Remark: Prescription dependence versus scale-variation coefficients; not a physical beta function

Prescription dependence. Changing $\mu$ changes the subtraction functions ${w^{(\rho ,\mu )}}_{\alpha }$ and therefore the extension ${\mathfrak{R}}_{\mathrm{L},\mu }$. The difference ${\mathfrak{R}}_{\mathrm{L},\mu }-{\mathfrak{R}}_{\mathrm{L},{\mu }_0}$ is always a finite contact combination (eq:L-scale-diff): it is a renormalization-prescription change in the sense of L.5, not a change of the off-diagonal physical kernel.

Scale-variation coefficients. The generator $A_{\mu }(u)$ of (eq:L-Amu) is the infinitesimal form of that prescription change. The contact coefficients $a_{\mu }(u,\gamma )$ in (eq:L-Amu-exp) (or their $\mu$-independent combinations, when they exist) are called scale-variation coefficients of the extension. They are determined by $\varpi$ and by the spherical behaviour of $u$; they are not free counterterm parameters once $\varpi$ and $\mu$ are fixed. The term scaling residue is reserved for a later anomaly expression of the schematic form $(E+4+N)R(u)$; no such anomaly evaluation is performed in XVI-L.

Not a physical beta function. The object varied here is the subtraction prescription for the numerical two-edge kernels of one selected graph, not a coupling in a complete gravitational effective action, and not a Callan–Symanzik operator for a physical correlator. No identification of $A_{\mu }$ with a gravitational beta function, a running of $\kappa$ or $\hslash$, or an RG flow of a diffeomorphism-invariant observable is claimed. Scale-dependent subtraction is kept strictly distinct from any such physical RG interpretation.