Definition: Regulated coordinate prescription

The coordinate functions ${{\mathsf{X}}_{\mathsf{q}}}^{\mu }=x^{\mu }+\kappa {{\mathsf{X}}_1}^{\mu }+{\kappa }^2{{\mathsf{X}}_2}^{\mu }$, $\mu =0,1,2,3$, are the formal solutions of

In the notation of the production plan, the operator is ${\mathcal{H}}_{g,\epsilon }[h]={\mathsf{D}}_{\epsilon }[q]$, the boundary operator is Cauchy restriction on $\Sigma$, and the data ${d^{\mu }}_{g,\epsilon }=(x^{\mu }{|}_{\Sigma },{{\delta }^{\mu }}_0)$ are independent of the configuration and of $g$.