TEBAC Yang–Mills Program IV: Uniform Spectral Coercivity in the Gauge-Invariant Observable Sector

This preprint is the final master version of Module IV in the TEBAC Yang–Mills program. The purpose of YM-IV is to close the uniform reconstructed mass-gap/coercivity module in the gauge-invariant observable sector. The paper works in the OS/GNS reconstructed physical Hilbert space exported by the preceding modules and proves a cutoff-uniform spectral-coercivity theorem of the form\[\langle H_{\Lambda_n,a_n}F,F\rangle_{\mathrm{OS}}\ge\Delta_*\|F\|_{\mathrm{OS}}^2,\qquadF\perp\Omega_{\Lambda_n,a_n},\qquad\Delta_*>0,\]for the reconstructed gauge-invariant sector along the prescribed cutoff trajectory. The manuscript consolidates the previous YM-IV development into a single referee-facing master synthesis. It incorporates coefficient leakage stability, flat-core holonomy rigidity, renormalized frame-and-holonomy stability, the cutoff-trajectory envelope, coherent-packet tail estimates, large-field summability, coordinate-residual summability, and a final non-circularity and claim-safety audit. The logical route is:\[\text{coefficient stability}+\text{holonomy rigidity}+\text{cutoff control}\Rightarrow\text{observable Poincare}\Rightarrow\text{spectral coercivity}\Rightarrow\text{mass gap}.\] This version supersedes the earlier YM-IV reduction manuscript titled “Uniform Spectral Coercivity, BSF Core Extraction, and Holonomy Pruning.” The earlier version reduced the problem to cutoff-stable coefficient leakage and non-abelian flat-core rigidity estimates; the present version consolidates the subsequent technical closure into a final YM-IV master module. Scope statement: this manuscript closes YM-IV as a uniform reconstructed mass-gap/coercivity module within the TEBAC Yang–Mills chain. It is not presented as the complete Clay-level solution by itself. The full Yang–Mills existence and mass gap theorem still requires the downstream YM-V assembly: continuum Schwinger functions, non-triviality, OS/Wightman axioms in the limit, regulator independence, and persistence of the gap in the continuum theory.