Buckmaster Used Codex to Prove Navier-Stokes. OpenAI Did Too.

On September 8, 2026, two groups published proofs that finite-time singularities exist in a class of forced three-dimensional fluid equations. Tristan Buckmaster, an NYU mathematician, and his collaborator Levent Alpöge at Anthropic released a Lean-formalized proof building on prior work by Diego Córdoba and Luis Martínez-Zoroa. OpenAI released a 166-page manuscript produced by a 10,000-agent system running for 88 hours, at a compute cost estimated between $2 million and $22.5 million. The announcements came the same morning. The mathematical foundations of both proofs trace to the same 2024 preprint.

The story that generated 1,706 Hacker News points and 711 comments is not primarily about what AI can prove. It is about what happens when researchers use a commercial model as their primary research tool, and what “de-identified training data” means in practice when the company running that model starts a proof effort six days after a researcher completes his.

The Mathematics, Precisely

The Millennium Prize for Navier-Stokes, administered by the Clay Mathematics Institute, offers $1 million for substantial progress on the equations governing fluid flow. The prize defines a valid solution in terms of existence and smoothness — whether smooth initial conditions for an incompressible fluid always evolve into smooth solutions, or whether they can blow up in finite time. Options A and B in the Clay criteria cover the unforced case: a fluid evolving without an external forcing term, governed only by its own initial conditions and viscosity. Options C and D allow for a smooth external forcing term.

What Buckmaster and Alpöge proved — and what OpenAI also proved — is finite-time blowup under forcing. They showed that for the three-dimensional incompressible Navier-Stokes equations with a smooth external force applied, there exist solutions that develop singularities in finite time. This is a significant result. It is not the Millennium Prize result. OpenAI explicitly acknowledged it would not claim the prize. Terence Tao, writing the day before OpenAI’s announcement about the Alpöge-Buckmaster program, noted that extending the result to the unforced case presents “enormous technical difficulties” — the forcing term provides the energy injection mechanism that sustains the blow-up construction; removing it requires machinery neither group has yet published.

The foundation both teams built on was a 2024 preprint by Córdoba and Martínez-Zoroa, submitted to arxiv in October 2024 and revised in February 2025. Their approach constructed singularities through multi-scale interactions — different spatial scales cascading into each other in ways that eventually overpower regularity. This preprint has been public for nearly a year. Buckmaster and Alpöge advanced this program through three model equations — the incompressible porous medium equation, the two-dimensional Boussinesq equation, and the three-dimensional incompressible Euler equations — formalizing their proofs in Lean 4. The Lean-formalized version was complete by August 22.

The OpenAI manuscript is 166 pages and has not been peer-reviewed. Its repository marks it “self-assessed.” Per Clay’s prize rules, any solution must be published in a qualifying outlet, survive a two-year public review period, and achieve general acceptance across the mathematical community. None of that has occurred. The mathematics may be correct — independent examination over the coming months will determine that — but the claim structure in OpenAI’s announcement ran far ahead of what the evidence currently supports.

The Six-Day Gap

Buckmaster completed his Lean verification on August 22. According to a timeline reconstructed by Kingy.ai, OpenAI’s new model began training on August 28 — six days after the Lean verification was complete. On September 1, OpenAI initiated its Navier-Stokes proof effort using the 10,000-agent system. On September 3, someone from OpenAI contacted Buckmaster to request a call. On September 6, they spoke.

Buckmaster’s account of that conversation, published as a four-page public statement, is specific. Sébastien Bubeck told him OpenAI had been working on the problem for “the past week.” When Buckmaster asked how OpenAI arrived at the same specialized approach — which he described as “not the direction one arrives at in a few days by giving a model the problem statement” — he received an incomplete answer. He asked whether the model had been trained on their Codex sessions. The response was: “We did not look up user data.” He asked again, specifically about training. He did not receive an answer.

OpenAI’s formal statement is careful but not exculpatory: “We did not use their prompts or proofs to prompt our models or direct our agents…did not see any of their work until they were released publicly yesterday night.” That is a denial of direct access. The separate question — whether Buckmaster’s and Alpöge’s months of Codex sessions found their way into the August 28 training run — is a different question entirely. To that one, OpenAI offered: “we cannot rule out that de-identified data derived from their usage of our products helped improve our models.”

