Theorem: Classical identity through κ 2

1. $\mathcal{C}$ is ${\mathcal{G}}_U$-invariant: if $(h,k,\varphi )\in \mathcal{C}$ and $(h^{\prime },k^{\prime })$ is the complete two-jet transform $h^{\prime }=h+K\xi$, $k^{\prime }=k+{\mathcal{L}}_{\xi }h+K\zeta +\frac{1}{2}{\mathcal{L}}_{\xi }K\xi$, $K\xi ={\mathcal{L}}_{\xi }\eta$, with ${\varphi }^{\prime }=F^{\ast }\varphi$, then $(h^{\prime },k^{\prime },{\varphi }^{\prime })\in \mathcal{C}$.

2. X-A's relational scalar and X-C's completed classical observable are ${\mathcal{G}}_U$-invariant through ${\kappa }^2$ for all smooth $(h,k)$, in particular on $\mathcal{C}$. The gravitational action enters neither object.

3. The varied-action identity of X-A's Route II holds for the total action with zero defect on the plateau. Off the plateau the defect is the right side of (eq:noether), recorded for all $(h,k)$, on or off $\mathcal{C}$.

Proof. (i) is Proposition prop:covariance: $F^{\ast }$ is invertible order by order and maps zero to zero. (ii) is X-A (iv) and X-C's fixed-regulator identity, both proved off shell in the metric; X-D's Corollary on the fixed algebra gives the explicit equation-ideal representatives. (iii) is Lemma lem:invariance together with Proposition prop:noether.