Landmark XIV-B The relative antifield complex on F W , and its trivial ghost sector on the free shell

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.

Figure fig:split-XIV-B records one view of the same separation. Figure fig:chain-XIV-B records one view of the same separation. Figure fig:pair-XIV-B records one view of the same separation. Figure fig:gate-XIV-B records one view of the same separation. The drawings add no hypothesis.

Figure fig:scope-XIV-B records that separation. The drawing adds no hypothesis.

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.}

The endings are Constructed, Proved obstructed, and 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 ${\mathcal{E}}_H$.

The following statements are copied and are not re-proved.

On the frozen sheet (XII-C's cubic-only $S_{\mathrm{D}}$, inverse ${G_{\mathrm{gr}}}^{\mathrm{R}}=-\mathcal{P}{E}_0^{\mathrm{R}}$ with coefficient $-1$, inert $g$ and ${\sigma }_{\epsilon }$), with the kinematic rule

($\mathcal{R}=K/2$, $c=2\xi$ recovers $K\xi$), and the extended action

one has, on the space $\mathfrak{F}$ of smooth multiplets with ghosts compactly supported in ${M_T}^{\circ }$,

with $w={\sigma }_{\epsilon }-1$ and the displayed XII-C densities. Every term of ${\mathcal{K}}_{\nu }$ carries $1-g$, $\partial g$, $w$, $\partial {\sigma }_{\epsilon }$, or ${\partial }^2{\sigma }_{\epsilon }$; hence ${\mathcal{K}}_{\nu }=0$ on $P_g\cap \{t>\epsilon \}$. On the unrestricted space the late-switch value is nonzero:

and every extension in the kinematic class ${\mathfrak{S}}_{\mathrm{kin}}$ has this antifield-degree-zero component; no $\hslash$ correction removes it. On the relative space ${\mathfrak{F}}_W$ (ghost $c\in {\mathfrak{g}}_W\otimes {\Lambda }^{\mathrm{odd}}$),

and XI-B's fermion $\Psi =-i\int {\bar{c}}_{\mu }(2D^{\mu }\gamma -b^{\mu })$ preserves both statements: $S_{\Psi }[\Phi ,{\Phi }^{\ast }]:=S_{\mathrm{ext}}[\Phi ,{\Phi }^{\ast }+\delta \Psi /\delta \Phi ]$ has $(S_{\Psi },S_{\Psi })=\mathfrak{K}$ on $\mathfrak{F}$, $0$ on ${\mathfrak{F}}_W$, and

The relative complex meets the free ghost shell only at zero: ${\mathfrak{g}}_W\cap \{\square c=0\}=\{0\}$ XIVA.

With $\mathcal{O}={{\Phi }^{(2)}}_{\mathrm{D}}+{\mathcal{A}}_1[f,g](r_g[\varphi ];\varphi )$ the unchanged lift, ${{\mathcal{C}}_{\mathrm{sol}}}^{\mathrm{bos}}=\{\square \gamma =0,P_m\varphi =0,b=D\gamma \}$ and ${{\mathcal{C}}_{\mathcal{W}}}^{\mathrm{bos}}=\{u\in {{\mathcal{C}}_{\mathrm{sol}}}^{\mathrm{bos}}:D^{\ast }b=0\text{ on }W\}$, under {(UCP${}_W$)}

and, for relative ghosts $c\in {\mathfrak{g}}_W\otimes {\Lambda }^{\mathrm{odd}}$ on ${{\mathcal{C}}_{\mathrm{sol}}}^{\mathrm{bos}}$,

with $⟨{{\mathcal{O}}^{\prime \prime }}_{\gamma \gamma },K\xi \otimes K\xi ⟩=0$ on ${{\mathcal{C}}_{\mathrm{sol}}}^{\mathrm{bos}}$. The master equation does not decide XII-F XIVA,XIIF.

The coefficient of $\lambda \hslash$ 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 ${R_1}^{(1)}$ XIIIC.

For every nonempty $W$ with $\overline{W}\subset \mathrm{int}\{g=1\}\cap \{\epsilon <t<T\}$, the coefficientwise vanishing space ${\mathcal{N}}_W=\{F\in {\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}:F{|}_{{\mathcal{C}}_W}=0\}$ satisfies: with $L_q=\int q^{\mu \nu }(D^{\ast }b{)}_{\mu \nu }$, $q\in {C_c}^{\infty }(W;{\mathrm{Sym}}^{2}TM)$, and $G_h=\int h{\mathrm{tr}}_{\eta }\gamma$, there are compact $q,h$ with

