Edge and switch defects outside the claim

For arbitrary compact slab parameters, passing (eq:variation) to the free algebra gives

Both mechanisms are visible. They must not be conflated.

If an excluded edge parameter is also supported in the already chosen plateau $P_g$, the switch operator vanishes and the coordinate edge term is isolated. If that plateau condition is absent, the switch term remains. The sheet does not assume that every chosen $g$ has an edge plateau.

For compact $\xi$ supported in $0<t<\epsilon$, the coordinate gap can also be written

The nonzero integral (eq:edge-gap) proves a coordinate failure. It does not, by itself, prove a nonzero total smeared on-shell response for every fixed $f,g$. The observation test can miss the witness, a free-field test can be an equation zero, or scalar switch terms can also be present. No identity is claimed for these excluded parameters, and ${\mathfrak{g}}_U$ is neither enlarged nor shrunk.

Bare dressings alone are not generally background invariant. For example, in a fixed quasi-free Hadamard representation their pairing involves $\hslash ⟨c_1[h],{\Delta }_{+}{c}_1[h^{\prime }]⟩$, which can change when $c_1[h]$ is shifted by $D_{\xi }f$. This does not contradict the invariance of the completed ${\hat{A}}_1$. No interacting two-point response is constructed or inferred.