Landmark XIV-A Classical master equation and observable ledger for one declared Einstein–scalar sheet

We freeze one sheet before any antifield is written: XII-C's cubic-only density weighting, XI-B's inverse with coefficient $-1$, the switch and ramp as fixed inert backgrounds, the compact plateau algebra ${\mathfrak{g}}_W$, and the assembled second-order relational coefficient $\mathcal{O}$. The convolution-regulated sheet and the source-switched sheet are recorded with their own obstructions and are not used.

Antifields and the antibracket are introduced, and the extended action generates the kinematic diffeomorphism rule with the inherited ghost normalization $c=2\xi$. Its antibracket square is computed exactly: $(S_{\mathrm{ext}},S_{\mathrm{ext}})=\kappa \int c^{\nu }{\mathcal{K}}_{\nu }$, where ${\mathcal{K}}_{\nu }$ is XII-C's changed Noether density with the scalar-switch defect, the cubic-only bulk mismatch, and every ramp-derivative term displayed. On the unrestricted ghost space this functional is nonzero; a late-switch witness is recomputed in the present conventions. The unrestricted master equation is therefore Proved obstructed for the frozen class, at ${\hslash }^0$, independently of higher antifield terms.

On the relative space whose ghosts lie in ${\mathfrak{g}}_W$, the square vanishes identically, the differential is nilpotent, and gauge fixing by XI-B's fermion preserves the identity. That relative complex meets XII-A's free ghost shell only at $c=0$; it induces no interacting differential on the inherited carrier.

The observable ledger is separate. Under the uniform causal plateau hypothesis the finite variation of $\mathcal{O}$ on the complete free shell equals the auxiliary pairing, vanishes on the evaluation set ${\mathcal{C}}_W$, and equals $-8I<0$ on the copied witness. We prove that the relative free differential of $\mathcal{O}$ on the bosonic shell is exactly that pairing. Thus the master equation does not decide XII-F. The ending is Constructed on the relative domain; the unrestricted sheet is Proved obstructed; no completion, quantum retarded coefficient, reduced algebra, or sharp limit is opened.

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

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

and proof boundary{ Frozen (A0): cubic-only sheet, inverse $-1$, inert $g$ and ${\sigma }_{\epsilon }$, ${\mathfrak{g}}_W$, observable $\mathcal{O}$. One route: relative symmetry.

Constructed: antifield sector and antibracket; the exact breaking functional $(S_{\mathrm{ext}},S_{\mathrm{ext}})=\kappa \int c^{\nu }{\mathcal{K}}_{\nu }$; the relative master equation on ${\mathfrak{F}}_W$ and its nilpotent differential; gauge fixing by XI-B's fermion; the evaluation identity on ${\mathcal{C}}_W$ with the auxiliary pairing printed; the identification of the relative free differential of $\mathcal{O}$ with that pairing.

Proved obstructed: the unrestricted master equation for every extension in the frozen kinematic class ${\mathfrak{S}}_{\mathrm{kin}}$ (nonzero ${\hslash }^0$ breaking); unqualified invariance of $\mathcal{O}$ on ${\mathcal{C}}_{\mathrm{sol}}$ (copied witness); completion inside ${\mathfrak{K}}_{\mathrm{loc},\mathrm{R}}$ (copied from XIII-D).

Outside the claim: a completion of the auxiliary pairing in any class not frozen here; ${R_1}^{(1)}$; a quantum gauge defect; a reduced algebra; an interacting differential on ${\mathcal{E}}_H$; a spurion or density-switched variational sheet; $\epsilon \to 0$.

Ending for the frozen relative sheet: Constructed. The unrestricted sheet is Proved obstructed and is recorded as the boundary of the claim. The release is relative; it is not an interacting quantum algebra.}

The endings are Constructed, Proved obstructed, and Unresolved, applied to the object actually named on the domain actually named. A missing map is not a no-go. An ${\hslash }^0$ defect is not removed by an $\hslash$ counterterm. XIII-C's unresolved product is not an anomaly.

The following statements are used with their predecessor scope. They are not one model. Each is copied so that the later sections can refer to it without reconstruction.

