Definition: Fixed-input Einstein jet

For a fixed free scalar ${\varphi }_0$, let ${\mathcal{J}}_g({\varphi }_0)$ consist of smooth formal coefficient families

such that the coefficients through ${\kappa }^2$ of $e=0$ and $E_{\varphi }=0$ vanish. Positive-order scalar coefficients have the X-A zero Cauchy data. No condition is imposed on metric Cauchy data.

This is the jet of the invariant Einstein equation, not of the gauge-fixed Euler equation. Requiring $D=0$ in addition is a possible further restriction, not a replacement of that equation.