Proposition: Record equality does not fix a residual instrument

Two instruments can produce the same record law for every input and nevertheless produce different residual states.

Proof. Use one deterministic outcome $\ast$. One instrument returns $|\ast ⟩⟨\ast |\otimes \rho$; the other returns $|\ast ⟩⟨\ast |\otimes {\Delta }_Z(\rho )$, where ${\Delta }_Z$ dephases a qubit. Both record laws are deterministic. For $\rho =|+⟩⟨+|$, the residual outputs are $|+⟩⟨+|$ and $I/2$, at trace distance $1/2$.