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% X-B-ii.tex
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% Build:
%   pdflatex -interaction=nonstopmode -halt-on-error X-B-ii.tex
%   pdflatex -interaction=nonstopmode -halt-on-error X-B-ii.tex
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% Role:
%   Closure document for four scope items of X-B (X-B-v3.tex).
%   C1 scope lemma; C2 one-contraction boundary (non-extension);
%   C3 normalization sheet (uniqueness relative to the classical
%   normalization; contact terms); C4 Branch B input list.
%
% Frozen field content (copied from X-B):
%   phi is the free quantum Klein--Gordon field.
%   h and k are classical, off-shell backgrounds.
%
% Constructed here: nothing beyond X-B. No time-ordered family,
% no graviton datum, no second-order map, no sharp limit.
%
% Debugging incorporated:
%   - all signs in X-B eq:contraction-check re-derived;
%   - kernel symmetry of Delta_F proved, not asserted;
%   - the degree-one part of the equation ideal identified;
%   - detection of free classes by spatially compact solutions
%     proved in both directions (X-B used one direction);
%   - classical normalization recorded on all of L_off, not only
%     on F_f, and shown to determine R_1;
%   - contact modifications classified relative to the sheet;
%   - the pointwise square of Delta_F shown to fail the product
%     criterion on the diagonal, so the one-contraction check does
%     not extend to a second pair by the tools of the sheet.
%
% Non-inputs: XII-B.tex (different track); retrieval summaries.
%
% No successful compilation, mechanized verification, or independent
% referee review is claimed.
%=====================================================================

\documentclass[11pt,a4paper]{article}

\usepackage[T1]{fontenc}
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\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}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}

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

\newcommand{\R}{\mathbb R}
\newcommand{\C}{\mathbb C}
\newcommand{\eps}{\varepsilon}
\newcommand{\dd}{\,\mathrm d}
\newcommand{\sig}{\sigma_\eps}
\newcommand{\Xone}{\mathsf X_1}
\newcommand{\Xtwo}{\mathsf X_2}
\newcommand{\Obs}{\mathcal O}
\newcommand{\ga}{\mathfrak 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{\Afone}{\mathfrak A_0^{(1)}}
\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{\Lone}{\mathsf L_{1,g}}
\newcommand{\Rone}{R_1}
\newcommand{\field}{\widehat\phi}
\newcommand{\pair}[2]{\left\langle #1,#2\right\rangle}
\newcommand{\Vp}{\overline V_+}
\newcommand{\Vm}{\overline V_-}
\newcommand{\DF}{\Delta_{\mathrm F}}
\newcommand{\DS}{\Delta_S}
\newcommand{\Dplus}{\Delta_+}
\newcommand{\Cf}{\mathcal C_{\mathrm F}}
\newcommand{\deltaD}{\delta_\Delta}
\newcommand{\Lop}{\mathsf L}

\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-ii}\\[0.4em]
Scope closure for the scalar-only retarded map:\\
the one-contraction boundary, the normalization sheet,\\
and the Branch B input list}
\author{Ngo Cheung}
\date{}

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

\begin{abstract}
This document closes four scope items left as statements in
Landmark X-B, where the free quantum field is the scalar \(\phi\)
and the metric coefficients \(h,k\) are classical off-shell
backgrounds. First, a scope proposition records that every object
constructed in X-B is a field-degree-one element of the free
algebra with polynomial classical background dependence and no
\(\hbar\); the non-goals are listed. Second, the one-contraction
check is placed on its exact boundary: the auxiliary kernel
\(\DF=\Dplus-i\EA\) is proved symmetric, the check is re-derived,
the pair passes the restricted causal-factorization and symmetry
requirements, and the pointwise square of \(\DF\) is proved to fail
the wavefront product criterion at every diagonal point, so the
tools of the sheet define no second pair and no time-ordered
family. Third, the normalization sheet is recorded on the whole
off-shell insertion domain and proved to determine the single map
\(\Rone(V_{1,g}[h];\cdot)\): a degree-one class in the free
algebra vanishes if and only if it is annihilated by every
spatially compact smooth solution, hence any contact modification
compatible with the classical normalization is zero in the free
algebra, and no contact can repair or break the first-order
identity. Fourth, the data that an independent Branch B would
require are listed, with a non-relabeling rule; no existence,
failure, or correction is asserted for that branch. Nothing is
constructed beyond X-B.
\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) {Named open items};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Identity checks};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {A new coefficient};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {Closing the four items does not construct a new object.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{Closure items construct nothing new. 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-ii}
\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) {Named open items};
\node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {A new coefficient};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {Closing the four items does not construct a new object.};
\draw[->] (a) -- (s);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{Closure items construct nothing new. 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-ii}
\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) {Named open items};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Identity checks};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {A new coefficient};
\node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {Closing the four items does not construct a new object.};
\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-ii}
\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) {Named open items};
\node[align=center,font=\small] (m) at (5.2,0) {not the same object};
\node[draw,rounded corners,align=center,text width=4.2cm,font=\small,inner sep=3pt] (b) at (10.4,0) {A new coefficient};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Identity checks. Closing the four items does not construct a new object.};
\draw (a) -- (m);
\draw (m) -- (b);
\draw[->] (m) -- (s);
\end{tikzpicture}
\caption{Named open items is not A new coefficient. 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-ii}
\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: Named open items};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Identity checks};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: A new coefficient};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {Closing the four items does not construct a new object.};
\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-ii}
\end{figure}

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


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



\scopebox{Release: four closure items, nothing new constructed}{%
C1 (scope): closed by Proposition~\ref{prop:scope}.
C2 (one-contraction boundary): closed by
Theorem~\ref{thm:nonextension}; the check is an identity for one
pair and does not extend by the sheet's tools.
C3 (normalization sheet): closed by Theorem~\ref{thm:unique}
and Corollary~\ref{cor:contact}; the classical normalization on
\(\Loff\) determines \(\Rone\), and sheet-compatible contact
modifications vanish in \(\Afree\).
C4 (Branch B): recorded as an input list
(Section~\ref{sec:branchB}); not opened.

\smallskip
No time-ordered family, graviton action, graviton propagator,
contraction domain for a quantum metric, second-order quantum map,
loop, state, or sharp limit is defined. X-B's release theorem is
neither enlarged nor withdrawn.}

\tableofcontents

%=====================================================================
\section{Closure items, endings, and evidence}
\label{sec:items}
%=====================================================================

\subsection{The four items}

X-B \cite{XB} constructs one normalized retarded map
\(\Rone(V_{1,g}[h];\cdot)\) for the scalar-only field content and
records four scope statements without separate proof documents.
They are closed here as follows.

\begin{enumerate}[label=\textbf{C\arabic*.}]
\item \emph{Scope.} Restate the boxed field content as a
proposition about the constructed objects and list the non-goals.
\item \emph{One-contraction boundary.} Decide whether the
one-contraction check (X-B, \texttt{eq:contraction-check}) is
promoted to a family of time-ordered maps. The ending chosen is
the explicit non-extension note: the check is proved for its one
pair, and it is proved that the tools of the sheet define no
second pair.
\item \emph{Normalization sheet.} Record the one map, state what
its transposed notation means, and settle the status of contact
terms relative to the classical normalization.
\item \emph{Branch B.} List, without opening, the data an
independent quantum-metric branch would require, and the
non-relabeling rule.
\end{enumerate}

\subsection{Permitted endings}

For C1--C3 the ending \textbf{Closed} means that the deliverable is
a proved statement about objects already defined in X-B. For C4
the only permitted ending is \textbf{Recorded}: a list is not a
construction, an obstruction, or a correction. No item is allowed
to end by defining a new product, a new vertex, or a new field.

