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

The fifth paper of this series closed the cell model, the integrable gauge action, and volume-uniform clustering, and reduced the remaining questions of the regulated sector of Absolute Frame Theory (AFT) to a phase question and a few control statements. This paper resolves the phase question and locates 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 with a parameter-uniform coefficient—the conditional variance scales as the square of the inverse capacity, by the bond's own determinant suppression, exactly cancelling the explicit inverse-square factor in the influence—whence the monotone-coupling recursion forces the extremal boundary states to coincide: 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, and 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 being 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 naming only the third as the Yang–Mills gauge wall.