Abstract

XV-C addresses the proposed instantiation of XV-B's relative-lifting framework for the frozen six-branch joint characteristic target. The supplied data retain the grouped $K_{47}$ contribution, both placements of $K_5$, and the supported sector $K_6+K_8+K_9$. They do not supply the complete cross-line operators, reconstruction space, correction class, or output definitions needed to construct the full graph operation.

We therefore take the explicitly permitted unresolved outcome, rather than inventing the missing graph data. The new mathematical results are local tests needed by the proposed application. We give a complete reconstruction-null criterion in a specified finite normal-jet module, collect the supplied supported-sector coefficients through normal order three, and prove an exact distributional extension criterion expressed by test-function estimates. This criterion yields a necessary and sufficient test for unrestricted supported realization of the supplied normal form. It also provides explicit failure witnesses and a support-sensitive description of the remaining ambiguity.

These results do not evaluate the unsupplied cross-line coefficients or establish membership in the frozen correction class. The full joint map remains unresolved before graph-level instantiation. No exhaustive obstruction, complete-graph wavefront theorem, or admitted quantum coefficient is asserted.