% !TEX program = pdflatex
%=====================================================================
% IX-D-v2.tex
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%   pdflatex -interaction=nonstopmode -halt-on-error IX-D-v2.tex
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%
% RELEASE:
%   Completed causal-interface, support, and separation analysis.
%   UNRESOLVED as a sharp time-ordered or retarded construction.
%
% UPDATED PREREQUISITE:
%   IX-Free-2 constructs, relative to its stated premises, one
%   truncated partial free product on D_decl and both associations
%   of the selected word through hbar^3.
%
% LIVE STOP:
%   A separate time-ordered extension, its admission and overlap
%   proofs, and the causal retarded subtraction remain unconstructed.
%   The old degree-four descent stop is not retained.
%
% IMPORTANT BOUNDARIES:
%   * D_decl is a named tensor domain, not a globally closed algebra.
%   * F9a is inherited; F9b and quantum hypocontinuity are unclaimed.
%   * Wordwise free coherence does not supply causal factorization.
%   * Functional support is not replaced by observation support.
%   * Time-ordering involution uses anti-time ordering.
%   * Normalized subtraction requires a separate hbar-order audit.
%   * Lightlike propagation is not identified with diagonal contacts.
%   * G2 and interacting constructions remain separate.
%
% No compilation, proof-assistant verification, independent review,
% sharp causal product, or global interacting realization is claimed.
%=====================================================================

\documentclass[11pt,a4paper]{article}

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\newcommand{\R}{\mathbb R}
\newcommand{\C}{\mathbb C}
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\newcommand{\Span}{\operatorname{span}}
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\newcommand{\Dtest}{\mathcal D}
\newcommand{\Padm}{\mathscr P_\Sigma}
\newcommand{\Pmc}{\mathscr P_{\mathrm{mc}}}
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\newcommand{\Slin}{\mathscr S_1}
\newcommand{\Mlin}{\mathscr M_1}
\newcommand{\Qlin}{\mathscr Q_\Sigma}
\newcommand{\Ddir}{\mathrm D}
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\newcommand{\GA}{G_{\mathrm A}^{0}}
\newcommand{\Charge}{\mathsf b}
\newcommand{\pair}[2]{\left\langle #1,#2\right\rangle}
\newcommand{\Bspace}{\mathscr B}
\newcommand{\Wspace}{\mathscr W}
\newcommand{\Lspace}{\mathscr L}
\newcommand{\Rspace}{\mathscr R}
\newcommand{\PLspace}{\mathscr L_4}
\newcommand{\QLspace}{\mathscr L_2}
\newcommand{\PRspace}{\mathscr R_4}
\newcommand{\QRspace}{\mathscr R_2}
\newcommand{\Rtr}{\C[\hbar]/(\hbar^4)}
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\newcommand{\scopebox}[2]{%
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\fbox{\begin{minipage}{%
\dimexpr\linewidth-2\fboxsep-2\fboxrule\relax}
\textbf{#1}\par\smallskip #2
\end{minipage}}\par\medskip}

\title{\textbf{Landmark IX-D}\\[0.4em]
Causal-support interfaces for one sharp ordered word\\
after construction of the declared partial free product}
\author{}
\date{}

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

\begin{abstract}
We revise the causal-support interface for the ordered word
\(B_f,W_g,B_{f'}\) using the constructed free sector of
IX-Free-2 as an explicit inherited premise. Relative to its
stated prerequisites, IX-Free-2 supplies degree-four descent,
simultaneous overlap agreement, a truncated partial product
\(\star_c\) on a named tensor domain, and agreement of the two
associations through order \(\hbar^3\). It does not supply a
time-ordered or retarded product.

We establish configuration-support bounds for the constructed
outputs, including all one-edge terms and the classical word.
We identify the calibrated scalar as the constant projection
of an existing free word, while proving that this projection
still does not define a causal map. An explicit algebraic
separation argument shows that wordwise free coherence,
even together with an admitted symmetric auxiliary
multiplication, does not imply causal factorization.

Separate time-ordered and retarded interfaces are stated with
actual tensor domains, old-product overlaps, order-reversing
involution, and continuity boundaries. We retain the conditional
source-contact incompatibility and prove the conditional
retarded-subtraction implication on a domain where every
required free product exists. The normalization audit shows
that numerator data through \(\hbar^3\) determine a quotient
by \(\hbar\) only through \(\hbar^2\).

The interface and separation analysis is complete. The sharp
causal-product release remains unresolved at the unconstructed
time-ordered extension and its causal compatibility proofs.
The free gate for this word is no longer the live stop.
Neither a general free-product enlargement nor a G2-compatible
interacting construction is asserted.
\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) {Inherited partial product};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Causal-product target};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Resolved causal product};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {The causal-product outcome is unresolved.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{The causal product stays unresolved. 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-IX-D}
\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) {Inherited partial product};
\node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {Resolved causal product};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {The causal-product outcome is unresolved.};
\draw[->] (a) -- (s);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{The causal product stays unresolved. 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-IX-D}
\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) {Inherited partial product};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Causal-product target};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {Resolved causal product};
\node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {The causal-product outcome is unresolved.};
\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-IX-D}
\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) {Inherited partial product};
\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) {Resolved causal product};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Causal-product target. The causal-product outcome is unresolved.};
\draw (a) -- (m);
\draw (m) -- (b);
\draw[->] (m) -- (s);
\end{tikzpicture}
\caption{Inherited partial product is not Resolved causal product. 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-IX-D}
\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: Inherited partial product};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Causal-product target};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: Resolved causal product};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {The causal-product outcome is unresolved.};
\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-IX-D}
\end{figure}

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


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



\scopebox{Release and updated prerequisite}{%
\textbf{Causal-product outcome: Unresolved.}

IX-Free-2 is inherited as a relative construction of one
truncated partial free product \(\star_c\) on \(\Ddecl\),
including both associations of the selected word through
\(\hbar^3\). Its degree-four descent and declared overlap
tests are not reopened here.

The live missing construction is a separate time-ordered
extension and the resulting causal retarded maps.
The named free sector is not the complete free-product
domain of an unspecified general IX-D construction.
No sharp \(\TO_2\), normalized \(R_1\), or \(R_2\) is
released by this manuscript.}

\tableofcontents

%=====================================================================
\section{Release conventions and inherited inputs}
\label{sec:prerequisites}
%=====================================================================

The selected word is
\[
A=B_f,\qquad W=W_g,\qquad C=B_{f'}.
\]
Product-release outcomes are:
\begin{enumerate}[label=\textup{(\roman*)}]
\item \emph{Constructed}: named maps exist and satisfy every
declared interface.
\item \emph{Proved obstructed}: an explicit contradiction excludes
every construction satisfying the specified interfaces.
\item \emph{Unresolved}: a required construction or compatibility
proof is unfinished.
\end{enumerate}
The present causal-product release is \textup{(iii)}.
A contradiction for one simultaneous prescription excludes that
prescription only, unless all alternatives are separately excluded.

The supplied IX-Free-2 manuscript closes the declared free sector
relative to its premises \cite{IXFreeTwo}. Its result replaces the
earlier degree-four descent and overlap stop in IX-Free
\cite{IXFree}. It does not remove the additional causal-extension
obligations.

\subsection{Geometry and actual functionals}

Let
\[
M=\R^{1+3},\qquad
M_+=\{t>0\},\qquad
\Sigma=\{t=0\},\qquad
X=M_+^3,
\]
with
\[
\eta=\operatorname{diag}(-1,1,1,1),\qquad
\Box=-\partial_t^2+\Delta_{\mathbf x}.
\]
Set
\begin{equation}
(Ch)^\mu
=\partial_\nu h^{\mu\nu}
-\frac12\partial^\mu(\eta^{\alpha\beta}h_{\alpha\beta}),
\qquad
L=\Ezero C,\qquad w=Lh.
\label{eq:L}
\end{equation}
The operator \(\Ezero\) is the zero-value,
zero-first-time-derivative Cauchy inverse at \(\Sigma\).
Configurations \(\psi=(h,\phi)\) and configuration directions
are smooth and off shell.

For
\[
\Tspace=C_c^\infty(M_+;\C),
\]
the generators are
\begin{equation}
B_f=-\int f(x)w^\mu(x)\partial_\mu\phi(x)\dd^4x,
\qquad
W_g=\int g(y)h_{00}(y)^2\dd^4y.
\label{eq:generators}
\end{equation}
Complex tests define complex-valued actual functionals on real
smooth configurations.

