Chapter 6: OpenAI solves the Navier-Stokes problem

Okay. Navier-Stokes, one of the Clay Millennium Prize problems, a grand challenge in math. For those who watched our New Year's predictions episode at the end of 2025, one of my predictions for this year was that AI would solve one of these Clay Millennium Prize problems. And it appears in the past 24 hours — actually in the past 6 hours, as of time of this recording — it has happened. OpenAI has a team. There were several academic teams that also either achieved it or came close. And there's probably going to be a bit of a second day story about whether there was some foul play between OpenAI and some of the frontier teams, who scooped whom, blah blah blah.

Punchline: Navier-Stokes, which is this grand challenge in math that deals with, as I put it when last we were discussing this, whether in some sense whether there's a way to stir a glass of liquid or water in such a way that a singularity pops out. Basically, can you stir a cup of coffee in such a way that you get a black hole out of it? It looks like in the continuum limit the answer is yes. And the OpenAI team, using a version that's not GPT-6 Astra but an internal more advanced model — appears it, uh, and I think we'll know more in 24 or 48 hours — appears the answer is yes. There is a way to stir a cup of coffee, an idealized cup of coffee, to get a black hole out of it. And they did it.

I can just see the headlines now. [unclear] proclaims black hole in your cup of coffee.

Surprised there's a singularity in the singularity.

Wait, can you also — like, Navier-Stokes is incredibly important for aeronautics, for submarines, for hydro. It's like artificial — it's not just about black holes and coffee. Let's talk about the —

No, but — no, but this is the point. The point of the Navier-Stokes conjecture is it hinges on whether it's possible in finite time with continuous initial conditions to achieve a singularity in an idealized fluid, and of course in the real world we don't actually have idealized fluid. Will you actually be able to create black holes in cups of coffee using this? No. But it teaches us something.

Yeah. Don't worry about black holes in cups of coffee yet. But this is just the first of many grand challenges. Like, Peter, you and I wrote in Solve Everything — like, now is the moment. You — I, if you have to pick a moment, I maintain singularity is, you know, an interval in time. But if you want to pick any sort of moment when the loud noises start, now is as good a time as any.

OpenAI reportedly was able to do this with only 10,000 agents in 88 hours with 130 billion tokens and approximately $6.5 million. 10,000 agents, 88 hours, 130 billion tokens, approximately $6 and a half million dollars of inference time compute, by at least one estimate, was all it took to solve a grand challenge in, call it, mathematical physics. And this will not be the last one. The era of grand challenges getting bulk solved by AI is here. It's now and it's going to be very exciting.

Yeah. Uh yeah, we've talked about cooking math, char-broiling math and physics. Everything is cooked at this point. Everything is cooked. It's just a matter of time.

Also, a lot of people are going to be intimidated by that $6 million number, but keep in mind that we're predicting 100x inference time — this is all inference time compute, not training time. And we're predicting 100x price-performance improvement between here and the end of the year. And it could be more like a million-x next year. So, take that — you know, it could be six bucks within a year and a half to do that exact same thing.

You want to talk about what Navier-Stokes means in real life to real people?

Yeah, I mean, kind of — Alex said, this is an idealized kind of scenario because it's not the full algebra as it were. So it's the idealized one where you go down and you drop a dimension from reality. Um, that causes a degenerate algebra which has things like commuting, time translations and others. But it's a very hard problem. It's one of the hardest of all time. Uh, Terence Tao, one of the top mathematicians, literally focused almost all his time on this.

Um, what happened is that there were rumors about a month — a week ago — that this had been solved and people [were asking], is it Anthropic, is it OpenAI and others. On August 28th, um, Noam Brown at OpenAI said in a reply to someone — they said, have you solved a Millennium Prize problem? He's like, no, we've thrown loads of compute at it, like nothing's happened. In the post today they said on August 28th we started training a new model that got really good at math, and then on September 1st they pointed it at this and then it solved it.

But more than that, I've been talking to buddies at OpenAI. This model has solved a lot more problems. It's solving problems quicker than anyone can ever see. Um, I think I shared a chart — again this is like live — that shows that on open math it literally doubles the solving rate, but it's figuring out things that nobody has seen before.

