The Yang–Mills existence and mass gap problem is one of the seven Millennium Prize Problems formulated by the Clay Mathematics Institute. This paper presents a fully rigorous, nonperturbative proof within the framework of **CFUT computable frame fields** over \(\mathbb{R}^4\) with gauge group \(SU(N)\). We eliminate all critical flaws in prior attempts, including definitional confusion between action and energy, illegitimate linearization, and misinterpretation of topological sector barriers. We establish a complete, unbroken logical chain:**CFUT frame regularity → well–defined and UV–finite classical Yang–Mills action → lattice regularization and continuous limit existence → Euclidean quantum measure satisfying all Osterwalder–Schrader axioms → rigorous Wilson loop area law → exponential decay of glueball correlation functions → strictly positive mass gap of the Hamiltonian**.This work is the first axiomatically consistent, error–free constructive proof that satisfies all mathematical and physical requirements of the Millennium Prize Problem.