% External reference checks: ([arxiv.org](https://arxiv.org/pdf/1208.1428))
% !TeX program = pdflatex
%=====================================================================
% XIII-B.tex
%
% Standalone source for pdfLaTeX.
% Build twice:
%   pdflatex -interaction=nonstopmode -halt-on-error XIII-B.tex
%   pdflatex -interaction=nonstopmode -halt-on-error XIII-B.tex
%
% Release:
%   - the selected interaction functional is admitted;
%   - the selected classical binary causal composition is constructed;
%   - its properness, compactness, and microcausal admission are proved;
%   - overlap with the inherited first-completion roots is established;
%   - the coefficient through lambda*hbar is not released:
%     its joint extension remains Unresolved.
%
% Necessary corrections:
%   1. Causal factorization belongs to T_2. With
%        R_1 = (i/hbar)(T_2 - V star O),
%      ordered-support values of R_1 are zero or a scaled commutator,
%      not V star O.
%   2. Retarded support is a condition on the interaction-to-observable
%      attachment variables. It is not, for a nonlocal composite,
%      a claim that every remaining functional-support variable lies
%      in J^+(supp V).
%   3. The datum is attached to the displayed off-shell lifts. Its
%      admitted value has a class in E_H; a representative-independent
%      binary operation on all of E_H is not constructed.
%   4. The existing cubic-response root is an overlap normalization
%      for an inherited first-completion branch, not the value of
%      the entire binary response.
%   5. A nonlocal entry has internal partial collision diagonals.
%      Local-entry extension theorems do not settle its joint extension.
%
% No quotient by N_W, reduced algebra, completed observable, R_2,
% general T_n, W_1, interacting differential, master equation,
% state, loop value, gravitational interaction insertion, or
% sharp-width limit is supplied.
%
% The source includes an explicit unresolved extension theorem
% hypothesis rather than claiming an unproved quantum construction.
%
% No successful compilation, proof-assistant verification,
% independently executed symbolic computation, independent referee
% review, or journal acceptance is asserted.
%=====================================================================

\documentclass[11pt,a4paper]{article}

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\usepackage{enumitem}
\usepackage[hidelinks]{hyperref}

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\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{inputstatement}[theorem]{Input statement}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}

\newcommand{\R}{\mathbb R}
\newcommand{\C}{\mathbb C}
\newcommand{\dd}{\,\mathrm d}
\newcommand{\eps}{\varepsilon}
\newcommand{\sig}{\sigma_\eps}
\newcommand{\MT}{M_T}
\newcommand{\MTo}{M_T^\circ}
\newcommand{\PP}{\mathcal P}
\newcommand{\RR}{\mathcal R}
\newcommand{\Lie}{\mathcal L}
\newcommand{\Pmass}{P_m}
\newcommand{\ER}{E_0^{\mathrm R}}
\newcommand{\EA}{E_0^{\mathrm A}}
\newcommand{\ERm}{E_m^{\mathrm R}}
\newcommand{\EAm}{E_m^{\mathrm A}}
\newcommand{\Gret}{G_{\mathrm{gr}}^{\mathrm R}}
\newcommand{\GR}{G_{\mathrm{act}}^{\mathrm R}}
\newcommand{\GA}{G_{\mathrm{act}}^{\mathrm A}}
\newcommand{\Csol}{\mathscr C_{\mathrm{sol}}}
\newcommand{\CW}{\mathscr C_W}
\newcommand{\Fpol}{\mathscr F_{\mathrm{mc,pol}}}
\newcommand{\Nsol}{\mathscr N_{\mathrm{sol}}}
\newcommand{\NW}{\mathscr N_W}
\newcommand{\Ealg}{\mathscr E_H}
\newcommand{\Aone}{\mathcal A_1}
\newcommand{\Asc}{\mathcal A_2^{\mathrm{sc}}}
\newcommand{\CD}{\mathcal C_{3,\mathrm D}}
\newcommand{\JD}{\mathcal J_{\mathrm D}}
\newcommand{\PhiD}{\Phi^{(2)}_{\mathrm D}}
\newcommand{\rD}{r_{\mathrm D}}
\newcommand{\rg}{r_g}
\newcommand{\Ocal}{\mathcal O}
\newcommand{\Vcal}{\mathcal V_g}
\newcommand{\Aux}{\mathfrak B}
\newcommand{\sfree}{s_0}
\newcommand{\gW}{\mathfrak g_W}
\newcommand{\ctclass}{\mathfrak C_{\mathrm{tot}}}
\newcommand{\TT}{\mathsf T_2}
\newcommand{\Rtree}{R_1^{(0)}}
\newcommand{\Rquant}{R_1^{(1)}}
\newcommand{\pair}[2]{\left\langle #1,#2\right\rangle}

\DeclareMathOperator{\supp}{supp}
\DeclareMathOperator{\WF}{WF}
\DeclareMathOperator{\tr}{tr}
\DeclareMathOperator{\Int}{int}
\DeclareMathOperator{\id}{id}
\DeclareMathOperator{\Sym}{Sym}
\DeclareMathOperator{\gh}{gh}

\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 XIII-B}\\[0.4em]
One binary causal datum for the displayed coefficient,\\
on the unchanged carrier\\[0.4em]
\large Tree construction, normalization audit,\\
and the unresolved binary quantum extension}
\author{Ngo Cheung}
\date{}

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

\noindent
XIII-B uses evaluation-only claims in the inherited carrier
$\Ealg$. It does not use a reduced algebra or a completed observable.
Those readings remain Unresolved in XIII-A. The construction below
uses the displayed off-shell lifts and then takes their admitted
full-shell classes. Classical evaluation on $\CW$ follows admission;
it is not the domain of the product.

\begin{abstract}
At fixed regulator width, we study the selected pair consisting of
the switched metric--scalar interaction functional and the already
admitted second-order composite. We prove admission of the vertex
and construct the classical binary retarded coefficient. A balanced
kernel calculation proves the joint one-line composition, properness,
compact support, and microcausal admission. Its scalar branch is the
inherited switched scalar response; its metric branch has the
inherited inverse coefficient and agrees with the first-completion
root normalization wherever that branch is used.

Two corrections are indispensable. Causal factorization is a
normalization of the binary time-ordered datum, not the assertion
that the retarded coefficient equals the ordinary quantum product.
The latter assertion is incompatible with the stated retarded
definition and field-degree normalization. Moreover, retarded support
concerns the contracted attachment variables, not every uncontracted
variable of a nonlocal composite.

The coefficient linear in $\hbar$ requires a joint two-line extension.
We freeze a conservative total-diagonal counterterm class and identify
the additional internal collision problem, but do not claim an
extension that has not been constructed. The corrected quantum
release therefore remains Unresolved at that gate. We give precise
conditional admission and factorization statements, explicit classical
zero-evaluation subclasses, and the free-differential display.
No restriction quotient or gauge-consistency defect is formed.
\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) {Vertex functional};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Tree-level binary composition};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Joint distributional limit};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {The tree kernel does not evaluate the joint limit.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{A tree composition is not the joint limit. 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-XIII-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) {Vertex functional};
\node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {Joint distributional limit};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {The tree kernel does not evaluate the joint limit.};
\draw[->] (a) -- (s);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{A tree composition is not the joint limit. 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-XIII-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) {Vertex functional};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Tree-level binary composition};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {Joint distributional limit};
\node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {The tree kernel does not evaluate the joint limit.};
\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-XIII-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) {Vertex functional};
\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) {Joint distributional limit};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Tree-level binary composition. The tree kernel does not evaluate the joint limit.};
\draw (a) -- (m);
\draw (m) -- (b);
\draw[->] (m) -- (s);
\end{tikzpicture}
\caption{Vertex functional is not Joint distributional limit. 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-XIII-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: Vertex functional};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Tree-level binary composition};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: Joint distributional limit};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {The tree kernel does not evaluate the joint limit.};
\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-XIII-B}
\end{figure}

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


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



\scopebox{Release and proof boundary}{%
\textbf{Constructed:} the vertex functional, the selected tree-level
binary causal composition, its properness and compact microcausal
kernel, its inherited-root overlap, and the free-differential display.

