When a Bigger U-Substitution Makes Calculus Easier

Grimmett and Welsh's prime-faced die exercise has a two-line proof and a hidden assumption. Here is where the prime does its work, and where the claim breaks.

A cyclotomic-style denominator and a ln x term look like a convergence problem but are not. The real work is tracking one parameter shift and one factor of 1/6 through a digamma-to-log-gamma step.

Paulin's conjugation-based Lie algebra needs Ad to be smooth before it can be differentiated. Here is where the partial differential comes from, and why separate smoothness is not enough.