Abstract

XV-E continues XV-D at its gate D0. Its first task is a source recovery pass: to obtain, from the supplied record, the definitions that XV-D found missing for the frozen six-branch joint target, and, if they are obtained, to define the preliminary full lift, evaluate its reconstruction-null defect, and test cancellation in the inherited correction class.

The recovery pass finds one change. Excerpts of XII-A and XII-D now in the record display tensor and kernel formulas that are candidates for the undefined symbols $\mathsf{A}$, $L$, and $R$ in the internal kernels. The record does not identify them with those symbols, and XIV-H's defining equation for $L$ is still not reproduced, so they are recorded as candidates and not used. All other items remain unavailable. No lift for the actual graph is definable, and no graph-specific defect, obstruction, or estimate is asserted. The graph-level outcome is Unresolved at E0.

The new mathematics concerns the one source-complete module, the finite normal-jet module with punctured reconstruction. We prove that the null defect of an extension lift is governed by the annihilator ideals of the punctured and extended coefficients, with $H/H_T\cong (\mathrm{Ann}{T}^{\circ }/\mathrm{Ann}T{)}^{M_r}$; that null relations divide into universal, flat-annihilator, and jet-detected relations, of which only the last produce a defect, and that the defect is locally a functional of a finite jet along the puncture. For a point puncture we prove that a descending extension exists whenever any extension exists, with an explicit correction by finitely many derivatives of the delta at the puncture; in one dimension every extension descends. For a general puncture we prove that a descending extension exists if and only if a stated relative estimate holds, and we leave open whether that estimate can fail. Finally we correct an overstatement in XV-D's coefficient proposition and show by examples exactly which implications between extendability of a trace and of its smooth multiple hold. None of these results is transferred to the inherited graph.