% !TeX program = pdflatex
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% Em01-C-v3.tex
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% Standalone pdfLaTeX source. No external files are required.
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%   pdflatex -interaction=nonstopmode -halt-on-error Em01-C-v3.tex
%   pdflatex -interaction=nonstopmode -halt-on-error Em01-C-v3.tex
%
% Revision C.3:
%   - makes the tensor basis, dual basis, and source conversion explicit;
%   - distinguishes the tensor Euler derivative from its coordinate
%     gradient and from the metric-valued source used in propagation;
%   - supplies a compact implementation inventory for the source rule;
%   - prints the unreduced coordinate-response expression and separates
%     its inherited physical interpretation from the proved reduction;
%   - provides an explicit finite response-kernel reduction;
%   - retains all seven internal rows and all three vertex types;
%   - states microlocal agreement with its actual convergence hypotheses;
%   - preserves the scalar singular-plus-finite expansion and its rate;
%   - corrects the documentary record without asserting reviewer access,
%     full-corpus inspection, or authentication.
%
% Scope:
%   Exact source specification, local plateau reduction, characteristic
%   geometry, complete-sum translation reduction, and scalar asymptotics.
%
%   The complete physical characteristic coefficient and its remainder
%   are not evaluated. No physical convergence, divergence, or quantum
%   admission is asserted.
%
% Verification:
%   No compiler, proof assistant, or symbolic algebra execution is claimed.
%=====================================================================

\documentclass[11pt,a4paper]{article}

\usepackage[T1]{fontenc}
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\usepackage{microtype}
\usepackage{amsmath,amssymb,amsthm,mathtools,mathrsfs}
\usepackage{array,booktabs,tabularx}
\usepackage{enumitem}
\usepackage[hidelinks]{hyperref}

\hypersetup{
  pdftitle={Em01-C: Characteristic Normal Forms and Complete-Sum Scale Reduction},
  pdfauthor={}
}

\allowdisplaybreaks
\setlength{\emergencystretch}{3em}
\setlist{leftmargin=*,itemsep=3pt,topsep=5pt}
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\numberwithin{equation}{section}
\setcounter{tocdepth}{1}

\newtheorem{theorem}{Theorem}[section]
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\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}

\newcommand{\R}{\mathbb R}
\newcommand{\C}{\mathbb C}
\newcommand{\dd}{\,\mathrm d}
\newcommand{\eps}{\varepsilon}
\newcommand{\sig}{\sigma_\eps}
\newcommand{\Test}{\mathcal D}
\newcommand{\Dist}{\mathcal D'}
\newcommand{\EDist}{\mathcal E'}
\newcommand{\PP}{\mathcal P}
\newcommand{\Amap}{\mathsf A}
\newcommand{\bmap}{\mathsf b}
\newcommand{\JD}{\mathcal J_{\mathrm D}}
\newcommand{\Kret}{K_{\mathrm{ret}}}
\newcommand{\Ksup}{K_{\mathrm{supp}}}
\newcommand{\Sresp}{S_{\mathrm{resp}}}
\newcommand{\Vertex}{\mathsf V}
\newcommand{\Fmap}{\mathsf F}
\newcommand{\Rec}{\mathsf{Rec}}
\newcommand{\Ucan}{U_{\mathrm{can}}}
\newcommand{\pair}[2]{\left\langle #1,#2\right\rangle}
\newcommand{\norm}[1]{\lVert #1\rVert}
\newcommand{\id}{\mathrm{id}}
\newcommand{\dirac}{\delta_{\mathrm D}}
\newcommand{\Gfib}{\mathbf G}

\DeclareMathOperator{\supp}{supp}
\DeclareMathOperator{\WF}{WF}
\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-C}\\[0.4em]
Characteristic Normal Forms and\\
Complete-Sum Scale Reduction\\[0.4em]
\large Explicit source conventions, exact local geometry,\\
and scalar singular asymptotics}
\author{}
\date{}

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

\begin{abstract}
We study a local regularization of the complete retained
Einstein--scalar contraction while keeping its internal kernel exact.
The aligned-null locus admits explicit normal coordinates in which
the three cone functions have differential rank two along that locus.
Their restriction to the supported response incidence has rank one,
with a definite quadratic transverse defect. We compute the coordinate
inverse, ambient density, and a weighted dilation adapted to these
equations.

The density-before-variation source has a finite symbolic
specification. An explicit tensor-to-component conversion and a
jet-polynomial Hessian determine all source coefficients without a
printed component list. On the retained chronological patch, the
source-ramp terms disappear by locality. A displayed unreduced
coordinate-response expression also reduces exactly by causal support
and the existing retarded convolution.

Transposition of the complete mixed vertex produces a compact,
test-dependent distribution in a relative translation variable.
The regularized contraction, dyadic increments, and logarithmic
scale derivatives are exactly its evaluations on specified compact
profiles. This construction retains every internal row and vertex
term and respects all reconstruction identities.

For a separate scalar cone regularization, we compute an order-zero
limit with an order-one convergence rate. Its response-normal
derivative has an explicit inverse-scale coefficient, an explicit
finite distribution, and an order-one remainder. A single fixed test
detects divergence whenever that coefficient is nonzero.

The scalar regularization differs from the physical
eight-relative-variable smoothing. The complete physical origin
coefficient and its controlled remainder remain unevaluated.
Accordingly, the results are geometric, symbolic, and analytic
reductions, not a convergence or failure theorem for the complete
physical family.
\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) {Characteristic coordinates};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Scalar inverse-scale example};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Physical complete-sum coefficient};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {The scalar calculation does not fill the physical coefficient.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{The scalar diagnostic is not the physical coefficient. 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-C}
\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) {Characteristic coordinates};
\node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {Physical complete-sum coefficient};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {The scalar calculation does not fill the physical coefficient.};
\draw[->] (a) -- (s);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{The scalar diagnostic is not the physical coefficient. 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-C}
\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) {Characteristic coordinates};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Scalar inverse-scale example};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {Physical complete-sum coefficient};
\node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {The scalar calculation does not fill the physical coefficient.};
\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-C}
\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) {Characteristic coordinates};
\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 complete-sum coefficient};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Scalar inverse-scale example. The scalar calculation does not fill the physical coefficient.};
\draw (a) -- (m);
\draw (m) -- (b);
\draw[->] (m) -- (s);
\end{tikzpicture}
\caption{Characteristic coordinates is not Physical complete-sum coefficient. 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-C}
\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: Characteristic coordinates};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Scalar inverse-scale example};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: Physical complete-sum coefficient};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {The scalar calculation does not fill the physical coefficient.};
\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-C}
\end{figure}

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


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



\scopebox{Submission scope}{%
The proved results concern the finite source specification, exact
local plateau reduction, characteristic coordinates and density,
complete-sum translation representation, and the explicitly defined
scalar asymptotics.

\smallskip
The source-specific characteristic estimate for
\(\Fmap_\delta(\Kret)\) is not proved. Neither a scalar diagnostic
nor a predecessor stopping label supplies that estimate. No quantum
admission is asserted.}

\tableofcontents

%=====================================================================
\section{The fixed-kernel problem and its specified inputs}
\label{sec:scope}
%=====================================================================

\subsection{The question}

The target is the existence of a limit
\begin{equation}
 t_\delta=\Fmap_\delta(\Kret),
 \qquad \delta\downarrow0,
 \label{eq:target}
\end{equation}
on arbitrary compact joint tests, including tests crossing the
aligned-null characteristic locus \(N\). The regularization is defined
explicitly in Section~\ref{sec:expression}.

This is a fixed-kernel problem. It is distinct from constructing a
continuous characteristic operator on an enlarged internal module:
a cancellation for the designated source need not persist under
arbitrary changes of its coefficient arrays.

A substantive physical conclusion would be one of:
\begin{enumerate}[label=\textup{(\roman*)}]
\item a complete-sum Cauchy estimate;
\item an evaluated complete-sum singular expansion with a controlled
remainder;
\item a fixed-test failure theorem for the specified unmodified
regularization.
\end{enumerate}
The present paper does not decide among these outcomes.

\subsection{Definition, evaluation, and provenance}

Three different qualifications are used below:
\begin{description}[style=nextline]
\item[Specified but unevaluated.]
A finite, unambiguous formula is available, but its component
expansion or its contraction in the characteristic calculation has
not been performed.

\item[Fixed input not instantiated here.]
An array or distribution is part of the mathematical data of the
paper. Its particular physical realization is not independently
derived here.

\item[Inherited interpretation.]
A predecessor identifies a displayed expression with a branch of the
physical graph. That interpretation is not inferred merely because
two manuscripts use similar notation.
\end{description}

A reproducible symbolic specification suffices to define coefficients.
An explicit component list is not inherently required. Conversely,
a component expansion does not replace a missing boundary-value
prescription.

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.25\linewidth}p{.24\linewidth}Y@{}}
\toprule
Object & Mathematical status & Treatment here\\
\midrule
Curvature and cubic density &
Finite specification &
Reproduced in Appendix~\ref{app:source}, without changing the
unintegrated representative.\\[0.3em]
Weighted Euler source &
Finite specification &
Derived by coefficient extraction and converted explicitly to
the propagated component convention.\\[0.3em]
\(q_{\alpha\beta}^{L,I,J}\) &
Specified jet coefficients &
Determined by \eqref{eq:jet-Hessian}; their complete graph
contraction is unevaluated.\\[0.3em]
\(\PP,\Amap,\bmap\) &
Explicit linear maps &
Their matrices and their component contractions are fixed below.\\[0.3em]
\(\mathsf D\) &
Fixed constant bilinear map &
Retained as an input; no physical component realization is
independently derived.\\[0.3em]
Crossed array \(C\) &
Fixed distributional input &
Its grouped tensor-product form is retained; physical component
boundary values are not evaluated.\\[0.3em]
Complete characteristic coefficient &
Open calculation &
Not supplied by the source rule or the scalar model.\\
\bottomrule
\end{tabularx}
\end{center}

The source and the covariance package are separate inputs.
Fixing the curvature convention does not instantiate the crossed
covariances.

\subsection{Self-contained setting and relation to predecessors}

The equations printed below define the mathematical instance studied
in this paper. Identifying that instance with a predecessor's frozen
graph requires the same source-output conversion, component pairings,
constant map \(\mathsf D\), crossed array, and vertex normalization.
Those identifications are not consequences of notation alone.

The source specification follows XII-C \cite{XIIC}.
The internal formulas follow the displayed retained architecture in
Em01-B-v3, XIV-H, and XIV-J \cite{Em01B,XIVH,XIVJ}.
All operations used in the new calculations are either defined below
or used with explicitly stated hypotheses.

