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% additional supplied XII-D admission excerpt.
% The latter is an explicit additional microlocal premise,
% not a result proved by XVI-H.
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% The manuscript constructs a reduced two-edge graph sector,
% proves its punctured-domain product and extension bounds,
% and constructs a family of root-attached distributions under
% the explicitly stated root wavefront hypothesis.
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% No normalized extension, interacting quantum coefficient,
% BRST/BV closure, or all-orders theorem is claimed.
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\hypersetup{
  pdftitle={Landmark XVI-I: Reduced insertion graphs and their extension problem},
  pdfsubject={Classical admission, explicit two-edge kernels, scaling degree, and conditional root attachment}
}

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\newcommand{\dd}{\,\mathrm d}
\newcommand{\ii}{\mathrm i}
\newcommand{\Dprime}{\mathcal D'}
\newcommand{\Eprime}{\mathcal E'}
\newcommand{\mc}{\mathscr F_{\mathrm{mc,pol}}}
\newcommand{\Isol}{\mathcal I_{\mathrm{sol}}}
\newcommand{\Unresolved}{\textbf{Unresolved}}

\newcommand{\scopebox}[2]{%
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      \dimexpr\linewidth-2\fboxsep-2\fboxrule\relax}
      \textbf{#1}\par\smallskip #2
    \end{minipage}%
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\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{assumption}[theorem]{Inherited input}
\newtheorem{problem}[theorem]{Remaining obligation}

\numberwithin{equation}{section}

\title{Landmark XVI-I\\[0.3em]
  \large Reduced insertion graphs, singular loci, and extension freedom\\
  \normalsize Revised proof manuscript\\
  Classical admission and graph existence under stated inputs;\\
  normalization and interacting observability unresolved}
\author{}
\date{}

\usepackage{tikz}
\begin{document}

\maketitle

\begin{abstract}
We distinguish the admitted classical prepared insertion from a
reduced graph sector involving that insertion and one compactly
switched cubic interaction. The supplied XII-D admission excerpt
concerns the same polarized kernel represented in revised XVI-H.
Under that additional inherited premise, direct microcausal
membership and a nonempty free-shell microcausal class follow.
This does not repair the different frozen class excluded by XI-G.

For the graph calculation, we choose an ambient massless
time-ordered graviton kernel and identify its restriction with the
combination \(H+\mathrm iG^{\mathrm A}\) in the stated
normalization. Its off-diagonal wavefront relation proves that the
two differentiated edges can be multiplied. The cubic Hessian and
the insertion source jets then define the reduced graph without
an unspecified tensor expansion. An explicit spacelike evaluation
proves nonvanishing of the open-index vertex block when the
interaction switch is nonzero; no nonvanishing of an arbitrary
source-contracted or root-attached sum is inferred.

The relative-coordinate building blocks have finite scaling degree.
Including both insertion and interaction derivatives gives an
upper bound of eight in four relative dimensions. Extensions exist,
but their local coefficients are not normalized. Under the
explicit root wavefront bound, the compressed root attachment
satisfies the distributional product criterion and has proper
support. It therefore yields a family of smeared coefficients
indexed by the extension choices, not a selected interacting
quantum observable. BRST/BV closure and the retarded interacting
coefficient remain unestablished.
\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) {Reduced insertion graphs};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Singular loci};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Extension freedom};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {Extension freedom is not a construction of the full graph.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{Reduced graphs are not the full insertion. 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-XVI-I}
\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) {Reduced insertion graphs};
\node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {Extension freedom};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {Extension freedom is not a construction of the full graph.};
\draw[->] (a) -- (s);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{Reduced graphs are not the full insertion. 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-XVI-I}
\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) {Reduced insertion graphs};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Singular loci};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {Extension freedom};
\node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {Extension freedom is not a construction of the full graph.};
\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-XVI-I}
\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) {Reduced insertion graphs};
\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) {Extension freedom};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Singular loci. Extension freedom is not a construction of the full graph.};
\draw (a) -- (m);
\draw (m) -- (b);
\draw[->] (m) -- (s);
\end{tikzpicture}
\caption{Reduced insertion graphs is not Extension freedom. 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-XVI-I}
\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: Reduced insertion graphs};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Singular loci};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: Extension freedom};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {Extension freedom is not a construction of the full graph.};
\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-XVI-I}
\end{figure}

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


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



\scopebox{Scope and release status.}{%
The ambient manifold is \(M=\mathbb R^4\).
The preparation width \(0<\varepsilon<T\) is fixed.
The preparation data \(f,g,\sigma_\varepsilon\) are those of
revised XVI-H.

The interaction switch
\(\chi_{\mathrm I}\in C_c^\infty(M;\mathbb R)\) is separate
from the preparation data. It is not a past-to-future switch.

\textbf{Established under explicit inherited inputs:}
classical microcausal membership of the matching prepared kernel;
the punctured two-edge distribution; its finite scaling-degree
bounds; extension existence; and a family of compressed
root-attached distributions.

\textbf{Not selected or established:}
extension normalization, a retarded interacting coefficient,
BRST/BV closure, or a physical quantum observable.

No new classical trisolution wavefront calculation is performed.
The classical cone consequence is an imported consequence, stated
with its hypotheses.

No sharp-width limit, global adiabatic limit, ultraviolet
completion, or all-orders finite-parameter claim is made.
No ultraviolet regulator-removal procedure is performed.
}

\section{Provenance, additional premises, and proof order}
\label{sec:provenance}

This manuscript revises the supplied \texttt{XVI-I.tex}.
Its classical references are the supplied revised
\texttt{XVI-H.tex}, revised \texttt{XVI-E.tex}, revised
\texttt{XVI-F.tex}, and current \texttt{XVI-G.tex}.

An additional supplied excerpt from XII-D,
\texttt{thm:admission} and \texttt{eq:t-bounds}, contains
\[
\WF(t_{\mathrm D})\subset Z_3
\]
and the associated microcausal admission statement.
That microlocal result was not a proved premise of revised
XVI-H. It is added here explicitly, rather than attributed
retroactively to that manuscript.

Complete XII-D, XI-G, and XI-B sources have not been independently
audited. References to their internal labels identify supplied
excerpts or formulas reproduced in the supplied manuscripts.

The proof has four separate stages:
\begin{enumerate}
\item match the classical kernel and state the narrow admission
      implication;
\item define and verify the ambient propagator and reduced
      two-edge distributions;
\item extend the relative-coordinate building blocks, without
      choosing their normalization;
\item test compressed root attachment by explicit pullback,
      product, and proper-pushforward conditions.
\end{enumerate}

The last stage advances beyond the original draft's unspecified
root-product stop: with the imported \(Z_2\) root bound, the
compressed product condition can be checked. This establishes
attachment existence for an extension family, not a normalized
quantum coefficient.

\section{Classical kernel, domains, and exact matching}
\label{sec:classical}

\subsection{Geometry and component conventions}

Let
\[
\eta=\operatorname{diag}(1,-1,-1,-1),
\qquad
\Box=\partial_t^2-\Delta,
\qquad
P_m=\Box+m^2,
\]
and set
\[
M_T=[0,T]\times\mathbb R^3,
\qquad
M_T^\circ=(0,T)\times\mathbb R^3.
\]
Fix
\[
0<\varepsilon<T,\quad m\ge0,\quad
f\in C_c^\infty(M_T^\circ;\mathbb R),\quad
g\in C_c^\infty(M_T^\circ;[0,1]).
\]
The fixed preparation ramp is
\[
\sigma_\varepsilon(t)=\chi(t/\varepsilon),
\qquad
\chi(s)=0\ (s\le0),\quad \chi(s)=1\ (s\ge1),
\]
with \(\chi\) real and smooth.

The metric perturbation is a covariant symmetric tensor.
Indices are raised only with \(\eta\).
We use a fixed basis of the ten-dimensional symmetric-tensor
fiber and its dual, with component indices \(I,J,K,L\).
Thus summation over field components counts independent
symmetric components once. It does not introduce a second
factor for off-diagonal spacetime components.

The fixed Minkowski fiber pairing identifies the tensor bundle
and its dual. Trace reversal is
\[
\mathcal P h=h-\frac12\eta\,\tr_\eta h,
\qquad
\mathcal P^2=\mathrm{id}.
\]
Its two-leg kernel matrix, with the fixed fiber identifications
understood, is denoted by \(\mathbb P^{IK}\).
All component contractions below use that convention.

The volume density is \(\dd^4x\).
Distributional pairings are bilinear, without conjugation.

