Lemma: Bounded-generator channel stability

For finite self-adjoint paths $H,\hat{H}$,

The induced channels, with references, fixed ancillas, and partial traces, have half-diamond distance bounded by this quantity, capped at one.

Compatible complete channel comparisons telescope: their half-diamond errors add, capped at one.

Proof. Variation of constants gives

The unitary factors have norm one. Expanding the induced channel difference in two terms bounds its complete trace norm by twice the unitary operator-norm difference. Pre- and postcomposition by channels contract that norm. Inserting one changed factor at a time proves telescoping.