% !TEX program = pdflatex
%=====================================================================
% X-C.tex
%
% Standalone source for pdfLaTeX.
%
% Build:
%   pdflatex -interaction=nonstopmode -halt-on-error X-C.tex
%   pdflatex -interaction=nonstopmode -halt-on-error X-C.tex
%
% Combined and corrected manuscript:
%   X-C-v2 supplies the release and the proof of record.
%   X-C-v1 supplies the polarization warning and rooted leg count.
%
% Frozen:
%   Fixed smooth ramp; scalar quantum; metric coefficients classical.
%   Actual off-shell insertion tests; unchanged plateau gauge algebra.
%
% Constructed:
%   R_1(V_{2,g}; F_c).
%   (1/2) R_2(V_{1,g},V_{1,g}; F_c) on linear insertions.
%   Normalization to the copied second scalar response.
%   Completed background Ward identity through kappa^2.
%
% Additional corrections:
%   - The Hessian response sign is written without an extra minus sign.
%   - Support statements for formal pullbacks are coefficientwise.
%   - Differentiated switch factors are retained at second order.
%   - An edge defect involves the second coordinate coefficient as well
%     as the first-order gap; no unsupported reduction is asserted.
%   - Reversing the second-vertex sign adds 2 phi_hat(L_2 E^A f).
%     It does not universally reduce to a numerical 3/2 coefficient.
%
% No successful compilation, mechanized verification, or independent
% referee review is claimed.
%
% External reference checks:
% Green operators and the exact sequence: ([arxiv.org](https://arxiv.org/pdf/0806.1036))
% Free Wick contraction presentation: ([arxiv.org](https://arxiv.org/pdf/1208.1428))
% Off-shell retarded-product terminology: ([arxiv.org](https://arxiv.org/abs/hep-th/0403213))
%=====================================================================

\documentclass[11pt,a4paper]{article}

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\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
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\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{premise}[theorem]{Premise}
\newtheorem{diagnostic}[theorem]{Diagnostic}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}

\DeclareMathOperator{\supp}{supp}
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\newcommand{\R}{\mathbb R}
\newcommand{\eps}{\varepsilon}
\newcommand{\dd}{\,\mathrm d}
\newcommand{\Lie}{\mathcal L}
\newcommand{\sig}{\sigma_\eps}
\newcommand{\Pm}{P_m}
\newcommand{\Xone}{\mathsf X_1}
\newcommand{\Xtwo}{\mathsf X_2}
\newcommand{\Obs}{\mathcal O}
\newcommand{\ga}{\mathfrak g_U}
\newcommand{\Gp}{\mathcal G_U}
\newcommand{\Dspace}{\mathcal D}
\newcommand{\Df}{\mathcal D_f}
\newcommand{\Bg}{\mathscr B}
\newcommand{\Loff}{\mathscr L_{\mathrm{off}}}
\newcommand{\Aoff}{\mathscr A_{\mathrm{off}}}
\newcommand{\Afree}{\mathfrak A_0}
\newcommand{\Jfree}{\mathfrak J_0}
\newcommand{\ER}{\mathsf E_m^{\mathrm R}}
\newcommand{\EA}{\mathsf E_m^{\mathrm A}}
\newcommand{\ERzero}{\mathsf E_0^{\mathrm R}}
\newcommand{\ERq}{\mathsf E_q^{\mathrm R}}
\newcommand{\EAq}{\mathsf E_q^{\mathrm A}}
\newcommand{\Lone}{\mathsf L_{1,g}}
\newcommand{\Ltwo}{\mathsf L_{2,g}}
\newcommand{\Lq}{\mathsf L[q]}
\newcommand{\Eg}{\mathsf E_g}
\newcommand{\Qop}{\mathsf Q}
\newcommand{\Nop}{\mathsf N}
\newcommand{\Rq}{\mathscr R_q}
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\newcommand{\Rtwo}{R_2}
\newcommand{\field}{\widehat\phi}
\newcommand{\formal}{\pmod{\kappa^3}}
\newcommand{\pair}[2]{\left\langle #1,#2\right\rangle}
\newcommand{\Fsharp}{F^\sharp}
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\newcommand{\SigmaF}{\Sigma_F}

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  \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-C}\\[0.4em]
The missing second-order retarded maps for the ramp\\
relational scalar, with the metric kept classical}
\author{Ngo Cheung}
\date{}

\usepackage{tikz}
\begin{document}
\maketitle

\begin{abstract}
At the fixed smooth regulator of Landmark X-B-v3, we construct the
second-vertex scalar map and the iterated first-vertex scalar map
on the frozen off-shell linear insertion tests.
The scalar alone is quantum; the metric coefficients remain
classical and off shell.
The maps have smooth compact retarded output, and their sum
equals the second scalar response copied from Landmark X-A.
They are the corresponding coefficients of a test-level
retarded M\o ller map, characterized by a Yang--Feldman recursion.

The completed relational coefficient through \(\kappa^2\) is
background-gauge invariant in the on-shell free algebra on the
unchanged ramp plateau algebra.
The proof of record is a three-term variation formula:
a compact free-equation term, a dressed coordinate defect,
and a dressed switch defect.
The switch derivatives are displayed before imposing the
plateau conditions.
The second-order variation retains propagation of the
first-order edge gap.

A complete pure-gauge two-jet verifies the operator signs and
closes the remainder left in X-B-v3, with the bare second dressing
retained.
A conformal massless diagnostic gives a classical number and
checks the sourced signs.
The maps have no positive-\(\hbar\) remainder in the specified
linear rooted prescription.
No general retarded-product hierarchy, sharp limit,
quantum metric, or two-point response is constructed.
\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) {Previous target};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Reassigned target};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {An inherited repair};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {Reassignment is not a transfer of an earlier ending.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{The reassigned target keeps its own ending. 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-C}
\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) {Previous target};
\node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {An inherited repair};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {Reassignment is not a transfer of an earlier ending.};
\draw[->] (a) -- (s);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{The reassigned target keeps its own ending. 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-C}
\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) {Previous target};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Reassigned target};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {An inherited repair};
\node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {Reassignment is not a transfer of an earlier ending.};
\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-C}
\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) {Previous target};
\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) {An inherited repair};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Reassigned target. Reassignment is not a transfer of an earlier ending.};
\draw (a) -- (m);
\draw (m) -- (b);
\draw[->] (m) -- (s);
\end{tikzpicture}
\caption{Previous target is not An inherited repair. 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-C}
\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: Previous target};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Reassigned target};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: An inherited repair};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {Reassignment is not a transfer of an earlier ending.};
\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-C}
\end{figure}

Figure~\ref{fig:split-X-C} records one view of the same separation. Figure~\ref{fig:chain-X-C} records one view of the same separation. Figure~\ref{fig:pair-X-C} records one view of the same separation. Figure~\ref{fig:gate-X-C} records one view of the same separation. The drawings add no hypothesis.


Figure~\ref{fig:scope-X-C} records that separation.
The drawing adds no hypothesis.



%=====================================================================
\section{Release, reassigned target, and the three endings}
\label{sec:release}
%=====================================================================

\scopebox{Release: Constructed, through \(\kappa^2\) at fixed \(\eps\)}{%
On the X-B-v3 sheet, the maps
\[
\Rone(V_{2,g};F_c)=\field(-\Ltwo\EA c),
\qquad
\tfrac12\Rtwo(V_{1,g},V_{1,g};F_c)
=\field(\Lone\EA\Lone\EA c)
\]
exist on the frozen off-shell linear tests, have smooth compact
retarded output, and carry no positive power of \(\hbar\).
Their sum on \(F_f\) is the copied response \(\int f\rho_{2,g}\),
where
\[
\rho_{2,g}[u]=\ER\Lone\ER\Lone u-\ER\Ltwo u.
\]
The completed coefficient
\[
\widehat A_2
=\Obs^{(2)}
+\Rone(V_{1,g};F_{c_1})
+\Rone(V_{2,g};F_f)
+\tfrac12\Rtwo(V_{1,g},V_{1,g};F_f)
\]
is invariant in \(\Afree\) on \(\ga\), with the free field held
fixed.
Together with the copied \(\widehat A_1\), this gives background
invariance through \(\kappa^2\).
The pure-gauge remainder is cancelled with \(\Obs^{(2)}\) retained.

