Proposition: Reference-stable channel error ledger

Suppose two implementations have common declared input and output spaces, joint initial-state discrepancy at most $e_{\mathrm{prep}}$, and successive channel or recorded-instrument half-diamond discrepancies $e_1,\dots ,e_k$. Then their final complete-record laws differ by at most

For adaptive steps, the allowances must be uniform over the admitted histories and settings.

Proof. Insert ideal intermediate outputs one operation at a time. Trace distance contracts under CPTP maps. Each half-diamond error bounds the discrepancy on every allowed joint input, including an untouched reference Watrous. The triangle inequality gives the sum. Final record readout contracts trace distance to total variation.