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\hypersetup{
  pdftitle={X-F-ST: Three fixed-width retarded trees and the background identity},
  pdfauthor={Ngo Cheung},
  pdfsubject={Supported transfer, background de Donder gauge,
  equation-level response, microcausal admission, and local Grade S}
}

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\title{\textbf{X-F-ST: Three fixed-width retarded trees\\
and the background identity}\\[5pt]
\large A supported transfer with a variational, support,\\
and equation-ideal audit}
\author{Ngo Cheung}
\date{}

\begin{document}
\maketitle

\begin{abstract}
We give a supported realization of Cheung's Landmark X-F
on the conical host and independently declared Minkowski
completion retained by X-A-ST, X-D-ST, and X-E-ST.
At fixed positive ramp width, background de Donder gauge
is applied to the quantum fluctuation, while the classical
metric coefficients remain unconstrained.

The mechanism is a typed commutator identity:
\[
 Q^{(1)}[K\xi]
   =\Lie_\xi^{\mathrm{dens}}\Pgr
                         -\Pgr\Lie_\xi^{\mathrm{cov}}.
\]
It yields a retarded second-order response whose background
variation is \(\Lie_\xi\rfield\) modulo the free graviton
equation ideal. The mixed tree consequently cancels the
one-tree defect of X-E-ST. Together with the matter and
self-interaction trees, the completed observable is
background invariant through order \(\kappa^2\hbar^0\),
with both free fields held fixed. The exact Einstein-density
covariance identity retains its off-shell Lie term; no
linearized Einstein constraint is imposed on the
gauge-fixed free graviton.

We construct all three kernels, prove compact support and
microcausal admission by a three-slot sum-zero argument,
and exhibit the graviton-slot equation representative
needed for the cancellation. The matter kernel retains
support \(C^3\); the mixed and self kernels can reach the
initial-layer causal reserve. Smooth scalar-slot contractions
lie in \(\DD_C\), whereas the complete kernels are
distributions in a polynomial carrier.

A necessary correction concerns interpretation: source
switching defines an equation-level retarded response,
not automatically a variational M{\o}ller map.
A local formal-adjoint test shows why that stronger
identification cannot be asserted for the unchanged
off-shell source law. Thus the direct tree identity
survives without claiming a switched-action or
Bogoliubov construction.

The principal tree mechanism is credited to X-F.
The contribution here is its supported proof and explicit
type, support, topology, and variational audit. This
provides a reproducible tree-level baseline for successors
without identifying it with loop admission or spacetime
emergence. A finite supplied-evaluation dictionary has a
complete destructive-clock instrument and regular
reference-preserving transport. The release is local
Grade S only.
\end{abstract}

\statementbox{Corrections fixed before the calculation}{%
The response constructed here is the unique retarded solution
of a prescribed source equation. ``Variational M{\o}ller map''
is not used as a synonym.

The two Lie derivatives in the commutator have different
tensor types.

The kernel before onset multiplication need not have compact
support in its source argument. Compactness is proved after
multiplication by \(g\), \(\sigma\), or compact generator
coefficients.

Only contractions leaving the scalar slot necessarily lie in
\(\DD_C\). A contraction leaving a graviton slot can have
support in the larger causal reserve.

Each tree is a coefficient. The observable multiplies their
sum by \(\kappa^2\) exactly once.}

\tableofcontents

%=====================================================================
\section{Purpose, source status, Enter, and the retained host}
\label{sec:host}
%=====================================================================

\subsection{What is transferred}

The analytical source is N.~Cheung's X-F, version 1,
which replaces the flat interaction gauge of X-E by a
background de Donder gauge and supplies the mixed and
self trees in addition to the matter tree
\cite{XF,XE}. The supplied X-E-ST and X-D-ST are
predecessor source records \cite{XEST,XDST}.
Their labels do not by themselves assert registration
or independent verification.

This paper states the inherited definitions needed by the
proof. It does not assume the parent all-tree theorem as a
premise. The direct commutator, admission, and equation-ideal
arguments are supplied below.

The supplied X-G correction distinguishes retarded
solution maps from variational M{\o}ller maps
\cite{XG}. That distinction is incorporated explicitly.
It does not change the three source expressions or the
direct cancellation proved here.

The identifier X-F-ST denotes this proposed manuscript.
No repository installation, external registration,
submission, acceptance, or successful compilation is asserted.

\subsection{Enter as an interface discipline}

The relevant Enter distinction is destructive classical
readout versus quantum return. Measured quantum clock names
are consumed; the complete returned interface contains a
classical record and the unmeasured residual
\cite{EnterI,EnterPhilosophy,En09}.

We retain four separations:
\begin{enumerate}
\item a supplied mathematical evaluation is not acquisition
of unknown continuum fields;
\item agreement of record probabilities is weaker than
agreement of complete instruments;
\item preservation of an untouched reference is not
generation of entanglement by gravity;
\item a protocol on a prescribed spacetime is not a
derivation of that spacetime or its dynamics.
\end{enumerate}
I-B supplies the prescribed-host clock and transport
interpretation \cite{IB}. It supplies neither the
Einstein--Hilbert action nor the three trees.

The finite operational results below use explicitly
specified Born instruments and prove their channel
identities directly. No abbreviated reconstruction of
Enter's characterization theorem is attempted.

\subsection{Exact retention of the host}

\begin{assumption}[The unchanged supported presentation]
\label{ass:host}
Retain the X-A-ST instance already used by X-D-ST and
the supplied X-E-ST:
\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
analytical equations retain \((+,-,-,-)\);
\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 used as a globally
hyperbolic Cauchy spacetime. Every global Green operator
and every free field in this paper belongs to
\(\widehat M\).
\end{assumption}

The singular set remains a worldsheet, not a worldline.
No second cone, global conical inverse, worldsheet
gluing, or host-metric evolution is introduced
\cite{IB,XAST}.

\begin{definition}[Local Grade S]
\label{def:grade}
The package consists of the inherited supported host;
named identities and response kernels with proved
normalization, support, and transformation laws;
a finite installed dictionary of supplied evaluations;
a complete destructive-clock instrument including
untouched references; and scheduled locality with a
nonempty regular reference-preserving route under
declared finite operational permissions.

Grade E, Einstein dynamics from records, and spacetime
emergence are not included.
\end{definition}

%=====================================================================
\section{Frozen analytical data and the scalar insertion target}
\label{sec:data}
%=====================================================================

\subsection{Supports and fixed onset}

Fix
\[
 m\ge0,\quad
 f\in C_c^\infty(\mathcal O;\RR),\quad
 g\in C_c^\infty((0,T_*)\times\RR^3;[0,1]).
\]
Choose a nonempty relatively compact open plateau \(U\)
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 \(\kappa\xi+\kappa^2\zeta\), where
\(\xi,\zeta\in\gp\).

Reserve \(K\) for the Killing operator
\[
 (K\xi)_{\mu\nu}
      =\partial_\mu\xi_\nu+\partial_\nu\xi_\mu
      =(\Lie_\xi\eta)_{\mu\nu}.
\]
For observation support write
\begin{equation}
 K_f=\supp f,\qquad
 S=\supp g\cap J_\eta^-(K_f),\qquad
 C=K_f\cup S,\qquad
 D_f=J_\eta^-(K_f)\cap M_{T_*}.
 \label{eq:sets}
\end{equation}
These sets are compact.

Fix \(\delta>0\) so that
\begin{equation}
 K_f\cup\supp g\cup\overline U
                     \subset\{2\delta<t<T_*\}.
 \label{eq:separation}
\end{equation}
Fix \(0<\varepsilon<\delta\), and set
\[
 \sigma(t)=\chi(t/\varepsilon),
\]
where \(\chi\) is smooth, zero for \(t\le0\), one
for \(t\ge1\), and flat at both endpoints.
The width remains positive throughout.

The classical coefficients \(h,k\) are arbitrary smooth
real symmetric tensors near the closed slab. No
equations or spatial decay are imposed. They are not
the host metric \(G\).

\subsection{Green operators and tensor types}

Let
\[
 \Box=\partial_t^2-\Delta,\qquad P_m=\Box+m^2.
\]
Use positive retarded and advanced inverses
\(\ER,\EA\) of \(P_m\), and
\(\ERzero,\EAzero\) of \(\Box\).
Thus \(P_m\ER=P_m\EA=1\) and
\(\ER^{\mathrm T}=\EA\). All existence,
finite-propagation, and continuity statements are
applied on the completion \cite{BGP}.

The fluctuation bundle is
\(E=\operatorname{Sym}^2T^*\widehat M\).
Its Euler derivatives are sections of
\(E^\vee=\operatorname{Sym}^2T\widehat M\otimes
|\Lambda^4T^*\widehat M|\).
The invariant pairing is integration of their
contraction.

Write
\begin{align}
 (\Lie_\xi^c r)_{\mu\nu}
   &=\xi^\lambda\partial_\lambda r_{\mu\nu}
       +r_{\lambda\nu}\partial_\mu\xi^\lambda
       +r_{\mu\lambda}\partial_\nu\xi^\lambda,
 \nonumber\\
 (\Lie_\xi^d A)^{\mu\nu}
   &=\partial_\lambda(\xi^\lambda A^{\mu\nu})
       -A^{\lambda\nu}\partial_\lambda\xi^\mu
       -A^{\mu\lambda}\partial_\lambda\xi^\nu.
 \label{eq:types}
\end{align}
They act respectively on \(E\) and \(E^\vee\), and
\begin{equation}
 \pair{r}{\Lie_\xi^d A}
                    =-\pair{\Lie_\xi^c r}{A}
 \label{eq:Lieadjoint}
\end{equation}
whenever the integrand has compact support.
An expression \([\Lie_\xi,Q]\), for
\(Q:E\to E^\vee\), always means
\(\Lie_\xi^d Q-Q\Lie_\xi^c\).

\subsection{Density coefficients and coordinate prescription}

Indices without an indicated metric are moved by \(\eta\).
For \(H=\tr_\eta l\), define
\begin{align}
 a_1^{\mu\nu}[l]&=\tfrac12H\eta^{\mu\nu}-l^{\mu\nu},
 &b_1[l]&=\tfrac12H,
 \nonumber\\
 b_2[l]&=\tfrac18H^2-\tfrac14l_{\alpha\beta}l^{\alpha\beta},
 \nonumber\\
 a_2^{\mu\nu}[l]
   &=l^\mu{}_\lambda l^{\lambda\nu}
      -\tfrac12Hl^{\mu\nu}+b_2[l]\eta^{\mu\nu},
 \nonumber\\
 A_2[h,k]&=a_1[k]+a_2[h],
 &B_2[h,k]&=b_1[k]+b_2[h].
 \label{eq:coefficients}
\end{align}
These are the coefficients of
\(J_qq^{\mu\nu}\) and \(J_q=\sqrt{-\det q}\) for
\(q=\eta+\kappa h+\kappa^2k\).

The zero-data coordinate coefficients are
\begin{align}
 \Box w^\rho[l]
    &=-\partial_\mu(\sigma a_1^{\mu\rho}[l]),
 \nonumber\\
 \Box v^\rho[h,k]
    &=-\partial_\mu\left(
       \sigma\{A_2^{\mu\rho}[h,k]
                      +a_1^{\mu\nu}[h]\partial_\nu w^\rho[h]\}\right).
 \label{eq:coordinates}
\end{align}
Their positions and velocities vanish at \(\Sigma\).
For nondecaying data these equations are defined locally
by finite-time Cauchy theory and finite propagation,
not by an unjustified integral over an infinite past.

