Chapter 37: The Yang–Mills Mass Gap

The τ-Gap Meta-Theorem (the relevant theorem, Ch. 39) is a machine: feed it an NF-discrete tower with a contractive defect functional and it returns a spectral gap. This chapter turns the machine on. We restrict the defect functional to the strong (C) sector, defining the strong defect functional Δ_C, and verify that τ-admissible gauge data (the relevant definition, Ch. 38) satisfies both hypotheses of the meta-theorem. The conclusion is the Yang–Mills Gap Theorem: the gap constant Γ^_s > 0 for the strong sector. We then draw the scope boundary with full precision. The τ-gap is proved for τ-admissible gauge data within Category τ; it is *not a solution to the Clay Millennium Problem, which requires identifying the τ strong sector with SU(3) Yang–Mills quantum field theory—a conjectural bridge deferred to Part X. The chapter closes with export contracts: the deliverables that downstream Parts and Books inherit from the Yang–Mills block.