% Em02-A-v6.tex
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% Suggested build:
%   pdflatex -interaction=nonstopmode -halt-on-error Em02-A-v6.tex
%   pdflatex -interaction=nonstopmode -halt-on-error Em02-A-v6.tex
%
% No external bibliography, figures, scripts, or included files.
%
% Revision boundary:
%   The supplied material does not determine the inherited physical
%   crossed array, component maps, source coefficients, root operation,
%   and observable normalization. This revision does not invent them.
%
% Main mathematical increment:
%   Explicit row--sector reductions and cone-tip localization calculations.
%   The scalar seed transition coefficients and their remainders are
%   calculated. Their localization powers are not physical smoothing powers.
%
% Repairs:
%   Scalar scale regularity is assumed in the singular-primitive theorem.
%   Non-tip chart coverage is restricted to its actual coordinate domains.
%   The smooth-data calculation is classified as a consistency benchmark.
%   Distributional orders and geometric localization powers are separated.
%
% Verification:
%   No successful compiler execution, machine proof, independent symbolic
%   computation, referee approval, or physical admission is claimed.

\documentclass[11pt,letterpaper]{article}

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

\hypersetup{
 pdftitle={Em02-A: Exact Sector Reductions and Local Scale Calculations},
 pdfauthor={}
}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{definition}
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\newcommand{\R}{\mathbb R}
\newcommand{\C}{\mathbb C}
\newcommand{\D}{\mathcal D}
\newcommand{\Dp}{\mathcal D'}
\newcommand{\Ep}{\mathcal E'}
\newcommand{\dd}{\,\mathrm d}
\newcommand{\ii}{\mathrm i}
\newcommand{\Rec}{\mathsf{Rec}}
\newcommand{\Kret}{K_{\mathrm{ret}}}
\newcommand{\Ksup}{K_{\mathrm{supp}}}
\newcommand{\Ucan}{U_{\mathrm{can}}}
\newcommand{\Irr}{\mathsf I_{\mathrm{RR}}}
\newcommand{\Iresp}{\mathsf I_{\mathrm{resp}}}
\newcommand{\Dscale}{D_\delta}
\newcommand{\Eulerh}{E_h}
\newcommand{\Ejet}{\mathsf E}
\newcommand{\Rows}{\mathcal B}
\newcommand{\pair}[2]{\langle #1,#2\rangle}
\newcommand{\Jet}[2]{\mathsf J_{#1,#2}}
\newcommand{\status}[1]{%
 \par\smallskip\noindent\textit{Status: #1.}\par\smallskip}

\DeclareMathOperator{\supp}{supp}
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\DeclareMathOperator{\rank}{rank}

\title{\textbf{Em02-A}\\[0.4em]
Exact Sector Reductions and Local Scale Calculations\\[0.4em]
\large Retarded seed transitions, complete translation pairings,\\
and the unresolved inherited physical specialization}
\author{}
\date{}

\begin{document}
\maketitle

\begin{abstract}
We reorganize the local fixed-kernel problem around the integrals
that must actually be calculated. The complete retarded inventory
is retained, while the crossed array is smoothed in eight flat
relative variables. Exact adjoints and the compact translation
representation are collected in supporting appendices.

The main text gives a row--sector contribution ledger, exact
non-tip integration formulas, and a cone-radius treatment of the
coincidence transition. For a fixed smooth localization profile,
the scalar seed measures supported by
$R(b-a)R(a-z)$ and $R(b-z)R(z-a)$ have explicitly calculated
order-two coincidence-localization coefficients. The seed
supported by $R(b-a)R(b-z)$ has an order-three coefficient.
Uniform remainders are proved directly. These powers concern an
independent internal localization parameter, not the physical
smoothing scale.

The same transition calculations are checked through the
translation-distribution construction. A complete derivative
certificate, if obtained for identified crossed data, determines
the singular primitive under an explicit scalar absolute
continuity hypothesis. The smooth-data expansion is retained
only as a consistency benchmark.

The supplied excerpts do not determine an executable inherited
physical input package. Accordingly, no physical singular-data
sector is evaluated, no physical complete coefficient or
integrable smoothing remainder is obtained, and no physical
limit, authorized correction, or quantum admission is asserted.
A completion ledger distinguishes these unresolved tasks from
the mathematical calculations proved here.
\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) {Displayed retarded seeds};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Cone-tip localization};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Physical singular coefficient};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {A seed transition is not the physical coefficient.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{A seed transition 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-Em02-A}
\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) {Displayed retarded seeds};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Cone-tip localization};
\node[draw,dashed,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Physical singular coefficient};
\draw[->] (a) -- (b);
\draw[->,dashed] (b) -- (c);
\end{tikzpicture}
\caption{A seed transition is not the physical coefficient. Reading downward is not a transfer. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:split-Em02-A}
\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) {Displayed retarded seeds};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Cone-tip localization};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Physical singular coefficient};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\end{tikzpicture}
\caption{Reading downward is not a transfer. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:chain-Em02-A}
\end{figure}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=3.4cm,font=\small,inner sep=3pt] (a) at (0,0) {Displayed retarded seeds};
\node[draw,rounded corners,align=center,text width=3.4cm,font=\small,inner sep=3pt] (b) at (6.2,0) {Physical singular coefficient};
\draw[dashed] (a) -- (b);
\end{tikzpicture}
\caption{Displayed retarded seeds is not Physical singular coefficient. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:pair-Em02-A}
\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) {Displayed retarded seeds};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Cone-tip localization};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Physical singular coefficient};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {A seed transition is not the physical coefficient.};
\end{tikzpicture}
\caption{Three statuses stay separate. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:gate-Em02-A}
\end{figure}

Figure~\ref{fig:scope-Em02-A}, Figure~\ref{fig:split-Em02-A}, Figure~\ref{fig:chain-Em02-A}, Figure~\ref{fig:pair-Em02-A}, and Figure~\ref{fig:gate-Em02-A} name the stop. Not a proof, and not an admission of a quantum coefficient.


\noindent\textbf{Scope of this revision.}
The requested physical completion cannot be produced from the
supplied input descriptions alone. This manuscript records that
stop at the beginning, rather than replacing the missing family
with diagnostic data. Its new calculations concern the displayed
retarded seed geometry.

\tableofcontents

%=====================================================================
\section{Identification first: the physical input remains unresolved}
\label{sec:identification}
%=====================================================================

The target is one local family
\begin{equation}
 t_\delta^\zeta(\Phi)
 =
 \pair{\mathsf T_{c_\delta}(\Kret)}{\zeta\Phi},
 \qquad
 \dot t_\delta^\zeta(\Phi)
 =
 \pair{\mathsf T_{\dot c_\delta}(\Kret)}{\zeta\Phi},
 \qquad
 \dot c_\delta=\delta\partial_\delta c_\delta.
 \label{eq:target}
\end{equation}
A physical calculation requires the values or complete symbolic
definitions of
\begin{equation}
 C,\quad \mathsf A,\quad\mathsf D,\quad\mathcal P,\quad
 \mathsf b,\quad q_{\alpha\beta},\quad
 f,g,\sigma_\varepsilon,\quad F,
 \label{eq:required-inputs}
\end{equation}
in one component and dual-pairing convention.

The supplied excerpts provide an internal formula inventory and
several conditional interfaces. They do not instantiate
\eqref{eq:required-inputs} for the inherited weighted observable.
In particular, a name for a propagator, source, or root operation
does not determine the component operation needed below.

\subsection{Required executable specification}

The following is an identification ledger, not an assumption
that the entries have been supplied.

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.18\linewidth}Yp{.18\linewidth}@{}}
\toprule
Input & Required specification & Present status\\
\midrule
$C$ &
Propagator factors; orientations of both relative slots;
component contractions; normalization; boundary-value
prescription; differentiated components. &
Unresolved\\[0.3em]
$\mathsf A,\mathsf D$ &
Finite component maps in the chosen tensor basis, including
the relationship, if any, between the two types of
$\mathsf A$. &
Unresolved\\[0.3em]
$\mathcal P,\mathsf b$ &
Complete response projection and scalar contraction maps. &
Unresolved\\[0.3em]
$q_{\alpha\beta}$ &
The actual weighted source operation, expanded in the ambient
source-jet convention. &
Unresolved\\[0.3em]
$f,g,\sigma_\varepsilon$ &
Specified functions or explicitly quantified local hypotheses,
with the inherited regulator width distinguished from $\delta$. &
Unresolved\\[0.3em]
$F$ &
The actual advanced-root operator, its domain, normalization,
and the resulting function on the chosen patch. &
Unresolved\\[0.3em]
Vertex &
Identification of the displayed vertex with the inherited
observable, including signs, factors, and derivative placement. &
Unresolved\\[0.3em]
Corrections &
The permitted family, support rules, tensor conditions,
normalization, and reference requirements. &
Unresolved\\
\bottomrule
\end{tabularx}
\end{center}

Finite symbolic definitions can fill this ledger without a
printed component table. The present manuscript does not fill
it by copying the arrays of a different application.

The documentary locations for the internal formulas are XIV-H
and XIV-I \cite{XIVH,XIVI}. XIV-J distinguishes reconstruction,
the supported-sector reference, and the missing full
characteristic map \cite{XIVJ}. These attributions concern the
supplied excerpts only.

\subsection{What may be transferred locally}

If an actual local identity
$k=\widetilde k$ has been proved on a neighborhood of
$\supp\zeta$, then
\[
 \pair{\mathsf T_c(k)}{\zeta\Phi}
 =
 \pair{\mathsf T_c(\widetilde k)}{\zeta\Phi}
\]
for every smooth crossed coefficient $c$. The vertex test and
all its derivatives remain supported there.

The supplied Em01-C plateau statement is such an internal
identity on its stated retained patch \cite{EmC}. Applying it
requires identifying the chosen patch, switch, and weighted
source with its hypotheses. It does not determine the advanced
root from local constancy of $f$, and it does not provide a
smoothing gain by itself.

A missing frozen presentation space is a separate operator
obligation. It does not prevent a fixed-kernel calculation once
the actual values have been identified \cite{PatchII}.

\subsection{Two targets, not one interchangeable patch}

\begin{description}
 \item[Target I: non-tip isolation.]
 A compact patch on which support excludes every row except
 one. This may decide failure for an identified family and a
 specified correction class.

 \item[Target II: coincidence.]
 A compact patch meeting $r=z-a=0$. All potentially contributing
 rows and the cone-tip transition must be retained.
\end{description}

Target I does not solve Target II. The support-isolated
Em01-G failure concerns its own identified application
\cite{EmG}; its coefficient is not imported here.

%=====================================================================
\section{The exact common pairing}
\label{sec:pairing}
%=====================================================================

Use $M=\R^4$, $W=M^4_{(a,b,z,y)}$, and
$\eta=\operatorname{diag}(1,-1,-1,-1)$.
The defining kernels and complete adjoints are displayed and
proved in Appendix~\ref{app:adjoints}. Their inventory is
\begin{equation}
 \Kret=
 K_{47}+K_5^{[a]}+K_5^{[z]}+K_6+K_8+K_{9d}+K_{9m},
 \qquad
 \Rows=\{47,5a,5z,6,8,9d,9m\}.
 \label{eq:inventory}
\end{equation}
There are six operational entries, since
$K_9=K_{9d}+K_{9m}$.

The mathematical crossed operation uses
\[
 \pi(w)=(U,V)=(y-a,y-b),
\]
\[
 h_I(c)=\eta^{\mu\nu}c_{\mu\nu|I},\qquad
 Z_I^\rho(c)
 =
 \partial_{V^\nu}c^{\rho\nu}{}_{|I}
             -\frac12\partial_V^\rho h_I(c),
\]
and
\begin{equation}
 \mathcal W_c^{I,J}[\phi]
 =
 g(y)\left[
 Z_I^\rho(c)(\pi w)\partial_{y^\rho}\phi_J(w)
 +\frac{m^2}{2}h_I(c)(\pi w)\phi_J(w)
 \right].
 \label{eq:vertex-test}
\end{equation}
Relative derivatives are taken before pullback and raising
uses $\eta$. Define
\begin{equation}
 \pair{\mathsf T_c(k)}\phi
 =
 -\ii\sum_{I,J}\pair{k^{I,J}}{\mathcal W_c^{I,J}[\phi]}.
 \label{eq:vertex}
\end{equation}
This fixes a mathematical operation. Its physical identification
is an unresolved entry of the ledger.