\subsection{Evidence contract}

The input is the full text of \texttt{X-B-v3.tex} \cite{XB}. The
classical premises that X-B copies from \texttt{X-A.tex} \cite{XA}
are used only as X-B states them. The retrieval summary that
motivated the four items is documentary context, not evidence.
\texttt{XII-B.tex}, which concerns the ramped cubic Einstein
source on a different track, is not an input and no statement of
it is used. Standard references are used only for standard facts:
Green operators and spatially compact solutions \cite{BGP},
wavefront calculus \cite{Hormander}, the Hadamard wavefront set
\cite{Radzikowski}, and the terminology of time-ordered and
retarded products \cite{FR,DF,BF}.

%=====================================================================
\section{Copied conventions}
\label{sec:conventions}
%=====================================================================

\begin{premise}[Free scalar data; copied from X-B]
\label{prem:free}
\(M=\R^4\), \(\eta=\diag(1,-1,-1,-1)\), \(\Box=\partial_t^2-\Delta\),
\(P_m=\Box+m^2\), \(m\geq0\). \(\Dspace=\Dspace(M)=C_c^\infty(M;\C)\)
with its LF topology. The retarded and advanced Green operators
\(\ER,\EA:\Dspace\to C^\infty(M)\) satisfy
\[
P_m\ER c=\ER P_mc=c,\qquad P_m\EA c=\EA P_mc=c,\qquad
\supp\ER c\subset J^+(\supp c),\qquad
\supp\EA c\subset J^-(\supp c),
\]
and \((\ER)^{\mathsf T}=\EA\), i.e.\
\(\pair{\EA c}{d}=\pair{c}{\ER d}\) for \(c,d\in\Dspace\).
Put \(\DS=\EA-\ER\); then \(\DS^{\mathsf T}=-\DS\), and
\(\DS\Dspace\) is exactly the space of smooth solutions of
\(P_mu=0\) with spatially compact support \cite[Thm.~3.4.7]{BGP}.
The sign conventions of X-B are
\[
\Delta_S^{\mathrm R}=-\ER,\qquad \Delta_S^{\mathrm A}=-\EA .
\]
A Hadamard bisolution \(\Dplus\in\Dspace'(M\times M)\) is fixed once,
with
\[
\Dplus-\Dplus^{\mathsf T}=i\DS,\qquad
\WF(\Dplus)\subset\{(x,\xi;y,-\eta):\ \xi\neq0,\ \eta\neq0\}
\]
where in addition the first covector lies in one closed half of the
null cone \cite[Thm.~5.1]{Radzikowski}; in particular no element of
\(\WF(\Dplus)\) has a zero covector in either slot.
\end{premise}

\begin{premise}[Algebra; copied from X-B]
\label{prem:algebra}
\(\Aoff\) is the algebra of polynomial functionals of
\(\varphi\in C^\infty(M)\) with compact derivative kernels, with the
Wick product
\(F\star G=\sum_{n\ge0}\frac{\hbar^n}{n!}
\pair{F^{(n)}}{\Dplus^{\otimes n}G^{(n)}}\).
For \(c\in\Dspace\), \(F_c(\varphi)=\int c\varphi\dd^4x\) and
\(\Loff=\{F_c\}\cong\Dspace\). \(\Jfree\) is the pointwise ideal
generated by \(\{F_{P_mu}:u\in\Dspace\}\); it is a star ideal;
\(\Afree=\Aoff/\Jfree\) and \(\field(c)=[F_c]\). Elements of
\(\Jfree\) vanish on smooth free solutions. Field degree is the
polynomial degree in \(\varphi\); \(\Aoff^{(n)}\) denotes the
homogeneous part of degree \(n\), and
\(\Afone\) the image of \(\Aoff^{(1)}\) in \(\Afree\), i.e.\
\(\Afone=\{\field(d):d\in\Dspace\}\).
\end{premise}

\begin{premise}[Branch A objects; copied from X-B]
\label{prem:branchA}
The ramp \(\sig\), switch \(g\in C_c^\infty((0,T)\times\R^3;[0,1])\),
label test \(f\), plateau \(P_g\), region \(U\), and algebra
\(\ga=C_c^\infty(U;TM)\) are those of X-B. Backgrounds \((h,k)\)
range over
\(\Bg=C^\infty(M;S^2T^*M)^2\). With \(a_1[h]\), \(b_1[h]\) as in X-B,
\begin{equation}
\Lone[h]u=\partial_\mu\bigl(g\,a_1^{\mu\nu}[h]\partial_\nu u\bigr)
+g\,m^2b_1[h]\,u,
\label{eq:Lone}
\end{equation}
a formally self-adjoint operator with smooth coefficients supported
in \(\supp g\). The vertex is
\(V_{1,g}[h,\varphi]=-\tfrac12\pair{\varphi}{\Lone[h]\varphi}\),
so that \(V_{1,g}'=-\Lone[h]\varphi\) and \(V_{1,g}''=-\Lone[h]\).
For smooth \(u\), \(\rho_{1,g}[h;u]=-\ER\Lone[h]u\).
The single normalized map is
\begin{equation}
\Rone(V_{1,g}[h];F_c)=\field(d_c[h]),\qquad
d_c[h]=-\Lone[h]\EA c,\qquad c\in\Dspace .
\label{eq:Rone}
\end{equation}
The frozen insertion tests \(c_0=f\), \(c_1[h]\), \(c_2[h,k]\) are
smooth, supported in \(K=\supp f\), and polynomial and continuous in
\((h,k)\) at fixed \(\eps\) (X-B, Proposition \texttt{prop:linear}).
The completed coefficient is
\(\widehat A_1[h]=\field(c_1[h]-\Lone[h]\EA f)\) and the retained
mixed term is \(\mathcal M[h]=\Rone(V_{1,g}[h];F_{c_1[h]})\).
The general insertion pair of X-B is
\begin{enumerate}[label=\textup{(A\arabic*)}]
\item \(V=-\tfrac12\pair{\varphi}{\Lop\varphi}\) with \(\Lop\)
formally self-adjoint with smooth compactly supported coefficients;
\item \(F_c\) with \(c\in\Dspace\).
\end{enumerate}
\end{premise}

%=====================================================================
\section{C1: the scope proposition}
\label{sec:scope}
%=====================================================================

\begin{definition}[Branch A sheet]
\label{def:sheet}
The Branch A sheet is the tuple
\[
\mathfrak S_A=\bigl(M,\eta,P_m;\ \sig,g,f,U;\ \Bg;\ \Dspace;\
\ER,\EA,\ERzero;\ \Dplus\bigr),
\]
together with the maps \((h,k)\mapsto c_j[h,k]\), \(j=0,1,2\), and
\(h\mapsto\Lone[h]\). Every object of X-B is built from
\(\mathfrak S_A\) by the operations: smooth multiplication,
differentiation, application of \(\ER\), \(\EA\), \(\ERzero\) to
compactly supported smooth sources, evaluation of background slots
on classical configurations, formation of \(F_c\), and passage to
\(\Afree\).
\end{definition}

