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% Em01-D.tex
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%   pdflatex -interaction=nonstopmode -halt-on-error Em01-D.tex
%   pdflatex -interaction=nonstopmode -halt-on-error Em01-D.tex
%
% Revision:
%   - makes the fixed-kernel problem the primary object;
%   - states and proves a conditional complete-sum decision theorem;
%   - gives a finite source specification and explicit output conversion;
%   - retains the seven internal rows and three vertex types;
%   - derives the shifted cone equations for the actual smoothing;
%   - proves an intrinsic translation representation;
%   - extracts coefficients from a declared homogeneous-germ certificate;
%   - separately proves both family and scale-derivative remainder bounds;
%   - treats coefficient uniqueness, support, profiles, and admissibility;
%   - moves the nonphysical diagnostic to an appendix;
%   - reduces every diagnostic profile moment to a one-dimensional integral.
%
% Scope:
% The physical covariance package is not supplied in the preparation
% materials. The main theorem is therefore conditional and is not an
% evaluated decision for the designated physical graph.
%
% No compiler, computer algebra system, or proof assistant was executed.
%=====================================================================

\documentclass[11pt,a4paper]{article}

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\usepackage{array,booktabs,tabularx}
\usepackage{enumitem}
\usepackage[hidelinks]{hyperref}

\hypersetup{
  pdftitle={Em01-D: Conditional Fixed-Kernel Characteristic Asymptotics},
  pdfauthor={}
}

\allowdisplaybreaks
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\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
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\newtheorem{definition}[theorem]{Definition}
\newtheorem{assumption}[theorem]{Assumption}
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\newtheorem{remark}[theorem]{Remark}

\newcommand{\R}{\mathbb R}
\newcommand{\C}{\mathbb C}
\newcommand{\N}{\mathbb N}
\newcommand{\dd}{\,\mathrm d}
\newcommand{\Test}{\mathcal D}
\newcommand{\Dist}{\mathcal D'}
\newcommand{\EDist}{\mathcal E'}
\newcommand{\PP}{\mathcal P}
\newcommand{\Amap}{\mathsf A}
\newcommand{\bmap}{\mathsf b}
\newcommand{\Dmap}{\mathsf D}
\newcommand{\JD}{\mathcal J_{\mathrm D}}
\newcommand{\Kret}{K_{\mathrm{ret}}}
\newcommand{\Vertex}{\mathsf V}
\newcommand{\Rec}{\mathsf{Rec}}
\newcommand{\pair}[2]{\langle #1,#2\rangle}
\newcommand{\norm}[1]{\lVert #1\rVert}
\newcommand{\deltaD}{\delta_{\mathrm D}}
\newcommand{\id}{\mathrm{id}}
\newcommand{\eps}{\varepsilon}

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\DeclareMathOperator{\WF}{WF}
\DeclareMathOperator{\tr}{tr}
\DeclareMathOperator{\EL}{EL}

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\newcommand{\scopebox}[2]{%
  \par\medskip\noindent
  \fbox{\begin{minipage}{%
    \dimexpr\linewidth-2\fboxsep-2\fboxrule\relax}
    \textbf{#1}\par\smallskip #2
  \end{minipage}}
  \par\medskip
}

\title{\textbf{Em01-D}\\[0.4em]
Conditional Fixed-Kernel Characteristic Asymptotics\\[0.4em]
\large Complete-sum coefficient extraction, controlled remainders,\\
and the distinction between raw limits and authorized extensions}
\author{}
\date{}

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

\begin{abstract}
We formulate the characteristic problem for one fixed retained
Einstein--scalar kernel, keeping all seven internal bookkeeping
rows, all three mixed-vertex terms, and the grouped crossed-line
difference. The actual eight-relative-variable smoothing is
represented exactly by a compact, test-dependent translation
distribution.

The main result is a conditional decision theorem. A specified
finite logarithmically homogeneous origin germ of the complete
translation distribution, with a uniformly H\"older continuous
remainder, determines every nonsummable coefficient and the finite
part. It yields quantitative bounds for both the family remainder
and its logarithmic scale derivative. The raw family converges
exactly when its complete nonsummable coefficients vanish;
otherwise one fixed compact joint test detects unboundedness.
Coefficient uniqueness makes the conclusion independent of
internal presentations. Support, profile independence, admissible
subtraction, and normalization are treated separately.

We reproduce the finite source rule and density derivative,
derive the shifted characteristic equations with their correct
second-slot sign, and retain the full coordinate density.
An appendix evaluates an exact nonphysical convolution diagnostic,
including all inverse powers and an order-eight remainder.

The supplied materials do not instantiate the designated physical
crossed covariances or verify the complete origin-germ hypothesis
for them. Thus the conditional theorem is proved here, but no
evaluated convergence, failure, or admissible-extension decision
for the designated physical graph is asserted.
\end{abstract}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (a) at (0,0) {Origin-germ certificate};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Conditional expansion};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Physical covariance package};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {The certificate is not verified for the designated physical graph.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{A conditional 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-D}
\end{figure}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (a) at (0,0) {Origin-germ certificate};
\node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {Physical covariance package};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {The certificate is not verified for the designated physical graph.};
\draw[->] (a) -- (s);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{A conditional 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-D}
\end{figure}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (a) at (0,0) {Origin-germ certificate};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Conditional expansion};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {Physical covariance package};
\node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {The certificate is not verified for the designated 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-D}
\end{figure}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=4.2cm,font=\small,inner sep=3pt] (a) at (0,0) {Origin-germ certificate};
\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 covariance package};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Conditional expansion. The certificate is not verified for the designated physical graph.};
\draw (a) -- (m);
\draw (m) -- (b);
\draw[->] (m) -- (s);
\end{tikzpicture}
\caption{Origin-germ certificate is not Physical covariance package. 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-D}
\end{figure}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (a) at (0,0) {Named: Origin-germ certificate};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Conditional expansion};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: Physical covariance package};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {The certificate is not verified for the designated 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-D}
\end{figure}

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


Figure~\ref{fig:scope-Em01-D} records that separation.
The drawing adds no hypothesis.



\scopebox{Scope of the revision}{%
The primary object is the named fixed-kernel family, not a map on
every internal presentation. The physical input identification and
the complete origin-germ estimate remain explicit hypotheses.

\smallskip
The nonphysical diagnostic is confined to
Appendix~\ref{app:diagnostic}; its assumptions and powers are not
used as physical inputs or physical asymptotic orders.}

\tableofcontents

%=====================================================================
\section{The fixed-kernel decision problem}
\label{sec:main}
%=====================================================================

For one fixed input package, the object is
\begin{equation}
 t_\delta
 =
 i\Vertex\left(
 \Kret,\rho_\delta*(\chi C)
 \right)\big|_U,
 \qquad
 \rho_\delta(h)=\delta^{-8}\rho(h/\delta).
 \label{eq:target}
\end{equation}
Here \(U\) is a relatively compact joint-coordinate patch,
\(\rho\in C_c^\infty(\R^8)\) has integral one, and the internal
kernel is kept exact.

The target is a statement on every compact joint test, including
tests meeting the aligned-null locus \(N\). No restriction to
factorized witnesses or normal annuli is made.

\subsection{Main theorem}

The exact translation distribution \(\Omega_\Phi\) is constructed
in Section~\ref{sec:translation}. Its complete origin-germ
certificate is defined in Section~\ref{sec:certificate}.
The certificate is a condition on the actual complete
distribution, not on a selected branch or a polynomial-degree
presentation.

\begin{theorem}[Conditional fixed-kernel decision]
\label{thm:main}
Fix the data of Sections~\ref{sec:inputs}--\ref{sec:contraction}
and a normalized compact profile \(\rho\).
Suppose the complete translation distribution has the
origin-germ certificate of Assumption~\ref{ass:germ}.

Then the family has the expansion
\begin{equation}
 t_\delta
 =
 S_\delta^\rho+T^\rho+R_\delta^\rho,
 \label{eq:mainexpansion}
\end{equation}
where
\begin{equation}
 S_\delta^\rho
 =
 \sum_{\lambda<0}\sum_{k=0}^{J_\lambda}
 \delta^\lambda(\log\delta)^k A_{\lambda,k}^\rho
 +
 \sum_{k=1}^{J_0}(\log\delta)^k A_{0,k}^\rho.
 \label{eq:singularsum}
\end{equation}
Absent exponents contribute no terms. All displayed coefficients
are determined by
\begin{align}
 A_{\lambda,k}^\rho(\Phi)
 &=
 \frac1{k!}
 \pair{H_{\lambda,J_\lambda-k}[\Phi]}{\rho},
 \label{eq:maincoeff}\\
 T^\rho(\Phi)
 &=
 W_\Phi(0)+A_{0,0}^\rho(\Phi).
 \label{eq:mainfinite}
\end{align}
If \(0\) is absent, \(A_{0,0}^\rho=0\).

For every compact \(L\Subset U\), there are
\(m_L,C_L>0\) and \(a_L\in(0,1]\) such that, for sufficiently
small scales and \(\supp\Phi\subset L\),
\begin{align}
 |R_\delta^\rho(\Phi)|
 &\le C_L\delta^{a_L}\|\Phi\|_{m_L},
 \label{eq:mainrem}\\
 \left|
 \left[
 \delta\partial_\delta(t_\delta-S_\delta^\rho)
 \right](\Phi)
 \right|
 &\le C_L\delta^{a_L}\|\Phi\|_{m_L}.
 \label{eq:maindrem}
\end{align}

Consequently:
\begin{enumerate}[label=\textup{(\roman*)}]
\item the specified raw family converges strongly if and only if
every coefficient in \eqref{eq:singularsum} is zero;
\item if one such coefficient is nonzero, one fixed compact
joint test detects an unbounded raw pairing;
\item subtraction of \eqref{eq:singularsum} always gives the
mathematical finite part \(T^\rho\) under the certificate,
but constitutes an admissible extension only under the
additional conditions of Section~\ref{sec:admissibility}.
\end{enumerate}
The proof does not require a continuous characteristic map on
the whole internal module.
\end{theorem}

Theorem~\ref{thm:main} is proved in
Section~\ref{sec:proofmain}. Its applicability to the designated
physical inputs is not assumed.

