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\hypersetup{
  pdftitle={X-E-ST: Einstein--Hilbert normalization and one matter-induced tree},
  pdfauthor={Ngo Cheung},
  pdfsubject={Supported fixed-width transfer, switched Noether identity,
  microcausal admission, and a one-tree background defect}
}

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\title{\textbf{X-E-ST: Einstein--Hilbert normalization\\
and one matter-induced tree}\\[5pt]
\large A supported fixed-width transfer on the inherited I-B host}
\author{Ngo Cheung}
\date{}

\begin{document}
\maketitle

\begin{abstract}
We give a supported fixed-width realization of Cheung's
Landmark X-E on the conical host and Minkowski completion
already retained by X-A-ST, X-C-ST, and X-D-ST.
The host is not reconstructed or evolved.

Starting from the Palatini identity, we derive the
Einstein--Hilbert quadratic form, its positive
transverse-traceless kinetic sign, the Feynman-weight
de Donder operator
\(\Pgr=-\frac14\Ptr\Box\), and its retarded inverse.
The switched total action has an explicit Noether defect.
Its invariant two-jet equations require
\((\partial_\mu g)T^{\mu}{}_\rho[\phi_0]=0\).
For every nonconstant fixed switch, an open dense set
of smooth free-scalar inputs violates a pointwise
necessary condition; their coupled invariant-jet
fibres are empty, although the gauge-fixed retarded
response exists.

We construct the isolated scalar-sourced tree by inserting
\(-2\Ptr\ERzero(g\Wick{T[\field]})\) into the affine
second-metric slot of the completed scalar observable.
An explicit three-slot distribution proves microcausal
admission of the resulting Wick cubic. We distinguish
its distributional kernel from its smooth contracted
tests, correct the support bound to \(C^3\), and prove
fixed-support continuity. Its coordinate response is
\[
 -2(\ERzero)^2\bigl((\partial_\nu g)
                            T^{\nu\mu}[\phi_0]\bigr).
\]
With the free graviton held fixed, the complete one-tree
expression is invariant through order \(\kappa\), but
has the order-\(\kappa^2\) defect
\(-\Aone[\Lie_\xi\rfield]\). A supported free-solution
witness proves that this defect is not an equation-zero
representative.

The principal action and tree constructions are credited
to X-E. The contribution of this transfer is an explicit
host-compatible proof and a corrected support, topology,
jet-order, and truncation audit. These distinctions provide
a usable baseline for subsequent tree completions without
mistaking one admitted scalar cubic for an interacting
quantum metric or a loop construction. A finite supplied
evaluation dictionary has a complete destructive-clock
instrument and reference-preserving scheduled transport.
The release is local Grade S only.
\end{abstract}

\statementbox{Four corrections made before the proof}{%
The full tree kernel is a compact distribution, not a
smooth test in \(\DD_C\). Its smooth partial contractions
do belong to \(\DD_C\).

The scalar-response part can have its first argument in
\(C\setminus K\), so the general two-point support is
\(C\times S\), not \(K\times S\).

The coefficient of the tree is denoted \(\tau\);
its contribution is \(\kappa^2\tau\), counted once.

The nonzero background defect is a defect of the specified
one-tree truncation. It is not a no-go theorem for
additional trees or a different gauge completion.}

\tableofcontents

%=====================================================================
\section{Purpose, attribution, and the unchanged host}
\label{sec:host}
%=====================================================================

\subsection{Source and exact target}

Landmark X-E fixes an Einstein--Hilbert action, a flat
gauge-fixed graviton operator, and one scalar-induced
retarded tree \cite{XE}. This manuscript transfers
those constructions to the supported presentation
retained by X-A-ST, X-C-ST, and the supplied X-D-ST
\cite{XAST,XCST,XDST}.

The action, its normalization, the scalar-sourced tree,
and the one-tree defect are attributed to X-E.
The proofs below are written on the completion rather
than asserted by analogy with the host. In particular,
the principal variation, free operator, cubic admission,
and nonzero witness are supplied explicitly.

The identifier X-E-ST denotes this proposed source.
It is not evidence of repository installation, external
registration, submission, acceptance, or independent
verification.

\subsection{Enter: readout is not geometric reconstruction}

Enter's relevant distinction is between destructive
classical readout and the return of a quantum system.
The measured clock names are consumed from the returned
quantum interface; the complete output retains the
classical code and the unmeasured residual
\cite{EnterI,EnterPhilosophy,En09}.

For this paper the operative separations are:
\begin{enumerate}
\item a supplied mathematical evaluation versus acquiring
an unknown continuum field;
\item a complete instrument versus its classical outcome law;
\item preservation of a reference versus generation of
entanglement by gravity;
\item a finite protocol on a prescribed spacetime versus
derivation of that spacetime or its equations.
\end{enumerate}
I-B supplies the prescribed-host interpretation,
calibration distinctions, and target-adapted transport
pattern \cite{IB}. None of these operational statements
derives the action used below.

\subsection{The inherited supported presentation}

\begin{assumption}[Exact host retention]
\label{ass:host}
Retain the following X-D-ST/X-A-ST instance without
modification:
\begin{enumerate}[label=\textup{(H\arabic*)}]
\item the regular time-oriented conical product
\((\mathcal M^\circ,G)\), its stipulated worldsheet
\(\mathcal W\), and its supplied future relation;
\item the supported injective developing chart
\[
 \iota:\mathcal L\longrightarrow\mathcal M^\circ
\]
and its retained compact reserve, disjoint from
\(\mathcal W\);
\item the independently declared analytical completion
\[
 \widehat M=\RR^{1+3},\qquad
 \eta=\operatorname{diag}(1,-1,-1,-1),\qquad
 M_{T_*}=[0,T_*]\times\RR^3,\qquad \Sigma=\{t=0\};
\]
\item the signature and time ledger
\[
 \iota^*G=-\eta,\qquad
 t=c_{\mathrm{host}}(T-T_\Sigma);
\]
the host retains signature \((-,+,+,+)\), and the
analysis retains \((+,-,-,-)\);
\item the observation region
\[
 \mathcal O\Subset(0,T_*)\times\RR^3,\qquad
 D_*:=J_\eta^-(\overline{\mathcal O})\cap M_{T_*}
                                              \Subset\mathcal L.
\]
Compact laboratory tests are transported and extended
by zero on \(\widehat M\).
\end{enumerate}
The regular conical product is not treated as a globally
hyperbolic Cauchy spacetime. All global free fields and
Green operators in this paper are on \(\widehat M\).
\end{assumption}

The conical singular set remains a worldsheet. There
is no substitution of a worldline, no global conical
inverse, and no host-metric update. The inherited
non-global-hyperbolicity diagnosis is not used as a
failure of the well-posed flat problems on the completion.

\begin{definition}[Local Grade S]
\label{def:grade}
The release consists of:
\begin{enumerate}[label=\textup{(S\arabic*)}]
\item the inherited supported host;
\item named action identities and response kernels with
proved normalization, support, and transformation laws;
\item a finite installed dictionary of supplied evaluations;
\item a complete destructive-clock instrument, including
untouched references;
\item scheduled locality and a nonempty regular
reference-preserving route under declared finite
operational permissions.
\end{enumerate}
It does not include Grade E, metric emergence, or
Einstein dynamics derived from records.
\end{definition}

%=====================================================================
\section{Frozen data and the inherited scalar completion}
\label{sec:frozen}
%=====================================================================

\subsection{Fixed width and supports}

Fix
\[
 m\ge0,\qquad
 f\in C_c^\infty(\mathcal O;\RR),\qquad
 g\in C_c^\infty((0,T_*)\times\RR^3;[0,1]).
\]
Let \(U\) be a nonempty relatively compact open set with
\[
 \overline U\Subset
       \mathcal L\cap\operatorname{int}\{g=1\},\qquad
 \gp=C_c^\infty(U;T\widehat M).
\]
The formal group \(\Gp\) is generated through second order
by \(\kappa\xi+\kappa^2\zeta\), with \(\xi,\zeta\in\gp\).
Set
\[
 K=\supp f,\qquad
 S=\supp g\cap J_\eta^-(K),\qquad C=K\cup S,
 \qquad D_f=J_\eta^-(K)\cap M_{T_*}.
\]
Choose \(\delta>0\) so that
\begin{equation}
 K\cup\supp g\cup\overline U
                     \subset\{2\delta<t<T_*\}.
 \label{eq:separation}
\end{equation}
Fix \(0<\varepsilon<\delta\), once and for all, and use
\[
 \sigma(t)=\chi(t/\varepsilon),
\]
where \(\chi\) is the smooth predecessor profile, zero
on \((-\infty,0]\), one on \([1,\infty)\), and flat
at both endpoints. No width-removal theorem is sought.

The backgrounds \(h,k\) are fixed smooth real symmetric
tensors on the completion. No equations or spatial
decay are imposed on them. Write
\[
 q=\eta+\kappa h+\kappa^2k.
\]
Formal identities for observables are understood modulo
\(\kappa^3\). Variations of the exact action are derived
before truncation; negative powers of \(\kappa\) in its
normalization are not manipulated in a nilpotent ring.

\subsection{Density coefficients}

All unspecified raising and lowering uses \(\eta\).
Let \(H=\eta^{\mu\nu}h_{\mu\nu}\), and define
\begin{align}
 a_1^{\mu\nu}[h]&=\tfrac12H\eta^{\mu\nu}-h^{\mu\nu},
 &b_1[h]&=\tfrac12H,
 \nonumber\\
 b_2[h]&=\tfrac18H^2-\tfrac14h_{\alpha\beta}h^{\alpha\beta},
 \nonumber\\
 a_2^{\mu\nu}[h]
   &=h^\mu{}_\lambda h^{\lambda\nu}
       -\tfrac12Hh^{\mu\nu}+b_2[h]\eta^{\mu\nu},
 \nonumber\\
 A^{\mu\nu}[h,k]&=a_1^{\mu\nu}[k]+a_2^{\mu\nu}[h],
 &B[h,k]&=b_1[k]+b_2[h].
 \label{eq:densities}
\end{align}
Thus, with \(J_q=\sqrt{-\det q}\),
\[
 J_qq^{\mu\nu}
   =\eta^{\mu\nu}+\kappa a_1^{\mu\nu}[h]
                    +\kappa^2A^{\mu\nu}[h,k]+O(\kappa^3),
\qquad
 J_q=1+\kappa b_1[h]+\kappa^2B[h,k]+O(\kappa^3).
\]

\subsection{Coordinates, scalar operators, and tests}

Let
\[
 \Box=\partial_t^2-\Delta,\qquad P_m=\Box+m^2.
\]
Use the positive retarded and advanced inverses
\(\ER,\EA\) of \(P_m\), and \(\ERzero,\EAzero\)
of \(\Box\). In particular,
\[
 P_m\ER=P_m\EA=1,\qquad \ER^{\mathrm T}=\EA.
\]
Their Cauchy, support, and continuity properties are
used only on the completion \cite{BGP}.

For a smooth symmetric tensor \(l\), write
\[
 w^\rho[l]=-\ERzero\partial_\mu
                              (\sigma a_1^{\mu\rho}[l]).
\]
The second coordinate coefficient solves
\begin{equation}
 \Box v^\rho[h,k]
  =-\partial_\mu\left\{
       \sigma\left(A^{\mu\rho}[h,k]
            +a_1^{\mu\nu}[h]\partial_\nu w^\rho[h]\right)
                    \right\},
 \label{eq:coordinates}
\end{equation}
with zero position and velocity at \(\Sigma\).
Retarded expressions for nondecaying backgrounds are
defined locally by the finite-time Cauchy problem.
Every retained dependence lies in \(D_f\).

The compactly switched Euler coefficients are
\begin{align}
 L_1[l]u
   &=\partial_\mu(g a_1^{\mu\nu}[l]\partial_\nu u)
                               +gm^2b_1[l]u,
 \nonumber\\
 L_2[h,k]u
   &=\partial_\mu(g A^{\mu\nu}[h,k]\partial_\nu u)
                               +gm^2B[h,k]u.
 \label{eq:scalarL}
\end{align}
Each divergence retains its \(\partial g\) terms.

Define
\begin{align}
 c_1[h]&=\partial_\mu(fw^\mu[h]),
 \nonumber\\
 c_2[h,k]
   &=-\partial_\mu\left(
       f(w^\nu[h]\partial_\nu w^\mu[h]-v^\mu[h,k])\right)
       +\tfrac12\partial_\mu\partial_\nu
                                      (fw^\mu[h]w^\nu[h]),
 \nonumber\\
 d_1[h]&=c_1[h]-L_1[h]\EA f,
 \nonumber\\
 d_2[h,k]&=c_2[h,k]-L_1[h]\EA c_1[h]
                     -L_2[h,k]\EA f+L_1[h]\EA L_1[h]\EA f.
 \label{eq:completed}
\end{align}
These are the completed tests of X-C-ST
\cite{XCST,XC}. They are restated to fix the insertion
target, not reconstructed as additional response maps.

\begin{lemma}[Affine second-metric slot]
\label{lem:affine}
There is a symmetric bilinear test map
\(\mathcal Q[h_1,h_2]\) such that
\begin{equation}
 d_2[h,k]=\mathcal Q[h,h]+d_1[k].
 \label{eq:affine}
\end{equation}
For fixed smooth backgrounds,
\[
 d_1[h],d_2[h,k]\in\DD_C.
\]
Their dependence on the backgrounds near the retained
causal reserve and on fixed-support observation tests
is continuous in the smooth topologies.
\end{lemma}

\begin{proof}
The \(k\)-linear parts of \(A,B,L_2\) are respectively
\(a_1[k],b_1[k],L_1[k]\). Equation
\eqref{eq:coordinates} and uniqueness give
\[
 v[h,k]=v[h,0]+w[k].
\]
The \(k\)-part of \(c_2\) is therefore \(c_1[k]\).
Subtracting \(L_1[k]\EA f\) proves the affine assertion;
polarize the remaining quadratic expression.