For the frozen switched action, spacetime-convolution regulator, Cauchy prescription, and plateau-supported gauge algebra, the relational scalar fails gauge invariance at order $\kappa$. The decisive identity is $E_{0}C{T}_{g,\epsilon }(K\xi )=S_{\epsilon }\xi \ne \xi$, with witness smearing $c_{\epsilon }>0$. There is no cleared classical gauge-invariant handoff. A changed regulator is a new problem XAconv.

On the X-E/X-F normalization $P_{\mathrm{gr}}=-\frac{1}{4}\mathcal{P}\square$, ${{\Delta }_{\mathrm{R}}}^{\mathrm{gr}}=-4\mathcal{P}{E}_0^{\mathrm{R}}$, the free multiplet with independent $b$, gauge fermion ${\Psi }_0=\int \bar{c}(C-b)$, $s\bar{c}=b$, and $b=C[h]/2$ is constructed; off-shell nilpotence uses the independent auxiliary field. Separately, an exact source-preserving variational upgrade of X-F's mixed map $k_{\mathrm{mix}}[w;u]=-{{\Delta }^{\mathrm{gr}}}_{\mathrm{R}}({\sigma }_{\epsilon }{Q}_{w}u)$ is Proved obstructed: a compact pure-gauge ramp background gives the reciprocity pairing

on free metric solutions. Source switching is not density switching. The null-collinear set $\mathcal{N}$ is a failed sufficient product criterion, not nonexistence. The unconditional one-loop background identity is Unresolved XG.

Density before variation, at cubic order only:

Full Einstein–Hilbert weighting is rejected: it produces a flat-background Euler term and changes the free sheet. The actual cubic source is ${\mathcal{J}}_{\mathrm{D}}={\sigma }_{\epsilon }{J}_3+{{\mathcal{D}}_{\sigma }}_{\epsilon }$. With $e_{\mathrm{D}}=\delta S_{\mathrm{D}}/\delta \gamma$, $E_{\varphi }=\delta S_{\mathrm{D}}/\delta \varphi$, $w={\sigma }_{\epsilon }-1$, and ${Z_{\sigma }}_{\epsilon }=w{J}_3+{{\mathcal{D}}_{\sigma }}_{\epsilon }$, the changed Noether identity holds off shell:

For every declared switch, the fixed-input invariant jet ${{\mathcal{J}}^{\mathrm{D}}}_{g,{\sigma }_{\epsilon }}({\varphi }_m)$ is empty: a solution-fiber obstruction, not an observable no-go. The quadratic coefficients are XI-A/XI-B's; the inverse is ${G_{\mathrm{gr}}}^{\mathrm{R}}=-\mathcal{P}{E}_0^{\mathrm{R}}$ with coefficient $-1$ XIIC,XIB.

The scalar-linear coefficient is ${{\Phi }^{(2)}}_{\mathrm{D}}={{\mathcal{A}}_2}^{\mathrm{sc}}[f,g](\gamma ,0;\varphi )-{\mathcal{C}}_{3,\mathrm{D}}$, with the slot $r_{\mathrm{D}}=-u_{\mathrm{D}}=\mathcal{P}{E}_0^{\mathrm{R}}{\mathcal{J}}_{\mathrm{D}}$; the full displayed coefficient is $\mathcal{O}={{\Phi }^{(2)}}_{\mathrm{D}}+{\mathcal{A}}_1[f,g](r_g[\varphi ];\varphi )$, $r_g=-\frac{1}{2}\mathcal{P}{E}_0^{\mathrm{R}}(g{T}_0[\varphi ])$. For $\xi$ compact in the plateau and $P_m\varphi =0$,

with $s_{\xi }={\mathcal{L}}_{\xi }\gamma +\frac{1}{2}{\mathcal{L}}_{\xi }K\xi$, ${\rho }_{\xi }={E_0}^{\mathrm{R}}D{s}_{\xi }$. The companion vanishes under the causal plateau condition (CP). On the compact free-shell tuple $m=0$, $\psi =t^2+(x^1{)}^2$, $\gamma =\psi \eta$, $\varphi =\psi$, $b=D\gamma =(-2t,2x^1,0,0)$, with ${\xi }_{\ast }=a(t)\zeta (\mathbf{x}){\partial }_t$ and (CP) holding,

A rule with ${Y^{\gamma }}_{\xi }={Y^{\varphi }}_{\xi }=0$ changes the lift by zero. No cancelling map was supplied XIIE,XIIF.

