The chapter walks through the mechanics of OpenAI's claimed proof that the forced Navier-Stokes equations can develop a finite-time singularity, built around an axisymmetric vortex whose radius shrinks faster than its height grows, causing spin velocity to run away to infinity at a specific time (t=1). Using a countdown time variable tau and scaling exponents, the hosts show the velocity blows up while total kinetic energy actually goes to zero, meaning the singularity comes from the fluid's own geometry rather than from energy being pumped in. They also note key caveats: the result covers only the forced version of the problem, has been checked by the Lean proof assistant but not yet confirmed by human mathematicians, and Princeton mathematician Stan Palasek found an obstruction that blocks the same construction from working in the unforced case.
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The Clay Mathematics Institute's official problem statement accepts a solution to any of four variants (forced or unforced, whole space or periodic), which is why solving the forced case still counts despite public pushback.
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The forcing mechanism pushes fluid outward at one localized spot and inward at another simultaneously, imparting spin without widening the vortex, rather than acting on the whole ring.
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OpenAI's paper lacks the detailed multi-agent transcripts that accompanied an earlier Anthropic result on a Riemann zeta related problem, making it harder to reconstruct how the proof was actually found.
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Lean verification checks the logical steps of the proof but does not itself guarantee the result is true, and Clay is described as waiting roughly two years for human mathematicians to confirm a 165-page proof before accepting it.
3,115 words · auto-generated from the episode video
2:31:13theorem proved by OpenAI constructs a smooth compactly supported external force that drives a fluid starting from rest into a singularity. >> That's what they're doing. They're they're solving C and D. >> Mhm. Right. >> Right. They're saying that I have found a way >> to force a singularity into we we found a we found a situation where the convective term beats the viscosity. >> Exactly. Yeah. Um and because the force is smooth and bounded, the singularity is not because the force is just infinite, right? It's literally the liquid the the fluid itself, the velocity field itself doing this. The force is is is trying to maintain some
2:31:56kind of structural tautness of that liquid. And the singularity happens at t= 1. It doesn't happen at t equals infinity. This is a finite time blow up. Okay. So let's get into some of the stuff that I understood from their paper. This is unlike the anthropic paper about Reman Zeta because in the anthropic version they gave us a full account of all of the agents and like even a transcript of the inner thoughts of these agents and how they coordinated with all these sub agents. So, I had this really nice story for our podcast where I was talking about how some of the agents like, you know, the something crashed and like they had saved a little lema of their proof on the hard disk or
2:32:38the best part. >> Yeah, it's like it's it was crazy that we got this like inner version of what the agents were doing. Here, OpenAI is not doing that. Um, but in any case, yeah, in any case, um, we can still kind of try to understand how they constructed the singularity. Okay, the core of the singularity is in a axiymmetric vortex that's centered at the origin. And this is figure one that they show. We start out with a vortex that's yay big and and yay tall. And the point is that as the vortex shrinks, the vortex is going to stretch, >> right? The whole thing is shrinking, but
2:33:19the radius is shrinking faster than the height. M that's the whole idea because be this is so funny because before we talked about the hurricane going to the tornado >> and the idea was the height was increasing >> so fast because of conservation of angular momentum and all they're just saying in this mathematical construction where you don't need the conservation of angular moment >> you still do you still do >> but they're hacking their way around it >> okay okay they're hacking their way around it >> okay okay >> they're they're saying that as the thing shrinks >> the radius is shrinking faster than my height, which you can kind of see in this example, right? >> The the the the blue part, the radius is
2:34:01kind of shrinking faster than the height, but as the radius shrinks, it's going to spin faster and faster and faster. That's how they're >> and I'm going to get an infinite amount of velocity. >> That's okay. I get it now. Yep. And um there was a really nice gift that was um prepared by Jay Padigar from the Wolframe community staff picks September 14, 2026. you should really check out their website and it shows sort of what's happening. Okay, in this case the camera you have to imagine is zooming in because the whole time this entire thing is shrinking >> right here it's kind of showing that like it's kind of like the same size the whole time but that's because the camera is zooming in with the vortex. >> Yes. >> Right. But but here what you're seeing let's let's look at this animation in a
2:34:42great bit of detail. >> Okay. >> The radius is going down and down. You can see >> Mhm. >> the the axial length is kind of going up and the aspect ratio >> Yeah. >> is going up. >> Yeah. >> That's because the radius is shrinking as the height increases, but it's not like, you know, it's it's not one of those powerpoints where you hold shift and you like and and and it preserves the shape. In this case, it's stretching. This is vortex stretching. And that inner vortex that you see, you see there's an inner vortex and outer vortex. That inner vortex is the thing that is causing the singularity. That inner vortex is the thing that is making stuff move at faster and faster speeds
2:35:22all the way to infinity as you approach time t= 1. That that makes total sense. And and it it it's it's so interesting because again, we've built this construction where I I can actually walk away now looking at this and understand why does what I'm looking at equal a finite time blow up of Navier Stokes. Um, and it's it's it's it's just it's funny because um there's we talk a lot on the show about how we're standing on the shoulders of so much understanding. Um, and I'll p