\subsection{The classical kernel input}

\begin{assumption}[Matching root and polarized kernel]
\label{input:kernel}
The classical completion is the stated XII-D insertion
\[
\mathcal C_{3,\mathrm D}(f,g)[h,\phi]
=\mathcal A_1[f,g](u_{\mathrm D}[h];\phi),
\]
with the response normalization
\(u_{\mathrm D}=-\mathcal P E_0^{\mathrm R}\mathcal J_{\mathrm D}\)
and preparation source
\[
\mathcal J_{\mathrm D}
=\sigma_\varepsilon J_3+\mathcal D_{\sigma_\varepsilon}.
\]

Its real polarization is the one represented in revised XVI-H
by a compact kernel \(t_{\mathrm D}\), symmetric in its two
tensor slots:
\[
\mathcal O_{\mathrm{pol}}(a,c;\phi)
=\langle t_{\mathrm D},a\otimes c\otimes\phi\rangle.
\]

In the chosen component frames, its general test-function action is
\begin{equation}
\begin{aligned}
\langle t_{\mathrm D},\Phi\rangle
=
\sum_{\substack{A,I,J\\|\alpha|+|\beta|\le2}}
(-1)^{|\alpha|+|\beta|}
\Big\langle B_A(y,x),\,
 c^A_{IJ;\alpha\beta}(y)
 (\partial_1^\alpha\partial_2^\beta\Phi_{IJ})(y,y,x)
\Big\rangle.
\end{aligned}
\label{eq:t-test}
\end{equation}
The coefficients are smooth and may be chosen symmetric:
\[
c^A_{IJ;\alpha\beta}=c^A_{JI;\beta\alpha}.
\]
The compact root kernel \(B_A(y,x)\) is the same root kernel
used in the XII-D source-root construction.
\end{assumption}

The superscript \(A\) labels a source-tensor component, not an
additional field species. The coefficients in
\eqref{eq:t-test} include the complete preparation correction.
They are not replaced by coefficients of \(\sigma_\varepsilon
J_3\) alone.

The action on arbitrary smooth tests is meaningful because
\(B\), and hence \(t_{\mathrm D}\), is compactly supported.
Revised XVI-H supplies finite-order continuity on \(C^\infty\),
not only on compactly supported configurations \cite{XVIH}.

\begin{proposition}[Diagonal polynomial and its derivatives]
\label{prop:Ft}
Define
\begin{equation}
F_t(h,\phi)
=\langle t_{\mathrm D},h\otimes h\otimes\phi\rangle.
\label{eq:Ft}
\end{equation}
Then, on the inherited real smooth completion domain,
\[
F_t(h,\phi)=\mathcal C_{3,\mathrm D}(f,g)[h,\phi].
\]
For real smooth tensor directions \(a,c\),
\begin{equation}
D_hF_t[a]
=2\langle t_{\mathrm D},a\otimes h\otimes\phi\rangle,
\qquad
D_h^2F_t[a,c]
=2\langle t_{\mathrm D},a\otimes c\otimes\phi\rangle.
\label{eq:Dh}
\end{equation}
There is no additional factorial in \(F_t\).
\end{proposition}

\begin{proof}
Degree-two homogeneity of the diagonal completion gives
\[
\mathcal O_{\mathrm{pol}}(h,h;\phi)
=\frac12\bigl(\mathcal C_{3,\mathrm D}[2h,\phi]
              -2\mathcal C_{3,\mathrm D}[h,\phi]\bigr)
=\mathcal C_{3,\mathrm D}[h,\phi].
\]
The kernel representation proves the first assertion.
Differentiating the quadratic polynomial and using tensor-slot
symmetry proves \eqref{eq:Dh}.
\end{proof}

\subsection{The additional admission and root-cone premises}

In Cartesian cotangent frames, define
\[
Z_n
=\{(x_1,\ldots,x_n;\xi_1,\ldots,\xi_n):
       \xi_1+\cdots+\xi_n=0,\ 
       (\xi_1,\ldots,\xi_n)\ne0\}.
\]
This uses the canonical identifications of cotangent fibers in
Minkowski space.

\begin{assumption}[Additional supplied microlocal statements]
\label{input:cones}
The supplied XII-D admission excerpt applies to the kernel in
Input~\ref{input:kernel}, with the same insertion, response
normalization, preparation data, scalar slot, and source
polarization. It gives
\begin{equation}
\WF(t_{\mathrm D})\subset Z_3.
\label{eq:tcone}
\end{equation}
The root statement reproduced in the supplied XVI-I gives
\begin{equation}
B\in\Eprime(M^2),
\qquad
\WF(B)\subset Z_2.
\label{eq:Bcone}
\end{equation}
The root represents a compact smooth root map:
\[
\int B_A(y,x)\phi(x)\dd^4x=a_{f,g}[\phi]_A(y)
\]
in the distributional kernel sense.

For the separate trisolution consequence only, the inherited
XI-G transport lemma is used in its stated compact-kernel
quantifier: it transports component covectors in Minkowski
frames, may move their base points, and restricts each surviving
entry to a null or zero covector.
\end{assumption}

These are additional inherited inputs, not new wavefront
calculations. The proof below does not derive
\eqref{eq:tcone} or \eqref{eq:Bcone} from compactness.

\begin{proposition}[Why the XII-D kernel is the matching kernel]
\label{prop:match}
Under Inputs~\ref{input:kernel} and~\ref{input:cones}, the
kernel to which the supplied XII-D admission statement applies
is exactly \(t_{\mathrm D}\) used here.
\end{proposition}

\begin{proof}
The diagonal insertion is the same
\(\mathcal C_{3,\mathrm D}=\mathcal A_1(u_{\mathrm D};\cdot)\).
Its polarization has the same factor \(1/2\).
The source polarization reproduces the same
\(\mathcal J_{\mathrm D}(a,c)\), including the ramp-derivative
terms. The root kernel and derivative-transfer signs are those
of \eqref{eq:t-test}.

Thus both constructions act on general tests by
\eqref{eq:t-test}. They define the same distribution.
There is no change of normalization, tensor species,
preparation, or real domain in this matching.
\end{proof}

\begin{theorem}[Narrow classical admission]
\label{thm:admission}
Under the additional microlocal input:
\begin{enumerate}
\item The species-symmetrized polynomial \(F_t\) is an element
      of the stated classical microcausal polynomial carrier.
\item The free-shell class of \(t_{\mathrm D}\) has a
      microcausal representative, namely \(t_{\mathrm D}\)
      itself.
\item Under the separately stated transport premise,
      \[
      \WF(\mathsf W_t)\cap
      \bigl(\widetilde V_+^{\,3}\cup
            \widetilde V_-^{\,3}\bigr)=\varnothing,
      \qquad
      \mathsf W_t
      =(\mathsf E_0\otimes\mathsf E_0\otimes\mathsf E_m)
          t_{\mathrm D}.
      \]
\end{enumerate}
These are classical kernel and free-shell conclusions.
They are not time-ordering, graph normalization, or BRST/BV
observability statements.
\end{theorem}

\begin{proof}
The closed future and past causal cones are pointed.
A nonzero tuple wholly in either cone cannot sum to zero.
Equation~\eqref{eq:tcone} therefore excludes the forbidden cone
products.

Embed the typed kernel in the direct-sum field bundle and apply
the normalized species symmetrization. Permutations preserve
compactness and the sum-zero covector condition. Evaluating
the symmetrized kernel on the repeated multiplet argument gives
the same polynomial \(F_t\); no extra factorial is inserted.
This proves item~1.

Define the free-shell null space in the matching compact-kernel
space by pairings with all smooth solutions of
\(\Box a=\Box c=0\), \(P_m\phi=0\).
The class \(t_{\mathrm D}+\Isol\) contains \(t_{\mathrm D}\).
Its own cone exclusion proves item~2 directly. It is unnecessary
to enlarge the frozen-kernel quantifier of XI-G's printed
criterion.

For item~3, the transport premise preserves the component
covectors in the fixed Minkowski frames. A surviving tuple
therefore still sums to zero and is nonzero. Pointedness again
excludes both cone products.
\end{proof}

\begin{corollary}[Compatibility with the frozen obstruction]
\label{cor:oldclass}
Suppose the matching frozen XI-G kernel \(u_0\), for data on
which its obstruction theorem applies, satisfies
\[
[u_0]_{\mathrm{sol}}\cap\mc=\varnothing.
\]
Then, under Theorem~\ref{thm:admission},
\[
t_{\mathrm D}-u_0\notin\Isol.
\]
Thus the admitted prepared class is not that excluded frozen
class.
\end{corollary}

\begin{proof}
If the difference were in \(\Isol\), the affine classes would
coincide. The admitted representative \(t_{\mathrm D}\) would
then contradict the frozen obstruction.
\end{proof}

The corollary is conditional on both sets of stated hypotheses,
including the matching frozen data. It is not a new evaluation
of the frozen witness, and does not extend the obstruction to
other data.