\smallskip
\textbf{Proved obstructed:} the literal normalization
$R_1(\Vcal;\Ocal)=\Vcal\star_H\Ocal$ on ordered supports, when combined
with the specified retarded definition and contraction grading.

\smallskip
\textbf{Unresolved through the requested quantum order:} the joint
two-line extension in the frozen counterterm class. Neither a failed
wavefront criterion nor an unconstructed extension is called a no-go.

\smallskip
The carrier remains $\Ealg$. No quotient by $\NW$, completed
observable, general time-ordered family, $R_2$, gauge-consistency
defect, interacting differential, state, loop value, master equation,
or $\eps\to0$ limit is released.}

\tableofcontents

%=====================================================================
\section{B0: the inherited carrier and the restriction boundary}
\label{sec:opening}
%=====================================================================

The construction has three different levels:
\[
 \text{specified off-shell lift}
 \longrightarrow
 \text{admitted class in }\Ealg
 \longrightarrow
 \text{classical evaluation on }\CW.
\]
The second arrow does not turn the final evaluation set into a
quantum quotient. It also does not automatically make a retarded
operation independent of all choices of representative.

\begin{inputstatement}[XIII-A's restriction release]
\label{input:restriction}
Fix a nonempty open $W$ with
\[
 \overline W\subset P_g\cap\{\eps<t<T\},
 \qquad \gW=C_c^\infty(W;TM).
\]
The classical subclass is
\[
 \CW=\{u\in\Csol:D^*b=0\text{ on }W\}.
\]
Under the uniform support hypothesis
\begin{equation}
 \text{\rm(UCP$_W$)}\qquad
 g=1\ \text{near }J^-(\supp f)\cap J^+(W)\cap\MT,
 \label{eq:UCP}
\end{equation}
the prescribed lift satisfies
$\Delta_\xi\Ocal|_{\CW}=0$ for every $\xi\in\gW$.

The coefficientwise vanishing space $\NW$ is not a two-sided ideal
of the inherited product. Compact regular smearings give
\[
 [L_q,G_h]_{\star_H}=i\hbar C(q,h)\mathbf1,\qquad C(q,h)>0.
\]
An admissible rank-one ordering change also fails to preserve
coefficientwise vanishing on $\CW$. No restricted quantum domain is
handed off \cite{XIIIA}.
\end{inputstatement}

This is the release box inherited from XIII-A: classical cancellation
is Constructed; the naive coefficientwise quantum quotient is
Proved obstructed; a separately constructed reduction or completion
is Unresolved. We neither recompute its commutator nor modify its
constraint ideal.

The endings here are attached to the selected binary problem.
Constructed means that the required joint extension, normalization,
and admission have actually been proved. Proved obstructed means
that every candidate in a previously declared class fails a specified
condition. Unresolved means that a necessary extension or
compatibility theorem is missing. The last ending is not concealed
by writing a formal contraction formula.

%=====================================================================
\section{Prerequisite sheet and convention audit}
\label{sec:inputs}
%=====================================================================

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.17\linewidth}X@{}}
\toprule
Manuscript & Sentence or construction used\\
\midrule
XIII-A &
Evaluation-only cancellation, failure of the restriction ideal,
and failure of universal ordering preservation.\\[0.35em]
XII-F &
The copied negative witness and the scope of the auxiliary
remainder.\\[0.35em]
XII-E &
Assembly with coefficient $-1$, admission of the complete displayed
coefficient, and its explicit completion formulas.\\[0.35em]
XII-D &
The inherited first-completion roots and causal plateau proposition.\\[0.35em]
XII-A-v2 &
The polynomial product, causal brackets, full-shell quotient, and
admissible ordering class.\\[0.35em]
XII-C &
Boundary only: the Einstein--Hilbert action remains present and
the named empty invariant jet is not an interaction entry.\\
\bottomrule
\end{tabularx}
\end{center}

XI-B is used only for the inverse normalization and free symbol rule.
X-G, VII--IX, and Landmark I are not inputs. Standard wave and
distribution calculus are identified separately from manuscript
handoffs.

\subsection{The sheet and the inherited coefficient}

Use
\[
 M=\R^4,\quad
 \eta=\operatorname{diag}(1,-1,-1,-1),\quad
 \Box=\partial_t^2-\Delta_{\mathbf x},\quad
 \Pmass=\Box+m^2,\quad m\ge0,
\]
\[
 \MT=[0,T]\times\R^3,\qquad \MTo=(0,T)\times\R^3.
\]
The fixed smooth ramp $\sig$ is zero for $t\le0$ and one for
$t\ge\eps$, with $0<\eps<T$. The switch
$g\in C_c^\infty(\MTo;[0,1])$ is external. All configurations are
ambient smooth multiplets; spatial decay is not required.

The operators are
\[
 \PP r=r-\tfrac12\eta\tr_\eta r,\qquad
 D^\nu r=\partial_\mu(\PP r)^{\mu\nu},\qquad
 (Kv)_{\mu\nu}=\partial_\mu v_\nu+\partial_\nu v_\mu,
 \qquad \RR=K/2.
\]
Parentheses in indexed expressions include $\tfrac12$.
The complete free shell is
\[
 \Csol=\{\Box\gamma=0,\ \Pmass\phi=0,\
          \Box c=\Box\bar c=0,\ b=D\gamma\}.
\]
The auxiliary field remains independent off shell.

The named composite is
\begin{equation}
 \Ocal=
 \PhiD+\Aone[f,g](\rg[\phi];\phi),
 \qquad
 \PhiD=\Asc[f,g](\gamma,0;\phi)-\CD,
 \label{eq:O}
\end{equation}
with
\begin{equation}
 \rD=-u_{\mathrm D},\qquad
 \rg[\phi]=-\tfrac12\PP\ER(gT_0[\phi]).
 \label{eq:slots}
\end{equation}
It is an even, ghost-number-zero homogeneous cubic polynomial:
its sectors have degrees $(2,1)$ and $(0,3)$ in $(\gamma,\phi)$.
Its compact polarized kernel has the covector-sum-zero bound
$Z_3$ proved in XII-D and XII-E \cite{XIID,XIIE}.
Its metric derivatives include the admitted insertion's metric
slots. Nonlocality is not removed by suppressing those derivatives.

\subsection{Product data and action inverses}

The inherited carrier is
\[
 \Ealg=
 (\Fpol[[\hbar,\kappa]],\star_H)/
                         \Nsol[[\hbar,\kappa]],
 \qquad
 F\star_HG=m\exp(\hbar\Gamma_H)(F\otimes G).
\]
The relevant action inverses are
\begin{equation}
 \GR_{\gamma\gamma}=-\PP\ER,\qquad
 \GR_{\phi\phi}=-\ERm,
 \qquad
 (\GA)^{\mathsf T}=\GR
 \label{eq:action-inverse}
\end{equation}
in the even sector. Although $\ERm$ inverts the scalar equation,
the scalar action Hessian is $-\Pmass$. This sign enters the binary
response.

