Proposition: Action route for the coordinate operator

For plateau transformations the finite variation of ${S_{\epsilon }}^{\mathrm{aux}}$ vanishes modulo ${\kappa }^3$. Variation in $Z^{\mu }$ then proves Lemma lem:plateaucoord without assuming it. Consequently Route II, together with Cauchy uniqueness, also yields Theorem thm:main.

Proof. For each $Z^{\mu }$ repeat (eq:varied) with $m=0$ and the multiplier $g$ replaced by $\sigma$. The terms contain $1-\sigma$ or $\partial \sigma$ on the compact gauge support, where both vanish. Differentiate this finite action identity with respect to compact variations of $Z^{\mu }$, transforming the variation by $F^{\ast }$. The Euler derivative is $-D_{\epsilon }[q]Z^{\mu }$, so the density-valued operator covariance follows by change of variables.

The coordinate and scalar equations obtained this way have the same fixed initial data as in Route I. Their unique solutions therefore have the same intertwiners. Relational inversion and composition give the observable identity.