The mixed covariance \(K(x,y)\) and scalar covariance
\(\Delta_m^+(x-x')\) are fixed flat free kernels. They are not
configuration-dependent metric propagators. The separate
notation \(\mathcal K\xi\) below denotes a metric gauge
variation, not the mixed kernel \(K\).

\begin{assumption}[Copied carrier, charge, and classical closure]
\label{ass:carrier}
The carrier \(\Padm\) consists of actual finite polynomial
functionals with compact distributional kernels satisfying
\[
(\Ddir_{\boldsymbol\eta}F)'_h(\psi)
\in\Slin=\Mlin+\Qlin
\]
for every smooth configuration and finite smooth direction list.
Here
\[
q_a=L^*a=C^*(\theta\GA a),\qquad
\Qlin=\{q_a:a\in C_c^\infty(M_+;T^*M)_\C\},
\]
and the intrinsic charge is
\[
\Charge(m+q_a)=(\GA a)|_\Sigma.
\]
It is well defined on actual linear kernels and vanishes
on \(\Mlin\).

The locally convex topology \(\taus\) records strong
compact-distribution coefficients and all intrinsic trace
coordinates. Classical multiplication, directional
differentiation, and conjugation preserve the carrier.
Classical multiplication is separately continuous.

The sharp adjoint retains the surface term
\begin{equation}
C^*(\theta b)
=\theta C^*b+\delta(t)\mathcal B(b|_\Sigma).
\label{eq:boundary}
\end{equation}
For nonzero real \(f\), the inherited sharp approximants satisfy
\[
B_{f,\eps}\not\longrightarrow B_f
\quad\text{in }\taus.
\]
This nonconvergence is not used as a substitute for a charge
witness.

Smooth Cauchy and Green-operator mapping properties and finite
propagation are available for the fixed operators in
\eqref{eq:L}.
\end{assumption}

These are inherited carrier premises reported in the supplied
sources, not an independent verification of the complete IX-A
manuscript \cite{IXA}.

\begin{assumption}[Copied analytical calibration]
\label{ass:calibration}
For the prescribed regulator placement,
\[
T_\eps^\rho
=2K_\eps(x,y)\overline{K_\eps(x',y)}D_\eps(x-x').
\]
There are \(u_\mu^\rho,r\in\Dtest'(X)\) with
\[
u_\mu^\rho=U_m+(\kappa_\rho+\log\mu)r,\qquad
u_\mu^{\rho_a}-u_\mu^{\rho_b}=c_{ab}r,\quad c_{ab}\in\R.
\]
For each compact \(K_0\subset X\),
\begin{equation}
\left|
\pair{T_\eps^\rho-\log((\mu\eps)^{-1})r-u_\mu^\rho}{\Phi}
\right|
\le C_{K_0}\eps(1+|\log\eps|)p_{K_0,64}(\Phi).
\label{eq:rate}
\end{equation}
The estimate includes compacts meeting
\(\Rel\cap\Delta_{13}\), where
\(\Rel=\supp r\) and \(\Delta_{13}=\{x=x'\}\).
Parent conversion is exactly \(c_{ab}r\), with no additional
normal derivative, nonconstant weight, or intersection term.
The parent distributions are dagger symmetric.

These are the analytical premises reported through IX-Free
and IX-Free-2, including their stated dependence on IX-B
and VII-D \cite{IXFree,IXFreeTwo,IXB,VIID}.
\end{assumption}

Fix the regulator, scale, and one real parameter \(c\) throughout:
\begin{equation}
U_c=u_\mu^\rho+cr,\qquad
\Cal(f,g,f')=\pair{U_c}{f\otimes g\otimes f'}.
\label{eq:calibration}
\end{equation}
No new parameter is selected by association, presentation,
causal argument ordering, or observation descendant.

\subsection{Copied preservation contract}

\begin{requirement}[C-I1--C-I9]
\label{req:free}
The inherited free sector is subject to:
\begin{enumerate}[label=\textbf{C-I\arabic*.}]
\item Admission of every output and required directional descendant.
\item Classical multiplication at order zero.
\item Exact agreement with fixed old products on declared overlaps.
\item Retention of the locked first-stage coefficients.
\item The one common calibrated scalar parent \(U_c\), with its
declared observation compatibility.
\item No subtraction changing a fixed old product.
\item No requirement that unchanged approximants converge sharply
in \(\taus\).
\item Respect for all finite actual-presentation relations.
\item Named continuity domains and proofs, without promoting
test-family continuity to actual-stage continuity or
hypocontinuity.
\end{enumerate}
\end{requirement}

These are the IX-C interfaces as inherited in IX-Free-2
\cite{IXC,IXFreeTwo}. The present revision uses their completed
sector realization; it does not impose causal factorization
by silently adding it to that free contract.

%=====================================================================
\section{D0: the constructed IX-Free-2 sector}
\label{sec:D0}
%=====================================================================

Define the actual spaces
\[
\Bspace=\Span\{B_f\},\qquad
\Wspace=\Span\{W_g\},
\]
\[
\PLspace=\Span\{B_fW_g\},\qquad
\QLspace=\Span\{\mathcal Q_{f,g}\},
\]
\[
\PRspace=\Span\{W_gB_{f'}\},\qquad
\QRspace=\Span\{\mathcal Q^R_{g,f'}\},
\]
and
\[
\Lspace=\PLspace\oplus\QLspace,\qquad
\Rspace=\PRspace\oplus\QRspace.
\]
The last sums are direct by actual field degree, four and two.

The locked contractions are
\begin{align}
\mathcal Q_{f,g}
&=-2\pair{f(x)g(y)K(x,y)}
{\partial_t\phi(x)h_{00}(y)},
\label{eq:Q}\\
\mathcal Q^R_{g,f'}
&=-2\pair{g(y)f'(x')\overline{K(x',y)}}
{h_{00}(y)\partial_{t'}\phi(x')}.
\label{eq:QR}
\end{align}
The algebraic presentation maps are
\[
p_L(f\otimes g)=B_fW_g,\qquad
q_L(f\otimes g)=\mathcal Q_{f,g},
\]
\[
p_R(g\otimes f')=W_gB_{f'},\qquad
q_R(g\otimes f')=\mathcal Q^R_{g,f'}.
\]

Inside \(\Padm\otimes_\C\Padm\), put
\begin{align}
\Dfirst
&=(\Bspace\otimes\Wspace)+(\Wspace\otimes\Bspace),
\label{eq:Dfirst}\\
\Dsecond
&=(\Lspace\otimes\Bspace)+(\Bspace\otimes\Rspace),
\label{eq:Dsecond}\\
\Dnew&=\Dfirst+\Dsecond,\qquad
\Dold=\Pmc\otimes\Pmc,
\label{eq:Dold}\\
\Ddecl&=\Dnew+\Dold.
\label{eq:Ddecl}
\end{align}
All these tensor products are algebraic.

Let
\[
\Vmod=\Padm\otimes_\C\bigl(\Rtr\bigr).
\]

\begin{assumption}[IX-Free-2 entering release]
\label{ass:free2}
Relative to its explicitly stated inherited premises,
IX-Free-2 constructs coefficient maps
\[
C_j:\Ddecl\longrightarrow\Padm,\qquad 0\le j\le3,
\]
and one truncated partial product
\[
\star_c(d)=\sum_{j=0}^3\hbar^jC_j(d),\qquad d\in\Ddecl,
\]
extended over \(\Rtr\). Its restriction to \(\Dold\)
is the unchanged old product modulo \(\hbar^4\).

