The Capacity Regulator of Absolute Frame Theory II: Infinite-Volume States, the Geometric Measure, and the Mass Gap as a Spectral Question

The companion paper proved exact, finite-volume reflection positivity for the capacity-regulated scalar and gauge sectors of Absolute Frame Theory (AFT), with the Shannon damping derived from the channel cell model and an invariant implementation requiring no lattice. This sequel carries the construction across its next three frontiers and sharpens the last one. First, the thermodynamic limit: on growing four-tori, the uniform boundedness of the channel observables makes compactness soft, and we prove that subsequential infinite-volume limits exist as states on the channel algebra, reflection-positive at every link plane, translation-invariant, with a nontrivial physical Hilbert space and a positive self-adjoint transfer contraction---the Osterwalder--Schrader data short of clustering, whose absence is delimited rather than blurred. Second, the geometric sector: a pointwise domination lemma shows that the derived per-cell damping tames density powers up to half the cell mode number for any base measure, suppresses super-capacity spikes exponentially in their volume, and dominates the induced-volume density through an arithmetic--geometric mean inequality that the programme's own mode count satisfies; consequently the Nambu--Goto-reweighted regulated measure exists as a probability measure at fixed physical resolution, with no renormalization. Third, dynamics as analysis: we give the dictionary that reformulates every dynamical question of the gauge sector as a property of objects that already exist, prove the equivalence between exponential clustering and a spectral gap of the existing transfer operator, and close the dictionary in the solvable venue---two-dimensional lattice gauge theory---where the capacity-damped weight yields an exact area law with a computable, softened string tension. Fourth, the geometric positivity question is answered in its mixture form: the Nambu--Goto density is not the Laplace transform of any positive measure on the cone of positive matrices, by an explicit negative mixed partial derivative on a diagonal slice; the matrix-mixture route to geometric reflection positivity is thereby closed analytically, in two and four dimensions alike, with a structural echo for string theory: the saddle-point equivalence between Nambu--Goto and Polyakov actions does not lift to a positive-mixture identity between measures. The remaining wall is stated precisely, together with the decoupling programme whose expansion parameter is physical: the inverse coherence bound $1/N_{\mathrm{crit}}=\kappa/\rho_c$.