Remark: Compatibility is separate from extending distributions

Extending the numerical kernels ${q^{\circ }}_{\lambda }\in {\mathcal{V}}_{\mathrm{adm}}$ to ${\mathfrak{R}}_{\mathrm{L}}({q^{\circ }}_{\lambda })\in {\mathcal{D}}^{\prime }({\mathbb{R}}^4)$ (L.4), classifying coherent contact changes (L.5), introducing a scale family (L.6), and attaching the imported root (L.7) are distributional constructions. Compatibility of those constructions with a BV gauge-fixed gravitational action — anomaly analysis, symmetry-restoring counterterms, ghost contributions — is a separate task. No result of L.1–L.7 is a BRST or BV theorem, and none is transferred from the background orientation named in Section sec:l0.