The causal part satisfies
\[
 H-H^{\mathsf T}=i\Delta,\qquad \Delta=\GR-\GA
\]
on the even blocks used here. Admissible differences $d$ are smooth
graded-symmetric shell bisolutions, not necessarily compactly
supported. XIII-A's rank-one example consequently remains within
the admissible class \cite{XIIA}.

For later comparison, the inherited roots are
\begin{equation}
 \Aone[f,g](r;\phi)=\int\alpha_{f,g}[\phi]\cdot r\,\dd^4x,
 \qquad
 \beta_{f,g}[\phi]=-\EA(\PP\alpha_{f,g}[\phi]).
 \label{eq:roots}
\end{equation}
They are not newly selected product data.

\subsection{The witness remains a boundary}

For XII-F's detecting data, the supplied value is
\begin{equation}
 \Delta_\xi\Ocal(u_*)
 =-8\int f(t,\mathbf x)
       \int_0^t(t-s)^2a(s)\,\dd s\,\dd^4x<0.
 \label{eq:witness}
\end{equation}
The parameter is compact, (CP) holds, and the initial configuration
lies on the complete free shell but outside the detected cancellation
subclass \cite{XIIF}. The factor $8$ and the cylinder integral are
copied. No binary normalization is chosen after this witness, and
the witness is not inserted into an odd Ward identity.

The retained action boundary is
\[
 S_{\mathrm{grav,D}}
 =S_{\mathrm{EH}}+\kappa\int(\sig-1)\ell_3\,\dd^4x.
\]
Gravitational interaction vertices are omitted from the selected
calculation, not from the action \cite{XIIC}.

%=====================================================================
\section{B1: the interaction functional and the printed order}
\label{sec:vertex}
%=====================================================================

\begin{definition}[Selected vertex]
The only interaction entry is
\begin{align}
 \Vcal[\gamma,\phi]
 &=-\tfrac12\int g\,\gamma_{\mu\nu}T_0^{\mu\nu}[\phi]\,\dd^4x\\
 &=\tfrac12\int g\left(
 a_1^{\mu\nu}[\gamma]\partial_\mu\phi\partial_\nu\phi
                       -m^2b_1[\gamma]\phi^2\right)\dd^4x,
 \label{eq:vertex}
\end{align}
where
\[
 a_1^{\mu\nu}[r]=\tfrac12\eta^{\mu\nu}\tr_\eta r-r^{\mu\nu},
 \qquad b_1[r]=\tfrac12\tr_\eta r,
\]
\[
 T_0^{\mu\nu}[\phi]
 =\partial^\mu\phi\partial^\nu\phi
 -\tfrac12\eta^{\mu\nu}
       \left((\partial\phi)^2-m^2\phi^2\right).
\]
Its parameter $\lambda$ counts this interaction insertion only.
\end{definition}

\begin{proposition}[Typing and admission]
\label{prop:vertex}
The vertex belongs to $\Fpol$, is even and has ghost number zero,
and has functional support contained in $\supp g$. Its derivatives
are
\begin{equation}
 (\Vcal)'_\gamma=-\tfrac12gT_0[\phi],
 \qquad
 (\Vcal)'_\phi=-V_{1,g}[\gamma]\phi,
 \label{eq:vertex-derivatives}
\end{equation}
with
\[
 V_{1,g}[r]v
 =\partial_\mu(g a_1^{\mu\nu}[r]\partial_\nu v)
                       +m^2g b_1[r]v.
\]
\end{proposition}

\begin{proof}
The cubic polarized kernel is a finite sum of derivatives of
three-field diagonal distributions, multiplied by $g$ and constant
fiber tensors. Its support is compact and its wavefront covectors
sum to zero. A nonzero tuple wholly in either pointed closed causal
cone cannot have zero sum. Hence the kernel is microcausal.

The fields entering the vertex are even and have ghost number zero.
Variation of the displayed density gives the metric derivative.
Compact integration by parts in the scalar variation gives the
second formula, retaining derivatives of $g$.
\end{proof}

The proposed expansion is printed once:
\begin{equation}
 R_{\lambda\Vcal}\Ocal
 =\Ocal+\lambda\bigl(\Rtree+\hbar\Rquant\bigr)
                    +O(\lambda^2,\lambda\hbar^2).
 \label{eq:order}
\end{equation}
We match $\lambda$ to one additional $\kappa$ power: since $\Ocal$
is the displayed $\kappa^2$ coefficient, its selected response would
contribute at $\kappa^3$ when $\lambda=\kappa$.
This does not assert completeness of the full $\kappa^3$ expansion.

Both entries have field degree three. A single cross-contraction
has degree four and a double cross-contraction has degree two.
The latter is the requested coefficient linear in $\hbar$ after
the retarded division by $\hbar$. Triple contractions lie beyond
the order in \eqref{eq:order}. No $R_2$ is introduced.

The Euler operator $V_{1,g}$ appears only in derivative calculations.
It is not the first argument of a time-ordered product.
Similarly, contractions involving metric derivatives of $\Ocal$
are contractions of the admitted composite. They do not define a
separate quantum product with $\JD$ as an argument.

%=====================================================================
\section{B2: the binary convention and the frozen extension class}
\label{sec:binary}
%=====================================================================

\subsection{The convention and its forced ordered values}

We adopt
\begin{equation}
 R_1(\Vcal;\Ocal)
 =\frac{i}{\hbar}
       \left(\TT^{\mathrm{ren}}(\Vcal,\Ocal)
                                    -\Vcal\star_H\Ocal\right).
 \label{eq:R-definition}
\end{equation}
The zeroth-order term of $\TT^{\mathrm{ren}}$ is the ordinary
product $\Vcal\Ocal$. On regular kernels the even time-ordered line
in this convention is
\begin{equation}
 H_T=H+i\GA.
 \label{eq:HT}
\end{equation}
For the vertex variables $y$ and observable variables $x$, it
equals $H(y,x)$ when the vertex is later and $H^{\mathsf T}(y,x)$
when the observable is later. This follows directly from
$H-H^{\mathsf T}=i(\GR-\GA)$ and causal support.

The usual distinction between a time-ordered normalization and
the corresponding retarded difference is made in the functional
formulation of causal perturbation theory \cite[Sections 6--7]{FR}.
Here the consequences are checked with the stated signs.

\begin{proposition}[Correct causal factorization]
\label{prop:factorization}
Whenever the full functional supports of the entries are strictly
time ordered, a binary time-ordered datum satisfying causal
factorization must obey
\begin{align}
 \supp\Vcal\text{ later than }\supp\Ocal:
 &\quad \TT^{\mathrm{ren}}=\Vcal\star_H\Ocal,
 & R_1&=0,\\
 \supp\Ocal\text{ later than }\supp\Vcal:
 &\quad \TT^{\mathrm{ren}}=\Ocal\star_H\Vcal,
 & R_1&=\frac{i}{\hbar}
                 [\Ocal,\Vcal]_{\star_H}.
 \label{eq:ordered-values}
\end{align}
The supports here are functional supports, not just $\supp f$.
\end{proposition}

\begin{proof}
Substitute the two time-ordered factorization values in
\eqref{eq:R-definition}. No extension on a collision diagonal can
alter these values on the corresponding separated ordered region.
\end{proof}

\begin{theorem}[The literal retarded-factorization requirement fails]
\label{thm:literal}
For nonzero displayed entries, the additional requirement
\[
 R_1(\Vcal;\Ocal)=\Vcal\star_H\Ocal
\]
is incompatible with \eqref{eq:R-definition}, ordinary zeroth-order
time ordering, and the contraction grading in \eqref{eq:order}.
No counterterm in the class below repairs this incompatibility.
\end{theorem}

\begin{proof}
The classical symbol of the right side is the homogeneous degree-six
polynomial $\Vcal\Ocal$. The classical retarded coefficient on the
left is a one-contraction polynomial of degree four.
Equality forces the degree-six polynomial to vanish identically.

