Proposition: No unique thermal ledger from the visible channel

Equality of the visible acquisition channels does not determine a unique bath heat, work cost, or cyclic reset cost.

Proof. Append to any implementation an independent fuel qubit, initially excited, and a designated Gibbs reservoir qubit, both with energy gap $ϵ>0$. At inverse temperature $\beta$, the reservoir excitation probability is $r=1/(1+e^{\beta ϵ})$. Swapping these two qubits changes the reservoir energy by $ϵ(1-r)$, while leaving the entire visible acquisition channel unchanged. Appending more such independent processes changes that designated ledger again.

These processes consume resources. They are not cyclic reset protocols unless the fuel, auxiliaries, and controls are also restored. The example proves underdetermination, not a violation of a thermodynamic inequality.