Lemma: Coverage and descent

The raw contacts span $\mathcal{V}$, every raw contact is allowed, and their evaluation descends to $\mathcal{V}$. Thus the catalogue satisfies all three clauses of Definition def:complete, with unrestricted coefficient space ${\mathbb{R}}^{15120}$.

Proof. If both derivatives act on one factor, integrate one derivative by parts. This gives a sum with derivatives on distinct factors, modulo a divergence. Relabel those factors into positions one and two. The remaining ten tensor slots are completely paired by metrics and hence occur in (eq:raw). Every even collection of orientation tensors reduces to metric pairings by repeated use of the determinant identity. Species words exhaust all assignments.

Conversely each raw expression has the declared locality, field degree, derivative degree, covariance, and parity.

After differentiation, a total derivative has the overall factor $i{\sum }_j{k}_{j\mu }=0$. Tensor identities vanish before and after evaluation. Hence action-equivalent representatives have identical responses.