OpenAI Claims AI Agents Solved the Navier-Stokes Millennium Problem in 88 Hours

On September 8, 2026, OpenAI announced that a system comprising approximately 10,000 AI Agents solved a version of the Navier–Stokes Millennium Problem in 88 hours. It constructed a special three-dimensional fluid: initially at complete rest, it developed localized velocities tending toward infinity in finite time after being subjected to smooth external forces, while maintaining finite total kinetic energy. This conclusion corresponds to the two “singularity existence” scenarios, C and D, outlined in the official problem statement by the Clay Mathematics Institute, and has been accompanied by a 166-page analytical proof and a Lean-formalized version. More astonishingly, the core reasoning was performed by an internal model still under training, which OpenAI described as “significantly stronger than GPT-6 Astra”; the entire Navier–Stokes project consumed about 130 billion output tokens and generated 2.7 million Agent messages, with computational costs reaching millions of dollars. However, the proof still requires peer-by-peer verification by the mathematical community, and Lean cannot automatically guarantee that the formalized proposition is entirely equivalent to the original problem. Meanwhile, mathematician Tristan Buckmaster of New York University questioned whether OpenAI indirectly benefited from unpublished research he and Levent Alpöge had deposited in Codex, accusing OpenAI of attempting to influence authorship; OpenAI denied accessing or using their specific data but acknowledged it could not fully rule out that anonymized use of such data may have helped improve the model.

Article author, source: OpenAI

The Navier-Stokes equations describe how fluids such as water and air move. Aircraft design, weather forecasting, blood flow, and oceanography all rely on these equations.

The equation treats fluids as continuous media rather than tracking individual molecules. A long-standing unresolved question is: if a three-dimensional fluid starts out very smooth, will it remain smooth forever, or might it develop a “singularity” within a finite time—where, in an increasingly small region, the velocity calculated by the equation keeps rising and eventually tends toward infinity?

Viscosity typically smooths out drastic changes. Honey flows more slowly than water precisely because it has higher viscosity. Therefore, proving that a viscous fluid can still develop singularities is far more difficult than creating runaway motion in an ideal, non-viscous environment.

Fluids in the real world cannot achieve infinite velocity. This conclusion primarily indicates that the continuous fluid model fails under certain conditions, not that aircraft, oceans, or human blood will suddenly exhibit infinite flow speeds.

The OpenAI system has not proven that singularities occur in all Navier-Stokes fluids, nor has it found a universal solution that can be directly applied to weather forecasting.

It constructs a counterexample: the fluid is initially at rest, and a smooth, finite-force is applied externally in both space and time. As the motion evolves, a rotational vortex continuously contracts toward the center while stretching axially, taking on a shape increasingly resembling a long strand of spaghetti.

The volume occupied by the vortex becomes smaller and smaller, while the internal velocity increases rapidly. At the designated singularity time, the maximum velocity approaches infinity; however, since the high-speed region simultaneously shrinks sharply, the total kinetic energy of the entire system remains finite.

The real challenge lies in not being able to artificially create a singularity by simply introducing an infinitely large external force. The model must cause acceleration, pressure, viscosity, and the fluid’s own momentum transport to become extremely large, yet cancel each other out in an extremely precise manner, so that the required external force remains smooth.

An online质疑 has emerged: OpenAI's proof used external forces, and therefore did not solve the "true" Millennium Problem.

According to the official problem statement published by the Clay Mathematics Institute,this statement is not accurate.

The official problem presents four optional routes. A and B require proving that all three-dimensional fluid flows satisfying the conditions remain smooth forever in the absence of external forces; C and D allow the use of external forces satisfying strict smoothness and decay conditions, requiring only the construction of an example in which no global smooth solution exists.

The OpenAI paper claims to simultaneously achieve C and D: one for infinite three-dimensional space and another for periodic three-dimensional space. The forces in the paper are not only smooth but also have compact support, meaning they act only within a finite range of space and time.