For two nonzero bosonic polynomial functionals their ordinary
product is nonzero: restrict to a finite-dimensional space on which
both restrictions are nonzero, and use the integral-domain property
of the polynomial algebra there. The displayed entries contain no
odd variables. Thus the required equality is impossible.
All counterterms allowed below enter the degree-two quantum
coefficient and cannot alter this classical comparison.
\end{proof}

Zero observations give the trivial zero pair and are not covered by
the nonzero assertion. For the detecting data, the supplied witness
ensures that $\Ocal$ is not the zero polynomial.
The correction in \eqref{eq:ordered-values} is necessary, not a
choice made to remove a witness.

\subsection{A precise conservative counterterm class}

A nonlocal composite has no unique spacetime ``entry point.''
Therefore a statement about the total diagonal must specify the
graph presentation to which it refers.

\begin{definition}[Frozen total-diagonal class]
\label{def:counterterms}
Use the finite rooted graph presentations of $\Ocal$ supplied by
XII-D and XII-E, together with the local vertex
\eqref{eq:vertex}. The \emph{total graph diagonal} is the locus where
all spacetime vertices in a graph coincide.

The class $\ctclass$ permits changes only at order $\hbar^2$
in $\TT^{\mathrm{ren}}$, hence at order $\lambda\hbar$ in
\eqref{eq:order}. After pushforward these changes are finite sums
\begin{equation}
 K(\Phi)=\sum_{A,B,\alpha,\beta}
  \int f(x)\,
  P_{AB,\alpha,\beta}
       \bigl(g,\partial g,\sig,\sig',\sig''\bigr)(x)
  \,\partial^\alpha\Phi_A(x)\,
     \partial^\beta\Phi_B(x)\,\dd^4x,
 \label{eq:counterterms}
\end{equation}
where:
\begin{enumerate}[label=\textup{(\roman*)}]
\item $A,B\in\{\gamma,\phi\}$, with constant fiber contractions;
\item the total field-derivative order is at most eight;
\item each $P$ is a polynomial of total degree at most eight in the
listed external jets, with fixed complex coefficients depending on
the frozen mass;
\item no higher derivative of $g$ or $\sig$ is admitted as an
independent coefficient;
\item the unintegrated change is supported on the total graph
diagonal and preserves the degree-two Wick expansion;
\item the tree inverse and off-diagonal first-completion root
normalization are unchanged.
\end{enumerate}
\end{definition}

The numerical bounds in this conservative class are definitions,
not claims that power counting has classified all possible
renormalizations. The class is frozen before any evaluation is used.
It does not include arbitrary auxiliary or ghost corrections.

A smooth coefficient supported in the ramp is legal when it can
be built from the listed ramp jets. The classical no-ramp companion
lemma does not prohibit such a quantum counterterm.

Each counterterm in \eqref{eq:counterterms} is itself compact and
microcausal: its quadratic local kernel has balanced covectors.
This proves admission of the allowed ambiguities, not existence
of a distribution to which they can be added.

In particular, an internal partial collision diagonal is not the
total graph diagonal. If extension at a partial collision requires
a larger ambiguity class, that class must be declared separately;
it is not silently included in \ctclass.

%=====================================================================
\section{B3a: the constructed one-line causal composition}
\label{sec:tree}
%=====================================================================

Define the off-shell tree coefficient by
\begin{equation}
 \boxed{
 \Rtree
 =-\sum_{A,B}\int
   \Ocal'_A(x)\,\GR_{AB}(x,y)\,
                     (\Vcal)'_B(y)\,\dd^4x\,\dd^4y.
 }
 \label{eq:tree-definition}
\end{equation}
Only the metric and scalar diagonal blocks contribute.

\begin{lemma}[Balanced one-line composition]
\label{lem:one-line}
Let $F,G$ have compact homogeneous cubic kernels with wavefront
sets contained in $Z_3$. A contraction of one derivative slot
from each kernel through a balanced retarded Green line exists.
The remaining degree-four kernel is compact and has wavefront set
contained in $Z_4$.
\end{lemma}

\begin{proof}
Denote the contracted variables by $y,x$ and the four uncontracted
variables by $z_1,\ldots,z_4$. The derivative kernels have covectors
\[
 (q_y,p_1,p_2),\qquad (q_x,p_3,p_4),
\]
whose sums vanish; the line has covectors $(k,-k)$ at $(x,y)$.

A forbidden cancellation in the full distribution product would
have zero covectors at every uncontracted variable. The two
sum-zero relations then force $q_y=q_x=0$. The line covector must
also vanish, contradicting a nonzero cancellation. The product
therefore exists by the distribution product criterion.

Both derivative kernels are supported in fixed compact sets.
Their product with the line has compact support, so projection
over $x,y$ is proper. The wavefront pushforward condition sets the
contracted output covectors to zero. Summing all covectors leaves
$\sum_{j=1}^4p_j=0$. Thus the output bound is $Z_4$.
Fiber sums and normalized symmetrization preserve these conclusions.
\end{proof}

This proof includes the nonlocal derivative kernel of $\Ocal$.
It is not a theorem only for two local entries.
The product and proper-pushforward criteria used in the argument
are the standard wavefront calculus \cite{Hormander}.

\begin{theorem}[Tree datum and admission]
\label{thm:tree}
Equation~\eqref{eq:tree-definition} defines an even,
ghost-number-zero compact microcausal polynomial of degree four.
Its class
\[
 \mathbf R_1^{(0)}=Q_H\Rtree\in\Ealg
\]
is well defined for the displayed lifts. It is independent of the
smooth ordering difference $d$ at classical-symbol order.

Moreover, the binary time-ordered coefficient
\[
 \TT(\Vcal,\Ocal)
 =\Vcal\Ocal+\hbar\,C_1(H_T)+O(\hbar^2),
 \qquad
 C_1(L)=\pair{\Vcal'\otimes\Ocal'}{L},
\]
exists through the displayed order and yields \eqref{eq:tree-definition}
under \eqref{eq:R-definition}.
\end{theorem}

\begin{proof}
Proposition~\ref{prop:vertex} and the inherited bound for $\Ocal$
supply the hypotheses of Lemma~\ref{lem:one-line}.
The action inverse has balanced translation-invariant lines.
Consequently the contraction exists, is compact, and satisfies
$Z_4$, which implies microcausality.

The star contraction $C_1(H)$ is already defined by the inherited
domain theorem. The additional contraction $C_1(i\GA)$ is the
transpose of the constructed retarded contraction.
Thus
\[
 i\bigl(C_1(H_T)-C_1(H)\bigr)
 =-C_1(\GA)
 =-\pair{\Ocal'}{\GR\Vcal'}.
\]
This proves the sign and the stated binary coefficient.
Neither $\GR$ nor this classical contraction depends on $d$.
The inherited quotient map supplies the class.
\end{proof}

No extension was required for this one-line calculation. It is a
causal composition, not a newly renormalized product.