The bare tests are supported in \(K\). Each term with
an \(L_j\) is supported in \(\supp g\), and advanced
support restricts it to \(J_\eta^-(K)\).
Finite-time smooth Cauchy estimates, Green continuity,
and finite differentiation prove continuity.
\end{proof}

\statementbox{The scalar response is not counted again}{%
The term
\(-L_2\EA f+L_1\EA L_1\EA f\) already represents the
second scalar response. The new operation is the
affine substitution \(k\mapsto k+k_1[\phi]\).
It adds \(d_1[k_1[\phi]]\) once, not another copy of
the scalar second-order pair.}

%=====================================================================
\section{E1: action, Palatini variation, and switched Noether identity}
\label{sec:action}
%=====================================================================

\subsection{Exact geometric conventions}

For a genuine Lorentzian metric \(q\), use
\begin{align*}
 \Gamma^\rho_{\mu\nu}
   &=\tfrac12q^{\rho\lambda}
       (\partial_\mu q_{\lambda\nu}
        +\partial_\nu q_{\lambda\mu}
        -\partial_\lambda q_{\mu\nu}),\\
 R^\rho{}_{\sigma\mu\nu}
   &=\partial_\mu\Gamma^\rho_{\nu\sigma}
      -\partial_\nu\Gamma^\rho_{\mu\sigma}
      +\Gamma^\rho_{\mu\lambda}\Gamma^\lambda_{\nu\sigma}
      -\Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\sigma},\\
 R_{\sigma\nu}&=R^\rho{}_{\sigma\rho\nu},
 \qquad R=q^{\sigma\nu}R_{\sigma\nu},
 \qquad G_{\mu\nu}=R_{\mu\nu}-\tfrac12q_{\mu\nu}R.
\end{align*}
The frozen Lagrangian is
\begin{align}
 S_{\mathrm{gr}}[q]
   &=-\frac1{2\kappa^2}\int J_qR[q]\,\dd^4x,
 \nonumber\\
 S_{\mathrm{sc},g}[q,\phi]
   &=\int\bigl((1-g)\mathcal L_\eta[\phi]
                         +g\mathcal L_q[q,\phi]\bigr)\dd^4x,
 \nonumber\\
 \mathcal L_q[q,\phi]
   &=\tfrac12J_q(q^{\mu\nu}\partial_\mu\phi\partial_\nu\phi
                                      -m^2\phi^2),
 \qquad S=S_{\mathrm{gr}}+S_{\mathrm{sc},g}.
 \label{eq:action}
\end{align}
The notation denotes local Lagrangian densities:
Euler derivatives are defined using compact variations.
No globally finite action integral or background decay
is assumed. This is the X-E action convention
\cite{XE}; its sign is checked in Section~\ref{sec:free}.

\begin{lemma}[Palatini first variation]
\label{lem:palatini}
For a compactly supported variation \(\delta q_{\mu\nu}\),
\begin{equation}
 \delta(J_qR)
   =-J_qG^{\mu\nu}\delta q_{\mu\nu}
       +\partial_\rho(J_qW^\rho),
 \qquad
 W^\rho=q^{\mu\nu}\delta\Gamma^\rho_{\mu\nu}
                    -q^{\rho\mu}\delta\Gamma^\nu_{\nu\mu}.
 \label{eq:palatini}
\end{equation}
Consequently
\[
 \frac{\delta S_{\mathrm{gr}}}{\delta q_{\mu\nu}}
       =\frac{J_q}{2\kappa^2}G^{\mu\nu}.
\]
\end{lemma}

\begin{proof}
The difference of two connections is a tensor, so
\(\delta\Gamma\) is tensorial. In normal coordinates,
and hence tensorially,
\[
 \delta R^\rho{}_{\sigma\mu\nu}
       =\nabla_\mu\delta\Gamma^\rho_{\nu\sigma}
                    -\nabla_\nu\delta\Gamma^\rho_{\mu\sigma}.
\]
Contraction and \(\nabla q=0\) give
\(q^{\mu\nu}\delta R_{\mu\nu}=\nabla_\rho W^\rho\).
Furthermore,
\[
 \delta J_q=\tfrac12J_qq^{\mu\nu}\delta q_{\mu\nu},
 \qquad
 \delta q^{\mu\nu}
       =-q^{\mu\alpha}q^{\nu\beta}\delta q_{\alpha\beta}.
\]
Substitution in \(\delta(J_qq^{\mu\nu}R_{\mu\nu})\)
gives \eqref{eq:palatini}; compact variation removes
the divergence from the Euler derivative.
\end{proof}

\subsection{Matter variation and the complete Euler equation}

Set
\[
 T^{\mu\nu}[q,\phi]
   =q^{\mu\alpha}q^{\nu\beta}\partial_\alpha\phi\partial_\beta\phi
       -\tfrac12q^{\mu\nu}
              (q^{\alpha\beta}\partial_\alpha\phi\partial_\beta\phi
                                                    -m^2\phi^2),
\]
and write \(T_0=T[\eta,\phi]\).

\begin{proposition}[Euler derivatives]
\label{prop:euler}
The tensor-density and scalar-density Euler derivatives are
\begin{align}
 \mathscr E^{\mu\nu}
   :=\frac{\delta S}{\delta q_{\mu\nu}}
   &=\frac{J_q}{2\kappa^2}
                      (G^{\mu\nu}[q]-\kappa^2gT^{\mu\nu}[q,\phi]),
 \label{eq:E-q}\\
 E_h^{\mu\nu}
   :=\frac{\delta S}{\delta h_{\mu\nu}}
   &=\kappa\mathscr E^{\mu\nu},
 &\frac{\delta S}{\delta k_{\mu\nu}}&=\kappa E_h^{\mu\nu},
 \nonumber\\
 E_\phi
   &=-\partial_\mu\left(
       [(1-g)\eta^{\mu\nu}+gJ_qq^{\mu\nu}]
                                      \partial_\nu\phi\right)
                -m^2(1-g+gJ_q)\phi.
 \label{eq:E-phi}
\end{align}
There are no derivatives of \(g\) in \(\mathscr E\);
all of them remain in \(E_\phi\) and in the divergence
identity below. In particular,
\[
 E_\phi=-(P_m+\kappa L_1[h]+\kappa^2L_2[h,k])\phi
                                                    +O(\kappa^3).
\]
\end{proposition}

\begin{proof}
Varying the displayed matter density algebraically gives
\[
 \delta_q\mathcal L_q
                 =-\tfrac12J_qT^{\mu\nu}[q,\phi]\delta q_{\mu\nu}.
\]
There are no metric derivatives in \(\mathcal L_q\).
Combine this with Lemma~\ref{lem:palatini}.
One scalar integration by parts gives
\eqref{eq:E-phi}, and \eqref{eq:densities} gives its
expansion.
\end{proof}

The first matter vertex is therefore
\begin{equation}
 V_{1,g}[l,\phi]
   =-\tfrac12\int g\,l_{\mu\nu}T_0^{\mu\nu}[\phi]\,\dd^4x,
 \qquad
 \frac{\delta V_{1,g}}{\delta l_{\mu\nu}}
       =-\tfrac12gT_0^{\mu\nu},
 \qquad
 \frac{\delta V_{1,g}}{\delta\phi}=-L_1[l]\phi.
 \label{eq:vertex}
\end{equation}
For later use, direct differentiation gives
\begin{equation}
 \partial_\mu T_0^{\mu\nu}[\phi]
                         =(P_m\phi)\partial^\nu\phi.
 \label{eq:stress-div}
\end{equation}

\subsection{A definite second-order metric equation}

The linearized Einstein tensor is
\begin{align}
 G_{\mathrm{lin},\mu\nu}[l]
   =\tfrac12\bigl(&
       \partial_\rho\partial_\mu l^\rho{}_\nu
       +\partial_\rho\partial_\nu l^\rho{}_\mu
       -\Box l_{\mu\nu}
       -\partial_\mu\partial_\nu\tr_\eta l
 \nonumber\\[-1mm]
       &-\eta_{\mu\nu}
          (\partial_\alpha\partial_\beta l^{\alpha\beta}
                                      -\Box\tr_\eta l)\bigr).
 \label{eq:Glin}
\end{align}
It satisfies \(\partial_\mu G_{\mathrm{lin}}^{\mu\nu}[l]=0\).

To specify the nonlinear coefficient without an
unspecified vertex, define
\[
 \mathcal G_2^{\mu\nu}[h,h]
  =\frac12\left.\frac{\dd^2}{\dd s^2}\right|_{s=0}
                  \left(J_{\eta+sh}G^{\mu\nu}[\eta+sh]\right).
\]
This is an explicit differential polynomial. It can
equivalently be calculated from
\[
 \Gamma_1^\rho{}_{\mu\nu}
   =\tfrac12\eta^{\rho\lambda}
       (\partial_\mu h_{\lambda\nu}
        +\partial_\nu h_{\lambda\mu}
        -\partial_\lambda h_{\mu\nu}),\qquad
 \Gamma_2^\rho{}_{\mu\nu}
                       =-h^\rho{}_\lambda\Gamma_1^\lambda{}_{\mu\nu},
\]
and
\[
 R_{2,\mu\nu}
   =\partial_\rho\Gamma_2^\rho{}_{\mu\nu}
       -\partial_\nu\Gamma_2^\rho{}_{\rho\mu}
       +\Gamma_1^\rho{}_{\rho\lambda}\Gamma_1^\lambda{}_{\mu\nu}
       -\Gamma_1^\rho{}_{\nu\lambda}\Gamma_1^\lambda{}_{\rho\mu},
\]
followed by the trace, inverse-metric, and density
expansions \eqref{eq:densities}.

The invariant metric equation through its second
geometric order is
\begin{align}
 &J_qG^{\mu\nu}[q]-\kappa^2gJ_qT^{\mu\nu}[q,\phi]
 \nonumber\\
 &\quad=\kappa G_{\mathrm{lin}}^{\mu\nu}[h]
       +\kappa^2\left(
           G_{\mathrm{lin}}^{\mu\nu}[k]
            +\mathcal G_2^{\mu\nu}[h,h]
                              -gT_0^{\mu\nu}[\phi_0]\right)
       +O(\kappa^3).
 \label{eq:metric-expansion}
\end{align}
Equivalently,
\[
 E_h^{\mu\nu}
  =\tfrac12G_{\mathrm{lin}}^{\mu\nu}[h]
      +\tfrac\kappa2
       (G_{\mathrm{lin}}^{\mu\nu}[k]
           +\mathcal G_2^{\mu\nu}[h,h]-gT_0^{\mu\nu}[\phi_0])
      +O(\kappa^2).
\]

\begin{remark}[Jet order is not shifted silently]
\label{rem:jetorder}
The coefficient of \(\kappa^2\) in \(E_h\) involves
the coefficient of \(\kappa^3\) in \(J_qG[q]\).
A covariant demand at that order requires a third
classical metric jet and its transformation law.
It is not a condition on an unchanged two-jet alone.
We use \eqref{eq:metric-expansion} for the invariant
two-jet surface, and keep the exact equation
\eqref{eq:E-q} when no truncation is intended.
\end{remark}

\subsection{The exact Noether identity}

For a contravariant tensor density of weight one,
\[
 \nabla_\mu\mathscr E^{\mu\nu}
      =\partial_\mu\mathscr E^{\mu\nu}
                       +\Gamma^\nu_{\mu\lambda}\mathscr E^{\mu\lambda}.
\]

\begin{theorem}[Switched Noether identity]
\label{thm:noether}
For every smooth pair \((q,\phi)\),
\begin{align}
 2q_{\rho\nu}\nabla_\mu\mathscr E^{\mu\nu}
   ={}&E_\phi\partial_\rho\phi
       -(\partial_\rho g)(\mathcal L_\eta-\mathcal L_q)
 \nonumber\\
      &-(\partial_\mu g)T_0^\mu{}_\rho[\phi]
          +(1-g)(P_m\phi)\partial_\rho\phi.
 \label{eq:noether}
\end{align}
The gravitational contribution has no switch defect.
On \(\operatorname{int}\{g=1\}\), the right-hand side
is \(E_\phi\partial_\rho\phi\).
\end{theorem}

\begin{proof}
Let \(\xi\) be an arbitrary compact vector field and
vary \(q,\phi\) by \(\Lie_\xi q,\xi^\rho\partial_\rho\phi\),
holding \(g\) and \(\eta\) fixed.
Naturality of \(J_qR\) makes its variation a divergence.
Naturality of the curved scalar density gives
\(\delta_\xi\mathcal L_q=\partial_\mu(\xi^\mu\mathcal L_q)\).

At \(q=\eta\), algebraic metric variation gives
\[
 \delta_\xi\mathcal L_\eta[\phi]
       =\partial_\mu(\xi^\mu\mathcal L_\eta)
                                      +T_0^{\mu\nu}\partial_\mu\xi_\nu.
\]
Therefore
\[
 \delta_\xi S
    =\int(\partial_\mu g)\xi^\mu(\mathcal L_\eta-\mathcal L_q)\dd^4x
         +\int(1-g)T_0^{\mu\nu}\partial_\mu\xi_\nu\,\dd^4x.
\]
Integrate the last term by parts and use
\eqref{eq:stress-div}.

The Euler form of the same variation is
\[
 \delta_\xi S
   =\int\xi^\rho\left(
      -2q_{\rho\nu}\nabla_\mu\mathscr E^{\mu\nu}
                          +E_\phi\partial_\rho\phi\right)\dd^4x.
\]
Here the metric term follows by substituting
\[
 (\Lie_\xi q)_{\mu\nu}
    =\xi^\lambda\partial_\lambda q_{\mu\nu}
           +q_{\lambda\nu}\partial_\mu\xi^\lambda
           +q_{\mu\lambda}\partial_\nu\xi^\lambda
\]
and integrating once. Equality for arbitrary \(\xi\)
proves \eqref{eq:noether}.
\end{proof}

This is the switch-defect mechanism of X-E
\cite[Noether identity]{XE}, now explicitly retained
on the completion. The generally noncompact set
\(\supp(1-g)\) is never replaced by a compact source.

