The Navier–Stokes Existence and Smoothness Problem: Motion is the Solution

This paper addresses the Navier--Stokes Existence and Smoothness Problem by introducing a motion-based framework grounded in directional movement persistence. The approach replaces traditional energy inequalities and time-based formulations with a structure defined by the persistence of motion m, a compression threshold Cₜ, and the collapse indicator EM. A solution is shown to remain smooth if the motion of the system satisfies m (t) Cₜ for all time. If this threshold is crossed and motion collapses, entropy appears in the form of EM 0, indicating a singularity. The Navier--Stokes equations are mapped into this framework by interpreting velocity as directional motion, viscosity as structural damping, and divergence-free flow as motion conservation. This allows the regularity problem to be reframed as a question of motion compression survivability. Comparative analysis with classical formulations is provided, and conditions for both smoothness and finite-time breakdown are derived within the new formal structure.