Theorem: Every symmetric point source has a formal matching jet

For $m\ge 3$, the map ${\mathsf{E}}_{\eta }:{\mathcal{J}}^2\to {\mathrm{Sym}}^2(V_D^{\ast })$ is surjective. For every symmetric tensor $b_{\mu \nu }$, there is a polynomial conformal perturbation with

Thus existence of an unrestricted matching jet at one point, without an independently extracted jet rule or compatibility conditions, is not a substantive dynamics-matching theorem.

Proof. Let $b={\eta }^{\mu \nu }{b}_{\mu \nu }$, and define the symmetric tensor

Its trace is

Set

Equation (eq:conformal-einstein) gives

Since the coupling is nonzero and $b$ was arbitrary, the map is surjective.