%=====================================================================
\section{E2: plateau covariance and the coupled-jet obstruction}
\label{sec:classical}
%=====================================================================

\subsection{Exact action invariance, not pointwise density equality}

\begin{proposition}[Plateau action and Euler covariance]
\label{prop:covariance}
For a diffeomorphism \(F\) supported in a compact subset
of \(U\), the density difference
\[
 \mathcal L[F^*q,F^*\phi]-\mathcal L[q,\phi]
\]
has compact support and integrates to zero.
The exact Euler densities obey
\[
 \mathscr E[F^*q,F^*\phi]=F^*\mathscr E[q,\phi],
 \qquad
 E_\phi[F^*q,F^*\phi]=F^*E_\phi[q,\phi].
\]
The first pullback is that of a contravariant
symmetric tensor density, and the second that of a
scalar density.
\end{proposition}

\begin{proof}
The curved scalar and Einstein--Hilbert densities are
natural. Their pullbacks have unchanged integrals
under a compactly supported change of variables.
Moreover, \(g\circ F^{-1}=g\), since \(g=1\) near
the support of \(F-\id\). The remaining flat scalar
term is unchanged pointwise where \(1-g\ne0\).

Differentiate this action equality after arbitrary
compact variations of \(q,\phi\). Change variables
in the resulting invariant tensor-density pairings.
Arbitrariness of the variations proves Euler covariance.
\end{proof}

The density itself need not be pointwise unchanged.
``Exact plateau invariance'' means invariance of the
local action modulo a compactly supported divergence,
not equality of its coordinate density values.

\subsection{The invariant two-jet surface}

\begin{definition}[Coupled invariant two-jet]
\label{def:coupled}
For a specified free scalar input \(\phi_0\), let
\(\mathfrak C^{[2]}_{\phi_0}\) consist of
\[
 q=\eta+\kappa h+\kappa^2k,\qquad
 \phi=\phi_0+\kappa\phi_1+\kappa^2\phi_2
\]
satisfying
\begin{equation}
 J_qG[q]-\kappa^2gJ_qT[q,\phi]=O(\kappa^3),
 \qquad E_\phi[q,\phi]=O(\kappa^3).
 \label{eq:coupled}
\end{equation}
Coordinate data remain \eqref{eq:coordinates}.
No gauge-fixed equation is substituted for the first
condition.
\end{definition}

The complete two-jet transformation is
\begin{align}
 h'&=h+K\xi,\qquad K\xi=\Lie_\xi\eta,
 \nonumber\\
 k'&=k+\Lie_\xi h+K\zeta+\tfrac12\Lie_\xi^2\eta,
 \qquad
 \phi'=F^*\phi,\qquad
 F^*=\exp(\kappa X_\xi+\kappa^2X_\zeta).
 \label{eq:jetlaw}
\end{align}
Proposition~\ref{prop:covariance} preserves
\eqref{eq:coupled}; truncating a natural equation
at the geometric order used there is legitimate.

\begin{theorem}[Necessary compatibility and generic empty fibres]
\label{thm:bianchi}
Every member of \(\mathfrak C^{[2]}_{\phi_0}\) satisfies
\begin{equation}
 (\partial_\mu g)T_0^\mu{}_\rho[\phi_0]=0.
 \label{eq:bianchi}
\end{equation}
If \(g\) is nonconstant, choose \(x_0\) with
\(\dd g(x_0)\ne0\). In the real smooth free-solution
space
\[
 \mathrm{Sol}_m=\{u\in C^\infty(\widehat M;\RR):P_mu=0\},
\]
the set
\[
 \{u:(\partial_\mu g)(x_0)T_0^\mu{}_\rho[u](x_0)
                           \ne0\text{ for some }\rho\}
\]
is open dense. Every corresponding coupled-jet fibre
is empty.

For \(m=0\), the particular input \(\phi_0=t\)
violates \eqref{eq:bianchi} wherever \(\dd g\ne0\).
\end{theorem}

\begin{proof}
The exact Bianchi identity gives
\(\nabla_\mu G^{\mu\nu}=0\). Expanding its density
version to second order on
\(G_{\mathrm{lin}}[h]=0\) yields
\[
 \partial_\mu\bigl(
    G_{\mathrm{lin}}^{\mu\nu}[k]
                      +\mathcal G_2^{\mu\nu}[h,h]\bigr)=0.
\]
Taking the divergence of \eqref{eq:metric-expansion}
and using \(P_m\phi_0=0\) proves \eqref{eq:bianchi}.
Equivalently, it is the order-zero switch condition
in \eqref{eq:noether}. Thus this condition follows
from the exact Noether identity, not from gauge fixing.

The pointwise expression is a continuous quadratic
polynomial in the first scalar jet. It is not identically
zero on \(\mathrm{Sol}_m\): choose a free solution \(v\)
with \(v(x_0)=0\) and
\(\partial_\mu v(x_0)=\delta_\mu^0\).
Such a solution follows from smooth Cauchy data, or
from \(\sin(m(t-t_0))/m\) when \(m>0\), with the
massless limit \(t-t_0\).
Its mixed stress has diagonal entries
\(1/2,-1/2,-1/2,-1/2\), so contraction with the
nonzero covector \(\dd g(x_0)\) is nonzero.
For any \(u\), at least one component of the expression
at \(u+\lambda v\) is a polynomial in \(\lambda\)
with nonzero quadratic coefficient. Its zeros have
empty interior. Arbitrarily small \(\lambda\) give
a violation. Continuity gives openness.

For \(u=t\),
\[
 T_0^{\mu\nu}[t]
       =n^\mu n^\nu-\tfrac12\eta^{\mu\nu},
 \qquad n=\dd t,
\]
and the same diagonal calculation proves the final
claim.
\end{proof}

This is an emptiness result for specified-input fibres.
The union of all fibres is not declared empty:
\(h=k=\phi_0=\phi_1=\phi_2=0\) is an immediate
compatible example. No impossibility theorem for an
unswitched Einstein--scalar theory follows.

\subsection{The inherited completed scalar identity}

Define
\[
 e[q]=f+\kappa d_1[h]+\kappa^2d_2[h,k],\qquad
 c[q]=f+\kappa c_1[h]+\kappa^2c_2[h,k].
\]
Then
\[
 e[q]=(1+L[q]\EA)^{-1}c[q],\qquad
 L[q]=\kappa L_1[h]+\kappa^2L_2[h,k].
\]

\begin{lemma}[Compact equation representatives]
\label{lem:scalaridentity}
For \(\xi,\zeta\in\gp\), let
\(Z_\xi=X_\zeta+\frac12X_\xi^2\). The completed tests satisfy
\begin{align}
 d_1[h']-d_1[h]
   &=P_m(X_\xi\EA f),
 \nonumber\\
 d_2[h',k']-d_2[h,k]
   &=P_m\bigl(X_\xi\EA d_1[h]+Z_\xi\EA f\bigr).
 \label{eq:scalaridentity}
\end{align}
All preimages displayed inside \(P_m\) are compactly
supported. The associated classical relational scalar
is invariant modulo \(\kappa^3\), including on
\(\mathfrak C^{[2]}_{\phi_0}\).
\end{lemma}

\begin{proof}
The ramp coordinate density transforms naturally on
the support of \(F-\id\), since \(\sigma=g=1\) there,
and is unchanged elsewhere. The transformation is the
identity near \(\Sigma\). Cauchy uniqueness gives
\(X[q']=X[q]\circ F\) coefficientwise.
The inverse-coordinate smearing density therefore
obeys \(c[q']=F^\sharp c[q]\).

Let \(E[q]=P_m+L[q]\) be the positive scalar Euler
operator, and let \(s=\EA e[q]\).
Then \(E[q]s=c[q]\). Plateau covariance gives
\[
 E[q']F^*=F^\sharp E[q].
\]
The advanced solution for the transformed source is
therefore \(s'=F^*s\), by coefficientwise advanced
uniqueness. Applying \(P_m\) gives
\[
 e[q']-e[q]=P_m((F^*-1)\EA e[q]).
\]
Expand through second order.
The generators have compact support, so the indicated
preimages are compact smooth functions.

For a smooth free input \(u\), pairing with \(u\)
annihilates these equation tests. The formal inverse
coordinates give the same completed scalar functional;
alternatively this is the retarded scalar recursion
encoded in \eqref{eq:completed}.
\end{proof}

This reproduces the fixed-algebra representative
mechanism retained in X-C-ST and X-D-ST
\cite{XCST,XDST}. It does not use an Einstein solution
to define the scalar observable.

%=====================================================================
\section{E3: the derived free graviton operator}
\label{sec:free}
%=====================================================================

\subsection{Quadratic form and sign}

\begin{proposition}[Quadratic Einstein--Hilbert coefficient]
\label{prop:quadratic}
Up to a total derivative,
\begin{equation}
 S_{\mathrm{FP}}[l]
      =\frac14\int l_{\mu\nu}G_{\mathrm{lin}}^{\mu\nu}[l]\,\dd^4x,
 \qquad
 \frac{\delta S_{\mathrm{FP}}}{\delta l_{\mu\nu}}
                              =\tfrac12G_{\mathrm{lin}}^{\mu\nu}[l].
 \label{eq:FP}
\end{equation}
This is the quadratic coefficient of
\eqref{eq:action}, not a separately imported free action.
For transverse-traceless spatial configurations,
\[
 S_{\mathrm{FP}}
      =\frac18\int\left(
         \partial_t l_{ij}\partial_t l_{ij}
             -\partial_a l_{ij}\partial_a l_{ij}\right)\dd^4x.
\]
Thus the physical transverse-traceless kinetic term
has positive sign.
\end{proposition}

\begin{proof}
For compact \(l\), set
\(I(s)=\int J_{\eta+sl}R[\eta+sl]\dd^4x\).
Lemma~\ref{lem:palatini} gives
\[
 I'(0)=0,\qquad
 I''(0)=-\int l_{\mu\nu}G_{\mathrm{lin}}^{\mu\nu}[l]\dd^4x.
\]
Multiplication by \(-1/(2\kappa^2)\), with \(s=\kappa\),
gives \eqref{eq:FP}. Symmetry of the second variation
gives its Euler derivative.
For trace-free divergence-free \(l\),
\(G_{\mathrm{lin}}[l]=-\frac12\Box l\).
One integration by parts gives the displayed kinetic form.
Compact variation gives the same local density result
without a decay assumption.
\end{proof}

A separate stationary normalization check agrees with
attraction. With \(\kappa^2=8\pi G_N\), a positive
static density \(\rho\) in the unswitched weak-field
diagnostic gives
\[
 \Delta\bar\gamma_{00}=2\kappa^2\rho,\qquad
 \bar\gamma_{00}=4\Phi_N,\qquad
 \Delta\Phi_N=4\pi G_N\rho,
\]
where
\(\Phi_N=-G_N\int\rho(y)/|\mathbf x-\mathbf y|\,\dd^3y\).
For the usual stationary nonrelativistic reduction,
\(\gamma_{00}=2\Phi_N\) and acceleration is
\(-\nabla\Phi_N\). This is a sign check on the frozen
completion action, not a static solution produced by
the compact-time experiment or a host update.

\subsection{Feynman-weight de Donder gauge fixing}

Define
\[
 (\Ptr l)_{\mu\nu}
     =l_{\mu\nu}-\tfrac12\eta_{\mu\nu}\tr_\eta l,
 \qquad
 C^\mu[l]=\partial_\nu(\Ptr l)^{\nu\mu},
 \qquad
 S_{\mathrm{gf}}[l]=\tfrac14\int C_\mu[l]C^\mu[l]\,\dd^4x.
\]

\begin{theorem}[Free operator and Green inverses]
\label{thm:free}
One has \(\Ptr^2=1\), and
\begin{align}
 G_{\mathrm{lin}}^{\mu\nu}[l]
   &=-\tfrac12\Box(\Ptr l)^{\mu\nu}
       +\partial^{(\mu}C^{\nu)}[l]
       -\tfrac12\eta^{\mu\nu}\partial_\rho C^\rho[l],
 \nonumber\\
 \frac{\delta(S_{\mathrm{FP}}+S_{\mathrm{gf}})}
                         {\delta l_{\mu\nu}}
   &=-\tfrac14\Box(\Ptr l)^{\mu\nu}
                       =:(\Pgr l)^{\mu\nu}.
 \label{eq:Pgr}
\end{align}
Consequently
\begin{equation}
 \Pgr=-\tfrac14\Ptr\Box,\qquad
 \DR=-4\Ptr\ERzero,\qquad
 \DA=-4\Ptr\EAzero.
 \label{eq:grgreen}
\end{equation}
For a compact source \(A\),
\[
 \supp\DR A\subset J_\eta^+(\supp A),\qquad
 C^\mu[\DR A]=-4\ERzero\partial_\nu A^{\nu\mu}.
 \label{eq:violation}
\]
\end{theorem}

\begin{proof}
In four dimensions trace reversal reverses the trace;
applying it twice gives the identity.
Substitution in \eqref{eq:Glin} gives the first line.

The gauge term varies as
\[
 \delta S_{\mathrm{gf}}
    =\int\delta l_{\mu\nu}\left(
       -\tfrac12\partial^{(\mu}C^{\nu)}
            +\tfrac14\eta^{\mu\nu}\partial_\rho C^\rho\right)\dd^4x.
\]
It cancels precisely the \(C\)-terms in
\(\frac12G_{\mathrm{lin}}\). This proves
\eqref{eq:Pgr}. The constant invertible trace-reversal
factor gives the inverse formulas and support.
Finally \(\Ptr\DR=-4\ERzero\), so differentiating proves
the last identity.
\end{proof}

Strictly, \(\Pgr\) is a constant invertible bundle factor
times a normally hyperbolic operator. Its Green theory
follows from that of \(\Box\); its principal symbol need
not be called the scalar identity symbol.

Gauge fixing specifies the free operator. It does
\emph{not} impose \(C[l]=0\) on every configuration in
its unrestricted covariant free-field carrier.
Indeed a retarded source with nonzero divergence
generally gives \(C[\DR A]\ne0\).

\begin{proposition}[No compact residual transformation]
\label{prop:residual}
If \(\xi\) is a compactly supported smooth vector field
and \(\Pgr K\xi=0\), then \(\xi=0\).
In particular,
\[
 \gp\cap\{\xi:K\xi\in\ker\Pgr\}=\{0\}.
\]
\end{proposition}

\begin{proof}
Since constant derivatives commute,
\[
 \Pgr K\xi=-\tfrac14\Ptr K(\Box\xi).
\]
Invertibility of \(\Ptr\) implies \(K(\Box\xi)=0\).
A Minkowski Killing field has affine coefficients:
differentiating its Killing equation and permuting
indices gives \(\partial_\alpha\partial_\beta v_\gamma=0\).
A compactly supported Killing field is therefore zero.
Thus \(\Box\xi=0\). Its Cauchy data vanish before its
compact support, so Cauchy uniqueness gives \(\xi=0\).
\end{proof}

Accordingly \(\gp\) acts on classical backgrounds in
the calculation below. It is not asserted to act as
a nontrivial residual gauge algebra of the free graviton.
The gauge-fixed action itself is not promoted to the
exact plateau-invariant action of
Proposition~\ref{prop:covariance}.

