% !TeX program = pdflatex %===================================================================== % XI-F.tex % % Standalone source for pdfLaTeX. % Build twice: % pdflatex -interaction=nonstopmode -halt-on-error XI-F.tex % pdflatex -interaction=nonstopmode -halt-on-error XI-F.tex % % Scope: % Two gates for the classical cubic Einstein response named in XI-E: % F1 admission of C_3 into XI-E's polynomial Hadamard algebra; % F2 the gauge-fixing companion of the cubic variation; % F3 the plateau sum of the second-order variations; % F4 what the sum is not. % % Corrections to the production sheet (recorded in the text): % 1. The admission obstruction is not finite rank. It is a timelike % conormal covector at the initial surface, produced by zero % Cauchy data for the second metric coefficient together with an % unswitched, unramped cubic source. F1 is proved obstructed for % the frozen representative; the on-shell class is left open. % 2. The gauge-fixing companion is the retarded response to 2D^*Ds. % It equals K zeta_gf with zeta_gf=-E_0^R Ds, pure gauge with zero % data, and pairs to zero on the plateau. It is not -I. % 3. XI-E's by-hand co-transformation and the covariant part of the % cubic response are one mechanism. The sheet's sum % delta_split + delta_0 C_3 + delta_gf = 0 + I + 0 counts it twice. % The corrected ledger is -I + I + 0 + 0 + 0 = 0. % 4. The quadratic flow term (1/2)L_xi K xi enters the cubic response % through J_3[K xi] = -A((1/2)L_xi K xi); it is not added by hand. % 5. The second-order parameter selected by gauge fixing is not in % g_U; the resulting remainder is a pure switching term, with no % ramp contribution, vanishing when J^-(supp f) cap J^+(supp xi) % lies in the plateau. % 6. The identity is stated with the Einstein-shell term % P E_0^R L^dens_xi(A gamma) retained and named; it vanishes when % A gamma = 0 on U, which includes the witness. % % Compile hygiene: no redefinition of predefined macros (XI-B's % \newcommand{\AA} would not compile); no theorem-header hacks. % % No interacting differential, quantum retarded product of the cubic % source, mixed loop, state, sharp-width limit, or Einstein-jet repair. % % No successful compilation, mechanized verification, independent % referee review, or journal acceptance is asserted. %===================================================================== \documentclass[11pt,a4paper]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage[margin=27mm]{geometry} \usepackage{microtype} \usepackage{amsmath,amssymb,amsthm,mathtools,mathrsfs} \usepackage{array,booktabs,tabularx} \usepackage{enumitem} \usepackage[hidelinks]{hyperref} \allowdisplaybreaks \setlength{\emergencystretch}{3em} \setlist{leftmargin=*,itemsep=3pt,topsep=5pt} \numberwithin{equation}{section} \setcounter{tocdepth}{1} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newcommand{\R}{\mathbb R} \newcommand{\C}{\mathbb C} \newcommand{\dd}{\,\mathrm d} \newcommand{\eps}{\varepsilon} \newcommand{\MT}{M_T} \newcommand{\MTo}{M_T^\circ} \newcommand{\sig}{\sigma_\eps} \newcommand{\PP}{\mathcal P} \newcommand{\RR}{\mathcal R} \newcommand{\Lie}{\mathcal L} \newcommand{\Ldens}{\mathcal L^{\mathrm{dens}}_\xi} \newcommand{\ga}{\mathfrak g_U} \newcommand{\Gp}{\mathcal G_U} \newcommand{\Afree}{\mathscr A_0} \newcommand{\Ealg}{\mathscr E_H} \newcommand{\Fpol}{\mathscr F_{\mathrm{mc,pol}}} \newcommand{\Csol}{\mathscr C_{\mathrm{sol}}} \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{\Ggr}{G_{\mathrm{gr}}^{\mathrm R}} \newcommand{\Pfree}{P_m} \newcommand{\sfree}{s_0} \newcommand{\Xone}{\mathsf X_1} \newcommand{\Xtwo}{\mathsf X_2} \newcommand{\Bcal}{\mathcal B} \newcommand{\Tcal}{\mathcal T} \newcommand{\Aone}{\mathcal A_1} \newcommand{\Atwo}{\mathcal A_2} \newcommand{\Atwothree}{\mathcal A_2^{(3)}} \newcommand{\Cthree}{\mathcal C_3} \newcommand{\Hcal}{\mathcal H} \newcommand{\Jcal}{\mathcal J} \newcommand{\Vone}{\mathsf V_{1,g}} \newcommand{\Vtwo}{\mathsf V_{2,g}} \newcommand{\Dfset}{\mathcal D_f} \newcommand{\Vp}{\widetilde V_+} \newcommand{\Vm}{\widetilde V_-} \newcommand{\sxi}{s_\xi} \newcommand{\zgf}{\zeta_{\mathrm{gf}}} \newcommand{\dgf}{\delta_{\mathrm{gf}}} \newcommand{\pair}[2]{\left\langle #1,#2\right\rangle} \DeclareMathOperator{\supp}{supp} \DeclareMathOperator{\WF}{WF} \DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\Int}{int} \DeclareMathOperator{\id}{id} \DeclareMathOperator{\diag}{diag} \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 XI-F}\\[0.4em] Admission of the cubic Einstein response,\\ and the plateau sum of the second-order variations} \author{Ngo Cheung} \date{} \usepackage{tikz} \begin{document} \maketitle \begin{abstract} Landmark XI-E named, and excluded from its release, the classical cubic Einstein response $u_3=\PP\ER(2\Hcal_2[\gamma])$ and its scalar-completed insertion $\Cthree=\Aone[u_3;\phi]$. We decide two gates for that object at fixed regulator width, in the normalization $\Ggr=-\PP\ER$. \emph{Admission.} The second metric coefficient carries the frozen zero Cauchy data on $\Sigma$, and the Einstein--Hilbert density carries neither the switch $g$ nor the ramp $\sig$. The slab kernel of $\Cthree$ therefore has the form $\theta(y^0)\beta(y)\,j_3(y;\cdot,\cdot)$: a sharp onset at the initial surface in a metric slot. For every frozen switch and every mass we exhibit admissible observation data for which this kernel has a timelike conormal covector $\pm\dd t$ in its wavefront set, so it leaves the polynomial microcausal class of XI-E. The admission map of XI-E does not apply to $\Cthree$. The failure is not the finite-rank mismatch and not a loop; it is the sharp-Cauchy singularity of VII-B, now in the gravitational slot, which the ramp regulates only in the coordinate operator. Whether the free-shell restriction of $\Cthree$ has some other microcausal representative is not decided. \emph{Plateau sum.} From XI-B's gauge fermion, the order-$\kappa$ variation of the de Donder term under the single-field law is $2D^*Ds$, with divergence $-\Box Ds$; its retarded response through the frozen inverse is the pure gauge $-2\RR\ER Ds=K\zgf$, with $\zgf=-\ER Ds$ the de Donder second-order parameter. Hence $u_3[\gamma+K\xi]-u_3[\gamma]=\sxi+K\zgf[\sxi]+\PP\ER\Ldens(A\gamma)$, $\sxi=\Lie_\xi\gamma+\tfrac12\Lie_\xi K\xi$: the complete second-order metric law of X-A, with $\zeta$ selected by gauge fixing. The cubic-completed second coefficient then shifts, for a free scalar and $A\gamma=0$ on $U$, by $\Aone[K\zgf;\phi_0]$: a pure switching term carrying $1-g$ or $\partial g$, with no ramp term, vanishing whenever $J^-(\supp f)\cap J^+(\supp\xi)$ lies in the plateau. On the XI-E witness the ledger reads $-I$ (held fixed) $+I$ (covariant cubic) $+0$ (gauge-fixing response) $+0+0$ (quadratic terms) $=0$. The production sheet's literal sum $\delta_{\mathrm{split}}+\delta_0\Cthree+\dgf =0+I+0$ double counts: XI-E's by-hand co-transformation and the covariant part of the cubic response are one mechanism. The corrected plateau sum vanishes; $\dgf$ is not $-I$, it is $0$. Neither result constructs an interacting differential, a quantum retarded product of the cubic source, a loop value, a sharp limit, or an Einstein-jet repair. \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) {Frozen cubic representative}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Admission gate}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {A repaired class}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {Gate 1 is proved obstructed for that representative, not for every later insertion.