Yang–Mills Mass Gap in Four Dimensions: Reduction and Closure for the BT–2 Route

This upload presents a single-manuscript reduction-and-closure framework for the four-dimensional SU(2) Yang-Mills mass-gap problem along the BT-2 route. Main idea The manuscript is organized in two parts: Part I (Reduction) develops the Osterwalder–Schrader (OS) window, the analytic and geometric infrastructure needed for the mass-gap transfer, and the reduction of 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 established through the following chain: δφ(t₀) > 0 λₘᵢₙ(-ΔM) ≥ κgeo\,δφ(t₀)² m₀ ≥ cdom\,λₘᵢₙ(-ΔM) m₀ ≥ cgap\,δφ² > 0, with the gap constant defined as cgap := cdomκgeo. Thus, the essential bottleneck is the target-side production of residual-density positivity. What Part I establishes Part I establishes the foundational analytic and geometric modules, including: The parameter-box feasibility theorem and the residual topology density δ_φ. The topological-coercivity-to-gap route. The FRD / KP / CM.1 / entropy-trace / EVI analytic modules. The strict interchange-of-limits package and OS reconstruction. The global operator-gap transfer. What Part II closes Part II closes the remaining shell-side theorem on one fixed canonical chart. It specifically 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 and a fixed-chart probability-floor transfer theorem. The canonical shell-side activation theorem. The representative local closure score on the canonical target $SU(2)$ family is summarized by: ΥSU(2) < 1. Downstream continuation after shell-side closure Once the shell-side theorem is in force, the remaining continuation is internal to the framework: shell-side closure codimension-one continuation target-law two-tail floor block-variance floor δφ(t₀) > 0 δφ > 0 m₀ ≥ cgapδφ² > 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 contain the supporting analytic, geometric, and target-specialization proofs underlying the main theorem chain.