Proposition: A noncompact classical kernel component

There are compact test densities $f$ for which ${\mathcal{H}}_f$ has noncompact spatial support in a field slot. This occurs even though the external smearing is compact.

Proof. Use two independent perturbation directions.

In the first direction take

Then $P\eta =0$, $Q_1=2D1$, and $B^i\eta ={\partial }_{i}D$.

In the second direction take

Then $P\theta =u$, $B^i\theta =0$. The local mixed term is zero: with both scalar perturbations zero, the lapse's local quadratic metric term has no mixed spatial-trace/time-time contribution.

Consequently the polarized quadratic form is

Integrating the compact $D$ source by parts gives its kernel coefficient

Choose $f_0(t,\mathbf{x})={\eta }_0(t)F(\mathbf{x})$, with $F\ge 0$, $\int F>0$, and let $u(t,\mathbf{x})=\beta (t)x^1$ on a neighborhood of its support, using a compact spatial cutoff outside that neighborhood. Then for a unit vector $\mathbf{n}$,

At times where ${\eta }_0\beta \ne 0$, this is nonzero in cones with $n_1\ne 0$ for arbitrarily large $R$. A compact $D$ supported near any such point detects the coefficient. Hence the kernel support is not spatially compact.