Proposition: Changes supported away from the initial surface

Let $Q_f$ be a smooth functional such that, for every smooth configuration, its metric derivative is supported in a fixed closed set disjoint from a neighbourhood of ${\Sigma }_0$. Then

is not microcausal.

In particular, adding a bilinear kernel supported on the ordinary diagonal, with its scalar projection contained in $\mathrm{supp}f$, cannot remove the obstruction.

Proof. At the witness configuration, the metric derivative of $Q_f$ vanishes in a neighbourhood of ${\Sigma }_0$. Therefore the derivative of the sum has exactly the same nonzero boundary wavefront component as (eq:metric-derivative).

For the final statement, a diagonal-supported kernel with scalar projection in $\mathrm{supp}f$ also has its metric projection in $\mathrm{supp}f$, which is a positive distance in time from ${\Sigma }_0$.