Distributions have complex-bilinear pairings. Unless indicated
otherwise, they are represented using flat coordinate densities.
A smooth coordinate-density conversion is displayed explicitly or
absorbed into a test or coefficient.

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

For wavefront-constrained operations we use the normal topology:
pullback and suitably supported pushforward are continuous, whereas
the relevant bilinear operations are hypocontinuous
\cite[Sections 3--6]{BDH}. Fixed compact tensor products and their
pushforwards below are defined independently of a characteristic
coincident-point product.

%=====================================================================
\section{Component conventions and the retained internal expression}
\label{sec:expression}
%=====================================================================

\subsection{Tensor basis, dual basis, and source-output conversion}

Use
\[
 M=\R^{1+3},\qquad
 \eta=\operatorname{diag}(1,-1,-1,-1),\qquad
 \Box=\partial_t^2-\Delta_{\mathbf x}.
\]
Let \(S=\operatorname{Sym}^2((\R^4)^*)\). Fix a basis
\(e_I\in S\) and its algebraic dual \(e^I\in S^*\), written as
symmetric contravariant tensors:
\begin{equation}
 (e^I)^{\mu\nu}(e_J)_{\mu\nu}=\delta^I_J,
 \qquad
 r_I=(e^I)^{\mu\nu}r_{\mu\nu},
 \qquad
 r_{\mu\nu}=\sum_I(e_I)_{\mu\nu}r_I.
 \label{eq:basis-duality}
\end{equation}
The algebraic dual \(e^I\) is not, in general, the tensor obtained by
raising both indices of \(e_I\) with \(\eta\).

One concrete implementation uses labels \(I=(i,j)\), \(0\le i\le j\le3\),
with
\[
 (e_{(i,j)})_{\mu\nu}
 =
 \frac{\delta_{\mu i}\delta_{\nu j}
       +\delta_{\mu j}\delta_{\nu i}}{1+\delta_{ij}},
\]
\[
 (e^{(i,j)})^{\mu\nu}
 =
 \frac12\left(
 \delta^\mu_i\delta^\nu_j+\delta^\mu_j\delta^\nu_i
 \right).
\]
Thus an off-diagonal basis tensor has two entries equal to one,
and its dual has two entries equal to \(1/2\).

The tensor Euler derivative is defined by
\[
 \delta\int \sig\ell_3
 =
 \int\JD^{\rho\sigma}\delta\gamma_{\rho\sigma}\dd^4x.
\]
For propagation we lower this output with the fixed background metric.
Its component polynomial is therefore
\begin{equation}
 \boxed{
 \mathcal J_x^L(X)
 =
 (e^L)^{\mu\nu}\eta_{\mu\rho}\eta_{\nu\sigma}
       \JD^{\rho\sigma}(x;X).
 }
 \label{eq:source-conversion}
\end{equation}
This is the source-output convention used in \(Q\) and \(T\) below.

To distinguish it from coordinate variation, let
\[
 \mathcal E_I=(e_I)_{\rho\sigma}\JD^{\rho\sigma},
 \qquad
 \Gfib_{IJ}=(e_I)_{\mu\nu}(e_J)^{\mu\nu}.
\]
Then
\begin{equation}
 \mathcal J^L=(\Gfib^{-1})^{LI}\mathcal E_I,
 \qquad
 (\Gfib^{-1})^{LI}
 =
 (e^L)^{\mu\nu}\eta_{\mu\rho}\eta_{\nu\sigma}(e^I)^{\rho\sigma}.
 \label{eq:source-gradient-conversion}
\end{equation}
The \(\mathcal E_I\) are the coordinate Euler components. They are
not interchangeable with the propagated components \(\mathcal J^L\).
In particular, off-diagonal factors must not be inserted twice.

The linear maps are
\begin{equation}
 \begin{split}
 (\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},\\
 \bmap^I&=\frac12\tr_\eta e_I.
 \end{split}
 \label{eq:linear-maps}
\end{equation}
Thus
\[
 a_1^{\mu\nu}[r]=\sum_I\Amap^{\mu\nu,I}r_I,
 \qquad
 b_1[r]=\sum_I\bmap^I r_I.
\]
The notations \(\Amap^{\mu\nu}{}_L\) and \(\bmap_L\), used when
contracting a response output, denote evaluation on \(e_L\):
\[
 \Amap^{\mu\nu}{}_L=a_1^{\mu\nu}[e_L],
 \qquad \bmap_L=b_1[e_L].
\]
Their index placement labels their role in the finite component
contraction; it does not instruct an additional multiplication by
\(\Gfib\).

Fix the constant bilinear map \(\mathsf D\) through
\[
 \dot a_2^{\alpha\mu}[G;h]
 =
 \sum_{I,J}\mathsf D^{\alpha\mu,I,J}h_I G_J.
\]
The general reduction results hold for every fixed such array.
Its designation as a particular physical density map remains an
input identification.

\subsection{The patch, source incidence, and retarded kernels}

Work on
\[
 W=A\times B\times Z\times Y,\qquad w=(a,b,z,y),
\]
where \(A,B,Y\) have separated closures, \(\overline Z\subset A\),
and
\[
 b^0>y^0>a^0>\eps,\qquad z^0>\eps,\qquad b^0>z^0.
\]
These inequalities hold throughout the patch.
The smooth ramp \(\sig\) equals one on \(t>\eps\).
The observation functions \(f,g\) are fixed and smooth.

Let \(\JD=\EL_\gamma(\sig\ell_3)\), with the finite specification
in Appendix~\ref{app:source}. Under
\eqref{eq:source-conversion}, its derivative has the local form
\begin{equation}
 (\mathcal J'[G]h)^L(v)
 =
 \sum_{|\alpha|+|\beta|\le2}
 q_{\alpha\beta}^{L,I,J}(v)
       \partial^\alpha G_J(v)\partial^\beta h_I(v).
 \label{eq:source-polynomial}
\end{equation}
Here and below Greek multiindices in \(\partial^\alpha\) are distinct
from Greek spacetime indices in indexed tensor contractions.

Define the source incidence
\begin{equation}
 Q^{L,I,J}(v;a,z)
 =
 \sum_{|\alpha|+|\beta|\le2}
 q_{\alpha\beta}^{L,I,J}(v)
 \partial_v^\alpha\dirac^{(4)}(v-z)
 \partial_v^\beta\dirac^{(4)}(v-a).
 \label{eq:Q}
\end{equation}
The superscript \((4)\) here denotes the dimension of the Dirac mass.
Primes, such as \(\dirac'(q)\), denote one-dimensional derivatives.

The kernels are
\begin{equation}
 R(t,\mathbf x)
 =\frac{\Theta(t)}{2\pi}\dirac(t^2-|\mathbf x|^2),
 \qquad
 B(t,\mathbf x)
 =\frac1{8\pi}\Theta(t)\Theta(t^2-|\mathbf x|^2).
 \label{eq:retarded-kernels}
\end{equation}
They satisfy \(B=R*R\). Indeed, the spatial transform of \(R\) is
\(\Theta(t)\sin(\rho t)/\rho\). Its causal time convolution is
\[
 \Theta(t)
 \frac{\sin(\rho t)-\rho t\cos(\rho t)}{2\rho^3},
\]
with value \(\Theta(t)t^3/6\) at \(\rho=0\).
Radial integration gives the same transform for the displayed \(B\).

Put
\[
 L^{\mu,I}(s,a)
 =\sig(a)\Amap^{\lambda\mu,I}\partial_{a^\lambda}R(s-a),
\]
\begin{equation}
 T^{L,I,J}(b;a,z)
 =
 \PP^L{}_M\int R(b-v)Q^{M,I,J}(v;a,z)\dd^4v.
 \label{eq:T}
\end{equation}
Fix also a smooth advanced observation root
\(F=E_m^{\mathrm A}f\), with \(m\ge0\).
The analysis below only uses its smoothness and its fixed value.

\subsection{The complete seven-row sum}

Define
\begin{equation}
 \Kret=
 K_{47}+K_{5,+}+K_{5,-}+K_6+K_8
       +K_{9,\mathrm{der}}+K_{9,\mathrm{mass}},
 \label{eq:Kret}
\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)\sig(a)\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)\sig(z)\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)\sig(a)\mathsf D^{\alpha\mu,I,J}
       \partial_{a^\alpha}R(b-a)\dirac^{(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 reproduce the retained architecture
\cite{Em01B,XIVH,XIVJ}. Its physical interpretation is that
\(K_{47}=K_4+K_7\); both placements of \(K_5\) and both terms of
\(K_9\) are displayed separately. The seven rows are bookkeeping
rows, not seven new interaction branches.

\begin{lemma}[Finite response reduction and internal order]
\label{lem:internal-order}
For \(D=R\) or \(D=B\),
\begin{equation}
 \begin{split}
 \int D(b-v)Q^{L,I,J}(v;a,z)\dd^4v
 =\sum_{\alpha,\beta}
 (-1)^{|\alpha|+|\beta|}
 \partial_z^\alpha\partial_a^\beta
 \Bigl[
 q_{\alpha\beta}^{L,I,J}(a)
 D(b-a)\dirac^{(4)}(a-z)
 \Bigr].
 \end{split}
 \label{eq:response-reduction}
\end{equation}
All derivatives, including those of smooth coefficients, are retained.

The internal distributions are defined. For every \(L\Subset W\),
\begin{equation}
 |\Kret^{I,J}(\Psi)|\le C_L\|\Psi\|_4,
 \qquad \supp\Psi\subset L.
 \label{eq:internal-order}
\end{equation}
\end{lemma}

\begin{proof}
Use
\[
 \partial_v^\alpha\dirac^{(4)}(v-z)
 =(-1)^{|\alpha|}\partial_z^\alpha\dirac^{(4)}(v-z),
\]
and the analogous identity for \(a\).
External derivatives commute with the distributional integration.
The undifferentiated integration is
\[
 \int q(v)D(b-v)
       \dirac^{(4)}(v-z)\dirac^{(4)}(v-a)\dd^4v
 =
 q(a)D(b-a)\dirac^{(4)}(a-z).
\]
This proves \eqref{eq:response-reduction}.

The source incidence is a differentiated pullback under the
submersion \((v,a,z)\mapsto(v-z,v-a)\).
Adding \(b-v\) supplies independent propagation variables.
Its integration is locally proper because the incidence forces
\(v=a=z\) on compact external test supports.

The \(K_{47}\) and \(K_5\) seeds are tensor-product pullbacks under
the invertible 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 response seeds in \eqref{eq:response-reduction} are likewise
tensor-product pullbacks.

