Companion I: Ultraviolet Gate and Route 1 Mass-Gap Chain

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 companion supplies the detailed proof home for the ultraviolet and Route 1 parts of the local Yang–Mills mass-gap theorem. Its role is to discharge the lattice/RG side of the construction before the sharp-local continuum endpoint is invoked. The companion proves the compact-simple A1 ultraviolet gate, the public group-scope export, the one-shot KP/Dobrushin entrance theorem, the tuned fixed-lattice OS gap lower bound, the same-scale Wilson-to-patched transport theorem, and the weak-window sufficiency certificate. It also records the bounded Wilson/dyadic OS limit, the continuum-time OS upgrade, the density and graph-core handoff, and the final QE3 continuum vacuum-gap transport theorem. The central contribution is to make the quantitative mass-gap chain conventional and auditable: the physical Wilson measure remains the unmodified lattice state; chart-truncation is used only as an analytic RG surrogate; de-patching returns the estimates to Wilson expectations at the exact theorem-facing level; and the fixed-lattice lower bound is transported to the continuum sharp-local Hilbert space without changing the theorem object. In the series architecture, this companion is the proof home for the quantitative Route 1 certificate whose lower-bound constant later appears in the core theorem.