% !TeX program = pdflatex %===================================================================== % XIV-B.tex % % Standalone source for pdfLaTeX. % Build twice: % pdflatex -interaction=nonstopmode -halt-on-error XIV-B.tex % pdflatex -interaction=nonstopmode -halt-on-error XIV-B.tex % % Continuation frozen (B0): % sheet XII-C cubic-only S_D, inverse -1, inert g and sigma_eps; % algebra g_W = C_c^infty(W;TM); % ghost space F_W only (c in g_W, antighost compact in the open slab); % observable the unchanged antifield-independent lift O; % route relative symmetry; no spurion, no source-switched % upgrade, no convolution sheet, no new completion class. % % Proved here: % - the graded algebra A_W of local functionals on F_W, the classical % antibracket on it, and the nilpotent differential s_ext=(S_ext, .) % (nilpotence from XIV-A's relative identity and Jacobi); % - on antifield-independent functionals s_ext is the kinematic rule % and is nilpotent off shell without the master equation; % - off F_W, s_ext^2 F = (1/2)(K,F) is nonzero on an explicit % antifield functional (copied late-switch witness); % - the Koszul--Tate sector of the gauge-fixed differential resolves % the equations of S_Psi, which agree with XII-A's free shell at % kappa^0 and differ at kappa^1 (explicit witness); % - the free-shell ghost theorem: F_W meets C_sol only at c = cbar = 0, % by zero-past uniqueness; no interacting ghost differential on E_H; % - the observable ledger copied with its domains, and a separation % lemma: (S_ext,S_ext)=0 on F_W while s_free O(u_*) = -8I theta != 0; % - two illegal lifts 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 C_W to a quotient of E_H. % % Not constructed: % R_1^(1), a quantum master equation, a Ward defect, s_hat_int, a % completion of the auxiliary pairing, a star-product on antifields, % a reduced algebra, a state, a loop, or a sharp-width limit. % % No successful compilation, proof-assistant verification, % independently executed symbolic computation, independent referee % review, or journal acceptance is asserted. %===================================================================== \documentclass[11pt,a4paper]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage[margin=27mm]{geometry} \usepackage{microtype} \usepackage{amsmath,amssymb,amsthm,mathtools,mathrsfs} \usepackage{array,booktabs,tabularx} \usepackage{enumitem} \usepackage[hidelinks]{hyperref} \allowdisplaybreaks \setlength{\emergencystretch}{3em} \setlist{leftmargin=*,itemsep=3pt,topsep=5pt} \numberwithin{equation}{section} \setcounter{tocdepth}{1} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{inputstatement}[theorem]{Input statement} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newcommand{\R}{\mathbb R} \newcommand{\C}{\mathbb C} \newcommand{\dd}{\,\mathrm d} \newcommand{\eps}{\varepsilon} \newcommand{\sig}{\sigma_\eps} \newcommand{\MT}{M_T} \newcommand{\MTo}{M_T^\circ} \newcommand{\PP}{\mathcal P} \newcommand{\RR}{\mathcal R} \newcommand{\Lie}{\mathcal L} \newcommand{\Pmass}{P_m} \newcommand{\ER}{E_0^{\mathrm R}} \newcommand{\EA}{E_0^{\mathrm A}} \newcommand{\Gret}{G_{\mathrm{gr}}^{\mathrm R}} \newcommand{\Csol}{\mathscr C_{\mathrm{sol}}} \newcommand{\Cbos}{\mathscr C_{\mathrm{sol}}^{\mathrm{bos}}} \newcommand{\CW}{\mathscr C_W} \newcommand{\CWbos}{\mathscr C_W^{\mathrm{bos}}} \newcommand{\Fpol}{\mathscr F_{\mathrm{mc,pol}}} \newcommand{\Nsol}{\mathscr N_{\mathrm{sol}}} \newcommand{\NW}{\mathscr N_W} \newcommand{\Ealg}{\mathscr E_H} \newcommand{\Aone}{\mathcal A_1} \newcommand{\Asc}{\mathcal A_2^{\mathrm{sc}}} \newcommand{\CD}{\mathcal C_{3,\mathrm D}} \newcommand{\JD}{\mathcal J_{\mathrm D}} \newcommand{\PhiD}{\Phi^{(2)}_{\mathrm D}} \newcommand{\rD}{r_{\mathrm D}} \newcommand{\rg}{r_g} \newcommand{\Ocal}{\mathcal O} \newcommand{\Aux}{\mathfrak B} \newcommand{\gW}{\mathfrak g_W} \newcommand{\Kbrk}{\mathfrak K} \newcommand{\Kden}{\mathcal K} \newcommand{\Sext}{S_{\mathrm{ext}}} \newcommand{\Sgf}{S_{\Psi}} \newcommand{\SD}{S_{\mathrm D}} \newcommand{\abr}[2]{\left(#1,#2\right)} \newcommand{\Fsp}{\mathfrak F} \newcommand{\FW}{\mathfrak F_W} \newcommand{\AW}{\mathfrak A_W} \newcommand{\Gfree}{\mathcal G_0} \newcommand{\Rampdef}{\mathfrak R} \newcommand{\Zsource}{Z_\sig} \newcommand{\Dsource}{\mathcal D_\sig} \newcommand{\Bsw}{\mathcal B} \newcommand{\Jcal}{\mathcal J} \newcommand{\Rquant}{R_1^{(1)}} \newcommand{\sfree}{s_0} \newcommand{\sext}{s_{\mathrm{ext}}} \newcommand{\sPsi}{s_{\Psi}} \newcommand{\Kclass}{\mathfrak K_{\mathrm{loc,R}}} \newcommand{\Sclass}{\mathfrak S_{\mathrm{kin}}} \newcommand{\Lag}{\mathfrak L} \newcommand{\pair}[2]{\left\langle #1,#2\right\rangle} \DeclareMathOperator{\supp}{supp} \DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\Int}{int} \DeclareMathOperator{\id}{id} \DeclareMathOperator{\gh}{gh} \newcommand{\scopebox}[2]{% \par\medskip\noindent \fbox{\begin{minipage}{% \dimexpr\linewidth-2\fboxsep-2\fboxrule\relax} \textbf{#1}\par\smallskip #2 \end{minipage}} \par\medskip } \title{\textbf{Landmark