${\mathcal{N}}_W[[\hslash ,\kappa ]]$ is not a two-sided ideal of ${\star }_H$, no associative product on $({\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}/{\mathcal{N}}_W)[[\hslash ,\kappa ]]$ makes restriction a unital homomorphism, a unital homomorphism killing every $L_q$ while preserving the brackets forces $\hslash 1=0$, and the admissible rank-one ordering change gives ${\alpha }_d({L_q}^2)={L_q}^2+\hslash C(q,h{)}^2$ XIIIA. Completion of the auxiliary pairing inside ${\mathfrak{K}}_{\mathrm{loc},\mathrm{R}}$ is Proved obstructed; the cochain ${\mathcal{C}}_{\xi }=-{\mathfrak{B}}_{\xi }[D\gamma ,\varphi ]$ is admitted and is not an integrated completion XIIID.

${\mathcal{E}}_H=({\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}[[\hslash ,\kappa ]],{\star }_H)/{\mathcal{N}}_{\mathrm{sol}}[[\hslash ,\kappa ]]$, with ${\mathcal{C}}_{\mathrm{sol}}=\{\square \gamma =0,P_m\varphi =0,\square c=\square \bar{c}=0,b=D\gamma \}$; ghosts in ${\mathcal{C}}_{\mathrm{sol}}$ are smooth ambient free solutions with arbitrary Cauchy data. The frozen free differential is $s_0\gamma =\mathcal{R}c$, $s_0\bar{c}=ib$, $s_0\varphi =s_{0}c=s_{0}b=0$, a nilpotent derivation of ${\star }_H$ descending to ${\mathcal{E}}_H$. A smooth solution of $\square u=0$ on the slab with zero Cauchy data on some $\{t=t_0\}$, $0\le t_0<T$, vanishes on $[t_0,T]\times {\mathbb{R}}^3$; the compact plateau residual intersection is $\{0\}$ XIIA,XIB,BGP.

The sheet is XIV-A's: XII-C's cubic-only $S_{\mathrm{D}}$, inverse ${G_{\mathrm{gr}}}^{\mathrm{R}}=-\mathcal{P}{E}_0^{\mathrm{R}}$ with coefficient $-1$, $g$ and ${\sigma }_{\epsilon }$ fixed and inert, ${\mathfrak{g}}_W={C_c}^{\infty }(W;TM)$ with $\overline{W}\subset P_g\cap \{\epsilon <t<T\}$, the data $(f,g,m,\epsilon ,T,W)$ fixed once and satisfying {(UCP${}_W$)}. The observable is the unchanged antifield-independent lift $\mathcal{O}$. The route is relative symmetry.

$\mathfrak{F}$ is the space of multiplets $(\gamma ,\varphi ,c,\bar{c},b)$ with $\gamma ,\varphi ,b$ smooth on a neighbourhood of $M_T$, no decay imposed, and $c,\bar{c}\in {C_c}^{\infty }({M_T}^{\circ };TM)\otimes {\Lambda }^{\mathrm{odd}}$ for a finite Grassmann algebra $\Lambda$, together with antifields ${\Phi }_A^{\ast }\in {C_c}^{\infty }({M_T}^{\circ })\otimes \Lambda$ of parity opposite to ${\Phi }^A$ and ghost number $-1-\mathrm{gh}{\Phi }^A$. The relative space is

One clarification of XIV-A is recorded. Its Definition of ${\mathfrak{F}}_W$ described the fields other than $c$ as “unrestricted smooth sections”, while its $\mathfrak{F}$ had both ghosts compactly supported and its antifields compactly supported. Nothing in XIV-A used a noncompact antighost or a noncompact antifield; the breaking functional (eq:breaking) depends on $c$ alone, and the fermion pairing $\int \bar{c}\cdot (2D\gamma -b)$ is finite for compact $\bar{c}$. We therefore fix ${\mathfrak{F}}_W\subset \mathfrak{F}$ as in Definition def:spaces. The only consequence below is that the antighost sector of the free-shell intersection is also trivial (Theorem thm:ghost).

The negative freeze is stated in the same breath. XIV-B does not select an antifield-dependent observable completion, a correction of the lower coefficient $\kappa {\mathcal{A}}_1$, a spurion law for $g$ or ${\sigma }_{\epsilon }$, a source-switched upgrade, the convolution sheet, or a new integration interface outside ${\mathfrak{K}}_{\mathrm{loc},\mathrm{R}}$. Those were left outside XIV-A's class. Reusing the witness $u_{\ast }$ after changing the observable would be a new contract and is not done. The unrestricted master equation is not reopened: its failure is copied as (eq:witnessA1).

