This paper is a standalone extension of the predecessor local-net Yang–Mills massgap theorem. The predecessor package constructs the bounded-region local gaugeinvariant Haag–Kastler/Wightman theory, its non-trivial vacuum representation, its exact strictly local endpoint, and its vacuum Hamiltonian gap. The present paper recalls that theorem package as a fixed input and proves the completion-control part of the Clay-scope proof chain. It fixes a theorem-scope class of extendedsupport line and su