Lemma: Raw contact coverage

The classes of the raw functionals (eq:raw) span $\mathcal{V}$. Every raw functional is an allowed contact.

Proof. If both derivatives act on one factor, integration by parts expresses the density modulo a divergence as a sum with derivatives on two distinct factors. Relabel those factors into positions one and two. There are then ten tensor slots: two derivative indices and eight field indices. Every complete metric contraction pairs those slots and is one of the perfect matchings.

A pair of Levi-Civita tensors reduces, by the determinant identity, to a signed sum of metric contractions. Iterating removes every even number of orientation tensors. This covers the declared parity-even contraction class. Species choices give every word in $\{1,2{\}}^4$.

Conversely, every raw expression has the declared field degree, derivative degree, locality, covariance, parity, and constant coefficients.