XIV-B}\\[0.4em] The relative antifield complex on $\mathfrak F_W$,\\ and its trivial ghost sector on the free shell} \author{Ngo Cheung} \date{} \usepackage{tikz} \begin{document} \maketitle \begin{abstract} XIV-A froze one sheet and proved the relative classical master equation $(\Sext,\Sext)=0$ on the space $\FW$ 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 $\FW$ with its antibracket, prove that $\sext=(\Sext,\,\cdot\,)$ is an odd, ghost-number-one, nilpotent derivation of it, and record that nilpotence fails off $\FW$ 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 $\FW$ 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 $\Ealg$. The observable ledger is copied with its domains. On $\FW$ the master identity holds while the relative free differential of the observable on the bosonic shell equals the auxiliary pairing, $-8I\,\vartheta\ne0$ 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 $\Ealg$. XIII-C's four obligations remain the unopened quantum gate. \end{abstract} \begin{figure}[ht] \centering \begin{tikzpicture}[x=1cm,y=1cm] \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (a) at (0,0) {XIV-A sheet}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Ghost-space calculation}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {A state or a BV charge}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {The ghost calculation does not produce a state.}; \draw[->] (a) -- (b); \draw[->] (b) -- (c); \draw[->] (b) -- (s); \end{tikzpicture} \caption{The ghost calculation stays on the XIV-A sheet. The three boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.} \label{fig:scope-XIV-B} \end{figure} \begin{figure}[ht] \centering \begin{tikzpicture}[x=1cm,y=1cm] \node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (a) at (0,0) {XIV-A sheet}; \node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {A state or a BV charge}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {The ghost calculation does not produce a state.}; \draw[->] (a) -- (s); \draw[->] (b) -- (s); \end{tikzpicture} \caption{The ghost calculation stays on the XIV-A sheet. The boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.} \label{fig:split-XIV-B} \end{figure} \begin{figure}[ht] \centering \begin{tikzpicture}[x=1cm,y=1cm] \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (a) at (0,0) {XIV-A sheet}; \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Ghost-space calculation}; \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {A state or a BV charge}; \node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {The ghost calculation does not produce a state.}; \draw[->] (a) -- (b); \draw[->] (b) -- (c); \draw[->] (c) -- (s); \end{tikzpicture} \caption{Reading downward is not a transfer. The boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.} \label{fig:chain-XIV-B} \end{figure} \begin{figure}[ht] \centering \begin{tikzpicture}[x=1cm,y=1cm] \node[draw,rounded corners,align=center,text width=4.2cm,font=\small,inner sep=3pt] (a) at (0,0) {XIV-A sheet}; \node[align=center,font=\small] (m) at (5.2,0) {not the same object}; \node[draw,rounded corners,align=center,text width=4.2cm,font=\small,inner sep=3pt] (b) at (10.4,0) {A state or a BV charge}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Ghost-space calculation. The ghost calculation does not produce a state.}; \draw (a) -- (m); \draw (m) -- (b); \draw[->] (m) -- (s); \end{tikzpicture} \caption{XIV-A sheet is not A state or a BV charge. The boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.} \label{fig:pair-XIV-B} \end{figure} \begin{figure}[ht] \centering \begin{tikzpicture}[x=1cm,y=1cm] \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (a) at (0,0) {Named: XIV-A sheet}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Ghost-space calculation}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: A state or a BV charge}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {The ghost calculation does not produce a state.}; \draw[->] (a) -- (b); \draw[->] (b) -- (c); \draw[->] (b) -- (s); \end{tikzpicture} \caption{Three statuses stay separate. The boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.} \label{fig:gate-XIV-B} \end{figure} Figure~\ref{fig:split-XIV-B} records one view of the same separation. Figure~\ref{fig:chain-XIV-B} records one view of the same separation. Figure~\ref{fig:pair-XIV-B} records one view of the same separation. Figure~\ref{fig:gate-XIV-B} records one view of the same separation. The drawings add no hypothesis. Figure~\ref{fig:scope-XIV-B} records that separation. The drawing adds no hypothesis. \scopebox{Release and proof boundary}{% \textbf{Frozen (B0):} the XIV-A sheet; ghost space $\FW\subset\Fsp$; unchanged antifield-independent observable $\Ocal$; relative route. \smallskip \textbf{Constructed:} the algebra $\AW$ of local functionals on $\FW$ with the classical antibracket; the nilpotent differential $\sext$ and its gauge-fixed form $\sPsi$; the Koszul--Tate identification; the free-shell ghost theorem; the copied observable ledger and the separation lemma. \smallskip \textbf{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 $\CW$ to a quotient of $\Ealg$ compatible with the inherited brackets. \smallskip \textbf{Outside the claim:} $\Rquant$; a quantum master equation; a Ward defect; $\widehat s_{\mathrm{int}}$; a completion of $\Aux_\xi$; a $\star_H$-product on antifields; a reduced algebra; $\eps\to0$. \smallskip \textbf{Ending:} Constructed for the relative algebra on $\FW$; Proved obstructed for the two lifts. It is not a quantum BV algebra and it does not license the joint-extension paper.