Abstract

XIV-A froze one sheet and proved the relative classical master equation $(S_{\mathrm{ext}},S_{\mathrm{ext}})=0$ on the space ${\mathfrak{F}}_W$ whose ghosts are supported where both backgrounds are trivial, together with its failure on the unrestricted space. This paper continues that complex and separates it from the inherited quantum carrier.

We construct the graded algebra of local classical functionals on ${\mathfrak{F}}_W$ with its antibracket, prove that $s_{\mathrm{ext}}=(S_{\mathrm{ext}},\cdot )$ is an odd, ghost-number-one, nilpotent derivation of it, and record that nilpotence fails off ${\mathfrak{F}}_W$ on an explicit antifield functional. The Koszul–Tate sector of the gauge-fixed differential resolves the equations of the gauge-fixed interacting action; these agree with XII-A's free shell at order ${\kappa }^0$ and differ at order $\kappa$.

The theorem of the paper is the free-shell ghost boundary: a ghost supported in $W$ that solves the free ghost equation is zero, by zero-past Cauchy uniqueness. Hence ${\mathfrak{F}}_W$ meets XII-A's complete shell only at $c=\bar{c}=0$, and the kinematic rule evaluated there reduces to the frozen free rule. The relative complex supplies no interacting ghost differential on ${\mathcal{E}}_H$.

The observable ledger is copied with its domains. On ${\mathfrak{F}}_W$ the master identity holds while the relative free differential of the observable on the bosonic shell equals the auxiliary pairing, $-8I\vartheta \ne 0$ on the copied witness. Two lifts are Proved obstructed: a support- or past-preserving embedding of a nonzero relative ghost into the free ghost shell, and the promotion of the evaluation set to a quotient of ${\mathcal{E}}_H$. XIII-C's four obligations remain the unopened quantum gate.