TEBAC Yang–Mills Program IV: Uniform Spectral Coercivity, BSF Core Extraction, and Holonomy-Pruning Front

This preprint is the fourth module of the TEBAC Yang--Mills program. It does not claim a completed proof of the Yang--Mills existence and mass gap problem. Its purpose is to isolate the mass-gap front after the finite-cutoff reflection-positive construction of YM-II and the Osterwalder--Schrader/GNS reconstruction layer of YM-III. The module formulates the mass-gap problem as a uniform spectral-coercivity problem on the vacuum-orthogonal gauge-invariant Hilbert sector. The central route is\[\text{uniform spectral coercivity}\Longrightarrow\text{no massless gauge-invariant packets}\Longrightarrow\text{positive Yang--Mills mass gap}.\]Earlier passes introduced the terminal-packing functional, gauge-invariant infrared depletion, finite-frame verification, terminal squeeze, observable Poincare inequalities, block-spin depletion, and a BSF/additive-energy uniformity route. The PASS7 version strengthens the proof-facing layer by isolating two precise hard theorem targets:\[\text{BSF core extraction}\]and\[\text{gauge-curvature / holonomy pruning with cutoff-stable constants}.\]It adds dyadic block-core extraction, fixed-cutoff regular core extraction, additive-energy amplification, coefficient leakage stability targets, coarse holonomy defects, fixed-cutoff flat-core pruning, flat-core rigidity constants, and a cutoff-stable flat-core rigidity theorem target. The module records the implication\[\text{cutoff-stable BSF extraction}+\text{cutoff-stable flat-core rigidity}\Longrightarrow\text{observable Poincare}\Longrightarrow\text{uniform spectral coercivity}\Longrightarrow\text{mass gap}.\] Claim-safety statement: YM-IV PASS7 is a proof-facing mass-gap reduction module. It strengthens and isolates the remaining analytic front, but it does not yet close the full Yang--Mills mass-gap theorem. The remaining hard fronts are cutoff-stable coefficient leakage stability and cutoff-stable flat-core rigidity. Final continuum-limit assembly remains downstream in YM-V.