Release, reassigned target, and the three endings

The target is reassigned to the missing scalar dynamics in X-B-v3's truncation. It is not the sharp-limit decision of an earlier pilot table. Taking $\epsilon \to 0$ before supplying the displayed ${\rho }_{2,g}$ would leave that truncation incomplete.

Three endings are fixed:

1. Constructed. Both maps exist on the frozen representatives, their sum is the copied response, and the completed coefficient is invariant.

2. Proved obstructed. A nonzero defect survives on the frozen algebra after both maps are included, or the required response normalization fails.

3. Unresolved. A vertex, sign, map, or insertion representative remains unnamed.

The ending reached is (1). A missing map is not a no-go. The previously displayed remainder is not repaired by a contact, by deleting a dressing term, or by shrinking the gauge algebra.

Both $\kappa$ and $\hslash$ are formal. Throughout the paper, operator and field identities involving the background expansion are understood modulo ${\kappa }^3$. Support assertions for coefficients of formal pullbacks are coefficientwise assertions, not claims about the support of an unexpanded finite pullback.