The scalar 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_2^{\mu\nu}[h,k]\partial_\nu u)
                                  +gm^2B_2[h,k]u.
 \label{eq:L}
\end{align}
Every switch derivative remains inside these divergences.

\subsection{Completed tests, counted once}

Set
\begin{align}
 c_1[l]&=\partial_\mu(fw^\mu[l]),
 \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[l]&=c_1[l]-L_1[l]\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 X-C-ST insertion representatives
\cite{XCST,XC}. They are actual off-shell tests before
any quotient is taken.

\begin{lemma}[Affine slot and compact scalar support]
\label{lem:affine}
There is a symmetric bilinear test map
\(\mathcal Q\) such that
\begin{equation}
 d_2[h,k]=\mathcal Q[h,h]+d_1[k].
 \label{eq:affine}
\end{equation}
For fixed smooth data,
\(d_1[l],d_2[h,k]\in\DD_C\), continuously in
the smooth data near the retained causal reserve.
\end{lemma}

\begin{proof}
The \(k\)-linear part of the second coordinate
equation is the first coordinate equation with
input \(k\). Uniqueness gives
\(v[h,k]=v[h,0]+w[k]\).
The \(k\)-part of \(c_2\) is therefore \(c_1[k]\);
the \(k\)-part of \(L_2\) is \(L_1[k]\).
This proves the affine identity, with polarization
of the remaining quadratic expression.

The bare representatives are supported in \(K_f\).
Every \(L_j\)-term is supported in \(\supp g\),
and advanced propagation restricts it to
\(J_\eta^-(K_f)\). Smooth Cauchy estimates and Green
continuity prove the stated continuity.
\end{proof}

The existing scalar pair
\(-L_2\EA f+L_1\EA L_1\EA f\) is not regenerated.
The new operation is only the affine insertion of a
derived second-order response.

%=====================================================================
\section{The retained action and the invariant-jet boundary}
\label{sec:action}
%=====================================================================

This section fixes the inherited action and records the
identities needed to prevent an equation-space
misidentification. It does not reopen X-E-ST's action
choice \cite{XEST,XE}.

Use the curvature convention
\[
 R^\rho{}_{\sigma\mu\nu}
  =\partial_\mu\Gamma^\rho_{\nu\sigma}
     -\partial_\nu\Gamma^\rho_{\mu\sigma}
     +\Gamma^\rho_{\mu\lambda}\Gamma^\lambda_{\nu\sigma}
     -\Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\sigma},
 \qquad R_{\sigma\nu}=R^\rho{}_{\sigma\rho\nu}.
\]
The local action 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).
 \label{eq:action}
\end{align}
These denote local Lagrangians with compact variations;
global finiteness of the action integral is not assumed.

\begin{lemma}[Retained first variation and free normalization]
\label{lem:action}
Let
\(\mathcal E[q]=J_qG^{\mu\nu}[q]\), regarded as an
\(E^\vee\)-valued density. Then
\begin{align}
 \delta(J_qR)
   &=-J_qG^{\mu\nu}\delta q_{\mu\nu}
                           +\partial_\rho(J_qW^\rho),
 \nonumber\\
 W^\rho
   &=q^{\mu\nu}\delta\Gamma^\rho_{\mu\nu}
                   -q^{\rho\mu}\delta\Gamma^\nu_{\nu\mu},
 \nonumber\\
 \mathcal E[\eta+\lambda l]
   &=\lambda G_{\mathrm{lin}}[l]
                                  +\lambda^2G_2[l,l]+O(\lambda^3),
 \label{eq:Eexp}\\
 S_{\mathrm{FP}}[l]
   &=\tfrac14\pair{l}{G_{\mathrm{lin}}[l]},
 &\frac{\delta S_{\mathrm{FP}}}{\delta l}
   &=\tfrac12G_{\mathrm{lin}}[l].
 \nonumber
\end{align}
The quadratic transverse-traceless spatial action is
\[
 \frac18\int\left(
    \partial_tl_{ij}\partial_tl_{ij}
              -\partial_al_{ij}\partial_al_{ij}\right)\dd^4x.
\]
In particular its kinetic sign is positive.
\end{lemma}

\begin{proof}
The Palatini identity
\(\delta R_{\mu\nu}
=\nabla_\rho\delta\Gamma^\rho_{\nu\mu}
-\nabla_\nu\delta\Gamma^\rho_{\rho\mu}\),
together with
\(\delta J_q=\frac12J_qq^{\mu\nu}\delta q_{\mu\nu}\)
and
\(\delta q^{\mu\nu}
=-q^{\mu\alpha}q^{\nu\beta}\delta q_{\alpha\beta}\),
gives the first formula.
Differentiating it once at \(\eta\) gives the quadratic
action and its symmetric Hessian. On trace-free,
divergence-free spatial fields
\(G_{\mathrm{lin}}[l]=-\frac12\Box l\);
integration by parts gives the last display.
\end{proof}

Here \(G_2\) is the symmetric polarization of the
coefficient in \eqref{eq:Eexp}, not the second
coefficient of a differently indexed Einstein tensor.
Explicitly,
\[
 G_2[l,l]=\frac12
       \left.\frac{\dd^2}{\dd s^2}\right|_{s=0}
                     (J_{\eta+sl}G^{\mu\nu}[\eta+sl]).
\]
It is an unambiguous constant-coefficient quadratic
differential polynomial of total derivative order two.
For a finite component basis it can be written
\begin{equation}
 G_2^I[r_1,r_2](y)
   =\sum_{\substack{J,L,\alpha,\beta\\
                         |\alpha|+|\beta|\le2}}
       b^{I,\alpha\beta}_{JL}
                   (\partial^\alpha r_1^J)(y)
                   (\partial^\beta r_2^L)(y).
 \label{eq:Boperator}
\end{equation}
The constants are fixed by the displayed derivative
definition; no independent cubic vertex is selected.

For completeness,
\[
 G_{\mathrm{lin},\mu\nu}[l]
 =\tfrac12\left(
   \partial_\rho\partial_\mu l^\rho{}_\nu
  +\partial_\rho\partial_\nu l^\rho{}_\mu
  -\Box l_{\mu\nu}-\partial_\mu\partial_\nu\tr_\eta l
  -\eta_{\mu\nu}(\partial_\alpha\partial_\beta l^{\alpha\beta}
                                      -\Box\tr_\eta l)\right).
\]

Define
\[
 T^{\mu\nu}[q,\phi]
 =q^{\mu\alpha}q^{\nu\beta}\partial_\alpha\phi\partial_\beta\phi
    -\tfrac12q^{\mu\nu}
       (q^{\alpha\beta}\partial_\alpha\phi\partial_\beta\phi
                                                     -m^2\phi^2),
 \qquad T_0=T[\eta,\phi].
\]
Then
\begin{align}
 \mathscr E_q
    &=\frac1{2\kappa^2}
                (\mathcal E[q]-\kappa^2gJ_qT[q,\phi]),
 \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:Euler}\\
 V_{1,g}[l,\phi]
    &=-\tfrac12\int g\,l_{\mu\nu}T_0^{\mu\nu}[\phi]\dd^4x,
 &\frac{\delta V_{1,g}}{\delta l}&=-\tfrac12gT_0[\phi].
 \nonumber
\end{align}

\begin{proposition}[Exact switch defect and the retained obstruction]
\label{prop:Noether}
With \(g,\eta\) fixed,
\begin{align}
 2q_{\rho\nu}\nabla_\mu\mathscr E_q^{\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}
A coupled invariant two-jet with free input \(P_m\phi_0=0\)
must satisfy
\begin{equation}
 (\partial_\mu g)T_0^\mu{}_\rho[\phi_0]=0.
 \label{eq:Bianchi}
\end{equation}
For nonconstant \(g\), an open dense set of smooth free
inputs violates a pointwise necessary condition, so its
specified-input coupled-jet fibres are empty.
\end{proposition}

\begin{proof}
Under \(\delta q=\Lie_\xi q\),
\(\delta\phi=\xi^\rho\partial_\rho\phi\), the curved
densities vary by divergences. At fixed \(\eta\),
\[
 \delta\mathcal L_\eta
   =\partial_\mu(\xi^\mu\mathcal L_\eta)
                                      +T_0^{\mu\nu}\partial_\mu\xi_\nu.
\]
Integrate against the switch factors and compare with
the Euler variation. Using
\(\partial_\mu T_0^{\mu\nu}=(P_m\phi)\partial^\nu\phi\)
gives \eqref{eq:Noether}.

Alternatively expanding the exact Bianchi identity on
\[
 G_{\mathrm{lin}}[h]=0,\qquad
 G_{\mathrm{lin}}[k]+G_2[h,h]=gT_0[\phi_0]
\]
immediately gives \eqref{eq:Bianchi}.

Choose \(x_0\) with \(\dd g(x_0)\ne0\). The violation
at \(x_0\) is a continuous quadratic polynomial in
the free scalar's first jet. A free solution \(v\)
with \(v(x_0)=0\) and \(\partial v(x_0)=\dd t\)
has mixed stress
\(\operatorname{diag}(1/2,-1/2,-1/2,-1/2)\);
such a first jet is realized by smooth Cauchy data.
Its contraction with \(\dd g(x_0)\) is nonzero.
For every \(u\), at least one component at \(u+\lambda v\)
has a nonzero quadratic coefficient in \(\lambda\).
Arbitrarily small \(\lambda\) therefore give a violation.
This proves density; continuity proves openness.
\end{proof}

The full invariant-jet family is not declared empty.
This obstruction is not removed by background gauge
fixing or by constructing retarded responses.

%=====================================================================
\section{F1: exact covariance with the Lie term retained}
\label{sec:F1}
%=====================================================================

\begin{lemma}[Naturality and typed second-order covariance]
\label{lem:cov2}
For every smooth vector field \(\xi\) and smooth
symmetric tensor \(r\),
\begin{equation}
 G_{\mathrm{lin}}[K\xi]=0,\qquad
 2G_2[K\xi,r]
       =\Lie_\xi^dG_{\mathrm{lin}}[r]
                       -G_{\mathrm{lin}}[\Lie_\xi^c r].
 \label{eq:cov2}
\end{equation}
No field equation is required.
\end{lemma}

\begin{proof}
The Einstein density is natural:
\(\mathcal E[F^*q]=F^*\mathcal E[q]\).
This follows either from the natural curvature
construction or by differentiating the invariant
Einstein--Hilbert action against compact variations.

Let \(F_s\) be the flow of a compact \(\xi\), and apply
naturality to \(q=\eta+\lambda r\).
The left input is
\[
 F_s^*(\eta+\lambda r)
   =\eta+sK\xi+\lambda r+s\lambda\Lie_\xi^c r
                                      +O(s^2)+O(s^2\lambda).
\]
The coefficient of \(s\) is
\(G_{\mathrm{lin}}[K\xi]=0\).
The coefficient of \(s\lambda\) is
\[
 G_{\mathrm{lin}}[\Lie_\xi^c r]+2G_2[K\xi,r]
                              =\Lie_\xi^dG_{\mathrm{lin}}[r].
\]
The expressions are local differential expressions,
so compact localization of \(\xi\) proves the identity
for arbitrary smooth germs.
\end{proof}

The exact identity is the F1 mechanism of X-F
\cite[Exact second-order covariance identity]{XF}.
The shortcut
\(G_{\mathrm{lin}}[\Lie_\xi r]=-2G_2[K\xi,r]\)
requires \(G_{\mathrm{lin}}[r]=0\).
A gauge-fixed free solution need not satisfy that
stronger equation.

