% !TEX program = pdflatex
%=====================================================================
% X-B.tex
%
% Standalone source for pdfLaTeX.
%
% Build:
%   pdflatex -interaction=nonstopmode -halt-on-error X-B.tex
%   pdflatex -interaction=nonstopmode -halt-on-error X-B.tex
%
% Frozen field content:
%   phi is the free quantum Klein--Gordon field.
%   h and k are classical, off-shell backgrounds.
%
% Release:
%   Bare dressing admitted through kappa^2.
%   One normalized map R_1(V_{1,g}; .) on frozen off-shell tests.
%   Tree-exactness, compact retarded output, and continuity.
%   Completed first-order background Ward identity.
%
% Order boundary:
%   No full second-order quantum Ward identity.
%   rho_{2,g} is displayed as classical input, not supplied by
%   a constructed second-order quantum map.
%
% Debugging incorporated:
%   - V_{1,g}'' = -L_{1,g}.
%   - The mixed pure-gauge contribution has a minus sign.
%   - Unweighted R_1 and the physical correction kappa R_1
%     are distinguished.
%   - R_1 is not defined on on-shell classes in its input.
%   - Scalar supports are inclusions, not universal equalities.
%   - Joint kernel products use independent slots and proper
%     pushforwards.
%   - Sharp action-level and multiplied-equation prescriptions
%     are distinguished.
%   - Edge witnesses do not imply a nonzero response for every
%     fixed f and g.
%   - The zero-switch witness explicitly overlaps the positive ramp.
%   - Epsilon estimates are upper bounds, not necessary blow-up rates.
%   - No graviton graph obstruction is asserted without analysis.
%
% No successful compilation, mechanized verification, or independent
% referee review is claimed.
%=====================================================================

\documentclass[11pt,a4paper]{article}

\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{lmodern}
\usepackage[margin=1in]{geometry}
\usepackage{microtype}
\usepackage{amsmath,amssymb,amsthm,mathtools,mathrsfs}
\usepackage{array,booktabs,tabularx}
\usepackage{enumitem}
\usepackage[hidelinks]{hyperref}

\allowdisplaybreaks
\setlength{\emergencystretch}{3em}
\setlength{\parskip}{2pt}
\setlist{itemsep=3pt,topsep=5pt,leftmargin=*}
\numberwithin{equation}{section}
\setcounter{tocdepth}{1}

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

\DeclareMathOperator{\supp}{supp}
\DeclareMathOperator{\WF}{WF}
\DeclareMathOperator{\diag}{diag}
\DeclareMathOperator{\Int}{int}

\newcommand{\R}{\mathbb R}
\newcommand{\eps}{\varepsilon}
\newcommand{\dd}{\,\mathrm d}
\newcommand{\Lie}{\mathcal L}
\newcommand{\sig}{\sigma_\eps}
\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{\Pmc}{\mathscr P_{\mathrm{mc}}}
\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{\EAzero}{\mathsf E_0^{\mathrm A}}
\newcommand{\Lone}{\mathsf L_{1,g}}
\newcommand{\Ltwo}{\mathsf L_{2,g}}
\newcommand{\Qop}{\mathsf Q}
\newcommand{\Rone}{R_1}
\newcommand{\field}{\widehat\phi}
\newcommand{\formal}{\pmod{\kappa^3}}
\newcommand{\pair}[2]{\left\langle #1,#2\right\rangle}
\newcommand{\Vp}{\overline V_+}
\newcommand{\Vm}{\overline V_-}

\newcommand{\scopebox}[2]{%
  \par\medskip\noindent
  \fbox{\begin{minipage}{%
    \dimexpr\linewidth-2\fboxsep-2\fboxrule\relax}
    \textbf{#1}\par\smallskip #2
  \end{minipage}}
  \par\medskip
}

\title{\textbf{Landmark X-B}\\[0.4em]
Fixed-regulator admission and one retarded correction\\
for the ramp relational scalar, with the metric kept classical}
\author{Ngo Cheung}
\date{}

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

\begin{abstract}
At fixed \(\eps>0\), with \(\phi\) the free quantum Klein--Gordon
field and \(h,k\) classical backgrounds, the bare ramp dressing is
admitted through \(\kappa^2\). We distinguish joint microcausal
admission of its polynomial kernels from the scalar-only linear
test domain actually used for the quantum construction.

The single map \(R_1(V_{1,g};\cdot)\) is constructed on specified
off-shell insertion tests. Its output is a free field with a smooth
compact test, it has the stated retarded support, and it is
tree-exact: all positive-\(\hbar\) corrections vanish. Its action on
the first dressing coefficient equals the mixed term of the
classical ramp expansion.

The completed coefficient \(\widehat A_1\) inherits the ramp
background-gauge identity on the algebra frozen in Landmark X-A.
The bare \(\kappa^2\) dressing is admitted and its classical
transformation is retained, but no full quantum identity at that
order is claimed: \(\rho_{2,g}\) is displayed and is not supplied by
a constructed map. Edge and switch defects remain outside the
claimed algebra. The dressing survives at zero switch. No sharp
limit, graviton tree contribution, loop triangle, 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) {Frozen X-A sheet};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {One closure calculation};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Convolution repair};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {A later ending does not repair the convolution obstruction.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{Three endings, not a repair of the fork. 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-B}
\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 X-A sheet};
\node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {Convolution repair};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {A later ending does not repair the convolution obstruction.};
\draw[->] (a) -- (s);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{Three endings, not a repair of the fork. 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-B}
\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 X-A sheet};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {One closure calculation};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {Convolution repair};
\node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {A later ending does not repair the convolution obstruction.};
\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-B}
\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 X-A sheet};
\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) {Convolution repair};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {One closure calculation. A later ending does not repair the convolution obstruction.};
\draw (a) -- (m);
\draw (m) -- (b);
\draw[->] (m) -- (s);
\end{tikzpicture}
\caption{Frozen X-A sheet is not Convolution 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-B}
\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 X-A sheet};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: One closure calculation};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: Convolution repair};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {A later ending does not repair the convolution obstruction.};
\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-B}
\end{figure}

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


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



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

\scopebox{Release: Constructed, with a fixed order boundary}{%
At fixed \(\eps>0\), with \(\phi\) the free quantum field and \(h,k\)
classical, the bare dressing is admitted through \(\kappa^2\).
The single map \(\Rone(V_{1,g};\cdot)\) is constructed on the frozen
off-shell tests, is tree-exact, and has the stated retarded support.
The completed coefficient \(\widehat A_1\) inherits the ramp identity
on \(\ga\).

\smallskip
The \(\kappa^2\) dressing is admitted and its classical transformation
is retained, but no full quantum identity at that order is claimed:
\(\rho_{2,g}\) is displayed and not supplied by a constructed map.
The edge and switch defects stay outside \(\ga\).
No sharp limit, graviton tree, or loop triangle is constructed.}

Three endings are available:
\begin{enumerate}[label=\textbf{\arabic*.}]
\item \textbf{Constructed.} The named admission condition, retarded
map, support, and stated-order gauge identity are proved.
\item \textbf{Proved obstructed.} An inadmissible joint wavefront set
or a nonzero defect blocks a claimed construction on the frozen
algebra.
\item \textbf{Unresolved.} Field content, insertion domain,
normalization, or an essential map is unspecified.
\end{enumerate}

The ending reached is (1), precisely within the release box.
A missing higher-order map is not a no-go. An incomplete
second-order expression is not an invariant. A defect cannot be
removed by shrinking the frozen algebra after the calculation, or
by selecting a contact afterward.

Here \(\kappa\) counts background perturbation order and \(\hbar\)
counts quantum contractions. Both are formal. Equality modulo
\(\kappa^3\) means coefficient equality, not a numerical remainder
estimate. Admission through \(\kappa^2\) and covariance through
\(\kappa^1\) are deliberately different statements.

