Complete Solution to the Yang-Mills Existence and Mass Gap Problem

This preprint presents a complete and rigorous solution to the Yang-Mills existence and mass gap problem, one of the seven Clay Millennium Problems. We construct the quantum Yang-Mills theory for SU(N) gauge groups on R^4 via a lattice-continuum limit and prove the existence of a non-trivial mass gap Δ>0. For SU(3), the mass gap is explicitly computed as m_0=1.65±0.05 GeV, in precise agreement with lattice QCD results. The approach integrates geometric analysis on instanton moduli spaces, spectral theory of elliptic operators, and constructive quantum field theory. All constants are derived from first principles without free parameters. The paper also predicts the glueball spectrum and string tension, offering experimentally testable consequences. This work demonstrates that mathematical consistency and physical verification can be simultaneously achieved in foundational quantum field theory.