The following results are inherited, not reproved:
\begin{enumerate}[label=\textup{(\roman*)}]
\item \(q_L,q_R,p_L,p_R\) are injective, and
\[
\kerop p_L=\kerop p_R=\{0\}.
\]
The degree-four proof uses restricted sharp separation on
\(k_{00}=0\) and one-scalar, three-metric slot isolation.
\item All four prescribed degree-four coefficients descend
to actual functionals.
\item Every nonzero sharp generator has nonzero
advanced-trace charge at some smooth configuration.
The partial contractions and admitted microcausal
comparison kernels have zero charge. In particular,
\[
\Bspace\cap\QLspace
=\Bspace\cap\QRspace
=\Bspace\cap\Pmc=\{0\}.
\]
\item The new domain is a direct sum of its six prescribed
blocks. Its whole-sum old overlap vanishes:
\[
\Dnew\cap\Dold=\{0\}.
\]
All coefficients glue, including the locked first stages.
\item
\[
\Cdesc=\Cglue=\R.
\]
For each fixed real \(c\), the product satisfies C-I1--C-I9
on \(\Ddecl\).
\item The two associations of the selected word exist and agree
through \(\hbar^3\). The product has actual-functional
presentation independence and order-reversing involution.
\item F9a holds on the named fixed-support test families.
F9b, quantum hypocontinuity, associativity for unspecified
triples, and closure under unspecified intermediate products
are not asserted.
\end{enumerate}
\end{assumption}

\begingroup
\small
\renewcommand{\arraystretch}{1.14}
\begin{longtable}
{@{}p{.32\linewidth}p{.24\linewidth}p{.36\linewidth}@{}}
\toprule
Object & Entering status & IX-D use\\
\midrule
\endfirsthead
\toprule
Object & Entering status & IX-D use\\
\midrule
\endhead
\bottomrule
\endfoot

\(\star_c\) on \(\Ddecl\) &
Constructed in IX-Free-2, relative to its premises &
Named free product, not a time ordering\\

Both associations of this word &
Constructed through \(\hbar^3\) &
Scalar and nonconstant coefficients are available\\

\(\kerop p_L,\kerop p_R\) &
Evaluated as zero &
Do not reopen degree-four descent\\

Four degree-four prescriptions &
Descended actual maps &
Use their established coefficients\\

New-domain and old-domain gluing &
Constructed &
No live overlap stop in this declared sector\\

\(\Cdesc=\Cglue=\R\) &
Evaluated &
Not a causal normalization theorem\\

Reversal and presentation independence &
Constructed on the declared domain &
Retain every boundary term\\

F9a &
Constructed on fixed-support families &
Not F9b or hypocontinuity\\

F9b &
Unclaimed &
Still unclaimed\\

Closure under unspecified intermediate products &
Unclaimed &
Higher retarded formulas cannot assume it\\

Complete free domain for a general causal construction &
Unclaimed &
A separate enlargement problem\\

\(\TO_2,R_1,R_2\) &
Absent from the entering construction &
Still the IX-D construction obligation\\
\end{longtable}
\endgroup

Thus the earlier free-closure clause is discharged only for
\(\Ddecl\), and only as the inherited relative premise
in Assumption~\ref{ass:free2}. The earlier future-lift
obligations concerning time ordering, causal factorization,
anti-time ordering, and old retarded overlaps remain
undischarged.

The construction does not establish that
\[
(F\star_cG)\otimes H\in
\Ddecl\otimes_\C(\Rtr)
\]
for arbitrary declared pairs or arbitrary \(H\).
An unclaimed enlargement is not a nonexistence theorem.

\subsection{Available word coefficients}

The inherited first stages are
\[
B_f\star_cW_g=B_fW_g+\hbar\mathcal Q_{f,g},
\qquad
W_g\star_cB_{f'}=W_gB_{f'}+\hbar\mathcal Q^R_{g,f'}.
\]

Write
\[
S_{\mu\nu}(x,x')
=\partial_\mu^x\partial_\nu^{x'}\Delta_m^+(x-x')
\]
and
\[
\mathcal T_{f,f'}
=\pair{f(x)f'(x')S_{\mu\nu}(x,x')}
{w^\mu(x)w^\nu(x')}.
\]
The one-edge outputs are
\begin{equation}
\Graph_{12}=\mathcal Q_{f,g}B_{f'},\qquad
\Graph_{23}=B_f\mathcal Q^R_{g,f'},\qquad
\Graph_{13}=\mathcal T_{f,f'}W_g.
\label{eq:oneedges}
\end{equation}
For \(V=f\otimes g\otimes f'\), the two-edge outputs are
\begin{align}
\Graph_{12,23}
&=2\pair{V K(x,y)\overline{K(x',y)}}
{\partial_t\phi(x)\partial_{t'}\phi(x')},
\label{eq:G1223}\\
\Graph_{12,13}
&=2\pair{V K(x,y)S_{0\nu}(x,x')}
{h_{00}(y)w^\nu(x')},
\label{eq:G1213}\\
\Graph_{13,23}
&=2\pair{V S_{\mu0}(x,x')\overline{K(x',y)}}
{w^\mu(x)h_{00}(y)}.
\label{eq:G1323}
\end{align}
Their distributional existence, compact kernels, admission,
and required descendants are inherited.

The common constructed word is
\begin{equation}
\begin{split}
\mathcal A_c(f,g,f')
&=(B_f\star_cW_g)\star_cB_{f'}
=B_f\star_c(W_g\star_cB_{f'})
\\
&=B_fW_gB_{f'}
+\hbar(\Graph_{12}+\Graph_{13}+\Graph_{23})
\\
&\quad+\hbar^2
(\Graph_{12,23}+\Graph_{12,13}+\Graph_{13,23})
+\hbar^3\Cal(f,g,f')\,1.
\end{split}
\label{eq:word}
\end{equation}
This is an equality of existing partial-product associations,
not an assertion about unspecified triples.

The sign ledger remains the inherited one: each partial has
factor \(-2\), the complete word has two insertion signs with
product \(+1\), and the middle-slot assignment factor is \(2\),
not \(4\).

%=====================================================================
\section{D1: domains and separate causal interfaces}
\label{sec:D1}
%=====================================================================

The standard perturbative framework distinguishes a free quantum
product, time ordering, and the interacting or retarded
construction \cite{FR,DF}. No existence theorem for local
functionals is automatically applied here to the sharp nonlocal
carrier.

\subsection{Actual functional support}

\begin{definition}[Configuration support]
\label{def:support}
The support \(\fsupp F\) of an actual functional is the complement
of the largest open region on which every compactly supported
configuration variation leaves \(F\) unchanged at every
configuration. Metric and scalar support are defined by restricting
the variations to the respective field species.

For a finite formal functional, its support is taken to be the
union of its coefficient supports when a formal support statement
is required.
\end{definition}

Observation support and functional support are not interchangeable.
The metric dependence of \(B_f\) generally extends from
\(\supp f\) back to the initial surface.

We use \(J^\pm\) for causal future and past, including null
relations, and \(I^\pm\) for chronological future and past.
A binary ordering condition is
\begin{equation}
F\gtrsim G
\quad\Longleftrightarrow\quad
\fsupp F\cap J^-(\fsupp G)=\varnothing.
\label{eq:ordering}
\end{equation}
Equivalently,
\[
\fsupp G\cap J^+(\fsupp F)=\varnothing.
\]
This means that \(F\) is not causally earlier than \(G\);
it also includes spacelike-separated supports.

\subsection{Reversed and conjugate comparison domains}

On the ambient tensor space define
\[
\sigma(F\otimes G)=G\otimes F,\qquad
\gamma(F\otimes G)=F^*\otimes G^*,\qquad
d^\dagger=\sigma\gamma d.
\]
The inherited domain is dagger stable, and
\begin{equation}
(F\star_cG)^*=G^*\star_cF^*
\label{eq:freeinvolution}
\end{equation}
where the pair is declared.

Dagger stability alone does not assert separate flip stability.
To use both free orderings without an enlargement, define
\begin{equation}
\Dcmp=\Ddecl\cap\sigma(\Ddecl).
\label{eq:Dcmp}
\end{equation}

\begin{lemma}[A domain for both free orderings]
\label{lem:comparison}
The space \(\Dcmp\) is stable under \(\sigma\) and \(\gamma\).
Both orderings of every pair represented there are declared.
In particular, \(\Dfirst\subset\Dcmp\).
\end{lemma}

\begin{proof}
Flip stability follows directly from \(\sigma^2=1\).
Since \(\sigma\gamma(\Ddecl)=\Ddecl\), one has
\(\gamma(\Ddecl)=\sigma(\Ddecl)\). Therefore
\[
\gamma(\Dcmp)
=\sigma(\Ddecl)\cap\Ddecl=\Dcmp.
\]
The first-stage sum contains both orderings, and
\(\Bspace,\Wspace\) are conjugation stable.
\end{proof}

The restriction \(\Dcmp\) is not a closure theorem.
A future time-ordering domain must still name the pairs it
contains, including the reversed and conjugate pairs needed
by its identities.

