Reflection Positivity Selects a Completely Monotone Regulator: A Channel-Capacity Damping of the Action Density for Scalar and Lattice-Gauge Measures

We prove that, for a Euclidean lattice measure whose two sides of a reflection plane communicate through a damping of the local action density, complete monotonicity of the damping is sufficient for reflection positivity at each fixed target dimension and necessary for the profile across all target dimensions. The Shannon capacity weight (1+ρ/ρ_c)^(−γ) is completely monotone, with an explicit Gamma-mixture representation. The saturated-barrier form is not, and fails already at N=1, hence at every target dimension. We establish exact finite-volume reflection positivity of the Shannon-damped measure about the symmetry plane of the lattice: for the scalar sector by a Fröhlich–Israel–Lieb–Simon factorization, and for lattice gauge theory with any compact Lie group by character positivity. The weight is bounded and polynomially suppressing: the lattice supplies the ultraviolet cutoff, and no ultraviolet improvement is claimed. The non-perturbative mass gap remains open and is delimited precisely.