%=====================================================================
\section{F2: background gauge, kinetic correction, and retarded jet law}
\label{sec:F2}
%=====================================================================

\subsection{The unchanged free operator}

Define
\[
 (\Ptr r)_{\mu\nu}
      =r_{\mu\nu}-\tfrac12\eta_{\mu\nu}\tr_\eta r,\qquad
 C^\mu[r]=\partial_\nu(\Ptr r)^{\nu\mu}.
\]
The inherited flat gauge fixing is
\(S_{\mathrm{gf}}^{(0)}=\frac14\int C_\mu C^\mu\).
Compact variation gives
\begin{equation}
 F^{(0)}=-\tfrac14\Ptr KC,\qquad
 \tfrac12G_{\mathrm{lin}}+F^{(0)}
      =\Pgr=-\tfrac14\Ptr\Box.
 \label{eq:P}
\end{equation}
Indeed,
\(G_{\mathrm{lin}}
 =\Ptr(-\frac12\Box+\frac12KC)\);
the \(KC\)-terms cancel.

Since \(\Ptr^2=1\) in four dimensions, the free
retarded and advanced inverses are
\begin{equation}
 \DR=-4\Ptr\ERzero,\qquad \DA=-4\Ptr\EAzero.
 \label{eq:Delta}
\end{equation}
These map tensor densities to covariant tensors;
constant \(\eta\)-identifications are implicit in
component notation.

The free equation is \(\Pgr r=0\), equivalently
\(\Box r=0\). It does not impose \(C[r]=0\) strongly.
The compact residual statement also remains unchanged:
if \(\xi\) is compact and \(\Pgr K\xi=0\), then
\(K\Box\xi=0\); the compact Killing field \(\Box\xi\)
is zero, and wave uniqueness gives \(\xi=0\).

\subsection{Background de Donder gauge}

Let \(\bar q=\eta+\kappa h\). For a covariant
fluctuation \(r\), put
\begin{align}
 \bar r^{\mu\nu}
   &=\bar q^{\mu\alpha}\bar q^{\nu\beta}r_{\alpha\beta}
       -\tfrac12\bar q^{\mu\nu}\bar q^{\alpha\beta}r_{\alpha\beta},
 \nonumber\\
 C_{\bar q}^{\mu}[r]&=\bar\nabla_\nu\bar r^{\nu\mu},
 \nonumber\\
 S_{\mathrm{gf}}[r;\bar q]
   &=\tfrac14\int J_{\bar q}\bar q_{\mu\nu}
                              C_{\bar q}^\mu[r]C_{\bar q}^\nu[r]\dd^4x,
 \nonumber\\
 F[r;\bar q]&=\frac{\delta S_{\mathrm{gf}}}{\delta r}
       =F^{(0)}r+\kappa F^{(1)}[h]r+O(\kappa^2).
 \label{eq:backgroundgauge}
\end{align}
Only the fluctuation is gauge fixed. Classical
backgrounds remain off shell and unconstrained.
No extra de Donder constraint ideal is imposed.

An explicit specification of \(F^{(1)}\) is useful.
Set \(\bar r_0=\Ptr r\), \(H=\tr_\eta h\), and
\begin{align}
 \beta^{\rho\nu}[h;r]
    &=-h^{\rho\alpha}r_\alpha{}^\nu
       -h^{\nu\alpha}r^\rho{}_\alpha
       +\tfrac12h^{\rho\nu}\tr_\eta r
       +\tfrac12\eta^{\rho\nu}h^{\alpha\beta}r_{\alpha\beta},
 \nonumber\\
 C_1^\nu[h;r]
    &=\partial_\rho\beta^{\rho\nu}
       +\tfrac12(\partial_\lambda H)\bar r_0^{\lambda\nu}
       +\Gamma_1^\nu{}_{\rho\lambda}[h]\bar r_0^{\rho\lambda},
 \nonumber\\
 \Gamma_1^\nu{}_{\rho\lambda}[h]
    &=\tfrac12\eta^{\nu\alpha}
       (\partial_\rho h_{\alpha\lambda}
         +\partial_\lambda h_{\alpha\rho}
         -\partial_\alpha h_{\rho\lambda}).
 \label{eq:C1}
\end{align}
Then \(F^{(1)}[h]r\) is the Euler derivative in \(r\)
of the fully specified quadratic density
\begin{equation}
 \mathcal L_{\mathrm{gf}}^{(1)}[h;r]
   =\tfrac14b_1[h]C_\mu C^\mu
       +\tfrac14h_{\mu\nu}C^\mu C^\nu
       +\tfrac12\eta_{\mu\nu}C^\mu C_1^\nu[h;r].
 \label{eq:gf1}
\end{equation}
Thus no unspecified gauge vertex is hidden in the notation.

\begin{proposition}[Kinetic operator and its support]
\label{prop:Q}
Define
\[
 Q[\bar q]r=\tfrac12D\mathcal E[\bar q]\,r+F[r;\bar q].
\]
It is the fluctuation-quadratic action Hessian, and
\begin{equation}
 Q[\eta+\kappa h]=\Pgr+\kappa Q^{(1)}[h]+O(\kappa^2),
 \qquad
 Q^{(1)}[h]r=G_2[h,r]+F^{(1)}[h]r.
 \label{eq:Q1}
\end{equation}
The operator \(Q^{(1)}[h]\) is formally self-adjoint
and of order at most two in \(r\).
Its coefficients are linear in the two-jet of \(h\), and
\[
 \supp(Q^{(1)}[h]r)\subset\supp h.
\]
Dependence on a background coefficient \(\kappa^2k\)
first changes \(Q\) at order \(\kappa^2\), hence affects
the observable beyond the order considered here.
\end{proposition}

\begin{proof}
Differentiate \eqref{eq:Eexp} in a fluctuation direction.
The term linear in \(r\) at order \(\kappa\) is
\(G_2[h,r]\).
Expansion of \eqref{eq:backgroundgauge} gives
\eqref{eq:C1}--\eqref{eq:gf1}, hence the other term.
Both arise as second variations of local quadratic
actions, proving formal self-adjointness.
Their explicit local differential formulas prove
order, coefficient dependence, and support.
\end{proof}

\subsection{The commutator identity}

\begin{theorem}[Background-gauge commutator]
\label{thm:key}
For every smooth \(\xi,r\),
\begin{align}
 F^{(1)}[K\xi]r
     &=\Lie_\xi^d(F^{(0)}r)-F^{(0)}(\Lie_\xi^c r),
 \nonumber\\
 Q^{(1)}[K\xi]r
     &=\Lie_\xi^d(\Pgr r)-\Pgr(\Lie_\xi^c r).
 \label{eq:key}
\end{align}
\end{theorem}

\begin{proof}
The background gauge-fixing density is natural under
simultaneous pullback of \(r,\bar q\).
Differentiating its invariant compact-variation
pairing gives
\[
 F[F_s^*r;F_s^*\eta]=F_s^*F[r;\eta].
\]
Its derivative at zero is the first identity.
Combine it with \eqref{eq:cov2} and
\eqref{eq:P}. Equivalently, differentiate
\(Q[F_s^*\eta](F_s^*r)=F_s^*(Q[\eta]r)\).
\end{proof}

This is the X-F commutator mechanism
\cite[Key identity]{XF}.
Both the Einstein-shell term and the gauge-fixing
correction are retained. No causal-shadow hypothesis
or extended pure-gauge representative is required.

\subsection{Derivation of the prescribed response equation}

Consider formally
\[
 q=\eta+\kappa(h+r)+\kappa^2k,\qquad
 S_{\mathrm{gr}}[q]+S_{\mathrm{gf}}[r;\eta+\kappa h]
                                      +S_{\mathrm{sc},g}[q,\phi].
\]
Its fluctuation Euler derivative through first order is
\begin{align}
 &\tfrac12G_{\mathrm{lin}}[h]+\Pgr r
 \nonumber\\
 &\quad+\kappa\left(
    \tfrac12G_{\mathrm{lin}}[k]+\tfrac12G_2[h,h]
       +Q^{(1)}[h]r+\tfrac12G_2[r,r]-\tfrac12gT_0[\phi]\right)
       +O(\kappa^2).
 \label{eq:fluctuation}
\end{align}
This follows directly from Lemma~\ref{lem:action},
polarization, and \eqref{eq:backgroundgauge}.

The background-field prescription removes the terms
independent of the fluctuation from its equation:
equivalently, it removes constant and linear
fluctuation terms from the local action expansion.
It does \emph{not} remove the linear kinetic equation.
The classical backgrounds \(h,k\) are not solved for.

Now write the fluctuation as \(r+\kappa k_q\), and
multiply the order-\(\kappa\) gravitational sources,
after Euler differentiation, by \(\sigma\).
The resulting definition is
\begin{align}
 \mathrm{Src}[h;r,\phi]
   &=\tfrac12gT_0[\phi]-\sigma Q^{(1)}[h]r
                                      -\tfrac12\sigma G_2[r,r],
 \nonumber\\
 k_q[h;r,\phi]&=\DR\,\mathrm{Src}[h;r,\phi].
 \label{eq:response}
\end{align}
For smooth inputs this is the unique zero-past
solution of \(\Pgr k_q=\mathrm{Src}\).

Every source in \eqref{eq:response} is past-compact:
its intersection with the causal past of a compact
set is compact. This follows from its vanishing for
\(t<0\) and flat finite propagation.
No spatial decay is needed \cite{BGP}.
Without onset multiplication the self source of a
general free field is not past-compact, and the
retarded Green operator cannot be applied on that
basis.

\subsection{The dynamical jet law}

\begin{lemma}[Retarded left inverse and transposition]
\label{lem:leftinverse}
For a smooth past-compact tensor \(u\),
\[
 \DR\Pgr u=u.
\]
For compact smooth tensor-density tests \(A\) and
past-compact smooth sources \(B\),
\[
 \pair{A}{\DR B}=\pair{\DA A}{B}.
\]
The same equalities hold distributionally on the
admissible compact kernel stages used below.
\end{lemma}

\begin{proof}
Use \(\Ptr^2=1\), constancy of \(\Ptr\), and
\(\ERzero\Box u=u\) on past-compact fields.
The second identity is retarded--advanced transposition.
Only the compact intersection
\(J^-(\supp A)\cap\supp B\) contributes.
Distributional versions follow by transposition on
the same proper-support stages.
\end{proof}

\begin{theorem}[The second-order law is a response identity]
\label{thm:jet}
For \(\xi\in\gp\), with \(r,\phi\) held fixed,
\begin{equation}
 k_q[h+K\xi;r,\phi]-k_q[h;r,\phi]
    =\Lie_\xi^c r-\DR\Lie_\xi^d\Pgr r.
 \label{eq:jet}
\end{equation}
For a smooth free \(r\), the second term is zero.
For the free quantum field it is an equation-ideal
term after admissible compact smearing.
\end{theorem}

\begin{proof}
The background dependence of the source is linear:
\[
 \delta_\xi\mathrm{Src}=-\sigma Q^{(1)}[K\xi]r
                                      =-Q^{(1)}[K\xi]r.
\]
The second equality holds because the operator
coefficients are supported in \(\supp\xi\subset U\),
where \(\sigma=1\).
By \eqref{eq:key},
\[
 -Q^{(1)}[K\xi]r
       =\Pgr\Lie_\xi^c r-\Lie_\xi^d\Pgr r.
\]
The field \(\Lie_\xi^c r\) has compact support.
Apply Lemma~\ref{lem:leftinverse}.