Both \(R\) and \(B\) are locally finite measures. For \(R\), the
future-cone radial measure is
\[
 \frac1{4\pi}t\,\dd t\,\dd\Omega.
\]
Every internal row has at most four derivatives of order-zero
seeds, multiplied by smooth coefficients. For \(K_8\), this bound
uses two source derivatives and two displayed output derivatives.
Transferring derivatives to the coefficient--test product proves
\eqref{eq:internal-order}.
\end{proof}

\subsection{The inherited unreduced coordinate map}

To make the causal plateau argument reproducible, display the
unreduced expression used for the coordinate-response branch in
XIV-H \cite[formula \texttt{eq:K8map}]{XIVH}:
\begin{equation}
 \begin{split}
 \widehat K_8^{I,J}
 =-\partial_{b^\mu}\Biggl[
 f(b)\int\!\!\int
 R(b-s)\partial_{s^\alpha}\Bigl(
 \sig(s)\Amap^{\alpha\mu}{}_L\PP^L{}_M
 R(s-v)Q^{M,I,J}(v;a,z)
 \Bigr)\dd^4s\,\dd^4v
 \Biggr].
 \end{split}
 \label{eq:K8-unreduced}
\end{equation}
Its identification as the physical coordinate-response branch is
inherited. The equality between this printed expression and
\eqref{eq:K8} on the retained patch is proved next.

\subsection{Exact plateau localization}

Let \(Q_3\) be the incidence constructed from
\(J_3=\EL_\gamma(\ell_3)\), using the same output conversion.
Let \(\Kret^{\mathrm{pl}}\) be the seven-row sum with
\(Q\) replaced by \(Q_3\) and the displayed
\(\sig(a),\sig(z)\) replaced by one.
No other data are changed.

\begin{proposition}[Exact local plateau reduction]
\label{prop:plateau}
On the retained patch,
\begin{equation}
 \Kret=\Kret^{\mathrm{pl}},
 \qquad
 \widehat K_8=K_8.
 \label{eq:plateau-identity}
\end{equation}
The first equality holds as an equality of internal distributions;
it therefore persists in every positive-scale regularized
contraction on tests in the patch.

Source-ramp derivative terms are exactly absent from this local
expression. No smoothing gain follows merely from their absence.
\end{proposition}

\begin{proof}
The weighted source is
\[
 \JD^{\mu\nu}
 =
 \sig J_3^{\mu\nu}
 +2C_2^{\mu\nu\alpha\beta}
                   \partial_\alpha\partial_\beta\sig
 -2M_2^{\mu\nu\rho}\partial_\rho\sig.
\]
The source incidence enforces \(v=a=z\). Near its retained support,
\(v^0>\eps\), so the source and all its coefficient jets agree
with the unweighted source there. The finite reduction
\eqref{eq:response-reduction} consequently gives identical response
distributions with \(Q\) and \(Q_3\).

The explicit factors \(\sig(a),\sig(z)\) are one on open
neighborhoods of their retained variables; all their derivatives
vanish there. This proves the first equality.

For \eqref{eq:K8-unreduced}, retarded support enforces
\[
 v=a=z,\qquad s\in J^+(a)\cap J^-(b).
\]
Hence \(s^0\ge a^0>\eps\). Replacing \(\sig(s)\) by one,
including inside the displayed derivative, leaves the
distribution unchanged. Differentiation does not enlarge support.

The intermediate integration is support-proper on compact endpoint
sets, since \(s\) lies in compact causal diamonds. Causal convolution
then gives
\[
 \int R(b-s)\partial_{s^\alpha}R(s-v)\dd^4s
 =
 \partial_{b^\alpha}B(b-v),
\]
which is exactly the reduced expression \eqref{eq:K8}.
\end{proof}

The proposition does not remove observation-switch derivatives,
change the frozen action, or identify a late response with an
unweighted response for unrelated source data.

\subsection{The crossed array and complete vertex}

Fix a symmetric crossed array
\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}
The four lines are fixed distributions; both tensor products exist.
The lower component label \(I\) denotes the dual input slot
contracted with the corresponding slot of \(K^{I,J}\).
No additional fiber metric is inserted in that contraction.

The linear submersion is
\[
 \mathfrak f(w)=(y-a,y-b).
\]
For smooth \(H\), 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 vertex is
\begin{equation}
 \Vertex(K,H)^J
 =
 \partial_{y^\nu}\left[g(y)\sum_I K^{I,J}Z_I^\nu(H)\right]
 -
 \frac{m^2}{2}g(y)\sum_I K^{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:vertex-test}
\end{equation}

This fixes the kinetic, trace-kinetic, and mass coefficients.
With the adopted polarization
\[
 T_{\mu\nu}(\phi,\psi)
 =
 \partial_{(\mu}\phi\,\partial_{\nu)}\psi
 -\frac12\eta_{\mu\nu}
 \left(
 \partial^\lambda\phi\,\partial_\lambda\psi-m^2\phi\psi
 \right),
\]
they are the coefficients obtained by transferring the unused
scalar derivative in the mixed Hessian
\(-\int g\,h^{\mu\nu}T_{\mu\nu}(\phi,\psi)\dd^4y\).
This is the normalization adopted for the retained vertex
\cite{Em01B}. The vertex mass term and the internal
\(K_{9,\mathrm{mass}}\) term are distinct.

Fix an open \(U\Subset W\), a cutoff
\(\chi\in C_c^\infty(\R^8)\) equal to one near
\(\mathfrak f(\overline U)\), and a profile
\(\rho\in C_c^\infty(\R^8)\) with \(\int\rho=1\). Define
\begin{equation}
 \rho_\delta(x)=\delta^{-8}\rho(x/\delta),
 \qquad
 C_\delta=\rho_\delta*(\chi C),
 \qquad
 t_\delta=\Fmap_\delta(\Kret)
          :=i\Vertex(\Kret,C_\delta)|_U.
 \label{eq:regularization}
\end{equation}
No positivity or symmetry of \(\rho\) is assumed.
Since \(C_\delta\) is smooth, every positive-scale expression exists.

%=====================================================================
\section{Exact characteristic normal coordinates}
\label{sec:geometry}
%=====================================================================

Define
\[
 q_R=(b-a)^2,\qquad q_1=(y-a)^2,\qquad q_2=(b-y)^2,
\]
\[
 \Sresp=\{q_R=0,\ a=z\},
\]
and
\begin{equation}
 N=
 \left\{
 \begin{array}{l}
 z=a,\quad y=a+\alpha\ell_{\mathbf n},\\
 b=a+\beta\ell_{\mathbf n},\quad
 0<\alpha<\beta,\quad
 \ell_{\mathbf n}=(1,\mathbf n),\quad \mathbf n\in S^2
 \end{array}
 \right\}.
 \label{eq:N}
\end{equation}

Fix \(w_0\in N\). On a sphere chart choose a smooth orthonormal
frame \(\mathbf e_1,\mathbf e_2\) of \(\mathbf n^\perp\), and set
\[
 \ell_{\mathbf n}=(1,\mathbf n),\qquad
 k_{\mathbf n}=(1,-\mathbf n),\qquad
 E_A=(0,\mathbf e_A).
\]
Then
\[
 \ell^2=k^2=0,\qquad \ell\cdot k=2,\qquad
 E_A\cdot E_B=-\delta_{AB}.
\]

\begin{theorem}[Characteristic coordinate normal form]
\label{thm:coordinates}
Near \(w_0\), the map
\begin{equation}
 \begin{split}
 z&=a+r,\\
 b&=a+\beta\ell_{\mathbf n}+u k_{\mathbf n},\\
 y&=a+\alpha\ell_{\mathbf n}
              +v k_{\mathbf n}+\xi^A E_A(\mathbf n)
 \end{split}
 \label{eq:coordinate-map}
\end{equation}
is a smooth coordinate map, with
\[
 \tau=(a,\mathbf n,\alpha,\beta),\qquad
 \nu=(r,u,v,\xi),\qquad \xi\in\R^2.
\]
There are eight tangential and eight normal variables:
\[
 N=\{\nu=0\},\qquad \Sresp=\{r=0,\ u=0\}.
\]
The cone equations are
\begin{align}
 q_R&=4\beta u,
 \label{eq:qR}\\
 q_1&=4\alpha v-|\xi|^2,
 \label{eq:q1}\\
 q_2&=4(\beta-\alpha)(u-v)-|\xi|^2.
 \label{eq:q2}
\end{align}
The ambient flat density is
\begin{equation}
 \begin{split}
 \dd^4a\,\dd^4b\,\dd^4z\,\dd^4y
 ={}&4(\beta-u)^2
 \dd^4a\,\dd\Omega\,\dd\alpha\,\dd\beta\\
 &\hspace{2em}\cdot
 \dd^4r\,\dd u\,\dd v\,\dd^2\xi .
 \end{split}
 \label{eq:coordinate-density}
\end{equation}
The map and inverse have bounded derivatives on compact chart sets
where \(\alpha,\beta-\alpha,\beta,\beta-u\) have positive lower
bounds and the sphere coordinate remains in a compact subchart.
\end{theorem}

\begin{proof}
For \(s=b-a\), with \(|\mathbf s|>0\), the inverse coordinates are
\[
 \mathbf n=\frac{\mathbf s}{|\mathbf s|},\qquad
 \beta=\frac{s^0+|\mathbf s|}{2},\qquad
 u=\frac{s^0-|\mathbf s|}{2}.
\]
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 invertibility.

The null-frame identities give the three exact cone equations.
They also identify \(N\) and \(\Sresp\).

The spatial radius of \(s\) is \(\beta-u\), and
\(s^0=\beta+u\). The absolute Jacobian of
\((\beta,u)\mapsto(s^0,|\mathbf s|)\) is two:
\[
 \dd^4s=2(\beta-u)^2\dd\beta\,\dd u\,\dd\Omega.
\]
At fixed \(\mathbf n\), the map
\((\alpha,v,\xi)\mapsto d\) has absolute determinant two.
The full transformation is block triangular after selecting
\(a,s\), so the determinants multiply.
Uniformity follows from the explicit smooth inverse.
\end{proof}

\begin{corollary}[Rank and definite quadratic defect]
\label{cor:rank}
Put \(h=\beta-\alpha>0\). Along \(N\),
\[
 \dd q_R=4\beta\,\dd u,\qquad
 \dd q_1=4\alpha\,\dd v,\qquad
 \dd q_2=4h(\dd u-\dd v).
\]
The three differentials have rank two there, and
\begin{equation}
 \dd q_2=\frac h\beta\dd q_R-\frac h\alpha\dd q_1.
 \label{eq:rank-relation}
\end{equation}
Throughout the coordinate chart,
\begin{equation}
 \frac{q_1}{\alpha}+\frac{q_2}{h}
 =
 \frac{q_R}{\beta}
 -\frac{\beta}{\alpha h}|\xi|^2.
 \label{eq:quadratic-relation}
\end{equation}
On \(\Sresp\), simultaneous vanishing of \(q_1,q_2\) is
equivalent to \(v=\xi=0\). Their restricted differentials have
rank one along \(N\), with the remaining constraint supplied by
the definite quadratic form in \eqref{eq:quadratic-relation}.
\end{corollary}

\begin{proof}
Differentiate the exact equations at \(u=v=\xi=0\).
Adding their divided forms gives the nonlinear relation.
On \(\Sresp\), \(q_R=0\); positivity of
\(\beta/(\alpha h)\) forces \(\xi=0\), and then \(v=0\).
\end{proof}

These rank statements are localized to \(N\); rank may increase away
from it. The flag has codimensions
\[
 \operatorname{codim}_W\Sresp=5,\qquad
 \operatorname{codim}_{\Sresp}N=3,\qquad
 \operatorname{codim}_W N=8.
\]
It is not a replacement numerical extension problem.