Raw failure and existence of a separately authorized extension
can occur together. They concern different constructions.

%=====================================================================
\section{The designated contraction and its input conventions}
\label{sec:inputs}
%=====================================================================

\subsection{Mathematical data versus physical identification}

A mathematical input package consists of
\begin{equation}
 \mathcal P=
 \left(
 m,\eps,\sigma,f,g,F,\Dmap,
 H_{T,\gamma},H_\gamma,H_{T,\phi},H_\phi,
 \rho,\chi,U
 \right).
 \label{eq:package}
\end{equation}
Every entry is held fixed as \(\delta\downarrow0\).

The four crossed entries are distributions, with the component
and relative-argument conventions printed below.
The functions \(f,g,F\) are fixed smooth functions; for the
designated scalar root its identification is
\[
 F=E_m^{\mathrm A}f,
\]
with the designated advanced Klein--Gordon operator.
The estimates proved here use the fixed smooth function \(F\),
not an alternative choice of its propagator.

The source switch \(\sigma\) is smooth, with
\(\sigma=1\) on \(x^0>\eps\), and \(\eps>0\) is fixed.
For the local analysis, observations and their derivatives are
bounded on all compact sets used.

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.28\linewidth}Y@{}}
\toprule
Entry & Identification needed for a physical application\\
\midrule
Metric cross-lines &
Actual component distributions, tensor propagator factor,
gauge, normalization, time ordering, boundary prescription,
and any state-dependent smooth terms.\\[0.3em]

Scalar cross-lines &
Actual mass-dependent distributions, state, normalization,
boundary prescription, and relations to the selected Green
operators.\\[0.3em]

Density map &
An identification of the designated
\(\dot a_2[G;h]\) with a derivative or polarization of a
specified quadratic density.\\[0.3em]

Source &
The unintegrated cubic density, curvature convention, and
propagated-output conversion of
Section~\ref{sec:source}.\\[0.3em]

Observations and root &
The frozen \(f,g\), the selected advanced root \(F\), and
the source-switch assumptions; no diagnostic observation
is substituted.\\[0.3em]

Corrections &
The actual allowed correction space, extension reference,
normalization conditions, and required output cone.\\
\bottomrule
\end{tabularx}
\end{center}

The supplied materials do not provide a complete identification
of the first three rows or the correction package. The theorem
therefore applies to a fixed mathematical package
\eqref{eq:package}, and applies to the physical graph only after
those identifications and its analytic certificate are proved.

\subsection{Tensor basis and propagated outputs}

Use Minkowski spacetime
\[
 M=\R^{1+3},\qquad
 \eta=\operatorname{diag}(1,-1,-1,-1),\qquad
 \Box=\partial_t^2-\Delta_{\mathbf x}.
\]
All distributions have complex-bilinear pairings and use flat
coordinate densities unless a density conversion is printed.

Let \(e_I\) be a basis of covariant symmetric tensors, with
algebraic dual \(e^I\):
\begin{equation}
 (e^I)^{\mu\nu}(e_J)_{\mu\nu}=\delta^I_J.
 \label{eq:duality}
\end{equation}
The dual basis is not obtained merely by metric index raising.

One concrete basis uses \(I=(i,j)\), \(0\le i\le j\le3\):
\[
 (e_{(i,j)})_{\mu\nu}
 =
 \frac{\delta_{\mu i}\delta_{\nu j}
       +\delta_{\mu j}\delta_{\nu i}}{1+\delta_{ij}},
\]
\[
 (e^{(i,j)})^{\mu\nu}
 =
 \frac12\left(
 \delta^\mu_i\delta^\nu_j+
 \delta^\mu_j\delta^\nu_i
 \right).
\]
Thus off-diagonal component factors are fixed by the duality,
not inserted independently.

Define
\[
 (\PP r)_{\mu\nu}
 =
 r_{\mu\nu}-\frac12\eta_{\mu\nu}\tr_\eta r,
\]
\[
 \PP^L{}_M=(e^L)^{\mu\nu}(\PP e_M)_{\mu\nu},
\]
\[
 \Amap^{\mu\nu,I}
 =
 \frac12\eta^{\mu\nu}\tr_\eta e_I-(e_I)^{\mu\nu},
 \qquad
 \bmap^I=\frac12\tr_\eta e_I.
\]
The notation \(\Amap^{\mu\nu}{}_L\) means evaluation on \(e_L\);
similarly \(\bmap_L=\bmap^L\). These role labels do not instruct
another fiber-metric multiplication.

For tests supported in \(L\Subset U\), put
\[
 \|\Phi\|_m=
 \max_{J,\ |\gamma|\le m}\sup|\partial^\gamma\Phi_J|.
\]
Strong distributional convergence means uniform convergence on
bounded test sets, which have common compact support and uniform
bounds on all derivatives.

%=====================================================================
\section{Finite source and density specifications}
\label{sec:source}
%=====================================================================

\subsection{The unintegrated cubic source}

For a formal metric \(q_s=\eta+s\gamma\), fix
\[
 \Gamma^\rho_{\mu\nu}(q)
 =
 \frac12q^{\rho\lambda}
 \left(
 \partial_\mu q_{\nu\lambda}
 +\partial_\nu q_{\mu\lambda}
 -\partial_\lambda q_{\mu\nu}
 \right),
\]
\[
 R_{\mu\nu}(q)
 =
 \partial_\rho\Gamma^\rho_{\mu\nu}
 -\partial_\nu\Gamma^\rho_{\mu\rho}
 +\Gamma^\rho_{\rho\lambda}\Gamma^\lambda_{\mu\nu}
 -\Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\rho},
\]
and \(R(q)=q^{\mu\nu}R_{\mu\nu}(q)\).
Declare
\begin{equation}
 \ell_3
 =
 -2[s^3]\left(
 \sqrt{-\det q_s}\,R(q_s)
 \right),
 \qquad
 \JD=\EL_\gamma(\sigma\ell_3).
 \label{eq:source}
\end{equation}
No integration-by-parts replacement of the density is made
before this variation.

Let \(X_{I,\gamma}\), \(|\gamma|\le2\), denote ordinary,
not factorial-normalized, derivative jets:
\[
 \partial^\gamma\gamma_{\mu\nu}
 =
 \sum_I(e_I)_{\mu\nu}X_{I,\gamma}.
\]
Mixed derivatives use their common multiindex.

The coordinate Euler components are
\[
 \mathcal E_I
 =
 \sum_{|\gamma|\le2}
 (-\partial)^\gamma
 \frac{\partial(\sigma\ell_3)}{\partial X_{I,\gamma}}.
\]
Let
\[
 \mathbf G_{IJ}=(e_I)_{\mu\nu}(e_J)^{\mu\nu}.
\]
The propagated source components are
\begin{equation}
 \mathcal J^L
 =
 (\mathbf G^{-1})^{LI}\mathcal E_I
 =
 (e^L)^{\mu\nu}\eta_{\mu\rho}\eta_{\nu\sigma}
 \JD^{\rho\sigma}.
 \label{eq:sourceconversion}
\end{equation}
Thus the coordinate Euler gradient is not substituted for the
metric-valued propagated source.

\begin{proposition}[Finite source Hessian]
\label{prop:sourcehessian}
The coefficients in
\[
 (\mathcal J'[G]h)^L(x)
 =
 \sum_{|\alpha|+|\beta|\le2}
 q_{\alpha\beta}^{L,I,J}(x)
 \partial^\alpha G_J(x)\partial^\beta h_I(x)
\]
are given by the finite formula
\begin{equation}
 q_{\alpha\beta}^{L,I,J}(x)
 =
 \left.
 \frac{\partial^2\mathcal J_x^L(X)}
 {\partial X_{J,\alpha}\partial X_{I,\beta}}
 \right|_{X=0}.
 \label{eq:sourcehessian}
\end{equation}
They satisfy
\[
 q_{\alpha\beta}^{L,I,J}
 =
 q_{\beta\alpha}^{L,J,I}.
\]
They vanish for \(|\alpha|+|\beta|>2\), and transform
covariantly under a simultaneous change of basis and dual basis.
\end{proposition}

\begin{proof}
The coefficient \(\ell_3\) is cubic in field jets, and each
monomial has total derivative weight two.
Euler differentiation removes one field factor. The derivatives
transferred by the Euler operator restore exactly the derivative
weight removed from that factor; they may also differentiate
\(\sigma\). The resulting source is quadratic and has total
derivative weight two when switch derivatives are included.
In particular the field derivative weight is at most two.

For any quadratic polynomial \(\mathcal J\),
\[
 D\mathcal J(G)[h]=D^2\mathcal J(0)[G,h].
\]
Expanding this identity in independent ordinary jet coordinates
gives \eqref{eq:sourcehessian}. Commutativity of polynomial
differentiation gives slot symmetry and all repeated-variable
factors. The underlying bilinear tensor map is unchanged by a
consistent component transformation.
\end{proof}

On a neighborhood where \(\sigma=1\), the full source and its
coefficient jets coincide with the unweighted source.
This is a local equality, not a gain in smoothing order.

