Release and proof boundary

and proof boundary{ Frozen (B0): the XIV-A sheet; ghost space ${\mathfrak{F}}_W\subset \mathfrak{F}$; unchanged antifield-independent observable $\mathcal{O}$; relative route.

Constructed: the algebra ${\mathfrak{A}}_W$ of local functionals on ${\mathfrak{F}}_W$ with the classical antibracket; the nilpotent differential $s_{\mathrm{ext}}$ and its gauge-fixed form $s_{\Psi }$; the Koszul–Tate identification; the free-shell ghost theorem; the copied observable ledger and the separation lemma.

Proved obstructed: every support-preserving or zero-past embedding of a nonzero relative ghost into the free ghost shell; every promotion of the evaluation set ${\mathcal{C}}_W$ to a quotient of ${\mathcal{E}}_H$ compatible with the inherited brackets.

Outside the claim: ${R_1}^{(1)}$; a quantum master equation; a Ward defect; ${\hat{s}}_{\mathrm{int}}$; a completion of ${\mathfrak{B}}_{\xi }$; a ${\star }_H$-product on antifields; a reduced algebra; $\epsilon \to 0$.

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