Proposition: Complete implementation-error ledger

For a fixed compatible sequence of ideal channels ${\Phi }_1,\dots ,{\Phi }_N$ and implementations ${\tilde{\Phi }}_1,\dots ,{\tilde{\Phi }}_N$,

Records, histories, failures, and work outputs must remain on the same comparison interfaces.

Proof. Subtract the compositions and replace one factor at a time. Each summand has one difference of corresponding factors with channels before and after it. Diamond-norm contraction under channel composition and the triangle inequality give the bound. Half-diamond distance between channels is at most one.