%=====================================================================
\section{The free carrier and fixed-width polynomial admission}
\label{sec:carrier}
%=====================================================================

\subsection{Separate backgrounds and quantum fields}

Use a free scalar \(\field\) and a free gauge-fixed
symmetric tensor \(\rfield\). The latter is not the
classical background \(h\). Its equations and commutator
are those of \(\Pgr\):
\[
 \field(P_mu)=0,\qquad \rfield(\Pgr v)=0,\qquad
 [\rfield(v),\rfield(w)]
       =i\hbar\pair{v}{(\DR-\DA)w}.
\]
The scalar commutator uses the scalar action Hessian
\(-P_m\), hence the causal inverse \(\EA-\ER\).

Fix a block-diagonal Hadamard Wick matrix with zero
scalar--graviton cross blocks. Its graviton block has
the form
\[
 H_{\mathrm{gr}}=-4\Ptr H_0,
\]
with the antisymmetric part chosen to match
\(\DR-\DA\). No positive physical state on this
covariant graviton carrier is assumed.
The Wick and microcausal framework is the usual
free functional construction \cite{FR}.

Write
\[
 z=h+\rfield,\qquad
 \Aone[l]=\field(d_1[l]),\qquad
 \Qhat[l_1,l_2]=\Wick{\field(\mathcal Q[l_1,l_2])}.
\]
When a metric argument is quantum, the notation means
the corresponding admitted mixed kernel, not a
scalar test evaluated at an operator-valued point.

\subsection{A finite-forest admission lemma}

Flat cotangent vectors are identified by the completion
coordinates. Define
\[
 Z_n=\left\{
  (x_1,p_1;\ldots;x_n,p_n):
                 \sum_{j=1}^np_j=0\right\}\setminus0.
\]
Let \(\overline V_\pm\) be the future/past causal covector
cones including zero.

\begin{lemma}[Admission of the displayed retarded forests]
\label{lem:forest}
Consider a finite forest of vertex positions with:
\begin{enumerate}
\item a retarded Klein--Gordon kernel or its derivatives
on each edge;
\item smooth multipliers at vertices;
\item a compact retained integration region;
\item external field copies attached to specified vertices,
allowing several copies at one vertex.
\end{enumerate}
The distribution obtained by integrating internal
vertices and inserting the external field copies exists.
Its external wavefront is contained in \(Z_n\).
It therefore avoids
\(\overline V_+^{\,n}\cup\overline V_-^{\,n}\).

This applies to the finitely many bare-coordinate and
scalar-response graphs in \eqref{eq:completed}, at fixed
width.
\end{lemma}

\begin{proof}
Choose a root in each component. Root positions and
oriented edge differences are independent linear
coordinates for a forest. In those coordinates the
unweighted edge distribution is a tensor product of
the existing retarded distributions, with constant
distributions in the root positions. It therefore
exists without multiplying dependent singular factors.

Its wavefront annihilates simultaneous translation of
all vertices of each component. Equivalently, the
sum of its vertex covectors is zero. Differentiation
and smooth multiplication preserve this bound
\cite{Hormander}.

Place every external copy at its assigned vertex.
The resulting map from vertex positions to the list
of external positions is smooth. Pushforward is defined
after the compact localization. Its wavefront formula,
including its possible normal covectors, still imposes
zero total external covector: the transpose differential
assigns each external covector to its vertex, and
preserves their total sum. Internal integrations set
the internal covectors to zero. This proves the \(Z_n\)
bound.

A collection of co-oriented causal covectors can have
zero sum only when all are zero, which has already
been excluded.

For the application, expand \eqref{eq:coordinates} and
\eqref{eq:completed} by the finite Leibniz rule.
The graphs are the single-coordinate line, the two
coordinate branches, the nested second-coordinate
line, the mixed coordinate--scalar branches, the
local quadratic metric insertion, and the iterated
scalar line. They are forests, not closed graphs.
Finite propagation, the initial cutoff \(\sigma\),
the compact switch, and the final observation test
place every retained vertex in a compact subset of
\(D_f\). A smooth compact localization equal to one
near that retained set justifies the pushforwards.
\end{proof}

The assertion concerns these explicitly generated
forests. It is not a theorem for arbitrary graph
products or for characteristic quantum cross-lines.

\begin{corollary}[The fixed-width bare multiplet polynomial]
\label{cor:bareadmission}
The coefficients
\[
 \field(f),\qquad \Aone[h+\rfield],\qquad
 \Qhat[h+\rfield,h+\rfield]+\Aone[k]
\]
are admitted fixed-width polynomial Wick functionals.
Their metric arguments can have support in \(D_f\),
not merely in \(C\).
\end{corollary}

\begin{proof}
Apply Lemma~\ref{lem:forest} componentwise, insert
classical backgrounds as smooth multipliers, and
symmetrize identical quantum slots.
\end{proof}

This free mixed-polynomial admission must not be denied
by saying simply that ``no quantum metric kernels are
admitted.'' What is not supplied is a complete
interacting second-order metric, a sharp mixed-metric
admission theorem, or a renormalized time-ordered
hierarchy.

\subsection{Equation ideal and the meaning of a witness}

Let \(\WW\) be the resulting microcausal polynomial Wick
carrier. Use its dynamical equation ideal \(\JJ\):
the two-sided ideal of admissible slotwise equation
images generated by \(P_m\) in scalar slots and
\(\Pgr\) in graviton slots, with the usual admissible
kernel completion. The Wick matrix is a bisolution.

\begin{lemma}[Two properties used below]
\label{lem:ideal}
An admissible compact kernel of the form \(P_m\) in
a scalar slot represents zero in \(\WW/\JJ\).
Every element of \(\JJ\) has zero normal symbol on
every smooth solution pair
\[
 P_mu=0,\qquad \Pgr r=0.
\]
Therefore a normal polynomial with a nonzero value on
one such pair is not in \(\JJ\).
\end{lemma}

\begin{proof}
The first statement is the slotwise equation relation.
When an equation generator is Wick-multiplied by any
functional, all contractions with that generator vanish
because the Wick matrix is a bisolution. The product
is consequently its ordinary normal-symbol product,
which vanishes on the free shell. The same holds for
admissible completed slotwise images, by continuity
of distributional pairing with fixed smooth configurations.
This proves the final separation criterion.
\end{proof}

Normal-symbol evaluation is not a state or an algebra
homomorphism for the Wick product. Only the vanishing
of the specified equation ideal is used.

%=====================================================================
\section{E4: the isolated matter-induced tree}
\label{sec:tree}
%=====================================================================

\subsection{Definition and normalization}

The flat Wick stress tensor is
\[
 \Theta^{\mu\nu}=\Wick{T_0^{\mu\nu}[\field]}.
\]
Define the auxiliary retarded quadratic response
\begin{equation}
 k_1[\field]
   =\DR\left(\tfrac12g\Theta\right)
   =-2\Ptr\ERzero(g\Theta).
 \label{eq:k1}
\end{equation}
It solves, in the distributional Wick sense,
\[
 \Pgr k_1+\frac{\delta V_{1,g}}{\delta r}=0
\]
with retarded support. It is the scalar-sourced
response used inside the observable, not a declaration
of the full second-order interacting metric.

\begin{definition}[One tree, counted once]
\label{def:tree}
Let
\begin{equation}
 \tau=\Wick{\Aone[k_1[\field]]},\qquad
 \mathcal T_{\mathrm{mat}}=\kappa^2\tau.
 \label{eq:tau}
\end{equation}
Total Wick ordering includes all three scalar letters.
The one-tree observable is
\begin{equation}
 \widehat\Phi_{\mathrm{one}}
   =\field(f)+\kappa\Aone[h+\rfield]
      +\kappa^2\left(
          \Qhat[h+\rfield,h+\rfield]+\Aone[k]+\tau\right).
 \label{eq:one}
\end{equation}
\end{definition}

The response \eqref{eq:k1} follows from
\eqref{eq:vertex} and \eqref{eq:grgreen}; its factor
\(-2\) is not adjusted by a Wick convention.
The tree is the isolated order-\(\hbar^0\) retarded
contraction along the graviton slot of the first
observable coefficient. It is not a definition of
the full retarded product.

The polynomial \(\tau\) is background independent.
No free graviton letter survives in it.

\subsection{An explicit two-point coefficient}

Let
\[
 F=\EA f,\qquad
 a^{\mu\nu,\alpha\beta}
   =\tfrac12\eta^{\mu\nu}\eta^{\alpha\beta}
      -\tfrac12(\eta^{\mu\alpha}\eta^{\nu\beta}
                         +\eta^{\mu\beta}\eta^{\nu\alpha}).
\]
Then
\(a_1^{\mu\nu}[l]=a^{\mu\nu,\alpha\beta}l_{\alpha\beta}\).
The mixed test kernel is
\begin{align}
 d_1[l](x)&=\int D^{\alpha\beta}(x,z)l_{\alpha\beta}(z)\,\dd^4z,
 \nonumber\\
 D^{\alpha\beta}(x,z)
  ={}&\partial_{x^\mu}\bigl(f(x)\Lambda^{\mu,\alpha\beta}(x,z)\bigr)
 \nonumber\\
     &-\partial_{x^\mu}\bigl(
        g(x)a^{\mu\nu,\alpha\beta}\partial_\nu F(x)
                                                   \delta(x-z)\bigr)
 \nonumber\\
     &-\tfrac12m^2g(x)F(x)\eta^{\alpha\beta}\delta(x-z),
 \nonumber\\
 \Lambda^{\mu,\alpha\beta}(x,z)
   ={}&-\sigma(z)a^{\nu\mu,\alpha\beta}
                                   \partial_{x^\nu}R(x-z),
 \qquad R=\ERzero.
 \label{eq:D}
\end{align}
Here \(R(x-z)\) denotes the retarded kernel, not a
pointwise value at its singular support. This is
the mixed first-coefficient formula of X-E
\cite[Kernel and support]{XE}, with its scalar-response
part retained.

Let \(R_2=R*R\), the retarded fundamental solution of
\(\Box^2\). The convolution is defined: on the product
of two future cones, addition has compact fibres over
compact sets. Equivalently, it is the twice-iterated
zero-past solution operator. Translation invariance
implies
\[
 \WF(R_j(x-y))\subset
       \{(x,p;y,-p):p\ne0\},\qquad j=1,2.
\]

Since \(g(y)\ne0\) implies \(y^0>2\delta\), every
retarded chain \(y\preceq z\preceq x\) contributing
to the tree has \(\sigma(z)=1\). Consequently
\begin{align}
 \tau
   &=\iint t^{\alpha\beta}(x,y)
                \Wick{\field(x)(\Ptr\Theta)_{\alpha\beta}(y)}
                                                     \dd^4x\,\dd^4y,
 \label{eq:twotree}\\
 t^{\alpha\beta}(x,y)
   ={}&2g(y)\partial_{x^\mu}
            \left(f(x)a^{\nu\mu,\alpha\beta}
                                      \partial_{x^\nu}R_2(x-y)\right)
 \nonumber\\
      &+2g(y)\partial_{x^\mu}
            \left(g(x)a^{\mu\nu,\alpha\beta}
                              \partial_\nu F(x)R(x-y)\right)
 \nonumber\\
      &+m^2g(y)g(x)F(x)\eta^{\alpha\beta}R(x-y).
 \label{eq:t}
\end{align}

\begin{proposition}[Existence and corrected support]
\label{prop:treesupport}
The formula \eqref{eq:t} defines a compact distribution
with
\begin{equation}
 \supp t\subset C\times S,\qquad
 \WF(t)\subset Z_2.
 \label{eq:treesupport}
\end{equation}
All its scalar external points lie in \(J_\eta^-(K)\).
The support claim \(K\times S\) need not hold for the
completed tree.
\end{proposition}

\begin{proof}
Insert \eqref{eq:k1} into the affine formula using
\eqref{eq:D}. The coordinate composition is
\[
 \int\Lambda^{\mu,\alpha\beta}(x,z)R(z-y)\,\dd^4z
       =-a^{\nu\mu,\alpha\beta}\partial_{x^\nu}R_2(x-y)
\]
on the relevant retarded support. The two local terms
in \eqref{eq:D} give the second and third lines of
\eqref{eq:t}, including their signs.

The first line is supported with \(x\in K\),
\(y\in\supp g\), and \(y\preceq x\).
The remaining lines have \(x\in\supp(gF)\subset S\),
\(y\in\supp g\), and \(y\preceq x\), hence \(y\in S\).
This proves the compact support in \eqref{eq:treesupport}.
There is no reason for \(\supp(gF)\) to be contained
in \(K\).

Each term is a derivative of a smooth multiplier times
a translation-invariant two-point distribution.
The wavefront multiplier and derivative rules give
the \(Z_2\) bound \cite{Hormander}.
\end{proof}

The formula also shows exact independence of this
particular coefficient from changes of the ramp below
the source onset, while that separation is maintained.
This is an equality between already defined
positive-width coefficients, not a sharp-width
admission theorem for \eqref{eq:one}.

\subsection{Three scalar slots, not just a two-point slogan}

Define the polarized trace-reversed stress
\[
 \widetilde T_{\alpha\beta}(v,w)
  =\tfrac12\bigl(T_{\mathrm{tr},\alpha\beta}[v+w]
                  -T_{\mathrm{tr},\alpha\beta}[v]
                  -T_{\mathrm{tr},\alpha\beta}[w]\bigr),
 \qquad T_{\mathrm{tr}}=\Ptr T_0.
\]
Thus
\(\widetilde T(u,u)=\Ptr T_0[u]\).
It is a constant-coefficient local bilinear
differential expression, including its mass term.

