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% Revision scope:
% - Separates analytic, documentary, witness, and admission certificates.
% - Makes the global finite asymptotic inventory explicit.
% - Corrects the reference-agreement hypothesis for authorization.
% - Exposes the complete inverse-scale normal-derivative cancellation.
% - Proves conormal wavefront bounds for the evaluated coefficients.
% - Defines cone kernels at their tips by locally finite measures.
% - Repairs labels on formerly unnumbered displays.
% - Treats unsupplied or unaudited predecessor results as documentary
%   context, not as proof premises.
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% The evaluated application is a complete degenerate mathematical
% specialization, not the physically identified Einstein--scalar graph.
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\hypersetup{
  pdftitle={Em01-E: Complete-Contraction Decision Templates and Verification Certificates},
  pdfauthor={}
}

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    \dimexpr\linewidth-2\fboxsep-2\fboxrule\relax}
    \textbf{#1}\par\smallskip #2
  \end{minipage}}
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}

\title{\textbf{Em01-E}\\[0.4em]
Complete-Contraction Decision Templates\\
and Verification Certificates\\[0.4em]
\large Fixed-kernel convergence, raw-prescription failure,\\
and authorized characteristic extensions}
\author{}
\date{}

\begin{document}
\maketitle

\begin{abstract}
We formulate and prove coefficient-level decision theorems for
a fixed retained contraction whose crossed array is regularized
in its actual eight relative variables. A complete finite
power--logarithm expansion with a compact-uniform vanishing
remainder determines raw convergence, fixed-test raw failure,
and the mathematical finite part. The nonsummable coefficients
and finite part are unique. The principal theorem does not
require a homogeneous translation germ.

Three sufficient verification routes are distinguished:
scale-integrability, direct complete-family expansion, and a
logarithmically homogeneous translation-origin certificate.
Localization and Taylor estimates connect explicit calculations
to these routes. Physical identification, witness evaluation,
reference agreement, subtraction authorization, normalization,
and microlocal output are separate records.

An evaluated complete degenerate specialization of the retained
seven-row architecture is calculated on arbitrary compact joint
tests. Five rows vanish by its declared observations; both
remaining internal rows and all three vertex types are retained.
The apparent order \(\delta^{-2}\) vanishes, a nonzero order
\(\delta^{-1}\) survives, and the subtracted family converges
strongly with an order-six \(O(\delta)\) remainder. A further
normal-derivative cancellation is exposed in the complete
inverse-scale coefficient. The singular coefficient and finite
part have normal orders at most two and four, respectively,
and wavefront sets contained in the conormal bundle of their
smooth attachment. The finite part is independent of the
specified even factorized profile class.

The specialization is not identified with the physical
Einstein--scalar contraction. A physical identification ledger
and a six-task verification protocol state what is still
required for that application. No physical characteristic
decision, authorized physical counterterm, or quantum
admission is inferred from the mathematical example.
\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) {Complete expansion};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Degenerate specialization};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Physical Einstein--scalar graph};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {The specialization is not the physical graph.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{A degenerate decision is not the physical verdict. 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-Em01-E}
\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) {Complete expansion};
\node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {Physical Einstein--scalar graph};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {The specialization is not the physical graph.};
\draw[->] (a) -- (s);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{A degenerate decision is not the physical verdict. 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-Em01-E}
\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) {Complete expansion};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Degenerate specialization};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {Physical Einstein--scalar graph};
\node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {The specialization is not the physical 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-Em01-E}
\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) {Complete expansion};
\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) {Physical Einstein--scalar graph};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Degenerate specialization. The specialization is not the physical graph.};
\draw (a) -- (m);
\draw (m) -- (b);
\draw[->] (m) -- (s);
\end{tikzpicture}
\caption{Complete expansion is not Physical Einstein--scalar graph. 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-Em01-E}
\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: Complete expansion};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Degenerate specialization};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: Physical Einstein--scalar graph};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {The specialization is not the physical 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-Em01-E}
\end{figure}

Figure~\ref{fig:scope-Em01-E} records that separation. Figure~\ref{fig:split-Em01-E} records one view of the same separation. Figure~\ref{fig:chain-Em01-E} records one view of the same separation. Figure~\ref{fig:pair-Em01-E} records one view of the same separation. Figure~\ref{fig:gate-Em01-E} records one view of the same separation. The drawings add no hypothesis.

\scopebox{Scope of the mathematical result}{%
This paper contains proved decision templates and one evaluated
complete mathematical specialization. It does not fill missing
physical inputs with convenient substitutes.

\smallskip
A singular finite distribution is not itself an obstruction.
Raw-prescription failure means that a nonzero complete coefficient
multiplies a scale factor with no finite limit.}

\tableofcontents

%=====================================================================
\section{Problem, contribution, and logical dependencies}
\label{sec:introduction}
%=====================================================================

The fixed-kernel problem is
\begin{equation}
 t_\delta
 =
 i\Vertex\left(
 \Kret,\rho_\delta*(\chi C)
 \right),
 \qquad
 \rho_\delta(h)=\delta^{-8}\rho(h/\delta),
 \qquad \delta\downarrow0.
 \label{eq:target}
\end{equation}
The internal distribution is kept exact. Only the grouped
crossed array is smoothed.

The supplied Em01-B and Em01-C excerpts describe positive-scale
regularization, moderate growth, characteristic coordinates,
and an exact translation representation
\cite{Em01B,Em01C}. The supplied project proposal attributes a
homogeneous-origin route to Em01-D-v2. These are statements of
project provenance: no unprovided predecessor proof is needed
for the results below. All defining formulas and analytic
arguments used here are stated in this manuscript.

The scale-integrability and homogeneous-origin methods are
infrastructure, not claimed as new analytic discoveries.
The contribution is their coefficient-level organization,
the operational verification procedures, and the evaluated
complete specialization.

The principal deliverables are:
\begin{enumerate}[label=\textup{(\roman*)}]
\item the complete-expansion decision theorem
(Theorem~\ref{thm:decision});
\item the exact-family verification protocol and its local-to-global
requirements (Sections~\ref{sec:routes} and \ref{sec:protocol});
\item the evaluated expansion, fixed detecting test, and conormal
coefficient bounds (Section~\ref{sec:application}).
\end{enumerate}

\subsection{What the certificates establish}

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.25\linewidth}Y Y@{}}
\toprule
Certificate & Establishes & Does not establish\\
\midrule
Verified wavefront noncancellation &
Applicability of the ordinary-product construction, with
the required maps and support hypotheses &
Physical identification of that product.\\[0.35em]
Verified cancelling wavefront configuration &
Failure of the displayed sufficient product criterion &
Nonexistence of every alternative product or extension.\\[0.35em]
Moving high-frequency witness &
Behavior on that specified test family &
Fixed-test raw divergence or full-kernel reconstruction.\\[0.35em]
Complete controlled expansion &
A decision for the specified regularized family &
Authorization of its subtraction.\\[0.35em]
Microlocal extension certificate &
Extension and wavefront control under its stated hypotheses &
Physical normalization or quantum admission.\\
\bottomrule
\end{tabularx}
\end{center}

The ordinary multiplication theorem used here is sufficient,
not a converse nonexistence theorem
\cite[Theorem~6.1]{BDH}. Thus the appropriate conclusion from
a verified cancellation configuration is:
\begin{quote}
The displayed wavefront configuration prevents direct application
of the ordinary-product criterion; the complete contraction and
any alternative definition require separate analysis.
\end{quote}

The algebraic distinction between an open and a closed attachment
pattern can be seen from
\[
 (k,-k,0)+(0,l,-l)=0
 \quad\Longrightarrow\quad k=l=0,
\]
whereas
\[
 (k,-k,0)+(0,l,-l)+(p,0,-p)=0
 \quad\Longleftrightarrow\quad k=l=-p.
\]
Whether the latter nonzero covectors actually belong to the
specified kernels' wavefront sets depends on their prescriptions
and endpoint orientations. This algebra is not a calculation of
the complete interaction coefficient.

\subsection{Dependency map}

The analytic and identification records are parallel:
\[
\begin{array}{c}
\text{specified mathematical package}\\
\downarrow\\
\text{exact positive-scale family}\\
\downarrow\\
\text{complete estimate or expansion}\\
\downarrow\\
\text{raw decision and mathematical finite part}
\end{array}
\qquad
\begin{array}{c}
\text{documented physical data}\\
\downarrow\\
\text{identification with the mathematical package}\\
\downarrow\\
\text{source and operator-domain compatibility}\\
\downarrow\\
\text{physical interpretation of the analytic decision}.
\end{array}
\]
After these stages, reference agreement, authorized subtraction,
normalization, overlaps, residual projection, and the required
quantum carrier must still be checked.

Raw convergence, raw failure, and an authorized extension are not
a mutually exclusive trichotomy. Raw failure and an independently
authorized extension may coexist.

%=====================================================================
\section{Fixed package, tests, conventions, and smoothing}
\label{sec:package}
%=====================================================================

\subsection{Spaces and topologies}

Use
\[
 M=\R^{1+3},
 \qquad
 \eta=\operatorname{diag}(1,-1,-1,-1),
 \qquad
 w=(a,b,z,y)\in M^4.
\]
Let \(U\Subset M^4\) be open. Let \(E_{\mathrm{out}}\) be a
fixed finite-dimensional complex vector space and set
\[
 V_{\mathrm{out}}=U\times E_{\mathrm{out}}.
\]
All component pairings are complex-bilinear, with a fixed
trivialization and the displayed flat coordinate densities.

The test space is
\[
 \Tests=\Test(U;E_{\mathrm{out}}^*).
\]
For \(\supp\Phi\subset L\Subset U\), define
\begin{equation}
 \|\Phi\|_{m,L}
 =
 \max_{J,\ |\gamma|\le m}
 \sup_U|\partial^\gamma\Phi_J|.
 \label{eq:seminorm}
\end{equation}
No restriction is made to factorized tests, selected witness
families, or normal annuli.

Strong convergence in \(\Tests'\) means uniform convergence on
bounded subsets of \(\Tests\). A bounded subset has common compact
support and bounded derivative seminorms of every order.

When wavefront-constrained spaces are used, the normal topology
means the strong distributional seminorms together with the
localized Fourier seminorms outside the declared cone
\cite[Section~3]{BDH}.

For local coordinate Fourier transforms, fix
\[
 \widehat{\zeta u}(\xi)
 =
 \pair{u}{\zeta(w)e^{-i w\cdot\xi}},
\]
where \(w\cdot\xi\) is the coordinate covector pairing.
Minkowski index raising is separately performed with \(\eta\).
An identification with another convention must record the
corresponding sign conversion.

\subsection{The indexed vertex}

Let \(I\) run over a fixed finite internal component set and let
\[
 K^{I,J}\in\Dist(U).
\]
For a smooth symmetric array \(H_I^{\mu\nu}\) on \(\R^8\), define
\[
 \mathfrak f(w)=(y-a,y-b),
\]
\[
 Z_I^\nu(H)
 =
 \mathfrak f^*\left(
 \partial_{2,\mu}H_I^{\mu\nu}
 -
 \frac12\eta_{\mu\lambda}
              \partial_2^\nu H_I^{\mu\lambda}
 \right),
 \qquad
 M_I(H)=\mathfrak f^*(\eta_{\mu\nu}H_I^{\mu\nu}).
\]
Fix a smooth observation \(g\) and mass \(m\ge0\). Define
\begin{equation}
 \Vertex(K,H)^J
 =
 \partial_{y^\nu}\left[
 g(y)\sum_IK^{I,J}Z_I^\nu(H)
 \right]
 -
 \frac{m^2}{2}g(y)\sum_IK^{I,J}M_I(H).
 \label{eq:vertex}
\end{equation}
Its test action is
\begin{equation}
 \pair{i\Vertex(K,H)}{\Phi}
 =
 -i\sum_{I,J}
 \pair{K^{I,J}}{
 g(y)\left[
 Z_I^\nu(H)\partial_{y^\nu}\Phi_J
 +
 \frac{m^2}{2}M_I(H)\Phi_J
 \right]}.
 \label{eq:vertextest}
\end{equation}

The three vertex types are kinetic, trace kinetic, and mass.
The internal row \(K_{9,\mathrm{mass}}\) and the crossed-vertex
mass term are different entries.

