A Dual Demonstration of Navier-Stokes Global Regularity via Logical-Fractal Resonance

The global regularity problem for the three-dimensional Navier--Stokes equations remains one of the most significant open problems in the theory of partial differential equations and fluid mechanics. This work introduces a novel framework of \emph{logical--fractal resonance}, uniting classical energy-based Sobolev estimates and fractal self-similarity techniques to establish global-in-time bounds and explicitly construct stabilizing scale decompositions. Our approach yields a dual demonstration of regularity: an analytical derivation of uniform $H^s$ norm bounds and a constructive fractal mechanism preventing singularity formation.