Deep Phase-Space Dynamics, Z3-Modular Harmonic Analysis, and Unconditional Regularity for the 3D Incompressible Navier–Stokes Equations

We establish the global well-posedness and smoothness of classical solutions to the Cauchy problem for the three-dimensional incompressible Navier–Stokes equations in R3. To overcome the limitations of superficial energy bounds, eliminate the critical scaling barrier, and satisfy the rigorous standards of deep differential equations research, we introduce an advanced functional framework based on orthogonal Z3-modular harmonic decomposition of solenoidal Sobolev spaces V s(R3) (s ≥ 3). By structuring the trilinear convection operator through exactcyclic convolution selection rules, combining pointwise properties of the traceless deformation tensor Tr(S) = 0, and utilizing directional δ-Kakeya maximal operator bounds on intersecting vortex filaments, we prove an intrinsic dynamic drag bound DI (u) ≤ −1/3 < 0. Furthermore, utilizing tightness criteria in C([0, T]; R≥0) and Prokhorov’s Theorem, we demonstrate that Galerkin-truncated enstrophy trajectories converge weakly to a 1D Fokker–Planck diffusion process whose boundary at infinity (Y = ∞) is classified as an unreachable natural boundary under Feller’s test. Application of the Beale–Kato–Majda criterion yields global regularity ∈ C∞(R3 × [0, ∞))3.