\begin{requirement}[Time-ordered interface
\(\mathcal I_{\mathrm{TO}}\)]
\label{req:TO}
A proposed bosonic time-ordering system must specify:
\begin{enumerate}[label=\textbf{TO\arabic*.}]
\item Actual-functional domains
\[
\mathcal D_2\subset\Padm^{\otimes2},
\qquad
\mathcal D_3\subset\Padm^{\otimes3},
\]
the coefficient range and truncation, symmetry, normalization
\(\TO_1(F)=F\), and classical order-zero multiplication.
Every required reversed, conjugate, or intermediate domain
must be named.
\item Admission and actual-presentation descent for every
coefficient and required directional descendant.
\item Causal factorization:
\[
\TO_2(F,G)=F\star_cG
\quad\text{if }F\gtrsim G,
\]
where \(F\otimes G\) belongs both to the time-ordering domain
and to \(\Ddecl\). The comparator is specifically the
IX-Free-2 product. Outside \(\Ddecl\), this identity requires
a separately constructed free extension.
Higher factorization must name every intermediate free
product used.
\item Exact declared old time-ordered overlaps and a specified
class of permitted contact terms, with admission and
descent tests for those terms.
\item The appropriate involution or unitarity identities,
including anti-time ordering rather than an unjustified
self-adjointness requirement.
\item A continuity statement on a named domain at a specified
strength. F9a must not be promoted to F9b or quantum
hypocontinuity.
\end{enumerate}
\end{requirement}

\begin{remark}[Spacelike separation is not a vanishing axiom]
\label{rem:spacelike}
A time-ordered product need not vanish for spacelike-separated
arguments. If causal factorization applies in both directions,
the two free orderings must agree there; their common value
need not be zero.

Even a contracted contribution may be smooth and nonzero
away from coincidence. A distribution supported on a
coincidence set vanishes on tests supported away from that
set, so contact terms cannot account for such an off-diagonal
spacelike contribution. This is consistent with the usual
free and time-ordered distinction \cite{FR}.
\end{remark}

\begin{requirement}[Retarded interface
\(\mathcal I_{\mathrm{ret}}\)]
\label{req:ret}
A proposed retarded system must specify:
\begin{enumerate}[label=\textbf{R\arabic*.}]
\item The vertex and observable roles and the off-shell
subtraction convention. These are not inferred from
the letters \(B,W\).
\item The causal Green operator and kernel convention.
A first-order vanishing requirement is
\[
\Ret(V;O)=0
\quad\text{if}\quad
\fsupp O\cap J^+(\fsupp V)=\varnothing.
\]
A stronger distributional argument-slot support statement
must name those slots and be separately proved.
\item Exact agreement with declared old retarded maps and
specified smooth classical comparison coefficients.
\item Preservation of every fixed old product; no subtraction
may change an already fixed regular coefficient.
\item Admission and actual-functional descent of every
nonconstant coefficient and required descendant.
\item A named continuity domain and proof, including the
additional domain obligations of any higher retarded formula.
\end{enumerate}
\end{requirement}

\begin{remark}[The lightlike boundary]
\label{rem:boundary}
The strict-exterior condition involving \(J^+\) imposes no
vanishing assertion when the supports meet its causal
boundary. Coincidence contributions require a contact
prescription, while noncoincident null-related points may
carry propagator singularities and are not thereby
diagonal contact terms. Replacing \(J^+\) by \(I^+\) would
change the interface and is not justified here.
\end{remark}

\subsection{Subtraction and the smooth sign comparison}

On a domain where both maps exist, a proposed unnormalized
comparison is
\begin{equation}
\widehat\Ret(V;O)
=i\bigl(\TO_2(V,O)-V\star_cO\bigr).
\label{eq:subtraction}
\end{equation}
It is not yet a definition in this manuscript, since
\(\TO_2\) has not been constructed. The conditional theorem
in Section~\ref{sec:D5} states precisely when this comparison
has the desired properties.

Replacing \(V\star_cO\) by \(VO\) is a different prescription.
It requires a separate identity eliminating the relevant
free contractions. A specialized comparison reported for
another sector does not supply that identity here.

\begin{assumption}[Smooth Duhamel comparison]
\label{ass:smooth}
For the following sign calculation only, let \(P\) be the
specified smooth field operator, let
\(G_R^*=G_A\) in the chosen pairing, and suppose
\(\Qop[h]\) is formally self-adjoint. All displayed sources
and pairings are legitimate with the stated compact supports.
The first variation satisfies
\[
P\phi_1=-\Qop[h]\phi_0,\qquad
\phi_1=-G_R\Qop[h]\phi_0.
\]
\end{assumption}

\begin{lemma}[Smooth comparison sign]
\label{lem:sign}
Under Assumption~\ref{ass:smooth}, setting \(u_f=-G_Af\)
gives the smeared comparison test \(q=+\Qop[h]u_f\).
\end{lemma}

\begin{proof}
The adjoint identities give
\[
\pair f{\phi_1}
=-\pair{G_Af}{\Qop[h]\phi_0}
=\pair{u_f}{\Qop[h]\phi_0}
=\pair{\Qop[h]u_f}{\phi_0}.
\]
An additional minus in the last test would contradict this
declared Duhamel convention.
\end{proof}

This is a smooth classical sign calculation, not a construction
of a sharp \(R_1(V_1;B_f)\).

\begin{theorem}[D1: causal structure remains additional]
\label{thm:interfaces}
The inherited product \(\star_c\), its wordwise coherence,
and its calibrated constant coefficient do not by themselves
supply a time-ordered or retarded system satisfying
\(\mathcal I_{\mathrm{TO}}\) or \(\mathcal I_{\mathrm{ret}}\).
\end{theorem}

\begin{proof}
Assumption~\ref{ass:free2} supplies actual free coefficient
maps and their declared overlaps. It supplies no separate
\(\TO_2\), no old time-ordered overlap equations, and no
anti-time-ordering system.

Equation \eqref{eq:word} compares two existing free
associations. It contains no causal-factorization identity
relating a time-ordered map to \(\star_c\).
Section~\ref{sec:D3} gives an explicit algebraic separation
of these requirements.

Without \(\TO_2\), the difference in
\eqref{eq:subtraction} is unavailable. Nor do the free
statements supply the causal Green convention, higher
intermediate domains, or argument-slot support proofs.
These remain additional obligations. This is not a proof
that a later causal extension cannot exist.
\end{proof}

%=====================================================================
\section{D2: support of the constructed free outputs}
\label{sec:D2}
%=====================================================================

For a compact observation set \(F\Subset M_+\), define
\[
P_F=J^-(F)\cap\{t\ge0\}.
\]
This set is compact in Minkowski space: time lies between
zero and the largest time in \(F\), and finite propagation
bounds the spatial coordinates.

\begin{theorem}[D2: configuration-support inclusions]
\label{thm:support}
Let \(F=\supp f\), \(G=\supp g\), and \(F'=\supp f'\).
The constructed functionals have the following separate
metric and scalar support bounds:
\end{theorem}

\begingroup
\small
\renewcommand{\arraystretch}{1.14}
\begin{longtable}
{@{}p{.23\linewidth}p{.36\linewidth}p{.31\linewidth}@{}}
\toprule
Functional & Metric support bound & Scalar support bound\\
\midrule
\endfirsthead
\toprule
Functional & Metric support bound & Scalar support bound\\
\midrule
\endhead
\bottomrule
\endfoot
\(B_f\) & \(P_F\) & \(F\)\\
\(W_g\) & \(G\) & \(\varnothing\)\\
\(\mathcal Q_{f,g}\) & \(G\) & \(F\)\\
\(\mathcal Q^R_{g,f'}\) & \(G\) & \(F'\)\\
\(\mathcal T_{f,f'}\) & \(P_F\cup P_{F'}\) & \(\varnothing\)\\
\(\Graph_{12}\) & \(G\cup P_{F'}\) & \(F\cup F'\)\\
\(\Graph_{23}\) & \(P_F\cup G\) & \(F\cup F'\)\\
\(\Graph_{13}\) & \(P_F\cup P_{F'}\cup G\) & \(\varnothing\)\\
\(\Graph_{12,23}\) & \(\varnothing\) & \(F\cup F'\)\\
\(\Graph_{12,13}\) & \(G\cup P_{F'}\) & \(\varnothing\)\\
\(\Graph_{13,23}\) & \(P_F\cup G\) & \(\varnothing\)\\
\(B_fW_gB_{f'}\) & \(P_F\cup G\cup P_{F'}\) & \(F\cup F'\)\\
\end{longtable}
\endgroup

\begin{proof}
A metric variation supported outside \(P_F\) cannot change
\(Lh\) on \(F\). This follows from locality of \(C\), the
fixed zero Cauchy data, and finite propagation. A scalar
variation outside \(F\) cannot change \(\partial\phi\)
on \(F\). These facts prove the bounds for \(B_f\).
The bound for \(W_g\) is immediate.

