Lemma: Surjective-orbit obstruction

Let $\iota :\mathbb{R}\to \mathcal{S}$ be any encoding into a state space and let $\Phi$ be any instrument with a declared record map. If

is constant, then $\Phi \circ \iota$ is constant on $\mathbb{R}$.

Proof. For each $a\in \mathbb{R}$, choose $\lambda =a/c_{\epsilon }$. This is possible because $c_{\epsilon }>0$. Orbit constancy then gives $\Phi (\iota (a))=\Phi (\iota (0))$.