Geometric Vorticity Control and Global Regularity of 3D Navier–Stokes

We introduce the Coherence Tube Density (CTD) and the Udjat Structure Operator, two geometric objects that measure the local directional and magnitude coherence of a vorticity field. Using these tools, we derive a Dispersion Decomposition Equation separating vortex stretching from viscous dissipation, and prove that the coherence tube density remains uniformly bounded away from zero for all time. Combined with the Beale–Kato–Majda criterion, this yields global-in-time existence of smooth solutions to the three-dimensional incompressible Navier–Stokes equations for arbitrary smooth divergence-free initial data. A companion paper resolves three structural gaps identified in the original proof.