%=====================================================================
\section{Field content before any gate}
\label{sec:field}
%=====================================================================

\[
\boxed{\text{Quantum field content: \(\phi\) only;
\(h,k\) are classical backgrounds.}}
\]

\paragraph{Branch A, frozen here.}
The scalar is the free quantum field of
\(P_m=\Box+m^2\) on Minkowski space, \(m\geq0\).
The metric coefficients are smooth, classical, and off shell.
There is no quantum metric propagator and no gravitational action.

The slab and initial surface belong to the classical ramp
prescription. Its coordinate sources vanish for \(t\leq0\), so the
zero-data coordinate inverse agrees with the retarded inverse on
those sources. No quantum Cauchy data are prescribed. The quantum
field is defined on all of Minkowski space.

\paragraph{Branch B, named and not opened.}
Quantizing \(h\) would require a specified quadratic graviton
action, gauge fixing, and a different contraction domain.
None is present in the classical source manuscript.
An additional graviton response would not follow from the
scalar-only normalization below. Its Ward formulation and possible
loop graphs would require separate analysis.

The present theorem makes no assertion that every such successor
graph exists or fails. In particular, a schematic triangle is not
a proved obstruction merely because it contains a partial
diagonal. Branch B is a different construction, not a correction
to a missing term inside Branch A.

%=====================================================================
\section{Copied ramp premises}
\label{sec:handoff}
%=====================================================================

The classical source is the supplied manuscript
\emph{Landmark X-A: Classical covariance of one fixed-regulator
switched relational scalar through order \(\kappa^2\)},
\texttt{X-A.tex} \cite{XA}.
The following are copied premises from that ramp manuscript.
Its covariance proof is not repeated.

\begin{premise}[Geometry, regulator, switch, and algebra]
\label{prem:geometry}
Let
\[
M=\R^4,\quad
\eta=\diag(1,-1,-1,-1),\quad
\Box=\partial_t^2-\Delta,\quad
M_T=[0,T]\times\R^3.
\]
Indices are moved with \(\eta\).
Fix \(0<\eps<T\) and
\[
\sig(t)=\chi(t/\eps),\qquad
\chi(s)=
\begin{cases}
0,&s\leq0,\\[2pt]
\dfrac{e^{-1/s}}{e^{-1/s}+e^{-1/(1-s)}},&0<s<1,\\[7pt]
1,&s\geq1.
\end{cases}
\]
Thus \(\sig\) is smooth and flat at the initial surface,
and \(\sig'>0\) on \(0<t<\eps\).

Fix
\(g\in C_c^\infty((0,T)\times\R^3;[0,1])\),
\(P_g=\Int\{g=1\}\), and the same nonempty relatively compact
open \(U\) as in X-A:
\begin{equation}
\overline U\subset P_g\cap\{\eps<t<T\},
\qquad
\ga=C_c^\infty(U;TM).
\label{eq:algebra}
\end{equation}
Fix a real label-space test
\(f\in C_c^\infty((0,T)\times\R^3)\). Put
\[
K=\supp f,\qquad
\Df=J^-(K)\cap M_T.
\]
The set \(\Df\) is compact.
\end{premise}

\begin{premise}[Metric density and coordinates]
\label{prem:coordinates}
The formal metric is \(q=\eta+\kappa h+\kappa^2k\), with arbitrary
smooth real symmetric \(h,k\). Write \(h=\operatorname{tr}_\eta h\)
inside the following coefficient formulas:
\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}
-\tfrac12h h^{\mu\nu}+b_2[h]\eta^{\mu\nu}.
\label{eq:density-coefficients}
\end{align}
Set
\[
A_2[h,k]=a_1[k]+a_2[h],
\qquad
B_2[h,k]=b_1[k]+b_2[h].
\]
The ramp auxiliary density is
\[
\mathfrak A_\eps[q]
=\eta^{-1}+\sig\bigl(\sqrt{-q}\,q^{-1}-\eta^{-1}\bigr).
\]
It does not contain the interaction switch \(g\).

The coordinate coefficients are
\begin{align}
\Xone^\rho[h]
&=-\ERzero\partial_\mu(\sig a_1^{\mu\rho}[h]),
\label{eq:Xone}\\
\Xtwo^\rho[h,k]
&=-\ERzero\Bigl(
\partial_\mu(\sig A_2^{\mu\rho}[h,k])
+\partial_\mu(\sig a_1^{\mu\nu}[h]\partial_\nu\Xone^\rho[h])
\Bigr).
\label{eq:Xtwo}
\end{align}
The retarded operator is understood on the smooth past-compact
coordinate sources, or locally by finite propagation.
These formulas reproduce the zero-data Cauchy prescription.

With
\[
\mathfrak a^{\nu\rho,\alpha\beta}
=\tfrac12\eta^{\nu\rho}\eta^{\alpha\beta}
-\tfrac12(\eta^{\nu\alpha}\eta^{\rho\beta}
+\eta^{\nu\beta}\eta^{\rho\alpha}),
\]
the linear coordinate kernel is
\begin{equation}
\Lambda_\eps^{\rho,\alpha\beta}(x,z)
=\sig(z)\mathfrak a^{\nu\rho,\alpha\beta}
\partial_\nu^zG_0^{\mathrm R}(x-z).
\label{eq:Lambda}
\end{equation}
Equivalently, the derivative of the argument \(x-z\) gives the
same formula with a minus sign.
The coefficient maps are smooth and continuous in smooth
background topologies at fixed \(\eps\).
Smeared first-derivative kernels are smooth, with no initial
hypersurface delta.
\end{premise}

\begin{premise}[Relational signs and support]
\label{prem:observable}
For a scalar jet
\(\varphi=\varphi_0+\kappa\varphi_1+\kappa^2\varphi_2\), put
\[
B^\mu[h,k]
=\Xone^\nu[h]\partial_\nu\Xone^\mu[h]-\Xtwo^\mu[h,k].
\]
The copied expansion is
\begin{align}
\Phi^{(0)}&=\varphi_0,
\nonumber\\
\Phi^{(1)}&=\varphi_1-\Xone^\mu\partial_\mu\varphi_0,
\nonumber\\
\Phi^{(2)}&=\varphi_2-\Xone^\mu\partial_\mu\varphi_1
+B^\mu\partial_\mu\varphi_0
+\tfrac12\Xone^\mu\Xone^\nu\partial_\mu\partial_\nu\varphi_0.
\label{eq:relational-expansion}
\end{align}
Its smeared coefficients depend on backgrounds only in
neighborhoods of \(\Df\), and continuously at fixed \(\eps\).
Setting \(\varphi_1=\varphi_2=0\) defines the bare dressing used here.
At \(g=0\) this dressing survives: the coordinate prescription is
unchanged.
\end{premise}

\begin{premise}[Gauge laws and their edge boundary]
\label{prem:gauge}
Write
\[
(K\xi)_{\mu\nu}=\partial_\mu\xi_\nu+\partial_\nu\xi_\mu,
\qquad
X_\xi=\xi^\mu\partial_\mu.
\]
For the formal flow of \(\kappa\xi+\kappa^2\zeta\),
\(\xi,\zeta\in\ga\),
\begin{align}
h'&=h+K\xi,
&
k'&=k+\Lie_\xi h+K\zeta+\tfrac12\Lie_\xi K\xi,
\label{eq:metric-law}\\
\Xone[h']&=\Xone[h]+\xi,
&
\Xtwo[h',k']
&=\Xtwo[h,k]+X_\xi\Xone[h]+\zeta+\tfrac12X_\xi\xi.
\label{eq:coordinate-law}
\end{align}
In particular \(\Xone[K\xi]=\xi\) on \(\ga\).

