% !TEX program = pdflatex %===================================================================== % X-G.tex % % Standalone source for pdfLaTeX. % % Build twice: % pdflatex -interaction=nonstopmode -halt-on-error X-G.tex % pdflatex -interaction=nonstopmode -halt-on-error X-G.tex % % Honest scope: % The normalized free BRST multiplet is constructed. % The switched-matter breaking is evaluated. % An exact source-preserving variational upgrade of X-F is % obstructed by an explicit pairing on free metric solutions. % The one-loop graph inventory is proved under stated conventions. % The local fish extension is constructed by the finite-scaling- % degree theorem. % A scalar model of the nonlocal mixed contraction is audited % microlocally. Failure of the sufficient product criterion is % NOT called nonexistence. % The requested one-loop background identity is conditional on % construction of its remaining operations. % % Corrections and distinctions: % - Retain P_gr=-(1/4) P Box and Delta_R=-4 P E_0^R. % - With s bar c=b, the fermion bar c(C-b) gives b=C/2. % - Independent b gives off-shell nilpotence. Eliminated b does not. % - Source switching is not density switching. % - A failed wavefront product criterion is not a no-go theorem. % - Bare and scalar-completed observable coefficients must not both % be fed through the scalar Moller map. % - The two mixed contraction graphs exhaust the non-tadpole % kappa^2 hbar sector only in the declared graph convention. % - Background-dependent one-point graphs require another vertex % and first feed the relational scalar at kappa^3 in that convention. % - General finite quantum counterterms are not classified by leg % counting alone. % - The first scalar slot of the completed D_1 kernel is not confined % to supp f. Its correct compact bound is supplied below. % - A constraint commutator proportional to hbar does not imply % 1 belongs to an ideal over C[[hbar]] without inverting hbar. % % Verification: % No pdflatex execution, proof-assistant check, independent referee % review, or journal acceptance is asserted. %===================================================================== \documentclass[11pt,a4paper]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage[margin=1in]{geometry} \usepackage{microtype} \usepackage{amsmath,amssymb,amsthm,mathtools,mathrsfs} \usepackage{array,booktabs,tabularx} \usepackage{enumitem} \usepackage[hidelinks]{hyperref} \allowdisplaybreaks \setlength{\emergencystretch}{3em} \setlength{\parskip}{2pt} \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{assumption}[theorem]{Assumption} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \DeclareMathOperator{\supp}{supp} \DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\Int}{int} \DeclareMathOperator{\WF}{WF} \DeclareMathOperator{\sd}{sd} \DeclareMathOperator{\id}{id} \newcommand{\R}{\mathbb R} \newcommand{\C}{\mathbb C} \newcommand{\dd}{\,\mathrm d} \newcommand{\eps}{\varepsilon} \newcommand{\Lie}{\mathcal L} \newcommand{\sig}{\sigma_\eps} \newcommand{\PP}{\mathcal P} \newcommand{\Pm}{P_m} \newcommand{\Pgr}{P_{\mathrm{gr}}} \newcommand{\Glin}{G_{\mathrm{lin}}} \newcommand{\Gtwo}{\mathcal G_2} \newcommand{\Qop}{\mathcal Q} \newcommand{\Fop}{\mathcal F} \newcommand{\ER}{\mathsf E_0^{\mathrm R}} \newcommand{\EA}{\mathsf E_0^{\mathrm A}} \newcommand{\ERm}{\mathsf E_m^{\mathrm R}} \newcommand{\EAm}{\mathsf E_m^{\mathrm A}} \newcommand{\DR}{\Delta_{\mathrm R}^{\mathrm{gr}}} \newcommand{\DA}{\Delta_{\mathrm A}^{\mathrm{gr}}} \newcommand{\ga}{\mathfrak g_U} \newcommand{\Gp}{\mathcal G_U} \newcommand{\hcl}{h_{\mathrm{cl}}} \newcommand{\kcl}{k_{\mathrm{cl}}} \newcommand{\qb}{\bar q} \newcommand{\hq}{\widehat h} \newcommand{\field}{\widehat\phi} \newcommand{\kq}{\widehat k} \newcommand{\Ahat}[1]{\widehat A_{#1}} \newcommand{\PhiF}{\widehat\Phi^{\mathrm F}} \newcommand{\Ideal}{\mathfrak J} \newcommand{\Wick}{\mathfrak W} \newcommand{\Afree}{\mathscr A_0} \newcommand{\sfree}{s_0} \newcommand{\Lag}{\mathfrak L} \newcommand{\Rprod}{\mathsf R} \newcommand{\Drec}{\mathscr D} \newcommand{\Nop}{\mathsf N} \newcommand{\Kf}{K_f} \newcommand{\Of}{O_f} \newcommand{\Cf}{C_f} \newcommand{\Lalpha}{\mathsf L_\alpha} \newcommand{\Lbeta}{\mathsf L_\beta} \newcommand{\Nnull}{\mathcal N} \newcommand{\pair}[2]{\left\langle #1,#2\right\rangle} \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 X-G}\\[0.4em] The normalized BRST multiplet and the one-loop background gate:\\ proved compatibility conditions, an exact-upgrade obstruction,\\ and the remaining nonlocal extension problem} \author{Ngo Cheung} \date{} \usepackage{tikz} \begin{document} \maketitle \begin{abstract} We examine the proposed one-loop continuation of the fixed-regulator relational scalar of Landmarks X-E and X-F. The graviton normalization is retained: $P_{\mathrm{gr}}=-\frac14\mathcal P\Box$ and $\Delta_{\mathrm R}^{\mathrm{gr}} =-4\mathcal P\mathsf E_0^{\mathrm R}$. The fermion $\int\bar c(C-b)$ constructs the corresponding free auxiliary--ghost multiplet. It gives $b=C[h]/2$, the required Feynman-weight gauge fixing, and the ghost term $-\bar c\Box c$. The independent auxiliary field makes the free differential nilpotent off shell. Its Green matrix and compatibility with the causal field algebra are proved. Two obstacles must be distinguished. The switched scalar action has an explicitly evaluated gauge-breaking coefficient. Separately, X-F multiplies gravitational sources by the ramp after variation. An exact variational upgrade retaining that field response on all allowed backgrounds is impossible: a compact pure-gauge background in the ramp yields a nonzero reciprocity pairing on genuine free metric solutions. This obstruction is classical and cannot be cancelled by a positive-quantum-order counterterm. Under a stated Wick graph convention, two non-tadpole contraction graphs exhaust the $\kappa^2\hbar$ sector. Their coefficients contain no classical background. This proves a conditional background-identity statement, not existence of the complete observable. The local scalar--graviton fish admits finite-scaling-degree extensions. A scalar model of the nonlocal coordinate contraction has a null-collinear exceptional set for the sufficient wavefront product criterion. Failure of that criterion is not called nonexistence. The required full retarded product is not supplied by local covariance alone. The free BRST construction and the exact-upgrade obstruction are released. An unconditional renormalized one-loop background identity for the unchanged X-F prescription is not claimed. \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) {Free graded multiplet}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Gauge fermion and Green operators}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Interacting quantum theory}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {Free-shell data are not an interacting completion.