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 $\Sigma \Delta m$, a compression threshold $C_t$, and the collapse indicator $E^M$. A solution is shown to remain smooth if the motion of the system satisfies $\Sigma \Delta m(t) > C_t$ for all time. If this threshold is crossed and motion collapses, entropy appears in the form of $E^M > 0$, 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.