Proposition: No thermal cost from a channel identity alone

A reduced channel does not determine its heat, work, or reusable-memory reset cost.

Proof. SWAP with an independent blank qubit implements reset of the observed memory while transferring its referenced input to the blank. With equal Hamiltonians $H_M=H_L=ϵ|1⟩⟨1|$, it conserves their sum and transfers energy $ϵp$ for excitation probability $p$. The same reset channel is obtained for every positive $ϵ$. Independent unobserved fuel or reservoir processes can also be appended without changing the observed channel. The blank and fuel resources must be restored before a cyclic cost is claimed.