OpenAI has announced a solution to one of the seven Millennium Prize Problems, presenting a proof regarding the existence and smoothness of solutions to the Navier-Stokes equations. The proof, developed entirely by an internal OpenAI system, demonstrates that the dynamics described by these fluid motion equations can produce singularities within a finite time frame. Alongside the announcement, the company has publicly released both the proof document and a formally verified version completed using the Lean theorem prover.
The Millennium Prize Problems, established by the Clay Mathematics Institute, represent some of the most profound questions at the frontier of mathematics. The question of whether three-dimensional smooth fluid motion can undergo "breakdown" has remained unresolved for approximately 90 years. A central objective of this research is to empower scientists in advancing scientific inquiry and technological development for the benefit of humanity. To tackle the Navier-Stokes problem, the team utilized an internal model with capabilities significantly surpassing GPT-6 Astra, underscoring the necessity of disclosing the pace of AI advancement and the capabilities achievable by next-generation models.
Understanding the Problem
Based on Newton's second law of motion (F=ma), the Navier-Stokes equations describe fluid movement. A key characteristic is that these equations treat fluid as a continuous medium rather than tracking individual molecules. These equations are applied across various fields including aircraft design, weather forecasting, and studying blood flow. A central unresolved question regarding these dynamic equations is whether the continuum approximation of fluid may break down. Specifically, for a three-dimensional incompressible fluid with constant density, can the Navier-Stokes equations produce a singularity, defined as the point where fluid velocity becomes infinite within finite time, even if the initial flow state is smooth? This singularity occurs despite the presence of viscosity, which typically has a smoothing effect on fluid flow. In physical reality, fluids cannot attain infinite velocity; such a singularity would indicate that the equations have failed to describe the fluid accurately. At that point, simulating the system would require tracking the motion of every individual particle.
The equations were derived in the 19th century by Claude-Louis Navier and George Gabriel Stokes. In 1934, Jean Leray proved the existence of solutions in a generalized sense, but the question of whether these solutions remain smooth at all times became a core unresolved issue. In 2000, the Clay Mathematics Institute designated the existence and smoothness of Navier-Stokes solutions as one of the seven Millennium Prize Problems.
Key Findings
Our system has provided an analytical proof, fully verified through Lean formalization: a fluid that is initially stationary and smooth can produce a singularity in finite time. The fluid is subjected to a smooth external force, and throughout the process from rest to singularity formation, energy remains finite. This conclusion proves Statement C (and also Statement D) from the Clay Mathematics Institute document, thereby resolving the Navier-Stokes Millennium Problem. The solution takes the form of a vortex: fluid rotates inward and stretches continuously, adopting a shape resembling spaghetti. The central region contracts persistently, velocity increases steadily, yet energy remains finite, consistent with physical laws. The challenge lies in the fact that the breakdown must arise from the fluid's own motion, not from artificially applied infinite external force. Mathematically, each term in the Navier-Stokes equations — acceleration, pressure gradient, momentum transport, and viscous terms — can attain large values, but they cancel out with precise balance. This delicate equilibrium ensures the external force remains smooth while fluid velocity approaches infinity. An illustration of local incompressible fluid motion shows that orange represents higher rotational angular velocity, while blue-green indicates slower rotation, with circulation velocity also dependent on radius. The streamline trajectories display inward spiral patterns with axial stretching.
How the Proof Was Discovered
Since August 28, training has been underway for a new internal model, which has demonstrated unprecedented performance across a curated benchmark set containing numerous mathematical problems. This model continues to undergo training, with capabilities still expanding. A comparison chart shows GPT-6 Astra and the internal model's performance on a selected set of public mathematics problems, with the horizontal axis representing test computation (logarithmic scale) and the vertical axis representing the pass rate.
On Tuesday, September 1, rumors surfaced regarding two Millennium Prize Problems being solved. Inspired by this news, combined with the leap in our internal model's capabilities, we initiated a testing phase to challenge the model with all unsolved Millennium Problems and several additional high-difficulty mathematical questions. We constructed a multi-agent collaborative system powered by our internal model. These agents can access tools, read cached internet information, and execute code. The agents were organized into groups capable of inter-group communication, with varying group sizes. The group ultimately responsible for the Navier-Stokes proof consisted of approximately 10,000 concurrent agents. Throughout the process, we maintained strict security protocols consistent with frontier model evaluations, including monitoring and isolation. For each problem, we provided different agent groups with various versions of the original question, covering all variants of the problem. For the Navier-Stokes problem, versions A and B (the direction of proving existence) and versions C and D (the direction of disproving) were assigned to separate groups.
Beyond the complete Millennium Problems, the multi-agent system was also tasked with a set of related, lower-difficulty problems. One of these involved the Navier-Stokes limit without the viscous term, namely the regularity question of the Euler equations. The agent team unexpectedly solved this particular problem. They completed a proof of blow-up for the Euler equations without external force: the fluid experiences no external forces. Nearly 100 agents worked collaboratively over approximately 50 hours to complete the disproof of Euler equation regularity. After observing the resolution of the Euler problem, we identified Navier-Stokes as the most promising objective. Consequently, we reallocated resources to focus on Navier-Stokes, shifting agents from other Millennium Problem tasks and inputting findings from the Euler results as prompts for the agents. During project progression, the internal model iterated to new versions, and we concurrently upgraded the underlying model for the agents. We encouraged diverse exploration strategies across different agent groups. Over time, using Codex, we integrated the most valuable ideas from each group, cross-pollinating the results. Subsequent prompts leveraged intermediate derivations produced by the agents themselves. The group that discovered the Navier-Stokes solution operated under this working model.
Approximately 88 hours after the initial launch of agents, on Saturday, September 5, the agent team obtained a solution to the Navier-Stokes problem. Following this, GPT-6 Astra completed the Lean formalization and verification, requiring an additional 17 hours. Across all attempted problem tasks, the agents collectively sent 4.9 million messages and consumed approximately 300 billion output tokens. For the Navier-Stokes problem specifically, agents sent 2.7 million messages and consumed about 130 billion output tokens.
Related Concurrent Work
Our project started on September 1, triggered by a rumor. Subsequently, we learned the rumor involved Anthropic researcher Leverent Alperg and New York University mathematics professor Tristan Buckmaster. After we fully completed our proof and finished Lean verification on September 6, we believed they might have also obtained a solution to the Navier-Stokes equations. We proactively reached out, suggesting simultaneous publication of results and offering recognition of their priority in a joint statement. Through communication, we discovered they had solved the Euler equations problem with external force. During our exchanges, we shared our complete set of prompts with them, and subsequently provided access to our full proof. We acknowledge their priority in the direction of Euler equations with external force, and we congratulate them on this outstanding mathematical achievement. Prior to the public release of their results, our research team and agents had no contact whatsoever with their research content. It is also important to note that no user data was accessed for the purpose of solving these problems. Although the likelihood is extremely low, we cannot entirely exclude the possibility that de-identified data from OpenAI product users may have influenced the model. However, the proofs are dramatically different; even in the case of the Euler equations, our conclusions differ (no external force versus external force).
Progress and Responsibility
We publish this result with the intention of reporting significant advancements in AI models. We have no intention of claiming the Millennium Prize award for this achievement. This milestone results from substantial work by mathematicians and AI researchers alike. This is a significant marker, yet it represents only a point-in-time snapshot in the ongoing progression of AI development.