Theorem: Tree datum and admission

Equation (eq:tree-definition) defines an even, ghost-number-zero compact microcausal polynomial of degree four. Its class

is well defined for the displayed lifts. It is independent of the smooth ordering difference $d$ at classical-symbol order.

Moreover, the binary time-ordered coefficient

exists through the displayed order and yields (eq:tree-definition) under (eq:R-definition).

Proof. Proposition prop:vertex and the inherited bound for $\mathcal{O}$ supply the hypotheses of Lemma lem:one-line. The action inverse has balanced translation-invariant lines. Consequently the contraction exists, is compact, and satisfies $Z_4$, which implies microcausality.

The star contraction $C_1(H)$ is already defined by the inherited domain theorem. The additional contraction $C_1(i{G}_{\mathrm{act}}^{\mathrm{A}})$ is the transpose of the constructed retarded contraction. Thus

This proves the sign and the stated binary coefficient. Neither ${G_{\mathrm{act}}}^{\mathrm{R}}$ nor this classical contraction depends on $d$. The inherited quotient map supplies the class.