Abstract

We freeze one sheet before any antifield is written: XII-C's cubic-only density weighting, XI-B's inverse with coefficient $-1$, the switch and ramp as fixed inert backgrounds, the compact plateau algebra ${\mathfrak{g}}_W$, and the assembled second-order relational coefficient $\mathcal{O}$. The convolution-regulated sheet and the source-switched sheet are recorded with their own obstructions and are not used.

Antifields and the antibracket are introduced, and the extended action generates the kinematic diffeomorphism rule with the inherited ghost normalization $c=2\xi$. Its antibracket square is computed exactly: $(S_{\mathrm{ext}},S_{\mathrm{ext}})=\kappa \int c^{\nu }{\mathcal{K}}_{\nu }$, where ${\mathcal{K}}_{\nu }$ is XII-C's changed Noether density with the scalar-switch defect, the cubic-only bulk mismatch, and every ramp-derivative term displayed. On the unrestricted ghost space this functional is nonzero; a late-switch witness is recomputed in the present conventions. The unrestricted master equation is therefore Proved obstructed for the frozen class, at ${\hslash }^0$, independently of higher antifield terms.

On the relative space whose ghosts lie in ${\mathfrak{g}}_W$, the square vanishes identically, the differential is nilpotent, and gauge fixing by XI-B's fermion preserves the identity. That relative complex meets XII-A's free ghost shell only at $c=0$; it induces no interacting differential on the inherited carrier.

The observable ledger is separate. Under the uniform causal plateau hypothesis the finite variation of $\mathcal{O}$ on the complete free shell equals the auxiliary pairing, vanishes on the evaluation set ${\mathcal{C}}_W$, and equals $-8I<0$ on the copied witness. We prove that the relative free differential of $\mathcal{O}$ on the bosonic shell is exactly that pairing. Thus the master equation does not decide XII-F. The ending is Constructed on the relative domain; the unrestricted sheet is Proved obstructed; no completion, quantum retarded coefficient, reduced algebra, or sharp limit is opened.