}; \draw[->] (a) -- (b); \draw[->] (b) -- (c); \draw[->] (b) -- (s); \end{tikzpicture} \caption{The frozen representative is obstructed. 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-XI-F} \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) {Frozen cubic representative}; \node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {A repaired class}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {Gate 1 is proved obstructed for that representative, not for every later insertion.}; \draw[->] (a) -- (s); \draw[->] (b) -- (s); \end{tikzpicture} \caption{The frozen representative is obstructed. 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-XI-F} \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) {Frozen cubic representative}; \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Admission gate}; \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {A repaired class}; \node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {Gate 1 is proved obstructed for that representative, not for every later insertion.}; \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-XI-F} \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) {Frozen cubic representative}; \node[align=center,font=\small] (m) at (5.2,0) {not the same object}; \node[draw,rounded corners,align=center,text width=4.2cm,font=\small,inner sep=3pt] (b) at (10.4,0) {A repaired class}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Admission gate. Gate 1 is proved obstructed for that representative, not for every later insertion.}; \draw (a) -- (m); \draw (m) -- (b); \draw[->] (m) -- (s); \end{tikzpicture} \caption{Frozen cubic representative is not A repaired class. 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-XI-F} \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: Frozen cubic representative}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Admission gate}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: A repaired class}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {Gate 1 is proved obstructed for that representative, not for every later insertion.}; \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-XI-F} \end{figure} Figure~\ref{fig:split-XI-F} records one view of the same separation. Figure~\ref{fig:chain-XI-F} records one view of the same separation. Figure~\ref{fig:pair-XI-F} records one view of the same separation. Figure~\ref{fig:gate-XI-F} records one view of the same separation. The drawings add no hypothesis. Figure~\ref{fig:scope-XI-F} records that separation. The drawing adds no hypothesis. \scopebox{Release: two gates, three endings, worst ending governs}{% \begin{description}[leftmargin=1.2em,itemsep=2pt] \item[Gate 1 (F1, admission): \textbf{Proved obstructed}.] The frozen representative of $\Cthree$ has a timelike conormal covector at $\Sigma$ in a metric slot and is not in $\Fpol$. XI-E's admission map does not apply. The on-shell class question is open. \item[Gate 2 (F2--F3, plateau sum): \textbf{Constructed}.] The de Donder companion is computed from XI-B's fermion with the coefficient $-1$; the finite transformation of $u_3$ is the X-A second-order law with $\zeta=\zgf$; the plateau sum is zero when $A\gamma=0$ on $U$, with the off-plateau remainder a switching term supported in $(\supp\dd g\cup\supp((1-g)f))\cap J^+(\supp\xi)$. \end{description} Overall ending: \textbf{Proved obstructed} (Gate 1). No quantum version of the plateau sum exists in $\Ealg$, because $\Cthree$ is not admitted. No $\widehat s_{\mathrm{int}}$, loop, $\eps\to0$ limit, or repair of $\Jcal_g(\phi_m)$.} \tableofcontents %===================================================================== \section{Scope, the two gates, and the corrections} \label{sec:scope} %===================================================================== XI-E closed three routes and left one object outside all of them: the cubic Einstein response $u_3$ and its insertion $\Cthree$ \cite{XIE}. XI-E computed the classical variation of $\Cthree$ on a witness and found a second nonzero number $I$, distinct from the $I$ that its split cancellation had already removed. It did not admit $\Cthree$ to its Hadamard algebra, and it did not compute the gauge-fixing companion of the cubic variation. Those are the two gates here. \begin{enumerate}[label=\textbf{F\arabic*.}] \item Admit $\Cthree$ into the polynomial on-shell Hadamard algebra $\Ealg$ of XI-E, or exhibit a wavefront direction that leaves its polynomial microcausal class. \item Compute the order-$\kappa$ variation of the de Donder term under the single-field transformation law, from XI-B's gauge fermion, and its retarded response through $\Ggr=-\PP\ER$. \item Assemble the second-order variations on the XI-E witness and decide whether their sum vanishes on the plateau, with the off-plateau support stated. \item Say what the result is not. \end{enumerate} The three endings of the series apply to each gate: \emph{Constructed}, \emph{Proved obstructed}, \emph{Unresolved}. The worst ending governs the release. \subsection{Six corrections to the production sheet} The sheet anticipated an admission proof by composition of retarded kernels, a gauge-fixing companion equal to $-I$, and a three-term sum $\delta_{\mathrm{split}}+\delta_0\Cthree+\dgf$. Each of these must be corrected before the gates can be decided. \begin{enumerate} \item The source $J_3=2\Hcal_2[\gamma]$ is local, unswitched and unramped, and the second coefficient has zero data on $\Sigma$. Its retarded response is $\ER J_3$ with the zero-data inverse, which truncates the source at $t=0$. The resulting kernel has a sharp onset at $\Sigma$ in a metric slot. That onset, not the finite-rank mismatch, is the obstruction (Section~\ref{sec:F1}). \item The gauge-fixing companion is the retarded response to the order-$\kappa$ de Donder variation $2D^*D\sxi$. It equals $K\zgf[\sxi]$ with $\zgf=-\ER D\sxi$: pure gauge, zero data, supported in $J^+(\supp\xi)$. Its $\Aone$-pairing on