Therefore, "using external force" itself does not disqualify the proof. What truly needs to be reviewed is: whether there are errors in the argument on page 166, whether the constructed force indeed satisfies all conditions, and whether the formalized proposition in Lean is exactly consistent with the paper and the official Clay problem statement.

OpenAI began training a new internal model on August 28. The company has not disclosed the name, parameter count, or training methodology, but stated that its mathematical capabilities have seen a leap forward and are “significantly stronger than GPT-6 Astra.”

On September 1, OpenAI heard rumors of potential breakthroughs on two of the Millennium Problems, so it decided to have its new model attempt all unsolved Millennium Problems, along with some relatively easier related questions.

Agents can access a cached version of the internet, execute code, and are divided into multiple groups that communicate internally. Different groups separately investigate proofs for the always-smooth routes A and B, as well as the construction of singularities via routes C and D.

The system first made progress on a related problem. Nearly 100 agents ran for approximately 50 hours, providing a proof of the possible formation of singularities in the three-dimensional Euler equations without external forces. The Euler equations can be understood as the Navier-Stokes equations without viscosity.

Researchers then concluded that the Navier-Stokes equations held the most promise, so they reallocated resources from other tasks, provided the results from the Euler equations to the new Agent, and swapped in newly trained model versions during operation.

Multiple teams continued exploring different approaches. OpenAI again used Codex to consolidate the valid intermediate conclusions generated by each team and cross-share this information with the other teams. Ultimately, the relevant Agent obtained the Navier–Stokes result on September 5, approximately 88 hours after the first Agents were launched.

Then, GPT-6 Astra spent another 17 hours converting and verifying the argument into a Lean formal proof.

The entire Navier-Stokes project generated approximately 2.7 million agent messages and consumed about 130 billion output tokens; when including all other mathematical problems attempted simultaneously, the total reached 4.9 million messages and approximately 300 billion output tokens. OpenAI research leads Mark Chen and Sébastien Bubeck both stated that the computational cost reached millions of dollars, but did not disclose the actual number of GPUs or internal cost details.

Lean is a formal proof assistant. As long as mathematical propositions and the assumptions used are accurately written, Lean can verify whether each step logically follows from the previous one, eliminating the kinds of logical gaps that are easily overlooked in manual proofs.

OpenAI's publicly released codebase provides results for both infinite and periodic spaces, along with instructions for independently building and verifying the proofs. This allows external experts to actually run the code, rather than relying solely on the company's press releases.

But Lean cannot decide for humans whether “the problem you entered is the original problem.”

Researchers still need to verify that the formalized version does not weaken the original proposition, does not introduce unwarranted assumptions, that the objects analyzed in the paper align with the Lean definitions, and that the formalized code does not treat the most critical mathematical conclusions as axioms. Quanta also notes that an indispensable step in human review is confirming that the statement proved in Lean is logically equivalent to the problem mathematicians actually aimed to solve.

Clay still lists the Navier-Stokes equations as an unsolved problem. According to its rules, a candidate solution must first be published in a qualified publication, undergo at least two years of rigorous scrutiny by the global mathematical community, and gain widespread acceptance before entering the formal verification process.

OpenAI stated that it will not apply for the $1 million prize. Regardless of whether the prize is claimed, the mathematical conclusion still requires the same peer review process.

This result was not invented by AI from scratch.

Spanish mathematicians Diego Córdoba and Luis Martínez-Zoroa have long studied constructing fluid singularities through layered vortices. Their general approach involves building an infinite stack of flows, each without singularities, and using strain at larger scales to progressively amplify vortices at smaller scales, ultimately forming an “infinite cascade.”

They had previously been able to create singularities in certain related equations, but the combined external forces were not smooth enough to satisfy the Millennium Prize conditions.

OpenAI's paper cites this line of research, incorporating oscillatory pulses, momentum transport, and stepwise error correction to ensure the final force remains smooth. Mathematician Charles Fefferman, who wrote the official title for Clay, told Quanta that Córdoba and Martínez-Zoroa are key figures in the history of this research.