\smallskip
Not claimed: the sharp limit, quantum metric, graviton tree,
triangle, \(\mathsf T_2\), \(\mathsf T_3\), nonlinear-insertion
\(\Rtwo\), or two-point responses.}

The target is reassigned to the missing scalar dynamics in
X-B-v3's truncation.
It is not the sharp-limit decision of an earlier pilot table.
Taking \(\eps\to0\) before supplying the displayed
\(\rho_{2,g}\) would leave that truncation incomplete.

Three endings are fixed:
\begin{enumerate}[label=\textbf{\arabic*.}]
\item \textbf{Constructed.}
Both maps exist on the frozen representatives, their sum is the
copied response, and the completed coefficient is invariant.
\item \textbf{Proved obstructed.}
A nonzero defect survives on the frozen algebra after both maps
are included, or the required response normalization fails.
\item \textbf{Unresolved.}
A vertex, sign, map, or insertion representative remains unnamed.
\end{enumerate}
The ending reached is (1).
A missing map is not a no-go.
The previously displayed remainder is not repaired by a contact,
by deleting a dressing term, or by shrinking the gauge algebra.

Both \(\kappa\) and \(\hbar\) are formal.
Throughout the paper, operator and field identities involving
the background expansion are understood modulo \(\kappa^3\).
Support assertions for coefficients of formal pullbacks are
coefficientwise assertions, not claims about the support of an
unexpanded finite pullback.

%=====================================================================
\section{Copied truncation and the pure-gauge remainder}
\label{sec:premises}
%=====================================================================

The supplied \texttt{X-A.tex} \cite{XA} and
\texttt{X-B-v3.tex} \cite{XB} supply the classical ramp prescription,
its covariance, admission, and the first scalar map.
Those covariance and cone proofs are not repeated.
The conventions needed to state the new construction are recorded
here so that its equations and domains are explicit.

\subsection{Geometry, backgrounds, regulator, and algebra}

Let
\[
M=\R^4,\qquad
\eta=\diag(1,-1,-1,-1),\qquad
\Box=\partial_t^2-\Delta,\qquad
\Pm=\Box+m^2,\quad m\geq0.
\]
The quantum field is defined on \(M\).
The slab \(M_T=[0,T]\times\R^3\) and its initial surface belong
to the classical coordinate prescription.

The classical backgrounds are arbitrary smooth real symmetric
tensors \(h,k\), and
\[
q=\eta+\kappa h+\kappa^2k.
\]
Indices are moved with \(\eta\).
Background topology means convergence of all derivatives on
compact subsets of \(M\).

Fix \(0<\eps<T\) and the same ramp
\[
\sig(t)=\chi(t/\eps),\qquad
\chi(s)=
\begin{cases}
0,&s\leq0,\\[2pt]
\dfrac{e^{-1/s}}{e^{-1/s}+e^{-1/(1-s)}},&0<s<1,\\[7pt]
1,&s\geq1.
\end{cases}
\]
Thus \(\sig\) is smooth and flat at zero.
Fix the same
\(g\in C_c^\infty((0,T)\times\R^3;[0,1])\), with
\(P_g=\Int\{g=1\}\), and
\begin{equation}
\overline U\subset P_g\cap\{\eps<t<T\},
\qquad
\ga=C_c^\infty(U;TM).
\label{eq:algebra}
\end{equation}
The formal group \(\Gp\) consists of pullbacks generated by
\(\kappa\xi+\kappa^2\zeta\), \(\xi,\zeta\in\ga\).

For the corresponding flow \(F\), the complete metric law is
\begin{align}
h'&=h+K\xi,
\nonumber\\
k'&=k+\Lie_\xi h+K\zeta+\tfrac12\Lie_\xi K\xi,
\qquad
(K\xi)_{\mu\nu}=\partial_\mu\xi_\nu+\partial_\nu\xi_\mu.
\label{eq:metric-law}
\end{align}
The second line is essential; \(k=0\) does not describe a
second-order pure-gauge jet when \(h=K\xi\).

\subsection{Coordinates and the bare insertion tests}

The coordinate coefficients copied from X-A are
\begin{align}
\Xone^\rho[h]
&=-\ERzero\partial_\mu(\sig a_1^{\mu\rho}[h]),
\nonumber\\
\Xtwo^\rho[h,k]
&=-\ERzero\left[
\partial_\mu(\sig A_2^{\mu\rho}[h,k])
+\partial_\mu\bigl(
\sig a_1^{\mu\nu}[h]\partial_\nu\Xone^\rho[h]\bigr)
\right].
\label{eq:coordinates}
\end{align}
The density coefficients are displayed in
Section~\ref{sec:sheet}.
Retarded propagation in \eqref{eq:coordinates} reproduces the
zero-data Cauchy prescription on the slab.

The copied coordinate laws are
\begin{align}
\Xone[h']&=\Xone[h]+\xi,
\nonumber\\
\Xtwo[h',k']
&=\Xtwo[h,k]+X_\xi\Xone[h]+\zeta+\tfrac12X_\xi\xi,
\qquad X_\xi=\xi^\mu\partial_\mu.
\label{eq:coordinate-laws}
\end{align}
In particular, \(\Xone[K\xi]=\xi\) on \(\ga\).
X-A's coordinate-only intertwiner also applies to compact
generators supported in \(\{\eps<t<T\}\), without requiring
an interaction plateau.

Fix a real
\(f\in C_c^\infty((0,T)\times\R^3)\), put \(K=\supp f\), and
\[
\Df=J^-(K)\cap M_T.
\]
For \(B^\mu=\Xone^\nu\partial_\nu\Xone^\mu-\Xtwo^\mu\), the bare
off-shell tests are
\begin{align}
c_0&=f,
&
c_1[h]&=\partial_\mu(f\Xone^\mu[h]),
\nonumber\\
c_2[h,k]&=-\partial_\mu(fB^\mu)
+\tfrac12\partial_\mu\partial_\nu
(f\Xone^\mu[h]\Xone^\nu[h]).
\label{eq:bare-tests}
\end{align}
Write \(F_c(u)=\int cu\,\dd^4x\).
Then \(\Obs^{(j)}=F_{c_j}\), and
\(\Obs_\kappa=F_f+\kappa F_{c_1}+\kappa^2F_{c_2}\).
The tests are smooth, supported in \(K\), and continuously
dependent on the backgrounds at fixed \(\eps\).
Their admission is copied from X-B-v3.

\subsection{Free algebra and the representative warning}

Use X-B-v3's free polynomial Wick algebra \(\Aoff\) and its
equation ideal \(\Jfree\), generated by \(F_{\Pm v}\) for
\(v\in\Dspace(M)=C_c^\infty(M;\mathbb C)\).
Set
\[
\Afree=\Aoff/\Jfree,\qquad
\field(c)=[F_c].
\]
The relevant relations are
\[
\field(\Pm v)=0,\qquad
[\field(c),\field(d)]
=i\hbar\pair{c}{(\EA-\ER)d}\,\mathbf1,
\]
where \(\ER,\EA\) are the Green operators of \(\Pm\).
The linear output topology is the quotient test topology.
The insertion domain is instead
\[
\Loff=\{F_c:c\in\Dspace(M)\},
\]
identified with actual off-shell tests.

This distinction cannot be suppressed.
The copied first map satisfies
\begin{equation}
\Rone(V_{1,g};F_c)=\field(-\Lone\EA c),
\qquad
\Rone(V_{1,g};F_{\Pm v})=\field(-\Lone v),
\label{eq:first-map}
\end{equation}
and the second expression need not vanish.
Consequently no iteration in this paper is performed through an
on-shell input class.