The dilation
\[
 \mathcal D_\lambda(\tau;r,u,v,\xi)
 =
 (\tau;\lambda r,\lambda^2u,\lambda^2v,\lambda\xi)
\]
satisfies
\[
 q_j\circ\mathcal D_\lambda=\lambda^2q_j,\qquad j=R,1,2,
\]
and has normal coordinate Jacobian \(\lambda^{10}\).
The weight of \(r\) is declared; the cone equations do not determine it.
The ambient density has the additional smooth ratio
\[
 \frac{(\beta-\lambda^2u)^2}{(\beta-u)^2}.
\]
This geometric dilation is distinct from the isotropic smoothing
in \eqref{eq:regularization}. No physical scaling degree is assigned
by this calculation alone. Extension frameworks require their
additional growth and microlocal hypotheses \cite{Dang,BF}.

%=====================================================================
\section{The complete contraction as a translation distribution}
\label{sec:translation}
%=====================================================================

\subsection{Transposition of the complete vertex}

Let \(K\) be a fixed internal distribution and
\(\Phi\in\Test(U;V_{\mathrm{out}}^*)\). Define compact relative
distributions
\begin{align}
 a_{I,\nu}[\Phi]
 &=
 \mathfrak f_*
 \left(
 \sum_J g(y)\partial_{y^\nu}\Phi_J(w)K^{I,J}(w)
 \right),
 \label{eq:aPhi}\\
 d_I[\Phi]
 &=
 \mathfrak f_*
 \left(
 \sum_J g(y)\Phi_J(w)K^{I,J}(w)
 \right).
 \label{eq:dPhi}
\end{align}
The pre-pushforward distributions have compact support.
Their pushforward is defined by transposition against smooth
relative functions.

Put
\begin{equation}
 \begin{split}
 \Lambda_{I,\mu\nu}[\Phi]
 ={}&
 \frac i2\left(
 \partial_{2,\mu}a_{I,\nu}[\Phi]
 +\partial_{2,\nu}a_{I,\mu}[\Phi]
 \right)\\
 &-\frac i2\eta_{\mu\nu}
                    \partial_2^\lambda a_{I,\lambda}[\Phi]
 -\frac{im^2}{2}\eta_{\mu\nu}d_I[\Phi].
 \end{split}
 \label{eq:Lambda}
\end{equation}

\begin{lemma}[Exact transposed vertex]
\label{lem:transpose}
For every smooth symmetric array \(H\),
\begin{equation}
 \pair{i\Vertex(K,H)}{\Phi}
 =
 \sum_{I,\mu,\nu}
 \pair{\Lambda_{I,\mu\nu}[\Phi]}{H_I^{\mu\nu}}.
 \label{eq:transpose}
\end{equation}
If \(K\) has local order at most four, then for
\(\supp\Phi\subset L\Subset U\),
\begin{equation}
 |\Lambda_{I,\mu\nu}[\Phi](\psi)|
 \le C_L\|\Phi\|_5\|\psi\|_5
 \label{eq:Lambda-order}
\end{equation}
on a fixed compact neighborhood of its relative support.
\end{lemma}

\begin{proof}
The kinetic part of the test action is
\[
 -i\sum_{I,\mu,\nu}
 a_{I,\nu}[\Phi](\partial_{2,\mu}H_I^{\mu\nu}).
\]
Transposition gives \(i\partial_{2,\mu}a_{I,\nu}\).
Symmetry of \(H\) gives the symmetrization in
\eqref{eq:Lambda}. The trace-kinetic coefficient \(i/2\) becomes
\(-i/2\), while the mass coefficient remains \(-im^2/2\).
This proves the identity.

The order-four estimate acts on a product of \(g\), a derivative
of \(\Phi\), and a pulled-back relative test. It uses at most five
derivatives of \(\Phi\). The additional relative derivative uses
at most five derivatives of \(\psi\).
\end{proof}

\subsection{The translation distribution}

Let
\[
 \mathfrak d(x,v)=x-v,\qquad
 \mathfrak d:\R^8\times\R^8\longrightarrow\R^8,
\]
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}
All factors are compactly supported. Thus \(\Omega_\Phi\) is a
compact distribution formed by defined tensor products and pushforward,
not by an undeclared coincident-point product.

A compact distribution acts on smooth functions by insertion of a
cutoff equal to one near its support. This makes the pairing with
\(\rho_\delta(x-v)\) precise even though that function need not be
compactly supported on \(\R^{16}\).

\begin{theorem}[Exact complete-sum translation representation]
\label{thm:translation}
For \(K=\Kret\),
\begin{equation}
 \boxed{t_\delta(\Phi)=\Omega_\Phi(\rho_\delta).}
 \label{eq:origin-evaluation}
\end{equation}
Every internal row and every vertex term is retained.

If \(\chi C\) has compact distributional order at most \(p\), then
\begin{equation}
 |\Omega_\Phi(\psi)|
 \le C_L\|\Phi\|_5\|\psi\|_{p+5},
 \qquad \supp\Phi\subset L,
 \label{eq:Omega-order}
\end{equation}
on a fixed compact neighborhood of the translation support.

Every equality of actual internal distributions is respected.
In particular, if \(\Rec A=0\), then the corresponding
\(\Omega_\Phi\) and all regularized values are zero.
\end{theorem}

\begin{proof}
Insert \(H=C_\delta\) into \eqref{eq:transpose}. The result is
\[
 \sum_{I,\mu,\nu}
 \left\langle
 \Lambda_{I,\mu\nu}[\Phi](x)\otimes\chi C_I^{\mu\nu}(v),
 \rho_\delta(x-v)
 \right\rangle,
\]
which is the pushforward definition of
\eqref{eq:origin-evaluation}.

Linearity gives
\[
 \Omega_\Phi=\sum_{b\in\mathcal B}\Omega_{b,\Phi},
 \qquad
 \mathcal B=\{47,5+,5-,6,8,9\mathrm{der},9\mathrm{mass}\}.
\]
Each row uses all terms of \eqref{eq:Lambda}.

The tensor pairing requires at most five \(x\)-derivatives and
at most \(p\) \(v\)-derivatives. On \(\psi(x-v)\) their combined
order is at most \(p+5\). The compact-support cutoffs have fixed
derivative bounds. This proves \eqref{eq:Omega-order}.

The construction begins with the actual \(K\), so a zero internal
distribution remains zero at every stage.
\end{proof}

The theorem identifies the complete object whose origin behavior must
be controlled. It does not establish regularity at that origin.

%=====================================================================
\section{Scale identities and the direct convergence target}
\label{sec:scale}
%=====================================================================

Define the compact profiles
\[
 \vartheta(h)=-8\rho(h)-h\cdot\nabla\rho(h)
             =-\operatorname{div}(h\rho(h)),
 \qquad
 \zeta(h)=2^8\rho(2h)-\rho(h).
\]
Then
\begin{equation}
 \delta\partial_\delta\rho_\delta=\vartheta_\delta,
 \qquad
 \rho_{\delta/2}-\rho_\delta=\zeta_\delta.
 \label{eq:profiles}
\end{equation}
Both have integral zero. Their moments satisfy
\[
 m_\gamma(\vartheta)=|\gamma|m_\gamma(\rho),\qquad
 m_\gamma(\zeta)=(2^{-|\gamma|}-1)m_\gamma(\rho).
\]
The first sign follows directly by integration by parts.

\begin{proposition}[Exact complete-sum scale identities]
\label{prop:scale}
The family \(t_\delta\) is smooth in the strong distributional topology
on compact subintervals of \((0,\infty)\).
Set
\[
 D_\delta=\delta\partial_\delta t_\delta,\qquad
 t_n=t_{2^{-n}},\qquad \Delta_n=t_{n+1}-t_n.
\]
Then
\begin{align}
 D_\delta(\Phi)&=\Omega_\Phi(\vartheta_\delta),
 \label{eq:Dorigin}\\
 \Delta_n(\Phi)&=\Omega_\Phi(\zeta_{2^{-n}}),
 \label{eq:increment}
\end{align}
and equivalently
\begin{align}
 D_\delta
 &=i\Vertex(\Kret,\vartheta_\delta*(\chi C)),
 \label{eq:Dvertex}\\
 \Delta_n
 &=i\Vertex(\Kret,\zeta_{2^{-n}}*(\chi C)).
 \label{eq:increment-vertex}
\end{align}
\end{proposition}

\begin{proof}
On a compact positive scale interval the profiles and their scale
derivatives have common compact support and are smooth in every
test seminorm. The estimate \eqref{eq:Omega-order} makes their
evaluations smooth uniformly on bounded joint test sets.
Differentiate or subtract \eqref{eq:origin-evaluation}.
\end{proof}

Only the auxiliary profiles vary with scale. The internal kernel,
source, switches, and grouped crossed distribution remain fixed.

The direct physical target is
\begin{equation}
 |\Omega_\Phi(\vartheta_\delta)|
 \le C_L\delta^\alpha\|\Phi\|_{m_L},
 \qquad \alpha>0.
 \label{eq:direct-target}
\end{equation}
This need not be proved through a function-valued regularity theorem
for \(\Omega_\Phi\).

