Theorem: Second-order classical covariance

At fixed $\epsilon >0$, for the frozen data above:

1. the coordinate and switched scalar coefficient problems have their stated unique solutions;

2. preflight is exact on ${\mathfrak{g}}_p$;

3. for every off-shell scalar jet and $F\in {\mathcal{G}}_p$,

   $${\Phi }_{g,\epsilon }[F^{\ast }q,F^{\ast }\varphi ]={\Phi }_{g,\epsilon }[q,\varphi ](\mathrm{mod}{\kappa }^3),$$

   and hence

   $${\mathcal{O}}_{f,g,\epsilon }[F^{\ast }q,F^{\ast }\varphi ]={\mathcal{O}}_{f,g,\epsilon }[q,\varphi ](\mathrm{mod}{\kappa }^3);$$

4. with fixed free input, the dynamical evaluation

   $${\mathcal{O}}_{f,g,\epsilon }[q,{\rho }_g[q;{\varphi }_0]]$$

   is invariant under the same group.

The metric is off shell throughout. No quantum object is part of this theorem.

Proof. Existence and preflight have been proved. Formal inversion of (eq:intertwiner) yields

Since $F^{\ast }\varphi =\varphi \circ F$,

Pairing with the unchanged $f$ proves the off-shell statement. Substitute (eq:scalarintertwiner) to prove the dynamical one.