Theorem: X-A classical covariance

Fix the data of Section sec:sheet: the slab $M_T$, the formal metric jet $q=\eta +\kappa h+{\kappa }^{2}k$, the switch $g$, the regulator ${\sigma }_{\epsilon }$, the plateau $U$ with $\overline{U}\subset P_g\cap \{\epsilon <t<T\}$, the gauge group ${\mathcal{G}}_U$, the switched scalar action ${S_g}^{\mathrm{sc}}$, the regulated coordinate prescription of Definition def:coordinates, and one label-space test $f\in {C_c}^{\infty }({M_T}^{\circ })$. Then:

1. the regulated coordinate problem has a unique formal solution ${\mathsf{X}}_q=x+\kappa {\mathsf{X}}_1+{\kappa }^2{\mathsf{X}}_2(\mathrm{mod}{\kappa }^3)$, with explicit zero-data Cauchy coefficients;

2. ${\mathsf{X}}_1[K\xi ]=\xi$ for every $\xi \in {\mathfrak{g}}_U$;

3. for every $F\in {\mathcal{G}}_U$, ${\mathsf{X}}_{F^{\ast }q}=F^{\ast }{\mathsf{X}}_q(\mathrm{mod}{\kappa }^3)$, with coefficient laws (eq:X1law)–(eq:X2law);

4. for every configuration $(h,k,\varphi )$ and every $F\in {\mathcal{G}}_U$,

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

5. the switched scalar solution with fixed free input satisfies ${\rho }_g[F^{\ast }q;{\varphi }_0]=F^{\ast }{\rho }_g[q;{\varphi }_0](\mathrm{mod}{\kappa }^3)$, and ${\mathcal{O}}_{f,g,\epsilon }[q,{\rho }_g[q;{\varphi }_0]]$ is invariant under ${\mathcal{G}}_U$;

6. the varied switched action is an explicit sum of terms carrying a factor $1-g$ or $\partial g$, and vanishes identically on ${\mathcal{G}}_U$;

7. the coefficients depend on the configuration only in ${\mathcal{D}}_f=J^{-}(\mathrm{supp}f)\cap M_T$, are continuous in the smooth topologies at fixed $\epsilon$, and their first functional derivatives at fixed configuration are smooth compactly supported kernels.

No quantum object is part of the statement.