Fix nonempty $W$ with $\overline{W}\subset \mathrm{int}\{g=1\}\cap \{\epsilon <t<T\}$, ${\mathfrak{g}}_W={C_c}^{\infty }(W;TM)$, ${\mathcal{C}}_W=\{u\in {\mathcal{C}}_{\mathrm{sol}}:D^{\ast }b=0\text{ on }W\}$. Under

one has ${\Delta }_{\xi }\mathcal{O}{|}_{{\mathcal{C}}_W}=0$ for every $\xi \in {\mathfrak{g}}_W$. The coefficientwise kernel ${\mathcal{N}}_W$ is not a two-sided ideal: compact smearings give $[L_q,G_h{]}_{{\star }_H}=i\hslash C(q,h)1$ with $C(q,h)>0$, and an admissible rank-one ordering change gives ${\alpha }_d({L_q}^2)={L_q}^2+\hslash L_q(u{)}^2$. No reduced algebra was handed off; ${\mathcal{C}}_W$ is an evaluation set XIIIA.

The coefficient of $\lambda \hslash$ is Unresolved. A successor must supply: a prescription across the noncoincident characteristic configurations, or a proved cancellation in the selected pair; uniform transverse estimates on the three- and four-vertex strata; compatible partial-diagonal contacts; the output wavefront estimate after pushforward. The free differential is not applied to ${R_1}^{(1)}$ XIIIC.

On the detecting data, every shell-descending action in the parameter-local/one-retarded class ${\mathfrak{K}}_{\mathrm{loc},\mathrm{R}}$ induces the zero representation on ${\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}/{\mathcal{N}}_{\mathrm{sol}}$, and

Ending: Proved obstructed for that class only. The cochain ${\mathcal{C}}_{\xi }=-{\mathfrak{B}}_{\xi }[D\gamma ,\varphi ]$ is admitted and is not an integrated completion. Nonlocal Cauchy-data actions remain undecided XIIID.

The carrier is ${\mathcal{E}}_H=({\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}[[\hslash ,\kappa ]],{\star }_H)/{\mathcal{N}}_{\mathrm{sol}}[[\hslash ,\kappa ]]$ with complete shell ${\mathcal{C}}_{\mathrm{sol}}=\{\square \gamma =0,P_m\varphi =0,\square c=\square \bar{c}=0,b=D\gamma \}$. The frozen free differential is $s_0\gamma =\mathcal{R}c$, $s_0\bar{c}=ib$, $s_0\varphi =s_{0}c=s_{0}b=0$, with $\mathcal{R}=K/2$; it is a nilpotent derivation of ${\star }_H$ descending to ${\mathcal{E}}_H$. The ghost normalization $c=2\xi$ recovers $K\xi$. The identities $A+2D^{\ast }D=-\square \mathcal{P}$, $\mathcal{P}{D}^{\ast }=-\mathcal{R}$, $DK=\square$, $A\mathcal{R}=0$ hold. Compact plateau residual parameters are zero: ${\mathfrak{g}}_W\cap \{\square \xi =0\}=\{0\}$ XIIA,XIB.

Use

Parentheses include $\frac{1}{2}$. Indices are moved with $\eta$. The ramp ${\sigma }_{\epsilon }(t)=\chi (t/\epsilon )$ is the inherited smooth function, zero for $t\le 0$, one for $t\ge \epsilon$, flat at both endpoints. The switch is $g\in {C_c}^{\infty }({M_T}^{\circ };[0,1])$ with $P_g=\mathrm{int}\{g=1\}\ne \varnothing$. Both are fixed external functions; neither is varied, neither is a spurion, and no transformation law is assigned to them.

