The copied switching display

The set (eq:sampling-support) is not a claim that the full functional dependence on $\gamma$ is localized there: ${\rho }_{\xi }$ was itself propagated from $S_{\xi }$. Nor is it a support bound for the scalar response before smearing.

The condition (CP) uses $\mathrm{supp}(K{\rho }_{\xi })$, not $\mathrm{supp}{\rho }_{\xi }$. Thus its cancellation need not arise by deleting each term of (eq:companion-display) separately. For example, a locally constant tail of ${\rho }_{\xi }$ can make the local term and a propagated switching term cancel.

A useful configuration-independent sufficient condition is

It implies (CP) for every $\gamma$ for this fixed parameter and observation. We use (UCP) in the decisive witness.