Theorem: Matched complete-channel stability

Suppose ideal and implemented stages are CPTP maps on matching complete interfaces, with

Then

For a matched finite adaptive tree with node allowances $e_v$, the bound is

Preparation and final-output allowances are added once on their actual comparison interfaces.

Proof. For a sequence, replace one factor at a time. Pre- and postcomposition by channels contract diamond norm; the triangle inequality gives the sum.

For a tree, a controlled continuation on classical direct-sum blocks has error at most the largest child error. Thus the recursive bound is

Backward induction gives the maximum root-to-leaf sum.