Proposition: Observation support is visible in the flat model

For the specified flat realization, $F\subset {\mathrm{supp}}_{\mathrm{fun}}{B_f}^{\mathrm{fl}}$.

Proof. At an interior point $x$ with $f(x)\ne 0$, choose $w\in \mathcal{D}(V)$, supported in a prescribed small neighbourhood, with $\mathrm{d}w(x)=0$, $Pw(x)\ne 0$. Set $h_{\alpha \beta }={\eta }_{\alpha \beta }Pw$. Then $Ch=-P\mathrm{d}w$; the compact zero-data identity for (eq:Eflat) gives $L_{\mathrm{fl}}h=-\mathrm{d}w$. The scalar derivative kernel is

whose value at $x$ is $-f(x)Pw(x)\ne 0$. Real test variations detect a nonzero complex kernel by decomposition if necessary. Taking closures proves the assertion.