}; \draw[->] (a) -- (b); \draw[->] (b) -- (c); \draw[->] (b) -- (s); \end{tikzpicture} \caption{The free multiplet is not the interacting theory. 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-X-G} \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) {Free graded multiplet}; \node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {Interacting quantum theory}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {Free-shell data are not an interacting completion.}; \draw[->] (a) -- (s); \draw[->] (b) -- (s); \end{tikzpicture} \caption{The free multiplet is not the interacting theory. 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-X-G} \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) {Free graded multiplet}; \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Gauge fermion and Green operators}; \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {Interacting quantum theory}; \node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {Free-shell data are not an interacting completion.}; \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-X-G} \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) {Free graded multiplet}; \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) {Interacting quantum theory}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Gauge fermion and Green operators. Free-shell data are not an interacting completion.}; \draw (a) -- (m); \draw (m) -- (b); \draw[->] (m) -- (s); \end{tikzpicture} \caption{Free graded multiplet is not Interacting quantum theory. 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-X-G} \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: Free graded multiplet}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Gauge fermion and Green operators}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: Interacting quantum theory}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {Free-shell data are not an interacting completion.}; \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-X-G} \end{figure} Figure~\ref{fig:split-X-G} records one view of the same separation. Figure~\ref{fig:chain-X-G} records one view of the same separation. Figure~\ref{fig:pair-X-G} records one view of the same separation. Figure~\ref{fig:gate-X-G} records one view of the same separation. The drawings add no hypothesis. Figure~\ref{fig:scope-X-G} records that separation. The drawing adds no hypothesis. \scopebox{Release and quantifiers}{% \textbf{Constructed:} the free graded multiplet, one gauge fermion, its Green operators, the free differential, the natural background ghost operator, the switched-matter breaking, and the conditional graph inventory. \smallskip \textbf{Proved obstructed:} a variational/Bogoliubov upgrade whose classical metric field map agrees exactly with X-F's source-switched mixed map for every allowed background and free metric input. \smallskip \textbf{Unresolved:} the complete nonlocal one-loop retarded product and an unconditional one-loop background-identity theorem for the requested prescription. \smallskip These statements have different quantifiers. The obstruction is not a universal no-go for perturbative gravity, nor a disproof of X-F's direct polynomial plateau identity.} \tableofcontents %===================================================================== \section{Scope and the inference that needs justification} \label{sec:scope} %===================================================================== X-F constructs three fixed-width tree symbols and proves their direct background identity through $\kappa^2$ at classical order \cite{XF}. It also states that its gravitational switch acts on sources, not on action densities. The latter distinction matters when a quantum retarded map is requested. The proposed continuation contains three logically separate tasks: \[ \begin{gathered} \text{construct a free graded carrier},\\ \text{specify a variational interacting retarded map},\\ \text{construct and normalize its nonlocal one-loop insertion}. \end{gathered} \] A tree kernel's admission proves none of the later tasks by itself. Local covariance of renormalized products is also different from comparison of two interacting splittings with the free input fields held fixed. The latter comparison requires an explicit quantum transport or perturbative-agreement statement. The additional normalization conditions in \cite{HW}, and the separate background transport analysis in \cite{TTZ}, illustrate this distinction. Neither paper is used as an automatic completion theorem for the present switched nonlocal observable. We give complete proofs of the finite algebraic and classical compatibility statements below. Conditional quantum statements are labelled as such. \begin{definition}[Exact source-preserving variational upgrade] \label{def:upgrade} An exact source-preserving upgrade of X-F consists of: \begin{enumerate}[label=\textup{(\alph*)}] \item its frozen free metric operator and causal relations; \item an even interaction functional with symmetric bosonic second variation, after elimination of the algebraic auxiliary field; \item a classical retarded metric field map agreeing with X-F's mixed response on every smooth free metric solution and every allowed smooth classical background; \item quantum corrections containing only nonnegative powers of $\hbar$. \end{enumerate} Agreement in (c) is equality of the field map, equivalently equality after every compact smooth linear metric smearing. It is stronger than agreement of one selected relational observable. \end{definition} This definition makes precise one natural meaning of a ``Bogoliubov completion of X-F''. Its failure does not exclude a weaker prescription assigning selected polynomial symbols to an algebra, or a new density-switched theory. %===================================================================== \section{Frozen conventions and a support correction} \label{sec:frozen} %===================================================================== \subsection{Geometry, switches, and tensor types} Retain \[ M=\R^4,\qquad \eta=\operatorname{diag}(1,-1,-1,-1),\qquad \Box=\partial_t^2-\Delta,\qquad \Pm=\Box+m^2,\quad m\geq0. \] Fix the X-E/X-F observation and scalar switch \[ f\in C_c^\infty((0,T)\times\R^3;\R),\qquad g\in C_c^\infty((0,T)\times\R^3;[0,1]). \] Let $U$ be relatively compact with \[ \overline U\subset P_g:=\Int\{g=1\},\qquad \ga=C_c^\infty(U;TM). \] Write $\Kf=\supp f$. The separation condition is \begin{equation} \Kf\cup\supp g\cup\overline U \subset\{2\delta0,\\ t,&m=0. \end{cases} \] It has $T_0^{00}=1/2$ and $T_0^{0i}=0$. At the leading metric $q=\eta$, the density difference is zero. Along a vertical line through a plateau point, $g$ rises from zero to one. Therefore $\partial_tg>0$ on some open interior region. Choose a nonnegative nonzero $\zeta\in C_c^\infty$ there and an odd Grassmann generator $\vartheta$. Set $c=\vartheta\zeta\partial_t$. Then \[ [sS_g^{\mathrm{sc}}]_{\kappa} =\frac{\vartheta}{2}\int\zeta\,\partial_tg\dd^4x\neq0. \] \end{proof} \begin{corollary}[Necessary master-action condition] \label{cor:CME} An ordinary canonical BV extension generating \eqref{eq:nonlinear} and having $S_{\mathrm{ref}}$ as its antifield-independent action cannot satisfy the classical master equation with the scalar switch and transformations unchanged. A correction starting strictly at $\hbar$ cannot cancel this classical breaking. \end{corollary} \begin{proof} The antifield-independent coefficient of the classical master equation requires $sS_{\mathrm{ref}}=0$. Theorem~\ref{thm:matter} violates this necessary condition. Coefficient extraction at $\hbar^0$ proves the final assertion. \end{proof} This is not a theorem that every observable quantum Ward coefficient is nonzero. Nor does it exclude a modified spurion rule, extra compensators, or a different local-coupling construction. Those would change the sheet. \subsection{Plateau invariance and the causal qualification} For a ghost test supported inside $P_g$, the two terms in \eqref{eq:sSfirst} vanish. The restricted symmetry therefore survives. Under a simultaneous background pullback supported in $U$, the metric, fluctuation, scalar, ghost, antighost, and auxiliary fields transform in their tensor types. The fermion and ghost density are natural, $g\circ F^{-1}=g$, and the flat scalar term is unchanged where its coefficient $1-g$ is nonzero. Thus the integrated reference action is exactly invariant under this simultaneous pullback. This background covariance is different from quantum-gauge invariance with $\qb$ held fixed. If an observation point belongs to $P_g$, its vertical past curve crosses a region where $g$ changes from zero. Hence $\supp\dd g$ meets its causal past. Without an observation-plateau or comparable causal hypothesis, compactness alone does not prove the asserted intersection. Even a nonempty causal intersection does not prove a nonzero ghost contraction with a particular observable. Propagator support, index contractions, and smearings still have to be evaluated. %===================================================================== \section{Source switching and an exact-upgrade obstruction} \label{sec:obstruction} %===================================================================== \subsection{The necessary reciprocity identity} For a fixed background $w$ define \[ \Nop_w=\sig Q_w. \] Since $Q_w$ is formally self-adjoint, \begin{equation} \Nop_w^*=Q_w\sig,\qquad \Nop_w-\Nop_w^*=[\sig,Q_w]. \label{eq:skew} \end{equation} Thus source multiplication generally destroys the symmetry of the action Hessian. The following on-shell test excludes an equation-zero representative repair. \begin{lemma}[Variational reciprocity on free inputs] \label{lem:reciprocity} Let an even variational interaction have reduced mixed metric Hessian $L_w$, with compact coefficients. If its classical response agrees with $-\DR\Nop_wu$ on all free metric solutions, then \begin{equation} \Drec_w(u,v) :=\int\bigl(u\cdot\Nop_wv-v\cdot\Nop_wu\bigr)\dd^4x=0 \label{eq:D} \end{equation} for all such solution pairs. \end{lemma} \begin{proof} Symmetry of second directional derivatives gives \[ \int u\cdot L_wv=\int v\cdot L_wu. \] Equality of the retarded responses, followed by application of $\Pgr$, gives $L_wu=\Nop_wu$ on every free input. Substitute into the symmetric pairing. Compactness of the coefficients removes all boundary terms. \end{proof} \subsection{A pure-gauge background in the ramp} A smooth ramp changing from zero to one has $\sig''\not\equiv0$ on $(0,\eps)$. Indeed, otherwise it would be affine there, and endpoint smoothness would force zero slope and hence a constant. Choose a nonnegative $\zeta\in C_c^\infty((0,\eps))$ supported where $\sig''\neq0$, with $\zeta\not\equiv0$, and put \[ \psi(t)=\zeta(t)\sig''(t). \] Choose \[ \chi_0\in C_c^\infty(\R^3),\qquad \chi_0\geq0,\qquad M_\chi:=\int\chi_0\dd^3x>0. \] Set \begin{equation} \xi=\psi(t)\chi_0(\mathbf x)\partial_t,\qquad w=K\xi. \label{eq:rampgauge} \end{equation} This is a permitted classical background. The vector field $\xi$ here is not asserted to belong to the plateau algebra: it parametrizes the chosen background, not an admitted plateau transformation. Let $e_{00}$ be the tensor whose only nonzero covariant component is $(e_{00})_{00}=1$, and take \[ u=e_{00},\qquad v=te_{00}. \] Both satisfy $\Pgr u=\Pgr v=0$. \begin{proposition}[Evaluated free-solution defect] \label{prop:Dsigned} For the background \eqref{eq:rampgauge}, \begin{equation} \boxed{ \Drec_w(u,v) =\frac{5M_\chi}{8}\int_0^\eps\sig''\psi\dd t =\frac{5M_\chi}{8}\int_0^\eps \zeta(\sig'')^2\dd t>0.