Admissible changes must be complete

At the selected order, let $C$ be the full allowed counterterm vector space after reconstruction relations, exchange symmetry, derivative identities, reality, support, and locked subgraph normalizations are imposed.

A bookkeeping decomposition

need not be direct. Integration by parts and field redefinitions are used only when authorized in this space.

A collision-supported tensor is not automatically the Euler derivative of a local bulk functional. Variational integrability and symmetry compatibility are separate tests CheungXVIL,CheungXVIM.