Global Smoothness of the Navier–Stokes Equation: A Projection Geometric Proof

We prove, within the ZFC axiom system, that the incompressible Navier-Stokes equation without external force admits a global smooth solution in any dimension. By introducing projection geometry and the symplectic capacity lower bound, we derive strict mathematical upper bounds for the velocity gradient and vorticity, fundamentally eliminating the formation of turbulence singularities. This proof rigorously resolves one of the Millennium Prize Problems and is fully compatible with recent finite-time blow-up results for equations with external forces, marking a significant breakthrough in fluid dynamics.