} \tableofcontents %===================================================================== \section{Release criteria and the copied premise sheet} \label{sec:inputs} %===================================================================== The endings are \textbf{Constructed}, \textbf{Proved obstructed}, and \textbf{Unresolved}, applied to the named object on the named domain. A missing quantum product is not a no-go. A constructed relative complex is not an interacting differential on $\Ealg$. \[ \boxed{\text{XIV-B continues }(\FW,\sext)\text{ and proves that it does not become a differential on }\Ealg.} \] The following statements are copied and are not re-proved. \begin{inputstatement}[XIV-A: the breaking functional and the two spaces] \label{input:XIVA} On the frozen sheet (XII-C's cubic-only $\SD$, inverse $\Gret=-\PP\ER$ with coefficient $-1$, inert $g$ and $\sig$), with the kinematic rule \begin{equation} s\gamma=\RR c+\tfrac\kappa2\Lie_c\gamma,\quad s\phi=\tfrac\kappa2c^\mu\partial_\mu\phi,\quad sc^\mu=\tfrac\kappa2c^\nu\partial_\nu c^\mu,\quad s\bar c_\mu=ib_\mu,\quad sb=0, \label{eq:kin} \end{equation} ($\RR=K/2$, $c=2\xi$ recovers $K\xi$), and the extended action \begin{equation} \Sext=\SD[\gamma,\phi]+\int\bigl[\gamma^{*\mu\nu}s\gamma_{\mu\nu} +\phi^*s\phi+c^*_\mu sc^\mu+\bar c^{*\mu}s\bar c_\mu\bigr]\dd^4x, \label{eq:Sext} \end{equation} one has, on the space $\Fsp$ of smooth multiplets with ghosts compactly supported in $\MTo$, \begin{equation} \abr\Sext\Sext=\Kbrk[\gamma,\phi,c] =\kappa\int_{\MT}c^\nu\Kden_\nu\dd^4x, \qquad \Kden_\nu=\Bsw_{g,\nu}+wQ_\nu(A\gamma)-\Rampdef_{\sig,\nu} -\kappa Q_\nu(\Zsource), \label{eq:breaking} \end{equation} with $w=\sig-1$ and the displayed XII-C densities. Every term of $\Kden_\nu$ carries $1-g$, $\partial g$, $w$, $\partial\sig$, or $\partial^2\sig$; hence $\Kden_\nu=0$ on $P_g\cap\{t>\eps\}$. On the unrestricted space the late-switch value is nonzero: \begin{equation} \Kbrk[0,\phi_m,\vartheta\zeta\partial_t] =\frac\kappa2\vartheta\int\zeta\,\partial_tg\dd^4x\ne0, \qquad 0\ne\zeta\in C_c^\infty(O),\ \zeta\ge0,\ \partial_tg\le-c_0<0 \text{ on }O, \label{eq:witnessA1} \end{equation} and every extension in the kinematic class $\Sclass$ has this antifield-degree-zero component; no $\hbar$ correction removes it. On the relative space $\FW$ (ghost $c\in\gW\otimes\Lambda^{\mathrm{odd}}$), \begin{equation} \abr\Sext\Sext=0\ \text{identically},\qquad \sext^2=0, \label{eq:relative} \end{equation} and XI-B's fermion $\Psi=-i\int\bar c_\mu(2D^\mu\gamma-b^\mu)$ preserves both statements: $\Sgf[\Phi,\Phi^*]:=\Sext[\Phi,\Phi^*+\delta\Psi/\delta\Phi]$ has $(\Sgf,\Sgf)=\Kbrk$ on $\Fsp$, $0$ on $\FW$, and \begin{equation} \Sgf\big|_{\Phi^*=0} =\SD+\int\bigl[2b_\mu D^\mu\gamma-b_\mu b^\mu +i\bar c_\mu(\Box c^\mu+\kappa D^\mu(\Lie_c\gamma))\bigr]\dd^4x . \label{eq:Sgf} \end{equation} The relative complex meets the free ghost shell only at zero: $\gW\cap\{\Box c=0\}=\{0\}$ \cite{XIVA}. \end{inputstatement} \begin{inputstatement}[XIV-A: the observable ledger] \label{input:ledger} With $\Ocal=\PhiD+\Aone[f,g](\rg[\phi];\phi)$ the unchanged lift, $\Cbos=\{\Box\gamma=0,\Pmass\phi=0,b=D\gamma\}$ and $\CWbos=\{u\in\Cbos:D^*b=0\text{ on }W\}$, under \textup{(UCP$_W$)} \begin{equation} \Delta_\xi\Ocal\big|_{\Cbos}=\Aux_\xi[b,\phi] =-2\int(D\Lie_\xi\beta_{f,g}[\phi])_\nu b^\nu\dd^4x, \qquad \Delta_\xi\Ocal\big|_{\CWbos}=0, \qquad \Delta_{\xi_*}\Ocal(u_*)=-8I<0, \label{eq:ledger} \end{equation} and, for relative ghosts $c\in\gW\otimes\Lambda^{\mathrm{odd}}$ on $\Cbos$, \begin{equation} \sfree\Ocal:=\pair{\Ocal'_\gamma}{\RR c}=\Aux_{c/2}[b,\phi], \qquad \sfree\Ocal(u_*)=-8I\,\vartheta\ne0\ \ (c=2\vartheta\xi_*), \label{eq:relO} \end{equation} with $\pair{\Ocal''_{\gamma\gamma}}{K\xi\otimes K\xi}=0$ on $\Cbos$. The master equation does not decide XII-F \cite{XIVA,XIIF}. \end{inputstatement} \begin{inputstatement}[XIII-C: the unopened gate] \label{input:XIIIC} The coefficient of $\lambda\hbar$ is Unresolved. A successor must supply a prescription through the noncoincident characteristic configurations or a proved cancellation; uniform transverse estimates on the three- and four-vertex strata; compatible partial contacts; the output wavefront theorem after proper pushforward. No free or extended differential is applied to $\Rquant$ \cite{XIIIC}. \end{inputstatement} \begin{inputstatement}[XIII-A and XIII-D: two closed