2:36:04I'll pause on this because we'll get to it in a minute, but it's so interesting um that we are accelerating our ability to build on the learnings of those who came before us. >> Yeah, it's it's really really cool. And there's a few more things that I want to talk about this open AAI um paper that I found as a physicist quite interesting because one of the key things that I love about physics and when I was doing statistical mechanics is like scaling arguments meaning like as I approach criticality criticality is a very big thing in statistical mechanics when we talk about like um um critical
2:36:44phenomenon magnetic materials things like that um It's all about the exponents, okay? It's about as I get closer and closer to the critical phenomenon, how does my stuff blow up and how does my stuff approach infinity? For example, um uh in a magnet, if I get close and closer to the cury temperature, which is where my magnet goes and remains like if I if I have a magnet and I cool it down to a certain temperature, then even when I remove the magnetic field on the outside, it'll remain a magnet. You know how like you take a paper clip and you you rub it, it becomes a magnet, but then like you leave it, it'll stop being
2:37:25a magnet. If you cool this paperclip down, >> it's going to remain magnetized because all of the spins are kind of frozen in. There's a specific temperature where that happens. And as my material approaches that temperature, stuff starts blowing up. In this case, the correlation length will start blowing up, meaning a spin can now talk to a spin that's infinitely far away. At least that's what the model suggests. And some of the stuff in this paper reminded me of that class that I was taking. Um, again, that was another class with Robin Bruins. It was a different class than fluid mechanics, but in any case, it was one of my favorite courses. Um, so let's take a look at some of these scaling arguments that the Open AI paper is claiming. So one thing that's very cool
2:38:07is they recast the time to instead of being just time as it's approaching one we we talk about tow which is a countdown time as I approach one. So you know a normal clock time would be like 0 seconds 0.5 seconds 0.7 seconds approaching one. But imagine taking one minus that time then I'm like 1 second away from singularity 0.5 seconds away from singularity. I'm counting down to zero. And zero means I've gotten to singularity. So, first thing they do is they recast time to become toao. And tow is like a a countdown. Okay? It's like a ticking thing that is going down to zero.
2:38:48>> And they say, what is happening to my radial and my length scales as I approach zero? Um, for one, the radial scale goes down like the square root of toao. >> Okay? So, it's still going to zero, right? Because square root of zero is still when tow goes to zero, the radial scale is going to zero. But it's going like the square root. On the other hand, the axial scale is going like to the 1/2 minus h where h is some nonzero number. And that h is key because that h is telling you that when tow goes to zero, I still have a little bit of stuff left. >> Okay? Like like it's the the aspect ratio is actually going to infinity. M
2:39:29>> as I squish this thing, this guy is not shrinking that fast. >> And so my height versus my radius, if I were to divide the two, that aspect ratio is actually going to infinity. >> Mhm. >> Mhm. >> Which is key. Okay. >> Because if the aspect ratio is going to infinity, >> which in this case, what we're saying that aspir aspect ratio is the vortex is becoming infinitely slender. >> Mhm. Yes. And so what's happening is the vortex is going like this. It's getting smaller and smaller. But because of incompressibility, >> the water, the fluid has to go somewhere. It's going to start shooting out. >> And the smaller this thing gets and the
2:40:10taller this thing gets, the faster it's going to shoot out. >> And so my azamutal velocity is going to go to infinity because it's going to shoot out of this vortex >> increasingly fast. Mhm. >> As I shrink this thing down and down and down. >> Mhm. >> Right. And this shows the characteristic velocity magnitudes. Again, you've got these scaling arguments where stuff goes like the square root, but then one of them goes like the square root minus h. And the h is not zero, right? The h is some h is some small number, but because it's negative, that means it's in the reciprocal. So when t goes to zero 1 / z
2:40:50you go to infinity >> right so the azamuthal velocities the the the shooting up and down is going to infinity >> right even though the radial velocity is on the order of going to zero >> that's so >> actually no in this case it's also going to infinity >> okay and and and it it's I'll pause I'll let's continue >> the next thing I want to show is is there infinite kinetic energy because if there is then we're >> then that's not good >> it doesn't yeah >> that doesn't That doesn't count. >> Well, let's look at it. The kinetic energy is determined by the the spatial integral of the velocity squared, right? Cuz it's 12 mv^ 2. So, it's just the velocity squared. Well, velocity squared goes like to the -1 - 2 h
2:41:32>> and the volume because the the it's the it's the radius multiplied by the height. Radius squar multiplied by the height. The volume goes like to the 3 minus h. You can just see the previous slides that I showed you and calculate the volume. You multiply the velocity squared by the volume and you get something that goes like to the 1/2 - 3h. When tow goes to zero, that thing goes to zero. >> Yeah. >> So the energy is going to zero, but the velocity is going to infinity. >> It's not an infinite. They're not putting infinite kinetic energy. >> They're not putting infinite kinetic energy into this thing >> system. It is a runaway singularity. Yes. That's not correlated to the amount of energy in the system. >> Yes. The energy in the system is not
2:42:14causing the singularity. The singularity is being caused by the fluid itself. >> The fluid dynamics itself. >> Yeah. The way that they've constructed this geometry and the tiny bit of forcing that they have. Let's look at the forcing. So that's figure two. Here you're looking at a bird's eye like sort of down the tornado hole >> of the thing. And you zoom in. The way they're forcing this thing is by pushing out and pushing in simultaneously. Okay? So you push out to the left and you push in to the right. What that does is impart the same angular momentum, but it cancels out kind of the radial thing. So all you're doing is turning this thing >> in some sense >> and not making it wider. >> Yeah. Yeah.