\begin{proposition}[Scope of the constructed objects]
\label{prop:scope}
Each of the objects
\[
\Obs_\kappa=F_{c_0}+\kappa F_{c_1}+\kappa^2F_{c_2},\qquad
\Rone(V_{1,g}[h];F_c),\qquad
\widehat A_1[h],\qquad
\mathcal M[h],\qquad
\delta_\xi\widehat A_1
\]
of X-B has the following properties.
\begin{enumerate}[label=\textup{(\roman*)}]
\item It is a finite sum of elements \(\field(d)\) (or \(F_d\)
before passage to the quotient) with \(d\in\Dspace\); its field
degree is one.
\item The test \(d\) depends polynomially on \((h,k)\), of degree at
most two in \(h\) and one in \(k\), and continuously in the smooth
background topology at fixed \(\eps\).
\item No \(\hbar\) occurs in \(d\). The constant \(\hbar\) and the
kernel \(\Dplus\) enter X-B only through the commutator
\([\field(c),\field(d)]=i\hbar\pair{c}{\DS d}\) and through the
one-contraction check of Section~\ref{sec:boundary}, in which
\(\Dplus\) cancels.
\item Every two-point kernel that occurs is a scalar Green kernel
(\(\ER\), \(\EA\), \(\ERzero\)) applied to a compactly supported
smooth source, or \(\Dplus\) as in \textup{(iii)}. Every background
slot is evaluated on a classical configuration; no background slot
is contracted against another background slot through a kernel.
\item No functional of \(h\) or \(k\) alone (in particular no
gravitational action) is formed.
\end{enumerate}
\end{proposition}

\begin{proof}
(i)--(ii) for \(\Obs_\kappa\): the \(c_j\) are the frozen tests of
X-B, Proposition \texttt{prop:linear}, each in \(\Dspace_K(M)\),
with \(c_1\) linear in \(h\) and \(c_2\) quadratic in \(h\) and
linear in \(k\) through \(\Xone[h]\) and \(\Xtwo[h,k]\); continuity
is the handoff statement. For \(\Rone\): \(d_c[h]=-\Lone[h]\EA c\)
is linear in \(h\) through the coefficients of \eqref{eq:Lone},
and \(\EA c\) is smooth, so \(d_c[h]\in\Dspace\) with support in
\(\supp g\); continuity is X-B, Theorem \texttt{thm:retarded}. For
\(\widehat A_1\) and \(\mathcal M\): sums and compositions of the
previous two with \(c\) replaced by \(f\) or \(c_1[h]\); the degree
in \(h\) of \(\mathcal M\) is two. For \(\delta_\xi\widehat A_1\):
X-B, \eqref{eq:defect-class-ref} below is the difference of two
such elements. Field degree one is the statement that each object
is a class of a functional \(F_d\), which has first derivative \(d\)
and vanishing higher derivatives.

(iii) The tests \(d\) are defined by \eqref{eq:Rone} and the
formulas for \(c_j\), none of which contains \(\hbar\) or
\(\Dplus\). X-B uses \(\Dplus\) in the Wick product and in the
one-contraction check; Proposition~\ref{prop:check} below shows
that the check is independent of \(\Dplus\).

(iv) By inspection of Definition~\ref{def:sheet}: the kernels
\(G_0^{\mathrm R}\) inside \(\Xone\), \(\Xtwo\), and
\(\Lambda_\eps\) are applied to the classical sources
\(\partial_\mu(\sig a_1^{\mu\rho}[h])\) and their second-order
analogues; \(\EA\) is applied to \(c\). The background slots of
\(c_2\) are evaluated on \((h,h)\) and \(k\): two slots of the same
kernel are both evaluated classically; neither is integrated against
a two-point distribution in the metric variables.

(v) Every object has field degree one in \(\varphi\); a functional
of \(h,k\) alone would have degree zero.
\end{proof}

For reference in the proof, X-B's defect formula is
\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-ref}
\end{equation}
both terms of which are elements \(\field(d)\).

\paragraph{Non-goals (recorded, not inferred).}
\begin{enumerate}[label=\textup{(N\arabic*)}]
\item No quantum metric propagator: no bidistribution in two metric
slots is present in \(\mathfrak S_A\).
\item No gravitational action and no graviton vertex: no functional
of degree zero in \(\varphi\) is formed.
\item No loop: no object contains a kernel contracted against
itself; the only two-point kernels are applied to smooth compact
sources.
\item No hidden contraction: by Proposition~\ref{prop:scope}(iii)
the constructed classes do not depend on \(\Dplus\).
\item No second-order quantum map: \(\rho_{2,g}\) is displayed in
X-B as classical input and is not an object of
Proposition~\ref{prop:scope}.
\item Branch B (Section~\ref{sec:branchB}) is a different
construction. Treating a graviton line as latent in \(\Rone\) would
contradict (N1)--(N4).
\end{enumerate}

%=====================================================================
\section{C2: the one-contraction boundary}
\label{sec:boundary}
%=====================================================================

\subsection{The auxiliary kernel}

\begin{lemma}[Properties of \(\DF\)]
\label{lem:DF}
Let \(\DF=\Dplus+i\Delta_S^{\mathrm A}=\Dplus-i\EA\), regarded as
the Schwartz kernel in \(\Dspace'(M\times M)\).
\begin{enumerate}[label=\textup{(\roman*)}]
\item \(\DF^{\mathsf T}=\DF\), i.e.\ \(\DF(x,y)=\DF(y,x)\).
\item \((P_m\otimes1)\DF=-i\,\deltaD\), where \(\deltaD\) is the
diagonal delta, \(\pair{\deltaD}{\psi}=\int\psi(x,x)\dd^4x\).
Equivalently \(P_m(\DF c)=-ic\) for \(c\in\Dspace\).
\item For \(c\in\Dspace\), both \(\Dplus c\) and \(\DF c\) are
smooth functions.
\item \(\WF(\DF)\supset\{(x,x;\xi,-\xi):\ x\in M,\ \xi\neq0\}\).
\item \(\DF-\Dplus=-i\EA\) and \(\DF-\Dplus^{\mathsf T}=-i\ER\).
\end{enumerate}
\end{lemma}

\begin{proof}
(i) Using \((\EA)^{\mathsf T}=\ER\) and
\(\Dplus^{\mathsf T}=\Dplus-i\DS\),
\[
\DF^{\mathsf T}=\Dplus^{\mathsf T}-i\ER
=\Dplus-i(\EA-\ER)-i\ER=\Dplus-i\EA=\DF .
\]
(ii) \(\Dplus\) is a bisolution, so \((P_m\otimes1)\Dplus=0\);
\(P_m\EA c=c\) for all \(c\) says \((P_m\otimes1)\EA=\deltaD\) for
the kernel of \(\EA\). Hence \((P_m\otimes1)\DF=-i\deltaD\).

(iii) \(\EA c\) is smooth by the Green operator property.
For \(\Dplus c\): since \(\WF(\Dplus)\) contains no point
\((x,\xi;y,0)\) with \(\xi\neq0\), the kernel maps \(\Dspace\) into
\(C^\infty\) \cite[Thm.~8.2.13]{Hormander}.

(iv) \(\deltaD\) is the pullback of \(\delta_0\in\Dspace'(\R^4)\)
under the submersion \((x,y)\mapsto x-y\), so
\(\WF(\deltaD)=\{(x,x;\xi,-\xi):\xi\neq0\}\)
\cite[Thm.~8.2.4]{Hormander}. A differential operator with smooth
coefficients does not enlarge the wavefront set
\cite[(8.1.11)]{Hormander}, so
\(\WF(\deltaD)=\WF((P_m\otimes1)\DF)\subset\WF(\DF)\).