${\mathfrak{A}}_W$ is the graded commutative algebra over $\mathbb{C}$ of local functionals

where $\mathcal{F}$ is a polynomial in finite jets of the fields and antifields whose coefficients are smooth functions of $x$ and of the backgrounds and are compactly supported in ${M_T}^{\circ }$, evaluated on ${\mathfrak{F}}_W$. Parity and ghost number are those of the integrand. Antifield degree is polynomial degree in the ${\Phi }_A^{\ast }$.

For $F,G$ functionals of fields and antifields the antibracket is

with the left/right Euler derivatives and the sign conventions of HT (Ch. 15). It has parity $|F|+|G|+1$ and ghost number $\mathrm{gh}F+\mathrm{gh}G+1$, and satisfies the graded Jacobi identity HT,GPS,FRBV.

For $F,G\in {\mathfrak{A}}_W$, $\left(F,G\right)\in {\mathfrak{A}}_W$. The extended action $S_{\mathrm{ext}}$ is a formal bulk functional, not an element of ${\mathfrak{A}}_W$, but for every $F\in {\mathfrak{A}}_W$ the bracket $\left(S_{\mathrm{ext}},F\right)$ is a well-defined element of ${\mathfrak{A}}_W$, and it depends on $F$ only through its restriction to ${\mathfrak{F}}_W$.

Proof. The density of $\left(F,G\right)$ is a finite sum of products of Euler derivatives of $\mathcal{F}$ and $\mathcal{G}$; each factor inherits the compact coefficient support, so the product is a local density with compact support. For $\left(S_{\mathrm{ext}},F\right)$, every term pairs a derivative of $S_{\mathrm{ext}}$ with a derivative of $F$, and the latter is compactly supported in ${M_T}^{\circ }$; the Euler derivatives of $S_{\mathrm{ext}}$ are local expressions in the fields and antifields (XII-C's $e_{\mathrm{D}},E_{\varphi }$ and the antifield-linear terms of (eq:Sext)), so the pairing is finite and local. Dependence on the restriction to ${\mathfrak{F}}_W$: the $c$-derivative of $F$ enters only paired with ${\delta }^{\mathrm{L}}{S}_{\mathrm{ext}}/\delta c^{\ast }=sc\in {\mathfrak{g}}_W\otimes \Lambda$, a direction tangent to ${\mathfrak{F}}_W$; all other slots of ${\mathfrak{F}}_W$ are unrestricted.

On ${\mathfrak{A}}_W$ define $s_{\mathrm{ext}}F:=\left(S_{\mathrm{ext}},F\right)$. Then:

1. $s_{\mathrm{ext}}$ is odd, has ghost number $+1$, and is a graded derivation: $s_{\mathrm{ext}}(FG)=(s_{\mathrm{ext}}F)G+(-1{)}^{|F|}F{s}_{\mathrm{ext}}G$;

2. ${s_{\mathrm{ext}}}^2=0$ on ${\mathfrak{A}}_W$;

3. on fields and antifields,

   $$s_{\mathrm{ext}}{\Phi }^A=s{\Phi }^A,s_{\mathrm{ext}}{\gamma }^{\ast \mu \nu }=\frac{{\delta }^{\mathrm{R}}{S}_{\mathrm{ext}}}{\delta {\gamma }_{\mu \nu }}={e_{\mathrm{D}}}^{\mu \nu }+(\text{terms linear in antifields}),$$

   and similarly $s_{\mathrm{ext}}{\varphi }^{\ast }=E_{\varphi }+(\dots )$, with $s$ the kinematic rule (eq:kin).

Proof. (i) $S_{\mathrm{ext}}$ is even of ghost number zero, so the grading of (eq:antibracket) gives the stated parity and ghost number; the derivation property is the graded Leibniz rule of the antibracket in its second slot HT (Ch. 15). (ii) For even $S$ the graded Jacobi identity gives $\left(S,\left(S,F\right)\right)=\frac{1}{2}\left(\left(S,S\right),F\right)$. By the copied relative identity (eq:relative), $\left(S_{\mathrm{ext}},S_{\mathrm{ext}}\right)$ vanishes on ${\mathfrak{F}}_W$ together with its functional derivatives along ${\mathfrak{F}}_W$ (XIV-A's proof: the $\gamma$- and $\varphi$-derivatives retain the factor $c^{\nu }$, and the $c$-derivative is $\kappa {\mathcal{K}}_{\nu }$ restricted to $W$, which is zero). Hence $\left(\left(S_{\mathrm{ext}},S_{\mathrm{ext}}\right),F\right)=0$ for every $F\in {\mathfrak{A}}_W$ and ${s_{\mathrm{ext}}}^2=0$. (iii) With $G={\Phi }^A(x)$ only the second term of (eq:antibracket) survives: $s_{\mathrm{ext}}{\Phi }^A=-{\delta }^{\mathrm{R}}{S}_{\mathrm{ext}}/\delta {\Phi }_A^{\ast }$. For even ${\Phi }^A$ the antifield is odd, and the right derivative of $\int {\Phi }_A^{\ast }s{\Phi }^A$ with respect to ${\Phi }_A^{\ast }$ is $-s{\Phi }^A$, so $s_{\mathrm{ext}}{\Phi }^A=s{\Phi }^A$; for odd ${\Phi }^A$ the antifield is even and the same result holds without the intermediate sign. With $G={\Phi }_A^{\ast }(x)$ only the first term survives, giving $s_{\mathrm{ext}}{\Phi }_A^{\ast }={\delta }^{\mathrm{R}}{S}_{\mathrm{ext}}/\delta {\Phi }^A$, whose antifield-independent part is the Euler derivative of $S_{\mathrm{D}}$.

