OpenAI published a research note on September 8, 2026, reporting that an internal AI system produced an analytical proof and a Lean formalization resolving the Navier–Stokes existence and smoothness question. The company says the proof shows that an initially smooth three‑dimensional incompressible fluid subject to a smooth force can develop a singularity in finite time while the total energy remains finite.
What the proof asserts
According to OpenAI, the solution establishes statement “C” (and also “D”) in the official Millennium Prize formulation. The reported construction is a vortex that spirals inward and stretches axially: a central region shrinks and accelerates while the fluid’s energy stays finite. The proof, OpenAI states, demonstrates the breakdown occurs through the fluid’s own motion rather than by introducing an infinite external force.
How the result was produced
OpenAI describes using a new internal model it characterizes as substantially more capable than GPT‑6 Astra. Training of that model began on August 28, and a coordinated evaluation effort launched on September 1. The Navier–Stokes group involved on the order of 10,000 concurrent agents within a multiagent system; OpenAI reports those agents produced about 2.7 million messages and roughly 130 billion output tokens for the Navier–Stokes work.
Earlier in the effort, nearly 100 agents worked for approximately 50 hours to produce a disproof for an unforced Euler regularity question. The Navier–Stokes resolution reportedly emerged about 88 hours after the first agents were launched on September 5, with a subsequent Lean formalization and verification performed over an additional 17 hours using GPT‑6 Astra. Across all attempted problems, OpenAI states the agents exchanged 4.9 million messages and used about 300 billion output tokens.
OpenAI also notes concurrent work by Levent Alpöge (Anthropic) and Tristan Buckmaster (NYU); the company says it recognized their priority on a forced Euler result and offered shared visibility into prompts and the proof. OpenAI stated it does not intend to claim the Millennium Prize for this result and emphasized ongoing work to study and govern the capabilities of the model used.
Original source: OpenAI News