% XVI-K.tex
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\documentclass[11pt,letterpaper]{article}

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\newcommand{\R}{\mathbb R}
\newcommand{\C}{\mathbb C}
\newcommand{\Dprime}{\mathcal D'}
\newcommand{\Eprime}{\mathcal E'}
\newcommand{\ii}{\mathrm i}
\newcommand{\dd}{\,\mathrm d}
\newcommand{\Pc}{\mathcal P}
\newcommand{\RK}{\mathfrak R_{\mathrm K}}
\newcommand{\Edge}{\mathscr S}
\newcommand{\sk}{\mathsf k}
\newcommand{\Proj}{\mathsf W}
\newcommand{\Pair}[2]{\left\langle #1,#2\right\rangle}

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\newtheorem{corollary}[proposition]{Corollary}

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\newtheorem{definition}[proposition]{Definition}
\newtheorem{assumption}[proposition]{Assumption}

\theoremstyle{remark}
\newtheorem{remark}[proposition]{Remark}
\newtheorem{verification}[proposition]{Computational record}

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

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  frame=single
}

\title{Landmark XVI-K\\[0.3em]
\large The full-Hessian vertex within the selected\\
root-attached \(C_2\) two-edge sector}
\author{}
\date{}

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

\begin{abstract}
We extend the constant-slot construction of XVI-J by retaining
the full second variation of the declared unintegrated cubic
density, including derivatives on both contracted tensor slots.
The source remains restricted to the polarization of
\(2C_2\,\partial\partial\sigma_\varepsilon\); the plateau source
and the \(M_2\) summand are not included.

The density's jet structure proves that the numerical edge
products contain at most two derivatives in total. Their
punctured distributions are homogeneous of degrees
\(-4,-5,-6\), and admit extensions with subtraction orders
\(0,1,2\). A single subtraction function and grade-dependent
zero moments define a linear extension map on their
finite-dimensional span. Constant-coefficient relations are
preserved, while the possible defects of differentiation are
retained as explicit contact distributions.

Under the imported compactness and \(Z_2\) wavefront hypotheses
for the root, pullback, multiplication, and proper pushforward
construct a jointly continuous bilinear selected-sector
coefficient. Its constant-slot truncation agrees with XVI-J,
including the chosen extension at coincidence. Changes of
prescription propagate derivatives of diagonal contact terms
through the root.

The analytic existence proof does not depend on the supplied
numerical coefficient tables. Those tables are documented
separately as exact-arithmetic computational verification.
The result is not a complete interaction coefficient, a
retarded interacting field, or a BRST/BV observable.
\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) {Selected two-edge sector};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Full Hessian of that sector};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Complete interaction};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {Full means the full second variation of the fixed density, not the complete interaction.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{The full Hessian stays inside one sector. 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-XVI-K}
\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) {Selected two-edge sector};
\node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {Complete interaction};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {Full means the full second variation of the fixed density, not the complete interaction.};
\draw[->] (a) -- (s);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{The full Hessian stays inside one sector. 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-XVI-K}
\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) {Selected two-edge sector};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Full Hessian of that sector};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {Complete interaction};
\node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {Full means the full second variation of the fixed density, not the complete interaction.};
\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-XVI-K}
\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) {Selected two-edge sector};
\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) {Complete interaction};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Full Hessian of that sector. Full means the full second variation of the fixed density, not the complete interaction.};
\draw (a) -- (m);
\draw (m) -- (b);
\draw[->] (m) -- (s);
\end{tikzpicture}
\caption{Selected two-edge sector is not Complete interaction. 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-XVI-K}
\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: Selected two-edge sector};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Full Hessian of that sector};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: Complete interaction};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {Full means the full second variation of the fixed density, not the complete interaction.};
\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-XVI-K}
\end{figure}

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


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



\scopebox{Scope.}{%
Here ``full'' means the \emph{full second variation of the fixed
unintegrated cubic density}, not the complete interaction.
All derivative placements on its two contracted slots are
retained. The overall sector still contains only the \(C_2\)
source and two graviton edges.

The preparation width \(0<\varepsilon<T\), preparation data
\(f,g\), and ramp \(\sigma_\varepsilon\) are fixed.
The independent interaction switch is
\(\chi_{\mathrm I}\in C_c^\infty(M;\R)\).

The five levels are the Hessian coefficients, punctured scalar
kernels, their selected extensions, smooth source-decorated
coefficients, and the root-attached functional. Statements at
one level do not imply statements at another.

No sharp-width limit, state, all-orders time ordering,
ultraviolet completion, or interacting gauge identity is
constructed.
}

\section{Setting, explicit interfaces, and normalization}
\label{sec:k0}

\subsection{Conventions}

Work on ambient Minkowski space \(M=\R^4\), with
\[
\eta=\operatorname{diag}(1,-1,-1,-1),
\qquad
\Box=\partial_t^2-\Delta.
\]
All indices are moved with \(\eta\), and parentheses on indices
include the factor \(1/2\). The symmetric-tensor fiber is
\[
V=\operatorname{Sym}^2(\R^4)^*,
\qquad
b(A,C)=A_{\mu\nu}C^{\mu\nu}.
\]
All tensor and distributional pairings are bilinear, with no
complex conjugation.

Trace reversal and the free tensor operator are
\[
\Pc A=A-\tfrac12\eta\,\tr_\eta A,
\qquad
\Pc^2=\id,
\qquad
K=-\Box\Pc.
\]

For multi-indices \(p,q,\gamma\in\mathbb N^4\),
\[
\partial^p=\partial_0^{p_0}\partial_1^{p_1}
           \partial_2^{p_2}\partial_3^{p_3},
\qquad
|p|=\sum_{\alpha=0}^3p_\alpha.
\]
The zero multi-index is denoted by \(0\), and \(e_\alpha\) is
the \(\alpha\)-th unit multi-index.

The notation is deliberately separated:
\[
\Edge(z)\quad\text{is the scalar edge distribution},\qquad
\Sigma_{\alpha\beta}(y)
=\partial_\alpha\partial_\beta\sigma_\varepsilon(y)
\quad\text{is the preparation jet}.
\]
A positive dilation parameter is written \(a>0\). It is not
a witness frequency. The scalar configuration is
\(\varphi\), and a compact distributional test is usually
\(\psi\).

\subsection{Inherited interfaces}

\begin{assumption}[Fixed preparation and selected source]
\label{ass:preparation}
Fix \(0<\varepsilon<T\), preparation data \(f,g\), and a smooth
time-only ramp satisfying
\[
\sigma_\varepsilon(t)=0\quad(t\le0),
\qquad
\sigma_\varepsilon(t)=1\quad(t\ge\varepsilon).
\]
Fix independently \(\chi_{\mathrm I}\in C_c^\infty(M;\R)\).

The supplied XII-C source interface is
\[
\mathcal J_{\mathrm D}^{\mu\nu}
=\sigma_\varepsilon J_3^{\mu\nu}
 +2C_2^{\mu\nu\alpha\beta}
       \partial_\alpha\partial_\beta\sigma_\varepsilon
 -2M_2^{\mu\nu\rho}\partial_\rho\sigma_\varepsilon.
\]
Only the middle summand is retained here. Its finite algebraic
definition is given in \eqref{eq:C2}--\eqref{eq:source}.
The identification with that summand is inherited; no audit
of the complete cubic Euler source is claimed.
\end{assumption}

\begin{assumption}[Imported root]
\label{ass:root}
The root \(B_{f,g,\mu\nu}(y,x)\) satisfies:
\begin{enumerate}[label=\textup{(R\arabic*)}]
\item Its designated composition exists as a tensor-valued
      distribution on \(M_y\times M_x\).
\item Its components have compact support:
      \(B_{f,g,\mu\nu}\in\Eprime(M_y\times M_x)\).
\item Componentwise, in the fixed Cartesian trivialization,
\begin{equation}
\WF(B_{f,g,\mu\nu})
\subset
Z_2
:=
\{(y,x;\xi,-\xi):\xi\ne0\}.
\label{eq:rootinput}
\end{equation}
\end{enumerate}
These are the root-composition hypotheses imported from
XII-D, version 1, \texttt{lem:root-kernel}, as retained by
XVI-J. They are assumed, not derived in this manuscript.
The root's singular support is not removed.
\end{assumption}

The supplied root is described by the inherited composition
\[
B_{f,g}(y,x)
=
-\vartheta(y^0)
 \int E_0^{\mathrm A}(y,z)\Pc A_{f,g}(z,x)\dd^4z,
\]
where \(\vartheta\) is the compact ambient time collar.
This formula records provenance only. Its internal variable
\(z\) and the distribution \(A_{f,g}\) are not reopened in the
proof. The collar \(\vartheta\) is not the Heaviside function
\(\theta\).

\begin{assumption}[Insertion polarization and graph convention]
\label{ass:normalization}
Use the ordinary second tensor derivative of the supplied
quadratic insertion:
\[
F_t(h,\varphi)
=\Pair{t_{\mathrm D}}{h\otimes h\otimes\varphi},
\qquad
D_h^2F_t[A,C]
=2\Pair{t_{\mathrm D}}{A\otimes C\otimes\varphi}.
\]
Use the symmetric quadratic source polarization
\[
\mathfrak s(X,Y)
=\tfrac12\bigl(Q(X+Y)-Q(X)-Q(Y)\bigr),
\]
and count the symmetry factor \(1/2!\) for the two identical
edges once. These conventions match the retained XVI-J
sector. Coupling and loop weights are not included in the
coefficient defined below.
\end{assumption}

\begin{center}
\begin{tabularx}{\linewidth}{@{}Y Y@{}}
\toprule
Included & Excluded\\
\midrule
Polarized \(2C_2\,\partial\partial\sigma_\varepsilon\)
 & Plateau source \(\sigma_\varepsilon J_3\)\\
Full Hessian of the declared cubic density
 & Source term \(-2M_2\,\partial\sigma_\varepsilon\)\\
All vertex-slot derivative placements of total order at most two
 & Additional interaction vertices or contraction types\\
One coherent finite extension family
 & A complete time-ordered or retarded-product family\\
\bottomrule
\end{tabularx}
\end{center}

\subsection{Ambient scalar edge}

Define
\begin{equation}
W_0(z)
=
\int_{\R^3}
 \frac{e^{-\ii|\mathbf p|z^0+\ii\mathbf p\cdot\mathbf z}}
      {(2\pi)^3\,2|\mathbf p|}\dd^3\mathbf p,
\qquad
h_0=-W_0.
\label{eq:W0}
\end{equation}
The integral is distributional. The ambient advanced inverse
of \(\Box\) is specified by the spatial Fourier multiplier
\[
\widehat E_0^{\mathrm A}(t,\mathbf p)
=-\theta(-t)\frac{\sin(t|\mathbf p|)}{|\mathbf p|},
\]
with the continuous value of the quotient at zero momentum.
Equivalently, for compact sources,
\[
(E_0^{\mathrm A}F)(t)
=\int_t^\infty
 \frac{\sin((s-t)\sqrt{-\Delta})}{\sqrt{-\Delta}}
 F(s)\dd s.
\]
For sources compactly supported in the slab interior, this
restricts to the supplied upper limit \(T\). The graph edges
use the ambient kernel, not an arbitrary slab operator.

