Proposition: Properties of

1. ${\mathfrak{R}}_{\mathrm{L}}:{\mathcal{V}}_{\mathrm{adm}}\to {\mathcal{D}}^{\prime }({\mathbb{R}}^4)$ is well defined and linear, and ${\mathfrak{R}}_{\mathrm{L}}(u){|}_{{\mathbb{R}}^4\setminus \{0\}}=u$ for every $u\in {\mathcal{V}}_{\mathrm{adm}}$.

2. For $u\in {\mathcal{H}}_k$, $\mathrm{sd}({\mathfrak{R}}_{\mathrm{L}}(u))\le k$, and $⟨{\mathfrak{R}}_{\mathrm{L}}(u),{w^{(k-4)}}_{\alpha }⟩=0$ for $|\alpha |\le k-4$.

3. Exact XVI-K restriction. ${\mathfrak{R}}_{\mathrm{L}}{|}_{{\mathcal{V}}_{\mathrm{K}}}={\mathfrak{R}}_{\mathrm{K}}$ as maps ${\mathcal{V}}_{\mathrm{K}}\to {\mathcal{D}}^{\prime }({\mathbb{R}}^4)$. In particular ${\mathfrak{R}}_{\mathrm{L}}(S^2)={\mathfrak{R}}_{\mathrm{K}}(S^2)={\bar{q}}^{\mathfrak{R}}$.

4. For each numerical label $\lambda$ contributing to (eq:QL2), ${\mathfrak{R}}_{\mathrm{L}}({q^{\circ }}_{\lambda })$ is an extension of ${q^{\circ }}_{\lambda }$ with $\mathrm{sd}\le 4+N_{\lambda }$; any other extension of the same scaling degree differs from it by ${\sum }_{|\gamma |\le N_{\lambda }}{c}_{\gamma }{\partial }^{\gamma }\delta$.

5. If ${\sum }_{\lambda }{r}_{\lambda }{q^{\circ }}_{\lambda }=0$ in ${\mathcal{D}}^{\prime }({\mathbb{R}}^4\setminus \{0\})$ with constant coefficients $r_{\lambda }$, then ${\sum }_{\lambda }{r}_{\lambda }{\mathfrak{R}}_{\mathrm{L}}({q^{\circ }}_{\lambda })=0$ in ${\mathcal{D}}^{\prime }({\mathbb{R}}^4)$.

Proof. (1)–(2) As in XVI-K, Proposition prop:RK(1)–(2): ${\mathcal{B}}_{\mathrm{adm}}$ is a basis; restriction to ${\mathbb{R}}^4\setminus \{0\}$ is linear and the identity on the basis; scaling degree and the moment conditions (eq:L-moments) are linear. (3) On each graded piece ${\mathcal{H}}_k$ with $k\le 6$ (the support of ${\mathcal{V}}_{\mathrm{K}}$), the data of Definition def:RL coincide with XVI-K, Definition def:RK: same $\varpi$, same $W_{\rho }$ for $\rho =k-4\le 2$, same zero moments. By Brunetti–Fredenhagen, Theorem 5.3, the extension of a fixed homogeneous distribution of scaling degree $k$ with these moments is unique, so ${\mathfrak{R}}_{\mathrm{L}}(u)={\mathfrak{R}}_{\mathrm{K}}(u)$ for every $u\in {\mathcal{V}}_{\mathrm{K}}$. (4) ${q^{\circ }}_{\lambda }\in {\mathcal{H}}_{4+N_{\lambda }}$ by Proposition prop:Lsd; apply (1), (2). The difference of two same-degree extensions is supported at $0$ and of scaling degree $\le 4+N_{\lambda }$, hence a combination of ${\partial }^{\gamma }\delta$ with $|\gamma |\le N_{\lambda }$. (5) The relation says an element of ${\mathcal{V}}_{\mathrm{adm}}$ is zero; ${\mathfrak{R}}_{\mathrm{L}}$ is linear, so it maps that element to $0$.