Theorem: XIV-B release

For the continuation frozen in Section sec:B0:

1. $({\mathfrak{A}}_W,s_{\mathrm{ext}})$ is a differential graded algebra of local classical functionals on ${\mathfrak{F}}_W$; $s_{\Psi }$ is its gauge-fixed form with XI-B's quadratic part; the Koszul–Tate sector resolves the equations of $S_{\Psi }$ and does not vanish on ${\mathcal{C}}_{\mathrm{sol}}$ at order $\kappa$.

2. Off ${\mathfrak{F}}_W$, ${s_{\mathrm{ext}}}^2{F}_v=\frac{\kappa }{2}\int v^{\nu }{\mathcal{K}}_{\nu }$, nonzero on the copied late-switch data.

3. ${\mathfrak{F}}_W\cap {\mathcal{C}}_{\mathrm{sol}}$ is the bosonic free shell with zero ghost sector; the relative complex induces on ${\mathcal{E}}_H$ nothing beyond the frozen $s_0$.

4. Under {(UCP${}_W$)} the copied ledger holds: zero on ${{\mathcal{C}}_{\mathcal{W}}}^{\mathrm{bos}}$, the pairing on ${{\mathcal{C}}_{\mathrm{sol}}}^{\mathrm{bos}}$, $-8I$ on $u_{\ast }$, and $s_0\mathcal{O}={\mathfrak{B}}_{c/2}$ for relative ghosts; this is compatible with the relative master equation and not decided by it.

5. Every linear map from relative ghosts to free ghost solutions that agrees with the ghost near $\overline{W}$ or has zero past data is zero; every promotion of ${\mathcal{C}}_W$ to a bracket-compatible quotient of ${\mathcal{E}}_H$ forces $\hslash 1=0$.

The ending is Constructed for the relative algebra on ${\mathfrak{F}}_W$ and Proved obstructed for the two lifts. It is not a constructed quantum BV algebra, and it does not license the joint-extension paper.

Proof. Theorems thm:sext, thm:ghost, thm:embedding, thm:quotient; Propositions prop:kinematic, prop:offFW, prop:gf, prop:KT; Lemma lem:separation; Corollaries cor:noint, cor:nopromotion.