Navier–Stokes: a new proof claim, and a dispute over credit
A fluid singularity, a Lean certificate, and two accounts of the race.
OpenAI · Tristan Buckmaster · Levent Alpöge
Start with the question
The problem
Can a three-dimensional incompressible viscous fluid that starts smoothly develop a singularity in finite time? Clay’s official formulation includes breakdown alternatives with smooth external forcing, as well as global-existence alternatives.
01
Read the claim precisely
On 8 September, OpenAI announced a construction intended to establish alternatives C and D of the Millennium Problem. It involves a smooth applied force. This does not establish blowup for the unforced viscous equation. The distinction is part of the problem statement, rather than a technicality to leave out of the story. [1][3]
02
From Euler to viscosity
OpenAI reports that coordinating agents first found an unforced Euler construction, then used that progress to pursue Navier–Stokes. The company describes an 88-hour search followed by 17 hours of formalization. Those timings are its own account. The repository supplies separate Euler and Navier–Stokes developments and comparator challenges. [1][2]
03
Why the announcement became contentious
Buckmaster’s statement raised concerns about research priority, private Codex sessions, and proposed authorship arrangements. OpenAI says neither its researchers nor its agents saw the pair’s work before publication and denies looking up specific user data. Its statement leaves open whether de-identified usage data contributed to model improvement. These are conflicting public accounts, not an established finding of misconduct. [4][1]
Inspect the evidence
The Lean formalization
Lean artifact released; independent assessment ongoing
The repository states whole-space and periodic results for every positive viscosity, with smooth forcing, and supplies comparator instructions. Checking the formal statement against Clay’s hypotheses is essential. We reviewed the documentation, not a local rebuild of the proof.
Open the Lean project ↗Sources & further reading
Reviewed 9 Sept 2026. X permalinks were recovered from indexed posts and linked discussions; direct X pages were not consistently accessible. They document the conversation, not proof correctness.
- 1.
- 2.
- 3.
- 4.
- 5.
- 6.
- 7.Levent Alpöge’s response ↗
Read alongside Buckmaster’s statement and OpenAI’s response. Allegations remain attributed to their authors.