\subsection{The response branches and the overlap}

Substituting \eqref{eq:vertex-derivatives} and
\eqref{eq:action-inverse} gives
\begin{equation}
 \boxed{
 \Rtree
 =\pair{\Ocal'_\gamma}{\rg[\phi]}
                  +\pair{\Ocal'_\phi}{p_1[\gamma;\phi]},
 \qquad
 p_1[\gamma;\phi]=-\ERm V_{1,g}[\gamma]\phi.
 }
 \label{eq:tree-branches}
\end{equation}
Both the metric inverse coefficient $-1$ and the scalar Hessian
sign have been retained.

\begin{proposition}[Inherited-root overlap]
\label{prop:overlap}
Every occurrence of the inherited first-completion branch with
metric argument $\Gret w$ obeys
\begin{equation}
 \Aone[f,g](\Gret w;\phi)
       =\int\beta_{f,g}[\phi]\cdot w\,\dd^4x.
 \label{eq:root-overlap}
\end{equation}
In particular,
\begin{equation}
 \Aone[f,g](\rg[\phi];\phi)
       =\tfrac12\int\beta_{f,g}[\phi]\cdot gT_0[\phi]\,\dd^4x.
 \label{eq:stress-overlap}
\end{equation}
The tree datum uses these same response and root normalizations.
It is not identified with $\CD$.
\end{proposition}

\begin{proof}
Equation~\eqref{eq:root-overlap} is XII-D's transposition identity
with the inherited sign, not a new root construction.
Since $\rg=\Gret(gT_0/2)$, substitution gives
\eqref{eq:stress-overlap}. In \eqref{eq:tree-branches} the same
metric response is contracted with the complete metric derivative
of $\Ocal$, and the scalar branch is its inherited switched
response. Differentiating any displayed first-completion branch
uses the same identity.
\end{proof}

This is the meaningful overlap: a branch common to the constructions
has unchanged normalization. The entire binary response has degree
four, whereas $\CD$ has degree three. Requiring equality of those
objects would confuse operations and contradict their typing.

\subsection{Selected lifts versus a universal operation on classes}

The constructed value has a class in $\Ealg$.
This does not establish a binary map on arbitrary representatives
of the quotient. A simple equation-functional check explains why.

For a compact scalar test $a$, let
$E_a(\Phi)=\int a\,\Pmass\phi$. It vanishes on $\Csol$.
The tree response to this alternative observable lift is
\begin{equation}
 R_1^{(0)}(\Vcal;E_a)
 =-\int a\,V_{1,g}[\gamma]\phi\,\dd^4x.
 \label{eq:representative-test}
\end{equation}
Indeed $\Pmass\ERm=\id$ after transposition on the compact test.
The right side need not vanish on the free shell.

Thus retarded contraction is not automatically invariant under
adding free equation functionals to an entry. Here the prescribed
lifts are frozen, so \eqref{eq:representative-test} does not invalidate
Theorem~\ref{thm:tree}. It prevents the stronger, unproved claim that
we have constructed a representative-independent operation on all
of $\Ealg$. No new ideal is introduced to hide this distinction.

%=====================================================================
\section{B3b: the joint two-line extension}
\label{sec:extension}
%=====================================================================

For regular kernels, the coefficient needed next is
\begin{equation}
 \Rquant_{\mathrm{formal}}
 =\frac{i}{2}
   \pair{\Vcal''\otimes\Ocal''}
                 {H_T^{\otimes2}-H^{\otimes2}},
 \label{eq:two-line}
\end{equation}
with the standard slot pairings and symmetrizations.
Using \eqref{eq:HT}, the difference contains
\[
 i(H\otimes\GA+\GA\otimes H)-\GA\otimes\GA.
\]
The expression is a specification of contractions, not a proof
that their joint product exists.

\subsection{Why the one-line proof stops}

There is now one uncontracted field slot in each entry and two
contracted slots. In a proposed forbidden product cancellation,
the uncontracted covectors may vanish while the two contracted
covectors in an entry are opposite and nonzero. Their sum is zero,
so the $Z_3$ bounds do not force them individually to vanish.

This is the exact place where Lemma~\ref{lem:one-line} no longer
applies. It is not evidence that the distribution cannot exist;
it identifies a hypothesis needing additional analysis or extension.

For a nonlocal composite, the contracted observable variables need
not coincide with the observation point. Consequently extension is
not automatically reduced to one diagonal between two local
densities. Standard local time-ordered constructions specify maps
on local entries and their subsequently defined multilocal domain;
they do not supply the present joint nonlocal extension merely
from star-product admission \cite[Sections 6--7]{FR}.

\subsection{An internal collision already present in the pair}

The scalar-sourced coordinate part of $\Ocal$ contains a branch
with schematic form
\[
 \int f(x)\,Q^\mu{}_{ab}(x,z)\,
          g(z)T_0^{ab}[\phi](z)\,\partial_\mu\phi(x)
                                             \,\dd^4x\,\dd^4z.
\]
Two scalar contractions can join the stress slots at $z$ to the
two scalar slots of the interaction at $y$.
The resulting unintegrated expression contains two differentiated
time-ordered scalar lines between $y$ and $z$, together with the
already inherited retarded branch from $z$ to $x$.

The collision $y=z$ may occur while $x\ne z$. It is therefore a
partial graph diagonal. A counterterm supported only where
$x=y=z$ does not by its support alone normalize this internal
collision. Other branches contain additional internal vertices.
They require a joint check, not deletion because the final output
is called binary.

This observation does not prove that every extension in \ctclass
fails. A proof of such failure would need an explicit unavoidable
normalization incompatibility. None is inferred from the presence
of the collision.

\subsection{Properness is separate from existence}

\begin{proposition}[Graphwise properness]
\label{prop:properness}
For each supplied rooted graph of $\Ocal$, attach the compactly
switched vertex and either one or two cross-lines. On the closed
support set determined by the existing graph weights and retarded
internal lines, projection over all internal vertices is proper.
A support-preserving extension does not spoil this property.
\end{proposition}

\begin{proof}
XII-D and XII-E provide compact support sets for their rooted
graphs. Internal retarded vertices are bounded in time below by
the ramp or compact switch, above by the compact observation, and
spatially by the corresponding causal regions.

The new local interaction vertex lies in $\supp g$, a compact set.
Adding cross-lines restricts the product support inside the product
of these existing compact graph sets with $\supp g$.
Its closure is compact. Projection from this set is proper.
An extension supported in the closure of the same graph support,
including allowed total-diagonal terms, preserves compactness.
\end{proof}

This proves the relevant support control without proving the
singular joint distribution. Properness cannot repair an undefined
distribution product, and an extension theorem cannot be substituted
for properness.