Fix $\zeta\in C_c^\infty(W)$, a relative cutoff
$\chi=1$ near $\pi(\supp\zeta)$, and one compact profile
$\rho\in C_c^\infty(\R^8)$ with $\int\rho=1$. Put
\begin{equation}
 C^\chi=\chi C,\qquad
 c_\delta=\rho_\delta*C^\chi,\qquad
 \rho_\delta(h)=\delta^{-8}\rho(h/\delta).
 \label{eq:smoothing}
\end{equation}
The differentiated profile is
\[
 \kappa(h)=-8\rho(h)-h\cdot\nabla\rho(h),\qquad
 \dot c_\delta=\kappa_\delta*C^\chi.
\]
It satisfies
\begin{equation}
 \int\kappa=0,\qquad
 \int h^\alpha\kappa(h)\dd h
 =|\alpha|\int h^\alpha\rho(h)\dd h.
 \label{eq:profile-moments}
\end{equation}
These identities follow by direct differentiation and integration
by parts.

For a two-line row define its smooth ambient amplitude by
\begin{equation}
 H_{e,c}^{I,J}[\Phi]
 =
 \left(
 \mathscr P_e^{I,J}
       [\mathcal W_c^{I,J}[\zeta\Phi]]
 \right)\circ\Gamma_e^{-1},
 \qquad e=47,5a,5z.
 \label{eq:ambient-amplitude}
\end{equation}
The inverse exists for these independent seed charts.
For response rows, retain the smooth restricted amplitude
\[
 H_{e,c}^{I,J}[\Phi](a,s,y)
 =
 \mathscr P_e^{I,J}
       [\mathcal W_c^{I,J}[\zeta\Phi]](a,s,y).
\]
Thus the derivatives have already been applied before any
geometric restriction below.

Let $\mu_e$ denote the scalar ambient seed measure:
\begin{align}
 \mu_{47}&=R(b-a)R(b-z)\,\dd^4a\,\dd^4b\,\dd^4z\,\dd^4y,
 \notag\\
 \mu_{5a}&=R(b-a)R(a-z)\,\dd^4a\,\dd^4b\,\dd^4z\,\dd^4y,
 \notag\\
 \mu_{5z}&=R(b-z)R(z-a)\,\dd^4a\,\dd^4b\,\dd^4z\,\dd^4y.
 \label{eq:scalar-seeds}
\end{align}
These expressions mean the independent tensor-product measures
of Appendix~\ref{app:adjoints}, not products formed after
coincident-variable restriction. Write $\mu_0$ for the response
measure
\[
 \pair{\mu_0}H
 =
 \int\dd^4a\,\dd R(s)\,\dd^4y\,H(a,a+s,a,y).
\]

\begin{proposition}[Exact seven-row starting value]
\label{prop:starting}
\status{Proved for displayed mathematical data}
The raw and differentiated complete pairings are
\begin{equation}
\begin{aligned}
 t_\delta^\zeta(\Phi)
 =-\ii\sum_{I,J}\bigg[
 &\sum_{e=47,5a,5z}
       \pair{\mu_e}{H_{e,c_\delta}^{I,J}[\Phi]}\\
 &+\sum_{e=6,8,9d,9m}
       \Iresp(H_{e,c_\delta}^{I,J}[\Phi])
 \bigg],
 \label{eq:exact-raw}\\
 \dot t_\delta^\zeta(\Phi)
 =-\ii\sum_{I,J}\bigg[
 &\sum_{e=47,5a,5z}
       \pair{\mu_e}{H_{e,\dot c_\delta}^{I,J}[\Phi]}\\
 &+\sum_{e=6,8,9d,9m}
       \Iresp(H_{e,\dot c_\delta}^{I,J}[\Phi])
 \bigg].
 \label{eq:exact-derivative}
\end{aligned}
\end{equation}
Every fixed-test pairing is smooth for $\delta>0$.
\end{proposition}

\begin{proof}
Apply the adjoint identities of
Theorem~\ref{thm:adjoints}. Distributional convolution may be
differentiated on compact positive-scale intervals because the
profiles form smooth families with bounded test seminorms.
Linearity of the vertex then gives
\eqref{eq:exact-derivative}.
\end{proof}

%=====================================================================
\section{Route B: geometry, support zeros, and the contribution ledger}
\label{sec:routeB}
%=====================================================================

Choose a local sphere frame
\[
 \ell=(1,\mathbf n),\quad k=(1,-\mathbf n),\quad
 E_A=(0,\mathbf e_A),\quad A=1,2,
\]
with $\ell\cdot k=2$ and
$E_A\cdot E_B=-\delta_{AB}$. Use
\begin{equation}
 z=a+r,\quad
 b=a+\beta\ell+u k,\quad
 y=a+\alpha\ell+v k+\xi^AE_A,
 \qquad
 r=\gamma\ell+\upsilon k+\omega^AE_A.
 \label{eq:chart}
\end{equation}
Fix a compact regular chart with positive lower bounds for
$\alpha$, $\beta-\alpha$, $\beta$, and $\beta-u$.
Where a non-tip formula requires $\beta-\gamma>0$, that
restriction is imposed separately.

Put
\[
 q_R=(b-a)^2,\quad q_1=(y-a)^2,\quad
 q_2=(b-y)^2,\quad q_z=(b-z)^2.
\]
Then
\begin{align}
 q_R&=4\beta u,&
 q_1&=4\alpha v-|\xi|^2,&
 q_2&=4(\beta-\alpha)(u-v)-|\xi|^2,
 \label{eq:cones}\\
 r^2&=4\gamma\upsilon-|\omega|^2,&
 q_z&=4(\beta-\gamma)(u-\upsilon)-|\omega|^2.
 \label{eq:internal-cones}
\end{align}
The density is
\begin{equation}
 \dd^4a\,\dd^4b\,\dd^4z\,\dd^4y
 =
 4(\beta-u)^2\dd\tau\,\dd^4r\,\dd u\,\dd v\,\dd^2\xi,
 \quad
 \dd\tau=\dd^4a\,\dd\Omega\,\dd\alpha\,\dd\beta,
 \label{eq:density}
\end{equation}
and $\dd^4r=2\dd\gamma\,\dd\upsilon\,\dd^2\omega$.
The inverse chart and density verification are given in
Appendix~\ref{app:geometry}.

\subsection{Seed constraints}

On the scalar supports:
\begin{align}
 47:\quad&
 u=0,\qquad
 \upsilon=-\frac{|\omega|^2}{4(\beta-\gamma)}
 \quad(\beta-\gamma>0);
 \label{eq:47-constraint}\\
 5a:\quad&
 u=0,\qquad r\text{ past null};
 \label{eq:5a-constraint}\\
 5z:\quad&
 r,\ b-z\text{ future null},\qquad
 \upsilon=\frac{\beta-\gamma}{\beta}u,\qquad
 |\omega|^2=\frac{4\gamma(\beta-\gamma)}{\beta}u
 \quad(0<\gamma<\beta);
 \label{eq:5z-constraint}\\
 \mathrm{response}:\quad&
 r=0,\qquad u=0.
 \label{eq:response-constraint}
\end{align}
For $5a$, directed nullness gives
$\gamma,\upsilon\le0$. For $5z$, it gives $u\ge0$.
The relations follow by equating the two expressions for
$|\omega|^2$ in \eqref{eq:internal-cones}.

\subsection{Support-isolated patches}

\begin{proposition}[Exact support isolation]
\label{prop:isolation}
\status{Proved for displayed mathematical data}
\begin{enumerate}[label=\textup{(\roman*)}]
 \item At
 \[
 b-a=\beta\ell,\qquad
 r=-\epsilon(1,\mathbf m),\quad
 \epsilon>0,\quad\mathbf m\ne\mathbf n,
 \]
 a sufficiently small neighborhood excludes every row except
 $5a$.

 \item If $b-a$ and $b-z$ are future null and nonparallel,
 a sufficiently small neighborhood excludes every row except
 $47$.
\end{enumerate}
The complete contraction on tests in either neighborhood equals
the surviving row contraction for every smooth crossed array.
No nonzero singular coefficient follows from isolation alone.
\end{proposition}

\begin{proof}
In (i),
\[
 (b-z)^2=2\beta\epsilon(1-\mathbf n\cdot\mathbf m)>0.
\]
This excludes the $47$ and $5z$ seed supports; $r\ne0$ excludes
the response rows.

In (ii), write $s=b-a$, $t=b-z$. Then
$r=s-t$ and $r^2=-2s\cdot t<0$.
Both $5$ supports require $r^2=0$, and the response rows require
$r=0$. Differentiation and smooth multiplication preserve
these local support exclusions.
\end{proof}

\subsection{A single partition convention}

Choose $b_0\in C^\infty([0,\infty);[0,1])$ with
$b_0=1$ on $[0,1]$ and $b_0=0$ on $[4,\infty)$.
A fixed coincidence partition is
\[
 \theta_{\mathrm{tip}}
 =b_0(|r|_{\mathrm E}^2/\epsilon_0^2),\qquad
 \theta_{\mathrm{away}}=1-\theta_{\mathrm{tip}},
\]
where $\epsilon_0$ is independent of $\delta$.

For $L>0$, let
\[
 a_{j,\delta}=b_0(q_j^2/(L^2\delta^2)),\quad j=1,2,
 \qquad a_{R,\delta}=b_0(q_R^2/(L^2\delta^2)).
\]
Use the exact five-sector partition
\begin{align}
 \theta_{00,\delta}&=a_{1,\delta}a_{2,\delta}a_{R,\delta},
 \notag\\
 \theta_{0R,\delta}&=a_{1,\delta}a_{2,\delta}(1-a_{R,\delta}),
 \notag\\
 \theta_{10,\delta}&=a_{1,\delta}(1-a_{2,\delta}),
 \notag\\
 \theta_{01,\delta}&=(1-a_{1,\delta})a_{2,\delta},
 \notag\\
 \theta_{11,\delta}&=(1-a_{1,\delta})(1-a_{2,\delta}).
 \label{eq:partition}
\end{align}
Their sum is one.

Throughout the main calculation, these functions multiply
\emph{already differentiated seed amplitudes}.
Thus
\begin{equation}
 v_{e,j,\delta}(\Phi)
 =
 -\ii\sum_{I,J}
 \mathsf I_e\left(
 \vartheta_{e,j,\delta}
 \mathscr P_e^{I,J}
       [\mathcal W_{\dot c_\delta}^{I,J}[\zeta\Phi]]
 \right),
 \qquad
 \sum_{e,j}v_{e,j,\delta}=\dot t_\delta^\zeta.
 \label{eq:postadjoint-partition}
\end{equation}
Here $\vartheta_{e,j,\delta}$ denotes the partition pulled back
to the appropriate seed chart, including the fixed tip/away
partition if used.

Individual $v_{e,j,\delta}$ are not declared to be derivatives
of raw sectors. No motion term is needed in
\eqref{eq:postadjoint-partition}. The different raw-sector
convention is recorded in Appendix~\ref{app:partitions}.

