The decision

Neither hypothesis is a shrinkage of ${\mathfrak{g}}_U$. (CP) is a condition on the observation and the switch; $A\gamma =0$ on $U$ is a condition on the configuration. The compact shift $K\xi$ does not preserve ${\mathcal{C}}_{\mathrm{sol}}$ (XI-B: ${\mathfrak{g}}_U\cap \{\square \xi =0\}=\{0\}$), so no action of ${\mathfrak{g}}_U$ by automorphisms of ${\mathcal{E}}_H$ is inferred, and no interacting differential or quantum Ward identity is hidden in Theorem thm:plateau.