What can be completed from the supplied material

The intended target is a map

for the actual frozen joint graph. Its value on the retained sum must use the prescribed grouped cross-lines and mixed vertex Hessian, agree with canonical composition on its proved domain, descend under reconstruction, and satisfy the required complete-graph analytic bounds.

The supplied XV-C plan expressly requires a prerequisite stop if the defining graph data cannot be recovered. That stop applies here. A construction or a nonexistence theorem for the actual full graph cannot be justified from operators and classes that have not been specified.

Accordingly, this manuscript has two distinct conclusions:

1. the local mathematical tests stated below are proved completely;

2. their application to the full frozen graph remains unresolved at the source-completeness gate.

The second conclusion is not a mathematical obstruction to existence.