Sharp Risk Bounds and a Shrinking-Variance Transition for Normal and Gamma Fits

Suppose lognormal data are fitted by an ordinary normal model and a shape-scale gamma model. If \(s>0\) is the log variance, the difference \(\Delta(s)\) between their optimized forward Kullback-Leibler risks satisfies \(2s/3<\Delta(s)<s\) and \(2/3<\Delta'(s)<1\); both constants are optimal. We establish the boundary behavior of the maximized gamma-to-normal log-likelihood ratio \(R_n\) when \(s=s_n\) tends to zero. If \(n s_n\) tends to \(c\in(0,\infty)\), \(R_n\) converges in distribution to \(N(2c/3,2c/3)\). If \(n s_n\) tends to infinity, gamma is selected with probability tending to one; if \(n s_n\) tends to zero, that probability tends to one half. An exact profile identity and uniform sample-moment expansion give the result. Fixed-variance likelihood and compact-prior marginal asymptotics are also included. The KL projection formulas and general misspecification theory are established prior ingredients. This is a scoped response to AIM Singular learning theory Problem 1.3, indexed as AIM-ALGEBRAIC_GEOMETRY-0271 in UnsolvedMath v1.6.0, not a resolution of every interpretation of the source prompt. The preprint is unrefereed and makes no absolute priority claim.