A Sharp Complete-Monotonicity Threshold for a Reciprocal Kummer Family

We establish the exact parameter range for complete monotonicity of the reciprocal Kummer family [F_λ(x)=[1+2λ x,{}_1F_1(1;3/2;x)]⁻¹, x>0, λ≥0.] The sharp threshold is (λ_=0.778361888421681…): complete monotonicity holds if and only if (λ=0) or (λ≥λ_). The central contribution is the proof of global sufficiency at the critical parameter (λ_<1), and throughout the interval (λ_≤λ<1). The proof establishes strict positivity of a critical causal Green density through spectral zero exclusion, boundary Laplace inversion, explicit cancellation of the first discrete-delay contribution, rigorous interval estimates, and analytic tail bounds. As an application, we obtain an exact positivity criterion for a truncated half-order memory–delay operator and determine its optimal weighted (L^p) inverse norm, including quadratic divergence at the critical exponential weight.