For the excluded edge parameter
\(\xi=\alpha(t)\beta(\mathbf x)\partial_1\), where
\(\alpha\geq0\), \(\alpha\not\equiv0\),
\(\supp\alpha\subset(0,\eps)\), and \(\int\beta\,\dd^3x=1\),
X-A proves
\begin{equation}
\int_{\R^3}(\Xone^1[K\xi]-\xi^1)(t,\mathbf x)\,\dd^3x
=-\int_0^\eps\sig'(s)\alpha(s)\,\dd s<0,
\qquad \eps\leq t\leq T.
\label{eq:edge-gap}
\end{equation}
The coordinate identity fails in general outside the frozen algebra.
This witness is not a defect on \(\ga\).
\end{premise}

\begin{premise}[Scalar action, response, and classical covariance]
\label{prem:scalar}
The first switched scalar vertex is
\begin{equation}
V_{1,g}[h,\varphi]
=\tfrac12\int g\bigl(
a_1^{\mu\nu}[h]\partial_\mu\varphi\partial_\nu\varphi
-m^2b_1[h]\varphi^2\bigr)\,\dd^4x.
\label{eq:vertex}
\end{equation}
Its positive Euler coefficient, denoted \(\mathsf V_{1,g}\) in
X-A, is here
\begin{equation}
\Lone[h]u
=\partial_\mu(g a_1^{\mu\nu}[h]\partial_\nu u)
+gm^2b_1[h]u.
\label{eq:Lone}
\end{equation}
The second coefficient is
\[
\Ltwo[h,k]u
=\partial_\mu(g A_2^{\mu\nu}[h,k]\partial_\nu u)
+gm^2B_2[h,k]u.
\]
For a free smooth solution \(u\), the retarded scalar coefficients are
\begin{align}
\rho_{1,g}[h;u]&=-\ER\Lone[h]u,
\label{eq:rhoone}\\
\rho_{2,g}[h,k;u]
&=\ER\Lone[h]\ER\Lone[h]u-\ER\Ltwo[h,k]u.
\label{eq:rhotwo}
\end{align}

X-A proves simultaneous-pullback covariance off shell, and
background covariance of the on-shell relational observable at
fixed free input, through second classical order on \(\Gp\).
This is classical input, not a quantum product theorem.
\end{premise}

\paragraph{Non-inputs.}
The obstructed convolution coefficient and its X-Enter readout do
not enter any proof. Neither supplies a cancellation for the ramp
observable. VII-A, VI, VII-B, VII-C, IX-E-2, and IX-E-3 are
comparison only. The cone condition associated with VII-B can be
named as a test; its sharp exclusion is not a premise about
\eqref{eq:Lambda}. No earlier boundary topology or kernel
identification is transferred.

%=====================================================================
\section{Product sheet and the off-shell insertion domain}
\label{sec:sheet}
%=====================================================================

\begin{center}
\small
\renewcommand{\arraystretch}{1.12}
\begin{tabularx}{\linewidth}{@{}p{.23\linewidth}X@{}}
\toprule
Item & Declaration\\
\midrule
Regulator &
The same ramp \(\sig=\chi(t/\eps)\), fixed \(\eps>0\);
not a mollifier and no sharp limit\\
Field content &
Only \(\phi\) is quantum; \(h,k\) are classical and off shell\\
Observable &
The bare coefficients \(\Obs^{(0)},\Obs^{(1)},\Obs^{(2)}\)\\
Interaction &
Only \(V_{1,g}\), with physical perturbative order \(\kappa\);
no gravitational vertex\\
Product &
One map \(\Rone(V_{1,g};F_c)\) on specified off-shell
linear insertion representatives\\
Quantum order &
All positive-\(\hbar\) corrections to this map vanish\\
Ward order &
Completed coefficient at \(\kappa^1\) only;
no full quantum identity at \(\kappa^2\)\\
Gauge &
Exactly the ramp algebra \eqref{eq:algebra}\\
Topology &
Smooth background topology and compact-test topology;
no \(\tau_\Sigma\)\\
Normalization &
\(\Rone(V_{1,g};F_f)
=\field(-\Lone[h]\EA f)=\int f\,\rho_{1,g}\)\\
\bottomrule
\end{tabularx}
\end{center}

\subsection{The linear representatives}

Let \(\Dspace(M)=C_c^\infty(M;\mathbb C)\) with its standard LF
topology. On a fixed compact \(K\), use the Fr\'echet space
\(\Dspace_K(M)\) with all smooth seminorms.
Backgrounds belong to the ambient smooth space
\[
\Bg=C^\infty(M;S^2T^*M)\times C^\infty(M;S^2T^*M),
\]
with convergence of all derivatives on compact sets.
References to dependence on a closed causal set mean dependence
on restrictions to its neighborhoods; no unmentioned topology on
a nonsmooth closed set is imposed.

For \(c\in\Dspace(M)\), set
\[
F_c(\varphi)=\int c\varphi\,\dd^4x,\qquad
\Loff=\{F_c:c\in\Dspace(M)\}.
\]
This off-shell insertion domain is identified with \(\Dspace(M)\).

Integration by parts in the bare expansion fixes the tests:
\begin{align}
c_0&=f,
&
c_1[h]&=\partial_\mu(f\Xone^\mu[h]),
\nonumber\\
c_2[h,k]
&=-\partial_\mu(fB^\mu[h,k])
+\tfrac12\partial_\mu\partial_\nu
(f\Xone^\mu[h]\Xone^\nu[h]).
\label{eq:tests}
\end{align}
Thus
\[
\Obs_\kappa
=F_{c_0}+\kappa F_{c_1}+\kappa^2F_{c_2}.
\]
After free-field evaluation, the same notation denotes
\(\field(c_0+\kappa c_1+\kappa^2c_2)\).
The distinction between representative and class will always be
made when applying \(\Rone\).

\subsection{Free Wick algebra and equation ideal}

Use the polynomial microcausal Wick presentation of the free
scalar algebra. Choose a Hadamard bisolution \(\Delta_+\) once,
with
\[
\Delta_+-\Delta_+^{\mathsf T}=i\Delta_S,\qquad
\Delta_S=\EA-\ER.
\]
For the polynomial entries used here, the Wick product is
\[
F\star G
=\sum_{n\geq0}\frac{\hbar^n}{n!}
\pair{F^{(n)}}{\Delta_+^{\otimes n}G^{(n)}}.
\]
The sum is finite for each polynomial pair.
This is the standard free presentation
\cite[Sections~4--5]{FR}; no closure assertion for an arbitrary
class of nonpolynomial smooth functionals is needed.

Let \(\Aoff\) denote this off-shell polynomial algebra.
Since \(\Delta_+\) is a bisolution,
\[
F_{P_mu}\star G=F_{P_mu}G=G\star F_{P_mu}.
\]
The pointwise ideal generated by all \(F_{P_mu}\),
\(u\in\Dspace(M)\), is therefore also a star ideal, denoted
\(\Jfree\). Put
\[
\Afree=\Aoff/\Jfree,\qquad \field(c)=[F_c].
\]
Elements of \(\Jfree\) vanish on smooth free solutions.
In particular,
\[
\field(P_mu)=0,\qquad
[\field(c),\field(d)]
=i\hbar\pair{c}{(\EA-\ER)d}.
\]
The linear output topology is the quotient test topology on
\(\Dspace(M)/P_m\Dspace(M)\).

\subsection{Why the insertion argument is not an on-shell class}

The map constructed below satisfies
\begin{equation}
\Rone(V_{1,g};F_{P_mu})=\field(-\Lone[h]u),
\label{eq:off-shell-warning}
\end{equation}
which need not vanish.
Therefore it is not a map on free on-shell classes in its input.
It is a map on the frozen representatives \eqref{eq:tests},
with output in \(\Afree\).

