Dyadic Geometric and Lyapunov Framework for Global Regularity of 3D Incompressible Navier–Stokes Equations

This work introduces a novel structural framework, called the Udjat framework, for studying the global regularity of the three-dimensional incompressible Navier-Stokes equations. By combining dyadic frequency decomposition, geometric analysis of vorticity, and higher-order differences, a closed dynamical system is constructed for key quantities controlling vorticity amplitude and directional gradients. A Lyapunov functional captures the dissipative effects of viscosity, allowing control over nonlinear growth at all relevant scales. The framework provides uniform bounds for vorticity and its derivatives, offering a potential approach toward globally smooth solutions for arbitrary bounded and regular initial data. This preprint presents the theoretical formulation and supporting estimates for further verification and development.