Yang–Mills Mass Gap in Four Dimensions and Canonical Shell-Side Activation for the BT–2 Route: A Synchronized Two-Paper Closure Package

Main idea The manuscript is organized in two parts. Part I (Reduction) develops the OS-window analytic and geometric infrastructure and reduces the remaining model-specific target-side burden to one canonical shell-side specialization theorem on a canonical $SU(2)$ family. Part II (Closure) proves that remaining shell-side theorem on one fixed canonical good packet chart and one fixed packet depth, and then follows the already prepared downstream continuation to the final mass-gap conclusion. Global theorem-facing chain The overall route is δφ(t₀) > 0 λₘᵢₙ(-ΔM) ≥ κgeo\,δφ(t₀)² m₀ ≥ cdom\,λₘᵢₙ(-ΔM) m₀ ≥ cgap\,δφ² > 0, with cgap := cdom\,κgeo. Thus the main bottleneck is the target-side production of residual-density positivity. What Part I proves Part I develops the OS-window framework, including: the parameter-box feasibility theorem, the residual topology density δφ, the topological-coercivity-to-gap route, the FRD / KP / CM.1 / entropy-trace / EVI modules, the strict interchange-of-limits package, OS reconstruction and the global operator-gap transfer. Within this framework, all downstream transfers from residual-density positivity to the mass-gap conclusion are theorem-derived. What Part II proves Part II closes the remaining shell-side theorem on one fixed canonical chart. More precisely, it proves: a shell-center principal separation theorem, a fixed-chart one-point shell-margin theorem via the affine donor chain, a near-good branch entry theorem, a fixed-chart probability-floor transfer theorem, the canonical shell-side activation theorem. These ingredients verify the canonical shell-side specialization theorem and yield ΥₛₕₑₗₗSU(2) < 1, together with Checklist A and the Stage I-Stage III shell-side closure chain on the canonical target family. A one-line target $SU(2)$ local closure interface A representative theorem-facing local closure score is $$ΥSU(2) := max \{ ΞBT2, {2 Eblk(_0)}{ρpack^{}v_{}SU(2),}, {2Cflowt_0γ}{δφ,lbSU(2),(t_0)} \},$$ and the local target closure criterion takes the one-line form $$ΥSU(2) < 1.$$ Here the witness-scale quantities are defined by $$cswSU(2), := 1/2\,pswSU(2), (ΔswSU(2),)^2$$ $$v_{}SU(2), := θ(1-θ)^2 {(1-ϑ)^6(cswSU(2),)^3} {C4, QSU(2),}$$ $$v_0SU(2),:=1/2ρpack^{}v_{}SU(2),, δφ,lbSU(2),(t_0):= √{1/2\,v_0SU(2),}.$$ The corresponding explicit witness radius is $$RₚₐₛₛSU(2), := max \{ RBT2, {128}{λcluster}, {24576π^2CEVI(LQ↑)^2} {λcluster^5v_0SU(2),}, {16}{λcluster} log ( {2048π^2CEVI(LQ↑)^2} {λcluster^4v_0SU(2),} ) \}.$$ This formula makes the local witness-producing interface mathematically explicit: once the local closure score is below $1$, choosing $R=RₚₐₛₛSU(2),$ yields a deterministic PASS instance of (E5-min) for the target $SU(2)$ local closure chain. Downstream continuation after shell-side closure Once the shell-side theorem is in force, the remaining continuation is internal: $$shell-side closure codimension-one continuation$$ $$ target-instantiated BT-2 closure (E5-min)$$ $$ δφ(t_0)>0 δφ>0 m_0 ≥ cgap\,δφ^2 > 0.$$ Reading guide This manuscript is intended to be read as a single-manuscript reduction-and-closure theorem package for the canonical target $SU(2)$ BT-2 route. The appendices retain the full analytic, geometric, and target-instantiation ledgers, including: proof packs, dependency maps, OS-window feasibility infrastructure, parameter and constants registries, the detailed BT-2 local proof map, the target $SU(2)$ instantiation bridge. Finite contract forms, packet-record forms, and auditable witness interfaces are retained as supplementary certification boundaries, rather than as the primary theorem-facing closure route. ------------------------ *Author: Lee Byoungwoo (leeclinic@protonmail.com)