No automatic descent to the unchanged free carrier

This is the same type of domain obstruction distinguished in X-H-ST, now exhibited without identifying compact relative ghosts with nonzero compact free ghost solutions XHST.

For the relative local comparison, one can choose a compact ghost coefficient equal to $t{\partial }_1$ near the compact observation test. It gives the same local non-descent diagnostic, but it is not a homogeneous free ghost on the full completion.

A compactly supported smooth solution of the homogeneous wave equation on spacetime is zero: take a Cauchy surface before its support and use uniqueness. Consequently the compact relative ghost sector and the unrestricted free ghost carrier cannot be identified.

The construction here is therefore an interacting off-shell classical relative BV complex. It is not an interacting derivation of the unchanged free causal algebra.