Proposition: Joint admission through κ 2

Regarded as polynomial functionals of the auxiliary multiplet $(h,k,\psi )$, the three bare coefficients

have compact derivative kernels of split type. They therefore satisfy the joint microcausal cone test on the completion.

This is a mathematical admission test on classical background slots. It does not quantize those slots.

Proof. The zeroth coefficient has a smooth first derivative. The first mixed kernel is a derivative, in its scalar slot, of

It is split by (eq:Lambda) and Lemma lem:split-calculus.

It remains to inspect the highest kernels at second order. With fixed tensor contractions and finite derivatives understood, they are generated by

The $k$-linear term is the corresponding two-slot coordinate kernel.

In the first line the factors share only $x$. In the second line the two deltas share only $w$; their product then meets the Green kernel only at $w$. In the third line attach the delta to the inner coordinate kernel at $w$, then attach the outer Green kernel at that single slot. Every step is covered by Lemma lem:split-calculus. Derivatives do not change this conclusion.

Every retarded path ends in $K_f$, and every coordinate source vertex carries $\sigma$. Consequently all effective background and internal vertices lie in

This set is compact. A smooth compact localization equal to one near it can be inserted before each internal pushforward; it does not change the kernel. The pushforwards are thus proper.

Lower derivative kernels are obtained by multiplying by smooth configurations and integrating the unused slots with that same localization. They remain split. Compactness follows from the same causal support argument. Finally, $\sigma$ is a smooth multiplier, including at $t=0$; no sharp boundary distribution has been inserted. Lemma lem:split-cone completes the proof.