Proposition: Exact further restriction to XVI-J

In the truncation of Proposition prop:LtoK, discard every term with $|p|+|q|>0$, i.e.\ keep only the constant-slot label $\{0,0\}$. The result is exactly XVI-J's distribution $U^{\mathfrak{R}}$ of XVI-J, eq:U, and the corresponding coefficient is exactly ${{\mathcal{C}}^{\mathfrak{R}}}_{2e}$ of XVI-J, eq:C.

Proof. Proposition prop:LtoK reduces to XVI-K; then XVI-K, Proposition prop:regression, reduces XVI-K to XVI-J as an identity of distributions including $y=w$. Composing the two exact restrictions gives the claim. Alternatively: $a_{\{0,0\}}={\chi }_{\mathrm{I}}\mathcal{K}$ with $\mathcal{K}$ of XVI-J, eq:Kdef (XVI-K, Proposition prop:hess(4)), and ${\mathfrak{R}}_{\mathrm{L}}(S^2)={\mathfrak{R}}_{\mathrm{K}}(S^2)={\bar{q}}^{\mathfrak{R}}$ (Proposition prop:RL(3)).