doors] \label{input:XIIIAD} For every nonempty $W$ with $\overline W\subset\Int\{g=1\}\cap\{\eps0; \label{eq:commutator} \end{equation} $\NW[[\hbar,\kappa]]$ is not a two-sided ideal of $\star_H$, no associative product on $(\Fpol/\NW)[[\hbar,\kappa]]$ makes restriction a unital homomorphism, a unital homomorphism killing every $L_q$ while preserving the brackets forces $\hbar\mathbf1=0$, and the admissible rank-one ordering change gives $\alpha_d(L_q^2)=L_q^2+\hbar C(q,h)^2$ \cite{XIIIA}. Completion of the auxiliary pairing inside $\Kclass$ is Proved obstructed; the cochain $\mathcal C_\xi=-\Aux_\xi[D\gamma,\phi]$ is admitted and is not an integrated completion \cite{XIIID}. \end{inputstatement} \begin{inputstatement}[XII-A and XI-B: carrier, free rule, uniqueness] \label{input:XIIA} $\Ealg=(\Fpol[[\hbar,\kappa]],\star_H)/\Nsol[[\hbar,\kappa]]$, with $\Csol=\{\Box\gamma=0,\Pmass\phi=0,\Box c=\Box\bar c=0,b=D\gamma\}$; ghosts in $\Csol$ are smooth ambient free solutions with arbitrary Cauchy data. The frozen free differential is $\sfree\gamma=\RR c$, $\sfree\bar c=ib$, $\sfree\phi=\sfree c=\sfree b=0$, a nilpotent derivation of $\star_H$ descending to $\Ealg$. A smooth solution of $\Box u=0$ on the slab with zero Cauchy data on some $\{t=t_0\}$, $0\le t_00$. Then $u=0$ on $\MT$. \end{lemma} \begin{proof} Work coefficientwise in $\Lambda$. Choose $t_0\in(0,t_1)$; $u$ vanishes on a neighbourhood of $\{t=t_0\}$, so $u$ and $\partial_tu$ vanish there. Uniqueness for the smooth homogeneous Cauchy problem on $[t_0,T]\times\R^3$ gives $u=0$ there \cite[Ch.~3]{BGP}; for $t\le t_0$, $u=0$ by hypothesis. \end{proof} \begin{theorem}[Free-shell ghost sector of $\FW$] \label{thm:ghost} \begin{enumerate}[label=\textup{(\roman*)}] \item A ghost $c\in\gW\otimes\Lambda^{\mathrm{odd}}$ with $\Box c=0$ is zero. An antighost $\bar c\in C_c^\infty(\MTo)\otimes\Lambda^{\mathrm{odd}}$ with $\Box\bar c=0$ is zero. \item Consequently \begin{equation} \FW\cap\Csol=\{(\gamma,\phi,0,0,b):\ (\gamma,\phi,b)\in\Cbos\}, \label{eq:intersection} \end{equation} the bosonic free shell with zero ghost sector. \item At every configuration in \eqref{eq:intersection} the kinematic rule \eqref{eq:kin} takes the values \[ s\gamma=0,\qquad s\phi=0,\qquad sc=0,\qquad s\bar c=ib,\qquad sb=0 . \] Every $\kappa$-dependent term of $s$ vanishes there, and the only nonzero component is the frozen free rule $\sfree\bar c=ib$. For an antifield-independent $F$, $(\sext F)(u)=\pair{F'_{\bar c}(u)}{ib}$ on the intersection; for $F$ without antighost argument, in particular for $\Ocal$, $(\sext F)(u)=0$ there. \end{enumerate} \end{theorem} \begin{proof} (i) $\supp c\subset\overline W$, compact in $\{t>\eps\}$, so $t_1:=\min_{\overline W}t>\eps>0$ and Lemma~\ref{lem:uniq} applies; $\supp\bar c$ is compact in the open slab, so $t_1>0$ and the lemma applies again. (ii) A configuration of $\FW$ lies in $\Csol$ exactly when its bosonic part lies in $\Cbos$ and its ghosts solve the free ghost equations; by (i) the latter forces $c=\bar c=0$. (iii) Each $\kappa$-dependent term of \eqref{eq:kin} is linear in $c$, as is $\RR c$; the antighost rule is $c$-independent. The last assertion is \eqref{eq:sF} evaluated at $c=0$. \end{proof} \begin{corollary}[No interacting ghost differential on $\Ealg$] \label{cor:noint} No nonzero relative ghost is a configuration of XII-A's shell. The restriction of the relative complex to $\FW\cap\Csol$ carries the inherited free rule $\sfree$ evaluated at $c=0$ and nothing else. Hence $(\AW,\sext)$ supplies no differential on $\Ealg$ beyond the frozen $\sfree$, and in particular no interacting ghost differential $\widehat s_{\mathrm{int}}$. \end{corollary} \begin{proof} Theorem~\ref{thm:ghost}(i)--(iii). $\Ealg$'s ghosts are free fields with arbitrary Cauchy data (Input statement~\ref{input:XIIA}); by (i) none of them, except zero, lies in $\gW\otimes\Lambda$. \end{proof} \begin{remark}[Tests versus configurations] A compact real vector field $v\in\gW$ is a legal smearing test for the ghost field of $\Ealg$: $c(v)=\int v_\mu c^\mu$ is an element of the carrier, with $\sfree c(v)=0$. That element is a functional, not a configuration, and $v$ is not thereby promoted to a free ghost solution. Likewise a compact ghost that is not a free solution is a legal element of $\FW$ and a legal argument of the breaking functional; it is not an element of $\Csol$. \end{remark} B2 ends \textbf{Proved}: the free-shell ghost sector of $\FW$ is $\{0\}$. %===================================================================== \section{B3: the observable ledger, copied, and the separation lemma} \label{sec:B3} %===================================================================== The identities \eqref{eq:ledger}--\eqref{eq:relO} are copied with their domains. The cylinder integral of XII-F and the uniform-plateau argument of XIII-A are not repeated. The release statement remains: evaluation on $\CWbos$, with the pairing printed on $\Cbos$; not closure on $\Csol$; the witness $u_*$ outside XIII-A's defect set; $\gW$ not shrunk after the witness. \begin{lemma}[Separation of the action identity from the observable identity] \label{lem:separation} On $\FW$ the following hold simultaneously and are compatible: \begin{enumerate}[label=\textup{(\alph*)}] \item $\abr\Sext\Sext=0$ and $\sext^2=0$ on $\AW$ (Theorem~\ref{thm:sext}); \item $s^2\Ocal=0$ off shell (Proposition~\ref{prop:kinematic}); \item for $c=2\vartheta\xi_*$ on the detecting tuple, $\sfree\Ocal(u_*)=\Aux_{\xi_*}(u_*)\vartheta=-8I\,\vartheta\ne0$, where $\sfree\Ocal$ is the $\kappa^0$ part of $\sext\Ocal=s\Ocal$ \eqref{eq:relO}. \end{enumerate} Neither (a) nor (b) implies that $s\Ocal$ vanishes on $\Cbos$; (c) shows that it does not. The master equation constrains the action; it does not decide the invariance of the given lift. \end{lemma} \begin{proof} (a) and (b) are proved above; (c) is copied from Input statement~\ref{input:ledger}. Compatibility: (a) is a statement about $\Sext$ and about $\sext$ on $\AW$; (b) is the statement that $s\Ocal$ is $s$-closed, which is automatic for an image of $s$; (c) is the value of that image at one point of $\Cbos$, outside the intersection \eqref{eq:intersection} because $c\ne0$ there. No statement among (a)--(c) cancels another. \end{proof} The $\kappa$-corrections are printed and left unevaluated, as in XIV-A: \begin{equation} s\Ocal=\sfree\Ocal+\frac\kappa2\Bigl[\pair{\Ocal'_\gamma}{\Lie_c\gamma} +\pair{\Ocal'_\phi}{c^\mu\partial_\mu\phi}\Bigr], \qquad \pair{\Ocal'_\phi}{Y}=\pair{(\PhiD)'_\phi}{Y} +\Aone\bigl(\rg{}'_\phi[Y];\phi\bigr)+\Aone\bigl(\rg;Y\bigr). \label{eq:kappa-corrections} \end{equation} Since $\Ocal$ is the $\kappa^2$ coefficient of the relational expansion, these terms enter at $\kappa^3$ there. A proposal that moves the lower coefficient $\kappa\Aone$ must display its own coupling power. $\Ocal$, which already contains the classical scalar response and the slot $\rD+\rg$, is not identified with a bare field observable of the BV complex; that is X-G's bare-versus-completed distinction and it is not made here. B3 ends with the copied evaluation on $\CWbos$ and the nonzero pairing on $\Cbos$. %===================================================================== \section{B4: two illegal lifts} \label{sec:B4} %===================================================================== Both lifts are excluded by explicit incompatibilities, not by absent formulas. \subsection{Ghost embedding} Let $\Gfree$ be the space of smooth $\Lambda$-valued free ghost solutions on the slab, $\Box c=0$, with arbitrary Cauchy data: the ghost configurations of $\Csol$. \begin{theorem}[No support- or past-preserving embedding of relative ghosts] \label{thm:embedding} Let $\iota:\gW\otimes\Lambda^{\mathrm{odd}}\to\Gfree$ be a linear map satisfying at least one of: \begin{enumerate}[label=\textup{(\alph*)}] \item $\iota(c)=c$ on a neighbourhood of $\overline W$ for every $c$; \item $\iota(c)$ has zero Cauchy data on some surface $\{t=t_0\}$ with $0\le t_0<\min_{\overline W}t$, for every $c$; \item $\supp\iota(c)\subset J^+(\supp c)\cap\MT$ for every $c$. \end{enumerate} Then $\iota=0$. In particular the inclusion $\FW\hookrightarrow\Fsp$ restricts to the zero map on the ghost sector of $\Csol$. \end{theorem} \begin{proof} (a) On the neighbourhood $N$ of $\overline W$, $\Box c=\Box\iota(c)=0$; on the complement of $\overline W$, $c=0$. Thus $\Box c=0$ on all of $\MT$ and $c=0$ by Theorem~\ref{thm:ghost}(i); then $\iota(c)=\iota(0)=0$. (b) A free solution with zero Cauchy data on $\{t=t_0\}$ vanishes on $[t_0,T]\times\R^3$ by uniqueness and on $[0,t_0]$ by the backward problem with the same data; hence $\iota(c)=0$. (c) $J^+(\supp c)\cap\MT\subset\{t\ge\min_{\overline W}t\}$, so $\iota(c)$ has zero Cauchy data on any earlier surface, and (b) applies. \end{proof} What is not excluded is recorded. A map that propagates the Cauchy data of $c$ from a surface crossing $W$ produces a nonzero free ghost; by (a) that ghost differs from $c$ near $W$, so the configuration on which the relative symmetry acts and the configuration in the carrier are different. Such a map is not an embedding of $\FW$'s ghost sector, its image is not in $\gW$, the relative identity \eqref{eq:relative} does not apply to it, and it lies outside the frozen contract. Whether the breaking functional vanishes on such ghosts is not decided here. \subsection{Evaluation quotient} \begin{theorem}[Copied: the evaluation set is not a quotient] \label{thm:quotient} For the frozen $W$, the coefficientwise vanishing space $\NW[[\hbar,\kappa]]$ is not a two-sided ideal of $\star_H$, and its image $(\NW/\Nsol)[[\hbar,\kappa]]$ is not a two-sided ideal of $\Ealg$. There is no unital $\C[[\hbar,\kappa]]$-algebra homomorphism from $\Ealg$ killing the classes of all $L_q$, $q\in C_c^\infty(W)$, and preserving the inherited brackets, except one with $\hbar\mathbf1=0$ in its target. Coefficientwise vanishing on $\CW$ is not preserved by every admissible ordering