Theorem: Surjective-orbit negative closure

Let $\mathsf{Enc}:\mathbb{R}\to \mathcal{S}$ be any encoding and let $\Phi :\mathcal{S}\to \mathcal{Y}$ be any declared output map. Under Assumption ass:convolution,

implies

Consequently no scalar-only readout is both invariant on the full witness orbit and informative about its input.

Proof. For any $a\in \mathbb{R}$, take $\lambda =a/c_{\epsilon }$. This is possible because $c_{\epsilon }>0$. Substitution proves the assertion.