Remark: Structural route, named

A second proof of Theorem thm:main would run: the switched gauge-fixed system is exactly ${\mathcal{G}}_U$-covariant when $(h_{\mathrm{cl}},k_{\mathrm{cl}},u,\varphi )$ all pull back (Lemma lem:natural and X-E's Lemma Exact plateau invariance, with ${\sigma }_{\epsilon }\circ F^{-1}={\sigma }_{\epsilon }$ since ${\sigma }_{\epsilon }=1$ on $U$); retarded uniqueness identifies the Møller solution at the transformed background with the pull-back of the Møller solution, both having the same free data on $t<2\delta$; X-C's off-shell invariance of the classical completed observable then gives invariance of the classical value on every solution pair; and the ${\hslash }^0$ part of the retarded products is the classical Møller composition DF. This would need the converse of Lemma lem:gslot's last statement (vanishing on all solution pairs implies membership in $\mathfrak{J}$) and would hold to all orders in $\kappa$ at ${\hslash }^0$. Neither ingredient is proved here; the direct proof is the one released.