The former blanket statement ``no admission is claimed'' is
therefore replaced by the narrow classical conclusion above.
The new insertion is admitted under the supplied XII-D premise;
the different frozen class remains excluded \cite{XIID,XIG}.

\subsection{Current witness matching}

Revised XVI-H, using revised XVI-E/F and current XVI-G, proves
\[
\mathcal O_{\mathbb C}
(\gamma_1^\lambda,\gamma_2^\lambda;\phi)
=I_\lambda[\phi]
=\mathcal A_1[f,g](R_\lambda;\phi)
\]
at each frozen \(\lambda>0\) for every real smooth scalar
evaluated there. It also proves the partial kernel contraction
\[
\langle t_{\mathrm D}^{\mathbb C},
  \gamma_1^\lambda\otimes\gamma_2^\lambda\otimes\phi\rangle
=\int j_\lambda\phi\dd^4x.
\]
The obsolete claim that this equality is unproved is not retained
\cite{XVIH,XVIE,XVIF,XVIG}.

The smooth density \(j_\lambda\) is a partial tensor evaluation,
not the full kernel and not the two-edge graph defined below.

\section{An ambient propagator in the selected normalization}
\label{sec:propagator}

\subsection{Ambient advanced kernel versus a slab inverse}

This section makes an explicit propagator choice.
It is not inferred merely from a free-shell equation.

Let \(W_0\) be the massless Minkowski positive-frequency
distribution
\begin{equation}
W_0(r)
=\int_{\mathbb R^3}
 \frac{e^{-\ii|\mathbf p|r^0+\ii\mathbf p\cdot\mathbf r}}
      {(2\pi)^3\,2|\mathbf p|}\dd^3\mathbf p.
\label{eq:W}
\end{equation}
The integral is distributional. Set \(h_0=-W_0\), in the
reported XI-E convention.

Let \(E_{0,\mathrm{amb}}^{\mathrm A}\) be the ambient advanced
fundamental solution of \(\Box\). Its spatial Fourier kernel is
\begin{equation}
\widehat E_{0,\mathrm{amb}}^{\mathrm A}(t,\mathbf p)
=-\theta(-t)\frac{\sin(t|\mathbf p|)}{|\mathbf p|},
\label{eq:EAamb}
\end{equation}
with the continuous value at \(\mathbf p=0\).
The retarded kernel is
\(\theta(t)\sin(t|\mathbf p|)/|\mathbf p|\).
These distributions are translation invariant on ambient \(M\).

For a source compactly supported in \(M_T^\circ\), its ambient
advanced solution restricted to the slab is
\[
(E_{0,\mathrm{amb}}^{\mathrm A}F)(u)
=\int_u^T
 \frac{\sin((s-u)\sqrt{-\Delta})}{\sqrt{-\Delta}}F(s)\dd s.
\]
The upper limit is \(T\) because the zero-extended source
vanishes above \(T\). This identifies the restriction with
the supplied slab sine integral on that source class.
It does not declare an arbitrary slab operator translation
invariant.

All graph edges below use the ambient kernel.
Thus an interaction point outside the slab is not evaluated by
an unjustified slab-only prescription.

\subsection{The tensor time-ordered kernel}

Define the ambient tensor distributions
\[
G^{\mathrm A}=-\mathcal P E_{0,\mathrm{amb}}^{\mathrm A},
\qquad
H=-\mathcal P h_0=\mathcal P W_0,
\]
and put
\begin{equation}
D_F=H+\ii G^{\mathrm A}
=\mathcal P d,
\qquad
d:=W_0-\ii E_{0,\mathrm{amb}}^{\mathrm A}.
\label{eq:DF}
\end{equation}

\begin{proposition}[Identification and inverse identity]
\label{prop:DF}
The scalar distribution \(d\) is
\begin{equation}
d(r)=
\begin{cases}
W_0(r),&r^0>0,\\
W_0(-r),&r^0<0,
\end{cases}
\label{eq:timeordered}
\end{equation}
with its globally defined distributional meaning fixed by
\eqref{eq:DF}. Consequently
\[
D_F^{\mathsf T}=D_F,
\qquad
(-\Box\mathcal P)D_F=\ii\,\mathrm{id}
\]
on ambient compact tensor tests.
In components,
\[
D_F^{IK}(y,w)=\mathbb P^{IK}d(y-w).
\]
\end{proposition}

\begin{proof}
For \(t<0\), the spatial Fourier expression is
\[
\frac{e^{-\ii|\mathbf p|t}}{2|\mathbf p|}
+\ii\frac{\sin(t|\mathbf p|)}{|\mathbf p|}
=\frac{e^{\ii|\mathbf p|t}}{2|\mathbf p|}.
\]
For \(t>0\), the advanced summand vanishes.
This proves \eqref{eq:timeordered} and symmetry under \(r\mapsto-r\).

The temporal derivative of the two Fourier branches has jump
\(-\ii\) at \(t=0\). Hence
\(\Box d=-\ii\delta\).
Using \(\mathcal P^2=\mathrm{id}\) gives
\[
(-\Box\mathcal P)(\mathcal P d)=\ii\delta\,\mathrm{id}.
\]
\end{proof}

This is an explicit ambient massless prescription, with the
factor \(\mathcal P\) fixed by the inherited inverse and
two-point conventions. No X-sheet normalization is imported.

The symbol \(i0\) below denotes its distributional boundary
condition, not the preparation width and not an ultraviolet
regulator-removal scheme.

\subsection{Wavefront relation and punctured products}

Use the mathematical Fourier convention
\(\widehat u(k)=\int e^{-\ii k\cdot r}u(r)\dd^4r\), with
the coordinate pairing in that formula.
Let
\[
q(r)=(r^0)^2-|\mathbf r|^2,
\qquad
\mathcal C^\circ=\{r\ne0:q(r)=0\}.
\]

\begin{lemma}[Off-origin form and oriented conormal bound]
\label{lem:WF}
On \(r\ne0\),
\begin{equation}
d(r)=-\frac1{4\pi^2}(q(r)-\ii0)^{-1},
\label{eq:rational}
\end{equation}
and
\begin{equation}
\WF(d|_{M\setminus\{0\}})
\subset
\{(r;\tau\,\dd q(r)):r\in\mathcal C^\circ,\ \tau<0\}.
\label{eq:WFd}
\end{equation}
Away from the nonzero light cone, \(d\) is smooth.
On a spacelike region,
\[
d(r)=\frac1{4\pi^2(|\mathbf r|^2-(r^0)^2)}.
\]
\end{lemma}

\begin{proof}
The angular Fourier integral in \eqref{eq:W} gives the
massless rational two-point distribution with the positive
frequency temporal boundary condition.
Selecting the two temporal branches in
\eqref{eq:timeordered} gives \eqref{eq:rational} off the origin.

At a nonzero null point, \(\dd q\ne0\), so \(q\) is a local
coordinate transverse to the cone.
For the stated Fourier convention,
\((s-\ii0)^{-1}\) has the single negative-frequency conormal
ray. Pulling it back by \(q\) gives \eqref{eq:WFd}.
At \(q\ne0\), the reciprocal is smooth.

This is also the massless specialization of the Feynman
wavefront description in \cite[Proposition 26]{BDHintro}.
That paper uses the opposite Fourier sign in its physical
wavefront convention; reversing the covector sign gives
\eqref{eq:WFd}. The one-ray product test is independent of
that sign choice.
\end{proof}

% Primary-source verification for the propagator wavefront statement:
% Brouder--Dang--Helein, Proposition 26, with Fourier-sign conversion. ([arxiv.org](https://arxiv.org/pdf/1404.1778))

\begin{proposition}[Two differentiated edges]
\label{prop:product}
For arbitrary multiindices \(\mu,\nu\), the product
\begin{equation}
p_{\mu\nu}^{\circ}(r)
=(\partial^\mu d)(r)(\partial^\nu d)(r)
\label{eq:p}
\end{equation}
exists on \(M\setminus\{0\}\).
Its pullback to
\[
\Omega=\{(y,w)\in M^2:y\ne w\}
\]
exists under \(r=y-w\).
Finite tensor contractions and smooth multipliers preserve
definedness.
\end{proposition}

\begin{proof}
Differentiation does not enlarge the wavefront set.
At each nonzero null point both factors have covectors on
the same open half-ray in \eqref{eq:WFd}.
Two such nonzero covectors cannot be negatives of each other.
The distributional product criterion therefore applies.
The difference map \((y,w)\mapsto y-w\) is a submersion, so
its pullback exists.

