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

Standard formulations of quantum gauge theories relying on a rigid Minkowski background implicitly enforce Special Relativity as a standalone framework, introducing structural kinematic inconsistencies. In this paper, we propose a complete resolution to the Yang-Mills Millennium Problem by incorporating the algebraic surrounding effect of General Relativity (GR) into quantum field theory. To remain fully compliant with the axiomatic setup on R4, GR is introduced purely algebraically via energy-density constraints and background metric fluctuations. Building upon the relational formulation established in Ref. [1, 2], local spacetime geometry is algebraically deformed by the global, multi-source energy distribution of the Universe, governed by a vector field D^μ(x) and an associated velocity ratio v/c. We demonstrate that the dynamic emergence of a strictly positive mass gap > 0 is generated either within a total global vacuum across the entire universe—ensuring a mass gap for any infinitesimal local energy concentration—or within a strictly local pure vacuum domain under a pointlike infinitesimal Dirac concentration. In both scenarios, scale-invariant saturation prevents localized field configurations from continuously decaying into zero-energy states (v/c does not go to 0). This non-zero spectral gap ensures the complete elimination of infrared (IR) divergences through exponential spatial damping (approx exp(-Delta * r)) and bounded transverse propagators. Concurrently, high-energy ultraviolet (UV) divergences are regularized through a rigorous framework combining stochastic metric dephasing (gμν = ημν + hμν)—yielding a universal Gaussian damping factor exp(-gamma * p^2 / Lambda_UV^2)—and topological causal disconnection driven by local light-speed saturation (v/c -> 1). This dual mechanism prevents short-distance contact singularities, ensuring that all n-point Wightman correlation functions remain well-defined tempered distributions in S'(R⁴ⁿ) and fully satisfying the axiomatic requirements formulated by the Clay Mathematics Institute.