What is imported from XVI-K, and how

XVI-K.tex is treated as a frozen dependency: it is cited, not edited. Its release is “Constructed, conditional on the imported root hypotheses, for the $C_2$ sector with full cubic vertex” (XVI-K.tex, title and Section sec:status). The following items are imported as stated there; nothing beyond the statements is used.

1. Edge kernel. $S:=h_0+i{E}_0^{\mathrm{A}}=-{{\Delta }_F}^{\mathrm{FR}}$, $D_F=-\mathcal{P}S$ (XVI-K, eq:S), a globally defined distribution of $z=y-w$, homogeneous of degree $-2$ (XVI-K, lem:homog).

2. Full cubic Hessian. XVI-K, Proposition prop:hess, eq:Lpq:

   $$D^2{\ell }_3(h)[A,C](w)={\sum _{i,j=0}}^9\sum _{|p|+|q|\le 2}{L^{pq}}_{ij}(w;h){\partial }^p{A}^i(w){\partial }^q{C}^j(w),$$

   with ${L^{pq}}_{ij}$ linear in ${\partial }^{r}h$, $|r|=2-|p|-|q|$, and ${L^{pq}}_{ij}={L^{qp}}_{ji}$.

3. Placement sign. XVI-K, eq:sign: ${{\partial }_w}^p[u(y-w)]=(-1{)}^{|p|}({\partial }^{p}u)(y-w)$, ${{\partial }_y}^p[u(y-w)]=({\partial }^{p}u)(y-w)$.

4. Extension map. XVI-K, Definition def:RK and Proposition prop:RK: the linear map ${\mathfrak{R}}_{\mathrm{K}}:{\mathcal{V}}_{\mathrm{K}}\to {\mathcal{D}}^{\prime }({\mathbb{R}}^4)$ built with the $\varpi$-projections $W_{\rho }$ (XVI-K, eq:Wrho) and zero declared moments (XVI-K, eq:moments).

5. Root attachment. XVI-K, Lemma lem:attachK and Theorem thm:main, conditional on (R1)–(R3).

6. Propagated ambiguity. XVI-K, Lemma lem:amb, eq:deltaCK.