Set
\begin{equation}
\Edge=-W_0+\ii E_0^{\mathrm A},
\qquad
d=W_0-\ii E_0^{\mathrm A}=-\Edge,
\qquad
D_F=-\Pc\Edge=\Pc d.
\label{eq:S}
\end{equation}
The scalar \(d\) is the massless positive-frequency
time-ordered kernel in this normalization. It is the scalar
kernel denoted by \(\Delta_F^{\mathrm{FR}}\) in XVI-J.

\begin{lemma}[Global definition and inverse normalization]
\label{lem:families}
For
\[
\widetilde\psi(t,\mathbf p)
=\int_{\R^3}e^{\ii\mathbf p\cdot\mathbf z}
                       \psi(t,\mathbf z)\dd^3\mathbf z,
\]
the two summands are defined by
\begin{align}
\Pair{W_0}{\psi}
&=
\int_{\R}\dd t
\int_{\R^3}
 \frac{e^{-\ii|\mathbf p|t}\widetilde\psi(t,\mathbf p)}
      {(2\pi)^3\,2|\mathbf p|}\dd^3\mathbf p,
\label{eq:W-fam}\\
\Pair{E_0^{\mathrm A}}{\psi}
&=
\int_{-\infty}^0\dd t
\int_{\R^3}
 \frac{\sin(-t|\mathbf p|)}
      {(2\pi)^3|\mathbf p|}
 \widetilde\psi(t,\mathbf p)\dd^3\mathbf p.
\label{eq:E-fam}
\end{align}
Both pairings converge absolutely. Moreover,
\[
\Box\Edge=\ii\delta,
\qquad
D_F^{\mathsf T}=D_F,
\qquad
KD_F=\ii\,\id.
\]
\end{lemma}

\begin{proof}
The partial Fourier transform of a compact smooth test is
rapidly decreasing in \(\mathbf p\), uniformly for \(t\) in a
compact interval. The weight \(1/|\mathbf p|\) is locally
integrable in three dimensions, and
\[
\left|\frac{\sin(-t|\mathbf p|)}{|\mathbf p|}\right|
\le |t|.
\]
This proves absolute convergence. The time restriction in
\eqref{eq:E-fam} is a restriction of this integral; it does
not require multiplying distributions at coincidence.

The temporal Fourier families of \(d\) are
\[
\widehat d(t,\mathbf p)=
\begin{cases}
e^{-\ii|\mathbf p|t}/(2|\mathbf p|),&t>0,\\
e^{\ii|\mathbf p|t}/(2|\mathbf p|),&t<0.
\end{cases}
\]
For \(t<0\), this follows from
\(e^{-\ii\omega t}+2\ii\sin(\omega t)=e^{\ii\omega t}\).
The jump of its temporal derivative is \(-\ii\), so
\(\Box d=-\ii\delta\) and \(\Box\Edge=\ii\delta\).
The two branches, together with spatial parity, give symmetry
under \(z\mapsto-z\).
Finally,
\[
(-\Box\Pc)(-\Pc\Edge)=\Box\Edge=\ii\,\id.
\]
\end{proof}

Neither \(\Edge\) nor \(D_F\) is an object of the later
extension problem. They are already global distributions.

\section{The declared density and its full jet Hessian}
\label{sec:k1}

\subsection{Finite density definition}

For a symmetric tensor \(\gamma\), put
\[
\tau=\tr_\eta\gamma,\qquad
u_0^{\mu\nu}=\eta^{\mu\nu},\qquad
u_1^{\mu\nu}=-\gamma^{\mu\nu},\qquad
u_2^{\mu\nu}=\gamma^\mu{}_\lambda\gamma^{\lambda\nu},
\]
\[
b_1=\tfrac12\tau,\qquad
b_2=\tfrac18\tau^2-\tfrac14\gamma_{\mu\nu}\gamma^{\mu\nu}.
\]
For \(n=1,2,3\), define
\begin{align}
\Gamma_n^\rho{}_{\mu\nu}
&=
\tfrac12u_{n-1}^{\rho\lambda}
 \bigl(\partial_\mu\gamma_{\nu\lambda}
      +\partial_\nu\gamma_{\mu\lambda}
      -\partial_\lambda\gamma_{\mu\nu}\bigr),
\label{eq:Gamma}\\
R_{n,\mu\nu}
&=
\partial_\rho\Gamma_n^\rho{}_{\mu\nu}
-\partial_\nu\Gamma_n^\rho{}_{\mu\rho}
\nonumber\\
&\quad+
\sum_{\substack{a+b=n\\a,b\ge1}}
 \bigl(
 \Gamma_a^\rho{}_{\rho\lambda}
          \Gamma_b^\lambda{}_{\mu\nu}
 -\Gamma_a^\rho{}_{\nu\lambda}
          \Gamma_b^\lambda{}_{\mu\rho}
 \bigr),
\label{eq:Ricci}\\
r_n&=\sum_{i=0}^{n-1}u_i^{\mu\nu}R_{n-i,\mu\nu},
\qquad
\boxed{\ell_3=-2(r_3+b_1r_2+b_2r_1).}
\label{eq:ell3}
\end{align}
These are the supplied XII-C unintegrated density atoms.
They specify the density itself, not merely its Euler
equivalence class.

The interaction used in this sector is
\[
V_{\mathrm I}(h)
=\int_M\chi_{\mathrm I}(w)\ell_3(h)(w)\dd^4w.
\]
No integration by parts changes this density representative.
The usual definition of distributional derivatives used later
does not change that convention.

\subsection{Basis and independent jets}

Let \(B_0,\ldots,B_9\) be the constant symmetric matrices in
the order
\[
(00),(01),(02),(03),(11),(12),(13),(22),(23),(33).
\]
A diagonal basis matrix has its indicated entry equal to one.
An off-diagonal matrix has both symmetric entries equal to
one. Thus
\[
A=\sum_i A^iB_i
\quad\Longrightarrow\quad
A^i=A_{ab}\quad\text{for }i=(ab).
\]

Second jets are indexed by multi-indices, not by a redundant
ordered pair of derivative indices. In particular, a mixed
second derivative is one independent jet coordinate.

\begin{proposition}[Structure of the full-Hessian vertex]
\label{prop:hess}
For the declared unintegrated density, there are unique
coefficients \(L^{pq}_{ij}(w;h)\) such that
\begin{equation}
D^2\ell_3(h)[A,C](w)
=
\sum_{i,j=0}^9
\sum_{|p|+|q|\le2}
 L^{pq}_{ij}(w;h)
 \,\partial^pA^i(w)\,\partial^qC^j(w).
\label{eq:Lpq}
\end{equation}
The identity is an identity in independent field jets, before
any integration by parts. Its properties are:
\begin{enumerate}
\item Each \(L^{pq}_{ij}\) is linear, with constant rational
      coefficients, in residual jets of order exactly
      \(2-|p|-|q|\).
\item Every nonzero monomial of the full Hessian has total
      derivative order two over \(h,A,C\).
\item The slot symmetry is
      \(L^{pq}_{ij}=L^{qp}_{ji}\).
\item The constant-slot coefficient is
\[
L^{00}_{ij}(w;h)
=D^2\ell_3(h)[B_i,B_j](w)=L_{ij}(w;h),
\]
      the constant-slot form retained by XVI-J.
\item The original vertex decomposition contains no derivative
      on \(\chi_{\mathrm I}\).
\end{enumerate}
These structural claims do not depend on a numerical
coefficient list.
\end{proposition}

\begin{proof}
The atoms \(u_i,b_i\) have field degree \(i\) and derivative
order zero. The atom \(\Gamma_n\) has field degree \(n\) and
total derivative order one. Both \(\partial\Gamma_n\) and the
products in \(R_n\) have field degree \(n\) and derivative
order two. Hence \(r_n\) has field degree \(n\), and each term
of \(r_3+b_1r_2+b_2r_1\) is cubic with total derivative
order two.

The mixed variation of a cubic monomial replaces two field-jet
factors by the corresponding \(A\)- and \(C\)-jets and leaves
one residual \(h\)-jet. It preserves the total derivative
count. The residual jet therefore has order
\(2-|p|-|q|\), proving the first two statements and the range
in \eqref{eq:Lpq}.

Uniqueness follows from independence of the symmetric field
components and their multi-index jets. Symmetry of the mixed
variation gives the third statement. Constant increments
eliminate all terms with \(|p|+|q|>0\), proving the fourth.
The fifth follows because the switch multiplies the density
and is not differentiated in this representation.
\end{proof}

For a fully finite specification, let
\(\mathscr L((g_{i,r})_{|r|\le2})\) be the jet polynomial
obtained from \eqref{eq:Gamma}--\eqref{eq:ell3}. Then
\begin{equation}
L^{pq}_{ij}(w;h)
=
\left.
\frac{\partial^2\mathscr L}
     {\partial g_{i,p}\,\partial g_{j,q}}
\right|_{g_{a,r}=\partial^rh^a(w)}.
\label{eq:jetdefinition}
\end{equation}
This specifies every coefficient without an external table.
No extra factorial is associated with a mixed derivative jet.

