The observation-smeared linear coordinate kernel

This is the precise smoothness statement in X-A XA ({:kernel}). It does not assert that the unsmeared operator kernel $L(x,y)$ is smooth. That kernel contains derivatives of the wave fundamental solution.

The distinction from the multiplied-equation prescription is explicit:

where

The divergence-form prescription has $\sigma C^{\ast }b$, not the extra term in (eq:multiplieddifference). The sharp prescription's surface delta is absent here. It would be inaccurate to insert a ${\sigma }^{\prime }{\mathcal{B}}_{\Sigma }$ term into (eq:smearedkernel) merely because it occurs for the different multiplied equation. No $\epsilon \downarrow 0$ limit is taken.