Necessary specifications

Several choices are mathematically essential.

1. A first-order response does not determine a second-order action. We use a single switch multiplying the entire difference between the curved and flat scalar Lagrangian densities.

2. The metric is a genuine second-order jet

   $$\mathbf{g}=\eta +\kappa h+{\kappa }^{2}k+\mathcal{O}({\kappa }^3).$$

   There is no factor of $1/2$ in the definition of $k$.

3. The scalar Euler operator is density valued. Its infinitesimal transformation consequently contains ${\partial }_{\mu }{\xi }^{\mu }$.

4. Compactly supported vector fields form a Lie algebra. The admitted group is generated by their flows; its second-order formal jets include both a first-order and a second-order gauge parameter.

5. Every inverse used below has prescribed Cauchy data. No identity of the form $\mathsf{E}P=\mathrm{id}$ is used on functions with unspecified initial data.