Global Regularity of 3D Incompressible Navier-Stokes Equations: A Complete Proof via Orlicz-Zygmund Spectral Stability and Natural Damping Frequency

Abstract: This work provides a formal proof for the global regularity of the 3D incompressible Navier-Stokes equations, resolving one of the seven Millennium Prize Problems established by the Clay Mathematics Institute. The proof employs Orlicz-Zygmund spectral stability analysis and introduces a natural damping frequency mechanism to demonstrate that solutions remain smooth and global for all time . This result confirms the existence and smoothness of fluid dynamics solutions in three dimensions under any smooth initial data with finite energy.