Theorem: Mathematical release

For the frozen contract:

1. The complete six-branch smooth-slot expression equals (eq:fullregroup), with its full scalar-response term given by (eq:B9root).

2. The characteristic lift with the residual metric free has codimension four, while its intersection with $a=z$ has codimension eight.

3. The elementary candidate phase (eq:phase) has rank-six auxiliary equations and a codimension-eight critical set. It is neither nondegenerate nor clean.

4. The transverse identity (eq:quadratic) exhibits the two relative directions missed by the first-order cone equations.

These conclusions establish an exact operator reduction and a specific phase-construction failure. They do not establish the joint characteristic distribution.

Proof. Theorem thm:regroup and Proposition prop:root give (i). Proposition prop:lifts gives (ii). Lemma lem:zerolocus and Theorem thm:nonclean give (iii). Proposition prop:quadratic gives (iv). The limitation follows from Proposition prop:limits.