Lemma: Complete scale derivative of the subtraction functions

For every $\rho \le 4$, every $\varpi$ with $\varpi (0)=1$, and $D=\mathrm{diag}(|\beta |)$:

with $T_k$ the Taylor polynomial of order $k$ at $0$. Equivalently ${w^{(\rho ,\mu )}}_{\alpha }(z)={\lambda }^{-|\alpha |}{w^{(\rho )}}_{\alpha }(\lambda z)$.

Proof. $M_{\beta \alpha }=\left(\genfrac{}{}{0ex}{}{\beta }{\alpha }\right){\lambda }^{|\beta |-|\alpha |}{\partial }^{\beta -\alpha }\varpi (0)$ (Leibniz), so $\mu {\partial }_{\mu }M=[D,M]$; differentiate $M{M}^{-1}=1$. For the basis functions $f_{\gamma }:=z^{\gamma }{\varpi }_{\mu }/\gamma !$, $\mu {\partial }_{\mu }{\varpi }_{\mu }=E{\varpi }_{\mu }$ (Proposition prop:M6-dil(1)) and $E{z}^{\gamma }=|\gamma |z^{\gamma }$ give $\mu {\partial }_{\mu }{f}_{\gamma }=(z^{\gamma }/\gamma !)E{\varpi }_{\mu }=(E-|\gamma |)f_{\gamma }$. With $\mu {\partial }_{\mu }(M^{-1}{)}_{\gamma \alpha }=(|\gamma |-|\alpha |)(M^{-1}{)}_{\gamma \alpha }$ (the entries of $D{M}^{-1}-M^{-1}D$),

The closed form holds because $z^{\alpha }{T}_k[1/{\varpi }_{\mu }]{\varpi }_{\mu }/\alpha !=z^{\alpha }/\alpha !+O(|z{|}^{\rho +1})$ and $M$ is invertible. Exact certificates (wpB/higher_grade_scaling.py, Section A): symbolic jets for $\rho \le 3$, two random rational jets for $\rho =3,4$, and an explicit non-radial $\varpi =e^q$ as functions, all $\rho \le 4$; regression_M.py re-checks the matrix identities independently for $\rho \le 2$ (symbolic) and $\rho =3$ (rational jets).