The next proof obligations

A construction must now do the following:

1. establish the actual two-edge distributions and their compact polynomial kernels;

2. check (eq:admission) for both metric-quadratic outputs, at every configuration and after the required descendants;

3. prove descent of all second-stage prescriptions through $p_L,q_L,p_R,q_R$;

4. evaluate the scalar relation tests and determine whether ${\mathcal{C}}_{\mathrm{desc}}$ is nonempty;

5. verify overlap agreement with the old bulk coefficients and between any two prescriptions for the same actual pair;

6. prove the named continuity claims without importing normal-stage sharp-admission convergence;

7. establish reversal of every nonconstant coefficient map.

An explicit relation with nonzero calibrated value would give a genuine factorization obstruction for the imposed parent class. A demonstrated failure of one proposed graph extension would exclude that candidate only, unless all permitted alternatives were separately excluded.