Define a symmetric cubic kernel \(t_3\) by
\begin{equation}
 \pair{t_3}{u_1\otimes u_2\otimes u_3}
   =\frac13\sum_{\mathrm{cyc}}
       \pair{t^{\alpha\beta}(x,y)}
             {u_1(x)\widetilde T_{\alpha\beta}(u_2,u_3)(y)}.
 \label{eq:t3}
\end{equation}

\begin{theorem}[Microcausal cubic admission]
\label{thm:treeadmission}
The definition \eqref{eq:t3} gives a compact symmetric
distribution on \(\widehat M^3\) satisfying
\begin{equation}
 \supp t_3\subset C^3,\qquad
 \WF(t_3)\subset Z_3,\qquad
 Z_3\cap(\overline V_+^{\,3}\cup\overline V_-^{\,3})
                                                   =\varnothing.
 \label{eq:WFtree}
\end{equation}
In particular,
\[
 \tau=\Wick{\field^{\otimes3}(t_3)}\in\WW
\]
is a well-defined totally Wick-ordered scalar cubic.
No product of singular quantum fields is evaluated
pointwise in its definition.
\end{theorem}

\begin{proof}
Begin with a component of \(t(x,y)\).
Insert two external variables at \(y\) by the
diagonal embedding
\[
 j(x,y)=(x,y,y).
\]
Pushforward \(j_*t\) is a defined compact distribution.
The derivatives in the polarized stress transfer to
external derivatives of \(j_*t\).
Thus \eqref{eq:t3} is an explicit distributional
definition.

For a wavefront covector
\((x,p;y,q_1;y,q_2)\) of this insertion, the
pushforward bound says that
\((p,q_1+q_2)\) either belongs to \(\WF(t)\) or
is zero. Both cases give
\[
 p+q_1+q_2=0.
\]
The second case includes the conormal of the inserted
diagonal; it is not discarded.
Derivatives and permutation symmetrization preserve
the bound. Proposition~\ref{prop:treesupport} gives
the support in \(C^3\). The causal-cone exclusion
follows as in Lemma~\ref{lem:forest}.
The microcausal Wick construction therefore admits
the kernel \cite{FR}.
\end{proof}

Proving a two-point wavefront bound alone would not
justify skipping the external coincidence.
The explicit insertion above is the required
three-slot step.

\subsection{Classical normalization and the switch-divergence tensor}

\begin{theorem}[Retarded response normalization]
\label{thm:normalization}
For a smooth free solution \(P_mu=0\), define
\[
 k_1[u]=-2\Ptr\ERzero(gT_0[u]).
\]
Then its coordinate response is
\begin{equation}
 w^\mu[k_1[u]]
      =-2(\ERzero)^2\left((\partial_\nu g)
                                      T_0^{\nu\mu}[u]\right).
 \label{eq:minus2}
\end{equation}
The classical normal-symbol value of the tree is
\begin{align}
 \tau[u]
  =\int f\Bigl(
       2\partial_\mu u\,(\ERzero)^2
              ((\partial_\nu g)T_0^{\nu\mu}[u])
                     -\ER L_1[k_1[u]]u\Bigr)\dd^4x.
 \label{eq:classicaltree}
\end{align}
Thus the normalization constant is \(-2\).
The scalar-response term is part of the same tree.
\end{theorem}

\begin{proof}
The response vanishes before the source onset, so
\(\sigma k_1[u]=k_1[u]\) on the coordinate source.
Since \(a_1[l]=-\Ptr l\),
\[
 w[k_1[u]]=\ERzero C[k_1[u]].
\]
Theorem~\ref{thm:free} and \eqref{eq:stress-div} give
\[
 C^\mu[k_1[u]]
    =-2\ERzero\partial_\nu(gT_0^{\nu\mu}[u])
    =-2\ERzero((\partial_\nu g)T_0^{\nu\mu}[u]).
\]
This proves \eqref{eq:minus2}.

Finally, self-adjointness of \(L_1\) on compact
pairings and retarded--advanced transposition give
\[
 \pair{d_1[l]}u
    =-\int f\,w^\mu[l]\partial_\mu u\,\dd^4x
                          -\int f\,\ER L_1[l]u\,\dd^4x.
\]
Substitute \(l=k_1[u]\).
\end{proof}

These are the X-E normalization and dressing
identifications \cite[Classical value and the dressing
formula]{XE}. Off the scalar shell the extra source
\(g(P_mu)\partial^\mu u\) must also be retained.

The same tensor
\((\partial_\nu g)T_0^{\nu\mu}[u]\) appears in the
necessary invariant-jet condition.
Nevertheless the gauge-fixed response always exists
for the compact smooth source.
Existence of \(k_1[u]\) does not establish a solution
of the invariant coupled equations.

As a global zero-past field identity, \(w[k_1[u]]=0\)
implies that tensor vanishes, by applying \(\Box^2\).
Vanishing of one observation pairing or vanishing
only on \(K\) does not imply global source conservation.

%=====================================================================
\section{Topology: smooth tests, distributional kernels, and free classes}
\label{sec:topology}
%=====================================================================

For a compact \(A\subset\widehat M\), use
\[
 \DD_A=\{a\in C_c^\infty(\widehat M;\CC):\supp a\subset A\},
 \qquad
 p_N(a)=\max_{|\alpha|\le N}\sup|\partial^\alpha a|.
\]
These are ambient smooth fixed-support test spaces,
including when \(A\) has a nonsmooth boundary.
The scalar degree-one quotient is
\[
 \Qm=\DD/P_m\DD,\qquad \DD=C_c^\infty(\widehat M;\CC).
\]

\begin{lemma}[Closed equation subspace]
\label{lem:closed}
The subspace \(P_m\DD\) is closed in \(\DD\).
Hence \(\Qm\) is Hausdorff, and convergence in
\(\DD_C\) passes to its continuous quotient map.
\end{lemma}

\begin{proof}
The continuous map
\(\EA-\ER:\DD\to C^\infty(\widehat M)\)
has kernel \(P_m\DD\).
If \(\EA a=\ER a=u\), its support lies in the
compact causal diamond
\(J^-(\supp a)\cap J^+(\supp a)\).
Thus \(u\in\DD\) and \(P_mu=a\).
The other inclusion follows from the Green identities.
\end{proof}

\subsection{The correct kernel space}

Write \(\EE'_\Gamma(A)\) for compact distributions
supported in \(A\) with wavefront in a specified
closed cone \(\Gamma\), equipped with the normal
microlocal topology. It is not identified with a
smooth Fr\'echet test space. Continuity of the
distributional operations used here is understood
in the sense of the normal topology
\cite{BDH}.

\begin{theorem}[Fixed-support continuity and contracted tests]
\label{thm:continuity}
Fix common compact support stages for \(f,g\), a common
plateau reserve, and the positive width. Then:
\begin{enumerate}[label=\textup{(\roman*)}]
\item the smooth coefficient arrays in
\eqref{eq:t} depend continuously on the input tests
in their fixed-support smooth Fr\'echet spaces;
\item the maps to \(t\in\EE'_{Z_2}(C\times S)\)
and \(t_3\in\EE'_{Z_3}(C^3)\) are continuous on
those stages;
\item for fixed smooth \(u,v\), the partial contraction
\[
 \ell_{u,v}=\pair{t_3}{u\otimes v\otimes\,\cdot}
\]
is an element of \(\DD_C\). It depends continuously
on the data and on \(u,v\) in their smooth topology
near \(C\);
\item the classes \([\ell_{u,v}]\in\Qm\) depend
continuously on the same data.
\end{enumerate}
The first derivative of the cubic normal symbol is
represented by \(3\ell_{u,u}\).
\end{theorem}

\begin{proof}
Green continuity makes \(F=\EA f\) depend smoothly
on \(f\) on every compact coefficient reserve.
The explicit expression \eqref{eq:t} consists of
a finite number of fixed distributions \(R,R_2\),
smooth coefficient multiplications, and derivatives.
The coefficient maps are continuous; multiplication
by a varying smooth coefficient on a fixed distribution
is continuous in the stated microlocal spaces.
The fixed diagonal insertion and proper pushforwards
are continuous in their normal topologies
\cite{BDH}. This proves (i)--(ii).

Smoothly testing two arguments of a distribution with
wavefront in \(Z_3\) leaves no possible nonzero covector
in the remaining argument: that would require
\((0,0,p)\in Z_3\), hence \(p=0\).
Thus the contraction is smooth. Its support is in \(C\).

For continuity into every smooth seminorm, the same
partial-pushforward wavefront argument applies with
all other variables tested. Its output cone is empty,
whose normal topology is the smooth topology.
On a fixed compact stage, bounded smooth test families
and their derivatives give the requisite estimates.
This also follows by contracting the finite formula
\eqref{eq:t} and moving derivatives to the smooth
factors. Lemma~\ref{lem:closed} proves (iv).
\end{proof}

If support sets vary, choose one common compact reserve
and causal shadow first. The theorem does not assert
continuity with uncontrolled escaping supports.

\statementbox{What X-D-ST does and does not transfer}{%
The \(\DD_C\) sheet and the closed scalar equation
subspace apply directly to fixed-background scalar
tests and to the smooth partial contractions above.

The whole cubic belongs to the polynomial on-shell
carrier \(\WW/\JJ\), not to the degree-one space
\(\Qm\). Its kernel is distributional. Neither a
fixed-support Fr\'echet theorem for smooth tests nor
their quotient convergence is a theorem about arbitrary
joint quantum metric kernels.}

No shrinking-layer estimate occurs here.
The width remains fixed, and no normal-topology limit
of a sharp mixed insertion is taken.
The limiting transmission prescription of X-D-ST is
also not identified with an unchanged zero-velocity
multiplied-equation insertion \cite{XDST,XD}.

\subsection{Supported transfer}

\begin{proposition}[Patch agreement]
\label{prop:patch}
Every retained coordinate, scalar-response, and matter-tree
pairing depends only on the backgrounds and sources
near the relevant compact subset of \(D_f\).
Completion extensions agreeing there give the same
supported result. Transport through \(\iota\) preserves
the pairings and their stated continuity.
\end{proposition}

\begin{proof}
Every displayed edge is retarded towards the observation
root, or advanced after transposition from that root.
All vertices remain in \(J_\eta^-(K)\), and the
zero-data coordinate problems stop at \(\Sigma\).
Thus finite propagation restricts their dependence
to \(D_f\). The explicit tree formula has both scalar
vertices in \(C\).

Distribution transport through a smooth embedding is
defined by the corresponding test-density pairing.
On the fixed compact reserve, the chart and its inverse
have bounded derivatives of every needed finite order.
The smooth test topologies and transported microlocal
bounds are consequently the corresponding equivalent
local descriptions.
\end{proof}

The flat bound \(\sum p_j=0\) is a completion-coordinate
description. On the host it is transported by the chart;
it is not a claim of a global cotangent addition law
around the conical worldsheet.

%=====================================================================
\section{The one-tree background identity and its nonzero defect}
\label{sec:defect}
%=====================================================================

\subsection{Transformation with the free fields held fixed}

Transform only the classical backgrounds:
\[
 h'=h+K\xi,\qquad
 k'=k+\Lie_\xi h+K\zeta+\tfrac12\Lie_\xi^2\eta,
 \qquad \xi,\zeta\in\gp.
\]
Both free fields, the Wick matrix, \(f,g\), and the width
remain fixed.

\begin{theorem}[Exact defect of the specified truncation]
\label{thm:defect}
In \(\WW/\JJ\), modulo \(\kappa^3\),
\begin{equation}
 \widehat\Phi_{\mathrm{one}}[h',k']
       -\widehat\Phi_{\mathrm{one}}[h,k]
          \equiv-\kappa^2\Aone[\Lie_\xi\rfield].
 \label{eq:defect}
\end{equation}
Thus the one-tree expression is invariant through order
\(\kappa\). The defect is independent of the classical
backgrounds and of \(\zeta\). The scalar-only tree
\(\tau\) contributes zero to this background variation.
\end{theorem}

\begin{proof}
Let \(z=h+r\) first be a sum of two smooth tensors.
For a full two-jet transform of \(z\), the required
second coefficient would be
\[
 k'_{\mathrm{full}}
    =k+\Lie_\xi z+K\zeta+\tfrac12\Lie_\xi^2\eta
    =k'+\Lie_\xi r.
\]
Lemma~\ref{lem:scalaridentity} is a polynomial identity
in \(z\) and \(k\). Polarization and fixed-width
admission permit replacement of \(r\) by \(\rfield\)
and total Wick ordering. Its right-hand sides are
compact scalar-slot \(P_m\)-images, hence belong
to \(\JJ\) by Lemma~\ref{lem:ideal}.

For the full transformed second slot the completed
bare multiplet is therefore invariant modulo \(\JJ\).
The actually frozen second slot is smaller by
\(\Lie_\xi\rfield\). The affine identity
\eqref{eq:affine} subtracts exactly
\(\Aone[\Lie_\xi\rfield]\) from the second coefficient.
Finally \(\tau\) has no classical-background dependence.
\end{proof}

This is the X-E background defect for the same
one-tree truncation \cite[Background gauge identity]{XE}.
It is not a statement about the all-tree expression
studied on another sheet.

The useful differential decomposition is
\begin{equation}
 (\Lie_\xi r)_{\mu\nu}
     =(K(r\cdot\xi))_{\mu\nu}
          -2\xi^\lambda\Gamma_{1,\lambda\mu\nu}[r],
 \qquad
 \Gamma_{1,\lambda\mu\nu}
      =\tfrac12(\partial_\mu r_{\lambda\nu}
                 +\partial_\nu r_{\lambda\mu}
                 -\partial_\lambda r_{\mu\nu}).
 \label{eq:LieGamma}
\end{equation}
Since \(r\cdot\xi\) has support in \(U\),
\eqref{eq:scalaridentity} removes its \(K\)-part modulo
\(\JJ\). Therefore
\[
 -\Aone[\Lie_\xi\rfield]
       \equiv2\Aone[\xi^\lambda\Gamma_{1,\lambda\,\cdot\,\cdot}
                                         [\rfield]]
                                            \pmod{\JJ}.
\]
This identifies the missing tensorial two-jet term;
it does not construct it.

\subsection{A supported equation-ideal separator}

\begin{theorem}[Existence of a nonzero supported defect]
\label{thm:witness}
Suppose \(m=0\). Inside any nonempty regular observation
region \(\mathcal O\), one can choose admissible
\(f,g,U,\xi\), fix them before evaluation, and find
smooth free fields \(u,r\) such that
\[
 P_0u=0,\qquad \Pgr r=0,\qquad
 \pair{d_1[\Lie_\xi r]}u\ne0.
\]
Consequently \(\Aone[\Lie_\xi\rfield]\notin\JJ\)
for that fixed instance.

