The Capacity Regulator of Absolute Frame Theory: Complete Monotonicity, Exact Reflection Positivity, and the Invariant Channel Sector

A companion paper established finite-volume reflection positivity (RP) for the bounded-routing sector of Absolute Frame Theory (AFT) and conjectured that the finite capacity of the $\mathcal{M}$--$\mathcal{A}$ channel furnishes a physical, Lorentz-invariant regulator for the constructive programme. This paper resolves the reflection-positivity content of that conjecture. We first prove two delimiting results: the unmollified hard capacity barrier assigns zero weight to almost every configuration, whence the regulator carries an intrinsic resolution fixed by the action-quantization envelope; and no Euclidean-invariant Gaussian covariance can regulate at all, because reflection positivity forces a positive spectral measure whose decay is bounded by the free one. The compatibility question then has an exact answer: in any implementation that couples the two sides of the reflection plane through a damping of the local action density, reflection positivity selects the completely monotone class, i.e. positive mixtures of Gaussian (tension-renormalization) weights. The interior barrier of the saturated sector fails this criterion and belongs to the modular (horizon) regime, while the Shannon cost $\gamma\ln(1+\rho/\rho_c)$ passes it exactly. We prove exact, finite-volume reflection positivity of the Shannon-damped measure on the temporal lattice for the scalar (embedding) sector and, by a character-positivity argument free of special functions, for lattice gauge theory with any compact group. We then derive the regulator rather than postulate it: tracing the unresolved modes of a channel cell whose stiffness is displaced by capacity sharing yields exactly the Shannon weight, with all constants fixed by the programme. Finally we exhibit the invariant implementation: vertex operators of the radially mollified embedding on the Herglotz shell, and bounded composite Wilson loops of the normal-bundle connection, both inheriting reflection positivity from the existing measure with no lattice and no renormalization; the derived substratum dimension $N=10$ enters constructively through a codimension count. The geometric (induced-metric) sector and the non-perturbative mass gap remain open and are delimited precisely.