the plateau is zero, not $-I$ (Section~\ref{sec:F2}). \item XI-E's cancellation theorem restored $\gamma\mapsto\gamma+\kappa\Lie_\xi\gamma$ by hand in a parametrization with an external second coefficient $k$. Once $k=u_3[\gamma]$ is generated by the response, the covariant part of $u_3[\gamma+K\xi]-u_3[\gamma]$ \emph{is} that shift. Adding $\delta_{\mathrm{split}}=0$, which already contains the by-hand $+I$, to $\delta_0\Cthree=I$ counts the same term twice (Section~\ref{sec:F3}). \item The quadratic flow term $\tfrac12\Lie_\xi K\xi$ of the single-field law is supplied by the response through the flatness identity $J_3[K\xi]=-A(\tfrac12\Lie_\xi K\xi)$. It is not added separately. \item The second-order parameter selected by gauge fixing is not in $\ga$. Its support is $J^+(\supp\xi)\cap\MT$. The resulting remainder is a switching term of X-A's and VI's type, with no ramp layer, and it vanishes under an explicit causal plateau condition. \item The identity is stated with the Einstein-shell term $\PP\ER\Ldens(A\gamma)$ retained. On the gauge-fixed free shell $A\gamma=-2D^*b$; the term vanishes when $D^*b=0$ on $U$, which holds for the witness and for every linearized Einstein solution. XI-A's distinction between the gauge-fixed and the Einstein shell is not erased. \end{enumerate} \subsection{Non-imports} X-E's normalization $\Delta_{\mathrm R}=-4\PP\ER$ with dressing $-2$, and X-F's three-tree cancellation in that normalization, are not premises. Their mechanism, that a held-fixed split can be a truncation, is the same mechanism that appears in Section~\ref{sec:F3}; no factor, flat-gauge residual, or sum-zero lemma is copied. VII-C's word, reflected residue, and hypotheses (E1)--(E4) stay out. The empty fixed-input Einstein jet $\Jcal_g(\phi_m)=\varnothing$ remains XI-A's result \cite{XIA}. %===================================================================== \section{Copied sheet} \label{sec:sheet} %===================================================================== \subsection{Conventions} \[ \MT=[0,T]\times\R^3,\qquad \MTo=(0,T)\times\R^3,\qquad \Sigma=\{t=0\}, \] \[ \eta=\diag(1,-1,-1,-1),\qquad \Box=\partial_t^2-\Delta,\qquad \Pfree=\Box+m^2,\qquad m\geq0. \] Indices are moved with $\eta$; parentheses include the factor $\tfrac12$. Classical fields are smooth on a neighbourhood of the closed slab; no spatial decay is imposed. The ramp $\sig(t)=\chi(t/\eps)$, $0<\eps0$ on $\{t>0\}$, and $\chi(1-s)=1-\chi(s)$. The switch $g\in C_c^\infty(\MTo;[0,1])$ has $P_g=\Int\{g=1\}\neq\varnothing$. The plateau $U$ is open, relatively compact, with $\overline U\subset P_g\cap\{\eps0\}$, $\ER\Box s=s$ and $\ER\partial_\mu s=\partial_\mu\ER s$. \begin{lemma}[Inverse of $A$ on compact plateau sources] \label{lem:GA} For $s$ smooth, compactly supported in $U$, \begin{equation} \Ggr As=s-2\RR\ER Ds . \label{eq:GA} \end{equation} \end{lemma} \begin{proof} $\Ggr As=(-\PP\ER)(-\Box\PP s)+(-\PP\ER)(-2D^*Ds) =\PP\PP s+2\PP D^*\ER Ds=s-2\RR\ER Ds$, using \eqref{eq:algebra}, $\ER\Box s=s$, and $\ER D^*=D^*\ER$ on $Ds$, which has zero data. \end{proof} \subsection{Coordinate operator, first and second coefficients} \[ a_1^{\mu\nu}[r]=\tfrac12\eta^{\mu\nu}\tr_\eta r-r^{\mu\nu},\qquad b_1[r]=\tfrac12\tr_\eta r,\qquad C^\mu[r]=-\partial_\alpha a_1^{\alpha\mu}[r]=D^\mu r, \] \begin{equation} \Xone[r]=-\ER\partial_\alpha\bigl(\sig a_1^{\alpha\cdot}[r]\bigr) =\ER\bigl(\sig Dr-\sig'a_1^{0\cdot}[r]\bigr). \label{eq:X1} \end{equation} \[ \Vone[r]u=\partial_\mu\bigl(ga_1^{\mu\nu}[r]\partial_\nu u\bigr)+gm^2b_1[r]u, \qquad \Aone[r;\phi]=\int_{\MT}f\bigl(-\ERm\Vone[r]\phi-\Xone[r]\cdot\partial\phi\bigr)\dd^4x . \] $\Atwo^{\mathrm{sc}}[H,k;\phi]$ is X-A's complete on-shell second coefficient, written out in XI-D: with $\rho_1=-\ERm\Vone[H]\phi$, $\rho_2=\ERm\Vone[H]\ERm\Vone[H]\phi-\ERm\Vtwo[H,k]\phi$, \[ \Atwo^{\mathrm{sc}}[H,k;\phi] =\int f\Bigl[\rho_2-\Xone[H]\cdot\partial\rho_1 +(\Xone[H]\cdot\partial\Xone[H]-\Xtwo[H,k])\cdot\partial\phi +\tfrac12\Xone^\mu[H]\Xone^\nu[H]\partial_\mu\partial_\nu\phi\Bigr]\dd^4x . \] Two copied facts about it: \begin{align} \Atwo^{\mathrm{sc}}[H,k+r;\phi]-\Atwo^{\mathrm{sc}}[H,k;\phi] &=\Aone[r;\phi] \qquad\text{(XI-D, affine lemma)}, \label{eq:affine}\\ \Aone[h';\phi_0]=\Aone[h;\phi_0],\quad \Atwo^{\mathrm{sc}}[h',k';\phi_0]&=\Atwo^{\mathrm{sc}}[h,k;\phi_0] \qquad\text{(X-A, Theorem (v))}, \label{eq:XAv} \end{align} for every free $\phi_0$, every smooth $(h,k)$, and \[ h'=h+K\xi,\qquad k'=k+\Lie_\xi h+K\zeta+\tfrac12\Lie_\xi K\xi,\qquad \xi,\zeta\in\ga . \] The free input $\phi_0$ is not transformed in \eqref{eq:XAv}; it is held fixed, exactly as XI-E held the scalar fixed in its cubic variation. \subsection{Free multiplet, algebra, and the inherited symbols} The independent multiplet is $(\gamma,\phi,c,\bar c,b)$ with the free rule $\sfree\gamma=\RR c$, $\sfree\phi=\sfree c=\sfree b=0$, $\sfree\bar c=ib$. The gauge fermion of XI-B is \begin{equation} \Psi=-i\int_{\MT}\bar c_\mu\bigl(2D^\mu\gamma-b^\mu\bigr)\dd^4x, \qquad \sfree\Psi=\int_{\MT}\bigl(2b_\mu D^\mu\gamma-b_\mu b^\mu+i\bar c_\mu\Box c^\mu\bigr)\dd^4x, \label{eq:fermion} \end{equation} and elimination of $b=D\gamma$ gives $S_{\mathrm{gf}}=\int D_\nu\gamma D^\nu\gamma$ with Euler derivative $E_{\mathrm{gf}}[\gamma]=2D^*D\gamma$ \cite{XIB,XIA}. The free shell $\Csol$ is $-\Box\PP\gamma=0$, $b=D\gamma$, $\Pfree\phi=0$, $\Box c=\Box\bar c=0$. XI-E's Route~1 constructed the polynomial on-shell Hadamard algebra $\Ealg=(\Fpol^{\mathrm{sol}}[[\hbar,\kappa]],\star_H)$ with block data $H_{\gamma\gamma}=-\PP h_0$, $H_{\gamma b}=D_yH_{\gamma\gamma}$, $H_{\phi\phi}=-h_m$, $H_{c\bar c}=-ih_0$, graded antisymmetric part equal to XI-B's causal matrix, and no state selected. Its domain is: \begin{definition}[XI-E's polynomial microcausal domain, copied] \label{def:domain} $\Fpol$ consists of finite graded polynomials $F(\Phi)=\sum_{n}\pair{u_n}{\Phi^{\otimes n}}$ on ambient configurations on $M=\R^4$, with compact, graded symmetric distributional kernels $u_n$ on $M^n$ satisfying \begin{equation} \WF(u_n)\cap\bigl(\Vp^{\,n}\cup\Vm^{\,n}\bigr)=\varnothing, \label{eq:mc} \end{equation} where $\Vp,\Vm$ are the closed future and past causal covector cones with zero included, and the total zero covector is never a wavefront point. Field indices are finite and do not change the criterion. Admission is kernel inclusion in $\Fpol$ followed by restriction to $\Csol$: $Q_H(F)=[F|_{\Csol}]\in\Ealg$. Slab functionals are viewed on $M$ through their slab restriction; for the inherited symbols the ramp is flat at $t=0$, so no boundary conormal appears. \end{definition} The inherited symbols $\Bcal_f=-\int f\Xone[\gamma]\cdot\partial\phi$ and $\Tcal(f,g)[\phi]=\int f(\Xone[r_g[\phi]])\cdot\partial\phi$, $r_g=-\tfrac12\PP\ER(gT_0[\phi])$, are