The Capacity Regulator of Absolute Frame Theory VI: Single-Phase Bulk, the Modular Horizon, and Euclidean Restoration

October 27th, 2026: updates following the development of the Absolute Frame Theory.

The fifth paper of this series closed the cell model, the integrable gauge action, and volume-uniform clustering. It thereby reduced the remaining questions of the regulated sector of Absolute Frame Theory (AFT) to a phase question and a few control statements. In this paper we resolve the phase question and locate the residue precisely. First, the bulk is single-phase. The conductance marginal of the regulated measure is FKG, and its single-site conditional-mean map is an ℓ∞-contraction whose coefficient is bounded across the control parameters on the physical range. The conditional variance scales as the square of the inverse capacity, up to a logarithmic correction at the marginal fibre exponent, by the bond's own determinant suppression, and this cancels the explicit inverse-square factor in the influence. Hence the monotone-coupling recursion forces the extremal boundary states to coincide, giving a unique Gibbs measure at every activity and up to strong spatial coupling. Second, the saturation boundary is not a Landau transition of the bulk but a change in the type of the local von Neumann algebra. The channel observable algebra and its commutant are the two factors of the tensor split into the tangent (accessible) and normal (the inaccessible, Gödelian fiber) directions of the derived dimension N=14=4+10, exactly at finite resolution. In the thermodynamic limit the infinite tensor product carries the Araki–Woods asymptotic ratio set of the channel thermal state to type III₁. Moreover the crossed product by the modular flow, with the saturation cap bounding the boost-charge spectrum, returns a type II₁ algebra with a finite trace. The commutant is, throughout, the inaccessible-fiber algebra: the operator-level content of the modular reading of black-hole information in this programme. Third, the angular KMS structure of the geometric sector survives the geometric weight as a discrete, collar-stepped modular semigroup of positive self-adjoint rotations, 2π-periodic. The continuous boost is obstructed at finite tension by the localized modular mark. Fourth, full Euclidean invariance is restored. The physical channel observables are exactly invariant in the continuum, with no lattice, while the lattice geometric sector restores rotational invariance as a dimension-six irrelevant flow controlled to all correlators by the cluster expansion. We close by decomposing the constructive residue into three classes (closed, AFT-specific but reduced at the physical resolution, and the single universal debt), and we name only the third as the Yang–Mills gauge wall.