Corollary: Vanishing on the frozen group; the Noether identity

For $F\in {\mathcal{G}}_U$, ${\delta }_F{S}_g^{\mathrm{sc}}[q,\varphi ]=0(\mathrm{mod}{\kappa }^3)$ identically on $\mathcal{C}$, off shell. Differentiating this identity in $\varphi$ recovers (eq:plateaueuler). In particular the first-order switch defect (eq:firstorder) vanishes because $1-g$ and $\partial g$ vanish on $\mathrm{supp}\xi \subset U$.

Proof. On $U\supset \mathrm{supp}(F-\mathrm{id})$, $1-g=0$ and $\partial g=0$, so both integrands in (eq:varied) vanish. The Euler identity follows by varying $\varphi$ with compact support and comparing Euler–Lagrange densities, which is the content of Lemma lem:plateaueuler.