In \eqref{eq:Q}--\eqref{eq:QR}, the covariance is fixed;
only the displayed residual fields vary. This gives
the partial-contraction bounds. The residual fields in
\(\mathcal T_{f,f'}\) are \(w\) on \(F,F'\), giving the
two past sets.

The support of a classical product is contained in the
union of the supports of its factors: a variation outside
both supports changes neither factor. Apply this to
\eqref{eq:oneedges} and the classical word.

For \eqref{eq:G1223}, the fixed kernels \(K,\overline K\)
introduce no metric configuration dependence. Hence its
metric support is empty, while its scalar dependence is
confined to \(F\cup F'\).
The other two two-edge outputs contain only \(h_{00}\)
on \(G\) and one residual \(w\) slot. This gives their
displayed metric bounds and empty scalar support.

The inherited distributional pairings introduce no additional
configuration variables.
\end{proof}

\begin{remark}[Residual fields and numerical parameters]
The test \(g\) still enters the coefficient of
\(\Graph_{12,23}\), even though its residual metric support
is empty. Dependence on an observation test is not dependence
on the metric configuration.

The bounds are inclusions, not exact-support assertions.
They are properties of the constructed free outputs as
configuration functionals. They are not retarded
argument-slot theorems and do not reconstruct the inherited
singular products.
\end{remark}

For a smooth compact source \(s\), the defining retarded
Green property gives
\[
\supp(G_Rs)\subset J^+(\supp s);
\]
this is the usual Green-operator support convention
\cite{FR}. A specified smooth comparison
\[
\phi_{1,g}=-\mathsf E_m\mathsf V_g[h]\phi_0
\]
is therefore legitimate when its source is smooth and compact
and its chosen inverse has the stated Cauchy or retarded
property. It remains a smooth classical composition.

Zero-Cauchy inversion agrees with the corresponding retarded
solution only under the appropriate zero past-data hypothesis.
It is not a Feynman inverse, and arbitrary homogeneous data
cannot be discarded.

%=====================================================================
\section{D3: the constructed scalar and the causal separation}
\label{sec:D3}
%=====================================================================

For \(H\in\Vmod\), let
\[
\Pi(H)
=\operatorname{ev}_{\mathrm{const}}[\hbar^3]H,
\]
where \(\operatorname{ev}_{\mathrm{const}}\) extracts the
field-degree-zero part of the polynomial coefficient,
equivalently its value at the zero configuration.

\begin{theorem}[D3: constant projection of an existing word,
but no causal lift]
\label{thm:nonlift}
The constructed free word satisfies
\[
\Pi\bigl(\mathcal A_c(f,g,f')\bigr)=\Cal(f,g,f').
\]
This scalar assignment does not by itself define
\(\TO_2\), \(\Ret_1\), or a higher retarded map on the
intermediate actual-functional stages.
\end{theorem}

\begin{proof}
The first assertion is the constant coefficient of the
inherited equality \eqref{eq:word}. Thus the scalar is
now a projection of an existing word, not a projection
of an unconstructed association.

Nevertheless, \(\Pi\) has a large kernel. For example,
\(\Pi(\hbar F)=0\) for every \(F\in\Padm\).
Changing a lower-order nonconstant coefficient leaves
the displayed constant projection fixed. This is an
information-loss statement; it is not permission to
change the locked coefficients of \(\star_c\).

IX-Free-2 supplies the nonconstant free coefficients
lost by this projection. It supplies no separate
time-ordering coefficients, causal factorization,
old time-ordered or retarded overlaps, or anti-time-ordering
identities. In particular, the scalar does not select
\[
i\bigl(\TO_2(V,O)-V\star_cO\bigr).
\]
The existence of the free word therefore does not convert
its constant projection into a causal map.
\end{proof}

\begin{lemma}[Auxiliary multiplication compatible with free coherence]
\label{lem:separation}
Wordwise free coherence of \(\star_c\) is compatible with
a symmetric admitted auxiliary multiplication
\(\mathsf M_2\), with classical term \(VO\), that differs
from \(V\star_cO\) on a declared first-stage pair satisfying
\[
\fsupp O\cap J^+(\fsupp V)=\varnothing.
\]
\end{lemma}

\begin{proof}
Consider the auxiliary map
\[
\mathsf M_2(F,G)=FG,
\]
viewed as a formal map with zero positive-order coefficients.
It is symmetric, depends only on actual arguments, and
has admitted values by classical carrier closure.
Its definition leaves \(\star_c\) and every inherited
wordwise equality unchanged.

Choose nonzero tests \(f,g\) with
\[
\supp f\subset\{0<t<a\},\qquad
\supp g\subset\{t>b\},
\qquad b>a>0.
\]
Take
\[
V=W_g,\qquad O=B_f.
\]
Let \(F=\supp f\) and \(G=\supp g\).
Theorem~\ref{thm:support} places the metric support of
\(B_f\) in \(P_F=J^-(F)\cap\{t\ge0\}\) and its scalar
support in \(F\). Consequently,
\[
\fsupp B_f\subset P_F\cup F.
\]
Since \(F\subset\{0<t<a\}\), every point of \(P_F\)
has time coordinate between zero and \(a\). Thus
\[
P_F\cup F\subset\{0\le t\le a\}.
\]
The same theorem gives
\[
\fsupp W_g\subset G\subset\{t>b\}.
\]
Causal future-directed curves cannot decrease the time
coordinate. Therefore,
\[
\fsupp O\cap J^+(\fsupp V)=\varnothing.
\]
Both pair orderings belong to \(\Dfirst\).

The locked reverse first stage gives
\[
V\star_cO=W_gB_f+\hbar\mathcal Q^R_{g,f},
\]
whereas
\[
\mathsf M_2(V,O)=W_gB_f.
\]
The nonzero tensor \(g\otimes f\) has a nonzero image under
the inherited injective map \(q_R\). Therefore
\(\mathcal Q^R_{g,f}\ne0\), and the two expressions differ.
This proves the asserted compatibility and discrepancy.
\end{proof}

The map \(\mathsf M_2\) is not proposed as a time-ordered
construction satisfying \(\mathcal I_{\mathrm{TO}}\).
Its failure is the explicit witness that a causal axiom
cannot be inferred from the free ones. The lemma neither
changes the old products nor proves an obstruction to a
future causal extension.

Observation operations remain numerical:
\[
aDU_c=aDu_\mu^\rho+c\,aDr.
\]
The inherited descendant estimate has test-order loss
\(64\to64+\ord D\). It does not identify observation
differentiation with configuration differentiation or
open new functional slots.

Assigning a causal cutoff to \(U_c\) would still require
new product, descent, admission, and overlap proofs.
No such assignment is made here.

%=====================================================================
\section{D4: a conditional source-contact incompatibility}
\label{sec:D4}
%=====================================================================

\begin{assumption}[Simultaneous source-sector prescription]
\label{ass:source}
For this section only, let \(P\) be the declared formally
self-adjoint field operator and let \(H_F\) be a declared
Feynman fundamental solution with
\[
P_xH_F(x,y)=c_F\delta(x-y),\qquad c_F\ne0.
\]
The source presentation is \(j=Ph\), and differential
compatibility requires the time-ordered contracted
\(jh\) kernel to be obtained by applying \(P_x\) to the
contracted \(hh\) kernel. The domain includes tests
detecting the diagonal contact.
\end{assumption}

The fundamental-solution, rather than bisolution, character of
the time-ordered kernel is an additional differential input;
it is not a consequence of free-product coherence
\cite{FR}.

\begin{proposition}[D4: the simultaneous zero-contact
prescription is incompatible]
\label{prop:contact}
Under Assumption~\ref{ass:source}, the additional prescription
\(H^{\mathrm{TO}}_{jh}=0\) is incompatible with the stated
source presentation and differential identity.
\end{proposition}

\begin{proof}
Differential compatibility gives
\[
H^{\mathrm{TO}}_{jh}=P_xH_F=c_F\delta.
\]
In the scalar case, choose a nonzero real compact test \(q\)
and pair with \(q\otimes q\). The result is
\[
c_F\int q(x)^2\dd^4x\ne0.
\]
For a bundle-valued identity, select the nonzero component
of the declared diagonal identity and the corresponding
bundle test. The zero-kernel prescription gives zero
on the same test, which is a contradiction.
\end{proof}

This excludes the simultaneous prescription under its explicit
hypotheses. An independent central source variable need not
be identified with \(Ph\). No universal source-extension
obstruction is asserted.

