Proposition: Stability of the named jet

The transformations (eq:metric-transform) and (eq:intertwiner-coeff) preserve ${{\mathcal{J}}_g}^2$. The relational coefficients therefore descend on its ${\mathcal{G}}_U$-orbit set.

Proof. The scalar equations and data are preserved by Proposition prop:intertwiners.

For the metric, pullback preserves the Levi–Civita connection because it preserves torsion freedom and metric compatibility. It therefore preserves the curvature and Einstein tensor. Taking coefficients gives

The last term vanishes on (eq:metric-equations). The leading scalar remains ${\varphi }_0$, so its order-two source is unchanged. The final assertion follows from Theorem thm:second.