Failure branches and resource claims

A deterministic instrument has completely positive branch maps ${\mathcal{F}}_h$ with

Its complete retained-record channel is

Deleting failures and normalizing a success branch changes the problem. If success probability depends on the unknown input, the normalized conditional rule is not a linear CPTP decoder on that input.

Example ex:assistance shows that a supplied classical record can assist a quantum correction. It establishes neither reusable thermodynamic garbage nor a heat advantage. Any such claim would additionally need the success and failure probabilities, side-information access, initial resources, switching work, heat convention, and the cost of restoring every resource used over repeated cycles.