On the 3D Navier-Stokes Regularity

This paper begins with the incompressible Navier–Stokes equations and derives the exact energy and enstrophy balances. The nonlinear term is then rewritten using vorticity and strain, isolating vortex stretching. We normalize enstrophy by kinetic energy to obtain a closed differential inequality. Estimating all remaining terms reduces the problem to a single time-integrated geometric quantity. Controlling this quantity yields bounded enstrophy and therefore global regularity. All provisional assumptions are introduced solely to propagate the argument to a single central lemma; proving this lemma retroactively validates each assumption in the chain, and control of the resulting integrated vortex stretching functional term yields uniform enstrophy bounds and hence global regularity.