Proof of the Navier-Stokes Existence and Smoothness Conjecture via Self-Adjoint Operator Spectral Theory

We prove the full Navier-Stokes existence and smoothness conjecture in three-dimensionalEuclidean 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 GapConjecture, we construct a sequence of finite-dimensional self-adjoint matrices from the energy functionalof the Navier-Stokes velocity field. We establish a strict spectral correspondence between the eigenvaluesof these matrices and the energy levels of the Navier-Stokes system. Using mathematical induction, themonotone convergence theorem for self-adjoint operators, and the same energy regularization method, weextend these results to the infinite-dimensional case, proving the existence of a global, smooth solution tothe Navier-Stokes equations for all smooth initial conditions, and that no finite-time blow-up occurs. Thisresult provides a rigorous mathematical foundation for fluid dynamics and resolves the fourth MillenniumPrize Problem. This paper is part of a unified self-adjoint operator spectral theory framework that has solved fourMillennium Prize Problems. For the complete series, see the author’s homepage: https: //zenodo. org/JianningYang. 2020 Mathematics Subject Classification. 35Q30; 76D05; 47B25; 47A10. Key words and phrases. Navier-Stokes equations; existence and smoothness; self-adjoint operators; spectral decomposition; Millennium Prize Problems.