For a compact smooth test \(A\),
\begin{equation}
 \pair{A}{\DR\Lie_\xi^d\Pgr\rfield}
       =-\rfield\bigl(\Pgr(\Lie_\xi^c\DA A)\bigr).
 \label{eq:smearedideal}
\end{equation}
The inner test is smooth and supported in \(\supp\xi\).
The bilinear version required by the observable is
proved in Section~\ref{sec:ideal}.
\end{proof}

Theorem~\ref{thm:jet} is dynamical: it follows from the
specified retarded equation. It neither constrains
the classical backgrounds nor removes
\eqref{eq:Bianchi}. It also does not transform the
free field itself by a pullback.

%=====================================================================
\section{A necessary distinction: source response versus variational map}
\label{sec:variational}
%=====================================================================

\begin{proposition}[A local variational test fails for the unchanged law]
\label{prop:nonvariational}
The source-switched mixed operator
\(\sigma Q^{(1)}[h]\) is not, for all smooth backgrounds
\(h\), the Hessian of a local action on unrestricted
off-shell fluctuations.

Consequently \eqref{eq:response} cannot be identified,
on the stated off-shell domain, with the first
variational M{\o}ller correction of an unspecified
switched action merely because its unswitched source
was obtained from \eqref{eq:fluctuation}.
\end{proposition}

\begin{proof}
For fixed \(h\), a quadratic-action Hessian is formally
self-adjoint under the tensor-density pairing.
Proposition~\ref{prop:Q} gives
\[
 (\sigma Q^{(1)}[h])^{\mathrm T}
                                      =Q^{(1)}[h]\sigma.
\]
There is no general equality with
\(\sigma Q^{(1)}[h]\).

An explicit admissible local example is \(h=\eta\).
For a positive constant \(c\), the four-dimensional
Einstein density obeys
\(\mathcal E[cq]=\mathcal E[q]\). Hence
\(D\mathcal E[c\eta]r=c^{-1}G_{\mathrm{lin}}[r]\).
The background gauge-fixing quadratic action at
\(c\eta\) also scales as \(c^{-1}\):
its density contributes \(c^2\), the lower metric
\(c\), and the two divergence factors \(c^{-4}\).
Therefore
\[
 Q[c\eta]=c^{-1}\Pgr,\qquad Q^{(1)}[\eta]=-\Pgr.
\]
In an open part of the ramp where \(\sigma'\ne0\),
\[
 [\Pgr,\sigma]r
       =-\tfrac14\Ptr
                    (2\sigma'\partial_t r+\sigma''r)
\]
is a nonzero differential operator. It is detected by
compact variations in that open set.
Thus \(-\sigma\Pgr\) is not formally self-adjoint.
\end{proof}

This is a concrete version of the distinction emphasized
in the supplied X-G record \cite{XG}.
It does not invalidate the retarded equation or the
direct identity \eqref{eq:jet}.
It excludes a stronger unqualified variational
interpretation of the unchanged off-shell source law.

Switching a local density before Euler differentiation
instead produces derivative-of-\(\sigma\) terms.
They are generally nonzero in the onset layer and
are not silently included in \eqref{eq:response}.
They can propagate to later observations.
What is used in the plateau variation is the separate
fact that \(Q^{(1)}[K\xi]\) has support where
\(\sigma=1\), not a claim that all onset effects vanish.

We therefore use the phrase \emph{equation-level
retarded response}. No variational Bogoliubov map,
Peierls-intertwining theorem, or time-ordered family
is released.

%=====================================================================
\section{Free polynomial carrier and the scalar background identity}
\label{sec:carrier}
%=====================================================================

\subsection{Ordering data and sign discipline}

Use a free scalar \(\field\) satisfying \(P_m\field=0\)
and a free tensor \(\rfield\) satisfying \(\Pgr\rfield=0\).
These are not the classical coefficients \(h,k\).

Fix scalar Hadamard bisolutions \(H_m,H_0\) with one
common microlocal orientation, and use zero cross
blocks and the stipulated product-kernel normalization
\begin{equation}
 H=\begin{pmatrix}
       H_m&0\\[1mm]0&-4\Ptr H_0
    \end{pmatrix}.
 \label{eq:Wickmatrix}
\end{equation}
Multiplication by a constant real matrix does not
reverse a wavefront orientation. The product and
commutators are defined by this frozen matrix:
\[
 F\star_HG
  =\sum_{n\ge0}\frac{\hbar^n}{n!}
      \pair{F^{(n)}}{H^{\otimes n}G^{(n)}},
 \qquad
 [\Phi(A),\Phi(B)]_{\star_H}
              =\hbar\pair{A}{(H-H^{\mathrm T})B}.
\]
Only finitely many terms occur for polynomials.

The Green inverses remain \eqref{eq:Delta}.
The action Hessian, equation operator, causal
antisymmetric kernel, and product kernel are not
interchanged by a sign convention. In particular,
no additional incompatible antisymmetric-part
condition is imposed on \eqref{eq:Wickmatrix}.
An identification with a separately signed Peierls
convention requires its own convention dictionary;
it is not needed for the tree proof.

This explicit choice is an algebraic free carrier,
not a positive physical graviton state. No strong
de Donder constraint quotient is added.
The standard free microcausal Wick construction is
used with its actual wavefront hypotheses
\cite{FR,Hormander}.

For \(n\ge2\), define
\begin{equation}
 Z_n=\left\{(x_1,p_1;\ldots;x_n,p_n):
                   \sum_{j=1}^np_j=0\right\}\setminus0.
 \label{eq:Zn}
\end{equation}
Covectors are identified by the flat completion
coordinates. A compact kernel with wavefront in
\(Z_n\) satisfies the microcausal condition, because
co-oriented causal covectors cannot have zero sum
unless all are zero.

Let \(\WW\) be the resulting microcausal polynomial
carrier, and \(\JJ\) its closed dynamical equation
ideal. It is generated by scalar \(P_m\)-images and
graviton \(\Pgr\)-images, with closure on the
admissible compact microlocal kernel stages.
The bisolution property makes these dynamical
relations compatible with \(\star_H\).

\subsection{Completed scalar identities as compact equation images}

Let \(X_\xi u=\xi^\mu\partial_\mu u\) and
\(Z_\xi=X_\zeta+\frac12X_\xi^2\).
The classical background transformation is
\begin{equation}
 h'=h+K\xi,\qquad
 k'=k+\Lie_\xi^c h+K\zeta+\tfrac12\Lie_\xi^cK\xi.
 \label{eq:backgroundlaw}
\end{equation}

\begin{lemma}[Inherited identity with its representative]
\label{lem:scalaridentity}
For fixed smooth backgrounds,
\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}
The displayed preimages are compactly supported.
\end{lemma}

\begin{proof}
Write
\[
 c=f+\kappa c_1+\kappa^2c_2,\quad
 e=f+\kappa d_1+\kappa^2d_2,\quad
 L=\kappa L_1+\kappa^2L_2.
\]
Then \(e=(1+L\EA)^{-1}c\).
On the frozen plateau, naturality of the coordinate
density and unchanged initial data give the formal
coordinate intertwiner
\(X[q']=X[q]\circ F\).
Thus the inverse-coordinate smearing density satisfies
\(c[q']=F^\sharp c[q]\).

Set \(s=\EA e\). It solves
\((P_m+L[q])s=c[q]\) with advanced support.
Scalar-density covariance on the plateau gives
\[
 (P_m+L[q'])F^*=F^\sharp(P_m+L[q]).
\]
Advanced uniqueness therefore gives \(s'=F^*s\).
Applying \(P_m\) yields
\(e'-e=P_m((F^*-1)\EA e)\).
Expansion gives \eqref{eq:scalaridentity}.
The generator coefficients have compact support.
\end{proof}

These are the X-C/X-D compact representative identities
\cite{XC,XCST,XDST}. No quotient is taken before the
response operators are applied.

\subsection{Polynomial substitution is legitimate at fixed width}

Define
\[
 \Aone[l]=\field(d_1[l]),\qquad
 \Qhat[l_1,l_2]=\Wick{\field(\mathcal Q[l_1,l_2])}.
\]
For quantum metric arguments these denote
distributional mixed kernels, not pointwise
operator-valued substitutions.

\begin{lemma}[Admission of the finite scalar-completion forests]
\label{lem:forest}
At fixed width the kernels of
\[
 \field(f),\qquad \Aone[h+\rfield],\qquad
 \Qhat[h+\rfield,h+\rfield]+\Aone[k]
\]
and the scalar-slot preimages obtained by polarizing
\eqref{eq:scalaridentity} are admitted compact
polynomial kernels.
Their metric variables can range over \(D_f\);
their scalar variable lies in \(C\).
\end{lemma}

\begin{proof}
Expand \eqref{eq:coordinates} and
\eqref{eq:completed} by the finite Leibniz rule.
The propagator graphs are the single coordinate
line, the two coordinate branches, the nested
coordinate response, the mixed coordinate--scalar
branches, a local quadratic metric insertion, and
the iterated scalar line. Each is a forest.

Root positions and oriented edge differences are
independent coordinates for a forest. Its unweighted
propagator distribution is a tensor product of
existing retarded kernels in those differences.
It is constant in simultaneous root translations,
so its wavefront has zero total vertex covector.
Smooth multipliers and derivatives preserve that
bound. Attaching several external field copies to
one vertex is a diagonal pushforward; its additional
normal covectors also have zero total sum.
Integrating internal vertices sets their covectors
to zero. Thus each external wavefront lies in \(Z_n\).

All contributing vertices are in the retained
finite-time causal past \(D_f\). Smooth compact
localizations equal to one near that compact region
make the pushforwards proper. They do not change the
retained kernels. The same argument applies with an
extra advanced transposition and compact generator
coefficients to the preimages in
\eqref{eq:scalaridentity}.
Standard tensor-product and pushforward wavefront
rules justify these operations \cite{Hormander}.
\end{proof}

Consequently the scalar completed expression with an
independent fixed free graviton has the defect
\begin{align}
 &\left[
  \field(f)+\kappa\Aone[h+\rfield]
       +\kappa^2\{\Qhat[h+\rfield,h+\rfield]+\Aone[k]\}
      \right]' -[\cdots]
 \nonumber\\
 &\hspace{30mm}\equiv
            -\kappa^2\Aone[\Lie_\xi^c\rfield]
                                  \pmod{\JJ,\kappa^3}.
 \label{eq:one-defect}
\end{align}
Indeed the full second slot for \(h+r\) would be
\(k'+\Lie_\xi^c r\).
Apply Lemma~\ref{lem:scalaridentity}, polarize,
and subtract that missing affine term.
This proves the X-E-ST truncation formula without
an Einstein-shell shortcut \cite{XEST,XE}.

%=====================================================================
\section{F3: three response kernels, support, and admission}
\label{sec:F3}
%=====================================================================

\subsection{The first insertion kernel}

Set \(F=\EA f\), and let
\[
 a^{\mu\nu,\alpha\beta}
 =\tfrac12\eta^{\mu\nu}\eta^{\alpha\beta}
     -\tfrac12(\eta^{\mu\alpha}\eta^{\nu\beta}
                              +\eta^{\mu\beta}\eta^{\nu\alpha}).
\]
Then
\(a_1^{\mu\nu}[l]=a^{\mu\nu,\alpha\beta}l_{\alpha\beta}\).
The exact kernel \(D\) of \(d_1\) 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\\
 &\quad-\partial_{x^\mu}\bigl(
      g(x)a^{\mu\nu,\alpha\beta}\partial_\nu F(x)\delta(x-z)\bigr)
 \nonumber\\
 &\quad-\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}
It follows directly by transposing the first coordinate
equation and expanding \(-L_1[l]F\).