Zero profile integral alone is insufficient. For example,
\[
 \dirac_0(\rho_\delta)=\delta^{-8}\rho(0),
\]
and a zero-integral scale profile may still have nonzero value at zero.

\begin{theorem}[Quantitative scale-integrability criterion]
\label{thm:Dini}
Let \(t_\delta\) be a strongly smooth distribution-valued family on
\(0<\delta\le\delta_0\), with
\(D_\delta=\delta\partial_\delta t_\delta\).
Suppose that for every \(L\Subset U\), there are \(m_L,C_L\) and a
nonnegative measurable \(a_L\) such that
\[
 \int_0^{\delta_0}a_L(s)\frac{\dd s}{s}<\infty
\]
and
\begin{equation}
 |D_\delta(\Phi)|
 \le C_La_L(\delta)\|\Phi\|_{m_L},
 \qquad \supp\Phi\subset L.
 \label{eq:Dini-hypothesis}
\end{equation}
Then \(t_\delta\) converges strongly to a distribution \(t_0\), with
\begin{equation}
 |(t_\delta-t_0)(\Phi)|
 \le C_L\|\Phi\|_{m_L}
             \int_0^\delta a_L(s)\frac{\dd s}{s}.
 \label{eq:Dini-remainder}
\end{equation}
If \(a_L(s)=s^\alpha\), \(\alpha>0\), then
\[
 |(t_\delta-t_0)(\Phi)|
 \le\frac{C_L}{\alpha}\delta^\alpha\|\Phi\|_{m_L},
\]
and, for dyadic scales within the parameter interval,
\[
 |\Delta_n(\Phi)|
 \le\frac{C_L}{\alpha}
       (1-2^{-\alpha})2^{-n\alpha}\|\Phi\|_{m_L}.
\]
The same integral controls parameters between adjacent dyadic scales.
\end{theorem}

\begin{proof}
For \(0<\delta_1<\delta_2\le\delta_0\),
\[
 (t_{\delta_2}-t_{\delta_1})(\Phi)
 =
 \int_{\delta_1}^{\delta_2}D_s(\Phi)\frac{\dd s}{s}.
\]
The hypothesis gives a Cauchy pairing for each test.
The limit is a distribution: locally its value is bounded by the
finite-order estimate for \(t_{\delta_0}\) plus the finite increment
bound, increasing the test order if necessary.
Taking \(\delta_1\downarrow0\) gives
\eqref{eq:Dini-remainder} uniformly on bounded test sets.
The positive-power estimates follow by integration.
\end{proof}

\begin{remark}[An optional stronger certificate]
If near zero the complete \(\Omega_\Phi\) is represented by a continuous
function \(W_\Phi\) with uniform local bounds
\[
 |W_\Phi(h)-W_\Phi(0)|
 \le C_L|h|^\alpha\|\Phi\|_{m_L},
 \qquad
 |W_\Phi(0)|\le C_L\|\Phi\|_{m_L},
\]
then
\[
 D_\delta(\Phi)
 =
 \int\vartheta(s)
 [W_\Phi(\delta s)-W_\Phi(0)]\dd s
 =O(\delta^\alpha\|\Phi\|_{m_L}).
\]
Also \(t_\delta(\Phi)\to W_\Phi(0)\), independently of the normalized
compact profile. This is sufficient, not a required route.
\end{remark}

The criterion is not verified for the physical complete sum.
An outcome classifier for an actually evaluated power--logarithm
expansion is retained in Appendix~\ref{app:asymptotics}.

%=====================================================================
\section{Explicit scalar cone asymptotics}
\label{sec:scalar}
%=====================================================================

\subsection{The scalar family}

Fix \(\alpha,h>0\), with \(\beta=\alpha+h\).
In the response-normal coordinate \(q=q_R=4\beta u\),
\[
 q_1=4\alpha v-|\xi|^2,\qquad
 q_2=\frac h\beta q-4hv-|\xi|^2.
\]
Choose nonnegative \(f_0,g_0\in C_c^\infty(\R)\) with
\(\int f_0=\int g_0=1\), and define
\begin{equation}
 P_\delta(q,v,\xi)
 =
 \delta^{-2}
 f_0\left(\frac{4\alpha v-|\xi|^2}{\delta}\right)
 g_0\left(
 \frac{(h/\beta)q-4hv-|\xi|^2}{\delta}
 \right).
 \label{eq:scalar-P}
\end{equation}
This is a cone-function regularization in four variables, relative
to \(\dd q\,\dd v\,\dd^2\xi\). It is not
\eqref{eq:regularization}.

For \(j=0,1\), put
\[
 F_j(s,X)
 =
 f_0(4\alpha s-|X|^2)g_0^{(j)}(-4hs-|X|^2),
\]
and define
\begin{equation}
 \mathfrak m_j=\int F_j,\qquad
 \mathfrak s_j=\int sF_j,\qquad
 \mathfrak q_j=\int |X|^2F_j,
 \label{eq:scalar-moments}
\end{equation}
with integration over \(\R\times\R^2\).

\begin{lemma}[Uniform anisotropic moment expansion]
\label{lem:scalar-moments}
The supports of \(F_j\), \(j=0,1\), are uniformly compact when
\(\alpha,h\) range over a compact subset of \((0,\infty)^2\).
For \(0<\delta\le1\),
\begin{equation}
 \begin{split}
 \int F_j(s,X)\psi(\delta s,\sqrt{\delta}X)\dd s\,\dd^2X
 ={}&\mathfrak m_j\psi(0,0)\\
 &+\delta\left[
 \mathfrak s_j\partial_v\psi(0,0)
 +\frac{\mathfrak q_j}{4}\Delta_\xi\psi(0,0)
 \right]\\
 &+O(\delta^2\|\psi\|_4),
 \end{split}
 \label{eq:moment-expansion}
\end{equation}
uniformly on those parameter sets.
\end{lemma}

\begin{proof}
Set
\[
 U=4\alpha s-Q,\qquad V=-4hs-Q,\qquad Q=|X|^2.
\]
Then
\[
 Q=-\frac{hU+\alpha V}{\beta}.
\]
Compact profile supports bound \(Q\) and \(s\), uniformly for
the stated parameters.

The functions \(F_j\) are radial in \(X\). Odd transverse moments
vanish, and
\[
 \int X_A X_BF_j(s,X)\dd^2X
 =
 \frac{\delta_{AB}}2\int |X|^2F_j(s,X)\dd^2X.
\]
Taylor-expand in \(\xi\) through cubic order at \((\delta s,0)\).
The odd terms integrate to zero; the fourth-order remainder is
bounded by \(C\delta^2|X|^4\|\psi\|_4\).
Expand the zero-transverse term to first order in \(v\), and
replace the transverse Hessian at \(v=\delta s\) by its value
at zero. These errors are also \(O(\delta^2\|\psi\|_4)\).
\end{proof}

\subsection{Order-zero incidence}

Define
\[
 S_\delta(\psi)
 =
 \int P_\delta(0,v,\xi)\psi(v,\xi)\dd v\,\dd^2\xi.
\]

\begin{theorem}[Scalar double-cone limit and rate]
\label{thm:scalar-limit}
The family converges strongly to
\[
 c_0(\alpha,h)\dirac(v)\dirac^{(2)}(\xi),
\]
where
\begin{equation}
 c_0(\alpha,h)
 =
 \frac{\pi}{4\beta}
 \int_{hU+\alpha V\le0}f_0(U)g_0(V)\dd U\,\dd V.
 \label{eq:c0}
\end{equation}
On compact test supports,
\begin{equation}
 |S_\delta(\psi)-c_0\psi(0,0)|
 \le C\delta\|\psi\|_2,
 \qquad
 |\delta\partial_\delta S_\delta(\psi)|
 \le C\delta\|\psi\|_2.
 \label{eq:scalar-rates}
\end{equation}
More precisely,
\[
 S_\delta(\psi)
 =
 c_0\psi(0,0)
 +\delta\left[
 \mathfrak s_0\partial_v\psi(0,0)
 +\frac{\mathfrak q_0}{4}\Delta_\xi\psi(0,0)
 \right]
 +O(\delta^2\|\psi\|_4).
\]
The constants are uniform on compact positive parameter sets.
If both profiles are even, then
\[
 c_0=\frac{\pi}{8\beta}.
\]
\end{theorem}

\begin{proof}
Rescale \(v=\delta s\), \(\xi=\sqrt{\delta}X\):
\begin{equation}
 S_\delta(\psi)
 =
 \int F_0(s,X)\psi(\delta s,\sqrt{\delta}X)
                    \dd s\,\dd^2X.
 \label{eq:scalar-rescale}
\end{equation}
The Jacobian cancels the \(\delta^{-2}\) factor.
Polar integration gives \(\dd^2X=\pi\dd Q\).
The absolute determinant of \((s,Q)\mapsto(U,V)\) is
\(4\beta\), and \(Q\ge0\) means \(hU+\alpha V\le0\).
Hence \(\mathfrak m_0=c_0\).

Reflection averaging in \(X\) bounds the test difference from
\(\psi(0,0)\) by
\[
 C\delta(|s|+|X|^2)\|\psi\|_2.
\]
This gives the first rate and strong convergence.
The refined expansion follows from
Lemma~\ref{lem:scalar-moments}.

For the scale derivative, differentiate
\eqref{eq:scalar-rescale}:
\[
 \delta\partial_\delta S_\delta(\psi)
 =
 \int F_0\left[
 \delta s\,\partial_v\psi(\delta s,\sqrt{\delta}X)
 +\frac{\sqrt{\delta}}2
 X\cdot\nabla_\xi\psi(\delta s,\sqrt{\delta}X)
 \right].
\]
Subtract \(\nabla_\xi\psi(\delta s,0)\) in the second term;
its integral against \(XF_0\) is zero. The remaining term
is \(O(\delta\|\psi\|_2)\).

When both profiles are even, simultaneous reflection
\((U,V)\mapsto(-U,-V)\) exchanges the two half-planes.
The boundary has measure zero, so each receives half the mass.
\end{proof}

The coefficient can depend on the normalized profiles.
Profiles supported in positive intervals give \(c_0=0\);
profiles supported in negative intervals give
\(c_0=\pi/(4\beta)\).

\subsection{Response-normal derivative}

For every \(\delta>0\), define
\[
 U_\delta=\dirac'(q)P_\delta(q,v,\xi).
\]
This means \(\dirac'(q)\otimes1\) multiplied by the smooth
function \(P_\delta\).
Let \(\delta_{\mathrm o}\) be the Dirac mass at
\((q,v,\xi)=(0,0,0)\), and put \(\kappa=h/\beta\).

\begin{theorem}[Scalar singular-plus-finite expansion]
\label{thm:scalar-derivative}
The family satisfies
\begin{equation}
 U_\delta
 =
 -\delta^{-1}c_1(\alpha,h)\delta_{\mathrm o}
 +T_{\mathrm{sc}}
 +O_{\mathrm{loc}}(\delta),
 \label{eq:scalar-expansion}
\end{equation}
with
\begin{equation}
 c_1(\alpha,h)
 =
 \frac{\pi h}{4\beta^2}
 \int_\R f_0(U)g_0\left(-\frac h\alpha U\right)\dd U,
 \label{eq:c1}
\end{equation}
\begin{equation}
 T_{\mathrm{sc}}
 =
 c_0\partial_q\delta_{\mathrm o}
 +\kappa\mathfrak s_1\partial_v\delta_{\mathrm o}
 -\frac{\kappa\mathfrak q_1}{4}\Delta_\xi\delta_{\mathrm o}.
 \label{eq:scalar-finite}
\end{equation}
The remainder satisfies
\[
 |O_{\mathrm{loc}}(\delta)(\Psi)|
 \le C\delta\|\Psi\|_4
\]
on compact test supports, uniformly on compact positive
parameter sets.