\subsection{Quadratic inverse-metric density}

Write \(t_r=\tr_\eta r\) and \(G:h=G_{\mu\nu}h^{\mu\nu}\).
For the declared density
\[
 \sqrt{-\det(\eta+sr)}(\eta+sr)^{\mu\nu}
 =
 \eta^{\mu\nu}+s a_1^{\mu\nu}[r]+s^2a_2^{\mu\nu}[r]+O(s^3),
\]
the quadratic coefficient is
\begin{equation}
 a_2^{\mu\nu}[r]
 =
 r^\mu{}_\lambda r^{\lambda\nu}
 -\frac12t_r r^{\mu\nu}
 +\left(\frac18t_r^2-\frac14r:r\right)\eta^{\mu\nu}.
 \label{eq:a2}
\end{equation}

\begin{proposition}[Density derivative]
\label{prop:density}
Its directional derivative is
\begin{align}
 Da_2[G][h]^{\mu\nu}
 ={}&
 h^\mu{}_\lambda G^{\lambda\nu}
 +G^\mu{}_\lambda h^{\lambda\nu}
 -\frac12\left(t_hG^{\mu\nu}+t_Gh^{\mu\nu}\right)
 \notag\\
 &+\left(\frac14t_Gt_h-\frac12G:h\right)\eta^{\mu\nu}.
 \label{eq:Da2}
\end{align}
If the designated \(\dot a_2[G;h]\) is identified with this
derivative, then
\[
 \Dmap^{\mu\nu,I,J}=Da_2[e_J][e_I]^{\mu\nu}.
\]
For every symmetric \(h\) and scalar \(\psi\),
\[
 a_2[\psi\eta]=0,
 \qquad
 Da_2[\psi\eta][h]=0.
\]
\end{proposition}

\begin{proof}
The inverse metric has coefficients
\(\eta^{\mu\nu},-r^{\mu\nu},
r^\mu{}_\lambda r^{\lambda\nu}\).
The determinant square root has coefficients
\[
 1,\qquad \frac12t_r,\qquad
 \frac18t_r^2-\frac14r:r.
\]
Their product gives \eqref{eq:a2}, and ordinary differentiation
gives \eqref{eq:Da2}.

For \(G=\psi\eta\), the first two terms in \eqref{eq:Da2}
give \(2\psi h^{\mu\nu}\); the next terms give
\(-2\psi h^{\mu\nu}-\psi t_h\eta^{\mu\nu}/2\), and the
last term gives \(\psi t_h\eta^{\mu\nu}/2\).
The sum is zero. The quadratic identity follows directly.
\end{proof}

The symmetric polarization
\[
 a_2[G,h]
 =
 \frac12\left(a_2[G+h]-a_2[G]-a_2[h]\right)
\]
satisfies \(Da_2[G][h]=2a_2[G,h]\).
This factor must be settled by the physical identification.
The density derivative is not assigned to the physical graph
merely because it is a familiar formula.

%=====================================================================
\section{The complete retained contraction}
\label{sec:contraction}
%=====================================================================

\subsection{Patch and internal distributions}

Work on
\[
 W=A\times B_0\times Z\times Y,\qquad w=(a,b,z,y),
\]
with separated endpoint regions and
\[
 b^0>y^0>a^0>\eps,\qquad
 z^0>\eps,\qquad b^0>z^0.
\]
The source switch is one throughout the retained variables.

Define
\[
 R(t,\mathbf x)
 =
 \frac{\Theta(t)}{2\pi}\deltaD(t^2-|\mathbf x|^2),
 \qquad
 B(t,\mathbf x)
 =
 \frac1{8\pi}\Theta(t)\Theta(t^2-|\mathbf x|^2).
\]
Put
\[
 L^{\mu,I}(s,a)
 =
 \Amap^{\lambda\mu,I}\partial_{a^\lambda}R(s-a),
\]
\[
 Q^{L,I,J}(v;a,z)
 =
 \sum_{|\alpha|+|\beta|\le2}
 q_{\alpha\beta}^{L,I,J}(v)
 \partial_v^\alpha\deltaD^{(4)}(v-z)
 \partial_v^\beta\deltaD^{(4)}(v-a),
\]
\[
 T^{L,I,J}(b;a,z)
 =
 \PP^L{}_M\int R(b-v)Q^{M,I,J}(v;a,z)\dd^4v.
\]

The fixed retained kernel is
\begin{equation}
 \Kret
 =
 K_{47}+K_{5,+}+K_{5,-}+K_6+K_8
 +K_{9,\mathrm{der}}+K_{9,\mathrm{mass}},
 \label{eq:Ksum}
\end{equation}
where
\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}
These formulas adopt the retained architecture supplied in
Em01-B and Em01-C \cite{Em01B,Em01C}. The seven rows are
bookkeeping rows, not seven new interaction branches.

\begin{lemma}[Exact response reduction and finite order]
\label{lem:internal}
All displayed internal distributions exist. For \(D=R,B\),
\begin{align}
 &\int D(b-v)Q^{L,I,J}(v;a,z)\dd^4v
 \notag\\
 &\qquad=
 \sum_{\alpha,\beta}
 (-1)^{|\alpha|+|\beta|}
 \partial_z^\alpha\partial_a^\beta
 \left[
 q_{\alpha\beta}^{L,I,J}(a)
 D(b-a)\deltaD^{(4)}(a-z)
 \right].
 \label{eq:response}
\end{align}
Every coefficient derivative generated by this formula is retained.
For \(L\Subset W\),
\[
 |\Kret^{I,J}(\Psi)|\le C_L\|\Psi\|_4,
 \qquad \supp\Psi\subset L.
\]
\end{lemma}

\begin{proof}
Move the derivatives on the two incidence deltas to their
external variables. The undifferentiated integration is
\[
 q(a)D(b-a)\deltaD^{(4)}(a-z),
\]
which gives \eqref{eq:response}.

The nonsupported seeds are tensor-product pullbacks under the
invertible linear 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 source incidence is the differentiated pullback under the
submersion \((v,a,z)\mapsto(v-z,v-a)\).
The response integrations are locally proper because the
incidence forces \(v=a=z\) on compact external supports.

Both \(R\) and \(B\) are locally finite measures.
Each row is a finite sum of at most four derivatives of such
tensor-product seeds, multiplied by smooth coefficients.
Transferring derivatives to coefficient--test products proves
the estimate.
\end{proof}

\subsection{Grouped crossed array}

Declare
\begin{equation}
 C_I^{\mu\nu}(x_1,x_2)
 =
 H_{T,\gamma}^{\mu\nu}{}_I(x_1)\otimes H_{T,\phi}(x_2)
 -
 H_\gamma^{\mu\nu}{}_I(x_1)\otimes H_\phi(x_2).
 \label{eq:C}
\end{equation}
Both tensor products exist. Their difference is kept grouped.
No undeclared mixed products are introduced.

The relative map is
\[
 \mathfrak f(a,b,z,y)=(y-a,y-b).
\]
For smooth symmetric \(H\), define
\[
 Z_I^\nu(H)
 =
 \mathfrak f^*\left(
 \partial_{2,\mu}H_I^{\mu\nu}
 -\frac12\eta_{\mu\lambda}\partial_2^\nu H_I^{\mu\lambda}
 \right),
 \qquad
 M_I(H)=\mathfrak f^*(\eta_{\mu\nu}H_I^{\mu\nu}).
\]
The complete vertex is
\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 arbitrary-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}

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.19\linewidth}p{.36\linewidth}Y@{}}
\toprule
Vertex type & Crossed operator & Coefficient and output\\
\midrule
Kinetic &
\(\partial_{2,\mu}H_I^{\mu\nu}\) &
\(\partial_{y^\nu}(igK_b^{I,J}\,\cdot)\).\\[0.3em]
Trace kinetic &
\(\partial_2^\nu H_I^{\mu\lambda}\) &
\(\partial_{y^\nu}(-ig\eta_{\mu\lambda}K_b^{I,J}\,\cdot/2)\).\\[0.3em]
Mass &
\(H_I^{\mu\nu}\) &
Multiplication by
\(-im^2g\eta_{\mu\nu}K_b^{I,J}/2\).\\
\bottomrule
\end{tabularx}
\end{center}

Taking the product of these three types with the seven rows
in \eqref{eq:Ksum} is the complete branch--vertex inventory.
The internal \(K_{9,\mathrm{mass}}\) and crossed-vertex mass
term are distinct and both remain present.

Fix \(U\Subset W\) and
\(\chi\in C_c^\infty(\R^8)\) equal to one near
\(\mathfrak f(\overline U)\).
Then \eqref{eq:target} is defined at every positive scale.

%=====================================================================
\section{Characteristic coordinates and actual smoothing}
\label{sec:geometry}
%=====================================================================

\subsection{Coordinates and full density}

Let
\[
 q_R=(b-a)^2,\qquad
 q_1=(y-a)^2,\qquad
 q_2=(b-y)^2.
\]
The aligned-null locus is
\[
 N=
 \left\{
 z=a,\quad
 y=a+\alpha\ell_{\mathbf n},\quad
 b=a+\beta\ell_{\mathbf n},\quad
 0<\alpha<\beta
 \right\},
\]
where \(\ell_{\mathbf n}=(1,\mathbf n)\), \(\mathbf n\in S^2\).