admitted in $\Ealg$; the physical one-graviton response of $\Bcal_f$ is $-\Tcal$ at coupling degree $\kappa^2$ \cite{XIC}. Their admission is not reopened here. \subsection{The cubic response and the witness} XI-A's unswitched action $S_{\mathrm{EH}}=-\frac2{\kappa^2}\int\sqrt{-q}R[q]$ with $q=\eta+\kappa\gamma$ gives $\sqrt{-q}\,G^{\mu\nu}(q)=\kappa\Hcal_1+\kappa^2\Hcal_2+\kappa^3\Hcal_3+O(\kappa^4)$, $\Hcal_1=G_{\mathrm{lin}}(\gamma)=\tfrac12A\gamma$, with the finite formulas of XI-A's appendix. The cubic vertex has metric Euler derivative $J_3[\gamma]=2\Hcal_2[\gamma]$, a quadratic local differential polynomial. XI-E defined \begin{equation} u_3[\gamma]=-\Ggr J_3[\gamma]=\PP\ER\bigl(2\Hcal_2[\gamma]\bigr), \qquad \Cthree(f,g)[\gamma,\phi]=\Aone[u_3[\gamma];\phi], \label{eq:C3} \end{equation} of field degree $(2,1)$, and proved, for $\delta_0\gamma=K\xi$, $\xi\in\ga$, \begin{equation} \delta_0J_3[\gamma]=\Ldens(A\gamma)-A(\Lie_\xi\gamma), \qquad \delta_0u_3=\Lie_\xi\gamma-2\RR\ER D\Lie_\xi\gamma\quad\text{if }A\gamma=0, \label{eq:XIEvar} \end{equation} where $(\Ldens v)^{\mu\nu}=\xi^\rho\partial_\rho v^{\mu\nu}-v^{\rho\nu}\partial_\rho\xi^\mu -v^{\mu\rho}\partial_\rho\xi^\nu+(\partial_\rho\xi^\rho)v^{\mu\nu}$. The witness (XI-C, XI-D, XI-E): $m=0$, \begin{equation} \gamma=t\eta,\qquad \phi=t^2+\tfrac13|\mathbf x|^2,\qquad \xi=a(t)\partial_t,\quad a\geq0,\ a\not\equiv0,\ \supp a\subset(\eps,T), \label{eq:witness} \end{equation} in a causal cylinder on which $g=1$, with a compact spatial cutoff on $\xi$ whose derivatives lie outside the observation dependence region, the plateau containing $\supp\xi$, and $f\geq0$, $f\not\equiv0$ observed after $\supp a$. Then $\Box\phi=0$, $-\Box\PP\gamma=0$, $b=D\gamma=(-2,0,0,0)$ is constant, $A\gamma=0$, and \begin{equation} I:=\Aone[\Lie_\xi\gamma;\phi] =4\int f(t,\mathbf x)\int_0^t(t-s)a(s)\dd s\dd^4x>0, \qquad \delta_0\Cthree=I . \label{eq:I} \end{equation} XI-E also proved the split cancellation $-\Aone[\Lie_\xi\gamma]+\Aone[\Lie_\xi\gamma]=0$ in the $(h,\gamma,k)$ parametrization with an external $k$, and recorded that the two $I$'s are different $I$'s. The two affine numbers, $S_\eps(t)$ for the bare insertion and $\eps/2$ for the complete word, are not touched here. \subsection{Normalization refusal} The coefficient of $\PP\ER$ is $-1$ throughout. Neither $-2\PP\ER$ nor $-4\PP\ER$ is used, and no graviton two-point function is selected. %===================================================================== \section{F1: the cubic kernel at the initial surface} \label{sec:F1} %===================================================================== \subsection{Kernel representation of the first coefficient} \begin{lemma}[Smooth kernel of $\Aone$] \label{lem:alpha} Fix $f$, $g$, and a smooth $\phi$ on the slab. There is a smooth symmetric contravariant tensor $\alpha=\alpha[f,g,\phi]$ on $\MT$ with $\supp\alpha\subset\Dfset$ such that \begin{equation} \Aone[r;\phi]=\int_{\MT}\alpha^{\rho\sigma}r_{\rho\sigma}\dd^4x \qquad\text{for every smooth }r . \label{eq:alpha} \end{equation} Explicitly, with $b=\EA(f\dd\phi)$, $b'=\EAm f$, and $(C^*b)^{\rho\sigma}=-\partial^{(\rho}b^{\sigma)}+\tfrac12\eta^{\rho\sigma}\partial_\mu b^\mu$, \begin{equation} \alpha^{\rho\sigma} =-\sig\,(C^*b)^{\rho\sigma} +g\Bigl[\tfrac12\eta^{\rho\sigma}\bigl(\partial^\mu b'\partial_\mu\phi-m^2\phi\,b'\bigr) -\partial^{(\rho}b'\,\partial^{\sigma)}\phi\Bigr]. \label{eq:alphaexplicit} \end{equation} \end{lemma} \begin{proof} The coordinate term $-\int f\Xone[r]\cdot\partial\phi$ is $-\Jc(f\dd\phi)[r]$ in X-A's notation, whose kernel is $\sig C^*(\EA(f\dd\phi))$ by X-A's kernel lemma; this gives the first term. For the scalar term, transposition gives $\int f\ERm\Vone[r]\phi=\int b'\Vone[r]\phi$, and integrating by parts in the divergence (legitimate because $g$ has compact interior support), \[ -\int b'\Vone[r]\phi =\int g\,a_1^{\mu\nu}[r]\partial_\mu b'\partial_\nu\phi-\int gm^2b_1[r]\phi\,b' . \] Inserting $a_1^{\mu\nu}[r]=\tfrac12\eta^{\mu\nu}\tr r-r^{\mu\nu}$, $b_1[r]=\tfrac12\tr r$, and symmetrizing gives the second term of \eqref{eq:alphaexplicit}. Supports: $b,b'$ are supported in $\Dfset$, and $\Dfset$ is past-closed in $\MT$. \end{proof} \begin{proposition}[Representation of $\Cthree$ and the onset at $\Sigma$] \label{prop:beta} Let $\beta:=\EA(\PP\alpha)$, a smooth symmetric tensor on $\MT$ with $\supp\beta\subset\Dfset$. Then \begin{equation} \Cthree(f,g)[\gamma,\phi]=\int_{[0,T]\times\R^3}\beta_{\mu\nu}(y)\,J_3^{\mu\nu}[\gamma](y)\dd^4y . \label{eq:C3beta} \end{equation} Viewed as a functional of ambient configurations through slab restriction, the degree-$(2,1)$ kernel $u$ of $\Cthree$ is supported in $\{y_1=y_2=y\in\Dfset\cap\{y^0\geq0\}\}\times\Dfset$ and has the form \begin{equation} u(y_1,y_2,x)=\int_{y^0\geq0}\widehat\beta_{\mu\nu}(y,x)\,j_3^{\mu\nu}(y;y_1,y_2)\dd^4y, \label{eq:ukernel} \end{equation} where $j_3$ is the constant-coefficient bilinear local kernel of $J_3$ (a finite sum of products of derivatives of $\delta(y-y_1)$ and $\delta(y-y_2)$) and $\widehat\beta$ is the $\phi$-slot kernel of $\beta$. The factor $\theta(y^0)$ is forced: the frozen Cauchy prescription gives the second coefficient zero data on $\Sigma$, so $\ER$ truncates every source at $t=0$, and the Einstein--Hilbert density carries no $g$ and no $\sig$ that could make $J_3[\gamma]$ vanish near $\Sigma$. \end{proposition} \begin{proof} $\Cthree=\int\alpha\cdot u_3=\int\alpha\cdot\PP\ER J_3$. Trace reversal is self-adjoint under the $\eta$-contraction, $\pair{\alpha}{\PP r}=\pair{\alpha}{r}-\tfrac12\tr\alpha\tr r=\pair{\PP\alpha}{r}$, and transposition (applicable because $\supp\PP\alpha\subset\Dfset$ vanishes near $t=T$ and $\ER J_3$ has zero data) gives \eqref{eq:C3beta}. The $y$-integral runs over the slab only, and $J_3[\gamma](y)$ depends on the jet of $\gamma$ at $y$; this is \eqref{eq:ukernel}. Compactness: $y\in\Dfset$, and the $\phi$-slot of $\alpha$ is supported in $\Dfset$ by Lemma~\ref{lem:alpha}. \end{proof} The onset $\theta(y^0)$ is the sharp step $\theta(t)$ that X-A replaced by $\sig$ in the coordinate operator (X-A, kernel lemma), and that VII-B found to carry a timelike conormal covector. Here it reappears in a metric slot, where the ramp does not act. Whether it is harmless depends on whether the coefficient $\beta$ vanishes on $\Sigma$. It does not. \subsection{Two tools} \begin{lemma}[Smearing] \label{lem:smear} Let $u$ be a compactly supported tensor-valued distribution on $M^3$ and $v,\psi$ smooth on $M$. Put $\ell:=\pair{u}{\,\cdot\otimes v\otimes\psi}$, a compactly supported distribution on $M$. Then \[ \WF(\ell)\subset\{(y_1,p):\ \exists\,y_2,y_3:\ (y_1,p;y_2,0;y_3,0)\in\WF(u)\}. \] In