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% Em01-G-v5.tex
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% EXECUTION RECORD
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% program, or proof assistant was executed in preparing this revision.
% The commands above are instructions, not execution reports.
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% NEW RESULTS AND CORRECTIONS
% - synchronized revision identity;
% - corrected spectator-derivative seminorm bookkeeping;
% - crossed derivative count q <= 3 for 47 and 5+;
% - evaluated cancellation of both non-tip inverse-cubic coefficients;
% - exact Klein--Gordon reduction of the 5+ crossed derivative order;
% - evaluated full inverse-quadratic coefficient of the non-tip 5+ row;
% - an explicit fixed test detecting that coefficient;
% - a complete-seven-row local failure certificate on an isolated
%   5+ neighborhood, for a nonempty class of admissible fixed data.
%
% REMAINING BOUNDARY
% The general plateau supported inverse-quadratic tensor sum is not
% evaluated here. It is retained as an explicit inventory.
% No global complete-family expansion, all-data raw-limit classification,
% residual-tip matching theorem, or quantum admission is asserted.
%=====================================================================

\documentclass[11pt,a4paper]{article}

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\hypersetup{
  pdftitle={Em01-G-v5: Supported Asymptotics and an Isolated Non-Tip Failure Certificate},
  pdfauthor={}
}

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\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{\Test}{\mathcal D}
\newcommand{\Dist}{\mathcal D'}
\newcommand{\PP}{\mathsf P}
\newcommand{\Amap}{\mathsf A}
\newcommand{\bmap}{\mathsf b}
\newcommand{\Dmap}{\mathsf D}
\newcommand{\Qbil}{\mathcal Q}
\newcommand{\Qsym}{\mathsf Q}
\newcommand{\Glin}{\mathcal G_1}
\newcommand{\Rlin}{\mathcal R_1}
\newcommand{\Tr}{\operatorname{tr}_{\eta}}
\newcommand{\Boxeta}{\Box_{\eta}}
\newcommand{\deltaD}{\delta_{\mathrm D}}
\newcommand{\Kret}{K_{\mathrm{ret}}}
\newcommand{\Ksupp}{K_{\mathrm{supp}}}
\newcommand{\Vertex}{\mathsf V}
\newcommand{\fmap}{\mathfrak f}
\newcommand{\pair}[2]{\langle #1,#2\rangle}
\newcommand{\norm}[1]{\lVert #1\rVert}
\newcommand{\Irr}{\mathsf I_{\mathrm{RR}}}
\newcommand{\Iresp}{\mathsf I_{\mathrm{resp}}}
\newcommand{\SeedOp}{\mathscr P}
\newcommand{\Dz}{\mathscr Z}
\newcommand{\Da}{\mathscr A}
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    \dimexpr\linewidth-2\fboxsep-2\fboxrule\relax}
    \textbf{#1}\par\smallskip #2
  \end{minipage}}
  \par\medskip
}

\title{\textbf{Em01-G-v5}\\[0.4em]
Supported-Sector Asymptotics\\
and an Isolated Non-Tip Failure Certificate\\[0.4em]
\large The Constant-Weight Einstein--Scalar Family}
\author{}
\date{}

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

\begin{abstract}
We study a fixed-profile regularization of a frozen seven-row
Einstein--scalar distributional family. The complete supported response
sector has an expansion with three possible inverse powers, a
degree-zero logarithm, and a finite distribution, with a compact-uniform
\(O(\delta(1+|\log\delta|))\) remainder. Its proof uses an exact
whole-support radial reduction, including the massive causal tail,
and an averaged resolvent estimate with explicit parameter seminorms.

The new coefficient calculation concerns the non-tip positive
\(K_5\) placement. Derivative placement reduces the possible crossed
order of both \(47\) and positive \(K_5\) to three, and their
inverse-cubic coefficients vanish by an explicit tensor contraction.
For positive \(K_5\), a Klein--Gordon identity further reduces the
exact crossed order to two. We evaluate its full inverse-quadratic
coefficient, including the mass vertex, and construct a compact joint
test detecting it.

On a noncollinear positive-\(K_5\) incidence, support geometry excludes
all six other rows. The detecting test therefore gives a
complete-seven-row nonconvergence certificate for a nonempty class of
admissible fixed data, without residual-tip matching. This is not an
all-data classification or a global complete-family expansion.
The general output-plateau supported inverse-quadratic tensor sum,
the aligned negative placement, and tip matching remain open.
\end{abstract}

\scopebox{Precise release}{%
The principal new result is an evaluated, geometrically isolated
inverse-quadratic coefficient and a fixed-test failure certificate
for the \emph{complete} chosen regularization under explicit hypotheses.

The supported theorem is a separate sector theorem. Its lower
output-plateau tensor inventory is not represented as evaluated.
No physical subtraction, profile-independent normalization,
gauge-invariant completion, or quantum admission is inferred.}


\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) {Constant-weight family};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Isolated K5+ test};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {All-data raw limit};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {One detecting test is not an all-data classification.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{An isolated failure is not the raw limit. 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-G}
\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) {Constant-weight family};
\node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {All-data raw limit};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {One detecting test is not an all-data classification.};
\draw[->] (a) -- (s);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{An isolated failure is not the raw limit. 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-G}
\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) {Constant-weight family};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Isolated K5+ test};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {All-data raw limit};
\node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {One detecting test is not an all-data classification.};
\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-G}
\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) {Constant-weight family};
\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) {All-data raw limit};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Isolated K5+ test. One detecting test is not an all-data classification.};
\draw (a) -- (m);
\draw (m) -- (b);
\draw[->] (m) -- (s);
\end{tikzpicture}
\caption{Constant-weight family is not All-data raw limit. 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-G}
\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: Constant-weight family};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Isolated K5+ test};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: All-data raw limit};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {One detecting test is not an all-data classification.};
\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-G}
\end{figure}

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

\tableofcontents

%=====================================================================
\section{Frozen family and principal statements}
\label{sec:main}
%=====================================================================

Let \(M=\R^{1+3}\), with
\[
 \eta=\operatorname{diag}(1,-1,-1,-1),\qquad
 \Boxeta=\partial_t^2-\Delta_{\mathbf x},\qquad
 P_m=\Boxeta+m^2.
\]
Spacetime index conversions use \(\eta\).
Distributional pairings are complex bilinear, with the displayed
densities.

Fix
\[
 m\ge0,\quad \lambda\in\R\setminus\{0\},\quad \hbar>0,
 \qquad f,g\in C_c^\infty(M;\R),\qquad F=E_m^{\mathrm A}f,
\]
a relatively compact separated coordinate patch \(U\subset M^4\), and
\[
 \rho\in C_c^\infty(\R^8;\R),\qquad \int\rho=1,
 \qquad
 \chi\in C_c^\infty(\R^8),\qquad
 \chi=1\ \text{near }\fmap(\overline U).
\]
Every datum is fixed before \(\delta\downarrow0\).
In particular \(F\) remains the advanced root of \(f\).

The positive-scale family is
\begin{equation}
 t_\delta^{\App}
 =
 i\lambda^3\Vertex\bigl(
 \Kret,\rho_\delta*(\chi C)\bigr)\big|_U,
 \qquad
 \rho_\delta(H)=\delta^{-8}\rho(H/\delta).
 \label{eq:target}
\end{equation}
Its arrays, source, kernels, and vertex are defined in
Appendices~\ref{app:inputs}--\ref{app:positive}. Set
\[
 \Ksupp=K_6+K_8+K_{9d}+K_{9m},\qquad
 \Kret=K_{47}+K_{5,+}+K_{5,-}+\Ksupp.
\]
The weights are identically one and there is no \(1/\hbar\)
response prefactor.

Tests belong to \(\Test(U;S^*)\), where
\(S=\operatorname{Sym}^2((\R^4)^*)\). For \(L\Subset U\), write
\[
 \|\Phi\|_{q,L}
 =
 \max_{J,|\gamma|\le q}\sup_L|\partial^\gamma\Phi_J|.
\]
Compactly supported tests are extended by zero when a coordinate
expression requires a fixed slightly larger neighborhood.

\begin{theorem}[Supported fixed-profile expansion]
\label{thm:supported}
There are distributions
\[
 A_{-3}^{\mathrm{supp}},\quad A_{-2}^{\mathrm{supp}},
 \quad A_{-1}^{\mathrm{supp}},\quad
 A_{\log}^{\mathrm{supp}},\quad T_{\mathrm{supp}}
 \in\Dist(U;S)
\]
such that
\begin{equation}
\begin{split}
 t_{\delta,\mathrm{supp}}^{\App}
 =\lambda^3\hbar^2\Big[
 &\delta^{-3}A_{-3}^{\mathrm{supp}}
 +\delta^{-2}A_{-2}^{\mathrm{supp}}
 +\delta^{-1}A_{-1}^{\mathrm{supp}}\\
 &+(\log\delta)A_{\log}^{\mathrm{supp}}
 +T_{\mathrm{supp}}\Big]+R_\delta^{\mathrm{supp}},
\end{split}
 \label{eq:supported-expansion}
\end{equation}
and
\begin{equation}
 |R_\delta^{\mathrm{supp}}(\Phi)|
 \le C_L|\lambda|^3\hbar^2
       \delta(1+|\log\delta|)\|\Phi\|_{40,L},
 \qquad \supp\Phi\subset L.
 \label{eq:supported-remainder}
\end{equation}
The four singular coefficients are supported on \(\Nmark\).
They are constructed by
\eqref{eq:scalar-coefficients} and \eqref{eq:supported-coefficients}.
The inverse-cubic coefficient is \eqref{eq:supported-leading}.
\end{theorem}

The marked set \(\Nmark\) and the moments \(M_2,M_3\) are defined below.

\begin{theorem}[Local complete-family failure]
\label{thm:failure-main}
Suppose \(U\) contains a marked positive-\(K_5\) incidence
\[
 b=a+\beta\ell_{\mathbf n},\qquad
 y=a+\alpha\ell_{\mathbf n},\qquad
 z=a-\epsilon(1,\omega),
\]
with \(0<\alpha<\beta\), \(\epsilon>0\), and
\(\omega\ne\mathbf n\). Assume \(f(b)g(y)\ne0\) and
\(M_2(\mathbf n)\ne0\) at that incidence.

There is an isolated neighborhood \(O\Subset U\) and a fixed
\(\Phi_*\in\Test(O;S^*)\) for which
\begin{equation}
 t_\delta^{\App}(\Phi_*)
 =
 \lambda^3\hbar^2
 \left[c_*\delta^{-2}
       +O\bigl(\delta^{-1}+|\log\delta|+1\bigr)\right],
 \qquad c_*\ne0.
 \label{eq:failure-main}
\end{equation}
Consequently the complete family has no raw distributional limit on
\(U\).

A nonempty class of admissible normalized real profiles satisfies
\(M_2(\mathbf n)>0\) for every \(\mathbf n\); it is given in
Corollary~\ref{cor:profile-class}.
\end{theorem}

The proof, including the evaluated detecting coefficient, is in
Sections~\ref{sec:positive-evaluation}--\ref{sec:witness}.
The theorem is conditional on explicit fixed data, not a universal
failure statement.

\subsection{Independence and dependencies}

``Intrinsic'' means uniquely determined by the fixed positive-scale
family, including its profile and scale convention. Three questions
are kept separate:
\begin{enumerate}
\item admissible relative-cutoff independence;
\item independence of auxiliary core or residual localizers;
\item profile independence.
\end{enumerate}
The first is exact at sufficiently small scales.
Whole-support sector coefficients do not use an auxiliary core cutoff.
A fixed residual localization defines its own localized family.
Profile independence is not asserted.

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.35\linewidth}Y@{}}
\toprule
Claim & Essential inputs\\
\midrule
Supported highest-jet cancellation &
Constant-slot identity and highest normal polynomial.\\
Supported pure second-normal coefficient &
Cartesian residual-order reduction and source divergence.\\
Supported leading coefficient &
Gauge-slot identity and density contraction.\\
Whole-support scalar expansion &
Exact cone-plus-tail reduction and averaged resolvent estimate.\\
Non-tip cubic cancellation &
Derivative placement and the null contraction of \(\Bkin\).\\
Positive-\(K_5\) quadratic coefficient &
Exact Klein--Gordon reduction and the second log-profile moment.\\
Complete-family detecting test &
Quadratic coefficient, test-jet evaluation, and support isolation.\\
\bottomrule
\end{tabularx}
\end{center}

%=====================================================================
\section{Characteristic coordinates and the supported normal form}
\label{sec:coordinates}
%=====================================================================

On a sphere subchart choose
\[
 \ell=(1,\mathbf n),\qquad k=(1,-\mathbf n),\qquad
 E_A=(0,\mathbf e_A),\quad A=1,2,
\]
with
\[
 \ell^2=k^2=0,\qquad \ell\cdot k=2,\qquad
 E_A\cdot E_B=-\delta_{AB}.
\]
Use
\begin{equation}
 z=a+r,\qquad b=a+\beta\ell+uk,\qquad
 y=a+\alpha\ell+vk+\xi^AE_A.
 \label{eq:chart}
\end{equation}
On \(\overline U\), assume positive lower bounds on
\(\alpha,\beta-\alpha,\beta-u\), and strict order and separation
\[
 b^0>y^0>a^0,\qquad b^0>z^0,\qquad b\ne a,\quad b\ne z.
\]
The inverse is
\[
 \mathbf n=\frac{\mathbf b-\mathbf a}{|\mathbf b-\mathbf a|},
 \quad
 \beta=\frac{b^0-a^0+|\mathbf b-\mathbf a|}{2},
 \quad
 u=\frac{b^0-a^0-|\mathbf b-\mathbf a|}{2},
\]
\[
 \alpha=\frac{y^0-a^0+\mathbf n\cdot(\mathbf y-\mathbf a)}2,
 \quad
 v=\frac{y^0-a^0-\mathbf n\cdot(\mathbf y-\mathbf a)}2,
 \quad
 \xi^A=\mathbf e_A\cdot(\mathbf y-\mathbf a).
\]
The cone functions and density are
\[
 q_R=4\beta u,\qquad q_1=4\alpha v-|\xi|^2,\qquad
 q_2=4(\beta-\alpha)(u-v)-|\xi|^2,
\]
\begin{equation}
 \dd^4a\,\dd^4b\,\dd^4z\,\dd^4y
 =
 4(\beta-u)^2
 \dd^4a\,\dd\Omega_{\mathbf n}\,\dd\alpha\,\dd\beta\,
 \dd^4r\,\dd u\,\dd v\,\dd^2\xi.
 \label{eq:density}
\end{equation}
The two null-coordinate Jacobians have absolute determinant two.