\subsection{The missing second scalar response}

The copied positive Euler coefficients are \(\Lone,\Ltwo\), and
for a smooth free input \(u\),
\begin{align}
\rho_{1,g}[u]&=-\ER\Lone u,
\nonumber\\
\rho_{2,g}[u]&=\ER\Lone\ER\Lone u-\ER\Ltwo u.
\label{eq:responses}
\end{align}
The switched Euler operator
\(\Eg[q]=\Pm+\kappa\Lone+\kappa^2\Ltwo\) is covariant under
\(\Gp\).
The unswitched curved density operator satisfies naturality
under every compactly supported diffeomorphism.
These are copied premises from X-A.

X-B-v3 constructs
\(\widehat A_1=\field(c_1-\Lone\EA f)\) and proves its invariance.
Its truncation is
\begin{align}
\Obs_\kappa+\kappa\Rone(V_{1,g};\Obs_\kappa)
={}&\Obs^{(0)}+\kappa\widehat A_1
\nonumber\\
&+\kappa^2\bigl(
\Obs^{(2)}+\Rone(V_{1,g};F_{c_1})\bigr).
\label{eq:old-truncation}
\end{align}
The mixed term has the fixed minus insertion sign:
\[
\Rone(V_{1,g};F_{c_1})
=-\int f\,\Xone\cdot\partial\rho_{1,g}\,\dd^4x.
\]
Its copied finite transformation on \(\ga\) is
\begin{align}
\mathcal M[h+K\xi]-\mathcal M[h]
={}&\field(-\Lone[h]\EA D_\xi f)
\nonumber\\
&-\field(D_\xi c_1[h])-\field(D_\xi^2f),
\qquad D_\xi r=\partial_\mu(\xi^\mu r).
\label{eq:mixed-law}
\end{align}
This is not an invariance statement.

The classical second coefficient has the additional
\(\int f\rho_{2,g}\).
On the complete pure-gauge jet
\(h=K\xi,\ k=\tfrac12\Lie_\xi K\xi\), the incomplete coefficient
in \eqref{eq:old-truncation} is
\(-\tfrac12\int fX_\xi^2\field\,\dd^4x\).
The missing classical response is
\(\tfrac12X_\xi^2u\).
Constructing that response is the present target.

%=====================================================================
\section{Non-inputs and unopened constructions}
\label{sec:noninputs}
%=====================================================================

The convolution coefficient and X-Enter are not inputs.
X-Enter's Option A concerns readout of the convolution coefficient
alone.
Its Option B reference is not supplied by the ramp observable,
and the ramp edge integral is not the convolution quantity
\(c_\eps\).
Reading \(\widehat A_2\) would preserve an upstream identity,
not prove it.

VII-A, VI, VII-C, IX-E-2, and IX-E-3 remain comparison.
No cone, token, sharp insertion, or earlier kernel is identified
with the ramp coordinate kernel.
VII-C's hypotheses (E1)--(E4) are not assumed.

Branch B, with a quantum metric, is a different construction.
There is no gravitational action, gauge-fixed graviton
propagator, or metric contraction in the present sheet.
A graviton tree or mixed triangle would require its own
definition and analysis.
Neither existence nor obstruction of those graphs follows from
the scalar-only maps below.

No time-ordered hierarchy is used to fill the missing coefficient.
The restricted maps are defined directly and then checked against
their dynamics, support, and equation-ideal variation.
This limited construction is sufficient for the stated target;
it is not a substitute for a product programme on nonlinear
insertions.

%=====================================================================
\section{Frozen sheet and C1: the copied second vertex}
\label{sec:sheet}
%=====================================================================

Write \(H=\operatorname{tr}_\eta h\).
The copied density coefficients are
\begin{align}
b_1[h]&=\tfrac12H,
&
b_2[h]&=\tfrac18H^2-\tfrac14h_{\alpha\beta}h^{\alpha\beta},
\nonumber\\
a_1^{\mu\nu}[h]&=\tfrac12H\eta^{\mu\nu}-h^{\mu\nu},
&
a_2^{\mu\nu}[h]
&=h^\mu{}_\alpha h^{\alpha\nu}
-\tfrac12Hh^{\mu\nu}+b_2[h]\eta^{\mu\nu}.
\label{eq:density}
\end{align}
Set
\[
A_2[h,k]=a_1[k]+a_2[h],\qquad
B_2[h,k]=b_1[k]+b_2[h].
\]
These are the \(\kappa^2\) coefficients of
\(\sqrt{-q}\,q^{-1}\) and \(\sqrt{-q}\), respectively.

The frozen switched density is
\((1-g)\mathfrak L_\eta+g\mathfrak L_q\).
Its two interaction coefficients are
\begin{align}
V_{1,g}[h,u]
&=\tfrac12\int g\bigl(
a_1^{\mu\nu}[h]\partial_\mu u\partial_\nu u
-m^2b_1[h]u^2\bigr)\,\dd^4x,
\nonumber\\
V_{2,g}[h,k,u]
&=\tfrac12\int g\bigl(
A_2^{\mu\nu}[h,k]\partial_\mu u\partial_\nu u
-m^2B_2[h,k]u^2\bigr)\,\dd^4x.
\label{eq:vertices}
\end{align}
The second vertex is therefore fixed by the supplied density
before either new map is defined.

The divergence-form coefficients, including switch derivatives,
are
\begin{align}
\Lone[h]u
={}&g\bigl[
\partial_\mu(a_1^{\mu\nu}[h]\partial_\nu u)
+m^2b_1[h]u\bigr]
+(\partial_\mu g)a_1^{\mu\nu}[h]\partial_\nu u,
\nonumber\\
\Ltwo[h,k]u
={}&g\bigl[
\partial_\mu(A_2^{\mu\nu}[h,k]\partial_\nu u)
+m^2B_2[h,k]u\bigr]
+(\partial_\mu g)A_2^{\mu\nu}[h,k]\partial_\nu u.
\label{eq:Euler-coefficients}
\end{align}
They are formally self-adjoint, with smooth compact coefficients.
Neither is obtained by multiplying an unswitched Euler equation
after variation by \(g\).

Compact integration by parts gives the sign ledger
\begin{equation}
S_0''=-\Pm,\qquad
V_{1,g}'[u]=-\Lone u,\quad V_{1,g}''=-\Lone,
\qquad
V_{2,g}'[u]=-\Ltwo u,\quad V_{2,g}''=-\Ltwo.
\label{eq:Hessians}
\end{equation}
The positive Euler coefficients are not the variational
Hessians.
This resolves the inconsistent Hessian wording in X-A's
handoff in accordance with its explicit Euler-sign audit.

\begin{center}
\small
\renewcommand{\arraystretch}{1.12}
\begin{tabularx}{\linewidth}{@{}p{.22\linewidth}X@{}}
\toprule
Item & Declaration\\
\midrule
Regulator & Same smooth ramp, same fixed \(\eps>0\)\\
Field content & Scalar quantum; \(h,k\) classical and off shell\\
Second vertex & \(V_{2,g}\) in \eqref{eq:vertices}, switched before variation\\
New maps &
\(\Rone(V_{2,g};F_c)\) and
\(\tfrac12\Rtwo(V_{1,g},V_{1,g};F_c)\), on linear representatives\\
Normalization & Their sum equals the copied \(\int c\rho_{2,g}\)\\
Ward order & Completed observable through \(\kappa^2\) on \(\ga\)\\
Topology & Smooth backgrounds and compact-test topology\\
Not opened &
Nonlinear-insertion \(\Rtwo\), \(\mathsf T_2\), \(\mathsf T_3\),
quantum metric, sharp limit, two-point responses\\
\bottomrule
\end{tabularx}
\end{center}

%=====================================================================
\section{C2 and C3: maps, transposition, signs, and polarization}
\label{sec:maps}
%=====================================================================

Put
\[
\Lq=\kappa\Lone+\kappa^2\Ltwo,\qquad
\Eg[q]=\Pm+\Lq.
\]
Every composition below is coefficientwise in the formal ring.

\begin{definition}[Test-level retarded M\o ller map]
\label{def:moller}
Define
\begin{equation}
\Rq=(1+\Lq\EA)^{-1}
=1-\kappa\Lone\EA
+\kappa^2(\Lone\EA\Lone\EA-\Ltwo\EA).
\label{eq:moller}
\end{equation}
For actual \(c\in\Dspace(M)\), the two missing maps are
\begin{align}
\Rone(V_{2,g};F_c)&=\field(-\Ltwo\EA c),
\label{eq:second-map}\\
\tfrac12\Rtwo(V_{1,g},V_{1,g};F_c)
&=\field(\Lone\EA\Lone\EA c).
\label{eq:iteration}
\end{align}
The notation \(\tfrac12\Rtwo\) names the second Taylor coefficient
for the first vertex alone, not a general retarded product.
\end{definition}

The advanced operator in these formulas is the transpose of
retarded propagation of the scalar.
Thus the terminology ``retarded map'' is consistent with
backward causal support of its test.

