Vanishing of the Local Monotonicity Limit and Global Regularity for the 3D Incompressible Navier–Stokes Equations

This paper completes a scale-invariant monotonicity program for thethree-dimensional incompressible Navier–Stokes equations. Building on a companion work that reduces the regularity problem to the behaviorof a localized monotonicity functional, we prove that the small-scale limit of thisfunctional must vanish at every spacetime point. The argument proceeds by contradiction via a scale-invariant blow-up analysis.Assuming a positive monotonicity limit yields a nontrivial ancient suitable weaksolution with constant monotonicity across all scales. We show that this forcesthe complete vanishing of the associated dissipation square term, leading to arigidity identity that implies spatial and temporal constancy of the solution. This contradiction excludes finite-time singularities and establishes globalsmoothness for the three-dimensional incompressible Navier–Stokes equations. This manuscript is intended as a companion and closure paper to“A Local Monotonicity Reduction of the 3D Navier–Stokes Regularity Problem”.