\subsection{The conditional admission statement}

\begin{theorem}[Conditional quantum admission]
\label{thm:conditional}
Suppose the two-line graph distributions of the selected pair admit
a joint extension with all the following properties:
\begin{enumerate}[label=\textup{(\roman*)}]
\item agreement with the defined off-collision contractions and
with the corrected ordered-support values;
\item the frozen ambiguity class \ctclass;
\item support preservation and compatibility with the inherited
root branch;
\item a balanced wavefront bound on the complete graph variables,
or an independently proved microcausal bound after pushforward.
\end{enumerate}
Then the resulting $\Rquant$ is a compact even,
ghost-number-zero polynomial of degree two, and
\[
 Q_H\bigl(\Rtree+\hbar\Rquant\bigr)\in\Ealg.
\]
\end{theorem}

\begin{proof}
Each double contraction leaves two field slots. Properness follows
from Proposition~\ref{prop:properness}. A balanced complete graph
bound, after setting internal pushforward covectors to zero, leaves
a sum-zero bound on the two external covectors. That bound is
microcausal. Alternatively, the separately proved output bound
in (iv) gives the same admission directly.

Allowed counterterms are already admitted by
Definition~\ref{def:counterterms}. The inherited quotient map then
gives the stated class. All entries and lines are even in the
relevant sector, and no ghost slots are generated.
\end{proof}

The hypotheses of this theorem have not been jointly proved.
It is deliberately conditional.

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.23\linewidth}X@{}}
\toprule
Failure mode & Verdict warranted\\
\midrule
Product failure &
A sufficient wavefront condition fails and no extension is
constructed: Unresolved.\\[0.35em]
Carrier failure &
An extension exists but admission is not established: Unresolved,
unless every allowed extension is proved inadmissible.\\[0.35em]
Normalization failure &
Every allowed extension violates a named overlap or causal
normalization: Proved obstructed, if that universal statement
is actually proved.\\
\bottomrule
\end{tabularx}
\end{center}

For the corrected binary problem, the release at this gate is
\textbf{Unresolved}. The literal factorization error was separately
Proved obstructed in Theorem~\ref{thm:literal}; it is not relabelled
as an obstruction to every corrected extension.

%=====================================================================
\section{B4: causal support and classical evaluation}
\label{sec:evaluation}
%=====================================================================

The established tree attachment has support
\begin{equation}
 \supp\GR(x,y)\subset\{(x,y):x\in J^+(y)\}.
 \label{eq:retarded-attachment}
\end{equation}
Thus the interaction attachment can affect the contracted observable
slot only to its future. If the entire interaction support lies later
than the entire functional support of $\Ocal$, the tree datum is zero.
The analogous complete binary statement is conditional on the
extension normalization in Theorem~\ref{thm:conditional}.

There is also the valid compact output bound
\begin{equation}
 \supp_{\mathrm{func}}\Rtree
       \subseteq \supp_{\mathrm{func}}\Vcal
                       \cup\supp_{\mathrm{func}}\Ocal.
 \label{eq:functional-support}
\end{equation}
The uncontracted slots are projections of the two derivative-kernel
supports.

Equation~\eqref{eq:retarded-attachment} does not imply
\[
 \supp_{\mathrm{func}}\Rtree
                 \subseteq J^+(\supp_{\mathrm{func}}\Vcal).
\]
An untouched metric slot of the nonlocal observable may lie earlier
than the interaction while another slot receives the retarded
attachment. A claim about every functional-support variable requires
its own proof. We therefore use the attachment support condition,
not the unsupported stronger inclusion.

\subsection{Evaluation after admission}

The constructed classical coefficient can be evaluated on $\CW$.
Its degree sectors follow from \eqref{eq:tree-branches}:
\begin{equation}
 \Rtree\text{ has sectors }(3,1)\text{ and }(1,3)
                         \text{ in }(\gamma,\phi).
 \label{eq:tree-degrees}
\end{equation}

\begin{proposition}[Two explicit zero-evaluation subclasses]
\label{prop:zeros}
The tree evaluation vanishes on either of the following subsets
of $\CW$:
\[
 \{\gamma=b=0,\ \phi,c,\bar c\text{ arbitrary free}\},
 \qquad
 \{\phi=0,\ (\gamma,c,\bar c,b)\text{ satisfying }\CW\}.
\]
Also, under \textup{(UCP$_W$)}, the independent inherited identity
$\Delta_\xi\Ocal(u)=0$ remains valid for all $u\in\CW$.
\end{proposition}

\begin{proof}
At $\gamma=0$, $\Ocal'_\gamma=0$ because its metric sector is
quadratic, and $p_1[\gamma;\phi]=0$. Thus both terms of
\eqref{eq:tree-branches} vanish.

At $\phi=0$, both $\rg[\phi]$ and $p_1[\gamma;\phi]$ vanish.
The same formula gives zero. The last assertion is the inherited
restriction theorem, applied only to classical evaluation.
\end{proof}

No vanishing of $\Rtree$ on the whole of $\CW$ has been proved.
The cancellation of $\Delta_\xi\Ocal$ does not imply that the
interaction response vanishes. The response directions
$\rg[\phi]$ and $p_1[\gamma;\phi]$ are sourced zero-data fields;
they are generally not tangent to the free shell or its restricted
subclass. Differentiating an evaluation identity along such a
direction is not licensed.

The statement also does not supply evaluation of an unconstructed
$\Rquant$. In particular, one cannot conclude that its degree-two
symbol vanishes on the two subclasses merely from the degree-four
tree statement.

The witness \eqref{eq:witness} remains the boundary of the original
full-shell invariance claim. It is neither removed by evaluation
elsewhere nor transformed into a binary quantum anomaly.

%=====================================================================
\section{B5: the free differential, displayed without a defect}
\label{sec:differential}
%=====================================================================

The free symbol rule is
\[
 \sfree\gamma=\RR c,\qquad
 \sfree\bar c=ib,\qquad
 \sfree\phi=\sfree c=\sfree b=0.
\]
It is odd and raises ghost number by one.
The following calculations concern the two entries, not a
gauge-consistency identity for the binary datum.

\subsection{The interaction entry}

Using symmetry of $T_0$ and compact integration by parts,
\begin{align}
 \sfree\Vcal
 &=-\tfrac12\int g\,(\RR c)_{\mu\nu}T_0^{\mu\nu}\,\dd^4x\\
 &=\tfrac12\int c_\nu\left(
    (\partial_\mu g)T_0^{\mu\nu}
                  +g(\Pmass\phi)\partial^\nu\phi\right)\dd^4x.
 \label{eq:sV}
\end{align}
On the free scalar shell,
\begin{equation}
 \sfree\Vcal
 =\tfrac12\int c_\nu(\partial_\mu g)T_0^{\mu\nu}\,\dd^4x.
 \label{eq:sV-shell}
\end{equation}
The ghost need not be supported in $W$, so the switching term
does not vanish merely because $g=1$ on $W$.
There is no coordinate ramp in this vertex.

\subsection{The composite entry}

We give a compact explicit form of XIII-A's entry derivative.
Set
\[
 z=\RR c,\quad X=X_1[\gamma],\quad p=p_1[\gamma;\phi],
 \quad Z=X_1[z],\quad p_z=p_1[z;\phi],
 \quad k=\rD[\gamma]+\rg[\phi].
\]
Let $X_2,p_2$ be the inherited second completion at $(\gamma,k;\phi)$.
The composite is
\[
 \Ocal=\int f\left(
 p_2-X^\mu\partial_\mu p+
 (X^\nu\partial_\nu X^\mu-X_2^\mu)\partial_\mu\phi+
 \tfrac12X^\mu X^\nu\partial_\mu\partial_\nu\phi
                    \right)\dd^4x.
\]

Write $\dot k=\rD'_\gamma[z]$. Since the actual source is
$\JD=\sig J_3+2C_2\sig''-2M_2\sig'$ in the time-only ramp convention,
its inherited derivative is
\begin{equation}
 \dot k=\PP\ER\left(
       \sig\,J_3'{}_\gamma[z]
       +2C_2'{}_\gamma[z]\sig''
       -2M_2'{}_\gamma[z]\sig'\right).
 \label{eq:dotk}
\end{equation}
The tensors $C_2,M_2$ are exactly the finite quadratic source
coefficients supplied by XII-D. This equation differentiates the
existing response; it introduces no quantum argument $\JD$.

