# Verification of the risk bounds and likelihood transition

The written proof establishes the positive digamma kernel, strict bounds
on its inverse, the sharp global risk inequalities and both endpoint
constants. The finite-sample identity is exact for every nonconstant
positive sample. Uniform Gaussian exponential-moment bounds justify the
shrinking-variance expansion along every deterministic shrinking sequence.
The first fluctuation is the centered Hermite polynomial Z^3 - 3Z, whose
variance is six. Its centering is essential to all three selection regimes.

The symbolic checker passes 42 exact conditions in normal and optimized
Python with identical reports. It also evaluates 31 high-precision risk
regressions and nine finite-sample remainder checks. A seeded Monte Carlo
regression illustrates the finite-sample behavior but is not an analytic
proof or a pass/fail criterion for the limit theorem.

The fixed-variance marginal result requires bounded proper prior densities
on compact subsets of the open parameter spaces, with positive continuous
densities at interior pseudo-true points. It does not cover arbitrary
Bayesian priors in the shrinking-variance regime. The proof has been
audited by the originating researcher, not an independent reviewer or
proof assistant. The broad source record remains unresolved as phrased.