Anthropic and an independent mathematician, a professor at New York, I think, um, had a solution to a smaller problem, which is the Euler blowup, which is kind of a slightly easier problem. It's still very, very hard. Um, and OpenAI thought that it could have been that, and so they reached out, and I talked to like Sebastien Bubeck, who kind of leads this. There's a whole hoo-ha about attribution this morning that's now getting cleared up.

But then when OpenAI kind of looked at it, they were like, well, that's that solution, this is the Navier-Stokes solution, and it's bitter-lessoned. You used to be able to get AI better by applying more compute to it. Now it seems like you can do that for any verifiable domain. I think that's the headline with this new type of model that only started training 9 days ago. So they figured out a new type of post-training, it seems. And again, people inside OpenAI are like, oh my God, what's happening now? These things are falling one after the other. It's like when you reach that capability threshold of going from a crap website to a good website from your coding agent. It's that, but for math and physics.

And you know, we've been hearing about it for the last few days. Like, I put out one of my physics results, a very beautiful one about chirality and the standard model. It's like very nice, but I was like, eh, I'm going to get that first before everything gets solved. And I think that it won't be a question of 2 years now. It'll be a question of much shorter than that.

The other thing that I think is very interesting is there was actually one more solution of a Euler blowup by, I think it was — was it Princeton? — using physics-inspired neural networks. That probably has more practical use than this one. But this one's an example of just now problems that are really complicated and have troubled us for ages and now tractable just by applying more compute to what seems to be a new breakthrough in model capability.

It really is — just maybe to develop that a bit more — it really is a bad day, I think, for Google DeepMind. So Google DeepMind had, purportedly, an entire team devoted to, as Emad said, physics-inspired neural networks, PINNs. They had been publishing incremental results towards Navier-Stokes. This was like the best, the worst kept secret from those like in the community of using AI and computational fluid dynamics type approaches to solve Navier-Stokes. Google DeepMind had an entire team devoted to this and they got trounced by a generalist model. It wasn't a PINN that solved Navier-Stokes in the end. It was just a generalist model that spent 88 hours reasoning from first principles without, as far as I can tell, any fine-tuning.

Yeah, such an important point, because I'm counting on Peter to solve this, you know, with you, Alex. But the problem that I think Google ran into is that everyone's inspired by Demis Hassabis, who solved protein folding using AI and then got a Nobel Prize and is now world famous and an awesome guy. So all the Google people are like, wow, I want to be the next Demis Hassabis. The problem is the prompt to solve this is like, give me thousands of GPUs and solve the problem. No one's going to give you a Nobel Prize for writing that prompt. It's entirely AI work.

And so — but I really feel like the post-training and, you know, the lining up of the problem is prize-worthy, because this is Nobel Prize. The other conversation we had this morning was the Nobel Prize is cooked, right? I mean — and Dave, you made the point very importantly that, you know, it's given out for work done 30 years ago. And now the work, this kind of Nobel laureate level prize work, is going to be happening every few weeks, every few days. And who gets it? The person who wrote the prompt or the model?

So I think this is actually really interesting in what happened this morning, because there was a whole hoo-ha, because the people who broke through on the Euler side — and they had a bunch of other kind of things extending out the work of Ortega and kind of others — wrote a letter saying OpenAI reached out to us and said, you know, give us priority and they would credit us, or they have this Navier-Stokes solution they'd put our name on and I could be the lead author if I drop my buddy from Anthropic. Oh, from that.

And so we were like — the mathematics community was like, oh my God, what's happened? In the end it turns out that, you know, there's subtlety in this, right? So Sebastien Bubeck, who leads it, is a great guy at OpenAI, has now posted his version of things, which just seems like a misunderstanding, because what OpenAI actually wanted to do is that they saw that there was something happening with Navier-Stokes related, and there was this solution that would have probably got to a Navier-Stokes solution by the Anthropic guy and this New York professor, and they said, look, we have solved it, but we will let you be lead author on this because you would have got there anyway, but we can't have an Anthropic person because that would be weird because it's our model.

So they were actually willing to give the credit to the original discoverers, but then it got a little bit political. And this is a question again of who did it, because they're like, you were the humans that did it, whereas our AI, we just pointed at condition C and D, which are the blowup solutions of Navier-Stokes that solved it. So we're not going to claim the Millennium Prize because it's not us, but you were the humans that took it the furthest before it got solved.