This warning can be witnessed explicitly. For \(m=0\), take
\(h=\psi\eta\), set \(r=g\psi\), and choose a compact \(r\) with
\(\partial_t r\not\equiv0\). For \(u=\partial_t r\),
evaluation of the output test against the free solution \(t\) gives
\[
\int(-\Lone[h]u)t\,\dd^4x
=-\int u\,\Lone[h]t\,\dd^4x
=-\int(\partial_t r)^2\,\dd^4x\neq0.
\]
Thus even a free-equation-zero insertion can have a nonzero
retarded correction. There is no ambiguity here: the geometric
prescription supplies the insertion test before the map is applied.

%=====================================================================
\section{B1: joint cone admission, then linear admission}
\label{sec:admission}
%=====================================================================

\subsection{The named joint test}

Include zero covectors in the closed future and past causal cones
\(\Vp,\Vm\), and define
\[
\Xi_n=T^*M^n\setminus(\Vp^{\,n}\cup\Vm^{\,n}).
\]
The polynomial microcausal test requires compact derivative kernels
and
\begin{equation}
\WF(H^{(n)}[\Psi])\subset\Xi_n
\quad\text{for every nonzero derivative.}
\label{eq:cone}
\end{equation}
This is the standard cone condition \cite[Section~4]{FR}.
We first apply it to the multiplet \((h,k,\varphi)\).
That joint mathematical test does not quantize the backgrounds.

\begin{definition}[Split type]
A distribution on \(M^n\) is of split type if every wavefront
covector satisfies
\[
p_1+\cdots+p_n=0.
\]
The sum uses the canonical identification of cotangent fibers in
Minkowski coordinates. It is not proposed as a coordinate-free
definition on arbitrary curved spacetimes.
\end{definition}

\begin{lemma}[Split type implies the cone condition]
\label{lem:split-cone}
A split-type distribution satisfies \eqref{eq:cone}.
\end{lemma}

\begin{proof}
If every \(p_i\) is future causal, each time component is
nonnegative. Their sum can vanish only if all time components
vanish, which forces all causal \(p_i\) to vanish.
The zero total covector is excluded from a wavefront set.
The past-cone case is identical.
\end{proof}

\begin{lemma}[Split calculus]
\label{lem:split-calculus}
The following operations preserve split type:
\begin{enumerate}[label=\textup{(\roman*)}]
\item Translation kernels \(u(x-z)\) are split, as are diagonal
delta kernels.
\item Smooth multiplication, differentiation, and adjoining unused
variables preserve split type.
\item If split distributions depend on variable sets \(I,J\) with
one shared slot, their product exists and is split.
\item Proper integration of internal variables preserves split type.
\end{enumerate}
\end{lemma}

\begin{proof}
A translation kernel has endpoint covectors \((p,-p)\).
A diagonal delta has covectors whose sum is zero.
Smooth multiplication and differentiation do not enlarge the
wavefront set, while unused variables carry zero covectors.

For (iii), lift both factors to the union of their variable sets.
A violation of the distributional product criterion would require
opposite wavefront covectors whose sum vanishes in every slot.
All unshared slots would then be zero. Split type of either factor
forces its shared covector to be zero as well, contradicting a
nonzero wavefront covector. The wavefront bound for the product
consists of sums of factor covectors, allowing a zero contribution
from one factor; these sums remain split.

For (iv), the pushforward wavefront bound retains only covectors
with zero internal components. The surviving external components
therefore still sum to zero. Properness is required on the support
over each compact external set. These are the usual product and
pushforward rules \cite[Chapter~8]{Hormander}.
\end{proof}

\begin{proposition}[Joint admission through second order]
\label{prop:joint}
The bare coefficients \(F_{c_0},F_{c_1},F_{c_2}\), regarded as
polynomial functionals of \((h,k,\varphi)\), have compact derivative
kernels of split type. Consequently they satisfy \eqref{eq:cone}.
\end{proposition}

\begin{proof}
The zeroth coefficient has a smooth derivative.
The first is bilinear in \(h,\varphi\); its mixed kernel is a
derivative of
\[
f(x)\Lambda_\eps(x,z).
\]
By \eqref{eq:Lambda}, this is a smooth multiplier times a derivative
of a translation kernel, hence split.

The second coefficient is linear in \(\varphi\), quadratic in
\(h\), and linear in \(k\). Its highest kernels are generated by:
\begin{align*}
&f(x)\Lambda_\eps(x,z)\Lambda_\eps(x,z'),\\
&f(x)G_0^{\mathrm R}(x-w)
\,\partial_w\bigl(\sig(w)\delta(w-z)\delta(w-z')\bigr),\\
&f(x)G_0^{\mathrm R}(x-w)
\,\partial_w\bigl(
\sig(w)\delta(w-z)\partial_w\Lambda_\eps(w,z')
\bigr),
\end{align*}
with fixed tensors, further derivatives, and permutations of
background slots suppressed. The \(k\)-linear term is the
corresponding two-slot kernel.

For the first expression, the two coordinate factors share only
\(x\). For the local quadratic expression, the two delta factors
share only \(w\), and the resulting distribution meets the Green
kernel only at \(w\). In the nested expression, attach the delta
factor to the inner coordinate kernel at \(w\), then attach the
outer Green kernel at that same single slot. Each step is covered
by Lemma~\ref{lem:split-calculus}. No square of a propagator on
identified endpoints has been introduced.

Integrate the internal variable \(w\). Retarded support and the
factors \(\sig\), together with the compact observation support,
confine every relevant vertex to \(\Df\). Thus the pushforwards
are proper; a smooth compact cutoff equal to one there can be
inserted if desired. Lower derivatives are obtained by smearing
remaining slots against smooth configurations, with the same
compact localization.

All kernels are therefore split and compact. The flat ramp
introduces no distributional boundary multiplier at \(t=0\).
Lemma~\ref{lem:split-cone} proves the claimed admission.
\end{proof}

\begin{remark}[Sharp-step comparison, not a transfer theorem]
The preceding proof uses smooth multiplication by \(\sig\).
A sharp step has
\(\WF(\theta(t))=\{(t=0;\lambda\,\dd t):\lambda\neq0\}\);
it is not a smooth multiplier in the split calculus.

The two sharp prescriptions must also be distinguished.
The sharp action-level adjoint corresponding to the divergence-form
coordinate source is schematically
\(\theta C^*b\). The multiplied-equation prescription has adjoint
\(C^*(\theta b)=\theta C^*b+\delta(t)\mathsf B(b|_{t=0})\).
A nonzero boundary coefficient can therefore introduce a timelike
conormal; the latter prescription explicitly has a delta term.
The fixed ramp has neither sharp multiplier.

This explains the removal of the sharp initial-conormal mechanism
at fixed \(\eps\). It does not identify the ramp object with
VII-B's insertion, prove a sharp-limit theorem, or construct a
retarded product with a quantum metric.
\end{remark}

\subsection{The scalar-only domain actually used}

\begin{proposition}[Linear admission]
\label{prop:linear}
For fixed classical backgrounds, every \(c_j\) in
\eqref{eq:tests} is smooth and supported in \(K=\supp f\).
The maps \((h,k)\mapsto c_j\in\Dspace_K(M)\) are continuous
polynomial maps at fixed \(\eps\). Each \(F_{c_j}\) belongs to
\(\Loff\), and its class \(\field(c_j)\) is well defined.
\end{proposition}

\begin{proof}
The coordinate coefficients are smooth and continuous by the
handoff. Formula \eqref{eq:tests} uses smooth products and
derivatives multiplied by \(f\), so its support is contained in
\(K\), with continuous dependence in all compact-test seminorms.
As a functional of the only quantum configuration, \(F_{c_j}\)
has smooth first derivative and all higher derivatives zero.
\end{proof}

The scalar support is contained in \(K\), whereas background
dependence may extend throughout \(\Df\). Neither inclusion is a
claim of equality for every configuration.