particular, if $u$ satisfies \eqref{eq:mc} with $n=3$, then $\WF(\ell)$ contains no nonzero covector in $\Vp\cup\Vm$. \end{lemma} \begin{proof} Smooth multiplication by $v\otimes\psi$ does not enlarge the wavefront set, and integration over $(y_2,y_3)$ of a compactly supported distribution is the pushforward estimate of \cite[Theorem~8.2.12]{Hormander}. If $(y_1,p;y_2,0;y_3,0)\in\WF(u)$ with $0\neq p\in\Vp$, then the triple lies in $\Vp^{\,3}$, contradicting \eqref{eq:mc}; similarly for $\Vm$. \end{proof} \begin{lemma}[Homogeneous reduction of the cubic source] \label{lem:ADM} Let $\gamma_{0\mu}=0$ and $\gamma_{ij}=\gamma_{ij}(t)$ on an open set $W\subset\MT$. Then on $W$, \begin{equation} J_3^{0i}[\gamma]=0,\qquad J_3^{00}[\gamma]=q(\dot\gamma,\dot\gamma),\qquad q(a,b):=\tfrac14\bigl(\tr a\,\tr b-a_{ij}b_{ij}\bigr), \label{eq:J300} \end{equation} with spatial traces and contractions taken with $\delta_{ij}$, and $\Hcal_1^{00}[\gamma]=0$. \end{lemma} \begin{proof} On $W$ the metric is $q=\dd t^2-h_{ij}(t)\dd x^i\dd x^j$ with $h=\delta-\kappa\gamma(t)$. The nonzero Christoffel symbols of XI-A's convention are $\Gamma^0_{ij}=\tfrac12\dot h_{ij}$ and $\Gamma^i_{0j}=K^i{}_j:=\tfrac12h^{ik}\dot h_{kj}$. Direct substitution into $R_{\mu\nu}=\partial_\rho\Gamma^\rho_{\mu\nu}-\partial_\nu\Gamma^\rho_{\mu\rho} +\Gamma^\rho_{\rho\lambda}\Gamma^\lambda_{\mu\nu}-\Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\rho}$ gives $R_{0i}=0$, $R_{00}=-\dot K-K^i{}_jK^j{}_i$ with $K=K^i{}_i$, and $h^{ij}R_{ij}=\dot K+K^2$, so that $R=R_{00}-h^{ij}R_{ij}$ and \[ G_{00}=R_{00}-\tfrac12R=\tfrac12\bigl(K^2-K^i{}_jK^j{}_i\bigr),\qquad G_{0i}=0 . \] Since $q^{00}=1$ and $q^{0i}=0$, $G^{00}=G_{00}$ and $G^{0i}=0$ for every $\kappa$; hence $J_3^{0i}=0$. Expanding $h^{ik}=\delta^{ik}+O(\kappa)$ and $\dot h=-\kappa\dot\gamma$ gives $K^i{}_j=-\tfrac12\kappa\dot\gamma_{ij}+O(\kappa^2)$, $K=-\tfrac12\kappa\tr\dot\gamma+O(\kappa^2)$, and $\sqrt{-q}=1+O(\kappa)$, so $\sqrt{-q}\,G^{00}=\tfrac18\kappa^2[(\tr\dot\gamma)^2-\dot\gamma_{ij}\dot\gamma_{ij}]+O(\kappa^3)$. Thus $\Hcal_1^{00}=0$ and $J_3^{00}=2\Hcal_2^{00}=q(\dot\gamma,\dot\gamma)$. The signature does not affect $G_{\mu\nu}$; the Friedmann check $G_{00}=3H^2$ for $h=a(t)^2\delta$ confirms the sign. \end{proof} \subsection{The obstruction} \begin{theorem}[The frozen cubic kernel leaves the polynomial microcausal class] \label{thm:F1} Fix $m\geq0$, the frozen $g$, and the frozen $\sig$. Let $t_g:=\min\{t:(t,\mathbf x)\in\supp g\}>0$. Choose $f\in C_c^\infty(\MTo)$, $f\geq0$, $f\not\equiv0$, with $\supp f\subset(0,t_g)\times\R^3$ and $\cos(mt)>0$ on $\supp f$. Let $\phi_m$ be XI-A's input, $\phi_m=\sin(mt)/m$ for $m>0$ and $\phi_m=t$ for $m=0$. Then the cubic kernel $u$ of $\Cthree(f,g)$ in \eqref{eq:ukernel} satisfies \[ \WF(u)\cap\bigl(\Vp^{\,3}\cup\Vm^{\,3}\bigr)\neq\varnothing . \] More precisely, with $\ell:=\pair{u}{\,\cdot\otimes v_2\otimes\phi_m}$ for the explicit $v_2$ below, there are points $y$ in an arbitrarily thin collar around $\Sigma$ with $(y,\pm\dd t)\in\WF(\ell)$. Consequently the ambient representative of $\Cthree(f,g)$ is not in $\Fpol$, and XI-E's admission map $Q_H$ does not apply to it. \end{theorem} \begin{proof} \emph{Step 1: the kernel for this $f$.} Since $\Dfset\subset\{t0$ for $t'>0$. Hence \begin{equation} \Lambda_f:=B(0)=\int_0^Tt'\,\sig(t')\,\partial_t\bar b(t')\dd t'<0 . \label{eq:Lambda} \end{equation} \emph{Step 2: homogeneous configurations and the reduced functional.} The spatial projection of $\Dfset$ lies in a ball $B_R$. Choose $\vartheta\in C_c^\infty(\R^3)$ with $\vartheta=1$ on $B_{R+1}$, and set, for $\lambda\in\R$, $\chi\in C_c^\infty((-\delta,\delta))$ with $\chi(0)=1$ and $0<\delta\eps\}$ outside $\supp g$ one has $\Box Y=1$, so $Y\not\equiv0$ and nonnegative bumps $f$ with $\Lambda_f\neq0$ exist above the ramp as well. The special choice $\supp f\subset\{t\eps\}$; it is not in $\ga$. \item $D\bigl(s+K\zgf[s]\bigr)=0$: $\zgf$ is the second-order parameter restoring the de Donder divergence of $s$. \item The divergence of the source is \eqref{eq:divgf}; the response satisfies $\Box\zgf=-Ds$. \end{enumerate} \end{proposition} \begin{proof} (a) $-\Ggr(2D^*Ds)=2\PP\ER D^*Ds=2\PP D^*\ER Ds=-2\RR\ER Ds=K(-\ER Ds)$, using \eqref{eq:algebra} and $\RR=\tfrac12K$. (b) Zero-data retarded solution of a source supported in $U\subset\{t>\eps\}$; its support is not compact in $U$. (c) $D(s+K\zgf)=Ds+\Box\zgf=Ds-Ds=0$. (d) is immediate. \end{proof} \subsection{Finite transformation of the cubic response} \begin{theorem}[The cubic response obeys the complete second-order law] \label{thm:u3law} For every smooth $\gamma$ and $\xi\in\ga$, \begin{equation} u_3[\gamma+K\xi]-u_3[\gamma] =\sxi+K\zgf[\sxi]+\PP\ER\Ldens(A\gamma), \qquad \sxi=\Lie_\xi\gamma+\tfrac12\Lie_\xi K\xi . \label{eq:u3law} \end{equation} If $A\gamma=0$ on $U$, the last term vanishes and \begin{equation} u_3[\gamma+K\xi]=u_3[\gamma]+\Lie_\xi\gamma+K\zgf[\sxi]+\tfrac12\Lie_\xi K\xi, \qquad D\,u_3[\gamma+K\xi]=D\,u_3[\gamma]. \label{eq:u3lawvac} \end{equation} \end{theorem} \begin{proof} By the proof of Proposition~\ref{prop:Etot}, $J_3[\gamma+K\xi]-J_3[\gamma]=\Ldens(A\gamma)-A\sxi$. Apply $-\Ggr=\PP\ER$ and Lemma~\ref{lem:GA} ($\sxi$ is compactly supported in $U$): $-\Ggr(-A\sxi)=\sxi-2\RR\ER D\sxi=\sxi+K\zgf[\sxi]$. If $A\gamma=0$ on $U\supset\supp\xi$, then $\Ldens(A\gamma)=0$. The divergence statement is Proposition~\ref{prop:gfprops}(c). \end{proof} Comparing with X-A's law $k'=k+\Lie_\xi h+K\zeta+\tfrac12\Lie_\xi K\xi$ at $(h,k)=(\gamma,u_3[\gamma])$: the response supplies the covariant term $\Lie_\xi\gamma$, the quadratic flow term through \eqref{eq:flat}, and a definite second-order parameter $\zeta=\zgf[\sxi]$. Gauge fixing does not break the second-order law; it selects $\zeta$. The selected $\zeta$ is not compactly supported in $U$, which is why a remainder survives in Section~\ref{sec:F3}. \subsection{The switching formula for a de Donder parameter} \begin{lemma}[Pure gauge with zero data: coordinate and switching terms] \label{lem:switch} Let $\zeta$ be smooth on $\MT$ with zero Cauchy data and $\supp\zeta\subset\{t\geq\eps\}$, and let $\phi_0$ be free. Then \begin{align} \Xone[K\zeta]&=\zeta, \label{eq:preflight}\\ \Aone[K\zeta;\phi_0] &=\int_{\MT}f\,(g-1)\,X_\zeta\phi_0\dd^4x -\int_{\MT}f\,\ERm\Bigl([\Pfree,g]X_\zeta\phi_0 +(\partial_\mu g)\,a_1^{\mu\nu}[K\zeta]\,\partial_\nu\phi_0\Bigr)\dd^4x, \label{eq:switch} \end{align} where $X_\zeta=\zeta^\mu\partial_\mu$ and $[\Pfree,g]u=(\Box g)u+2(\partial^\mu g)\partial_\mu u$. Every term carries a factor $g-1$ or $\partial g$ evaluated on $\supp\zeta\cap\Dfset$. In particular \begin{equation} \Aone[K\zeta;\phi_0]=0\qquad\text{whenever }\Dfset\cap\supp\zeta\subset P_g . \label{eq:switchvanish} \end{equation} \end{lemma} \begin{proof} \eqref{eq:preflight}: $a_1[K\zeta]$ is supported in $\supp\zeta$, where $\sig=1$ and $\sig'=0$, so $\Box\Xone[K\zeta]=-\partial_\alpha a_1^{\alpha\cdot}[K\zeta]=C[K\zeta]=\Box\zeta$, and both sides have zero data; this is X-A's preflight argument, which uses only the support and the data, not compactness. \eqref{eq:switch}: by X-A's lemma on the pure-gauge variation, $\Vone[K\zeta]\phi_0=g(D_\zeta\Pfree\phi_0-\Pfree X_\zeta\phi_0)+(\partial g)a_1[K\zeta]\partial\phi_0 =-g\Pfree X_\zeta\phi_0+(\partial g)a_1[K\zeta]\partial\phi_0$. Since $gX_\zeta\phi_0$ has zero data, $\ERm(g\Pfree X_\zeta\phi_0)=gX_\zeta\phi_0-\ERm[\Pfree,g]X_\zeta\phi_0$. Subtract the coordinate term $\int f\zeta\cdot\partial\phi_0$. The support statement follows by transposing $\ERm$ onto $f$: $\int f\ERm(\cdot)=\int(\EAm f)(\cdot)$ with $\EAm f$ supported in $\Dfset$. If $g=1$ on a neighbourhood of $\Dfset\cap\supp\zeta$, then $\partial g=0$ there as well, and all terms vanish. \end{proof} %===================================================================== \section{F3: the plateau sum} \label{sec:F3} %===================================================================== \subsection{The cubic-completed second coefficient} \begin{definition}[Cubic-completed coefficient] \label{def:A23} For a free $\phi_0$ and a smooth $\gamma$, \begin{equation} \Atwothree[\gamma;\phi_0] :=\Atwo^{\mathrm{sc}}\bigl[\gamma,u_3[\gamma];\phi_0\bigr]-\Tcal(f,g)[\phi_0]. \label{eq:A23} \end{equation} \end{definition} The second metric coefficient of the gauge-fixed equation with the scalar source is $u_3[\gamma]+r_g[\phi_0]$; by \eqref{eq:affine} the $r_g$ term contributes $\Aone[r_g[\phi_0];\phi_0]=-\Tcal-\Tcal_{\mathrm{sc}}$, with $\Tcal_{\mathrm{sc}}:=\int f\ERm\Vone[r_g]\phi_0$ the scalar-response companion of XI-C's coordinate tree. Both are independent of $\gamma$ and inert under every operation below; we follow the sheet and keep only $-\Tcal$. Nothing in this section changes if $-\Tcal_{\mathrm{sc}}$ is added. \begin{theorem}[Variation of the cubic-completed coefficient] \label{thm:F3} For every smooth $\gamma$, every free $\phi_0$, and every $\xi\in\ga$, \begin{align} \Aone[\gamma+K\xi;\phi_0]-\Aone[\gamma;\phi_0]&=0, \label{eq:F3first}\\ \Atwothree[\gamma+K\xi;\phi_0]-\Atwothree[\gamma;\phi_0] &=\Aone\bigl[K\zgf[\sxi];\phi_0\bigr] +\Aone\bigl[\PP\ER\Ldens(A\gamma);\phi_0\bigr]. \label{eq:F3} \end{align} \end{theorem} \begin{proof} \eqref{eq:F3first} is \eqref{eq:XAv} at first order. For \eqref{eq:F3}, $\Tcal$ cancels, and we split the difference in two steps. First, with $\gamma+K\xi$ fixed in the first slot, \eqref{eq:affine} and Theorem~\ref{thm:u3law} give \[ \Atwo^{\mathrm{sc}}[\gamma+K\xi,u_3[\gamma+K\xi]]-\Atwo^{\mathrm{sc}}[\gamma+K\xi,u_3[\gamma]] =\Aone[\sxi]+\Aone[K\zgf[\sxi]]+\Aone[\PP\ER\Ldens(A\gamma)]. \] Second, apply \eqref{eq:XAv} with $(h,k)=(\gamma,u_3[\gamma])$, $F$ the flow of $\kappa\xi$ ($\zeta=0$), for which $k'=u_3[\gamma]+\sxi$: $\Atwo^{\mathrm{sc}}[\gamma+K\xi,u_3[\gamma]+\sxi]=\Atwo^{\mathrm{sc}}[\gamma,u_3[\gamma]]$. By \eqref{eq:affine}, \[ \Atwo^{\mathrm{sc}}[\gamma+K\xi,u_3[\gamma]]-\Atwo^{\mathrm{sc}}[\gamma,u_3[\gamma]] =-\Aone[\sxi]. \] Adding the two steps gives \eqref{eq:F3}. ($u_3[\gamma]$ is smooth on the closed slab and extends smoothly across its boundary; X-A's identities use only values on $\MT$.) \end{proof} \begin{corollary}[Plateau sum] \label{cor:plateau} If $A\gamma=0$ on $U$, then \begin{equation} \Atwothree[\gamma+K\xi;\phi_0]-\Atwothree[\gamma;\phi_0] =\Aone\bigl[K\zgf[\sxi];\phi_0\bigr], \label{eq:plateausum} \end{equation} a pure switching term of the form \eqref{eq:switch}, and \begin{equation} \Atwothree[\gamma+K\xi;\phi_0]=\Atwothree[\gamma;\phi_0] \qquad\text{whenever}\qquad \Dfset\cap J^+(\supp\xi)\subset P_g . \label{eq:plateauzero} \end{equation} \end{corollary} \begin{proof} Theorem~\ref{thm:F3}, Proposition~\ref{prop:gfprops}(b), and Lemma~\ref{lem:switch}: $\supp\zgf[\sxi]\subset J^+(\supp\xi)\cap\MT\subset\{t\geq\eps\}$, with zero data. \end{proof} \subsection{The ledger on the witness} Take \eqref{eq:witness}. Then $A\gamma=0$ everywhere, so Corollary~\ref{cor:plateau} applies, and $g=1$ on the whole observation dependence cylinder, so $\Dfset\cap J^+(\supp\xi)\subset P_g$ and the sum is zero. The sheet asks that the contributions be printed before they are added. In the cylinder, with $r=\Lie_\xi\gamma$, $s_2=\tfrac12\Lie_\xi K\xi$, and $p:=aa''+2a'^2$, \[ r_{00}=a+2ta',\quad r_{ij}=-a\delta_{ij},\qquad (s_2)_{00}=p,\quad (s_2)_{0i}=(s_2)_{ij}=0, \] \[ D^0r=ta'',\qquad D^0s_2=\tfrac12p',\qquad D^ir=D^is_2=0 . \] \paragraph{L1, held-fixed cross term, linear in $\xi$.} XI-E recomputed $\Xone^0[r]=ta-2\int_0^ta$, $\Vone[r]\phi=-4ta'-2t^2a''$, $-\ER\Vone[r]\phi=2t^2a-4\int_0^tsa$, whence the density $4\int_0^t(t-s)a(s)\dd s$ and \[ \text{L1}=-\Aone[r;\phi]=-I . \] \paragraph{L2, held-fixed cross term, quadratic in $\xi$.} $a_1^{00}[s_2]=-\tfrac12p$, $a_1^{ij}[s_2]=-\tfrac12p\delta^{ij}$, $\Xone^0[s_2]=\ER(\tfrac12p')=\tfrac12P(t)$ with $P(t)=\int_0^tp$, $\Vone[s_2]\phi=\partial_0(-pt)+a_1^{ij}\partial_i\partial_j\phi=-2p-tp'$, and $-\ER\Vone[s_2]\phi=\int_0^t(t-s)(2p+sp')\dd s=tP(t)$. The density is $tP-\tfrac12P\cdot2t=0$: \[ \text{L2}=-\Aone[s_2;\phi]=0 . \] \paragraph{L3, covariant part of the cubic response, linear.} \[ \text{L3}=+\Aone[\Lie_\xi\gamma;\phi]=+I . \] \paragraph{L4, covariant part of the cubic response, quadratic.} \[ \text{L4}=+\Aone[\tfrac12\Lie_\xi K\xi;\phi]=0 . \] \paragraph{L5, gauge-fixing response.} $\zgf[\sxi]=z(t)\partial_t$ with $z=-\ER(ta''+\tfrac12p')=-(ta-2\int_0^ta+\tfrac12P)$, $z(0)=z'(0)=0$. For any such $\zeta=z\partial_t$ in the $g=1$ cylinder, $(K\zeta)_{00}=2z'$, $a_1^{00}[K\zeta]=-z'$, $a_1^{ij}[K\zeta]=-z'\delta^{ij}$, $\Vone[K\zeta]\phi=-4z'-2tz''$, $-\ER\Vone[K\zeta]\phi=2tz$, $\Xone[K\zeta]=\zeta$, and the density is $2tz-z\,\partial_0\phi=2tz-2tz=0$. This is Lemma~\ref{lem:switch} verified by hand: \[ \text{L5}=\Aone[K\zgf[\sxi];\phi]=0 . \] (For $t$ beyond $\supp a$, $z$ is the constant $2\int a-\tfrac12\int a'^2$: the de Donder parameter ends as a rigid time translation. This is a remark, not an input.) \paragraph{L6, Einstein-shell term.} \[ \text{L6}=\Aone[\PP\ER\Ldens(A\gamma);\phi]=0\qquad(A\gamma=0). \] \paragraph{L7, tree.} $\Tcal$ has no metric argument: $\text{L7}=0$. \begin{equation} \boxed{\; \text{L1}+\text{L2}+\text{L3}+\text{L4}+\text{L5}+\text{L6}+\text{L7} =-I+0+I+0+0+0+0=0 .