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_\lambda(x)=\bigl[1+2\lambda x,{}_1F_1(1;3/2;x)\bigr]^{-1},\qquad x>0,\quad \lambda\geq0.] The sharp threshold is (\lambda_=0.778361888421681\ldots): complete monotonicity holds if and only if (\lambda=0) or (\lambda\geq\lambda_). The central contribution is the proof of global sufficiency at the critical parameter (\lambda_<1), and throughout the interval (\lambda_\leq\lambda<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.