Proposition: Conditional pushforward admission

Suppose the complete two-line graphs admit a joint extension in ${\mathfrak{C}}_{\mathrm{part}}$, with the stated normalizations, and suppose their full graph wavefront sets have a balanced bound preserved under all specified internal compositions. Then the pushed-forward coefficient is a compact even, ghost-number-zero degree-two polynomial satisfying

Consequently

Proof. Properness gives a compact distributional pushforward. For an output wavefront point the internal covectors are zero. Summing the balanced complete-graph covectors leaves $p_1+p_2=0$ on the two residual field slots. A nonzero tuple lying wholly in either pointed closed causal cone cannot have zero sum, so the output is microcausal.

Every double contraction leaves two even, ghost-number-zero slots. The inherited quotient map supplies the stated class.