Set
\[
 \Sigma=\{u=v=\xi=0\},\qquad
 \Sresp=\{r=u=0\},\qquad
 \Nmark=\{r=u=v=\xi=0\},
\]
\[
 \vartheta=(a,\mathbf n,\alpha,\beta),\qquad
 \tau=(\vartheta,v,\xi),\qquad
 \dd\vartheta=\dd^4a\,\dd\Omega_{\mathbf n}\,\dd\alpha\,\dd\beta.
\]
Write \(\dd\tau=\dd\vartheta\,\dd v\,\dd^2\xi\).

For a residual multiindex \(\zeta\), define
\[
 \Enormal_{\zeta,j}(c)(\varphi)
 =
 (-1)^{|\zeta|+j}
 \int c(\tau)\partial_r^\zeta\partial_u^j
                      \varphi(\tau,0,0)\dd\tau.
\]

\begin{proposition}[Unique supported normal form]
\label{prop:normal}
The complete supported sector has a unique representation
\begin{equation}
 \Ksupp^{I,J}
 =
 \sum_{|\zeta|+j\le3}
       \Enormal_{\zeta,j}(c_{\zeta,j}^{I,J}),
 \qquad c_{\zeta,j}^{I,J}\in C^\infty.
 \label{eq:normal}
\end{equation}
For the ordered transformed seed operator
\[
 \widehat{\SeedOp}_{\mathrm{supp}}^{I,J}
 =
 \sum_{\zeta,j,\gamma}
 a_{\zeta,j,\gamma}^{I,J}(\tau,r,u)
      \partial_\tau^\gamma\partial_r^\zeta\partial_u^j,
\]
the coefficients are
\begin{equation}
 c_{\zeta,j}^{I,J}
 =
 (-1)^{|\zeta|+j}
 \sum_\gamma(\partial_\tau^\gamma)^t
 \left[\frac{\beta}{2\pi}
       a_{\zeta,j,\gamma}^{I,J}(\tau,0,0)\right].
 \label{eq:normal-extraction}
\end{equation}
The transpose uses \(\dd\tau\).
\end{proposition}

\begin{proof}
On the separated future patch,
\(R(b-a)=\deltaD(u)/(8\pi\beta)\). Thus
\[
 \pair{R(b-a)\deltaD^{(4)}(a-z)}{\varphi}
 =
 \frac1{2\pi}\int\beta\,\varphi(\tau,0,0)\dd\tau.
\]
The supported seed operator has order at most three.
Restrict its ordered coefficients and transpose tangential derivatives
to obtain \eqref{eq:normal-extraction}.
A test \(r^\zeta u^j\omega(r,u)\psi(\tau)/(\zeta!j!)\), with
\(\omega=1\) near zero, isolates a selected normal coefficient.
This proves uniqueness.
\end{proof}

%=====================================================================
\section{Supported algebraic coefficients}
\label{sec:algebra}
%=====================================================================

Set \(h_\mu=\ell_\mu/2\) and \(\mathfrak m=\beta/(2\pi)\).
The source symbol has its first momentum in the \(G\)-slot:
\[
 \Qsym(H,G;p,q)
 =e^{-(p+q)\cdot x}\Qbil(He^{q\cdot x},Ge^{p\cdot x}),
 \qquad
 \Qsym^{I,J}=\Qsym(e_I,e_J;\,\cdot\,,\,\cdot\,).
\]

For constant \(H\), define
\[
 T_H(q)^{\mu\nu}
 =2e^{-q\cdot x}\Glin(He^{q\cdot x})^{\mu\nu},
 \qquad
 C_H(q)^\mu_{\nu\rho}
 =e^{-q\cdot x}C(He^{q\cdot x})^\mu_{\nu\rho}.
\]

\begin{lemma}[Source identities]
\label{lem:source}
The printed source obeys
\begin{align}
 q_\nu\Qsym(H,G;0,q)^{\mu\nu}&=0,
 \label{eq:constant-slot}\\
 (p+q)_\nu\Qsym(H,G;p,q)^{\mu\nu}
 &=-C_H(q)^\mu_{\nu\rho}T_G(p)^{\nu\rho}
   -C_G(p)^\mu_{\nu\rho}T_H(q)^{\nu\rho}.
 \label{eq:mixed-divergence}
\end{align}
Also
\begin{equation}
 \Qsym(H,G;0,q)=0
 \quad\text{if}\quad H=q\otimes t+t\otimes q.
 \label{eq:gauge-slot}
\end{equation}
\end{lemma}

\begin{proof}
Consider the raised Einstein density
\[
 J(g)^{\mu\nu}
 =
 2\sqrt{-\det g}\,g^{\mu\rho}g^{\nu\sigma}
 \left(R(g)_{\rho\sigma}
 -\frac12g_{\rho\sigma}g^{\alpha\beta}R(g)_{\alpha\beta}\right).
\]
The mixed coefficient of \(J(\eta+sH+tG)\), lowered with
\(\eta\), is \eqref{eq:source}.
The connection and Ricci mixed coefficients are \(D,U\);
the scalar and Einstein coefficients are \(s,E\).
The determinant derivative supplies the trace terms, and the two
inverse-metric derivatives supply the four index-conversion terms.

The density Bianchi identity
\[
 \partial_\nu J(g)^{\mu\nu}
 +\Gamma(g)^\mu_{\nu\rho}J(g)^{\nu\rho}=0
\]
gives \eqref{eq:mixed-divergence} at mixed order.
At \(p=0\), its right side vanishes.

For a direct constant-background check, the linearized Ricci symbol
at \(g_t=\eta+tG\) is
\[
 \frac12\left[
 q_\mu q_{g_t}^{\alpha}H_{\alpha\nu}
 +q_\nu q_{g_t}^{\alpha}H_{\alpha\mu}
 -q_{g_t}^2H_{\mu\nu}
 -q_\mu q_\nu\operatorname{tr}_{g_t}H\right].
\]
Its Einstein contraction is transverse.
For \(H=q\otimes t+t\otimes q\), the symbol is zero at every
\(g_t\), as is seen by substitution; the corresponding vector
amplitude is \(g_t^{-1}t\).
Differentiate the raised density in \(t\) to obtain
\eqref{eq:constant-slot} and \eqref{eq:gauge-slot}.
\end{proof}

These are classical source identities, not quantum Ward identities.

\begin{proposition}[Highest normal polynomial]
\label{prop:top}
The degree-three normal polynomial is
\begin{equation}
\begin{split}
 \sum_{|\zeta|+j=3}c_{\zeta,j}^{I,J}z^\zeta w^j
 =
 \mathfrak m w
 \left[
 \frac{f(b)\beta}{4}\ell_\mu\ell_\nu
 +\frac{g(b)}2(\partial_\mu F)(b)\ell_\nu
 \right]\Qsym^{I,J}(z,-z-hw)^{\mu\nu}.
\end{split}
 \label{eq:top}
\end{equation}
In particular \(c_{0,3}^{I,J}=0\), and there are no pure residual
degree-three jets.
\end{proposition}

\begin{proof}
The highest normal symbols of the response seed fields are
\[
 \Dz_\mu\mapsto z_\mu,\qquad
 \Da_\mu\mapsto-z_\mu-h_\mu w,\qquad
 \partial_{s^\mu}\mapsto h_\mu w.
\]
The order-three insertion sign is negative.
The identity
\(\Amap^{\rho\mu}{}_L\PP^L{}_M=-(e_M)^{\rho\mu}\)
cancels it. The \(K_8\) multiplier is
\(s_\rho/2=\beta\ell_\rho/2\).
The other degree-three row is the derivative-root row.
This gives \eqref{eq:top}.
Set \(z=0\) and use \eqref{eq:constant-slot}; the factor \(w\)
excludes pure residual terms.
\end{proof}

Thus \(j\le2\) can be used in the supported normal form.

\begin{theorem}[Pure second-normal coefficient]
\label{thm:c02}
The pure \(u^2\) coefficient is
\begin{equation}
\begin{split}
 c_{0,2}^{I,J}
 =\mathfrak m\Big[
 &-f(b)\Dmap^{\alpha\mu,I,J}h_\alpha h_\mu\\
 &+\partial_{b^\nu}\bigl(g(b)\partial_{b^\mu}F(b)\bigr)
                    \Qsym^{I,J}(0,h)^{\mu\nu}\\
 &+\frac{m^2g(b)F(b)}2\,\Tr\Qsym^{I,J}(0,h)\Big].
\end{split}
 \label{eq:c02}
\end{equation}
The \(K_8\) contribution is zero.
\end{theorem}

\begin{proof}
In Cartesian response coordinates \((c,s,r,y)\), all row weights
are independent of \(r\).
The subsequent coordinate change does not mix \(r\) with other
coordinates. Residual order zero therefore uses only source
terms with first source multiindex zero.

Let \(\mathscr Q^{\mu\nu}(\partial_s)
=\Qsym^{I,J}(0,\partial_s)^{\mu\nu}\).
Its divergence in either output index is zero.
The residual-order-zero \(K_8\) part is
\[
 \partial_{s^\mu}
 \left[f(b)\partial_{s^\rho}
                 \mathscr Q^{\rho\mu}(\partial_s)B(s)\right]
                                         \deltaD^{(4)}(r)=0.
\]
The derivative-root part reduces to
\[
 \partial_{b^\nu}\bigl(g(b)\partial_{b^\mu}F(b)\bigr)
             \mathscr Q^{\mu\nu}(\partial_s)R(s)\deltaD^{(4)}(r).
\]
Since \(\bmap(\PP Q)=-\Tr Q/2\), the mass-root part is
\[
 \frac{m^2g(b)F(b)}2
             \Tr\mathscr Q(\partial_s)R(s)\deltaD^{(4)}(r).
\]
The density part is
\[
 -\Dmap^{\alpha\mu,I,J}
       \partial_{s^\mu}\bigl(f(b)\partial_{s^\alpha}R(s)\bigr)
                                         \deltaD^{(4)}(r).
\]
All remaining propagation expressions have order at most two.
Their pure second-normal symbol, times \(\mathfrak m\), is
\eqref{eq:c02}.
\end{proof}

Set \(P_J(\mathbf n)=(e_J)_{\rho\sigma}\ell^\rho\ell^\sigma\).

\begin{lemma}[Projected supported coefficients]
\label{lem:projection}
One has
\begin{align}
 \sum_I(e^I)^{\mu\nu}\ell_\mu c_{0,2}^{I,J}
 &=-\frac{\mathfrak m f(b)}8P_J(\mathbf n)\ell^\nu,
 \label{eq:projection0}\\
 \sum_I(e^I)^{\mu\nu}\ell_\mu c_{\zeta,2}^{I,J}
 &=0,\qquad |\zeta|=1.
 \label{eq:projection1}
\end{align}
\end{lemma}

\begin{proof}
Contract with an arbitrary \(t_\nu\).
Completeness replaces the first input by
\(H=(\ell_\flat\otimes t+t\otimes\ell_\flat)/2\).
Equation~\eqref{eq:gauge-slot} kills both source terms in
\eqref{eq:c02}. Direct substitution in the density array gives
\[
 \ell_\alpha\ell_\mu
 D_{\mathrm{dens}}^{\alpha\mu}(e_J,H)
 =\frac12(\ell\cdot t)P_J.
\]
This proves \eqref{eq:projection0}.

For the mixed coefficient, set \(w=1\) in \eqref{eq:top}.
The total source momentum is \(K=-h\), and \(T_H(K)=0\).
As \(z\to0\),
\[
 T_{e_J}(z)=O(|z|^2),\quad C_{e_J}(z)=O(|z|),\quad
 T_H(K-z)=O(|z|).
\]
The mixed divergence identity makes the contraction with \(K\)
of the source \(O(|z|^2)\). Its linear term is zero, proving
\eqref{eq:projection1}.
\end{proof}

%=====================================================================
\section{Exact radial reduction of the entire crossed trace}
\label{sec:radial}
%=====================================================================

On the actual relative time region,
\[
 D=W_0\otimes\Delta_m,\qquad
 C_I^{\mu\nu}=i\hbar^2N_I^{\mu\nu}D.
\]
For a compact smooth profile \(\nu\), let
\(d_\delta^\nu=\nu_\delta*D\) locally.
Admissible relative cutoffs are eventually irrelevant.