\;} \label{eq:ledger} \end{equation} This is Corollary~\ref{cor:plateau} evaluated, and the positive witness pairing \eqref{eq:I} verifies the sign: the held-fixed $-I$ is removed by the covariant part of the response generated by the cubic vertex, not by the gauge-fixing response, which pairs to zero. \subsection{The sheet's three symbols, and the double count} The production sheet asked for $\delta_{\mathrm{split}}+\delta_0\Cthree+\dgf$, with XI-E's values. In the ledger above these are: \begin{align*} \delta_{\mathrm{split}}&=\text{L1}+\text{L2}+(\text{by-hand }+I)=-I+0+I=0 &&\text{(XI-E, cancellation theorem)},\\ \delta_0\Cthree&=\text{L3}+\text{L5}\big|_{\text{linear}}=I+0=I &&\text{(XI-E, cubic variation)},\\ \dgf&=\text{L5}=0 &&\text{(Proposition~\ref{prop:gfprops}, this paper)}. \end{align*} Their literal sum is $0+I+0=I$. That number is not an obstruction. The ``by-hand $+I$'' inside $\delta_{\mathrm{split}}$ is the shift $\gamma\mapsto\gamma+\kappa\Lie_\xi\gamma$ restored by XI-E in a parametrization with an external second coefficient $k$ and an independent field $\gamma$; L3 is the shift $\Lie_\xi\gamma$ generated by the cubic response once $k=u_3[\gamma]$. They are one term in two notations. XI-E's theorem is correct in its own parametrization, and XI-E's $\delta_0\Cthree=I$ is correct; what is incorrect is the sheet's instruction to add them. Removing the duplicate gives \eqref{eq:ledger}. Nothing is dropped: $\Cthree$ is retained, and so is its $I$. The sheet's expectation $\dgf=-I$ is replaced by the computed value $\dgf=0$. \subsection{The Einstein-shell term, named} On the gauge-fixed free shell $-\Box\PP\gamma=0$ one has $A\gamma=-2D^*D\gamma=-2D^*b$, which vanishes exactly when $D^*b=0$; constant $b$ (the witness) and $b=0$ (the de Donder Einstein shell) are examples, and every linearized Einstein solution in any gauge satisfies $A\gamma=0$. For gauge-fixed-shell configurations with $D^*b\neq0$ on $U$, Theorem~\ref{thm:F3} retains the term $\Aone[\PP\ER\Ldens(-2D^*b);\phi_0]$. It is an explicit classical functional; its value on the plateau is not computed here and is not asserted to vanish. This is XI-A's distinction between the gauge-fixed wave equation and the Einstein equation, met again at second order: the cubic source transforms as a density only modulo $A\gamma$. %===================================================================== \section{Support of the remainder} \label{sec:support} %===================================================================== \begin{proposition}[Support of the plateau remainder] \label{prop:support} Let $A\gamma=0$ on $U$ and $\xi\in\ga$. The remainder $R_\xi:=\Aone[K\zgf[\sxi];\phi_0]$ in \eqref{eq:plateausum} is given by \eqref{eq:switch} with $\zeta=\zgf[\sxi]$. As a functional of $\zeta$ its kernel is supported in \begin{equation} \bigl(\supp\dd g\ \cup\ \supp((1-g)f)\bigr)\ \cap\ J^+(\supp\xi)\ \cap\ \Dfset . \label{eq:remsupport} \end{equation} There is no ramp term: $\zgf[\sxi]$ vanishes on $\{t\leq\eps\}$ and $\Xone[K\zgf]=\zgf$ exactly. With the observation-plateau hypothesis $\supp f\subset P_g$ the second set is empty and the kernel is supported in $\supp\dd g\cap J^+(\supp\xi)\cap\Dfset$. Without it, $\supp((1-g)f)$ remains, as in XI-E's Route~2. \end{proposition} \begin{proof} Lemma~\ref{lem:switch} and Proposition~\ref{prop:gfprops}(b). The first term of \eqref{eq:switch} is supported in $\supp f\cap\{g\neq1\}\cap\supp\zeta$; the second, after transposition onto $\EAm f$, in $\supp\partial g\cap\supp\zeta\cap\Dfset$. \end{proof} Compare XI-E's support formula for arbitrary compact ghosts, which had the additional layer $\{0\leq t\leq\eps\}$: an unrestricted ghost can enter the ramp, while the de Donder parameter selected by the cubic response cannot, because it is retarded from $U$. The remainder is a genuine switching defect: for an observation whose dependence region meets $\supp\dd g$ inside $J^+(\supp\xi)$, the second-order parameter forced by gauge fixing reaches the switching region, where the switched scalar action is not covariant. It is zero on the witness and under \eqref{eq:plateauzero}. %===================================================================== \section{F4: what the sum is not} \label{sec:F4} %===================================================================== \begin{itemize} \item Equation \eqref{eq:plateauzero} is a classical coefficient identity on off-shell plateau parameters $\xi\in\ga$, for a free scalar held fixed. It is not nilpotence of an interacting differential, not a quantum master equation, and not a quantum Ward identity. No $\widehat s_{\mathrm{int}}$, antibracket, or antifield term is constructed; XI-B's antifields remain slots. \item The co-transformation $\Lie_\xi\gamma$, now generated by the response rather than inserted by hand, is still not free: on the witness its spatial wave operator contains $-a''\delta_{ij}$. It is not an automorphism of $\Afree$, and the ghost rule $\sfree\gamma=\tfrac12Kc$ is not that shift. \item Because $\Cthree$ is not admitted (Theorem~\ref{thm:F1}), there is no element of $\Ealg$ whose symbol is $\Atwothree$, and no quantum version of \eqref{eq:ledger} exists in the frozen algebra. The ledger is classical. \item The two affine numbers are untouched: the bare insertion still fails by $S_\eps(t)$, the complete coordinate word still leaves $\eps/2$. Neither enters the cubic ledger; neither is erased by it. \item No mixed loop is evaluated; no state is selected; no $\eps\to0$ limit is taken; $\Jcal_g(\phi_m)=\varnothing$ stands. \end{itemize} %===================================================================== \section{Decision, exclusions, and handoff} \label{sec:decision} %===================================================================== \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{.10\linewidth}p{.27\linewidth}X@{}} \toprule Gate & Ending & Exact scope\\ \midrule F1 & \textbf{Proved obstructed} & The frozen representative of $\Cthree(f,g)$ has $\pm\dd t\in\WF$ at $\Sigma$ in a metric slot (Theorem~\ref{thm:F1}); it leaves $\Fpol$, and XI-E's admission map does not apply. Mechanism: zero Cauchy data for the second coefficient together with an unswitched, unramped cubic source. Not finite rank; not a loop. The on-shell class is open.\\[0.4em] F2 & \textbf{Constructed} & De Donder variation $2D^*D\sxi$ from XI-B's fermion, divergence $-\Box D\sxi$, response $K\zgf[\sxi]$, $\zgf=-\ER D\sxi$, coefficient $-1$ (Lemma~\ref{lem:gf}, Proposition~\ref{prop:gfprops}). Ghost term has no metric leg at this order.\\[0.4em] F3 & \textbf{Constructed} & $u_3$ obeys the X-A second-order law with $\zeta=\zgf$ (Theorem~\ref{thm:u3law}); the cubic-completed coefficient shifts by $\Aone[K\zgf]$ plus the named Einstein-shell term (Theorem~\ref{thm:F3}); zero when $A\gamma=0$ on $U$ and $\Dfset\cap J^+(\supp\xi)\subset P_g$; ledger $-I+0+I+0+0+0+0=0$ on the witness; remainder supported in \eqref{eq:remsupport}, no ramp term.