We present the solution to the Yang-Mills existence and mass gap problem by incorporating full General Relativity into Quantum Field Theory via its surrounding equation. By replacing the idealized single-source, asymptotically flat approximation with a multi-source surrounding energy distribution vector D^μ(x), the local field operator calibration is dynamically governed through the purely algebraic structure of the global background environment via the scale-invariant ratio v/c. We show that this ratio possesses an intrinsic scale invariance property under global energy rescalings λ ∈ ℝ*, preventing field configurations from collapsing into zero-energy or infinitesimal states. In low-density vacuum environments, the surrounding background dynamically amplifies the effective gauge interaction energy in inverse proportion to the background density, guaranteeing a strictly positive mass gap Δ > 0. Furthermore, at high energy scales and short distances, the accumulation of local surrounding energy in D^μ(x) induces an intrinsic saturation of the v/c ratio, dynamically damping high-momentum interaction amplitudes and providing a non-perturbative UV regularization scheme without ad hoc cutoffs or counterterm subtractions. This algebraic embedding links full relativistic background constraints directly to the resolution of fundamental open problems in subatomic quantum field theory.