In particular,
\begin{equation}
 \supp D\subset\{(x,z)\in C\times D_f:z\preceq x\},
 \qquad \WF(D)\subset Z_2.
 \label{eq:Dsupport}
\end{equation}
The local \(gF\)-terms explain why the first support
factor is \(C\), not generally \(K_f\).

Suppressing fixed tensor indices, define
\begin{equation}
 N_0(x,y)=(D\circ\DR)(x,y)
       =\int D(x,z)\DR(z,y)\dd^4z.
 \label{eq:N}
\end{equation}
It is covariant in the source-density argument \(y\).

\begin{lemma}[Sum-zero operations actually needed]
\label{lem:sumzero}
The following operations are defined:
\begin{enumerate}[label=\textup{(\alph*)}]
\item differentiation and multiplication by smooth functions
preserve the bound \(Z_n\);
\item if \(\WF(A(x,y))\subset Z_2\), the insertion
\[
 A(x,y)\delta(y-y')
\]
exists and has wavefront in \(Z_3\);
\item if \(A(x,z)\) is compactly supported with wavefront
in \(Z_2\), and \(B(z,y)\) has wavefront in \(Z_2\),
then \(A\circ B\) exists and has wavefront in \(Z_2\).
\end{enumerate}
Consequently \(N_0\) is defined and
\[
 \WF(N_0)\subset Z_2,\qquad
 \supp N_0\subset\{(x,y):x\in C,\ y\preceq x\}.
\]
The source argument of \(N_0\) need not have a lower
time bound.
\end{lemma}

\begin{proof}
Part (a) is the ordinary differential and multiplier
wavefront rule.

For (b), the lifted covectors of the two factors have
forms
\[
 (p,-p,0),\qquad (0,q,-q).
\]
Their sum can vanish only when both are zero, so the
product criterion applies. Every possible product
covector, including a zero covector from either
factor, has zero total sum.

For (c), an opposing pair in the product
\(A(x,z)B(z,y)\) would have zero external \(x\)- and
\(y\)-covectors. The \(Z_2\) condition then forces
its internal covectors to vanish as well.
Compact support of \(A\) makes the integration in
\(z\) proper over compact external test sets.
For the pushforward, the internal covector is zero;
the two edge relations therefore give \(p_x+p_y=0\).
There are no exceptional one-slot wavefronts because
a zero covector in one argument of a \(Z_2\) relation
forces the other to be zero.
Apply this to \(D\) and the constant matrix times \(R\)
in \eqref{eq:Delta}.
The support follows from \(y\preceq z\preceq x\).
\end{proof}

The sum-zero calculus is the parent admission
mechanism \cite[Sum-zero calculus]{XF}.
Notice that compactness of \(D\) does not make
\(J^-(\supp D)\) compact. The onset factors in the
next step are essential.

\subsection{Definitions without duplicate prefactors}

Let
\[
 \Theta=\Wick{T_0[\field]},\qquad
 \kfield=\DR\left(
       \tfrac12g\Theta-\sigma Q^{(1)}[h]\rfield
                       -\tfrac12\sigma\Wick{G_2[\rfield,\rfield]}
                         \right).
\]
This is an auxiliary derived response, defined after
compact smearing by transposition. It is not an
independent free field or a full interacting metric.

Define the three coefficient functionals
\begin{align}
 T_{\mathrm{mat}}
   &=\Wick{\Aone[\DR(\tfrac12g\Theta)]},
 \nonumber\\
 T_{\mathrm{mix}}
   &=\Wick{\Aone[\DR(-\sigma Q^{(1)}[h]\rfield)]},
 \nonumber\\
 T_{\mathrm{self}}
   &=\Wick{\Aone[\DR(-\tfrac12\sigma G_2[\rfield,\rfield])]}.
 \label{eq:trees}
\end{align}
All scalar and tensor letters are totally Wick ordered.
The degrees in \((\field,\rfield)\) are respectively
\((3,0),(1,1),(1,2)\).
For the mixed term zero cross blocks make its Wick
ordering the ordinary scalar--tensor product.

Use the all-tree expression
\begin{align}
 \widehat\Phi_F[h,k]
  ={}&\field(f)+\kappa\Aone[h+\rfield]
 \nonumber\\
    &+\kappa^2\left\{
      \Qhat[h+\rfield,h+\rfield]+\Aone[k]
        +T_{\mathrm{mat}}+T_{\mathrm{mix}}+T_{\mathrm{self}}
                \right\}.
 \label{eq:alltree}
\end{align}
By \eqref{eq:affine}, it is precisely the normal
polynomial obtained from the completed tests at
\[
 q=\eta+\kappa(h+\rfield)+\kappa^2(k+\kfield)
\]
through this formal order. No full scalar Bogoliubov
map is applied a second time.

\subsection{Explicit distributional kernels}

Choose finite component bases for tensors.
Let \(\mathsf T\) be the constant-coefficient
bidifferential expression for the polarized flat
stress, and \(\mathsf B\) the expression fixed by
\eqref{eq:Boperator}. Define their transposes by
\[
 \pair{\mathsf T^{\mathrm T}\Psi}{u\otimes v}
       =\pair{\Psi}{\mathsf T(u,v)},\qquad
 \pair{\mathsf B^{\mathrm T}\Psi}{r_1\otimes r_2}
       =\pair{\Psi}{G_2[r_1,r_2]}.
\]
The notation includes the indicated repeated source
point. All component contractions are finite.

The tree kernels are
\begin{align}
 t_{\mathrm{mat}}
    &=\Sym_3\left\{
      \tfrac12\mathsf T^{\mathrm T}
          [g(y)N_0(x,y)\delta(y-y')]\right\},
 \nonumber\\
 M(x,y)
    &=-(Q^{(1)}[h]_y)^{\mathrm T}
                              [\sigma(y)N_0(x,y)],
 \nonumber\\
 t_{\mathrm{self}}
    &=-\tfrac12\Sym_{y,y'}\left\{
       \mathsf B^{\mathrm T}
          [\sigma(y)N_0(x,y)\delta(y-y')]\right\}.
 \label{eq:treekernels}
\end{align}
Thus
\[
 T_{\mathrm{mat}}
     =\Wick{\field^{\otimes3}(t_{\mathrm{mat}})},\quad
 T_{\mathrm{mix}}=(\field\otimes\rfield)(M),\quad
 T_{\mathrm{self}}
     =\Wick{(\field\otimes\rfield\otimes\rfield)(t_{\mathrm{self}})}.
\]
Here \(\Sym_3\) is normalized symmetrization over
the identical scalar slots. It changes no cubic
normal-symbol value. Only the two graviton slots
are symmetrized in the self kernel.

\begin{theorem}[Correct supports and microcausal admission]
\label{thm:admission}
All three kernels are compact distributions, with
\begin{align}
 \supp t_{\mathrm{mat}}
     &\subset\bigcup_{\pi\in\mathfrak S_3}
                         \pi(C\times S\times S)\subset C^3,
 \nonumber\\
 \supp M&\subset C\times D_f,
 \nonumber\\
 \supp t_{\mathrm{self}}&\subset C\times D_f\times D_f.
 \label{eq:treesupport}
\end{align}
Before normalized scalar symmetrization, the source
points of each cubic agree: \(y=y'\).

Their wavefronts satisfy
\begin{equation}
 \WF(t_{\mathrm{mat}})\subset Z_3,\qquad
 \WF(M)\subset Z_2,\qquad
 \WF(t_{\mathrm{self}})\subset Z_3.
 \label{eq:treeWF}
\end{equation}
Consequently the three tree coefficients belong to
\(\WW\). All their external points lie in
\(J_\eta^-(K_f)\).
\end{theorem}

\begin{proof}
The distribution \(g(y)N_0(x,y)\) is compactly
supported in \(C\times S\).
The distribution \(\sigma(y)N_0(x,y)\) is compactly
supported in \(C\times D_f\): its source time is
bounded below by zero, and \(y\preceq x\) with
\(x\in C\).
This argument includes the support closure at
\(t=0\); smooth flatness of \(\sigma\) introduces
no surface distribution.

The smooth multipliers and differential operators
in \eqref{eq:treekernels} preserve \(Z_2\).
Lemma~\ref{lem:sumzero}(b) handles the external
coincidence of the two source fields and gives
\(Z_3\). It retains conormal covectors of that
coincidence. It is not an appeal to a two-point
chain as a substitute for the cubic test.
Differential operators and permutations do not
enlarge the stated supports.

Finally \(Z_n\) avoids the two co-oriented causal
cones, so these compact kernels meet the defining
microcausal criterion \cite{FR,Hormander}.
Their partially contracted derivatives meet the
same criterion.
\end{proof}

A self source is local and quadratic in the
graviton, but this does not mean that a product of
singular quantum fields has been evaluated
pointwise. Its meaning is exactly the three-slot
distribution \eqref{eq:treekernels}.

\subsection{Matter normalization and classical values}

\begin{proposition}[Classical values and the retained coefficient \(-2\)]
\label{prop:classical}
For smooth inputs \(u,r\), the normal-symbol values
of the three trees are
\begin{align}
 T_{\mathrm{mat}}[u]
    &=\pair{d_1[-2\Ptr\ERzero(gT_0[u])]}u,
 \nonumber\\
 T_{\mathrm{mix}}[h;u,r]
    &=\pair{d_1[-\DR(\sigma Q^{(1)}[h]r)]}u,
 \nonumber\\
 T_{\mathrm{self}}[u,r]
    &=\pair{d_1[-\tfrac12\DR(\sigma G_2[r,r])]}u.
 \label{eq:classical}
\end{align}
Together they are the completed affine insertion
\(d_1[k_q[h;r,u]]\), counted once.

If \(P_mu=0\), the matter coordinate response is
\begin{equation}
 w^\mu[-2\Ptr\ERzero(gT_0[u])]
      =-2(\ERzero)^2
                      ((\partial_\nu g)T_0^{\nu\mu}[u]).
 \label{eq:minus2}
\end{equation}
\end{proposition}

\begin{proof}
Pair the kernels \eqref{eq:treekernels} with smooth
fields and transpose their indicated differential
operators. This gives the three sources in
\eqref{eq:response}. The coefficient \(-2\) follows
from \(\DR(\frac12gT_0)=-2\Ptr\ERzero(gT_0)\).

The matter response vanishes before \(\supp g\);
there \(\sigma=1\) on every contributing coordinate
source. Since \(a_1[l]=-\Ptr l\),
\[
 w[l]=\ERzero C[l],\qquad
 C^\mu[-2\Ptr\ERzero(gT_0[u])]
          =-2\ERzero\partial_\nu(gT_0^{\nu\mu}[u]).
\]
Use \(\partial_\nu T_0^{\nu\mu}[u]=0\) on the
free scalar shell.
\end{proof}

These values are evaluations of normal polynomial
symbols, not quantum expectation values.
They require no state. In particular, they do not
supply positivity or a physical graviton Hilbert
space.