} \label{eq:Dnonzero} \end{equation} \end{proposition} \begin{proof} By \eqref{eq:commutator}, on a free input, $Q_wu=-\Pgr\Lie_\xi u$. Directly, \[ (\Lie_\xi u)_{00}=2\psi'\chi_0,\qquad (\Lie_\xi u)_{0i}=\psi\,\partial_i\chi_0,\qquad (\Lie_\xi u)_{ij}=0, \] and \[ (\Lie_\xi v)_{00}=(\psi+2t\psi')\chi_0,\qquad (\Lie_\xi v)_{0i}=t\psi\,\partial_i\chi_0,\qquad (\Lie_\xi v)_{ij}=0. \] For a tensor with no spatial--spatial components, \[ (\Pgr a)^{00}=-\tfrac18\Box a_{00}. \] It follows that \[ (Q_wu)^{00}=\tfrac18\Box(2\psi'\chi_0),\qquad (Q_wv)^{00} =\tfrac18\Box((\psi+2t\psi')\chi_0). \] After spatial integration, the Laplacian terms vanish. Since \[ (\psi+2t\psi')''-t(2\psi')''=5\psi'', \] we obtain \[ \Drec_w(u,v) =\frac{5M_\chi}{8}\int\sig\psi''\dd t. \] Twice integrating by parts gives \eqref{eq:Dnonzero}. \end{proof} The homogeneous solutions can be replaced by free solutions with compact Cauchy data. Choose their initial data equal to those of $u,v$ on a ball containing the initial causal shadow of $\supp w$, and multiply by a larger compact cutoff. Finite propagation preserves equality on a neighbourhood of $\supp w$. Since $Q_w$ is local and supported there, the value \eqref{eq:Dnonzero} is unchanged. Together with $b=C[u]/2$ or $C[v]/2$ and zero scalar and ghost fields, these are configurations of the full free multiplet. \begin{theorem}[No exact source-preserving variational upgrade] \label{thm:obstruction} There is no upgrade satisfying Definition~\ref{def:upgrade} on X-F's unrestricted background class. The contradiction remains if off-shell representatives are changed by terms vanishing on all free configurations. It cannot be removed by the algebraic auxiliary field, ghost loops, or counterterms beginning strictly at quantum order. \end{theorem} \begin{proof} Linearize the claimed reduced action response at zero fluctuation, zero scalar, and zero ghosts, in the background \eqref{eq:rampgauge}. Its mixed metric Hessian must satisfy Lemma~\ref{lem:reciprocity}. Proposition~\ref{prop:Dsigned} contradicts that condition. Terms vanishing on all free configurations do not change the witness. Algebraic auxiliary elimination yields another action, whose second variation is still symmetric. Ghost loops and positive-$\hbar$ counterterms do not change the classical coefficient used in the contradiction. \end{proof} \subsection{Relation to density switching} The quadratic functional \[ V_{\mathrm{sym}}[u] =\tfrac12\pair{u}{\sig Q_wu} \] has Hessian \begin{equation} V_{\mathrm{sym}}'' =\tfrac12(\sig Q_w+Q_w\sig), \label{eq:symmetrized} \end{equation} not $\sig Q_w$. Actual switching of a selected density before variation similarly supplies ramp-derivative terms, with details depending on the density representative. These are classical corrections. They cannot be hidden inside a finite renormalization that starts at $\hbar$. For $w=K\xi$ with $\xi\in\ga$, the relevant support lies where $\sig=1$. The witness of this section does not occur in X-F's particular plateau variation. Therefore the direct three-tree identity of X-F is not refuted. What fails is the stronger identification of its entire source-switched field map with the classical part of one variational Bogoliubov map for all permitted backgrounds. %===================================================================== \section{The one-loop coefficient and its graph inventory} \label{sec:inventory} %===================================================================== \subsection{Bare versus completed observables} For an actual variational theory, write \[ V=\kappa V_1+\kappa^2V_2+O(\kappa^3), \] and let the \emph{bare} relational insertion be \[ O=F_f+\kappa O_1+\kappa^2O_2+O(\kappa^3), \qquad F_f=\int f\phi\dd^4x. \] At first order, \[ O_1=\phi(c_1[\hcl+h]), \qquad c_1[u]=\partial_\mu(fX_1^\mu[u]). \] The scalar completion $-\phi(\mathsf L_{1,g}[u]\EAm f)$ is generated by the scalar response. It must not also be inserted as an independent bare term and then regenerated by the same scalar retarded map. With \[ R_V(O)=\sum_{n\geq0}\frac1{n!} \Rprod_n(V^{\otimes n};O), \] the coefficient at order $\kappa^2$ is \begin{equation} O_2+\Rprod_1(V_1;O_1)+\Rprod_1(V_2;F_f) +\tfrac12\Rprod_2(V_1,V_1;F_f). \label{eq:fullcoefficient} \end{equation} This identity is combinatorial. It does not identify a density-switched tree coefficient with X-F's source-switched one. \subsection{The precise graph convention} \begin{assumption}[Graph convention for the inventory] \label{ass:graphs} For the inventory below: \begin{enumerate}[label=\textup{(\alph*)}] \item free cross-blocks join only matching field species, apart from the ghost--antighost block; \item the auxiliary field has been eliminated algebraically; \item interaction vertices and insertion monomials use a fixed normal-ordering convention with no self-contractions at a single vertex; \item connected contraction graphs are expanded coefficientwise in the flat free carrier; backgrounds occur through the explicit vertex coefficients; \item no additional explicit quantum counterterm is included in the inventory. \end{enumerate} \end{assumption} At order $\kappa$, the relevant leg types are \[ \begin{array}{c|c} \text{vertex}&\text{quantum legs}\\ \hline V_{1,g}[h,\phi]&h\phi\phi\\ V_{1,g}[\hcl,\phi]&\phi\phi\\ \text{mixed metric quadratic vertex}&hh\\ S_3[h]&hhh\\ \text{background ghost quadratic vertex}&\bar c c\\ \text{ghost--metric vertex}&h\bar c c \end{array} \] Only $h\phi\phi$ carries both a metric and a scalar leg without a background coefficient. \begin{proposition}[Two non-tadpole graph types] \label{prop:inventory} Under Assumption~\ref{ass:graphs}, the connected $\kappa^2\hbar$ coefficient of \eqref{eq:fullcoefficient} has exactly two contraction types: \begin{enumerate}[label=\textup{(\alph*)}] \item $\Lalpha$: two lines between the $h\phi$ part of $O_1$ and one $h\phi\phi$ vertex; \item $\Lbeta$: the $F_f$ scalar line attaches to one $h\phi\phi$ vertex, and two lines, one metric and one scalar, join that vertex to a second $h\phi\phi$ vertex. \end{enumerate} Each type has one external scalar leg. Its unextended flat contraction coefficient contains no $\hcl,\kcl$, ghost field, gravitational cubic vertex, or background gauge-fixing coefficient. There are no connected contraction terms at $\kappa^2\hbar^r$ with $r\geq2$ under these assumptions. \end{proposition} \begin{proof} For $n$ interaction vertices and $E$ contraction lines, the quantum power is $E-n$. Connectedness requires $E\geq n$. For $n=0$, the insertion is normal ordered, so no quantum contraction occurs. For $n=1$, the order-one insertion has either one scalar leg or one scalar and one metric leg. Quantum order one requires $E=2$. Both lines must end on the two-leg insertion. The only order-one vertex with the required species is $h\phi\phi$. It leaves one scalar leg. A single $V_2$ attached to the one-leg $F_f$ has at most one line and cannot contribute at quantum order one. For $n=2$, the insertion is $F_f$ and has degree one. Quantum order one requires $E=3$. One line attaches $F_f$ to a scalar-bearing vertex; the other two lines must run between the two interaction vertices. The vertex attached to $F_f$ must therefore have two remaining legs. The background $\phi\phi$ vertex does not. The $h\phi\phi$ vertex does, leaving $h\phi$; the only second vertex matching both species is another $h\phi\phi$ vertex. Again one scalar leg