Resolution of the Yang-Mills Millennium Problem: The Solution via Full General Relativity and its Surrounding Equation

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 Dμ(x) and the velocity ratio v/c. We show that the algebraic cancellation λ/λ 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 (Δ > 0). Simultaneously, high-energy divergences are inherently regularized: ultraviolet self-interactions remain strictly bounded by background saturation through the infinite interaction time delay (t' → ∞) 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 ∫ D A exp(i S[A]) and ensures that the n-point Wightman correlation functions <0| A(x₁) ... A(xₙ) |0> remain well-defined tempered distributions across all of R⁴. Furthermore, we demonstrate that the algebraic deformation generated by Dμ(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.