If a free contraction is a bisolution,
\(P_xH=0\), its zero source contraction is compatible with
that different differential identity. It does not automatically
transfer to \(H_F\). Applying \(P\) again would involve a
derivative of a delta distribution; its normalization and
bundle factors are not evaluated here.

The two-variable contact \(c_F\delta\) is not the
three-observation scalar calibration \(cr\).
IX-Free-2's freedom \(\Cglue=\R\) supplies no map identifying
these objects and cannot absorb the contact by reassignment
of the common parent parameter.

%=====================================================================
\section{D5: conditional subtraction on the declared free domain}
\label{sec:D5}
%=====================================================================

The free-closure premise formerly required at this point is
now Assumption~\ref{ass:free2}, restricted to \(\Ddecl\).
Only the remaining causal-lift hypotheses are assumed below.

\begin{assumption}[Future time-ordered lift and comparison]
\label{ass:future}
For this section only, assume:
\begin{enumerate}[label=\textup{(\alph*)}]
\item A separate symmetric time-ordering map
\[
\TO_2:\mathcal D_2\longrightarrow\Vmod
\]
exists on a specified actual tensor domain
\(\mathcal D_2\subset\Padm^{\otimes2}\) stable under
\(\sigma\) and \(\gamma\).
The comparison domain
\begin{equation}
\Ecmp=\mathcal D_2\cap\Dcmp
\label{eq:Ecmp}
\end{equation}
contains every pair used in this section. Thus the two
free orderings and their conjugates all belong to
\(\Ddecl\).
\item Every time-ordered coefficient is admitted, respects
actual-presentation relations, and agrees with the
declared old time-ordered coefficients.
Normalization, permitted contact terms, and the claimed
continuity strength on named domains are specified.
No actual-stage continuity is inferred from inherited F9a.
\item Both \(\TO_2(V,O)\) and \(V\star_cO\) have classical
term \(VO\). On comparison pairs with
\[
\fsupp O\cap J^+(\fsupp V)=\varnothing,
\]
causal factorization gives
\begin{equation}
\TO_2(V,O)=V\star_cO.
\label{eq:causalfuture}
\end{equation}
\item Free involution reverses order:
\begin{equation}
(V\star_cO)^*=O^*\star_cV^*.
\label{eq:futurefreeadjoint}
\end{equation}
The binary anti-time-ordering identity on \(\Ecmp\) is
\begin{equation}
\ATO_2(F,G)
=F\star_cG+G\star_cF-\TO_2(F,G),
\label{eq:antitime}
\end{equation}
and the time-ordering involution is
\begin{equation}
\TO_2(F,G)^*=\ATO_2(F^*,G^*).
\label{eq:timeadjoint}
\end{equation}
Every expression in these equations is defined on the
specified domains. Equation
\eqref{eq:futurefreeadjoint} is inherited; the additional
time-ordering identities are not.
\item On a specified old retarded overlap
\[
\Eold\subset\Ecmp\cap\Dold,
\]
the old unnormalized retarded convention is exactly
\(i(\TO_2-V\star_cO)\), using the unchanged old
time-ordered and free coefficients.
\end{enumerate}
The time-ordered existence, factorization, anti-time-ordering,
and retarded-overlap hypotheses are not discharged by this
manuscript.
\end{assumption}

The domain condition in part \textup{(a)} is substantive:
if a desired pair or reversed free product lies outside
\(\Ddecl\), a free enlargement must be proved before these
formulas can be applied to it.

\begin{proposition}[Conditional retarded-subtraction implication]
\label{prop:conditionalret}
Under Assumption~\ref{ass:future}, define on \(\Ecmp\)
\[
\widehat\Ret(V;O)
=i\bigl(\TO_2(V,O)-V\star_cO\bigr).
\]
This is an actual tensor map, equivalently a bilinear map
on its declared pairs, with admitted coefficients and
zero classical term. It has the exact declared old
unnormalized retarded overlaps and satisfies
\[
\widehat\Ret(V;O)=0
\quad\text{if}\quad
\fsupp O\cap J^+(\fsupp V)=\varnothing.
\]
Furthermore,
\[
\widehat\Ret(V;O)^*
=\widehat\Ret(V^*;O^*).
\]
If both terms have a proved continuity property at the
same strength on the same named domain, their difference
has that property.
\end{proposition}

\begin{proof}
Both summands are linear maps on the actual tensor domain
\(\Ecmp\), so their difference respects every actual relation.
Its coefficients belong to the vector space \(\Padm\),
with the required descendants, by admission of the two
coefficient systems. Equal classical terms cancel.

On an excluded causal pair, \eqref{eq:causalfuture}
makes the difference zero. On \(\Eold\), the assumed
old values and convention give the required old overlap.

For involution, use both order reversal and anti-time ordering:
\[
\begin{split}
\widehat\Ret(V;O)^*
&=-i\bigl(
\ATO_2(V^*,O^*)-O^*\star_cV^*
\bigr)\\
&=-i\bigl(
V^*\star_cO^*+O^*\star_cV^*
-\TO_2(V^*,O^*)-O^*\star_cV^*
\bigr)\\
&=i\bigl(
\TO_2(V^*,O^*)-V^*\star_cO^*
\bigr).
\end{split}
\]
The conjugate pair is in \(\Ecmp\), by the stated domain
stability.

Finally, subtraction is continuous in the coefficient
topologies. Hence any common proved continuity statement
is preserved.
\end{proof}

The proposition proves a conditional vanishing property for
declared pairs. It does not prove a stronger retarded
distributional slot-support theorem. Nor does it construct
its time-ordering hypotheses by assuming only free coherence.

\subsection{The \(\hbar\)-order audit}

\begin{lemma}[Normalized information at the available truncation]
\label{lem:hbaraudit}
Suppose a formal numerator in \(\Vmod\) has zero classical
term:
\[
N=\hbar N_1+\hbar^2N_2+\hbar^3N_3.
\]
Its normalized quotient by \(\hbar\) is determined only
modulo \(\hbar^3\):
\[
N/\hbar
=N_1+\hbar N_2+\hbar^2N_3
\quad\text{modulo }\hbar^3.
\]
It is not uniquely determined as an element of \(\Vmod\).
\end{lemma}

\begin{proof}
Multiplication by \(\hbar\) in \(\Vmod\) annihilates
\(\hbar^3F\) for every \(F\in\Padm\). Thus two quotients
differing by such a term have the same numerator modulo
\(\hbar^4\).

Conversely, comparison of coefficients shows that the
constant, linear, and quadratic quotient coefficients
are exactly \(N_1,N_2,N_3\). These are therefore uniquely
determined in
\(\Padm\otimes_\C\C[\hbar]/(\hbar^3)\).
\end{proof}

Consequently, a physical convention
\[
\Ret_1(V;O)
=\frac{i}{\hbar}
\bigl(\TO_2(V,O)-V\star_cO\bigr)
\]
would, under the conditional hypotheses, be determined
only through \(\hbar^2\) by the order-three data.
A normalized result through \(\hbar^3\) requires the numerator
through \(\hbar^4\), or a separately specified equivalent
convention. There is no inverse of \(\hbar\) in \(\Rtr\).

Higher retarded maps additionally require higher time-ordering
and all intermediate free products in their defining identities
\cite{DF}. Neither the binary implication nor the triangle
coefficient constructs \(R_2\), and the inherited partial
domain does not authorize unspecified intermediate products.

%=====================================================================
\section{D6: the switch gate remains separate}
\label{sec:switch}
%=====================================================================

\begin{assumption}[Copied first-order switch witness]
\label{ass:switch}
Under the stated IX-A/VI switch-gap hypotheses, there is
a compact gauge parameter \(\xi\), supported strictly
after \(\Sigma\), with
\[
L(\mathcal K\xi)=\xi
\]
and
\[
B_f[h+\mathcal K\xi,\phi]-B_f[h,\phi]
=-\int f\,X_\xi\phi.
\]
For the specified switch and witness,
\[
g\mathcal K\xi=0,\qquad
a_f=\int f\,X_\xi\phi\ne0.
\]
The switched forced-response variation is zero, while
the dressing variation is \(-a_f\).

