Published September 9, 2026 · 8 min read · By Yash Khare

OpenAI's Navier-Stokes Math Claim Ignites a Credit Fight

Two mathematicians spent nearly a year quietly using AI models to chip away at one of fluid dynamics' hardest open problems. Days after word of their unpublished progress reportedly reached OpenAI, the company says an internal, unreleased model solved a related $1 million Millennium Prize problem outright. What should have been one of the biggest AI-for-math stories of the year has instead turned into a public fight over who deserves the credit — and whether OpenAI tried to erase a rival researcher from it because he works at Anthropic.

What Buckmaster and Alpöge actually proved

On September 8, 2026, Tristan Buckmaster (an NYU mathematician) and Levent Alpöge (a researcher at Anthropic, working on this entirely as a personal project unconnected to his employer) publicly posted three results: finite-time blowup — the point where a smooth, well-behaved solution suddenly develops a singularity — with smooth forcing for the incompressible porous media equation, the 2D Boussinesq system, and the 3D incompressible Euler equations. All three came with a formal, computer-checked verification in Lean.

In his own public statement, Buckmaster was careful to credit the underlying idea to mathematicians Diego Córdoba and Luis Martínez-Zoroa, who had spent years developing the approach with rougher forcing. "We took their work as a starting point, using Large Language Models to push their program to completion," he wrote, adding that he believes Martínez-Zoroa "deserves a Fields Medal" for the foundational idea. The two used Anthropic's Claude and OpenAI's Codex (running GPT-5.6 Sol and, later, Astra) throughout — Buckmaster admitted the first LLM-generated proof Alpöge sent him was "the most horrendous I have ever read," and that the eventual Euler write-up is "closer to model output under human direction than to a human-written paper."

The much bigger prize hanging over all this

The reason this matters beyond one paper: the same family of equations connects directly to Navier-Stokes, one of the seven Clay Mathematics Institute Millennium Prize Problems, open since 1934 and worth $1 million to whoever resolves it. Buckmaster and Alpöge's next (unreleased) target was hypo-dissipative Navier-Stokes — described in his statement as "suggestive of a path to unforced Euler."

Separately, OpenAI announced on September 8, 2026 — hours after Buckmaster and Alpöge's post — that an internal, unreleased next-generation model ("significantly more capable than GPT-6 Astra," per the company) had produced a proof of finite-time blowup for the forced version of Navier-Stokes: a roughly 100-page proof generated by a swarm of around 10,000 concurrent AI agents, consuming an estimated 300 billion output tokens (reportedly around $22.5 million in compute) over about 88 hours.

How the dispute unfolded

According to Buckmaster's own detailed public statement, on September 3 — with rumors already circulating that a major open problem had been resolved — he emailed a prominent OpenAI mathematician to get ahead of the confusion, explicitly noting this was a personal collaboration with no institutional backing. The reply offered OpenAI's help and compute "to avoid competing." Buckmaster proposed talking the following week.

Instead, on Sunday, September 6, he was asked to meet that same day. Sebastien Bubeck joined the call and told him an internal OpenAI model had already produced a full proof for forced Navier-Stokes — a route Buckmaster says "almost nobody" outside his own team was pursuing, and "not the direction one arrives at in a few days by giving a model the problem statement." Over the course of the call, it emerged that an entire OpenAI team — not just a model given a bare problem statement — had been working on it, and that the first prompt had reportedly been sent only in the days after information about Buckmaster and Alpöge's progress had reached OpenAI. Buckmaster says he asked directly whether the model had accessed his team's Codex sessions, where they had been storing project drafts; he says he was told the model "did not look up user data," but got no answer on training.

OpenAI then offered two options, per Buckmaster: post simultaneously with credit shared, or let Buckmaster alone author a paper crediting OpenAI's internal model — while, he says, Bubeck twice pushed to drop Alpöge from authorship, citing his employment at Anthropic. When Buckmaster said he'd go public if OpenAI proceeded that way, he says the reply was, "Why would you ruin your career?" — followed by, "If you don't want me to be nice, then I don't have to be nice." Buckmaster declined both proposals and published his account alongside the mathematics.

OpenAI's response

Bubeck publicly called Buckmaster's allegations "false and inflammatory." OpenAI researchers Sholto Douglas and Will Depue separately stated the company never accessed or trained on the pair's private Codex sessions, calling that scenario essentially impossible under OpenAI's data controls — OpenAI has said it "did not see any of their work through any means until they released it publicly." Douglas also publicly acknowledged discomfort with how the situation was handled. As of publication, neither Alpöge nor Anthropic had issued a separate public statement.

What's still unverified

Both sides' proofs are brand new and unreviewed. OpenAI's claimed result also covers a modified, forced version of Navier-Stokes — not the exact, unforced formulation the Clay Institute's official problem statement asks for — and Buckmaster says he has not personally seen OpenAI's proof. Even a genuine, verified resolution of the original problem would need to survive roughly two years of publication and community acceptance before the Clay Institute would consider awarding its $1 million prize. Buckmaster was explicit on this point: "I am not accusing anyone of anything. I am stating what I was told, when, and what was proposed to me."

Why this matters beyond the drama

Strip away the credit fight, and there's a real story underneath: a mathematician and an AI model reportedly closed decades-old gaps in fluid dynamics in about a month of work — what Buckmaster himself called "a Deep Blue-Kasparov moment" for mathematics. That's a genuinely significant signal about how far AI-assisted formal proof work has come, echoing Claude's recent formalization of Fermat's Last Theorem. But it also previews a messier future: as frontier labs race each other using rumors and leaked signals about rivals' unpublished work, questions about data access, authorship, and who a breakthrough actually belongs to are going to keep getting harder to untangle — not easier.

Frequently asked questions

Sources: Tristan Buckmaster's public statement (PDF), TechCrunch, OpenAI's official X post, Scientific American (all September 2026).

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