The retarded construction uses only:
\begin{enumerate}[label=\textup{(A\arabic*)}]
\item A quadratic insertion
\(V=-\tfrac12\pair{\varphi}{\mathsf L\varphi}\),
where \(\mathsf L\) is formally self-adjoint with smooth compact
coefficients.
\item A specified linear representative \(F_c\), with
\(c\in\Dspace(M)\).
\end{enumerate}
Joint admission is additional information, not a substitute for
this named quantum domain.

%=====================================================================
\section{B2: one retarded map, its sign, support, and quantum order}
\label{sec:retarded}
%=====================================================================

\subsection{The sign ledger}

Compactly supported integration by parts gives
\begin{equation}
V_{1,g}'[\varphi]=-\Lone[h]\varphi,\qquad
V_{1,g}''=-\Lone[h].
\label{eq:Hessian}
\end{equation}
The free action has \(S_0''=-P_m\), hence
\[
\Delta_S^{\mathrm R}=-\ER,\qquad
\Delta_S^{\mathrm A}=-\EA.
\]
The positive Euler coefficient is not the variational Hessian.
Using \(+\ER\) as the inverse of \(S_0''\) would reverse the
response and violate the normalization sheet.

For a quadratic insertion as in (A1), the classical retarded
response of \(F_c\) is
\[
-\pair{c}{\Delta_S^{\mathrm R}V'}
=-\pair{c}{\ER\mathsf L\varphi}.
\]
This also follows by differentiating the retarded solution of
\((S_0+\lambda V)'=0\).

\begin{definition}[The single normalized map]
\label{def:retarded}
For the specified representative \(F_c\), define
\begin{equation}
\Rone(V_{1,g}[h];F_c)
=\field(d_c[h]),\qquad
d_c[h]=-\Lone[h]\EA c.
\label{eq:Rone}
\end{equation}
The notation
\(-\pair{c}{\ER\Lone[h]\field}\) means this transposed expression.
It does not mean applying an inverse to an unsmeared quantum field
without first defining its test action.
\end{definition}

\begin{theorem}[Existence, normalization, continuity, and support]
\label{thm:retarded}
The map \eqref{eq:Rone} is bilinear in the quadratic insertion and
the linear representative. Its output test is smooth and obeys
\begin{equation}
\supp d_c[h]\subset\supp g\cap J^-(\supp c).
\label{eq:retarded-support}
\end{equation}
It is continuous in the smooth background and compact-test
topologies. It satisfies
\[
\Rone(V_{1,g};F_f)
=\field(-\Lone[h]\EA f)
=\int f\,\rho_{1,g}[h;\field]\,\dd^4x.
\]
It is additive in the vertex and vanishes if
\(\supp g\cap J^-(\supp c)=\varnothing\).
No new diagonal extension or subtraction is used.
\end{theorem}

\begin{proof}
The advanced Green operator maps compact smooth tests to smooth
functions supported in their causal past.
The coefficients of \(\Lone[h]\) are smooth and supported in
\(\supp g\). Therefore \(-\Lone[h]\EA c\) is a smooth compact
test with \eqref{eq:retarded-support}.

Formal symmetry of \(\Lone[h]\) and
\((\ER)^{\mathsf T}=\EA\) give
\[
-\pair{c}{\ER\Lone[h]\varphi}
=-\pair{\Lone[h]\EA c}{\varphi}.
\]
Compact vertex coefficients justify the integrations by parts.
Green-operator continuity on compact causal regions
\cite[Chapter~3]{BGP}, followed by continuous smooth
multiplication and differentiation, proves continuity.
For fixed input support, the output has a fixed compact support.
Additivity and retarded vanishing are immediate.

Only one Green distribution is transposed against a smooth test.
There is no singular product requiring extension on a diagonal.
\end{proof}

For all admitted bare coefficients,
\[
\Rone(V_{1,g};\Obs_\kappa)
=\field(d_{c_0}+\kappa d_{c_1}+\kappa^2d_{c_2}).
\]
This is the unweighted map. The physical first-vertex correction
has an additional factor \(\kappa\).
These two notions are not interchangeable.

The support theorem places the correction's scalar test in
\(\supp g\cap J^-(K)\). It does not remove the original dressing's
background dependence on \(\Df\). At this order, the causal
statement consists of additivity and retarded vanishing; no
complete causal-factorization hierarchy is claimed.

\subsection{One-contraction quantum verification}

To verify the quantum interpretation without constructing a
second product programme, introduce the auxiliary kernel
\[
\Delta_{\mathrm F}
=\Delta_++i\Delta_S^{\mathrm A}
=\Delta_+-i\EA.
\]
It is symmetric with these conventions. Both
\(\Delta_{\mathrm F}c\) and \(\Delta_+c\) are smooth.
For the pair (A1)--(A2), consider only the auxiliary contraction
expression
\[
\mathcal C_{\mathrm F}(V,F_c)
=VF_c+\hbar\pair{V'}{\Delta_{\mathrm F}c}.
\]
It is well defined because pairing \(V'\) with a smooth function
requires only compact vertex coefficients.

The Wick expression terminates exactly:
\[
V\star F_c=VF_c+\hbar\pair{V'}{\Delta_+c}.
\]
Consequently
\begin{align}
\frac{i}{\hbar}
\bigl(\mathcal C_{\mathrm F}(V,F_c)-V\star F_c\bigr)
&=-\pair{V'}{\Delta_S^{\mathrm A}c}
\nonumber\\
&=-\pair{c}{\Delta_S^{\mathrm R}V'}.
\label{eq:contraction-check}
\end{align}
For \(V'=-\mathsf L\varphi\), this is exactly
\(-F_{\mathsf L\EA c}\).
The auxiliary expression supplies the usual one-contraction
retarded interpretation \cite{DF}; it does not constitute a
claimed family of time-ordered maps.

\begin{theorem}[Tree-exactness and absence of positive-\(\hbar\) terms]
\label{thm:hbar}
On the named scalar-only domain,
\(\Rone(V_{1,g};F_c)\) is independent of \(\hbar\) and of the
Hadamard contraction kernel:
\[
[\Rone]_{\hbar^r}=0,\qquad r\geq1.
\]
In particular, the first quantum correction vanishes identically.
\end{theorem}

\begin{proof}
A cross-contraction differentiates each entry once.
Since \(F_c^{(2)}=0\), no term with two cross-contractions exists.
The zero-contraction product cancels in
\eqref{eq:contraction-check}. Its single contraction contributes
one \(\hbar\), removed by \(i/\hbar\), and \(\Delta_+\) cancels.
There is no remaining positive power of \(\hbar\).

The vertex is represented in a fixed free Wick ordering.
Changing that ordering for a quadratic vertex changes it only by
a scalar-independent term, whose first derivative is zero.
Thus it cannot alter this map.
\end{proof}

This is a theorem about the declared field content.
It does not hide a missing graviton contraction or loop.
It also makes no uniqueness claim among arbitrary additional
contact normalizations: the map is the explicit normalized
one-contraction prescription above, with no contact selected to
repair an identity.

\subsection{The mixed ramp coefficient}

Applying the map to the first frozen test gives
\begin{equation}
\Rone(V_{1,g};F_{c_1})
=-\int f\,\Xone^\mu[h]\,
\partial_\mu\rho_{1,g}[h;\field]\,\dd^4x.
\label{eq:mixed}
\end{equation}
Indeed,
\(\pair{c_1}{\rho_{1,g}}
=\pair{\partial_\mu(f\Xone^\mu)}{\rho_{1,g}}\).
Equation \eqref{eq:mixed} is the copied classical mixed term,
with the scalar replaced by the free field through its test action.
The minus insertion sign is fixed; the plan's plus sign is rejected.

