Theorem: Scale derivative equals dilation defect at every grade

For every $N\in \{0,\dots ,4\}$, $b\in {\mathcal{B}}_N$, $\mu >0$ and admissible $\varpi$,

1. Top order ($|\gamma |=N$): $a_{N,b,\gamma }=\frac{(-1{)}^N}{\gamma !}{\int }_{S^3}b{\omega }^{\gamma }\mathrm{d}\sigma$, independent of $\mu$ and $\varpi$; it is the intrinsic residue of $b$ (common to all extensions of degree $4+N$).

2. Lower orders ($k=N-|\gamma |\ge 1$):

   $$a_{N,b,\gamma }(\mu )=\frac{(-1{)}^{|\gamma |+1}}{\gamma !}(\frac{\mu }{{\mu }_0}{)}^k⟨b{|}_{S^3}{\omega }^{\gamma },{\nu }_k[\varpi ]{⟩}_{S^3},{\nu }_k[\varpi ](\omega )=k{{\int }_0}^{\infty }{r}^{-k-1}(T_k[1/\varpi ]\varpi (r\omega )-1)\mathrm{d}r.$$

The identity $B_{\mu }=0$ is pure prescription calculus; the explicit top-order values below use Theorem thm:wpA-Sbv.

Proof. $A_{\mu }$: differentiate $⟨b,{W^{(\mu )}}_N\phi ⟩$; $\mu {\partial }_{\mu }{w}_{\gamma }$ vanishes to order $N$ at $0$ (since ${\partial }^{\beta }{w}_{\gamma }(0)={{\delta }^{\beta }}_{\gamma }$ for all $\mu$), so the pairing is the unique extension; Lemma lem:wpB-dw. ${\mathcal{A}}_{\mu }$: by homogeneity ${\mathcal{A}}_{\mu }(b)$ is supported at $0$ of order $\le N$; its coefficients are $(-1{)}^{|\gamma |}⟨{\mathfrak{R}}_{\mathrm{L},\mu }b,(N-E)w_{\gamma }⟩=(-1{)}^{|\gamma |}⟨{\mathfrak{R}}_{\mathrm{L},\mu }b,(N-|\gamma |)w_{\gamma }-(E-|\gamma |)w_{\gamma }⟩$; the first term is a zero moment, the second vanishes to order $N$. Hence the coefficients agree; $B_{\mu }=0$. Polar coordinates, $r^{-|\gamma |}{w}_{\gamma }(r\omega )=\frac{{\omega }^{\gamma }}{\gamma !}{T}_k[1/{\varpi }_{\mu }]{\varpi }_{\mu }(r\omega )$ and one integration by parts give (1)–(2); for $k=0$ the radial integral telescopes to $-\varpi (0)=-1$. Certificates: top order by two independent exact routes (contour residues; Hadamard finite part plus $i\pi {\delta }^{(m-1)}$ in $x=Q{|}_{S^3}$), equal on every slot; finite exact end-to-end tests with an explicit compactly supported non-radial $\varpi$ for $N=0,1,2$ (Euclidean toy kernels, Section B of the script).