The scalar-response part of the matter tree remains
present:
\[
 \pair{d_1[l]}u
   =-\int f\,w^\mu[l]\partial_\mu u\,\dd^4x
                              -\int f\,\ER L_1[l]u\,\dd^4x.
\]
Its support need not be \(K_f\) in the transposed
scalar slot. This is the support correction already
made in X-E-ST \cite{XEST}.

%=====================================================================
\section{F4: the graviton-slot ideal and direct cancellation}
\label{sec:ideal}
%=====================================================================

\subsection{The needed equation-ideal extension}

\begin{lemma}[Compact mixed equation images]
\label{lem:gslot}
Let \(v(x,y)\) be compactly supported with
\(\WF(v)\subset Z_2\), scalar-density valued in \(x\)
and covariant tensor valued in \(y\).
Then the mixed polynomial with kernel
\[
 (1\otimes\Pgr_y)v
\]
is admitted and belongs to \(\JJ\).

Every element of \(\JJ\) has zero normal symbol on
all smooth free pairs \(P_mu=0,\Pgr r=0\).
No converse is used.
\end{lemma}

\begin{proof}
Differentiation preserves support and \(Z_2\), so
the equation image is admitted.

For a finite sum of smooth tensor products
\(v=\sum_i a_i(x)b_i(y)\), its polynomial is
\[
 \sum_i\field(a_i)\star_H\rfield(\Pgr b_i).
\]
The cross blocks of \(H\) are zero, so this is also
the ordinary product. It is in the two-sided equation
ideal.

A general compact \(v\) is approximated by smooth
kernels in the admissible microlocal topology on
one slightly larger compact support stage.
One may first regularize and then approximate each
smooth kernel by finite sums of compact tensor-product
tests. Their smooth convergence implies convergence
in that microlocal topology. Differentiation and the
kernel-to-polynomial map are continuous, and \(\JJ\)
is closed on these stages. Thus the same conclusion
holds for \(v\) \cite{Hormander,BDH}.

For an equation generator, all Wick contractions
with another functional vanish by the bisolution
property of \(H\). Its product normal symbol
therefore vanishes on free pairs.
This persists under the specified closure, since
pairing compact kernels with fixed smooth fields
is continuous.
\end{proof}

This argument supplies the mixed graviton slot needed
in the cancellation; a scalar-slot equation relation
alone would be insufficient. It is not a general
converse theorem identifying every free-shell
vanishing functional with an equation image.

\subsection{An explicit compact preimage of the remaining term}

\begin{proposition}[The response remainder is in the ideal]
\label{prop:remainderideal}
For \(\xi\in\gp\),
\[
 \mathcal R_\xi
     :=\Aone[\DR\Lie_\xi^d\Pgr\rfield]\in\JJ.
\]
A compact mixed preimage is
\begin{equation}
 v_\xi(x,y)=-\Lie_{\xi,y}^c N_0(x,y),\qquad
 \supp v_\xi\subset C\times\supp\xi,\qquad
 \WF(v_\xi)\subset Z_2,
 \label{eq:preimage}
\end{equation}
with
\[
 \mathcal R_\xi
     =(\field\otimes\rfield)
                          ((1\otimes\Pgr_y)v_\xi).
\]
\end{proposition}

\begin{proof}
The expression before final field pairing is
\[
 \int N_0(x,y)(\Lie_\xi^d\Pgr r)(y)\dd^4y.
\]
Apply \eqref{eq:Lieadjoint} in \(y\), then formal
self-adjointness of \(\Pgr\).
The coefficients of \(\Lie_\xi^c\) vanish outside
\(\supp\xi\). Thus \(v_\xi\) is compact even though
\(N_0\) alone need not be compact in \(y\).
Its wavefront is still in \(Z_2\), so
Lemma~\ref{lem:gslot} applies.
\end{proof}

\subsection{The direct all-tree theorem}

\begin{theorem}[Background identity through \(\kappa^2\hbar^0\)]
\label{thm:main}
Let \(\xi,\zeta\in\gp\) act on the classical backgrounds
by \eqref{eq:backgroundlaw}, and hold
\(\field,\rfield,H,f,g,\sigma\) fixed. Then
\begin{equation}
 \widehat\Phi_F[h',k']-\widehat\Phi_F[h,k]
                              \equiv0\pmod{\JJ,\kappa^3}.
 \label{eq:main}
\end{equation}
This is the direct identity for the specified
tree-order normal polynomial. It is not an
interacting quantum Ward or BRST theorem.
\end{theorem}

\begin{proof}
The completed scalar part with independent free
graviton has variation \eqref{eq:one-defect}.
The matter and self trees have no classical-background
dependence and hence are inert.

The mixed tree depends linearly on \(h\). By
Theorem~\ref{thm:jet},
\begin{align*}
 T_{\mathrm{mix}}[h+K\xi]-T_{\mathrm{mix}}[h]
    &=\Aone[\Lie_\xi^c\rfield]
                       -\Aone[\DR\Lie_\xi^d\Pgr\rfield]\\
    &\equiv\Aone[\Lie_\xi^c\rfield]\pmod{\JJ},
\end{align*}
where Proposition~\ref{prop:remainderideal} handles
the last term.

Its first term cancels the negative term in
\eqref{eq:one-defect}. The scalar-slot representatives
in that identity are already in \(\JJ\), by
Lemma~\ref{lem:forest}. No second-order parameter
acts on a quantum field at this order.
\end{proof}

The mechanism and direct theorem are credited to
X-F \cite[Background identity of the all-tree
observable]{XF}. Their supported proof here does
not use a variational M{\o}ller identification.

\begin{corollary}[The one-tree defect is a truncation defect]
\label{cor:truncation}
Delete \(T_{\mathrm{mix}}\) and \(T_{\mathrm{self}}\)
from \eqref{eq:alltree}. The remaining matter-only
expression has defect
\[
 -\kappa^2\Aone[\Lie_\xi^c\rfield]
                                      \pmod{\JJ,\kappa^3}.
\]
It is the negative of the background variation of
the omitted mixed tree. A nonzero witness for that
truncation is not an obstruction to
Theorem~\ref{thm:main}.
\end{corollary}

\begin{proof}
The matter and self terms are inert; the mixed
variation is the cancelling term computed above.
\end{proof}

At \(h=0\), \(T_{\mathrm{mix}}\) vanishes, but its
first background variation generally does not.
Dropping a function because its value vanishes at
one background would lose that variation.

An all-order route based on simultaneous pullback,
retarded uniqueness, and a converse free-shell
separation theorem is not proved.
In the source-switched setting its variational
interpretation would also require an additional
construction. Only the direct finite-order proof
above is released.

%=====================================================================
\section{F5: comparison with flat interaction gauge}
\label{sec:F5}
%=====================================================================

Replace the background gauge fixing by its flat form
at every background. This does not change the free
operator, but sets \(F^{(1)}=0\), so
\[
 Q_{\mathrm{flat}}^{(1)}[h]r=G_2[h,r].
\]
Keep all other source and insertion prescriptions
unchanged.

\begin{proposition}[Admitted flat-gauge residual]
\label{prop:flat}
The resulting expression satisfies
\begin{align}
 \widehat\Phi_{F,\mathrm{flat}}[h',k']
       -\widehat\Phi_{F,\mathrm{flat}}[h,k]
    \equiv\kappa^2\Aone\left[
           \DR(\Lie_\xi^dF^{(0)}-F^{(0)}\Lie_\xi^c)\rfield
                                  \right]
                            \pmod{\JJ,\kappa^3},
 \label{eq:flat}
\end{align}
where \(F^{(0)}=-\frac14\Ptr KC\).
The residual is admitted. Its class modulo \(\JJ\)
is not decided here.
\end{proposition}

\begin{proof}
By \eqref{eq:cov2} and \eqref{eq:P},
\[
 Q_{\mathrm{flat}}^{(1)}[K\xi]
       =[\Lie_\xi,\Pgr]-[\Lie_\xi,F^{(0)}].
\]
The first term gives the same cancellation as in
Theorem~\ref{thm:main}, with its same compact
equation remainder. The second gives
\eqref{eq:flat}.

Its local differential source operator has coefficients
supported in \(\supp\xi\). The kernel is obtained by
applying its formal transpose to \(N_0\) in the
second argument. It therefore has compact support
in \(C\times\supp\xi\) and wavefront in \(Z_2\).
\end{proof}

This is the F5 comparison of X-F
\cite[Flat-gauge residual]{XF}.
It is not the functional detected by X-E-ST's
one-tree witness.

For example, on a free configuration with \(C[r]=0\),
the first term in the commutator vanishes and the
response of the second is
\[
 -K\ERzero C[\Lie_\xi^c r].
\]
The vector parameter is generally supported in
\(J^+(\supp\xi)\), not compactly inside \(U\).
The compact plateau formula for \(d_1[K\xi]\)
cannot be applied to it without a separate support
argument. For \(C[r]\ne0\), an additional constraint
term remains. Neither case is decided by simply
reusing the one-tree witness.

%=====================================================================
\section{Topology, support transport, and diagnostic checks}
\label{sec:topology}
%=====================================================================

\subsection{The scalar fixed-support sheet}

For compact \(A\subset\widehat M\), define
\[
 \DD_A=\{a\in C_c^\infty(\widehat M;\CC):\supp a\subset A\},
 \qquad
 p_N(a)=\max_{|\alpha|\le N}\sup|\partial^\alpha a|.
\]
These are ambient fixed-support smooth spaces, not
spaces of functions defined only on the compact set.

The scalar degree-one quotient is
\[
 \Qm=\DD/P_m\DD,\qquad \DD=C_c^\infty(\widehat M;\CC).
\]

\begin{lemma}[Closed equation subspace and reduction]
\label{lem:closed}
The subspace \(P_m\DD\) is closed in \(\DD\).
The quotient \(\Qm\) is Hausdorff.
For fixed smooth backgrounds,
\[
 L_1\EA,\quad L_2\EA,\quad L_1\EA L_1\EA
\]
are continuous from tests supported in \(K_f\)
to \(\DD_C\). In particular,
\[
 \delta d_1=\delta c_1,\qquad
 \delta d_2=\delta c_2-L_1\EA\delta c_1
\]
when only the coordinate representatives vary.
\end{lemma}

\begin{proof}
The continuous map
\(\EA-\ER:\DD\to C^\infty(\widehat M)\)
has kernel \(P_m\DD\).
If \(\EA a=\ER a=u\), then \(u\) is supported in
the compact causal diamond of \(\supp a\), so
\(u\in\DD\) and \(P_mu=a\).
The converse follows from the Green identities.

Advanced support, compact coefficients of \(L_j\),
and Green continuity give the mapping assertions.
Subtracting \eqref{eq:completed} gives the final
identities.
\end{proof}

These are the directly applicable X-D-ST topology
and reduction facts \cite{XDST}. Its width-removal
theorem is not used.

