Proposition: The descendant qualification

Deleting the label $Q$ does not remove the virial current if the declared field list still contains ${\partial }_{\mu }Q$. A list omitting both $Q$ and the needed bilinear descendants can fail to contain the displayed physical virial current.

Proof. The identity

proves the first assertion.

For the second, a linear field-label list consisting only of the identity, the linear field $\varphi$ and its descendants, and $T_{\mu \nu }$ and its descendants need not contain the bilinear dimension-three vector $:\varphi {\partial }_{\mu }\varphi :$. It is neither a linear-field descendant nor a dimension-four stress-tensor descendant. This is a restricted label space, not a claim that it is closed under Wick products or coincident products.