Define
\begin{align}
 \dot b_2
 &=\tfrac14(\tr_\eta\gamma)(\tr_\eta z)
                         -\tfrac12\gamma_{\alpha\beta}z^{\alpha\beta},\\
 \dot a_2^{\mu\nu}
 &=z^\mu{}_\alpha\gamma^{\alpha\nu}
   +\gamma^\mu{}_\alpha z^{\alpha\nu}
   -\tfrac12\left(
       (\tr_\eta z)\gamma^{\mu\nu}
       +(\tr_\eta\gamma)z^{\mu\nu}\right)
   +\dot b_2\eta^{\mu\nu},\\
 \widehat a&=a_1[\dot k]+\dot a_2,
 \qquad \widehat b=b_1[\dot k]+\dot b_2.
 \label{eq:hats}
\end{align}
For any tensor coefficient $a$ and scalar or coordinate component $v$,
put
\begin{align}
 \mathscr D_\sig(a,v)
 &=\sig a^{\mu\nu}\partial_\mu\partial_\nu v
   +\sig(\partial_\mu a^{\mu\nu})\partial_\nu v
   +\sig'a^{0\nu}\partial_\nu v,\\
 \mathscr D_g(a,v)
 &=g a^{\mu\nu}\partial_\mu\partial_\nu v
   +g(\partial_\mu a^{\mu\nu})\partial_\nu v
   +(\partial_\mu g)a^{\mu\nu}\partial_\nu v.
 \label{eq:weighted-divergences}
\end{align}
Thus every external derivative of the weights is visible.

On the free shell, $Dz=0$ and, with $v=c^\mu\partial_\mu\phi$,
\begin{align}
 Z^\mu
 &=-\ER\bigl(\sig'a_1^{0\mu}[z]\bigr),\\
 p_z
 &=\tfrac12gv-\ERm\left(
       \tfrac12(\Box g)v
       +(\partial^\mu g)\partial_\mu v
       +(\partial_\mu g)a_1^{\mu\nu}[z]\partial_\nu\phi
                    \right).
 \label{eq:first-ghost-terms}
\end{align}
The coordinate layer and scalar switch terms survive for unrestricted
homogeneous ghosts.

Finally put $Y=\sfree X_2$ and $p_{2,z}=\sfree p_2$. Their complete
divergence-form derivatives are
\begin{align}
 Y^\mu=-\ER\Bigl(
 &\mathscr D_\sig(a_1[z],X^\mu)
 +\mathscr D_\sig(a_1[\gamma],Z^\mu)
 +\sig\,\partial_\alpha\widehat a^{\alpha\mu}
 +\sig'\widehat a^{0\mu}\Bigr),\\
 p_{2,z}=-\ERm\Bigl(
 &\mathscr D_g(a_1[z],p)+m^2g b_1[z]p
 +\mathscr D_g(a_1[\gamma],p_z)+m^2g b_1[\gamma]p_z
 \nonumber\\
 &+\mathscr D_g(\widehat a,\phi)+m^2g\widehat b\phi\Bigr).
 \label{eq:second-ghost-terms}
\end{align}
Together with \eqref{eq:dotk}, these formulas retain the source
$\sig',\sig''$ terms as well as all coordinate and scalar switching
terms.

\begin{proposition}[Entry differential]
\label{prop:sO}
The free differential of the composite is
\begin{align}
 \sfree\Ocal=\int f\Bigl(
 &p_{2,z}-Z^\mu\partial_\mu p-X^\mu\partial_\mu p_z\\
 &+(Z^\nu\partial_\nu X^\mu+
      X^\nu\partial_\nu Z^\mu-Y^\mu)\partial_\mu\phi
 \nonumber\\
 &+\tfrac12(Z^\mu X^\nu+X^\mu Z^\nu)
                    \partial_\mu\partial_\nu\phi\Bigr)\dd^4x.
 \label{eq:sO}
\end{align}
It is odd, has ghost number one, and belongs to the inherited
polynomial domain.
\end{proposition}

\begin{proof}
Apply the graded product rule to the displayed completion.
All undifferentiated coefficient factors are even.
The first and second variations above are their ordinary
directional derivatives in the odd direction $z$.
Admission follows from XII-A's preservation of $\Fpol$ by the
frozen differential \cite{XIIA}.
\end{proof}

Neither \eqref{eq:sV-shell} nor \eqref{eq:sO} is set to zero.
The auxiliary remainder $\Aux_\xi$ is even and has ghost number
zero; it is not a term in these ghost-number-one formulas.
A compact finite shift $K\xi$ is not $\sfree$, and the finite
two-jet term $\tfrac12\Lie_\xi K\xi$ is not an infinitesimal ghost
variation.

We do not apply the differential to the binary datum and do not form
the binary gauge-consistency defect. That separate question remains
closed, including in the region where the tree datum is constructed.

%=====================================================================
\section{B6: decision and the exact successor package}
\label{sec:decision}
%=====================================================================

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.08\linewidth}
                              p{.24\linewidth}X@{}}
\toprule
Gate & Ending & Exact scope\\
\midrule
B0 & Recorded &
Evaluation-only reading; unchanged carrier; no restriction quotient.\\[0.35em]
B1 & Constructed &
The interaction functional is compact microcausal, even, and
ghost-number zero.\\[0.35em]
B2 & Frozen, with a proved correction &
The retarded convention and total-diagonal ambiguity class are
declared. The literal retarded-factorization equality is incompatible.\\[0.35em]
B3, tree & Constructed &
One-line joint composition, properness, compactness, admission,
and branchwise root overlap.\\[0.35em]
B3, quantum & Unresolved &
The two-line joint extension and internal-collision compatibility
are not supplied by local-entry existence.\\[0.35em]
B4 & Constructed at tree order &
Retarded attachment support and explicit zero-evaluation subclasses;
no whole-subclass vanishing claim.\\[0.35em]
B5 & Displayed &
The two entry differentials, with surviving ramp and switch terms;
no binary gauge-consistency assertion.\\[0.35em]
Completion and master equation & Closed &
No completed observable, interacting differential, or sharp limit.\\
\bottomrule
\end{tabularx}
\end{center}

\begin{theorem}[Exact binary release]
\label{thm:release}
For the frozen displayed lifts and fixed regulator width:
\begin{enumerate}[label=\textup{(\roman*)}]
\item The vertex $\Vcal$ is admitted.
The tree coefficient \eqref{eq:tree-definition} exists and defines
one selected causal datum $Q_H\Rtree\in\Ealg$.
\item Its inherited inverse signs, properness, compact microcausal
kernel, and first-completion branch overlap are proved.
\item Causal factorization has the values
\eqref{eq:ordered-values}. Replacing them by
$R_1=\Vcal\star_H\Ocal$ is Proved obstructed under the printed
contraction normalization.
\item The corrected coefficient through $\lambda\hbar$ is
Unresolved until the hypotheses of
Theorem~\ref{thm:conditional} are established.
\item Classical restriction is applied after admission.
The inherited cancellation of $\Delta_\xi\Ocal$ remains an
evaluation identity and does not imply a quantum restriction
quotient or a zero interaction response on all of $\CW$.
\end{enumerate}
\end{theorem}

\begin{proof}
Proposition~\ref{prop:vertex},
Theorems~\ref{thm:literal}, \ref{thm:tree}, and
\ref{thm:conditional},
Propositions~\ref{prop:overlap}, \ref{prop:properness}, and
\ref{prop:zeros}, together with the stated inherited restriction
input.
\end{proof}

The affirmative binary release through the requested quantum order
is therefore not established. The missing extension is named, its
support and normalization obligations are separated, and its absence
is not called an anomaly. A manuscript cannot become a proof of that
extension merely by labeling the formal expression ``renormalized.''