On a sphere chart choose an orthonormal transverse frame
\(E_A=(0,\mathbf e_A)\), and put \(k_{\mathbf n}=(1,-\mathbf n)\).
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:coordinates}
\end{equation}

\begin{proposition}[Exact local geometry]
\label{prop:geometry}
These are coordinates near a retained point of \(N\), with
\[
 N=\{r=u=v=\xi=0\}.
\]
Writing \(d=\beta-\alpha>0\), one has
\begin{align}
 q_R&=4\beta u,&
 q_1&=4\alpha v-|\xi|^2,&
 q_2&=4d(u-v)-|\xi|^2.
 \label{eq:cones}
\end{align}
The ambient flat density is
\begin{equation}
 \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:density}
\end{equation}
Along \(N\), the three cone differentials have rank two.
Throughout the chart,
\begin{equation}
 \frac{q_1}{\alpha}+\frac{q_2}{d}
 =
 \frac{q_R}{\beta}
 -\frac{\beta}{\alpha d}|\xi|^2.
 \label{eq:defect}
\end{equation}
\end{proposition}

\begin{proof}
For \(s=b-a\), the inverse is
\[
 \mathbf n=\frac{\mathbf s}{|\mathbf s|},\qquad
 \beta=\frac{s^0+|\mathbf s|}{2},\qquad
 u=\frac{s^0-|\mathbf s|}{2}.
\]
For \(x=y-a\),
\[
 \alpha=\frac{x^0+\mathbf n\cdot\mathbf x}{2},\qquad
 v=\frac{x^0-\mathbf n\cdot\mathbf x}{2},\qquad
 \xi^A=\mathbf e_A\cdot\mathbf x.
\]
Also \(r=z-a\). These formulas prove invertibility.

The null-frame identities give \eqref{eq:cones}.
The \((\beta,u)\) Jacobian contributes
\(2(\beta-u)^2\), and the \((\alpha,v,\xi)\) determinant
contributes two, proving \eqref{eq:density}.
Differentiating \eqref{eq:cones} at \(N\) gives the rank
statement. Their divided sum gives \eqref{eq:defect}.
\end{proof}

\subsection{Shifted equations for the actual relative variables}

The convolution is
\[
 C_\delta(x)
 =
 \pair{\chi C(v)}{\rho_\delta(x-v)}.
\]
Its displacement belongs to \(\R^4\times\R^4\), not to a
space of independently regularized cone functions.

Write a displacement as \(\delta(H_1,H_2)\), where
\[
 H_i=A_i\ell_{\mathbf n}+B_i k_{\mathbf n}+Z_i^AE_A.
\]

\begin{proposition}[Exact shifted cone equations]
\label{prop:shifted}
With \(x_1=y-a\), \(x_2=y-b\), the shifted equations are
\begin{align}
 (x_1-\delta H_1)^2
 &=
 4(\alpha-\delta A_1)(v-\delta B_1)
 -|\xi-\delta Z_1|^2,
 \label{eq:shift1}\\
 (x_2-\delta H_2)^2
 &=
 4(d+\delta A_2)(u-v+\delta B_2)
 -|\xi-\delta Z_2|^2.
 \label{eq:shift2}
\end{align}
On the bounded rescaled core
\[
 u=\delta U,\qquad v=\delta V,\qquad
 \xi=\sqrt{\delta}\,\Xi,
\]
their divided forms are
\begin{align}
 \delta^{-1}(x_1-\delta H_1)^2
 &=
 4(\alpha-\delta A_1)(V-B_1)
 -|\Xi-\sqrt{\delta}Z_1|^2,
 \label{eq:scaledshift1}\\
 \delta^{-1}(x_2-\delta H_2)^2
 &=
 4(d+\delta A_2)(U-V+B_2)
 -|\Xi-\sqrt{\delta}Z_2|^2,
 \label{eq:scaledshift2}\\
 \delta^{-1}q_R&=4\beta U.
 \label{eq:scaledouter}
\end{align}
The coordinate density in \((u,v,\xi)\) contributes
\(\delta^3\), with its full smooth factor
\(4(\beta-\delta U)^2\).

All these algebraic formulas and their required derivatives
are uniform on compact positive tangential parameter ranges
and bounded rescaled cores.
\end{proposition}

\begin{proof}
In \eqref{eq:coordinates},
\[
 x_2=-d\ell_{\mathbf n}+(v-u)k_{\mathbf n}+\xi^AE_A.
\]
Subtracting \(\delta H_2\) and taking the square gives
\eqref{eq:shift2}; the first equation is analogous.
Substitution gives the divided equations.

The two longitudinal normal variables contribute \(\delta^2\),
and the two transverse variables contribute \(\delta\).
Uniformity follows directly from the polynomial expressions
and the bounded coordinate transformations.
\end{proof}

In invariant notation the second sign is
\[
 (y-b-\delta H_2)^2
 =
 (b-y)^2+2\delta(b-y)\cdot H_2+\delta^2H_2^2.
\]
It must not be replaced by the first-slot sign.

The bounded core above does not establish a finite sector cover
for the physical boundary values. In particular:
\begin{itemize}
\item supported rows impose \(r=0\);
\item \(K_{47}\) also contains the cone of \(b-z\);
\item the two \(K_5\) rows contain the cone-tip factors
\(R(a-z)\) and \(R(z-a)\).
\end{itemize}
No common physical weight for \(r\), no complete sector
decomposition, and no physical scaling degree follows solely
from Proposition~\ref{prop:shifted}.

Nor does the convolution permit pairing \(K\) with every
individual translated singular crossed factor. The exact
distributional representation used next avoids that additional
product assumption.

%=====================================================================
\section{An intrinsic complete-sum translation representation}
\label{sec:translation}
%=====================================================================

For \(K=\Kret\) and a compact joint test \(\Phi\), define
\[
 a_{I,\nu}[\Phi]
 =
 \mathfrak f_*\left(
 \sum_Jg(y)\partial_{y^\nu}\Phi_J(w)K^{I,J}(w)
 \right),
\]
\[
 d_I[\Phi]
 =
 \mathfrak f_*\left(
 \sum_Jg(y)\Phi_J(w)K^{I,J}(w)
 \right).
\]
The pre-pushforward distributions have compact support, so
these pushforwards are defined by transposition.

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}
Let \(\mathfrak d(x,v)=x-v\), and define
\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{theorem}[Exact complete-contraction representation]
\label{thm:translation}
The compact distribution \(\Omega_\Phi\) is well defined, and
\begin{equation}
 t_\delta(\Phi)=\Omega_\Phi(\rho_\delta).
 \label{eq:exacttranslation}
\end{equation}
All seven rows and all three vertex types are retained.

If \(\chi C\) has order at most \(p\), then locally
\begin{equation}
 |\Omega_\Phi(\psi)|
 \le C_L\|\Phi\|_5\|\psi\|_{p+5}
 \label{eq:Omegaorder}
\end{equation}
on a fixed compact neighborhood of its support.

Every equality of actual internal distributions is respected.
In particular, a reconstruction-null presentation gives
\(\Omega_\Phi=0\) for every test.
\end{theorem}

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

Insert \(H=\rho_\delta*(\chi C)\). The resulting tensor pairing
is
\[
 \sum_{I,\mu,\nu}
 \pair{
 \Lambda_{I,\mu\nu}[\Phi](x)\otimes\chi C_I^{\mu\nu}(v)
 }{\rho_\delta(x-v)}.
\]
The factors are compact, so a compact cutoff can be inserted
near their product support. Pushforward by \(\mathfrak d\)
gives \eqref{eq:exacttranslation}.

The internal order-four estimate gives order at most five for
\(\Lambda[\Phi]\), with a factor \(\|\Phi\|_5\).
Combining at most five \(x\)-derivatives and \(p\)
\(v\)-derivatives on \(\psi(x-v)\) gives
\eqref{eq:Omegaorder}.

Every operation starts from the actual \(K\) and is linear.
A zero reconstructed distribution therefore remains zero.
\end{proof}

\begin{corollary}[Exact scale derivative]
\label{cor:scale}
Put
\[
 \theta(h)=-8\rho(h)-h\cdot\nabla\rho(h)
          =-\operatorname{div}(h\rho(h)).
\]
Then
\[
 \int\theta=0,\qquad
 \delta\partial_\delta\rho_\delta=\theta_\delta,
\]
and
\begin{equation}
 \delta\partial_\delta t_\delta(\Phi)
 =
 \Omega_\Phi(\theta_\delta).
 \label{eq:scaled}
\end{equation}
The family is strongly smooth on compact positive-scale
intervals.
\end{corollary}

\begin{proof}
Differentiate the compact profile in its test-function topology.
On a compact positive-scale interval, all profiles and scale
derivatives have common support and uniformly bounded
seminorms. Estimate \eqref{eq:Omegaorder} justifies
differentiation uniformly on bounded joint test sets.
\end{proof}

This representation concerns the complete contraction before
any conclusion about its origin behavior.

%=====================================================================
\section{The complete origin-germ certificate}
\label{sec:certificate}
%=====================================================================

\subsection{Distributional dilation}

For \(u\in\Dist(\R^8)\) and \(s>0\), define
\[
 \pair{u(s\,\cdot)}{\psi}
 =
 s^{-8}\pair{u}{\psi(\,\cdot\,/s)}.
\]
In particular,
\[
 \pair{u(s\,\cdot)}{\rho}=u(\rho_s).
\]
This convention fixes the signs and powers in the coefficient
calculation.

\begin{definition}[Logarithmically homogeneous chain]
A chain \(H_0,\ldots,H_J\in\Dist(\R^8)\) of degree
\(\lambda\in\R\) satisfies
\begin{equation}
 H_j(s\,\cdot)
 =
 s^\lambda
 \sum_{k=0}^j\frac{(\log s)^k}{k!}H_{j-k},
 \qquad s>0.
 \label{eq:chain}
\end{equation}
The chain is defined as distributions, including at the origin.
A merely off-origin homogeneous expression is not enough.
\end{definition}

\begin{assumption}[Complete origin-germ certificate]
\label{ass:germ}
For the fixed complete translation map \(\Phi\mapsto\Omega_\Phi\),
assume:
\begin{enumerate}[label=\textup{(\arabic*)}]
\item A finite set \(\mathcal E\subset(-\infty,0]\) of distinct
real degrees and integers \(J_\lambda\ge0\) is specified.

