A Conditional Regularity Theorem for the 3D Navier–Stokes Equations via Cross‑Product Trapping

This paper formulates a geometric hypothesis on the Fourier‑space structure of the Navier–Stokes nonlinearity and proves a conditional regularity theorem. As stated in the abstract, “we formulate a geometric hypothesis on the Fourier‑space structure of the Navier‑Stokes nonlinearity: the existence of a cross‑closed subspace that traps the solution for all times,” and “if the initial data belong to such a subspace, the 3D Navier‑Stokes equations are globally well‑posed.” Analytical and numerical evidence is presented suggesting that such cross‑closed subspaces exist and are generic. The work builds on twelve previous papers on geometric frustration and represents a concrete step toward resolving the Clay Millennium Problem.