\subsection{Fixed contraction package}

\begin{definition}[Fixed contraction package]
\label{def:package}
A package is
\begin{equation}
 \mathcal P
 =
 \left(
 U,V_{\mathrm{out}},\Kret,C,\Vertex,\rho,\chi,
 (\Vref,t^\circ),\Call,\Gamma_{\mathrm{out}}
 \right),
 \label{eq:package}
\end{equation}
with the following fixed data.
\begin{enumerate}[label=\textup{(\arabic*)}]
\item \(\Kret\) is an actual complete internal distribution,
with tensor slots and component duality specified.

\item The grouped crossed array is
\begin{equation}
 C_I^{\mu\nu}
 =
 H_{T,\gamma}^{\mu\nu}{}_I\otimes H_{T,\phi}
 -
 H_\gamma^{\mu\nu}{}_I\otimes H_\phi.
 \label{eq:crossed}
\end{equation}
Its four input distributions, relative arguments, prescriptions,
and tensor conventions are specified. No undeclared mixed
products are introduced.

\item The vertex is \eqref{eq:vertex}, including fixed \(m,g\).

\item \(\rho\in C_c^\infty(\R^8)\) is real and has integral one.
Nonnegativity is not required. The cutoff
\(\chi\in C_c^\infty(\R^8)\) equals one near
\(\mathfrak f(\overline U)\).

\item When a reference is available, \(\Vref\subset U\) is open
and \(t^\circ\in\Dist(\Vref;E_{\mathrm{out}})\) is specified.
Any claimed agreement with it requires a proof, not merely
its designation as a reference.

\item \(\Call\) specifies permitted scale-dependent subtractions,
finite additions, and normalization rules. These are not
inferred from the existence of a finite part.

\item A requested output operation includes its spaces, support
conditions, maps, residual bundle, and required closed
wavefront cone \(\Gamma_{\mathrm{out}}\).
\end{enumerate}
Every entry is independent of \(\delta\).
\end{definition}

Set
\[
 C_\delta=\rho_\delta*(\chi C),
 \qquad
 t_\delta=i\Vertex(\Kret,C_\delta).
\]
This is the fixed-kernel family also denoted
\(\mathsf F_\delta(\Kret)\), with \(C,\rho,\chi\) held fixed.

\begin{proposition}[Positive scale and intrinsicness]
\label{prop:positive}
For every \(\delta>0\), \(t_\delta\) is defined on all of
\(\Tests\). It is strongly smooth on compact positive-scale
intervals.

It depends on the actual internal distribution, not on a
presentation. In particular, reconstruction-null presentations
give the zero positive-scale family.
\end{proposition}

\begin{proof}
The convolution of the compact distribution \(\chi C\) with a
compact smooth profile is smooth. Formula \eqref{eq:vertextest}
therefore pairs each internal distribution with a smooth compact
test.

On a compact positive-scale interval, the profiles and all
scale derivatives have common compact support and bounded
test seminorms. Their convolutions have locally bounded smooth
seminorms. Local finite-order estimates for \(K\) justify
differentiation, uniformly on bounded joint test sets.

All operations start from the actual \(K\). A zero reconstructed
distribution has zero pairing with every resulting test.
\end{proof}

%=====================================================================
\section{Physical identification and documentary status}
\label{sec:identification}
%=====================================================================

The regularized mathematical family can exist when its
identification with a proposed physical construction remains
unproved. This distinction is part of the package, not an
analytic defect.

A physical identification record must identify the source,
propagators, observations, root operator, density derivative,
tensor duality, and correction rules. A required relation such as
\[
 F=E_m^{\mathrm A}f
\]
cannot be replaced by an independently convenient \(F\) while
retaining the same physical claim.

\subsection{Identification ledger}

For each row below, a completed physical record must give:
\begin{enumerate}[label=\textup{(\arabic*)}]
\item an exact defining equation or component formula;
\item a manuscript identifier, exact version, and passage;
\item its proof status and source/operator domain;
\item the map identifying it with the mathematical entry.
\end{enumerate}

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.22\linewidth}Y Y@{}}
\toprule
Entry & Mathematical specification & Physical verification required\\
\midrule
Source coefficients &
The actual finite array \(q_{\alpha\beta}^{L,I,J}\) in
Appendix~\ref{app:kernels} &
Full component formula, polarization, action--source sign,
and any preparation corrections.\\[0.35em]
Propagators and crossed lines &
\(R,B\), and the four inputs in \eqref{eq:crossed} &
Prescription, relative arguments, orientations, and domains;
no Hadamard/Feynman replacement without identification.\\[0.35em]
Observations &
The fixed \(f,g\) used in the internal rows and vertex &
Exact switch and observation data and their support rules.\\[0.35em]
Scalar root &
The fixed \(F\) in \(K_9\) &
The required root relation, including boundary/Cauchy data.\\[0.35em]
Density map &
The fixed \(\Dmap\) in \(K_6\) &
Derivation from the actual density and its slot conventions.\\[0.35em]
Tensor and Fourier conventions &
Fixed algebraic duality, \(\eta\), and Fourier sign &
An explicit conversion from the source conventions.\\[0.35em]
Corrections and normalization &
The class \(\Call\) &
Allowed contacts, support, coherence, normalization equations,
and overlap rules.\\
\bottomrule
\end{tabularx}
\end{center}

For the intended physical graph, these rows are not completed
by the supplied excerpts alone. For the mathematical application
below, they are replaced by explicit mathematical definitions,
and that replacement is stated openly.

\subsection{Predecessor claims are not silently imported}

The preparation materials contain excerpts labelled XI-A, XI-B,
XI-C, X-G, and XVI-A through later XVI identifiers, with some
claims depending on additional versions or unprovided sources.
Here they are documentary context, not independently verified
premises.

In particular, this paper does not assume that:
\begin{itemize}
\item a gauge-fixed free inverse establishes an invariant
sourced Einstein jet;
\item a classical tree construction establishes a closed
time-ordered contraction;
\item admission of a Wick insertion establishes admission of
its insertion--vertex product;
\item one witness contraction determines the complete kernel;
\item a source modification changes its free-shell class;
\item a selected two-edge sector is the full retained contraction.
\end{itemize}
Any future use of a predecessor theorem must first match its
actual hypotheses, definitions, version, and operation domain.

Registration and a retrieval restatement are not proof checks.
Conversely, the absence of registration is not a mathematical
nonexistence theorem.

%=====================================================================
\section{The complete-expansion decision theorem}
\label{sec:decision}
%=====================================================================

\begin{assumption}[Verified complete expansion]
\label{ass:expansion}
For one fixed package and profile,
\begin{equation}
 t_\delta=S_\delta+T+R_\delta,
 \label{eq:expansion}
\end{equation}
where
\begin{equation}
 S_\delta
 =
 \sum_{\lambda\in\mathcal E_-}
 \sum_{k=0}^{J_\lambda}
 \delta^\lambda(\log\delta)^kA_{\lambda,k}
 +
 \sum_{k=1}^{J_0}(\log\delta)^kA_{0,k}.
 \label{eq:singular}
\end{equation}
The following requirements hold.
\begin{enumerate}[label=\textup{(\arabic*)}]
\item \(\mathcal E_-\subset(-\infty,0)\) is finite, with distinct
real exponents. The exponent and logarithmic-degree inventory
is one fixed finite inventory on \(U\), independent of the
compact test support.

\item Every \(J_\lambda,J_0\) belongs to \(\Nzero\).

\item The coefficients and \(T\) are actual distributions in
\(\Tests'\), not unverified formal expressions.

\item For every \(L\Subset U\), there are \(m_L\in\Nzero\),
\(C_L>0\), and \(\omega_L(\delta)\ge0\), tending to zero, such that
\begin{equation}
 |R_\delta(\Phi)|
 \le C_L\omega_L(\delta)\|\Phi\|_{m_L,L},
 \qquad \supp\Phi\subset L,
 \label{eq:remainder}
\end{equation}
for all sufficiently small scales, uniformly over these tests.

\item The calculation proves equality for the complete
positive-scale family: every retained row, every vertex type,
both terms of \eqref{eq:crossed}, and every required localization
sector are included.
\end{enumerate}
Equal scalar scale factors have been collected.
\end{assumption}

\begin{theorem}[Complete-expansion decision]
\label{thm:decision}
Under Assumption~\ref{ass:expansion}:
\begin{enumerate}[label=\textup{(\roman*)}]
\item \(t_\delta-S_\delta\to T\) strongly in \(\Tests'\), with
the compact-local rate \eqref{eq:remainder};

\item the raw family converges strongly exactly when all
coefficients in \eqref{eq:singular} vanish;

\item if a complete nonsummable coefficient survives, there are
one fixed compact joint test \(\Phi_*\), an index
\((\lambda_*,k_*)\), and \(c\ne0\) such that
\begin{equation}
 t_\delta(\Phi_*)
 =
 c\,\delta^{\lambda_*}(\log\delta)^{k_*}(1+o(1)),
 \label{eq:dominant}
\end{equation}
where either \(\lambda_*<0\), or
\(\lambda_*=0,\ k_*\ge1\). Hence
\(|t_\delta(\Phi_*)|\to\infty\);

\item the nonsummable coefficients and \(T\) are unique in
this finite real power--logarithm class and intrinsic to
the actual positive-scale family.
\end{enumerate}
\end{theorem}

\begin{proof}
Every bounded test set has common compact support and bounded
\(\|\cdot\|_{m_L,L}\). Thus \eqref{eq:remainder} proves (i),
and proves raw convergence when all nonsummable coefficients
vanish.

Otherwise choose the smallest exponent having a nonzero
coefficient, treating the degree-zero logarithms as exponent
zero. Within that exponent choose the largest logarithmic degree
with a nonzero coefficient. Its distribution is detected by one
fixed compact test. Higher exponents are negligible relative to
its scalar factor, since positive powers dominate every fixed
power of a logarithm. Lower logarithmic degrees at the same
exponent are also negligible. The finite term and remainder
are negligible. This proves \eqref{eq:dominant}, and excludes
raw convergence even in the weak distributional sense.

For uniqueness, subtract two expansions and combine their finite
inventories. A nonzero nonsummable difference would have an
unbounded fixed pairing, while the difference of the finite terms
and remainders has a finite limit. Thus all nonsummable differences
vanish. Taking the limit identifies the finite distributions.
\end{proof}

\begin{corollary}[Raw-limit theorem]
\label{cor:rawlimit}
If all complete nonsummable coefficients vanish, then
\[
 t_\delta\longrightarrow T
 \quad\text{strongly in }\Dist(U;E_{\mathrm{out}}).
\]
This alone gives neither profile independence, global overlap
compatibility, a required wavefront cone, nor quantum admission.
\end{corollary}

\begin{corollary}[Raw-prescription failure theorem]
\label{cor:rawfailure}
A surviving complete nonsummable coefficient proves failure
of the specified unmodified family on one fixed test.
It does not prove failure of every extension or every other
construction.
\end{corollary}

\begin{corollary}[Reconstruction-safe coefficient extraction]
\label{cor:descent}
Presentations of the same actual internal distribution have
the same extracted coefficients and finite part whenever
Assumption~\ref{ass:expansion} holds. A reconstruction-null
presentation has all these coefficients equal to zero.
\end{corollary}

\begin{proof}
Their positive-scale families agree by
Proposition~\ref{prop:positive}. Apply uniqueness.
\end{proof}

\begin{remark}[Singular limits are allowed]
A finite limit \(a(\tau)\delta_N\), or a finite transverse
derivative of such a distribution, is compatible with raw
distributional convergence. A singular coefficient is an
obstruction here only when it is nonzero and multiplied by
one of the nonsummable scale factors in \(S_\delta\).
The finite part itself may be singular.
\end{remark}

%=====================================================================
\section{Three sufficient verification routes}
\label{sec:routes}
%=====================================================================

The routes below are alternatives. They do not classify every
possible scale behavior. Failure to verify one route does not
prove failure of the contraction.

