Proposition: Residual-source separation with a robust gap

Equal record channels need not determine residual source functionals. There are qubit instruments with identical records and residual trace distance $1/2$ on one input. Implementation errors ${\epsilon }_1,{\epsilon }_2$ leave a gap at least $1/2-{\epsilon }_1-{\epsilon }_2$.

Proof. Use one coarse outcome and compare the identity residual channel with computational dephasing ${\Delta }_Z$. Both record channels are constant. On $|+⟩⟨+|$, the residuals are $|+⟩⟨+|$ and $I/2$; their difference is $X/2$, with eigenvalues $\pm 1/2$. The residual observable $X$ has expectations one and zero. The error statement is the triangle inequality.