Proposition: No reset cost from channel equality

A reduced reset channel or partial trace does not determine a unique heat, work, or reusable-memory reset cost.

Proof. SWAP a memory with an independent blank qubit. The memory is reset and its referenced input is transferred to the other qubit. For equal gaps $H_M=H_L=\epsilon |1⟩⟨1|$, SWAP conserves their sum and transfers energy $\epsilon p$, where $p$ is the initial excitation probability. The same reset channel is obtained for every positive $\epsilon$. Independent fuel, bath, or actuator processes may also change the implementation ledger without changing that channel. The consumed blank and fuel must be restored before a cyclic cost is claimed.