\subsection{Route A: scale-integrability and Cauchy control}

\begin{theorem}[Scale-integrability route]
\label{thm:scale}
Suppose for every \(L\Subset U\),
\begin{equation}
 |(\delta\partial_\delta t_\delta)(\Phi)|
 \le C_L b_L(\delta)\|\Phi\|_{m_L,L},
 \qquad
 \int_0^{\delta_0}b_L(s)\frac{\dd s}{s}<\infty,
 \label{eq:Dini}
\end{equation}
with \(b_L\ge0\). Then \(t_\delta\to t_0\) strongly for
some distribution \(t_0\), and
\[
 |(t_\delta-t_0)(\Phi)|
 \le C_L\|\Phi\|_{m_L,L}
       \int_0^\delta b_L(s)\frac{\dd s}{s}.
\]
\end{theorem}

\begin{proof}
Positive-scale smoothness gives
\[
 (t_{\delta_2}-t_{\delta_1})(\Phi)
 =
 \int_{\delta_1}^{\delta_2}
 (s\partial_s t_s)(\Phi)\frac{\dd s}{s}.
\]
The estimate makes the pairings Cauchy uniformly on bounded
test sets. A fixed positive-scale member plus the integrable
increments supplies a finite-order local bound for the limit.
It is therefore a distribution. Letting \(\delta_1\downarrow0\)
gives the tail estimate.
\end{proof}

This is the scale criterion described in Em01-C
\cite{Em01C}. It can be preferable to coefficient extraction
when complete cancellation is estimated directly.

\begin{proposition}[Dyadic verification with interval filling]
\label{prop:dyadic}
Put \(\delta_n=2^{-n}\delta_0\). Suppose for each compact \(L\)
there are \(e_n,h_n\ge0\) such that
\begin{align*}
 |(t_{\delta_{n+1}}-t_{\delta_n})(\Phi)|
 &\le C_L e_n\|\Phi\|_{m_L,L},
 &\sum_ne_n&<\infty,\\
 \sup_{\delta_{n+1}\le\delta\le\delta_n}
 |(t_\delta-t_{\delta_n})(\Phi)|
 &\le C_L h_n\|\Phi\|_{m_L,L},
 &h_n&\longrightarrow0.
\end{align*}
Then the full family converges strongly. On that scale interval,
its error is bounded by
\[
 C_L\left(h_n+\sum_{j\ge n}e_j\right)\|\Phi\|_{m_L,L}.
\]
\end{proposition}

\begin{proof}
Sum the dyadic increments. Their finite-order bounds give
a distributional limit and uniform convergence on bounded
test sets along the dyadic sequence. The interval estimate
connects every intervening scale to that sequence.
\end{proof}

Dyadic convergence without interval control is not convergence
of the full family.

\subsection{Route B: direct expansion of the complete family}

Route B verifies Assumption~\ref{ass:expansion} directly from
an exact integral or defined distributional representation.
Changes of variables, derivative transposition, and tail
control must concern the actual smoothing.

\begin{lemma}[Weighted Taylor verification on a compact support]
\label{lem:weighted}
Fix \(L\Subset U\). Suppose the exact localized complete family
has a finite representation
\begin{equation}
\begin{split}
 t_\delta(\Phi)
 =
 \sum_\ell&
 \delta^{\beta_\ell}(\log\delta)^{k_\ell}
 \int_{D_\ell}a_\ell(z)\\
 &{}\times F_\ell[\Phi]\left(
 z,\delta^{w_{\ell,1}}x_{\ell,1}(z),\ldots,
 \delta^{w_{\ell,d_\ell}}x_{\ell,d_\ell}(z)
 \right)\dd z,
\end{split}
 \label{eq:weightedrep}
\end{equation}
where \(\beta_\ell\in\R\) and \(k_\ell\in\Nzero\).
Assume:
\begin{enumerate}[label=\textup{(\arabic*)}]
\item each \(D_\ell\) is a fixed compact integration domain,
\(a_\ell\) is integrable, and the \(x_{\ell,i}\) are bounded;
\item every \(w_{\ell,i}>0\);
\item \(F_\ell[\Phi]\) is linear in \(\Phi\), smooth in its
last variables on neighborhoods containing the Taylor
segments, and its derivatives through order \(N_\ell+1\)
are uniformly bounded by \(C_L\|\Phi\|_{m_L,L}\);
\item
\[
 \beta_\ell+w_{\ell,\min}(N_\ell+1)>0,
 \qquad
 w_{\ell,\min}=\min_iw_{\ell,i}.
\]
\end{enumerate}
Taylor expansion gives, on this fixed support, a finite
power--logarithm expansion with continuous coefficient
functionals and a remainder
\(C_L\delta^{a_L}\|\Phi\|_{m_L,L}\), for some \(a_L>0\).

A global expansion additionally requires compatible local
coefficient distributions and one common finite inventory.
\end{lemma}

\begin{proof}
The Taylor polynomial is
\[
 \sum_{|\gamma|\le N_\ell}
 \frac{\partial_x^\gamma F_\ell[\Phi](z,0)}{\gamma!}
 \prod_i
 \delta^{w_{\ell,i}\gamma_i}x_{\ell,i}(z)^{\gamma_i}.
\]
Its uniform remainder is bounded by
\[
 C_L\delta^{w_{\ell,\min}(N_\ell+1)}
       \|\Phi\|_{m_L,L}.
\]
Integration against \(a_\ell\) preserves the bound.
The prefactor leaves a positive remainder power, with at
most a fixed logarithm.

Collect negative exponents and degree-zero logarithms into
\(S_\delta\), degree-zero constants into \(T\), and positive
powers into the remainder. A smaller positive power absorbs
every finite logarithmic factor. The test-jet bounds give
coefficient continuity on the fixed-support test space.
\end{proof}

The lemma does not justify \eqref{eq:weightedrep}; obtaining
that exact representation is a separate task. It also does
not control omitted sectors.

\subsection{Route C: an exact translation-origin certificate}

For a compact joint test, define compact relative distributions
\[
 a_{I,\nu}[\Phi]
 =
 \mathfrak f_*\left(
 \sum_Jg(y)\partial_{y^\nu}\Phi_JK^{I,J}
 \right),
 \qquad
 d_I[\Phi]
 =
 \mathfrak f_*\left(
 \sum_Jg(y)\Phi_JK^{I,J}
 \right).
\]
Their pushforwards are defined because the distributions being
pushed forward are compactly supported. Set
\begin{align}
 \Lambda_{I,\mu\nu}[\Phi]
 ={}&
 \frac i2\left(
 \partial_{2,\mu}a_{I,\nu}[\Phi]
 +
 \partial_{2,\nu}a_{I,\mu}[\Phi]
 \right)
 \notag\\
 &-\frac i2\eta_{\mu\nu}
       \partial_2^\lambda a_{I,\lambda}[\Phi]
 -\frac{im^2}{2}\eta_{\mu\nu}d_I[\Phi].
 \label{eq:Lambda}
\end{align}
For \(\mathfrak d(x,v)=x-v\), put
\begin{equation}
 \Omega_\Phi
 =
 \sum_{I,\mu,\nu}
 \mathfrak d_*
 \left(
 \Lambda_{I,\mu\nu}[\Phi]\otimes\chi C_I^{\mu\nu}
 \right).
 \label{eq:Omega}
\end{equation}

\begin{proposition}[Exact translation and scale profiles]
\label{prop:translation}
The compact distribution \(\Omega_\Phi\) exists, and
\begin{equation}
 t_\delta(\Phi)=\Omega_\Phi(\rho_\delta).
 \label{eq:translation}
\end{equation}
For
\[
 \theta(h)=-8\rho(h)-h\cdot\nabla\rho(h),
\]
one has
\begin{equation}
 \int\theta=0,\qquad
 \delta\partial_\delta t_\delta(\Phi)
 =
 \Omega_\Phi(\theta_\delta).
 \label{eq:scaleprofile}
\end{equation}
These are complete-family identities and respect reconstruction.
\end{proposition}

\begin{proof}
Transposing the derivatives in \eqref{eq:vertextest} gives
\[
 \pair{i\Vertex(K,H)}{\Phi}
 =
 \sum_{I,\mu,\nu}
 \pair{\Lambda_{I,\mu\nu}[\Phi]}{H_I^{\mu\nu}}.
\]
The kinetic sign becomes \(+i\), the trace-kinetic sign
\(-i/2\), and the mass sign remains \(-im^2/2\).
Symmetry accounts for the first-line symmetrization.

Substitute \(H=\rho_\delta*(\chi C)\).
The tensor pairing is against \(\rho_\delta(x-v)\).
Both tensor factors are compactly supported, so the
pushforward in \eqref{eq:Omega} is defined and yields
\eqref{eq:translation}.

Differentiation gives
\(\delta\partial_\delta\rho_\delta=\theta_\delta\).
Integration of \(-\operatorname{div}(h\rho)\) gives
\(\int\theta=0\). Proposition~\ref{prop:positive} justifies
the scale identity. Every construction is linear in the
actual internal distribution.
\end{proof}

\begin{definition}[Logarithmically homogeneous chain]
For a distribution \(H\) on \(\R^8\), set
\[
 H(s\,\cdot)(\psi)=s^{-8}H(\psi(\,\cdot\,/s)).
\]
A chain \(H_0,\ldots,H_J\) of real degree \(\lambda\) satisfies
\[
 H_j(s\,\cdot)
 =
 s^\lambda
 \sum_{k=0}^j\frac{(\log s)^k}{k!}H_{j-k}.
\]
The chain consists of distributions defined at the origin as
well as away from it.
\end{definition}

\begin{proposition}[Homogeneous-origin route]
\label{prop:origin}
Suppose, on a common neighborhood of zero for tests supported
in each fixed \(L\),
\[
 \Omega_\Phi
 =
 \sum_{\lambda\in\mathcal E}
 H_{\lambda,J_\lambda}[\Phi]+W_\Phi,
 \qquad
 \mathcal E\subset(-\infty,0]\text{ finite}.
\]
Use one common finite inventory on \(U\). Assume:
\begin{enumerate}[label=\textup{(\arabic*)}]
\item the \(H_{\lambda,j}[\Phi]\) are linear, with local
finite-order bounds in the joint-test and translation-test
arguments, and form the displayed homogeneous chains;
\item \(W_\Phi\) is a continuous function near zero, linear in
\(\Phi\), and for some \(a_L>0\),
\[
 |W_\Phi(0)|+
 \sup_{0<|h|<r_L}
 \frac{|W_\Phi(h)-W_\Phi(0)|}{|h|^{a_L}}
 \le C_L\|\Phi\|_{m_L,L}.
\]
\end{enumerate}
Then Assumption~\ref{ass:expansion} follows with
\[
 A_{\lambda,k}(\Phi)
 =
 \frac1{k!}H_{\lambda,J_\lambda-k}[\Phi](\rho),
\]
and
\[
 T(\Phi)=W_\Phi(0)+H_{0,J_0}[\Phi](\rho),
\]
where the degree-zero term is omitted when absent.
The family remainder is \(O_L(\delta^{a_L})\).

A separately calculated logarithmic scale derivative of
the subtracted family has the same power bound.
\end{proposition}

\begin{proof}
For small scales the profile samples only the specified germ.
Each chain contributes its exact finite power--logarithm
polynomial. The continuous remainder contributes
\[
 W_\Phi(0)+
 \int\rho(h)\bigl(W_\Phi(\delta h)-W_\Phi(0)\bigr)\dd h.
\]
The H\"older estimate bounds the latter integral.

For the derivative statement use \eqref{eq:scaleprofile},
not differentiation of a remainder inequality. The chain
terms differentiate their exact polynomials, while the
remaining term is
\[
 \int\theta(h)\bigl(W_\Phi(\delta h)-W_\Phi(0)\bigr)\dd h.
\]
It has the same bound since \(\int\theta=0\).
\end{proof}

This route is stronger than the principal family-level
hypothesis. The application below verifies Route B directly;
no homogeneous-origin certificate is claimed for it.

