Prime-Faced Dice and Independence: Why Fairness Matters

Four integrals, each worked both ways, and the chain-rule test that tells you when to reach for the larger chunk.

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.