a volume of machine-assisted results that is about to swamp the limited reviewers who exist. Multi-agent neutral review is not a replacement for a Tao reading your paper. It is a way to make sure the messy spots get found before a human has to spend their afternoon on them.

Science is changing. This week it changed faster than most weeks. I expect this to be the trend. The right response is not to be dazzled by the token count, and it is not to pretend nothing happened. It is to read carefully, credit honestly, and keep the humans where they are still faster.

To all of you independent researchers out there: your work matters. The world needs understanding. Keep writing. Keep learning. Keep using AI, and keep thinking, because the ideas that bridge the gap are going to come from you.

Sources