%=====================================================================
\section{Localization and complete-sum verification}
\label{sec:protocol}
%=====================================================================

\subsection{Strictly separated characteristic coordinates}

For
\[
 \ell_{\mathbf n}=(1,\mathbf n),\qquad
 k_{\mathbf n}=(1,-\mathbf n),
\]
and an orthonormal spatial transverse frame \(E_A(\mathbf n)\),
use
\begin{equation}
 z=a+r,\qquad
 b=a+\beta\ell_{\mathbf n}+uk_{\mathbf n},\qquad
 y=a+\alpha\ell_{\mathbf n}+vk_{\mathbf n}+\xi^AE_A.
 \label{eq:geometry}
\end{equation}
Work on a sphere subchart with positive lower bounds on
\(\alpha,\beta-\alpha,\beta,\beta-u\). The aligned-null locus is
\[
 N=\{r=u=v=\xi=0\}.
\]
The exact cone equations and density are
\begin{align}
 q_R&=4\beta u,&
 q_1&=4\alpha v-|\xi|^2,&
 q_2&=4(\beta-\alpha)(u-v)-|\xi|^2,
 \label{eq:cones}\\
 \dd^4a\,\dd^4b\,\dd^4z\,\dd^4y
 &=
 4(\beta-u)^2
 \dd^4a\,\dd\Omega\,\dd\alpha\,\dd\beta\,
 \dd^4r\,\dd u\,\dd v\,\dd^2\xi.
 \label{eq:geometrydensity}
\end{align}

For \(s=b-a\), the inverse has
\[
 \mathbf n=\frac{\mathbf s}{|\mathbf s|},\qquad
 \beta=\frac{s^0+|\mathbf s|}{2},\qquad
 u=\frac{s^0-|\mathbf s|}{2}.
\]
For \(d=y-a\),
\[
 \alpha=\frac{d^0+\mathbf n\cdot\mathbf d}{2},\qquad
 v=\frac{d^0-\mathbf n\cdot\mathbf d}{2},\qquad
 \xi^A=\mathbf e_A\cdot\mathbf d,\qquad r=z-a.
\]
These formulas prove the local coordinate property.
The null-frame identities prove \eqref{eq:cones}.
The two endpoint Jacobians are \(2(\beta-u)^2\) and \(2\);
the full map is block triangular. This proves
\eqref{eq:geometrydensity} and compact-local derivative bounds.
The convention agrees with the supplied Em01-C coordinate
excerpt \cite{Em01C}.

Strict separation of \(a,y,b\) does not remove \(z=a\).
The supported incidence must remain in the complete calculation.

\subsection{The actual smoothing displacement}

Write
\[
 H_i=A_i\ell_{\mathbf n}+B_i k_{\mathbf n}+Z_i^AE_A,
 \qquad d=\beta-\alpha.
\]
For the true displacement \(\delta(H_1,H_2)\),
\begin{align}
 (y-a-\delta H_1)^2
 &=
 4(\alpha-\delta A_1)(v-\delta B_1)
 -
 |\xi-\delta Z_1|^2,
 \label{eq:shift1}\\
 (y-b-\delta H_2)^2
 &=
 4(d+\delta A_2)(u-v+\delta B_2)
 -
 |\xi-\delta Z_2|^2.
 \label{eq:shift2}
\end{align}
In particular,
\[
 (y-b-\delta H_2)^2
 =
 (b-y)^2+2\delta(b-y)\cdot H_2+\delta^2H_2^2.
\]
The sign must be converted when the second cone is written
in terms of \(b-y\).

The suggested core
\[
 u,v=O(\delta),\qquad \xi=O(\sqrt\delta)
\]
is not a complete localization proof. It supplies no common
scaling for \(r\), no tail estimate, and no physical
asymptotic coefficient by itself.

Nor is this weighted scaling automatically Dang's Euler
scaling. An Euler field acts as the identity on the normal
ideal modulo its square \cite[Section~1]{Dang}. On the
\((u,v,\xi)\) slice the weighted generator
\[
 u\partial_u+v\partial_v+\frac12\xi^A\partial_{\xi^A}
\]
sends \(\xi^A\) to \(\xi^A/2\), not to \(\xi^A\) modulo
quadratic terms. Applying an ordinary Euler-scaling extension
theorem therefore requires an additional transfer argument.

\subsection{The complete-sum calculation protocol}

\paragraph{P1: keep the full inventory.}
Retain
\[
 \{47,5+,5-,6,8,9_{\mathrm{der}},9_{\mathrm{mass}}\}
 \times
 \{\mathrm{kinetic},\mathrm{trace\ kinetic},\mathrm{mass}\},
\]
with both crossed terms. The row definitions are printed in
Appendix~\ref{app:kernels}.

\paragraph{P2: cover all sectors meeting the patch.}
Besides the rescaled core, account for the supported response
incidence, the \(b-z\) cone in \(K_{47}\), the \(a-z\) and
\(z-a\) cone tips in the two \(K_5\) placements, and the
complementary sectors on each compact joint support.
A core or annular calculation is not a complete estimate.

\paragraph{P3: standardize conventions.}
Use one tensor pairing, derivative placement, density,
relative-slot sign, and coefficient-distribution representation.
For example,
\[
 \partial_{a^\alpha}R(b-a)\deltaD^{(4)}(a-z)
\]
does not mean differentiation of the whole incidence product.

\paragraph{P4: calculate and collect before deciding.}
Produce exact local representations, extract candidate orders,
transpose to one common test-jet convention, and sum all
row--vertex contributions at each order.
A nonzero branch coefficient is not a complete obstruction.

\paragraph{P5: certify the complete remainder.}
State one finite derivative order on each fixed compact support.
Include the tails and any partition derivatives.
A leading algebraic zero without a vanishing complete
remainder does not prove convergence.

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.20\linewidth}Y Y@{}}
\toprule
Order & Coefficient deliverable & Estimate deliverable\\
\midrule
Negative power--logarithm &
An evaluated sum in a common distributional convention &
Controlled error after all contributions are collected.\\[0.35em]
Degree-zero logarithm &
The complete coefficient, with both crossed terms &
No unaccounted finite or tail contribution.\\[0.35em]
Finite order &
A continuous distribution \(T\), possibly singular &
A compact-uniform remainder tending to zero.\\
\bottomrule
\end{tabularx}
\end{center}

\subsection{Local assembly and global compatibility}

\begin{lemma}[Finite localization assembly]
\label{lem:assembly}
Let \(L\Subset U\), and let a fixed finite smooth partition
\(\sum_{\ell=1}^N\zeta_\ell=1\) hold near \(L\).
Suppose each exact localized complete family
\(t_\delta(\zeta_\ell\Phi)\) has an expansion with a
compact-uniform vanishing finite-order remainder.

Their sum, with equal factors collected, gives such an expansion
on tests supported in \(L\). Its inventory is finite on this
support.
\end{lemma}

\begin{proof}
Sum the exact localized identities. Multiplication by the fixed
cutoffs is bounded in every required test seminorm. Take the
maximum derivative order and sum the finitely many remainder
envelopes. The resulting envelope tends to zero.
\end{proof}

\begin{proposition}[Globalization with a common finite inventory]
\label{prop:global}
Let an open cover of \(U\) carry verified expansions of the
restricted actual family. Suppose all local expansions use one
common finite power--logarithm inventory, allowing zero
coefficients, and have compact-local bounds of the form
\eqref{eq:remainder}.

Then the local coefficients and finite parts agree on overlaps,
define distributions on \(U\), and give
Assumption~\ref{ass:expansion} globally.
\end{proposition}

\begin{proof}
On an overlap, uniqueness in Theorem~\ref{thm:decision}, applied
to compact tests there, identifies the coefficient restrictions.
The distributions therefore glue.

For any compact \(L\), select a finite subordinate localization
near \(L\). Lemma~\ref{lem:assembly} gives the local remainder
bound for the glued expansion. The common inventory is already
finite and independent of \(L\).
\end{proof}

Overlap uniqueness does not manufacture a common global finite
inventory from inventories that are only locally finite.
A scale-dependent partition also requires separate uniform
estimates.

%=====================================================================
\section{Witnesses, bounded sets, and six verification tasks}
\label{sec:tasks}
%=====================================================================

\subsection{Three distinct test certificates}

A verification record must distinguish:
\begin{enumerate}[label=\textup{(\arabic*)}]
\item \textbf{Fixed-test certificate:}
one \(\Phi_*\in\Tests\), independent of scale, with a proved
dominant asymptotic such as \eqref{eq:dominant}.

\item \textbf{Bounded-set certificate:}
a specified bounded set \(B\subset\Tests\) and uniform control
of \(\sup_{\Phi\in B}|(t_\delta-T)(\Phi)|\), or a proved
failure of that control.

\item \textbf{Moving-witness calculation:}
a family \(\Phi_\lambda\), with its support and derivative
growth recorded, and the behavior of its pairings.
\end{enumerate}

For example, if
\[
 \Phi_\lambda(w)=b(w)e^{i\lambda \xi\cdot w},
 \qquad b\in C_c^\infty(U),\quad \xi\ne0,
\]
then its derivatives have polynomial frequency growth.
For nonzero \(b\), differentiation in a direction pairing
nontrivially with \(\xi\) shows that this is not a bounded
test family.

The distinction is not cosmetic. On \(\R\), take
\(A=\partial_x\deltaD\), and \(\psi=1\) near zero.
The constant family \(t_\delta=A\) converges strongly, but
\[
 A(\psi e^{i\lambda x})=-i\lambda.
\]
Thus growth on an unbounded moving test family can coexist
with strong convergence of the distributional family.

Witness identities can check signs and selected coefficients.
A specified partial contraction is not automatically injective
on kernels: already a single nonzero test functional on
\(C_c^\infty\) has a nontrivial kernel. Adding a tensor product
with a function in that kernel preserves the partial evaluation
while changing the full kernel.

\subsection{Six verification tasks for a physical application}

These tasks do not add assumptions to the decision theorem.
They specify how to supply its existing hypotheses.

\begin{enumerate}[label=\textbf{V\arabic*.}]
\item \textbf{Logical certificate audit.}

\emph{Deliverable:} the table of
Section~\ref{sec:introduction}, with each claimed conclusion
assigned to its actual certificate.

\emph{Limit:} a failed sufficient product test is not a
nonexistence theorem.

\item \textbf{Physical identification freeze.}

\emph{Deliverable:} the completed ledger of
Section~\ref{sec:identification}, including exact formulas,
versions, domains, and proof status.

\emph{Acceptance condition:} no required physical entry is
replaced by a convenient independent mathematical choice.

\item \textbf{Narrow but complete localization.}

\emph{Deliverable:} an exact localized representation on
arbitrary joint tests in one strictly separated characteristic
patch.

\emph{Acceptance condition:} every sector meeting the patch
is included, including the incidence \(z=a\).

\item \textbf{Complete analytic calculation.}

\emph{Deliverable:} a complete expansion with a compact-uniform
vanishing remainder, or a complete scale-integrability estimate.

\emph{Limit:} Route C is not required if Route A or B suffices;
a branch calculation or moderate-growth bound is not a decision.

\item \textbf{Correct test certificate.}

\emph{Deliverable:} if divergence survives, one fixed compact
joint test with a justified dominant asymptotic. Record any
bounded-set or moving-witness claims separately.

\emph{Limit:} selected witness agreement is not full-kernel
reconstruction.

\item \textbf{Extension and admission audit.}

\emph{Deliverable:} reference agreement, subtraction support,
wavefront bounds, authorized corrections, normalization,
overlaps, and residual projection.

\emph{Limit:} a mathematical finite part is not automatically
an admitted quantum coefficient.
\end{enumerate}

The next physical milestone is deliberately narrow:
\begin{quote}
For one physically frozen package on one strictly separated
characteristic patch, derive an exact complete representation
and prove raw convergence or fixed-test raw divergence with
a compact-uniform remainder.
\end{quote}
The supplied excerpts do not complete that milestone.
The application below demonstrates the procedure for its own
explicit mathematical inputs.

%=====================================================================
\section{Evaluated complete degenerate specialization}
\label{sec:application}
%=====================================================================

\subsection{All inputs}

This section decides one complete family. It does not change
its density map to isolate a difference. Its degeneracy is
explicit: five internal rows and the second crossed tensor
product are identically zero.