remains. The same degree bounds give $E\leq2$ for $n=1$ and $E\leq3$ for $n=2$. Thus $E-n\leq1$. Orders $n\geq3$ require at least $\kappa^3$. \end{proof} \begin{remark}[What the proposition does not prove] The proposition does not construct either required nonlocal retarded operation. It also does not classify every allowed finite renormalization. An explicitly added order-$\kappa^2\hbar$ counterterm is outside Assumption~\ref{ass:graphs} and must be tested separately. The absence of higher-loop contraction graphs is therefore not an unconditional all-$\hbar$ theorem about every possible normalized observable. \end{remark} \subsection{Tadpoles and the background-dependent boundary} In the declared convention, a one-vertex tadpole is absent. A background-dependent graviton, ghost, or scalar one-point graph requires a second vertex. Feeding that order-$\kappa^2$ metric one-point response through the order-$\kappa$ relational metric insertion first contributes at $\kappa^3$. For example, the possible pairs are \[ (hhh,\;\hcl hh),\qquad (h\bar c c,\;\hcl\bar c c),\qquad (h\phi\phi,\;\hcl\phi\phi). \] The two legs between the vertices form the loop, while the metric leg reaches the relational insertion. A different normal-ordering convention can generate one-vertex tadpole normalization terms. Such terms must be included explicitly, rather than declaring the two-graph inventory universal. In particular, the word ``one-point function'' must not silently mean an expectation value in an unspecified state. The main lesson is narrower than the original plan: ghosts are necessary for the BRST carrier, but the non-tadpole scalar coefficient at $\kappa^2\hbar$ does not contain a ghost loop. %===================================================================== \section{The local fish and the nonlocal product criterion} \label{sec:kernels} %===================================================================== \subsection{The ordinary two-vertex fish} Let $D_F^\mu$ denote flat scalar Feynman product data and \[ D_F^{\mathrm{gr}}=-4\PP D_F^0. \] The metric--scalar fish between two matter vertices has finite sums of component kernels of the form \begin{equation} t_{\alpha\beta}(r) =D_F^0(r)\, \partial^\alpha\partial^\beta D_F^m(r), \qquad |\alpha|+|\beta|\leq2, \qquad r\neq0. \label{eq:fish} \end{equation} Tensor coefficients and external scalar derivatives are suppressed. \begin{proposition}[Finite-scaling-degree fish extensions] \label{prop:fish} The kernels \eqref{eq:fish} are well defined away from $r=0$ and have scaling degree at most six in four dimensions. They admit extensions with that scaling-degree bound. Two such extensions differ by \begin{equation} \sum_{|\gamma|\leq2}a_\gamma\partial^\gamma\delta(r), \label{eq:fishambiguity} \end{equation} with the relevant tensor coefficients and lower-degree restrictions for the less singular terms. \end{proposition} \begin{proof} Away from coincidence, null-related Feynman singularities in the two factors have the same first-slot orientation, so their nonzero wavefront covectors cannot add to zero. At spacelike separation both factors are smooth. A scalar Feynman kernel has scaling degree two. At most two derivatives occur on the contracted scalar line, so the product has scaling degree at most $2+2+2=6$. The finite-scaling-degree extension theorem gives an extension without increasing that bound \cite{BF}. A difference is supported at $r=0$. A point-supported distribution of scaling degree at most six in four dimensions has the form \eqref{eq:fishambiguity}. \end{proof} The derivative order two in \eqref{eq:fishambiguity} is the \emph{internal extension ambiguity}. The external scalar factors may already carry derivatives. It would be incorrect to infer that every resulting local quadratic density has total derivative order at most two. \begin{lemma}[Smoothness after compact kernel smearing] \label{lem:smoothing} Let $k(y,y')$ be a distribution with $\WF(k)\subset\Sigma_2$, where \[ \Sigma_2=\{(y,\eta;y',\eta'):\eta+\eta'=0,\ (\eta,\eta')\neq0\}. \] For a compact smooth $a(y)$, the properly supported pairing $\int k(y,y')a(y)\dd y$ is smooth in $y'$. \end{lemma} \begin{proof} The pushforward wavefront bound can only produce an output covector $\eta'$ from a kernel covector with $\eta=0$. The sum-zero condition then forces $\eta'=0$, which is not a wavefront covector. \end{proof} A translation-invariant extension of \eqref{eq:fish}, lifted as $k(y,y')=t(y-y')$, has wavefront set in $\Sigma_2$. Applying Lemma~\ref{lem:smoothing} to the compact scalar switch and the smoothly smeared insertion leg shows that the isolated local two-vertex contribution has the form \[ \Lbeta=\field(\ell_\beta),\qquad \ell_\beta\in C_c^\infty(M), \] with compact support supplied by the remaining vertex switch. This is a local graph construction with the stated finite normalization freedom. It is not a construction of the nonlocal mixed retarded product. \subsection{A scalar model of the coordinate contraction} The coordinate part of the bare insertion contains a derivative of $\ER(x,z)$. Consequently a representative time-ordered scalar contraction has the underlying product \begin{equation} k_T(x,z,y) =\ER(x,z)D_F^0(z,y)D_F^m(x,y). \label{eq:triangle} \end{equation} The actual insertion has derivatives, tensor contractions, and smooth coefficient functions. Equation \eqref{eq:triangle} is used only to audit the scalar product criterion, not to assert the full loop coefficient equals this one term. Choose the Hadamard orientation so that the first-slot Feynman covector is future-pointing at the later endpoint and past-pointing at the earlier endpoint. Reversing the Fourier convention reverses all these signs without changing the argument. Define \begin{equation} \Nnull= \left\{ \begin{array}{l} (x,z,y):\ y=x-\lambda\ell,\ z=x-\mu\ell,\\ \ell\neq0\ \text{future null},\ 0\leq\lambda\leq\mu \end{array} \right\}. \label{eq:nullset} \end{equation} It includes the corresponding partial coincidences and the total diagonal. \begin{proposition}[The exceptional scalar wavefront configuration] \label{prop:nullWF} For the scalar kernels in \eqref{eq:triangle}, the sufficient wavefront product criterion holds off $\Nnull$. At a strictly ordered null-collinear triple $z\prec y\prec x$ it fails. This failure does not prove nonexistence of a distributional extension, nor nonexistence of the fully contracted retarded coefficient. \end{proposition} \begin{proof} The three pulled-back wavefront