\subsection{Row--sector support table}

In the next table, $P$ means ``not excluded by scalar support.''
It does not mean a proved nonzero contribution. A zero is an
exact support zero in the post-adjoint convention.

\begin{center}
\small
\begin{tabular}{@{}lcccccc@{}}
\toprule
Row & $00$ & $0R$ & $10$ & $01$ & $11$
    & Away from $r=0$\\
\midrule
$47$ & $P$ & $0$ & $P$ & $P$ & $P$ & $P$\\
$5a$ & $P$ & $0$ & $P$ & $P$ & $P$ & $P$\\
$5z$ & $P$ & $P$ & $P$ & $P$ & $P$ & $P$\\
$6$  & $P$ & $0$ & $P$ & $P$ & $P$ & $0$\\
$8$  & $P$ & $0$ & $P$ & $P$ & $P$ & $0$\\
$9d$ & $P$ & $0$ & $P$ & $P$ & $P$ & $0$\\
$9m$ & $P$ & $0$ & $P$ & $P$ & $P$ & $0$\\
\bottomrule
\end{tabular}
\end{center}

\begin{proof}[Proof of the zeros]
On the $47$, $5a$, and response scalar seeds, $q_R=0$, so
$a_{R,\delta}=1$ and $\theta_{0R,\delta}=0$.
The response seeds have $r=0$, where
$\theta_{\mathrm{away}}=0$.
Because the partition is inserted after the adjoints,
no derivative of these vanishing factors has been generated.
\end{proof}

The remaining calculation information is:

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.10\linewidth}Yp{.13\linewidth}Y@{}}
\toprule
Row & Non-tip coordinates & Maximal crossed order
 & Unresolved physical remainder task\\
\midrule
$47$ &
\eqref{eq:47-constraint}; $\beta-\gamma$ bounded below &
$3$ &
Crossed boundary values and angular concentration.\\[0.3em]
$5a$ &
$\gamma<0$ or $\upsilon<0$, bounded away from zero &
$3$ &
Identified crossed factor and the $r$-tip junction.\\[0.3em]
$5z$ &
Two charts in Section~\ref{sec:nontip}; their stated denominators
bounded below &
$5$ &
The $0R$ sector, angular concentration, and the $r$-tip junction.\\[0.3em]
$6$ &
Response incidence &
$3$ &
Actual crossed normal traces and complete cancellation.\\[0.3em]
$8$ &
Response incidence; ambient source jets &
$4$ &
Same, with the coordinate-response multiplier.\\[0.3em]
$9d$ &
Response incidence; ambient source jets &
$4$ &
Same, with the actual advanced-root derivative.\\[0.3em]
$9m$ &
Response incidence; ambient source jets &
$3$ &
Same, with the actual mass-root term.\\
\bottomrule
\end{tabularx}
\end{center}

Every crossed factor in this table is currently unidentified for
the inherited family. Consequently no physical scaling exponent
is entered.

\subsection{Where the core substitution is valid}

The identity
\[
 (\beta-\alpha)q_1+\alpha q_2
 =
 4\alpha(\beta-\alpha)u-\beta|\xi|^2
\]
implies, on the fixed compact chart,
\[
 q_R,q_1,q_2=O(\delta)
 \quad\Longrightarrow\quad
 u,v=O(\delta),\quad \xi=O(\sqrt\delta).
\]
Thus the $00$ sector permits
$(u,v,\xi)=(\delta u_*,\delta v_*,\sqrt\delta X)$.
The translated equations, including their higher-order terms,
are retained in Appendix~\ref{app:geometry}.

Neither this implication nor simultaneous crossed cones forces
$r=O(\delta)$. Moreover, the $5z$ support admits
$q_1=q_2=0$ with $q_R>0$ fixed; see
Proposition~\ref{prop:counterexample}.
The $0R$ sector therefore cannot be removed by the core argument.

%=====================================================================
\section{Route B: the exact non-tip integrals}
\label{sec:nontip}
%=====================================================================

This section identifies the actual integrals remaining after
the adjoints. It does not insert an unspecified physical
boundary value.

Let
\[
 \widehat H_{e,\delta}
 =
 H_{e,\dot c_\delta}[\Phi]
\]
expressed in the characteristic coordinates; component sums
are suppressed temporarily. Let $\Theta_{j,\delta}$ be a
chosen post-adjoint sector factor.

\begin{proposition}[Non-tip scalar reductions]
\label{prop:nontip}
\status{Proved for displayed mathematical data}
On compact supports within the indicated coordinate domains,
the two-line differentiated integrals are the following exact
scalar integrals.

For $47$, put
$\upsilon_{47}=-|\omega|^2/[4(\beta-\gamma)]$. Then
\begin{equation}
 \int
 \frac{\beta}{8\pi^2(\beta-\gamma)}
 \bigl(\Theta_{j,\delta}\widehat H_{47,\delta}\bigr)
 (\tau;\gamma,\upsilon_{47},\omega,0,v,\xi)
 \dd\tau\,\dd\gamma\,\dd^2\omega\,\dd v\,\dd^2\xi.
 \label{eq:47-integral}
\end{equation}

For $5a$ on $\gamma<0$, put
$\upsilon_{5a}=|\omega|^2/(4\gamma)$. Then
\begin{equation}
 \int
 \frac{\beta}{8\pi^2|\gamma|}
 \bigl(\Theta_{j,\delta}\widehat H_{5a,\delta}\bigr)
 (\tau;\gamma,\upsilon_{5a},\omega,0,v,\xi)
 \dd\tau\,\dd\gamma\,\dd^2\omega\,\dd v\,\dd^2\xi.
 \label{eq:5a-integral}
\end{equation}
On $\upsilon<0$, the alternative formula eliminates
$\gamma=|\omega|^2/(4\upsilon)$ and uses density
$\beta/(8\pi^2|\upsilon|)$.

For $5z$ on $0<\gamma<\beta$, put
\[
 \upsilon_\omega=\frac{|\omega|^2}{4\gamma},\qquad
 u_\omega=\frac{\beta|\omega|^2}
                     {4\gamma(\beta-\gamma)}.
\]
Then the integral is
\begin{equation}
 \int
 \frac{(\beta-u_\omega)^2}
      {8\pi^2\gamma(\beta-\gamma)}
 \bigl(\Theta_{j,\delta}\widehat H_{5z,\delta}\bigr)
 (\tau;\gamma,\upsilon_\omega,\omega,u_\omega,v,\xi)
 \dd\tau\,\dd\gamma\,\dd^2\omega\,\dd v\,\dd^2\xi.
 \label{eq:5z-integral}
\end{equation}
On $\upsilon>0$, an alternative eliminates
\[
 \gamma_\omega=\frac{|\omega|^2}{4\upsilon},\qquad
 u_\omega=\frac{\beta\upsilon}{\beta-\gamma_\omega}
\]
and has density
\[
 \frac{(\beta-u_\omega)^2}
      {8\pi^2\upsilon(\beta-\gamma_\omega)}
\]
in $\dd\tau\,\dd\upsilon\,\dd^2\omega\,\dd v\,\dd^2\xi$.

The response measure has density
\begin{equation}
 \frac{\beta}{2\pi}\dd\tau\,\dd v\,\dd^2\xi
 \quad\text{at }r=0,\ u=0.
 \label{eq:response-density}
\end{equation}
Its amplitude is the already computed response adjoint,
not a newly chosen tangential jet.
\end{proposition}

\begin{proof}
Away from a cone vertex,
$R(x)=(2\pi)^{-1}\Theta(x^0)\delta(x^2)$.
The full null-frame density is
\[
 8(\beta-u)^2
 \dd\tau\,\dd\gamma\,\dd\upsilon\,\dd^2\omega\,
 \dd u\,\dd v\,\dd^2\xi.
\]
For $47$, the constraint Jacobian in $(u,\upsilon)$ is
$16\beta(\beta-\gamma)$.
For $5a$, it is $16\beta|\gamma|$, or
$16\beta|\upsilon|$ in the alternative chart.
For $5z$, the two Jacobians are respectively
$16\gamma(\beta-\gamma)$ and
$16\upsilon(\beta-\gamma)$.
Combining these with $(2\pi)^{-2}$ proves the formulas.
The directed support conditions select the indicated domains.

For the response seed, integrate $\delta^{(4)}(r)$ in the
original $r$ density, and then
$\delta(4\beta u)/(2\pi)$. The result is $\beta/(2\pi)$.
\end{proof}

\begin{remark}[Corrected chart coverage]
The two $5a$ charts cover compact past-null non-tip regions
after a finite partition: at least one of $|\gamma|$ and
$|\upsilon|$ is bounded below.

For $5z$, the formulas above cover the non-tip region
\emph{within the additional domain}
$\beta-\gamma\ge c_0>0$.
Separation of $b-z$ alone does not imply that inequality:
a nonzero future-null $b-z$ can be parallel to $k$.
Outside this domain one must use another chart or the
independent cone-radius formula below. No global coverage is
claimed from these two denominators alone.
\end{remark}

To finish a singular non-tip calculation, the actual
$\dot c_\delta$ must now be inserted into
\eqref{eq:47-integral}--\eqref{eq:5z-integral}. Its boundary
values, derivatives, contractions, and remainder are not
determined by the scalar densities.

%=====================================================================
\section{Route B: an explicit coincidence-transition calculation}
\label{sec:transition}
%=====================================================================

The coincidence transition is treated in independent cone
coordinates rather than by extrapolating the non-tip
denominators.

For $T\ge0$ and $\mathbf m\in S^2$, write
$n_{\mathbf m}=(1,\mathbf m)$. The exact cone measure is
\begin{equation}
 \dd R(Tn_{\mathbf m})
 =
 \frac1{4\pi}T\,\dd T\,\dd\mathbf m.
 \label{eq:radial-cone}
\end{equation}
For a compact smooth ambient scalar amplitude $H$,
\begin{align}
 \pair{\mu_{5a}}H
 &=
 \frac1{16\pi^2}
 \int S T\,
 H(a,a+Sn_{\mathbf n},a-Tn_{\mathbf m},y)
 \dd^4a\,\dd^4y\,\dd S\,\dd T\,
                      \dd\mathbf n\,\dd\mathbf m,
 \label{eq:radial-5a}\\
 \pair{\mu_{5z}}H
 &=
 \frac1{16\pi^2}
 \int S T\,
 H(z-Tn_{\mathbf m},z+Sn_{\mathbf n},z,y)
 \dd^4z\,\dd^4y\,\dd S\,\dd T\,
                      \dd\mathbf n\,\dd\mathbf m.
 \label{eq:radial-5z}
\end{align}
Both radial variables range over $[0,\infty)$.
These formulas include $T=0$ without introducing a
$1/|\gamma|$ denominator.

Introduce a second parameter $\epsilon>0$, independent of
$\delta$, and define
\[
 \theta_\epsilon(r)=b_0(|r|_{\mathrm E}^2/\epsilon^2).
\]
Set
\begin{equation}
 c_R=\int_0^\infty t\,b_0(2t^2)\dd t
     =\frac14\int_0^\infty b_0(s)\dd s,
 \qquad
 c_{47}=\frac1{2\sqrt2\,\pi}
               \int_0^\infty t^2b_0(t^2)\dd t.
 \label{eq:transition-constants}
\end{equation}
Both constants are finite and strictly positive.

In the characteristic chart define the incidence measure
$\nu_0$ by
\[
 \pair{\nu_0}H
 =
 \int H(a,b,a,y)\big|_{u=0}
                           \dd\tau\,\dd v\,\dd^2\xi.
\]
Thus, locally,
\begin{equation}
 \dd\mu_0=\frac{\beta}{2\pi}\dd\nu_0.
 \label{eq:incidence-measures}
\end{equation}

\begin{theorem}[Calculated scalar seed transition coefficients]
\label{thm:transition}
\status{Proved for displayed mathematical data}
Fix a compact characteristic localization as above. For
amplitudes $H$ supported there, and sufficiently small $\epsilon$,
\begin{align}
 \pair{\mu_{5a}}{\theta_\epsilon H}
 &=
 c_R\epsilon^2\pair{\mu_0}H
       +O(\epsilon^3)\|H\|_{1,K},
 \label{eq:tip-5a}\\
 \pair{\mu_{5z}}{\theta_\epsilon H}
 &=
 c_R\epsilon^2\pair{\mu_0}H
       +O(\epsilon^3)\|H\|_{1,K},
 \label{eq:tip-5z}\\
 \pair{\mu_{47}}{\theta_\epsilon H}
 &=
 c_{47}\epsilon^3\pair{\nu_0}H
       +O(\epsilon^4)\|H\|_{1,K}.
 \label{eq:tip-47}
\end{align}
Here $K$ is a fixed enlarged compact neighborhood, and the
constants are uniform for the stated support class.
For the response measure,
\[
 \pair{\mu_0}{\theta_\epsilon H}=\pair{\mu_0}H
\]
exactly.