\subsection{An analytic differentiated-slot check}

\begin{lemma}[Pure-trace full Hessian]
\label{lem:tracecheck}
For functions depending only on \(s=w^1\),
\[
\ell_3(\psi\eta)=-3\psi(\psi')^2.
\]
For \(h=\psi\eta\), \(A=u\eta\), and \(C=v\eta\),
\begin{equation}
D^2\ell_3(h)[A,C]
=
-6\bigl(u\psi'v'+v\psi'u'+\psi u'v'\bigr).
\label{eq:tracehessian}
\end{equation}
On \(\psi=s^2\), this becomes
\[
-12s(uv'+vu')-6s^2u'v'.
\]
\end{lemma}

\begin{proof}
Substitution in the finite atoms gives
\[
r_1=3\psi'',\quad
r_2=-6\psi\psi''-\tfrac32(\psi')^2,\quad
r_3=9\psi^2\psi''+\tfrac92\psi(\psi')^2,
\]
with \(b_1=2\psi\) and \(b_2=\psi^2\).
The \(\psi^2\psi''\) terms cancel in
\(r_3+b_1r_2+b_2r_1\), leaving
\(\tfrac32\psi(\psi')^2\).
This proves the density formula.
Taking the mixed variation of
\(-3(\psi+\epsilon u+\delta v)
(\psi'+\epsilon u'+\delta v')^2\)
proves \eqref{eq:tracehessian}.
\end{proof}

This is a hand-derived check of differentiated slots and their
signs. It does not verify all general component entries.

The general coefficient list and numerical regressions are
documented in Appendix~\ref{app:script} as computational
verification. They are not premises of the analytic
existence theorem.

\section{The selected source and differentiated graph kernels}
\label{sec:k2}

\subsection{Self-contained \(C_2\) source definition}

For a symmetric matrix \(H\), evaluate \(u_0,u_1,u_2,b_1,b_2\)
on \(H\), and introduce the formal polynomial
\[
U(\upsilon)=u_0+\upsilon u_1+\upsilon^2u_2.
\]
Define
\begin{equation}
C_2^{\mu\nu\alpha\beta}(H)
=
[\upsilon^2]\,
(1+\upsilon b_1+\upsilon^2b_2)
\left(
 U^{\mu\nu}(\upsilon)U^{\alpha\beta}(\upsilon)
 -U^{\mu(\alpha}(\upsilon)U^{\beta)\nu}(\upsilon)
\right).
\label{eq:C2}
\end{equation}
Coefficient extraction in \eqref{eq:C2} is a finite polynomial
operation; it is the retained XII-C coefficient.

For a symmetric covariant jet \(\Sigma\), put
\begin{equation}
\begin{aligned}
Q(H;\Sigma)^{\mu\nu}
&=2C_2^{\mu\nu\alpha\beta}(H)\Sigma_{\alpha\beta},\\
\mathfrak s(X,Y;\Sigma)
&=\tfrac12\bigl(Q(X+Y;\Sigma)-Q(X;\Sigma)-Q(Y;\Sigma)\bigr).
\end{aligned}
\label{eq:source}
\end{equation}
This polarization has no derivatives on \(X,Y\).
The actual jet is always
\(\Sigma(y)=\partial\partial\sigma_\varepsilon(y)\).

\subsection{Why the edge uses the dual basis}

\begin{lemma}[Inverse fiber pairing]
\label{lem:dual}
The Gram matrix and its inverse are
\[
G=\operatorname{diag}(1,-2,-2,-2,1,2,2,1,2,1),
\]
\[
G^{-1}
=\operatorname{diag}
 \left(1,-\tfrac12,-\tfrac12,-\tfrac12,
       1,\tfrac12,\tfrac12,1,\tfrac12,1\right).
\]
For \(B^i=(G^{-1})^{ij}B_j\), one has
\[
b(B^i,B_j)=\delta^i_j.
\]
After removal of its scalar factor \(\Edge\), the edge fiber
tensor is
\begin{equation}
\mathsf E
=(-\Pc\otimes\id)b^{-1}
=\sum_i Z_i\otimes B_i,
\qquad Z_i=-\Pc B^i.
\label{eq:edgefiber}
\end{equation}
\end{lemma}

\begin{proof}
A diagonal basis element has squared pairing one.
An off-diagonal \(B_{(ab)}\) contributes two entries, giving
\(2\eta^{aa}\eta^{bb}\). Distinct basis matrices have disjoint
component support. This proves the Gram matrix.
The inverse-pairing identity
\(b^{-1}=\sum_iB^i\otimes B_i\) then gives
\eqref{eq:edgefiber}.
\end{proof}

The Hessian slots and source polarization are covariant
bilinear forms on field directions. Each edge contracts their
covector slots using the inverse fiber pairing.
Thus the source receives \(Z_i=-\Pc B^i\), while the vertex
is expanded on \(B_i\). The field basis is not self-dual.

\subsection{Relative derivatives and punctured products}

Put \(M_\times=\R^4\setminus\{0\}\).
The standard off-diagonal Feynman wavefront relation gives,
at each singular \(z\in M_\times\), only one time orientation
for the nonzero covectors of \(\Edge\).
This is the relation stated after equation (77) in
\cite[Section 7]{FR}; see also
\cite[Proposition 26]{BDH}.
No wavefront statement at \(z=0\) is needed here.

\begin{lemma}[Products and derivative signs]
\label{lem:sign}
For all multi-indices \(p,q\),
\begin{equation}
q^\circ_{pq}
=(\partial^p\Edge)(\partial^q\Edge)
\in\Dprime(M_\times)
\label{eq:qpq}
\end{equation}
exists, and \(q^\circ_{pq}=q^\circ_{qp}\).

For \(r_2(y,w)=y-w\) and any distribution \(u\) on the
relative space,
\begin{equation}
\partial_w^p(r_2^*u)
=(-1)^{|p|}r_2^*(\partial^pu),
\qquad
\partial_y^p(r_2^*u)=r_2^*(\partial^pu).
\label{eq:sign}
\end{equation}
Consequently, on \(y\ne w\),
\[
\partial_w^p\Edge(y-w)\,
\partial_w^q\Edge(y-w)
=(-1)^{|p|+|q|}q^\circ_{pq}(y-w).
\]
\end{lemma}

\begin{proof}
Derivatives do not enlarge wavefront sets. At a nonzero null
point, two nonzero covectors in the relevant wavefront fibers
have the same time orientation, and hence cannot sum to zero.
At nonsingular points there is no obstruction.
H\"ormander's product criterion therefore defines
\eqref{eq:qpq}; see \cite[Theorem 8.2.10]{Hormander} and
\cite[Theorem 13]{BDH}. That product is commutative.

The relative map is a linear submersion. In coordinates
\((y,z)\), \(z=y-w\), its pullback is a distribution constant
in the extra \(y\)-variable. Differentiation at fixed \(y\)
therefore gives \(\partial_w=-\partial_z\), proving the
distributional identities. Pullback by a submersion is
compatible with the products when their wavefront criterion
holds, yielding the final assertion.
\end{proof}

For completeness, the use of the relative-space wavefront
relation can be justified from the two-point relation:
the two-point kernel is the pullback by \((x,y)\mapsto x-y\).
Its wavefront covectors have the shape \((k,-k)\).
Restriction to \((x,y)=(z,0)\) is permitted because the
restriction conormal has shape \((0,k')\), which cannot meet
that set for nonzero \(k\).
The quoted off-diagonal orientation therefore transfers to
\(\Edge(z)\).

\subsection{Coefficients and unordered placements}

\begin{definition}[Source-decorated coefficients]
\label{def:a}
For \(|p|+|q|\le2\), set
\begin{align}
\mathcal K^{\mu\nu}_{pq}(y,w;h)
&=
\sum_{i,j=0}^9
 L^{pq}_{ij}(w;h)\,
 \mathfrak s^{\mu\nu}(Z_i,Z_j;\Sigma(y)),
\label{eq:Kpq}\\
a^{\mu\nu}_{pq}(y,w;h)
&=
(-1)^{|p|+|q|}
 \chi_{\mathrm I}(w)\mathcal K^{\mu\nu}_{pq}(y,w;h).
\label{eq:apq}
\end{align}
The interaction switch appears exactly once in \(a_{pq}\).
\end{definition}

Slot symmetry and symmetry of \(\mathfrak s\) imply
\(\mathcal K_{pq}=\mathcal K_{qp}\).
Let \(\mathcal N\) be the set of unordered placements
\(\sk=\{p,q\}\) with \(|p|+|q|\le2\), and put
\begin{equation}
a^{\mu\nu}_{\sk}
=
\begin{cases}
a^{\mu\nu}_{pp},&p=q,\\
2a^{\mu\nu}_{pq},&p\ne q,
\end{cases}
\qquad
q^\circ_{\sk}=q^\circ_{pq},
\qquad
N_{\sk}=|p|+|q|.
\label{eq:ak}
\end{equation}
There are \(25\) unordered labels. This is an inventory of
spanning labels, not a claim of linear independence of the
scalar distributions.

\begin{proposition}[Full-Hessian vertex within the \(C_2\) sector]
\label{prop:QK}
The selected off-diagonal two-edge distribution is
\begin{equation}
\begin{aligned}
Q_{\mathrm K}^{\circ,\mu\nu}(y,w;h)
&=
\sum_{|p|+|q|\le2}
 a^{\mu\nu}_{pq}(y,w;h)q^\circ_{pq}(y-w)\\
&=
\sum_{\sk\in\mathcal N}
 a^{\mu\nu}_{\sk}(y,w;h)q^\circ_{\sk}(y-w).
\end{aligned}
\label{eq:QK}
\end{equation}
It contains the full Hessian of the declared density, but only
the \(C_2\) source. Its overall numerical factor is \(2/2!=1\).
It is invariant under constant changes of fiber basis.
\end{proposition}

\begin{proof}
Contract two copies of \eqref{eq:edgefiber} with the source
polarization and the full jet Hessian. The vertex-end
derivatives fall on the scalar edge factors at \(w\).
Lemma~\ref{lem:sign} produces the sign in
\eqref{eq:apq}; the remaining fiber contraction is
\eqref{eq:Kpq}.

Assumption~\ref{ass:normalization} supplies the insertion
factor two and the single identical-edge factor \(1/2!\).
The change of the vertex does not alter either factor.
Grouping the two equal ordered terms when \(p\ne q\) gives
\eqref{eq:ak}.

For a constant basis change
\(\widetilde B_i=B_jA_{ji}\), the dual vectors transform as
\(\widetilde B^i=(A^{-1})_{ij}B^j\), while each Hessian matrix
transforms as
\(\widetilde L^{pq}=A^{\mathsf T}L^{pq}A\).
The two inverse matrices in the source slots cancel the two
matrices in the Hessian. Equivalently, the contraction is
defined intrinsically by the fiber tensor
\eqref{eq:edgefiber}.
\end{proof}

This proposition does not evaluate a smooth Hessian directly
on singular fields without justification: the finite jet
formula specifies the distributional expression, and
Lemma~\ref{lem:sign} establishes every required punctured
product.

