An exact two-mean law for singleton-avoidance probabilities

We determine the infimum of Pr(X=1,Z=0) over finitely supported nonnegative integer-valued pairs with real-stable probability generating polynomial and prescribed means (alpha,b). In the positive region 0<alpha<2 and 0<=b<min(1,2-alpha), the answer is the smaller of a degree-two value and an explicit categorical-Poisson limit. For alpha<=1 the limiting categorical trial has no X mass; for alpha>1 and b>0 it necessarily has positive X mass. A determinantal pinching preserves the coefficient and both means exactly. Three perturbations of a hypothetical finite global minimizer exclude all intermediate Bernoulli block sizes, using an analytic estimate for large blocks and three elementary polynomial factorizations for the remaining sizes. The bound is sharp even for finite determinantal measures with Hermitian positive-contraction kernels, counted on two disjoint blocks. The formula concerns unrestricted finite degree and one singleton target; no application to a TSP approximation constant is asserted.