Proposition: A reduced channel does not determine a thermal cost

Equality of reduced channels, including a reset channel or partial trace, does not determine heat, work, or cyclic reusable-memory reset cost.

Proof. Let $M$ be a qubit and $L$ an independent blank qubit. For every referenced input,

The reduced channel on $M$ is always reset. Choose

The swap conserves their sum. If the initial excitation probability is $p$, the energy changes are

The same reset channel therefore accompanies different energy transfers as $ϵ$ varies.

Even with these Hamiltonians fixed, append an independent fuel qubit initially excited and a Gibbs reservoir qubit with the same gap. Swapping them changes reservoir energy by $ϵ(1-r)$, where $r=(1+e^{\beta ϵ}{)}^{-1}$, without changing the observed memory channel. Such spectator processes alter an implementation ledger while leaving its reduced channel unchanged.

These examples consume resources. They are not cyclic erasure protocols until blanks, fuel, and controls are restored.