transport. \end{theorem} \begin{proof} XIII-A's non-ideal theorem, its corollary, and its rank-one ordering calculation, copied as Input statement~\ref{input:XIIIAD}. The mechanism is \eqref{eq:commutator}: the formal scalar $i\hbar C(q,h)\mathbf1$ with $C(q,h)>0$ does not vanish on $\CW$, which contains the zero configuration. \end{proof} \begin{corollary}[The evaluation identity cannot be promoted] \label{cor:nopromotion} The identity $\Delta_\xi\Ocal|_{\CWbos}=0$ of \eqref{eq:ledger} cannot be realized as an identity in a quotient of $\Ealg$ by the classes of functionals vanishing on $\CW$, compatibly with the inherited brackets and a nonzero formal unit. The auxiliary pairing itself belongs to $\NW\setminus\Nsol$: it vanishes on $\CW$ and has nonzero class in $\Ealg$ by the copied witness. Its vanishing is an evaluation on $\CWbos$, and remains one. \end{corollary} \begin{proof} Such a promotion is a unital homomorphism whose kernel contains $(\NW/\Nsol)[[\hbar,\kappa]]$, hence every $\bar L_q$; Theorem~\ref{thm:quotient} forces $\hbar\mathbf1=0$. Membership of $\Aux_\xi$ in $\NW$ is the vanishing on $\CW$ in \eqref{eq:ledger}; non-membership in $\Nsol$ is the value $-8I\ne0$ at $u_*\in\Csol$. \end{proof} \subsection{The cochain stays outside} The admitted cochain $\mathcal C_\xi=-\Aux_\xi[D\gamma,\phi]$ cancels the pairing algebraically on $\Cbos$ and is not an integrated completion in $\Kclass$ (Input statement~\ref{input:XIIIAD}). It is not inserted into $\Ocal$: that would change the observable and void the reuse of $u_*$ (Section~\ref{sec:B0}). It is not inserted into $\Sext$: the only ghost-dependent form it could take is $\mathcal C_{c/2}$, which is odd of ghost number one and cannot be a summand of an even, ghost-number-zero action; a parameter-indexed family is not a term of one functional. B4 ends \textbf{Proved obstructed} for both lifts. %===================================================================== \section{B5: the recorded quantum gate} \label{sec:B5} %===================================================================== XIII-C's four obligations are repeated unchanged (Input statement~\ref{input:XIIIC}): a prescription through the noncoincident characteristic configurations or a proved cancellation in the selected pair; uniform transverse estimates on the three- and four-vertex strata and their intersections; compatible extensions with the permitted partial contacts; the output wavefront estimate after proper pushforward. No coefficient of $\lambda\hbar$ exists. Accordingly: neither $\sext$, $\sPsi$, nor $\sfree$ is applied to $\Rquant$; no Hadamard line is contracted against $\JD$; no binary Ward defect is formed. The classical breaking functional $\Kbrk$ is the antibracket square of the extended action on the unrestricted ghost space at $\hbar^0$; it is not the anomaly of an unconstructed quantum defect, and \eqref{eq:offFW-witness} is not a quantum statement. X-G's reciprocity pairing on the source-switched sheet with inverse $-4$, and X-A's convolution defect $S_\eps\xi\ne\xi$, remain non-identifications: neither sheet is the one frozen here, and no obstruction proved on either is imported. XII-C's empty coupled jet $\Jcal^{\mathrm D}_{g,\sig}(\phi_m)=\varnothing$ is a solution-fiber statement, not a failure of $\Ocal$; the slot $\rD+\rg$ solves a gauge-fixed forcing equation and does not fill it. The XI-G classes remain excluded; $\Ocal$ is not a representative. %===================================================================== \section{Decision, release, and handoff} \label{sec:decision} %===================================================================== \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{.07\linewidth} p{.27\linewidth}X@{}} \toprule Gate & Ending & Exact scope\\ \midrule B0 & Frozen & XIV-A sheet; $\FW\subset\Fsp$ with compact antighost; unchanged $\Ocal$; relative route; no completion, spurion, or new sheet.\\[0.35em] B1 & Constructed & $\AW$ with classical antibracket; $\sext$ odd, $\gh+1$, nilpotent; $\sext^2F=\tfrac12(\Kbrk,F)\ne0$ off $\FW$; gauge-fixed $\sPsi$; Koszul--Tate sector resolves $\Sgf$'s shell, equal to $\Csol$ at $\kappa^0$, different at $\kappa^1$.\\[0.35em] B2 & Proved & $\FW\cap\Csol$ has $c=\bar c=0$; the kinematic rule there is $\sfree$ at $c=0$; no interacting ghost differential on $\Ealg$.\\[0.35em] B3 & Copied; separation proved & Evaluation on $\CWbos$; $\Aux_\xi$ printed on $\Cbos$; $-8I$ on $u_*$; $(\Sext,\Sext)=0$ compatible with $\sfree\Ocal(u_*)\ne0$.\\[0.35em] B4 & Proved obstructed & No support- or past-preserving ghost embedding; no evaluation quotient; cochain outside $\Ocal$ and $\Sext$.\\[0.35em] B5 & Recorded & XIII-C unopened; no differential on $\Rquant$; non-identifications retained.