Vortex-Line Curvature as the Geometric Obstruction to Navier-Stokes Regularity: A Conditional Regularity Criterion

This paper develops a geometric framework for the three-dimensional incompressible Navier-Stokes equations on the periodic torus T³. The central objects are the vortex stretching scalar Φ = ξ·Sξ, its nodal surface N = Φ = 0, and the vortex-line curvature κ = | (ξ·∇) ξ|. The main result is a conditional regularity theorem: if the Filamentation Condition holds — that is, if ∫₀ᵀ‖κ (t) ‖²L∞ dt is finite for all finite T — then global regularity follows via a legitimate Grönwall inequality with coefficient in L¹ from the energy identity alone, and the Prodi-Serrin criterion. The paper proves that a pointwise maximum principle for κ fails in the transverse-strain regime, derives the exact κ evolution equation from NS, and identifies the η² vs η³ superlevel-set dichotomy as the precise geometric threshold for Grönwall closure. The Filamentation Condition is shown to be equivalent in analytical difficulty to the original regularity problem.