Quantum Corrections and Finite Gribov Uniqueness in a Non-local Gauge Theory: An Essential Addendum to the Proof of the Yang-Mills Mass Gap

This supplementary paper brings to mathematical completion the proof of the mass gap presented in the preprint "A Rigorous Proof of the Mass Gap in SU(N) Yang-Mills Theory - Version v2" DOI:10.5281/ZENODO.14975444 (hereafter referred to as Agawa (v2)). We resolve a critical contradiction concerning the treatment of quantum corrections, which was discovered in a past draft of the supplement. By rigorously re-deriving the non-local vertex functions from first principles and re-evaluating the loop orientation average, we demonstrate that the theory preserves asymptotic freedom. This establishes the validity of the cluster expansion in Agawa (v2). Furthermore, we add a deliberation on the finite Gribov copy problem, resolving it via a proof of "probabilistic uniqueness", to ensure the completeness of the proof. Hereby, the proof of the Yang-Mills mass gap by Agawa is finalized as a self-consistent work.