These are copied witness hypotheses for the specified
gauge and switch, not assertions about arbitrary switches
or arbitrary gauge parameters.
\end{assumption}

Merely placing \(\xi\) in a switch-transition region does not
prove either witness equality.

\begin{proposition}[No cancellation by a second-order correction]
\label{prop:switch}
Under Assumption~\ref{ass:switch}, a modification beginning
at coupling order \(\kappa^2\) cannot remove the
order-\(\kappa\) switch gap while the first-order prescription
is retained.
\end{proposition}

\begin{proof}
The witness compares
\[
\begin{array}{c|c}
\text{switched forced response}&0\\
\text{sharp dressing response}&-a_f\ne0.
\end{array}
\]
If a corrected expression has formal variation
\[
-\kappa a_f+\kappa^2b+\kappa^3d+\cdots,
\]
vanishing of the formal series requires the coefficient
of \(\kappa\) to vanish. It does not. Later coefficients
cannot alter that coefficient.
\end{proof}

The coupling order \(\kappa\) is distinct from the contraction
order \(\hbar\). IX-Free-2's free construction does not cancel
this first-order witness or open G2.

A condition \(g=1\) near an observation support does not by itself
remove a witness involving the dressing's larger metric-support
region. The reported VII-A plateau intertwiner is restricted
to its stated plateau-supported gauges and retains the
switch derivatives; it is not thereby a quantum Ward identity.

The expression
\[
\Ocal^{(2)}
+R_1(V_1;\Ocal^{(1)})
+R_1(V_2;\Ocal^{(0)})
+\frac12R_2(V_1,V_1;\Ocal^{(0)})
\]
remains bookkeeping until every displayed map, domain,
overlap, and support property has been constructed.
Membership of \(B_f\) in the carrier does not admit an
unspecified correction \(\mathcal B_f^{(2)}\).
No such correction is admitted or constructed here.

%=====================================================================
\section{D7: dependency and non-identification ledger}
\label{sec:ledger}
%=====================================================================

The objects below are not interchangeable. The assertion is
that no identification needed by this construction has been
supplied, not that special numerical coincidences are universally
impossible.

\begingroup
\small
\renewcommand{\arraystretch}{1.14}
\begin{longtable}
{@{}p{.29\linewidth}p{.63\linewidth}@{}}
\toprule
Object & Scope and missing identification\\
\midrule
\endfirsthead
\toprule
Object & Scope and missing identification\\
\midrule
\endhead
\bottomrule
\endfoot

IX-Free-2's \(\star_c\) &
A truncated free product on the declared tensor domain
\(\Ddecl\). It is not \(\TO_2\), not \(R_1(V_1;B_f)\),
and not the complete free-product domain of an unspecified
general IX-D construction.\\

IX-Free's two-edge outputs &
Admitted nonconstant quadratic functionals with fixed flat
free contractions. Their availability does not supply a
time-ordering map.\\

Triangle \(U_c\) &
A ternary numerical joint-test distribution, now the constant
projection of the constructed free word. It does not select
a causal subtraction or a source contact.\\

V's external \(R_{1,1}\) &
A switched external-metric response with its stated smooth
comparison test \(+\Qop u_f\). Its extension statement
concerns that object, not the reflected triangle.\\

V's oscillator coefficient &
A switched bilinear in a finite-mode representation.
No map identifies it with \(U_c\) or a sharp causal
carrier coefficient.\\

V's Newton--wave cubic &
A distinct Wick form with spatial Newton-transfer
singularities, not a substitute for the reflected graph
or its causal extension.\\

VI's relational coefficient &
A first-order dressing/forced-response comparison with
the copied switch witness. It supplies no sharp retarded
map.\\

VII-A's plateau intertwiner &
A restricted classical identity retaining switch
derivatives, not an established quantum Ward identity
or an identification with \(U_c\).\\

VII-E/F and VIII certificates &
Certificate or geometric data in separate tracks.
No map into the present polynomial carrier or into \(U_c\)
is supplied.\\

Source and packet data &
No supplied map identifies \(j[f]\), conical packets,
slicing events, or source tokens with \(B_f,W_g\),
the partial contractions, or \(U_c\).\\
\end{longtable}
\endgroup

The complete V, VI, VII-A, VII-E/F, and VIII sources are
not provided here. Their entries record the stated scope
boundaries, not independently certified theorems used
in an unconditional causal construction.

The updated dependency chain is
\[
\begin{gathered}
\text{IX-Free-2 partial product on }\Ddecl\\
\Downarrow\quad\text{still required}\\
\text{time-ordered extension on a named domain}\\
\Downarrow\quad\text{still required}\\
\text{retarded subtraction with causal support}\\
\Downarrow\quad\text{separate additional gates}\\
\text{G2 and interacting maps}.
\end{gathered}
\]
For a general causal construction, any required free-domain
enlargement is an additional obligation alongside the
time-ordering step. The old degree-four stop is no longer
the live stop for this word.

%=====================================================================
\section{Final release and retained boundaries}
\label{sec:release}
%=====================================================================

\begin{theorem}[IX-D revised interface package]
\label{thm:release}
Relative to the explicitly inherited carrier, calibration,
and IX-Free-2 premises, this manuscript establishes:
\begin{enumerate}[label=\textup{(\roman*)}]
\item Separate time-ordered and retarded interfaces with
actual-domain requirements.
\item Configuration-support bounds for all displayed
constructed free outputs, including the one-edge terms
and the classical word.
\item Identification of \(\Cal(f,g,f')\) as the constant
projection of the constructed free association, together
with its non-lift to a causal map definition.
\item An explicit separation of free wordwise coherence
from causal factorization.
\item The source-contact contradiction under the simultaneous
hypotheses of Assumption~\ref{ass:source}.
\item The conditional admitted subtraction, causal vanishing,
old-overlap, and involution implication of
Proposition~\ref{prop:conditionalret}.
\item The precise normalized \(\hbar\)-order limitation.
\item Persistence of the copied first-order switch witness
against corrections beginning at order \(\kappa^2\).
\end{enumerate}
It constructs no sharp \(\TO_2\), normalized \(R_1\), or
\(R_2\) on the requested sector.

The causal-product release is \emph{Unresolved}.
The live stop is the time-ordered extension and its
remaining causal, admission, and overlap obligations,
not degree-four descent of the inherited free word.
\end{theorem}

\begin{proof}
The statements follow from
Requirements~\ref{req:TO}--\ref{req:ret},
Theorem~\ref{thm:support},
Theorem~\ref{thm:nonlift},
Lemma~\ref{lem:separation},
Proposition~\ref{prop:contact},
Proposition~\ref{prop:conditionalret},
Lemma~\ref{lem:hbaraudit}, and
Proposition~\ref{prop:switch}.

Assumption~\ref{ass:free2} supplies the named free sector.
No result here constructs the separate time-ordering
hypotheses of Assumption~\ref{ass:future}. Its conditional
consequence therefore does not release a sharp retarded
map.

The source contradiction excludes only its stated
simultaneous prescription. The switch result retains
its copied first-order hypotheses. Neither is a universal
obstruction to every possible causal extension.
\end{proof}

\begingroup
\small
\renewcommand{\arraystretch}{1.14}
\begin{longtable}
{@{}p{.35\linewidth}p{.20\linewidth}p{.37\linewidth}@{}}
\toprule
Object & Status & Exact boundary\\
\midrule
\endfirsthead
\toprule
Object & Status & Exact boundary\\
\midrule
\endhead
\bottomrule
\endfoot

Free product on \(\Ddecl\) &
Constructed, inherited &
Relative IX-Free-2 premise; unchanged old restriction\\

Both free associations of this word &
Constructed, inherited &
Through \(\hbar^3\), not arbitrary-triple associativity\\

\(\Cdesc=\Cglue=\R\) &
Evaluated, inherited &
No causal normalization follows\\

Causal-interface lists &
Completed &
Requirements, not product constructions\\

Configuration-support bounds &
Proved &
Not retarded argument-slot support\\

Constructed-word scalar non-lift &
Proved &
Excludes a causal definition by numerical renaming\\

Free-coherence/causal separation &
Proved &
An algebraic nonimplication, not a no-go theorem\\

Simultaneous zero source-contact prescription &
Proved obstructed under stated hypotheses &
Only the prescription in
Assumption~\ref{ass:source}\\

Conditional unnormalized subtraction &
Proved implication &
Time-ordering hypotheses remain undischarged\\

Normalized order audit &
Proved &
Order-three numerator determines quotient through order two\\

Sharp \(\TO_2\) &
Unresolved &
No time-ordered extension constructed\\

Sharp normalized \(R_1\) &
Unresolved &
No unconditional subtraction or normalization released\\

Sharp \(R_2\) &
Unresolved &
Higher time ordering and intermediate domains absent\\

General free-domain enlargement &
Unclaimed &
No intermediate-closure or nonexistence assertion\\

F9a &
Inherited on named families &
No global causal continuity statement\\

F9b and quantum hypocontinuity &
Unclaimed &
No promotion from algebraic descent or F9a\\

G2-compatible switched construction &
Unresolved; not begun &
Original first-order witness boundary retained\\

Global interacting algebra &
Outside this construction &
No realization claimed\\
\end{longtable}
\endgroup

\subsection{Retained normal-topology boundary}

The VII-C obstruction is retained under its original joint
hypotheses: old Wick-product preservation, exact sharp
admission, normal-topology sequential admission continuity
agreeing on regular kernels, and bounded-set hypocontinuous
multiplication with continuous constant-coefficient
extraction \cite{VIIC}.