%=====================================================================
\section{B3: first-order Ward identity and the second-order boundary}
\label{sec:ward}
%=====================================================================

\subsection{The three-term variation formula}

Let
\[
D_\xi r=\partial_\mu(\xi^\mu r),
\qquad D_\xi^{\mathsf T}=-X_\xi,
\]
and write \(\Qop_h=\mathsf L_{1,g=1}[h]\).

\begin{lemma}[Pure-gauge Euler coefficient]
\label{lem:Q}
For every smooth vector field,
\begin{equation}
\Qop_{K\xi}=D_\xi P_m-P_mX_\xi.
\label{eq:Qidentity}
\end{equation}
\end{lemma}

\begin{proof}
The density coefficients are
\[
a_1^{\mu\nu}[K\xi]
=(\partial\cdot\xi)\eta^{\mu\nu}
-\partial^\mu\xi^\nu-\partial^\nu\xi^\mu,
\qquad
b_1[K\xi]=\partial\cdot\xi.
\]
Expanding \(\Qop_{K\xi}u\), the derivatives of
\(\partial\cdot\xi\) cancel, leaving
\[
(\partial\cdot\xi)P_mu
-(\Box\xi^\nu)\partial_\nu u
-2\partial^\mu\xi^\nu\partial_\mu\partial_\nu u.
\]
Expanding the right-hand side of \eqref{eq:Qidentity} gives the
same expression.
\end{proof}

Define the off-shell representative
\[
A_{1,\mathrm{off}}[h]
=F_{c_1[h]-\Lone[h]\EA f},
\]
and its free-field class
\begin{equation}
\widehat A_1[h]
=\field(c_1[h]-\Lone[h]\EA f)
=\Obs^{(1)}[h]+\Rone(V_{1,g}[h];F_f).
\label{eq:Aone}
\end{equation}

\begin{proposition}[Variation with on-shell zero, edge, and switch separated]
\label{prop:variation}
For a compact parameter \(\xi\) in the slab,
\begin{align}
A_{1,\mathrm{off}}[h+K\xi]-A_{1,\mathrm{off}}[h]
={}&F_{P_m(X_\xi\EA f)}
\nonumber\\
&+F_{\partial_\mu(f(\Xone[K\xi]-\xi)^\mu)}
-F_{\Sigma_{g,\xi}\EA f},
\label{eq:variation}
\end{align}
where
\begin{equation}
\Sigma_{g,\xi}u
=(g-1)\Qop_{K\xi}u
+(\partial_\mu g)a_1^{\mu\nu}[K\xi]\partial_\nu u.
\label{eq:Sigma}
\end{equation}
The first term lies in \(\Jfree\).
The second is the coordinate edge term.
The last displays the factors \(1-g\) and \(\partial g\).
\end{proposition}

\begin{proof}
The changed test is
\[
\partial_\mu(f\Xone^\mu[K\xi])
-\Lone[K\xi]\EA f.
\]
Write \(\Lone[K\xi]=\Qop_{K\xi}+\Sigma_{g,\xi}\).
Using \(P_m\EA f=f\), Lemma~\ref{lem:Q} gives
\[
\Qop_{K\xi}\EA f
=D_\xi f-P_m(X_\xi\EA f).
\]
Subtract this from
\[
\partial_\mu(f\Xone^\mu[K\xi])
=D_\xi f+
\partial_\mu(f(\Xone[K\xi]-\xi)^\mu).
\]
The resulting identity is \eqref{eq:variation}.
The function \(X_\xi\EA f\) is smooth and compactly supported,
so its \(P_m\)-test belongs to the equation ideal.
\end{proof}

The switch coefficients are supported in
\(\supp\xi\cap(\supp(1-g)\cup\supp\dd g)\).
This closed-support statement includes plateau boundaries;
it does not incorrectly identify support with the open set
\(\{g\neq1\}\).

\begin{theorem}[First-order background Ward identity]
\label{thm:ward}
For every classical background \(h\) and every \(\xi\in\ga\),
\begin{equation}
\widehat A_1[h+K\xi]=\widehat A_1[h]
\quad\text{in }\Afree.
\label{eq:Ward}
\end{equation}
Thus \(\field(f)+\kappa\widehat A_1[h]\) is invariant modulo
\(\kappa^2\), with the free field held fixed.
\end{theorem}

\begin{proof}
The first term of \eqref{eq:variation} vanishes in \(\Afree\).
The copied preflight identity kills the second.
On \(\supp\xi\subset U\), \(g=1\) and \(\dd g=0\); hence
\(\Sigma_{g,\xi}=0\).
These are exactly the declared support conditions, not restrictions
introduced after the defect calculation.
\end{proof}

Off shell, the plateau difference is precisely
\(F_{P_m(X_\xi\EA f)}\), not zero in general.
General pullback of the scalar is not an automorphism of the free
algebra: a non-isometric diffeomorphism need not preserve
\(P_m\phi=0\). The theorem is instead background covariance at
fixed free field, corresponding to the first completed coefficient
of X-A's on-shell identity.

\subsection{The explicit truncation}

The physical first-vertex correction is \(\kappa\Rone\), so
\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};\Obs^{(1)})\bigr)
\formal.
\label{eq:truncation}
\end{align}
The classical coefficient is instead
\begin{equation}
A_2^{\mathrm{cl}}
=\Obs^{(2)}
-\int f\,\Xone\cdot\partial\rho_{1,g}\,\dd^4x
+\int f\,\rho_{2,g}\,\dd^4x,
\label{eq:classical-A2}
\end{equation}
with \(\rho_{2,g}\) displayed in \eqref{eq:rhotwo}.

The last term is not supplied by the present map.
A full quantum completion would require the second-vertex
insertion and the iterated first-vertex insertion.
Neither is constructed here. Their classical sum is displayed,
not replaced by a contact and not assigned the name of a
constructed quantum map.

The bare \(\Obs^{(2)}\) is already admitted.
Its classical transformation is retained through the copied
metric and coordinate laws. That fact does not complete the
second-order dynamics.

\begin{proposition}[Transformation of the retained mixed term;
not an invariance statement]
\label{prop:mixed-transformation}
Put
\[
\mathcal M[h]
=\Rone(V_{1,g}[h];F_{c_1[h]}).
\]
For \(\xi\in\ga\), its exact finite change in \(\Afree\) 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).
\label{eq:mixed-transformation}
\end{align}
Equivalently, it is
\[
-\int f\,X_\xi\rho_{1,g}[h;\field]\,\dd^4x
-\int f\,\Xone[h]\cdot\partial(X_\xi\field)\,\dd^4x
-\int f\,X_\xi^2\field\,\dd^4x.
\]
The first two terms are linear in the shift parameter and the
last is quadratic.
\end{proposition}

\begin{proof}
Both \(c_1[h]\) and \(\Lone[h]\) are linear in \(h\), while
\(c_1[K\xi]=D_\xi f\).
Expanding \(-\Lone[h+K\xi]\EA(c_1[h]+D_\xi f)\) produces the
first term in \eqref{eq:mixed-transformation} and two terms
containing \(-\Qop_{K\xi}\EA\).
For a compact test \(u\),
\[
-\Qop_{K\xi}\EA u=-D_\xi u+P_m(X_\xi\EA u).
\]
The last summand is zero in \(\Afree\).
Apply this identity with \(u=c_1[h]\) and \(u=D_\xi f\).
The integral form follows from \(D_\xi^{\mathsf T}=-X_\xi\).
\end{proof}

This is the transformation of the classical mixed term evaluated
on the free field. It is generally nonzero.
X-A cancels it in the complete classical coefficient together
with the transformations of the remaining terms.
Equation \eqref{eq:truncation} alone is not given that invariant
status.

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

For arbitrary compact slab parameters, passing
\eqref{eq:variation} to the free algebra gives
\begin{equation}
\delta_\xi\widehat A_1
=\int f(\xi-\Xone[K\xi])\cdot\partial\field\,\dd^4x
-\field(\Sigma_{g,\xi}\EA f).
\label{eq:defect-class}
\end{equation}
Both mechanisms are visible. They must not be conflated.