\item For each \(\lambda\), linear maps
\[
 H_{\lambda,j}:\Test(U;V_{\mathrm{out}}^*)\longrightarrow
 \Dist(\R^8),\qquad 0\le j\le J_\lambda,
\]
give chains satisfying \eqref{eq:chain}.
For every compact joint support \(L\) and compact relative
support \(B\), there are finite orders \(m,n\) and a constant
such that
\[
 |H_{\lambda,j}[\Phi](\psi)|
 \le C_{L,B}\|\Phi\|_m\|\psi\|_n,
 \quad \supp\Phi\subset L,\quad \supp\psi\subset B.
\]

\item For every compact \(L\Subset U\), a radius \(r_L>0\)
and linear continuous-function representatives \(W_\Phi\)
on \(B(0,r_L)\) are specified such that
\begin{equation}
 \Omega_\Phi
 =
 \sum_{\lambda\in\mathcal E}
 H_{\lambda,J_\lambda}[\Phi]+W_\Phi
 \quad\text{on }B(0,r_L),
 \qquad \supp\Phi\subset L.
 \label{eq:germ}
\end{equation}

\item For some \(a_L\in(0,1]\), finite order \(m_L\), and
constant \(C_L\),
\begin{align}
 |W_\Phi(0)|
 &\le C_L\|\Phi\|_{m_L},\\
 |W_\Phi(h)-W_\Phi(0)|
 &\le C_L|h|^{a_L}\|\Phi\|_{m_L},
 \qquad |h|<r_L.
 \label{eq:Holder}
\end{align}
\end{enumerate}
The certificate concerns the entire \(\Omega_\Phi\), including
all sectors and all joint tests. Its existence is not inferred
from the cone rank, the source rule, or moderate growth.
\end{assumption}

The certificate is sufficient, not asserted to be necessary.
It supplies one concrete route to a decision without assuming
a power--logarithm expansion solely for convenience.

Its maps must be computed from the fixed package when an
evaluated application is claimed. Formula \eqref{eq:maincoeff}
is coefficient extraction from that computed germ; it is not
itself a physical evaluation of the germ.

\subsection{Bookkeeping and complete summation}

If a rowwise certificate has independently been proved, rows
can be collected into one common distributional germ:
\[
 H_{\lambda,j}
 =
 \sum_{\text{row }b,\ \text{vertex type }v}
 H_{\lambda,j;b,v}.
\]
Missing chain entries are handled by first collecting the
full distribution of each degree and forming its chain.

The conclusion depends on the resulting complete
\(A_{\lambda,k}^\rho\), not on the nonvanishing of one
\(H_{\lambda,j;b,v}\). In particular the two tensor products
in \eqref{eq:C} remain part of the complete summation.

%=====================================================================
\section{Coefficient calculation and remainder proof}
\label{sec:proofmain}
%=====================================================================

\subsection{Expansion of the family}

For a sufficiently small scale,
\(\delta\supp\rho\subset B(0,r_L)\).
Using \eqref{eq:germ} and \eqref{eq:chain},
\begin{align}
 t_\delta(\Phi)
 &=
 \sum_{\lambda\in\mathcal E}
 \delta^\lambda
 \sum_{k=0}^{J_\lambda}
 \frac{(\log\delta)^k}{k!}
 H_{\lambda,J_\lambda-k}[\Phi](\rho)
 \notag\\
 &\quad+
 \int\rho(h)W_\Phi(\delta h)\dd h.
 \label{eq:germexpansion}
\end{align}
Because \(\int\rho=1\), the continuous contribution is
\[
 W_\Phi(0)+
 \int\rho(h)\left(W_\Phi(\delta h)-W_\Phi(0)\right)\dd h.
\]
Its remainder satisfies
\begin{equation}
 \left|
 \int\rho(h)\left(W_\Phi(\delta h)-W_\Phi(0)\right)\dd h
 \right|
 \le
 C_L\delta^{a_L}\|\Phi\|_{m_L}
 \int|\rho(h)||h|^{a_L}\dd h.
 \label{eq:explicitrem}
\end{equation}
These identities establish
\eqref{eq:mainexpansion}--\eqref{eq:mainrem}.
The coefficient estimates in Assumption~\ref{ass:germ}
make every \(A_{\lambda,k}^\rho\) and \(T^\rho\) an actual
distribution on \(U\).

\subsection{A separately proved differentiated remainder}

The scale derivative is evaluated directly on
\(\theta_\delta\), rather than obtained by differentiating
\eqref{eq:explicitrem}.

By \eqref{eq:chain}, the homogeneous contributions to
\(\Omega_\Phi(\theta_\delta)\) are exactly the logarithmic
scale derivatives of their terms in
\eqref{eq:germexpansion}. The degree-zero constant term has
zero scale derivative. Therefore
\[
 \left[
 \delta\partial_\delta(t_\delta-S_\delta^\rho)
 \right](\Phi)
 =
 \int\theta(h)W_\Phi(\delta h)\dd h.
\]
Since \(\int\theta=0\),
\begin{align}
 \left|
 \left[
 \delta\partial_\delta(t_\delta-S_\delta^\rho)
 \right](\Phi)
 \right|
 &\le
 C_L\delta^{a_L}\|\Phi\|_{m_L}
 \int|\theta(h)||h|^{a_L}\dd h.
 \label{eq:explicitdrem}
\end{align}
This proves \eqref{eq:maindrem} by an independent argument.
Its bound is integrable with respect to \(\dd\delta/\delta\).

\subsection{Raw convergence or fixed-test failure}

If all coefficients in \eqref{eq:singularsum} vanish,
\eqref{eq:mainrem} gives strong convergence to \(T^\rho\).

Conversely, choose the smallest degree with a nonzero
nonsummable coefficient and, within that degree, the highest
logarithmic power whose coefficient is nonzero.
There is one fixed compact joint test \(\Phi_*\) detecting it.
For that test, the corresponding scalar term dominates all
larger degrees and all lower logarithmic powers at the same
degree.

If the degree is negative, an inverse power times a nonzero
logarithmic polynomial is unbounded. If the degree is zero,
the highest surviving logarithmic degree is at least one and
is also unbounded. The finite part and the remainder cannot
cancel this growth.

This proves all raw-family assertions of
Theorem~\ref{thm:main}. The admissibility statement is addressed
in Section~\ref{sec:admissibility}.

\subsection{Uniqueness and reconstruction-safe extraction}

\begin{proposition}[Uniqueness of the extracted asymptotics]
\label{prop:uniqueness}
For finite real negative degrees, nonnegative integer logarithmic
orders, a degree-zero logarithmic part, and a remainder tending
to zero on every test, the nonsummable coefficients and the
finite distribution are unique.

Consequently, the coefficients in Theorem~\ref{thm:main}
are intrinsic to the fixed regularized family, even if the
origin-germ decomposition itself is not unique.
\end{proposition}

\begin{proof}
Subtract two expansions and combine equal exponents.
If a nonsummable coefficient in the difference is nonzero,
the dominance argument above gives one unbounded fixed
pairing, contradicting the vanishing remainder.
Thus all nonsummable differences are zero.
Taking the limit then makes the finite distributions equal.
\end{proof}

\begin{corollary}[Reconstruction descent]
\label{cor:descent}
Two presentations of the same actual internal kernel have
identical extracted nonsummable coefficients and finite parts
whenever the theorem applies.

If a presentation reconstructs to zero, all its extracted
nonsummable coefficients and its finite part are zero.
\end{corollary}

\begin{proof}
The positive-scale families are identical by
Theorem~\ref{thm:translation}. Apply
Proposition~\ref{prop:uniqueness}.
\end{proof}

This avoids extracting an intrinsic coefficient from the highest
polynomial degree of an internal presentation. For example,
\[
 q\deltaD'(q)+\deltaD(q)=0
\]
is an exact distributional identity, although its two summands
are separately nonzero.

\subsection{An alternative certificate without a homogeneous germ}

\begin{proposition}[Direct scale-integrability]
\label{prop:Dini}
Suppose, instead of Assumption~\ref{ass:germ}, that on each
compact \(L\Subset U\),
\[
 |\Omega_\Phi(\theta_\delta)|
 \le C_L b_L(\delta)\|\Phi\|_{m_L},
 \qquad
 \int_0^{\delta_0}b_L(s)\frac{\dd s}{s}<\infty.
\]
Then the raw fixed-kernel family converges strongly, with
\[
 |(t_\delta-t_0)(\Phi)|
 \le
 C_L\|\Phi\|_{m_L}
 \int_0^\delta b_L(s)\frac{\dd s}{s}.
\]
\end{proposition}

\begin{proof}
For \(0<\delta_1<\delta_2\),
\[
 (t_{\delta_2}-t_{\delta_1})(\Phi)
 =
 \int_{\delta_1}^{\delta_2}
 \Omega_\Phi(\theta_s)\frac{\dd s}{s}.
\]
The estimate gives a Cauchy pairing and the displayed tail bound.
The limit is a distribution because a fixed positive-scale
member plus the integrable increments has a local finite-order
bound. The same estimate is uniform on bounded test sets.
\end{proof}

This alternative does not presuppose a power--logarithm form.
Like the origin-germ certificate, it still requires a complete
physical calculation when applied to the physical package.

%=====================================================================
\section{Support, profiles, admissibility, and output}
\label{sec:admissibility}
%=====================================================================

\subsection{Support requires a proved reference domain}

\begin{proposition}[Support of nonsummable coefficients]
\label{prop:support}
Suppose Theorem~\ref{thm:main} applies and the same family
has a raw limiting reference on an open set \(V\subset U\).
Then every nonsummable coefficient restricts to zero on \(V\),
and
\[
 \supp A_{\lambda,k}^\rho\subset U\setminus V.
\]
Moreover \(T^\rho|_V\) equals that reference.
\end{proposition}

\begin{proof}
Apply the fixed-test dominance argument to tests supported
in \(V\). A nonzero restricted nonsummable coefficient would
contradict convergence there. After those coefficients vanish,
the remainder estimate identifies the limit with \(T^\rho|_V\).
\end{proof}

Support in \(N\) follows only if the reference has been proved
throughout \(U\setminus N\), or if a direct localization proof
establishes that support. The supported-sector reference is not
automatically a reference for the complete kernel.