The tensor matrices are constant, and contraction is a finite
component sum. Smooth multiplication is always defined.
\end{proof}

% Primary-source verification for the product criterion:
% Brouder--Dang--Helein, Theorem 13. ([arxiv.org](https://arxiv.org/pdf/1404.1778))

The conclusion includes noncoincident null-related points.
The light cone remains singular there, but is not deleted from
the distributional domain.

\section{The cubic interaction and its exact jet Hessian}
\label{sec:vertex}

\subsection{A declared density representative}

Retain the supplied cubic density representative
\begin{equation}
\ell_3(h)=-2(r_3+b_1r_2+b_2r_1),
\label{eq:ell3}
\end{equation}
with the following atoms:
\[
u_0^{\mu\nu}=\eta^{\mu\nu},
\qquad
u_1^{\mu\nu}=-h^{\mu\nu},
\qquad
u_2^{\mu\nu}=h^\mu{}_\rho h^{\rho\nu},
\]
\[
b_1=\frac12\tr_\eta h,
\qquad
b_2=\frac18(\tr_\eta h)^2-\frac14h_{\mu\nu}h^{\mu\nu},
\]
\[
\Gamma_n^\rho{}_{\mu\nu}
=\frac12u_{n-1}^{\rho\lambda}
  (\partial_\mu h_{\nu\lambda}
   +\partial_\nu h_{\mu\lambda}
   -\partial_\lambda h_{\mu\nu}),
\qquad n=1,2,3,
\]
\[
\begin{aligned}
R_{n,\mu\nu}
={}&\partial_\rho\Gamma_n^\rho{}_{\mu\nu}
   -\partial_\nu\Gamma_n^\rho{}_{\mu\rho}\\
 &+\sum_{\substack{a+b=n\\a,b\ge1}}
 \left(
 \Gamma_a^\rho{}_{\rho\lambda}\Gamma_b^\lambda{}_{\mu\nu}
 -\Gamma_a^\rho{}_{\nu\lambda}\Gamma_b^\lambda{}_{\mu\rho}
 \right),
\end{aligned}
\]
and
\[
r_n=\sum_{i=0}^{n-1}u_i^{\mu\nu}R_{n-i,\mu\nu}.
\]
Only coefficients appearing in these finite expressions are
used. This is the unintegrated representative supplied in the
original XVI-I through XII-C, not an Euler-class replacement.

Each monomial has three field-jet factors and total field
derivative order two. The density has jet order at most two.
The distinction between total derivative order and the maximum
jet order of an individual factor will be used below.

Choose
\[
\chi_{\mathrm I}\in C_c^\infty(M;\mathbb R),
\qquad
V(h)=\int_M\chi_{\mathrm I}(w)\ell_3(h)(w)\dd^4w.
\]
The preparation correction belongs to the insertion kernel.
It is not added again as an interaction vertex.

\subsection{Jet-coordinate definition of the Hessian}

Write \(h_{K,a}=\partial^a h_K\), where the multiindices label
independent symmetric jet coordinates.
Define
\begin{equation}
L_{KL;ab}[h](w)
=
\left.
\frac{\partial^2\ell_3}
     {\partial h_{K,a}\,\partial h_{L,b}}
\right|_{j^2h(w)}.
\label{eq:L}
\end{equation}
Absent jet variables have zero coefficient.

\begin{proposition}[Exact bilinear differential operator]
\label{prop:Hessian}
For smooth tensor directions \(A,C\),
\begin{equation}
D^2\ell_3(h)[A,C](w)
=
\sum_{K,L,a,b}
L_{KL;ab}[h](w)
(\partial^aA_K)(w)(\partial^bC_L)(w).
\label{eq:Hessian}
\end{equation}
The coefficients satisfy
\[
L_{KL;ab}=L_{LK;ba},
\qquad
|a|+|b|\le2
\]
whenever they are nonzero.
They are linear differential expressions in the remaining
field \(h\). For such a term, the derivative order on that
remaining factor is at most \(2-|a|-|b|\).
Moreover,
\[
D^2V(h)[A,C]
=\int\chi_{\mathrm I}D^2\ell_3(h)[A,C]\dd^4w.
\]
No integration by parts is made in this definition.
\end{proposition}

\begin{proof}
The chain rule in the independent jet coordinates gives
\eqref{eq:Hessian}. Equality of mixed partial derivatives gives
the symmetry.

Differentiating a cubic jet monomial removes two field factors.
The remaining coefficient is linear in one field jet.
The original total derivative order was two, so the derivatives
on the two varied factors have sum at most two, and the remaining
factor has the stated residual order.

The interaction switch is fixed under field differentiation.
This proves the last formula without derivatives of
\(\chi_{\mathrm I}\).
\end{proof}

This defines the full tensor Hessian unambiguously.
A flattened table is unnecessary, and no unspecified
bidifferential operator is substituted for \eqref{eq:L}.

\section{The reduced graph and its combinatorial normalization}
\label{sec:graph}

\subsection{Two vertices and two labelled contractions}

For two distinct vertices, let the cross-contraction operator
use \(D_F\), with one derivative at each vertex per edge.
The second term of its exponential carries \(1/2!\).
The insertion derivative is \(D_h^2F_t=2t_{\mathrm D}\).
The interaction derivative is the ordinary second derivative
\eqref{eq:Hessian}, with no divided-power convention.

Therefore the numerical multiplier of the two-edge contribution
is
\[
\frac1{2!}\times2=1.
\]
The sums over the two labelled legs are retained in the Hessian
and the two kernel slots. There is no second division for
tensor-slot symmetry.

This is the graph-expansion convention of
\cite[Eq.~(82)]{FR}, adopted here only for this numerical
sector. Coupling powers, the vertex multiplier in an
\(S\)-matrix, and \(\hbar^2\) are not included in the reduced
kernel. Thus the object below is not called a complete first
quantum coefficient.

% Primary-source verification for the identical-edge factorial:
% Fredenhagen--Rejzner, equation (82). ([arxiv.org](https://arxiv.org/html/1208.1428v2))

\subsection{Open tensor legs and source-contracted reduction}

\begin{definition}[Reduced open-index blocks]
\label{def:blocks}
For \(|\alpha|+|\beta|\le2\), define on \(\Omega\)
\begin{equation}
\begin{aligned}
\mathcal V^{\alpha\beta}_{IJ}[h](y,w)
=
(-1)^{|\alpha|+|\beta|}\chi_{\mathrm I}(w)
\sum_{K,L,a,b}L_{KL;ab}[h](w)\,
 &\bigl(\partial_y^\alpha\partial_w^a
          D_F^{IK}(y,w)\bigr)\\
 {}\times&
 \bigl(\partial_y^\beta\partial_w^b
          D_F^{JL}(y,w)\bigr).
\end{aligned}
\label{eq:Vblock}
\end{equation}
The source-contracted reduced kernel is
\begin{equation}
U_A^\circ[h](y,w)
=
\sum_{I,J,\alpha,\beta}
c^A_{IJ;\alpha\beta}(y)
\mathcal V^{\alpha\beta}_{IJ}[h](y,w).
\label{eq:U}
\end{equation}
It still has the source/root index \(A\).
The scalar insertion slot is not contracted.
\end{definition}

The derivatives \(\partial_y^\alpha\) and
\(\partial_y^\beta\) arise from the differentiated source
deltas in \eqref{eq:t-test}. Their displayed sign is the
distributional derivative-transfer sign.

In \eqref{eq:Vblock}, derivatives act only on their designated
edge. They do not act on \(c\), \(\chi_{\mathrm I}\), or \(L\).
This fixes the recipe before any extension or integration.

\begin{theorem}[Reduced graph on the punctured domain]
\label{thm:reduced}
For every smooth real \(h\), all blocks
\(\mathcal V^{\alpha\beta}_{IJ}[h]\) and all
\(U_A^\circ[h]\) are distributions on \(\Omega\).
They are linear in \(h\).
Their coefficients and derivative placements are exactly
specified by \eqref{eq:L}, \eqref{eq:Vblock}, and
\eqref{eq:U}.

