Finite Depth Entrance versus Stable Manifold Renormalization Group Control

This note is a structural companion to the technical “Route 1” construction of a strictly positive continuum mass gap for four-dimensional Yang–Mills on a bounded macroscopic observable algebra O_ℓ at fixed positive scale ℓ > 0. It isolates the minimal renormalization-group (RG) control that is logically sufficient for the macroscopic spectral-gap statement and contrasts it with the classical stable-manifold RG template. Instead of constructing a global RG trajectory that remains on (or enters) a stable manifold forever, the note shows that a single finite-depth entrance condition suffices: after an explicitly bounded number of blocking steps, the exact blocked lattice specification lies inside a uniform Dobrushin contraction basin (δ(Γ) < 1). From that point onward, uniqueness of the infinite-volume Gibbs state and exponential clustering of truncated correlations follow from standard Dobrushin technology; Osterwalder–Schrader reconstruction then converts the Euclidean exponential decay into a strictly positive lower bound on the spectrum of the reconstructed Hamiltonian. The note makes three contributions that can be cited independently: (C1) An abstract “entrance-to-gap” theorem (Theorem 2) for any local net of bounded gauge-invariant observables at fixed macroscopic scale ℓ, stated with complete proof using only OS positivity, translation/reflection covariance, and finite-depth Dobrushin contractivity. (C2) A self-contained elementary lemma chain (Appendix A) transferring uniform one-site conditional χ² control to Dobrushin contractivity via total-variation bounds and influence matrices. (C3) A dictionary table explicitly mapping every module and estimate of the Route 1 manuscript (heat-kernel majorant, L² transfer budget, multiplicative accumulation, explicit blocking depth B(β), etc.) onto the abstract primitives used here, with logical status indicated for each entry. No new model-dependent estimates are introduced. The discussion is written in standard mathematical-physics language and clarifies precisely which RG ingredients are (and are not) required for a rigorous macroscopic mass-gap result: full stable-manifold geometry, relevant/irrelevant decompositions, or infinite-scale trajectory control are unnecessary once a one-shot entrance inequality is available. The ultraviolet identification and OS-positivity requirements remain external inputs, exactly as in the companion technical work. This provides a streamlined logical skeleton that separates the hard analytic entrance step from the standard consequences used to extract the continuum spectral gap.