Therefore, even if OpenAI’s achievement is fully validated, it cannot be simply summarized as “AI thought independently for 88 hours and solved a problem humans never came close to.” A more accurate description is that the new model completed a long-overdue critical step, building upon existing human research pathways, team guidance, massive parallel search, and formalized tools.

The dispute involves New York University mathematician Tristan Buckmaster and Anthropic researcher Levent Alpöge, who stated that this was a personal collaboration, not an official Anthropic project.

Building on the work of Córdoba and Martínez-Zoroa, they used Claude, Codex, GPT‑5.6 Sol, and Astra to assist with their research. On August 15, they obtained results including the Euler equations with smooth external forces, and completed Lean verification on August 22; originally, they planned to spend several weeks organizing the model-generated proofs into a paper more accessible to humans.

Buckmaster said that after the news was leaked prematurely, they were forced to publicly announce their research on September 7. He explicitly acknowledged that at the time, the two had not completed the full Navier-Stokes Millennium Problem, but had only achieved results related to the Euler equations and other associated systems.

But he believes that "constructing singularities by smoothing external forces" is a fairly uncommon approach, and OpenAI only invested substantial resources into researching the same direction after hearing rumors of their breakthrough. Since the two had fed drafts and research processes into Codex over the past year, he asked whether OpenAI’s new model had been exposed to or trained on this content.

OpenAI stated that neither researchers nor agents viewed each other's proofs prior to public release, nor did they access any specific user data to solve this problem. The company also acknowledged that, while unlikely, it cannot rule out the possibility that anonymized data generated by the two individuals using the product may have contributed to model improvements.

These two statements are not entirely contradictory: not directly reading a private draft during the problem-solving process does not mean the draft never indirectly influenced the model through earlier data processing. However, there is currently no public evidence proving that such an influence actually occurred, nor can the specific training source be inferred from the model’s final generated proof.

The dispute later expanded to issues of authorship and communication. Buckmaster alleged that Sébastien Bubeck had proposed that he serve as the lead author of OpenAI’s rewritten proof, but declined due to Alpöge’s affiliation with Anthropic, which he did not want to include; he also claimed that Bubeck pressured him with remarks such as, “Why would you ruin your career?”

Bubeck denied requesting that Alpöge be removed from his own Euler equation paper. He explained that the discussion concerned involving Buckmaster in rewriting OpenAI’s Navier-Stokes proof, and that having an employee from a competing company co-author work from OpenAI’s internal models would create a conflict of interest. Bubeck also apologized for the phrasing regarding his career, stating that it was extremely inappropriate but not a threat.

The descriptions of the context and true intentions of both parties' dialogue remain different, and existing public information is insufficient to determine which version is complete or accurate.

Traditional mathematical achievements typically allow for a relatively clear timeline to be established through paper drafts, emails, lectures, and preprints. With the introduction of AI, research trajectories are now scattered across prompts, model weights, agent messages, training data, code repositories, and internal company experiment logs.

A mathematician might propose a key insight, which a model expands into a proof; another model might absorb similar clues from anonymized training data; the research team then searches for the complete result using tens of thousands of agents and millions of messages. The final paper can be verified by Lean, but Lean can only answer whether the proof is valid, not where the original idea came from.

This incident also illustrates that large-scale parallel reasoning is becoming a new type of research instrument. OpenAI did not rely on a single conversation to find an answer; instead, it invested millions of dollars to have multiple agents explore numerous failed paths, then continuously filter, aggregate, and reallocate resources.

Its meaning is closer to a virtual research institution powered by a language model than to an AI mathematician who suddenly has a flash of insight.

If the proof withstands scrutiny, it will be one of the most significant mathematical achievements involving AI to date, signaling that state-of-the-art models have progressed from organizing literature and formalizing existing proofs to completing original research left unfinished by humans.

But the breakthrough also leaves significant questions unresolved: when the research process relies on closed models, unverifiable training data, and computational scales beyond the reach of ordinary scholars, existing institutions have yet to provide mature answers to how results should be reproduced, unpublished work protected, and academic credit allocated.