(v) The first is the definition. For the second,
\(\DF-\Dplus^{\mathsf T}=\Dplus-i\EA-\Dplus+i\DS=-i\EA+i\EA-i\ER=-i\ER\).
\end{proof}

\subsection{The check, re-derived}

For the pair (A1)--(A2) define, as in X-B,
\[
\Cf(V,F_c)=VF_c+\hbar\pair{V'}{\DF c},\qquad
\Cf(F_c,V)=F_cV+\hbar\pair{c}{\DF V'} .
\]
Both pairings are of a compactly supported smooth function against
a smooth function: \(V'(\varphi)=-\Lop\varphi\) is smooth with
support in the compact set carrying the coefficients of \(\Lop\),
and \(\DF c\), \(\DF V'\) are smooth by Lemma~\ref{lem:DF}(iii)
and because \(V'\) is itself a compact smooth test for each
\(\varphi\).

\begin{lemma}[Wick termination]
\label{lem:wick}
For the pair \textup{(A1)}--\textup{(A2)},
\[
V\star F_c=VF_c+\hbar\pair{V'}{\Dplus c},\qquad
F_c\star V=VF_c+\hbar\pair{V'}{\Dplus^{\mathsf T}c}.
\]
\end{lemma}

\begin{proof}
\(F_c^{(1)}=c\) and \(F_c^{(n)}=0\) for \(n\geq2\), so the Wick sum
has only the terms \(n=0,1\). For the second formula,
\(\pair{c}{\Dplus V'}=\pair{V'}{\Dplus^{\mathsf T}c}\).
\end{proof}

\begin{proposition}[The one-contraction identity]
\label{prop:check}
For the pair \textup{(A1)}--\textup{(A2)},
\begin{equation}
\frac{i}{\hbar}\bigl(\Cf(V,F_c)-V\star F_c\bigr)
=\pair{V'}{\EA c}
=-\pair{V'}{\Delta_S^{\mathrm A}c}
=-\pair{c}{\Delta_S^{\mathrm R}V'} ,
\label{eq:check}
\end{equation}
and for \(V'=-\Lop\varphi\) the right-hand side equals
\(-F_{\Lop\EA c}(\varphi)\). In particular, for
\(V=V_{1,g}[h]\) it equals \(F_{d_c[h]}(\varphi)\), whose class is
\(\Rone(V_{1,g}[h];F_c)\). The identity is independent of \(\Dplus\)
and of \(\hbar\). Moreover \(\Cf(V,F_c)=\Cf(F_c,V)\).
\end{proposition}

\begin{proof}
By Lemma~\ref{lem:wick} and Lemma~\ref{lem:DF}(v),
\(\Cf(V,F_c)-V\star F_c=\hbar\pair{V'}{(\DF-\Dplus)c}
=-i\hbar\pair{V'}{\EA c}\). Multiplying by \(i/\hbar\) gives
\(\pair{V'}{\EA c}\), since \(i\cdot(-i)=1\). The second equality is
\(\Delta_S^{\mathrm A}=-\EA\). The third is transposition,
\(\pair{V'}{\EA c}=\pair{\ER V'}{c}=-\pair{c}{\Delta_S^{\mathrm R}V'}\),
legitimate because \(V'\) is compactly supported and smooth.
For \(V'=-\Lop\varphi\): \(\pair{-\Lop\varphi}{\EA c}
=-\pair{\varphi}{\Lop\EA c}\) by formal self-adjointness of \(\Lop\)
with compact coefficients and smoothness of \(\EA c\); this is
\(-F_{\Lop\EA c}(\varphi)\), and with \(\Lop=\Lone[h]\) it is
\(F_{d_c[h]}(\varphi)\). Independence of \(\Dplus\) and \(\hbar\) is
visible in the result. Symmetry: by Lemma~\ref{lem:DF}(i),
\(\pair{c}{\DF V'}=\pair{\DF^{\mathsf T}c}{V'}=\pair{V'}{\DF c}\),
and \(VF_c=F_cV\) pointwise.
\end{proof}

\begin{proposition}[Restricted causal factorization for the one pair]
\label{prop:causal}
Let \(V\) be as in \textup{(A1)} with coefficient support
\(S=\supp\Lop\subset M\) compact, and \(c\in\Dspace\).
\begin{enumerate}[label=\textup{(\roman*)}]
\item If \(S\cap J^-(\supp c)=\varnothing\), then
\(\Cf(V,F_c)=V\star F_c\), and \(\Rone(V;F_c)=0\) for
\(V=V_{1,g}[h]\).
\item If \(S\cap J^+(\supp c)=\varnothing\), then
\(\Cf(V,F_c)=F_c\star V\).
\end{enumerate}
\end{proposition}

\begin{proof}
(i) \(\Cf(V,F_c)-V\star F_c=-i\hbar\pair{V'}{\EA c}\), and
\(\supp\EA c\subset J^-(\supp c)\) is disjoint from
\(\supp V'\subset S\). The vanishing of \(\Rone\) is then
\eqref{eq:check}, or directly \(\Lone[h]\EA c=0\).
(ii) By Lemma~\ref{lem:wick} and Lemma~\ref{lem:DF}(v),
\(\Cf(V,F_c)-F_c\star V=\hbar\pair{V'}{(\DF-\Dplus^{\mathsf T})c}
=-i\hbar\pair{V'}{\ER c}\), and
\(\supp\ER c\subset J^+(\supp c)\) is disjoint from \(S\).
\end{proof}

Propositions~\ref{prop:check} and~\ref{prop:causal} say exactly
what X-B says: the auxiliary expression supplies the usual
one-contraction retarded interpretation for this pair, with the
symmetry and the two causal-factorization properties that a
time-ordered product would have to satisfy on this pair.

\subsection{What a family would require, and why the sheet stops}

A family of time-ordered maps in the sense of \cite{DF,BF,FR}
consists of multilinear maps \(T_n\) on \(n\)-tuples of admissible
functionals, satisfying at least
\begin{enumerate}[label=\textup{(T\arabic*)}]
\item symmetry in the entries;
\item causal factorization: \(T_{n+k}(A,B)=T_n(A)\star T_k(B)\)
whenever the support of \(A\) is disjoint from the causal past of
that of \(B\);
\item a Wick-type expansion in which the \(j\)-contraction term of
\(T_2(A,B)\) is \(\frac{\hbar^j}{j!}\pair{A^{(j)}}{\DF^{\otimes j}B^{(j)}}\)
wherever this is defined, with a specified extension where it is
not;
\item \(T_1=\mathrm{id}\), and normalization conditions for the
extensions in \textup{(T3)}.
\end{enumerate}
A retarded product is then \(R(A;B)=\frac{i}{\hbar}(T_2(A,B)-A\star B)\).

\begin{theorem}[Non-extension of the one-contraction check]
\label{thm:nonextension}
\begin{enumerate}[label=\textup{(\roman*)}]
\item The expression \(\Cf(V,F_c)\) is defined by the sheet for
every pair \textup{(A1)}--\textup{(A2)}, and
Propositions~\ref{prop:check}--\ref{prop:causal} are the
restrictions of \textup{(T1)}--\textup{(T3)} to this pair; the
resulting retarded expression is \eqref{eq:Rone}.
\item For the pair \((V,V)\) with \(V\) as in \textup{(A1)} and
\(\Lop\neq0\), the two-contraction term that \textup{(T3)} would
prescribe,
\(\frac{\hbar^2}{2}\pair{V''}{\DF^{\otimes2}V''}\), contains the
pointwise product \(\DF\cdot\DF\) of the kernel with itself on
\(M\times M\). This product is not defined by the wavefront product
criterion: at every point of the diagonal, \(\WF(\DF)\) contains
both \((x,x;\xi,-\xi)\) and \((x,x;-\xi,\xi)\), whose sum is zero.
\item By contrast the corresponding Wick term
\(\frac{\hbar^2}{2}\pair{V''}{\Dplus^{\otimes2}V''}\) is defined by
the same criterion, consistently with the definition of \(\Aoff\).
\item Consequently the sheet of X-B defines no \(T_2(V,V)\), no
\(R(V;V)\), and no \(T_n\) for \(n\geq2\) on general polynomial
pairs. The one-contraction check is an identity for the pair
\textup{(A1)}--\textup{(A2)} and is not a family of time-ordered
maps. Passing the check for this pair does not pass the
requirements \textup{(T1)}--\textup{(T4)} for any other pair.
\end{enumerate}
\end{theorem}

\begin{proof}
(i) is Propositions~\ref{prop:check} and~\ref{prop:causal}.

