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I want to talk about this Navier Stokes result for a minute because I see alot of people sharing it and not nearly enough people explaining it. A share for this one would be appreciated.

The Navier-Stokes equations are just Newton's second law of motion, F=ma, adapted for liquids and gases. Instead of tracking the acceleration of a solid object, the equations track how a bit of fluid moves, deforms, and flows as different forces push and pull on it. The reason these equations are so notorious, and the subject of a million dollar mathematics prize, is because of turbulence. Smooth, layered flow, or laminar flow, is relatively straightforward to calculate. But once the fluid speeds up and turns into a chaotic turbulence mess, the mathematics become incredibly non linear. The fluid's movement feeds back into itself, creating unpredictable, swirling eddies that become so complex you baisically need a super computer to simulate it.

Ok so what the hell is the millennium prize then?
The millennium prize is for proof that the equations can result in a singularity or what you see referred to as a "blow up". This is where the equations essentially go to infinity.

Ok so why doesnt your cup of tea blow up or turn into a singularity that swallows your house? The Navier Stokes prize isnt about proving that nature makes blow ups or singularities possible, its about proving the limitations of the model's mathematics.

Ok so why can a model allow for things we dont observe? This is THE SAME type of mathmatical allowance that general relativity gives when the model breaks down around black holes. The chance that a black hole is an actual singularity where everything goes to infinity is basically zero. Thats not how the universe actually works.

Ok so WHY is this possible in the model? The Navier Stokes equations assumes fluid is an infinitely divisible, perfectly smooth substance. In the math, you can zoom in forever, and it remains a continuous fluid. But that's NOT how a real fluid in reality is structured. Real fluids are made of discrete, individual molecules bumping into each other. Theres a real limit to how far you can zoom in. In the proof and the model it means that as a vortex shrinks and speeds up, it eventually gets smaller than the distance between individual water molecules. This cant happen in reality.

So the millennium prize isnt about finding if nature allows for infinite turbulence or eddies until a singularity forms, its about FINDING THE LIMITATION OF THE NAVIER STOKES EQUATIONS, proving global regularity false.

Ok so why all the drama with the Open AI solution? It used a method called "forcing". In the simulation they essentially used a continuous force to keep the vortex spinning faster and faster until it blows up, the vortex is smaller than the actual molecules that the fluid is made of. It goes to infinity in a time period that ISNT infinity. While the millennium prize doesnt specifically PROHIBIT forcing, most mathematicians would agree that they are specifically interested in the unforced proof, the spontaneous blow up. But that doesnt mean the result isnt legitimate.

I hope this clears up some of the misunderstanding.

BREAKING: OpenAI might have stolen another major proof.

In a detailed Mastodon post, which I report in full in the comments, Andreas Thom presents several pieces of evidence suggesting that OpenAI may have trained Astra on conversations in which he and Gábor Kun were working on Gromov’s soficity conjecture, one of the ten problems OpenAI later announced Astra had solved.

I know Andreas. We met several times early in our careers. He is an exceptional mathematician, a leading expert on sofic and hyperlinear groups, and one of the most respected scholars in the field. He has spent two decades working on this problem.

If his account is correct, this is not a minor dispute over attribution. It would mean that unpublished human work was absorbed into a model and then presented to the world as a breakthrough by the model itself.

And if the allegations raised by Levent Alpöge, Tristan Buckmaster, and now Andreas Thom are all substantiated, we are no longer looking at isolated incidents.

We may be looking at one of the greatest intellectual scandals in the history of science.

AI is not discovering new mathematics.

AI is stealing human discovery.

unfortunately i contacted them on the night of wednesday, september 2nd making the following points

1. i had intel they had information about math work i was doing and a few days before they had spun up a group to try to compete.
2. i emphasized and reemphasized mine was a strictly personal mathematical collaboration
3. that this was a collaboration that had been going on for a year
4. that in my personal collaboration we were happy users of codex as well as obviously claude (and i later complimented astra explicitly after it came out for its help, and reassured i would obviously be sprinkling acknowledgments for it, previous models, and codex liberally in the writeups)
5. that it would look terrible if openai were competing against mathematician consumers, and my collaborator (Tristan) wanted to email about this explicitly
6. that i expected none but in the worst case intended to refuse marketing around the result we were working towards, being the fruit of a personal external academic collaboration, and wanted to express this because i figured worries about such marketing by their leadership was why a team was spun up to extraordinarily directly compete.

sebastien and i have in the past months had our fun pushing each other on twitter, but as i process it all i keep feeling bad about Tristan.

1. neither of us could sleep Sunday night, so we went on a walk around the city. i finally asked him why fluids, and he told me it was the navier-stokes problem. he told me parts of the story about his classic nonuniqueness result with Vicol i hadn’t heard yet, and also about how he too grew up writing shaders as a kid and then switched to math
2. tristan is a mathematician’s mathematician. he kept rock solid to his principles during the tense calls and negotiations, immediately turning down a career dream and million dollar prize for me, the sort of strange excited unclear-quasiuseful confused idiot in our collaboration. fifteen minutes later, his feet were up on his desk with three toes per foot peeking out of ripped socks
3. our periodic call after i’d texted him euler blowup was just laughing for an hour or two. just laughing. i’d seen fable deal correctly with monge ampere in the equality case of ehrhart’s conjecture, so in some call before that i’d really pushed him for the craziest shit he could think of that i might try. i texted him the pdf as an update on that “diego luis 3” line, in which following his incredible expression of intuition i hodgepodge slapped together a bunch of our recent work, in particular correcting a bunch of better and lesser known papers in the literature, as well as on diego and luis’ ipm paper, and hoped an analogous construction for boussinesq (thus axisym euler) might work out. i managed to wade through enough confusion to insist to the model it stop telling me stupid regularity stuff about kelvin waves, and a rayleigh-taylor ansatz stuck. like i said, just laughing. when i texted him the pdf, the worst thing either of us had seen so far, i didn’t even comment on the euler blowup claim, only explaining why skepticism should remain for boussinesq, because i found it so shocking. i mean what the hell did i know, i must’ve confused myself by misunderstanding one adjective like i was used to with algebraic geometry. im so lucky to have been able to live that
4. he managed all this this entire time with unclear sleep, timestamps continue to amaze me, due to just recently having a kid

it is obvious he deserves the same sort of praise he heaped upon luis, perhaps mutatis mutandis:)