The intended compressed attachment is
\[
B_A(y,x)U_A^\circ[h](y,w)
\]
on the full variables \((y,w,x)\), followed by integration in
\(y\). Its definition is tested separately in
Section~\ref{sec:attachment}.
\end{theorem}

\begin{proof}
Using \(r=y-w\),
\[
\partial_y^\alpha\partial_w^aD_F^{IK}(y,w)
=
(-1)^{|a|}\mathbb P^{IK}
(\partial^{\alpha+a}d)(r).
\]
Thus each summand is a constant tensor contraction of one
of the products in Proposition~\ref{prop:product}, multiplied
by smooth coefficients. Every sum is finite.
The Hessian coefficients are linear in \(h\).
\end{proof}

The entire switched kernel is generally not translation
invariant. Only its numerical edge building blocks are
distributions of \(r=y-w\).
Restoring the smooth coefficients in \(y,w\) must not be
mistaken for a new translation-invariance assertion.

\section{An explicit off-diagonal nonvanishing test}
\label{sec:nonzero}

\begin{proposition}[A nonzero open-index vertex block]
\label{prop:nonzero}
Suppose \(\chi_{\mathrm I}\not\equiv0\).
There are a smooth real background \(h\), tensor contractions
of the open indices of \(\mathcal V^{00}_{IJ}[h]\), and a
compact test supported in a spacelike subset of \(\Omega\),
for which the resulting pairing is nonzero.

This does not imply that every source-contracted kernel
\(U_A^\circ[h]\), or the root-attached sum, is nonzero.
\end{proposition}

\begin{proof}
Choose \(w_0\) with \(\chi_{\mathrm I}(w_0)\ne0\), and let
\[
h_{\mu\nu}(w)=(w^1-w_0^1)^2\eta_{\mu\nu}.
\]
At \(w_0\), the field and its first derivatives vanish, while
\[
\partial_1^2h_{\mu\nu}(w_0)=2\eta_{\mu\nu}.
\]
Take constant tensor directions
\[
A_{\mu\nu}=\eta_{\mu\nu},
\qquad
C_{11}=1,\quad C_{\mu\nu}=0\text{ otherwise}.
\]

We calculate the Hessian at this jet rather than infer it from
scaling degree. For the metric
\[
q_\kappa=\eta+\kappa(h+sA+tC),
\]
put \(a=1+\kappa s\), \(b=a-\kappa t\).
At \(w_0\),
\[
q_\kappa=\operatorname{diag}(a,-b,-a,-a),
\qquad
\partial q_\kappa=0,
\qquad
\partial_1^2q_\kappa=2\kappa\eta.
\]
The supplied Ricci convention gives
\[
R(q_\kappa)(w_0)=\frac{6\kappa}{ab},
\qquad
\sqrt{-q_\kappa}\,R(q_\kappa)(w_0)
=6\kappa\sqrt{\frac ab}.
\]
Therefore the coefficient of \(\kappa\) in the signed density
\(-2\kappa^{-2}\sqrt{-q_\kappa}R\) is
\[
\ell_3(h+sA+tC)(w_0)=6st-\frac92t^2.
\]
Indeed,
\[
\sqrt{\frac ab}
=1+\frac{\kappa t}{2}
 +\kappa^2\left(-\frac{st}{2}+\frac{3t^2}{8}\right)
 +O(\kappa^3).
\]
Thus
\[
D^2\ell_3(h)[A,C](w_0)=6.
\]

Every Hessian coefficient with \(|a|+|b|>0\) has its remaining
field differentiated at most once. Those coefficients vanish
at \(w_0\), because \(h(w_0)=\partial h(w_0)=0\).
Consequently all vertex derivative-edge corrections vanish at
that point, not merely in a scaling estimate.

Choose open-index contractions whose images under the
invertible matrix \(\mathbb P\) are \(A,C\).
Choose
\[
y_0=w_0+(0,R,0,0),\qquad R>0.
\]
At this spacelike pair the selected component of
\(\mathcal V^{00}\) is exactly
\[
6\chi_{\mathrm I}(w_0)d(y_0-w_0)^2
=\frac{6\chi_{\mathrm I}(w_0)}{16\pi^4R^4}\ne0.
\]
The graph is a smooth real function on a neighborhood of this
spacelike pair. By continuity, a sufficiently small nonnegative
test supported there has a nonzero pairing with that component.
\end{proof}

If \(\chi_{\mathrm I}=0\), every reduced block is zero.
Even for nonzero \(\chi_{\mathrm I}\), the preparation data may
make the later contracted sum vanish; for example \(f=0\)
makes the root vanish. No universal nonvanishing claim is made.

\section{Relative scaling degree and the diagonal extension}
\label{sec:extension}

\subsection{Scaling is applied to numerical building blocks}

For \(u^\circ\in\Dprime(\mathbb R^4\setminus\{0\})\), use the
usual scaling degree at the origin:
\[
\sd(u^\circ)
=\inf\{\omega:\lambda^\omega u^\circ_\lambda\to0
                    \text{ as }\lambda\downarrow0\},
\]
with the normalized rescaling of tests.

The diagonal \(y=w\) has codimension four.
Use coordinates \((r,w)=(y-w,w)\); \(w\) is tangential.
The smooth interaction and preparation coefficients are not
part of the translation-invariant numerical kernel.

\begin{proposition}[Termwise scaling bounds]
\label{prop:sd}
The nonzero numerical products \(p_{\mu\nu}^\circ\) of
\eqref{eq:p} are homogeneous on the punctured domain of degree
\[
-4-|\mu|-|\nu|,
\]
and therefore have scaling degree
\[
4+|\mu|+|\nu|.
\]
For zero products the corresponding inequality suffices.

The vertex-only blocks with \(\alpha=\beta=0\) have local
transverse scaling degree at most six.
Including the insertion source derivatives gives
\begin{equation}
\sd_{y=w}(U_A^\circ[h])\le8
\label{eq:sd8}
\end{equation}
locally, uniformly on compact tangential sets.
This is an upper bound before root attachment, not a sharp
degree of the total tensor sum.
\end{proposition}

\begin{proof}
The massless \(d\) is homogeneous of degree \(-2\).
Its derivative of order \(|\mu|\) has degree
\(-2-|\mu|\). The punctured product is already defined by
Proposition~\ref{prop:product}; rescaling respects that product.
Hence its degree is the sum of the two degrees.

For \(\alpha=\beta=0\), the Hessian satisfies
\(|a|+|b|\le2\), so the numerical edge degree is at most six.
With the insertion derivatives included,
\[
|\alpha+a|+|\beta+b|
\le |\alpha|+|\beta|+|a|+|b|\le4.
\]
This gives the bound eight.

In relative coordinates, all remaining coefficients are smooth
functions of \((r,w)\). Their derivatives are bounded on the
compact sets used in a local transverse scaling test.
Smooth multiplication cannot increase the bound there.
Finite summation completes the estimate.
\end{proof}

This proof does not deduce a pointwise-product estimate merely
from a derivative lemma. It first proves the product, then uses
its explicit homogeneity.

\subsection{Extension family}

\begin{theorem}[Point-reduced extension freedom]
\label{thm:extension}
For the finitely many numerical products occurring in the graph,
extensions
\[
\bar p_{\mu\nu}\in\Dprime(\mathbb R^4)
\]
exist without increasing their scaling degree.

For a fixed pair with \(n=|\mu|+|\nu|\), two such extensions
differ by
\begin{equation}
\sum_{|\gamma|\le n}
a_{\mu\nu;\gamma}\,\partial^\gamma\delta(r).
\label{eq:ambiguity}
\end{equation}
In particular, all building blocks in the source-decorated
sector have ambiguity order at most four.

Replacing the numerical products in
\eqref{eq:Vblock}--\eqref{eq:U} by these extensions defines
distributions \(\bar{\mathcal V}^{\alpha\beta}_{IJ}[h]\) and
\(\bar U_A[h]\) on all \(M^2\).
No coefficients \(a_{\mu\nu;\gamma}\) are chosen.
\end{theorem}

\begin{proof}
The point-extension theorems for finite scaling degree apply
in four relative dimensions \cite[Theorems 5.2--5.3]{BF}.
The difference of two extensions is supported at \(r=0\).
A point-supported distribution is a finite sum of derivatives
of \(\delta\), and
\[
\sd(\partial^\gamma\delta)=4+|\gamma|.
\]
The scaling bound \(4+n\) therefore permits only
\(|\gamma|\le n\).