(ii) The kernel of \(V''=-\Lop\) is \(-\Lop_x\deltaD(x,y)\). In
\(\pair{V''}{\DF^{\otimes2}V''}
=\int V''(x_1,x_2)\DF(x_1,y_1)\DF(x_2,y_2)V''(y_1,y_2)\),
integrating the first delta kernel against \(x_2\) identifies the
first slots of the two \(\DF\) factors (with \(\Lop\) acting on one
of them), and integrating the second delta kernel against \(y_2\)
identifies their second slots. The result is the pairing of the
compactly supported coefficient distribution of
\(\Lop\otimes\Lop\) against the restriction to
\(\{x_1=x_2,\ y_1=y_2\}\) of \(\DF(x_1,y_1)\DF(x_2,y_2)\), that is,
against a derivative of the pointwise product \(\DF\cdot\DF\) on
\(M\times M\). The sufficient condition for the existence of a
product \(uv\) of distributions is that there be no
\((z,\zeta)\in\WF(u)\) with \((z,-\zeta)\in\WF(v)\)
\cite[Thm.~8.2.10]{Hormander}. With \(u=v=\DF\) and
\(z=(x,x)\), \(\zeta=(\xi,-\xi)\), Lemma~\ref{lem:DF}(iv) gives
both \((z,\zeta)\) and \((z,-\zeta)\) in \(\WF(\DF)\). The criterion
fails at every diagonal point.

(iii) Every element of \(\WF(\Dplus)\) has a nonzero first covector
in one fixed closed half of the null cone
(Premise~\ref{prem:free}). If \((z,\zeta)\) and \((z,-\zeta)\) were
both in \(\WF(\Dplus)\), the first components \(\xi\) and \(-\xi\)
would both lie in that closed half cone, forcing \(\xi=0\), which is
excluded. Hence the criterion holds and \(\Dplus\cdot\Dplus\) is
defined \cite[Thm.~8.2.10]{Hormander}.

(iv) A map \(T_2\) satisfying (T3) on \((V,V)\) requires a
distribution extending \(\DF\cdot\DF\) across the diagonal; the
sheet contains no such extension and no normalization condition
(T4) selecting one. The existence of such extensions and their
freedom is the renormalization problem of \cite{BF}, which is not
opened here. Without \(T_2(V,V)\) there is no \(R(V;V)\); without
\(T_2\) on general pairs there is no family. The last sentence is
the content of (i)--(iii): the pair (A1)--(A2) never meets the
undefined product because \(F_c^{(2)}=0\).
\end{proof}

\begin{remark}[Where the boundary lies]
The boundary is not at \(\DF\) itself, which is a well-defined
distribution and a fundamental solution up to the factor \(-i\)
(Lemma~\ref{lem:DF}(ii)); it is at the first pointwise product of
\(\DF\) with itself, i.e.\ at the first pair of entries both of
field degree at least two. X-B's wording ``it does not constitute a
claimed family of time-ordered maps'' is therefore exact. No
statement of this section asserts that such a family does not
exist; it asserts that the sheet does not define one.
\end{remark}

%=====================================================================
\section{C3: the normalization sheet}
\label{sec:normalization}
%=====================================================================

\subsection{The degree-one part of the free algebra}

\begin{lemma}[Graded ideal]
\label{lem:graded}
\(\Jfree\) is graded by field degree, and
\(\Jfree\cap\Aoff^{(1)}=\{F_{P_mv}:v\in\Dspace\}\). Consequently,
for \(d\in\Dspace\), \(\field(d)=0\) in \(\Afree\) if and only if
\(d\in P_m\Dspace\).
\end{lemma}

\begin{proof}
A general element of the pointwise ideal is a finite sum
\(\sum_iG_iF_{P_mu_i}\) with \(G_i\in\Aoff\). Decomposing each
\(G_i\) into homogeneous parts and using that the pointwise product
of homogeneous functionals of degrees \(p,q\) is homogeneous of
degree \(p+q\), every homogeneous component of the sum is again in
\(\Jfree\); so \(\Jfree\) is graded. Its degree-one part consists of
\(\sum_i\lambda_iF_{P_mu_i}=F_{P_m(\sum_i\lambda_iu_i)}\),
\(\lambda_i\in\C\). Finally \(F_d\in\Jfree\) iff
\(F_d\in\Jfree\cap\Aoff^{(1)}\) iff \(F_d=F_{P_mv}\) for some
\(v\), iff \(d=P_mv\), since \(F_d\) determines \(d\) (test on
\(\varphi\in\Dspace\)).
\end{proof}

\begin{lemma}[Detection by spatially compact solutions]
\label{lem:detect}
For \(d\in\Dspace\) the following are equivalent:
\begin{enumerate}[label=\textup{(\alph*)}]
\item \(d\in P_m\Dspace\), i.e.\ \(\field(d)=0\);
\item \(\DS d=0\);
\item \(\pair{d}{u}=0\) for every \(u\in\DS\Dspace\) (every smooth
spatially compact solution);
\item \(\pair{d}{u}=0\) for every smooth solution \(u\) of \(P_mu=0\).
\end{enumerate}
\end{lemma}

\begin{proof}
(a)\(\Rightarrow\)(d): \(\pair{P_mv}{u}=\pair{v}{P_mu}=0\), the
integration by parts being justified by compactness of
\(\supp v\). (d)\(\Rightarrow\)(c) is trivial. (c)\(\Rightarrow\)(b):
for \(e\in\Dspace\), \(\pair{d}{\DS e}=\pair{\DS^{\mathsf T}d}{e}
=-\pair{\DS d}{e}\); vanishing for all \(e\) forces the smooth
function \(\DS d\) to vanish. (b)\(\Rightarrow\)(a): if
\(\EA d=\ER d=:v\), then
\(\supp v\subset J^+(\supp d)\cap J^-(\supp d)\), which is compact
in Minkowski space (global hyperbolicity), so \(v\in\Dspace\) and
\(P_mv=P_m\ER d=d\). The parenthetical identification in (c) is
\cite[Thm.~3.4.7]{BGP}.
\end{proof}

\begin{definition}[Classical evaluation]
\label{def:ev}
For a smooth solution \(u\) of \(P_mu=0\), define
\(\ev_u:\Afone\to\C\) by \(\ev_u\field(d)=\pair{d}{u}\). This is
well defined by Lemma~\ref{lem:detect}, (a)\(\Rightarrow\)(d), and
the family \(\{\ev_u:u\in\DS\Dspace\}\) separates points of
\(\Afone\) by Lemma~\ref{lem:detect}, (c)\(\Rightarrow\)(a).
\end{definition}

