Proposition: Record equality does not fix the residual instrument

Two complete one-outcome instruments can have identical record laws for all inputs while producing different residual states.

Proof. Take the channels ${\mathrm{id}}_2$ and ${\Delta }_Z(\rho )=P_0\rho P_0+P_1\rho P_1$, each with the single reported outcome $\ast$. Both report $\ast$ with probability one. For $\rho =|+⟩⟨+|$, their residual states are $|+⟩⟨+|$ and $I_2/2$, at trace distance $1/2$.