Fix \(\alpha>0\), \(d=\beta-\alpha>0\), and the frame.
For a compact amplitude \(A(v,\xi)\), supported in the separated
time region, set
\begin{equation}
 \Js_\delta^\nu[A]
 =
 \int A(v,\xi)d_\delta^\nu
  (\alpha\ell+vk+\xi E,\,-d\ell+vk+\xi E)\dd v\,\dd^2\xi.
 \label{eq:scalar-trace}
\end{equation}
Compact smooth spectator dependence is allowed.

Write the smoothing displacement as
\[
 H_i=A_i\ell+B_i k+Z_iE,\qquad
 \widetilde\nu_{\mathbf n}(H)=4\nu(H_1,H_2).
\]
The factor four is the joint frame Jacobian.
Below \(\dd H\) is frame-coordinate measure.

Define
\begin{align}
 a_\delta&=\alpha-\delta A_1,&
 b_\delta&=d+\delta A_2,&
 e_\delta&=a_\delta+b_\delta,&
 c_\delta&=a_\delta/e_\delta,\notag\\
 Z_\delta&=(b_\delta Z_1+a_\delta Z_2)/e_\delta,\notag\\
 \kappa_\delta&=4a_\delta b_\delta/e_\delta,&
 D_\delta&=a_\delta b_\delta|Z_1-Z_2|^2/e_\delta^2,\notag\\
 L_\delta&=\kappa_\delta(B_2-B_1)-\delta D_\delta.
 \label{eq:radial-parameters}
\end{align}
All denominators have fixed positive lower bounds on the compact
parameter ranges under consideration.

Set
\begin{equation}
\begin{split}
 \mathcal A_{\delta,H}(t,z)
 =\frac1{2\pi}\int_0^{2\pi}
 A\left(
 \delta B_2-\frac{
 |\sqrt t\,\omega_\varphi+\delta(Z_\delta-Z_2)|^2+z}
 {4b_\delta},
 \sqrt t\,\omega_\varphi+\delta Z_\delta
 \right)\dd\varphi,
\end{split}
 \label{eq:angular-amplitude}
\end{equation}
where \(\omega_\varphi=(\cos\varphi,\sin\varphi)\).
Let
\[
 h_m(z)=-\frac{m^2}{8\pi}
       \sum_{j=0}^\infty\frac{(-m^2z/4)^j}{j!(j+1)!},
\]
and
\begin{equation}
 \mathcal F_{\delta,H}(t)
 =
 \mathcal A_{\delta,H}(t,0)
 +\frac{2\pi}{c_\delta}\int_0^t
 h_m\left(\frac{t-r}{c_\delta}\right)
 \mathcal A_{\delta,H}
       \left(r,\frac{t-r}{c_\delta}\right)\dd r.
 \label{eq:radial-amplitude}
\end{equation}

\begin{theorem}[Whole-support radial identity]
\label{thm:radial}
At sufficiently small positive scales,
\begin{equation}
 \Js_\delta^\nu[A]
 =
 -\frac1{32\pi^2}\int
 \frac{\widetilde\nu_{\mathbf n}(H)}{e_\delta}
 \int_0^\infty
 \frac{\mathcal F_{\delta,H}(t)}
      {t-\delta L_\delta+i0}\dd t\,\dd H.
 \label{eq:radial}
\end{equation}
The right side is an averaged distributional pairing.
The radial amplitudes are smooth up to \(t=0\), have common compact
radial support, and satisfy uniform finite-order estimates.
\end{theorem}

\begin{proof}
The mode conventions in Appendix~\ref{app:inputs} give, on the
separated relative region,
\[
 W_0(x_1)=-\frac1{4\pi^2}(x_1^2-i0)^{-1},
 \qquad
 \Delta_m(x_2)=-\frac1{2\pi}\deltaD(x_2^2)
                         -\Theta(x_2^2)h_m(x_2^2).
\]
The series for \(h_m\) obeys
\(4zh_m''+8h_m'+m^2h_m=0\) and
\(4h_m(0)=-m^2/(2\pi)\), matching the causal normalization.
At \(m=0\), it vanishes identically.

For the cone part,
\[
 q_{2,\delta}
 =4b_\delta(-v+\delta B_2)-|\xi-\delta Z_2|^2.
\]
Its delta sets
\[
 v=\delta B_2-\frac{|\xi-\delta Z_2|^2}{4b_\delta},
 \qquad\text{Jacobian }(4b_\delta)^{-1}.
\]
Completion of the square gives
\[
 q_{1,\delta}
 =-\frac{e_\delta}{b_\delta}
       \bigl(|\xi-\delta Z_\delta|^2-\delta L_\delta\bigr).
\]
The negative multiplier changes \(-i0\) to \(+i0\).
Before angular integration the coefficient is
\(-1/(32\pi^3e_\delta)\).
The substitution
\(\xi-\delta Z_\delta=\sqrt t\,\omega_\varphi\), with
\(\dd^2\xi=\frac12\dd t\,\dd\varphi\), gives the cone term of
\eqref{eq:radial}.

For the tail, put \(z=q_{2,\delta}\ge0\). Then
\[
 v=\delta B_2-\frac{|\xi-\delta Z_2|^2+z}{4b_\delta},
 \qquad \dd v=-\frac{\dd z}{4b_\delta},
\]
and
\[
 q_{1,\delta}
 =-\frac{e_\delta}{b_\delta}
          (t+c_\delta z-\delta L_\delta).
\]
The tail integral is
\[
 -\frac1{16\pi e_\delta}
 \int_0^\infty\int_0^\infty
 \frac{h_m(z)\mathcal A_{\delta,H}(t,z)}
      {t+c_\delta z-\delta L_\delta+i0}\dd t\,\dd z.
\]
Change to \(w=t+c_\delta z\).
The inner Jacobian is \(1/c_\delta\), and the ratio of tail and
cone prefactors is \(2\pi\). This yields
\eqref{eq:radial-amplitude}.

Circular averaging eliminates odd Cartesian Taylor terms and is
smooth in the squared radius. A radial derivative costs at most
two Cartesian amplitude derivatives. The corresponding integral
remainders are uniform on compact parameter sets.

Compact amplitude support bounds \(\xi,v\).
The displayed substitutions bound both nonnegative radial variables.
Thus the collapsed amplitude has common compact support.

For the distributional justification, replace \(i0\) by
\(i\varepsilon\), perform the ordinary changes, pair in \(H\),
and let \(\varepsilon\downarrow0\).
The map \(H\mapsto L_\delta\) is a submersion in \(B_2-B_1\).
The averaged boundary values therefore exist without assigning an
individual-displacement crossed product.
\end{proof}

The integral covers the whole fixed physical support. In particular
the rescaled tail \(t/\delta\to\infty\) is included.

%=====================================================================
\section{Averaged resolvent estimate and corrected parameter ledger}
\label{sec:resolvent}
%=====================================================================

For \(F\in C_c^\infty([0,\infty))\), put
\(f_j=F^{(j)}(0)/j!\), and define
\begin{equation}
 \FP_j(F)=\lim_{\epsilon\downarrow0}
 \left[
 \int_\epsilon^\infty\frac{F(t)}{t^{j+1}}\dd t
 -\sum_{l=0}^{j-1}\frac{f_l}{j-l}\epsilon^{l-j}
 +f_j\log\epsilon\right].
 \label{eq:finite-part}
\end{equation}
The sum is empty for \(j=0\).

On fixed sets \(0\le t\le T\), \(H\in K\), \(\theta\in P\),
\(|\delta|\le\delta_0\), define
\begin{equation}
 \mathcal P_{N,r}(F)
 =
 \max_{\substack{i\le N+2,\ j\le N+1\\
                 |\gamma|\le N+2+r,\ |\omega|\le r}}
 \sup\left|
 \partial_t^i\partial_\delta^j
 \partial_H^\gamma\partial_\theta^\omega
                         F_{\delta,H,\theta}(t)\right|.
 \label{eq:parameter-seminorm}
\end{equation}
All derivative orders are nonnegative.
For a compact smooth weight \(w\), define
\[
 \mathcal W_{N,r}(w)
 =
 \max_{\substack{j\le N+1,\ |\gamma|\le N+2+r\\|\omega|\le r}}
 \sup|\partial_\delta^j\partial_H^\gamma
                            \partial_\theta^\omega w|.
\]
The additional \(r\) in the \(H\)-order is essential: spectator
differentiation of a transformed smoothing coordinate can
differentiate the original weight or amplitude in \(H\).

Use
\[
 \log(-L+i0)=\log|L|+i\pi\mathbf1_{\{L>0\}},
 \qquad
 \partial_L\log(-L+i0)=(L-i0)^{-1}.
\]

\begin{lemma}[Uniform averaged resolvent expansion]
\label{lem:resolvent}
For the radial parameters \eqref{eq:radial-parameters}, a smooth
common-support family \(F_{\delta,H,\theta}\), and a smooth compact
weight \(w_{\delta,\theta}(H)\), one has after pairing in \(H\)
\begin{equation}
\begin{split}
 \int_0^\infty\frac{F_{\delta,H,\theta}(t)}
                       {t-\delta L_\delta+i0}\dd t
 =
 \sum_{j=0}^N\delta^jL_\delta^j
 \left[
 \FP_j(F_{\delta,H,\theta})
 -f_{j,\delta,H,\theta}
      \{\log\delta+\log(-L_\delta+i0)\}\right]
 +\mathcal R_{N,\delta,H,\theta}.
\end{split}
 \label{eq:resolvent-expansion}
\end{equation}
The averaged remainder, with spectator derivatives through order \(r\),
is bounded by
\begin{equation}
 C_{N,r}\delta^{N+1}(1+|\log\delta|)
                \mathcal W_{N,r}(w)\mathcal P_{N,r}(F).
 \label{eq:resolvent-bound}
\end{equation}
Constants depend on fixed supports and bounded parameter derivatives
of the coordinate changes, including a positive lower bound on
\(\kappa_\delta\).
\end{lemma}

\begin{proof}
Choose a fixed compact cutoff \(\omega_0=1\) near zero.
Write
\[
 F(t)=\omega_0(t)\sum_{j=0}^Nf_jt^j+t^{N+1}G(t).
\]
The quotient \(G\) is smooth, has common compact support, and
\(\|G\|_{C^1}\le C_N\|F\|_{C^{N+2}}\), also after the displayed
parameter derivatives.

Polynomial division gives
\[
 \int\frac{t^j\omega_0(t)}{t-z+i0}\dd t
 =
 \sum_{l=0}^{j-1}z^l\int t^{j-1-l}\omega_0(t)\dd t
 +z^j\int\frac{\omega_0(t)}{t-z+i0}\dd t,
\]
\[
 \int\frac{t^{N+1}G(t)}{t-z+i0}\dd t
 =
 \sum_{l=0}^Nz^l\int t^{N-l}G(t)\dd t
 +z^{N+1}\int\frac{G(t)}{t-z+i0}\dd t.
\]
Integrate explicitly where \(\omega_0=1\), and expand the
denominator away from zero. This gives
\[
 \int_0^\infty\frac{\omega_0(t)}{t-z+i0}\dd t
 =
 -\log(-z+i0)+\sum_{l=0}^Nz^l\FP_l(\omega_0)+O(|z|^{N+1}).
\]
The remainder after removal of the logarithm is smooth near \(z=0\).
Combining these identities with \(z=\delta L_\delta\) gives the
coefficients in \eqref{eq:resolvent-expansion}.

To estimate the remaining resolvent, choose
\(C_0>\sup|L_\delta|\), and a smooth \(\chi_{\mathrm{in}}\)
equal one on \([0,2C_0]\), zero on \([4C_0,\infty)\).
Split by \(\chi_{\mathrm{in}}(t/\delta)\).

On the outer support,
\(|t-\delta L_\delta|\ge t/2\), so the integral is
\(O(1+|\log\delta|)\) in a zeroth-order norm.

For the inner part put \(t=\delta x\), and let
\[
 \eta=(A_1,B_1,Z_1,A_2,Z_2).
\]
The explicit coordinate change is
\[
 B_2=B_1+\frac{L+\delta D_\delta}{\kappa_\delta},
 \qquad
 \left|\frac{\partial B_2}{\partial L}\right|
 =\kappa_\delta^{-1}.
\]
The quantities \(D_\delta,\kappa_\delta\) are independent of
\(B_1,B_2\). The transformed inner test is
\[
 \frac{w_\delta(H_\delta(L,\eta,\theta))}{\kappa_\delta}\,
 L^{N+1}G_{\delta,H_\delta(L,\eta,\theta),\theta}(\delta x)
                                      \chi_{\mathrm{in}}(x).
\]
It has common compact support.
The chain rule gives uniformly bounded first \(L\)-derivatives.
Spectator derivatives of order \(r\) can add at most \(r\)
\(H\)-derivatives. These are covered by
\eqref{eq:parameter-seminorm} and \(\mathcal W_{N,r}\).

Uniformly for compact \(x\)-ranges,
\[
 \left|\pair{(x-L+i0)^{-1}}{\psi(L)}\right|
 \le C\max_{j\le1}\|\partial_L^j\psi\|_\infty.
\]
Indeed, subtract \(\psi(x)\) times a fixed cutoff in the
principal-value integral; the remaining numerator is controlled
by a first derivative. The delta part is controlled by
\(|\psi(x)|\). Integrating in \(x,\eta\) gives an \(O(1)\)
inner bound.

