Vacuum-sector mass gap for the local gauge-invariant sharp-local Yang–Mills net

This series presents the QFT/PDE-facing reconstruction of an AASC-native solution: the admissible Yang–Mills endpoint is first identified as the bounded-region local gauge-invariant vacuum net, after which Wilson/RG, OS/Wightman, and sharp-local analytic machinery are used to reconstruct that endpoint in conventional constructive-field-theoretic terms. This paper is the core theorem statement of that reconstruction. It proves a vacuum-sector mass-gap theorem for the bounded-region local gauge-invariant sharp-local Yang–Mills net generated by flowed curvature composites, for each compact connected simple gauge group (G). The theorem-level object is deliberately local: it is the Haag–Kastler/Wightman vacuum representation of the bounded-region gauge-invariant net, not an assertion about extended-support line or surface sectors. The proof is organized through a finite public theorem spine: a compact-simple ultraviolet gate, a Route 1 lattice-side one-shot entrance and fixed-lattice mass-gap chain, a Lane A sharp-local construction, and a QE3 transport step carrying the fixed-lattice lower bound into the continuum sharp-local OS Hilbert space. The result establishes the existence of the local gauge-invariant Yang–Mills vacuum theory, non-triviality, OS/Wightman reconstruction strength, and a strictly positive vacuum Hamiltonian gap. Faithful Wilson regularizations are shown to identify the same continuum local theory qualitatively within the admissible gradient-flow renormalization class and coherent normalization, while the explicit quantitative lower-bound certificate remains indexed by the auxiliary faithful regularization.