Let $F$ be a smooth functional of the fields alone, local or nonlocal, with compactly supported derivative kernels (for instance $\mathcal{O}$, or any element of ${\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$). Then on ${\mathfrak{F}}_W$

and $s^{2}F=0$ off shell, on all of $\mathfrak{F}$, by kinematic nilpotence alone. This nilpotence does not use the master equation.

Proof. For antifield-independent $F$ only the second term of (eq:antibracket) survives, giving (eq:sF); the pairing is finite because $s{\Phi }^A$ is compactly supported for compact ghosts. Since $s{\Phi }^A$ contains no antifields, $sF$ is again antifield-independent, and $s^{2}F={\sum }_A⟨{F^{\prime }}_A,s^2{\Phi }^A⟩+{\sum }_{A,B}⟨{F^{\prime \prime }}_{AB},s{\Phi }^A\otimes s{\Phi }^B⟩$; the first sum vanishes by XIV-A's kinematic nilpotence lemma, and the second vanishes by graded symmetry of $F^{\prime \prime }$ against the odd directions.

The distinction between Theorem thm:sext(ii) and Proposition prop:kinematic is the point of the relative space: nilpotence on antifield-dependent functionals requires $\left(S_{\mathrm{ext}},S_{\mathrm{ext}}\right)=0$. The next statement records, from copied data only, that this requirement is not idle.

On $\mathfrak{F}$, for every local functional $F$ of fields and antifields,

For $F_v=\int c_{\mu }^{\ast }{v}^{\mu }{\mathrm{d}}^{4}x$ with real $v\in {C_c}^{\infty }({M_T}^{\circ };TM)$,

If $\mathrm{supp}v\subset W$, then ${s_{\mathrm{ext}}}^2{F}_v=0$ identically, consistently with Theorem thm:sext.

Proof. The first equality is the Jacobi identity with (eq:breaking); the second holds because $\mathfrak{K}$ is antifield-independent. For $F_v$, ${\delta }^{\mathrm{L}}{F}_v/\delta c_{\mu }^{\ast }=v^{\mu }$ and ${\delta }^{\mathrm{R}}\mathfrak{K}/\delta c^{\nu }=\kappa {\mathcal{K}}_{\nu }$ (the density ${\mathcal{K}}_{\nu }$ is even, so left and right derivatives agree). Evaluating at $\gamma =0$, $\varphi ={\varphi }_m$, where XIV-A computed ${\mathcal{K}}_{\nu }=({\partial }_{\mu }g){T_0}^{\mu }{}_{\nu }[{\varphi }_m]$ with ${T_0}^{00}=\frac{1}{2}$, ${T_0}^{0i}=0$, gives (eq:offFW-witness); the integral is strictly negative by the copied falling-switch lemma. For $\mathrm{supp}v\subset W$, ${\mathcal{K}}_{\nu }=0$ on $W$ by (eq:breaking).

This is not a reopening of the unrestricted master equation: no new Noether calculation is made, and the value is the copied late-switch witness read through the Jacobi identity.

Let $S_{\Psi }$ be the gauge-fixed action of Input statement input:XIVA and $s_{\Psi }:=\left(S_{\Psi },\cdot \right)$. Then $s_{\Psi }$ is an odd, ghost-number-one, nilpotent derivation of ${\mathfrak{A}}_W$; $s_{\Psi }{\Phi }^A=s{\Phi }^A$ for every field; and the antifield-degree-zero part of $S_{\Psi }$ is (eq:Sgf), whose quadratic part is XI-B's free gauge-fixed action with auxiliary equation $b=D\gamma$ and ghost term $i\bar{c}\square c$.

