Proposition: Finite-scaling-degree fish extensions

The kernels (eq:fish) are well defined away from $r=0$ and have scaling degree at most six in four dimensions. They admit extensions with that scaling-degree bound. Two such extensions differ by

with the relevant tensor coefficients and lower-degree restrictions for the less singular terms.

Proof. Away from coincidence, null-related Feynman singularities in the two factors have the same first-slot orientation, so their nonzero wavefront covectors cannot add to zero. At spacelike separation both factors are smooth.

A scalar Feynman kernel has scaling degree two. At most two derivatives occur on the contracted scalar line, so the product has scaling degree at most $2+2+2=6$.

The finite-scaling-degree extension theorem gives an extension without increasing that bound BF. A difference is supported at $r=0$. A point-supported distribution of scaling degree at most six in four dimensions has the form (eq:fishambiguity).