These are localization asymptotics in $\epsilon$. They are
not smoothing asymptotics in $\delta$.
\end{theorem}

\begin{proof}
For $5a$, use \eqref{eq:radial-5a}.
On the $r=-Tn_{\mathbf m}$ cone,
$|r|_{\mathrm E}^2=2T^2$. Replace the amplitude by its value at
$T=0$. The removed cone factor has total mass
\[
 \frac1{4\pi}\int_{S^2}\int_0^\infty
          T b_0(2T^2/\epsilon^2)\dd T\,\dd\mathbf m
 =c_R\epsilon^2.
\]
The remaining integral is $\pair{\mu_0}H$.
The amplitude difference is at most
$C T\|H\|_{1,K}$, and
\[
 \int_0^\infty T^2b_0(2T^2/\epsilon^2)\dd T
 =O(\epsilon^3).
\]
The remaining cone and passive measures have bounded mass on
a fixed compact set. This proves \eqref{eq:tip-5a}.

For $5z$, use \eqref{eq:radial-5z}.
At $T=0$, the ambient point is $(z,z+Sn_{\mathbf n},z,y)$,
again giving $\mu_0$ after renaming the base variable.
The same radial mass and Lipschitz estimate prove
\eqref{eq:tip-5z}.

For $47$, a sufficiently small coincidence neighborhood
satisfies $\beta-\gamma\ge c_0>0$. Use
\eqref{eq:47-integral} without a smoothing amplitude.
On its seed,
\[
 u=0,\qquad
 \upsilon=-\frac{|\omega|^2}{4(\beta-\gamma)},\qquad
 |r|_{\mathrm E}^2=2\gamma^2+2\upsilon^2+|\omega|^2.
\]
Set $\gamma=\epsilon G$, $\omega=\epsilon W$. Then
\[
 \upsilon
 =-\epsilon^2\frac{|W|^2}{4(\beta-\epsilon G)},
\]
and the cutoff confines $(G,W)$ to a fixed bounded set.
The density becomes
\[
 \epsilon^3\frac{\beta}{8\pi^2(\beta-\epsilon G)}.
\]
Its leading factor is $\epsilon^3/(8\pi^2)$.
The cutoff differs from $b_0(2G^2+|W|^2)$ by $O(\epsilon^2)$,
uniformly on that bounded set. The amplitude differs from its
incidence value by $O(\epsilon)\|H\|_{1,K}$, and the density
factor differs from $1/(8\pi^2)$ by $O(\epsilon)$.
Therefore the leading coefficient is
\[
 \frac1{8\pi^2}\int_{\R\times\R^2}
                 b_0(2G^2+|W|^2)\dd G\,\dd^2W.
\]
Changing $\sqrt2G$ to a Euclidean coordinate and then using
three-dimensional polar coordinates identifies this constant
as $c_{47}$ in \eqref{eq:transition-constants}.
All errors are $O(\epsilon^4)\|H\|_{1,K}$.
Finally $\theta_\epsilon=1$ at $r=0$, proving the response
identity.
\end{proof}

\begin{remark}[What was calculated]
The coefficients and remainders in
Theorem~\ref{thm:transition} are explicit scalar seed
calculations through the coincidence transition.
The $47$ seed has no internal cone vertex there, whereas the
$5$ seeds contain a cone radius tending to zero.
Their different localization powers are derived from their
measures, not imposed as a common $r$-dilation.
\end{remark}

\subsection{Insertion into the differentiated complete pairing}

At positive $\delta$, all amplitudes
$H_{e,\dot c_\delta}^{I,J}[\Phi]$ are smooth, so the theorem
applies to them. Define
\[
 V_{e,\delta,\epsilon}(\Phi)
 =
 -\ii\sum_{I,J}
 \pair{\mu_e}
      {\theta_\epsilon H_{e,\dot c_\delta}^{I,J}[\Phi]},
 \qquad e=47,5a,5z.
\]
Then
\begin{align}
 V_{5a,\delta,\epsilon}
 &=
 -\ii c_R\epsilon^2
       \sum_{I,J}\pair{\mu_0}
                         {H_{5a,\dot c_\delta}^{I,J}}
       +\mathcal E_{5a,\delta,\epsilon},
 \notag\\
 V_{5z,\delta,\epsilon}
 &=
 -\ii c_R\epsilon^2
       \sum_{I,J}\pair{\mu_0}
                         {H_{5z,\dot c_\delta}^{I,J}}
       +\mathcal E_{5z,\delta,\epsilon},
 \notag\\
 V_{47,\delta,\epsilon}
 &=
 -\ii c_{47}\epsilon^3
       \sum_{I,J}\pair{\nu_0}
                         {H_{47,\dot c_\delta}^{I,J}}
       +\mathcal E_{47,\delta,\epsilon}.
 \label{eq:transition-insertion}
\end{align}
The ambient derivatives in these amplitudes precede their
incidence evaluation.

If $C^\chi$ has distributional order at most $h$, the elementary
convolution estimate and the derivative-placement ledger give
\begin{align}
 |\mathcal E_{5a,\delta,\epsilon}(\Phi)|
 &\le C_H\epsilon^3\delta^{-h-12}\|\Phi\|_{6,H},
 \notag\\
 |\mathcal E_{5z,\delta,\epsilon}(\Phi)|
 &\le C_H\epsilon^3\delta^{-h-14}\|\Phi\|_{6,H},
 \notag\\
 |\mathcal E_{47,\delta,\epsilon}(\Phi)|
 &\le C_H\epsilon^4\delta^{-h-12}\|\Phi\|_{6,H}.
 \label{eq:two-parameter-errors}
\end{align}
Indeed, one additional ambient derivative of each smooth
amplitude adds at most one crossed derivative and one test
derivative to the bounds in Appendix~\ref{app:bounds}.

These estimates do not provide an integrable complete
$\delta$-remainder. Substituting $\epsilon=\epsilon(\delta)$
would also require control of the complementary region.
The incidence expressions in \eqref{eq:transition-insertion}
still contain the unidentified crossed coefficient and have
not been evaluated as physical smoothing coefficients.

%=====================================================================
\section{Route A: the complete origin contribution and its cross-check}
\label{sec:routeA}
%=====================================================================

For a smooth crossed array $H_0$, let
\[
 \Lambda_\Phi^\zeta(H_0)
 =
 \pair{\mathsf T_{H_0}(\Kret)}{\zeta\Phi}.
\]
Resolving this finite functional into crossed components gives
compact distributions $\Lambda_{\Phi,A}^\zeta$ in the relative
variables. With $\mathfrak d(x,v)=x-v$, define
\begin{equation}
 \Omega_\Phi^\zeta
 =
 \sum_A\mathfrak d_*
           (\Lambda_{\Phi,A}^\zeta\otimes C_A^\chi).
 \label{eq:Omega}
\end{equation}

\begin{proposition}[Exact complete translation bridge]
\label{prop:translation}
\status{Proved for displayed mathematical data}
The translation distribution is compact and satisfies
\begin{equation}
 \pair{\Omega_\Phi^\zeta}\psi
 =
 -\ii\sum_{e,I,J}\mathsf I_e
 \left(
 \mathscr P_e^{I,J}
 [\mathcal W_{\psi*C^\chi}^{I,J}[\zeta\Phi]]
 \right).
 \label{eq:translation-bridge}
\end{equation}
Consequently
\[
 t_\delta^\zeta(\Phi)=\pair{\Omega_\Phi^\zeta}{\rho_\delta},
 \qquad
 \dot t_\delta^\zeta(\Phi)
                      =\pair{\Omega_\Phi^\zeta}{\kappa_\delta}.
\]
\end{proposition}

\begin{proof}
The compact-local internal order-four estimate and the vertex
give order at most five for each relative functional.
Both tensor factors are compact, so their difference
pushforward exists. Tensor evaluation gives
$\pair{\Omega_\Phi^\zeta}\psi
=\Lambda_\Phi^\zeta(\psi*C^\chi)$.
The adjoints prove \eqref{eq:translation-bridge}.
A detailed order and support estimate appears in
Appendix~\ref{app:translation}.
\end{proof}

Choose a translation cutoff $\vartheta=1$ near zero.
For sufficiently small $\delta$, both profiles sample only
$\vartheta\Omega_\Phi^\zeta$ exactly.

The potential row fibers are
\[
 \{(w,v):
 w\in\Sigma_e\cap\supp\zeta,\quad
 v\in\supp C^\chi,\quad \pi(w)-v=h\},
\]
where $\Sigma_e$ is the corresponding scalar seed support.
They are potential contributors only. Classifying their
singular, regular, and degenerating pieces requires the actual
crossed singular structure. The equation $\pi(w)=v$ alone does
not impose $r=0$.

\subsection{An actual cross-check of the transition calculation}

For fixed $\epsilon$, define a row functional by inserting
$\theta_\epsilon$ \emph{after} its adjoint, exactly as in
$V_{e,\delta,\epsilon}$. Construct its translation distribution
$\Omega_{e,\Phi,\epsilon}$ by the same component tensor
pushforward. Then
\begin{equation}
 V_{e,\delta,\epsilon}(\Phi)
 =
 \pair{\Omega_{e,\Phi,\epsilon}}{\kappa_\delta}.
 \label{eq:transition-translation}
\end{equation}

To check the coefficients rather than merely the representation,
take an arbitrary compact translation profile $\psi$ and put
$c=\psi*C^\chi$. Theorem~\ref{thm:transition} yields
\begin{align}
 \pair{\Omega_{5a,\Phi,\epsilon}}\psi
 &=
 -\ii c_R\epsilon^2
       \sum_{I,J}\pair{\mu_0}{H_{5a,c}^{I,J}[\Phi]}
       +O(\epsilon^3)\sum_{I,J}\|H_{5a,c}^{I,J}\|_{1,K},
 \notag\\
 \pair{\Omega_{5z,\Phi,\epsilon}}\psi
 &=
 -\ii c_R\epsilon^2
       \sum_{I,J}\pair{\mu_0}{H_{5z,c}^{I,J}[\Phi]}
       +O(\epsilon^3)\sum_{I,J}\|H_{5z,c}^{I,J}\|_{1,K},
 \notag\\
 \pair{\Omega_{47,\Phi,\epsilon}}\psi
 &=
 -\ii c_{47}\epsilon^3
       \sum_{I,J}\pair{\nu_0}{H_{47,c}^{I,J}[\Phi]}
       +O(\epsilon^4)\sum_{I,J}\|H_{47,c}^{I,J}\|_{1,K}.
 \label{eq:transition-crosscheck}
\end{align}
Thus the same constants and incidence functionals occur in
the seed and translation descriptions for every such $\psi$,
not only the chosen mollifier.

This verifies the calculated internal localization coefficients.
It does not supply the missing physical origin-germ expansion.

\subsection{The weakest sufficient physical certificate}

A full homogeneous germ decomposition is unnecessary if one
can directly prove
\begin{equation}
 \pair{\Omega_\Phi^\zeta}{\kappa_\delta}
 =
 \sum_{\lambda,k}
       \delta^\lambda(\log\delta)^k B_{\lambda,k}(\Phi)
 +\mathcal R_\delta(\Phi),
 \label{eq:desired-certificate}
\end{equation}
with evaluated complete coefficients and
\[
 |\mathcal R_\delta(\Phi)|
 \le\omega_H(\delta)\|\Phi\|_{j,H},
 \qquad
 \int_0^{\delta_*}\omega_H(t)\frac{\dd t}{t}<\infty.
\]
For the inherited family, neither the coefficients nor this
remainder have been calculated.

