Declared local contact class

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

Equivalently, the declared tensor-contraction class uses complete metric contractions and even numbers of Levi-Civita tensors. The metric-pairing description is the standard orthogonal invariant-tensor framework Brauer; no independence of those pairings is assumed.

Let ${\mathfrak{P}}_{10}$ be the perfect matchings of $\{1,\dots ,10\}$. For $I=(I_1,I_2,I_3,I_4)\in \{1,2{\}}^4$ and $P\in {\mathfrak{P}}_{10}$, define

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