The polynomial-division remainder contains
\(\delta^{N+1}\). Combining both pieces proves
\eqref{eq:resolvent-bound}. To differentiate with respect to
spectators, use the \(L\)-coordinate representation first.
Its singular factor is fixed; differentiation acts only on the
transformed smooth test. This also justifies differentiation
through the pairing.
\end{proof}

For the diagnostic amplitudes \(\omega_0,t\omega_0,t^2\omega_0\),
the exact polynomial-division identities place the first logarithms
at orders \(0,1,2\), respectively.
For amplitudes vanishing near zero, ordinary denominator expansion
recovers the same finite-part coefficients and no logarithms.
These checks hold for weights crossing \(L=0\).

%=====================================================================
\section{Scalar coefficients, profile moments, and locality}
\label{sec:scalar}
%=====================================================================

Let \(f_{j,\delta,H}=\partial_t^j\mathcal F_{\delta,H}(0)/j!\).
Define
\begin{align}
 U_j^\nu(\delta)[A]
 &=\frac1{32\pi^2}
   \int\widetilde\nu_{\mathbf n}
             \frac{L_\delta^jf_{j,\delta,H}}{e_\delta}\dd H,
 \notag\\
 V_j^\nu(\delta)[A]
 &=-\frac1{32\pi^2}
   \int\widetilde\nu_{\mathbf n}\frac{L_\delta^j}{e_\delta}
   \left[\FP_j(\mathcal F_{\delta,H})
       -f_{j,\delta,H}\log(-L_\delta+i0)\right]\dd H,
 \notag\\
 u_n^\nu[A]
 &=\sum_{j=0}^n\frac1{(n-j)!}\partial_\delta^{n-j}U_j^\nu(0)[A],
 \qquad
 v_n^\nu[A]
 =\sum_{j=0}^n\frac1{(n-j)!}\partial_\delta^{n-j}V_j^\nu(0)[A].
 \label{eq:scalar-coefficients}
\end{align}

\begin{theorem}[Whole-support scalar expansion]
\label{thm:scalar}
For every \(N\ge0\),
\begin{equation}
 \Js_\delta^\nu[A]
 =
 \sum_{n=0}^N\delta^n
       \bigl[(\log\delta)u_n^\nu[A]+v_n^\nu[A]\bigr]
 +\mathcal R_{N,\delta}^\nu[A],
 \label{eq:scalar-expansion}
\end{equation}
with
\[
 |\mathcal R_{N,\delta}^\nu[A]|
 \le C_N\delta^{N+1}(1+|\log\delta|)\|A\|_{4N+24}.
\]
Spectator derivatives through order \(r\) are controlled by
\(\|A\|_{4N+24+2r}\), on fixed compact supports.

If \(\nu=D_H\rho\) for a homogeneous constant-coefficient operator
of order \(q\), then \(u_n^\nu=0\) for \(n<q\).
Every \(u_n^\nu[A]\) depends only on jets of \(A\) of order at most
\(2n\) at \((v,\xi)=0\).
For \(n<q\), the same locality holds for \(v_n^\nu[A]\).
For \(q>0\),
\begin{equation}
 v_0^\nu[A]
 =\frac{A(0,0)}{32\pi^2\beta}
   \int\widetilde\nu_{\mathbf n}
                         \log(B_1-B_2+i0)\dd H.
 \label{eq:v0}
\end{equation}
\end{theorem}

\begin{proof}
The radial identity and resolvent lemma give an expansion with
smooth \(U_j,V_j\). Smoothness of the logarithmic pairing follows
by changing \(H\) to \((L,\eta)\), leaving the fixed locally
integrable function \(\log(-L+i0)\).
Taylor expansion through order \(N-j\) gives
\eqref{eq:scalar-coefficients} and the stated remainder.

The derivative ledger is as follows.
Radial order \(N+2\) costs at most \(2N+4\) amplitude derivatives.
Parameter Taylor differentiation costs at most \(N+1\) more.
The \(H\)-test order costs at most \(N+2+r\), and spectator
differentiation costs at most \(r\).
Thus \(4N+7+2r\) is sufficient for this part of the proof.
The larger \(4N+24+2r\) bound conservatively includes all
fixed-support quotient, coordinate, and tail estimates.

The scaling relations
\[
 \mathcal F_{\delta,sH}=\mathcal F_{s\delta,H},\qquad
 L_\delta(sH)=sL_{s\delta}(H)
\]
show that the coefficient of \(\delta^{n-j}\) in
\(L_\delta^jf_{j,\delta,H}/e_\delta\) is a homogeneous polynomial
of degree \(n\) in \(H\). A profile derivative of order \(q>n\)
annihilates it.

A radial jet of order \(j\) uses at most \(2j\) amplitude derivatives
at the shifted radial origin. Taking \(n-j\) parameter derivatives
and setting \(\delta=0\) uses jets of order at most
\(2j+n-j\le2n\) at zero. This proves locality of every \(u_n\).
For \(n<q\), the nonlocal finite-part portion of \(v_n\) is again
polynomial of smoothing degree \(n\) and vanishes. Its remaining
logarithmic-profile portion uses those same finite radial jets.

At \(\delta=0\),
\[
 e_0=\beta,\quad
 L_0=\frac{4\alpha d}{\beta}(B_2-B_1),\quad
 \mathcal F_{0,H}(0)=A(0,0).
\]
The finite part is independent of \(H\).
The zero mass \(\int\nu=0\) removes it and the constant logarithm
of \(4\alpha d/\beta\), giving \eqref{eq:v0}.
\end{proof}

Write \(S=B_2-B_1\), and define
\begin{equation}
 M_j(\mathbf n)
 =
 \int\widetilde\rho_{\mathbf n}(H)(S-i0)^{-j}\dd H,
 \qquad j=2,3.
 \label{eq:Mj}
\end{equation}
These are pullbacks by the submersion \(H\mapsto S\), paired with
the compact joint profile.

Since
\(\partial_{H_i^\mu}B_i=\ell_\mu/2\), one obtains
\begin{align}
 v_0^{\partial_{H_2^\mu}\partial_{H_2^\kappa}\rho}[A]
 &=-\frac{M_2}{128\pi^2\beta}\ell_\mu\ell_\kappa A(0,0),
 \label{eq:second-moment}\\
 v_0^{\partial_{H_2^\mu}\partial_{H_2^\kappa}
                     \partial_{H_2^\sigma}\rho}[A]
 &=-\frac{M_3}{128\pi^2\beta}
                  \ell_\mu\ell_\kappa\ell_\sigma A(0,0).
 \label{eq:third-moment}
\end{align}
These follow by integration by parts from
\[
 \partial_S^2\log(-S+i0)=-(S-i0)^{-2},\qquad
 \partial_S^3\log(-S+i0)=2(S-i0)^{-3}.
\]

%=====================================================================
\section{Supported assembly and its leading coefficient}
\label{sec:supported-assembly}
%=====================================================================

At small scales the exact combined test is
\begin{equation}
 \mathcal U_\delta^{I,J}[\Phi]
 =
 g(y)\left[
 (e^I)^{\mu\nu}\fmap^*(\partial_{2,\mu}d_\delta^\rho)
                                  \partial_{y^\nu}\Phi_J
 -\frac{m^2}{2}\tauI\fmap^*d_\delta^\rho\,\Phi_J\right].
 \label{eq:combined-test}
\end{equation}
The crossed factor is independent of \(r\).
At fixed characteristic tangential coordinates,
\(\partial_u=-k^\mu\partial_{2,\mu}\) on that factor.
The normal derivatives commute with Cartesian output derivatives.

Let
\[
 w(\vartheta;v,\xi)
 =(a,a+\beta\ell,a,a+\alpha\ell+vk+\xi E).
\]
For \(0\le b\le j\le2\), use the amplitudes
\begin{align}
 A_{\zeta,j,b,I,J,\mu,\nu}^{\mathrm K}
 &=
 (-1)^{|\zeta|+j}\binom jb c_{\zeta,j}^{I,J}
 (e^I)^{\mu\nu}g(y)
 (\partial_r^\zeta\partial_u^{j-b}
                         \partial_{y^\nu}\Phi_J)(w),
 \notag\\
 A_{\zeta,j,b,I,J}^{\mathrm M}
 &=
 (-1)^{|\zeta|+j}\binom jb c_{\zeta,j}^{I,J}
 \left(-\frac{m^2}{2}\tauI\right)g(y)
 (\partial_r^\zeta\partial_u^{j-b}\Phi_J)(w),
 \label{eq:supported-amplitudes}
\end{align}
and profiles
\[
 \nu_{b,\mu}^{\mathrm K}
 =(-k\cdot\nabla_{H_2})^b\partial_{H_2^\mu}\rho,
 \quad q_{\mathrm K}=b+1,
 \qquad
 \nu_b^{\mathrm M}=(-k\cdot\nabla_{H_2})^b\rho,
 \quad q_{\mathrm M}=b.
\]
For the resulting finite list \(\mathcal L\), the exact expression is
\begin{equation}
 t_{\delta,\mathrm{supp}}^{\App}(\Phi)
 =
 \lambda^3\hbar^2\sum_{\sigma\in\mathcal L}
 \int\delta^{-q_\sigma}
             \Js_\delta^{\nu_\sigma}[A_\sigma]\dd\vartheta,
 \qquad q_\sigma\le3.
 \label{eq:exact-supported}
\end{equation}
Both root rows and both vertex terms are included.

The coefficients are
\begin{align}
 A_{-p}^{\mathrm{supp}}(\Phi)
 &=
 \sum_{\substack{\sigma\in\mathcal L\\q_\sigma\ge p}}
 \int v_{q_\sigma-p}^{\nu_\sigma}[A_\sigma]\dd\vartheta,
 \quad p=1,2,3,\notag\\
 A_{\log}^{\mathrm{supp}}(\Phi)
 &=\sum_{\sigma\in\mathcal L}
       \int u_{q_\sigma}^{\nu_\sigma}[A_\sigma]\dd\vartheta,
 \notag\\
 T_{\mathrm{supp}}(\Phi)
 &=\sum_{\sigma\in\mathcal L}
       \int v_{q_\sigma}^{\nu_\sigma}[A_\sigma]\dd\vartheta.
 \label{eq:supported-coefficients}
\end{align}

\begin{proof}[Proof of Theorem~\ref{thm:supported}]
Leibniz's rule gives \eqref{eq:exact-supported}; crossed derivatives
are transferred to profile derivatives using
\[
 \partial_x^\gamma(\rho_\delta*D)
 =\delta^{-|\gamma|}((\partial^\gamma\rho)_\delta*D).
\]
Apply Theorem~\ref{thm:scalar} with \(N=q_\sigma\).
Moment cancellation removes inverse-power logarithms.
Each scaled remainder is
\(O(\delta(1+|\log\delta|))\).
Scalar amplitude order is at most \(36\), and normal plus output
test derivatives add at most four, giving
\eqref{eq:supported-remainder}.

All singular coefficients are finite jets at \(v=\xi=0\);
the supported normal form already imposes \(r=u=0\).
They are therefore supported on \(\Nmark\).
Uniform compact-stage estimates prove distributional continuity.

No bounded-core cutoff occurred.
Relative-cutoff independence follows from
Appendix~\ref{app:positive}.
Uniqueness follows by subtracting two expansions with vanishing
remainders and successively testing their most singular nonzero
difference. This also identifies the finite coefficient and gives
coordinate-overlap consistency.
\end{proof}

\begin{theorem}[Evaluated supported inverse-cubic coefficient]
\label{thm:supported-leading}
For \(w_0=(a,a+\beta\ell,a,a+\alpha\ell)\),
\begin{equation}
\begin{split}
 A_{-3}^{\mathrm{supp}}(\Phi)
 =
 -\frac1{512\pi^3}\int
 M_3(\mathbf n)f(a+\beta\ell)
 \sum_JP_J(\mathbf n)
 (\ell^\nu\partial_\nu g)(a+\alpha\ell)
                         \Phi_J(w_0)\dd\vartheta.
\end{split}
 \label{eq:supported-leading}
\end{equation}
\end{theorem}

\begin{proof}
Only \(j=b=2\) kinetic terms have \(q=3\).
Three profile integrations by parts give
\[
 \int\widetilde\nu_{2,\mu}^{\mathrm K}
                       \log(B_1-B_2+i0)\dd H=-\ell_\mu M_3.
\]
Equations~\eqref{eq:v0}, \eqref{eq:projection0}, and
\eqref{eq:projection1} leave
\[
 \frac1{512\pi^3}\int M_3 f(a+\beta\ell)
      \sum_JP_Jg(a+\alpha\ell)
                     \partial_\alpha[\Phi_J(w_0)]\dd\vartheta.
\]
Compact tangential integration by parts in \(\alpha\) yields
\eqref{eq:supported-leading}.
\end{proof}

The supported raw family converges exactly when its four singular
coefficients vanish. Its mathematical subtracted family converges
strongly to \(\lambda^3\hbar^2T_{\mathrm{supp}}\).
These sector statements alone do not decide the complete family.