If both routes establish \eqref{eq:desired-certificate}, their
complete sums agree by \eqref{eq:translation-bridge}.
Individual fibers need not correspond to individual sectors.
After scalar absolute continuity and an anchor are used to
integrate the certificate, uniqueness of finite real
power--logarithm expansions identifies the singular
coefficients and finite part; see
Appendix~\ref{app:decisions}.

%=====================================================================
\section{Remainder audit, local verdict, and completion ledger}
\label{sec:verdict}
%=====================================================================

The remaining physical estimates are not hidden inside the
word ``complete.''

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.25\linewidth}Yp{.17\linewidth}@{}}
\toprule
Region or operation & Required calculation & Status\\
\midrule
Isolated non-tip row &
Actual crossed boundary value, component contraction, fixed-test
coefficient, and remainder. &
Unresolved\\[0.3em]
$r=0$ transition &
Apply the actual crossed factors to the cone-radius integrals,
with compatible angular and radial estimates. &
Unresolved\\[0.3em]
$0R$ sector &
A $5z$ estimate without assuming $q_R=O(\delta)$. &
Unresolved\\[0.3em]
Single crossed cone &
Regularity and boundary values of the identified crossed array
on the retained region. &
Unresolved\\[0.3em]
Complementary regions &
Tail estimates justified by that array's actual singular
structure. &
Unresolved\\[0.3em]
Complete collection &
Convert all coefficients to one ambient test convention and
sum across rows and sectors. &
Unresolved\\[0.3em]
Origin remainder &
Test-uniform integrable control of every residual fiber
contribution. &
Unresolved\\
\bottomrule
\end{tabularx}
\end{center}

The mathematical transition remainders in
\eqref{eq:two-parameter-errors} are proved, but they do not
resolve any of these complete physical smoothing estimates.

\subsection{Completion ledger}

The status categories used here are:
\begin{itemize}
 \item \textbf{proved for displayed mathematical data};
 \item \textbf{physically identified};
 \item \textbf{physically evaluated};
 \item \textbf{conditional};
 \item \textbf{unresolved}.
\end{itemize}
No claim in the present revision reaches either of the two
physical completion categories.

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}Yp{.34\linewidth}@{}}
\toprule
Principal claim & Status in this revision\\
\midrule
Exact kernels and all seven adjoints &
Proved for displayed mathematical data\\[0.3em]
Row--sector support zeros and non-tip scalar integrals &
Proved for displayed mathematical data\\[0.3em]
Scalar coincidence-localization coefficients
$c_R$ and $c_{47}$, with uniform remainders &
Proved for displayed mathematical data\\[0.3em]
Translation cross-check of those localization coefficients &
Proved for displayed mathematical data\\[0.3em]
Executable inherited package and local transfer hypotheses &
Unresolved\\[0.3em]
Singular physical non-tip coefficient &
Unresolved\\[0.3em]
Physical coincidence coefficient and complete remainder &
Unresolved\\[0.3em]
Dini and singular-primitive decisions &
Conditional; their implications are proved\\[0.3em]
Smooth-data expansion &
Proved mathematical consistency benchmark\\[0.3em]
Physical subtraction authorization &
Unresolved\\[0.3em]
Inherited operator and output microlocal upgrade &
Unresolved\\
\bottomrule
\end{tabularx}
\end{center}

\subsection{Local verdict}

No inherited physical local limit or failure theorem is released.
The requested physical completion criterion is not met by the
supplied inputs or by this revision.

The immediate missing material is the executable specification
in Section~\ref{sec:identification}. Once supplied, the shortest
first calculation is an admissible isolated non-tip row, if
such a patch exists. Its result may guide, but cannot replace,
the coincidence and origin analyses.

%=====================================================================
\section{Corrections and subsequent obligations}
\label{sec:subsequent}
%=====================================================================

Authorization begins only after a surviving complete singularity
or a constructed limit is known.

A mathematical singular primitive does not become permitted
because it removes a divergent test pairing. The supplied XV-B
lifting theorem tests membership in a specified correction
image; it does not enlarge that image \cite{XVB}.
The supplied Em01-E authorization theorem also requires
reference, support, tensor, growth, microlocal, and overlap
conditions \cite{EmE}. They have not been discharged here.

A fixed-kernel calculation is not an operator theorem.
An operator domain must preserve the actual physical
relationships responsible for cancellation and must support
continuous input seminorm estimates. The reconstruction
criterion and an operator Dini interface are retained in
Appendix~\ref{app:upgrades}.

A local limit does not by itself give the later joint section,
contact compatibility, proper pushforward, output wavefront
bound, or observable normalization. In particular,
\[
 Q_HR_1^{(1)}\in\mathscr E_H
\]
is not established.

The supplied source comparisons do not identify the application
data of Em01-F or Em01-G with the inherited weighted family
\cite{EmF,EmG}. The En01 recorded-circuit and decoherence
materials are a different track and are not used as input
definitions or premises.

\paragraph{Verification boundary.}
The source below is self-contained and intended for pdfLaTeX.
The proofs are mathematical arguments for the displayed objects.
No compiler execution, machine verification, independent
symbolic computation, exhaustive corpus search, registration
verification, referee approval, or acceptance is claimed.

%=====================================================================
\appendix
\section{Conventions, defining kernels, and exact adjoints}
\label{app:adjoints}
%=====================================================================

\begin{convention}
Pairings are complex bilinear and use coordinate densities.
For a trivial finite-dimensional bundle $E$,
$\D(X;E^*)=C_c^\infty(X;E^*)$ has its usual test topology and
$\Dp(X;E)=\D(X;E^*)'$.
A bounded test set has common compact support and bounds on
every fixed derivative order. Strong distribution seminorms
are $\sup_{\Phi\in\mathcal A}|\pair u\Phi|$ for bounded
$\mathcal A$.

Spacetime indices range over $\{0,1,2,3\}$ and are summed when
repeated. Component labels $I,J,L,M$ range over
$\{1,\ldots,10\}$; finite component contractions are understood
where indicated. Source multiindices are explicitly summed.
The internal array has components $(I,J)$ and the output has
components $J$.

For compact $H$, $\|\Phi\|_{j,H}$ is the maximum over components
and derivatives of order at most $j$. All estimates concern
tests supported in the indicated compact set; enlarged fixed
neighborhoods may be used for coefficient norms.
\end{convention}

Define
\[
 \pair R\psi
 =
 \frac1{4\pi}\int_0^\infty t
        \int_{S^2}\psi(t,t\mathbf n)\dd\mathbf n\,\dd t,
 \qquad
 B(t,\mathbf x)=\frac1{8\pi}\Theta(t-|\mathbf x|).
\]
The first formula fixes the cone vertex convention.

\begin{lemma}
\label{lem:B}
In $\Dp(M)$, $\partial_\mu B=x_\mu R/2$.
\end{lemma}
\begin{proof}
Time differentiation gives
$(8\pi)^{-1}\int_{\R^3}\psi(|\mathbf x|,\mathbf x)\dd^3x$.
Spatial differentiation gives the same integral multiplied by
$-x^i/|\mathbf x|$. These equal the actions of $tR/2$ and
$x_iR/2$. The integrals are locally integrable at the vertex;
there is no additional point mass.
\end{proof}

Fix arbitrary constant arrays
$\mathsf A^{\lambda\mu,I}$,
$\mathsf A^{\kappa\mu}{}_L$,
$\mathsf D^{\lambda\mu,I,J}$,
$\mathcal P^L{}_M$, $\mathsf b_L$;
$m\ge0$; smooth $f,g,F,\sigma$; and smooth source coefficients
$q_{\alpha\beta}^{L,I,J}$ for $|\alpha|+|\beta|\le2$.
These choices define a mathematical class. They do not
instantiate the inherited family.

Put
\[
 L^{\mu,I}(s,a)
 =
 \sigma(a)\mathsf A^{\lambda\mu,I}
                        \partial_{a^\lambda}R(s-a).
\]
Define
\begin{align}
 K_5^{[a],I,J}
 &=
 \partial_{b^\mu}\left[
 f(b)\sigma(a)\partial_{a^\lambda}R(b-a)
 \mathsf A^{\lambda\theta,I}
                  \partial_{a^\theta}L^{\mu,J}(a,z)
 \right],
 \label{eq:K5a}\\
 K_5^{[z],I,J}
 &=
 \partial_{b^\mu}\left[
 f(b)\sigma(z)\partial_{z^\lambda}R(b-z)
 \mathsf A^{\lambda\theta,J}
                  \partial_{z^\theta}L^{\mu,I}(z,a)
 \right],
 \label{eq:K5z}\\
 K_{47}^{I,J}
 &=
 -\partial_{b^\mu}\left[
 \left(
 f(b)\partial_{b^\nu}L^{\mu,J}(b,z)
 -\delta^\mu_\nu\partial_{b^\rho}(f(b)L^{\rho,J}(b,z))
 \right)L^{\nu,I}(b,a)
 \right],
 \label{eq:K47}\\
 K_6^{I,J}
 &=
 \partial_{b^\mu}\left[
 f(b)\sigma(a)\mathsf D^{\lambda\mu,I,J}
       \partial_{a^\lambda}R(b-a)\delta^{(4)}(a-z)
 \right].
 \label{eq:K6}
\end{align}
In $K_6$, the displayed $a$ derivative acts on $R$ only.

The source
\[
 Q^{L,I,J}(v;a,z)
 =
 \sum_{|\alpha|+|\beta|\le2}
 q_{\alpha\beta}^{L,I,J}(v)
 \partial_v^\alpha\delta^{(4)}(v-z)
 \partial_v^\beta\delta^{(4)}(v-a)
\]
exists in the independent linear coordinates
$(v,v-z,v-a)$. Its action is
\[
 \pair Q\Psi
 =
 \sum_{\alpha,\beta}\int q_{\alpha\beta}(v)
       (\partial_z^\alpha\partial_a^\beta\Psi)(v,v,v)
       \dd^4v.
\]
The source signs cancel under endpoint transposition.

For $S=R$ or $B$, use the precise response definition
\begin{equation}
 \int S(b-v)Q(v;a,z)\dd^4v
 :=
 \sum_{\alpha,\beta}(-1)^{|\alpha|+|\beta|}
 \partial_z^\alpha\partial_a^\beta
 [q_{\alpha\beta}(a)S(b-a)\delta^{(4)}(a-z)].
 \label{eq:response-definition}
\end{equation}
The bracket is a seed in the independent coordinates
$(a,b-a,z-a)$.

Set
\begin{align}
 T^{L,I,J}
 &=\mathcal P^L{}_M\int R(b-v)Q^{M,I,J}(v;a,z)\dd^4v,
 \notag\\
 K_8^{I,J}
 &=
 -\partial_{b^\mu}\left[
 f(b)\mathsf A^{\kappa\mu}{}_L\mathcal P^L{}_M
 \partial_{b^\kappa}
       \int B(b-v)Q^{M,I,J}(v;a,z)\dd^4v
 \right],
 \label{eq:K8}\\
 K_{9d}^{I,J}
 &=
 -\partial_{b^\nu}\left[
 g(b)\partial_{b^\mu}F(b)
                    \mathsf A^{\mu\nu}{}_L T^{L,I,J}
 \right],
 \label{eq:K9d}\\
 K_{9m}^{I,J}
 &=-m^2g(b)F(b)\mathsf b_L T^{L,I,J}.
 \label{eq:K9m}
\end{align}
All rows are extended independently of $y$.