Pull back each extension under the submersion
\((y,w)\mapsto y-w\), multiply by the displayed smooth
coefficients, and take the finite component sums. These
operations define the claimed global distributions.
\end{proof}

% Primary-source verification for extension existence and the jet ambiguity:
% Brunetti--Fredenhagen, Theorems 5.2 and 5.3. ([arxiv.org](https://arxiv.org/pdf/math-ph/9903028))

The finite collection of point-extension constants is finite
dimensional before additional relations. Edge-exchange symmetry
can be retained by choosing
\(\bar p_{\mu\nu}=\bar p_{\nu\mu}\), for example by averaging.
That algebraic restriction does not fix the remaining constants.

A subtraction projection can be specified by functions
\(w_\gamma(r)\) satisfying
\[
\partial^\beta w_\gamma(0)=\delta^\beta_\gamma,
\qquad |\beta|,|\gamma|\le n,
\]
and by the values of the extension on these functions.
No such values, covariance conditions, Ward conditions, or
renormalization scale are selected here.

There are two different ambiguity statements:
\begin{itemize}
\item The numerical translation-invariant building blocks have
      the constant-coefficient point ambiguities
      \eqref{eq:ambiguity}.
\item An unrestricted extension of the full distribution in
      \((r,w)\) can have coefficients depending on \(w\);
      transverse order alone does not make that larger freedom
      finite dimensional.
\end{itemize}
The present construction uses the first, explicitly specified
extension family. It does not classify every possible
tangential or fully root-attached ambiguity.

\subsection{Strata and what the extension covers}

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{0.23\linewidth}Y Y@{}}
\toprule
Locus & Treatment & Remaining issue\\
\midrule
\(y=w\)
 & Point extension in four relative dimensions;
   source-decorated degree bounded by eight
 & Normalization constants unchosen\\
\addlinespace
\(q(y-w)=0,\ y\ne w\)
 & Included in the punctured distribution;
   one-ray wavefront product verified
 & Not a new coincidence extension\\
\addlinespace
Root-side singularities in \(B(y,x)\)
 & Retained in the imported root kernel
 & Compressed product tested below;
   singular support not declared empty\\
\addlinespace
Unfolded root collisions involving \(z\)
 & Already part of the inherited construction of \(B\)
 & No new independently unfolded product is claimed\\
\addlinespace
Insertion diagonal \(y_1=y_2\)
 & Included in the derivative-transfer formula
   \eqref{eq:t-test}
 & Not an additional bubble variable\\
\bottomrule
\end{tabularx}
\end{center}

\section{Root attachment: maps, wavefront test, and properness}
\label{sec:attachment}

\subsection{The compressed route}

Let
\[
X=M_y\times M_w\times M_x,
\]
and define the projections
\[
p_B(y,w,x)=(y,x),
\qquad
p_U(y,w,x)=(y,w),
\qquad
\pi(y,w,x)=(w,x).
\]
They are submersions.
The proposed compressed graph distribution is
\begin{equation}
\mathcal G[h]
=\sum_A(p_B^*B_A)(p_U^*\bar U_A[h]),
\qquad
R[h]=\pi_*\mathcal G[h].
\label{eq:attachment}
\end{equation}

For a general root, the product would require
\begin{equation}
\WF(p_B^*B_A)
\cap\bigl(-\WF(p_U^*\bar U_A[h])\bigr)=\varnothing
\label{eq:root-condition}
\end{equation}
for every component, followed by properness of \(\pi\) on the
product support.

\begin{theorem}[Compressed attachment under the root cone input]
\label{thm:attach}
Under \eqref{eq:Bcone}, condition
\eqref{eq:root-condition} holds for every distribution
\(\bar U_A[h]\) constructed above, regardless of its detailed
wavefront set.

The product \(\mathcal G[h]\) is compactly supported.
Its pushforward \(R[h]\in\Eprime(M_w\times M_x)\) is defined.
The same construction works on the off-diagonal domain before
extension.

No normalized extension is thereby selected.
\end{theorem}

\begin{proof}
The lifted root wavefront covectors have the form
\[
(y,w,x;p,0,-p),\qquad p\ne0,
\]
by \eqref{eq:Bcone}.
The lifted reduced distribution has covectors
\[
(y,w,x;k,\ell,0),
\]
with \((k,\ell)\ne0\).
For these two covectors to cancel, the \(x\)-component would
force \(p=0\), contrary to the nonzero root covector.
Thus the product criterion is satisfied.

The root is compact in \((y,x)\).
Every reduced term contains the interaction switch
\(\chi_{\mathrm I}(w)\), including the diagonal extension
terms. Hence its \(w\)-support is contained in
\(\supp\chi_{\mathrm I}\).
The product support is a closed subset of the compact set
\[
\{(y,w,x):(y,x)\in\supp B,\
                    w\in\supp\chi_{\mathrm I}\}.
\]
The projection \(\pi\) is therefore proper on that support,
and the pushforward is defined.

Neither argument requires the root to be smooth or the
reduced graph to be smooth at a null-related pair.
\end{proof}

% Primary-source verification for the pullback/product/proper-pushforward interface:
% Brouder--Dang--Helein, Propositions/Theorems in Sections 5--6. ([arxiv.org](https://arxiv.org/pdf/1409.7662))

The decisive property is that the root has no nonzero wavefront
covector with zero scalar-slot covector:
\[
(y,x;p,0)\notin\WF(B)\quad(p\ne0).
\]
The stronger \(Z_2\) input guarantees it.
If that input were withheld, the generic test
\eqref{eq:root-condition} would remain an unresolved attachment
obligation. Compactness alone would not replace it.

This compressed argument does not separately multiply every
factor of an unfolded formula
\[
E_0^{\mathrm A}(y,z)A_{f,g}(z,x).
\]
That internal composition is used only through the already
constructed root \(B\). No new claim about all unfolded
partial-diagonal products is made.

\subsection{Smeared coefficients form a family}

\begin{corollary}[Unnormalized root-attached coefficients]
\label{cor:coefficient}
For each extension tuple \(\{\bar p_{\mu\nu}\}\), each smooth
\(h\), and each real smooth scalar \(\phi\), the number
\begin{equation}
C_{\bar p}(h,\phi)
=\langle R[h],1\otimes\phi\rangle
\label{eq:coefficient}
\end{equation}
is well defined by compact support.
It is linear in \(h\) and in \(\phi\).
Equivalently,
\begin{equation}
C_{\bar p}(h,\phi)
=
\sum_A
\left\langle\bar U_A[h](y,w),
                    a_{f,g}[\phi]_A(y)\right\rangle,
\label{eq:smearfirst}
\end{equation}
where compact cutoffs on the fixed effective support are
understood.
\end{corollary}

\begin{proof}
Theorem~\ref{thm:attach} gives a compact output distribution,
which acts on arbitrary smooth functions.
The Hessian coefficients are linear in \(h\), and the root is
linear in \(\phi\).

To obtain \eqref{eq:smearfirst}, multiply the root kernel by
the smooth scalar, push forward its \(x\)-slot, and use the
inherited root-kernel identity. The resulting
\(a_{f,g}[\phi](y)\) is smooth. The wavefront test just proved,
together with proper support, permits the corresponding
distributional composition and interchange of these
pushforwards. Pairing against the reduced distribution gives
\eqref{eq:smearfirst}.
\end{proof}

Thus root attachment is no longer left at an unspecified
product obstruction under the stated \(Z_2\) premise.
What remains unresolved is the selection of the extension
tuple and the normalization of the resulting coefficient.

The family \(C_{\bar p}\) is not asserted to be independent
of its extension constants. Its differences are diagonal
contact contributions propagated through the root.
Locality of a subtraction in \(r\) does not make
\(a_{f,g}[\phi]\), or the final coefficient as a functional of
the original fields, local.

No identification with ordinary bulk gravitational counterterms
is proved.

\section{Off-shell representatives, free variation, and quantum limits}
\label{sec:limits}

\subsection{The off-shell representative is fixed}

Every graph formula uses the particular off-shell
\(t_{\mathrm D}\) of \eqref{eq:t-test}.
Free-shell equivalence is not used to change it.

An ideal term such as \(\Box_1 a\) pairs trivially with a free
tensor solution. It need not pair trivially with the chosen
propagator, since
\[
\Box_yD_F=-\ii\mathcal P\delta(y-w).
\]
Such changes can create contact terms in a graph calculation.
It is incorrect to conclude that every nonzero ideal term
produces a nonzero graph; the assertion is only that free-shell
invisibility does not guarantee time-ordered invisibility.

