Abstract

We study the compatibility between distributional extension and reconstruction descent for linear constructions on presentations. Given a canonical map on a proved open domain, a preliminary lift, and a specified affine space of permitted corrections, we prove that a reconstruction-descending candidate exists exactly when the preliminary lift's null defect vanishes in the corresponding relative obstruction quotient. Contact-supported corrections cannot cancel a null defect detected away from their support.

For a specified finite normal-jet module, we obtain an explicit normal form and a complete distributional null-relation criterion. Supported extension preserving the normal-order bound is equivalent to extension of its unique coefficients; the complete unrestricted supported ambiguity is also determined. We give a general closed-set extension criterion, distinguish universal and flat-annihilator relations from potentially jet-detected defects, and construct a simultaneous annihilator-preserving correction at a point puncture by finitely many derivatives of a delta distribution. The point-puncture result is not asserted for a positive-dimensional characteristic locus.

The application retains the full six-branch target $K_{47}+K_5+K_6+K_8+K_9$, including both placements of $K_5$. The interface results do not construct its missing characteristic contraction, establish its complete-graph estimates, or admit a quantum coefficient.