An alternative extension of the punctured numerical kernel

There is also an explicit extension route when one starts with the raw punctured equal-time contraction.

Let $\mathbf{r}=\mathbf{x}-\mathbf{y}$. Off $\mathbf{r}=0$, the differentiated Hadamard contraction in the rod term is smooth. Its spatial singularity has an upper bound of the form

on compact tangential sets. Indeed, the four-dimensional Hadamard germ has leading degree two, the differentiated field factor has at most two derivatives, and the dipole kernel has degree two.

The bound is conservative: cancellations or equal-time parity can lower the actual degree.