Choose a symmetric-tensor basis with \(e_0=\eta\), completed
by the six off-diagonal symmetric tensors and the three
trace-free diagonal tensors
\[
 d_i{}_{\mu\nu}
 =
 \delta_{\mu0}\delta_{\nu0}
 +
 \delta_{\mu i}\delta_{\nu i},
 \qquad i=1,2,3.
\]
Use its algebraic dual. The residual fiber is \(\C^{10}\).

Fix
\[
 \ell=(1,0,0,1),\qquad
 k=(1,0,0,-1),\qquad
 \kappa(x)=x^0-x^3.
\]
Then
\[
 \kappa(\ell)=0,\qquad
 \kappa(k)=2,\qquad
 \kappa^\sharp=\ell.
\]

Let \(U\) be a sufficiently small relatively compact
neighborhood of
\[
 a_*=(3,0,0,0),\qquad
 b_*=a_*+4\ell,\qquad
 z_*=a_*,\qquad
 y_*=a_*+2\ell.
\]
Require disjoint closures of the \(b\)- and \(y\)-projections,
the time inequalities
\(b^0>y^0>a^0\), \(b^0>z^0\), and, in the fixed-frame
coordinates below,
\[
 \frac72<B<\frac92,\qquad
 \frac32<A<\frac52.
\]
Thus \(A<B\).

Choose fixed compact smooth \(\zeta_b,\zeta_y\), equal to
one near the respective endpoint projections, with disjoint
supports. Set
\begin{equation}
 m=1,\qquad
 f=0,\qquad
 F(x)=x^0\zeta_b(x),\qquad
 g(x)=\zeta_b(x)+x^0\zeta_y(x),\qquad
 \Dmap=0.
 \label{eq:appobservations}
\end{equation}
Hence \(F(b)=b^0\), \(g(b)=1\), and \(g(y)=y^0\) on
the relevant neighborhoods.

Define the quadratic source
\[
 \mathcal J[\gamma]=\frac12\gamma_0^2e_0.
\]
Its derivative is
\[
 \mathcal J'[G]h=G_0h_0e_0.
\]
The source array used here encodes this derivative, not its
half-polarization. Its sole nonzero coefficient is
\[
 q_{00}^{0,0,0}=1,
\]
with both derivative multiindices zero.

Choose the even normalized bump
\[
 p(s)=
 \begin{cases}
 c\exp\!\left(-\dfrac1{1-s^2}\right),& |s|<1,\\
 0,& |s|\ge1,
 \end{cases}
 \qquad \int p=1,
\]
and set
\[
 \psi=p,\qquad \omega(X)=p(X^1)p(X^2).
\]
For \(h=A\ell+uk+X\), \(X\in\R^2\), define
\begin{equation}
 \rho_4(h)=p(2u)\psi(A)\omega(X),
 \qquad
 \rho=\rho_4\otimes\rho_4.
 \label{eq:appprofile}
\end{equation}
Since \(\dd^4h=2\dd A\,\dd u\,\dd^2X\), these profiles
have integral one.

The crossed inputs are
\begin{align}
 H_{T,\gamma}^{\mu\nu}{}_I(x)
 &=\delta_{I0}\eta^{\mu\nu}\deltaD(\kappa(x)),
 &
 H_{T,\phi}(x)&=\deltaD(\kappa(x)),
 \notag\\
 H_\gamma&=0,
 &
 H_\phi&=0.
 \label{eq:applines}
\end{align}
Fix any compact \(\chi=1\) near
\(\mathfrak f(\overline U)\).

Define the smooth attachment and reference domain by
\begin{equation}
\begin{split}
 S_\kappa
 &=
 \{z=a,\ b=a+B\ell,\ y=a+A\ell+Y\}\cap U,
 \qquad Y\in\R^2,\\
 \Vref&=U\setminus S_\kappa.
\end{split}
 \label{eq:appsupport}
\end{equation}
The reference is zero, as proved below.
Set \(\Call=\{0\}\): no nonzero scale-dependent subtraction
or finite addition is authorized for this package.
No restrictive quantum output condition is imposed.

\begin{remark}[Identification fails by design]
This is not the physical Einstein--scalar source or crossed
prescription. In particular, its independently specified
\(F(b)=b^0\) is not \(E_m^{\mathrm A}f\), since \(f=0\).
The specialization is complete for its declared data, but
does not exercise a nonzero seven-row physical graph.
\end{remark}

\subsection{Complete internal and vertex ledger}

The tensor maps of Appendix~\ref{app:kernels} give
\[
 \PP e_0=-e_0,\qquad
 \Amap^{\mu\nu,0}=\eta^{\mu\nu},\qquad
 \bmap^0=2.
\]
Put
\[
 D(w)=R(b-a)\deltaD^{(4)}(a-z).
\]
The source response is \(T^{0,0,0}=-D\). Therefore
\begin{equation}
 K_{9,\mathrm{der}}^{0,0}=\partial_{b^0}D,
 \qquad
 K_{9,\mathrm{mass}}^{0,0}=2b^0D,
 \label{eq:appK}
\end{equation}
and all other components vanish. The first five internal
rows vanish exactly because \(f=0\). Thus
\[
 \Kret^{0,0}=\partial_{b^0}D+2b^0D.
\]

The actual convolution has the exact local value
\begin{equation}
 (C_\delta)_I^{\mu\nu}(x_1,x_2)
 =
 \delta_{I0}\eta^{\mu\nu}\delta^{-2}
 p\!\left(\frac{\kappa(x_1)}{\delta}\right)
 p\!\left(\frac{\kappa(x_2)}{\delta}\right).
 \label{eq:appC}
\end{equation}
Indeed the \(\kappa\)-marginal of \(\rho_4\) is \(p\).
The product profile gives the two four-variable convolutions.
For sufficiently small scales, the cutoff is invisible
near \(\mathfrak f(\overline U)\), including for derivatives.

Set
\[
 s=\kappa(y-a),\qquad t=\kappa(b-a),
\]
\[
 A_\delta=\delta^{-3}p(s/\delta)p'((s-t)/\delta),
 \qquad
 E_\delta=\delta^{-2}p(s/\delta)p((s-t)/\delta).
\]
The kinetic crossed term is \(+\ell^\nu A_\delta\), the
trace-kinetic term is \(-2\ell^\nu A_\delta\), and the
trace is \(4E_\delta\).

The \(\ell\)-derivative annihilates both profile arguments.
Here \(K\) is independent of \(y\), and
\(\ell^\nu\partial_\nu g(y)=1\). Consequently
\[
 t_\delta=-i\Kret^{0,0}A_\delta
          -2iy^0\Kret^{0,0}E_\delta.
\]

All six nonzero row--vertex cells are:
\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.23\linewidth}p{.17\linewidth}Y@{}}
\toprule
Internal row & Vertex type & Exact transposed contribution\\
\midrule
\(9_{\mathrm{der}}\) & Kinetic &
\(-i\langle D,\partial_{b^0}(A_\delta\Phi_0)\rangle\).\\
\(9_{\mathrm{der}}\) & Trace kinetic &
\(2i\langle D,\partial_{b^0}(A_\delta\Phi_0)\rangle\).\\
\(9_{\mathrm{der}}\) & Mass &
\(2i\langle D,y^0\partial_{b^0}(E_\delta\Phi_0)\rangle\).\\
\(9_{\mathrm{mass}}\) & Kinetic &
\(2i\langle D,b^0A_\delta\Phi_0\rangle\).\\
\(9_{\mathrm{mass}}\) & Trace kinetic &
\(-4i\langle D,b^0A_\delta\Phi_0\rangle\).\\
\(9_{\mathrm{mass}}\) & Mass &
\(-4i\langle D,b^0y^0E_\delta\Phi_0\rangle\).\\
\bottomrule
\end{tabularx}
\end{center}

The other fifteen cells vanish by the fixed data \(f=0\).
The second crossed term vanishes by \eqref{eq:applines}.
Neither vanishing is an asymptotic omission.

Expanding the \(b^0\)-derivatives gives
\begin{equation}
\begin{split}
 t_\delta(\Phi)=-i\Big\langle D,\ &
 \delta^{-4}p(s/\delta)p''((s-t)/\delta)\Phi_0\\
 &+\delta^{-3}p(s/\delta)p'((s-t)/\delta)
 \left[-\partial_{b^0}\Phi_0+2(b^0+y^0)\Phi_0\right]\\
 &-2\delta^{-2}p(s/\delta)p((s-t)/\delta)y^0
 \left[\partial_{b^0}\Phi_0-2b^0\Phi_0\right]
 \Big\rangle .
\end{split}
 \label{eq:appexact}
\end{equation}

\subsection{Exact integral and density}

Use the fixed-frame coordinates
\[
 b=a+B\ell+uk+X,\qquad
 y=a+A\ell+vk+Y,\qquad
 z=a+r.
\]
Write
\[
 \tau=(a,B,A,Y),\qquad
 \nu=(r,u,v,X),\qquad
 \dd\tau=\dd^4a\,\dd B\,\dd A\,\dd^2Y.
\]
This linear coordinate system has constant absolute Jacobian
four. The attachment is \(\nu=0\), of codimension eight.
Moreover,
\[
 (b-a)^2=4Bu-|X|^2,\qquad s=2v,\qquad t=2u.
\]

Define ambient test expressions
\begin{align*}
 \mathcal F_0[\Phi]&=\Phi_0,\\
 \mathcal F_1[\Phi]&=-\partial_{b^0}\Phi_0
                         +2(b^0+y^0)\Phi_0,\\
 \mathcal F_2[\Phi]&=-2y^0
                         \left(\partial_{b^0}\Phi_0-2b^0\Phi_0\right),
\end{align*}
and then evaluate them:
\begin{equation}
 F_j(\tau;u,v,X)
 =
 \mathcal F_j[\Phi]
 \left(a,a+B\ell+uk+X,a,a+A\ell+vk+Y\right).
 \label{eq:appF}
\end{equation}
Ambient derivatives are taken before evaluation. Tests are
extended by zero outside their compact support in \(U\).

\begin{lemma}[Exact rescaled complete integral]
\label{lem:appintegral}
For all sufficiently small scales,
\begin{equation}
\begin{split}
 t_\delta(\Phi)
 =-\frac i2\sum_{j=0}^2\delta^{j-2}
 &\int\dd\tau\int_\R\dd S\int_0^\infty\dd T\
 p(S)p^{(2-j)}(S-T)\\
 &{}\times\frac1{2\pi}\int_0^{2\pi}
 F_j\left(
 \tau;\frac{\delta T}{2},\frac{\delta S}{2},
 \sqrt{2B\delta T}\,e_\theta
 \right)\dd\theta ,
\end{split}
 \label{eq:appintegral}
\end{equation}
where \(e_\theta=(\cos\theta,\sin\theta)\). The tangential
integration is over the fixed coordinate range of the patch;
zero extension of the tests makes its relevant support compact.
\end{lemma}

\begin{proof}
Integrate \(z\) against the incidence. The endpoint densities
are
\[
 \dd^4b=2\dd B\,\dd u\,\dd^2X,\qquad
 \dd^4y=2\dd A\,\dd v\,\dd^2Y.
\]
On the patch the future retarded kernel is
\[
 \frac1{2\pi}\deltaD(4Bu-|X|^2).
\]
Its \(u\)-integration contributes \(1/(4B)\). The combined
factor is therefore
\[
 4\cdot\frac1{2\pi}\cdot\frac1{4B}
 =
 \frac1{2\pi B}.
\]

With angular averaging and \(Q_X=|X|^2\),
\(\dd^2X=\pi\dd Q_X\). Set \(t=Q_X/(2B)\) and \(s=2v\).
The resulting measure factor is exactly
\(\dd s\,\dd t/2\). Finally \(s=\delta S,t=\delta T\)
contributes \(\delta^2\). Substitution in
\eqref{eq:appexact} proves the identity.
\end{proof}

There are no omitted scale sectors for this specialization.
The exact profile support gives
\[
 |S|\le1,\qquad |S-T|\le1,\qquad 0\le T\le2.
\]
The rescaled integration domain is fixed and compact.
For fixed compact test support, all relevant tangential
parameters stay in a fixed compact range. The identically
zero rows require no further cone-tip calculation for these
declared data.

