Proof

Proof. Let the carrier be a qubit and $K$ a classical bit. An earlier allowed setting $h\in \{0,1\}$ writes $K=h$. Replace the carrier by $|0⟩⟨0|$, leaving $K$ untouched. Finally apply $X^K$ and measure the carrier in the computational basis. The outcome equals $h$.

Immediately after replacement, both carrier states are $|0⟩⟨0|$, but the joint states are

Each enlarged update is a specified Markov channel on $S\otimes K$. The later carrier law depends on history through $K$ despite the identical causal-break carrier state.

For the reset channel $\mathcal{R}(X)=|0⟩⟨0|\mathrm{Tr}X$, carrier trace distance becomes zero in each sector. The label remains perfectly distinguishable.