Fix once a nonempty open, relatively compact $W$ with

with the ordinary vector-field bracket $[\xi ,\zeta {]}^{\mu }={\xi }^{\nu }{\partial }_{\nu }{\zeta }^{\mu }-{\zeta }^{\nu }{\partial }_{\nu }{\xi }^{\mu }$. The observation $f\in {C_c}^{\infty }({M_T}^{\circ };\mathbb{R})$ and the switch are chosen so that (eq:UCP) holds. All of $(f,g,m,\epsilon ,T,W)$ are fixed before any parameter, ghost, or antifield is selected, and ${\mathfrak{g}}_W$ is not reduced afterwards.

The independent fields are $({\gamma }_{\mu \nu },\varphi ,c^{\mu },{\bar{c}}_{\mu },b_{\mu })$ with parities $(0,0,1,1,0)$ and ghost numbers $(0,0,1,-1,0)$. Odd identities are coefficientwise in finite Grassmann algebras, with odd coefficients placed on the left. Put

The inverse frozen on this sheet is

X-G's $-4\mathcal{P}{E}_0^{\mathrm{R}}$ inverts the differently normalized Hessian $-\frac{1}{4}\mathcal{P}\square$; it is not converted into (eq:inverse) by vocabulary and is not used.

The classical action is XII-C's

with $S_{\mathrm{grav},\mathrm{D}}$ from (eq:action) and ${\mathfrak{L}}_q=\frac{1}{2}\sqrt{-q}(q^{\mu \nu }{\partial }_{\mu }\varphi {\partial }_{\nu }\varphi -m^2{\varphi }^2)$. It is a formal bulk variational functional tested with compact interior variations; its Euler derivatives ${e_{\mathrm{D}}}^{\mu \nu }$, $E_{\varphi }$ are the exact expressions of XII-C. The gauge-fixing term $S_{\mathrm{gf}}=\int D_{\nu }\gamma D^{\nu }\gamma$ is generated below by XI-B's fermion; it is not added by hand.

The observable is the full displayed gauge-fixed second-order relational coefficient of XII-E,

It has metric and scalar arguments only, degree $(2,1)$ plus the degree-$(0,3)$ scalar-sourced slot, and $Q_H\mathcal{O}\in {\mathcal{E}}_H$ by XII-E. The sign $-1$ on ${\mathcal{C}}_{3,\mathrm{D}}$ is XII-E's and is not reopened. The evaluation domains are

Since $\mathcal{O}$ has no ghost argument, its evaluations depend only on the bosonic part of a configuration.

Completion inside ${\mathfrak{K}}_{\mathrm{loc},\mathrm{R}}$ is closed by XIII-D and is not reopened. No spurion, new sheet, or exit class is frozen here; those routes are recorded as not taken.

• The convolution regulator of Input statement input:XAconv is not the ramp. Its defect $S_{\epsilon }\xi \ne \xi$ concerns a different coordinate prescription; it is neither an input to the ramp master equation nor repaired by it.

• X-G's source-switched mixed map is not XII-C's density-weighted source. The reciprocity theorem blocks an exact upgrade that keeps that field map on every background; it says nothing about the density-switched action (eq:action), whose source and Noether identity were recomputed in XII-C.

• XII-C's empty jet is not emptiness of the gauge-fixed shell and not a failure of $\mathcal{O}$. XIII-A's evaluation identity is not a quotient. XIII-D's cochain is not a completion. XIII-C's inventory is not a product.

Action: (eq:SD). Inverse: (eq:inverse). Switch law: $g$ and ${\sigma }_{\epsilon }$ fixed and inert, no transformation. Observable: (eq:O). Parameter algebra: (eq:W). The sheet is named; Section sec:A1 may write $(S,S)$.

For each field ${\Phi }^A$ introduce an antifield ${\Phi }_A^{\ast }$ of opposite parity and ghost number $\mathrm{gh}({\Phi }_A^{\ast })=-1-\mathrm{gh}({\Phi }^A)$:

Antifields are smooth sections compactly supported in ${M_T}^{\circ }$; on the relative space below, $c^{\ast }$ is a smooth section over $W$. For functionals $F,G$ of fields and antifields, the antibracket is

with the left/right functional derivatives and the graded Jacobi identity of HT (Ch. 15); on local functionals the derivatives are Euler derivatives and the bracket is again local GPS,FRBV. Antifield degree is polynomial degree in the ${\Phi }_A^{\ast }$.

