Intrinsic Operational Gradients and the Global Regularity of 3D Navier–Stokes Equations

We present a novel geometric framework for establishing global regularity of three-dimensional incompressible Navier–Stokes equations by interpreting angular entropy in Fourier space as a threading aggregate—a coherent operational pattern studied in operational geometry. We prove that incompressibility constrains energy to thread through two-dimensional subspaces perpendicular to each wavevector, creating a threading deficit that decays exponentially under viscous dissipation. By showing that angular entropy growth requires activating non-coplanar triads (which maximally oppose both the incompressibility projection and the operational gradient from the Intrinsic Operational Gradient Theorem), we establish that angular complexity cannot grow unboundedly in finite time. This yields a conditional regularity result: if angular entropy remains bounded, smooth solutions extend globally. We argue this bound follows from the threading coherence structure, closing a critical gap in prior angular entropy approaches.