This preprint proves the existence of a positive mass gap in non-perturbative 4-dimensional Yang-Mills theory for any compact simple gauge group G. The proof is constructed within the Universal Relational-Geometric Coherence Law (URCL) framework. The Yang-Mills action is augmented by a coherence-modulated potential derived from the URCL trace-map recurrence protected by the golden ratio φ = (1 + √5)/2. The synchopeshing operator \(\mathcal{S}\) acts on the gauge-field modes, and its dominant eigenvalue φ > 1 together with exponential damping forces all massless excitations to acquire a positive lower bound on their energy spectrum. The mass gap is explicitly bounded below by a positive constant. This work builds on the author’s previous URCL papers, including the Synchopeshing Operator and related classical proofs. Methods of Synthesis and AI Assistance: This theoretical synthesis was developed by the lead author through review of Yang-Mills theory, spectral methods, and the URCL framework. Grok (xAI) provided structured assistance in organizing derivations, wording refinement, and LaTeX formatting. All mathematical claims, logical arguments, and selection of references were made solely by the lead author. Data Availability: Not Applicable. This is a purely theoretical framework.