Abstract We establish the global smooth regularity of the three-dimensional incompressible Navier-Stokes equations on R³ for smooth, rapidly decaying, divergence-free initial data with finite energy. Our approach proceeds by contradiction, assuming a hypothetical finite-time singularity. Applying the Caffarelli-Kohn-Nirenberg criterion, we identify the emergence of non-small, scale-invariant cylinders which, via a guarded one-mouth extraction, generate a non-trivial ancient vorticity profile. This analysis is formalized within the Topological-Renormalization and Endpoint Persistence (TREP) framework, which allows for a rigorous ledger-based bookkeeping of critical channels. The core of the proof lies in the Projective Null-Form shell estimate for vortex stretching. We demonstrate that the strain kernel's geometric properties induce a second-order vanishing of the inner symbol, resulting in an angular gain of (/Q) ². The weighted absorption is closed through scalar-amplitude diffusion, avoiding the limitations of weighted Bernstein inequalities. We establish that the vorticity amplitude dissipation | |² = | |² + ² | |² effectively bounds the angular defect. Under the Navier-Stokes-TREP verification package, we show that the ancient endpoint must have zero projective defect, reducing the system to a smooth two-dimensional-three-component profile. Finally, strong critical convergence transfers the smallness condition back to the prelimit active branch, inducing a contradiction. This result excludes all potential finite-time singularities, including those escaping to spatial infinity, thereby proving global smooth regularity for the Navier-Stokes system.
Juan José Chelía (Sat,) studied this question.