Theorem: XIV-A release

For the sheet frozen in Section sec:sheet:

1. On the unrestricted field space $\mathfrak{F}$, every extension in the kinematic class ${\mathfrak{S}}_{\mathrm{kin}}$ has antibracket square $\kappa \int c^{\nu }{\mathcal{K}}_{\nu }$ at antifield degree zero, with ${\mathcal{K}}_{\nu }$ given by (eq:Kden); this functional is nonzero (Proposition prop:witnessA1). The unrestricted classical master equation is Proved obstructed in ${\mathfrak{S}}_{\mathrm{kin}}$ at ${\hslash }^0$.

2. On the relative space ${\mathfrak{F}}_W$ of $W$-supported ghosts, $(S_{\mathrm{ext}},S_{\mathrm{ext}})=0$ exactly, $s_{\mathrm{ext}}$ is nilpotent, and XI-B's gauge fixing preserves the identity. The relative complex meets XII-A's free ghost shell only at $c=0$.

3. Under {(UCP${}_W$)}, the finite variation of $\mathcal{O}$ on ${{\mathcal{C}}_{\mathrm{sol}}}^{\mathrm{bos}}$ equals the auxiliary pairing ${\mathfrak{B}}_{\xi }[b,\varphi ]$; it vanishes on ${{\mathcal{C}}_{\mathcal{W}}}^{\mathrm{bos}}$ and equals $-8I<0$ on the copied witness. The relative free differential of $\mathcal{O}$ on ${{\mathcal{C}}_{\mathrm{sol}}}^{\mathrm{bos}}$ is ${\mathfrak{B}}_{c/2}[b,\varphi ]$.

The ending for the frozen relative sheet, namely $({\mathfrak{F}}_W,s_{\mathrm{ext}})$ together with evaluation of $\mathcal{O}$ on ${{\mathcal{C}}_{\mathcal{W}}}^{\mathrm{bos}}$, is Constructed. The unrestricted versions of both interfaces are Proved obstructed and are the boundary of the claim. The auxiliary pairing is restricted to an evaluation set; it is neither cancelled nor completed.

Proof. Theorems thm:breaking, thm:obstructA1, thm:relative, thm:eval; Propositions prop:gf, prop:carrier, prop:relO.