%=====================================================================
\section{Non-tip derivative placement and cubic cancellation}
\label{sec:nontip}
%=====================================================================

Write \(r=A\ell+Bk+ZE\). Then
\[
 r^2=4AB-|Z|^2,\qquad
 (b-z)^2=4(\beta-A)(u-B)-|Z|^2,
 \qquad \dd^4r=2\dd A\,\dd B\,\dd^2Z.
\]
The \(47\) graph charts are
\[
 B=u-\frac{|Z|^2}{4(\beta-A)},\quad \beta-A>0,
 \qquad
 A=\beta-\frac{|Z|^2}{4(u-B)},\quad u-B>0.
\]
They cover compact nonzero future residual cones after finite
localization. On the first graph, with \(u=0\), the seed density is
\[
 \frac{\beta}{8\pi^2(\beta-A)}
        \dd\vartheta\,\dd A\,\dd^2Z\,\dd v\,\dd^2\xi.
\]
For \(5+\), put
\[
 -r=\epsilon j_\omega,\qquad
 j_\omega=(1,\omega),\qquad \epsilon>0.
\]
The seed density is
\begin{equation}
 \frac{\beta\epsilon}{8\pi^2}
       \dd\vartheta\,\dd\epsilon\,\dd\Omega_\omega\,
       \dd v\,\dd^2\xi,\qquad u=0.
 \label{eq:positive-density}
\end{equation}

\begin{lemma}[Crossed derivative placement]
\label{lem:placement}
On either \(47\) or \(5+\), every term in the exact differentiated
seed expression has crossed derivative order at most three.
The \(t\)-derivatives in the seed operators never differentiate the
crossed factor.
\end{lemma}

\begin{proof}
For \(47\),
\[
 a=c-s,\quad b=c,\quad z=c-t,\qquad
 x_1=y-c+s,\quad x_2=y-c.
\]
On the crossed factor,
\[
 \partial_t=0,\qquad \partial_s=\partial_1,\qquad
 \Db=\partial_c+\partial_s+\partial_t=-\partial_2.
\]
Each printed operator term has one \(\partial_s\) and one \(\Db\).
The kinetic vertex adds one crossed derivative.

For \(5+\),
\[
 a=c,\quad b=c+s,\quad z=c-t,\qquad
 x_1=y-c,\quad x_2=y-c-s.
\]
Thus \(\partial_t=0\) and \(\partial_s=-\partial_2\).
There are two possible \(\partial_s\)-actions on the crossed factor,
and the kinetic vertex adds one.
All other derivatives act on coefficients or joint tests.
\end{proof}

Define the constant kinetic completeness tensor
\begin{equation}
 \Bkin^{\alpha\beta\kappa\nu}
 =
 \sum_I\Amap^{\alpha\beta,I}(e^I)^{\kappa\nu}
 =
 \frac12\eta^{\alpha\beta}\eta^{\kappa\nu}
 -\frac12\bigl(
 \eta^{\alpha\kappa}\eta^{\beta\nu}
 +\eta^{\alpha\nu}\eta^{\beta\kappa}\bigr).
 \label{eq:Bkin}
\end{equation}
Directly,
\begin{align}
 \Bkin^{\alpha\beta\kappa\nu}\ell_\alpha\ell_\kappa&=0,
 \label{eq:Bkin-null}\\
 \Bkin^{\alpha\beta\kappa\nu}
             \partial_{2,\alpha}\partial_{2,\kappa}
 &=-\frac12\eta^{\beta\nu}\Box_{\eta,2}.
 \label{eq:Bkin-box}
\end{align}

\begin{theorem}[Sharpened non-tip expansions]
\label{thm:nontip}
For a fixed compact non-tip localization of either \(47\) or \(5+\),
there are distributions such that
\begin{equation}
 t_{\delta,j}^{\App,\mathrm{loc}}
 =
 \lambda^3\hbar^2
 \left[
 \delta^{-2}A_{-2,j}
 +\delta^{-1}A_{-1,j}
 +(\log\delta)A_{\log,j}+T_j\right]+R_{\delta,j},
 \quad j=47,5+,
 \label{eq:nontip-expansion}
\end{equation}
with
\[
 |R_{\delta,j}(\Phi)|
 \le C_L|\lambda|^3\hbar^2
          \delta(1+|\log\delta|)\|\Phi\|_{44,L}.
\]
The singular coefficients are supported on the corresponding
localized seed incidence intersected with \(\Sigma\).
Both inverse-cubic coefficients vanish:
\[
 A_{-3,47}=A_{-3,5+}=0.
\]
Inverse-quartic and inverse-quintic terms are absent already by
derivative placement.
\end{theorem}

\begin{proof}
Localize the complete joint test before the seed operator acts;
all localization derivatives are retained.
Expand by the finite Leibniz rule, transfer crossed derivatives
to homogeneous derivative profiles, and use the exact seed measures.
The residual variables are compact smooth spectators.
Lemma~\ref{lem:placement} gives \(q\le3\).
Theorem~\ref{thm:scalar} with \(N=q\) yields possible inverse powers
through three and the stated logarithmic and finite terms.

At inverse-cubic order, every crossed derivative contributes a
longitudinal \(\ell\)-factor in the log-profile moment.
In \(47\), all three printed operator terms contain the input
contraction
\(\Amap^{\sigma\nu,I}\ell_\sigma(e^I)^{\kappa\gamma}\ell_\kappa\).
In \(5+\), it is
\(\Amap^{\alpha\beta,I}\ell_\alpha
 (e^I)^{\kappa\nu}\ell_\kappa\).
Both vanish by \eqref{eq:Bkin-null}.
The mass vertex has crossed order at most two and cannot contribute
at inverse-cubic order.

The scalar bound costs at most \(36\) amplitude derivatives.
At most five pre-existing test derivatives occur.
Order \(44\) is conservative.
Locality sets \(v=\xi=0\), while the seed sets \(u=0\).
No condition \(r=0\) follows.
\end{proof}

%=====================================================================
\section{Exact positive-placement reduction and evaluated coefficient}
\label{sec:positive-evaluation}
%=====================================================================

Work in the \(5+\) chart
\[
 \kappa_+(c,s,t,y)=(c,c+s,c-t,y).
\]
For a joint test write \(\widehat\Phi_J=\Phi_J\circ\kappa_+\), and set
\[
 T_{J,\beta\rho}
 =\partial_{t^\beta}\partial_{t^\rho}\widehat\Phi_J,
 \qquad
 T_{J,\beta\rho\nu}
 =\partial_{t^\beta}\partial_{t^\rho}
                       \partial_{y^\nu}\widehat\Phi_J.
\]
Every \(t\)-derivative acts on the test only.
The coefficient \(f\) below means \(f(c+s)\).

\begin{lemma}[Exact crossed-order-two reduction]
\label{lem:positive-reduction}
At sufficiently small scales, the complete \(5+\) seed numerator
can be written as a sum of crossed-order-two terms
\begin{equation}
\begin{split}
 \mathcal V_{\delta,2}
 =-g(y)\sum_J\Amap^{\rho\mu,J}\Big[
 &\Bkin^{\alpha\beta\kappa\nu}
       (\partial_{2,\mu}\partial_{2,\kappa}d_\delta^\rho)
                   \partial_{s^\alpha}(fT_{J,\beta\rho\nu})\\
 &+\frac{m^2}{2}f\eta^{\alpha\beta}
       (\partial_{2,\alpha}\partial_{2,\mu}d_\delta^\rho)
                                      T_{J,\beta\rho}\Big]
\end{split}
 \label{eq:positive-q2}
\end{equation}
and a finite sum of crossed-order-zero and crossed-order-one terms.
The relative arguments of \(d_\delta^\rho\) are
\((y-c,y-c-s)\).
\end{lemma}

\begin{proof}
On every required compact relative neighborhood, eventually
\[
 (\Box_{\eta,2}+m^2)d_\delta^\rho=0.
 \label{eq:local-KG}
\]
Indeed, \(P_{m,2}(W_0\otimes\Delta_m)=0\).
The cutoff commutator is supported away from the relative image,
so its convolution vanishes there at small scales.

The kinetic part after the \(t\)-derivatives is
\[
 g(y)\sum_J\Bkin^{\alpha\beta\kappa\nu}
 \Amap^{\rho\mu,J}
 \partial_{s^\alpha}
    \left[f\partial_{s^\mu}
       \bigl((\partial_{2,\kappa}d_\delta^\rho)
                                  T_{J,\beta\rho\nu}\bigr)\right].
\]
Because \(\partial_s=-\partial_2\) on the crossed factor,
its six Leibniz terms consist of:
\begin{enumerate}[label=\textup{(\roman*)}]
\item the apparent crossed-order-three term
\(f\partial_{2,\alpha}\partial_{2,\mu}
                       \partial_{2,\kappa}d_\delta^\rho\,T\);
\item two terms combining to
\(-\partial_{2,\mu}\partial_{2,\kappa}d_\delta^\rho
                         \partial_{s^\alpha}(fT)\);
\item a term
\(-f\partial_{2,\alpha}\partial_{2,\kappa}d_\delta^\rho
                                      \partial_{s^\mu}T\);
\item two terms of crossed order one.
\end{enumerate}
After contraction by \(\Bkin\), item (i) is crossed order one
by \eqref{eq:Bkin-box} and the local Klein--Gordon equation.
Item (iii) is crossed order zero.
Item (ii) gives the first line of \eqref{eq:positive-q2}.

For the mass vertex,
\[
 \sum_I\Amap^{\alpha\beta,I}\tauI=\eta^{\alpha\beta}.
\]
Its crossed-order-two term is the second line of
\eqref{eq:positive-q2}; all its remaining terms have crossed order
at most one. This proves an exact reduction, not merely a
leading-symbol cancellation.
\end{proof}

For the marked positive incidence, define
\[
 w_+(\vartheta,\epsilon,\omega)
 =
 (a,a+\beta\ell,a-\epsilon j_\omega,a+\alpha\ell),
\qquad
 R_J^\rho(\mathbf n)=\Amap^{\rho\mu,J}\ell_\mu.
\]
The notation \([\,\cdot\,]_+\) means:
apply the displayed seed-chart derivatives first, then evaluate at
\[
 c=a,\quad s=\beta\ell,\quad t=\epsilon j_\omega,\quad
 y=a+\alpha\ell.
\]
Use
\[
 \dd\mu_+
 =\dd\vartheta\,\dd\epsilon\,\dd\Omega_\omega.
\]

\begin{theorem}[Evaluated full positive-placement inverse-quadratic coefficient]
\label{thm:positive-quadratic}
On every fixed non-tip positive-placement localization,
\begin{equation}
\begin{split}
 A_{-2,5+}(\Phi)
 =
 \frac1{1024\pi^4}\int
 \epsilon M_2(\mathbf n)g(a+\alpha\ell)
 \sum_JR_J^\rho(\mathbf n)
 \Big[
 &\Bkin^{\alpha\beta\kappa\nu}\ell_\kappa
              \partial_{s^\alpha}(fT_{J,\beta\rho\nu})\\
 &+\frac{m^2}{2}f\ell^\beta T_{J,\beta\rho}
 \Big]_+\dd\mu_+ .
\end{split}
 \label{eq:positive-quadratic}
\end{equation}
Here the spacetime index \(\alpha\) inside the bracket is summed;
the characteristic coordinate \(\alpha\) in the integration density
is a separate scalar variable.
All kinetic and mass contributions at inverse-quadratic order are
included. There are no root-row contributions to this row.
\end{theorem}

\begin{proof}
By Lemma~\ref{lem:positive-reduction}, inverse-quadratic order
comes only from \eqref{eq:positive-q2}.
Apply \eqref{eq:second-moment}.
Its negative sign cancels the outer negative sign of that numerator.
The first term becomes
\[
 \frac{M_2}{128\pi^2\beta}g
 R_J^\rho\Bkin^{\alpha\beta\kappa\nu}\ell_\kappa
                         \partial_{s^\alpha}(fT_{J,\beta\rho\nu}).
\]
The mass term has the same prefactor times
\((m^2/2)fR_J^\rho\ell^\beta T_{J,\beta\rho}\).
Multiplication by the seed density
\(\beta\epsilon/(8\pi^2)\) gives
\eqref{eq:positive-quadratic}.
All remaining exact terms have crossed order at most one and
cannot contribute at inverse-quadratic order.
\end{proof}

This is a full tensor-contracted local coefficient, not an inventory
in terms of unspecified normal coefficients.

%=====================================================================
\section{Support isolation and a fixed complete-family detecting test}
\label{sec:witness}
%=====================================================================

\begin{proposition}[Isolated non-tip neighborhoods]
\label{prop:isolation}
The following two kinds of marked incidences admit neighborhoods
on which the complete internal family equals the indicated single row:
\begin{enumerate}
\item A \(47\) graph point with \(\beta-A>0\) and \(Z\ne0\):
the complete family equals \(47\).
\item A \(5+\) point with \(\epsilon>0\) and \(\omega\ne\mathbf n\):
the complete family equals \(5+\).
\end{enumerate}
The equalities are equalities of internal distributions restricted
to the neighborhood; hence they hold for the regularized vertex
on tests supported there.
\end{proposition}

\begin{proof}
At the first graph, with \(u=0\),
\[
 B=-\frac{|Z|^2}{4(\beta-A)},\qquad
 r^2=-\frac{\beta|Z|^2}{\beta-A}<0.
\]
Both \(K_5\) placements require a null residual attachment.
The supported rows require \(r=0\).
A small neighborhood excludes all these closed supports.

