Remark: Extent of the advance

Theorem thm:reference proves a reference on the local complement of $N_U$ for the supported sector only.

It does not prove

for the complete six-branch sum. In particular, it supplies no characteristic composition of $K_{47}$ or either $K_5$ placement with ${\mathcal{C}}_{\mathrm{cross}}$. Those kernels have different support geometry and remain in the target.