Use the determinant-one independent charts
\begin{align}
 \Gamma_{47}(b,s,t,y)&=(b-t,b,b-s,y),\notag\\
 \Gamma_{5a}(a,s,t,y)&=(a,a+s,a-t,y),\notag\\
 \Gamma_{5z}(z,s,t,y)&=(z-t,z+s,z,y),\notag\\
 \Gamma_{\mathrm{resp}}(a,s,y)&=(a,a+s,a,y).
 \label{eq:seed-charts}
\end{align}
Let
\[
 \Irr(H)=\int H(x,s,t,y)\dd^4x\,\dd R(s)\,\dd R(t)\,\dd^4y,
 \qquad
 \Iresp(H)=\int H(a,s,y)\dd^4a\,\dd R(s)\,\dd^4y.
\]
The base $x$ is $b,a,z$ for the three two-line charts.
Write $\Psi_e=\Psi\circ\Gamma_e$ and
\[
 \Jet{\alpha}{\beta}\Psi(a,s,y)
 =
 (\partial_z^\alpha\partial_a^\beta\Psi)(a,a+s,a,y).
\]
Ambient derivatives precede restriction.

Put
\[
 \mathcal B_\mu
 =\partial_{b^\mu}+\partial_{s^\mu}+\partial_{t^\mu},
 \quad
 h_{47}=\sigma(b-t)\sigma(b-s),\quad f_\rho=\partial_\rho f.
\]
The exact adjoints are
\begin{align}
 \mathscr P_{47}^{I,J}[\Psi]
 ={}&
 -\mathsf A^{\lambda\mu,J}\mathsf A^{\theta\nu,I}
 \partial_{s^\nu}\partial_{s^\lambda}\partial_{t^\theta}
 [h_{47}f(b)\mathcal B_\mu\Psi_{47}]
 \notag\\
 &-\mathsf A^{\lambda\rho,J}\mathsf A^{\theta\mu,I}
 \partial_{s^\lambda}\partial_{t^\theta}
 [h_{47}f_\rho(b)\mathcal B_\mu\Psi_{47}]
 \notag\\
 &+\mathsf A^{\lambda\rho,J}\mathsf A^{\theta\mu,I}
 \partial_{s^\rho}\partial_{s^\lambda}\partial_{t^\theta}
 [h_{47}f(b)\mathcal B_\mu\Psi_{47}],
 \label{eq:P47}\\
 \mathscr P_{5a}^{I,J}[\Psi]
 ={}&
 \mathsf A^{\lambda\theta,I}\mathsf A^{\rho\mu,J}
 \partial_{s^\lambda}\partial_{t^\theta}\partial_{t^\rho}
 [f(a+s)\sigma(a)\sigma(a-t)\partial_{s^\mu}\Psi_{5a}],
 \label{eq:P5a}\\
 \mathscr P_{5z}^{I,J}[\Psi]
 ={}&
 \mathsf A^{\lambda\theta,J}\mathsf A^{\rho\mu,I}
 \partial_{s^\lambda}\partial_{t^\theta}\partial_{t^\rho}
 [f(z+s)\sigma(z)\sigma(z-t)\partial_{s^\mu}\Psi_{5z}],
 \label{eq:P5z}\\
 \mathscr P_6^{I,J}[\Psi]
 ={}&
 -\mathsf D^{\lambda\mu,I,J}
 \partial_{s^\lambda}
 [f(a+s)\sigma(a)\partial_{s^\mu}\Psi_{\mathrm{resp}}],
 \label{eq:P6}\\
 \mathscr P_8^{I,J}[\Psi]
 ={}&
 \sum_{\alpha,\beta,L,M}
 q_{\alpha\beta}^{M,I,J}(a)
 \mathsf A^{\kappa\mu}{}_L\mathcal P^L{}_M\frac{s_\kappa}{2}
 \Jet{\alpha}{\beta}[f(b)\partial_{b^\mu}\Psi],
 \label{eq:P8}\\
 \mathscr P_{9d}^{I,J}[\Psi]
 ={}&
 \sum_{\alpha,\beta,L,M}
 q_{\alpha\beta}^{M,I,J}(a)\mathcal P^L{}_M
 \mathsf A^{\mu\nu}{}_L
 \Jet{\alpha}{\beta}
 [g(b)\partial_{b^\mu}F(b)\partial_{b^\nu}\Psi],
 \label{eq:P9d}\\
 \mathscr P_{9m}^{I,J}[\Psi]
 ={}&
 -m^2\sum_{\alpha,\beta,L,M}
 q_{\alpha\beta}^{M,I,J}(a)\mathcal P^L{}_M\mathsf b_L
 \Jet{\alpha}{\beta}[g(b)F(b)\Psi].
 \label{eq:P9m}
\end{align}
All source sums satisfy $|\alpha|+|\beta|\le2$.

\begin{theorem}[Exact adjoints and existence]
\label{thm:adjoints}
\status{Proved for displayed mathematical data}
For compact ambient $\Psi$,
\[
 \pair{K_e^{I,J}}\Psi
 =
 \begin{cases}
 \Irr(\mathscr P_e^{I,J}[\Psi]),&e=47,5a,5z,\\
 \Iresp(\mathscr P_e^{I,J}[\Psi]),&e=6,8,9d,9m.
 \end{cases}
\]
Their compact-local distributional orders are at most
$4,4,4,2,3,3,2$, respectively.
The response rows are supported in
$\{a=z,\ b-a\in\partial J^+\}$.
\end{theorem}

\begin{proof}
For $5a$, $s=b-a$, $t=a-z$ give
\[
 \partial_{a^\lambda}R(b-a)=-\partial_{s^\lambda}R(s),
 \quad
 \partial_{a^\theta}L^{\mu,J}(a,z)
 =-\sigma(z)\mathsf A^{\rho\mu,J}
                 \partial_{t^\theta}\partial_{t^\rho}R(t).
\]
The internal signs cancel. Transposing the outer divergence
and three seed derivatives gives \eqref{eq:P5a}.
The $5z$ proof is identical.

For $47$, substitute
\[
 L^{\mu,J}(b,z)
 =-\sigma(z)\mathsf A^{\lambda\mu,J}\partial_{s^\lambda}R(s),
 \quad
 L^{\nu,I}(b,a)
 =-\sigma(a)\mathsf A^{\theta\nu,I}\partial_{t^\theta}R(t).
\]
The bracket has three terms with signs $+,-,-$, containing
respectively three, two, and three seed derivatives.
The outer negative divergence transposes positively; the
remaining signs are $-,-,+$. The outer ambient $b$ derivative
becomes $\mathcal B_\mu$, proving \eqref{eq:P47}, including
the derivative-of-$f$ term.

For $6$, transpose the outer divergence and the separate
$-\partial_sR$ factor without differentiating the incidence
delta. This gives \eqref{eq:P6}.

For the response rows, source-transposition signs cancel
those in \eqref{eq:response-definition}. Outer negative
divergences transpose positively. Lemma~\ref{lem:B} gives
the $s_\kappa R/2$ factor in $8$. This proves the remaining
formulas, including both root terms.

Every integral is against a locally finite seed measure
with a compact smooth differential test amplitude.
Counting its test derivatives proves existence, continuity,
and the orders. Differentiation preserves the stated support.
\end{proof}

This audit is against the displayed defining kernels.
It is not an audit of an unidentified physical component
conversion or observable.

%=====================================================================
\section{Derivative placement and conservative bounds}
\label{app:bounds}
%=====================================================================

In the independent charts,
\[
\begin{array}{c|cc}
 e&U&V\\
\hline
 47&y-b+t&y-b\\
 5a&y-a&y-a-s\\
 5z&y-z+t&y-z-s.
\end{array}
\]
Thus $s$ derivatives in $47$ and $t$ derivatives in $5a$
cannot reach the crossed coefficient. In $47$,
$\mathcal B_\mu U=0$ and
$\mathcal B_\mu V=-e_\mu$.

If $C^\chi$ has compact distributional order at most $h$,
then for each multiindex $\gamma$,
\[
 \sup_x|\partial^\gamma c_\delta(x)|
 +\sup_x|\partial^\gamma\dot c_\delta(x)|
 \le C_\gamma\delta^{-8-h-|\gamma|}.
\]
This is the finite-order distribution estimate applied to
translated differentiated profiles.

The resulting row bounds are
\[
 |t_{\delta,e}^\zeta(\Phi)|
 +|\dot t_{\delta,e}^\zeta(\Phi)|
 \le C_{H,e}\delta^{-h-d_e}\|\Phi\|_{j_e,H},
\]
with
\[
\begin{array}{c|rrrrrrr}
 e&47&5a&5z&6&8&9d&9m\\
\hline
 d_e&11&11&13&11&12&12&11\\
 j_e&5&5&5&3&4&4&3.
\end{array}
\]
The maximal crossed derivative orders are
$3,3,5,3,4,4,3$. The vertex contributes at most one crossed
derivative; source $z$ derivatives cannot reach it.
Finite mass of the localized seed measures completes the
estimate.

The complete bound is
$C_H\delta^{-h-13}\|\Phi\|_{5,H}$.
It is a moderate bound, not a leading exponent or a Dini
certificate. This distinction is also explicit in the supplied
Em01-B estimate \cite{EmB}.

%=====================================================================
\section{Characteristic chart and exact translated equations}
\label{app:geometry}
%=====================================================================

For $s=b-a$ and $d_0=y-a$, the chart inverse is
\[
 \mathbf n=\frac{\mathbf s}{|\mathbf s|},\quad
 \beta=\frac{s^0+|\mathbf s|}{2},\quad
 u=\frac{s^0-|\mathbf s|}{2},
\]
\[
 \alpha=\frac{d_0^0+\mathbf n\cdot\mathbf d_0}{2},\quad
 v=\frac{d_0^0-\mathbf n\cdot\mathbf d_0}{2},\quad
 \xi^A=\mathbf e_A\cdot\mathbf d_0,\quad r=z-a.
\]
The Jacobian in $(\beta,u)$ is two; spatial polar coordinates
contribute $(\beta-u)^2$. At fixed $\mathbf n$, the
$(\alpha,v,\xi)$ Jacobian is two. Block triangularity proves
\eqref{eq:density}.

The aligned locus and response incidence are
\[
 N=\{r=u=v=\xi=0\},\qquad
 S_{\mathrm{resp}}=\{r=0,u=0\}.
\]
Along $N$,
\[
 \dd q_R=4\beta\dd u,\quad
 \dd q_1=4\alpha\dd v,\quad
 \dd q_2=4(\beta-\alpha)(\dd u-\dd v).
\]
Their rank is two. On $T S_{\mathrm{resp}}$ along $N$,
the crossed pair has rank one. This is not a rank statement
throughout the incidence.

For bounded translation variables $p,q$,
\[
 U=\alpha\ell+v k+\xi^AE_A,\quad
 V=(\alpha-\beta)\ell+(v-u)k+\xi^AE_A.
\]
Under $u=\delta u_*$, $v=\delta v_*$,
$\xi=\sqrt\delta X$,
\begin{align}
 (U-\delta p)^2
 ={}&
 \delta(4\alpha v_*-|X|^2-2\alpha\ell\cdot p)
 \notag\\
 &-2\delta^{3/2}X^AE_A\cdot p
 +\delta^2(p^2-2v_*k\cdot p),
 \notag\\
 (V-\delta q)^2
 ={}&
 \delta\bigl(
 4(\alpha-\beta)(v_*-u_*)-|X|^2
                   -2(\alpha-\beta)\ell\cdot q\bigr)
 \notag\\
 &-2\delta^{3/2}X^AE_A\cdot q
 +\delta^2(q^2-2(v_*-u_*)k\cdot q).
 \label{eq:translated-cones}
\end{align}
The normal Jacobian is $\delta^3$.

