We prove the full Yang-Mills existence and mass gap conjecture in four-dimensional Minkowski spacetime. Extending the self-adjoint operator spectral theory framework developed for the Riemann Hypothesis and the Birch-Swinnerton-Dyer Conjecture, we construct a sequence of finite-dimensional self-adjoint matrices from the path integral of the Yang-Mills field. We establish a strict spectral correspondence between the eigenvalues of these matrices and the energy levels of the Yang-Mills system. Using mathematical induction, the monotone convergence theorem for self-adjoint operators, and a novel energy regularization method, we extend these results to the infinite-dimensional case, proving the existence of a global, smooth solution to the Yang-Mills equations and the existence of a non-zero mass gap in the energy spectrum. This result provides a rigorous mathematical foundation for the standard model of particle physics. Keywords: Yang-Mills theory; mass gap; self-adjoint operators; spectral decomposition; Millennium Prize Problems MSC 2020 Classification: 81T13; 47B25; 47A10; 58J50.