Proposition: Equal records leave a residual gap

Two instruments can have identical classical records on every input while their residual states have trace distance $1/2$ on one qubit input. Their residual source expectations can therefore differ.

Proof. Use one coarse outcome and compare identity with computational dephasing. Both record channels are constant. On $|+⟩⟨+|$, their residuals are $|+⟩⟨+|$ and $I/2$. The difference is $X/2$, with eigenvalues $1/2,-1/2$. The trace distance is $1/2$, and the observable $X$ has expectations one and zero.