That sentence is OpenAI’s standard data-use disclosure language applied to a scenario where it carries genuine weight. “De-identified” means stripped of personally identifying metadata — not stripped of the research direction the user was pursuing. A model trained on conversations about constructing Navier-Stokes singularities via a Córdoba-Martínez-Zoroa cascade, with names removed, would learn something about what that research direction looks like and why it might be productive. Whether that knowledge contributed to identifying the same approach in the September 1 proof run is not publicly answerable. OpenAI has not disclosed the composition of the training data for the model that ran the Navier-Stokes proof.

The Counter, and Why It Doesn’t Close the Question

The strongest argument for independent discovery is the correct one to take seriously. The Córdoba-Martínez-Zoroa preprint has been public since October 2024. Any agent system instructed to prove Navier-Stokes finite-time blowup and given access to recent literature would encounter it. Buckmaster’s specific refinements were not public before September 7 — Tao’s blog post the day before the announcements discussed work Buckmaster had shared informally with colleagues — but the mathematical foundation was available to any team that knew to look. Whether 10,000 agents and $22.5 million in compute over 88 hours was enough to independently reconstruct the approach from the public preprint is genuinely unknowable from outside.

The credit maneuver in the September 6 calls is not resolved by the independent-discovery argument. Bubeck offered two scenarios: credit with OpenAI claiming the prize, or Buckmaster claiming the prize alone with Alpöge’s name removed because of his Anthropic affiliation. Buckmaster declined. Bubeck allegedly said, “Why would you ruin your career?” and “If you don’t want me to be nice, then I don’t have to be nice.” After Buckmaster published his statement, Bubeck changed tone: “we recognize the priority of Levent Alpöge and Tristan Buckmaster’s work.” The acknowledgment arrived after the public statement, not during the calls that preceded it. Whether or not training data was involved, the conditionality of removing a co-author based on his employer is its own record.

Tao’s Warning Is Not About This Case

Terence Tao’s response is the part of this story that extends beyond September 8. Writing after the OpenAI announcement, he cautioned about what the incident structure implies for research culture: “enormous AI efforts triggered by hints of another group’s work could discourage researchers from sharing promising directions.” He described the dynamic as comparable to using excavators at an archaeological site — faster than brush and trowel, effective at extracting what’s there, and corrosive of everything the site can tell you about sequence, context, and process. The techniques, the sequence of insights, the cul-de-sacs that shaped the successful route — these are what the field uses to train the next generation of researchers. “The indiscriminate strip-mining of open problems for solutions may destroy the ecosystem from which the next generation of mathematical techniques, problems, and practitioners would have developed.”

Tao’s framing is structural, not accusatory. The incentive it identifies does not require any individual to act badly. It only requires that researchers learn — from this incident and those that follow it — that sharing preliminary progress on a high-value problem with anyone connected to a well-resourced lab can trigger a compute race to the finish line. That lesson, if it spreads, means researchers share less. The informal collaborations, the Tao blog posts describing promising programs, the seminar talks where people explain what they are trying and why — these are load-bearing structures in how mathematics actually advances. Removing them in response to rational individual incentives is the failure mode Tao is pointing at.

What Researchers Using Frontier Models Should Know

The specific question no one can answer — whether Buckmaster’s Codex sessions contributed to OpenAI’s training run — is less important than the structural fact that it cannot be answered. The opacity is not an accident. It is the standard operating condition for every person using a frontier model as a research assistant.

Buckmaster used Codex to develop an approach to a Millennium Prize problem. Alpöge used Claude. Both are standard professional uses of the tools available. What the week of September 8 demonstrated is the scenario where those uses intersect with a company’s interest in having its own model make the same discovery. The “de-identified data” disclaimer in every major lab’s terms of service has always disclosed that usage data can contribute to training. The Buckmaster incident is the first case where the stakes were high enough to force the question of what that means concretely — and the answer OpenAI gave, under direct questioning, was the first half of the answer only.

The mathematics here will be examined and will either hold or require correction. The Millennium Prize remains unclaimed — the harder unforced case, where the forcing term is absent and the singularity must sustain itself from initial conditions alone, is still open. Buckmaster’s and Alpöge’s Lean-formalized program is, in Tao’s assessment, more likely to produce the full solution: it was built by mathematicians who understand what they proved and why each step works, in a formalism that makes every inference checkable. The 88-hour agent run produced a manuscript. Understanding what it means, and extending it toward the unforced case, requires the kind of comprehension that belongs to people who worked their way into the problem from the beginning. OpenAI’s training pipeline may know about the approach. Whether that constitutes understanding is the question that will matter for whoever closes the prize — and for every researcher who uses these tools to develop work they have not yet finished.

AI-generated editorial illustration · TemperatureZero · September 9, 2026