Proof. The antifield shift ${\Phi }^{\ast }\mapsto {\Phi }^{\ast }+\delta \Psi /\delta \Phi$ is a canonical transformation of (eq:antibracket) HT (Ch. 18); it preserves ${\mathfrak{F}}_W$ because $\Psi$ does not depend on $c$, so $c^{\ast }$ is unchanged, and the other shifts are compactly supported for compact $\bar{c}$. Hence $(S_{\Psi },S_{\Psi })$ is $(S_{\mathrm{ext}},S_{\mathrm{ext}})$ at shifted antifields, which is zero on ${\mathfrak{F}}_W$ by (eq:relative), and nilpotence follows as in Theorem thm:sext. The field rule is unchanged because $\delta S_{\Psi }/\delta {\Phi }^{\ast }=\delta S_{\mathrm{ext}}/\delta {\Phi }^{\ast }$ under a translation of the antifields. The display (eq:Sgf) is copied.

At antifield degree zero, $s_{\Psi }{\Phi }_A^{\ast }={\delta }^{\mathrm{R}}{S}_{\Psi }/\delta {\Phi }^A{|}_{{\Phi }^{\ast }=0}$ is the Euler derivative of the gauge-fixed interacting action (eq:Sgf). Its ${\kappa }^0$ coefficient is the system

which is XII-A's complete shell ${\mathcal{C}}_{\mathrm{sol}}$. Its ${\kappa }^1$ coefficient is not zero on ${\mathcal{C}}_{\mathrm{sol}}$: at the configuration $\gamma =0$, $b=0$, $\varphi ={\varphi }_m$, $c=\bar{c}=0$, which lies in ${\mathcal{C}}_{\mathrm{sol}}$,

Hence the Koszul–Tate differential of the relative complex does not vanish on ${\mathcal{C}}_{\mathrm{sol}}$; it resolves the equations of $S_{\Psi }$, which differ from the free shell at order $\kappa$. The ghost equation of $S_{\Psi }$, $\square c+\kappa D({\mathcal{L}}_c\gamma )=0$, is likewise not $\square c=0$.

Proof. The ${\kappa }^0$ system is read from (eq:Sgf): the cubic-only modification does not change the free Hessian (XII-C), the auxiliary and ghost terms are XI-B's, and XI-B proves $A\gamma +2D^{\ast }b=0$, $b=D\gamma$ is equivalent to $-\square \mathcal{P}\gamma =0$, $b=D\gamma$. For the witness, XII-C's exact metric Euler derivative is $e_{\mathrm{D}}=\frac{2}{\kappa }\sqrt{-q}G-\frac{\kappa }{2}g\sqrt{-q}{T}_q+\kappa {Z_{\sigma }}_{\epsilon }$; at $\gamma =0$ the Einstein tensor and ${Z_{\sigma }}_{\epsilon }$ vanish and $T_q=T_0$, so $e_{\mathrm{D}}=-\frac{\kappa }{2}g{T}_0[{\varphi }_m]$ exactly, and ${T_0}^{00}[{\varphi }_m]=\frac{1}{2}$. The gauge-fixing contribution $2D^{\ast }b$ vanishes at $b=0$ and the ghost contribution at $c=\bar{c}=0$.

Thus ${\mathfrak{A}}_W$ carries a classical nilpotent differential whose Koszul–Tate part resolves a $\kappa$-deformed shell. It is not equipped with ${\star }_H$, is not quotiented by ${\mathcal{N}}_{\mathrm{sol}}$, and its antifields are not quantum fields. B1 ends Constructed.

Let $u$ be a smooth $\Lambda$-valued section on a neighbourhood of $M_T$ with $\square u=0$ and $\mathrm{supp}u\subset \{t\ge t_1\}$ for some $t_1>0$. Then $u=0$ on $M_T$.

Proof. Work coefficientwise in $\Lambda$. Choose $t_0\in (0,t_1)$; $u$ vanishes on a neighbourhood of $\{t=t_0\}$, so $u$ and ${\partial }_{t}u$ vanish there. Uniqueness for the smooth homogeneous Cauchy problem on $[t_0,T]\times {\mathbb{R}}^3$ gives $u=0$ there BGP (Ch. 3); for $t\le t_0$, $u=0$ by hypothesis.

1. A ghost $c\in {\mathfrak{g}}_W\otimes {\Lambda }^{\mathrm{odd}}$ with $\square c=0$ is zero. An antighost $\bar{c}\in {C_c}^{\infty }({M_T}^{\circ })\otimes {\Lambda }^{\mathrm{odd}}$ with $\square \bar{c}=0$ is zero.

