Proposition: Three-term variation: proof of record

For every classical background and arbitrary compact generators in the open slab, set

Then

The first term belongs coefficientwise to $P_m\mathcal{D}(M)$. The other two are the dressed coordinate and switch defects. In particular,

Proof. Write $w^{\prime }={{\mathsf{E}}_{{\mathsf{q}}^{\prime }}}^{\mathrm{A}}c[q^{\prime }]$. By advanced uniqueness, ${\mathsf{E}}_g[q]w=c[q]$, ${\mathsf{E}}_g[q^{\prime }]w^{\prime }=c[q^{\prime }]$, and $P_{m}w=e[q]$, $P_m{w}^{\prime }=e[q^{\prime }]$.

By definition of the defects,

Hence

This is a compact source. The difference is advanced-type: formal compactly supported pullback changes $w$ only by compactly supported coefficients, so no causal-preservation assumption about $F$ is needed.

Lemma lem:advanced therefore gives

Apply $P_m$, and use $P_m{{\mathsf{E}}_{{\mathsf{q}}^{\prime }}}^{\mathrm{A}}={\mathcal{R}}_{q^{\prime }}$, to obtain (eq:variation).

Every coefficient of $(F^{\ast }-1)w$ has compact generator support. Thus its $P_m$-image is an explicit equation-ideal test. Expand $w_0={{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}f$, $w_1={{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}{e}_1$, and ${\mathcal{R}}_{q^{\prime }}=1-\kappa {\mathsf{L}}_{1,g}[h^{\prime }]{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}+O({\kappa }^2)$ to obtain (eq:var1)–(eq:var2).