For ordinary-product agreement, one sufficient microlocal
package consists of localized nonopposing closed wavefront
bounds for each actual internal/crossed pairing and normal-topology
convergence of the smoothed crossed factors in those bounds.
Fixed-factor product convergence then gives the ordinary
contraction on that verified domain \cite{BDH}.
% Primary-source verification: ([arxiv.org](https://arxiv.org/abs/1409.7662))

A failed sufficient product criterion is not, by itself, a
nonexistence theorem.

\subsection{Fixed-profile versus profile-independent limits}

\begin{proposition}[Profile comparison under the certificate]
\label{prop:profiles}
For two normalized compact profiles using the same input package,
\begin{align}
 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).
\end{align}
The second expression is zero if degree zero is absent.
It compares finite parts; if the raw families converge,
it compares their raw limits.
\end{proposition}

\begin{proof}
Subtract \eqref{eq:maincoeff} and \eqref{eq:mainfinite}.
The value \(W_\Phi(0)\) is independent of the profile.
\end{proof}

\begin{corollary}[All-profile criterion]
\label{cor:allprofiles}
Consider all real normalized compact smooth profiles.
Under a common certificate:
\begin{enumerate}[label=\textup{(\roman*)}]
\item raw convergence for every such profile implies
\(H_{\lambda,J_\lambda}[\Phi]=0\) for every
\(\lambda<0\) and every \(\Phi\);
\item it also implies
\(H_{0,j}[\Phi]=0\) for \(j<J_0\);
\item given these conditions, profile independence of the
raw limit holds exactly when
\(H_{0,J_0}[\Phi]\) is a constant distribution in the
translation variable, for every \(\Phi\).
\end{enumerate}
The statements about degree zero are omitted when that degree
is absent.
\end{corollary}

\begin{proof}
A distribution vanishing on all normalized compact tests is zero:
differences of normalized tests span the zero-integral test
subspace, and adjoining one normalized test spans the full
test space. Apply this observation to the coefficients whose
vanishing is required by raw convergence.

For profile independence, a distribution has equal values on
all normalized compact tests exactly when it annihilates all
zero-integral tests. It is then a constant multiple of the
integration functional.
\end{proof}

Convergence for one profile does not imply any of these
all-profile statements.

\subsection{Authorized subtraction and normalization}

Let \(Z=U\setminus V\), where \(V\) is a proved reference domain.
Suppose the frozen problem supplies an actual linear ambiguity
space
\[
 \mathcal A\subset\Dist_Z(U;V_{\mathrm{out}})
\]
with all required tensor, support, growth, and wavefront
restrictions. This notation does not specify that physical
space; it exposes the needed input.

\begin{proposition}[Conditional admissible finite part]
\label{prop:authorized}
Suppose:
\begin{enumerate}[label=\textup{(\arabic*)}]
\item Theorem~\ref{thm:main} applies.
\item The reference on \(V\) is proved.
\item The scale-dependent subtraction \(S_\delta^\rho\) is
authorized by the actual correction rules.
\item The finite part \(T^\rho\) satisfies the required
intermediate microlocal and compatibility conditions.
\end{enumerate}
Then \(T^\rho\) is an extension of that reference obtained by the
authorized subtraction, and its candidate normalized extensions
in the supplied ambiguity class are
\[
 T^\rho+\mathcal A.
\]

For a specified linear normalization map \(\mathcal N\),
a normalization
\[
 \mathcal N(T^\rho+a)=0,\qquad a\in\mathcal A,
\]
exists exactly when
\[
 -\mathcal N(T^\rho)\in\mathcal N(\mathcal A).
\]
When a solution exists, it is unique exactly when
\[
 \ker(\mathcal N|_{\mathcal A})=\{0\}.
\]
\end{proposition}

\begin{proof}
Theorem~\ref{thm:main} constructs the subtracted limit.
Proposition~\ref{prop:support} gives agreement on \(V\).
Authorization and the stated compatibility conditions supply
the additional admissibility requirements.

The normalization equation is the linear equation
\(\mathcal N(a)=-\mathcal N(T^\rho)\).
Its solvability and uniqueness are precisely the displayed
image and kernel conditions.
\end{proof}

No finite-dimensionality of \(\mathcal A\) is assumed.
A normal-order bound alone does not establish it.

Distribution-extension theory requires its own scaling or
growth hypotheses \cite{BF}; it does not identify a subtraction
as an allowed physical correction.
% Primary-source verification: ([arxiv.org](https://arxiv.org/abs/math-ph/9903028))
Microlocal extension results additionally impose hypotheses
controlling the extension's wavefront set \cite{Dang}.
% Primary-source verification: ([arxiv.org](https://arxiv.org/abs/1412.2808))

\begin{proposition}[Corrections vanishing on the patch]
\label{prop:contact}
If every permitted correction restricts to zero on \(U\), then
a fixed-test raw divergence supported in \(U\) cannot be removed
by those corrections.
\end{proposition}

\begin{proof}
Every such correction pairs to zero with every test supported
in \(U\), at every scale. It leaves the detecting pairing
unchanged.
\end{proof}

\subsection{Fourier bounds and the output operation}

The finite-order remainder estimates alone do not establish a
wavefront bound.

\begin{proposition}[An explicit microlocal certificate]
\label{prop:Fourier}
Let
\[
 v_\delta=t_\delta-S_\delta^\rho,
 \qquad v_\delta\longrightarrow T^\rho
\]
strongly as proved above. Fix a closed joint cone
\(\Gamma\subset T^*U\setminus0\).
Suppose, for each localization \(\zeta\) and closed frequency
cone \(V_0\) with
\[
 (\supp\zeta\times V_0)\cap\Gamma=\varnothing,
\]
and each integer \(r\ge0\), that
\[
 \sup_{\xi\in V_0}
 (1+|\xi|)^r|\widehat{\zeta v_\delta}(\xi)|
 \le C_{r,\zeta,V_0}
\]
uniformly for small \(\delta\).

Then \(\WF(T^\rho)\subset\Gamma\), and \(v_\delta\) converges
in the normal topology associated with \(\Gamma\).
\end{proposition}

\begin{proof}
For each fixed \(\xi\), strong convergence gives convergence
of the localized Fourier transform. Passing to the limit in
every weighted bound gives rapid decay off \(\Gamma\), hence
the wavefront inclusion.

For convergence of an order-\(r\) Fourier seminorm, use the
uniform order-\(r+1\) bound to control frequencies above a
radius \(R\) by \(C/(1+R)\). Below that radius the localized
exponential tests form a bounded test set, so strong
convergence is uniform. First choose \(R\) large, then take
\(\delta\) small.
\end{proof}

After multiplying an existing joint distribution by a compact
local cutoff, the projection
\[
 \pi(a,b,z,y)=(z,y)
\]
has a defined pushforward. This operation cannot construct a
missing joint product or supply an unproved output cone.
Overlap agreement and quantum admission remain separate from
the fixed-kernel limit.

%=====================================================================
\section{Completed results and the remaining physical application}
\label{sec:conclusion}
%=====================================================================

The main mathematical result is
Theorem~\ref{thm:main}: a conditional decision for one fixed,
complete family, with intrinsic coefficient extraction and
separately proved family and differentiated remainders.

The revision also supplies:
\begin{enumerate}[label=\textup{(\roman*)}]
\item the finite source generator and propagated-output conversion;
\item the density derivative, with its identification condition;
\item the full seven-row, three-vertex inventory;
\item exact characteristic coordinates, density, and shifted cones;
\item a complete translation representation using only defined
tensor products and compact pushforwards;
\item coefficient uniqueness, reference-dependent support,
profile comparison, and normalization criteria;
\item an evaluated nonphysical diagnostic in the appendix.
\end{enumerate}

For the designated physical graph, two distinct tasks remain:
\begin{enumerate}[label=\textup{(\arabic*)}]
\item identify the actual covariance and density-map package,
without substituting diagnostic data;
\item prove a complete origin-germ certificate, direct
scale-integrability bound, or another controlled complete
asymptotic calculation for that package.
\end{enumerate}
Neither task is discharged by relabeling an arbitrary parameter
as physical. In particular, the appendix's
\(\delta^{-3},\delta^{-2},\delta^{-1}\) powers are not assigned
to the physical family.

Thus this manuscript releases a self-contained conditional
theorem and a completed diagnostic, but not an evaluated
physical convergence or failure verdict, an authorized physical
extension, or a normalized quantum coefficient.

%=====================================================================
\appendix
\section{Exact nonphysical eight-variable diagnostic}
\label{app:diagnostic}
%=====================================================================

This appendix is a methodological check of convolution,
derivative placement, signs, density factors, and remainder
control. Its crossed distributions and density arrays are
explicitly nonphysical.

\subsection{Data and actual convolution}

Fix
\[
 \ell=(1,0,0,1),\qquad
 k=(1,0,0,-1),\qquad
 \kappa=(1,0,0,-1)\in(\R^4)^*.
\]
Then
\[
 \kappa(x)=x^0-x^3,\quad
 \kappa(\ell)=0,\quad
 \kappa(k)=2,\quad
 \kappa^\sharp=\ell.
\]

Choose nonnegative \(p\in C_c^\infty(\R)\) of integral one,
and compact smooth \(\psi,\omega\), of integral one on
\(\R,\R^2\), respectively.
For \(h=A\ell+uk+X\), put
\[
 \rho_4(h)=p(2u)\psi(A)\omega(X),
 \qquad
 \rho=\rho_4\otimes\rho_4.
 \label{eq:diagnosticprofile}
\]
Since \(\dd^4h=2\dd A\,\dd u\,\dd^2X\), the profile is
normalized and its \(\kappa\)-marginal is \(p\).

