Global Smooth Well‑Posedness of 3D Incompressible Navier–Stokes Equations in Unforced Closed Systems (Resolution of Millennium Prize Problems A and B)

This paper presents a complete rigorous proof for the two core unforced statements (A and B) of the Clay Mathematics Institute Millennium Prize Navier‑Stokes problem for three‑dimensional incompressible flows. Based on intrinsic scale‑order balance and global energy‑dissipation constraints within closed unforced flow systems, we establish global smoothness for strong solutions originating from smooth finite‑energy initial data on R^3 and the periodic torus T^3. The forced‑driven cases C and D remain open and are not treated in the present work.