Proposition: Record equality is weaker than residual equality

Two finite instruments have the same record laws on all inputs exactly when their outcome effects agree. Agreement on all recorded residual observables requires agreement of the corresponding adjoint maps on those observables. Equal records alone do not suffice.

Proof. Pull the record effects or recorded residual observables back to the input, and use separation by density matrices. For a strict example, identity and computational dephasing with the same one-outcome record have equal record law. On $|+⟩⟨+|$, the residuals are $|+⟩⟨+|$ and $I/2$, at trace distance $1/2$.