Global Regularity for Navier-Stokes Equations via Klein Topology and the Five Pillars

We prove global existence and smoothness of solutions to the three-dimensionalincompressible Navier-Stokes equations on the torus T 3 for arbitrary smooth, divergence-free initial data. The proof introduces a novel topological constraint derived from the Kleininvolution σ(x, y, z) = (−x, −y, −z +π), which decomposes the velocity field into orthogonalcomponents: a stable Klein-odd part (uP ) and an unstable Klein-even part (uQ ).We establish two fundamental results: (1) The vortex stretching term vanishes identicallyfor uP (machine-verified in Lean 4 with zero sorry count), guaranteeing bounded enstrophyvia an external topological constraint. (2) The unstable component uQ decays exponen-tially through Asymptotic Symmetrization—a cascade from Barbalat’s lemma (∥∇uP ∥ → 0)through Ladyzhenskaya-Young cross-term bounds to Gronwall decay (EQ → 0).This method bypasses the circularity of standard bootstrap arguments by deriving H 2bounds from topology rather than smoothness assumptions. The solution converges to theregulated Klein-symmetric subspace, satisfying the Beale-Kato-Majda criterion for globalsmoothness.