Proposition: Reference-stable implementation ledger

Suppose each input preparation has trace-distance error at most $e_{\mathrm{prep}}$, and the successive common-space channels or complete recorded instruments have half-diamond errors $e_1,\dots ,e_k$. The final complete-law error is at most

truncated at one.

For bounded generators with CPTP propagators,

Proof. Insert intermediate ideal outputs one map at a time. Trace distance contracts under channels, while a half-diamond error bounds every normalized input, including referenced inputs. The triangle inequality gives the telescoping ledger. Final measurement contracts it to total variation.

For the generator estimate, variation of constants gives

Channel composition has diamond norm one. Take norms and divide by two Watrous.