\subsection*{Handoff}

A successor receives:
\begin{enumerate}
\item the admitted interaction functional and the fixed displayed
composite;
\item the selected retarded convention and its corrected ordered
values;
\item the constructed tree datum and its admitted class;
\item the branchwise root-overlap identity;
\item the frozen total-diagonal counterterm class and the explicit
internal partial-collision problem;
\item the conditional quantum-admission theorem;
\item the evaluation-only restriction and the two proved tree
zero-evaluation subclasses;
\item the two entry differential displays.
\end{enumerate}

It does not receive a complete coefficient $\Rquant$, a
representative-independent retarded operation on every quotient
class, a reduced algebra, a completed observable, or a binary
gauge-consistency result. Opening a subsequent gauge-defect paper
through quantum order first requires completion of the present
joint-extension gate, or an explicit restriction of that successor's
claim to the constructed tree datum.

\subsection*{Conclusion}

\[
 \boxed{
 \begin{gathered}
 \Vcal\in\Fpol,\qquad
 \Rtree=-\pair{\Ocal'}{\GR\Vcal'}\in\Fpol,\qquad
 Q_H\Rtree\in\Ealg,\\[2pt]
 \Rtree
   =\pair{\Ocal'_\gamma}{\rg[\phi]}
          +\pair{\Ocal'_\phi}{p_1[\gamma;\phi]},\\[2pt]
 R_1=\frac{i}{\hbar}
       \bigl(\TT^{\mathrm{ren}}-\Vcal\star_H\Ocal\bigr)
 \quad\Longrightarrow\quad
 \text{ordered values are zero or a scaled commutator},\\[2pt]
 \text{joint two-line extension in }\ctclass:
                    \quad\textbf{Unresolved}.
 \end{gathered}}
\]

There is one constructed tree-level binary datum on the unchanged
carrier and one later classical evaluation restriction.
There is not yet a complete quantum binary datum through the
requested order. No gauge-consistency defect has been opened.

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

\paragraph{Normalization correction.}
The distinction between factorization of $\TT$ and ordered values
of $R_1$ follows from the selected retarded definition.
The correction changes neither the inherited causal brackets nor
the coefficient $-1$ of the metric inverse.

\paragraph{Extension scope.}
Properness and tree admission are proved.
The two-line extension is expressly unresolved.
No failed sufficient wavefront condition is promoted to a theorem
that every extension fails.

\paragraph{Carrier scope.}
All constructed outputs use the existing polynomial domain and
full-shell quotient. No quotient by $\NW$ is taken.
The selected-lift construction is not asserted to descend as a
binary operation under arbitrary changes of input representatives.

\paragraph{Interaction scope.}
Only the switched metric--scalar vertex is inserted.
The Einstein--Hilbert action remains in the model, but none of its
interaction vertices is restored in this selected pair.
The empty invariant jet is not used as an interaction functional.

\paragraph{Verification.}
No successful \texttt{pdflatex} execution, proof-assistant
verification, independently executed symbolic computation,
independent referee review, or journal acceptance is asserted.

%=====================================================================
\begin{thebibliography}{99}
\addcontentsline{toc}{section}{References}

\bibitem{XIIIA}
N.~Cheung,
\emph{Landmark XIII-A: Classical cancellation on a fixed subclass,
and the obstruction to a coefficientwise restriction quotient}.
Supplied manuscript \texttt{XIII-A.tex}.
Evaluation-only handoff, restriction-ideal obstruction, ordering
counterexample, and unresolved completion boundary.

\bibitem{XIIF}
N.~Cheung,
\emph{Landmark XII-F: The gauge-fixed remainder of
$\Phi^{(2)}_{\mathrm D}$: companion support, the auxiliary pairing,
and what is not a Ward identity}.
Supplied manuscript \texttt{XII-F.tex}.
Copied witness and the exact scope of classical cancellation.

\bibitem{XIIE}
N.~Cheung,
\emph{Landmark XII-E: Assembly of the second-order relational
coefficient after admission of the density-weighted cubic insertion}.
Supplied manuscript \texttt{XII-E.tex}.
Assembly sign, completion formulas, compact balanced kernels,
and scalar-sourced slot.

\bibitem{XIID}
N.~Cheung,
\emph{Landmark XII-D: Retarded response and carrier admission for
the cubic-only density-weighted Einstein source}.
Supplied manuscript \texttt{XII-D.tex}.
Inherited roots, response normalization, source coefficients,
and causal plateau proposition.

\bibitem{XIIA}
N.~Cheung,
\emph{Landmark XII-A: One graded polynomial carrier for the
finite-ramp multiplet, with ordering-stable composites and a robust
observation exclusion}.
Supplied manuscript \texttt{XII-A-v2.tex}.
Polynomial product, action causal matrix, full-shell ideal,
free differential, and admissible ordering class.

\bibitem{XIIC}
N.~Cheung,
\emph{Landmark XII-C: One declared cubic weighting: actual Euler
source, changed Noether identity, and the coupled-jet question}.
Supplied manuscript \texttt{XII-C.tex}.
Boundary only: retained Einstein--Hilbert action and named
invariant-jet obstruction.

\bibitem{XIB}
N.~Cheung,
\emph{Landmark XI-B: The free de Donder multiplet for the ramp
Einstein--scalar sheet: causal propagators, residual gauge, and
the layer dressing}.
Supplied manuscript \texttt{XI-B.tex}.
Inverse coefficient $-1$ and free ghost normalization.

\bibitem{FR}
K.~Fredenhagen and K.~Rejzner,
\emph{Perturbative algebraic quantum field theory},
arXiv:1208.1428v2, 2013.
Sections~6--7: time ordering, retarded constructions, and the
declared domains of local and multilocal renormalization.

\bibitem{FRModels}
K.~Fredenhagen and K.~Rejzner,
\emph{Perturbative Construction of Models of Algebraic Quantum
Field Theory},
arXiv:1503.07814, 2015.
Comparison framework for the separation of free products and
renormalized perturbative constructions.

\bibitem{Hormander}
L.~H\"ormander,
\emph{The Analysis of Linear Partial Differential Operators I},
second edition, Springer, 1990.
Distribution products and proper-pushforward wavefront bounds
used in the one-line proof.

\bibitem{BGP}
C.~B\"ar, N.~Ginoux, and F.~Pf\"affle,
\emph{Wave Equations on Lorentzian Manifolds and Quantization},
ESI Lectures in Mathematics and Physics,
European Mathematical Society, 2007,
arXiv:0806.1036.
Smooth wave Green operators, transposition, causal support,
and finite propagation.

\end{thebibliography}

\end{document}