Theorem: Compact microcausal completed kernel

The product and pushforward in (eq:t-kernel) exist. The kernel is compact and symmetric in its metric slots, with

After species symmetrization it defines ${\mathcal{C}}_{3,\mathrm{D}}(f,g)\in {\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$.

Proof. A forbidden cancellation in the product would have zero external covectors in $x,y_1,y_2$. The root bound forces its $y$ covector to be zero. The source bound then forces its $y$ covector to be zero. There is no nonzero total cancellation, so the product exists.

The root is compact and the source is supported on the local diagonal, making the $y$ projection proper. For an output wavefront point, write the root covectors as $(p,-p)$ and the source covectors as $(q,p_1,p_2)$. Internal cancellation gives $q=-p$; the local source relation gives $p_1+p_2=p$. The scalar output covector is $-p$. Thus the three output covectors sum to zero.

The support assertion follows from the root support, the diagonal source support, and its nonnegative-time coefficients. Permutation of the three species slots preserves the sum-zero bound. A nonzero tuple wholly in either causal cone cannot sum to zero. This is precisely XII-A's microcausal exclusion.