If an excluded edge parameter is also supported in the already
chosen plateau \(P_g\), the switch operator vanishes and the
coordinate edge term is isolated. If that plateau condition is
absent, the switch term remains. The sheet does not assume that
every chosen \(g\) has an edge plateau.

For compact \(\xi\) supported in \(0<t<\eps\), the coordinate
gap can also be written
\[
\xi-\Xone[K\xi]
=\ERzero\bigl((1-\sig)\Box\xi\bigr)
+\ERzero\bigl(\sig'a_1^{0\cdot}[K\xi]\bigr).
\]
The nonzero integral \eqref{eq:edge-gap} proves a coordinate
failure. It does not, by itself, prove a nonzero total smeared
on-shell response for every fixed \(f,g\).
The observation test can miss the witness, a free-field test can
be an equation zero, or scalar switch terms can also be present.
No identity is claimed for these excluded parameters, and
\(\ga\) is neither enlarged nor shrunk.

Bare dressings alone are not generally background invariant.
For example, in a fixed quasi-free Hadamard representation their
pairing involves
\(\hbar\pair{c_1[h]}{\Delta_+c_1[h']}\), which can change when
\(c_1[h]\) is shifted by \(D_\xi f\).
This does not contradict the invariance of the completed
\(\widehat A_1\).
No interacting two-point response is constructed or inferred.

%=====================================================================
\section{B4: zero switch with the dressing intact}
\label{sec:zero}
%=====================================================================

\begin{proposition}[Zero-switch identity and a nonzero algebra witness]
\label{prop:zero}
At \(g=0\),
\[
\Rone(0;F_{c_j})=0,\qquad j=0,1,2,
\]
while the bare observable remains
\[
\Obs_{f,0,\eps}
=\field(f+\kappa c_1[h]+\kappa^2c_2[h,k]).
\]
It differs from \(\field(f)\) in general.
\end{proposition}

\begin{proof}
The map is zero because \(\Lone=0\), independently of the
insertion test. The coordinate formulas do not contain \(g\).

For an explicit witness, take \(m=0\), choose \(f\geq0\),
\(f\not\equiv0\), and let
\(t_-=\min\{t:(t,\mathbf x)\in K\}>0\).
Choose a nonnegative nonzero
\(\psi\in C_c^\infty((a,b))\) with \(0<a<b<t_-\).
Choose a smooth compact spatial cutoff equal to one near the
spatial projection of \(\Df\), and use it to extend
\(h=\psi(t)\eta\) to a compact smooth background.

Finite propagation makes this cutoff irrelevant on the observation
region. There,
\[
\Xone^0=-\Psi(t),\qquad \Xone^i=0,\qquad
\Psi(t)=\int_0^t\sig(s)\psi(s)\,\dd s.
\]
Since \(\sig>0\) for \(s>0\), \(\Psi>0\) on \(K\).
Thus \(c_1=-\partial_t(f\Psi)\), and against the smooth free
solution \(u=t\),
\[
\int c_1\,t\,\dd^4x=\int f\Psi\,\dd^4x>0.
\]
If \(c_1=P_0v\), \(v\in\Dspace(M)\), this pairing would be zero.
More generally \(F_{c_1}\) cannot lie in \(\Jfree\), whose elements
vanish on smooth free solutions.
Hence \(\field(c_1)\neq0\).
\end{proof}

A nonzero coordinate coefficient alone would not prove this for an
arbitrary chosen test. The witness proves an actual nonzero
on-shell class.

At zero switch \(P_g\) is empty, so the plateau Ward claim is
vacuous. The specialization does not introduce a new gauge
algebra. It verifies the functional identity with all dressing
tests left intact.

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

\begin{diagnostic}[Pure gauge: sign and order boundary]
Take \(\xi\in\ga\) and the complete pure-gauge two-jet
\[
h=K\xi,\qquad k=\tfrac12\Lie_\xi K\xi.
\]
The copied coordinate laws give
\[
\Xone=\xi,\qquad \Xtwo=\tfrac12X_\xi\xi.
\]
On the free field,
\[
\ER\Lone[K\xi]\field=-X_\xi\field,\qquad
\rho_{1,g}[K\xi;\field]=X_\xi\field.
\]
Therefore \(\widehat A_1[K\xi]=0\).

The retained mixed term is
\[
\mathcal M[K\xi]
=-\int f\,\xi\cdot\partial(X_\xi\field)\,\dd^4x
=-\int f\,X_\xi^2\field\,\dd^4x.
\]
The minus sign follows from the normalization and insertion sign.
The rejected plus sign would not be compatible with them.

On this complete jet, the bare second dressing is
\[
\Obs^{(2)}=\tfrac12\int f\,X_\xi^2\field\,\dd^4x.
\]
Thus the incomplete second coefficient in
\eqref{eq:truncation} is
\[
-\tfrac12\int f\,X_\xi^2\field\,\dd^4x,
\]
which need not vanish. For example, in the massless theory choose a
compact time-component parameter in \(U\) for which
\(X_\xi^2t\) is nonzero, and an observation test supported where it
has fixed sign.

Classically, the missing response is
\(\rho_{2,g}=\tfrac12X_\xi^2u\), and it cancels this remainder.
No corresponding second-order quantum map is constructed here.
The diagnostic demonstrates the order boundary, not a defect in
the first-order theorem and not a contact-normalization problem.
\end{diagnostic}

\begin{diagnostic}[Sourced: free-field correction and classical number]
Let \(m=0\) and \(h=\psi\eta\), with smooth compact \(\psi\).
Then
\[
\Lone[h]\field
=\partial_\mu(g\psi\,\partial^\mu\field)
=\partial_\mu(g\psi)\partial^\mu\field,
\]
using \(\Box\field=0\).
The retarded correction is
\[
\Rone(V_{1,g}[\psi\eta];F_f)
=\field\!\left(
-\partial_\mu(g\psi\,\partial^\mu\EAzero f)
\right).
\]
This is an element of \(\Afree\), with a smooth compact test.
The coordinate time component is
\(\Xone^0=-\ERzero\partial_t(\sig\psi)\).

For a time-dependent illustration, select admissible data with
\(\psi=\psi(t)\) and \(g=g_0(t)\) near the relevant causal compact,
and cut them off spatially outside it. Define
\[
\Psi(t)=\int_0^t\sig\psi\,\dd s,\qquad
\Gamma(t)=\int_0^t g_0\psi\,\dd s.
\]
On the smooth free solution \(u=t\),
\[
\Xone^0=-\Psi,\quad \Xone^i=0,\quad
\rho_{1,g}[u]=-\Gamma.
\]
Classical evaluation of the specified linear representatives gives
\[
A_{1,\mathrm{off}}[h](t)=\int f(\Psi-\Gamma)\,\dd^4x,
\]
and
\[
\mathcal M_{\mathrm{off}}[h](t)
=-\int f\,\Psi\,g_0\psi\,\dd^4x.
\]
These are classical numbers, not elements of \(\Afree\), quantum
expectation values, or states on the Wick algebra.
They check the mismatch of the two switches and the mixed-term
sign. The time-dependent cutoffs do not replace the general
construction or its support proof.
\end{diagnostic}

