Proposition: Exact finite diagonal comparison

For a start at one label and $n\le B-k_0$, let ${\Delta }_Z\rho =P_0\rho P_0+P_1\rho P_1$. Then

The geometric history laws agree exactly.

Proof. In each history block the difference is a Hermitian off-diagonal pair. Its trace-distance contribution is the absolute value of one off-diagonal entry. Each source step multiplies that entry by $r_{\eta }/2$; feedback changes only its phase. There are $2^n$ active histories. Summing gives (eq:diagonal-comparison). Off-diagonal entries have zero trace in every history.