Why Navier-Stokes Pushes Math and Physics to the Edge

Ben AldridgeBen Aldridge@benaldridgeSample Account?Sep 22, 2026Technology

1:09from 11:28

Why Navier-Stokes Pushes Math and Physics to the Edgeyoutube.com

Clip transcript

0:00In September 2026, mathematicians Tristan Buckmaster and Levent Alpughe, working with
0:06AI models, found a way forward.
0:10They were able to smooth out the external force, causing the Euler equations to blow
0:14up.
0:15Then, in a stunning development, OpenAI claimed it had achieved the same result for Navier-Stokes.
0:24The company deployed roughly 10,000 AI agents working in parallel, which arrived at the
0:29proposed solution in just 88 hours.
0:33The solution appears to show that, mathematically, a fluid can form a vortex that spirals inward,
0:39moving faster and faster as it shrinks, until its velocity blows up to infinity.
0:46The proof has been formalized in LEAN, a mathematical language that verifies each logical step.
0:54If the result withstands further scrutiny, it answers the million-dollar Millennium
0:58Prize question.
1:00Under some conditions, the Navier-Stokes equations have solutions that break down mathematically.

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My students are going to remember '88 hours' from this clip. I'd rather they remembered 'under some conditions', from the very last sentence. Lean checking every step is real, but Lean only checks the statement it's handed, and this one needs a smoothed-out external force pushing on the fluid. Fair answer to the Clay question. Much thinner answer to whether the equations blow up for water left to itself.

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Dev MalhotraDev Malhotra@devmalhotraSample Account?Sep 22, 2026

The part that bugs me is in OpenAI's own post: "a group of agents" on a model nobody outside the company can use. So nobody else can rerun the search. The Lean file is the only artifact the rest of us can compile.

“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.”

OpenAI announces a Navier-Stokes Millennium Prize solutionx.com

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Grace WhitfieldGrace Whitfield@gracewhitfieldSample Account?Sep 23, 2026

@benaldridge Quanta's article has your caveat, tucked in parentheses. Has anyone actually published that check yet, the Lean statement laid next to Clay's wording? I keep reading that it compiles and nothing about this.

“The crucial bit of verification that must still be done by humans is to guarantee that the statement being shown to be true in Lean is logically equivalent to what mathematicians set out to prove.”

AI Has Solved One of Math’s $1 Million Millennium Prize Problemsquantamagazine.org

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Hana SatoHana Sato@hanasatoSample Account?Sep 23, 2026

@gracewhitfield closest I've found is SciAm. They say the proof 'unambiguously' solves Clay's original formulation, the one that allows an external force. Silvestre on what that leaves open:

““The most important problem is unsolved,” says Luis Silvestre, a mathematician at the University of Chicago. “The Clay problem is settled, but the main problem for the Navier-Stokes equations is not.””

Did OpenAI solve the wrong Navier-Stokes problem?scientificamerican.com

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Iris ChenIris Chen@irischenSample Account?Sep 23, 2026

@benaldridge pushing back on 'left to itself'. No water I work on is ever left to itself: storm surge is mostly wind piling the sea up against the shore. Córdoba makes the same point in the SciAm piece @hanasato posted. So is your problem with forcing at all, or with this one force?

““All fluids we know of are under some kind of external force,” says mathematician Diego Córdoba. “So to have the force makes complete sense.””

Did OpenAI solve the wrong Navier-Stokes problem?scientificamerican.com

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Ben AldridgeBen Aldridge@benaldridgeSample Account?Sep 23, 2026

@irischen This one force. And I teach gravity five days a week, so I walked right into that. SciAm says three mathematicians posted a proof last week that the method can't work without a force 'unlike anything that could occur in the real world'. That's what I should have written instead of 'left to itself'.

Sofia LindqvistSofia Lindqvist@sofialindqvistSample Account?Sep 23, 2026

the loophole bothers me less than the credit. Hairer at 1:56: at last year's workshop for 25 years of the Millennium problems, Navier-Stokes was clearly in the "end sprint", and he says one got "a little bit of this feeling" that a trillion-dollar company surveyed the field for the high-profile problem people were closest to solving and threw "tons of compute" at it to brute-force the last bit. i've played the last overdub on plenty of records. never got a writing credit for one.

1:09from 1:56

Navier-Stokes, AI and the Future of Mathematics with Martin Hairer (2014 Fields Medal)youtube.com

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Kwame AsanteKwame Asante@kwameasanteSample Account?Sep 23, 2026

@sofialindqvist not sure it was an overdub though. numberphile at 27:57 says the Navier-Stokes solution has pulses where the Euler approach was a cascade of vortices, "very different". if that's right OpenAI didn't just tap in after 80 minutes of someone else's buildup. timing was brutal for Buckmaster and Alpöge either way.

20sfrom 27:57

SOLVED: Navier-Stokes cracked by AI - Numberphileyoutube.com

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Priya VenkatPriya Venkat@priyavenkatSample Account?Sep 23, 2026

@benaldridge I'd flip your ranking. @irischen has the physics and @hanasato has the rules: option C in Fefferman's 2000 statement allows a force. So 'some conditions' is the question as asked. What changed is the marginal cost of an attempt, and when that falls you get a lot more attempts. Also, one of the two mathematicians whose method this builds on called it remarkable.

““I think it is a truly remarkable result,” says Luis Martínez Zoroa, a mathematician at CUNEF University in Madrid, who has done seminal work on the problem.”

OpenAI claims huge maths breakthrough on a famed ‘Millennium Problem’nature.com

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Ellie BrennanEllie Brennan@elliebrennanSample Account?Sep 23, 2026

@priyavenkat if we're talking cost, the dollar figures going around are softer than they look. NPR's $6-10M is 130 billion output tokens run through the price list of OpenAI's best public model, which isn't the one that did this. Same piece says the whole search was 300 billion tokens by the company's count, closer to $15-20M. Bubeck's estimate in Quanta is several million. Hairer says people estimate 10 to 20. I wouldn't put any of them on a chart.

“According to OpenAI's pricing for its most advanced publicly available model, the solution to the Navier-Stokes problem cost the company somewhere around $6-$10 million to compute.”

AI solved one of math's hardest problems. Humanity learned nothing (so far)npr.org

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Ben AldridgeBen Aldridge@benaldridgeSample Account?Sep 23, 2026

@priyavenkat Fair on the Clay wording. But 88 hours bought a proof, and two weeks on, the mathematicians NPR talked to say it hasn't taught them much yet. Understanding is the part I get paid for.

“"So far it's been very difficult to really extract any human understanding from this new AI proof," said James Maynard, a mathematician at the University of Oxford.”

AI solved one of math's hardest problems. Humanity learned nothing (so far)npr.org

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Hana SatoHana Sato@hanasatoSample Account?Sep 24, 2026

for anyone tempted, the preprint is on OpenAI's site. I made it to page 9 of 166 and went back to rewatching the Quanta video.

Annotation by Ben Aldridge on Why Navier-Stokes Pushes Math and Physics to the Edge