\\ \bottomrule \end{tabularx} \end{center} \begin{theorem}[XIV-B release] \label{thm:release} For the continuation frozen in Section~\ref{sec:B0}: \begin{enumerate}[label=\textup{(\roman*)}] \item $(\AW,\sext)$ is a differential graded algebra of local classical functionals on $\FW$; $\sPsi$ is its gauge-fixed form with XI-B's quadratic part; the Koszul--Tate sector resolves the equations of $\Sgf$ and does not vanish on $\Csol$ at order $\kappa$. \item Off $\FW$, $\sext^2F_v=\tfrac\kappa2\int v^\nu\Kden_\nu$, nonzero on the copied late-switch data. \item $\FW\cap\Csol$ is the bosonic free shell with zero ghost sector; the relative complex induces on $\Ealg$ nothing beyond the frozen $\sfree$. \item Under \textup{(UCP$_W$)} the copied ledger holds: zero on $\CWbos$, the pairing on $\Cbos$, $-8I$ on $u_*$, and $\sfree\Ocal=\Aux_{c/2}$ for relative ghosts; this is compatible with the relative master equation and not decided by it. \item 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 $\CW$ to a bracket-compatible quotient of $\Ealg$ forces $\hbar\mathbf1=0$. \end{enumerate} The ending is \textbf{Constructed} for the relative algebra on $\FW$ and \textbf{Proved obstructed} for the two lifts. It is not a constructed quantum BV algebra, and it does not license the joint-extension paper. \end{theorem} \begin{proof} Theorems~\ref{thm:sext}, \ref{thm:ghost}, \ref{thm:embedding}, \ref{thm:quotient}; Propositions~\ref{prop:kinematic}, \ref{prop:offFW}, \ref{prop:gf}, \ref{prop:KT}; Lemma~\ref{lem:separation}; Corollaries~\ref{cor:noint}, \ref{cor:nopromotion}. \end{proof} \subsection*{What is not added} $\Ealg$ is not enlarged by declaring antifields quantum. No $\widehat s_{\mathrm{int}}$ is written. $\Aux_\xi$ is not completed. No triangle residue is subtracted, no cone imported, no Enter record encoded. $\eps\to0$ is not taken. The relative master equation is not treated as a repair of the unrestricted breaking identity; the latter is copied, and Proposition~\ref{prop:offFW} records its consequence. \subsection*{Handoff} A successor receives: \begin{enumerate} \item the frozen continuation: XIV-A's sheet, $\FW\subset\Fsp$, unchanged $\Ocal$, relative route; \item the algebra $\AW$, the antibracket \eqref{eq:antibracket}, the differential $\sext$ with \eqref{eq:sext-generators}, its gauge-fixed form, and the Koszul--Tate identification with witness \eqref{eq:KT-witness}; \item the failure of nilpotence off $\FW$, \eqref{eq:offFW-witness}; \item the free-shell ghost theorem \eqref{eq:intersection} and the evaluation of the kinematic rule on the intersection; \item the copied ledger \eqref{eq:ledger}--\eqref{eq:relO}, the separation lemma, and the printed, unevaluated corrections \eqref{eq:kappa-corrections}; \item the two obstructions of Section~\ref{sec:B4}; \item XIII-C's four obligations, unchanged. \end{enumerate} It receives no interacting differential on $\Ealg$, no cleared observable on $\Cbos$, no Ward defect, no reduced algebra, and no completed quantum binary datum. If a quantum paper is written next, it is the joint-extension paper, and XIV-B does not license it. \subsection*{Conclusion} \[ \boxed{ \begin{gathered} (\AW,\sext),\quad \sext=\abr\Sext{\,\cdot\,},\quad \sext^2=0\ \text{on }\FW; \qquad \sext^2F_v=\tfrac\kappa2\!\int v^\nu\Kden_\nu\ne0\ \text{off }\FW;\\[2pt] \gW\otimes\Lambda\cap\Gfree=\{0\},\qquad \FW\cap\Csol=\Cbos\times\{c=\bar c=0\},\qquad s\big|_{\FW\cap\Csol}=\sfree\big|_{c=0};\\[2pt] \abr\Sext\Sext=0\ \text{on }\FW\quad\text{and}\quad \sfree\Ocal(u_*)=-8I\,\vartheta\ne0\ \text{on }\Cbos;\\[2pt] \text{ghost embedding: }\iota=0;\qquad \text{evaluation quotient: }\hbar\mathbf1=0 . \end{gathered}} \] The relative complex exists and is nilpotent where the backgrounds are trivial. Its ghosts never reach the free shell. The observable's variation is an evaluation on $\CWbos$ and a nonzero pairing on $\Cbos$. Neither fact is an interacting quantum algebra, and the quantum gate in front of any Ward defect is still XIII-C's. %===================================================================== \section*{Declarations} \addcontentsline{toc}{section}{Declarations} \paragraph{Classical scope.} All constructions are classical, at $\hbar^0$. The antibracket is the classical antibracket of local functionals on smooth configuration spaces. No quantum antibracket, quantum master equation, $\star_H$ on antifields, state, or loop is constructed. \paragraph{Domain scope.} The relative complex lives on $\FW$, whose ghosts are supported where both backgrounds are trivial and whose antighosts and antifields are compactly supported in the open slab. Its intersection with XII-A's shell has zero ghost sector. The observable identities are evaluations on the named bosonic sets, not quotients and not closure on $\Csol$. \paragraph{Debugging recorded.} XIV-A's $\FW$ is fixed as a subspace of its $\Fsp$, with compact antighost; nothing in XIV-A used a noncompact antighost. The free-shell ghost theorem is stated for the ghost $c$ by the $W$-support and for the antighost by compact support in the open slab; the antighost rule on the intersection is the inherited $\sfree\bar c=ib$, not zero. The Koszul--Tate sector is identified with the equations of the gauge-fixed interacting action, which coincide with the free shell only at order $\kappa^0$. \paragraph{Dependency scope.