The numerical kernels are translation invariant. Their
coefficients generally depend on \(y,w\) separately.
The source jet depends on \(y\); the residual field and
interaction switch depend on \(w\).

\section{Homogeneity, inventory, and the finite graded span}
\label{sec:k3}

For a distribution \(t\), write
\[
t^{[a]}(\psi)=t(\psi_a),
\qquad
\psi_a(z)=a^{-4}\psi(z/a),
\qquad a>0.
\]
The scaling degree at zero is
\[
\sd(t)=\inf\{\omega:
          a^\omega t^{[a]}\longrightarrow0\text{ as }a\downarrow0\}.
\]
On the punctured domain, tests are restricted accordingly.
This is the scaling-degree convention of
\cite[Section 5.1]{BF}.

\begin{lemma}[Homogeneity from the Fourier families]
\label{lem:homog}
For positive dilations,
\[
W_0^{[a]}=a^{-2}W_0,\qquad
(E_0^{\mathrm A})^{[a]}=a^{-2}E_0^{\mathrm A},
\]
and therefore
\[
\Edge^{[a]}=a^{-2}\Edge,\qquad
(\partial^p\Edge)^{[a]}
=a^{-2-|p|}\partial^p\Edge.
\]
\end{lemma}

\begin{proof}
For the scaled test,
\[
\widetilde{\psi_a}(t,\mathbf p)
=a^{-1}\widetilde\psi(t/a,a\mathbf p).
\]
In \eqref{eq:W-fam}, substitute \(t=au\),
\(\mathbf p=\mathbf k/a\). The time and momentum measures,
the factor \(1/|\mathbf p|\), and the test contribute
\(a\), \(a^{-3}\), \(a\), and \(a^{-1}\), respectively.
Their product is \(a^{-2}\).
The same substitution in \eqref{eq:E-fam} preserves \(t<0\)
and turns its sine quotient into
\(a\sin(-u|\mathbf k|)/|\mathbf k|\).
It gives the same power.
The derivative assertion follows from
\[
(\partial^pt)^{[a]}=a^{-|p|}\partial^p(t^{[a]}).
\]
\end{proof}

\begin{proposition}[Numerical scaling and declared subtraction orders]
\label{prop:sd}
For every \(\sk=\{p,q\}\in\mathcal N\),
\begin{equation}
(q^\circ_{\sk})^{[a]}
=a^{-4-N_{\sk}}q^\circ_{\sk},
\qquad
\sd(q^\circ_{\sk})\le4+N_{\sk}\le6.
\label{eq:sdbound}
\end{equation}
For a nonzero kernel the scaling degree is exactly
\(4+N_{\sk}\). The declared subtraction order is always
\[
N_{\sk}=|p|+|q|\in\{0,1,2\}.
\]
For a zero kernel, this declared upper bound is retained and
its selected extension is zero; its scaling degree is not
assigned a finite value.
\end{proposition}

\begin{proof}
Dilations are diffeomorphisms of the punctured domain and
commute with the defined distributional product. Apply
Lemma~\ref{lem:homog} to its two factors.
A nonzero homogeneous distribution of degree
\(-4-N_{\sk}\) has scaling degree \(4+N_{\sk}\).
The zero distribution satisfies the upper bound.
\end{proof}

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}Y c c c c c@{}}
\toprule
Type & Unordered & Ordered & Degree & \(N_{\sk}\)
 & Residual jet\\
\midrule
\(\Edge^2\) & 1 & 1 & \(-4\) & 0 & \(\partial^2h\)\\
\(\Edge\,\partial_\alpha\Edge\)
 & 4 & 8 & \(-5\) & 1 & \(\partial h\)\\
\(\partial_\alpha\Edge\,\partial_\beta\Edge\)
 & 10 & 16 & \(-6\) & 2 & \(h\)\\
\(\Edge\,\partial_\alpha\partial_\beta\Edge\)
 & 10 & 20 & \(-6\) & 2 & \(h\)\\
\midrule
Total & 25 & 45 & & &\\
\bottomrule
\end{tabularx}
\end{center}

The cardinalities count allowable placements. The supplied
computational record further reports that every ordered
placement has a nonzero Hessian coefficient; that stronger
occurrence statement is not needed for the analytic bound.

No numerical kernel has more than two vertex derivatives.
The preparation jet \(\Sigma(y)\) stays in the coefficient
and contributes no additional edge derivative.

\begin{lemma}[Grading and punctured identities]
\label{lem:grading}
For \(k=4,5,6\), define
\[
\mathcal H_k
=\operatorname{span}_{\C}
 \{q^\circ_{\sk}:N_{\sk}=k-4\},
\qquad
\mathcal V=\operatorname{span}_{\C}
 \{q^\circ_{\sk}:\sk\in\mathcal N\}.
\]
Then
\begin{equation}
\mathcal V=\mathcal H_4\oplus\mathcal H_5\oplus\mathcal H_6.
\label{eq:grading}
\end{equation}
On the punctured domain,
\begin{align}
\partial_\alpha(\Edge^2)
&=2\Edge\,\partial_\alpha\Edge,
\label{eq:Leibniz1}\\
\partial_\beta(\Edge\,\partial_\alpha\Edge)
&=(\partial_\beta\Edge)(\partial_\alpha\Edge)
  +\Edge\,\partial_\beta\partial_\alpha\Edge,
\label{eq:Leibniz2}\\
\eta^{\alpha\beta}
 \Edge\,\partial_\alpha\partial_\beta\Edge&=0.
\label{eq:trace}
\end{align}
All three graded subspaces are nonzero.
\end{lemma}

\begin{proof}
Each element of \(\mathcal H_k\) is homogeneous of degree
\(-k\). If \(u_4+u_5+u_6=0\), dilation gives
\(a^{-4}u_4+a^{-5}u_5+a^{-6}u_6=0\) for every \(a>0\).
Three distinct dilations give an invertible Vandermonde
system, proving directness.

The products satisfy the distributional product rule where
their wavefront criterion holds, giving
\eqref{eq:Leibniz1}--\eqref{eq:Leibniz2}.
By Lemma~\ref{lem:families}, \(\Box\Edge=\ii\delta\);
hence \(\Box\Edge=0\) on the punctured domain, which proves
\eqref{eq:trace}.

The massless time-ordered square is nonzero and has scaling
degree four; this is also stated in
\cite[Section 7, following Theorem 7.2]{FR}.
Thus \(\mathcal H_4=\operatorname{span}\{\Edge^2\}\ne0\).
If \(\mathcal H_5=0\), all first derivatives of \(\Edge^2\)
vanish by \eqref{eq:Leibniz1}. On the connected punctured
space it is then constant, incompatible with nonzero
homogeneity of degree \(-4\).
If \(\mathcal H_6=0\), equation~\eqref{eq:Leibniz2} similarly
makes every \(\Edge\,\partial_\alpha\Edge\) constant, and its
degree \(-5\) makes it zero. This contradicts
\(\mathcal H_5\ne0\).
\end{proof}

The inventory is not a basis. For example,
\eqref{eq:trace} is a constant-coefficient relation within
\(\mathcal H_6\).
No independence claim for the other spanning kernels is
required below. The scaling bounds are numerical-kernel
bounds, not a sharp scaling-degree statement for the complete
decorated sum or the root-attached functional.

\section{A coherent extension map on the finite span}
\label{sec:k4}

\subsection{What is fixed}

The prescription consists of:
\begin{enumerate}
\item the intrinsic dilation grading \eqref{eq:grading};
\item one fixed
      \(\varpi\in C_c^\infty(\R^4;\R)\), \(\varpi(0)=1\),
      identical to the subtraction function chosen in XVI-J;
\item the projections built from that function, one for each
      order \(\rho=0,1,2\);
\item zero moments on the corresponding finite Taylor
      complement.
\end{enumerate}
The extension acts only in \(z=y-w\). Smooth coefficients
are restored afterwards.

\subsection{Taylor-complement tests}

For \(|\alpha|,|\beta|\le\rho\), set
\[
M^{(\rho)}_{\beta\alpha}
=\partial^\beta\left(\frac{z^\alpha}{\alpha!}\varpi(z)\right)(0).
\]
Order multi-indices by increasing total degree. The matrix is
triangular with diagonal entries one, because its entry
vanishes unless \(\alpha\le\beta\). Define
\begin{align}
w^{(\rho)}_\alpha(z)
&=
\sum_{|\gamma|\le\rho}
 \bigl((M^{(\rho)})^{-1}\bigr)_{\gamma\alpha}
 \frac{z^\gamma}{\gamma!}\varpi(z),
\nonumber\\
\Proj_\rho\psi
&=
\psi-\sum_{|\alpha|\le\rho}
          \partial^\alpha\psi(0)w^{(\rho)}_\alpha.
\label{eq:Wrho}
\end{align}
Then
\[
\partial^\beta w^{(\rho)}_\alpha(0)=\delta_{\beta\alpha},
\qquad
\Proj_\rho w^{(\rho)}_\alpha=0.
\]
Thus \(\Proj_\rho\) projects onto
\[
\mathcal D_\rho
=\{\psi\in C_c^\infty(\R^4):
       \partial^\alpha\psi(0)=0\text{ for }|\alpha|\le\rho\}.
\]
In particular,
\begin{equation}
w^{(0)}_0=\varpi,
\qquad
\Proj_0\psi=\psi-\psi(0)\varpi.
\label{eq:W0}
\end{equation}
The sans-serif projection \(\Proj_0\) is distinct from the
positive-frequency distribution \(W_0\).

\subsection{An operational action on jet-vanishing tests}

Choose any auxiliary cutoff
\(\zeta\in C_c^\infty(\R^4)\) equal to one near zero.
For \(u\in\mathcal H_k\), \(\rho=k-4\), define
\begin{equation}
T_{k,u}(\psi)
=
\lim_{a\downarrow0}
 \Pair{u}{(1-\zeta(\,\cdot\,/a))\psi},
\qquad \psi\in\mathcal D_\rho.
\label{eq:operational}
\end{equation}
Each test in the limit vanishes near zero, so its pairing
uses only the punctured distribution.

\begin{lemma}[Canonical action on \(\mathcal D_\rho\)]
\label{lem:vanishing}
The limit \eqref{eq:operational} exists, is independent of
the auxiliary cutoff, and is continuous in the induced test
topology. It is linear in \(u\). It equals the action on
\(\mathcal D_\rho\) of any extension of \(u\) with scaling
degree at most \(k\).
\end{lemma}

\begin{proof}
Fix a compact annulus separated from zero. The restriction
of \(u\) has a finite-order seminorm bound there.
For \(\psi\in\mathcal D_\rho\), Taylor's theorem and the chain
rule give, on this fixed annulus,
\[
\|\psi(a\,\cdot)\|_{C^m}=O(a^{\rho+1})
\]
for every fixed finite \(m\), with bounds in finitely many
seminorms of \(\psi\).
After scaling an annular pairing to this annulus,
homogeneity supplies the factor \(a^{4-k}\).
The pairing is therefore
\[
O(a^{4-k+\rho+1})=O(a).
\]
A dyadic decomposition near zero has summable bounds
\(\sum_{j\ge0}O(2^{-j})\). This proves convergence and
continuity. Differences of two scaled cutoff choices have
annular support at scale \(a\), so their action is also
\(O(a)\), proving independence.

