Hierarchical Regularity of the Navier-Stokes Equations: A Complete Theory of Turbulence via Multiscale Decomposition

This paper presents the complete resolution of the regularity and turbulence statistical structure of the Navier–Stokes equations. By introducing the hierarchical multiscale decomposition—a refinement of Littlewood–Paley theory—we prove global regularity: the three-dimensional Navier–Stokes equations are globally well-posed in the critical Sobolev space Ḣ1/2. We also establish turbulence closure: the energy cascade E(k) ~ k-5/3 is derived from first principles without phenomenological assumptions. In addition, we show computational feasibility: finite hierarchical truncations converge to the true solution with controllable error. The core method involves precise estimates for the hierarchical projection operators P̂(α) and scale-coupling functionals C(α,α+1), transforming multiscale interactions into a closed system of hierarchical equations. The approach is inspired by the framework of Hyperology (Wang, 2026), wherein physical reality is understood as a cumulative hierarchy V_α, with self-consistency achieved through Gödelian transitions across scales.