If $q_R=O(\delta)$ and both translated squares are $O(\delta)$,
compactness and bounded translations first give
$q_1,q_2=O(\delta)$. The quadratic defect then gives
$u,v=O(\delta)$ and $\xi=O(\sqrt\delta)$.
This proves the translated core estimate without dropping the
higher-order terms.

\begin{proposition}[Failure of automatic internal-core localization]
\label{prop:counterexample}
\status{Proved for displayed mathematical data}
The core conditions do not force $r=O(\delta)$.
Simultaneous crossed cones on the $5z$ seed do not force
$q_R=O(\delta)$.
\end{proposition}

\begin{proof}
Take $u=v=\xi=0$ and fixed $r=\epsilon\ell$ with
$0<\epsilon<\beta$. This lies on the $47$ and $5z$ scalar
supports while $r/\delta$ is unbounded.
A fixed past-null $r$ gives the analogous $5a$ example.

For the second assertion, fix small $u_0>0$ and put
\[
 v=\frac{\beta-\alpha}{\beta}u_0,\qquad
 |\xi|^2=\frac{4\alpha(\beta-\alpha)}{\beta}u_0.
\]
Then $q_1=q_2=0$, but $q_R=4\beta u_0>0$.
Choose $0<\gamma<\beta$ and set
\[
 \upsilon=\frac{\beta-\gamma}{\beta}u_0,\qquad
 |\omega|^2=\frac{4\gamma(\beta-\gamma)}{\beta}u_0.
\]
Both internal vectors are future null, giving a $5z$ seed
point. It remains fixed as $\delta\downarrow0$.
\end{proof}

%=====================================================================
\section{Partition identities and complete assembly}
\label{app:partitions}
%=====================================================================

If raw ambient sectors are used instead of the main
post-adjoint convention, define
\[
 t_{j,\delta}(\Phi)
 =
 \pair{\mathsf T_{c_\delta}(\Kret)}
                  {\theta_{j,\delta}\zeta\Phi}.
\]
Then
\[
 \Dscale t_{j,\delta}
 =
 \pair{\mathsf T_{\dot c_\delta}(\Kret)}
                  {\theta_{j,\delta}\zeta\Phi}
 +
 \pair{\mathsf T_{c_\delta}(\Kret)}
                  {(\Dscale\theta_{j,\delta})\zeta\Phi}.
\]
All spatial derivatives of the cutoff generated by the vertex
and adjoints are retained. If the partition sums to one near
$\supp\zeta$, its summed motion terms cancel.
For the cutoffs in \eqref{eq:partition},
\[
 \Dscale a_{j,\delta}
 =
 -2\frac{q_j^2}{L^2\delta^2}
           b_0'\!\left(\frac{q_j^2}{L^2\delta^2}\right).
\]

For raw sectors defined after an adjoint,
$\I_e(\vartheta_{e,j,\delta}\mathcal L_{e,c_\delta})$,
differentiation also produces
$\I_e((\Dscale\vartheta_{e,j,\delta})\mathcal L_{e,c_\delta})$.
Moving a cutoff through an adjoint generates additional
spatial commutator terms. These conventions are not termwise
interchangeable.

If the chosen exact differentiated partition gives a finite
sum $v_{j,\delta}$, and each term has a verified finite
power--logarithm expansion with an integrable remainder, then
the complete coefficients are the sums of those coefficients.
The remainder is bounded by the sum of the majorants and
the maximum of the finite test orders. This is finite
addition, not a sector estimate supplied by the partition.

%=====================================================================
\section{Translation estimates and optional germ certificates}
\label{app:translation}
%=====================================================================

The order-four internal estimate gives
\[
 |\Lambda_\Phi^\zeta(H_0)|
 \le C_H\|\Phi\|_{5,H}
       \|H_0\|_{5,\pi(\supp\zeta\cap H)}.
\]
Thus the relative component functionals in
\eqref{eq:Omega} have order at most five and compact support.
If $C^\chi$ has order at most $h$, tensor evaluation gives
\[
 |\pair{\Omega_\Phi^\zeta}\psi|
 \le C_H\|\Phi\|_{5,H}\|\psi\|_{h+5,D},
\]
where $D$ is a fixed compact neighborhood of the difference
support. Moreover,
\[
 \supp\Omega_\Phi^\zeta
 \subseteq\pi(\supp(\zeta\Phi))-\supp C^\chi.
\]
The pushforward uses a cutoff equal to one near the compact
tensor support, so it is independent of the choice of that
cutoff. No unsmoothed same-variable product is defined.

\begin{proposition}[Regular-origin certificate]
\label{prop:regular-germ}
\status{Conditional; implication proved for displayed mathematical data}
Suppose the complete $\Omega_\Phi^\zeta$ has a continuous
representative $W_\Phi$ on a common origin neighborhood for
tests supported in each compact $H$, with
\[
 |W_\Phi(0)|\le C_H\|\Phi\|_{j,H},\qquad
 |W_\Phi(h)-W_\Phi(0)|
 \le C_H|h|^a\|\Phi\|_{j,H},\quad a>0.
\]
Then $t_\delta^\zeta$ converges strongly to
$W_\bullet(0)$, and both its raw difference and logarithmic
derivative are $O_H(\delta^a)\|\Phi\|_{j,H}$.
\end{proposition}

\begin{proof}
At small scale,
$t_\delta^\zeta(\Phi)=\int\rho(s)W_\Phi(\delta s)\dd s$.
For the derivative use
$\int\kappa(s)[W_\Phi(\delta s)-W_\Phi(0)]\dd s$.
Normalization and zero mean give the estimates.
Uniqueness of continuous representatives gives linearity;
the test bound gives distributional continuity.
\end{proof}

For a distribution $G$ in $\R^8$, let $\Eulerh=h\cdot\partial_h$.
If
\[
 (\Eulerh-\lambda)G_j=G_{j-1},\qquad G_{-1}=0,
\]
then
\[
 \pair{G_J}{\rho_\delta}
 =
 \delta^\lambda\sum_{k=0}^J
       \frac{(\log\delta)^k}{k!}\pair{G_{J-k}}\rho.
\]
Indeed the transpose of $\Eulerh$ is $-(8+\Eulerh)$, so these
pairings satisfy a finite triangular ordinary differential
system in $\log\delta$. Its solution gives the formula.
The $\kappa_\delta$ pairing is its exact logarithmic derivative.

This optional chain calculation determines powers from a
specified distributional identity. The existence of the
interface is not evidence that the physical germ has such
a decomposition \cite{EmD,EmE}.

%=====================================================================
\section{Dini limits, singular primitives, and uniqueness}
\label{app:decisions}
%=====================================================================

\begin{definition}[Scalar scale regularity]
A family $u_\delta$ has scalar scale regularity if each
fixed-test pairing is locally absolutely continuous as a
function of $\log\delta$. Its logarithmic derivative is
understood almost everywhere in that scalar sense.
\end{definition}

\begin{theorem}[Fixed-kernel Dini criterion]
\label{thm:Dini}
\status{Conditional; implication proved for displayed mathematical data}
Let $u_\delta\in\Dp(X;E)$ have scalar scale regularity.
Suppose, on every compact support $H$,
\[
 |\pair{\Dscale u_\delta}\Phi|
 \le\omega_H(\delta)\|\Phi\|_{j_H,H},
 \qquad
 \int_0^{\delta_*}\omega_H(t)\frac{\dd t}{t}<\infty.
\]
Then $u_\delta$ has a strong distributional limit $u_0$, with
\[
 |\pair{u_\delta-u_0}\Phi|
 \le\|\Phi\|_{j_H,H}
                 \int_0^\delta\omega_H(t)\frac{\dd t}{t}.
\]
\end{theorem}

\begin{proof}
Scalar absolute continuity gives
\[
 \pair{u_\delta-u_\eta}\Phi
 =
 \int_\eta^\delta
       \pair{\Dscale u_t}\Phi\,\frac{\dd t}{t}.
\]
The bound makes every pairing Cauchy. The anchor
$u_{\delta_*}$ has a finite-order compact-local estimate;
adding the derivative integral gives such an estimate for
the limit. It is therefore a distribution. The tail estimate
is uniform on bounded test sets and proves strong convergence.
\end{proof}

\begin{theorem}[Singular primitive with explicit regularity]
\label{thm:primitive}
\status{Conditional; implication proved for displayed mathematical data}
Let $u_\delta\in\Dp(X;E)$ have scalar scale regularity.
Suppose its logarithmic derivative has the decomposition
\[
 \Dscale u_\delta
 =
 \sum_{\lambda\in\mathcal E}
       \delta^\lambda P_\lambda(\log\delta)+H_\delta,
 \qquad
 \mathcal E\subset(-\infty,0]\text{ finite},
\]
in scalar pairings almost everywhere. The polynomial
coefficients are distributions and $H_\delta$ satisfies the
Dini bounds of Theorem~\ref{thm:Dini}.

For $\lambda<0$, define
\[
 Q_\lambda(t)
 =
 \sum_{j=0}^{\deg P_\lambda}
       (-1)^j\lambda^{-j-1}P_\lambda^{(j)}(t),
\]
and define $Q_0(t)=\int_0^tP_0(s)\dd s$ when present.
Absent terms are zero. Then
\[
 S_\delta=
 \sum_{\lambda<0}\delta^\lambda Q_\lambda(\log\delta)
                       +Q_0(\log\delta)
\]
has the stated singular derivative and
$u_\delta-S_\delta$ has a strong finite limit $T$, with the
Dini tail rate.

The raw family converges if and only if every coefficient of
$S_\delta$ vanishes. Otherwise one fixed compact test detects
an unbounded raw pairing.
\end{theorem}

\begin{proof}
The finite formula telescopes to
$\lambda Q_\lambda+Q_\lambda'=P_\lambda$; also
$Q_0'=P_0$. Thus
$\Dscale(u_\delta-S_\delta)=H_\delta$.
Scalar scale regularity permits integration, and
Theorem~\ref{thm:Dini} gives the finite limit.

For a surviving singular coefficient, choose the smallest
real exponent with a nonzero coefficient and the largest
logarithmic degree at that exponent. A fixed compact test
detects it. Its scalar factor dominates all remaining
singular terms, the finite term, and the vanishing remainder.
Its absolute value tends to infinity.
If all singular coefficients vanish, the Dini estimate gives
strong convergence.
\end{proof}

The scalar regularity hypothesis is essential: an
almost-everywhere derivative identity alone does not justify
the integral identity used in the proof.

The finite part has the anchor formula
\[
 T=
 u_{\delta_*}-S_{\delta_*}
       -\int_0^{\delta_*}H_t\,\frac{\dd t}{t},
\]
understood in scalar pairings with the proved compact-local
bounds. Therefore two derivative calculations must also use
the same anchor before comparing finite parts.

\begin{proposition}[Uniqueness of complete coefficients]
\label{prop:uniqueness}
Suppose two expansions of the same family have finite real
negative-power logarithmic polynomials, positive logarithmic
powers at exponent zero, finite distributional terms, and
remainders tending strongly to zero. Their collected singular
coefficients and finite terms agree.
\end{proposition}

\begin{proof}
Subtract the expansions and combine their finite exponent
inventories. A nonzero singular difference would be unbounded
on a fixed detecting test, whereas the remaining difference
has a finite limit. All singular differences therefore vanish;
taking the limit identifies the finite terms.
\end{proof}

Dini integrability is sufficient, not necessary.
For a fixed distribution $D$,
$\delta\sin(\delta^{-2})D\to0$ strongly, although its derivative
contains $-2\delta^{-1}\cos(\delta^{-2})D$ with nonintegrable
absolute logarithmic-scale pairing on a detecting test.

%=====================================================================
\section{Consistency benchmark: explicit complete smooth-data expansion}
\label{app:smooth}
%=====================================================================

This benchmark is not a singular physical coefficient
calculation.