The assertion is existential in the observation and
switch data. It does not say that every previously
fixed smearing detects a defect.
\end{theorem}

\begin{proof}
Choose nonempty \(V\Subset\mathcal O\), with its closure
away from \(\Sigma\), and a nonzero real
\(\chi_0\in C_c^\infty(V)\).
Choose a plateau \(U\) containing \(\overline V\)
within the regular reserve and a compact switch equal
to one near \(\overline U\).
Take
\[
 \xi=\chi_0\partial_1,\qquad
 r_{\mu\nu}=x^1\delta^2_\mu\delta^2_\nu,\qquad
 u=tx^1.
\]
Both fields are allowed nondecaying free configurations:
\(\Box r=0\) and \(\Box u=0\).
They are not required to satisfy the additional
strong condition \(C[r]=0\).

Because \(r_{1\mu}=0\),
\[
 l:=\Lie_\xi r=\chi_0\,\delta^2\otimes\delta^2.
\]
On its support, \(g=\sigma=1\). Set
\(w_*^\mu=\partial_\nu a_1^{\nu\mu}[l]\). Then
\[
 w_*^0=-\tfrac12\partial_t\chi_0,\qquad
 w_*^1=\tfrac12\partial_1\chi_0,\qquad
 w^\mu[l]=-\ERzero w_*^\mu.
\]
The first scalar response is
\[
 \rho_1[l;u]
      =-\ERzero\partial_\mu(a_1^{\mu\nu}[l]\partial_\nu u).
\]
For this \(u,l\), the term with the Hessian of \(u\)
vanishes, and the source is
\(x^1w_*^0+tw_*^1\).

The retarded commutator formulas
\[
 \ERzero(x^1v)=x^1\ERzero v
                         +2\ERzero\partial_1\ERzero v,
 \qquad
 \ERzero(tv)=t\ERzero v
                         -2\ERzero\partial_t\ERzero v
\]
follow by applying \(\Box\) and zero-past uniqueness.
They give
\begin{equation}
 \rho_1[l;u]-w^\mu[l]\partial_\mu u
       =2(\ERzero)^2\partial_1\partial_t\chi_0.
 \label{eq:witnessvalue}
\end{equation}

Let
\(\Psi=\partial_1\partial_t(\ERzero)^2\chi_0\).
It is smooth, and
\(\Box^2\Psi=\partial_1\partial_t\chi_0\).
If \(\Psi\) vanished throughout \(V\), this derivative
would vanish there, hence everywhere since
\(\chi_0\) is supported in \(V\).
A compactly supported function with
\(\partial_1\partial_t\chi_0=0\) is zero:
\(\partial_t\chi_0\) is independent of \(x^1\) and
compactly supported, and then \(\chi_0\) is independent
of \(t\) and compactly supported. This is a contradiction.

There is therefore a sign-definite nonzero patch of
\(\Psi\) in \(V\). Choose a nonnegative observation
test \(f\) supported there. Then
\[
 \pair{d_1[l]}u=2\int f\Psi\,\dd^4x\ne0.
\]
Choose \(\delta\) and then \(\varepsilon<\delta\)
to satisfy \eqref{eq:separation}. All data are now fixed.
Lemma~\ref{lem:ideal} proves nonmembership in \(\JJ\).
\end{proof}

The free-solution witness is the mechanism used in
X-E \cite[Nonzero defect witness]{XE}, localized here
so that the detector lies in the retained observation
region.

\subsection{What the witness does not exclude}

The proposed replacement
\(\rfield\mapsto\rfield+\kappa\Lie_\xi\rfield\)
is generally not an automorphism of the unchanged
free carrier: for general \(\xi\),
\[
 [\Box,\Lie_\xi]\ne0,
\]
so a free solution need not map to another free
solution.

However, this does not prove that every completion is
impossible. A successor can change the declared
interaction/gauge completion and supply additional
tree terms. The supplied X-F record explicitly treats
a different background-gauge, all-tree problem
\cite{XF}. Nothing here imports that construction,
and nothing here refutes its stated narrower repair.

%=====================================================================
\section{Diagnostics and named omissions}
\label{sec:diagnostics}
%=====================================================================

\subsection{A nonempty scalar-source diagnostic}

Let \(m=0\), \(u=t\), and let \(g=g_0(t)\) on a causal
cylinder containing all relevant retained vertices.
Use spatial cutoffs outside that cylinder.
Define
\[
 G_1(t)=\int_0^tg_0(s)\dd s,\qquad
 G_2(t)=\int_0^tG_1(s)\dd s,\qquad
 G_3(t)=\int_0^tG_2(s)\dd s.
\]
Since \(\Ptr T_0[t]=n\otimes n\), direct zero-data
integration gives
\[
 k_1[t]=-2G_2\,n\otimes n,\qquad
 C^0[k_1[t]]=-G_1,\qquad
 w^0[k_1[t]]=-G_3.
\]
Also
\[
 -\ER L_1[k_1[t]]t
                     =-\int_0^t g_0(s)G_2(s)\dd s.
\]
Therefore
\begin{equation}
 \tau[t]
   =\int f(x)\left(
       \int_0^{t_x}(1-g_0(s))G_2(s)\dd s\right)\dd^4x.
 \label{eq:sourcecheck}
\end{equation}
This is nonnegative for \(f\ge0\), and is nonzero
for suitable smooth switching and observation choices.
It checks the \(-2\) response normalization, the
coordinate sign, and inclusion of the scalar-response
part.

This same free scalar can obstruct the invariant
coupled jet by Theorem~\ref{thm:bianchi}. A defined
gauge-fixed tree and an empty invariant-jet fibre can
therefore coexist without contradiction.

\subsection{Complete classical pure gauge}

For \(\xi\in\gp\), use
\[
 h=K\xi,\qquad k=\tfrac12\Lie_\xi^2\eta.
\]
The scalar-only completed coefficients have zero
variation modulo the scalar equation ideal by
\eqref{eq:scalaridentity}.
With an independent fixed free graviton present,
the extra defect is exactly
\eqref{eq:defect}. Thus the classical two-jet check
and the one-tree quantum-background check have
different conclusions.

\subsection{Conserved sources}

For a conserved compact symmetric source \(A\),
\[
 C[\DR A]=0,\qquad w[\DR A]=0
\]
when its support is above the ramp. The scalar-response
part of \(d_1[\DR A]\) need not vanish.
Hence a zero dressing is not a theorem that the whole
tree is zero.

\subsection{Not constructed here}

The following are named, not supplied:
\begin{enumerate}
\item the gravitational self-interaction tree associated
with the cubic Einstein--Hilbert density;
\item the other tree terms required by a different
background-gauge completion;
\item a complete second-order interacting quantum metric
carrying the missing tensorial transformation;
\item the mixed scalar--graviton loop at
\(\kappa^2\hbar\);
\item a time-ordered hierarchy, normalization family,
ghost sector, or loop subtraction.
\end{enumerate}

If \(S_3[r]\) denotes the cubic Einstein--Hilbert
coefficient, the self-response has the schematic
affine insertion
\[
 \kappa^2\Wick{\Aone[
              \DR(-\delta S_3/\delta r)]}.
\]
This term is of tree order, not excluded by a leg
count. Its construction is not undertaken here.

Likewise the insertion \(\Aone[\rfield]\) is already
bilinear in free fields. A second cross-contraction
with \(V_{1,g}[\rfield,\field]\) can produce a loop
coefficient. Tree exactness of an earlier
scalar-linear response rule is not transferred to
that mixed insertion. No claim of a vanishing
positive-\(\hbar\) remainder is made.

%=====================================================================
\section{E5: VII-C and the non-transferred analytic problems}
\label{sec:VII}
%=====================================================================

\subsection{The comparison requirements}

VII-C's obstruction concerns its specified sharp
mixed insertion and Wick matrix \cite{VIIC}.
For comparison, its extension requires:
\begin{enumerate}[label=\textup{(E\arabic*)}]
\item a Hausdorff locally convex formal algebra extending
its fixed multiplet Wick algebra;
\item an admitted insertion agreeing with the unchanged
sharp functional on all smooth configurations;
\item sequentially continuous admission of its regular
approximants in its specified mixed-kernel normal
topology;
\item bounded-set hypocontinuous multiplication and
continuous extraction of the field-independent
order-\(\hbar^3\) coefficient of the required word.
\end{enumerate}
Its detecting word is
\(B_f\star_H W_g\star_H B_{f'}\), with
\(W_g=\int g\,h_{00}^2\).
The supplied logarithmic coefficient is a conclusion
for that package, not a universal obstruction to all
fixed-width free-polynomial constructions.

\begin{center}
\small
\renewcommand{\arraystretch}{1.12}
\begin{tabularx}{\linewidth}{@{}p{.08\linewidth}p{.39\linewidth}X@{}}
\toprule
Item & VII-C requires & Present construction\\
\midrule
(E1) &
Its exact de Donder multiplet and fixed Wick matrix &
A free scalar and free gauge-fixed tensor with the
normalization \(-4\Ptr H_0\); this block is not absent,
but is not identified with VII-C's block\\
(E2) &
Its unchanged sharp zero-Cauchy mixed insertion &
A completed fixed-width observable and its isolated
scalar-sourced cubic\\
(E3) &
The stated normal-topology approximating family &
No width-removal family for the mixed insertion\\
(E4) &
The mandatory triple word and its fully contracted
\(\hbar^3\) extraction &
An isolated retarded tree coefficient; no \(W_g\)
word or corresponding extraction\\
\bottomrule
\end{tabularx}
\end{center}

VII-C consequently does not apply and is not contradicted.
Unlike X-D-ST, this paper does contain a free graviton
block, so that part of the audit has been redone rather
than copied as ``absent.'' Its exact kernel, topology,
normalization, and mandatory word still do not match.

\subsection{Other non-identifications}

The fixed-support smooth topology of \(\DD_C\), the
degree-one quotient \(\Qm\), and the normal topology
of a specified distributional kernel are not identified.
Nor is any of them identified with IX-A-ST's
boundary-sensitive sharp-insertion topology
\cite{IXAST}.

The general inclusion
\[
 \WF(au)\subseteq\WF(u),\qquad a\in C^\infty,
\]
does not select a subtraction, construct an undefined
product, or provide a loop regulator
\cite{Hormander}. Fixed positive width is not an
ultraviolet completion.

The polynomial smoothing-growth record of Em01-B
and the scalar diagnostic record of Em01-C concern
different families and do not supply a characteristic
composition theorem \cite{EmRecords}.
The separated raw-family divergence of Em02-D is
not repaired by constructing this tree. These records
are comparisons, not premises.

No part of the full characteristic-map programme at
historical H2 or full I3/J2 is discharged by the
forest calculation above. Its propagator variables
are independent tree variables, not the unresolved
cross-line product of that programme.

%=====================================================================
\section{Finite supplied evaluations and the Enter instrument}
\label{sec:Enter}
%=====================================================================

\subsection{What is installed}

Choose finitely many supplied smooth free configurations
or polarized triples and fixed tests satisfying the
supported contract. A dictionary entry can be
\[
 a_i=\pair{t_{3,i}}{u_{i,1}\otimes u_{i,2}\otimes u_{i,3}},
\]
or a supplied expectation whose state and existence
have been separately specified.

This paper uses the first, normal-symbol evaluation
option. It does not construct a detector for
\(\tau\), a state of the graviton, or a procedure
for acquiring arbitrary field data.

By Theorem~\ref{thm:continuity}, these finitely many
evaluations are continuous on their fixed data stages.
The installed implementation stores rational reports
or certified enclosures for them. Arbitrary real
numbers and arbitrary smooth fields are not assumed
to have finite exact descriptions.

\subsection{Complete destructive clock}

Let \(C_{\mathrm{clk}}\cong\CC^{2^{n_C}}\), \(n_C\ge1\),
have a calibrated orthonormal basis
\(\{\ket{\tau_k}\}\), and let \(A\) contain all surviving
finite registers. Put
\[
 D_k=\bra{\tau_k}_{C_{\mathrm{clk}}}\otimes I_A,
 \qquad P_k=\proj{\tau_k}.
\]

\begin{theorem}[Complete instrument, including references]
\label{thm:clock}
The map
\begin{equation}
 \mathcal M(X)
   =\sum_k\proj{k}_R\otimes D_kXD_k^\dagger
   =\sum_k\proj{k}_R\otimes
       \Tr_{C_{\mathrm{clk}}}
       \bigl[(P_k\otimes I_A)X(P_k\otimes I_A)\bigr]
 \label{eq:clock}
\end{equation}
is CPTP and satisfies
\[
 \Tr_R\mathcal M(X)=\Tr_{C_{\mathrm{clk}}}X.
\]
Both identities hold after adjoining any untouched
reference. All measured clock names are consumed;
no quantum clock output is returned.
\end{theorem}

\begin{proof}
The Kraus maps are \(\ket{k}_R\otimes D_k\), and
\[
 \sum_kD_k^\dagger D_k
      =\sum_kP_k\otimes I_A=I.
\]
Expand
\(X=\sum_{j,l}\kb{\tau_j}{\tau_l}\otimes X_{jl}\).
Both expressions in the \(k\)-th recorded block equal
\(X_{kk}\); their sum is the partial trace.
The same calculation includes any reference factor.
Consumption is the declared destructive-readout
interface: its quantum codomain contains \(A\),
not \(C_{\mathrm{clk}}\).
\end{proof}

This is the complete-channel pattern of I-B and
En09-A, with its proof given for this finite interface
\cite{IB,En09}. Conditional reference states may be
steered by selection of an outcome, while the
unconditional reference marginal remains unchanged.

If distinct calibrated labels are required to equal
the time of one fixed readout event on every positive
branch, only one label can occur. Nontrivial
multi-outcome histories instead use a separate map
from codes to labelled events.
The clock encoding is finite; the geometric time
coordinate survives its destruction.

No stationarity or clock-covariance assumption is
used to infer a Schr\"odinger evolution from arbitrary
conditioning. The instrument identity alone would
not justify that further interpretation.

