The application package

To instantiate the relative-lifting theorem for the retained graph, the following package must be provided in one consistent signature and domain.

1. Carrier and reconstruction. Specify the full presentation space, reconstructed image, all relations, and the topology if continuity is required. A local test module is not this identification.

2. Joint operations. Specify the complete grouped cross-line operators, mixed vertex Hessian, tensor contractions, and all six branches in the same attachment variables. Recovered symbolic formulas and grouping are relevant inputs, not a characteristic composition theorem.

3. Canonical map and domain. Give the canonical map $m^{\circ }$ and its proved open domain. No equality

   $$U_{\mathrm{can}}=W\setminus \overline{N}$$

   is inferred from internal reconstruction or from the supported-sector reference.

4. A permitted preliminary lift. Construct $F_0$ acting on tests meeting the characteristic locus, or prove extension estimates that actually produce such a lift with the intended analytic restrictions.

5. The correction family. Determine the permitted map-valued corrections. If they are parametrized by $F_0+\mathcal{A}$, prove the affine parametrization and its compatibility with the frozen contact class, normalizations, topology, and microlocal requirements.

6. The actual null defect. Compute $d_0=F_0{|}_{\ker \mathsf{Rec}}$ and either give an admissible correction solving

   $$d_0+\partial a=0$$

   or evaluate a nonzero obstruction in a family proved exhaustive for the claimed exclusion.

7. Complete analytic output. Establish the complete-graph wavefront theorem, controlled remainder, uniform transverse estimates, contact compatibility, properly defined pushforward, output-space operations, and the normalized admission statement.

The supplied record does not discharge this package. This is a statement about what is implemented and proved here, not a claim that no relevant definition exists elsewhere in the corpus. Recovering a definition does not construct an admissible preliminary lift or evaluate its defect.