Suppose $C$ is a smooth crossed array near
$\pi(\supp\zeta)$, with $\chi=1$ on a larger neighborhood.
Any smooth extension agreeing there may be used in the
localized formulas. Fix an even normalized $\rho$ and put
\[
 M_\alpha=\int s^\alpha\rho(s)\dd s,\qquad
 C_{[n]}=
 \sum_{|\alpha|=n}\frac{(-1)^nM_\alpha}{\alpha!}
                                  \partial^\alpha C.
\]
Odd arrays vanish. Define the complete moment distributions
\[
 A_n^\zeta(\Phi)
 =
 \pair{\mathsf T_{C_{[n]}}(\Kret)}{\zeta\Phi}.
\]
Their explicit seed actions are given by
\eqref{eq:exact-raw} with $C_{[n]}$ replacing $c_\delta$.

\begin{theorem}[Explicit complete smooth-data expansion]
\label{thm:smooth}
\status{Proved mathematical consistency benchmark}
For every $N\ge0$,
\[
 t_\delta^\zeta
 =
 \sum_{j=0}^N\delta^{2j}A_{2j}^\zeta+R_{N,\delta},
 \qquad
 \dot t_\delta^\zeta
 =
 \sum_{j=1}^N2j\,\delta^{2j}A_{2j}^\zeta+\dot R_{N,\delta},
\]
and on compact joint test supports
\[
 |R_{N,\delta}(\Phi)|+|\dot R_{N,\delta}(\Phi)|
 \le C_{H,N}\delta^{2N+2}\|\Phi\|_{5,H}.
\]
In particular the complete family has limit $A_0^\zeta$
with an order-two rate.
\end{theorem}

\begin{proof}[Route B]
At small scale,
$c_\delta(x)=\int\rho(s)C(x-\delta s)\dd s$.
Taylor-expand through degree $2N+1$, uniformly with crossed
derivatives through order five. Evenness removes odd terms
and gives the raw expansion with a
$C^5$ remainder of order $\delta^{2N+2}$.
The moments of $\kappa$ give the differentiated expansion
and the same remainder order independently.
Insert both into the finite exact adjoints. Their localized
seed measures have finite mass, proving the test estimates.
\end{proof}

\begin{proof}[Route A]
Near the translation origin,
\[
 \Omega_\Phi^\zeta=W_\Phi,\qquad
 W_\Phi(h)=\Lambda_\Phi^\zeta(C(\,\cdot-h)).
\]
This follows by testing the convolution representation on
profiles supported in that neighborhood.
The smooth germ has derivatives
\[
 \partial_h^\alpha W_\Phi(0)
 =
 (-1)^{|\alpha|}\Lambda_\Phi^\zeta(\partial^\alpha C),
\]
uniformly bounded by $\|\Phi\|_{5,H}$.
Taylor expansion against $\rho$ and separately against
$\kappa$ gives the same coefficients and errors.
\end{proof}

The benchmark verifies signs, profile moments, complete row
retention, and the two-route comparison. It does not identify
the inherited covariance with smooth data.

%=====================================================================
\section{Authorization, descent, and separate microlocal interfaces}
\label{app:upgrades}
%=====================================================================

\begin{proposition}[Specified correction-family obstruction]
Let one fixed compact test have a nonconvergent raw pairing.
If every correction in a specified permitted family has a
convergent pairing on that test, no corrected family in that
class has a weak limit.
\end{proposition}
\begin{proof}
Subtract the convergent correction pairing from a proposed
corrected limit.
\end{proof}
Support-separated tests annihilate corrections supported in
a disjoint closed set. This excludes the specified class,
not every possible construction.

Let $P$ be Hausdorff locally convex, let
$\Rec:P\to V_{\mathrm{int}}$ be continuous and linear with
Hausdorff target, and give $K=\im\Rec$ the topology of
$P/\ker\Rec$.

\begin{proposition}[Continuous reconstruction descent]
A continuous linear $G:P\to\Dp(X;E)$ factors uniquely through
a continuous map on $K$ if and only if $G(\ker\Rec)=0$.
\end{proposition}
\begin{proof}
Define the descended map by
$M(\Rec A)=G(A)$. Null annihilation makes it well defined;
the quotient topology gives continuity.
\end{proof}

Canonical agreement on an open set gives only the restricted
null identity. It does not detect distributions supported in
the complement.

\begin{proposition}[Operator Dini interface]
Let continuous linear $G_\delta:P\to\Dp(X;E)$ have scalar scale
regularity and annihilate $\ker\Rec$. Suppose
\[
 |\pair{\Dscale G_\delta(A)}\Phi|
 \le\omega_H(\delta)p_H(A)\|\Phi\|_{j_H,H},
 \qquad
 \int_0^{\delta_*}\omega_H(t)\frac{\dd t}{t}<\infty,
\]
and
\[
 |\pair{G_{\delta_*}(A)}\Phi|
 \le q_H(A)\|\Phi\|_{k_H,H},
\]
for continuous input seminorms $p_H,q_H$.
Then the limit is continuous and linear, uniform on bounded
input and test sets, and descends continuously to $K$.
\end{proposition}
\begin{proof}
The fixed-kernel integration estimate carries the factor
$p_H(A)$. Together with the anchor it bounds every strong
output seminorm by a continuous input seminorm.
Continuous seminorms are bounded on bounded input sets.
Null annihilation passes to the limit.
\end{proof}

For a finite supported correction
\[
 C(A)=\sum_{|\gamma|\le r}\Ejet_\gamma B_\gamma(A),
 \qquad
 \pair{\Ejet_\gamma b}\Phi
 =(-1)^{|\gamma|}\pair b{\partial_n^\gamma\Phi(s,0)},
\]
the tests
$(-1)^{|\gamma|}n^\gamma\vartheta(n)\psi(s)/\gamma!$,
with $\vartheta=1$ near zero, extract the separate coefficients.
Thus coherence is coefficientwise.
A finite normal order does not make the tangential ambiguity
finite-dimensional.

Restriction is strongly continuous, being the transpose of
extension by zero of compact tests. Limits inherit a separately
proved reference. Equal references on $O$ imply difference
support in $X\setminus O$, not specifically in $N$ unless that
stronger support conclusion is proved.

Compatible local maps glue by a locally finite partition of
unity. Only finitely many terms meet each compact test support.
Overlap compatibility is a hypothesis, not supplied by a local
limit.

Finally fix a closed conic $\Gamma\subset T^*X\setminus0$.
Suppose $u_\delta\in\Dp_\Gamma(X;E)$, $u_\delta\to u$ strongly,
and for every compact cutoff $\theta$, closed frequency cone
$\Omega$ disjoint from $\Gamma$ over $\supp\theta$, and every $N$,
\[
 \sup_{\xi\in\Omega}(1+|\xi|)^N
                   |\widehat{\theta u_\delta}(\xi)|
\]
is uniformly bounded. Passing to the limit gives rapid decay
outside $\Gamma$, hence $\WF(u)\subset\Gamma$.
For convergence of an order-$N$ Fourier seminorm, the uniform
order-$N+1$ bound controls frequencies above $R$ by
$2C/(1+R)$; below $R$, strong convergence is uniform on the
bounded family of localized exponential tests.
This is a separate Fourier certificate, not a consequence of
the finite-order Dini estimate.

%=====================================================================
\section{Revision and debugging record}
\label{app:debug}
%=====================================================================

\begin{enumerate}[label=\textup{D\arabic*.}]
 \item The manuscript begins with the unresolved physical
 input ledger. No diagnostic package is relabeled physical.

 \item The adjoints, generic decision theorems, operator
 interfaces, and smooth benchmark have been moved out of
 the main calculation.

 \item The singular-primitive theorem now explicitly assumes
 scalar scale regularity. Its proof uses absolute continuity,
 not merely an almost-everywhere derivative identity.

 \item The main partition multiplies already differentiated
 seed amplitudes. Raw-sector motion terms are retained only
 in the separate raw-sector convention.

 \item The $0R$ zeros for $47$, $5a$, and response rows are
 proved in precisely that post-adjoint convention.

 \item Source derivatives remain ambient derivatives before
 incidence restriction. The $6$ derivative is never
 redistributed onto its incidence delta.

 \item All three $47$ terms, including the derivative of $f$,
 and both $9$ root terms are retained.

 \item The non-tip $5z$ chart coverage is not inferred from
 separation of $b-z$ alone. Its additional denominator
 condition is stated.

 \item Cone-tip calculations use $T\,\dd T\,\dd\mathbf m$.
 Non-tip denominators are not extrapolated through $T=0$.

 \item The calculated powers $\epsilon^2$ and $\epsilon^3$
 describe internal localization. They are not assigned to
 the smoothing scale $\delta$.

 \item The transition cross-check tests the translation
 construction against arbitrary compact profiles. It
 establishes the same geometric coefficients in both
 descriptions.

 \item The two-parameter errors do not constitute a complete
 physical smoothing remainder. Complementary regions remain
 on the audit ledger.

 \item The smooth-data theorem is a consistency benchmark,
 not a singular physical evaluation.

 \item Finite-order estimates, physical authorization,
 reconstruction descent, wavefront control, and admission
 remain distinct.
\end{enumerate}

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

\bibitem{XIVH}
N.~Cheung,
\emph{Landmark XIV-H}, version 1.
Supplied excerpts from \texttt{XIV-H.tex}, including the
source incidence and exact $K_5$, $K_8$, and $K_9$ formulas.
Used at supplied-excerpt scope.

\bibitem{XIVI}
N.~Cheung,
\emph{Landmark XIV-I}, version 1.
Supplied excerpts from \texttt{XIV-I.tex}, including
\texttt{eq:K47}, \texttt{eq:K6},
\texttt{prop:precomposition}, \texttt{thm:I2}, and
\texttt{thm:I3}.
Used for the internal inventory and its distinction from
the joint characteristic operation.

\bibitem{XIVJ}
N.~Cheung,
\emph{Landmark XIV-J}, version 1.
Supplied excerpts from \texttt{XIV-J.tex}, including
\texttt{prop:descent}, \texttt{lem:null},
\texttt{thm:reference}, and \texttt{thm:J2}.
Used for reconstruction and supported-reference boundaries.

\bibitem{EmB}
N.~Cheung,
\emph{Em01-B}, version 3.
Supplied excerpts from \texttt{Em01-B-v3.tex}, including
\texttt{thm:regularization} and \texttt{thm:moderate}.
Used to distinguish positive-scale regularization and
moderate growth from a characteristic limit.

\bibitem{EmC}
N.~Cheung,
\emph{Em01-C}, version 3.
Supplied excerpts from \texttt{Em01-C-v3.tex}, including
the characteristic chart, translation representation, and
\texttt{prop:plateau}.
Local physical transfer requires identification of its
stated patch and hypotheses.

\bibitem{EmD}
N.~Cheung,
\emph{Em01-D}, version 2.
Supplied excerpts from \texttt{Em01-D-v2.tex}, including
the complete translation and conditional origin-germ decision.
No physical germ certificate is imported.

\bibitem{EmE}
N.~Cheung,
\emph{Em01-E}, version 2.
Supplied excerpts from \texttt{Em01-E-v2.tex}, including
the origin route, decision interface, and authorization theorem.
Their physical hypotheses remain unverified here.

\bibitem{EmF}
N.~Cheung,
\emph{Em01-F}, version 3.
Supplied excerpts from \texttt{Em01-F-v3.tex}.
Its specified application is not identified with the
inherited weighted observable.

\bibitem{EmG}
N.~Cheung,
\emph{Em01-G}, version 5.
Supplied excerpts from \texttt{Em01-G-v5.tex}, including
\texttt{thm:failure-main}.
Its support-isolated failure concerns its own family and
is not transferred to the inherited coincidence problem.

\bibitem{XVB}
N.~Cheung,
\emph{Landmark XV-B}, version 1.
Supplied excerpt from \texttt{XV-B.tex},
\texttt{thm:lifting}.
Used for membership in a specified correction image,
not for automatic authorization.

\bibitem{PatchII}
N.~Cheung,
\emph{XV-F-patch-ii}, version 1.
Supplied excerpt from \texttt{XV-F-patch-ii.tex},
\texttt{prop:K47}.
Used for the operator/value distinction in the
presentation-space audit.

\end{thebibliography}

\end{document}