Stein Solution Factors via Mills Ratios: Affine Birth Rates and Symmetric Potential Distributions

We study Stein solution factors for indicator test functions by a common Mills-ratio method in discrete and continuous settings. In the discrete case, a general birth–death formulation gives exact solution envelopes and recovers the first-increment theory of Brown and Xia. For binomial and Poisson targets, and for negative binomial targets with shape parameter $r\ge 1$, the additional algebraic structure of affine birth rates yields improved uniform bounds for the Stein solution. In the continuous case, we consider symmetric densities proportional to $e^{-V}$. Under natural structural assumptions on the potential $V$, Mills ratios yield explicit finite bounds for the indicator Stein solution and show that the uniform derivative and drift Stein factors have the sharp value $1$. Under an additional one-crossing condition, the solution-factor optimization can be carried out exactly and gives the optimal value $1\mathrm{/}(4p(0))$. The even-power targets arising in statistical mechanics, including the quartic critical Curie–Weiss law, provide the original examples, and the calculation extends to the wider Subbotin family. For $1<\beta <2$ in the Subbotin family, the exact envelope has two off-center maximizers characterized by a unique incomplete-gamma equation, while the two first-order factors remain equal to $1$. The results exhibit a common discrete–continuous mechanism behind improved and, in the continuous sharp regime, optimal Stein solution factors.