We demonstrate that algebraically integrating General Relativity into quantum field theory via the Surrounding Matter framework provides a rigorous resolution to the Yang--Mills mass gap problem. Within this framework, the local energy-momentum structure is governed by the multi-source vector field $D^mu(x)$ and the velocity ratio $v/c$. We show that the algebraic cancellation $lambda/lambda$ within the surrounding ratio ensures that an infinitesimal energy distribution generates the exact same space-time calibration as a finite one across all gauge generators. Consequently, gauge field interactions in a non-empty Universe cannot decay into zero-energy states; for localized excitations in vacuum environments, the surrounding background dynamically enhances their rest mass, establishing a strictly positive lower bound for the ground-state spectrum ($Delta > 0$). Simultaneously, high-energy divergences are inherently regularized: ultraviolet self-interactions remain strictly bounded by background saturation through the infinite interaction time delay ($t' to infty$) and the dynamic attenuation driven by the algebraic relativistic factor. This uniform upper bound guarantees the unconditional convergence of the non-perturbative Feynman path integral $int mathcal{D}A , e^{iS[A]}$ and ensures that the $n$-point Wightman correlation functions $langle 0 | A(x_1) dots A(x_n) | 0 angle$ remain well-defined tempered distributions across all of $mathbb{R}^4$. Furthermore, we demonstrate that the algebraic deformation generated by $D^mu(x)$ resolves the historical obstacle of infinite static field energy without introducing metric singularities, removing the necessity for a rigid Minkowski background. Building upon a rigorous gravitational baseline, these results reveal that General Relativity, through its global algebraic surrounding effect, plays an active and indispensable role in governing quantum gauge field dynamics at subatomic scales.