Proposition: No unique implementation cost from a visible channel

Equality of a selected visible channel does not determine a unique heat or work ledger, even after a conserving spectator Hamiltonian is fixed.

Proof. Append independent qubits $W,L$ with $H_W=H_L=ϵ|1⟩⟨1|$, $ϵ>0$. Prepare $W$ excited and $L$ in ${\gamma }_L=\mathrm{diag}(1-p,p)$, $0<p<1$. Doing nothing and applying SWAP both conserve $H_W+H_L$ and leave the selected visible channel unchanged. The first has designated reservoir energy change zero, while the second has $Q_L=ϵ(1-p)>0$, supplied by the decrease of $W$'s energy. The consumed resource differs. These are not cyclic erasure protocols unless the spectator resources are restored.