Proposition: Observation support is visible in the scalar slot

For the flat realization,

Every nonempty open set meeting $F$ contains scalar dependence at a suitable off-shell metric configuration.

Proof. Choose an interior point $x$ with $f(x)\ne 0$ in the given open set. Take $w$ compactly supported in a smaller interior neighbourhood with

For example, a temporal quadratic multiplied by a cutoff equal to one near $x$ has these properties. Set

Equation (eq:C-flat) gives

Because $w$ and all its derivatives have zero initial data, uniqueness of the zero-Cauchy problem gives

The scalar derivative is therefore

At $x$, its smooth kernel equals $-f(x)Pw(x)\ne 0$. It is detected by a scalar variation in the prescribed open set.

If $f$ is complex, the nonzero complex kernel is still detected by a real test, or by its real and imaginary decomposition. Taking closures gives all of $F$, including its boundary points. Configurations with $Lh=0$ do not defeat this argument: functional support uses all configurations, and the displayed configuration is a detector.