\subsection{What the transposed notation means}

\begin{lemma}[Transposition]
\label{lem:transpose}
For \(c\in\Dspace\), \(h\in\Bg\), and every smooth \(u\) (not
necessarily a solution),
\begin{equation}
-\pair{c}{\ER\Lone[h]u}=\pair{d_c[h]}{u}.
\label{eq:transpose}
\end{equation}
If \(u\) is a solution, the left side is
\(\int c\,\rho_{1,g}[h;u]\dd^4x\) and the right side is
\(\ev_u\Rone(V_{1,g}[h];F_c)\). Thus the notation
\(-\pair{c}{\ER\Lone[h]\field}\) denotes the element
\(\field(d_c[h])\in\Afone\), characterized by its classical
evaluations; it does not denote the application of \(\ER\) to an
unsmeared quantum field.
\end{lemma}

\begin{proof}
\(\Lone[h]u\) is smooth with support in the compact set \(\supp g\),
so \(\ER\Lone[h]u\) is defined and smooth, and
\(\pair{c}{\ER w}=\pair{\EA c}{w}\) for compact \(w\). Then
\(\pair{\EA c}{\Lone[h]u}=\pair{\Lone[h]\EA c}{u}\) by formal
self-adjointness of \(\Lone[h]\), legitimate because its
coefficients are compactly supported and \(\EA c\), \(u\) are
smooth. Hence \(-\pair{c}{\ER\Lone[h]u}=\pair{-\Lone[h]\EA c}{u}
=\pair{d_c[h]}{u}\). The rest is Definition~\ref{def:ev} and
\(\rho_{1,g}[h;u]=-\ER\Lone[h]u\).
\end{proof}

\subsection{The sheet determines the map}

\begin{definition}[Classical normalization on \(\Loff\)]
\label{def:sheetN}
Fix \(h\in\Bg\). A map \(\mathsf R:\Loff\to\Afone\) satisfies the
\emph{classical normalization sheet} for \(V_{1,g}[h]\) if
\begin{equation}
\ev_u\mathsf R(F_c)=\int c\,\rho_{1,g}[h;u]\dd^4x
=-\pair{c}{\ER\Lone[h]u}
\qquad\text{for all }c\in\Dspace,\ u\in\DS\Dspace .
\label{eq:sheet}
\end{equation}
No linearity or continuity of \(\mathsf R\) is assumed in this
definition.
\end{definition}

X-B's product sheet writes the normalization row for the single
test \(f\); the proof of X-B, Theorem \texttt{thm:retarded}, is for
arbitrary \(c\). Definition~\ref{def:sheetN} records the
normalization on the whole domain \(\Loff\), which is what that
proof establishes.

\begin{theorem}[The classical normalization determines \(\Rone\)]
\label{thm:unique}
For every \(h\in\Bg\), the map \(\Rone(V_{1,g}[h];\cdot)\) of
\eqref{eq:Rone} satisfies \eqref{eq:sheet}, and it is the only map
\(\Loff\to\Afone\) that does.
\end{theorem}

\begin{proof}
Existence is Lemma~\ref{lem:transpose}. If \(\mathsf R\) satisfies
\eqref{eq:sheet}, write \(\mathsf R(F_c)=\field(d'_c)\) with
\(d'_c\in\Dspace\). Then \(\pair{d'_c-d_c[h]}{u}=0\) for all
\(u\in\DS\Dspace\), so \(d'_c-d_c[h]\in P_m\Dspace\) by
Lemma~\ref{lem:detect}, and \(\field(d'_c)=\field(d_c[h])\) by
Lemma~\ref{lem:graded}.
\end{proof}

\begin{corollary}[Contact modifications relative to the sheet]
\label{cor:contact}
Let \(N:\Dspace\to\Dspace\) be any map and
\(\mathsf R^N(F_c)=\field(d_c[h]+Nc)\).
\begin{enumerate}[label=\textup{(\roman*)}]
\item \(\mathsf R^N\) satisfies \eqref{eq:sheet} if and only if
\(\DS Nc=0\) for all \(c\), if and only if \(\field(Nc)=0\) for
all \(c\); in that case \(\mathsf R^N=\Rone(V_{1,g}[h];\cdot)\) as
maps into \(\Afree\).
\item Example of a sheet-compatible modification: \(Nc=P_m(\chi c)\)
with \(\chi\in C^\infty(M)\); it is invisible in \(\Afree\).
\item Example of a sheet-incompatible modification: \(Nc=wc\) with
\(w\in C^\infty(M)\) real and \(w\not\equiv0\); in particular
\(w=\lambda\,g\,b_1[h]\) with \(\lambda\neq0\) whenever
\(g\,b_1[h]\not\equiv0\).
\end{enumerate}
\end{corollary}

\begin{proof}
(i) \(\ev_u\mathsf R^N(F_c)-\ev_u\Rone(F_c)=\pair{Nc}{u}\); apply
Lemma~\ref{lem:detect} to \(Nc\) for each \(c\), and
Lemma~\ref{lem:graded}. (ii) \(P_m(\chi c)\in P_m\Dspace\).
(iii) Choose \(x_0\) with \(w(x_0)\neq0\). The Cauchy problem with
smooth compactly supported data \(u|_{t=x_0^0}=\psi\),
\(\partial_tu|_{t=x_0^0}=0\), \(\psi(\mathbf x_0)\neq0\), has a
unique smooth solution, spatially compact by finite propagation
\cite[Thm.~3.2.11]{BGP}; hence \(u\in\DS\Dspace\)
\cite[Thm.~3.4.7]{BGP}, and \(u(x_0)\neq0\). On a small ball around
\(x_0\) the real function \(\operatorname{Re}(w\bar u)\) or
\(\operatorname{Im}(w\bar u)\) has a fixed sign; choose \(c\geq0\),
\(c\not\equiv0\), supported in that ball with the corresponding
phase so that \(\pair{wc}{u}\neq0\). Thus \(\DS(wc)\neq0\) and (i)
fails.
\end{proof}

\begin{corollary}[No contact enters the first-order identity]
\label{cor:ward}
Let \(N\) be sheet-compatible as in Corollary~\ref{cor:contact}(i)
and \(\widehat A_1^N[h]=\Obs^{(1)}[h]+\mathsf R^N(F_f)\). Then
\(\widehat A_1^N[h]=\widehat A_1[h]\) in \(\Afree\) for every
\(h\), so X-B's Theorem \texttt{thm:ward} holds for
\(\widehat A_1^N\) exactly when it holds for \(\widehat A_1\). No
contact term is selected to repair the identity, and none could
break it without violating \eqref{eq:sheet}.
\end{corollary}

\begin{proof}
Corollary~\ref{cor:contact}(i) with \(c=f\).
\end{proof}

\begin{remark}[Scope of the uniqueness statement]
\label{rem:scope-unique}
Theorem~\ref{thm:unique} is uniqueness relative to the classical
normalization on \(\Loff\) and to the target \(\Afone\). It
replaces the sentence of X-B ``makes no uniqueness claim among
arbitrary additional contact normalizations'' by a precise one:
every modification of \(\Rone\) with values in \(\Afone\) either
changes a classical evaluation or is zero in \(\Afree\). Two
remarks delimit this.

(1) Under the usual grading in which a term with \(j\) contractions
has field degree lowered by \(2j\) and carries \(\hbar^j\)
\cite{DF,FR}, a finite renormalization of a would-be
\(T_2(V,F_c)\) by a local term respecting that grading has field
degree \(3-2j\) with \(j\geq1\), hence \(j=1\) and degree one;
after multiplication by \(i/\hbar\) it is an element of \(\Afone\)
of the form \(\field(Nc)\) with \(N\) local. All such terms are
covered by Corollary~\ref{cor:contact}. This remark presupposes a
\(T_2\), which the sheet does not define
(Theorem~\ref{thm:nonextension}); it explains why no additional
freedom is hidden at this order, and it is not a theorem about a
family.

