A declared subtraction germ

Fix the native bundle-valued Hadamard subtraction germ of XX-C, with the action normalization ${\mathcal{A}}^{-1}$ in the source slot. Only its following property is used:

This is a transport-parametrix normalization statement, not a conclusion from wavefront upper bounds alone.

Let $H_{\mathrm{sub}}$ be a real symmetric extension of the symmetric part of that germ, using a smooth symmetric cutoff equal to one in a neighborhood of the diagonal relevant to the compact support of $f$. Its extension away from that neighborhood is part of the prescription.

Define

Then $D$ is smooth near the full diagonal.

For the equal-time map

all differentiated entries of $D$ used below have smooth pullbacks: near $\mathbf{x}=\mathbf{y}$ this follows from full-diagonal smoothness; away from coincidence, the two points on this homogeneous slice are spacelike separated, so neither $W$ nor the chosen near-diagonal subtraction has a null singularity there.

Derivatives in the two spacetime slots are taken before equal-time restriction.