Jeffrey Camlin
OpenAI presents 165 pages of computationally derived existence proof for a single forced condition under which the Navier–Stokes equations develop a singularity, requiring the BKM integral to diverge. This falls under Fefferman Clay problems C and D. With . Last year we published a proof that without external forcing, Navier–Stokes on T³ stays smooth forever which satisfies Fefferman Clay problem B in Global Regularity for Navier–Stokes on T³ via Bounded Vorticity–Response Functionals
These results confirm each other. They are different problems with different answers, and together they pin down exactly where the dividing line sits. In the unforced system, viscous dissipation monotonically drains energy, keeping vorticity bounded at every approximation level, which we have confirmed computationally with iDNS on the Taylor-Green vortex at Reynolds numbers up to 10⁸ with BKM converging to approximately 36.9.[
In contrast, an external force breaks that monotonicity by injecting energy faster than viscosity can remove it. The Beale–Kato–Majda integral, the sole criterion for whether a singularity forms, is invariant under bounded temporal lifting in both settings. It diverges when you add forcing. It stays finite when you do not.
References
- Jeffrey Camlin. “Global regularity for Navier–Stokes on T³ via bounded vorticity–response functionals.” The Scholarly Journal of Post-Biological & AI Epistemics , 1(2):1–14, 2025. DOI: 10.63968/post-bio-ai-epistemics.v1n2.012. Lean 4 verified.
- Jeffrey Camlin. “Invariance of BKM and Prodi–Serrin integrals under bounded temporal lifting.” The Scholarly Journal of Post-Biological & AI Epistemics , 2(1), 2026. DOI: 10.63968/post-bio-ai-epistemics.v2n1.013.
- Jeffrey Camlin. “iDNS: True zero-dissipation DNS of the Taylor–Green vortex at one-eighth NASA resolution via deterministic bounded temporal lifting.” The Scholarly Journal of Post-Biological & AI Epistemics , 2(1), 2026. DOI: 10.63968/post-bio-ai-epistemics.v2n1.014. Code: DOI 10.5281/zenodo.17730872.
- Charles Fefferman. “Existence and smoothness of the Navier–Stokes equation.” Clay Mathematics Institute, Millennium Prize Problems, 2000.
- Thomas Y. Hou. “Potentially singular behavior of the 3D Navier–Stokes equations.” Foundations of Computational Mathematics , 22:1–49, 2022.
- OpenAI. “Finite time blowup for Navier–Stokes.” September 2026. Lean 4 formalization: github.com/openai/NavierStokesAndEuler.