} XIV-A supplies the sheet, the kinematic rule, the extended action, the breaking functional and its witness, the relative identity, the gauge-fixed form, and the observable ledger. XIII-A supplies the non-ideal theorem and ordering counterexample. XIII-D supplies the class exclusion and the cochain. XII-A and XI-B supply the carrier, the free rule, and the uniqueness input. XII-C supplies the exact metric Euler derivative used in the Koszul--Tate witness. The algebra $\AW$, the nilpotence and its failure off $\FW$, the Koszul--Tate identification, the free-shell ghost theorem, the separation lemma, and the two obstructions are proved here. \paragraph{Verification.} No successful \texttt{pdflatex} execution, proof-assistant check, independently executed symbolic computation, independent referee review, or journal acceptance is asserted. The value $-8I$ is copied from XII-F and not recomputed. %===================================================================== \begin{thebibliography}{99} \addcontentsline{toc}{section}{References} \bibitem{XIVA} N.~Cheung, \emph{Landmark XIV-A: Classical master equation and observable ledger for one declared Einstein--scalar sheet}. Supplied manuscript \texttt{XIV-A.tex}. Frozen sheet, kinematic rule, extended action, breaking functional, late-switch witness, relative master equation, gauge fixing, carrier boundary, observable ledger. \bibitem{XIIIA} N.~Cheung, \emph{Landmark XIII-A: Classical cancellation on a fixed subclass, and the obstruction to a coefficientwise restriction quotient}. Supplied manuscript \texttt{XIII-A.tex}. Non-ideal theorem, no-quotient corollary, ordering counterexample, uniform plateau hypothesis. \bibitem{XIIIC} N.~Cheung, \emph{Landmark XIII-C: The joint two-line extension of one selected binary pair, on the unchanged carrier}. Supplied manuscript \texttt{XIII-C.tex}. Unresolved coefficient and the four successor obligations. \bibitem{XIIID} N.~Cheung, \emph{Landmark XIII-D: One bounded classical completion of the auxiliary pairing, or its obstruction on the unchanged sheet}. Supplied manuscript \texttt{XIII-D.tex}. Class exclusion and the admitted cochain. \bibitem{XIIF} N.~Cheung, \emph{Landmark XII-F: The gauge-fixed remainder of $\Phi^{(2)}_{\mathrm D}$: companion support, the auxiliary pairing, and what is not a Ward identity}. Supplied manuscript \texttt{XII-F.tex}. Auxiliary pairing, shell rewriting, witness value, nonzero carrier class. \bibitem{XIIC} N.~Cheung, \emph{Landmark XII-C: One declared cubic weighting: actual Euler source, changed Noether identity, and the coupled-jet question}. Supplied manuscript \texttt{XII-C.tex}. Selected action, exact metric Euler derivative, changed Noether identity, empty jet. \bibitem{XIIA} N.~Cheung, \emph{Landmark XII-A: One graded polynomial carrier for the finite-ramp multiplet, with ordering-stable composites and a robust observation exclusion}. Supplied manuscript \texttt{XII-A-v2.tex}. Carrier, complete shell, frozen free differential. \bibitem{XIB} N.~Cheung, \emph{Landmark XI-B: The free de Donder multiplet for the ramp Einstein--scalar sheet: causal propagators, residual gauge, and the layer dressing}. Supplied manuscript \texttt{XI-B.tex}. Inverse $-1$, gauge fermion, auxiliary equations, ghost normalization, compact residual intersection. \bibitem{XG} N.~Cheung, \emph{Landmark X-G: The normalized BRST multiplet and the one-loop background gate}. Supplied manuscript \texttt{X-G.tex}. Reciprocity obstruction and inverse $-4$, recorded as non-identifications; bare-versus-completed distinction. \bibitem{XAconv} N.~Cheung, \emph{Landmark X-A: Classical covariance of a fixed-regulator switched relational scalar through order $\kappa^2$: an explicit regulator obstruction}. Supplied manuscript \texttt{X-A-conv.tex}. Convolution-regulator defect, recorded as a non-identification. \bibitem{HT} M.~Henneaux and C.~Teitelboim, \emph{Quantization of Gauge Systems}, Princeton University Press, 1992. Chapters~15, 17, 18: antibracket, Jacobi identity, master equation, canonical transformations and gauge fixing. \bibitem{GPS} J.~Gomis, J.~Par\'is, and S.~Samuel, ``Antibracket, antifields and gauge-theory quantization,'' \emph{Physics Reports} \textbf{259} (1995), 1--145, arXiv:hep-th/9412228. Antifield-number decomposition and Koszul--Tate sector. \bibitem{FRBV} K.~Fredenhagen and K.~Rejzner, ``Batalin--Vilkovisky formalism in the functional approach to classical field theory,'' \emph{Communications in Mathematical Physics} \textbf{314} (2012), 93--127, arXiv:1101.5112. Antibracket of local functionals on smooth configuration spaces. \bibitem{BGP} C.~B\"ar, N.~Ginoux, and F.~Pf\"affle, \emph{Wave Equations on Lorentzian Manifolds and Quantization}, ESI Lectures in Mathematics and Physics, European Mathematical Society, 2007, arXiv:0806.1036. Chapter~3: smooth Cauchy uniqueness and finite propagation. \end{thebibliography} \end{document}