This paper presents a construction of four-dimensional Euclidean Yang–Mills theory with a strictly positive mass gap for any compact simple gauge group G, addressing the Clay Mathematics Institute Millennium Problem. The argument identifies a volume-uniform Poincaré inequality for the lattice Yang–Mills Gibbs measure as the key mechanism linking multi-scale renormalization group control to exponential clustering and the emergence of a mass gap in the continuum limit. Starting from lattice Yang–Mills theory with Wilson action at sufficiently weak coupling, the analysis combines axial gauge reduction, finite-range propagator decomposition, and scale-by-scale spectral estimates driven by asymptotic freedom. The resulting functional inequality yields Dobrushin–Shlosman complete analyticity and enables the construction of a continuum quantum field theory satisfying the Osterwalder–Schrader axioms with a strictly positive mass gap. Nontriviality of the continuum limit is demonstrated through a susceptibility-growth mechanism. Tracking a gauge-invariant observable across renormalization scales yields an exponential susceptibility bound χ(β) ≥ c e²ᶜ⁰ᵝ, preventing the physical mass from diverging in the continuum limit. Non-Gaussianity follows from the non-abelian Schwinger–Dyson identity at the effective scale, where the running coupling becomes order one and generates a non-zero truncated four-point function. The construction applies uniformly to all compact simple Lie groups, including SU(N), SO(N), Sp(N), G₂, F₄, E₆, E₇, and E₈. A technical supplement provides proofs of the renormalization-group step, the Martinelli finite-volume bootstrapping argument establishing the volume-uniform Poincaré inequality, and the continuum reconstruction. This manuscript is in preparation for submission to a peer-reviewed journal.