covectors have the forms \[ (k,-k,0),\qquad (0,l,-l),\qquad (p,0,-p) \] in the slot order $(x,z,y)$. Their sum vanishes precisely when \[ l=k=-p. \] No two nonzero factors alone can give a zero sum because one endpoint would have an unpaired nonzero covector. For distinct singular endpoints, all three differences must be null. Pairwise null-separated points in Minkowski space are collinear along a null line. Retarded support puts $z$ before $x$. The Feynman signs permit $l=-p$ only when $y$ lies between $z$ and $x$. At such a triple, choose $p$ future null along the common line, $k=l=-p$, and use the two-sided null wavefront of the retarded kernel. The sum is zero, so the criterion fails. The partial-coincidence cases give the boundary parameters in \eqref{eq:nullset}; $x=z\neq y$ does not allow cancellation of the two equal-orientation Feynman covectors. Thus the product theorem certifies the complement, but not the exceptional configurations. A sufficient theorem's failed hypothesis is not a nonexistence proof. \end{proof} The corresponding Hadamard product \[ k_H(x,z,y)=\ER(x,z)H_0(z,y)H_m(x,y) \] satisfies the product criterion everywhere: $l$ and $p$ have the same fixed Hadamard orientation, so $l=-p$ is impossible. It also has a sum-zero wavefront bound in three slots. This distinction explains why admission of a fixed-width Wick monomial is not admission of its time-ordered product with a local vertex. The ordinary small-diagonal extension of the local fish does not settle \eqref{eq:triangle}: $\Nnull$ has points away from the total diagonal. Derivative and tensor cancellations, or cancellations between terms in the complete retarded product, may change the answer. They have not been evaluated here. Accordingly, the full nonlocal $\Lalpha$ is recorded as unconstructed, not as proved impossible. %===================================================================== \section{What background covariance follows conditionally} \label{sec:conditional} %===================================================================== \subsection{The restricted graph statement} Suppose the remaining operations are constructed in a fixed flat graph prescription, and their normalized coefficients preserve the background independence of Proposition~\ref{prop:inventory}. Suppose also that the normal-ordering and finite insertion terms are specified. Then the loop coefficient has the form \[ \field(\ell),\qquad \ell\in C_c^\infty(M), \] where $\ell$ depends on the observation, switches, mass, ordering data, and normalization constants, but not on $(\hcl,\kcl)$. \begin{theorem}[Conditional fixed-background-input identity] \label{thm:conditional} Assume that such an admitted coefficient $\ell$ has been constructed. Define \begin{equation} \Phi^{\mathrm G}(f) =\PhiF(f)+\kappa^2\hbar\,\field(\ell) \pmod{\kappa^3,\hbar^2}. \label{eq:PhiG} \end{equation} If the inherited X-F polynomial identity holds in the corrected carrier, then under \eqref{eq:twojet}, with the free fields held fixed, \[ \Phi^{\mathrm G}(f)[\hcl',\kcl'] -\Phi^{\mathrm G}(f)[\hcl,\kcl] =0\pmod{\Ideal,\kappa^3,\hbar^2}. \] \end{theorem} \begin{proof} The tree difference belongs to $\Ideal$ by the inherited identity. The added coefficient has no dependence on the two classical backgrounds, so its difference is identically zero. \end{proof} The existence and admissibility hypothesis is substantive. The theorem is not an unconditional completion of X-G. It also does not identify \eqref{eq:PhiG} with the Bogoliubov image of one unchanged interaction. Such an identification would require the additional compatibility contradicted by Theorem~\ref{thm:obstruction}. \subsection{The general variational background gate} For a new density-switched variational sheet, a more general proof must construct: \begin{enumerate}[label=\textup{(\alph*)}] \item retarded products on all actual local vertices and the nonlocal relational insertion, including every coefficient in \eqref{eq:fullcoefficient}; \item quantum comparison maps for the changed background splittings, with the required perturbative agreement; \item transport of causal and Hadamard data, rather than an unproved free pullback automorphism; \item equation-ideal compatibility and the relevant finite Ward normalization. \end{enumerate} If these data produce the finite intertwiner \[ R_{V_{B'}}(O_{B'}) =R_{V_B}(O_B)\pmod{\Ideal}, \] then extraction of its $\kappa^2\hbar$ coefficient gives the requested background identity. Coefficient extraction is immediate; construction of the intertwiner is not. The distinction is precisely why general local covariance and the classical structural argument cannot substitute for the missing quantum operation \cite{HW,TTZ}. \subsection{Renormalization freedom and anomalies} Graph counting does not determine every admissible finite quantum normalization. Additional local terms can involve external coupling jets, observation-test jets, scalar fields, and background curvature data. The candidate defect is not exhausted by one schematic expression $\int(\partial\xi)\mathcal R[\hcl]$. A proof of anomaly removability must: \begin{enumerate} \item construct the actual defect; \item derive its consistency condition; \item classify the admissible finite normalization terms; \item solve the cancellation equation within that class. \end{enumerate} None of these steps follows from a failed scalar product criterion or from the absence of a particular unrenormalized background graph. The explicit reciprocity defect of Section~\ref{sec:obstruction} is different. It is a classical obstruction to an exact source-preserving field-map identification, not an evaluated one-loop anomaly. %===================================================================== \section{Corrections to the proposed continuation} \label{sec:debug} %===================================================================== \paragraph{The three readings of the X-E defect.} X-F's exact identity \[ 2\Gtwo(K\xi,u) =\Lie_\xi\Glin[u]-\Glin[\Lie_\xi u] \] must replace the on-shell shortcut everywhere. A free gauge-fixed metric need not satisfy $\Glin[u]=0$. The full background gauge correction packages both terms into \eqref{eq:commutator}. \paragraph{The strength of ``Moller response''.} A retarded solution map for a prescribed equation and a variational Moller map are not synonyms. Theorem~\ref{thm:obstruction} separates them for the frozen source law. The direct tree identity can remain valid without the stronger variational identification. \paragraph{The graph-order correction.