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

Abstract

Version of August 21st, 2026, following the development within the Absolute Frame Theory. A companion paper established finite-volume reflection positivity (RP) for the bounded-routing sector of Absolute Frame Theory (AFT). That paper also conjectured that the finite capacity of the ℳ–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. Moreover, 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. The Shannon cost γ ln(1+ρ/ρ_c), by contrast, passes it exactly. We then prove exact, finite-volume reflection positivity of the Shannon-damped measure on the temporal lattice for the scalar (embedding) sector. A character-positivity argument free of special functions extends that result to lattice gauge theory with any compact group. We next 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 inherit reflection positivity from the existing measure with no lattice and no renormalization. The derived substratum dimension N=14 enters constructively through a codimension count. The geometric (induced-metric) sector and the non-perturbative mass gap remain open and are delimited precisely.