(2) Uniqueness relative to the axioms (T1)--(T4) of a family is not
addressed, because no family is defined.
\end{remark}

\subsection{The normalization sheet, recorded}

\begin{center}
\small
\renewcommand{\arraystretch}{1.12}
\begin{tabularx}{\linewidth}{@{}p{.26\linewidth}X@{}}
\toprule
Item & Record\\
\midrule
Domain &
\(\Loff=\{F_c:c\in\Dspace\}\), off-shell representatives; not
on-shell classes (X-B, \texttt{eq:off-shell-warning})\\
Codomain &
\(\Afone\subset\Afree\), field degree one\\
The one map &
\(\Rone(V_{1,g}[h];F_c)=\field(-\Lone[h]\EA c)\)\\
Transposed notation &
\(-\pair{c}{\ER\Lone[h]\field}\) means \(\field(d_c[h])\),
characterized by \eqref{eq:transpose}\\
Classical normalization &
\eqref{eq:sheet} on all of \(\Loff\); determines the map
(Theorem~\ref{thm:unique})\\
Contact terms &
Sheet-compatible ones vanish in \(\Afree\); none selected; none
needed (Corollaries~\ref{cor:contact}, \ref{cor:ward})\\
Quantum order &
\(\hbar\)-free; \(\Dplus\)-independent (Proposition~\ref{prop:check})\\
Causal properties &
Additivity in the vertex; vanishing when
\(\supp g\cap J^-(\supp c)=\varnothing\)
(Proposition~\ref{prop:causal})\\
Not claimed &
A time-ordered family; uniqueness relative to (T1)--(T4)\\
\bottomrule
\end{tabularx}
\end{center}

%=====================================================================
\section{C4: Branch B input list}
\label{sec:branchB}
%=====================================================================

This section records data. It defines none of them, asserts
neither their existence nor their failure, and corrects nothing in
Branch A. X-B's statement is retained: a graviton response does not
follow from the scalar-only normalization, and a schematic triangle
with a partial diagonal is not an obstruction theorem.

\paragraph{Data a quantum-metric branch would have to supply.}
\begin{enumerate}[label=\textup{(G\arabic*)}]
\item A quadratic graviton action on the ramp sheet, its gauge
fixing, and its ghost sector, with the normalization of the
graviton Euler operator stated (the distinction between an
equation inverse and an action inverse belongs to another track
and is not imported).
\item Retarded and advanced Green operators for the gauge-fixed
graviton operator, with their transposition conventions and
support properties.
\item A graviton Hadamard two-point matrix compatible with
\textup{(G2)}: the shell equations, the antisymmetric part, and the
wavefront condition in the metric slots.
\item A contraction domain: a class of functionals of
\((h,\phi,\text{ghosts})\) with the cone condition imposed in the
metric slots, including the status of \(k\) and of the classical
coordinate kernel \(\Lambda_\eps\) once \(h\) is quantum.
\item The vertex \(V_{1,g}\) with \(h\) quantum: its field degree in
\(h\), the Wick ordering of mixed monomials, and its \(\hbar\),
\(\kappa\) grading.
\item The extension problem for the mixed kernel
\(f(x)g(y)K_{\mathrm{coord}}(z,x)\Delta^{\mathrm{gr}}(y,z)
\partial\partial\Delta(y,x)\) at \(y=x\), with its wavefront
analysis, and the corresponding problem for any graviton tree
\(\Xone[\Delta^{\mathrm{gr}}_{\mathrm R}(\tfrac12gT[\phi])]\).
\item A Ward formulation: the BRST operator, its action on
\(\field\), \(h\), and ghosts, and the status of \(\ga\) under it.
\item A comparison rule stating what agreement with the scalar-only
\(\Rone\) at order \(\kappa^1\) would mean, if anything, once
\textup{(G1)}--\textup{(G7)} are supplied.
\end{enumerate}

\paragraph{Non-relabeling rule.}
\begin{enumerate}[label=\textup{(R\arabic*)}]
\item \(\Lambda_\eps\) and \(G_0^{\mathrm R}\) are classical
coordinate kernels applied to classical sources; they are not
\textup{(G2)}.
\item \(\Dplus\) is the scalar two-point kernel; it is not
\textup{(G3)}.
\item The joint cone admission of X-B, Proposition
\texttt{prop:joint}, is a test on classical background slots; it is
not \textup{(G4)}.
\item \(V_{1,g}[h,\varphi]\) with classical \(h\) is not
\textup{(G5)}.
\item \(\Rone(V_{1,g}[h];\cdot)\) has field degree one in
\(\phi\) and degree zero in any quantum metric; it is not a
graviton response, and nothing in Proposition~\ref{prop:scope}
is a missing term of such a response.
\end{enumerate}

\begin{remark}
If \textup{(G1)}--\textup{(G8)} were supplied, the pair
\((V_{1,g},V_{1,g})\) with quantum \(h\) would meet a product of
two-point kernels at identified endpoints; the scalar version of
that boundary is Theorem~\ref{thm:nonextension}(ii). This is a
pointer to where the analysis would begin, not a prediction of its
outcome.
\end{remark}

%=====================================================================
\section{Debugging record relative to X-B}
\label{sec:debug}
%=====================================================================

\begin{enumerate}[label=\textup{D\arabic*.}]
\item \textbf{Signs in the check.} The chain
\(\Cf-V\star F_c=-i\hbar\pair{V'}{\EA c}\), times \(i/\hbar\),
equals \(\pair{V'}{\EA c}=-\pair{V'}{\Delta_S^{\mathrm A}c}
=-\pair{c}{\Delta_S^{\mathrm R}V'}\), and for \(V'=-\Lop\varphi\)
equals \(-F_{\Lop\EA c}\). This agrees with X-B's
\texttt{eq:contraction-check} and with \(d_c=-\Lone\EA c\).
\item \textbf{Symmetry of \(\DF\).} Asserted in X-B, proved in
Lemma~\ref{lem:DF}(i); it is kernel symmetry
\(\DF(x,y)=\DF(y,x)\), not a Hermiticity statement.
\item \textbf{Both directions of detection.} X-B's zero-switch
proof uses ``elements of \(\Jfree\) vanish on smooth free
solutions'' (one direction, sufficient there). Lemma~\ref{lem:detect}
proves the converse, which is what makes Theorem~\ref{thm:unique}
possible. X-B's argument is unchanged.
\item \textbf{Graded ideal.} The identification
\(\field(d)=0\iff d\in P_m\Dspace\) is used implicitly in X-B; it
is Lemma~\ref{lem:graded} here.
\item \textbf{Normalization row.} X-B's sheet states the row for
\(F_f\); its theorem proves it for all \(c\). Recorded on \(\Loff\)
in Definition~\ref{def:sheetN}.
\item \textbf{Uniqueness wording.} X-B's ``no uniqueness claim
among arbitrary additional contact normalizations'' is sharpened to
Theorem~\ref{thm:unique} and Remark~\ref{rem:scope-unique}; no
theorem of X-B is withdrawn.
\item \textbf{Lemma \texttt{lem:Q}.} Re-expanded:
\(\partial_\mu a_1^{\mu\rho}[K\xi]=-\Box\xi^\rho\), the
\(\partial(\partial\cdot\xi)\) terms cancel, and both sides equal
\((\partial\cdot\xi)P_mu-(\Box\xi^\nu)\partial_\nu u
-2\partial^\mu\xi^\nu\partial_\mu\partial_\nu u\). Consistent.
\item \textbf{Zero-switch witness.} With \(h=\psi(t)\eta\),
\(a_1=\psi\eta\), and the zero-data inverse of a time-only source,
\(\Xone^0=-\int_0^t\sig\psi\), \(c_1=-\partial_t(f\Psi)\),
\(\int c_1t=\int f\Psi>0\). Consistent.
\item \textbf{Edge gap formula.} \(\xi-\Xone[K\xi]
=\ERzero((1-\sig)\Box\xi)+\ERzero(\sig'a_1^{0\cdot}[K\xi])\)
re-derived from \(\partial_\mu(\sig a_1^{\mu\rho})
=\sig'a_1^{0\rho}-\sig\Box\xi^\rho\) and \(\xi=\ERzero\Box\xi\).
Consistent.
\item \textbf{Product criterion.} Theorem~\ref{thm:nonextension}(ii)
states failure of H\"ormander's sufficient criterion, not
nonexistence of every extension; (iii) confirms that the Wick
product of X-B is unaffected.
\item \textbf{Non-inputs.} \texttt{XII-B.tex} and the retrieval
summaries are not used. No statement of X-A beyond those copied in
X-B is used.
\item \textbf{Compilation.} All macros defined; \texttt{mathrsfs}
loaded for \verb|\mathscr|; all \verb|\ref| and \verb|\cite|
targets exist; no external files.
\end{enumerate}

