Proposition: Limiting defect identity for fixed generators

For fixed compact interior generators, the coordinate defect tests converge in a fixed compact-test space. Let their limits be ${\theta }_{j,0}$. With $L_1^{\prime }=L_{1,g}[h^{\prime }]$, one has

For this fixed compact interior generator class, ${\theta }_{1,0}={\theta }_{2,0}=0$. Switch defects can nevertheless remain.

Proof. Convergence of the bare tests for both fixed background pairs and continuity of the density-pullback coefficients give convergence in (eq:theta). The compact sources ${\Sigma }_F{\mathsf{E}}_{\mathrm{A}}{e}_{\epsilon }$ also converge, since their coefficient supports are fixed. The operator ${\mathcal{R}}_{q^{\prime }}$ is independent of $\epsilon$ and continuous on each required compact source stage. Thus (eq:variation) passes to the limit. Extracting its two coefficients gives (eq:defect1)–(eq:defect2).

A fixed compact subset of the open slab has positive distance from $\Sigma$. For sufficiently small $\epsilon$, the coordinate transformation is therefore supported where ${\sigma }_{\epsilon }=1$. Coordinate naturality, which does not require an interaction plateau, gives ${\Theta }_{F,\epsilon }=0$ for those widths. Hence its limits vanish. The same argument does not force $g=1$ on those supports, so it does not remove ${\Sigma }_F$.