Statement on OpenAI Navier–Stokes priority dispute (Buckmaster, 2026-09-08)

TL;DR

Tristan Buckmaster (NYU / Courant) publishes a first-person statement announcing three finite-time blowup results (smooth-force incompressible porous media, Boussinesq, and 3D incompressible Euler) with concurrent Lean verification, joint with Levent Alpöge (Anthropic, personal capacity). Buckmaster credits the underlying analytic program to Diego Córdoba and Luis Martínez-Zoroa, extended from rough to smooth forcing using LLM assistance (Claude, Codex with GPT-5.6 Sol, Astra for writeup/auditing). Approximately half the document is Buckmaster’s account of September 3–7 correspondence with OpenAI: he alleges that after their team learned of the parallel work through a rumor, an OpenAI research group produced (within days) a 100-page proof of forced Navier–Stokes blowup, and that during two September 6 calls involving Sébastien Bubeck, OpenAI proposed publication arrangements Buckmaster interpreted as pressuring him to drop Alpöge from authorship (because Alpöge works at Anthropic). Buckmaster reports being told “why would you ruin your career?” when he threatened to go public. He is explicit that he has not seen OpenAI’s proof and is not accusing anyone of anything — he is reporting what was said and proposed. See On the Navier–Stokes Millennium Prize Problem for OpenAI’s public release the same day.

Key claims

Buckmaster and Alpöge publicly release three finite-time blowup results with smooth forcing (incompressible porous media, Boussinesq, 3D incompressible Euler); they claim they also have blowup for hypo-dissipative Navier–Stokes but are withholding release pending Lean verification [§Opening paragraphs].
The underlying analytic program (forced blowup constructions) is credited to Diego Córdoba and Luis Martínez-Zoroa; Buckmaster and Alpöge extended it from rough to smooth forcing using LLM assistance (Claude, Codex with GPT-5.6 Sol, Astra for writeup/auditing) [§Program credit].
The Euler writeup is described by Buckmaster himself as “much closer to what models produce under human direction than to a paper written by a person” and characterized as “AI slop”; the reasons given for shipping it in that state involve external time pressure [§Presentation quality].
Timeline of the OpenAI interaction as reported: (1) Sep 3 — rumor circulating that Anthropic had resolved a major open problem; Buckmaster emails an OpenAI mathematician directly; (2) Sep 4 — declines to meet that day; (3) Sep 6 12:45 — asked to meet “at any point today”; two calls that afternoon with Bubeck [§Timeline].
Buckmaster’s account of the September 6 calls: OpenAI reported an internal model had produced a ~100-page proof of forced Navier–Stokes blowup, targeting Fefferman options (c) and (d) — the exact route Buckmaster/Alpöge had been quietly pursuing; the prompt “shown to me had been written by prompting Codex,” and it emerged during the call that “an entire team had been working on the problem” over “the past few days, after information about our work had reached OpenAI” [§Timeline].
Two proposals Buckmaster reports were offered by OpenAI: (a) simultaneous posting of both results, (b) Buckmaster alone (with Alpöge removed) writes a paper presenting the Navier–Stokes result with attribution to OpenAI’s internal model; Buckmaster reports being told twice that Bubeck “wanted Levent removed from authorship” and that “it would all be simple if only it were not the case that … Levent works at Anthropic” [§Timeline].
When Buckmaster said he would go public if OpenAI released as proposed, he reports being asked “Why would you ruin your career?” and, later, “If you don’t want me to be nice, then I don’t have to be nice.” [§Timeline].
Buckmaster asks whether the OpenAI model was trained on, or had access to, the Codex sessions in which he and Alpöge were drafting the work; reports being told “the model did not look up user data” but no answer on training [§Timeline].
Buckmaster is explicit on scope: he has not seen OpenAI’s proof, does not know what their model did or how, does not know whether his data was used, and “is not accusing anyone of anything” — the document reports what was said and proposed rather than making causal claims [§Disclaimer].

Method

This is a personal statement / open letter, not a research paper. Its evidentiary claims fall into three buckets: (1) the mathematical work itself is available as three separate preprints referenced but not attached, (2) the timeline of correspondence quotes one full email from Buckmaster verbatim and paraphrases replies from OpenAI, and (3) descriptions of two phone calls with Sébastien Bubeck are Buckmaster’s first-person recollection with no transcript. The document’s own framing acknowledges the asymmetry: Buckmaster explicitly notes he did not see the OpenAI proof, does not know their training-data provenance, and is reporting what was said rather than making causal accusations.

Results

No quantitative results — this is a statement, not a paper. Two qualitative artifacts:

Three math preprints announced (smooth-force blowup: incompressible porous media, Boussinesq, 3D incompressible Euler) with Lean verification; a fourth (hypo-dissipative Navier–Stokes) withheld pending Lean completion.
A public timeline of Sep 3–7 correspondence with OpenAI, released the same day OpenAI announced its own Navier–Stokes solution (On the Navier–Stokes Millennium Prize Problem).

Why it’s interesting

This is a rare public artifact for the AI-for-AI Research concept: a first-person timeline from a domain expert on the inside of a frontier-lab priority dispute, with the counterpart artifact (On the Navier–Stokes Millennium Prize Problem) released the same day. Two things it independently corroborates: (a) both efforts targeted the same narrow analytic route (Córdoba–Martínez-Zoroa smooth-forcing program) and OpenAI’s public timeline agrees that its work began September 1 after hearing the rumor, which matches Buckmaster’s account of when tips about their progress started leaking; (b) OpenAI’s stated compute (~10,000 agents, 88h, 300B output tokens) matches Buckmaster’s characterization of “an entire team … an insane amount of compute … one of a number of things that was tried.” What the two accounts diverge on is (a) whether the model was given the problem statement cold vs. supplied with iteratively-refined prompts drafted using Codex, and (b) what was proposed during the September 6 calls regarding authorship — Buckmaster’s account of proposals to remove Alpöge is not addressed in OpenAI’s public release. Both artifacts together are the cleanest filed case study for the wiki on how frontier-lab AI-for-AI-research systems interact with parallel human research programs, and — as the OpenAI post itself notes — on the norms question of what happens when Codex sessions used for private drafting may or may not become training data.