Conventions

Use $\eta =\mathrm{diag}(1,-1,-1,-1)$ and $\square ={{\partial }_t}^2-{\Delta }_{\mathbf{x}}$. Let $R(x-v)$ denote the kernel of ${E_0}^{\mathrm{R}}$. Define constant fiber maps by

where $I$ ranges over symmetric metric components. All sums over $I$ use the inherited tensor pairing.

The coordinates $(a,z)$ are the two metric field slots: $h_I(a)$ is the contracted direction and $G_J(z)$ the residual direction. They are not replaced by a smooth residual test in the kernel formulas.