\subsection{Distributional kernels and typed contractions}

Write \(\EE'_\Gamma(A)\) for compact distributions
supported in \(A\) with wavefront in the closed cone
\(\Gamma\), with the normal microlocal topology.
It is not identified with a smooth Fr\'echet stage.
Pullbacks and proper pushforwards are continuous on
their stated stages; general distribution products
have the appropriate hypocontinuity properties
\cite{BDH}.

\begin{theorem}[Continuity on the generated kernel stages]
\label{thm:continuity}
Fix common compact support reserves, a common plateau,
and the positive width. Then:
\begin{enumerate}[label=\textup{(\roman*)}]
\item the three kernel constructions are continuous in
their smooth coefficient data on these reserves,
into their stated compact normal microlocal stages;
\item contracting all quantum arguments except the
scalar output in \(M\) or \(t_{\mathrm{self}}\)
against fixed smooth fields gives an element of
\(\DD_C\), continuously in those fields and data;
\item every two-slot smooth contraction of the symmetric
matter cubic leaves a smooth test in \(\DD_C\);
\item a contraction leaving a graviton slot gives a
smooth compact tensor test supported in \(D_f\)
in general, not necessarily in \(C\);
\item the scalar tests in (ii)--(iii) pass continuously
to \(\Qm\).
\end{enumerate}
The full mixed and cubic kernels belong to the
polynomial carrier quotient \(\WW/\JJ\), not to
the degree-one quotient \(\Qm\).
\end{theorem}

\begin{proof}
The generated maps consist of finite smooth coefficient
operations, fixed propagator distributions, the
proper compositions of Lemma~\ref{lem:sumzero},
and fixed diagonal insertions. Their wavefront
conditions and supports are already proved.
The normal-topology operations therefore give
continuity for these fixed-distribution,
smooth-coefficient constructions \cite{BDH}.
No joint-continuity theorem for arbitrary varying
singular kernels is asserted.

For smoothness of a one-slot remainder after partial
contraction, its only possible wavefront covector
would be \((0,\ldots,0,p)\) in \(Z_n\). This forces
\(p=0\), so there is no wavefront. The support is
the relevant projection of \eqref{eq:treesupport}.
Normal-topology pushforward with empty output cone
is smooth convergence on compact sets. This proves
the corresponding continuity into every smooth
seminorm.

The quotient assertion follows from
Lemma~\ref{lem:closed}.
\end{proof}

There is no \(\varepsilon\to0\) argument here.
In particular, no convergence of arbitrary quantum
metric kernels follows from the X-D-ST compact-test
theorem.

\subsection{Supported agreement}

\begin{proposition}[Supported patch agreement]
\label{prop:patch}
All retained pairings depend only on the backgrounds
and sources near the contributing compact subset of
\(D_f\). Completion extensions agreeing there give
the same supported result.
Transport through \(\iota\) preserves the pairings,
their continuity, and the transported wavefront bounds.
\end{proposition}

\begin{proof}
Every edge in the displayed trees is retarded towards
the observation root or advanced after transposition.
All retained source times are nonnegative, and all
vertices remain in \(J_\eta^-(K_f)\).
Finite propagation therefore restricts the
calculation to \(D_f\).
Nested scalar and coordinate responses obey the same
causal restriction.

Distribution transport is defined by transposition on
compact tests and densities. The chart and its inverse
have bounded derivatives of every required finite
order on the compact reserve.
The ordinary test topologies and transported
microlocal stages give the same local pairings.
\end{proof}

The completion formula \(\sum p_j=0\) is transported
by the chart. It is not a claim of global cotangent
addition or global translation symmetry on the cone.

\subsection{Diagnostics with exact scope}

\paragraph{Pure-gauge background.}
At \(h=0\), \(T_{\mathrm{mix}}=0\), but
Theorem~\ref{thm:jet} gives
\[
 \delta_\xi k_q=\Lie_\xi^c r
\]
for every smooth free \(r\). This checks the sign
of the cancelling term without transforming the free
carrier.

\paragraph{Zero free graviton.}
On \(r=0\), the mixed and self classical values vanish.
The matter coefficient reduces exactly to X-E-ST's
tree, with its \(-2\) normalization and full scalar
response retained.

\paragraph{Zero scalar.}
Every displayed observable tree vanishes at \(u=0\).
This is a statement about these coefficients, not
vacuum energy or a graviton state.

\paragraph{Gauge-fixed versus invariant shell.}
A free \(\Pgr r=0\) field can have
\(G_{\mathrm{lin}}[r]\ne0\). The exact F1 identity
and background gauge correction are both necessary.
No proof step replaces the gauge-fixed shell by the
linearized Einstein shell.

\paragraph{Classical tadpole.}
The fluctuation-independent terms removed from
\eqref{eq:fluctuation} would induce classical shifts
of the fluctuation. Their retarded completion and its
background bookkeeping are not part of this release.
No second copy of the classical two-jet law is added.

%=====================================================================
\section{F6: VII-C audit and the separate analytic boundaries}
\label{sec:F6}
%=====================================================================

VII-C concerns a specified sharp mixed insertion,
a fixed Wick matrix, a specified normal-topology
approximating family, and continuous extraction of
the fully contracted coefficient of
\[
 B_f\star_H W_g\star_H B_{f'},\qquad
 W_g=\int g\,h_{00}^2.
\]
Its logarithmic obstruction retains that joint
hypothesis package \cite{VIIC}.

\begin{center}
\small
\renewcommand{\arraystretch}{1.12}
\begin{tabularx}{\linewidth}{@{}p{.08\linewidth}p{.39\linewidth}X@{}}
\toprule
Item & VII-C requires & X-F-ST constructs\\
\midrule
(E1) &
Its exact de Donder multiplet and fixed Wick matrix &
A declared flat free tensor carrier with block
\(-4\Ptr H_0\); background gauge enters
\(Q^{(1)}[h]\), not a new free product\\
(E2) &
Its unchanged sharp zero-Cauchy mixed insertion &
Completed ramp insertions at fixed positive width\\
(E3) &
Admission continuous on its specified normal-topology
approximating stages &
Three fixed-width coefficient kernels; no such
approximating family\\
(E4) &
The required triple word and continuous
field-independent \(\hbar^3\) extraction &
Three isolated \(\hbar^0\) trees; no required word
or extraction\\
\bottomrule
\end{tabularx}
\end{center}

The theorem therefore does not apply and is not
contradicted. The presence of a self tree does not
identify its field content, kernel, or contraction
with that word.

The omitted mixed quantum contractions are a possible
later comparison point, but even there VII-C would
require an actual hypothesis match. Its obstruction
is not automatically a theorem about every mixed
loop.

\paragraph{No product-hierarchy inference.}
The earlier scalar one-contraction boundary does
not construct a general \(\mathsf T_2\), and the
three trees do not construct one either
\cite{XBST,XCST}.
No loop kernel square, extension prescription,
contact catalogue, or subtraction is selected.

\paragraph{Different transmission and boundary domains.}
VII-I's finite-width free algebra and changed sharp
transmission data are not this prescription.
IX-A-ST's boundary-sensitive carrier and IX-D-ST's
causal-support interface do not supply a time-ordered
lift for these kernels, and these trees do not solve
the IX-E problem \cite{ComparisonRecords}.

\paragraph{Growth is not convergence.}
Em01-B's polynomial smoothing growth is not a
characteristic limit or a rate for the present
kernels. Em01-C's aligned-null scalar family is
different. Em02-D's raw-family nonconvergence is
a contrast, not a repair \cite{EmRecords}.

\paragraph{Smooth multiplication is not subtraction.}
For fixed smooth \(a\),
\(\WF(au)\subseteq\WF(u)\).
This fact does not construct an undefined product,
choose a finite part, or supply uniform estimates
for arbitrary scale-dependent multipliers
\cite{Hormander,XVIA}.

\paragraph{Characteristic-map boundary.}
The present sum-zero calculation is for the explicitly
constructed internal trees. It does not compose
their kernels with the separate characteristic quantum
cross-lines. Historical H2 and the full I3/J2
obligations are not discharged here; XIV-I's narrower
internal-amplitude construction is not withdrawn
\cite{BoundaryRecords}.

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

\subsection{Installed evaluations are not acquired fields}

A finite dictionary specifies fixed data, a tree label,
and supplied smooth free inputs. Its values are, for
example,
\[
 a_i=T_{\mathrm{mat},i}[u_i],\qquad
 a_j=T_{\mathrm{mix},j}[h_j;u_j,r_j],\qquad
 a_l=T_{\mathrm{self},l}[u_l,r_l].
\]
Polarized evaluations are also allowed.
Theorem~\ref{thm:continuity} makes each finite
evaluation continuous on its declared data stage.

The table stores rational reports or certified
enclosures. Arbitrary smooth fields and arbitrary
real numbers are not assumed to have finite exact
descriptions. No detector coupling to these
continuum functionals, field-to-qubit preparation,
or unknown-field acquisition is supplied.

\subsection{Complete destructive clock}

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

\begin{theorem}[Complete clock instrument]
\label{thm:clock}
The map
\begin{equation}
 \mathcal M(X)
   =\sum_j\proj{j}_R\otimes D_jXD_j^\dagger
   =\sum_j\proj{j}_R\otimes
       \Tr_{C_{\mathrm{clk}}}
          [(P_j\otimes I_A)X(P_j\otimes I_A)]
 \label{eq:clock}
\end{equation}
is CPTP, and
\[
 \Tr_R\mathcal M(X)=\Tr_{C_{\mathrm{clk}}}X.
\]
These are identities on all operators, including
every untouched reference extension.
The clock names are consumed; no quantum clock
factor is returned.
\end{theorem}

\begin{proof}
The recorded Kraus maps are
\(\ket j_R\otimes D_j\), whose squared moduli sum
to \(\sum_jP_j\otimes I_A=I\).
Expand
\(X=\sum_{j,l}\kb{\tau_j}{\tau_l}\otimes X_{jl}\).
Both \(j\)-th blocks equal \(X_{jj}\);
their sum is the partial trace.
The identical calculation includes any reference.
Consumption is the declared destructive interface,
whose quantum codomain contains \(A\), not the
measured clock.
\end{proof}

This is the I-B/En09-A pattern, proved on the present
finite interface \cite{IB,En09}.
Conditional reference steering is compatible with the
unchanged unconditional reference marginal.
A positive branch is normalized by its own probability;
a zero branch is not assigned a normalized state.

If distinct calibrated labels must equal the time
of one fixed readout event on every positive branch,
only one label can occur. A nontrivial history-label
readout uses a separate calibration map.
Neither operation consumes the geometric time axis.
The complete extraction identity alone also does not
imply relational unitary dynamics; additional clock
covariance and stationarity hypotheses would be needed.

\subsection{An explicit report encoding and its error}

Normalize finitely many supplied reports to
\(v_j\in[-1,1]\), using one installed scale.
For \(y\in\{-1,1\}\), set
\[
 p_y(v)=\frac{1+yv}{2},\qquad
 \mathcal I_v(X)
    =\sum_{j,y}\proj{j,y}_R\otimes p_y(v_j)D_jXD_j^\dagger.
\]

\begin{proposition}[Reference-stable report comparison]
\label{prop:reports}
The channel \(\mathcal I_v\) is CPTP, and
\[
 \ddia(\mathcal I_v,\mathcal I_{\widetilde v})
       =\tfrac12\max_j|v_j-\widetilde v_j|,
 \qquad
 \ddia(\Lambda,\Gamma)=\tfrac12\|\Lambda-\Gamma\|_\diamond.
\]
\end{proposition}

\begin{proof}
Use the complete Kraus family
\(\sqrt{p_y(v_j)}\ket{j,y}_R\otimes D_j\).
For an arbitrary input operator with reference,
the difference has orthogonal blocks
\(y(v_j-\widetilde v_j)X_{jj}/2\).
Its trace norm is bounded by
\[
 \max_j|v_j-\widetilde v_j|\sum_j\|X_{jj}\|_1
       \le\max_j|v_j-\widetilde v_j|\,\|X\|_1.
\]
The last inequality is pinching contraction.
A clock-basis input at a maximizing index saturates
the estimate.
\end{proof}

This is a statement about the chosen probabilistic
encoding. Nearby numbers encoded as different
orthogonal deterministic words need not produce
nearby channels.