The inner test \(\Lone\EA c\) is an actual smooth compact test.
Iteration may be written as applying the first-map rule to
\(F_{-\Lone\EA c}\), but it must not be written as applying that
rule to an unspecified representative of its on-shell class.
Equation \eqref{eq:first-map} explains why: changing an insertion
by \(F_{\Pm v}\) can change its response.

\begin{theorem}[Existence, support, continuity, and normalization]
\label{thm:maps}
The two maps \eqref{eq:second-map}--\eqref{eq:iteration} have
smooth compact output tests supported in
\[
\supp g\cap J^-(\supp c).
\]
They are continuous in compact-test and smooth background
topologies.
For every smooth \(u\), and hence for every smooth free \(u\),
\begin{equation}
\pair{\Lone\EA\Lone\EA c-\Ltwo\EA c}{u}
=\pair{c}{\ER\Lone\ER\Lone u-\ER\Ltwo u}.
\label{eq:normalization}
\end{equation}
Their sum on \(F_f\) is therefore \(\int f\rho_{2,g}\), understood
as the free field with the displayed compact test.
No diagonal extension or subtraction is used.
\end{theorem}

\begin{proof}
The Green operators act continuously on compact smooth tests,
with their respective causal supports; transposition exchanges
the retarded and advanced operators
\cite[Chapter~3]{BGP}.
Consequently \(\EA c\) is smooth, and applying \(\Ltwo\) produces
a compact test supported in \(\supp g\cap J^-(\supp c)\).

The intermediate support
\(K_1=\supp(\Lone\EA c)\) is contained in the same set.
The iterated output is supported in
\[
\supp g\cap J^-(K_1)
\subset\supp g\cap J^-(\supp c).
\]
Continuous differentiation and multiplication give continuity.
For input tests in a fixed compact support, every output lies
in a fixed compact subset of \(\supp g\).

Formal self-adjointness and transposition give
\begin{align*}
\pair{c}{\ER\Lone\ER\Lone u}
&=\pair{\Lone\EA\Lone\EA c}{u},\\
\pair{c}{\ER\Ltwo u}
&=\pair{\Ltwo\EA c}{u}.
\end{align*}
All source factors are compact because of \(g\).
Every Green operation is applied to a smooth source or test;
there is no singular product requiring extension.
\end{proof}

\subsection{The factorial and the relative sign}

For arbitrary smooth input \(u\), the scalar response polynomial is
\begin{equation}
r_q u
=u-\kappa\ER\Lone u
+\kappa^2(\ER\Lone\ER\Lone u-\ER\Ltwo u).
\label{eq:scalar-moller}
\end{equation}
It satisfies
\[
(\Pm+\Lq)r_q u=\Pm u.
\]
For a free input it reproduces the copied retarded coefficients.

With the free Hessian inverse \(\Delta_0^{\mathrm R}=-\ER\),
the first response is
\[
-\pair{c}{\Delta_0^{\mathrm R}V_{1,g}'[u]}
=-\pair{c}{(-\ER)(-\Lone u)}
=-\pair{c}{\ER\Lone u}.
\]
No further minus sign is inserted.
The second vertex enters once with the same negative response
sign.
Two successive first responses give a positive iteration.

For the first vertex alone, replacing \(\kappa\) by \(\lambda\)
in \((1+\lambda\ER\Lone)^{-1}u\) shows that the second derivative
at zero is twice \(\ER\Lone\ER\Lone u\).
This explains the coefficient \(\tfrac12\Rtwo\).
The normalization is dynamical, not a contact chosen after a
gauge calculation.

\subsection{Polarization and the ban on diagonal additivity}

The second-vertex map is linear in its quadratic vertex.
The diagonal iteration is quadratic, not additive, in \(V_{1,g}\).

For two specified quadratic vertices \(V,W\), with positive
Euler coefficients \(L_V,L_W\), its polarization is the
bookkeeping formula
\begin{equation}
R_2^{\mathrm{book}}(V,W;F_c)
=\field\bigl(
L_V\EA L_W\EA c+L_W\EA L_V\EA c
\bigr).
\label{eq:polarization}
\end{equation}
It is bilinear in these two coefficients and reduces on the
diagonal to twice \eqref{eq:iteration}.
Replacing \(V\) by \(V+W\) therefore retains the mixed terms.
Dropping them under a claim of diagonal additivity would be
incorrect.

Formula \eqref{eq:polarization} does not construct a hierarchy
on arbitrary entries.
The causal statements made here are the displayed chain support,
linearity or polarization, and vanishing when the vertex
supports cannot form a causal chain into the insertion support.
In particular, both new maps vanish if
\(\supp g\cap J^-(\supp c)=\varnothing\).
No full causal-factorization hierarchy is claimed.

%=====================================================================
\section{Yang--Feldman characterization and quantum order}
\label{sec:yf}
%=====================================================================

Define the linear interacting field and formal Green operators by
\begin{align}
\field_q(c)&=\field(\Rq c),
\nonumber\\
\ERq&=(1+\ER\Lq)^{-1}\ER
=\ER(1+\Lq\ER)^{-1},
\nonumber\\
\EAq&=(1+\EA\Lq)^{-1}\EA
=\EA(1+\Lq\EA)^{-1}.
\label{eq:interacting-operators}
\end{align}
These are finite formal expansions at the declared order.
No assertion about numerical inversion of the perturbed
operator is needed.

\begin{theorem}[Linear Yang--Feldman characterization]
\label{thm:yf}
The following identities hold:
\begin{enumerate}[label=\textup{(\roman*)}]
\item \(\Rq=1-\Rq\Lq\EA\).
It is the unique formal series with leading term \(1\)
satisfying this recursion.
Equivalently,
\(\field_q(c)=\field(c)-\field_q(\Lq\EA c)\).
\item \(\field_q(\Eg[q]v)=0\) for every \(v\in\Dspace(M)\).
\item If \(\supp c\cap J^+(\supp g)=\varnothing\), then
\(\field_q(c)=\field(c)\).
\item The algebraic commutator is
\[
[\field_q(c),\field_q(d)]
=i\hbar\pair{c}{(\EAq-\ERq)d}\,\mathbf1.
\]
\end{enumerate}
\end{theorem}

\begin{proof}
The inverse identity
\((1+A)^{-1}=1-(1+A)^{-1}A\), with \(A=\Lq\EA\), proves (i).
Comparing coefficients determines
\(-\Lone\EA\) at first order and
\(\Lone\EA\Lone\EA-\Ltwo\EA\) at second order uniquely.

For compact \(v\), \(\EA\Pm v=v\), so
\[
\Eg[q]v=(1+\Lq\EA)\Pm v,\qquad
\Rq\Eg[q]v=\Pm v.
\]
Its class vanishes, proving (ii).
The support condition in (iii) gives \(\Lq\EA c=0\).

For (iv), transposition gives
\(\Rq^{\mathsf T}=(1+\ER\Lq)^{-1}=1-\ERq\Lq\), while
\(\Rq=1-\Lq\EAq\).
Using the free commutator,
\[
\pair{\Rq c}{(\EA-\ER)\Rq d}
=\pair{c}{(1-\ERq\Lq)(\EA-\ER)(1-\Lq\EAq)d}.
\]
The operator in the middle equals
\[
(\EAq-\ERq\Lq\EAq)-(\ERq-\ERq\Lq\EAq)
=\EAq-\ERq.
\]
This proves the commutator identity.
\end{proof}

The theorem characterizes the declared linear-field construction.
It does not produce a state or an interacting two-point response.
The commutator is an algebraic consistency check only.

\begin{proposition}[Tree-exactness and the local leg count]
\label{prop:hbar}
The maps \eqref{eq:second-map} and \eqref{eq:iteration} contain no
positive power of \(\hbar\) and no Hadamard contraction kernel.
They are tree-exact in the specified linear rooted prescription.
\end{proposition}

\begin{proof}
Algebraically this follows from \eqref{eq:moller}: its coefficients
are ordinary test operators, with neither \(\hbar\) nor a
Hadamard kernel.
The quantum statement concerns these coefficients, not the
\(\hbar\) in the field commutator.