\subsection{Angular Taylor expansion and profile moments}

For \(n\ge0\), define
\begin{equation}
 \mathcal L_n(B,S,T)
 =
 \sum_{\substack{a+b+c=n\\a,b,c\ge0}}
 \frac{(T/2)^a(S/2)^b(BT/2)^c}
      {a!\,b!\,(c!)^2}
 \partial_u^a\partial_v^b\Delta_X^c.
 \label{eq:appLn}
\end{equation}
A subscript \(0\) below means \(u=v=X=0\).

\begin{lemma}[Uniform angular Taylor expansion]
\label{lem:angular}
For \(M\in\Nzero\), on the compact parameter ranges above,
\begin{equation}
\begin{split}
 &\frac1{2\pi}\int_0^{2\pi}
 F\left(
 \frac{\delta T}{2},\frac{\delta S}{2},
 \sqrt{2B\delta T}\,e_\theta
 \right)\dd\theta\\
 &\qquad=
 \sum_{n=0}^M\delta^n[\mathcal L_n(B,S,T)F]_0
 +
 O\left(\delta^{M+1}\|F\|_{2M+2}\right),
\end{split}
 \label{eq:angular}
\end{equation}
uniformly for smooth \(F\) with the displayed derivative bound
on a fixed neighborhood of the Taylor segments.
\end{lemma}

\begin{proof}
Taylor-expand in \(X\) through order \(2M+1\).
Odd angular moments vanish. At degree \(2c\), the averaged
term is
\[
 \frac{(R^2/4)^c}{(c!)^2}\Delta_X^cF,
 \qquad R^2=2B\delta T.
\]
This follows by integration of the even monomials on the
unit circle. Expand its coefficient in \((u,v)\) through
degree \(M-c\). Collecting \(a+b+c=n\) gives
\eqref{eq:appLn}.

The transverse remainder is \(O(R^{2M+2})\).
Each longitudinal remainder is \(O(\delta^{M-c+1})\)
multiplied by \(\delta^c\). Thus every error is
\(O(\delta^{M+1})\). The indicated \(2M+2\) derivative
bound controls all these terms.
\end{proof}

Put
\[
 Q_p=\int_\R p(S)^2\dd S>0,\qquad
 P(S)=\int_{-\infty}^S p(V)\dd V,
\]
and
\[
 M_{d,a,b}
 =
 \int_\R\int_0^\infty
 S^aT^bp(S)p^{(d)}(S-T)\dd T\,\dd S.
\]
For the required moments, integration by parts gives
\[
 M_{d,a,b}
 =
 b!\int S^ap(S)p^{(d-b-1)}(S)\dd S
 \quad (b<d),
\]
and
\[
 M_{d,a,d}
 =
 d!\int S^ap(S)P(S)\dd S.
\]
The latter includes \(d=0\).

Evenness and normalization evaluate all moments through
finite order:
\begin{center}
\small
\begin{tabular}{@{}cccccc@{}}
\toprule
\(M_{2,0,0}\) & \(M_{2,0,1}\) & \(M_{2,1,0}\) &
\(M_{2,0,2}\) & \(M_{2,1,1}\) & \(M_{2,2,0}\)\\
\midrule
\(0\) & \(Q_p\) & \(-Q_p/2\) & \(1\) & \(0\) & \(0\)\\
\bottomrule
\end{tabular}

\medskip
\begin{tabular}{@{}cccc@{}}
\toprule
\(M_{1,0,0}\) & \(M_{1,0,1}\) &
\(M_{1,1,0}\) & \(M_{0,0,0}\)\\
\midrule
\(Q_p\) & \(1/2\) & \(0\) & \(1/2\)\\
\bottomrule
\end{tabular}
\end{center}
For example,
\[
 \int pp'=0,\qquad
 \int Spp'=-Q_p/2,\qquad
 \int pP=1/2.
\]
The other zero moments are odd integrals.
No two-variable moment needed for the expansion is left
unevaluated.

Define
\begin{equation}
 \mathcal A(\Phi)
 =
 -\frac{iQ_p}{2}
 \int\dd\tau
 \left[
 \frac12\partial_uF_0
 -\frac14\partial_vF_0
 +\frac B2\Delta_XF_0+F_1
 \right]_0,
 \label{eq:appA}
\end{equation}
and
\begin{equation}
\begin{split}
 \mathcal T(\Phi)
 =-\frac i2\int\dd\tau\Bigg[
 &\left(
 \frac18\partial_u^2+
 \frac B4\partial_u\Delta_X+
 \frac{B^2}{16}\Delta_X^2
 \right)F_0\\
 &+\left(\frac14\partial_u+\frac B4\Delta_X\right)F_1
 +\frac12F_2
 \Bigg]_0.
\end{split}
 \label{eq:appT}
\end{equation}

\begin{theorem}[Evaluated complete-family decision]
\label{thm:application}
For the fixed specialization above,
\begin{equation}
 \boxed{
 t_\delta=\delta^{-1}\mathcal A+\mathcal T+\mathcal R_\delta,
 \qquad
 |\mathcal R_\delta(\Phi)|
 \le C_L\delta\|\Phi\|_{6,L}.
 }
 \label{eq:appdecision}
\end{equation}
The bound holds uniformly on arbitrary tests supported in
each fixed \(L\Subset U\), for all sufficiently small scales.

The apparent \(\delta^{-2}\) coefficient is zero.
The complete inverse-scale coefficient is
\eqref{eq:appA}, and the finite part is \eqref{eq:appT}.
Both are supported in \(S_\kappa\), and
\(\mathcal A\ne0\).

Consequently one fixed compact joint test has an unbounded
raw pairing, whereas
\[
 t_\delta-\delta^{-1}\mathcal A
 \longrightarrow\mathcal T
\]
strongly, with the displayed order-six \(O(\delta)\) rate.
\end{theorem}

\begin{proof}
Apply Lemma~\ref{lem:angular} to the \(j\)-term of
\eqref{eq:appintegral}, with \(M=2-j\).
The Taylor error is
\[
 O\left(\delta^{3-j}\|F_j\|_{6-2j}\right).
\]
Multiplication by \(\delta^{j-2}\) leaves \(O(\delta)\).
Here \(F_0\) contains no ambient test derivative, while
\(F_1,F_2\) contain at most one. Their coefficients and
coordinate derivatives are bounded on the relevant compact
range. Hence ambient test order six controls every remainder.
The fixed compact integration ranges preserve uniformity.

At order \(\delta^{-2}\), only \(j=0,n=0\) contributes.
Its moment is \(M_{2,0,0}=0\).
At order \(\delta^{-1}\), the pairs \((j,n)=(0,1),(1,0)\)
give \eqref{eq:appA}. At finite order, the pairs
\((0,2),(1,1),(2,0)\) give \eqref{eq:appT}.

The formulas are continuous finite-jet functionals along
\(S_\kappa\), so they define distributions supported there.

For nonvanishing, choose nonnegative nonzero
\(\Psi(\tau)\) compactly supported in a small tangential
subchart. Choose a compact normal cutoff equal to one near
\(\nu=0\), and let \(\Phi_{*,0}\) be their product, with
all other components zero. The support can be chosen inside
\(U\). Along the attachment,
\[
 \partial_uF_0=\partial_vF_0=\Delta_XF_0=0,
 \qquad
 \partial_{b^0}\Phi_{*,0}=\frac12\partial_B\Psi,
\]
because
\(\partial_{b^0}=(\partial_B+\partial_u)/2\).
Tangential integration kills \(\partial_B\Psi\). Therefore
\begin{equation}
 \mathcal A(\Phi_*)
 =
 -iQ_p\int
 (2a^0+A+B)\Psi(\tau)\dd\tau\ne0.
 \label{eq:detect}
\end{equation}
The coefficient in parentheses is positive on the patch.

Thus, with \(c=\mathcal A(\Phi_*)\ne0\),
\[
 t_\delta(\Phi_*)
 =
 c\,\delta^{-1}+\mathcal T(\Phi_*)+O(\delta).
\]
The complete-expansion theorem gives the claimed raw failure
and strong subtracted convergence.
\end{proof}

\subsection{An additional complete-coefficient cancellation}

\begin{corollary}[Cancellation of the inverse-scale \(u\)-jet]
\label{cor:appcancel}
Write \(\varphi(\tau;u,v,X)=F_0(\tau;u,v,X)\).
Then
\begin{equation}
 \boxed{
 \mathcal A(\Phi)
 =
 -\frac{iQ_p}{2}\int\dd\tau
 \left[
 -\frac14\partial_v\varphi
 +\frac B2\Delta_X\varphi
 +2(2a^0+A+B)\varphi
 \right]_0 .
 }
 \label{eq:appAsimple}
\end{equation}
The normal derivative \(\partial_u\varphi\) cancels in the
complete inverse-scale coefficient. This cancellation does
not remove its nonzero zeroth-order term.
\end{corollary}

\begin{proof}
The fixed-frame identity gives
\[
 F_1
 =
 -\frac12(\partial_B+\partial_u)\varphi
 +2(b^0+y^0)\varphi.
\]
Thus \(\frac12\partial_uF_0+F_1\) equals
\[
 -\frac12\partial_B\varphi+2(b^0+y^0)\varphi.
\]
The first term integrates to zero in \(B\), and on the
attachment \(b^0+y^0=2a^0+A+B\).
\end{proof}

\subsection{Normal order and conormal wavefront}

Let \(\delta_{S_\kappa}\) use the tangential density \(\dd\tau\):
\[
 \delta_{S_\kappa}(\Phi)=\int\Phi(\tau,0)\dd\tau
\]
in the relevant component. This convention specifies the
attachment density; it is not an implicit ambient-density
normalization.

\begin{proposition}[Coefficient normal jets and wavefront bounds]
\label{prop:appWF}
Locally, \(\mathcal A\) and \(\mathcal T\) are finite normal
jets along \(S_\kappa\), with smooth tangential coefficients.
Their normal orders are at most two and four, respectively.
Moreover,
\[
 \WF(\mathcal A)\subset N^*S_\kappa,
 \qquad
 \WF(\mathcal T)\subset N^*S_\kappa,
\]
where the zero section is omitted from \(N^*S_\kappa\).
These conclusions do not assert normal-topology convergence
of the subtracted family.
\end{proposition}

\begin{proof}
Formula \eqref{eq:appAsimple} involves only the normal jets
\(\partial_v\varphi,\Delta_X\varphi,\varphi\), with smooth
coefficients. Its normal order is at most two.

In \eqref{eq:appT}, the only tangential test derivatives arise
from
\(\partial_{b^0}=(\partial_B+\partial_u)/2\) in \(F_1,F_2\).
Integrate all \(\partial_B\)-terms by parts. Their coefficients
are smooth. The remaining derivatives are normal, with order
at most four. Thus both distributions have local representations
\[
 \sum_{|\gamma|\le m}
 c_\gamma(\tau)\partial_\nu^\gamma\deltaD(\nu),
 \qquad c_\gamma\in C^\infty,
\]
with \(m=2\) or \(4\), and the specified tangential density.

For completeness, localize this representation by a compact
smooth cutoff. Multiplication by the cutoff produces another
finite normal jet with compact smooth tangential coefficients.
Its Fourier transform is a finite sum
\[
 (i\xi_\nu)^\gamma\widehat{c_\gamma}(\xi_\tau).
\]
On a closed cone separated from
\(\{\xi_\tau=0,\ \xi_\nu\ne0\}\),
\(|\xi_\tau|\) is bounded below by a positive multiple of the
total frequency. Rapid tangential Fourier decay dominates the
finite normal polynomial. This proves the stated conormal
containment. The coordinate change here is linear, so it
transports the bound directly to the original joint variables.

This describes the coefficients, not uniform Fourier estimates
for the approximating distributions.
\end{proof}

The smooth-coefficient normal-jet/conormal relation is also
described in Dang's supported-distribution analysis
\cite[Lemma~8.1]{Dang}. The direct proof above is sufficient
for this application.