%=====================================================================
\section{Release}
\label{sec:release}
%=====================================================================

\begin{center}
\small
\renewcommand{\arraystretch}{1.12}
\begin{tabularx}{\linewidth}{@{}p{.07\linewidth}p{.19\linewidth}X@{}}
\toprule
Item & Ending & Content\\
\midrule
C1 & \textbf{Closed} &
Proposition~\ref{prop:scope}: degree-one, classical-polynomial,
\(\hbar\)-free, \(\Dplus\)-independent objects; non-goals
(N1)--(N6).\\[0.3em]
C2 & \textbf{Closed} &
Theorem~\ref{thm:nonextension}: the check is an identity for one
pair with restricted (T1)--(T3); \(\DF\cdot\DF\) fails the product
criterion on the diagonal; no family is defined.\\[0.3em]
C3 & \textbf{Closed} &
Theorem~\ref{thm:unique}, Corollaries~\ref{cor:contact},
\ref{cor:ward}: the classical normalization on \(\Loff\) determines
\(\Rone\); sheet-compatible contacts vanish in \(\Afree\).\\[0.3em]
C4 & \textbf{Recorded} &
Input list (G1)--(G8) and non-relabeling rule (R1)--(R5); nothing
opened.\\
\bottomrule
\end{tabularx}
\end{center}

\begin{theorem}[X-B-ii release]
\label{thm:release}
For the frozen scalar-only field content of X-B:
\begin{enumerate}[label=\textup{(\roman*)}]
\item every constructed object is a field-degree-one element of
\(\Afree\) with polynomial classical background dependence and no
\(\hbar\) (Proposition~\ref{prop:scope});
\item the one-contraction check is an identity for the pair
\textup{(A1)}--\textup{(A2)}, symmetric and causally factorizing on
that pair, and the tools of the sheet define no second pair: the
pointwise square of \(\DF\) fails the wavefront product criterion
at every diagonal point (Theorem~\ref{thm:nonextension});
\item the single map \(\Rone(V_{1,g}[h];\cdot)\) is the unique map
\(\Loff\to\Afone\) with the classical normalization
\eqref{eq:sheet}; every sheet-compatible contact modification is
zero in \(\Afree\), and the first-order identity of X-B is
independent of any such modification (Theorem~\ref{thm:unique},
Corollaries~\ref{cor:contact}, \ref{cor:ward});
\item the Branch B data are listed and not supplied
(Section~\ref{sec:branchB}).
\end{enumerate}
No time-ordered family, graviton datum, second-order quantum map,
loop, state, or sharp limit is constructed, and none is excluded.
\end{theorem}

\begin{proof}
Each item cites the result that proves it.
\end{proof}

\[
\boxed{
\begin{gathered}
\text{C1, C2, C3 closed as scope theorems; C4 recorded.}\\[2pt]
\tfrac{i}{\hbar}\bigl(\Cf(V,F_c)-V\star F_c\bigr)=\pair{V'}{\EA c}
\ \text{for one pair};\qquad
\DF\cdot\DF\ \text{undefined by the sheet on the diagonal};\\[2pt]
\ev_u\mathsf R(F_c)=-\pair{c}{\ER\Lone[h]u}\ \forall c,u
\ \Longrightarrow\ \mathsf R=\Rone(V_{1,g}[h];\cdot).
\end{gathered}}
\]

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

\paragraph{Construction scope.}
Nothing is constructed beyond X-B. The theorems here are about
objects defined in X-B and about what the X-B sheet does and does
not define.

\paragraph{Predecessor scope.}
X-B is used in its supplied form \texttt{X-B-v3.tex}; X-A only as
copied there. Two of X-B's sentences are sharpened
(Section~\ref{sec:debug}, D3, D6); none is withdrawn.

\paragraph{No-go scope.}
Theorem~\ref{thm:nonextension} does not assert that a time-ordered
family does not exist; it asserts that the sheet does not define
one. Section~\ref{sec:branchB} asserts neither existence nor
failure of a quantum-metric branch.

\paragraph{Verification status.}
No successful \texttt{pdflatex} execution, proof-assistant check,
or independent referee review is claimed. The source uses standard
theorem environments and defines all internal reference targets.

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

\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 Full text used. Objects: Premises
\texttt{prem:geometry}--\texttt{prem:scalar}; Definition
\texttt{def:retarded}; Theorems \texttt{thm:retarded},
\texttt{thm:hbar}, \texttt{thm:ward}; Propositions
\texttt{prop:joint}, \texttt{prop:linear}, \texttt{prop:zero};
equations \texttt{eq:Rone}, \texttt{eq:contraction-check},
\texttt{eq:off-shell-warning}, \texttt{eq:defect-class}.

\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 only through the premises copied in X-B.

\bibitem{FR}
K.~Fredenhagen and K.~Rejzner.
\newblock \emph{Perturbative algebraic quantum field theory}.
\newblock arXiv:1208.1428.
\newblock Terminology for the free Wick presentation, the cone
condition, and time-ordered products.

\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{Rev.\ Math.\ Phys.} \textbf{16} (2004), 1291--1348.
\newblock Terminology for retarded products and the
\(\hbar\)/field-degree grading of renormalization terms.

\bibitem{BF}
R.~Brunetti and K.~Fredenhagen.
\newblock Microlocal analysis and interacting quantum field theories:
renormalization on physical backgrounds.
\newblock \emph{Comm.\ Math.\ Phys.} \textbf{208} (2000), 623--661.
\newblock Reference for the extension problem of products of
Feynman-type kernels across diagonals; not opened 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 Green operators, transposition, the Cauchy problem
(Thm.~3.2.11), and spatially compact solutions as
\(\DS\Dspace\) (Thm.~3.4.7).

\bibitem{Hormander}
L.~H\"ormander.
\newblock \emph{The Analysis of Linear Partial Differential Operators I}.
\newblock Second edition. Springer, 1990.
\newblock Wavefront set under differential operators (8.1.11),
pullbacks (Thm.~8.2.4), products (Thm.~8.2.10), and kernels
(Thm.~8.2.13).

\bibitem{Radzikowski}
M.~J.~Radzikowski.
\newblock Micro-local approach to the Hadamard condition in quantum
field theory on curved space-time.
\newblock \emph{Comm.\ Math.\ Phys.} \textbf{179} (1996), 529--553.
\newblock Wavefront set of Hadamard two-point distributions
(Thm.~5.1).

\end{thebibliography}

\end{document}