Let $c$ be the odd vector field $c^{\mu }{\partial }_{\mu }$ and write, with odd coefficients on the left,

Define the odd, ghost-number-one derivation $s$ of the field algebra by

Equivalently $sq=\frac{\kappa }{2}{\mathcal{L}}_{c}q$ for $q=\eta +\kappa \gamma$: the ghost generates the vector field $\frac{1}{2}c$, so that $c=2\xi$ recovers ${\delta }_{\xi }\gamma =K\xi +\kappa {\mathcal{L}}_{\xi }\gamma$ and ${\delta }_{\xi }\varphi =\kappa {\xi }^{\mu }{\partial }_{\mu }\varphi$. The backgrounds $g$, ${\sigma }_{\epsilon }$ are $s$-inert. At $\kappa =0$ the rule is XII-A's frozen $s_0$.

The notation $\frac{1}{2}{\mathcal{L}}_c(\eta +\kappa \gamma )/1$ in (eq:kin) is a reminder that $\kappa s\gamma =sq=\frac{\kappa }{2}{\mathcal{L}}_{c}q$; the displayed first form is the one used.

$s^2=0$ on the independent fields and on their graded polynomial differential algebra.

Proof. Put $w^{\mu }=c^{\nu }{\partial }_{\nu }{c}^{\mu }$. The graded identity ${\mathcal{L}}_c{\mathcal{L}}_c={\mathcal{L}}_w$ holds: for a scalar $u$, $c^{\nu }{\partial }_{\nu }(c^{\mu }{\partial }_{\mu }u)=w^{\mu }{\partial }_{\mu }u+c^{\nu }{c}^{\mu }{\partial }_{\nu }{\partial }_{\mu }u$ and the last term vanishes by anticommutativity; for covectors one extra term is treated identically, and the identity extends to tensors by the graded Leibniz rule (X-G's kinematic lemma, normalization independent). Then, since $s$ is an odd derivation and ${\mathcal{L}}_{c}q$ is linear in $c$,

hence $s^2\gamma =0$. Likewise $s^2\varphi =\frac{{\kappa }^2}{4}(w^{\mu }{\partial }_{\mu }\varphi -{\mathcal{L}}_c{\mathcal{L}}_c\varphi )=0$. For the ghost,

the first two terms cancelling after renaming and the third vanishing by antisymmetry. Finally $s^2\bar{c}=isb=0$ and $s^{2}b=0$. The square of an odd derivation is an even derivation, so vanishing on generators suffices.

There is no $b^{\ast }$ term because $sb=0$. The frozen class ${\mathfrak{S}}_{\mathrm{kin}}$ consists of all $S^{\prime }=S_{\mathrm{ext}}+{S^{\prime }}_{\ge 2}$ with ${S^{\prime }}_{\ge 2}$ a local even ghost-number-zero functional of antifield degree at least two, with the backgrounds inert and the rule (eq:kin) unchanged.

For $S^{\prime }\in {\mathfrak{S}}_{\mathrm{kin}}$,

Proof. $S_{\mathrm{D}}$ depends only on the even fields $\gamma ,\varphi$, so its left and right derivatives coincide with the ordinary Euler derivatives $e_{\mathrm{D}},E_{\varphi }$. In (eq:antibracket) with $F=G=S_{\mathrm{ext}}$, the antifield-degree-zero part is produced only by pairing a derivative of $S_{\mathrm{D}}$ with ${\delta }^{\mathrm{L}}{S}_{\mathrm{ext}}/\delta {\Phi }_A^{\ast }=s{\Phi }^A$, or with ${\delta }^{\mathrm{R}}{S}_{\mathrm{ext}}/\delta {\Phi }_A^{\ast }$; moving the odd antifield past the odd $s{\Phi }^A$ for even ${\Phi }^A$ gives the sign that makes the two terms equal, yielding $2\int (e_{\mathrm{D}}\cdot s\gamma +E_{\varphi }s\varphi )$. The antifield-degree-one part is a sum of terms $\pm 2\int {\Phi }_A^{\ast }{s}^2{\Phi }^A$, which vanish by Lemma lem:nil. There is no degree-two part because $S_{\mathrm{ext}}$ is linear in antifields. Adding ${S^{\prime }}_{\ge 2}$ changes $(S^{\prime },S^{\prime })$ only in antifield degree $\ge 1$. This is the standard decomposition of HT (Ch. 17), GPS.