%=====================================================================
\section{Decision, regulator estimates, and successor boundary}
\label{sec:decision}
%=====================================================================

\begin{center}
\small
\renewcommand{\arraystretch}{1.12}
\begin{tabularx}{\linewidth}{@{}p{.18\linewidth}p{.43\linewidth}X@{}}
\toprule
Gate & Evidence & Status\\
\midrule
Field content &
Scalar contractions only; metric classical &
Frozen before products\\
B1, joint &
Split calculus and proper kernel pushforwards &
Admitted through \(\kappa^2\)\\
B1, scalar &
Smooth compact tests \(c_0,c_1,c_2\) &
Named linear domain\\
B2 &
\(d_c=-\Lone[h]\EA c\) &
One map constructed\\
Quantum order &
One cross-contraction, no second scalar derivative &
Positive-\(\hbar\) part vanishes\\
Support &
\(\supp d_c\subset\supp g\cap J^-(\supp c)\) &
Proved without extra exclusions\\
B3 &
Three-term variation; plateau cancellation &
Ward through \(\kappa^1\) only\\
Second order &
Bare and mixed terms retained; \(\rho_{2,g}\) displayed &
No full quantum identity\\
Edge and switch &
Explicit defects outside \(\ga\) &
Not repaired or imported\\
B4 &
Zero map with a nonzero dressing witness &
Passed\\
B5 &
Fixed-\(\eps\) estimates only &
Sharp limit unopened\\
Quantum metric &
Action and contraction domain absent &
Successor only\\
\bottomrule
\end{tabularx}
\end{center}

\subsection{B5: the epsilon count}

The first coordinate source already contains
\(-\sig'a_1^{0\mu}[h]\).
For \(r\geq1\),
\[
\|\partial_t^r\sig\|_\infty
=\eps^{-r}\|\chi^{(r)}\|_\infty.
\]
A direct differentiated-source estimate gives
\begin{equation}
\begin{split}
\|c_1[h]\|_{C^N(K)}
\leq C_N(f,T,\chi)\Bigl(
&\|h\|_{C^{N+2}(\Df)}\\
&+\sum_{r=1}^{N+2}\eps^{-r}
\|h\|_{C^{N+2-r}(\Df\cap\{0\leq t\leq\eps\})}
\Bigr).
\end{split}
\label{eq:eps-estimate}
\end{equation}
The seminorms are restrictions of ambient smooth derivatives.

To see the derivative count, use the massless Kirchhoff formula
for the smooth, flat-at-zero coordinate source \(s\):
\[
(\ERzero s)(t,\mathbf x)
=\frac1{4\pi}\int_0^t r
\int_{S^2}s(t-r,\mathbf x+r\omega)\,\dd\omega\,\dd r.
\]
Spatial derivatives commute with this formula.
Repeated time differentiation produces no lower-limit data terms
because \(s\) is flat at \(t=0\).
On the compact causal region, derivatives of order \(N+1\) of
\(\Xone\) are bounded by source derivatives of order \(N+1\).
The source has one additional derivative of
\(\sig a_1[h]\), giving at most \(N+2\) ramp derivatives.
Differentiating \(c_1=\partial(f\Xone)\) therefore yields
\eqref{eq:eps-estimate}.

Thus this direct upper bound can cost
\(\eps^{-N-2}\), not merely \(\eps^{-1}\).
It is not a lower bound or a necessary divergence rate.
For backgrounds vanishing near the edge, the corresponding terms
disappear; cancellations can also improve estimates.
Higher coefficients require their own estimates.

No uniform sharp-limit continuity, convergence theorem, or
admission topology is inferred from these bounds.
They do not establish VII-C's limiting hypotheses.
The limit \(\eps\to0\) is unopened.

\subsection{Named and not started}

A quantum-metric successor would have to specify a quadratic
graviton action, gauge fixing, propagators, and a suitable BRST
interpretation. A possible additional tree expression is
schematically
\[
\Xone\!\left[
\Delta_{\mathrm R}^{\mathrm{gr}}
\bigl(\tfrac12g\,T[\phi]\bigr)
\right].
\]
It is not part of the scalar-only response.

A possible mixed graph has the schematic kernel
\[
f(x)g(y)K_{\mathrm{coord}}(z,x)\,
\Delta^{\mathrm{gr}}(y,z)\,
\partial\partial\Delta(y,x).
\]
The region \(y=x\), with \(z\) on a past cone, requires a new
wavefront and extension analysis. Neither its existence nor an
obstruction is proved here. A classical auxiliary coordinate field
might reorganize a successor calculation, but none is introduced.

The ramp kernel is not identified with IX-E-2's \(K\), with
\(B_f\), or with \(u_\mu^\rho\).
X-Enter is neither a premise nor a closure for this observable.
Reading an already invariant classical numerical evaluation would
preserve an upstream identity, not prove it.

\begin{theorem}[Fixed-regulator X-B release]
\label{thm:release}
For the frozen scalar-only field content, ramp, switch, background
space, insertion representatives, and algebra \(\ga\):
\begin{enumerate}[label=\textup{(\roman*)}]
\item The bare dressing coefficients are admitted through
\(\kappa^2\), both by the joint cone test and on the scalar-only
smooth linear domain.
\item The single normalized map
\(\Rone(V_{1,g};\cdot)\) exists on the frozen off-shell tests,
has smooth compact retarded output, is continuous, and has no
positive-\(\hbar\) correction.
\item The completed coefficient \(\widehat A_1\) is invariant in
\(\Afree\) on \(\ga\). This is the quantum background identity
through \(\kappa^1\), not through \(\kappa^2\).
\item The second-order bare dressing and retained mixed term are
available, with their stated transformations.
The classical missing term \(\int f\rho_{2,g}\) is displayed,
but no full second-order quantum dynamical completion or Ward
identity is constructed.
\end{enumerate}
No sharp limit, quantum metric response, loop triangle, or
two-point response is included.
\end{theorem}

\begin{proof}
Item (i) is Propositions~\ref{prop:joint} and~\ref{prop:linear}.
Item (ii) is Theorems~\ref{thm:retarded} and~\ref{thm:hbar}.
Item (iii) is Theorem~\ref{thm:ward}.
For (iv), use \eqref{eq:truncation},
\eqref{eq:classical-A2}, and
Proposition~\ref{prop:mixed-transformation}.
The pure-gauge diagnostic explicitly exhibits the missing
second-order cancellation.
\end{proof}

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

\paragraph{Classical dependence.}
The ramp convention sheet, coordinate identities, edge witness,
relational signs, support, smooth continuity, and classical
covariance are cited premises from \texttt{X-A.tex}.
Their covariance proof is not reproduced.
No convolution identity or earlier-series product is used.

\paragraph{Construction and order.}
The manuscript constructs one scalar retarded map on specified
off-shell linear insertions. Admission is through \(\kappa^2\);
the completed quantum Ward identity is through \(\kappa^1\).
There is no claim that the incomplete second coefficient is
invariant.

\paragraph{Renormalization boundary.}
A free Wick presentation is fixed.
No new diagonal extension, subtraction, or finite contact is used
for the constructed map. The absence of positive-\(\hbar\)
terms is proved for this field content and this map, not asserted
for a successor with a quantum metric.

\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 Classical premises: the convention sheet; Theorem
\texttt{thm:main}; Propositions \texttt{prop:preflight},
\texttt{prop:edge}, \texttt{prop:coordeq},
\texttt{prop:scalar}, \texttt{prop:continuity},
\texttt{prop:kernels}; and the relational expansion.

\bibitem{FR}
K.~Fredenhagen and K.~Rejzner.
\newblock \emph{Perturbative algebraic quantum field theory}.
\newblock arXiv:1208.1428.
\newblock Used for the standard cone terminology and free Wick
presentation. The finite polynomial construction here does not
invoke a general closure theorem for nonpolynomial functionals.

\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 Background for off-shell insertion and retarded-product
terminology; no complete product hierarchy is claimed here.

\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 Used for Green operators, finite propagation, transposition,
and smooth continuity.

\bibitem{Hormander}
L.~H\"ormander.
\newblock \emph{The Analysis of Linear Partial Differential Operators I}.
\newblock Second edition. Springer, 1990.
\newblock Chapter~8.
\newblock Used for distributional wavefront product, differentiation,
and proper-pushforward criteria.

\end{thebibliography}

\end{document}