On September 7th, just before OpenAI announced their proof of NS, Tristan Buckmaster put out a proof of blowup in the 3d Euler equations with smooth forcing, along with a statement in which he accused OpenAI of having stolen his work through his Codex sessions (https://cims.nyu.edu/\~tristanb/statement.pdf). In it, he talks specifically about how the version of NS that OpenAI had proven was a red flag for him that OpenAI had used his work:
I was told that an internal OpenAI model had produced a proof of finite time blowup for the forced Navier-Stokes equations. When Levent asked by text for the precise statement, the answer was: "Existence of forced blowup in R³ and T³," with "the forcing function is smooth option c and d in Fefferman." I was told the proof is about 100 pages. I have not seen it.
I should say here why I interpreted their statement the way I did, the interpretation I will discuss below. The route to the Clay problem through a smooth force, options c and d in Fefferman's statement of the problem, is the route Luis and Diego opened and the one Levent and I had quietly chosen to attack. Almost nobody else I know of was working on it. It is not the direction one arrives at in a few days by giving a model the problem statement. When I heard "forced," it was a bright red flag.
Numberphile recently released an interview with him (https://youtu.be/Mzbtj5nkMXI?si=XJMP5f6YYk9S9X0_&t=945) where he makes a similar statement.
The Clay Math institute's official statement of the Navier-Stokes problem (https://www.claymath.org/wp-content/uploads/2022/06/navierstokes.pdf) was written by Fefferman, and lays out 4 possible statements which, if proven, would constitute a solution to the millennium prize problem. Options A and B are to show that smooth initial data always lead to smooth solutions, and in this case the forcing function may be taken to be zero. Options C and D are essentially the opposite - showing that a blowup solution exists, but in this case there can be a non-zero smooth forcing function. (Options A and C are posed in R\^3 while options B and D are posed in R\^3/Z\^3).
I can't make heads or tails of his claim that the fact that OpenAI's proof used a forcing function was somehow a "red flag", since that's simply the statement of NS put out by the Clay institute. I'm not an expert on NS by any means, but I've done work on other non-linear PDEs and attended many conference talks about NS and I can't understand what Buckmaster is getting at here. If anyone more knowledgeable about NS could explain what he might be trying to say here I would really appreciate it.
(To be clear, I think a lot of the other claims he made about how OpenAI handled this situation should be taken very seriously, but this one completely baffles me)