Definition
The Navier-Stokes existence and smoothness problem is one of the Clay Mathematics Institute’s seven Millennium Prize Problems, concerning whether solutions to the Navier-Stokes equations for fluid motion remain smooth or develop singularities.
Key Points
-
2026-10-09: Cambridge team documents NL vs Lean mismatch in Lemma 8.6 (m+4 vs m+5 derivatives) — challenges “verified” marketing of ai-autoformalization without line-by-line audit (2026-10-09-navier-stokes-lean-mismatch-arxiv, 2026-10-08-openai-navier-stokes-lean-proof-mismatch)
-
2026-09-08: openai claims internal system resolved the problem in ~88 hours with Lean formalization — establishing statements C and D in the official formulation (2026-09-21-openai-navier-stokes-openai-blog)
OpenAI's Navier-Stokes solution remains unverified by independent mathematicians. Lean compilation does not prove NL↔Lean fidelity; Cambridge work does not by itself refute the blow-up claim.
- Controversy over attribution, pace of AI-driven discoveries, and academic verification standards
- Triggered formation of IAS-hosted math advisory group (Sep 21, 2026)
- 25 Fields Medalists signed open letter earlier in September 2026 criticizing AI lab competition on famous problems
Related
- ai-for-science
- ai-for-mathematics
- automated-theorem-proving
- ai-autoformalization
- lean-formalization
- openai
- institute-for-advanced-study
- frontier-ai-governance