\subsection{Free derivation with preparation held fixed}

Declare the preparation data and \(t_{\mathrm D}\) fixed under
the free variation. Adopt the inherited free field formulas
\[
s_0h=\mathcal R c,\qquad
s_0\phi=0,\qquad s_0c=0,\qquad
s_0\bar c=\ii b,\qquad s_0b=0,
\]
with
\[
(\mathcal R c)_{\mu\nu}=\partial_{(\mu}c_{\nu)}.
\]
Ghost directions may be treated as formal Grassmann-valued
smooth directions in the following polynomial identity.

\begin{proposition}[Free variation of the prepared polynomial]
\label{prop:s0}
With that convention,
\begin{equation}
s_0F_t(h,\phi)
=2\langle t_{\mathrm D},
               (\mathcal R c)\otimes h\otimes\phi\rangle.
\label{eq:s0}
\end{equation}
No vanishing of the right-hand side is asserted.
\end{proposition}

\begin{proof}
Only the two tensor arguments vary.
The scalar variation is zero, and tensor-slot symmetry makes
the two tensor contributions equal.
Compact support makes the pairing with smooth directions
well defined.
\end{proof}

If preparation data are assigned transformations, their
variation contributes an additional term.
It is not computed here.

The chain-rule identity is not a Ward identity or a proof of
BRST/BV closure. No interacting differential, BV antibracket,
master action, or compensating graph family is constructed.

\subsection{The frozen scalar density is not this graph}

For a specified scalar two-point distribution \(\mathsf H_2\),
\[
\langle\mathsf H_2,j_\lambda\otimes j_\mu\rangle
\]
is definable because \(j_\lambda\otimes j_\mu\) is a compact
smooth test density.
That is a scalar pairing, not the tensor two-edge product,
and it does not determine the interaction Hessian,
normalization, or graph extension.

Here the tensor edge prescription has been explicitly chosen.
It does not provide a positive state for the full gauge
multiplet, ghost and scalar time-ordered blocks, or a complete
family of time-ordered products.

\subsection{What is still missing}

No Bogoliubov or retarded interacting coefficient is assembled
from the family \eqref{eq:coefficient}.
No extension is selected by a covariance, field-equation,
unitarity, Ward, or renormalization condition.
No scale derivative is assigned to an unchosen normalized
extension.

The boundary-value notation defining \(d\) is not a temporary
ultraviolet regularization of the graph.
The extension problem is treated directly in distribution
space.

No global adiabatic limit, sharp preparation-width limit,
ultraviolet completion, or all-orders finite-parameter
renormalizability theorem is part of this manuscript.

\section{Consolidated theorem}
\label{sec:main}

\begin{theorem}[Classical admission and the reduced extension family]
\label{thm:main}
Retain the exact classical matching inputs, the supplied
classical and root cone bounds, the ambient propagator
prescription, and the declared cubic density representative.

Then:
\begin{enumerate}
\item The classical prepared polynomial has direct microcausal
      membership, and its free-shell class contains its own
      microcausal kernel.
\item The oriented off-diagonal wavefront relation of the
      selected propagator defines every differentiated
      two-edge product occurring in the explicit reduced
      sector.
\item For nonzero \(\chi_{\mathrm I}\), the open-index
      vertex block has a nonzero spacelike component for a
      specified smooth background. No general nonvanishing
      of the full contracted sum follows.
\item The source-decorated reduced sector has local transverse
      scaling degree at most eight in four relative
      dimensions.
\item Its numerical building blocks admit point extensions;
      their ambiguity orders are at most four. No
      normalization constants are chosen.
\item Under the root \(Z_2\) premise, compressed root attachment
      and proper pushforward exist for each extension choice.
      The resulting smeared coefficients form a family.
\end{enumerate}
No member of that family is selected as a normalized retarded
interacting coefficient or a BRST/BV observable.
\end{theorem}

\begin{proof}
Theorem~\ref{thm:admission} proves item~1.
Proposition~\ref{prop:product} and
Theorem~\ref{thm:reduced} prove item~2.
Proposition~\ref{prop:nonzero} proves item~3.
Proposition~\ref{prop:sd} proves item~4.
Theorem~\ref{thm:extension} proves item~5.
Theorem~\ref{thm:attach} and
Corollary~\ref{cor:coefficient} prove item~6.
The unchosen normalizations and unconstructed interacting
identities are recorded in Section~\ref{sec:limits}.
\end{proof}

\section{Status ledger and corrections}
\label{sec:status}

\small
\renewcommand{\arraystretch}{1.12}
\begin{longtable}{@{}p{0.30\linewidth}p{0.65\linewidth}@{}}
\toprule
Object or result & Current status\\
\midrule
\endfirsthead

\toprule
Object or result & Current status, continued\\
\midrule
\endhead

\bottomrule
\endfoot

Classical polarized kernel
 & Established under the revised XVI-H root/source inputs.
   Its general test action is \eqref{eq:t-test}.\\
\addlinespace

Matching XII-D admission kernel
 & Exact matching under the explicitly supplied domain and
   normalization premises:
   Proposition~\ref{prop:match}.\\
\addlinespace

Direct classical admission
 & Established as an imported-premise consequence:
   Theorem~\ref{thm:admission}.
   The old blanket non-admission disclaimer is removed.\\
\addlinespace

Prepared free-shell class
 & Contains the admitted representative \(t_{\mathrm D}\).
   On data with the frozen obstruction, it cannot be the
   excluded frozen class:
   Corollary~\ref{cor:oldclass}.\\
\addlinespace

Trisolution cone exclusion
 & Consequence of the separately stated transport premise
   and imported \(Z_3\) bound.
   No new classical wavefront calculation is performed.\\
\addlinespace

Witness matching to \(j_\lambda\)
 & Established in revised XVI-H using current XVI-E/F/G.
   It is partial evaluation, not the graph kernel.\\
\addlinespace

Ambient tensor propagator
 & Explicitly chosen and identified:
   Proposition~\ref{prop:DF}.
   \(D_F=\mathcal P d\), with
   \((-\Box\mathcal P)D_F=\ii\,\mathrm{id}\).
   The slab restriction is justified on interior sources.\\
\addlinespace

Off-diagonal edge product
 & Proved for the chosen propagator, including nonzero
   null-related pairs:
   Proposition~\ref{prop:product}.\\
\addlinespace

Cubic Hessian and derivative recipe
 & Defined by exact jet differentiation:
   \eqref{eq:L}--\eqref{eq:Hessian}.
   Total vertex derivative order is at most two.\\
\addlinespace

Reduced graph sector
 & Defined and proved distributional on \(y\ne w\):
   \eqref{eq:Vblock}--\eqref{eq:U},
   Theorem~\ref{thm:reduced}.\\
\addlinespace

Nonvanishing
 & Proved for an open-index vertex block when the switch is
   nonzero:
   Proposition~\ref{prop:nonzero}.
   Not asserted for every contracted or attached sum.\\
\addlinespace

Scaling degree
 & Vertex-only bound six; source-decorated bound eight.
   Local transverse upper bounds, not a sharp total degree:
   Proposition~\ref{prop:sd}.\\
\addlinespace

Diagonal extension
 & Existence and point-building-block ambiguity established:
   Theorem~\ref{thm:extension}.
   Normalization constants unchosen.\\
\addlinespace

Compressed root attachment
 & Product and proper-pushforward tests proved under
   \(\WF(B)\subset Z_2\):
   Theorem~\ref{thm:attach}.
   No root singular support is deleted.\\
\addlinespace

Smeared root-attached coefficient
 & A family exists for extension choices:
   Corollary~\ref{cor:coefficient}.
   No normalized member is selected.\\
\addlinespace

Unfolded root graph
 & No separate product theorem for every internal
   \(z\)-factor is claimed.
   The compressed inherited root is the route used.\\
\addlinespace

Free variation
 & Chain-rule formula proved with fixed preparation:
   Proposition~\ref{prop:s0}.
   Closure is not asserted.\\
\addlinespace

Retarded interacting coefficient
 & Not assembled.
   Time-ordered normalization and the interacting identity
   remain unresolved.\\
\addlinespace

BRST/BV observability
 & Not established.
   No interacting differential, Ward identity, or master
   construction is supplied.\\
\addlinespace

Sharp-width and ultraviolet claims
 & Outside scope.
   No regulator-removal or all-orders theorem is performed.\\

