Lemma: The reconstruction rule and its coefficient class

1. The two XIV-J values $\mathsf{Rec}(M_{\mathrm{R}})=\delta (q)R(r)$, $\mathsf{Rec}(i\theta M_{\mathrm{R}})={\delta }^{\prime }(q)R(r)$ Jnull are instances of (eq:Recrule) with $c_{00}=1$, resp.\ $c_{10}=1$, all other coefficients zero.

2. If $\mathsf{Rec}$ is linear over multiplication by the chart function $q$ (a property not reproduced), then (eq:Recrule) annihilates the null relation $(qi\theta +1)M_{\mathrm{R}}$ of Jnull: $q{\delta }^{\prime }(q)R(r)+\delta (q)R(r)=0$ by Lemma lem:taylor.

3. Suppose the coefficients $c_{j\beta }$ are independent of $(q,r)$. Then the family $\{{\delta }^{(j)}(q){{\partial }_r}^{\beta }R(r)\}$ is linearly independent over that class, ${\mathcal{K}}_{\mathrm{int}}$ is free on it, and $\mathsf{Rec}(A)=0$ iff all $c_{j\beta }(A)=0$. In particular $\{{{\partial }_r}^{\beta }R{\}}_{\beta }$ is linearly independent over $\mathbb{C}$.

4. Suppose the coefficient class contains the chart functions $q$ or $r^2$. Then the family is linearly dependent over that class:

   $$q{\delta }^{(j)}(q){{\partial }_r}^{\beta }R(r)=-j{\delta }^{(j-1)}(q){{\partial }_r}^{\beta }R(r),r^{2}R(r)=0,r^2{\partial }_{\lambda }R(r)=-2r_{\lambda }R(r).$$

   If $\mathsf{Rec}$ is linear over the coefficient class, each of these is a null relation of ${\mathcal{A}}^m(V)$; XIV-J's relation is the first with $j=1$, $\beta =0$. In this case $\ker \mathsf{Rec}$ strictly contains the coefficient kernel, and ${\mathcal{K}}_{\mathrm{int}}$ is not free on the family.

Proof. (i) is read off; (ii) is Lemma lem:taylor. (iii) First, $\{{{\partial }^{\beta }}_{r}R{\}}_{\beta }$ is independent over $\mathbb{C}$: if ${\sum }_{\beta }{a}_{\beta }{\partial }^{\beta }R=0$, apply the Fourier transform on ${\mathbb{R}}^4$ ($R$ is tempered). From $\square R={\delta }_0$, $\hat{R}$ coincides on the open set $\Omega =\{{p_0}^2\ne |\mathbf{p}{|}^2\}$ with a smooth nowhere-vanishing function. The relation gives $P(p)\hat{R}=0$ with $P(p)={\sum }_{\beta }{a}_{\beta }(\pm ip{)}^{\beta }$, so $P=0$ on $\Omega$, hence $P\equiv 0$ and all $a_{\beta }=0$. Now pair a vanishing sum $\sum c_{j\beta }(\xi ){\delta }^{(j)}(q){\partial }^{\beta }R(r)$ with $\psi (\xi )\chi (q)\theta (r)$, where $\xi$ denotes the chart variables other than $(q,r)$: ${\sum }_{j,\beta }⟨c_{j\beta },\psi ⟩(-1{)}^j{\chi }^{(j)}(0)⟨{\partial }^{\beta }R,\theta ⟩=0$. Choose $\chi$ with ${\chi }^{(j)}(0)={\delta }_{j{j}_0}$ for the finitely many $j$ occurring; then ${\sum }_{\beta }⟨c_{j_0\beta },\psi ⟩{\partial }^{\beta }R=0$, so $⟨c_{j_0\beta },\psi ⟩=0$ for all $\beta ,\psi$. (iv) The first identity is Lemma lem:taylor. $r^{2}R(r)=\frac{1}{2\pi }\Theta (r^0)r^2\delta (r^2)=0$ by $q\delta (q)=0$ pulled back along $r\mapsto r^2$ on $r\ne 0$, and $R$ has no mass at $r=0$ by Lemma lem:R(i). Differentiating $r^{2}R=0$ gives $2r_{\lambda }R+r^2{\partial }_{\lambda }R=0$. The remaining assertions are restatements.