For any fixed compact test equal to one near the origin,
\begin{equation}
 U_\delta(\Psi)=-c_1\delta^{-1}
 \label{eq:fixed-test-exact}
\end{equation}
for every sufficiently small \(\delta\).
Consequently the raw scalar family converges exactly when
\(c_1=0\). If \(f_0,g_0\) are strictly positive near zero,
then \(c_1>0\), and a single fixed test detects divergence.
\end{theorem}

\begin{proof}
The derivative-delta action gives
\begin{equation}
 \begin{split}
 U_\delta(\Psi)
 ={}&
 -\int P_\delta(0,v,\xi)
                    \partial_q\Psi(0,v,\xi)\dd v\,\dd^2\xi\\
 &-\kappa\delta^{-3}
 \int f_0\left(\frac{4\alpha v-|\xi|^2}{\delta}\right)
 g_0'\left(\frac{-4hv-|\xi|^2}{\delta}\right)
                  \Psi(0,v,\xi)\dd v\,\dd^2\xi.
 \end{split}
 \label{eq:scalar-derivative-action}
\end{equation}
The second integral becomes
\[
 -\kappa\delta^{-1}
 \int F_1(s,X)\Psi(0,\delta s,\sqrt{\delta}X)\dd s\,\dd^2X.
\]
Its zeroth moment is
\[
 \mathfrak m_1
 =
 \frac{\pi}{4\beta}
 \int_{hU+\alpha V\le0}f_0(U)g_0'(V)\dd U\,\dd V
 =
 \frac{\pi}{4\beta}
 \int f_0(U)g_0(-hU/\alpha)\dd U.
\]
Therefore \(\kappa\mathfrak m_1=c_1\).

Lemma~\ref{lem:scalar-moments} yields
\[
 \begin{split}
 -\kappa\delta^{-1}\int F_1\Psi(0,\delta s,\sqrt{\delta}X)
 ={}&
 -c_1\delta^{-1}\Psi(0,0,0)\\
 &-\kappa\mathfrak s_1\partial_v\Psi(0,0,0)
 -\frac{\kappa\mathfrak q_1}{4}\Delta_\xi\Psi(0,0,0)\\
 &+O(\delta\|\Psi\|_4).
 \end{split}
\]
The first integral in \eqref{eq:scalar-derivative-action} is
\[
 -c_0\partial_q\Psi(0,0,0)+O(\delta\|\Psi\|_3).
\]
First derivatives of a Dirac mass act with a minus sign;
the transverse Laplacian acts with a plus sign.
This gives \eqref{eq:scalar-finite}.

For a fixed test constant one near the origin, the shrinking
rescaled supports lie in its constant region and its
\(q\)-derivative is zero. This proves
\eqref{eq:fixed-test-exact}.
If \(c_1=0\), \eqref{eq:scalar-expansion} gives strong convergence
to \(T_{\mathrm{sc}}\); otherwise the fixed pairing diverges.
Strict positivity near zero makes the integral defining \(c_1\)
strictly positive.
\end{proof}

The equivalent convergence of
\(U_\delta+c_1\delta^{-1}\delta_{\mathrm o}\) to \(T_{\mathrm{sc}}\)
is a statement about this explicitly defined scalar family.
It does not select a counterterm for the physical graph.

\subsection{Tangential parameters and the original density}

The estimates are uniform when \(\alpha,h\) vary over compact
tangential parameter sets, so they may be integrated against
compact tangential tests.

Adding the order-zero factors
\(\dirac(q)\dirac^{(4)}(r)\) gives the order-zero concentration on
\[
 q=r=v=\xi=0,
\]
namely \(N\). Adding \(\dirac^{(4)}(r)\) to the derivative family
gives its inverse-scale concentration there.
All attachment variables are retained.

The coordinate-density conversion is treated in
Appendix~\ref{app:density}. A smooth positive density factor changes
the coefficient through its jets but cannot remove a nonzero
inverse-scale coefficient by contributing only a finite term.

\begin{remark}[No transfer to the physical complete sum]
The coefficients \(c_0,c_1,T_{\mathrm{sc}}\) belong to the scalar
regularization \eqref{eq:scalar-P}. No tensor/source identification
has made them coefficients of \eqref{eq:regularization}.
They prove neither survival nor cancellation of a singular
coefficient in the full graph.
\end{remark}

%=====================================================================
\section{Localization, agreement, and the remaining physical calculation}
\label{sec:ledger}
%=====================================================================

\subsection{Cutoff locality}

\begin{proposition}[Eventual exact cutoff independence]
\label{prop:cutoff}
Let \(\chi_1,\chi_2\) equal one near \(\mathfrak f(L)\),
\(L\Subset U\). For a fixed compact profile, their regularized
values on tests in \(L\) agree at every sufficiently small scale.
The same holds for logarithmic scale derivatives and dyadic increments.
\end{proposition}

\begin{proof}
The distribution \((\chi_1-\chi_2)C\) vanishes near the compact
relative image. For small scales, convolution with
\(\rho_\delta,\vartheta_\delta,\zeta_\delta\), including the required
derivatives, vanishes near that image.
The vertex test action gives equality.
\end{proof}

For fixed \(\chi\), two profiles satisfy
\[
 (t_\delta^\rho-t_\delta^{\widetilde\rho})(\Phi)
 =
 \Omega_\Phi(\rho_\delta-\widetilde\rho_\delta).
\]
Profile independence requires control of this complete expression.

\subsection{Microlocal agreement with its convergence hypotheses}

Let \(V\subset U\) be an open domain. A sufficient localized
agreement hypothesis is the following: after the needed localization,
each pair consisting of an internal component and a pulled-back
crossed vertex component has closed wavefront bounds
\(\Gamma_K,\Gamma_C\) satisfying
\[
 \Gamma_K\cap(-\Gamma_C)=\varnothing,
\]
and the smoothed crossed components converge to the unsmoothed
ones in the normal topology with that bound \(\Gamma_C\).

Under these hypotheses, the fixed-factor product converges to the
ordinary product by hypocontinuity
\cite[Theorem 6.1]{BDH}. Smooth weights, output derivatives, and the
finite sums preserve strong distributional convergence. Hence any
existing characteristic limit agrees on \(V\) with the ordinary
vertex contraction.

The supplied Em01-B-v3 excerpt reports such localized ordinary-product
convergence on its verified domain \cite{Em01B}.
That report is used only with its stated localization and topology
hypotheses; the mere existence of an ordinary product is not treated
as convergence in an unspecified stronger topology.

If two characteristic limits have the same reference on \(V\),
their difference is supported in \(U\setminus V\). Support in \(N\)
requires a common reference throughout \(U\setminus N\).
No equality
\[
 \Ucan=W\setminus N
 \quad\text{or}\quad
 \mathcal O_{\Kret}=U\setminus N
\]
is asserted.

The supported-sector reference of XIV-J concerns
\(\Ksup=K_6+K_8+K_9\), not the full retained sum
\cite{XIVJ}. It does not identify the latter agreement domain.

\subsection{Evaluated and unevaluated entries}

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.27\linewidth}Y@{}}
\toprule
Task & Exact status\\
\midrule
Source conventions and \(q\) &
Explicit tensor conversion and finite coefficient rule supplied;
complete contracted characteristic coefficient unevaluated.\\[0.3em]
Source-ramp contribution on the retained patch &
Exactly absent by Proposition~\ref{prop:plateau};
no smoothing exponent follows from that fact.\\[0.3em]
Unreduced coordinate response &
Printed and reduced distributionally; its physical branch
interpretation is inherited.\\[0.3em]
Complete translation sum &
Defined exactly with all seven rows and three vertex types.\\[0.3em]
Physical leading origin coefficient &
Not evaluated.\\[0.3em]
Physical characteristic remainder &
No summable or scale-uniform estimate proved.\\[0.3em]
Scalar coefficients and remainder &
Explicit \(c_0,c_1,T_{\mathrm{sc}}\), with quantitative rates
for the stated scalar regularization.\\
\bottomrule
\end{tabularx}
\end{center}

The next calculation must instantiate the designated crossed
boundary values and the physical constant map \(\mathsf D\), collect
the complete branch--vertex sum in the characteristic coordinates,
and bound its remainder on arbitrary fixed compact joint tests.

A leading algebraic zero is insufficient without that remainder
estimate. A nonzero branch coefficient is insufficient without proving
that it survives the complete sum.

\subsection{Wavefront, contacts, and admission}

A wavefront theorem for a physical limit requires a stated closed cone
and uniform localized Fourier bounds outside it.
Finite-scale smoothness of \(C_\delta\) does not provide those bounds.

Properness controls pushforward after a distribution exists.
Extension results require their own growth and microlocal conditions
\cite{Dang,BF}; they do not by themselves identify a raw smoothing
limit or its physical normalization.

Under the inherited contact hypothesis, collision-supported contacts
vanish on the retained separated patch. An eventual nonzero physical
fixed-test defect there could not be removed by those contacts.
No such physical defect is proved by the scalar family.

%=====================================================================
\section{Exact mathematical release}
\label{sec:release}
%=====================================================================

\begin{theorem}[Geometric, source, and analytic release]
\label{thm:release}
For the explicitly specified setting:
\begin{enumerate}[label=\textup{(\roman*)}]
\item the propagated source convention is
\eqref{eq:source-conversion}, and its finite coefficient rule is
\eqref{eq:jet-Hessian};
\item the internal response reduction and local order bound are
\eqref{eq:response-reduction} and \eqref{eq:internal-order};
\item the plateau equalities \eqref{eq:plateau-identity} hold on
the retained patch;
\item the characteristic coordinates, density, rank relation, and
quadratic defect are those of Theorem~\ref{thm:coordinates} and
Corollary~\ref{cor:rank};
\item the complete regularized contraction has the exact translation
and scale-profile representations
\eqref{eq:origin-evaluation}--\eqref{eq:increment-vertex}, with
reconstruction descent;
\item an integrable complete-sum scale bound gives the quantitative
fixed-kernel limit of Theorem~\ref{thm:Dini};
\item the scalar order-zero and derivative families have the explicit
asymptotics and fixed-test conclusions of
Theorems~\ref{thm:scalar-limit} and \ref{thm:scalar-derivative};
\item cutoff locality is exact, and microlocal agreement has the
explicit hypotheses stated in Section~\ref{sec:ledger}.
\end{enumerate}
\end{theorem}

\begin{proof}
Use Proposition~\ref{prop:source-rule},
Lemma~\ref{lem:internal-order},
Proposition~\ref{prop:plateau},
Theorem~\ref{thm:coordinates},
Corollary~\ref{cor:rank},
Theorem~\ref{thm:translation},
Proposition~\ref{prop:scale},
Theorem~\ref{thm:Dini},
the scalar theorems, and Proposition~\ref{prop:cutoff}.
The stated microlocal agreement is the fixed-factor application
specified in Section~\ref{sec:ledger}.
\end{proof}

The physical complete-sum origin coefficient and its controlled
remainder remain unevaluated. Thus no convergence or failure theorem
for \(\Fmap_\delta(\Kret)\) is released.