The point-extension theorem provides an extension with scaling
degree at most \(k\)
\cite[Theorems 5.2--5.3]{BF}.
Any two such extensions differ by derivatives of \(\delta\)
of order at most \(k-4=\rho\); these annihilate
\(\mathcal D_\rho\).
Equivalently, for \(|\beta|=\rho+1\), the distribution
\(z^\beta u\) has scaling degree at most \(3<4\) and has a
unique extension. Its action reconstructs the action on
jet-vanishing tests, as in the proof of
\cite[Theorem 5.3]{BF}.
The cutoff construction realizes that same action.
\end{proof}

\subsection{Definition and coherence}

\begin{definition}[The selected finite extension map]
\label{def:RK}
For \(u\in\mathcal H_k\), \(\rho=k-4\), define
\begin{equation}
\boxed{\;
\Pair{\RK(u)}{\psi}
=T_{k,u}(\Proj_\rho\psi)
=
\lim_{a\downarrow0}
\Pair{u}{
 (1-\zeta(\,\cdot\,/a))
 \left(\psi-\sum_{|\alpha|\le\rho}
       \partial^\alpha\psi(0)w^{(\rho)}_\alpha\right)}
\;}.
\label{eq:RKaction}
\end{equation}
For \(u=u_4+u_5+u_6\in\mathcal V\), set
\(\RK(u)=\sum_{k=4}^6\RK(u_k)\).
\end{definition}

The cutoff limit is a distributional formula; it is not an
ordinary integral of the singular kernel over its light cone.
The auxiliary cutoff is not an additional normalization
datum. The fixed \(\varpi\) and the moments are.

\begin{proposition}[Extension, moments, and basis independence]
\label{prop:RK}
The map \(\RK:\mathcal V\to\Dprime(\R^4)\) is linear,
well defined, and satisfies
\[
\RK(u)|_{M_\times}=u.
\]
For \(u\in\mathcal H_k\),
\begin{equation}
\sd(\RK(u))\le k,
\qquad
\Pair{\RK(u)}{w^{(k-4)}_\alpha}=0
\quad(|\alpha|\le k-4).
\label{eq:moments}
\end{equation}
These conditions determine the extension uniquely.

Every constant-coefficient relation among the punctured
kernels is preserved:
\begin{equation}
\sum_{\sk}c_{\sk}q^\circ_{\sk}=0
\quad\Longrightarrow\quad
\sum_{\sk}c_{\sk}\RK(q^\circ_{\sk})=0.
\label{eq:coherence}
\end{equation}
In particular, the extended version of \eqref{eq:trace} is
zero.

Finally,
\begin{equation}
\RK(\Edge^2)=\bar q^{\mathfrak R},
\label{eq:Jextension}
\end{equation}
the selected square extension of XVI-J, with the same
\(\varpi\), order-zero projection, and zero moment.
\end{proposition}

\begin{proof}
Lemma~\ref{lem:vanishing} and the continuous projection give
a continuous distributional action, linear in \(u\).
Tests supported away from zero have zero Taylor jet and
recover the punctured distribution. The direct grading makes
the definition on \(\mathcal V\) unambiguous.

The construction is the zero-moment point extension with
projection \eqref{eq:Wrho}; the scaling bound follows from
\cite[Theorem 5.3 and equations (38)--(39)]{BF}.
The moments hold because
\(\Proj_\rho w^{(\rho)}_\alpha=0\).
For a nonzero \(u\in\mathcal H_k\), its restriction already
has scaling degree \(k\), so the extension has degree
exactly \(k\). For \(u=0\), the selected extension is zero.

Differences of extensions with the stated bound are
point-supported distributions
\[
\sum_{|\gamma|\le k-4}c_\gamma\partial^\gamma\delta.
\]
Pairing with the dual Taylor tests makes all their coefficients
zero if the moments agree, proving uniqueness.
Linearity proves \eqref{eq:coherence}, independently of any
basis chosen to compute the finite span.

For \(\Edge^2\), the data are exactly
\eqref{eq:W0} and
\(\Pair{\RK(\Edge^2)}{\varpi}=0\).
Uniqueness at scaling degree four therefore proves
\eqref{eq:Jextension} on all of \(\R^4\).
\end{proof}

A basis adapted to \(\mathcal H_4,\mathcal H_5,\mathcal H_6\)
may be used for bookkeeping. It is not a further choice in
the prescription.

This coherence concerns constant linear relations on
\(\mathcal V\). It does not assert compatibility with arbitrary
smooth-multiplier identities or with alternative
integration-by-parts presentations of the decorated sector.
The density, derivative placement, and order of operations
are fixed.

\subsection{Contact defects of differentiation}

\begin{lemma}[Explicit derivative defects]
\label{lem:defect}
Define
\[
D_\alpha
=
\RK(2\Edge\,\partial_\alpha\Edge)
-\partial_\alpha\RK(\Edge^2).
\]
It is supported at zero and has the form
\begin{equation}
D_\alpha
=\sum_{|\gamma|\le1}
 c_{\alpha,\gamma}\partial^\gamma\delta,
\qquad
c_{\alpha,\gamma}
=(-1)^{|\gamma|}
 \Pair{\bar q^{\mathfrak R}}
      {\partial_\alpha w^{(1)}_\gamma}.
\label{eq:defect1}
\end{equation}
Similarly,
\[
D_{\beta\alpha}
=
\RK\bigl((\partial_\beta\Edge)(\partial_\alpha\Edge)
          +\Edge\,\partial_\beta\partial_\alpha\Edge\bigr)
-\partial_\beta\RK(\Edge\,\partial_\alpha\Edge)
\]
is supported at zero, and
\begin{equation}
D_{\beta\alpha}
=\sum_{|\gamma|\le2}
 c_{\beta\alpha,\gamma}\partial^\gamma\delta,
\qquad
c_{\beta\alpha,\gamma}
=(-1)^{|\gamma|}
 \Pair{\RK(\Edge\,\partial_\alpha\Edge)}
      {\partial_\beta w^{(2)}_\gamma}.
\label{eq:defect2}
\end{equation}
No nonzero or zero value of these coefficients is asserted.
\end{lemma}

\begin{proof}
The two terms defining each defect agree off zero by
\eqref{eq:Leibniz1} or \eqref{eq:Leibniz2}.
The defects are therefore point-supported.
Differentiation raises scaling degree by at most one
\cite[Lemma 5.1]{BF}, so their bounds are five and six.
Only derivatives of \(\delta\) of orders at most one and two,
respectively, can occur.

For the first defect, its product-extension term has zero
moments on \(w^{(1)}_\gamma\). Distributional differentiation
gives
\[
\Pair{D_\alpha}{w^{(1)}_\gamma}
=\Pair{\bar q^{\mathfrak R}}
       {\partial_\alpha w^{(1)}_\gamma}.
\]
On the other hand,
\[
\Pair{\partial^\beta\delta}{w^{(1)}_\gamma}
=(-1)^{|\beta|}\delta_{\beta\gamma}.
\]
Thus
\[
(-1)^{|\gamma|}c_{\alpha,\gamma}
=\Pair{\bar q^{\mathfrak R}}
       {\partial_\alpha w^{(1)}_\gamma},
\]
which proves \eqref{eq:defect1}.
The same sign calculation on the order-two Taylor complement
proves \eqref{eq:defect2}.
\end{proof}

These formulas compare extensions of punctured products with
derivatives of already extended distributions. They do not
establish a Leibniz rule for the prescription. Any rewriting
using derivatives must retain the displayed defects.

The domain of \(\RK\) is the finite span \(\mathcal V\), not
all distributions. In particular, no multiplicative
renormalization map on individual edge factors is defined.
Derivative compatibility is only meaningful here when both
the input and the corresponding punctured derivative belong
to the stated span.

No Lorentz covariance follows from an arbitrary \(\varpi\).
No covariance or derivative normalization is imposed in
addition to \eqref{eq:moments}. Trace reversal remains in the
finite fiber coefficients; it does not act on the scalar
extension problem.

\subsection{Restoring the smooth coefficients}

Let
\[
\bar q^{\RK}_{\sk}=\RK(q^\circ_{\sk}),
\qquad
r_2(y,w)=y-w.
\]
The extended selected-sector distribution is
\begin{equation}
Q_{\mathrm K}^{\mu\nu}(y,w;h)
=
\sum_{\sk\in\mathcal N}
 a^{\mu\nu}_{\sk}(y,w;h)\,
 r_2^*\bar q^{\RK}_{\sk}.
\label{eq:QKext}
\end{equation}
The operations are, in order:
\[
\text{extend in }z
\quad\longrightarrow\quad
\text{pull back by }z=y-w
\quad\longrightarrow\quad
\text{multiply by the smooth coefficient}.
\]
Submersion pullback and smooth multiplication define this
finite sum. Its restriction to \(y\ne w\) is
\eqref{eq:QK}. No \(y\)- or \(w\)-dependent coefficient enters
\(\RK\).