2. Consequently

   $${\mathfrak{F}}_W\cap {\mathcal{C}}_{\mathrm{sol}}=\{(\gamma ,\varphi ,0,0,b):(\gamma ,\varphi ,b)\in {{\mathcal{C}}_{\mathrm{sol}}}^{\mathrm{bos}}\},$$

   the bosonic free shell with zero ghost sector.

3. At every configuration in (eq:intersection) the kinematic rule (eq:kin) takes the values

   $$s\gamma =0,s\varphi =0,sc=0,s\bar{c}=ib,sb=0.$$

   Every $\kappa$-dependent term of $s$ vanishes there, and the only nonzero component is the frozen free rule $s_0\bar{c}=ib$. For an antifield-independent $F$, $(s_{\mathrm{ext}}F)(u)=⟨{F^{\prime }}_{\bar{c}}(u),ib⟩$ on the intersection; for $F$ without antighost argument, in particular for $\mathcal{O}$, $(s_{\mathrm{ext}}F)(u)=0$ there.

Proof. (i) $\mathrm{supp}c\subset \overline{W}$, compact in $\{t>\epsilon \}$, so $t_1:={\min }_{\overline{W}}t>\epsilon >0$ and Lemma lem:uniq applies; $\mathrm{supp}\bar{c}$ is compact in the open slab, so $t_1>0$ and the lemma applies again. (ii) A configuration of ${\mathfrak{F}}_W$ lies in ${\mathcal{C}}_{\mathrm{sol}}$ exactly when its bosonic part lies in ${{\mathcal{C}}_{\mathrm{sol}}}^{\mathrm{bos}}$ and its ghosts solve the free ghost equations; by (i) the latter forces $c=\bar{c}=0$. (iii) Each $\kappa$-dependent term of (eq:kin) is linear in $c$, as is $\mathcal{R}c$; the antighost rule is $c$-independent. The last assertion is (eq:sF) evaluated at $c=0$.

No nonzero relative ghost is a configuration of XII-A's shell. The restriction of the relative complex to ${\mathfrak{F}}_W\cap {\mathcal{C}}_{\mathrm{sol}}$ carries the inherited free rule $s_0$ evaluated at $c=0$ and nothing else. Hence $({\mathfrak{A}}_W,s_{\mathrm{ext}})$ supplies no differential on ${\mathcal{E}}_H$ beyond the frozen $s_0$, and in particular no interacting ghost differential ${\hat{s}}_{\mathrm{int}}$.

Proof. Theorem thm:ghost(i)–(iii). ${\mathcal{E}}_H$'s ghosts are free fields with arbitrary Cauchy data (Input statement input:XIIA); by (i) none of them, except zero, lies in ${\mathfrak{g}}_W\otimes \Lambda$.

A compact real vector field $v\in {\mathfrak{g}}_W$ is a legal smearing test for the ghost field of ${\mathcal{E}}_H$: $c(v)=\int v_{\mu }{c}^{\mu }$ is an element of the carrier, with $s_{0}c(v)=0$. 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 ${\mathfrak{F}}_W$ and a legal argument of the breaking functional; it is not an element of ${\mathcal{C}}_{\mathrm{sol}}$.

B2 ends Proved: the free-shell ghost sector of ${\mathfrak{F}}_W$ is $\{0\}$.

The identities (eq:ledger)–(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 ${{\mathcal{C}}_{\mathcal{W}}}^{\mathrm{bos}}$, with the pairing printed on ${{\mathcal{C}}_{\mathrm{sol}}}^{\mathrm{bos}}$; not closure on ${\mathcal{C}}_{\mathrm{sol}}$; the witness $u_{\ast }$ outside XIII-A's defect set; ${\mathfrak{g}}_W$ not shrunk after the witness.

On ${\mathfrak{F}}_W$ the following hold simultaneously and are compatible:

1. $\left(S_{\mathrm{ext}},S_{\mathrm{ext}}\right)=0$ and ${s_{\mathrm{ext}}}^2=0$ on ${\mathfrak{A}}_W$ (Theorem thm:sext);

2. $s^2\mathcal{O}=0$ off shell (Proposition prop:kinematic);

3. for $c=2\vartheta {\xi }_{\ast }$ on the detecting tuple, $s_0\mathcal{O}(u_{\ast })={\mathfrak{B}}_{{\xi }_{\ast }}(u_{\ast })\vartheta =-8I\vartheta \ne 0$, where $s_0\mathcal{O}$ is the ${\kappa }^0$ part of $s_{\mathrm{ext}}\mathcal{O}=s\mathcal{O}$ (eq:relO).

Neither (a) nor (b) implies that $s\mathcal{O}$ vanishes on ${{\mathcal{C}}_{\mathrm{sol}}}^{\mathrm{bos}}$; (c) shows that it does not. The master equation constrains the action; it does not decide the invariance of the given lift.

