Proposition: Record agreement does not determine residuals

Two channels can have identical classical records for every input and different surviving residual states.

Proof. For a qubit, let

and define channels to a one-label record and a surviving qubit by

Both records are always $\ast$. For $|+⟩=(|0⟩+|1⟩)/\sqrt{2}$, the residual states are $|+⟩⟨+|$ and $I/2$. Their difference is the Pauli matrix $X_{\mathrm{P}}/2$, with eigenvalues $\pm 1/2$. Thus their trace distance is $1/2$. The residual observable $X_{\mathrm{P}}$ distinguishes them.