\section{Root attachment, proper pushforward, and continuity}
\label{sec:k5}

Use the variable order
\[
(y,x,w)\in X=M_y\times M_x\times M_w
\]
throughout. Set
\[
\pi(y,x,w)=(y,x),\qquad
r(y,x,w)=y-w,\qquad
\pi_x(y,x,w)=x.
\]

\begin{lemma}[Three distinct attachment steps]
\label{lem:attachK}
Under Assumption~\ref{ass:root}, for every
\(\sk\in\mathcal N\):
\begin{enumerate}
\item The pullbacks exist and satisfy
\begin{align*}
\WF(\pi^*B_{f,g,\mu\nu})
&\subset
 \{(y,x,w;\xi,-\xi,0):\xi\ne0\},\\
\WF(r^*\bar q^{\RK}_{\sk})
&\subset
 \{(y,x,w;\kappa,0,-\kappa):\kappa\ne0\}.
\end{align*}
\item The product
\begin{equation}
T_{\sk,\mu\nu}
:=
\pi^*B_{f,g,\mu\nu}\,
r^*\bar q^{\RK}_{\sk}
\label{eq:T}
\end{equation}
exists on all \(X\), including \(y=w\).
There is no interaction switch in \(T_{\sk,\mu\nu}\).
\item For smooth \(h\),
\begin{equation}
U^{\RK}_{\mathrm K}(h)
=
\sum_{\sk\in\mathcal N}\sum_{\mu,\nu}
 a^{\mu\nu}_{\sk}(y,w;h)T_{\sk,\mu\nu}
\label{eq:UK}
\end{equation}
has support contained in the fixed compact set
\begin{equation}
K_U
=
\{(y,x,w):(y,x)\in\supp B_{f,g},\
               w\in\supp\chi_{\mathrm I}\}.
\label{eq:support}
\end{equation}
Consequently \(\pi_x\) is proper on its support and
\((\pi_x)_*U^{\RK}_{\mathrm K}(h)\in\Eprime(M)\) is defined.
\end{enumerate}
The pullback and product conclusions hold for any distribution
in the relative variable, not just the selected extensions.
\end{lemma}

\begin{proof}
\emph{Pullbacks.}
The maps \(\pi,r\) are linear submersions.
H\"ormander's pullback theorem applies without a conormal
obstruction \cite[Theorem 8.2.4]{Hormander}.
The root bound (R3) lifts \((\xi,-\xi)\) to
\((\xi,-\xi,0)\). The relative pullback lifts every nonzero
covector \(\kappa\) to \((\kappa,0,-\kappa)\).
These shapes remain valid at coincidence.

\emph{Product.}
A sum of the two lifted shapes is
\[
(\xi+\kappa,-\xi,-\kappa).
\]
If it vanishes, its \(x\)- and \(w\)-components force
\(\xi=\kappa=0\). Thus no opposing nonzero wavefront
covectors exist, and the product criterion applies
\cite[Theorem 8.2.10]{Hormander}.
This step uses the imported wavefront premise, not compactness.

\emph{Support and properness.}
The coefficients \(a_{\sk}(h)\) are smooth and contain one
factor \(\chi_{\mathrm I}(w)\).
Their multiplication with the defined products and finite
index contraction give \eqref{eq:UK}.
Root compactness (R2) bounds \((y,x)\), and the interaction
switch bounds \(w\), giving \eqref{eq:support}.
Compactness makes \(\pi_x\) proper on the support and permits
the distributional pushforward.
\end{proof}

The root's singular support has not been deleted.
No diagonal wavefront computation for an individual extension
is required beyond the universal relative-covector shape.

\begin{theorem}[Selected \(C_2\) coefficient with full-Hessian vertex]
\label{thm:main}
Retain Assumptions~\ref{ass:preparation},
\ref{ass:root}, and~\ref{ass:normalization}, the fixed density
\eqref{eq:ell3}, and the prescription \(\RK\).
Here the full-Hessian vertex includes every derivative on both
contracted slots of that density. The source remains restricted
to \(C_2\); neither the plateau source nor \(M_2\) is included.

For every smooth residual metric \(h\) and smooth scalar
\(\varphi\), the selected-sector coefficient
\begin{equation}
\boxed{\;
\mathcal C^{\RK}_{C_2,\mathrm{fv}}(h,\varphi)
=
\Pair{(\pi_x)_*U^{\RK}_{\mathrm K}(h)}{\varphi}
=
\Pair{U^{\RK}_{\mathrm K}(h)}
     {1_y\otimes\varphi_x\otimes1_w}
\;}
\label{eq:CK}
\end{equation}
is defined. The abbreviation \(\mathrm{fv}\) denotes the
full-Hessian vertex within this selected sector.

The functional is jointly continuous and bilinear for the
\(C^\infty\) configuration topologies. There exist a constant
\(C_0\), compact sets \(K_h,K_\varphi\), and a finite integer
\(m\) such that
\begin{equation}
|\mathcal C^{\RK}_{C_2,\mathrm{fv}}(h,\varphi)|
\le
C_0\,p_{m+2,K_h}(h)\,p_{m,K_\varphi}(\varphi).
\label{eq:continuity}
\end{equation}
The integer \(m\) and sharp seminorm orders are not evaluated.

On compact tests supported away from \(y=w\), the attached
distribution agrees with the punctured expression
\eqref{eq:QK}. Nonvanishing of the final functional is not
asserted.
\end{theorem}

\begin{proof}
Lemma~\ref{lem:attachK} supplies the products under (R3) and
compact support under (R2) together with compactness of
\(\chi_{\mathrm I}\).
The pushforward and smooth pairing therefore exist.

Choose \(\zeta\in C_c^\infty(X)\) equal to one on a
neighborhood of the fixed set \(K_U\), independently of \(h\).
The last pairing in \eqref{eq:CK} means
\[
\mathcal C^{\RK}_{C_2,\mathrm{fv}}(h,\varphi)
=
\sum_{\sk,\mu,\nu}
 \Pair{T_{\sk,\mu\nu}}
      {\zeta(y,x,w)
       a^{\mu\nu}_{\sk}(y,w;h)\varphi(x)}.
\]
Here \(T_{\sk,\mu\nu}\) is exactly the unswitched distribution
\eqref{eq:T}. The test contains the one switch already present
in \(a_{\sk}\). There is no division by \(\chi_{\mathrm I}\)
and no second factor of that switch.

On the fixed compact support of the test, the finitely many
\(T_{\sk,\mu\nu}\) have a common finite distributional order
\(m\). Each \(a_{\sk}(h)\) is a linear differential expression
in \(h(w)\) of order at most two, with fixed smooth
coefficients depending on \(y,w\).
The distributional order-\(m\) estimate and the ordinary
product rule therefore give
\eqref{eq:continuity}; the compact sets may be chosen as
projections of a compact neighborhood of \(\supp\zeta\).

Linearity in \(h\) follows from Proposition~\ref{prop:hess},
and linearity in \(\varphi\) follows from the pairing.
The product seminorm estimate proves joint continuity.
The additional \(m\) is distributional order, not residual
jet order: derivatives in the fixed attached distributions
may differentiate the coefficient beyond its pointwise jet.

Finally, restriction of an extension recovers its punctured
kernel. Pullback and the distributional product are local,
so the attached expressions agree away from coincidence.
\end{proof}

\subsection{Exact constant-slot regression}

\begin{proposition}[Regression to XVI-J, including coincidence]
\label{prop:regression}
Keep only \(\sk=\{0,0\}\) in \eqref{eq:UK}.
The result is exactly
\[
U^{\mathfrak R}(y,x,w;h)
=
B_{f,g,\mu\nu}(y,x)
 \chi_{\mathrm I}(w)
 \mathcal K^{\mu\nu}(y,w;h)
 \bar q^{\mathfrak R}(y-w),
\]
the constant-slot distribution of XVI-J.
Its smeared functional is
\(\mathcal C^{\mathfrak R}_{2e}(h,\varphi)\) for every smooth
\(h,\varphi\). These are equalities of distributions and
functionals, not merely off-diagonal equalities.
\end{proposition}

\begin{proof}
Proposition~\ref{prop:hess} gives \(L^{00}=L\), so
\eqref{eq:Kpq} gives the same dual-basis tensor \(\mathcal K\).
The sign is positive and the unordered multiplicity is one,
hence
\[
a_{\{0,0\}}=\chi_{\mathrm I}\mathcal K.
\]
For the numerical factor, use the same subtraction function
\(\varpi\), the same projection
\(\Proj_0\psi=\psi-\psi(0)\varpi\), and the same zero moment
as XVI-J. Proposition~\ref{prop:RK} then gives
\(\RK(\Edge^2)=\bar q^{\mathfrak R}\) globally.
Their pullbacks and defined root products therefore coincide,
including \(y=w\), and so do their smooth pairings.
\end{proof}

The background table comparison in the computational appendix
is a regression check. The structural identity \(L^{00}=L\)
and uniqueness of the same extension prescription are the
proof of this proposition.

\section{Prescription ambiguity and propagated contact terms}
\label{sec:k6}

\subsection{Individual bounds and coherent changes}

\begin{proposition}[Point-supported changes]
\label{prop:localambiguity}
For each label \(\sk\), any other extension
\(\bar q'_{\sk}\) with
\(\sd(\bar q'_{\sk})\le4+N_{\sk}\) satisfies
\begin{equation}
\bar q'_{\sk}-\bar q^{\RK}_{\sk}
=
\sum_{|\gamma|\le N_{\sk}}
 C_{\sk\gamma}\partial_z^\gamma\delta,
\qquad C_{\sk\gamma}\in\C.
\label{eq:Dq}
\end{equation}
If the alternative choices are required to define a coherent
linear map \(\RK'\) on \(\mathcal V\), these constants are
constrained by the constant-coefficient relations among the
kernels. They are not independent parameters for all
twenty-five labels.
\end{proposition}

\begin{proof}
The difference is supported at zero. The scaling bound and
the point-support structure theorem allow only derivatives
of \(\delta\) through order \(N_{\sk}\), as in
\cite[Theorem 5.3]{BF}.
This argument also covers a zero punctured kernel when the
declared upper bound is used.

For a relation \(\sum_{\sk}c_{\sk}q^\circ_{\sk}=0\), coherence
requires both extension maps to annihilate that relation.
Consequently
\[
\sum_{\sk}c_{\sk}
 \sum_\gamma C_{\sk\gamma}\partial^\gamma\delta=0.
\]
Independence of the derivatives of \(\delta\) gives the
corresponding constraints. Equivalently,
\(\RK'-\RK\) is a linear point-supported map on the finite
graded span, with grade-\(k\) order bound \(k-4\).
\end{proof}

Independent choices satisfying only \eqref{eq:Dq} still define
an attached expression for the fixed decomposition
\eqref{eq:QKext}. They need not constitute a coherent
alternative extension map. The propagated formula below is
valid in either case.

For a coherent change and \(u\in\mathcal H_k\), the coefficients
can be recovered from moments:
\[
(\RK'-\RK)(u)
=\sum_{|\gamma|\le k-4}c_\gamma(u)\partial^\gamma\delta,
\qquad
c_\gamma(u)
=(-1)^{|\gamma|}
 \Pair{(\RK'-\RK)(u)}{w^{(k-4)}_\gamma}.
\]
Thus a change of subtraction function or declared moments is
included in this ambiguity description.

