Local transverse degree

The recalled chart (eq:density) represents the residue as a smooth tangential coefficient times a delta distribution in four normal variables $v=(\sigma ,\mathbf{w})$. Its transverse scaling degree is four. A nonzero normal derivative of order $k$ with smooth tangential coefficient has transverse scaling degree $4+k$, since

and each normal derivative adds one inverse power of $\lambda$.

This degree calculation is not a complete ambiguity classification for ${u_{\mu }}^{\rho }$ or later graphs. A normal-degree bound alone does not restrict arbitrary tangential weights.