Two proofs went public this week. The dispute is about affiliation, not mathematics.
The Navier-Stokes equations mathematically describe how fluids move, and there's been a $1m bounty on proving their solutions since 2000. Twenty-six years later, those tough math problems became the subject of a complicated discussion on ownership in the age of AI.
On September 8, Tristan Buckmaster, Levent Alpöge, and Matei Coiculescu published proofs showing that several closely related equations, including 3D incompressible Euler, can blow up in finite time when you push them with a smooth external force. "Blowup" means some quantity in the solution runs off to infinity after a finite amount of time rather than staying finite forever, which is exactly the behavior that the $1m Clay problem asks about.
They verified the proofs in Lean (a programmatic proof assistant), so the argument is machine-checked rather than resting on a referee's reading. Terence Tao wrote up the results on his own blog, which is a fair proxy for how seriously the field is taking them.
The work took about a year, and most of it was slow going. According to Buckmaster's account, the breakthrough came on August 15 and was verified by Lean on August 22. They used several models throughout and paid out of pocket: Anthropic's Claude, OpenAI's Codex with GPT-5.6 Sol, and more recently Astra for writeups and auditing arguments. Every draft of the project went into their Codex sessions.
