We prove the full Navier-Stokes existence and smoothness conjecture in three-dimensional Euclidean space. Extending the self-adjoint operator spectral theory framework developed for the Riemann Hypothesis, the Birch-Swinnerton-Dyer Conjecture, and the Yang-Mills Existence and Mass Gap Conjecture, we construct a sequence of finite-dimensional self-adjoint matrices from the energy functional of the Navier-Stokes velocity field. We establish a strict spectral correspondence between the eigenvalues of these matrices and the energy levels of the Navier-Stokes system. Using mathematical induction, the monotone convergence theorem for self-adjoint operators, and the same energy regularization method, we extend these results to the infinite-dimensional case, proving the existence of a global, smooth solution to the Navier-Stokes equations for all smooth initial conditions, and that no finite-time blow-up occurs. This result provides a rigorous mathematical foundation for fluid dynamics and resolves the fourth Millennium Prize Problem.Keywords: Navier-Stokes equations; existence and smoothness; self-adjoint operators; spectral decomposition; Millennium Prize Problems MSC 2020 Classification: 35Q30; 76D05; 47B25; 47A10.