The ordinary two-vertex fish

A translation-invariant extension of (eq:fish), lifted as $k(y,y^{\prime })=t(y-y^{\prime })$, has wavefront set in ${\Sigma }_2$. Applying Lemma lem:smoothing to the compact scalar switch and the smoothly smeared insertion leg shows that the isolated local two-vertex contribution has the form

with compact support supplied by the remaining vertex switch. This is a local graph construction with the stated finite normalization freedom. It is not a construction of the nonlocal mixed retarded product.