Theorem: Conditional microcausal admission

Suppose a complete joint distribution $T$ has been constructed, all required vertex operations are defined, and its support is proper over the two surviving field slots. Let $\pi$ be that slot projection. Assume the output wavefront estimate proves

Then its appropriately symmetrized compact output kernel defines a quadratic element of ${\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$, and hence a class under ${Q_H}^{\mathrm{if}}$. Its normalization must be supplied by the selected binary retarded-coefficient convention.

Proof. Proper support makes the pushforward defined and compact. For the projection,

The assumed exclusion gives microcausality. Symmetrization preserves compactness and the same cone exclusion. Apply the defined quotient map Hormander,FredenhagenRejzner.