Contact class and three-clause completeness

Let $\mathcal{V}$ be the real vector space of local quartic density classes with exactly two derivatives, constant coefficients, no extra fields, no inverse operators, and no background tensors other than $\eta$ and the orientation tensor. The densities are translation invariant, Lorentz scalar, and parity even. The quotient is by total derivatives and genuine tensor identities, not by equations of motion or field redefinitions.

For a species word $\mathbf{I}=(I_1,I_2,I_3,I_4)\in \{1,2{\}}^4$ and a perfect matching $P$ of ten slots, define

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