Remark: Effect versus integrated functional

A parameter-indexed family $C_{\xi }$ is not automatically the variation of one functional $F$. Calling $C_{\xi }=-{\mathfrak{B}}_{\xi }$ a “counterterm” does not establish ${\bar{\delta }}_{\xi }\bar{F}=-{\bar{\mathfrak{B}}}_{\xi }$. The latter is the integrated interface tested here.