Dipole Green tensor angular cancellation ($K=\frac{I}{3b}\delta -\mathrm{pv}\frac{3n\otimes n-I}{4\pi b{r}^3}$)

For the three-dimensional reference Green function $g(r)=1/(4\pi br)$, the distributional mixed Hessian $K={\nabla }_x{\nabla }_{x^{\prime }}g$ has the displayed contact and principal-value parts. Since the spherical mean of $n\otimes n$ is $I/3$, its ordinary dipole part has zero angular integral. Radial weighting therefore isolates the contact term, provided boundary truncation is absent or negligible in the bulk limit.