At the positive-placement point,
\(r=-\epsilon j_\omega\), and
\[
 (b-z)^2
 =(\beta\ell+\epsilon j_\omega)^2
 =2\beta\epsilon(1-\mathbf n\cdot\omega)>0.
\]
This excludes the \(47\) and negative-placement seeds,
which require \(b-z\) to be null.
Since \(r\ne0\), it excludes every supported row.
Differentiation does not enlarge seed support.
Small compact neighborhoods preserve the strict inequalities.
\end{proof}

Fix such an isolated positive neighborhood \(O\).
Let
\[
 q=r^2=(z-a)^2.
\]
Near the non-tip positive cone use the smooth residual coordinates
\[
 \epsilon=-r^0,\qquad
 \omega=-\frac{\mathbf r}{|\mathbf r|},\qquad q=r^2.
\]
They form local coordinates: the spatial radius is recovered from
\(|\mathbf r|^2=\epsilon^2-q\).

Let
\[
 \eta_J=(e^J)^{\mu\nu}\eta_{\mu\nu},
 \qquad
 \sum_J\eta_Je_J=\eta.
\]
Choose a compact smooth \(\psi(\vartheta,\epsilon,\omega)\) in a
small tangential subchart, and a compact normal cutoff
\(\zeta(u,v,\xi,q)\), equal one near zero.
Choose their supports so that the following test is supported in \(O\):
\begin{equation}
 (\Phi_*)_J
 =
 \eta_J\,\frac{q^2}{2}\,\frac{|\xi|^2}{2}\,
            \psi(\vartheta,\epsilon,\omega)\zeta(u,v,\xi,q).
 \label{eq:witness-test}
\end{equation}

\begin{theorem}[Evaluated detecting coefficient]
\label{thm:witness}
For \eqref{eq:witness-test},
\begin{equation}
\begin{split}
 A_{-2,5+}(\Phi_*)
 =
 -\frac1{256\pi^4}
 \int
 \frac{\alpha\epsilon^3}{\beta}
 M_2(\mathbf n)f(a+\beta\ell)g(a+\alpha\ell)
 (1-\mathbf n\cdot\omega)^2
               \psi(\vartheta,\epsilon,\omega)\dd\mu_+ .
\end{split}
 \label{eq:witness-value}
\end{equation}
The mass-vertex contribution to this test coefficient is zero.
The formula holds for every \(m\ge0\).
\end{theorem}

\begin{proof}
In the positive seed chart, \(r=-t\).
At \(q=0\),
\[
 \partial_{t^\beta}\partial_{t^\rho}(q^2/2)
 =4r_\beta r_\rho.
\]
Thus the two \(t\)-derivatives must both fall on \(q^2/2\)
for a contribution at the incidence. The remaining factor
\(|\xi|^2/2\) makes \(T_{J,\beta\rho}=0\), so the mass term
in \eqref{eq:positive-quadratic} vanishes.

Also \(T_{J,\beta\rho\nu}=0\).
Therefore the derivative of \(f\) contributes zero, and the
surviving kinetic term has one \(s\)-derivative and one output
derivative on \(|\xi|^2/2\).
At \(\Sigma\), with \(c,t,y\) fixed during \(s\)-differentiation,
\[
 \partial_{s^\alpha}\xi^A
 =\frac{\alpha}{\beta}(E_A)_\alpha,\qquad
 \partial_{y^\nu}\xi^A=-(E_A)_\nu.
 \label{eq:transverse-derivatives}
\]
The scalar \(\alpha/\beta\) here is the ratio of characteristic
coordinates. Consequently
\[
 \partial_{s^\alpha}\partial_{y^\nu}(|\xi|^2/2)
 =-\frac{\alpha}{\beta}
                  \sum_{A=1}^2(E_A)_\alpha(E_A)_\nu.
\]
The explicit tensor \eqref{eq:Bkin} gives
\[
 \Bkin^{\alpha\beta\kappa\nu}\ell_\kappa
             \sum_A(E_A)_\alpha(E_A)_\nu=\ell^\beta.
\]
Hence the bracketed kinetic derivative in
\eqref{eq:positive-quadratic}, after summing the output direction,
becomes
\[
 -\frac{4\alpha}{\beta}
       f\psi\,(\ell\cdot r)
       \sum_J\eta_JR_J^\rho r_\rho.
\]
Since \(\Amap(\eta)^{\rho\mu}=\eta^{\rho\mu}\),
\[
 \sum_J\eta_JR_J^\rho r_\rho=\ell\cdot r.
\]
Finally
\(\ell\cdot r=-\epsilon(1-\mathbf n\cdot\omega)\).
Substitute into \eqref{eq:positive-quadratic} to obtain
\eqref{eq:witness-value}.

No cutoff or tangential-amplitude derivative survives this evaluation:
a nonzero incidence jet must use both \(t\)-derivatives on \(q^2/2\)
and the \(s,y\)-derivatives on \(|\xi|^2/2\).
This also verifies that no omitted localization derivative affects
the coefficient.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:failure-main}]
The hypotheses and continuity permit a sufficiently small isolated
tangential neighborhood where the continuous complex density
\[
 M_2(\mathbf n)f(a+\beta\ell)g(a+\alpha\ell)
\]
is nonzero and has values in a fixed open half-plane after multiplication
by one complex phase.
Choose a nonnegative nonzero \(\psi\) there.
All other factors in \eqref{eq:witness-value} are positive:
\(\alpha,\beta,\epsilon>0\) and
\(1-\mathbf n\cdot\omega>0\).
Thus \(c_*=A_{-2,5+}(\Phi_*)\ne0\).

Support isolation makes the complete pairing equal to the \(5+\)
pairing. The controlled non-tip expansion gives
\eqref{eq:failure-main}. Its inverse-quadratic term cannot be canceled
by lower orders. The fixed complete-family pairing is unbounded.
A distributional limit would have finite pairings on every fixed test,
so no raw limit exists.
\end{proof}

\begin{corollary}[An explicit nonempty profile class]
\label{cor:profile-class}
Let \(\rho\ge0\), \(\int\rho=1\), be compactly supported in
\begin{equation}
 H_2^0-H_1^0>
 |\mathbf H_2-\mathbf H_1|+2s_0
 \quad\text{for some }s_0>0.
 \label{eq:profile-class}
\end{equation}
Then \(M_2(\mathbf n)>0\) for every \(\mathbf n\).
Such profiles exist.

For these profiles, any isolated positive-placement neighborhood
with \(f>0\) at the response point and \(g>0\) at the observation
point admits the complete-family detecting test.
In particular one may impose an output plateau \(g=1\).
\end{corollary}

\begin{proof}
In frame coordinates,
\[
 S=B_2-B_1
 =\frac12\ell_{\mathbf n}\cdot(H_2-H_1)
 \ge\frac12\left(
 H_2^0-H_1^0-|\mathbf H_2-\mathbf H_1|\right)>s_0.
\]
Thus the boundary value in \(M_2\) is the ordinary positive function
\(S^{-2}\), and its normalized nonnegative-profile integral is
strictly positive.
A small nonnegative normalized bump centered at
\(H_1=0,H_2=(T,\mathbf0)\), with \(T>2s_0\), supplies an example.
The test integral is then strictly negative for nonnegative
nonzero \(\psi\).
\end{proof}

\begin{remark}[What the certificate does and does not require]
The root \(F=E_m^{\mathrm A}f\) is unchanged. No independent root jet
is selected, and no response plateau is needed.
The root rows vanish on the detecting neighborhood by their internal
support, not by a root identity.
The certificate concerns the actual seven-row sum on a fixed test.
It does not require a global coefficient assembly or tip matching,
because all other rows are exactly absent on that test support.
\end{remark}

%=====================================================================
\section{The supported plateau inventory and remaining boundaries}
\label{sec:remaining}
%=====================================================================

\begin{definition}[Two different plateaux]
On \(U_0\Subset U\), an \emph{output plateau} means \(g=1\) near
the \(y\)-projection of \(\overline U_0\).
A \emph{response plateau} additionally means \(g=1\) near the
\(b\)-projection of \(\overline U_0\).
The first removes output-switch derivatives; the second simplifies
root weights at \(b\). Neither changes \(F=E_m^{\mathrm A}f\).
\end{definition}

On an output plateau,
\(A_{-3}^{\mathrm{supp}}|_{U_0}=0\).
The general supported inverse-quadratic tensor calculation is not
completed by the isolated positive-placement calculation.
For precision, its explicit radial-profile inventory is retained in
Appendix~\ref{app:plateau}.

For the negative placement, on
\(a_0\le A\le\beta-a_0\), the two future cone equations give
\[
 B=(1-A/\beta)u,\qquad
 |Z|^2=4A(1-A/\beta)u,\qquad u\ge0.
\]
Its exact measure is
\[
 \frac{(\beta-u)^2}{4\pi^2\beta}
 \dd\vartheta\,\dd A\,\dd u\,\dd\varphi\,\dd v\,\dd^2\xi.
\]
Unlike \(47\) and \(5+\), \(u\) remains integrated.
The \(u=0\) trace theorem is not a theorem for this sector.
At non-tip \(A\), \(u=O(\delta)\) gives \(Z=O(\sqrt\delta)\);
if \(A=O(\delta)\) as well, the equation instead gives
\(Z=O(\delta)\). The intermediate regimes and endpoint matching
remain separate obligations.

The outstanding tasks for a global complete-family expansion are:
\begin{enumerate}
\item evaluation of the supported plateau tensor inventory;
\item lower coefficient evaluation of \(47\);
\item the complete negative-placement asymptotic;
\item residual-tip and non-tip overlap cancellation;
\item common-coordinate complete scale assembly and a vanishing
compact-uniform remainder.
\end{enumerate}

\subsection{Reference agreement and topology}

Away from nonzero relative cones the crossed lines are smooth;
on the cones their wavefront covectors are conormal.
Pulled-back crossed covectors have form
\[
 (-k_1,-k_2,0,k_1+k_2).
\]
A nonzero one can have zero \(y\)-component only on \(\Sigma\).
Internal rows have zero \(y\)-covector.
Thus the ordinary product is defined on \(U\setminus\Sigma\).
On compact subdomains, small relative translations preserve
nonopposition. Pullback continuity, compact-parameter bounds, and
fixed-factor multiplication apply under these verified hypotheses
\cite[Proposition~5.1, Theorems~5.9 and~6.1]{BDH}.

The supported reference exists on \(U\setminus\Nmark\), and
\(T_{\mathrm{supp}}\) agrees with it there.
No maximal canonical-domain equality is asserted.
Extension theory has additional scaling and microlocal hypotheses
\cite{Dang}; it is not used to infer the raw limit.

The displayed compact-stage remainder bounds imply strong
distributional convergence for the corresponding subtracted sector
families: bounded test sets have common compact support and bounded
finite seminorms.
No wavefront-normal convergence theorem is asserted.

%=====================================================================
\section{Verification record}
\label{sec:verification}
%=====================================================================

The new written coefficient audit consists of:
\begin{enumerate}
\item seed-chart crossed derivative placement;
\item the explicit completeness tensor \(\Bkin\);
\item its null and d'Alembertian contractions;
\item the local Klein--Gordon identity after cutoff smoothing;
\item all crossed-order-two kinetic and mass terms of \(5+\);
\item the second profile moment, including its sign;
\item the positive seed density and combined factor \(1/(1024\pi^4)\);
\item the residual-square and transverse-square test jets;
\item support isolation for the complete seven-row sum;
\item a positive admissible profile class.
\end{enumerate}
The proof of the detecting coefficient does not use an unexecuted
symbolic result or a theorem from the unrelated manuscript excerpts
supplied with the revision request.

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.43\linewidth}Y@{}}
\toprule
Check & Actual status\\
\midrule
First and second pdfLaTeX passes & Not executed.\\
Generated log / equation layout inspection & No generated log available.\\
Computer-algebra component audit & Not executed.\\
Numerical integration & Not executed.\\
Proof-assistant verification & Not executed.\\
Primary microlocal references &
The cited BDH paper and Dang reference were consulted;
the radial and coefficient arguments are proved here.\\
Independent referee certification & Not asserted.\\
Submission or acceptance & Not asserted.\\
\bottomrule
\end{tabularx}
\end{center}

Actual two-pass compilation and log inspection remain necessary before
submission. The source is arranged for pdfLaTeX, but successful
compilation is not reported as an observed event.

%=====================================================================
\appendix
\section{Frozen component arrays, source, and crossed kernels}
\label{app:inputs}
%=====================================================================

For \(I=(i,j)\), \(0\le i\le j\le3\), define
\[
 (e_I)_{\mu\nu}
 =\frac{\delta_{\mu i}\delta_{\nu j}
       +\delta_{\mu j}\delta_{\nu i}}{1+\delta_{ij}},
 \qquad
 (e^I)^{\mu\nu}
 =\frac12(\delta_i^\mu\delta_j^\nu+\delta_j^\mu\delta_i^\nu).
\]
Then \(e^I(e_J)=\delta^I_J\).
The dual basis is algebraic, not the raised basis.

Set
\[
 \PP H=H-\frac12\eta\,\Tr H,\qquad \PP^2=1,
\]
\[
 \Amap^{\mu\nu,I}
 =\frac12\eta^{\mu\nu}\Tr e_I-(e_I)^{\mu\nu},
 \qquad
 \bmap_I=\frac12\Tr e_I,
 \qquad
 \PP^L{}_M=(e^L)^{\mu\nu}(\PP e_M)_{\mu\nu}.
\]
A lower component label means evaluation on its basis tensor.

