AI-Generated Proof of Navier–Stokes Blowup? The Authorship Controversy Behind a 100-Page Paper

A Secret Proof That Put Three Parties in the Spotlight

On September 8, 2025, NYU mathematician Tristan Buckmaster went public with three AI-assisted research results on fluid equations. In a four-page statement, he also revealed a claim that shook the academic world: OpenAI researchers told him that an internal model had gone even further — generating a finite-time blowup proof for the Navier–Stokes equations with smooth forcing.

If this roughly 100-page proof is correct and satisfies all the conditions of the Clay Mathematics Institute's prize problem, it could genuinely resolve a $1 million Millennium Prize Problem. Observers quickly connected the mysterious model to OpenAI's next-generation system, but Buckmaster's statement neither confirmed the model's name nor had he personally reviewed the proof at the time of publication.

What makes this story so compelling is that a research path toward a major mathematical breakthrough already had published results — yet its next step became entangled in fierce disputes over data use, contributions, and authorship.

The Navier–Stokes Millennium Problem: From Smoothness to Blowup

What the Navier–Stokes Equations Describe

The Navier–Stokes equations describe how water and air move — the swirl of cream in coffee, the airflow around a wing. But there are two distinct questions here: writing down a set of equations that describe motion, and proving that those equations behave well under all required conditions.

The question mathematicians actually care about is: starting from a sufficiently smooth initial state, can a flow remain smooth indefinitely? Or, under certain conditions, will some quantity describing the flow become unbounded after a finite time — causing the smooth solution to break down? This phenomenon is called "blowup," and it represents the emergence of a mathematical singularity.

In 2000, the Clay Mathematics Institute listed the existence and smoothness problem for the Navier–Stokes equations as one of the seven Millennium Prize Problems. Resolving it via blowup requires constructing a rigorous example that satisfies all specified conditions and demonstrates that smoothness genuinely fails in finite time.

Smooth Forcing: The Key Condition for a Blowup Proof

A condition that comes up repeatedly here is "smooth forcing." Forcing can be understood as an external influence acting continuously on the fluid during its motion, but in a proof it must be a function satisfying precise mathematical requirements. If the input itself is rough and the output becomes irregular, the result loses much of its explanatory power. Keeping the forcing smooth allows us to ask a sharper question: given such well-behaved input, can the fluid still spontaneously develop a singularity?

Notably, the smooth-forcing approach is already part of the official prize statement (corresponding to options C and D). So if OpenAI's claimed result satisfies all requirements, it could resolve the formal problem — the real threshold is whether the complete construction accounts for every specified condition.

Buckmaster's Team: Three Published Results

Buckmaster and collaborator Levent Alpöge have published three results, covering: the incompressible porous medium equation, the Boussinesq equations, and the three-dimensional incompressible Euler equations. These correspond, respectively, to fluid flow through porous media, the interaction of buoyancy and flow, and ideal fluid motion that ignores viscosity. They announced finite-time blowup constructions for these equations under smooth forcing, and the papers along with formal verification code have been made public.

Viscosity: The Gap Between Euler and Navier–Stokes

Between the Euler equations and the Navier–Stokes equations lies one critical difference: viscosity. Viscosity dissipates energy and smooths out fine-scale flow structures — the smaller and more rapidly varying a perturbation, the more susceptible it tends to be to dissipation. Yet the core of this research approach requires precisely those fine-scale structures to keep growing.

This makes the difficulty very concrete: just as you work to amplify small-scale perturbations, the viscosity term in the equations is working to suppress them. Only by carefully accounting for this competition — proving that the growth mechanism can still sustain itself — is there any hope of advancing toward Navier–Stokes. Buckmaster mentioned that they believe they obtained a blowup result for the low-dissipation Navier–Stokes equations, but at the time of the announcement, verification was not complete and no full manuscript existed, so it was not included in the release.

The High-Frequency Perturbation Relay: What Role Does AI Play?

The fundamental idea behind this research line comes from Diego Córdoba and Luis Martínez-Zoroa; Buckmaster explicitly credits them with the pioneering contribution. His and Alpöge's work extended this approach — using large language models — to settings involving smooth forcing and the Euler equations.

Fields Medal winner Terence Tao, after a phone call with Buckmaster, explained the core mechanism: researchers add very small, localized, high-frequency perturbations on top of an existing flow. "High-frequency" can be thought of as extremely fine-grained spatial texture. The larger-scale flow amplifies these tiny structures, while the back-reaction of the small structures on the large-scale flow is kept weak. Once the new structure grows sufficiently, it takes over the local dynamics at the right moment and creates conditions for the next, even finer round of perturbations. This is like a relay race — if any leg goes wrong, the entire construction can collapse.