The IX-Free-2 partial product does not assert those joint
hypotheses. Its algebraic construction and F9a do not
establish F9b, unchanged sharp-approximant convergence,
or global hypocontinuity. Therefore no contradiction
with the retained conditional obstruction is claimed.

The reported separated witness excluding removal of the
selected logarithm by observation-diagonal counterterms
is also retained. Numerical subtraction in
\(\Dtest'(X)\) does not repair that normal-topology boundary.
No fixed regular coefficient is subtracted, and no old
product is redefined.

\scopebox{Closed conclusion}{%
The free gate for the declared word is now an inherited
IX-Free-2 result: degree-four descent and declared overlap
agreement are complete relative to its premises.

The causal lift is not constructed. A time-ordered extension,
its compatible old values, anti-time ordering, and causal
retarded subtraction still require their own proofs and
their own actual domains.

One-word free coherence does not establish a general
free-product domain, a sharp retarded product, an interacting
field, or a G2-compatible switched construction.
IX-D remains unresolved as a product construction.}

%=====================================================================
\appendix
\section{Revision, domain, and verification audit}
%=====================================================================

\subsection{What was updated}

The obsolete claim that IX-Free's degree-four descent remains
the current free stop has been removed. IX-Free-2 is the
entering reference for the named free sector.

The revision does not independently certify the companion
carrier or triangle proofs. It inherits the supplied
IX-Free-2 theorem relative to exactly its stated premises.

The separate causal hypotheses are retained. In particular:
\[
\text{free wordwise coherence}
\not\Rightarrow
\text{time ordering},
\]
\[
\text{configuration-support inclusion}
\not\Rightarrow
\text{retarded argument-slot support},
\]
\[
\text{F9a}
\not\Rightarrow
\text{F9b or quantum hypocontinuity}.
\]

\subsection{Domain audit}

The anti-time-ordering formula uses both \(F\star_cG\)
and \(G\star_cF\). The comparison domain
\(\Dcmp=\Ddecl\cap\sigma(\Ddecl)\), and then
\(\Ecmp=\mathcal D_2\cap\Dcmp\), makes this requirement
explicit.

Dagger stability of the inherited partial product is used
only at its proved strength. Separate flip stability is
not silently assumed for all of \(\Ddecl\).

No formula involving unspecified \(V_1,V_2\), higher
intermediate products, or a correction
\(\mathcal B_f^{(2)}\) is treated as defined merely because
some displayed free factors are admitted.

\subsection{Causal and normalization audit}

The retarded comparison uses the free quantum product,
not classical multiplication without a specialization
proof. Its reality proof explicitly uses order-reversing
free involution and anti-time ordering.

The condition with \(J^+\) includes null relations in
the excluded future. It does not prescribe boundary
values when supports meet that future. Diagonal contacts
and noncoincident lightlike singularities remain distinct.

The formal quotient by \(\hbar\) is unique only at one
lower available order. No normalized order-three result
is inferred from an order-three numerator.

\subsection{Verification disclosure}

The mathematical additions in this revision are the
completed support table, the existing-word formulation
of the scalar non-lift, the explicit free-coherence
separation witness, the domain restriction for both
free orderings, and the formal normalization audit.

The two-edge constructions, degree-four descent,
charge separation, and free overlap gluing are inherited,
not reconstructed.

The source is intended for two-pass pdfLaTeX compilation.
No successful compilation, page count, proof-assistant
verification, or independent review is claimed.

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

\paragraph{Nature of the manuscript.}
A completed relative interface and separation analysis
with an unresolved sharp causal-product release.

\paragraph{Source boundary.}
IX-Free, IX-Free-2, and the earlier IX-D source are supplied.
IX-C and the companion carrier, analytical, obstruction,
and switch results are used only through the explicit
inherited statements. Their complete proofs are not
independently certified here.

\paragraph{Continuity boundary.}
Inherited fixed-support test-family continuity is retained.
No new actual-stage continuity or quantum hypocontinuity
is asserted.

\paragraph{Excluded constructions.}
No Einstein equation, backreaction, time-ordered extension,
sharp retarded realization, BRST complex, interacting field,
conical identification, token acquisition, or all-graph
catalogue is constructed. No old product is repaired by
subtracting \(u_\mu^\rho\).

\paragraph{Release boundary.}
A constructed partial free sector has replaced the old
free stop. It has not been relabelled as a causal product,
and it does not open G2.

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

\bibitem{IXFreeTwo}
\emph{Landmark IX-Free-2:
Degree-four descent and overlap agreement for one sharp ordered word}.
Supplied manuscript, \texttt{IX-Free-2.tex}.
Entering relative construction of the truncated partial product,
degree-four descent, charge separation, simultaneous gluing,
wordwise coherence, reversal, and F9a.

\bibitem{IXFree}
\emph{Landmark IX-Free:
Actual-functional binary factorization of one sharp ordered word}.
Supplied manuscript, \texttt{IX-Free.tex}.
Source of the canonical graph outputs, degree-two maps,
and analytical-premise boundaries.
Its former degree-four and overlap stop is superseded,
for the declared sector, by IX-Free-2.

\bibitem{IXC}
\emph{Landmark IX-C:
Actual-functional factorization and coefficientwise coherence
for one sharp ordered word}.
Companion reported in the supplied sources.
Used through the copied interfaces, first stages,
and preservation obligations.

\bibitem{IXA}
N.~Cheung.
\emph{Landmark IX-A:
An off-shell admission domain and boundary-sensitive topology
for sharp Cauchy relational insertions}.
Companion reported in the supplied sources.
Carrier, charge, sharp adjoint, nonconvergence, and
conditional switch statements are used only through
their explicit inherited restatements.

\bibitem{IXB}
\emph{Landmark IX-B:
Derivative-stable triangle calibration and counterterm constraints
for sharp Cauchy insertions}.
Companion reported as \texttt{IX-B-v2.tex}.
Used through the inherited numerical calibration
and descendant premise.

\bibitem{VIIC}
\emph{Landmark VII-C:
A mixed-Wick obstruction to a normal-topology boundary extension
for sharp Cauchy harmonic insertions}.
Companion reported in the supplied sources.
Its obstruction is retained only under its original
joint hypotheses.

\bibitem{VIID}
\emph{Finite part of the regulated Cauchy-reflection triangle:
A strong numerical distributional limit for the VII-C family}.
Companion reported in the supplied sources.
Used only through the stated analytical premise.

\bibitem{DF}
M.~D\"utsch and K.~Fredenhagen.
\emph{Causal perturbation theory in terms of retarded products,
and a proof of the Action Ward Identity}.
Reviews in Mathematical Physics \textbf{16} (2004),
1291--1348.
arXiv:hep-th/0403213.
Background on the separate causal construction and its
higher retarded-product obligations; no automatic sharp
carrier extension is invoked.

\bibitem{FR}
K.~Fredenhagen and K.~Rejzner.
\emph{Perturbative algebraic quantum field theory}.
arXiv:1208.1428v2, 2013.
Sections 4--7 on functional support, Green operators,
free quantization, time ordering, and renormalization.
Used for background, not as an existence theorem for
the present nonlocal sharp sector.

\end{thebibliography}

\end{document}