I want to take home for a second to what Dave said earlier, just for people listening, like, what the heck is Navier-Stokes and what is all this math stuff? Navier-Stokes is about fluid dynamics, and this is where we improve uh aircraft design, submarine design, even in an artificial heart, how blood flows, how do we make it, you know, able to be more efficient and not clot. Uh, there's going to have massive implications to everyday life. Uh, and I just want to make that point here. Um, who was — who else was jumping in, Alex?

Yeah. So, two points. One just quickly about killer apps — uh, singularity-esque sci-fi apps of Navier-Stokes. As mentioned previously, in principle if you can get finite time singularities, as appears to be the case with idealized fluids, one of the killer apps that Terry Tao had flagged previously was, in principle you could, if you created the right initial conditions in a fluid, you could create a self-replicating machine that creates smaller and smaller copies of itself.

So, we talk from time to time on this pod, where's nanotech? Nanotech, at least Drexler and nanotech, never showed up. Well, imagine a fluid-based — not diamondoid — a fluid-based nanotech, self-replicating machines where if you could craft the right initial and/or boundary conditions of a fluid, creating a self-replicating machine made entirely out of a fluid. That's potentially one killer app of solving the singularity program in Navier-Stokes.

Secondly, just on the hubbub surrounding priority, who solved it, who didn't solve it, specifically with reference to the Euler problem: when OpenAI in the past few hours announced it, I found especially concerning at the bottom of OpenAI's announcement where, presumably they were trying to give appropriate credit to other teams, including teams pursuing the Euler approach, they did add a disclaimer that I thought was a bit of a head-scratcher, saying they can't rule out — the OpenAI team can't rule out the possibility that maybe some of the other teams' work might have been incorporated into the training of the OpenAI model that was used to solve Navier-Stokes.

So I think, to the extent that that's the case, that that should be viewed like a — OpenAI, if you're listening, please, like, it's a deterrent to everyone else using your models if that is indeed the case, if you're training on everyone else's research. Yeah, that's a problem. Please fix it. But just in general, I think we want to be able to live in a world where researchers benefit from frontier capabilities without worrying that the frontier platform is going to compete with them.

I think it's worth saying here just quickly, this is clearly an RSI model. You can't train something in that type of period unless you've got a recursive self-improvement loop going. And so they've definitely got one there. And they were saying like, oh yeah, we've anonymized it. Well, anonymizing someone else's research and then potentially scooping them is insufficient. So, again, not sure what the ground truth is.

Yeah.

But I will say this, I've reviewed kind of the two approaches and I've done quite a bit of work on that. They are actually very different, but it can still learn, and again, this thing can leak. Sorry, Salim.

All right. Salim, close us out here, please.

Just — if I — I want to — I think this is such a powerful commentary and exemplar of the broader thesis, right? That for 500 years we've scaled science with how many brilliant scientists we could train up and point at various problems. Now we've just demonstrated absolutely, for real and unequivocally, that we can spin up 10,000 researchers on a Tuesday afternoon. Uh, and now we go from staff on demand to intelligence on demand. Really, really — intelligence — breakthroughs on demand. And we talked about this on the last episode and the episode before that. It purely comes down to what is the imagination of the problem you want to go after now.

Yeah. There is no — this becomes really — how fun is the world right now? No ceiling. Yeah. It's an amazing time.

I just — let me take — let me draft on that to say to everybody here: whatever you thought you could do, think bigger and then go even bigger than that. You're given incredible superpowers. Don't limit yourself by what your parents did or your school teachers or your colleagues do. You know, you have no limits on your abilities.

Emad, you want to jump on that, please?

I say, Alex, which one's next Millennium?

Probably — probably Yang-Mills.

I'd agree. I think we're on the same page there.

Yang-Mills is the next to fall. You want a — I guess a September prediction uh for the next six months? I think Yang-Mills is the next Millennium Prize to fall. But interestingly, it's just because that's the next chosen target. You know, you choose a different target, it'll be the next to fall.

They did have a different target. They were targeting Riemann and P versus NP before, and they switched to Navier-Stokes. But I'd agree with Alex. Yang-Mills will be the next one.

For everybody who's not Alex, it's a conjecture in particle theory concerning the existence of a mass gap under certain conditions.

There's something about mathematical physics that is both sexy from a let's-spend-inference-time-compute-on-it and also tractable. So I'd bet Yang-Mills mass gap.

Yeah, you might think of it like the resolution of the universe, perhaps. Okay.

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