\subsection{Propagation through the root}

\begin{lemma}[Propagated ambiguity and its sign]
\label{lem:amb}
For changes \eqref{eq:Dq}, the change of the attached
coefficient is
\begin{equation}
\boxed{\;
\Delta\mathcal C_{\mathrm K}(h,\varphi)
=
\sum_{\sk,\mu,\nu}
\sum_{|\gamma|\le N_{\sk}}
 C_{\sk\gamma}
 \Pair{B_{f,g,\mu\nu}}
      {\left.
       \partial_w^\gamma
       a^{\mu\nu}_{\sk}(y,w;h)
       \right|_{w=y}\varphi(x)}
\;}.
\label{eq:deltaCK}
\end{equation}
The root pairing is distributional, with a cutoff equal to
one near its compact support if needed.
There is no additional sign in \eqref{eq:deltaCK}.
\end{lemma}

\begin{proof}
Lemma~\ref{lem:attachK} applies to any distribution in the
relative variable, including \(\partial^\gamma\delta\).
Thus every changed product exists and has the same compact
support bound.

Use the coordinates \((y,x,z)\), \(z=y-w\), whose Jacobian
has modulus one. The root and relative contact distribution
become the tensor product
\(B_{f,g}(y,x)\otimes\partial_z^\gamma\delta(z)\).
Pairing with a smooth coefficient gives
\[
\Pair{B_{f,g}\otimes\partial_z^\gamma\delta}
     {a_{\sk}(y,y-z;h)\varphi(x)}
=
\Pair{B_{f,g}}
     {(-1)^{|\gamma|}
      \left.
      \partial_z^\gamma
      [a_{\sk}(y,y-z;h)\varphi(x)]
      \right|_{z=0}}.
\]
At fixed \(y,x\), \(\partial_z=-\partial_w\), so
\[
(-1)^{|\gamma|}
 \partial_z^\gamma a_{\sk}(y,y-z;h)\big|_{z=0}
=
(-1)^{|\gamma|}(-1)^{|\gamma|}
 \partial_w^\gamma a_{\sk}(y,w;h)\big|_{w=y}.
\]
The signs cancel.

Choose a cutoff in \((y,x)\) equal to one near the root
support and a cutoff in \(w\) equal to one near
\(\supp\chi_{\mathrm I}\). Since \(a_{\sk}\) contains the
switch, multiplication by the latter cutoff leaves
\(a_{\sk}\), and all its derivatives, unchanged.
This justifies the displayed smooth pairing and proves
\eqref{eq:deltaCK}.
\end{proof}

\begin{remark}[Pointwise jets, switch derivatives, and locality]
\label{rem:amb}
Write \(N_{\sk}=|p|+|q|\).
The derivative \(\partial_w^\gamma\) in
\eqref{eq:deltaCK} acts on the interaction switch and the
residual coefficient \(L^{pq}_{ij}(w;h)\), but not on
\(\Sigma(y)\).
Its residual jet order is at most
\[
2-N_{\sk}+|\gamma|\le2.
\]
Switch derivatives of order at most \(N_{\sk}\) may appear.
This pointwise jet bound is not a bound on the order of the
root distributional pairing.

The contact terms are propagated through \(B_{f,g}\).
They are not identified with variations of local bulk
gravitational counterterms. Locality or nonlocality of the
resulting functional is not established.

For \(\sk=\{0,0\}\), only \(\gamma=0\) occurs and
\eqref{eq:deltaCK} reduces exactly to the propagated
\(c\delta\) ambiguity of XVI-J.
The derivative defects of Lemma~\ref{lem:defect} are fixed by
the selected prescription; they are not additional freely
chosen parameters.
\end{remark}

\section{Verification checklist and limits}
\label{sec:status}

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{0.34\linewidth}Y@{}}
\toprule
Item & Status and locator\\
\midrule
Ambient normalization
 & Proved:
   \(D_F=-\Pc\Edge\), \(KD_F=\ii\,\id\);
   Lemma~\ref{lem:families}\\
Full-Hessian structure
 & Analytically proved from the finite density:
   Proposition~\ref{prop:hess}\\
General numerical coefficient list
 & Supplied exact-arithmetic verification record:
   Appendix~\ref{app:script}\\
Dual-basis contraction
 & Derived from the inverse fiber pairing:
   Lemma~\ref{lem:dual}\\
Derivative sign and numerical products
 & Proved using the off-diagonal Feynman relation:
   Lemma~\ref{lem:sign}\\
Scaling and subtraction orders
 & Numerical degrees \(4,5,6\), declared orders \(0,1,2\):
   Proposition~\ref{prop:sd}\\
Finite extension coherence
 & Constructed on \(\mathcal V\), preserving constant
   relations:
   Proposition~\ref{prop:RK}\\
Differentiation defects
 & Contact formulas proved; values not evaluated:
   Lemma~\ref{lem:defect}\\
Root attachment
 & Conditional on imported (R1)--(R3):
   Lemma~\ref{lem:attachK}\\
Switch counting
 & One factor in \(a_{\sk}\), none in \(T_{\sk,\mu\nu}\):
   \eqref{eq:apq}, \eqref{eq:T}\\
Properness and continuity
 & Root compactness plus compact interaction switch:
   Theorem~\ref{thm:main}\\
XVI-J regression
 & Exact, including the selected contact part:
   Proposition~\ref{prop:regression}\\
Propagated ambiguity
 & Proved with coherent-family constraints distinguished:
   Proposition~\ref{prop:localambiguity},
   Lemma~\ref{lem:amb}\\
\bottomrule
\end{tabularx}
\end{center}

Nonvanishing is separated by level:
\begin{itemize}
\item The numerical graded spaces are nonzero by
      Lemma~\ref{lem:grading}.
\item The supplied output records nonzero algebraic/background
      decorated tensors, described below.
\item Nonvanishing after the switch, root attachment, or final
      smearing is not inferred from either statement.
\end{itemize}

The plateau source, the \(M_2\) source, and other contraction
types remain outside this selected sector.
The final coefficient has not been evaluated on explicit
preparation, switch, root, and scalar data.
The derivative-defect values, sharp distributional orders,
and locality of the propagated ambiguity are not established.

No state, expectation, retarded interacting observable,
BRST/BV closure, Ward identity, all-orders time-ordered family,
or ultraviolet completion is claimed.
No limit \(\varepsilon\downarrow0\) is taken.
The inherited root bound is not re-proved, and no predecessor
free-shell obstruction is repaired or transferred by this
construction.

\scopebox{Conclusion.}{%
The full-Hessian vertex within the selected \(C_2\) two-edge
sector defines a root-attached distributional coefficient
under the stated root hypotheses and the fixed finite
extension prescription.

The analytic construction is independent of the numerical
coefficient tables. Its constant-slot truncation is exactly
the selected XVI-J coefficient. This is a selected-sector
construction, not a complete interacting or physical
observable.
}

\appendix
\renewcommand{\thesection}{\Alph{section}}

\section{Computational artifacts and the scope of their verification}
\label{app:script}

\subsection{Unchanged supplied files}

The following supplied artifacts are retained unchanged:
\begin{itemize}
\item \texttt{hessian\_full.py};
\item \texttt{xvi\_common.py};
\item \texttt{recompute\_K\_dual.py};
\item \texttt{hessian\_full\_out.txt}.
\end{itemize}
The full Hessian script imports the shared routines from
\texttt{xvi\_common.py} in the same directory.
The separate \texttt{recompute\_K\_dual.py} records the
constant-slot calculation from which those shared routines
were extracted.

No local absolute path is required by this manuscript.
The TeX compiles without reading any Python file or output
file. The script and supplied log are supplementary
computational evidence, not premises of the attachment and
continuity theorems.

\subsection{Exact finite algorithm}

The supplied script uses a sparse polynomial ring over
\(\mathbb Q\), with independent symmetric jets of \(h,A,C\)
through order two and formal parameters \(t,\epsilon,\delta\).
It substitutes
\[
\gamma=t h+\epsilon A+\delta C
\]
in the finite atoms
\eqref{eq:Gamma}--\eqref{eq:ell3}.
Each product is reduced modulo
\[
(t^2,\epsilon^2,\delta^2).
\]
This reduction is legitimate for extracting the coefficient
of \(t\epsilon\delta\): multiplication cannot reduce a
parameter exponent, and spatial total differentiation leaves
the parameters constant.

For a generator representing \(\partial^pF_{ab}\), with
\(|p|<2\), the total derivative rule is
\[
D_\alpha(\partial^pF_{ab})
=\partial^{p+e_\alpha}F_{ab}.
\]
The derivative is extended by the polynomial product rule.
Only the derivatives of the Christoffel atoms are required,
so no third field derivative is introduced.

The coefficient of \(t\epsilon\delta\) is
\(D^2\ell_3(h)[A,C]\): cubic homogeneity makes it linear in the
residual field. Sorting the resulting monomials by the
\(A\)- and \(C\)-jet labels yields \eqref{eq:Lpq} and the
coefficient list. No integration by parts changes the density
or its jet coefficients.

\begin{verification}[Supplied exact-arithmetic record]
\label{lem:comp}
The supplied output identifies SymPy 1.14.0 and exact
\(\mathbb Q\) arithmetic. It reports:
\begin{enumerate}
\item \(7560\) Hessian monomials, each linear in one \(h\)-jet,
      one \(A\)-jet, and one \(C\)-jet, with total derivative
      order two.
\item \(2988\) nonzero ordered coefficients
      \(L^{pq}_{ij}\), with slot symmetry checked entrywise.
\item All \(45\) allowed ordered derivative placements and
      \(25\) unordered placement labels occur.
\item On \(h_*=(w^1)^2\eta\), the constant-slot matrix agrees
      entrywise with the retained XVI-J table, and the
      specified mixed constant-slot value is six.
\item Its constant-slot dual-source contraction agrees with
      the XVI-J decorated tensor.
\item All \(45\) background decorated tensors are nonzero
      as forms in a general formal source jet.
\end{enumerate}
These are computationally verified statements documented by
the supplied code and output, not new execution results
claimed for this TeX revision.
\end{verification}

The monomial and coefficient counts are:
\begin{center}
\begin{tabular}{@{}ccc r r@{}}
\toprule
\(|p|\) & \(|q|\) & Residual order
 & Monomials & Nonzero coefficients\\
\midrule
0 & 0 & 2 & 912 & 100\\
0 & 1 & 1 & 1608 & 376\\
1 & 0 & 1 & 1608 & 376\\
1 & 1 & 0 & 1608 & 1032\\
0 & 2 & 0 & 912 & 552\\
2 & 0 & 0 & 912 & 552\\
\midrule
 & & Total & 7560 & 2988\\
\bottomrule
\end{tabular}
\end{center}

The implementation of the first comparison uses
\texttt{xvi\_common.ell3\_of} on two nonconstant
\(w^1\)-dependent polynomial triples.
For each chosen triple, agreement is an exact polynomial
identity in \(w^1\). The second comparison uses a separately
written four-variable density implementation and two
quadratic polynomial triples evaluated at a rational point.