\subsection{An explicit finite-report error}

For reports \(v_k\in[-1,1]\), define
\[
 p_y(v)=\frac{1+yv}{2},\qquad y\in\{-1,1\},
\]
and
\[
 \mathcal I_v(X)
    =\sum_{k,y}\proj{k,y}_R\otimes p_y(v_k)D_kXD_k^\dagger.
\]

\begin{proposition}[Complete-report stability]
\label{prop:report}
The map \(\mathcal I_v\) is CPTP, and, with half-diamond
distance
\(\ddagger(\Lambda,\Gamma)=\frac12\|\Lambda-\Gamma\|_\diamond\),
\begin{equation}
 \ddagger(\mathcal I_v,\mathcal I_{\widetilde v})
                      =\tfrac12\max_k|v_k-\widetilde v_k|.
 \label{eq:report}
\end{equation}
The formula includes all untouched references.
\end{proposition}

\begin{proof}
Use Kraus maps
\(\sqrt{p_y(v_k)}\ket{k,y}_R\otimes D_k\).
For any input operator, including a reference, the
difference has orthogonal blocks
\[
 \frac{y(v_k-\widetilde v_k)}2X_{kk}.
\]
The output trace norm is bounded by
\[
 \max_k|v_k-\widetilde v_k|\sum_k\|X_{kk}\|_1
       \le\max_k|v_k-\widetilde v_k|\,\|X\|_1.
\]
The last inequality is contraction of pinching,
an average of unitary conjugations.
A clock-basis input at a maximizing \(k\) saturates
the bound.
\end{proof}

Thus a supplied enclosure of a normalized report gives
a complete-channel allowance. This relies on the
declared probabilistic encoding. Encoding distinct
approximations as different deterministic orthogonal
words would not have the same continuity property.

The finite report is not an acquisition of the field
that generated its mathematical value.

%=====================================================================
\section{Scheduled locality, regular transport, and resources}
\label{sec:transport}
%=====================================================================

\subsection{Support of the complete finite network}

At each tick let the channel factor over a prescribed
partition of finite sites into blocks. Include every
retained record and control wire. For final sites \(Y\),
define
\[
 N_{t,t}(Y)=Y,\qquad
 N_{s,t}(Y)=
  \bigcup\{B:\ B\text{ is a block at tick }s+1,\
                         B\cap N_{s+1,t}(Y)\ne\varnothing\}.
\]

\begin{theorem}[Scheduled locality]
\label{thm:locality}
For the complete network,
\[
 (\Phi_{s+1}^*\circ\cdots\circ\Phi_t^*)(\mathcal A(Y))
                               \subseteq\mathcal A(N_{s,t}(Y)).
\]
Equal input reductions on that backwards region give
equal corresponding output reductions.
Interventions outside the complete backwards region
do not change the corresponding joint output law.

If each dependency edge has a separately certified
regular future-causal host route, ancestry is contained
in the prescribed metric causal relation.
\end{theorem}

\begin{proof}
A channel adjoint is unital. A block disjoint from the
current support therefore acts on an identity factor.
A meeting block enlarges support only within that
block. Induct backwards through the ticks.
Testing every output observable gives the reduction
and intervention claims. References are retained in
the same argument. Concatenation of the certified
routes proves metric containment.
\end{proof}

This is the scheduled-support mechanism of VII-E
\cite{VIIE}. An ancestor is in the metric past of
its output. An operational dependency bound is not
an identification with an entire physical causal
past, and scheduler consistency does not derive a
Lorentz cone \cite{En08A}.

\subsection{A nonempty regular transport route}

Choose a straight developed spatial segment of length
\(\ell>0\) in the inherited regular convex tube.
After readout, lookup, and a positive processing
buffer, depart at developed time \(t_{\mathrm d}\).
Choose \(n\ge1\) and stations
\[
 \mathbf x_j=\mathbf x_0+
          \frac jn(\mathbf x_n-\mathbf x_0),\qquad j=0,\ldots,n.
\]
Every station has a \(p\)-qubit payload register
\(A_j\), and the registers \(A_1,\ldots,A_n\) begin
in independent blank states \(\omega\).
At hop \(j\), SWAP \(A_{j-1}\) with \(A_j\), with
developed tick duration \(\ell/n\).

The two-register tick-boundary SWAP is a declared
finite operational permission, not a derived
microscopic interaction Hamiltonian. Cross-port
dependencies are null, and same-station dependencies
are timelike. Classical report wires have separate
causal certificates.

\begin{theorem}[Reference-preserving regular route]
\label{thm:transport}
For every payload--reference state \(\rho_{A_0E}\),
after hop \(j\) the joint state is
\[
 \rho_{A_jE}\otimes
    \bigotimes_{\substack{0\le i\le n\\i\ne j}}\omega_{A_i}.
\]
The logical payload channel is the identity with every
reference included. Classical reports can be forwarded
with their entire joint law.
Arrival occurs at
\[
 t_{\mathrm d}+\ell,\qquad
 T_{\mathrm d}+\ell/c_{\mathrm{host}},
\]
and all certified routes stay in the regular part.
\end{theorem}

\begin{proof}
A SWAP moves the whole payload tensor factor,
including its reference correlations, into the next
blank and leaves the old factor blank. Induction gives
the formula. Forwarding diagonal classical records
preserves their joint law. The target-adapted station
spacing and tick duration give the stated null
cross-port displacements inside the regular tube.
\end{proof}

This is the supplied I-B/En08-B route
\cite{IB,En08B}. It preserves already existing
correlations; it does not show that gravity creates
entanglement.

\subsection{Declared resources and errors}

A conservative preallocated reserve for the clock,
station payloads, \(b\)-bit report reserves, lookup
workspace, and one optional report probe is
\[
 n_C+(n+1)p+(n+1)b+w_{\mathrm{lookup}}+1.
\]
Description registers, classical tables, references,
supplier holdings, reset allowances, and returned
outputs are separately included when present.
The route uses \(n\) register SWAPs and \(n\)
classical transmissions. Basis change, readout,
lookup, buffers, and all durations are charged.

For compatible complete implementations,
\begin{equation}
 \ddagger(\widetilde\Gamma,\Gamma)
  \le\min\left\{1,\
       \epsilon_{\mathrm{enc}}+\epsilon_{\mathrm{dec}}
       +\epsilon_{\mathrm{table}}+\epsilon_{\mathrm{clock}}
                              +\sum_j\epsilon_j\right\}.
 \label{eq:error}
\end{equation}
This follows by telescoping channel replacement,
diamond-norm contraction under channels, and the
triangle inequality \cite{Watrous}.
Equation \eqref{eq:report} supplies one explicit
table allowance. An arrival or failure probability
is covered only if its record belongs to the compared
complete output.

No table-generation algorithm, field detector, gate
synthesis rate, or exact causality of a faulty
implementation is inferred from \eqref{eq:error}.

%=====================================================================
\section{Release, non-claims, and successor package}
\label{sec:release}
%=====================================================================

\begin{theorem}[X-E-ST local Grade S release]
\label{thm:release}
For the fixed supported host and the fixed-width
sheet declared before the calculation:
\begin{enumerate}[label=\textup{(\roman*)}]
\item the Palatini variation, switched metric and scalar
Euler equations, and exact Noether defect are
constructed on the completion;
\item the plateau local action is invariant and its
Euler densities are covariant; the inherited
classical scalar identity holds through
\(\kappa^2\), including on the invariant two-jet
surface when that specified-input fibre is
nonempty;
\item the Bianchi condition \eqref{eq:bianchi} is necessary;
for a nonconstant switch, an open dense set of
free scalar inputs has an empty coupled-jet fibre;
\item the Einstein--Hilbert quadratic form and
Feynman-weight de Donder operator have the stated
sign and retarded normalization; the compact
residual algebra is trivial;
\item the isolated matter tree is the admitted scalar
Wick cubic \(\tau\), with support in \(C^3\),
wavefront in \(Z_3\), fixed-support continuity,
and normalization \(-2\);
\item the one-tree background identity holds through
order \(\kappa\), with the precise order-\(\kappa^2\)
defect \eqref{eq:defect}; that class is nonzero
for the supported instance of
Theorem~\ref{thm:witness};
\item the installed finite dictionary has a complete
destructive-clock instrument, scheduled locality,
a regular reference-preserving route, and a
declared resource/error ledger.
\end{enumerate}
This is local Grade S, not spacetime emergence.
\end{theorem}

\begin{proof}
Lemmas~\ref{lem:palatini} and
Theorem~\ref{thm:noether}, with
Proposition~\ref{prop:euler}, prove (i).
Proposition~\ref{prop:covariance} and
Lemma~\ref{lem:scalaridentity} prove (ii).
Theorem~\ref{thm:bianchi} proves (iii).
Proposition~\ref{prop:quadratic},
Theorem~\ref{thm:free}, and
Proposition~\ref{prop:residual} prove (iv).
Proposition~\ref{prop:treesupport},
Theorems~\ref{thm:treeadmission},
\ref{thm:normalization}, and
\ref{thm:continuity} prove (v).
Theorems~\ref{thm:defect} and
\ref{thm:witness} prove (vi).
Sections~\ref{sec:Enter}--\ref{sec:transport}
prove (vii), completing
Definition~\ref{def:grade}.
\end{proof}

\begin{center}
\small
\renewcommand{\arraystretch}{1.12}
\begin{tabularx}{\linewidth}{@{}p{.30\linewidth}p{.22\linewidth}X@{}}
\toprule
Object & Status & Exact boundary\\
\midrule
Fixed-width matter tree &
Constructed &
One affine-slot insertion, counted once\\
Tree external support &
Proved &
\(C^3\subset(J_\eta^-(K))^3\), not generally \(K\times S\)
for its two-point coefficient\\
Smooth scalar contractions &
Continuous &
\(\DD_C\), then the degree-one quotient \(\Qm\)\\
Whole cubic &
Admitted &
A distributional polynomial kernel in \(\WW/\JJ\),
not an element of \(\Qm\)\\
Background identity &
Through \(\kappa\) &
A nonzero \(\kappa^2\) one-tree truncation defect\\
Free graviton &
Specified &
Gauge-fixed free carrier; no physical-state claim\\
Complete interacting second-order metric &
Not constructed &
The auxiliary scalar-sourced response is not the full metric\\
VII-C word &
Not formed &
Its exact admission/topology/word package is unmatched\\
Self-interaction and other completion trees &
Not constructed here &
No universal obstruction asserted\\
Loop or time-ordered hierarchy &
Not constructed &
No extension or subtraction prescription\\
Spacetime emergence &
Not constructed &
The host and action are supplied inputs\\
\bottomrule
\end{tabularx}
\end{center}

\subsection{Potential use and novelty boundary}

The main usable result is not a new claim of
Einstein--Hilbert dynamics emerging from records.
It is a consistently normalized supported example
where:
\[
 \text{invariant-jet compatibility},\quad
 \text{gauge-fixed retarded solvability},\quad
 \text{free polynomial admission},\quad
 \text{background covariance}
\]
have separately proved and different verdicts.

The action and one-tree mechanism are X-E's
\cite{XE}. The support correction, explicit external
three-slot admission, distinction between kernel and
test topologies, geometric jet-order audit, and
host-localized defect witness prevent several
otherwise consequential overstatements in using that
mechanism.

\subsection{Successor package}

A successor receives:
\begin{enumerate}
\item the unchanged host, chart, completion, and signature
ledger;
\item the action variation, exact switch defect, and
invariant two-jet compatibility condition;
\item the derived free operator, retarded inverse, and
compact-residual result;
\item the single scalar-sourced tree, its coefficient
\(-2\), its complete scalar-response part,
its corrected support, and its cubic wavefront
and continuity theorems;
\item the exact one-tree background defect and the
nonzero equation-ideal witness;
\item the finite supplied-evaluation and complete-channel
encoding contract.
\end{enumerate}
It receives no constructed self-interaction tree,
complete interacting quantum metric, VII-C word,
time-ordered hierarchy, characteristic quantum
coefficient, loop regulator, or spacetime-emergence
map.

The sharp scalar limit remains the separate X-D-ST
result. It is not inherited as sharp metric admission.

\subsection{Standing boundaries}

No quantum coefficient of the inherited characteristic
graph is admitted. That sentence does not deny the
free graviton or the specific scalar cubic actually
admitted here.

O.1 remains \loc{NOT_CLOSED} in the inherited
prescription scope used by this proposal.
No assertion about a different later prescription's
subsector is imported. XVI-N is not discharged.
XV-B is not opened as the downstream
quantum-admission exit.

Historical H2 and the full I3/J2 characteristic-map
obligations are not discharged here.
XIV-I's narrower internal-amplitude result is not
withdrawn or equated with the missing full map
\cite{BoundaryRecords}.

The host metric, worldsheet, time coordinate, and
future relation remain fixed throughout.
Neither action variation nor clock readout produces
a host-update map.

