Lemma: Order-four reconstruction estimate

For every compact $L⋐W$, there are a continuous coefficient seminorm $p_{L,4}$ and a constant $C_L$ such that

Consequently reconstruction is strongly continuous, $\ker \mathsf{Rec}$ is closed, and the quotient is Hausdorff.

Proof. The retarded kernel $R$ is a locally finite positive measure. Indeed its integral on a compact future cone segment has the radial form

Thus the tensor-product seeds are locally order zero. The normal-delta seeds are likewise locally finite measures. Smooth changes of coordinates preserve order-zero estimates on compact subsets, with their density factors included.

Every reconstructed generator is a derivative of order at most four of one of these measures, multiplied by a smooth coefficient. Transferring derivatives to the product of the coefficient and test proves (eq:internal-order). The seminorm may be taken as the maximum of coefficient derivatives through order four on a slightly enlarged fixed compact chart set.

A bounded test set has common support and bounded derivatives, so the estimate proves strong continuity. Closedness of the kernel and Hausdorffness of the quotient follow.