The density mixed derivative is
\begin{equation}
\begin{split}
 D_{\mathrm{dens}}^{\mu\nu}(G,H)={}&
 \left[\frac14(\Tr G)(\Tr H)
       -\frac12G_{\rho\sigma}H^{\rho\sigma}\right]\eta^{\mu\nu}\\
 &-\frac12[(\Tr G)H^{\mu\nu}+(\Tr H)G^{\mu\nu}]\\
 &+G^{\mu\rho}H_\rho{}^\nu+H^{\mu\rho}G_\rho{}^\nu,
 \qquad
 \Dmap^{\mu\nu,I,J}=D_{\mathrm{dens}}^{\mu\nu}(e_J,e_I).
\end{split}
 \label{eq:density-array}
\end{equation}
It is the mixed derivative, not a half-polarization.
The stipulated free tensor Hessian and inverse are
\(-\Boxeta\PP\) and \(-\PP E_0^{\mathrm R}\).

For symmetric fields \(H,G\), define
\[
 S(H)_{\sigma\mu\nu}
 =\partial_\mu H_{\sigma\nu}+\partial_\nu H_{\sigma\mu}
                         -\partial_\sigma H_{\mu\nu},
 \qquad
 C(H)^\rho_{\mu\nu}
 =\frac12\eta^{\rho\sigma}S(H)_{\sigma\mu\nu},
\]
\[
 D(H,G)^\rho_{\mu\nu}
 =-\frac12[H^{\rho\sigma}S(G)_{\sigma\mu\nu}
                         +G^{\rho\sigma}S(H)_{\sigma\mu\nu}],
\]
\[
 \Rlin(H)_{\mu\nu}
 =\partial_\rho C(H)^\rho_{\mu\nu}
                         -\partial_\nu C(H)^\rho_{\mu\rho},
 \qquad
 r_1(H)=\eta^{\mu\nu}\Rlin(H)_{\mu\nu},
 \qquad
 \Glin(H)=\Rlin(H)-\frac12\eta r_1(H),
\]
\begin{align*}
 U(H,G)_{\mu\nu}={}&
 \partial_\rho D(H,G)^\rho_{\mu\nu}
 -\partial_\nu D(H,G)^\rho_{\mu\rho}\\
 &+C(H)^\rho_{\rho\alpha}C(G)^\alpha_{\mu\nu}
  +C(G)^\rho_{\rho\alpha}C(H)^\alpha_{\mu\nu}\\
 &-C(H)^\rho_{\nu\alpha}C(G)^\alpha_{\mu\rho}
  -C(G)^\rho_{\nu\alpha}C(H)^\alpha_{\mu\rho},
\end{align*}
\[
 s(H,G)=\eta^{\mu\nu}U(H,G)_{\mu\nu}
       -H^{\mu\nu}\Rlin(G)_{\mu\nu}
       -G^{\mu\nu}\Rlin(H)_{\mu\nu},
\]
\[
 E(H,G)_{\mu\nu}
 =U(H,G)_{\mu\nu}-\frac12\eta_{\mu\nu}s(H,G)
 -\frac12[H_{\mu\nu}r_1(G)+G_{\mu\nu}r_1(H)].
\]
The frozen source is
\begin{equation}
\begin{split}
 \Qbil(H,G)_{\mu\nu}=2\Big[
 &E(H,G)_{\mu\nu}
 +\frac12(\Tr H)\Glin(G)_{\mu\nu}
 +\frac12(\Tr G)\Glin(H)_{\mu\nu}\\
 &-H_\mu{}^\alpha\Glin(G)_{\alpha\nu}
  -H_\nu{}^\alpha\Glin(G)_{\mu\alpha}\\
 &-G_\mu{}^\alpha\Glin(H)_{\alpha\nu}
  -G_\nu{}^\alpha\Glin(H)_{\mu\alpha}\Big].
\end{split}
 \label{eq:source}
\end{equation}
Its component coefficients are
\[
 q_{\alpha\beta}^{L,I,J}
 =[p^\alpha q^\beta](e^L)^{\mu\nu}
                         \Qsym(e_I,e_J;p,q)_{\mu\nu}.
\]
The symbol is homogeneous of total degree two:
\[
 \Qbil(H,G)^L
 =\sum_{\alpha,\beta,I,J}
 q_{\alpha\beta}^{L,I,J}
       (\partial^\alpha G_J)(\partial^\beta H_I).
\]
Multiindex derivatives are coordinate derivatives.

The advanced root is
\[
 F(t,\mathbf x)
 =\int\frac{\dd^3\mathbf p}{(2\pi)^3}e^{i\mathbf p\cdot\mathbf x}
 \int_t^\infty
 \frac{\sin(\omega_{\mathbf p}(s-t))}{\omega_{\mathbf p}}
                         \widehat f(s,\mathbf p)\dd s,
 \quad \omega_{\mathbf p}=\sqrt{|\mathbf p|^2+m^2}.
\]
The zero-frequency quotient is \(s-t\).
Differentiation gives \(P_mF=f\) and zero future data.

The retarded massless measure and its convolution square are
\[
 R(\psi)=\frac1{4\pi}\int_{\R^3}
       \frac{\psi(|\mathbf x|,\mathbf x)}{|\mathbf x|}
                                               \dd^3\mathbf x,
 \qquad
 B(x)=\frac1{8\pi}\mathbf1_{\{x^0\ge|\mathbf x|\}}.
\]
They obey
\[
 \Boxeta R=\deltaD^{(4)},\qquad R*R=B,\qquad
 \partial_{x^\mu}B(x)=\frac{x_\mu}{2}R(x)
                              \quad(x^0>0,\ x\ne0).
\]
Their spatial transforms are
\[
 \widehat R(t,\mathbf p)
 =\Theta(t)\frac{\sin(|\mathbf p|t)}{|\mathbf p|},
 \qquad
 \widehat B(t,\mathbf p)
 =\Theta(t)\frac{\sin(|\mathbf p|t)
           -|\mathbf p|t\cos(|\mathbf p|t)}{2|\mathbf p|^3}.
\]

The scalar lines are
\[
 W_m(t,\mathbf x)
 =\int\frac{\dd^3\mathbf p}{(2\pi)^3\,2\omega_{\mathbf p}}
                  e^{-i\omega_{\mathbf p}t+i\mathbf p\cdot\mathbf x},
\]
\[
 D_{T,m}(t,\mathbf x)
 =\int\frac{\dd^3\mathbf p}{(2\pi)^3\,2\omega_{\mathbf p}}
                  e^{-i\omega_{\mathbf p}|t|+i\mathbf p\cdot\mathbf x}.
\]
The conventions are
\[
 P_mD_{T,m}=-i\deltaD^{(4)},\qquad
 W_m(x)-W_m(-x)=-i\Delta_m(x),\qquad P_m\Delta_m=0.
\]
Set
\[
 \Pi_{\alpha\beta\rho\sigma}
 =\frac12(\eta_{\alpha\rho}\eta_{\beta\sigma}
          +\eta_{\alpha\sigma}\eta_{\beta\rho}
          -\eta_{\alpha\beta}\eta_{\rho\sigma}),
\]
\[
 N_I^{\mu\nu}
 =\eta^{\mu\alpha}\eta^{\nu\beta}
           \Pi_{\alpha\beta\rho\sigma}(e^I)^{\rho\sigma},
 \qquad \tauI=\eta_{\rho\sigma}(e^I)^{\rho\sigma}.
\]
The grouped crossed array is
\[
 C_I^{\mu\nu}
 =\hbar^2N_I^{\mu\nu}
       (D_{T,0}\otimes D_{T,m}-W_0\otimes W_m).
\]

%=====================================================================
\section{Frozen rows and complete seed transpositions}
\label{app:rows}
%=====================================================================

Define
\[
 L^{\mu,I}(s,a)=\Amap^{\alpha\mu,I}\partial_{a^\alpha}R(s-a),
\]
\[
 Q^{L,I,J}(v;a,z)
 =\sum_{\alpha,\beta}q_{\alpha\beta}^{L,I,J}
       \partial_v^\alpha\deltaD^{(4)}(v-z)
       \partial_v^\beta\deltaD^{(4)}(v-a),
\]
\[
 T^{L,I,J}(b;a,z)
 =\PP^L{}_M\int R(b-v)Q^{M,I,J}(v;a,z)\dd^4v.
\]
The source incidence is the submersion
\((v,a,z)\mapsto(v-z,v-a)\).
The rows are
\begin{align*}
 K_{47}^{I,J}
 &=-\partial_{b^\mu}\Big[
 \big(f(b)\partial_{b^\nu}L^{\mu,J}(b,z)
 -\delta^\mu_\nu\partial_{b^\rho}(f(b)L^{\rho,J}(b,z))\big)
                                          L^{\nu,I}(b,a)\Big],\\
 K_{5,+}^{I,J}
 &=\partial_{b^\mu}\Big[
 f(b)\partial_{a^\alpha}R(b-a)\Amap^{\alpha\beta,I}
                            \partial_{a^\beta}L^{\mu,J}(a,z)\Big],\\
 K_{5,-}^{I,J}
 &=\partial_{b^\mu}\Big[
 f(b)\partial_{z^\alpha}R(b-z)\Amap^{\alpha\beta,J}
                            \partial_{z^\beta}L^{\mu,I}(z,a)\Big],\\
 K_6^{I,J}
 &=\partial_{b^\mu}\Big[
 f(b)\Dmap^{\alpha\mu,I,J}
       \partial_{a^\alpha}R(b-a)\deltaD^{(4)}(a-z)\Big],\\
 K_8^{I,J}
 &=-\partial_{b^\mu}\Big[
 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\Big],\\
 K_{9d}^{I,J}
 &=-\partial_{b^\nu}\Big[
 g(b)\partial_{b^\mu}F(b)\Amap^{\mu\nu}{}_L
                                       T^{L,I,J}(b;a,z)\Big],\\
 K_{9m}^{I,J}
 &=-m^2g(b)F(b)\bmap_LT^{L,I,J}(b;a,z).
\end{align*}
The internal \(y\)-slot is the smooth constant one.

For \(D_0=R,B\),
\[
 \int D_0(b-v)Q^{L,I,J}(v;a,z)\dd^4v
 =
 \sum_{\alpha,\beta}(-1)^{|\alpha|+|\beta|}
 \partial_z^\alpha\partial_a^\beta
 [q_{\alpha\beta}^{L,I,J}D_0(b-a)\deltaD^{(4)}(a-z)].
\]
The incidence makes this pushforward locally proper.

The scalar seeds are
\[
 R(b-a)R(b-z),\quad R(b-a)R(a-z),\quad
 R(b-z)R(z-a),\quad R(b-a)\deltaD^{(4)}(a-z).
\]
They are tensor-product pullbacks under invertible coordinate changes,
including the residual tips. Their row derivative orders are at most
\(4,4,4,2,3,3,2\).

Use the charts
\[
 \begin{aligned}
 \kappa_{47}(c,s,t,y)&=(c-s,c,c-t,y),\\
 \kappa_{5,+}(c,s,t,y)&=(c,c+s,c-t,y),\\
 \kappa_{5,-}(c,s,t,y)&=(c-t,c+s,c,y),\\
 \kappa_{\mathrm{resp}}(c,s,r,y)&=(c,c+s,c+r,y),
 \end{aligned}
\]
and fields
\[
 \Db_\mu=\partial_{c^\mu}+\partial_{s^\mu}+\partial_{t^\mu},
 \qquad \Dz_\mu=\partial_{r^\mu},
 \qquad \Da_\mu=\partial_{c^\mu}-\partial_{s^\mu}-\partial_{r^\mu}.
\]
The complete transposed operators are
\begin{align*}
 \SeedOp_{47}^{I,J}W={}&
 -\Amap^{\rho\mu,J}\Amap^{\sigma\nu,I}
 \partial_{s^\sigma}\partial_{t^\nu}\partial_{t^\rho}
                                           (f(c)\Db_\mu W)\\
 &-\delta^\mu_\nu\Amap^{\rho\alpha,J}\Amap^{\sigma\nu,I}
 \partial_{s^\sigma}\partial_{t^\rho}
                              ((\partial_\alpha f)(c)\Db_\mu W)\\
 &+\delta^\mu_\nu\Amap^{\rho\alpha,J}\Amap^{\sigma\nu,I}
 \partial_{s^\sigma}\partial_{t^\alpha}\partial_{t^\rho}
                                           (f(c)\Db_\mu W),\\
 \SeedOp_{5,+}^{I,J}W={}&
 \Amap^{\alpha\beta,I}\Amap^{\rho\mu,J}
 \partial_{s^\alpha}\partial_{t^\beta}\partial_{t^\rho}
                                  (f(c+s)\partial_{s^\mu}W),\\
 \SeedOp_{5,-}^{I,J}W={}&
 \Amap^{\alpha\beta,J}\Amap^{\rho\mu,I}
 \partial_{s^\alpha}\partial_{t^\beta}\partial_{t^\rho}
                                  (f(c+s)\partial_{s^\mu}W),\\
 \SeedOp_6^{I,J}W={}&
 -\Dmap^{\alpha\mu,I,J}
           \partial_{s^\alpha}(f(c+s)\partial_{s^\mu}W),\\
 \SeedOp_8^{I,J}W={}&
 \sum q_{\alpha\beta}^{M,I,J}\PP^L{}_M\Amap^{\rho\mu}{}_L
 \frac{s_\rho}{2}\Dz^\alpha\Da^\beta
                                  (f(c+s)\partial_{s^\mu}W),\\
 \SeedOp_{9d}^{I,J}W={}&
 \sum q_{\alpha\beta}^{M,I,J}\PP^L{}_M\Amap^{\mu\nu}{}_L
 \Dz^\alpha\Da^\beta
               (g(c+s)\partial_\mu F(c+s)\partial_{s^\nu}W),\\
 \SeedOp_{9m}^{I,J}W={}&
 -m^2\sum q_{\alpha\beta}^{M,I,J}\PP^L{}_M\bmap_L
 \Dz^\alpha\Da^\beta(g(c+s)F(c+s)W).
\end{align*}
Every derivative acts on its full indicated parenthesis.