Choose one tensor-component label \(I_*\) for both incoming
and residual slots, and set
\[
 H_{T,\gamma}^{\mu\nu}{}_I(x)
 =
 \delta_{I,I_*}\eta^{\mu\nu}\deltaD(\kappa(x)),
 \qquad
 H_{T,\phi}(x)=\deltaD(\kappa(x)),
\]
\[
 H_\gamma=0,\qquad H_\phi=0.
 \label{eq:diagnosticlines}
\]
Set \(m=0\), take
\[
 f=1\ \text{near }\overline{B_0},
 \qquad
 g(y)=y^0\ \text{near }\overline Y,
\]
and keep the source and all remaining internal data fixed.
Compare \(\Dmap_0=0\) with the array whose sole nonzero
entry is
\[
 \Dmap_1^{00,I_*,I_*}=1.
 \label{eq:diagnosticD}
\]
Using the same slot label also makes this diagnostic array
symmetric in its two input slots.

\begin{lemma}[Exact smoothed crossed array]
\label{lem:diagnosticC}
For sufficiently small scales on the relevant relative image,
\[
 (C_\delta)_I^{\mu\nu}(x_1,x_2)
 =
 \delta_{I,I_*}\eta^{\mu\nu}\delta^{-2}
 p\left(\frac{\kappa(x_1)}{\delta}\right)
 p\left(\frac{\kappa(x_2)}{\delta}\right).
 \label{eq:diagnosticC}
\]
\end{lemma}

\begin{proof}
The \(\kappa\)-marginal of \((\rho_4)_\delta\) is
\(\delta^{-1}p(\,\cdot\,/\delta)\). Hence
\[
 (\rho_4)_\delta*\deltaD(\kappa(\,\cdot\,))
 =
 \delta^{-1}p(\kappa(\,\cdot\,)/\delta).
\]
The product profile makes the eight-variable convolution the
tensor product of these convolutions.

The relative cutoff is one near the compact image needed by
the test. Compact profile support gives exact local equality
with the uncut convolution at sufficiently small scales,
including its required derivatives.
\end{proof}

\subsection{Difference of complete families}

Define
\[
 u_\delta=t_\delta^{\Dmap_1}-t_\delta^{\Dmap_0}.
\]
Only \(K_6\) differs, but both sides are complete seven-row
families.

\begin{lemma}[Exact diagnostic difference]
\label{lem:diagnosticdifference}
Put
\[
 s=\kappa(y-a),\qquad t=\kappa(b-a).
\]
Then
\begin{equation}
 u_\delta(\Phi)
 =
 i\pair{R(b-a)\deltaD^{(4)}(a-z)}{
 \partial_{b^0}^{\,2}
 \left[
 \delta^{-3}p(s/\delta)p'((s-t)/\delta)
 \Phi_{I_*}(a,b,z,y)
 \right]}.
 \label{eq:diagnosticdifference}
\end{equation}
\end{lemma}

\begin{proof}
The kernel difference is
\[
 K_6^{I_*,I_*}(\Dmap_1)-K_6^{I_*,I_*}(\Dmap_0)
 =
 -\partial_{b^0}^{\,2}R(b-a)\deltaD^{(4)}(a-z).
\]
The derivative \(\partial_{a^0}\) in \(K_6\) acts on \(R\)
alone, not on the incidence.

For \eqref{eq:diagnosticC}, the kinetic/trace-kinetic combination
is
\[
 Z_{I_*}^\nu(C_\delta)
 =
 -\ell^\nu\delta^{-3}p(s/\delta)p'((s-t)/\delta).
\]
The kinetic tensor contraction gives \(\ell^\nu\), while the
trace contribution subtracts \(2\ell^\nu\).
The derivative along \(\ell\) annihilates both profile arguments
and the \(y\)-independent internal kernel.
Since \(\ell^\nu\partial_\nu g=1\), the output divergence
leaves exactly the stated product. Transposing the two
\(b^0\)-derivatives proves \eqref{eq:diagnosticdifference}.
\end{proof}

\subsection{Coordinates and the exact measure}

Use fixed-frame coordinates
\[
 b=a+B\ell+uk+X,\qquad
 y=a+A\ell+vk+Y,\qquad z=a+r,
\]
with \(B\) in a compact positive range on the patch.
Then
\[
 (b-a)^2=4Bu-|X|^2,\qquad t=2u,\qquad s=2v.
\]
Set
\[
 \tau=(a,B,A,Y),\qquad
 \dd\tau=\dd^4a\,\dd B\,\dd A\,\dd^2Y,
\]
and
\[
 S_\kappa
 =
 \left\{
 z=a,\quad b=a+B\ell,\quad y=a+A\ell+Y
 \right\}\cap U.
\]
This has codimension eight, but is not the physical
aligned-null locus \(N\): \(Y\) need not vanish.

Tests are extended by zero outside their compact support in
\(U\). Define
\[
 F_j(\tau;u,v,X)
 =
 (\partial_{b^0}^{\,j}\Phi_{I_*})
 \left(
 a,a+B\ell+uk+X,a,a+A\ell+vk+Y
 \right),
 \qquad j=0,1,2.
\]
The \(b^0\)-derivatives are ambient derivatives taken before
evaluation. Put
\[
 c_0=1,\qquad c_1=-2,\qquad c_2=1,
\]
\[
 P_j(S,T)=p(S)p^{(3-j)}(S-T),\qquad T\ge0.
\]

\begin{lemma}[Exact rescaled integral]
\label{lem:diagnosticrescale}
For sufficiently small scales,
\begin{align}
 u_\delta(\Phi)
 =\frac{i}{2}\sum_{j=0}^2c_j\delta^{j-3}
 \int\dd\tau\int_\R\dd S\int_0^\infty\dd T\
 P_j(S,T)
 \notag\\
 {}\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.
 \label{eq:diagnosticrescale}
\end{align}
\end{lemma}

\begin{proof}
Expansion of the derivative in
\eqref{eq:diagnosticdifference} gives
\[
 \sum_{j=0}^2c_j\delta^{j-5}
 p(s/\delta)p^{(3-j)}((s-t)/\delta)
 \partial_{b^0}^{\,j}\Phi_{I_*}.
\]
The incidence integrates out \(z\).

The two endpoint coordinate densities contribute
\[
 \dd^4b=2\dd B\,\dd u\,\dd^2X,\qquad
 \dd^4y=2\dd A\,\dd v\,\dd^2Y.
\]
The future retarded cone is
\((2\pi)^{-1}\deltaD(4Bu-|X|^2)\).
Its \(u\)-integration gives the remaining measure
\[
 \frac1{2\pi B}\dd^2X\,\dd v
\]
in addition to \(\dd\tau\).

Write \(Q=|X|^2\) and include angular averaging, so
\(\dd^2X=\pi\dd Q\).
Set \(t=Q/(2B)\) and \(s=2v\). The exact resulting factor is
\(\dd s\,\dd t/2\).
Finally \(s=\delta S,t=\delta T\) supplies \(\delta^2\).
This proves \eqref{eq:diagnosticrescale}.
\end{proof}

\subsection{All diagnostic coefficients}

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:diagnosticLn}
\end{equation}

\begin{lemma}[Uniform angular Taylor formula]
\label{lem:diagnosticTaylor}
For \(M\ge0\),
\begin{align}
 &\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
 \notag\\
 &\qquad=
 \sum_{n=0}^M\delta^n[\mathcal L_n(B,S,T)F]_0
 +O\left(\delta^{M+1}\|F\|_{2M+2}\right),
 \label{eq:diagnosticTaylor}
\end{align}
uniformly on compact positive \(B\)-ranges and compact
\(S,T\)-ranges with \(T\ge0\).
\end{lemma}

\begin{proof}
Taylor-expand in \(X\) through order \(2M+1\).
Odd angular moments vanish. The averaged order-\(2c\)
term is
\[
 \frac{(R^2/4)^c}{(c!)^2}\Delta_X^cF,
 \qquad R^2=2B\delta T.
\]
This follows by integrating the even monomials on the unit
circle.

Expand each such coefficient in \((u,v)\) through total
order \(M-c\). Collecting scale order \(n=a+b+c\)
gives \eqref{eq:diagnosticLn}.
The transverse remainder is \(O(R^{2M+2})\).
The longitudinal remainder for the \(c\)-term is
\(O(\delta^{M-c+1})\) multiplied by \(\delta^c\).
All errors are \(O(\delta^{M+1})\) with the stated order.
\end{proof}

For \(e=-3,-2,-1,0\), define
\begin{align}
 \mathcal B_e(\Phi)
 =
 \frac{i}{2}
 \sum_{\substack{0\le j\le2\\n=e+3-j\ge0}}
 c_j\int\dd\tau\int_\R\dd S\int_0^\infty\dd T\
 P_j(S,T)
 [\mathcal L_n(B,S,T)F_j]_0.
 \label{eq:diagnosticcoeff}
\end{align}
For these exponents, \(n\le3-j\). Thus the formula is finite.