} In the declared non-tadpole normal convention, background-dependent metric one-point loops first feed the relational scalar at $\kappa^3$, not $\kappa^2$. The order-$\kappa^2\hbar$ scalar sector contains the two mixed contraction types of Proposition~\ref{prop:inventory}. This is a scheme-qualified statement. \paragraph{The scalar completion must be counted once.} Applying the full scalar Bogoliubov map to already completed $d_1,d_2$ regenerates scalar corrections. A complete calculation must begin from a bare insertion or explicitly subtract the duplicated scalar response. \paragraph{The completed support bound.} The $g\EAm f$ local terms extend the scalar slot beyond $\supp f$. The corrected compact set is $\Of$ in \eqref{eq:correctsets}. \paragraph{The Hadamard sign.} No $+4$ replacement is made. The action-Hessian, equation, causal, and product-kernel conventions are distinguished in Section~\ref{sec:free}. A uniform joint Hadamard orientation must be checked, not inferred from a coefficient's sign. \paragraph{The wavefront logical direction.} Failure of a sufficient product criterion neither constructs a residue nor proves nonexistence. No reflected light-ray residue is identified here. \paragraph{The formal constraint-ideal qualification.} Suppose a constraint commutator gives $i\hbar a\mathbf1$ with $a\neq0$. A two-sided ideal containing that constraint then contains $\hbar\mathbf1$. Over $\C[[\hbar]]$ this does not by itself imply it contains $\mathbf1$. At fixed nonzero $\hbar$, or after adjoining $\hbar^{-1}$, the conclusion changes. No extra de Donder constraint quotient is used here. \paragraph{The matter support correction.} The unrestricted off-shell breaking includes $(1-g)(\Pm\phi)\partial\phi$. Its leading free-shell coefficient is supported where $g$ changes. This does not justify an unevaluated observable Ward-defect claim. \paragraph{The ordering data at quantum order.} Unlike the classical tree coefficient, the loop contractions use the frozen Hadamard/Feynman product data. Such data are not a chosen state. Their transport and finite-ordering changes must be tracked, rather than declared irrelevant. %===================================================================== \section{Decision and final release} \label{sec:decision} %===================================================================== \begin{center} \small \renewcommand{\arraystretch}{1.12} \begin{tabularx}{\linewidth}{@{}p{.22\linewidth}X@{}} \toprule Gate & Result and scope\\ \midrule Free BRST carrier & Constructed: independent auxiliary field, normalized fermion, ghost action, Green matrix, and causal differential.\\ Background ghost geometry & Constructed: natural operator, curvature formula, and pure-gauge commutator identity.\\ Unrestricted master action & The usual unchanged canonical extension is obstructed by the evaluated classical matter breaking. No full quantum master operation is constructed.\\ Exact source-preserving upgrade & Proved obstructed on the unrestricted X-F background class, with a nonzero pairing on free metric solutions.\\ G1: local fish & Finite-scaling-degree extensions constructed, with bounded local ambiguity.\\ G1: nonlocal insertion & Unresolved: the full mixed retarded product and its exceptional-set cancellations or extension are not constructed.\\ G2: one-loop background identity & Conditional on construction of the coefficient or quantum transport. Not released unconditionally.\\ G3: finite normalization & The required complete classification and anomaly-cancellation equation are not constructed.\\ G4: physical representation & Positivity, physical cohomology representation, and Kugo--Ojima analysis are not opened.\\ G5: changed coupling prescription & Whole-Lagrangian switching, spurions, compensators, and an adiabatic construction are not opened.\\ \bottomrule \end{tabularx} \end{center} \begin{theorem}[Qualified X-G release] \label{thm:release} For the frozen coefficients and background class: \begin{enumerate}[label=\textup{(\roman*)}] \item The free BRST multiplet and differential of Section~\ref{sec:free} are constructed and normalized to X-E/X-F's graviton operator. \item The unrestricted scalar gauge variation is \eqref{eq:B}; its leading coefficient has the nonzero free-shell witness of Theorem~\ref{thm:matter}. \item No exact source-preserving variational upgrade satisfying Definition~\ref{def:upgrade} exists. The explicit obstruction is \[ \frac{5M_\chi}{8} \int_0^\eps\zeta(\sig'')^2\dd t>0. \] \item Under Assumption~\ref{ass:graphs}, the two mixed contraction types exhaust the non-tadpole $\kappa^2\hbar$ inventory. The local fish admits the extensions of Proposition~\ref{prop:fish}. \item The scalar nonlocal product criterion has the exceptional configuration of Proposition~\ref{prop:nullWF}. Its failure is not a nonexistence theorem. \item Theorem~\ref{thm:conditional} is a conditional background-identity result. The existence assumptions needed for an unconditional one-loop completion have not been replaced by an asserted covariance principle. \end{enumerate} \end{theorem} \begin{proof} Item (i) is Proposition~\ref{prop:gfnorm}, Proposition~\ref{prop:Ghb}, and Theorem~\ref{thm:freeBRST}. Item (ii) is Proposition~\ref{prop:breaking} and Theorem~\ref{thm:matter}. Item (iii) is Theorem~\ref{thm:obstruction}. Item (iv) is Proposition~\ref{prop:inventory} and Proposition~\ref{prop:fish}. Item (v) is Proposition~\ref{prop:nullWF}. Item (vi) is the coefficientwise proof of Theorem~\ref{thm:conditional} and its stated hypotheses. \end{proof} \subsection*{The three endings} The free algebraic construction reaches \textbf{Constructed}. The exact source-preserving variational interpretation reaches \textbf{Proved obstructed}, with the quantifiers of Definition~\ref{def:upgrade}. The originally requested unconditional one-loop background completion reaches \textbf{Unresolved} on the supplied prescription. Calling it constructed would require either: \begin{enumerate} \item a non-variational quantum prescription with all its operations and identities proved; or \item a newly declared variational sheet, including its classical ramp corrections and the full nonlocal quantum transport construction. \end{enumerate} Neither is silently inserted into X-G. A missing operation is recorded as missing. An explicit classical incompatibility is recorded as an obstruction. A failed sufficient microlocal