\subsection{Reference, profile class, and authorization}

\begin{proposition}[Application reference and profile comparison]
\label{prop:appconsequences}
For this specialization:
\begin{enumerate}[label=\textup{(\roman*)}]
\item \(t_\delta\) is eventually zero on each compact subset
of \(U\setminus S_\kappa\), so the proved reference there is zero;

\item replacing \(p,\psi,\omega\) by fixed compact smooth
normalized profiles of the same factorized form, using one
common even nonnegative longitudinal \(p\) for the two
four-variable factors, preserves \(\mathcal T\);

\item the inverse-scale coefficient depends on
\(Q_p=\int p^2\), and is not profile independent;

\item \(\delta^{-1}\mathcal A\) is a mathematical subtraction
but is not authorized by the declared class \(\Call=\{0\}\).
\end{enumerate}
No uniformity over all profiles is asserted.
\end{proposition}

\begin{proof}
The support of the exact expression satisfies
\[
 r=0,\qquad |v|=O(\delta),\qquad
 0\le u=O(\delta),\qquad |X|=O(\sqrt\delta).
\]
All tangential parameters stay in compact positive ranges.
Thus the support approaches \(S_\kappa\) uniformly and
eventually misses every compact set in its complement.
Distributional differentiation does not enlarge support.

The finite-part moment values are \(1,1/2,0\), independent
of the even normalized \(p\). The other factors affect the
convolution only through their normalized marginals.
For a different fixed compact longitudinal support, the
rescaled integration range remains compact, although its
size and the remainder constants may change.

The inverse-scale formula contains \(Q_p\); normalized even
profile dilations change this number. Since
\(\mathcal A\ne0\), the proposed subtraction is nonzero
and is not in \(\{0\}\).
\end{proof}

The attachment \(S_\kappa\) is not the physical aligned-null
locus \(N\): its transverse variable \(Y\) need not vanish.
The correction class forbids this subtracted construction,
but does not exclude independently defined extensions of
the zero reference, such as the zero distribution.

%=====================================================================
\section{References, profiles, and authorized extensions}
\label{sec:extensions}
%=====================================================================

\subsection{Reference-dependent support}

\begin{proposition}[Support and reference agreement]
\label{prop:support}
Suppose Assumption~\ref{ass:expansion} holds and the complete
raw family converges distributionally to \(t^\circ\) on
\(\Vref\). Then every nonsummable coefficient vanishes there,
\[
 \supp A_{\lambda,k}\subset U\setminus\Vref,
\]
including the degree-zero logarithmic coefficients, and
\[
 T|_{\Vref}=t^\circ.
\]
\end{proposition}

\begin{proof}
A nonzero restricted nonsummable coefficient would produce
an unbounded fixed pairing supported in \(\Vref\), contradicting
the proved reference convergence. After these coefficients
vanish, the remainder bound identifies the restricted limit.
\end{proof}

Support only in \(N\) requires a complete reference throughout
\(U\setminus N\), or a direct support proof. A reference for a
supported sub-sum is not a reference for the full retained sum.

\subsection{Cutoffs and profiles}

\begin{proposition}[Cutoff locality]
\label{prop:cutoff}
If two compact cutoffs equal one near \(\mathfrak f(L)\),
their families have identical pairings on tests supported in
\(L\) at sufficiently small scales. Whenever expansions exist,
their coefficient restrictions and finite parts agree on \(L\).
\end{proposition}

\begin{proof}
Compact profile support implies that the convolution and its
required derivatives at \(\mathfrak f(L)\) sample only a small
neighborhood of that set. The cutoff difference is zero there.
Then use coefficient uniqueness.
\end{proof}

One-profile convergence does not imply profile independence.
For two verified expansions, compare their finite distributions
or prove convergence to zero of the difference of their
subtracted families.

Under Route C the comparison is explicitly
\[
 A_{\lambda,k}^{\rho}-A_{\lambda,k}^{\widetilde\rho}
 =
 \frac1{k!}
 H_{\lambda,J_\lambda-k}[\cdot](\rho-\widetilde\rho),
\]
\[
 T^\rho-T^{\widetilde\rho}
 =
 H_{0,J_0}[\cdot](\rho-\widetilde\rho),
\]
with the last expression zero when the degree-zero chain is absent.
The application proves only its stated factorized-profile result,
not independence for arbitrary eight-variable profiles.

\subsection{Authorization and normalization}

Let \(Z=U\setminus\Vref\).
Specify:
\begin{itemize}
\item a class \(\mathcal B\) of authorized scale-dependent
subtractions;
\item a vector space
\(\mathcal A\subset\Dist_Z(U;E_{\mathrm{out}})\) of permitted
scale-independent additions;
\item the required tensor, symmetry, growth, overlap, and
microlocal conditions;
\item a linear normalization map \(\mathcal N\), defined on
the vector space containing \(T+\mathcal A\), and a target \(n_*\).
\end{itemize}

Here a ``finite addition'' is scale independent; it need not
belong to a finite-dimensional vector space. A transverse-order
bound limits the normal derivative types, not the dimension over
\(\C\) of the coefficient freedom. Already
\(a(\tau)\delta_S\), with arbitrary compact smooth \(a\), gives
infinite-dimensional freedom at normal order zero.

\begin{theorem}[Authorized-extension theorem]
\label{thm:authorized}
Suppose:
\begin{enumerate}[label=\textup{(\arabic*)}]
\item Assumption~\ref{ass:expansion} is verified;
\item the raw family has proved reference convergence
\(t_\delta|_{\Vref}\to t^\circ\);
\item \(S_\delta\in\mathcal B\), with the authorized subtraction
sign \(t_\delta-S_\delta\);
\item \(T\) satisfies the required tensor, growth, support,
microlocal, and overlap conditions;
\item additions from \(\mathcal A\) preserve those conditions.
\end{enumerate}
Then \(T\) is an authorized subtracted extension of \(t^\circ\).
Its normalized extensions in this ambiguity class are exactly
\[
 T+a,\qquad a\in\mathcal A,\qquad
 \mathcal N(T+a)=n_*.
\]
They exist if and only if
\[
 n_*-\mathcal N(T)\in\mathcal N(\mathcal A).
\]
If a solution exists, it is unique if and only if
\[
 \ker(\mathcal N|_{\mathcal A})=\{0\}.
\]
\end{theorem}

\begin{proof}
The complete-expansion theorem supplies the subtracted limit.
Proposition~\ref{prop:support} gives its reference agreement.
The authorization and admissibility hypotheses supply the
additional extension requirements.

The normalization equation is
\(\mathcal N(a)=n_*-\mathcal N(T)\).
Its solvability is the image condition. Differences of
solutions form the kernel of the restricted map.
\end{proof}

Microlocal extension results require their own scaling and
wavefront hypotheses \cite{Dang}. They do not identify the
physical source or authorize a frozen counterterm.

\begin{proposition}[Corrections invisible on a detecting patch]
\label{prop:invisible}
If every permitted correction restricts to zero on an open
patch containing a fixed detecting test, those corrections
cannot remove that test's raw divergence.
\end{proposition}

\begin{proof}
Every such correction has zero pairing with that test at
every scale. The original pairing is unchanged.
\end{proof}

%=====================================================================
\section{Microlocal output as a separate layer}
\label{sec:microlocal}
%=====================================================================

\subsection{Ordinary-product agreement}

A sufficient localized reference package has closed bounds
\(\Gamma_K,\Gamma_C\) satisfying
\[
 \Gamma_K\cap(-\Gamma_C)=\varnothing,
\]
and normal-topology convergence of the pulled-back smoothed
crossed components in the fixed cone \(\Gamma_C\).
Then fixed-factor multiplication converges by hypocontinuity
\cite[Theorem~6.1]{BDH}. The actual pullbacks and supports
must already be verified.

Smooth weights, derivatives, and finite sums preserve the
resulting strong distributional convergence. This identifies
an existing limit with the ordinary contraction on that
proved domain. Merely having an ordinary product does not
supply convergence in an unspecified stronger topology.

\subsection{Uniform Fourier certificate}

\begin{theorem}[Microlocal upgrade]
\label{thm:microlocal}
Let \(v_\delta=t_\delta-S_\delta\to T\) strongly, and let
\(\Gamma\subset T^*U\setminus0\) be closed and conic.
Assume \(v_\delta\in\Dist_\Gamma(U;E_{\mathrm{out}})\).
For every \(\zeta\in C_c^\infty(U)\), closed frequency cone
\(V_0\subset\R^{16}\setminus0\) with
\[
 (\supp\zeta\times V_0)\cap\Gamma=\varnothing,
\]
and integer \(r\ge0\), suppose
\[
 \sup_{\xi\in V_0}
 (1+|\xi|)^r
 |\widehat{\zeta v_\delta}(\xi)|
 \le C_{r,\zeta,V_0}
\]
uniformly for small scales, componentwise.

Then
\[
 \WF(T)\subset\Gamma,
\]
and \(v_\delta\to T\) in the normal topology associated with
\(\Gamma\).
\end{theorem}

\begin{proof}
Strong convergence gives convergence of each localized
Fourier value. Passing to the limit in the uniform rapid-decay
bounds proves \(\WF(T)\subset\Gamma\).

For the order-\(r\) Fourier seminorm of the difference,
the uniform order-\(r+1\) bounds make the tail above radius
\(R\) bounded by \(C/(1+R)\), for both \(v_\delta\) and \(T\).
Within that radius, the localized exponential tests form
a bounded test set, so convergence is uniform.
First choose \(R\) large, then let \(\delta\downarrow0\).
Together with the strong seminorms, this proves normal
convergence.
\end{proof}

The finite-order remainder bound alone does not verify this
certificate. In particular, conormal bounds for the application's
limit coefficients do not provide these uniform estimates for
its regularized family.

\subsection{Residual projection}

For a compact joint cutoff \(\zeta\), let
\[
 \pi(a,b,z,y)=(z,y).
\]
The compactly supported pushforward \(\pi_*(\zeta T)\) exists.
For a declared joint cone define
\[
 \pi_*\Gamma_\zeta
 =
 \left\{
 (\pi(w);\eta):
 \eta\ne0,\ w\in\supp\zeta,\
 (w;d\pi_w^{\,T}\eta)\in\Gamma
 \right\}.
\]
Because \(\pi\) is a submersion, \(d\pi_w^{\,T}\eta\ne0\)
for \(\eta\ne0\).

Proper-support pushforward is continuous in the normal topology
with its projected cone bound
\cite[Theorem~6.3]{BDH}. Thus a verified joint normal limit
can be projected. To meet a required output cone one must
also check
\[
 \pi_*\Gamma_\zeta\subset\Gamma_{\mathrm{out}}.
\]

Projection of an existing joint distribution does not construct
an absent joint product. It does not supply global overlaps,
physical normalization, an interacting identity, or membership
in a specified quantum carrier.

%=====================================================================
\section{Release, audit record, and stopping conditions}
\label{sec:release}
%=====================================================================

\begin{theorem}[Mathematical release]
\label{thm:release}
The results established in this manuscript are:
\begin{enumerate}[label=\textup{(\roman*)}]
\item the complete-expansion raw decision, coefficient
uniqueness, and intrinsic extraction;

\item the three sufficient verification routes, including the
separately proved differentiated-origin estimate;

\item compact-local Taylor verification, finite assembly,
and globalization under a common finite inventory;

\item the evaluated complete degenerate specialization
\eqref{eq:appdecision}, including its cancelled
\(\delta^{-2}\) coefficient, surviving \(\delta^{-1}\)
coefficient, fixed detecting test, and order-six remainder;

\item the additional inverse-scale \(u\)-jet cancellation,
normal-order bounds, conormal coefficient wavefront bounds,
zero off-attachment reference, and the stated finite-part
profile comparison;

\item conditional authorization, normalization, and microlocal
output results, each with its separate hypotheses.
\end{enumerate}
\end{theorem}

\begin{proof}
Apply Theorem~\ref{thm:decision}, the results of
Sections~\ref{sec:routes} and \ref{sec:protocol},
Theorem~\ref{thm:application},
Corollary~\ref{cor:appcancel},
Propositions~\ref{prop:appWF} and
\ref{prop:appconsequences}, and
Theorems~\ref{thm:authorized} and \ref{thm:microlocal}.
\end{proof}

\paragraph{Analytic checks made in the written calculation.}
The application checks profile normalization, actual
eight-variable smoothing, source derivative convention,
trace reversal, all nonzero row--vertex signs, every declared
zero row, cone integration, Jacobians, fixed rescaled support,
profile moments, uniform Taylor errors, coefficient continuity,
one fixed detecting test, and coefficient conormal bounds.