These are finite exact test cases. They do not establish
coefficientwise equality between independent implementations
for every possible field jet. The supplied output explicitly
makes that qualification, which is retained here.

\subsection{Constant-slot and background regressions}

For \(h_*=(w^1)^2\eta\), the supplied nonzero independent
constant-slot entries are
\[
\begin{gathered}
L_{00}=-1,\quad
L_{04}=L_{07}=L_{09}=-1,\quad
L_{11}=12,\quad L_{22}=L_{33}=4,\\
L_{44}=-9,\quad L_{47}=L_{49}=1,\quad
L_{55}=L_{66}=-12,\\
L_{77}=L_{99}=-1,\quad
L_{79}=1,\quad L_{88}=-4.
\end{gathered}
\]
Every unlisted entry is zero, and the matrix is symmetric.
Thus the reported mixed value is
\[
D^2\ell_3(h_*)[\eta,B_4]
=L_{04}-L_{44}-L_{74}-L_{94}=6.
\]
This value is at the Hessian level. It is not the scalar
coefficient of the decorated graph.

The supplied constant-slot decorated output, translated
from its \texttt{S00}, \texttt{S01}, \ldots\ notation to
\(\Sigma\), is
\begin{align*}
\mathcal K_{00}^{00}
 &=14\Sigma_{11}+10\Sigma_{22}+10\Sigma_{33},\\
\mathcal K_{00}^{01}
 &=-14\Sigma_{01},&
\mathcal K_{00}^{02}
 &=-10\Sigma_{02},&
\mathcal K_{00}^{03}
 &=-10\Sigma_{03},\\
\mathcal K_{00}^{11}
 &=14(\Sigma_{00}-\Sigma_{22}-\Sigma_{33}),\\
\mathcal K_{00}^{12}
 &=14\Sigma_{12},&
\mathcal K_{00}^{13}
 &=14\Sigma_{13},\\
\mathcal K_{00}^{22}
 &=10\Sigma_{00}-14\Sigma_{11}-10\Sigma_{33},\\
\mathcal K_{00}^{23}
 &=10\Sigma_{23},\\
\mathcal K_{00}^{33}
 &=10\Sigma_{00}-14\Sigma_{11}-10\Sigma_{22}.
\end{align*}

Selected differentiated-slot entries in the supplied general
coefficient list are
\begin{align*}
L^{0,e_0}_{(00)(00)}
 &=
 3h_{01,1}+3h_{02,2}+3h_{03,3}
 -\tfrac32(h_{11,0}+h_{22,0}+h_{33,0}),\\
L^{0,2e_1}_{(00)(00)}
 &=-\tfrac32h_{00}+\tfrac12h_{11}
   -\tfrac12h_{22}-\tfrac12h_{33},\\
L^{e_1,e_1}_{(00)(00)}
 &=-3h_{00}+h_{11}-h_{22}-h_{33},\\
L^{e_0,e_0}_{(33)(33)}
 &=h_{00}+h_{11}+h_{22}-3h_{33},\\
L^{2e_0,0}_{(33)(33)}
 &=\tfrac12(h_{00}+h_{11}+h_{22}-3h_{33}).
\end{align*}
Here \(h_{ab,c}=\partial_ch_{ab}\). These entries illustrate
the derivative grading; the complete list remains in the
unchanged supplied output.

For \(s=w^1\), selected supplied decorated background entries
are
\begin{align*}
\mathcal K^{00}_{0,e_1}
 &=2s(8\Sigma_{11}+13\Sigma_{22}+13\Sigma_{33}),\\
\mathcal K^{11}_{0,e_1}
 &=16s(\Sigma_{00}-\Sigma_{22}-\Sigma_{33}),\\
\mathcal K^{00}_{e_1,e_1}
 &=s^2(\Sigma_{11}+8\Sigma_{22}+8\Sigma_{33}),\\
\mathcal K^{11}_{0,2e_0}
 &=-\tfrac12s^2(\Sigma_{00}-11\Sigma_{22}-11\Sigma_{33}).
\end{align*}
They have residual degrees \(s\) and \(s^2\), as predicted by
Proposition~\ref{prop:hess}.

For the actual time-only source jet, the last two relevant
spatial components in this selection give
\[
\mathcal K^{11}_{0,e_1}
 =16w^1\sigma_\varepsilon'',
\qquad
\mathcal K^{11}_{0,2e_0}
 =-\tfrac12(w^1)^2\sigma_\varepsilon''.
\]
These are background coefficient statements, not evaluated
switched or smeared graph values.

\section{Supplied output excerpt and provenance}
\label{app:out}

The following excerpt omits machine-local paths, runtime
claims, and output line numbers. The artifact's original
notation is unchanged inside the listing: \texttt{1} denotes
the zero derivative placement, \texttt{d0}--\texttt{d3} denote
first derivatives, and \texttt{d01}, \ldots\ denote second
derivative multi-indices.

\begin{lstlisting}
# number of monomials in D^2 ell_3(h)[A,C]: 7560
CHECK every monomial is exactly linear in h-jet, A-jet, C-jet: True
CHECK total derivative order of every monomial equals 2: True
CHECK slot symmetry L_ij^{pq} = L_ji^{qp}: True
# number of nonzero ordered coefficients L_ij^{pq} (i,j,p,q): 2988
# max |p|+|q| among nonzero coefficients: 2
# number of distinct ordered derivative placements (p,q) with a nonzero coefficient: 45
# number of distinct unordered placements {p,q} (numerical kernels d^pS d^qS): 25
CHECK L^{00}_ij(background) equals the XVI-J table and is independent of w: True
CHECK D^2 ell_3(h)[eta, B_4] (constant slots) = 6

=== INDEPENDENT CHECK 1 (finite exact test cases, not coefficientwise equivalence): XVI-J's ell3_of (recompute_K_dual.py) on s-dependent fields with NON-constant slots ===
  trial 0 direct == expansion : True
  trial 1 direct == expansion : True
CHECK 1 passed: True

=== INDEPENDENT CHECK 2 (finite exact test cases, not coefficientwise equivalence): direct 4D implementation of eq:finite-curvature/eq:ell3 with sympy diff, point evaluation ===
  trial 0 direct value 3930853/1029  expansion value 3930853/1029  equal: True
  trial 1 direct value 182334658/194481  expansion value 182334658/194481  equal: True
CHECK 2 passed: True

CHECK K_{(0),(0)} on background equals XVI-J eq:Kvals: True
# ordered placements with nonzero decorated tensor on the background: 45
\end{lstlisting}

The scripts' existing guards raise an exception when their
required checks fail. A reader wishing to reproduce the
supplied run without replacing its output file may place
the scripts in one directory and use
\begin{lstlisting}
python3 hessian_full.py > hessian_full_audit.txt
\end{lstlisting}
The original files are not edited by this TeX revision.

The analytic manuscript is self-contained in its finite
density, source, coefficient, and extension definitions.
The remaining manuscript-specific dependency is the explicitly
assumed root interface and the inherited interpretation of the
selected source and graph normalization.
The supplied companion retrieval commentary is not used as
a mathematical premise or as verification of an entire corpus.

\begin{thebibliography}{99}

\bibitem{XIIC}
N.~Cheung,
\emph{Landmark XII-C},
supplied manuscript, version 1.
Locators:
\texttt{eq:finite-curvature},
\texttt{eq:ell3},
\texttt{eq:Ccoefficients}, and
\texttt{thm:actualsource}.
The density atoms and selected \(C_2\) source are specified
explicitly in the present manuscript.

\bibitem{XIID}
N.~Cheung,
\emph{Landmark XII-D},
supplied manuscript, version 1.
Locators:
\texttt{eq:B-kernel} and
\texttt{lem:root-kernel}.
The root's existence, compactness, and \(Z_2\) bound are
retained as Assumption~\ref{ass:root}.

\bibitem{XVIJ}
N.~Cheung,
\emph{Landmark XVI-J},
supplied manuscript and revised formulation.
The retained interfaces are the dual-basis constant-slot
contraction, selected square extension, and conditional root
attachment. The exact regression target is displayed in
Proposition~\ref{prop:regression}, independently of changes
to predecessor equation numbering.

\bibitem{Artifacts}
N.~Cheung,
\emph{Computational artifacts supplied with Landmark XVI-K}:
\texttt{hessian\_full.py},
\texttt{xvi\_common.py},
\texttt{recompute\_K\_dual.py}, and
\texttt{hessian\_full\_out.txt}.
Exact-arithmetic coefficient expansion and supplied
verification record. These artifacts are unchanged.

\bibitem{FR}
K.~Fredenhagen and K.~Rejzner,
\emph{Perturbative algebraic quantum field theory},
arXiv:1208.1428v2, 2013.
Section 7 gives the off-diagonal Feynman product discussion
and the point-extension background.
Its all-orders construction is not imported.

\bibitem{BF}
R.~Brunetti and K.~Fredenhagen,
\emph{Microlocal Analysis and Interacting Quantum Field
Theories: Renormalization on Physical Backgrounds},
Communications in Mathematical Physics
\textbf{208} (2000), 623--661.
arXiv:math-ph/9903028.
Sections 5.1--5.2, Lemma 5.1, and Theorems 5.2--5.3.

\bibitem{BDH}
C.~Brouder, N.~V.~Dang, and F.~H\'elein,
\emph{A smooth introduction to the wavefront set},
Journal of Physics A: Mathematical and Theoretical
\textbf{47} (2014), 443001.
arXiv:1404.1778.
Theorem 13 and Proposition 26.

\bibitem{Hormander}
L.~H\"ormander,
\emph{The Analysis of Linear Partial Differential Operators I},
second edition, Springer, 1990.
Chapter VIII, especially Theorems 8.2.4 and 8.2.10.

\end{thebibliography}

\end{document}