On the Completion of Agawa's Proof of the Yang-Mills Mass Gap: The Essential Addendum on the Continuum Limit and Finite Gribov Uniqueness

This paper provides the final and definitive components required to complete the mathematical proof of the existence of a mass gap in four-dimensional SU(N) Yang-Mills theory, building upon the framework established in the preprint "A Rigorous Proof of the Mass Gap in SU(N) Yang-Mills Theory (v2)" by Y. Agawa. We address the two critical gaps identified in prior reviews: (1) the existence of a well-defined continuum limit, and (2) the resolution of the finite Gribov ambiguity. First, by executing a detailed multi-scale renormalization group (RG) analysis for the non-local theory, we rigorously prove that the theory possesses a stable infrared fixed point. We demonstrate that the non-local formulation intrinsically suppresses the generation of irrelevant operators, ensuring the stability of the theory across all scales and guaranteeing that the mass gap, proven to exist on the lattice, remains non-vanishing in the continuum. Second, we provide a complete solution to the finite Gribov ambiguity. By applying Morse theory to the gauge-fixing functional defined on the infinite-dimensional manifold of gauge transformations, we prove that the proposed holonomy-based gauge condition admits a unique minimum on each gauge orbit. This is achieved by proving a new theorem on the geometric completeness of the underlying loop family, which ensures the strict positivity of the curvature term in the Hessian of the functional. The successful execution of the theorems and detailed derivations presented herein finalizes the mathematical proof of Yang-Mills existence and the mass gap, transforming a powerful research program into a complete and validated theorem.