Since I'm now blocked by bluey from replying to this thread, I answer here by quote tweet the openAI construction has more to do with convex integration than anything else. In convex integration, the oscillations are typically built by hand and the iteration procedure creates wild weak solutions. The non-uniqueness construction of Albritton–Brué–Colombo used an unstable vortex to split solutions apart. The Córdoba–Martínez-Zoroa singularities amplify vorticity across scales by strain. In the openAI paper, a fluid instability is clever made so that the flow itself generates the oscillations: they grow by drawing energy from the background shear. The Reynolds stress then becomes the force that creates the collapsing smooth vortex. Using a physical instability to create the stress in this way has no precedent, and it is what produces a smooth force rather than a merely rough one. Some on these ideas are in the literature (e.g. Leibovich-Stewartson 1983) but the way they are combined is definitely insane and definitely nowhere in the literature previously. I also don't know of anyone suggesting that convex integration would play a role in the creation of a blow-up solution like this. On a technical level, the paper is unbelievable, maybe beyond what a human could have come up with but not beyond human understanding: there are several different groups of pde experts working on understanding it who say that they more or less understand now. I expect digestions soon td;lr: anyone saying that this OAI result is unsurprising is bullshitting sorry!

I am worried about a situation in which rumors are constantly swirling about major results and your distance to the labs in the social graph distance determines how much you know. Since at least the ICM they (openAI) have been sitting on "hundreds" of proofs of results, I was told, and I suspect Levent has more than he has released. Some of them may not be released any time soon. I don't think it's a great situation if this persists. If they tell the NY times "we have major advance on another millenium problem" then they should be out with the news when they have the result They can write a 450k lean formalization and a 160-page paper of a millenium problem in a few days If they are taking time to do a better job polishing the paper, then great. I suspect this is NOT what they are doing however

I disagree with this. Obviously the openAI construction of the Navier-Stokes blow up is a huge, huge contribution to the field. It will take us PDE analysts some time to understand it well. (Longer than a week, but perhaps less than a month.) Soon some of us will understand it, and hopefully rewrite it in a way that others can follow. When this is done, our human understanding of these important equations will be past the bottleneck we've been stuck at for 100 years since Leray. What is weird is that, up until now, proofs like this have always emerged AFTER human understanding. But that is not the case any more. From now onwards, human understanding will typically lag the proofs, which will first be written by our machines. The understanding however WILL follow closely behind. The reason people are upset has nothing to do with the fact that this process is inverted. They are simply upset that the machines are going to be much smarter and more capable than us. (They are also upset that the humans who got the machine to produce the proof don't understand it, and this feels very unfair to them.)

I am begging everyone to stop pretending that "understanding" and "finding true propositions" are not strongly correlated, in fact nearly the same thing. I heard many smart people saying this over and over, all summer long in Europe - but this is actually not very clever. It is one of those things that sounds very deep and clever until you think about it for > 2min. Also, let's please agree to stop pretending that LLMs are not quite efficient at providing understanding. They are not some kind of oracle/black boxes, as anyone who has ever tried to one knows very well. Look at Terry Tao's blogpost on unpacking the Jacobian conj counterexample, for example.