Theorem: Present joint-extension verdict

The results above do not construct the full ${R_1}^{(1)}$ for the selected pair in ${\mathfrak{C}}_{\mathrm{part}}$. They also do not exclude every extension in that class. The exact ending is Unresolved.

Proof. The full contraction inventory contains the triangle of Lemma lem:pinch. A joint distribution on its complete off-collision domain has not been constructed by the available sum-zero argument. The larger-cycle transverse estimates and intersection compatibility have likewise not been proved.

The primitive extension theorem treats only the specified parallel-line factors. It does not establish those missing hypotheses. Conversely, failure of the sufficient product criterion is not a theorem that a joint distribution or a cancellation prescription cannot exist. No universal normalization or admission incompatibility has been proved. The release is therefore Unresolved.