Proposition: Exact restriction to XVI-K

Two separate identities are recorded; neither substitutes for the other.

1. Source-sector truncation. In the ordered decomposition (eq:UL)/(eq:QL), retain only the summand $A=C_2$ (equivalently, in the sector-resolved grouped form, retain only the coefficients ${a^{C_2}}_{\lambda }$). The resulting root-attached distribution is exactly ${U^{{\mathfrak{R}}_{\mathrm{K}}}}_{\mathrm{K}}$ of XVI-K, eq:UK, as an element of ${\mathcal{D}}^{\prime }(M^3)$ (extension included), and the corresponding truncation of (eq:CL) is exactly ${{\mathcal{C}}^{{\mathfrak{R}}_{\mathrm{K}}}}_{C_2,\mathrm{fv}}$ of XVI-K, eq:CK.

2. Extension restriction. Independently, ${\mathfrak{R}}_{\mathrm{L}}{|}_{{\mathcal{V}}_{\mathrm{K}}}={\mathfrak{R}}_{\mathrm{K}}$ as maps ${\mathcal{V}}_{\mathrm{K}}\to {\mathcal{D}}^{\prime }({\mathbb{R}}^4)$ (Proposition prop:RL(3)).

Discarding numerical labels outside ${\mathcal{V}}_{\mathrm{K}}$ is not equivalent to the source-sector truncation: other sectors can contribute to labels in ${\mathcal{V}}_{\mathrm{K}}$ (e.g.\ $N=1,2$), while the order-$0$ label $\{0,0\}$ arises only in $C_2$.

Proof. (1) On the $C_2$, $a=b=0$ sector, Definition def:La reduces to XVI-K, Definition def:a: $P_{C_2}$ is the polarization of $2C_2\partial \partial {\sigma }_{\epsilon }$ (L.1, check S3), and ${a^{C_2}}_{00,pq}$ coincides with $a_{pq}$ of XVI-K, eq:apq. The numerical kernels are ${q^{\circ }}_{\{p,q\}}$ of XVI-K. Combined with (2), the Hörmander products with $B_{f,g}$ coincide with those of XVI-K, Lemma lem:attachK, and the smeared coefficients coincide. (2) Proposition prop:RL(3).