Proof. (a) and (b) are proved above; (c) is copied from Input statement input:ledger. Compatibility: (a) is a statement about $S_{\mathrm{ext}}$ and about $s_{\mathrm{ext}}$ on ${\mathfrak{A}}_W$; (b) is the statement that $s\mathcal{O}$ is $s$-closed, which is automatic for an image of $s$; (c) is the value of that image at one point of ${{\mathcal{C}}_{\mathrm{sol}}}^{\mathrm{bos}}$, outside the intersection (eq:intersection) because $c\ne 0$ there. No statement among (a)–(c) cancels another.

The $\kappa$-corrections are printed and left unevaluated, as in XIV-A:

Since $\mathcal{O}$ is the ${\kappa }^2$ coefficient of the relational expansion, these terms enter at ${\kappa }^3$ there. A proposal that moves the lower coefficient $\kappa {\mathcal{A}}_1$ must display its own coupling power. $\mathcal{O}$, which already contains the classical scalar response and the slot $r_{\mathrm{D}}+r_g$, 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 ${{\mathcal{C}}_{\mathcal{W}}}^{\mathrm{bos}}$ and the nonzero pairing on ${{\mathcal{C}}_{\mathrm{sol}}}^{\mathrm{bos}}$.

Both lifts are excluded by explicit incompatibilities, not by absent formulas.

Let ${\mathcal{G}}_0$ be the space of smooth $\Lambda$-valued free ghost solutions on the slab, $\square c=0$, with arbitrary Cauchy data: the ghost configurations of ${\mathcal{C}}_{\mathrm{sol}}$.

Let $\iota :{\mathfrak{g}}_W\otimes {\Lambda }^{\mathrm{odd}}\to {\mathcal{G}}_0$ be a linear map satisfying at least one of:

1. $\iota (c)=c$ on a neighbourhood of $\overline{W}$ for every $c$;

2. $\iota (c)$ has zero Cauchy data on some surface $\{t=t_0\}$ with $0\le t_0<{\min }_{\overline{W}}t$, for every $c$;

3. $\mathrm{supp}\iota (c)\subset J^{+}(\mathrm{supp}c)\cap M_T$ for every $c$.

Then $\iota =0$. In particular the inclusion ${\mathfrak{F}}_W\hookrightarrow \mathfrak{F}$ restricts to the zero map on the ghost sector of ${\mathcal{C}}_{\mathrm{sol}}$.

Proof. (a) On the neighbourhood $N$ of $\overline{W}$, $\square c=\square \iota (c)=0$; on the complement of $\overline{W}$, $c=0$. Thus $\square c=0$ on all of $M_T$ and $c=0$ by Theorem thm:ghost(i); then $\iota (c)=\iota (0)=0$. (b) A free solution with zero Cauchy data on $\{t=t_0\}$ vanishes on $[t_0,T]\times {\mathbb{R}}^3$ by uniqueness and on $[0,t_0]$ by the backward problem with the same data; hence $\iota (c)=0$. (c) $J^{+}(\mathrm{supp}c)\cap M_T\subset \{t\ge {\min }_{\overline{W}}t\}$, so $\iota (c)$ has zero Cauchy data on any earlier surface, and (b) applies.

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 ${\mathfrak{F}}_W$'s ghost sector, its image is not in ${\mathfrak{g}}_W$, the relative identity (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.

For the frozen $W$, the coefficientwise vanishing space ${\mathcal{N}}_W[[\hslash ,\kappa ]]$ is not a two-sided ideal of ${\star }_H$, and its image $({\mathcal{N}}_W/{\mathcal{N}}_{\mathrm{sol}})[[\hslash ,\kappa ]]$ is not a two-sided ideal of ${\mathcal{E}}_H$. There is no unital $\mathbb{C}[[\hslash ,\kappa ]]$-algebra homomorphism from ${\mathcal{E}}_H$ killing the classes of all $L_q$, $q\in {C_c}^{\infty }(W)$, and preserving the inherited brackets, except one with $\hslash 1=0$ in its target. Coefficientwise vanishing on ${\mathcal{C}}_W$ is not preserved by every admissible ordering transport.

Proof. XIII-A's non-ideal theorem, its corollary, and its rank-one ordering calculation, copied as Input statement input:XIIIAD. The mechanism is (eq:commutator): the formal scalar $i\hslash C(q,h)1$ with $C(q,h)>0$ does not vanish on ${\mathcal{C}}_W$, which contains the zero configuration.