criterion is recorded as neither. %===================================================================== \appendix \section{Coefficient and logical audit} \label{app:audit} %===================================================================== \subsection{The free coefficients} The fixed chain is \[ \Psi_0=\int\bar c(C-b),\qquad b=C/2,\qquad S_{\mathrm{gf}}=\tfrac14\int C^2, \] \[ \Pgr=-\tfrac14\PP\Box,\qquad \DR=-4\PP\ER,\qquad \Delta_{hb}=K\delta_0,\qquad \Delta_{c\bar c}=-\delta_0. \] It produces the free cancellation \[ \{Kc,\bar c\}+[h,b]=0. \] Changing $b=C/2$ to $b=C$ while leaving the other entries unchanged breaks this chain. \subsection{Why the obstruction is isolated} For the witness in Section~\ref{sec:obstruction}, the scalar and ghost fields are zero. Linearization is at zero metric fluctuation. The gravitational self source is quadratic in that fluctuation and therefore does not contribute to the linearized mixed response. Thus neither an omitted matter tree nor an unevaluated self-tree can cancel the pairing \eqref{eq:Dnonzero}. \subsection{Why it does not contradict plateau covariance} The vector field defining the witness is supported in the ramp, not in $U$. It specifies an allowed background $w=K\xi$. The no-go tests the response on all backgrounds; it does not apply a forbidden parameter as a plateau symmetry. Conversely, X-F's direct identity uses parameters supported in $U$, where $\sig=1$. These are distinct tests. \subsection{Why conditional background independence is easy} If a completed coefficient is literally $\field(\ell)$ with $\ell$ independent of the backgrounds, its held-fixed variation is zero. This observation does not construct $\ell$, prove that it is the intended retarded coefficient, or impose perturbative agreement. These are the three missing inferences avoided in Section~\ref{sec:conditional}. \section*{Declarations} \addcontentsline{toc}{section}{Declarations} \paragraph{Dependencies.} The signed quadratic action, scalar switch, completed test formula, and tensor conventions are taken from the supplied \texttt{X-E.tex}. The background gauge operator, exact commutator, source prescription, and direct tree identity are taken from the supplied \texttt{X-F.tex}. Earlier series results are premises where explicitly identified, not independently verified here. \paragraph{New proofs.} The normalized auxiliary--ghost Green matrix and free causal differential; the background ghost operator; the evaluated switched-matter coefficient; the free-solution reciprocity obstruction using a compact pure-gauge ramp background; the qualified contraction inventory; the corrected completed support bound; and the scalar nonlocal product-criterion audit. \paragraph{Quantum scope.} No interacting quantum BRST differential, complete master equation, full nonlocal loop value, anomaly cancellation, state, positivity theorem, physical representation, sharp-width limit, or adiabatic limit is constructed. \paragraph{VII-C.} The word $B_f\star W_g\star B_{f'}$ and its extension hypotheses are not constructed. No residue identification or subtraction is made. The present scalar exceptional-set audit does not apply VII-C's theorem. \paragraph{Einstein jet.} The earlier switched-source conservation condition is not changed. No claim is made that the generically empty coupled Einstein jet has been filled. \paragraph{Verification status.} The source is standalone and requires no external figures, bibliography database, or executable code. The source has been organized to avoid unresolved local labels and external cross-reference commands. No successful \texttt{pdflatex} execution, proof-assistant check, numerical verification, independent referee review, or journal acceptance is asserted. %===================================================================== \begin{thebibliography}{99} \addcontentsline{toc}{section}{References} \bibitem{XE} N.~Cheung. \newblock \emph{Landmark X-E: A specified gravitational action for the ramp relational scalar, and the first graviton tree}. \newblock Supplied manuscript \texttt{X-E.tex}. \bibitem{XF} N.~Cheung. \newblock \emph{Landmark X-F: The second-order quantum metric as the retarded Moller response, and the closure of the $\kappa^2$ background identity}. \newblock Supplied manuscript \texttt{X-F.tex}. \bibitem{BF} R.~Brunetti and K.~Fredenhagen. \newblock Microlocal analysis and interacting quantum field theories: renormalization on physical backgrounds. \newblock \emph{Communications in Mathematical Physics} 208 (2000), 623--661. \newblock arXiv:math-ph/9903028. \newblock Used for the finite-scaling-degree extension theorem, not as a construction of the nonlocal mixed retarded product. \bibitem{FR} K.~Fredenhagen and K.~Rejzner. \newblock \emph{Perturbative algebraic quantum field theory}. \newblock arXiv:1208.1428v2, 2013. \newblock Background for polynomial Hadamard products, retarded products, and ordering equivalences. \bibitem{HW} S.~Hollands and R.~M.~Wald. \newblock Conservation of the stress tensor in perturbative interacting quantum field theory in curved spacetimes. \newblock \emph{Reviews in Mathematical Physics} 17 (2005), 227--312. \newblock arXiv:gr-qc/0404074v2. \newblock Used to distinguish perturbative agreement from the preceding covariance and normalization conditions. Its scalar-field theorem is not imported as a graded gravitational completion. \bibitem{TTZ} M.~Taslimi Tehrani and J.~Zahn. \newblock Background independence in gauge theories. \newblock \emph{Annales Henri Poincare} 21 (2020), 1135. \newblock arXiv:1804.07640v3. \newblock Comparison for quantum background transport and possible obstructions. No anomaly-removal theorem for the present switched model is inferred. \bibitem{BFR} R.~Brunetti, K.~Fredenhagen, and K.~Rejzner. \newblock Quantum gravity from the point of view of locally covariant quantum field theory. \newblock \emph{Communications in Mathematical Physics} 345 (2016), 741--779. \newblock arXiv:1306.1058. \newblock Comparison framework for locally covariant perturbative gravity and renormalized BV constructions. It does not identify source switching with density switching. \bibitem{Hor} L.~H\"ormander. \newblock \emph{The Analysis of Linear Partial Differential Operators I}, second edition. \newblock Springer, 1990. \newblock Used for the wavefront product and pushforward criteria. A failed sufficient product criterion is not used as a nonexistence theorem. \end{thebibliography} \end{document}