\\[0.4em] F4 & Recorded & Not a Ward identity; not an automorphism; no quantum version in $\Ealg$; affine numbers untouched.\\ \bottomrule \end{tabularx} \end{center} \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{.08\linewidth}X X X@{}} \toprule Gate & Constructed only if & Obstructed if & Result\\ \midrule F1 & $\Cthree\in\Ealg$ with the XI-E embedding & kernel leaves the copied wavefront class & \textbf{Obstructed}: timelike conormal at $\Sigma$\\ F2 & de Donder variation computed in the coefficient $-1$ & --- & \textbf{Constructed}\\ F3 & plateau sum proved, support stated & interior pairing nonzero on the witness & \textbf{Constructed}: sum $=0$; the sheet's $I$ was a double count\\ \bottomrule \end{tabularx} \end{center} The worst ending among the two gates governs: \textbf{Proved obstructed}, at the admission gate. The plateau sum is constructed as a classical identity and could not be quantized in $\Ealg$ even if its sum were nonzero. \subsection*{Exclusions that persist} \begin{itemize} \item No state, Hadamard selection beyond XI-E's product data, or expectation value. \item No mixed-loop value; the cubic source is not contracted against a Hadamard line. \item No interacting differential, antibracket, or master identity. \item No sharp-width limit; no identification with VII-B's or IX-A's sharp insertions, although the covector found in Theorem~\ref{thm:F1} is the same timelike conormal type. \item No repair of $\Jcal_g(\phi_m)=\varnothing$; no switching of the Einstein--Hilbert density; no ramp on the cubic source. \item X-E's and X-F's normalizations and identities are not premises. \end{itemize} \subsection*{What a successor receives} \begin{enumerate} \item The kernel representation \eqref{eq:C3beta}--\eqref{eq:ukernel} and the obstruction Theorem~\ref{thm:F1}, with the open on-shell question stated. \item The de Donder companion $K\zgf$ and the finite transformation law \eqref{eq:u3law}, including the Einstein-shell term. \item The variation identity \eqref{eq:F3}, the plateau corollary, the ledger \eqref{eq:ledger}, and the support formula \eqref{eq:remsupport}. \item The correction that XI-E's by-hand co-transformation and the covariant part of the cubic response are one mechanism. \item Two named routes that would change the model: a ramped cubic source, or metric data off $\Sigma$. Neither is taken here. \end{enumerate} \subsection*{Conclusion} \[ \boxed{ \begin{gathered} \Cthree\notin\Fpol:\ \text{the frozen cubic kernel carries }\pm\dd t\text{ at }\Sigma;\\[2pt] u_3[\gamma+K\xi]-u_3[\gamma]=\Lie_\xi\gamma+\tfrac12\Lie_\xi K\xi+K\zgf+\PP\ER\Ldens(A\gamma), \quad \zgf=-\ER D\sxi;\\[2pt] \Atwothree[\gamma+K\xi]-\Atwothree[\gamma]=\Aone[K\zgf;\phi_0]\ \ (A\gamma=0\text{ on }U), \quad -I+I+0=0\text{ on the witness.} \end{gathered}} \] A reader can now see that $\Cthree$ is not an element of the frozen algebra, and that the second $I$ of the cubic variation is cancelled by the held-fixed term it was always paired with, while the gauge-fixed companion contributes zero. A zero sum does not construct the interacting class XI-E withheld. %===================================================================== \section*{Declarations} \addcontentsline{toc}{section}{Declarations} \paragraph{Proof scope.} The kernel representation, the wavefront witness, the fermion computation, the finite transformation law, the variation identity, the ledger, and the support statement are proved on smooth configurations in the displayed conventions. The on-shell representative question for $\Cthree$ is stated as open. \paragraph{Dependencies.} X-A supplies the Cauchy facts, kernel lemma, preflight argument, and Theorem~(v). XI-A supplies the signed action and the gauge-fixed versus Einstein distinction. XI-B supplies the fermion, normalization, and free shell. XI-C and XI-D supply the tree sign and the affine lemma. XI-E supplies the algebra, its domain, the cubic variation, and the witness. Hörmander supplies the wavefront tools. \paragraph{Corrections.} The admission mechanism, the value of the gauge-fixing companion, the double count in the three-term sum, the origin of the quadratic flow term, the non-plateau support of the de Donder parameter, and the retained Einstein-shell term are corrected relative to the production sheet. \paragraph{Verification.} No successful \texttt{pdflatex} execution, proof-assistant check, executed symbolic computation, independent referee review, or journal acceptance is asserted. All nonzero and zero witnesses are analytic. %===================================================================== \begin{thebibliography}{9} \addcontentsline{toc}{section}{References} \bibitem{XA} N.~Cheung, \emph{Landmark X-A: Classical covariance of one fixed-regulator switched relational scalar through order $\kappa^2$}. Supplied manuscript \texttt{X-A.tex}. Used for the Cauchy lemma, transposition, kernel lemma, preflight argument, pure-gauge variation lemma, and Theorem~(v). \bibitem{XIA} N.~Cheung, \emph{Landmark XI-A: A switched Einstein--scalar action for the ramp relational scalar}. Supplied manuscript \texttt{XI-A.tex}. Used for the signed action, $E_{\mathrm{gf}}=2D^*D\gamma$, the breaking $-\Box D$, and the empty fixed-input jet. \bibitem{XIB} N.~Cheung, \emph{Landmark XI-B: The free de Donder multiplet for the ramp Einstein--scalar sheet}. Supplied manuscript \texttt{XI-B.tex}. Used for the gauge fermion, the normalization $-\PP\ER$, and the free shell. \bibitem{XIC} N.~Cheung, \emph{Landmark XI-C: One fixed-regulator graviton tree for the ramp relational scalar}. Supplied manuscript \texttt{XI-C.tex}. Used for the tree sign $-\Tcal$ and the held-fixed defect. \bibitem{XID} N.~Cheung, \emph{Landmark XI-D: Admission of the fixed-width graviton tree and one ghost-number-zero identity}. Supplied manuscript \texttt{XI-D.tex}. Used for the written-out second coefficient and the affine lemma. \bibitem{XIE} N.~Cheung, \emph{Landmark XI-E: On-shell admission of the fixed-width tree, a localized first-order identity, and the held-fixed defect}. Supplied manuscript \texttt{XI-E.tex}. Used for the Hadamard algebra and its domain, the cubic response, its variation, the split cancellation, and the witness. \bibitem{Hormander} L.~H\"ormander, \emph{The Analysis of Linear Partial Differential Operators I}, Grundlehren der mathematischen Wissenschaften 256, second edition, Springer, 1990. Used for Definition~8.1.2, Proposition~8.1.3, and Theorem~8.2.12. \bibitem{BGP} C.~B\"ar, N.~Ginoux, and F.~Pf\"affle, \emph{Wave Equations on Lorentzian Manifolds and Quantization}, ESI Lectures in Mathematics and Physics, European Mathematical Society, 2007. arXiv:0806.1036. Used for smooth Cauchy theory and finite propagation. \end{thebibliography} \end{document}