Remark: What ``no subtraction needed'' can mean

For the azimuthal current computed by time splitting at $\Delta \phi =0$, the direct parametrix has zero azimuthal derivative difference. No extra subtraction appears in that particular calculation.

It is false that the derivative difference of $(4{\pi }^2{{\sigma }_E}^2{)}^{-1}$ vanishes along every splitting. For example a spatial derivative of that singular function is generally nonzero and divergent. Equation (eq:current-prescription), using the regular kernel, is the splitting-independent definition.