2:42:55>> Oh, that's it's kind of nice. >> That's nice. >> Yeah. And also the pushes are localized. >> They're not in a total ring. Like if you zoom in on that part of the ring, they're localized to these little tiny spots. >> This goes back to the convective term versus the field, the vector field. like the changes are happening at the very Yeah. at a very smallest part and then those changes are then rock ricocheting into the smaller and smaller y >> vortices to create that thing. Right. >> Okay. This is good. >> I mean it's it's very cool and and this is on the backs of mathematicians over the ages. But I still think like this definitively shows that there are situations where the convective term wins no matter how hard the viscosity
2:43:36tries. Right. the the vendetta was not su successful by >> exactly and and so you would think that this is the end of Navier Stokes not quite it is the end of the millennium problem because the clay mathematics institute um Charles Fefferman had a very stringent criteria and this thing meets that stringent criteria it's been verified by lean crucially that does not mean that it is true right lean could have errors so we're still waiting and the claim mathematics institute actually is also waiting for 2 years before it says says, "Yep, this is done." So, there needs to be human mathematicians that actually understand this 165page proof um and and tell us, "Yep, we're good to go." Um, but the Navier Stokes
2:44:18problem itself is actually also not fully put to rest because they have solved the forced version, right? This this required a little bit of paddling in some sense. You took a ore and you like paddled the vortex to become what it is. Um, what about the unforced scenario, right? That is still an open question and Princeton mathematician Stan Palisk actually identified um a problem with this particular proof that would make it impossible to apply to an unforced scenario. >> So like this geometry is not going to work for the unforced. This only works because you have those little paddling terms that that are imparting the finite amount of energy into that cascade. Um,
2:45:00so without that hand of God forcing term, >> yeah, >> it's not it's not it doesn't work. And so that part at least is still an open question. This natural unforced setting and and this is as I've seen some of the dialogue about this. This is where you know so Scientific American had an article about this and they were like you know open AAI did not solve Navier Stokes and you know what they were sort of trying to argue is they didn't solve A and B >> they solved C and D like that's the argument that they were making but they made it seem like it was complicated. No, but the claim mathematics institute said any of the four. Correct. Like and
2:45:42this is this is where again I'm just trying to talk about the nuance of what actually was done versus sometimes the way in which it gets then presented in media downstream. >> Yeah. >> And you know some people now are arguing well the way it's defined by clay mathematics is the problem. >> Yeah. >> And it's like okay you can make that argument. >> You weren't arguing that before. >> Right. But why are we It's funny that now that it starts when it's been solved by something that people that >> either makes people feel uncomfortable or they feel like it's by an entity that didn't give credit to those before them or whatever it might be. But I just it has been very interesting to see the dialogue over the last two and a half
2:46:24weeks knowing we were going to go through this. >> I focused on the larger environment. >> Yeah. And now having walked through this, you know, with nuances applied 6 months ago or a year ago. >> Yeah. >> If you said that a model, even given all of the human advancement, had figured out finite time blow up for Navier Stokes, people would have laughed in your face. >> Yeah. >> They would have laughed at you. >> Yeah. I mean, those are guys who are like, "Yeah, wake me up when AI solves a millennium problem." And now the millennium problems are illdefined. You know, I don't know. I don't know. Um
2:47:06this so I want to let people know here this is the best explanation of Navier Stokes in the context of OpenAI's announcement and the history of this subject at a level that is accessible to most general people but is not meant to necessarily be the deepest mathematical uh >> debate about the nuances. Although it's quite good. >> I I did the best I could. It's quite good. >> I'm not a mathematician. >> I think it's quite good. Um but we will um and it's funny and I just noticed this.
2:47:46You're still listening right now. Our lights have an automatic setting at a certain time in the night that turn it to bedtime mode. And so our lights have actually gotten a little bit darker. I was wondering >> in in the video playback, but because we're 3 hours in and we started a little bit late tonight and so we're going to do a quick reset for our lighting and we're going to come back and finish up with now taking a look at >> now that we understand what has been done >> although we didn't get the same detail as with anthropic. >> Yeah. >> What is the environment in which this happened in how do we think about where this is all