Remark: General N

For $u\in {\mathcal{H}}_{4+N}$ and $\rho =N$, differentiating the zero-moment formula yields the contact expansion

(with the pairing understood via ${W^{(\mu )}}_{\rho }$). Equality $A_{\mu }=(E+4+N){\mathfrak{R}}_{\mathrm{L},\mu }$ for $N\ge 1$, previously Unresolved here, is now resolved: the $\mu$-dependence of $M^{(\rho ,\mu )}$ is accounted for by $\mu \mathrm{d}{M}^{-1}=D{M}^{-1}-M^{-1}D$, and $B_{\mu }=0$ for all $N\le 4$ (Lemma lem:wpB-dw, Theorem thm:wpB-B0).