The local contraction count confirms the scope.
Use the fixed polynomial Wick presentation of X-B-v3.
In that presentation each cross-contraction differentiates both
entries once and supplies one factor of \(\hbar\)
\cite[Sections~4--5]{FR}.
A quadratic vertex with a linear insertion admits only one
cross-contraction, as in X-B-v3's first-map verification.

For two quadratic vertices and one linear insertion there are
five scalar legs.
A connected rooted graph joining the three entries needs two
edges, consumes four legs, and leaves one scalar leg.
Since the insertion has degree one, the graph is a chain ending
at that insertion.
Three contractions would require six legs and are impossible.
The two possible vertex orderings give the bookkeeping
polarization \eqref{eq:polarization}.

The two contraction factors are removed by the second-order
retarded prefactor.
A vacuum component between the two vertices disconnected from
the insertion is not part of the rooted relative response.
Self-contractions are absent in the fixed Wick presentation.
Thus the rooted coefficients have no positive-\(\hbar\)
remainder.
\end{proof}

The leg count does not construct time-ordered products or license
unanalysed diagonal extensions.
The maps are defined by \eqref{eq:moller} and characterized by
Theorem~\ref{thm:yf}; the count explains why that restricted
construction has the stated quantum order.
No cancellation theorem for an unconstructed
\(\mathsf T_2,\mathsf T_3\) programme is used.

Scalar-independent changes of quadratic Wick ordering have zero
first field derivative and do not change these linear responses.
Nor does tree-exactness make \(\Rq\) a map on free on-shell inputs:
\[
\Rq(\Pm v)=\Pm v-\Lq v+\Lq\EA\Lq v,
\]
whose class can be nonzero.
The warning about actual intermediate tests remains in force.

%=====================================================================
\section{C4: completed test, three-term variation, and cancellation}
\label{sec:ward}
%=====================================================================

\subsection{The completed Yang--Feldman test}

For \(c[q]=f+\kappa c_1+\kappa^2c_2\), set
\begin{align}
e[q]&=\Rq c[q]=f+\kappa e_1+\kappa^2e_2,
\nonumber\\
e_1&=c_1-\Lone\EA f,
\nonumber\\
e_2&=c_2-\Lone\EA e_1-\Ltwo\EA f.
\label{eq:completed-test}
\end{align}
The last line is the coefficient recursion obtained from
\((1+\Lq\EA)e=c\).
Expanding it gives
\[
e_2=c_2-\Lone\EA c_1-\Ltwo\EA f+\Lone\EA\Lone\EA f.
\]
Consequently
\begin{equation}
\widehat A[q]=\field(e[q])
=\field(f)+\kappa\widehat A_1+\kappa^2\widehat A_2
\label{eq:completed-field}
\end{equation}
has precisely the four second-order terms in the release box.

Every \(e_j\) is a smooth compact test, with
\[
\supp e_j\subset
K\cup(\supp g\cap J^-(K)).
\]
The background dependence is continuous at fixed \(\eps\).
By transposition and the copied bare expansion,
\(\pair{e_2}{u}\) is X-A's complete classical second coefficient
for every smooth free input \(u\).
This observation is a normalization check, not yet the Ward
proof.

\subsection{Scalar and density pullbacks}

For arbitrary compact \(\xi,\zeta\) in the open slab, put
\[
F^*=1+\kappa X_\xi
+\kappa^2\bigl(X_\zeta+\tfrac12X_\xi^2\bigr),
\qquad
\Fsharp=1+\kappa D_\xi
+\kappa^2\bigl(D_\zeta+\tfrac12D_\xi^2\bigr).
\]
The first acts on scalars; the second is density pullback,
including the Jacobian.

\begin{lemma}[Formal duality]
\label{lem:duality}
For compact formal tests \(c\) and smooth formal scalars \(\psi\),
\[
\pair{\Fsharp c}{F^*\psi}=\pair{c}{\psi}.
\]
\end{lemma}

\begin{proof}
\(D_\xi^{\mathsf T}=-X_\xi\).
Thus \((\Fsharp)^{\mathsf T}\) is the inverse scalar pullback
\(1-\kappa X_\xi+
\kappa^2(-X_\zeta+\tfrac12X_\xi^2)\).
Multiplying by \(F^*\) gives the identity through second order.
\end{proof}

\begin{definition}[Coordinate and switch defects]
\label{def:defects}
For \(q'=F^*q\), define
\[
\ThetaF[q]=c[q']-\Fsharp c[q],
\qquad
\SigmaF[q]=\Eg[q']F^*-\Fsharp\Eg[q].
\]
Write
\(\ThetaF=\kappa\Theta^{(1)}+\kappa^2\Theta^{(2)}\) and
\(\SigmaF=\kappa\Sigma^{(1)}+\kappa^2\Sigma^{(2)}\).
\end{definition}

The first defect is a compact test.
The second is a differential operator with coefficients
supported on the compact generator support.

\begin{lemma}[Coordinate defect]
\label{lem:theta}
Coefficientwise, \(\supp\Theta^{(j)}\subset K\), and
\begin{align}
\Theta^{(1)}
&=\partial_\mu\bigl(
f(\Xone[K\xi]-\xi)^\mu\bigr),
\nonumber\\
\Theta^{(2)}
&=c_2[h',k']-c_2[h,k]
-D_\xi c_1[h]
-\bigl(D_\zeta+\tfrac12D_\xi^2\bigr)f.
\label{eq:theta}
\end{align}
For compact generators supported in \(\{\eps<t<T\}\),
\(\ThetaF=0\).
\end{lemma}

\begin{proof}
Differential operators in a formal pullback do not enlarge the
support of each test coefficient; hence the support statement.
Expansion of the definition gives \eqref{eq:theta}.

On the coordinate plateau, the copied off-shell coordinate
identity gives
\(\pair{c[q']}{F^*\psi}=\pair{c[q]}{\psi}\) for every scalar jet.
Formal duality makes the right side
\(\pair{\Fsharp c[q]}{F^*\psi}\).
Since \(F^*\) is formally invertible, the tests agree.
This uses the copied intertwiner, not a new proof of ramp
covariance.
\end{proof}

\subsection{Switch factors displayed through second order}

Let \(\Qop_j\) denote the unswitched positive Euler coefficients,
and set
\(\Nop_j=\Qop_j-\mathsf L_{j,g}\).
Then
\begin{align}
\Nop_1[h]u
&=(1-g)\Qop_1[h]u
-(\partial_\mu g)a_1^{\mu\nu}[h]\partial_\nu u,
\nonumber\\
\Nop_2[h,k]u
&=(1-g)\Qop_2[h,k]u
-(\partial_\mu g)A_2^{\mu\nu}[h,k]\partial_\nu u.
\label{eq:N}
\end{align}
These operators need not have compact coefficients.
Their transformation differences below do.

\begin{lemma}[Switch defect and its retained derivatives]
\label{lem:sigma}
The switch defect has coefficients supported in
\(C=\supp\xi\cup\supp\zeta\), and
\begin{align}
\Sigma^{(1)}
&=-\Nop_1[K\xi]
=(g-1)\Qop_{K\xi}
+(\partial_\mu g)a_1^{\mu\nu}[K\xi]\partial_\nu,
\nonumber\\
\Sigma^{(2)}
&=-\bigl(\Nop_2[h',k']-\Nop_2[h,k]\bigr)
-\Nop_1[h']X_\xi+D_\xi\Nop_1[h].
\label{eq:sigma}
\end{align}
If \(g=1\) on a neighbourhood of \(C\), then \(\SigmaF=0\).
\end{lemma}

\begin{proof}
The copied naturality of the unswitched curved operator
\(\mathsf E_1[q]\), and
\(\Eg[q]=\mathsf E_1[q]-\kappa\Nop_1-\kappa^2\Nop_2\), give
\[
\SigmaF
=-\bigl(\kappa\Nop_1[h']+\kappa^2\Nop_2[h',k']\bigr)F^*
+\Fsharp\bigl(\kappa\Nop_1[h]+\kappa^2\Nop_2[h,k]\bigr).
\]
Expansion gives \eqref{eq:sigma}.
Outside \(C\), the transformation is locally the identity, so
the operator coefficients vanish there.

