OpenAI says an internal system produced an analytical proof and a Lean formalization that 3D incompressible Navier-Stokes can develop a singularity in finite time with smooth forcing and finite energy, claiming this resolves statement C (and D) of the Navier-Stokes Millennium Prize Problem; the company says the model is 'significantly more capable than GPT-6 Astra' (OpenAI, 8 Sep, self-reported and not peer-reviewed). src

The OpenAI Navier-Stokes claim is contested: Terence Tao called the parallel finite-time blowup result for fluid equations by Alpoge and Buckmaster (with significant AI input, formalized in Lean) 'a remarkable achievement' while noting they 'had to release far earlier than planned due to external events'; a statement by the rival researchers quoted on Hacker News alleges OpenAI threatened careers ('Why would you ruin your career?') and dangled the USD 1m prize over contamination claims (Terence Tao on mathstodon; Hacker News thread, 8 Sep). src

France 24 reports OpenAI said an unreleased model solved the Navier-Stokes problem in about 88 hours using 'millions of dollars' worth of computing power' (roughly 1,000 times what the company spent on earlier mathematical results, per researcher Sebastien Bubeck), with 10,000 AI agents working on the problem at once at the final stage; chief research officer Mark Chen called it 'a significant milestone for AI research', and OpenAI says it will not claim the USD 1 million Millennium prize money, which under Clay Mathematics Institute rules requires publication in a peer-reviewed journal and two years of acceptance review — Institute president Martin Bridson told AFP 'The process of evaluation is deliberately unhurried, and we shall ensure that it is absolutely rigorous' (France 24/AFP, 9 Sep). src [!claim] asserted NYU mathematician Tristan Buckmaster and Anthropic researcher Levent Alpoge, whose own AI-assisted work on three related equations was released hours before OpenAI's claim, allege OpenAI only took up the problem after word of their research spread and pursued the same unusual approach they had spent months developing; OpenAI's researchers denied having had any access to the rival work ('we did not use their prompts or proofs to prompt our models or direct our agents'), but said in a later post on X that 'while unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models' — a contamination question the Clay prize rules do not obviously settle (France 24/AFP, 9 Sep). src [!claim] asserted Terence Tao argues the collection of good, fruitful open problems is now being 'mined in a non-renewable fashion' — one can generate any number of open problems at will (like finding the 10^10^10th digit of pi), but the vast majority are not worth attention, and the difficulty is knowing which questions are genuinely fruitful; in the current AI era, he says, there are no definitive frontiers separating 'AI-feasible' from 'AI-hard' problems, in part because AI companies refuse to disclose their models' capabilities (Terence Tao on mathstodon, 8-9 Sep). src