D. Four interpretive counterterm categories

The local collision distributions (eq:L-deltaQ), together with the fixed defects of Lemma lem:Ldefect, carry four interpretive categories (attributes), which are kept distinct and are not asserted to form a disjoint partition of summands:

1. (1)

   Candidate bulk metric counterterms. Summands in which no derivative hits ${\chi }_{\mathrm{I}}$, and whose residual-metric jet dependence can be read as a candidate local density in $h$ (after diagonal restriction $w=y$). Preparation factors, if present, are evaluated on the collision diagonal; no claim is made that any such summand equals the variation of a bulk gravitational action.

2. (2)

   Preparation-dependent insertions. Summands whose coefficient retains nontrivial dependence on the ${\sigma }_{\epsilon }$-jet in $P_A(y)$. These are insertion/preparation counterterms, not pure bulk metric densities.

3. (3)

   Switch-derivative terms. Summands carrying ${\partial }^{{\gamma }^{\prime }}{\chi }_{\mathrm{I}}$ with $|{\gamma }^{\prime }|\ge 1$. These depend on the choice of interaction switch.

4. (4)

   Fixed differential-normalization defects. The distributions $D_{\alpha ,u}$ of Lemma lem:Ldefect. They are determined by $\varpi$ and the moment prescription (eq:L-moments); they are not free parameters of a coherent change $\Delta {\mathfrak{R}}_{\mathrm{L}}$.

Categories (1)–(3) parametrize coherent changes; category (4) is fixed once ${\mathfrak{R}}_{\mathrm{L}}$ is fixed. A single summand may carry more than one attribute. Do not identify all four with bulk gravitational action counterterms: only category (1) is even a candidate for such an identification, and no such identification is proved here (cf.\ XVI-K, Remark rem:amb(2)).