For an explicit display, put \(X=X_\xi\), \(D=D_\xi\), and
\(\Delta A_2=A_2[h',k']-A_2[h,k]\).
The second line of \eqref{eq:sigma}, acting on \(u\), is
\begin{align}
\Sigma^{(2)}u={}&
(1-g)\left[
-(\Qop_2[h',k']-\Qop_2[h,k])u
-\Qop_1[h']Xu+D\Qop_1[h]u
\right]
\nonumber\\
&+(\partial_\mu g)\Delta A_2^{\mu\nu}\partial_\nu u
+(\partial_\mu g)a_1^{\mu\nu}[h']\partial_\nu(Xu)
\nonumber\\
&-(Xg)\Qop_1[h]u
-\partial_\rho\left[
\xi^\rho(\partial_\mu g)a_1^{\mu\nu}[h]\partial_\nu u
\right].
\label{eq:sigma-expanded}
\end{align}
The last divergence retains derivatives of the switch; expanding
it can produce second derivatives of \(g\).
No such term is silently removed.

On a neighbourhood where \(g=1\), \(1-g\) and all positive-order
derivatives of \(g\) vanish.
Every displayed term is therefore zero on \(C\), and the
operator is zero outside \(C\) already.
\end{proof}

The familiar density-level factors involving \(g\circ F\) are
consistent with \eqref{eq:N}--\eqref{eq:sigma-expanded}.
They are not used as a second residual derivation.
The proof of record below uses the defined defects themselves.

\subsection{Advanced uniqueness and the three-term variation}

Call a smooth formal series advanced-type if each coefficient
is supported in \(J^-(K_0)\) for some compact \(K_0\).

\begin{lemma}[Advanced solution and compact completed test]
\label{lem:advanced}
For \(d\in\Dspace(M)[[\kappa]]/(\kappa^3)\), the equation
\(\Eg[q]w=d\) has a unique advanced-type solution,
\[
w=\EAq d=\EA\Rq d,
\qquad
\Pm w=\Rq d.
\]
\end{lemma}

\begin{proof}
Coefficientwise the equations are
\[
\Pm w_0=d_0,\quad
\Pm w_1=d_1-\Lone w_0,\quad
\Pm w_2=d_2-\Lone w_1-\Ltwo w_0.
\]
Every right side is compact because the interaction coefficients
are compact.
Applying \(\EA\) recursively produces the stated series.

A homogeneous advanced-type solution vanishes on a sufficiently
late Cauchy surface; Cauchy uniqueness makes it zero.
This proves uniqueness inductively.
Finally,
\(\Pm\EA\Rq d=\Rq d\), since the latter is a compact test.
\end{proof}

\begin{proposition}[Three-term variation: proof of record]
\label{prop:variation}
For every classical background and arbitrary compact generators
in the open slab, set
\[
w=\EAq c[q]=\EA e[q].
\]
Then
\begin{equation}
e[q']-e[q]
=\Pm\bigl((F^*-1)w\bigr)
+\mathscr R_{q'}\ThetaF[q]
-\mathscr R_{q'}\bigl(\SigmaF[q]w\bigr).
\label{eq:variation}
\end{equation}
The first term belongs coefficientwise to
\(\Pm\Dspace(M)\).
The other two are the dressed coordinate and switch defects.
In particular,
\begin{align}
e_1'-e_1
={}&\Pm(X_\xi\EA f)
+\Theta^{(1)}-\Sigma^{(1)}\EA f,
\label{eq:var1}\\
e_2'-e_2
={}&\Pm\left[
X_\xi\EA e_1+
\bigl(X_\zeta+\tfrac12X_\xi^2\bigr)\EA f
\right]
\nonumber\\
&+\Theta^{(2)}-\Sigma^{(2)}\EA f
-\Sigma^{(1)}\EA e_1
\nonumber\\
&-\Lone[h']\EA
\bigl(\Theta^{(1)}-\Sigma^{(1)}\EA f\bigr).
\label{eq:var2}
\end{align}
\end{proposition}

\begin{proof}
Write \(w'=\mathsf E_{q'}^{\mathrm A}c[q']\).
By advanced uniqueness,
\(\Eg[q]w=c[q]\), \(\Eg[q']w'=c[q']\), and
\(\Pm w=e[q]\), \(\Pm w'=e[q']\).

By definition of the defects,
\[
\Eg[q']F^*w
=\Fsharp c[q]+\SigmaF w
=c[q']-\ThetaF+\SigmaF w.
\]
Hence
\[
\Eg[q'](w'-F^*w)=\ThetaF-\SigmaF w.
\]
This is a compact source.
The difference is advanced-type: formal compactly supported
pullback changes \(w\) only by compactly supported coefficients,
so no causal-preservation assumption about \(F\) is needed.

Lemma~\ref{lem:advanced} therefore gives
\[
w'-F^*w
=\mathsf E_{q'}^{\mathrm A}(\ThetaF-\SigmaF w).
\]
Apply \(\Pm\), and use
\(\Pm\mathsf E_{q'}^{\mathrm A}=\mathscr R_{q'}\), to obtain
\eqref{eq:variation}.

Every coefficient of \((F^*-1)w\) has compact generator support.
Thus its \(\Pm\)-image is an explicit equation-ideal test.
Expand \(w_0=\EA f\), \(w_1=\EA e_1\), and
\(\mathscr R_{q'}=1-\kappa\Lone[h']\EA+O(\kappa^2)\) to obtain
\eqref{eq:var1}--\eqref{eq:var2}.
\end{proof}

The last line of \eqref{eq:var2} is essential.
It propagates the first-order coordinate and switch defects
through the retarded response.
No plateau condition has been used in deriving this formula.

\begin{theorem}[Completed background Ward identity]
\label{thm:ward}
For every \(F\in\Gp\), with the free field held fixed,
\[
\widehat A[F^*q]=\widehat A[q]\quad\text{in }\Afree\formal.
\]
Equivalently, \(\widehat A_1[h']=\widehat A_1[h]\) and
\(\widehat A_2[h',k']=\widehat A_2[h,k]\).
\end{theorem}

\begin{proof}
The coordinate plateau gives \(\ThetaF=0\) by
Lemma~\ref{lem:theta}.
The interaction plateau gives \(\SigmaF=0\) by
Lemma~\ref{lem:sigma}: \(g=1\) and \(\dd g=0\) on a neighbourhood
of both generator supports.

Equation \eqref{eq:variation} reduces to
\[
e[q']-e[q]=\Pm((F^*-1)w),
\]
with a compact function in each coefficient.
Every difference therefore lies in the specified equation ideal.
Passing to free-field classes proves the theorem.
\end{proof}

The cancellation uses exactly the frozen algebra.
Off shell the difference is the displayed equation-ideal
representative, not generally zero.
Pullback of the scalar is not asserted to be an automorphism of
the unchanged free shell.
The statement is background covariance at fixed free field.

\begin{corollary}[Route II cross-check, not the proof of record]
\label{cor:routeII}
The normalization to X-A's classical coefficient and its on-shell
covariance provide an independent pairing cross-check.
\end{corollary}

\begin{proof}
If a compact test \(d\) annihilates every smooth free solution,
take the solution \(u=(\EA-\ER)d'\) for arbitrary compact \(d'\).
Antisymmetry gives \((\EA-\ER)d=0\).
The Green-operator exact sequence then gives
\(d\in\Pm\Dspace(M)\)
\cite[Theorem~3.4.7]{BGP}.