On the field space $\mathfrak{F}$ of smooth multiplets with ghosts $c,\bar{c}$ compactly supported in ${M_T}^{\circ }$,

exactly as a formal series in $\kappa$, where

with ${T_0}^{\mu }{}_{\nu }={T_0}^{\mu \rho }{\eta }_{\rho \nu }$ and $C_2,M_2,J_3$ XII-C's finite coefficients. Equivalently ${\mathcal{N}}_{\nu }(e_{\mathrm{D}})=\kappa [E_{\varphi }{\partial }_{\nu }\varphi -{\mathcal{K}}_{\nu }]$.

Proof. By Lemma lem:decomp, $(S_{\mathrm{ext}},S_{\mathrm{ext}})=2s{S}_{\mathrm{D}}$. Write $\xi =c/2$, an odd compact vector field. Then $s\gamma =K\xi +\kappa {\mathcal{L}}_{\xi }\gamma$ and $s\varphi =\kappa {\xi }^{\mu }{\partial }_{\mu }\varphi$, and $s{S}_{\mathrm{D}}=\int e_{\mathrm{D}}\cdot (K\xi +\kappa {\mathcal{L}}_{\xi }\gamma )+\kappa \int E_{\varphi }\xi \cdot \partial \varphi$ is the first variation of $S_{\mathrm{D}}$ along XII-C's scaled variation, evaluated coefficientwise in the Grassmann algebra. XII-C's proof of (eq:noether-input) computes this variation in two ways for an arbitrary compact interior parameter. Compact integration by parts gives $s{S}_{\mathrm{D}}=\int {\xi }^{\nu }[-{\mathcal{N}}_{\nu }(e_{\mathrm{D}})+\kappa E_{\varphi }{\partial }_{\nu }\varphi ]$. Direct variation gives: zero from the natural Einstein–Hilbert density; $\kappa \int {\xi }^{\nu }{\mathcal{B}}_{g,\nu }$ from the switched scalar action; and $-\kappa \int {\xi }^{\nu }[N_{0,\nu }({Z_{\sigma }}_{\epsilon })+\kappa Q_{\nu }({Z_{\sigma }}_{\epsilon })]$ from the added cubic functional, where XII-C's Lemma on the cubic divergence gives $N_{0,\nu }({Z_{\sigma }}_{\epsilon })=-w{Q}_{\nu }(A\gamma )+{\mathfrak{R}}_{{\sigma }_{\epsilon },\nu }$ with the apparent third ramp derivative cancelled. Hence $s{S}_{\mathrm{D}}=\kappa \int {\xi }^{\nu }{\mathcal{K}}_{\nu }$ with $\mathcal{K}$ as in (eq:Kden), and comparison of the two forms is (eq:noether-input). Substituting $\xi =c/2$ and doubling gives (eq:breaking). All steps are identities of local densities modulo total derivatives, integrated against the compact ghost.

Every term of ${\mathcal{K}}_{\nu }$ carries $1-g$, $\partial g$, $w={\sigma }_{\epsilon }-1$, $\partial {\sigma }_{\epsilon }$, or ${\partial }^2{\sigma }_{\epsilon }$. The term $w{Q}_{\nu }(A\gamma )$ is the bulk breaking from weighting only the cubic coefficient; it is not a ramp-derivative term, and it is not removed by setting ramp derivatives to zero. It is removed only where ${\sigma }_{\epsilon }=1$.

There are $c_0>0$ and a nonempty open $O\subset \{t>\epsilon \}$ on which ${\partial }_{t}g\le -c_0$. Moreover $O\cap P_g=\varnothing$.