The measures are
\[
 \Irr(V)=\frac1{16\pi^2}\int
 \frac{V(c,(|\mathbf x|,\mathbf x),(|\mathbf x'|,\mathbf x'),y)}
      {|\mathbf x|\,|\mathbf x'|}
       \dd^4c\,\dd^4y\,\dd^3\mathbf x\,\dd^3\mathbf x',
\]
\[
 \Iresp(V)=\frac1{4\pi}\int
 \frac{V(c,(|\mathbf x|,\mathbf x),0,y)}{|\mathbf x|}
                              \dd^4c\,\dd^4y\,\dd^3\mathbf x.
\]
The exact actions are
\[
 \pair{K_j^{I,J}}{W}
 =
 \begin{cases}
 \Irr(\SeedOp_j^{I,J}[W\circ\kappa_j]),&j=47,5+,5-,\\
 \Iresp(\SeedOp_j^{I,J}[W\circ\kappa_{\mathrm{resp}}]),
                                      &j=6,8,9d,9m.
 \end{cases}
\]
They follow by integration by parts against locally finite seed
measures. The two \(K_5\) line signs cancel; the outer derivative
supplies the final transposition sign.
In \(K_6\), the original attachment derivative acts only on the
retarded line. The \(K_8\) multiplier remains outside the
source-transposed derivatives. The derivative-root outer sign
becomes positive, whereas the mass-root sign stays negative.

%=====================================================================
\section{Positive-scale vertex, local equations, and cutoffs}
\label{app:positive}
%=====================================================================

The relative map is
\[
 \fmap(a,b,z,y)=(y-a,y-b).
\]
For smooth \(H\), define
\[
 \Vertex(K,H)^J
 =
 \partial_{y^\nu}\left[g(y)\sum_IK^{I,J}Z_I^\nu(H)\right]
 -\frac{m^2}{2}g(y)\sum_IK^{I,J}M_I(H),
\]
\[
 Z_I^\nu(H)
 =
 \fmap^*\left(
 \partial_{2,\mu}H_I^{\mu\nu}
 -\frac12\eta_{\mu\alpha}\partial_2^\nu H_I^{\mu\alpha}\right),
 \qquad
 M_I(H)=\fmap^*(\eta_{\mu\nu}H_I^{\mu\nu}).
\]
On \(x_1^0>0,x_2^0<0\),
\[
 D_{T,0}(x_1)=W_0(x_1),\qquad D_{T,m}(x_2)=W_m(-x_2),
\]
so the crossed array is \(i\hbar^2N(W_0\otimes\Delta_m)\).
Also
\[
 N_I^{\mu\nu}=(e^I)^{\mu\nu}
              -\frac12\eta^{\mu\nu}\tauI,\qquad
 \eta_{\mu\nu}N_I^{\mu\nu}=-\tauI.
\]
The trace-kinetic terms cancel.
The two factors \(i\) multiply to \(-1\); transposition of the
output derivative yields
\[
 t_\delta^{\App}(\Phi)
 =
 \lambda^3\hbar^2\sum_{I,J}
                  \pair{\Kret^{I,J}}{\mathcal U_\delta^{I,J}[\Phi]}.
\]

At positive scale the convolution is smooth, so all products with
the internal distributions are defined.
If two relative cutoffs equal one near the required compact image,
their difference vanishes on a fixed neighborhood.
For small \(\delta\), every translated smoothing test and every
differentiated-profile test centered near that image stays in the
neighborhood. The crossed jets and complete family therefore agree
exactly at small scales.

The same support argument applies to local differential identities.
For instance,
\[
 (\Box_{\eta,2}+m^2)(\chi D)
 =[(\Box_{\eta,2}+m^2),\chi]D
\]
has support away from the required relative neighborhood.
Its convolution is zero there at sufficiently small scales, proving
the local Klein--Gordon identity used in
Lemma~\ref{lem:positive-reduction}.

All operations act on reconstructed kernels; the identity
\(q\deltaD'(q)+\deltaD(q)=0\) is therefore preserved.
No profile factorization or unrestricted-component positivity is used.

%=====================================================================
\section{Supported output-plateau inverse-quadratic inventory}
\label{app:plateau}
%=====================================================================

This appendix preserves the precise unfinished supported calculation.
It is not needed for the complete-family certificate.

Put
\[
 S=B_2-B_1,\qquad E=A_2-A_1,\qquad
 \kappa=\frac{4\alpha d}{\beta},\qquad
 Z_0=\frac{dZ_1+\alpha Z_2}{\beta},
\]
and write \(\langle P\rangle_\rho=\int\widetilde\rho_{\mathbf n}P\dd H\).
In addition to \(M_2\), define
\[
 B_3=\langle B_2(S-i0)^{-3}\rangle_\rho,\quad
 Z_{0,3}^A=\langle Z_0^A(S-i0)^{-3}\rangle_\rho,\quad
 E_3=\langle E(S-i0)^{-3}\rangle_\rho,
\]
\[
 D_3^A=\langle(Z_1^A-Z_2^A)(S-i0)^{-3}\rangle_\rho,\qquad
 Q_4=\langle|Z_1-Z_2|^2(S-i0)^{-4}\rangle_\rho.
\]
For an amplitude \(A\), evaluate its displayed jets at zero and define
\begin{equation}
\begin{split}
 \Lcontact_\mu[A]={}&
 \ell_\mu\Bigg[
 \left(\frac32M_2-B_3\right)\partial_vA
 -Z_{0,3}^C\partial_{\xi^C}A
 +\frac{\kappa M_2}{2}
 \left(\frac14\Delta_\xi A-\frac1{4d}\partial_vA
                   -\frac{m^2\beta}{4\alpha}A(0)\right)\\
 &\hspace{17mm}
 +\left(\frac{E_3}{\beta}-\frac{3Q_4}{4\beta}\right)A(0)
 \Bigg]
 -\frac{\alpha M_2}{\beta}(E_C)_\mu\partial_{\xi^C}A
 +\frac{D_3^C}{\beta}(E_C)_\mu A(0)
 -\frac{M_2}{2\beta}k_\mu A(0).
\end{split}
 \label{eq:contact-operator}
\end{equation}
Here \(C=1,2\) is summed.

\begin{proposition}[Scalar contact correction]
For the supported derivative profiles,
\[
 v_0^{\nu_{1,\mu}^{\mathrm K}}[A]
 =\frac{\ell_\mu M_2}{64\pi^2\beta}A(0),\qquad
 v_0^{\nu_2^{\mathrm M}}[A]
 =-\frac{M_2}{32\pi^2\beta}A(0),
\]
\[
 v_1^{\nu_{2,\mu}^{\mathrm K}}[A]
 =\frac1{32\pi^2\beta}\Lcontact_\mu[A].
\]
\end{proposition}

\begin{proof}
The first two formulas follow from \eqref{eq:v0}.
For the third, at \(\delta=0\),
\[
 \partial_\delta f_{0,\delta,H}
 =B_2\partial_vA+Z_0^C\partial_{\xi^C}A,
\]
\[
 f_{1,0,H}
 =\frac14\Delta_\xi A-\frac1{4d}\partial_vA
                         -\frac{m^2\beta}{4\alpha}A(0),
\]
where the last term is the massive-tail jet.
Also
\[
 e'_0=E,\qquad
 L'_0=\kappa\left(-\frac{A_1}{\alpha}
                 +\frac{A_2}{d}-\frac{E}{\beta}\right)S
              -\frac{\alpha d}{\beta^2}|Z_1-Z_2|^2.
\]
The finite-part terms of \(V_0'(0)+V_1(0)\) have smoothing
degree one and vanish against the order-three profile.
After deleting the polynomial terms annihilated by that profile,
the remaining pairing is
\[
 \frac1{32\pi^2\beta}\int\widetilde\nu_{2,\mu}^{\mathrm K}
 \left[
 \left(B_2\partial_vA+Z_0^C\partial_{\xi^C}A
       -\frac E\beta A(0)+\kappa Sf_{1,0,H}\right)\log(-S+i0)
 -\frac{|Z_1-Z_2|^2}{4\beta}(S-i0)^{-1}A(0)\right]\dd H.
\]
Use
\[
 \partial_{H_2^\mu}
 =\frac{k_\mu}{2}\partial_{A_2}
  +\frac{\ell_\mu}{2}\partial_{B_2}
  -(E_C)_\mu\partial_{Z_2^C},
 \qquad
 \nu_{2,\mu}^{\mathrm K}
 =\partial_{B_2}^2\partial_{H_2^\mu}\rho.
\]
Three integrations by parts evaluate the pairing as
\eqref{eq:contact-operator}.
\end{proof}

On an output plateau define
\[
 \mathcal H_{\zeta,j,J}^{\nu}
 =
 (\partial_r^\zeta\partial_u^{j-1}
                       \partial_{y^\nu}\Phi_J)(w_0),
 \qquad j=1,2,
\]
\[
 \mathcal B_{\zeta,I,J}^{\nu}(v,\xi)
 =
 (-1)^{|\zeta|}c_{\zeta,2}^{I,J}(\vartheta,v,\xi)
          (\partial_r^\zeta\partial_{y^\nu}\Phi_J)(w(\vartheta;v,\xi)).
\]

\begin{proposition}[Exhaustive supported plateau inventory]
\label{prop:plateau-inventory}
For tests supported in an output plateau,
\begin{equation}
\begin{split}
 A_{-2}^{\mathrm{supp}}(\Phi)
 =\frac1{32\pi^2}\int\frac1\beta\Bigg\{
 &\frac{M_2}{2}
 \sum_{\substack{I,J,\zeta,j\\j=1,2,\ |\zeta|+j\le3}}
 (-1)^{|\zeta|+j}j\,
 c_{\zeta,j}^{I,J}(\vartheta,0,0)
 (e^I)^{\mu\nu}\ell_\mu\mathcal H_{\zeta,j,J}^{\nu}\\
 &+\frac{m^2M_2}{2}
 \sum_{\substack{I,J,\zeta\\|\zeta|\le1}}
 (-1)^{|\zeta|}\tauI c_{\zeta,2}^{I,J}(\vartheta,0,0)
                            (\partial_r^\zeta\Phi_J)(w_0)\\
 &+\sum_{\substack{I,J,\zeta,\mu,\nu\\|\zeta|\le1}}
 (e^I)^{\mu\nu}\Lcontact_\mu[\mathcal B_{\zeta,I,J}^{\nu}]
 \Bigg\}\dd\vartheta.
\end{split}
 \label{eq:plateau-inventory}
\end{equation}
\end{proposition}

\begin{proof}
Inverse-quadratic supported order consists exactly of
\(q=2,n=0\) and \(q=3,n=1\) in
\eqref{eq:supported-coefficients}.
These are respectively the \(b=1\) kinetic terms, the \(b=2\)
mass terms, and the first correction of the \(b=2\) kinetic terms.
Insert the preceding scalar contact formulas with the signs and
binomial factors in \eqref{eq:supported-amplitudes}.
\end{proof}

\scopebox{Unfinished supported tensor sum}{%
The lower coefficients \(c_{\zeta,1}^{I,J}\) and their combined
kinetic contractions still have to be evaluated from
\eqref{eq:normal-extraction}. Formula~\eqref{eq:plateau-inventory}
is an exhaustive local recipe, not a proof of cancellation or
nonvanishing of the general supported plateau coefficient.

The new complete-family failure theorem uses the evaluated isolated
positive-placement coefficient instead, and does not silently replace
this remaining task.}

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

\bibitem{EmF}
\emph{Em01-F-v3: Constant-Weight Einstein--Scalar Substitution
and Exact Positive-Scale Regrouping}.
Unpublished supplied manuscript.
All frozen arrays and rows used here are repeated in the appendices.

\bibitem{EmG4}
\emph{Em01-G-v4.tex}.
Supplied predecessor attachment, internally titled
``Em01-G-v3.1: Supported-Sector Asymptotics and Non-Tip
Residual-Cone Reductions.''
This revision synchronizes the identity, corrects parameter
bookkeeping, sharpens the crossed derivative count, and evaluates
an isolated positive-placement coefficient.

\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.
Proposition~5.1 treats continuous pullback in the normal topology;
Theorem~5.9 treats parameter families of pullbacks;
Theorem~6.1 treats hypocontinuous multiplication under nonopposition.
These results are not substitutes for the radial remainder proved here.

\bibitem{Dang}
N.~V.~Dang,
\emph{The extension of distributions on manifolds,
a microlocal approach},
arXiv:1412.2808v1 (2014).
Cited to distinguish scaling-based extension hypotheses from evaluation
of the chosen raw regularization.

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