By normalization and X-A's on-shell identity, the compact test
\(e_2[q']-e_2[q]\) annihilates every smooth free solution.
It therefore lies in \(\Pm\Dspace(M)\).
\end{proof}

This pairing argument does not display switch or edge terms.
It confirms the result but is not the proof of record;
Proposition~\ref{prop:variation} supplies that proof.

%=====================================================================
\section{Edge and switch defects retained outside the claim}
\label{sec:defects}
%=====================================================================

For arbitrary compact generators, passing \eqref{eq:variation}
to the free algebra gives
\[
\widehat A[q']-\widehat A[q]
=\field\bigl(\mathscr R_{q'}(\ThetaF-\SigmaF w)\bigr).
\]
The two mechanisms remain separate.

For an excluded edge parameter, the copied first-coordinate gap is
\[
\xi-\Xone[K\xi]
=\ERzero((1-\sig)\Box\xi)
+\ERzero(\sig'a_1^{0\cdot}[K\xi]).
\]
X-A's witness
\(\xi=\alpha(t)\beta(\mathbf x)\partial_1\), with
\(\alpha\geq0\), nonzero and supported in \((0,\eps)\), and
\(\int\beta\,\dd^3x=1\), gives
\[
\int_{\R^3}(\Xone^1[K\xi]-\xi^1)(t,\mathbf x)\,\dd^3x
=-\int_0^\eps\sig'\alpha\,\dd s<0
\]
for \(\eps\leq t\leq T\).

If the generators also lie in an existing interaction plateau,
then \(\SigmaF=0\), while
\[
\delta\widehat A_1
=\field(\Theta^{(1)}),\qquad
\delta\widehat A_2
=\field\bigl(\Theta^{(2)}-\Lone[h']\EA\Theta^{(1)}\bigr).
\]
The second coefficient retains the propagated first-order gap.
Its independent coordinate contribution is the explicit test
difference in \eqref{eq:theta}, involving the second coordinate
coefficient; it is not asserted to reduce to the first gap alone.

A nonzero coordinate witness does not prove a nonzero total
smeared on-shell response for every fixed \(f,g\).
The observation test may miss it, an output test may be an
equation zero, and switch terms may also be present.
No second-order map can repair a genuinely nonzero first-order
coefficient.

For generators outside the interaction plateau,
\(\Sigma^{(1)},\Sigma^{(2)}\) and their dressed terms in
\eqref{eq:var1}--\eqref{eq:var2} are retained.
They are not absorbed into a contact or into a revised gauge
algebra.
On \(\ga\), by contrast, both defects vanish before taking
classes, so the stop condition of C4 is not triggered.

At \(g=0\), both Euler coefficients and both new maps vanish.
The bare coordinate dressing remains.
Since the interaction plateau is empty, this specialization
does not introduce a new Ward algebra.

%=====================================================================
\section{Two diagnostics}
\label{sec:diagnostics}
%=====================================================================

\begin{diagnostic}[Pure-gauge operator check on the complete two-jet]
\label{diag:puregauge}
Let \(\xi\in\ga\), \(\zeta=0\), and take
\[
h=K\xi,\qquad k=\tfrac12\Lie_\xi K\xi.
\]
Write \(X=X_\xi\), \(D=D_\xi\), \(L=\Lone[K\xi]\), and
\(M_2=\Ltwo[K\xi,\tfrac12\Lie_\xi K\xi]\).
Expanding the copied plateau Euler covariance on the flat
configuration gives
\begin{align}
L&=D\Pm-\Pm X,
\nonumber\\
M_2&=\tfrac12D^2\Pm-\tfrac12\Pm X^2-LX
=\tfrac12D^2\Pm+\tfrac12\Pm X^2-D\Pm X.
\label{eq:pg-operators}
\end{align}

Using \(\Pm\EA f=f\) and \(\EA\Pm v=v\) for compact \(v\),
\begin{align*}
L\EA f&=Df-\Pm(X\EA f),\\
L\EA L\EA f
&=D^2f-\Pm(X\EA Df)-D\Pm(X\EA f)
+\Pm(X^2\EA f),\\
M_2\EA f
&=\tfrac12D^2f+\tfrac12\Pm(X^2\EA f)
-D\Pm(X\EA f).
\end{align*}
Therefore
\[
L\EA L\EA f-M_2\EA f
=\tfrac12D^2f+
\Pm\bigl(\tfrac12X^2\EA f-X\EA Df\bigr).
\]
The function inside \(\Pm\) is compact because of \(X\).
The new maps consequently sum in \(\Afree\) to
\[
\tfrac12\field(D^2f)
=\tfrac12\int fX_\xi^2\field\,\dd^4x.
\]

The copied bare second dressing is
\(\tfrac12\int fX_\xi^2\field\), and the mixed term is
\(-\int fX_\xi^2\field\).
Thus
\[
\widehat A_2[K\xi,\tfrac12\Lie_\xi K\xi]
=\bigl(\tfrac12-1+\tfrac12\bigr)
\int fX_\xi^2\field\,\dd^4x
=0.
\]
The new maps cancel the incomplete remainder with
\(\Obs^{(2)}\) retained.
Taking \(k=0\) instead would not test a pure-gauge two-jet.
\end{diagnostic}

\paragraph{Correction to the reversed-sign shorthand.}
Reversing only the second-vertex sign gives
\(L\EA L\EA f+M_2\EA f\), whose class is
\[
\tfrac32\field(D^2f)-2\field(D\Pm(X\EA f)).
\]
The second term is not generally an equation zero.
Thus a universal assertion that the reversed new sum is simply
\(\tfrac32\int fX_\xi^2\field\) is incorrect.
What is universal is that reversal adds
\(2\field(M_2\EA f)\) to the correctly completed coefficient.
The operator computation above, rather than the shorthand,
tests the sign.

For an explicit sign-failure diagnostic, choose admissible
massless data with a sufficiently large plateau cylinder,
\(\xi=a(t)\partial_t\) near the relevant causal compact,
a spatial cutoff outside it, and nonzero compact \(a\) supported
before the observation times.
On the free solution \(u=t\),
\[
M_2t=\partial_t\bigl(\tfrac12(a')^2-\tfrac12aa''\bigr),
\qquad
\ER M_2t=\int_0^t(a')^2\,\dd s-\tfrac12aa'.
\]
After the support of \(a\), reversal therefore produces
\(2\int(a')^2\,\dd s>0\), although \(X^2t=0\) there.
A nonnegative observation bump detects this nonzero classical
evaluation.
This also shows why the extra term cannot be discarded.

\begin{diagnostic}[Sourced conformal evaluation: a classical number]
\label{diag:sourced}
Let \(m=0\), \(h=\psi\eta\), and \(k=0\).
Then
\[
a_1[h]=\psi\eta,\qquad b_1[h]=2\psi,\qquad
b_2[h]=\psi^2,\qquad a_2[\psi\eta]=0.
\]
Indeed, for the formal conformal metric the density
\(\sqrt{-q}\,q^{-1}=(1+\kappa\psi)\eta\) is exactly linear.
Thus \(\Ltwo=0\) in this massless example:
only the iteration supplies the second scalar response.

Take \(\psi=\psi(t)\), with smooth support away from zero, and
\(g=g_0(t)\) near the relevant causal compact.
Spatial cutoffs outside that compact make the backgrounds and
switch admissible without affecting the evaluation.
Define
\[
\Psi=\int_0^t\sig\psi\,\dd s,\qquad
\Psi_2=\int_0^t(\sig\psi)^2\,\dd s,\qquad
\Gamma=\int_0^tg_0\psi\,\dd s,\qquad
\Gamma_2=\int_0^t(g_0\psi)^2\,\dd s.
\]
On the smooth free solution \(u=t\),
\[
\Xone^0=-\Psi,\quad \Xtwo^0=\Psi_2,\quad
\rho_{1,g}=-\Gamma,\quad \rho_{2,g}=\Gamma_2,
\]
and the spatial coordinate coefficients vanish in the observation
region.

