The Capacity Regulator of Absolute Frame Theory VII: Channel Cyclicity, the Effective Gauge Measure, and the Regulated Yang–Mills Mass Gap under the Physical-Continuum Hypothesis

The sixth paper of this series resolved the phase question of the regulated sector of Absolute Frame Theory (AFT) and named the single remaining debt: the Yang–Mills gauge wall. This paper makes two contributions. First, it closes the two remaining control residues of the channel and gauge sectors: the channel vacuum is cyclic for the quasi-free channel algebra—the identifiability thesis cast as an Osterwalder–Schrader structure, a Reeh–Schlieder property of the channel net—and the composite Wilson-loop sector pushes forward to a well-defined, gauge-invariant, bounded, reflection-positive effective gauge measure. Second, it confronts the gauge wall under a declared physical-continuum hypothesis A*: the capacity regulator at the envelope resolution ε*∼ℓP is the physical ultraviolet completion of the gauge sector, not a lattice to be removed—the action is quantized, not spacetime (Axiom I, the information bottleneck). Under A* the mass gap of the composite gauge sector is the spectral gap of the physical transfer operator—positive by Perron–Frobenius and equal to the clustering rate—hence its existence follows from A*, and we prove the two theorems that make that conditional gap robust—upgrading it from a fixed-volume existence to a volume-uniform, Euclidean-restored gap, while A* itself remains a declared hypothesis: (1) the regulated gauge gap is volume-uniform, by local strong-coupling data and a volume-independent polymer expansion; (2) the Euclidean restoration preserves it—the rotation breaking is the removable scaffold anisotropy, a dimension-six irrelevant flow that vanishes with the scaffold scale, the directional gap becomes isotropic as the square of the scaffold resolution, and the gap stays bounded below and never closes. We place the four routes to the wall—the dimensional ladder, the cluster expansion on the effective measure, the spectral/modular reduction, and the epistemic decomposition—and state what A* does and does not provide: at fixed ε* the regulated gauge theory carries a volume-uniform, Euclidean-restored, nonzero mass gap; the strict zero-spacing continuum, the Millennium problem, is not crossed and, under A*, is not the programme's physical question. A* is falsifiable through its consequences—because the only discreteness is of the scalar action, it predicts no local Lorentz violation (an exactly-zero anisotropic prediction), a global cosmological frame rather than a local one, and any discreteness signature of Lorentz-invariant type; a gapless or deconfined gauge-invariant sector, or an ultraviolet completion scale far from ℓP, also refute it. The same hypothesis closes a second regulator residue: the positive lower bound on the bond conductance underlying the single-phase bulk of the companion, which is the physical gradient variance, not a lattice artifact. The bright line is kept: the gap's existence is structural, only its scale is identified.