\paragraph{Documentary boundary.}
The supplied excerpts explain the project context but do not
constitute a complete audit of their source manuscripts or
registry versions. No XI/X/XVI theorem is imported as a premise
for the evaluated application. The homogeneous-origin route is
proved here without relying on an unavailable Em01-D-v2 proof.

\paragraph{Execution boundary.}
No pdfLaTeX compiler, symbolic algebra system, numerical
integration program, or proof assistant was run.
This is a standalone two-run pdfLaTeX source.
Successful compilation, machine verification, independent
referee approval, and acceptance are not reported as observed
results.

\paragraph{Physical stopping condition.}
The physical ledger must be completed before the present
mathematical specialization is replaced by a physical claim.
That physical family must then receive an exact complete
localized representation and a controlled estimate on arbitrary
joint tests. Selected witness matching, source-level identities,
or a partial-sector extension cannot substitute for this task.

\scopebox{Final distinction}{%
A rank defect identifies where special analysis is required.
A complete estimate or controlled expansion decides the
specified regularized family.
An admissibility certificate determines whether an extension
is authorized.

\smallskip
This manuscript decides its complete mathematical specialization.
It does not decide the physically identified Einstein--scalar
characteristic contraction or admit its quantum coefficient.}

%=====================================================================
\appendix
\section{Retained internal formulas and local existence}
\label{app:kernels}
%=====================================================================

This appendix states the retained architecture as mathematical
definitions. Its identification with any physical Hessian is
a separate ledger entry.

Let \(e_I\) be a basis of covariant symmetric tensors and \(e^I\)
its algebraic dual:
\[
 (e^I)^{\mu\nu}(e_J)_{\mu\nu}=\delta^I_J.
\]
All spacetime index raising uses \(\eta\). Define
\[
 (\PP r)_{\mu\nu}
 =
 r_{\mu\nu}-\frac12\eta_{\mu\nu}\tr_\eta r,
\]
\[
 \Amap^{\mu\nu,I}
 =
 \frac12\eta^{\mu\nu}\tr_\eta e_I-(e_I)^{\mu\nu},
 \qquad
 \bmap^I=\frac12\tr_\eta e_I,
\]
\[
 \PP^L{}_M=(e^L)^{\mu\nu}(\PP e_M)_{\mu\nu}.
\]
The notation \(\Amap^{\mu\nu}{}_L\) means evaluation on
\(e_L\), not a newly introduced component metric.
Similarly \(\bmap_L=\bmap^L\).

Fix smooth \(f,g,F\), a constant array \(\Dmap\), and finite
smooth source coefficients
\(q_{\alpha\beta}^{L,I,J}\), with
\(|\alpha|+|\beta|\le2\).

\subsection{Cone kernels, including their tips}

Define the locally finite retarded cone measure by
\begin{equation}
 R(\psi)
 =
 \frac1{4\pi}\int_{\R^3}
 \frac{\psi(|\mathbf x|,\mathbf x)}{|\mathbf x|}
 \dd^3\mathbf x.
 \label{eq:Rmeasure}
\end{equation}
The integrand is locally integrable at \(\mathbf x=0\).
Away from the cone tip this is the familiar notation
\[
 R(t,\mathbf x)
 =
 \frac{\Theta(t)}{2\pi}\deltaD(t^2-|\mathbf x|^2).
\]
Equation \eqref{eq:Rmeasure}, rather than an unjustified
submersion pullback at the tip, defines the global distribution.

Also define the locally integrable function
\[
 B(t,\mathbf x)=\frac1{8\pi}\mathbf1_{\{t\ge|\mathbf x|\}}.
\]
Its boundary value on the cone is irrelevant to its
distribution. Both \(R\) and \(B\) have local order zero.

\subsection{Source incidence and retained rows}

Set
\[
 L^{\mu,I}(s,a)
 =
 \Amap^{\lambda\mu,I}\partial_{a^\lambda}R(s-a),
\]
\[
 Q^{L,I,J}(v;a,z)
 =
 \sum_{\alpha,\beta}
 q_{\alpha\beta}^{L,I,J}(v)
 \partial_v^\alpha\deltaD^{(4)}(v-z)
 \partial_v^\beta\deltaD^{(4)}(v-a),
\]
\[
 T^{L,I,J}(b;a,z)
 =
 \PP^L{}_M\int R(b-v)Q^{M,I,J}(v;a,z)\dd^4v.
\]

The seven retained rows are
\begin{align}
 K_{47}^{I,J}
 &=-\partial_{b^\mu}\Bigl[
 \bigl(
 f(b)\partial_{b^\nu}L^{\mu,J}(b,z)
 -\delta^\mu_\nu\partial_{b^\rho}
              (f(b)L^{\rho,J}(b,z))
 \bigr)L^{\nu,I}(b,a)\Bigr],
 \label{eq:K47}\\
 K_{5,+}^{I,J}
 &=\partial_{b^\mu}\Bigl[
 f(b)\partial_{a^\alpha}R(b-a)
 \Amap^{\alpha\beta,I}
 \partial_{a^\beta}L^{\mu,J}(a,z)
 \Bigr],
 \label{eq:K5plus}\\
 K_{5,-}^{I,J}
 &=\partial_{b^\mu}\Bigl[
 f(b)\partial_{z^\alpha}R(b-z)
 \Amap^{\alpha\beta,J}
 \partial_{z^\beta}L^{\mu,I}(z,a)
 \Bigr],
 \label{eq:K5minus}\\
 K_6^{I,J}
 &=\partial_{b^\mu}\Bigl[
 f(b)\Dmap^{\alpha\mu,I,J}
 \partial_{a^\alpha}R(b-a)\deltaD^{(4)}(a-z)
 \Bigr],
 \label{eq:K6}\\
 K_8^{I,J}
 &=-\partial_{b^\mu}\Bigl[
 f(b)\Amap^{\alpha\mu}{}_L\PP^L{}_M
 \partial_{b^\alpha}
 \int B(b-v)Q^{M,I,J}(v;a,z)\dd^4v
 \Bigr],
 \label{eq:K8}\\
 K_{9,\mathrm{der}}^{I,J}
 &=-\partial_{b^\nu}\Bigl[
 g(b)\partial_{b^\mu}F(b)
 \Amap^{\mu\nu}{}_L T^{L,I,J}(b;a,z)
 \Bigr],
 \label{eq:K9der}\\
 K_{9,\mathrm{mass}}^{I,J}
 &=-m^2g(b)F(b)\bmap_L T^{L,I,J}(b;a,z).
 \label{eq:K9mass}
\end{align}
Define
\[
 \Kret=
 K_{47}+K_{5,+}+K_{5,-}+K_6+K_8
 +K_{9,\mathrm{der}}+K_{9,\mathrm{mass}}.
\]
These definitions reproduce the retained convention attributed
to Em01-B and Em01-C in the supplied materials
\cite{Em01B,Em01C}. No physical plateau or full source theorem
is inferred merely from the printed formulas.

\begin{lemma}[Response reduction and local order]
\label{lem:internal}
The rows above are defined distributions. For \(D_0=R,B\),
\begin{equation}
\begin{split}
 &\int D_0(b-v)Q^{L,I,J}(v;a,z)\dd^4v\\
 &\quad=
 \sum_{\alpha,\beta}
 (-1)^{|\alpha|+|\beta|}
 \partial_z^\alpha\partial_a^\beta
 \left[
 q_{\alpha\beta}^{L,I,J}(a)
 D_0(b-a)\deltaD^{(4)}(a-z)
 \right].
\end{split}
 \label{eq:response}
\end{equation}
On every compact joint support, the retained components have
order at most four.
\end{lemma}

\begin{proof}
The source deltas are a differentiated tensor-product pullback
under the submersion
\((v,a,z)\mapsto(v-z,v-a)\).
Including \(D_0(b-v)\) introduces another independent relative
variable. Thus the response product is defined in independent
relative coordinates before integration.

Move the \(v\)-derivatives on the deltas to \(z,a\).
The undifferentiated integration is
\[
 q(a)D_0(b-a)\deltaD^{(4)}(a-z).
\]
The incidence gives local properness over compact external
supports, proving \eqref{eq:response}. All derivatives of
the smooth coefficient generated by this identity remain
in the expression.

The nonsupported seeds are tensor-product pullbacks under
the invertible linear coordinate maps
\[
 (a,b,z,y)\mapsto(b,b-a,b-z,y),
\]
\[
 (a,b,z,y)\mapsto(a,b-a,a-z,y),
 \qquad
 (a,b,z,y)\mapsto(z,b-z,z-a,y).
\]
The supported response seed is likewise a tensor product
in independent relative variables. These seeds have local
order zero. Each retained row uses at most four derivatives
of such seeds with smooth multipliers. Transposition gives
the order-four compact-local estimate.
\end{proof}

Every coefficient derivative in \eqref{eq:response} is retained.
The derivative of \(R\) in \(K_6\) is not the derivative of
its entire incidence product.

%=====================================================================
\section{A family remainder is not a derivative remainder}
\label{app:debug}
%=====================================================================

For any fixed nonzero distribution \(A\), let
\[
 r_\delta=\delta\sin(\delta^{-2})A.
\]
Then \(r_\delta=O(\delta)\) in every relevant local finite-order
bound, but
\[
 \delta\partial_\delta r_\delta
 =
 \left[
 \delta\sin(\delta^{-2})
 -2\delta^{-1}\cos(\delta^{-2})
 \right]A.
\]
Thus one cannot differentiate a vanishing family remainder
inequality and obtain a summable scale estimate.

The evaluated application establishes a Route B family
expansion. No differentiated application remainder is claimed.
The derivative statement in Proposition~\ref{prop:origin}
was proved independently from the exact scale profile.

%=====================================================================
\begin{thebibliography}{99}

\bibitem{Em01B}
N.~Cheung,
\emph{Em01-B: Intrinsic Regularization of the Joint
Characteristic Map}.
Excerpts attributed to \texttt{Em01-B-v3.tex}, version~3,
supplied in the preparation materials.
Documentary source for the retained convention and reported
regularization results; not an independently audited full source.
Enter Calculus Companion, manuscript identifier Em01-B,
\url{https://enter.grok.me}.

\bibitem{Em01C}
N.~Cheung,
\emph{Em01-C: Characteristic Normal Forms and
Complete-Sum Scale Reduction}.
Excerpts attributed to \texttt{Em01-C-v3.tex}, version~3,
supplied in the preparation materials.
Documentary source for the coordinate, translation, and
scale-integrability conventions; the arguments used here
are given in this manuscript.
Enter Calculus Companion, manuscript identifier Em01-C,
\url{https://enter.grok.me}.

\bibitem{BDH}
C.~Brouder, N.~V.~Dang, and F.~H\'elein,
``Continuity of the fundamental operations on distributions
having a specified wave front set
(with a counter example by Semyon Alesker),''
\emph{Studia Mathematica} \textbf{232} (2016), 201--226.
doi:10.4064/sm8316-3-2016.
arXiv:1409.7662v2.
\url{https://arxiv.org/abs/1409.7662v2}.
% Primary source consulted:
% https://arxiv.org/pdf/1409.7662v2
% Section 3; Theorems 6.1 and 6.3.

\bibitem{Dang}
N.~V.~Dang,
\emph{The extension of distributions on manifolds,
a microlocal approach}.
arXiv:1412.2808v1 (2014).
\url{https://arxiv.org/abs/1412.2808v1}.
% Primary source consulted:
% https://arxiv.org/pdf/1412.2808v1
% Section 1; supported-distribution discussion and Lemma 8.1.

\end{thebibliography}

\end{document}