The varied action (Route II)

This is the classical Noether identity of the frozen prescription. It is not a quantum Ward identity. Its first-order defect (eq:firstorder) for parameters leaving the plateau is the same structure as the switched Noether defect of VI, in the present signature and switching convention.

The coordinate prescription is not action-derived in the sense of (eq:action); its covariance was proved in Lemma lem:plateaucoord. The two routes are independent: Route I proves invariance of the observable through the defining equations and data; Route II proves that the dynamics selecting the on-shell evaluation is itself invariant, with all switch terms displayed.