Proposition: Recovery of at the reference scale

${\mathfrak{R}}_{\mathrm{L},{\mu }_0}={\mathfrak{R}}_{\mathrm{L}}$ as maps ${\mathcal{V}}_{\mathrm{adm}}\to {\mathcal{D}}^{\prime }({\mathbb{R}}^4)$. In particular ${\mathfrak{R}}_{\mathrm{L},{\mu }_0}{|}_{{\mathcal{V}}_{\mathrm{K}}}={\mathfrak{R}}_{\mathrm{K}}$.

Proof. At $\mu ={\mu }_0$, Definition def:RLmu uses the same $\varpi$, the same $W_{\rho }$, and the same zero moments as Definition def:RL; uniqueness of the Brunetti–Fredenhagen extension of fixed degree with these moments gives equality. The restriction identity is Proposition prop:RL(3).