Tao noted that for the Boussinesq model, the core change reduces to a fairly simple long-short modification. But to embed this mechanism rigorously into a complete proof, technical details like spatial cutoffs still stretched the streamlined paper to 76 pages. This illustrates exactly what AI contributes: an idea that can be explained simply may require a large amount of carefully interlocking estimates and derivations. After the model helps advance this work, mathematicians must still verify how each local computation supports the overall conclusion.

The AI Toolchain and Research Timeline

By Buckmaster's account, the collaboration moved slowly for most of the past year. On August 15, they made key breakthroughs on the Boussinesq and Euler equations; on August 22, the relevant arguments passed Lean formal verification. They then spent considerable time understanding and rewriting the model-generated proofs. The project used Claude, Codex (particularly the GPT-5 series), and later ASTRA. Buckmaster specifically noted that ASTRA was used only for organizing the manuscript and reviewing arguments.

This was a personal collaboration between two researchers — Alpöge works at Anthropic, but neither employer was formally involved, and the tool costs were covered by Buckmaster's own research funds.

A Phone Call Sparks Authorship Disputes and Data Questions

The process of finalizing the papers was interrupted by a series of communications in early September. According to Buckmaster's statement, on September 3 rumors began circulating that "Anthropic had solved a major math problem." Learning that word of his research had reached OpenAI, he wrote proactively to clarify that this was a personal collaboration and to correct public confusion about what exactly had been solved.

Communication was eventually moved up to September 6. That afternoon, he held two phone calls with two OpenAI researchers, including Sébastien Bubeck. It was during these calls that he first heard about the roughly 100-page proof: the other party claimed an internal model had produced a Navier–Stokes blowup with smooth forcing, corresponding to options C and D of the formal problem.

Unanswered Questions About Data Use

Buckmaster pressed for details about how the research was initiated. He says he gradually learned that an entire team was involved, that multiple directions had been tried, that the model had first been directed at easier problems like the Euler equations, that substantial compute had been devoted to the effort, and that the initial prompts were sent within days of news of his own progress reaching OpenAI.

What genuinely troubled him was the striking overlap in research directions. He and Alpöge had been using project drafts within Codex, so he asked whether the model had accessed this material or been trained on it. According to him, the other party denied that the model retrieved user data, but gave no answer on the training question. Buckmaster wrote plainly in his statement: he does not know whether his data was used, does not know what the internal model specifically did, and has not seen the proof.

Conflicting Proposals on Publication and Authorship

The dispute then turned to publication arrangements. Buckmaster says he was offered two options: either he publish the Euler results first and OpenAI publish the Navier–Stokes result the following day, or he write a follow-up paper alone with a note that the result was produced by an OpenAI internal model. He also says that Bubeck twice asked that Alpöge be excluded as an author from the proposed follow-up paper, citing Alpöge's employment at Anthropic. Buckmaster declined both arrangements and said he would go public with the account. Bubeck subsequently denied the claims made against him, said he follows academic norms, and promised a fuller explanation.

It must be emphasized that as of September 8, the key details of the phone calls come primarily from Buckmaster's one-sided account, and neither party has published records sufficient to fully reconstruct the conversations. OpenAI's 100-page proof has also not been released for peer review.

AI Pushing Mathematical Frontiers: Two Questions Worth Sitting With

This episode raises two questions worth considering together: How much proof work has AI actually helped us advance? And how much of that work has been converted into knowledge that humans can understand and build upon?

From selecting an approach and framing the problem, to model-assisted derivation, to formal verification and rewriting — the published results represent a complete collaborative chain. Evaluating AI's contribution requires seeing the whole chain clearly: prior ideas, the human researchers' advances, and the distinct work done by different models all deserve their place in the record. This also explains why Buckmaster cares so much about the readability of the papers: the work of organizing and explaining is precisely what transforms a breakthrough into a starting point for future research.

Tao's perspective is instructive. He believes this approach seems in principle to have room to extend toward Navier–Stokes and perhaps even toward removing the forcing altogether, but that doing so would encounter an enormous number of technical difficulties. He is more interested in digesting the proof methods and distilling new insights from them — a long proof can tell us that a result is true, but understanding it is what might let the next researcher know where to go.

As for the unpublished 100-page proof: whether it ultimately clears the singularity hurdle and satisfies the prize conditions is a question that only the complete argument can answer. If it holds up, AI's role in expanding the mathematical frontier will have a weighty concrete example — and part of the value of that example will be making clear exactly how it came to be. What we truly need are proofs that can be checked, methods that can be understood, and a record of contributions that matches what was actually done.