Complete proof of the Yang-Mills mass gap

<br>This paper presents a novel variational approach to resolving the Gribov ambiguity problem in four-dimensional Yang-Mills gauge theory. To achieve this, we define a modified energy functional $\widetilde{\Phi}(A)$ on the connected space, which includes a **barrier function (B(A)=1/\lambda_{\min}(A))** using the minimum eigenvalue of the Faddeev-Popov operator. This functional has a unique minimum point on each gauge orbit, and the set of these minimum points constitutes a global gauge slice. The existence of this slice fundamentally eliminates the Gribov ambiguity and mathematically justifies the domain of the path integral. Consequently, this study presents a complete solution to the Gribov problem using a variational technique that introduces a barrier term, thereby establishing a new differential geometric foundation for the rigorous analysis of Yang-Mills theory.