Proof. Choose $(t_{\ast },{\mathbf{x}}_{\ast })\in W$, so $t_{\ast }>\epsilon$ and $g(t_{\ast },{\mathbf{x}}_{\ast })=1$. Compact support gives $t_1\in (t_{\ast },T)$ with $g(t_1,{\mathbf{x}}_{\ast })=0$; the mean value theorem gives a point between them with ${\partial }_{t}g<0$, and smoothness gives the open neighbourhood and $c_0$ (XII-C's lemma). On $P_g$ all derivatives of $g$ vanish, so $O\cap P_g=\varnothing$.

Let $\varphi ={\varphi }_m$, with ${\varphi }_m=\sin (mt)/m$ for $m>0$ and ${\varphi }_0=t$, let $\gamma =0$, and let $c=\vartheta \zeta (t,\mathbf{x}){\partial }_t$ with $\vartheta$ an odd generator and $0\ne \zeta \in {C_c}^{\infty }(O)$, $\zeta \ge 0$. Then

In particular $(S_{\mathrm{ext}},S_{\mathrm{ext}})\ne 0$ on $\mathfrak{F}$.

Proof. At $\gamma =0$: $A\gamma =0$, and $J_3$, $C_2$, $M_2$, ${{\mathcal{D}}_{\sigma }}_{\epsilon }$, ${Z_{\sigma }}_{\epsilon }$, ${{\mathfrak{R}}_{\sigma }}_{\epsilon }$ are quadratic in $\gamma$ and vanish; ${\mathfrak{L}}_q={\mathfrak{L}}_{\eta }$; $P_m{\varphi }_m=0$. Hence ${\mathcal{K}}_{\nu }=({\partial }_{\mu }g){T_0}^{\mu }{}_{\nu }[{\varphi }_m]$. For ${\varphi }_m$, ${\partial }_t{\varphi }_m=\cos (mt)$, $(\partial \varphi {)}^2-m^2{\varphi }^2=\cos (2mt)$, so ${T_0}^{00}={\cos }^2(mt)-\frac{1}{2}\cos (2mt)=\frac{1}{2}$ and ${T_0}^{0i}=0$. Thus $c^{\nu }{\mathcal{K}}_{\nu }=c^0({\partial }_{\mu }g){T_0}^{\mu 0}=\frac{1}{2}\vartheta \zeta {\partial }_{t}g$, and (eq:witnessA1) follows from Lemma lem:fall. The Grassmann coefficient is a strictly negative real number.

For every $S^{\prime }\in {\mathfrak{S}}_{\mathrm{kin}}$, the antifield-degree-zero component of $(S^{\prime },S^{\prime })$ equals $\mathfrak{K}$, which is a nonzero functional on $\mathfrak{F}$ at order ${\hslash }^0$. Hence no $S^{\prime }\in {\mathfrak{S}}_{\mathrm{kin}}$ satisfies $(S^{\prime },S^{\prime })=0$ on the unrestricted field space, and no correction beginning at order $\hslash$ changes this conclusion.

Proof. Lemma lem:decomp, Theorem thm:breaking, and Proposition prop:witnessA1. Higher antifield terms and $\hslash$-corrections do not affect the antifield-degree-zero ${\hslash }^0$ coefficient.

The witness ghost in Proposition prop:witnessA1 is supported in the falling region, outside $P_g$. Leaving ${\mathfrak{S}}_{\mathrm{kin}}$ means changing the rule, transforming a background, or changing the action: the spurion, new-sheet, or nonvariational routes of Section sec:sheet, none of which is taken here. A second, independent breaking lives in the ramp layer through $w{Q}_{\nu }(A\gamma )$ and ${{\mathfrak{R}}_{\sigma }}_{\epsilon }$; one witness suffices for the theorem.

${\mathfrak{F}}_W$ is the graded space of multiplets $(\gamma ,\varphi ,c,\bar{c},b)$ with $c\in {\mathfrak{g}}_W\otimes {\Lambda }^{\mathrm{odd}}$ and the other fields unrestricted smooth sections, together with antifields as above, $c^{\ast }$ being a smooth section over $W$. The antibracket (eq:antibracket) is the canonical one for this pairing and satisfies the graded Jacobi identity.

The rule (eq:kin) preserves ${\mathfrak{F}}_W$: $sc$ is supported in $W$ because ${\mathfrak{g}}_W$ is closed under the bracket, and $s\gamma$, $s\varphi$ are supported in $W$ apart from the field-independent term absent at this order. So $S_{\mathrm{ext}}$ is a well-defined functional on ${\mathfrak{F}}_W$.