The profile integrals can all be reduced explicitly.
Let
\[
 P(S)=\int_{-\infty}^S p(V)\dd V,
\]
and for \(d\ge1\), \(0\le b\le d\), set
\[
 \mathcal M_d^{a,b}
 =
 \int_\R\int_0^\infty
 S^aT^b p(S)p^{(d)}(S-T)\dd T\,\dd S.
\]

\begin{lemma}[Evaluation of every required profile moment]
\label{lem:diagnosticmoments}
The moments are
\begin{equation}
 \mathcal M_d^{a,b}
 =
 \begin{cases}
 \displaystyle
 b!\int_\R S^ap(S)p^{(d-b-1)}(S)\dd S,
 &b<d,\\[0.8em]
 \displaystyle
 d!\int_\R S^ap(S)P(S)\dd S,
 &b=d.
 \end{cases}
 \label{eq:diagnosticmoments}
\end{equation}
Every moment in \eqref{eq:diagnosticcoeff} is of this type,
with \(d=3-j\), \(a\) the power of \(S\), and \(b\) the
power of \(T\).
\end{lemma}

\begin{proof}
For \(b=0\),
\[
 \int_0^\infty p^{(d)}(S-T)\dd T=p^{(d-1)}(S).
\]
For \(b>0\), integration by parts gives the recursion
\[
 \int_0^\infty T^bp^{(d)}(S-T)\dd T
 =
 b\int_0^\infty T^{b-1}p^{(d-1)}(S-T)\dd T.
\]
If \(b<d\), iterate \(b\) times and use the first identity.
If \(b=d\), iterate \(d\) times, leaving \(d!P(S)\).
Multiplication by \(S^ap(S)\) gives
\eqref{eq:diagnosticmoments}.

In \(\mathcal L_n\), the \(T\)-power is at most \(n\),
and \(n\le3-j=d\), proving the final assertion.
\end{proof}

Equivalently, each summand of \(\mathcal B_e\) is obtained by
replacing its polynomial \(S,T\)-factor in
\eqref{eq:diagnosticLn} by the corresponding moment
\eqref{eq:diagnosticmoments}. This evaluates all coefficients
without a two-dimensional residual profile integral.

\begin{theorem}[Complete diagnostic expansion]
\label{thm:diagnostic}
The difference of complete families satisfies
\begin{equation}
 u_\delta
 =
 \delta^{-3}\mathcal B_{-3}
 +\delta^{-2}\mathcal B_{-2}
 +\delta^{-1}\mathcal B_{-1}
 +\mathcal B_0+\mathcal R_\delta,
 \label{eq:diagnosticexpansion}
\end{equation}
with coefficients \eqref{eq:diagnosticcoeff} and their
evaluated moments \eqref{eq:diagnosticmoments}.
For every \(L\Subset U\),
\[
 |\mathcal R_\delta(\Phi)|
 \le C_L\delta\|\Phi\|_8,
 \qquad \supp\Phi\subset L,
\]
at sufficiently small scales.

Every coefficient is supported in \(S_\kappa\). In particular,
\begin{equation}
 \mathcal B_{-3}(\Phi)
 =
 -\frac{i}{2}\|p'\|_{L^2}^2
 \int
 \Phi_{I_*}(a,a+B\ell,a,a+A\ell+Y)\dd\tau.
 \label{eq:diagnosticleading}
\end{equation}
This leading coefficient is nonzero on every chart containing
a nonempty portion of \(S_\kappa\).
\end{theorem}

\begin{proof}
The profiles \(P_j\) have a common compact \(S,T\)-support:
\(S\) lies in \(\supp p\), and \(S-T\) lies in the support
of a derivative of \(p\). Thus \(T\) is bounded.

Apply Lemma~\ref{lem:diagnosticTaylor} to the \(j\)-term in
\eqref{eq:diagnosticrescale}, with \(M=3-j\).
Before multiplication, the error is
\[
 O\left(\delta^{4-j}\|F_j\|_{8-2j}\right).
\]
Multiplication by \(\delta^{j-3}\) gives \(O(\delta)\).
Since \(F_j\) contains \(j\) ambient derivatives, the total
test order is at most \(8-j\), hence at most eight.
For small scales the tangential support stays in a fixed
compact set; this follows from compact test support, bounded
profile support, and the positive bounded \(B\)-range.
The estimate is therefore uniform for all tests supported
in \(L\).

Collecting \(e=j-3+n\) gives
\eqref{eq:diagnosticcoeff}. All coefficient functionals use
finite jets along \(S_\kappa\), proving support there.

For \(e=-3\), only \(j=n=0\) occurs. Its moment is
\[
 \int_\R p(S)\int_0^\infty p'''(S-T)\dd T\,\dd S
 =
 \int_\R p(S)p''(S)\dd S
 =
 -\|p'\|_{L^2}^2.
\]
This proves \eqref{eq:diagnosticleading}.
A compact normalized smooth profile cannot have derivative
identically zero. A test with positive nonzero restriction
to the attachment detects the coefficient.
\end{proof}

\begin{corollary}[Fixed-test failure and finite part]
\label{cor:diagnostic}
There is one fixed compact test \(\Phi_*\) for which
\[
 |u_\delta(\Phi_*)|\longrightarrow\infty.
\]
Hence the two complete mathematical families cannot both
have raw limits.

The explicitly subtracted difference
\[
 u_\delta
 -\delta^{-3}\mathcal B_{-3}
 -\delta^{-2}\mathcal B_{-2}
 -\delta^{-1}\mathcal B_{-1}
\]
converges strongly to \(\mathcal B_0\), with the stated
order-eight \(O(\delta)\) rate.
\end{corollary}

\begin{proof}
Choose \(\Phi_*\) detecting \(\mathcal B_{-3}\).
Then
\[
 \delta^3u_\delta(\Phi_*)
 \longrightarrow\mathcal B_{-3}(\Phi_*)\ne0.
\]
The difference could not diverge if both original families
converged. The finite-part conclusion follows directly from
the remainder estimate.
\end{proof}

All common internal rows cancel exactly in this comparison
at every positive scale. Thus it is a complete-family
comparison, not a proof of survival of an isolated branch
in the physical sum.

The coefficients are supported in \(S_\kappa\), not asserted
to be supported only in \(N\). Their finite subtraction is
not automatically a permitted physical correction.

%=====================================================================
\section{Verification, debugging, and documentary record}
\label{app:verification}
%=====================================================================

\subsection{Finite source implementation}

The source specification can be reproduced without selecting
a different density:
\begin{enumerate}[label=\textup{(\arabic*)}]
\item choose the basis and algebraic dual in \eqref{eq:duality};
\item expand \(q_s^{-1}\), \(\sqrt{-\det q_s}\), and the
declared curvature only through the order needed for
\([s^3]\);
\item identify commuting mixed field derivatives by ordinary
multiindices;
\item form the coordinate Euler derivatives of
\(\sigma\ell_3\), retaining switch derivatives;
\item apply \eqref{eq:sourceconversion};
\item extract \eqref{eq:sourcehessian};
\item impose the plateau only after the local source has been
formed;
\item contract the full branch--vertex inventory in a common
density and test convention.
\end{enumerate}

The mathematical checks are:
slot symmetry, total derivative weight, repeated-variable
factors, basis covariance, and
\[
 D\mathcal J(G)[h]=D^2\mathcal J(0)[G,h].
\]
No executed component output is claimed.

\subsection{Derivative placement}

The expression
\[
 \partial_{a^\alpha}R(b-a)\deltaD^{(4)}(a-z)
\]
is not
\[
 \partial_{a^\alpha}
 \left[R(b-a)\deltaD^{(4)}(a-z)\right].
\]
The difference is an incidence derivative. This distinction
is essential in the diagnostic and must be preserved in
the physical calculation.

Leibniz derivatives in \eqref{eq:response}, on the other hand,
are part of the exact response and may not be discarded.

\subsection{Remainder differentiation}

A bound \(r_\delta=O(\delta)\) alone does not imply a small
logarithmic scale derivative. For example, for a fixed
nonzero distribution \(A\),
\[
 r_\delta=\delta\sin(\delta^{-2})A
\]
has an \(O(\delta)\) family bound, whereas
\[
 \delta\partial_\delta r_\delta
 =
 \left[
 \delta\sin(\delta^{-2})
 -2\delta^{-1}\cos(\delta^{-2})
 \right]A.
\]
The differentiated estimate in the main theorem was therefore
proved directly using the exact scale profile.
No differentiated remainder theorem is inferred for the
appendix merely from its \(O(\delta)\) remainder.

\subsection{Execution record}

No pdfLaTeX compiler, symbolic algebra system, numerical
integration program, or proof assistant was executed.
The file is arranged for a standalone two-run pdfLaTeX build.
Successful compilation, mechanically certified identities,
independent referee verification, and journal acceptance are
not represented as observed results.

The supplied manuscripts are used with their printed domains
and scope restrictions. Registry summaries are not treated as
physical input instantiations or independent proof verification.

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

\bibitem{Em01B}
\emph{Em01-B: Intrinsic Regularization of the Joint
Characteristic Map}.
Supplied manuscript source \texttt{Em01-B-v3.tex}.
Complete retained kernel architecture, vertex, regularization,
moderate-growth estimate, and conditional limit criteria.

\bibitem{Em01C}
\emph{Em01-C: Characteristic Normal Forms and
Complete-Sum Scale Reduction}.
Supplied manuscript source \texttt{Em01-C-v3.tex}.
Source-output convention, finite source rule, local plateau
reduction, characteristic coordinates, translation representation,
and separately defined scalar asymptotics.

\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.
arXiv:1409.7662.
doi:10.4064/sm8316-3-2016.
Normal topology and continuity or hypocontinuity of fundamental
distributional operations under their wavefront hypotheses.

\bibitem{Dang}
N.~V.~Dang,
\emph{The extension of distributions on manifolds,
a microlocal approach}.
arXiv:1412.2808.
Geometric scaling and microlocal conditions for extension
across an embedded submanifold.

\bibitem{BF}
R.~Brunetti and K.~Fredenhagen,
``Microlocal Analysis and Interacting Quantum Field Theories:
Renormalization on Physical Backgrounds,''
\emph{Communications in Mathematical Physics}
\textbf{208} (2000), 623--661.
arXiv:math-ph/9903028.
doi:10.1007/s002200050004.
Distribution-extension and perturbative renormalization framework.

\end{thebibliography}

\end{document}