OpenAI Researchers Report an AI-Found Singularity in Navier–Stokes Equations

Summary

OpenAI researchers announced that roughly 10,000 autonomous AI agents, running on an internal model unavailable to the public, had found a singularity in the three-dimensional Navier–Stokes equations. The equations describe how viscous fluids such as air and water flow, and the corresponding Millennium Prize problem asks whether their solutions can remain smooth or develop an infinitesimally concentrated blowup in unbounded three-dimensional space. OpenAI says the agents obtained a result after 88 hours, then used another model for 17 more hours to formalize it in Lean; the agents exchanged nearly 5 million messages, at an estimated cost of several million dollars. Lean verification establishes that the encoded statement follows formally, but human mathematicians must still confirm that it matches the intended Navier–Stokes claim and assess the proof’s mathematical significance. The work builds on an unconventional analytic strategy developed by Diego Córdoba and Luis Martínez-Zoroa, whose earlier constructions used infinite cascades of smooth solutions but struggled to preserve a smooth forcing function. A separate team led by Tristan Buckmaster and Levent Alpöge announced related AI-assisted results shortly beforehand, including a Lean-verified result for the Euler equations and an unverified claim concerning an easier Navier–Stokes variant. OpenAI claims priority for the Navier–Stokes result, while Buckmaster’s account suggests the timelines and possible access to earlier work remain disputed. The reported proof could become the most important AI-assisted mathematical result so far if it survives scrutiny, but the article stresses that formal verification does not by itself settle the priority dispute or guarantee acceptance by mathematicians. The singularity has no immediate physical consequence because real fluids are molecular rather than perfectly continuous, yet it would clarify a fundamental and counterintuitive property of idealized fluid dynamics.