Scale derivative versus dilation defect at every grade (work package B)

top-order data. All top-order values are $i{\pi }^2{c}^2\times \mathbb{Q}$, and with the certified $c$ the unit is $i{\pi }^2{c}^2=i/(16{\pi }^2)$ (the real, finite-part contributions vanish identically). $N=0$: $a=-2i{\pi }^2{c}^2=-i/(8{\pi }^2)$, the M.6 residue $r$ of Theorem thm:M6-S2. $N=1$: $a_{1,\{(),\mu \},\mu }=-i{\pi }^2{c}^2=-i/(16{\pi }^2)$, others $0$. Nonzero top slots per grade: $1,4,22,68,194$ of $1,16,100,400,1225$ ($289$ in total). In the audited equivariant coordinates of M.7 ($C_{N,b,\gamma }=(-1{)}^{{\gamma }_1+{\gamma }_2+{\gamma }_3}[Q^k{\square }^k{n}_b{]}_{\gamma }$), the residue map is, in units $i{\pi }^2{c}^2=i/(16{\pi }^2)$, $-2{\varphi }_{00}$; $\frac{1}{2}{\varphi }_{00}$; $-\frac{1}{12}{\varphi }_{00}-\frac{1}{48}{\varphi }_{11}$; $\frac{1}{96}{\varphi }_{00}+\frac{1}{384}{\varphi }_{11}$; $-\frac{1}{960}{\varphi }_{00}-\frac{1}{3840}{\varphi }_{11}-\frac{1}{61440}{\varphi }_{22}$ for $N=0,\dots ,4$. Tests where the hypotheses hold: Lorentz equivariance on all six generators (pull-back action, intertwiner $z^{\mu }\leftrightarrow {\eta }^{\mu \nu }{\partial }_{\nu }\delta$); $\mathrm{Res}({\partial }_{\alpha }u)={\partial }_{\alpha }\mathrm{Res}(u)$ for all $340$ label/direction pairs with $N\le 3$; every one of the $185$ relation vectors annihilates the top-order, the radial-specialization and the general-$\varpi$ coefficient vectors; all $255$ labels reconstructed and checked against direct evaluation. The derivative and Lorentz identities are not used for, and do not hold for, the lower-order coefficients ($\varpi$ is neither covariant nor Leibniz-compatible).

dependence. $B_{\mu }=0$ does not remove prescription dependence: it resides in the lower-order coefficients, common to $A_{\mu }$ and ${\mathcal{A}}_{\mu }$, which scale as $(\mu /{\mu }_0{)}^{N-|\gamma |}$ and depend on ${\nu }_k[\varpi ]$, $k=1,\dots ,4$. Missing datum: XVI-L fixes $\varpi \in {C_c}^{\infty }$, $\varpi (0)=1$, but does not specify it; the values of ${\nu }_k[\varpi ]$ (equivalently the functionals ${\Lambda }_k(P)=⟨P(Q-i0{)}^{-6}{|}_{S^3},{\nu }_k⟩$ on ${\mathrm{Poly}}_{8-k}$) are what is missing. For a reflection-even $\varpi$ the odd-$k$ coefficients vanish. The named specialization ${\mathfrak{R}}_{\mathrm{L},\mu }[{\varpi }_{\mathrm{rad}}]$ (Euclidean-radial $\varpi$; not asserted by XVI-L) has constant ${{\nu }_k}^{\mathrm{rad}}$, only $k=2,4$ survive, and its contact difference from ${\mathfrak{R}}_{\mathrm{L},\mu }[\varpi ]$ has coefficients $-(-1{)}^{|\gamma |}⟨b,w_{\gamma }[\varpi ]-w_{\gamma }[{\varpi }_{\mathrm{rad}}]⟩$, i.e.\ $\frac{(-1{)}^{|\gamma |+1}}{k\gamma !}⟨b{\omega }^{\gamma },{\nu }_k[\varpi ]-{{\nu }_k}^{\mathrm{rad}}⟩$ for $k\ge 1$ and $\frac{(-1{)}^{N+1}}{\gamma !}⟨b{\omega }^{\gamma },{{\int }_0}^{\infty }{r}^{-1}(\varpi -{\varpi }_{\mathrm{rad}})(r\omega )\mathrm{d}r⟩$ for $k=0$.