Proposition: Observation indistinguishability obstruction

Suppose a rule's geometric output must factor through a declared observation map $\mathrm{Obs}$. If there exist admissible configurations $x,x^{\prime }$ such that

but the target requires different geometric outputs at $x$ and $x^{\prime }$, then no rule with that observation boundary realizes the target on both configurations.

Proof. If the output is $\Phi (\mathrm{Obs}(x))$, equality of the observations forces equality of the outputs. This contradicts the target requirement.