The Capacity Regulator of Absolute Frame Theory IV: Phase Structure, Quantitative Non-Degeneracy, and the Angular KMS Germ

The third paper of this series established exponential clustering for the capacity-regulated scalar sector of Absolute Frame Theory (AFT) in its physical window, chessboard estimates for the regulated measure, and the collar theorem for the geometric sector. This fourth paper closes, reduces, or quantifies the principal questions that remained. First, phase structure: the chessboard suppression of super-capacity regions holds with volume-independent constants and therefore passes to every infinite-volume state of the torus class—no saturated phase exists anywhere in the class; free energy and correlators are real-analytic in the routing coupling and the inverse capacity inside the window—no transition there; and the one-dimensional venue has no transition at any activity, by a positivity-improving argument. The regulator is self-stabilizing, compressing absolute density and suppressing hot events ever more strongly as capacity shrinks, while relative saturation grows; the single-phase conjecture this supports is stated as such. Second, quantitative non-degeneracy: at every point the mollified Jacobian is an exactly independent Gaussian matrix, the induced metric is Wishart, and the negative moments of its smallest eigenvalue are finite precisely below the threshold 11/2—half the codimension of the rank-deficient locus: the derived substratum dimension N=14, which makes the immersion almost sure, also grades integrability, and the transfer to the regulated measure holds with an explicit conditional bound, volume-uniformly inside the physical window. Consequently, the second fundamental form has finite moments below order 11/4, with the Hölder split landing exactly on the programme's mode budget, and the lowered-index normal curvature—the composite gauge field—has finite second moments; the first moment of the fully contracted Yang–Mills scalar is the named residue. Third, the saturation point is reduced: a Perron–Frobenius argument gives, at every activity and fixed spatial volume, a simple principal transfer eigenvalue, a positive gap, and a unique temporal limit, leaving open only the spatial-volume uniformity of the gap. Fourth, the angular germ: for the channel sector, angular Osterwalder–Schrader positivity, the KMS reflection symmetry at angle 2π, and positivity of the angular spectrum are derived, and the analytic continuation to the wedge-boost KMS structure is obtained from the virtual-representation theory of symmetric spaces; the angular collar and its composition defect transport the geometric sector verbatim. Fifth, the cell model is reduced to a single law: the per-cell dampings compatible with reflection positivity at every mode count are exactly the Bernstein-class exponents, and the remaining input is the linear water-level response of the channel noise—the canonical Bernstein function. Sixth, Euclidean restoration is quantified: the rotation-violating part of the regulated correlators dies as (ε/r)² up to a logarithm in the window, the dimension-six irrelevance that the programme's no-Lorentz-violation prediction requires. The frontier is restated in prioritized form: no existence questions remain open in the regulated and geometric sectors; the live residues are control.