\end{longtable}
\normalsize

The principal corrections to the original XVI-I are:
\begin{enumerate}
\item \textbf{Admission implications are stated honestly.}
      The additional \(Z_3\) premise concerns the matching
      prepared kernel and implies narrow classical admission.
      It is not simultaneously imported and denied.
\item \textbf{The ambient propagator is actually specified.}
      Its translation-invariant definition is separated from
      a slab inverse. Algebraic inversion alone is not used
      as a wavefront theorem.
\item \textbf{The two-edge recipe includes insertion derivatives.}
      The source-delta derivatives and their signs are included
      in \eqref{eq:Vblock}, not omitted from the power count.
\item \textbf{Total derivative order is audited.}
      The cubic Hessian places at most two derivatives in total
      on its two varied legs, not independently four.
      The insertion can add another two, yielding the
      conservative bound eight.
\item \textbf{Nonvanishing has an off-diagonal test.}
      The switch must be nonzero, and the derivative
      corrections vanish at the chosen jet.
      Scaling degree alone is not a nonvanishing proof.
\item \textbf{Point and tangential ambiguities are separated.}
      A finite point-extension family is not a classification
      of arbitrary tangential counterterms.
\item \textbf{Root attachment is tested rather than merely withheld.}
      The compressed product condition follows from the
      supplied root cone bound. The normalized coefficient
      nevertheless remains unselected.
\item \textbf{Version references are synchronized.}
      The current witness chain is not described using
      superseded XVI-E/F/G stops.
\end{enumerate}

\section{Remaining obligations and final release}
\label{sec:remaining}

\begin{problem}[Normalization of the selected sector]
Choose the extension constants and impose the desired
covariance, derivative identities, field equations, exchange
symmetry, and any Ward conditions.
Prove compatibility of those choices before calling
\(C_{\bar p}\) a normalized insertion coefficient.

Extension existence does not choose these conditions or their
solutions.
Status: \Unresolved.
\end{problem}

\begin{problem}[Interacting and gauge interpretation]
Specify the complete perturbative normalization, coupling and
\(\hbar\) orders, relevant time-ordered and retarded products,
and companion sectors needed for an interacting identity.
Then formulate and prove the appropriate gauge or BRST/BV
statement.

The admitted classical insertion and one reduced graph do not
supply that construction.
Status: not established.
\end{problem}

No selected counterterm table, scale derivative, retarded
interacting coefficient, or physical observable is asserted.
No repair of the frozen XI-G obstruction is claimed.

The manifold, preparation width, graph prescription, component
conventions, and extension domains have been fixed explicitly.
The manuscript is a proof of the stated reduced construction
and its extension family, not a proof of the unresolved
interacting programme.

No successful compiler execution, full-series search, or
machine verification is represented.

\scopebox{Final release statement.}{%
The additional supplied XII-D premise admits the matching
classical prepared kernel; the old frozen class remains excluded
on its own hypotheses.

The chosen ambient propagator defines the reduced two-edge
sector off coincidence. Its source-decorated relative scaling
degree is bounded by eight. Point extensions exist with
unfixed local coefficients.

Under the root \(Z_2\) bound, compressed attachment and smearing
are defined for each extension choice. The result is a family,
not a selected normalized quantum coefficient.

Normalization, the retarded interacting construction, and
BRST/BV observability remain \Unresolved.
}

\begin{thebibliography}{99}

\bibitem{XVIH}
\emph{Landmark XVI-H}.
Supplied revised \texttt{XVI-H.tex},
``Compact polarized kernel and witness recovery.''
Locators:
\texttt{thm:Opol}, \texttt{thm:trep},
\texttt{prop:estimate}, \texttt{prop:tmatch},
and \texttt{thm:main}.
Used for the compact classical representation, continuity,
and current witness matching.
Its unproved full-kernel wavefront gate is not retroactively
reported as proved there.

\bibitem{XVIE}
\emph{Landmark XVI-E}.
Supplied revised \texttt{XVI-E.tex},
``Polarized root representation and witness-level completion
compatibility.''
Locators:
\texttt{lem:TD}, \texttt{lem:TC}, and
\texttt{thm:compatibility}.
Used for matching the polarization convention and current
witness compatibility.

\bibitem{XVIF}
\emph{Landmark XVI-F}.
Supplied revised \texttt{XVI-F.tex},
``Auxiliary/root compatibility and direct witness completion.''
Locators:
\texttt{prop:compat}, \texttt{thm:compatibility},
and \texttt{cor:family-pointwise}.
Used for the current auxiliary/root and direct-completion
record, not a superseded residual stop.

\bibitem{XVIG}
\emph{Landmark XVI-G}.
Supplied current \texttt{XVI-G.tex},
``Frequency-dependent witness testing and resonant asymptotics.''
Locators:
\texttt{prop:import}, \texttt{prop:kernel-slot},
and \texttt{prop:resonant}.
Its scalar density is not substituted for the graph kernel.

\bibitem{XIID}
\emph{Manuscript XII-D}, version 1.
Supplied identifier \texttt{e3e3f67d}.
Additional supplied excerpts:
\texttt{def:completed}, lines 556--569;
\texttt{thm:root}, lines 628--674;
\texttt{thm:admission} and \texttt{eq:t-bounds},
lines 805--841.
The root \(Z_2\) statement is reproduced in the supplied
original XVI-I as \texttt{lem:root-kernel}.
These interfaces are used as explicit inherited premises;
the complete source is not independently audited.

\bibitem{XIIC}
\emph{Manuscript XII-C}, version 1.
Supplied identifier \texttt{e0611fd7}.
Density formulas reproduced in the supplied original XVI-I:
\texttt{eq:ell3}, \texttt{eq:finite-curvature},
and \texttt{eq:J3}.
Additional source excerpt:
\texttt{thm:actualsource}, lines 636--678.
The declared unintegrated density representative is retained.

\bibitem{XIG}
\emph{Manuscript XI-G}, version 1.
Supplied identifier \texttt{3cae2be9}.
Supplied excerpts:
\texttt{def:class}, lines 494--507;
\texttt{thm:criterion}, lines 954--987;
\texttt{thm:obstruction}, lines 1107--1134.
The compact-kernel transport premise is reproduced in the
supplied original XVI-I.
The printed frozen-kernel criterion is not silently promoted
to a universal theorem.

\bibitem{XIB}
\emph{Manuscript XI-B}, version 1.
Supplied identifier \texttt{862627cd}.
Interfaces reproduced in the supplied XVI documents:
the quadratic metric operator, \texttt{lem:Green},
\texttt{thm:graviton}, and the free differential
\texttt{eq:s}.
Used for the normalization and free-variation convention.

\bibitem{XIE}
\emph{Manuscript XI-E}.
Two-point formulas reproduced in the supplied original XVI-I:
\texttt{eq:W}, \texttt{eq:h},
\texttt{eq:Heven}, and \texttt{eq:antisym}.
The massless \(W_0\) and \(h_0=-W_0\) conventions are retained.
No complete XI-E source verification is asserted.

\bibitem{BF}
R.~Brunetti and K.~Fredenhagen,
\emph{Microlocal Analysis and Interacting Quantum Field
Theories: Renormalization on Physical Backgrounds},
Communications in Mathematical Physics \textbf{208}
(2000), 623--661;
arXiv:math-ph/9903028.
Section 5.1 and Theorems 5.2--5.3 are used for point extensions
of finite scaling degree. The full interacting construction
is not imported.

\bibitem{FR}
K.~Fredenhagen and K.~Rejzner,
\emph{Perturbative algebraic quantum field theory},
arXiv:1208.1428v2.
Equation (82) fixes the adopted identical-edge factorial
convention. Sections 6--7 provide graph and extension
background; no complete time-ordered-product family is
transferred to the present nonlocal insertion.

\bibitem{BDHintro}
C.~Brouder, N.~V.~Dang, and F.~H\'elein,
\emph{A smooth introduction to the wavefront set},
Journal of Physics A: Mathematical and Theoretical
\textbf{47} (2014), 443001;
arXiv:1404.1778.
Theorem 13 and Proposition 26 supply the product criterion
and the Feynman wavefront description.
The Fourier-sign conversion is stated explicitly in
Lemma~\ref{lem:WF}.

\bibitem{BDHops}
C.~Brouder, N.~V.~Dang, and F.~H\'elein,
\emph{Continuity of the fundamental operations on distributions
having a specified wave front set
(with a counter example by Semyon Alesker)},
arXiv:1409.7662v2.
Sections 5--6 give the pullback, product, and proper-pushforward
interfaces used in the compressed root-attachment test.
No stronger unrestricted continuity theorem is asserted.

\end{thebibliography}

\end{document}