Matching questions for each candidate collision term

Write $\Delta Q_{\mathrm{L}}$ in the four interpretive categories of Section sec:l5D, and let $T$ be any individual summand (or any coherent linear combination that one proposes to match). Before any claim that $T$ is a gravitational or insertion counterterm, the following questions are to be answered affirmatively or negatively with a written argument; none is answered here.

1. (Q1)

   Local metric functional? Can $T$ arise as the metric variation (or as a contribution to the metric variation) of a local functional of the residual metric $h$, after the collision diagonal $w=y$ is taken and after preparation and switch factors are specialized? Only class (1) of Section sec:l5D is even a candidate; classes (2)–(4) are not bulk metric densities by construction.

2. (Q2)

   Composite-insertion renormalization? Is $T$ instead a renormalization of the composite insertion carried by the complete source ${\mathcal{J}}_{\mathrm{D}}$ (preparation-dependent counterterms of class (2)), rather than a bulk gravitational action counterterm?

3. (Q3)

   {Essential ${\sigma }_{\epsilon }$ dependence?} Does the coefficient of $T$ depend essentially on the preparation ${\sigma }_{\epsilon }$ (or on its jet), in the sense that it fails to factor through a ${\sigma }_{\epsilon }$-independent local density of $h$ after the fixed preparation of the Scope box is restored? Essential ${\sigma }_{\epsilon }$ dependence places $T$ in class (2), not in class (1).

4. (Q4)

   {${\chi }_{\mathrm{I}}$ derivatives?} Does $T$ involve derivatives ${\partial }^{{\gamma }^{\prime }}{\chi }_{\mathrm{I}}$ with $|{\gamma }^{\prime }|\ge 1$ (class (3))? Such terms depend on the choice of interaction switch and are not bulk metric counterterms.

5. (Q5)

   Removable only by IBP / field-equation reduction? Is $T$ removable only by an integration-by-parts identity or by a field-equation reduction that the present off-shell setup forbids? XVI-L works with an off-shell smooth residual metric $h$ and with the printed density representative of the cubic vertex (no IBP; cf.\ XVI-K, Proposition prop:hess, and the artifact boundary of Appendix app:script). A term that vanishes only after IBP or after imposing the linearized Einstein equation is not identified with zero in this note, and is not thereby matched to a bulk on-shell counterterm.

6. (Q6)

   Additional graphs and ghosts for a symmetry test? What additional graphs, gauge-fixing data, and ghost (or antifield) contributions would be required before a gravitational symmetry test — diffeomorphism Ward identity, BRST cohomology statement, or BV master-equation compatibility — could be stated for the sector containing this graph? Answering (Q6) is a prerequisite for any claim that a matched counterterm is symmetry-compatible; it is not supplied by extending numerical distributions.