Lemma: The remaining contraction Ward defect is smooth

The kernel (eq:symmetric-defect) is a smooth mixed bisolution. Its vanishing is not asserted.

Proof. Before Hermitian averaging, the advanced Green contributions cancel exactly, leaving a constant multiple of

The operator identities imply

Its wavefront relation is contained in the Feynman relation. A homogeneous two-slot solution cannot have a nonzero Feynman wavefront: propagation in the first slot along the full null bicharacteristic would carry a directed Feynman singularity through coincidence to the forbidden time orientation. Non-null diagonal directions are excluded by elliptic regularity. Thus $T$ is smooth. This is also the uniqueness-modulo-smooth part of the distinguished-parametrix construction IS. Adjunction and averaging preserve smoothness.