The identity ${\Delta }_{\xi }\mathcal{O}{|}_{{{\mathcal{C}}_{\mathcal{W}}}^{\mathrm{bos}}}=0$ of (eq:ledger) cannot be realized as an identity in a quotient of ${\mathcal{E}}_H$ by the classes of functionals vanishing on ${\mathcal{C}}_W$, compatibly with the inherited brackets and a nonzero formal unit. The auxiliary pairing itself belongs to ${\mathcal{N}}_W\setminus {\mathcal{N}}_{\mathrm{sol}}$: it vanishes on ${\mathcal{C}}_W$ and has nonzero class in ${\mathcal{E}}_H$ by the copied witness. Its vanishing is an evaluation on ${{\mathcal{C}}_{\mathcal{W}}}^{\mathrm{bos}}$, and remains one.

Proof. Such a promotion is a unital homomorphism whose kernel contains $({\mathcal{N}}_W/{\mathcal{N}}_{\mathrm{sol}})[[\hslash ,\kappa ]]$, hence every ${\bar{L}}_q$; Theorem thm:quotient forces $\hslash 1=0$. Membership of ${\mathfrak{B}}_{\xi }$ in ${\mathcal{N}}_W$ is the vanishing on ${\mathcal{C}}_W$ in (eq:ledger); non-membership in ${\mathcal{N}}_{\mathrm{sol}}$ is the value $-8I\ne 0$ at $u_{\ast }\in {\mathcal{C}}_{\mathrm{sol}}$.

The admitted cochain ${\mathcal{C}}_{\xi }=-{\mathfrak{B}}_{\xi }[D\gamma ,\varphi ]$ cancels the pairing algebraically on ${{\mathcal{C}}_{\mathrm{sol}}}^{\mathrm{bos}}$ and is not an integrated completion in ${\mathfrak{K}}_{\mathrm{loc},\mathrm{R}}$ (Input statement input:XIIIAD). It is not inserted into $\mathcal{O}$: that would change the observable and void the reuse of $u_{\ast }$ (Section sec:B0). It is not inserted into $S_{\mathrm{ext}}$: 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 Proved obstructed for both lifts.

XIII-C's four obligations are repeated unchanged (Input statement 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 $\lambda \hslash$ exists.

Accordingly: neither $s_{\mathrm{ext}}$, $s_{\Psi }$, nor $s_0$ is applied to ${R_1}^{(1)}$; no Hadamard line is contracted against ${\mathcal{J}}_{\mathrm{D}}$; no binary Ward defect is formed. The classical breaking functional $\mathfrak{K}$ is the antibracket square of the extended action on the unrestricted ghost space at ${\hslash }^0$; it is not the anomaly of an unconstructed quantum defect, and (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 $S_{\epsilon }\xi \ne \xi$, remain non-identifications: neither sheet is the one frozen here, and no obstruction proved on either is imported. XII-C's empty coupled jet ${{\mathcal{J}}^{\mathrm{D}}}_{g,{\sigma }_{\epsilon }}({\varphi }_m)=\varnothing$ is a solution-fiber statement, not a failure of $\mathcal{O}$; the slot $r_{\mathrm{D}}+r_g$ solves a gauge-fixed forcing equation and does not fill it. The XI-G classes remain excluded; $\mathcal{O}$ is not a representative.

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.

${\mathcal{E}}_H$ is not enlarged by declaring antifields quantum. No ${\hat{s}}_{\mathrm{int}}$ is written. ${\mathfrak{B}}_{\xi }$ is not completed. No triangle residue is subtracted, no cone imported, no Enter record encoded. $\epsilon \to 0$ is not taken. The relative master equation is not treated as a repair of the unrestricted breaking identity; the latter is copied, and Proposition prop:offFW records its consequence.

A successor receives:

1. the frozen continuation: XIV-A's sheet, ${\mathfrak{F}}_W\subset \mathfrak{F}$, unchanged $\mathcal{O}$, relative route;

2. the algebra ${\mathfrak{A}}_W$, the antibracket (eq:antibracket), the differential $s_{\mathrm{ext}}$ with (eq:sext-generators), its gauge-fixed form, and the Koszul–Tate identification with witness (eq:KT-witness);

3. the failure of nilpotence off ${\mathfrak{F}}_W$, (eq:offFW-witness);

4. the free-shell ghost theorem (eq:intersection) and the evaluation of the kinematic rule on the intersection;

5. the copied ledger (eq:ledger)–(eq:relO), the separation lemma, and the printed, unevaluated corrections (eq:kappa-corrections);

6. the two obstructions of Section sec:B4;

7. XIII-C's four obligations, unchanged.

It receives no interacting differential on ${\mathcal{E}}_H$, no cleared observable on ${{\mathcal{C}}_{\mathrm{sol}}}^{\mathrm{bos}}$, 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.