Lemma: Completeness of history retention

The map ${\mathcal{H}}_n$ is a channel. Summing its blocks with a common terminal label gives the ordinary terminal hybrid state.

Proof. Each path map is completely positive. Summing probabilities over the final label uses (eq:instrument-complete); repeating this sum backwards over the path leaves ${\sum }_{g_0}\mathrm{Tr}{\rho }_{g_0}=1$. This proves trace preservation. The terminal summation is precisely repeated substitution in (eq:hybrid-map).