%=====================================================================
\appendix
\section{Debugging and quantifier audit}
\label{app:audit}
%=====================================================================

\begin{longtable}{@{}p{.35\linewidth}p{.60\linewidth}@{}}
\toprule
Potential overstatement & Correction in this source\\
\midrule
\endfirsthead
\toprule
Potential overstatement & Correction in this source\\
\midrule
\endhead

A new spacetime is constructed &
The I-B/X-A-ST chart and host are retained.\\[4pt]

Green operators live on the regular cone &
They live on the separate Minkowski completion.\\[4pt]

The singular defect is a worldline &
The inherited product defect is a worldsheet.\\[4pt]

The Einstein--Hilbert quadratic action is copied &
It is derived from the first Palatini variation.\\[4pt]

Plateau invariance means identical density values &
It means invariant local action modulo a compact
divergence; the density can change.\\[4pt]

Gauge fixing is part of the invariant classical action &
Only the free quantum graviton receives the declared
flat gauge-fixing term.\\[4pt]

A gauge-fixed solution solves the invariant equations &
The switch-divergence compatibility condition can fail.\\[4pt]

The entire coupled surface is empty &
Specified-input fibres violating a necessary condition
are empty; compatible zero data remain possible.\\[4pt]

A geometric two-jet fixes \(E_h\) through \(\kappa^2\) &
That coefficient needs a third metric jet.
The invariant two-jet condition is stated using
\(JG-\kappa^2gJT\) modulo \(\kappa^3\).\\[4pt]

The covariant free carrier imposes \(C[r]=0\) strongly &
Its equation is \(\Pgr r=0\); this distinction is
essential for the witness.\\[4pt]

There is no quantum graviton block &
A free block is present. It is not the VII-C block
and is not a full interacting metric.\\[4pt]

All tree coefficients are smooth tests in \(\DD_C\) &
The complete cubic kernel is a compact distribution.
Its smooth partial contractions lie in \(\DD_C\).\\[4pt]

The cubic belongs to the degree-one quotient \(\Qm\) &
Only its contracted linear tests do. The cubic belongs
to the polynomial carrier quotient.\\[4pt]

The normal microlocal space is a smooth Fr\'echet stage &
No such identification is made.\\[4pt]

The tree has first-slot support in \(K\) &
Its scalar-response part can have first-slot support
in \(S\); the safe bound is \(C\times S\).\\[4pt]

A two-point bound proves cubic admission automatically &
The repeated scalar point is inserted explicitly and
its conormal covectors are retained.\\[4pt]

The tree includes \(\kappa^2\) twice &
\(\tau\) is the coefficient; the contribution is
\(\kappa^2\tau\).\\[4pt]

The scalar second-order response is added again &
It remains the single pair in \eqref{eq:completed}.\\[4pt]

The dressing formula omits off-shell stress divergence &
The extra \(g(P_mu)\partial u\) is removed only on
the free scalar shell.\\[4pt]

One vanishing pairing identifies source conservation &
Only a global zero-past field identity permits that
inference by applying \(\Box^2\).\\[4pt]

The defect is nonzero for every fixed test &
The identity holds for every admissible instance;
nonzero detection is an explicitly quantified
existence theorem.\\[4pt]

The defect obstructs every possible tree completion &
It is a witness for the declared one-tree truncation,
not for later all-tree constructions.\\[4pt]

Smooth multiplication supplies a loop subtraction &
It neither defines an absent product nor selects a
renormalization prescription.\\[4pt]

The table acquires an unknown field &
It encodes supplied evaluations or enclosures.\\[4pt]

Clock destruction removes geometric time &
Only finite quantum clock names are consumed.\\[4pt]

All clock labels are actual times of one event &
Distinct event-time calibration forces a single
positive label; histories require a different map.\\[4pt]

Scheduled ancestry is the physical causal past &
Only containment follows from independently certified
routes.\\[4pt]

SWAP transport derives microscopic gravity &
Finite endpoint operations and causal routes are
declared permissions.\\[4pt]

Reference preservation is gravity-generated entanglement &
The protocol preserves correlations already present.\\[4pt]

A proposed integration list certifies registration &
No repository edits or tests have been executed here.\\

\bottomrule
\end{longtable}

%=====================================================================
\section{Documentary comparisons, not mathematical premises}
\label{app:comparisons}
%=====================================================================

The requested OpenAI catalogue comparisons have the
following documentary scope:
\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.13\linewidth}p{.39\linewidth}X@{}}
\toprule
Family & Supplied topic description & No transfer to this paper\\
\midrule
282 &
Scale-to-conformal and free-scalar comparisons &
Not a metric admission or emergence theorem\\
362 &
Relativistic Vlasov--Maxwell smoothness &
Not the Einstein--Hilbert variation or this tree\\
376 &
Forced Navier--Stokes analysis &
Not a loop regulator or ultraviolet completion\\
\bottomrule
\end{tabularx}
\end{center}

These are attributed to the separate OpenAI catalogue
\cite{OpenAICatalogue}. Their proofs are not verified
here and none is a premise. The comparison does not
assert that a reported catalogue claim is established
external mathematics.

Similarly, the supplied Em-series growth and
asymptotic records concern their named distribution
families. Moderate growth is not characteristic
convergence, an isolated divergent raw family is not
the present tree, and a scalar diagnostic is not a
complete quantum-graph coefficient
\cite{EmRecords}.

%=====================================================================
\section{Proposed companion integration contract}
\label{app:integration}
%=====================================================================

This appendix records the requested integration path,
not executed changes. The source remains proposed.

A repository maintainer may:
\begin{enumerate}
\item install this file as
\loc{content/manuscripts/X-E-ST.tex};
\item add the corresponding import and file-list entry
to \loc{sources.server.ts};
\item add \loc{ManuscriptMeta} in \loc{registry.ts},
with the actual file version and hash, and the
routing note below;
\item add bearings and boundary text to
\loc{qg-problems.ts};
\item add separation edges in \loc{connections.ts}
to X-E, X-C-ST, X-D-ST, I-B, and VII-C, without
identifying their domains;
\item add the scoped emergence section and the
unconstructed successor obligations in
\loc{proposals.ts};
\item run the repository's actual source, reference,
metadata, connection, and proposal tests, and
record their real results.
\end{enumerate}

\paragraph{Proposed routing note.}
X-E-ST is a local Grade S supported realization of a
fixed-width action and one scalar-sourced retarded
tree on the inherited conical host and declared flat
completion. Its scalar cubic is admitted with corrected
compact support and topology. Its one-tree background
identity stops at a displayed second-order truncation
defect. It does not admit a complete interacting
quantum metric, construct a loop regulator, derive
Einstein dynamics from records, or construct spacetime
emergence.

\paragraph{Proposed metadata summary.}
\begin{verbatim}
identifier: X-E-ST
sourceStatus: proposed
grade: local Grade S
primaryAnalyticSource: X-E@1
hostRetainedFrom:
  - I-B
  - X-A-ST
  - X-C-ST
  - X-D-ST
relationshipRole: supported-transfer-and-audit
widthStatus: fixed-positive-width
constructed:
  - Palatini and switched Noether identities
  - gauge-fixed retarded inverse
  - one scalar-sourced Wick cubic
  - fixed-support kernel and contracted-test continuity
  - one-tree background defect and supported witness
  - finite supplied-evaluation instrument and route
notConstructed:
  - complete interacting second-order quantum metric
  - self-interaction or all-tree completion
  - VII-C mandatory word
  - time-ordered hierarchy or loop subtraction
  - sharp mixed-metric admission
  - spacetime emergence
registryEditsExecuted: false
repositoryTestsExecuted: false
compilationAsserted: false
\end{verbatim}

No invented registry schema, generated hash, passing
test report, or external registration is represented
by this summary.

%=====================================================================
\section*{Declarations}
\addcontentsline{toc}{section}{Declarations}

\paragraph{Authorship and attribution.}
The author name follows the supplied predecessor.
The action and principal one-tree results are credited
to N.~Cheung's X-E. The supported transfer and the
explicit corrections identified in this file are not
presented as priority claims for those parent results.

\paragraph{Dependency scope.}
The inherited scalar formulae and supported host are
stated explicitly. Standard Green theory, wavefront
calculus, normal-topology operations, and the free
Wick framework are cited where used.
The unrelated characteristic-map programme and the
OpenAI catalogue are not premises.

\paragraph{Quantum scope.}
The admitted free graviton and scalar cubic are not
a complete interacting quantum metric, a physical
graviton state, a quantum master equation, or the
inherited characteristic coefficient. No loop
calculation or ultraviolet completion is claimed.

\paragraph{Operational scope.}
The finite interface encodes supplied evaluations.
It is not a field detector or an acquisition theorem.
The supplied geometry and geometric time remain
after clock consumption.

\paragraph{Verification.}
No successful \loc{pdflatex} execution,
proof-assistant check, symbolic-program execution,
numerical validation, repository integration,
independent referee review, experiment, or journal
acceptance is asserted. The mathematical release
consists of the definitions and written proofs above.

%=====================================================================
\begin{thebibliography}{99}
\addcontentsline{toc}{section}{References}

\bibitem{XE}
N.~Cheung,
\emph{Landmark X-E: A specified gravitational action
for the ramp relational scalar, and the first graviton tree}.
\loc{X-E.tex}, version 1.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/X-E}.
Parent action, Palatini variation, free normalization,
matter tree, background defect, and witness.

\bibitem{XAST}
N.~Cheung,
\emph{X-A-ST: Supported classical covariance of a
fixed-ramp relational scalar}.
Supplied manuscript \loc{X-A-ST.tex}, version 1.
Inherited host, completion, signature ledger, coordinate
prescription, and supported classical handoff.

\bibitem{XCST}
N.~Cheung,
\emph{X-C-ST: Second-order scalar completion on the
inherited X-A-ST/X-B-ST host}.
Supplied manuscript \loc{X-C-ST.tex}, version 1.
Completed scalar tests, affine slot, counting, and
background identity.

\bibitem{XDST}
N.~Cheung,
\emph{X-D-ST: The sharp-width limit of the completed
scalar tests}.
Supplied \loc{X-D-ST.tex}.
Fixed-support scalar topology, quotient, completion-normal
transmission, and quantifier audit.
The source supplied with this request is the referenced
version; no external registration is inferred.

\bibitem{XC}
N.~Cheung,
\emph{Landmark X-C: The missing second-order retarded
maps for the ramp relational scalar, with the metric
kept classical}.
\loc{X-C-v3.tex}, version 3.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/X-C}.

\bibitem{XD}
N.~Cheung,
\emph{Landmark X-D: Convergence of the completed ramp
relational scalar as the regulator width tends to zero}.
\loc{X-D-v3.tex}, version 3.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/X-D}.
Separate fixed-background scalar sharp-limit result.

\bibitem{XF}
N.~Cheung,
\emph{Landmark X-F}.
\loc{X-F.tex}, version 1.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/X-F}.
Documentary comparison: background-gauge and all-tree
completion on a different declared sheet; its construction
is not imported.

\bibitem{IB}
N.~Cheung,
\emph{I-B: A Finite Enter Spacetime:
Calibrated Destructive Time Readout on a Prescribed
Conical Product}.
\loc{I-B-v1.tex}, version 1.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/I-B}.
Prescribed host, worldsheet, clock calibration, and
target-adapted regular transport.

\bibitem{EnterI}
N.~Cheung,
\emph{Manuscript I}.
\loc{I.tex}, version 2.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/I}.
Destructive-readout characterization within its stated
operational class; contextual attribution only.

\bibitem{EnterPhilosophy}
N.~Cheung,
\emph{The Philosophy of Enter: Destructive Readout,
Quantum Recovery, and the Refusal of Identification}.
\loc{EntPhil-v2.tex}, version 2.
Enter Calculus Companion.
\url{https://enter.grok.me/enter}.
Interpretive background, not an action or field-theory
premise.

\bibitem{BGP}
C.~B\"ar, N.~Ginoux, and F.~Pf\"affle,
\emph{Wave Equations on Lorentzian Manifolds and Quantization}.
ESI Lectures in Mathematics and Physics,
European Mathematical Society, 2007.
\url{https://arxiv.org/abs/0806.1036}.
Cauchy theory, finite propagation, Green operators,
and their continuity; used on the completion.

\bibitem{FR}
K.~Fredenhagen and K.~Rejzner,
\emph{Perturbative algebraic quantum field theory}.
arXiv:1208.1428, version 2, 2013.
\url{https://arxiv.org/abs/1208.1428}.
Free microcausal functionals, Wick products, and
equation ideals. No interacting hierarchy is imported.

\bibitem{Hormander}
L.~H\"ormander,
\emph{The Analysis of Linear Partial Differential Operators I:
Distribution Theory and Fourier Analysis},
second edition, Springer, 1990;
Classics in Mathematics reprint, 2003.
Tensor products, wavefront calculus, and pushforwards.

\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 counterexample by Semyon Alesker),''
\emph{Studia Mathematica} \textbf{232} (2016), 201--226.
\url{https://doi.org/10.4064/sm8316-3-2016}.
Normal-topology continuity and hypocontinuity
interfaces.

\bibitem{VIIC}
N.~Cheung,
\emph{VII-C}.
\loc{VII-C.tex}, version 1.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/VII-C}.
Specified normal-topology sharp Wick-extension
obstruction; comparison only.

\bibitem{IXAST}
N.~Cheung,
\emph{IX-A-ST}.
\loc{IX-A-ST.tex}, version 1.
Supplied manuscript record,
Enter Calculus Companion.
\url{https://enter.grok.me/browse/IX-A-ST}.
Boundary-sensitive unchanged sharp-insertion topology;
not identified with the topology used here.

\bibitem{VIIE}
N.~Cheung,
\emph{VII-E}.
\loc{VII-E.tex}, version 1.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/VII-E}.
Complete scheduled-support and readout-dependence
mechanism.

\bibitem{En08A}
N.~Cheung,
\emph{En08-A}.
\loc{En08-A.tex}, version 1.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/En08-A}.
Scheduler consistency and the separate geometric
reconstruction question.

\bibitem{En08B}
N.~Cheung,
\emph{En08-B}.
Version 1, Enter Calculus Companion.
\url{https://enter.grok.me/browse/En08-B}.
Finite reference-preserving transport relative to a
prescribed null boundary.

\bibitem{En09}
N.~Cheung,
\emph{En09-A: Relational Enter}.
\loc{En09-A.tex}, version 1.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/En09-A}.
Complete clock instrument, subnormalized conditioning,
and reference extension.

\bibitem{Watrous}
J.~Watrous,
\emph{The Theory of Quantum Information}.
Cambridge University Press, 2018.
Finite instruments, completely bounded trace norm,
and channel comparison.

\bibitem{EmRecords}
N.~Cheung,
\emph{Em01-B}, version 3;
\emph{Em01-C}, version 3;
and \emph{Em02-D}, version 4.
Version-specific supplied manuscript excerpts,
Enter Calculus Companion.
\url{https://enter.grok.me/browse}.
Separate smoothing-growth, scalar-diagnostic, and
raw-family asymptotic comparisons; none is an analytic
premise of this paper.

\bibitem{OpenAICatalogue}
OpenAI,
\emph{Mathematics catalogue}.
Supplied documentary descriptions of families
282, 362, and 376 in the Enter Calculus Companion.
\url{https://enter.grok.me/browse}.
Separate catalogue attribution only; its proofs are
not verified or used here.

\bibitem{BoundaryRecords}
N.~Cheung,
\emph{XIV-H}, \emph{XIV-I}, and \emph{XIV-J},
version 1; technical \emph{XV-B}, version 2;
and \emph{XVI-N}, revision r12.
Version-specific supplied manuscript records,
Enter Calculus Companion.
\url{https://enter.grok.me/browse}.
Historical and full characteristic-map boundaries,
separate from the fixed-width tree constructed here.

\end{thebibliography}

\end{document}