Definition: Partial-diagonal ambiguity class; copied

${\mathcal{C}}_{\mathrm{part}}$ consists of finite sums of graphwise contact changes at order ${\hslash }^2$ in the referenced time-ordered product (line convention copied from XIII-B XIIIB), which before pushforward have the form

with $r_I$ transverse to ${\Delta }_I$, $S_{\Gamma /I}$ an inherited spectator graph coefficient with two remaining field slots, $w_{\Gamma }$ built from $g,\partial g,{\sigma }_{\epsilon },{\sigma }_{\epsilon }^{\prime },{\sigma }_{\epsilon }^{\prime \prime }$ and the smooth external graph weights already present, and $D_{\Gamma }$ “the specified finite graph differential operator.” Each change is even, of ghost number zero and field degree two, supported in the existing compact graph closure, and leaves the tree datum and inherited observable-subgraph normalizations unchanged. Only well-defined products and pushforwards are members XIIICpart. The bound six is a normal multi-index bound.