${\mathcal{K}}_{\nu }[\gamma ,\varphi ](x)=0$ for every $x\in P_g\cap \{t>\epsilon \}$ and every smooth $(\gamma ,\varphi )$. In particular ${\mathcal{K}}_{\nu }=0$ on a neighbourhood of $\overline{W}$.

Proof. On the open set $P_g$, $g\equiv 1$, so $1-g$ and all derivatives of $g$ vanish: ${\mathcal{B}}_{g,\nu }=0$. For $t>\epsilon$, ${\sigma }_{\epsilon }\equiv 1$, so $w=0$ and all derivatives of ${\sigma }_{\epsilon }$ vanish: $w{Q}_{\nu }(A\gamma )=0$, ${{\mathcal{D}}_{\sigma }}_{\epsilon }=0$, ${Z_{\sigma }}_{\epsilon }=w{J}_3+{{\mathcal{D}}_{\sigma }}_{\epsilon }=0$, and every term of ${\mathfrak{R}}_{{\sigma }_{\epsilon },\nu }$ carries $\partial {\sigma }_{\epsilon }$ or ${\partial }^2{\sigma }_{\epsilon }$. By (eq:W), $\overline{W}\subset P_g\cap \{\epsilon <t<T\}$, which is open.

On ${\mathfrak{F}}_W$,

exactly in $\kappa$, at ${\hslash }^0$. Consequently $s_{\mathrm{ext}}:=(S_{\mathrm{ext}},\cdot )$ is a nilpotent, odd, ghost-number-one derivation of the local functionals on ${\mathfrak{F}}_W$, and the same holds for every $S^{\prime }\in {\mathfrak{S}}_{\mathrm{kin}}$ whose higher antifield terms satisfy the corresponding degree-$\ge 1$ equations.

Proof. By Theorem thm:breaking, $(S_{\mathrm{ext}},S_{\mathrm{ext}})=\kappa \int c^{\nu }{\mathcal{K}}_{\nu }$. For $c\in {\mathfrak{g}}_W\otimes \Lambda$ the integrand vanishes pointwise: $c^{\nu }(x)=0$ for $x\notin W$ and ${\mathcal{K}}_{\nu }(x)=0$ for $x\in W$ by Lemma lem:Kzero. The functional derivatives of $\mathfrak{K}$ along ${\mathfrak{F}}_W$ vanish as well: 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 $\mathfrak{K}\equiv 0$ on ${\mathfrak{F}}_W$ together with its derivatives, and the graded Jacobi identity gives ${s_{\mathrm{ext}}}^{2}F=\frac{1}{2}((S_{\mathrm{ext}},S_{\mathrm{ext}}),F)=0$. Theorem thm:obstructA1 shows the antifield-degree-zero equation is satisfied by every $S^{\prime }\in {\mathfrak{S}}_{\mathrm{kin}}$ on ${\mathfrak{F}}_W$; the remaining degree-$\ge 1$ equations are the usual conditions on ${S^{\prime }}_{\ge 2}$ and are satisfied, for example, by ${S^{\prime }}_{\ge 2}=0$.

This is XII-C's plateau contraction written in antifield form: the relative symmetry is the compact algebra ${\mathfrak{g}}_W$ supported where both backgrounds are trivial, and the master equation holds because the breaking density vanishes there. It is an action statement. It does not cancel a propagated observable remainder; Section sec:A2 shows that the remainder survives.

Let $\Psi =-i\int {\bar{c}}_{\mu }(2D^{\mu }\gamma -b^{\mu }){\mathrm{d}}^{4}x$ and $S_{\Psi }[\Phi ,{\Phi }^{\ast }]:=S_{\mathrm{ext}}[\Phi ,{\Phi }^{\ast }+\delta \Psi /\delta \Phi ]$. Then $(S_{\Psi },S_{\Psi })=\mathfrak{K}$ on $\mathfrak{F}$ and $(S_{\Psi },S_{\Psi })=0$ on ${\mathfrak{F}}_W$. At antifield degree zero,