Proposition: No implementation cost from channel equality

Equality of a selected visible channel does not determine a unique reservoir heat, work-storage change, or reusable-memory reset ledger, even when the appended complete dynamics conserves a specified total energy.

Proof. Take any fixed visible implementation and append independent qubits $W,L$ with

Prepare $W$ in $|1⟩$ and $L$ in its Gibbs state ${\gamma }_L=\mathrm{diag}(1-p,p)$, with $0<p<1$. Compare doing nothing on $WL$ with swapping $W$ and $L$. Both spectator operations conserve $H_W+H_L$, and neither changes the selected visible channel.

The first operation has $Q_L=0$. The swap leaves $L$ excited and has $Q_L=ϵ(1-p)>0$. The battery's energy decreases by the same amount. The two implementations therefore have different heat and resource-return ledgers despite the same visible channel. The swap consumes a resource; it is not claimed to be a cyclic cost optimum. Precisely that missing resource-restoration specification prevents a reset-cost inference from channel equality.