Even a subsequent fixed-kernel limit would remain distinct from a
continuous map on the full reconstructed module, profile independence,
global gluing, the output wavefront theorem, and normalized quantum
admission.

\[
 \boxed{
 \begin{gathered}
 t_\delta(\Phi)=\Omega_\Phi(\rho_\delta),\\
 D_\delta(\Phi)=\Omega_\Phi(\vartheta_\delta),\\
 \Delta_n(\Phi)=\Omega_\Phi(\zeta_{2^{-n}}),\\
 \text{source specification: explicit and reproducible},\\
 \text{physical complete-sum origin asymptotics: unevaluated}.
 \end{gathered}}
\]

%=====================================================================
\appendix
\section{Finite source specification and implementation inventory}
\label{app:source}
%=====================================================================

\subsection{Curvature and the fixed unintegrated density}

Use the XII-C convention \cite{XIIC}
\[
 R_{\mu\nu}
 =
 \partial_\rho\Gamma^\rho_{\mu\nu}
 -\partial_\nu\Gamma^\rho_{\mu\rho}
 +\Gamma^\rho_{\rho\lambda}\Gamma^\lambda_{\mu\nu}
 -\Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\rho}.
\]
Parentheses on two indices include a factor \(1/2\).
No integration-by-parts replacement of the density is made.

For \(\tau=\tr_\eta\gamma\), put
\[
 u_0^{\mu\nu}=\eta^{\mu\nu},\qquad
 u_1^{\mu\nu}=-\gamma^{\mu\nu},\qquad
 u_2^{\mu\nu}=\gamma^\mu{}_\lambda\gamma^{\lambda\nu},
\]
\[
 b_0=1,\qquad b_1=\frac12\tau,\qquad
 b_2=\frac18\tau^2-\frac14\gamma_{\mu\nu}\gamma^{\mu\nu}.
\]
For \(n=1,2,3\), define
\begin{align}
 \Gamma_n^\rho{}_{\mu\nu}
 &=
 \frac12u_{n-1}^{\rho\lambda}
 \left(
 \partial_\mu\gamma_{\nu\lambda}
 +\partial_\nu\gamma_{\mu\lambda}
 -\partial_\lambda\gamma_{\mu\nu}
 \right),\\
 R_{n,\mu\nu}
 &=
 \partial_\rho\Gamma_n^\rho{}_{\mu\nu}
 -\partial_\nu\Gamma_n^\rho{}_{\mu\rho}
 \nonumber\\
 &\quad+
 \sum_{\substack{a+b=n\\a,b\ge1}}
 \left(
 \Gamma_a^\rho{}_{\rho\lambda}\Gamma_b^\lambda{}_{\mu\nu}
 -\Gamma_a^\rho{}_{\nu\lambda}\Gamma_b^\lambda{}_{\mu\rho}
 \right),\\
 r_n&=\sum_{i=0}^{n-1}u_i^{\mu\nu}R_{n-i,\mu\nu}.
 \label{eq:finite-curvature}
\end{align}
The density is
\begin{equation}
 \ell_3=-2(r_3+b_1r_2+b_2r_1).
 \label{eq:ell3}
\end{equation}
Equivalently,
\[
 \ell_3=-2[s^3]\left(
 \sqrt{-\det(\eta+s\gamma)}R[\eta+s\gamma]
 \right),
\]
where \(s\) is a formal parameter, not a smoothing scale.

Set
\[
 V_{1,\mu\nu}=R_{1,\mu\nu}-\frac12\eta_{\mu\nu}r_1,
\]
\[
 V_{2,\mu\nu}=R_{2,\mu\nu}-\frac12\eta_{\mu\nu}r_2
                                      -\frac12\gamma_{\mu\nu}r_1.
\]
Then
\begin{equation}
 J_3^{\mu\nu}
 =
 2\left(
 V_2^{\mu\nu}+b_1V_1^{\mu\nu}
 -\gamma^{\mu\alpha}V_{1,\alpha}{}^\nu
 -\gamma^{\nu\alpha}V_{1,\alpha}{}^\mu
 \right).
 \label{eq:J3}
\end{equation}

\subsection{Layer coefficients and the weighted source}

Define
\[
 \mathcal U^{\mu\nu}(s)=u_0^{\mu\nu}+su_1^{\mu\nu}+s^2u_2^{\mu\nu},
 \qquad
 \mathcal D(s)=1+sb_1+s^2b_2,
\]
and
\begin{equation}
 C_j^{\mu\nu\alpha\beta}
 =
 [s^j]\mathcal D(s)
 \left(
 \mathcal U^{\mu\nu}(s)\mathcal U^{\alpha\beta}(s)
 -\mathcal U^{\mu(\alpha}(s)\mathcal U^{\beta)\nu}(s)
 \right),
 \quad j=0,1,2.
 \label{eq:C-coefficient-rule}
\end{equation}
Put
\begin{equation}
 M_2^{\mu\nu\rho}
 =
 C_0^{\mu\nu\alpha\beta}\Gamma_2^\rho{}_{\alpha\beta}
 +C_1^{\mu\nu\alpha\beta}\Gamma_1^\rho{}_{\alpha\beta}.
 \label{eq:M2}
\end{equation}
The full weighted tensor source is
\begin{equation}
 \boxed{
 \JD^{\mu\nu}
 =
 \sig J_3^{\mu\nu}
 +2C_2^{\mu\nu\alpha\beta}
                 \partial_\alpha\partial_\beta\sig
 -2M_2^{\mu\nu\rho}\partial_\rho\sig.
 }
 \label{eq:actualsource}
\end{equation}

For completeness, the coefficient argument is as follows.
For \(q=\eta+s\gamma\), compact covariant metric variation gives
\[
 \frac{\delta}{\delta\gamma_{\mu\nu}}
 \left(
 -\frac2{s^2}\int\sigma\sqrt{-q}R[q]\dd^4x
 \right)
 =
 \frac2s\sqrt{-q}
 \left[
 \sigma G^{\mu\nu}
 +q^{\mu\nu}\Box_q\sigma-\nabla^\mu\nabla^\nu\sigma
 \right].
\]
Indeed, if \(h=\delta q\),
\[
 \delta R
 =-R^{\mu\nu}h_{\mu\nu}
 +\nabla_\mu\nabla_\nu h^{\mu\nu}
 -\Box_q(q^{\mu\nu}h_{\mu\nu}),
\]
and two compact integrations by parts yield the displayed variation.

The order-\(s\) coefficient on the left is
\(\EL_\gamma(\sigma\ell_3)\).
The Einstein term on the right gives \(\sigma J_3\), by expanding
\(\sqrt{-q}\) and the two inverse metrics raising \(G_{\mu\nu}\).
The remaining term is
\[
 \frac2s C^{\mu\nu\alpha\beta}(q)
 \left(
 \partial_\alpha\partial_\beta\sigma
 -\Gamma^\rho_{\alpha\beta}(q)\partial_\rho\sigma
 \right),
\]
whose order-\(s\) coefficient is the layer part of
\eqref{eq:actualsource}. This calculation does not select a
full-density weighted gravitational action.

\subsection{Jet-polynomial extraction}

Let \(X_{I,\gamma}\), \(|\gamma|\le2\), be ordinary derivative jets:
\[
 \partial^\gamma\gamma_{\mu\nu}
 =
 \sum_I(e_I)_{\mu\nu}X_{I,\gamma}.
\]
They are not factorial-normalized.
Mixed derivatives are represented by their common multiindex;
the jet derivations commute:
\[
 \mathbb D_\mu X_{I,\gamma}=X_{I,\gamma+e_\mu}.
\]

At a fixed \(x\), construct \(\JD^{\rho\sigma}(x;X)\) from the
displayed finite formulas, treating \(\sig(x)\) and its derivatives
as external coefficients. Apply
\eqref{eq:source-conversion} to obtain \(\mathcal J_x^L(X)\).

\scopebox{Implementation inventory}{%
\begin{enumerate}[label=\textup{(\arabic*)},itemsep=2pt,topsep=2pt]
\item Choose \(e_I,e^I\) satisfying \eqref{eq:basis-duality};
the concrete unordered-pair basis is one admissible choice.
\item Use \(\eta=\operatorname{diag}(1,-1,-1,-1)\), the displayed
curvature convention, and symmetrization with factor \(1/2\).
\item Represent ordinary field derivatives by \(X_{I,\gamma}\);
identify commuting mixed derivatives before taking polynomial
derivatives.
\item Retain \(\sig,\partial\sig,\partial^2\sig\) as fixed coefficients.
Do not vary the switch or discard layer terms before localization.
\item Construct \(J_3,C_2,M_2\), then the tensor source
\eqref{eq:actualsource}; finally convert its output by
\eqref{eq:source-conversion}.
\item Differentiate the resulting quadratic jet polynomial with
respect to the two indicated jet coordinates and evaluate at zero.
The \(\alpha\)-slot belongs to \(G_J\) at \(z\); the \(\beta\)-slot
belongs to \(h_I\) at \(a\).
\end{enumerate}}

\begin{proposition}[Finite source-coefficient rule]
\label{prop:source-rule}
The source coefficients are
\begin{equation}
 \boxed{
 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:jet-Hessian}
\end{equation}
They vanish for \(|\alpha|+|\beta|>2\).
Repeated-variable factors and the factors from differentiating a
quadratic source are included automatically.

The rule is covariant under a simultaneous transformation of the
basis, dual basis, input components, and output conversion.
\end{proposition}

\begin{proof}
The source is homogeneous quadratic in field jets.
The \(J_3\) part contains only derivative-weight patterns
\(X_0X_2\) and \(X_1X_1\). The \(C_2\) part contains \(X_0X_0\);
the \(M_2\) part contains \(X_0X_1\).
Thus total field-derivative weight is at most two.