The bare second dressing gives \(\Psi\sig\psi-\Psi_2\);
the mixed term gives \(-\Psi g_0\psi\);
the iteration gives \(\Gamma_2\).
Consequently
\[
\pair{e_2}{t}
=\int f\bigl(\Gamma_2-\Psi_2+\Psi\psi(\sig-g_0)\bigr)\,\dd^4x.
\]
The first coefficient similarly gives
\(\int f(\Psi-\Gamma)\,\dd^4x\).
Both numbers vanish if \(g_0\psi=\sig\psi\) throughout the
contributing past interval.

These are classical evaluations of the fixed representatives,
not elements of \(\Afree\), states on the Wick algebra, or quantum
expectation values.
The calculation checks signs and the retained iteration;
it is not an extension or Ward proof.
\end{diagnostic}

%=====================================================================
\section{Decision and final release}
\label{sec:decision}
%=====================================================================

\begin{center}
\small
\renewcommand{\arraystretch}{1.12}
\begin{tabularx}{\linewidth}{@{}p{.15\linewidth}p{.48\linewidth}X@{}}
\toprule
Gate & Evidence & Status\\
\midrule
C1 &
Copied \(V_{2,g}\), Hessian \(-\Ltwo\), retained switch derivatives &
Frozen\\
C2 &
Output \(-\Ltwo\EA c\); support and continuity &
Constructed\\
C3 &
Output \(\Lone\EA\Lone\EA c\); actual intermediate test &
Constructed on linear insertions\\
Polarization &
Diagonal quadratic dependence; mixed bookkeeping retained &
No diagonal additivity\\
Normalization &
Transposition to the copied \(\rho_{2,g}\) &
Passed\\
Quantum order &
Yang--Feldman test recursion and rooted five-leg count &
Tree-exact on the named domain\\
C4 &
Three-term variation; explicit compact equation-ideal term &
Ward through \(\kappa^2\)\\
Pure gauge &
Complete two-jet; operator calculation;
\(\tfrac12-1+\tfrac12=0\) &
Closed\\
Outside \(\ga\) &
Dressed switch and edge defects, including propagated gap &
Retained\\
C5 &
Fixed \(\eps\) only &
Sharp limit unopened\\
Branch B &
No metric action or contraction domain &
Not opened\\
\bottomrule
\end{tabularx}
\end{center}

The \(\eps^{-r}\) upper bounds of X-B-v3 are not VII-C's
hypotheses (E1)--(E4); no \(\eps\to0\) limit is taken.

The quantum metric, graviton tree, and triangle remain separate
successor problems.
A partial diagonal in a schematic graph is not by itself a
proved obstruction.
No statement about those graphs is imported into this release.

\begin{theorem}[Fixed-regulator X-C release]
\label{thm:release}
On the frozen X-B-v3 sheet, with the scalar quantum and \(h,k\)
classical:
\begin{enumerate}[label=\textup{(\roman*)}]
\item The maps
\[
\Rone(V_{2,g};F_c)=\field(-\Ltwo\EA c),\qquad
\tfrac12\Rtwo(V_{1,g},V_{1,g};F_c)
=\field(\Lone\EA\Lone\EA c)
\]
exist on the frozen off-shell linear tests.
Their outputs are smooth compact tests supported in
\(\supp g\cap J^-(\supp c)\), continuously dependent on the
backgrounds, with no positive-\(\hbar\) remainder.
\item Their sum on \(F_f\) equals the copied
\(\int f\rho_{2,g}\).
\item The completed \(\widehat A_2\) is background-gauge invariant
in \(\Afree\) on \(\ga\), with the free field held fixed.
Together with \(\widehat A_1\), it makes the completed observable
invariant through \(\kappa^2\).
\end{enumerate}
The pure-gauge remainder is cancelled with \(\Obs^{(2)}\)
retained.
No sharp limit, quantum metric, graviton tree, triangle,
\(\mathsf T_2\), \(\mathsf T_3\), nonlinear-insertion \(\Rtwo\),
or two-point response is constructed.
\end{theorem}

\begin{proof}
Theorem~\ref{thm:maps} gives existence, support, continuity, and
normalization.
Theorem~\ref{thm:yf} and Proposition~\ref{prop:hbar} give the
restricted linear-field characterization and quantum order.
Proposition~\ref{prop:variation} is the proof of record for the
variation, and Theorem~\ref{thm:ward} cancels its defects on the
frozen algebra.
Diagnostic~\ref{diag:puregauge} verifies the missing cancellation
by an explicit operator computation.
\end{proof}

The release box, variation theorem, and final theorem have the
same order and domain.
The Route II pairing remains a cross-check, not the proof.

%=====================================================================
\section*{Declarations}
\addcontentsline{toc}{section}{Declarations}

\paragraph{Dependence on supplied manuscripts.}
The ramp convention sheet, coordinate laws, complete metric
pullback, density coefficients, scalar responses, coordinate
intertwiner, unswitched naturality, and plateau covariance are
premises from \texttt{X-A.tex}.
Admission, the free algebra, bare tests, first scalar map,
off-shell warning, mixed transformation, and old truncation are
premises from \texttt{X-B-v3.tex}.
Neither ramp covariance nor the joint cone proof is repeated.

\paragraph{Construction and normalization.}
The new maps are coefficients of the explicitly defined
test-level retarded M\o ller map.
The second vertex and signs are fixed before the Ward
calculation.
No contact, subtraction, or diagonal extension is selected to
enforce covariance.
No uniqueness among arbitrary unconstructed retarded-product
hierarchies is asserted.

\paragraph{Corrections incorporated.}
Formal support statements are coefficientwise.
Differentiated switch factors are retained.
The first-order gap is propagated at second order.
Intermediate tests are actual representatives.
Diagonal iteration is quadratic rather than additive.
The reversed-sign pure-gauge shorthand is replaced by its full
operator expression and an explicit nonzero diagnostic.

\paragraph{Verification status.}
No successful \texttt{pdflatex} execution, proof-assistant check,
numerical verification, or independent referee review is claimed.
The source uses no external figures, bibliography database, or
executable code.

%=====================================================================
\begin{thebibliography}{9}
\addcontentsline{toc}{section}{References}

\bibitem{XA}
N.~Cheung.
\newblock \emph{Landmark X-A: Classical covariance of one
fixed-regulator switched relational scalar through order
\(\kappa^2\)}.
\newblock Supplied manuscript \texttt{X-A.tex}.
\newblock Used for the frozen action, density and coordinate
coefficients, naturality, plateau covariance, classical responses,
and edge witness.

\bibitem{XB}
N.~Cheung.
\newblock \emph{Landmark X-B: Fixed-regulator admission and one
retarded correction for the ramp relational scalar, with the
metric kept classical}.
\newblock Supplied manuscript \texttt{X-B-v3.tex}.
\newblock Used for admission, the free algebra, insertion tests,
first map, off-shell warning, mixed transformation, and
second-order truncation.

\bibitem{BGP}
C.~B\"ar, N.~Ginoux, and F.~Pf\"affle.
\newblock \emph{Wave Equations on Lorentzian Manifolds and Quantization}.
\newblock European Mathematical Society, 2007.
\newblock arXiv:0806.1036.
\newblock Chapter~3 supplies Green-operator and Cauchy facts;
Theorem~3.4.7 supplies the exact sequence used in the pairing
cross-check.

\bibitem{FR}
K.~Fredenhagen and K.~Rejzner.
\newblock \emph{Perturbative algebraic quantum field theory}.
\newblock arXiv:1208.1428.
\newblock Used here for the finite polynomial Wick contraction
presentation in the local leg count, not for a general product
or closure theorem.

\bibitem{DF}
M.~D\"utsch and K.~Fredenhagen.
\newblock Causal perturbation theory in terms of retarded products,
and a proof of the Action Ward Identity.
\newblock \emph{Reviews in Mathematical Physics}
\textbf{16} (2004), 1291--1348.
\newblock arXiv:hep-th/0403213.
\newblock Comparison background for off-shell retarded-product
terminology; no complete hierarchy from that programme is
claimed as constructed here.

\end{thebibliography}

\end{document}