Proposition: Observable geometric response is a stronger witness

If some $O_G\in {\mathcal{O}}_G$, with ${\Vert O\Vert }_G\le 1$, and some code states satisfy

then the complete residual is nonconstant and exact whole-code recovery from $R$ is impossible.

The converse is false.

Proof. Lift the observable to $O_G\otimes I_E$ and apply Lemma lem:witness and Theorem thm:exact.

For the converse, take

The complete residual recovers the diary and is nonconstant, but ${\mathcal{C}}_G={\mathcal{T}}_{|\mathrm{vac}⟩⟨\mathrm{vac}|}$. Every declared geometric observable is input independent.