For a quadratic polynomial,
\[
 D\mathcal J_x(G)[h]=D^2\mathcal J_x(0)[G,h].
\]
Expanding the right-hand side in independent jet coordinates gives
\eqref{eq:jet-Hessian} and \eqref{eq:source-polynomial}.
Ordinary differentiation supplies a factor two when the same
quadratic variable occurs twice. A basis change transforms the
Hessian and its two arguments consistently, leaving the underlying
bilinear map unchanged.
\end{proof}

This is a finite symbolic specification. It is not a claim that
a component program was executed or that a physical characteristic
coefficient was contracted from its output.

\subsection{A pure-trace layer check}

For \(\gamma=\psi\eta\),
\[
 u_1=-\psi\eta^{-1},\qquad u_2=\psi^2\eta^{-1},\qquad
 b_1=2\psi,\qquad b_2=\psi^2.
\]
The coefficient rule gives \(C_1=C_2=0\), and
\[
 \Gamma_2=-\psi\Gamma_1,\qquad
 \Gamma_1^\rho{}_{\mu\nu}
 =
 \frac12\left(
 \delta^\rho_\nu\partial_\mu\psi
 +\delta^\rho_\mu\partial_\nu\psi
 -\eta_{\mu\nu}\partial^\rho\psi
 \right).
\]
Therefore
\[
 \mathcal D_\sigma^{\mu\nu}[\psi\eta]
 =
 -\psi\left[
 \eta^{\mu\nu}(\partial_\rho\sigma)\partial^\rho\psi
 +(\partial^\mu\sigma)\partial^\nu\psi
 +(\partial^\nu\sigma)\partial^\mu\psi
 \right],
\]
and
\[
 \eta_{\mu\nu}\mathcal D_\sigma^{\mu\nu}[\psi\eta]
 =
 -6(\partial_\rho\sigma)\psi\partial^\rho\psi.
\]
For \(G=\psi\eta\), \(h=\zeta\eta\), its derivative is
\[
 \eta_{\mu\nu}
 D\mathcal D_\sigma^{\mu\nu}[\psi\eta](\zeta\eta)
 =
 -6(\partial_\rho\sigma)
       \left(\zeta\partial^\rho\psi+\psi\partial^\rho\zeta\right).
\]
This checks an aggregate source coefficient and the layer
normalization. It is not a full graph cancellation.

%=====================================================================
\section{An evaluated power--logarithm expansion}
\label{app:asymptotics}
%=====================================================================

The following is an outcome classifier for coefficients actually
computed for a family. Its hypotheses are not asserted for the
physical contraction.

\begin{theorem}[Integration of a nonsummable scale expansion]
\label{thm:powerlog}
Suppose
\[
 D_\delta
 =
 \sum_{j=1}^J\delta^{\lambda_j}P_j(\log\delta)+R_\delta,
\]
where the real exponents \(\lambda_j\le0\) are distinct,
each \(P_j\) is a distribution-valued polynomial, and locally
\[
 |R_\delta(\Phi)|
 \le C_L\delta^\alpha\|\Phi\|_{m_L},
 \qquad \alpha>0.
\]
For \(M_j=\deg P_j\), set
\[
 Q_j(t)=
 \begin{cases}
 \displaystyle
 \sum_{r=0}^{M_j}
 \frac{(-1)^r}{\lambda_j^{r+1}}P_j^{(r)}(t),
       &\lambda_j<0,\\[1.2em]
 \displaystyle\int_0^tP_j(s)\dd s,
       &\lambda_j=0.
 \end{cases}
\]
Then
\[
 t_\delta
 =
 \sum_j\delta^{\lambda_j}Q_j(\log\delta)
 +T+O_{\mathrm{loc}}(\delta^\alpha)
\]
for a distribution \(T\), with the stated finite-order remainder.
The raw family converges if and only if every \(P_j\) is zero.
Otherwise a fixed compact test detects unboundedness.
\end{theorem}

\begin{proof}
The finite polynomials solve
\[
 (\lambda_j+\partial_t)Q_j=P_j.
\]
The scale derivative of
\(t_\delta-\sum_j\delta^{\lambda_j}Q_j(\log\delta)\)
is \(R_\delta\). Theorem~\ref{thm:Dini} gives the constant
distribution \(T\) and the remainder.

If a polynomial remains, select the smallest exponent with a
nonzero distributional polynomial and then its highest nonzero
coefficient. A fixed test detects that coefficient.
Its inverse-power term dominates larger exponents and lower
polynomial powers. When the smallest exponent is zero, its
primitive has positive logarithmic degree. The fixed scalar
pairing is unbounded in either case.
\end{proof}

If a limit is independently proved on an open \(V\subset U\),
all nonsummable coefficients restrict to zero there. This follows
by applying the same dominance argument successively to tests
supported in \(V\). Their supports lie in \(U\setminus V\).
Support in \(N\) requires the proved choice \(V=U\setminus N\).

%=====================================================================
\section{Density and vertex audits}
\label{app:density}
%=====================================================================

\subsection{Response-normal density}

With \(q=q_R\),
\[
 u=\frac q{4\beta}.
\]
The ambient density becomes
\[
 \frac{(\beta-q/(4\beta))^2}{\beta}
 \dd^4a\,\dd\Omega\,\dd\alpha\,\dd\beta\,
 \dd^4r\,\dd q\,\dd v\,\dd^2\xi.
\]
Its \(q=0\) factor is \(\beta>0\).

For a smooth factor \(J(q,\tau)\), use \(J\Psi\) as the scalar test.
The inverse-scale coefficient becomes \(J(0,\tau)c_1\).
The finite distribution is multiplied by \(J\); in particular,
\[
 J\partial_q\delta_{\mathrm o}
 =
 J(0,\tau)\partial_q\delta_{\mathrm o}
 -\partial_qJ(0,\tau)\delta_{\mathrm o}.
\]
The derivative of \(J\) changes the finite part, not the
inverse-scale coefficient.

\subsection{Branch--vertex inventory}

For every internal row \(K_b\), the three terms are
\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.18\linewidth}p{.34\linewidth}Y@{}}
\toprule
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}
Their product with the seven rows is the complete finite inventory.

Output derivatives act after multiplication and therefore retain
derivatives of \(g\), the internal distribution, and both relative
crossed dependences. By contrast, \(\partial_2\) acts only on the
crossed scalar slot before pullback.

The transposed signs are:
kinetic \(-i\) becomes \(+i\);
trace-kinetic \(+i/2\) becomes \(-i/2\);
mass remains \(-im^2/2\).
Symmetrization introduces no extra physical polarization factor.

%=====================================================================
\section{Provenance and verification record}
\label{app:verification}
%=====================================================================

\paragraph{Status of documentary statements.}
The predecessor materials cited below are provenance records for
the formulas adopted in this revision. Their availability to an
author, reviewer, retrieval system, or reader are separate questions.
No claim of access or inspection is inferred from a reviewer's
description.

The preparation package for this revision includes the supplied
Em01-C versions, source texts attributed to XII-C, XIV-H, and XIV-J,
and attributed excerpts or retrieval passages from Em01-A, Em01-B-v3,
and XIV-I. This statement does not certify complete-source inspection
of Em01-A, Em01-B-v3, or XIV-I, source authentication, a registry
state, or a scan of the entire series.

The calculations needed for the new source, geometric, translation,
and scalar results are reproduced in this file.
An inherited physical identification remains distinguished from
those calculations.

\paragraph{Revision corrections.}
The algebraic dual basis is distinguished from metric index raising.
The source output is converted explicitly before extracting its
coefficients. The coordinate Euler gradient is not substituted for
the propagated metric-valued source.
The source implementation inventory fixes symmetrization, jet
normalization, switch treatment, slot order, and coefficient order.
The unreduced coordinate map is printed, with its physical provenance
separate from the proved causal reduction.
Microlocal agreement is stated with localized normal-convergence
hypotheses rather than inferred from weak convergence.

\paragraph{Execution limits.}
No pdfLaTeX compiler, proof assistant, symbolic algebra program,
or numerical experiment was executed in producing this revision.
Successful compilation, pagination, mechanically certified identities,
independent referee verification, and journal acceptance are not
asserted. The file is designed for the two-run build in its header.

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

\bibitem{Em01A}
\emph{Em01-A: Conditional Compatibility of Linear Representatives
and the Limits of Witness-to-Graph Inference}.
Attributed manuscript passages supplied in the preparation materials.
Cited for provenance concerning reconstruction and
witness-to-distribution distinctions, not as a complete-source audit.

\bibitem{Em01B}
\emph{Em01-B: Intrinsic Regularization of the Joint Characteristic Map}.
Attributed passages from \texttt{Em01-B-v3.tex}.
Retained branch formulas, vertex normalization, regularization, and
the reported localized ordinary-product convergence.
The relevant defining formulas are reproduced in this paper.

\bibitem{XIIC}
N.~Cheung,
\emph{Landmark XII-C: One declared cubic weighting:
actual Euler source, changed Noether identity,
and the coupled-jet question}.
Supplied manuscript source \texttt{XII-C.tex}.
Curvature, cubic density, and actual weighted Euler source.
Registered manuscript identifier: XII-C, Enter Calculus Companion,
\url{https://enter.grok.me}.

\bibitem{XIVH}
N.~Cheung,
\emph{Landmark XIV-H: Internal kernel relations on the operator route}.
Supplied manuscript source \texttt{XIV-H.tex}.
Source incidence, both \(K_5\) placements, and the unreduced
coordinate-response map.
Registered manuscript identifier: XIV-H, Enter Calculus Companion,
\url{https://enter.grok.me}.

\bibitem{XIVI}
N.~Cheung,
\emph{Landmark XIV-I: Uniform internal retarded amplitudes at the
diagonal--flow-out transition}.
Attributed excerpts and reproduced handoff statements.
The internal retarded multiplier is not used as a prescription for
the unresolved crossed graph product.
Registered manuscript identifier: XIV-I, Enter Calculus Companion,
\url{https://enter.grok.me}.

\bibitem{XIVJ}
N.~Cheung,
\emph{Landmark XIV-J: The joint characteristic map on the unchanged carrier}.
Supplied manuscript source \texttt{XIV-J.tex}.
Reconstruction descent and the supported-sector reference, with its
restricted domain.
Registered manuscript identifier: XIV-J, Enter Calculus Companion,
\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}.
Normal topology and the fundamental operation theorems.

\bibitem{Dang}
N.~V.~Dang,
\emph{The extension of distributions on manifolds,
a microlocal approach}.
arXiv:1412.2808,
\url{https://arxiv.org/abs/1412.2808}.
Comparison framework for geometric scaling and microlocal extension.

\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,
\url{https://arxiv.org/abs/math-ph/9903028}.
Comparison framework for distributional extension and renormalization.

\end{thebibliography}

\end{document}