En10-A concerns a separate source-weight acquisition
contract \cite{En10A}. It is not supplied merely by
installing the evaluation table above.

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

At each tick let the complete finite channel factor
over a prescribed partition into finite-site blocks.
All classical records and control wires are included.
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}[Complete scheduled locality]
\label{thm:locality}
The network satisfies
\[
 (\Phi_{s+1}^*\circ\cdots\circ\Phi_t^*)(\mathcal A(Y))
                              \subseteq\mathcal A(N_{s,t}(Y)).
\]
Equal input reductions there give equal corresponding
output reductions. An intervention outside the complete
backwards region cannot change the corresponding joint
record law.

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

\begin{proof}
A channel adjoint is unital. A block not meeting the
current observable support acts on an identity factor.
A meeting block enlarges support only within itself.
Backward induction proves containment.
Testing all output observables proves the reduction
and intervention statements, with references retained.
Concatenate the independently certified routes for
the geometric containment.
\end{proof}

This is the prescribed-schedule support mechanism
\cite{VIIE}. An ancestor lies in the metric past
of its output. Scheduled ancestry is an upper bound
on dependence, not the entire physical causal past.
Scheduler consistency alone does not derive a
Lorentz cone \cite{En08A}.

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 \(t_{\mathrm d}\).
For \(n\ge1\), place stations
\[
 \mathbf x_j=\mathbf x_0+\frac jn(\mathbf x_n-\mathbf x_0),
       \qquad j=0,\ldots,n.
\]
Each has a \(p\)-qubit register \(A_j\).
Initialize \(A_1,\ldots,A_n\) in independent blank
states \(\omega\), and SWAP \(A_{j-1}\) with \(A_j\)
at hop \(j\), using developed duration \(\ell/n\).

The tick-boundary two-register SWAP is a declared
finite permission. Cross-port dependencies are null,
and same-station dependencies are timelike.
A microscopic interaction Hamiltonian is not inferred.

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

\begin{proof}
Each SWAP moves the complete payload factor, including
its correlations with \(E\), into the next independent
blank and leaves the previous factor blank.
Induction proves the state formula.
Classical forwarding preserves diagonal record blocks
and their joint correlations.
Station spacing and tick duration give the certified
null cross-port displacement inside the regular tube.
\end{proof}

This is the I-B/En08-B transport pattern
\cite{IB,En08B}. It preserves existing correlations;
it is not gravity-generated entanglement.

A conservative preallocated reserve is
\[
 n_C+(n+1)p+(n+1)b+w_{\mathrm{lookup}}+1,
\]
for the clock, station payloads, \(b\)-bit report
reserves, workspace, and one optional report probe.
Tables, descriptions, references, supplier storage,
reset permissions, and returned factors are separately
charged. The route uses \(n\) register SWAPs and
\(n\) classical transmissions; all basis changes,
readout, lookup, buffers, and durations are included.

For compatible complete implementations,
\begin{equation}
 \ddia(\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:errors}
\end{equation}
Replace one factor at a time, use diamond-norm
contraction under channels, and sum the allowances
\cite{Watrous}. A failure or arrival probability is
covered only if its record is part of the compared
complete output.

No hardware synthesis rate, continuum detector, or
exact causal containment of a faulty implementation
is inferred from this error bound.

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

\begin{theorem}[X-F-ST local Grade S release]
\label{thm:release}
On the fixed supported host and at fixed positive width:
\begin{enumerate}[label=\textup{(\roman*)}]
\item the inherited action, normalization, scalar
completion, and invariant-jet switch obstruction
remain those explicitly stated above;
\item background de Donder gauge on the fluctuation gives
the typed F1 covariance and F2 commutator identities;
\item the prescribed second-order response is uniquely
retarded, satisfies the dynamical jet law, and
introduces no independent field;
\item the matter, mixed, and self trees are constructed
with the corrected supports, normalization, and
two- or three-slot microcausal bounds;
\item their generated kernels and typed smooth contractions
have the continuity properties of
Theorem~\ref{thm:continuity};
\item the all-tree observable is background invariant
through \(\kappa^2\hbar^0\) modulo \(\JJ\), with
the explicit graviton-slot preimage
\eqref{eq:preimage};
\item the flat-gauge residual is admitted and displayed,
with its class left open;
\item the finite supplied-evaluation interface has a
complete destructive clock, scheduled locality,
regular reference-preserving transport, and the
declared resource/error ledger.
\end{enumerate}
The retarded source law is not promoted to a
variational M{\o}ller or Bogoliubov construction.
These statements form local Grade S only.
\end{theorem}

\begin{proof}
Lemmas~\ref{lem:affine} and \ref{lem:action}, and
Proposition~\ref{prop:Noether}, prove (i).
Lemma~\ref{lem:cov2}, Proposition~\ref{prop:Q},
and Theorem~\ref{thm:key} prove (ii).
Past-compact Green theory and
Theorem~\ref{thm:jet} prove (iii).
Theorem~\ref{thm:admission} and
Proposition~\ref{prop:classical} prove (iv).
Theorem~\ref{thm:continuity} proves (v).
Lemma~\ref{lem:gslot},
Proposition~\ref{prop:remainderideal}, and
Theorem~\ref{thm:main} prove (vi).
Proposition~\ref{prop:flat} proves (vii).
Sections~\ref{sec:instrument}--\ref{sec:transport}
prove (viii).
Proposition~\ref{prop:nonvariational} supplies the
interpretive restriction.
\end{proof}

\begin{center}
\small
\renewcommand{\arraystretch}{1.12}
\begin{tabularx}{\linewidth}{@{}p{.31\linewidth}p{.21\linewidth}X@{}}
\toprule
Object & Status & Exact boundary\\
\midrule
Background de Donder interaction gauge &
Constructed &
Quantum fluctuation only; free carrier unchanged\\
Derived \(k_q\) &
Constructed &
Equation-level retarded response, not an independent
or complete interacting metric\\
Three \(\kappa^2\hbar^0\) trees &
Constructed &
Matter tree retained once; mixed and self added once\\
All-tree background identity &
Proved &
Through \(\kappa^2\), modulo the equation ideal\\
X-E-ST one-tree defect &
Truncation corollary &
Not an all-tree field-content obstruction\\
Flat-gauge residual &
Displayed and admitted &
Its class is not decided\\
Variational switched-source identification &
Not supplied &
The unrestricted source law fails the local Hessian test\\
VII-C word and time-ordered hierarchy &
Not constructed &
No mandatory word, causal-product family, or extraction\\
Loops, ghosts, BRST/BV, positivity &
Not constructed &
No quantum Ward, physical state, or master equation\\
Sharp-width metric admission &
Not constructed &
X-D-ST remains a separate scalar theorem\\
Spacetime emergence &
Not constructed &
Host and action are supplied inputs\\
\bottomrule
\end{tabularx}
\end{center}

\subsection{Significance without priority inflation}

The principal identity and three-tree mechanism belong
to Cheung's X-F \cite{XF}. The supported transfer
adds an explicit local implementation of that proof
and audits the places where a stronger conclusion
would otherwise be inferred.

The most useful distinction is
\[
 \text{retarded solvability}
 \ \ne\ \text{variational integrability}
 \ \ne\ \text{microcausal admission}
 \ \ne\ \text{an interacting quantum identity}.
\]
Here retarded solvability, the specific polynomial
admission, and the direct tree background identity
are proved. A general variational identification is
not. That separation gives a successor a precise
baseline rather than an overstated completion.

\subsection{Successor package}

A successor towards an X-G transfer receives:
\begin{enumerate}
\item the unchanged host, chart, signature, and completion;
\item the retained action variation and switched Noether defect;
\item the explicit background gauge and kinetic correction;
\item the typed commutator and dynamical response law;
\item the three kernels, corrected support sets, matter
coefficient \(-2\), and sum-zero admission;
\item the direct all-tree identity and compact graviton-slot
equation representative;
\item the truncation corollary and admitted flat-gauge
residual with its class undecided;
\item the source-versus-density switching distinction and
the limitation on variational terminology;
\item the finite supplied-evaluation encoding contract.
\end{enumerate}

It receives no complete interacting quantum metric,
higher self tree, VII-C word, time-ordered hierarchy,
loop regulator, \(\hbar^1\) coefficient, ghosts,
BRST construction, all-order theorem, physical-state
selection, or spacetime-emergence map.
It does not receive X-D-ST's sharp scalar limit as
metric admission.

\subsection{No external status update}

In the inherited problem ledger, Problems 1, 2, and 8
remain unfinished, Problem 12 remains unresolved,
and Problems 3--7 and 9--11 are not reached by this
release. These are documentary labels, not newly
defined or solved problems of this paper.

O.1 stays \loc{NOT_CLOSED} in the inherited
prescription scope. No unrelated later subsector
is used to change that label.
XV-B is not opened as the downstream admission exit;
XVI-N is not discharged.
Historical H2 and the missing full I3/J2 maps are
not discharged here, while XIV-I's narrower amplitude
result is not withdrawn \cite{BoundaryRecords}.

No quantum coefficient of the inherited characteristic
graph is admitted. This does not deny the free carrier
or the three explicitly admitted tree coefficients.

The host metric, worldsheet, time coordinate, and
future relation remain fixed. A regulator-description
change or table update returns no new host metric.

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

\begin{longtable}{@{}p{.35\linewidth}p{.60\linewidth}@{}}
\toprule
Potential error & Correction\\
\midrule
\endfirsthead
\toprule
Potential error & Correction\\
\midrule
\endhead

Constructing a second cone &
The I-B/X-A-ST host and chart remain unchanged.\\[4pt]

Using global Cauchy theory on the regular product &
All global analytical problems are on the declared
Minkowski completion.\\[4pt]

Calling the conical defect a worldline &
It is the inherited worldsheet.\\[4pt]

Swapping signatures in Euler equations &
The host and analytical signature ledger is retained.\\[4pt]

Using one Lie derivative on both sides of an Euler map &
Input tensors and output tensor densities have the
different Lie derivatives in \eqref{eq:types}.\\[4pt]

Dropping \(\Lie_\xi G_{\mathrm{lin}}[r]\) &
The exact F1 identity is used off shell.\\[4pt]

Imposing \(G_{\mathrm{lin}}[r]=0\) on every free graviton &
The carrier equation is \(\Pgr r=0\); no strong
de Donder constraint is added.\\[4pt]

Deleting the linear kinetic equation in the
background-field prescription &
Only the fluctuation-independent Euler terms are removed.\\[4pt]

Calling every retarded solution a variational M{\o}ller map &
Proposition~\ref{prop:nonvariational} gives a concrete
formal-adjoint obstruction to that unqualified claim.\\[4pt]

Switching a source as though it were a density &
Euler differentiation of a switched density produces
additional onset derivatives.\\[4pt]

Saying onset terms cannot affect a later observation &
They can propagate. The direct variation instead uses
compact generator support where \(\sigma=1\).\\[4pt]

Applying a retarded Green operator to an unprepared
self source &
The prescribed smooth onset is needed for the
past-compact source domain.\\[4pt]

Treating \(N_0=D\circ\DR\) as globally compact &
Its source argument can extend into the infinite past;
\(gN_0\), \(\sigma N_0\), and generator-localized
versions have the compact supports actually used.\\[4pt]

Putting the first scalar argument only in \(K_f\) &
The \(g\EA f\) local terms extend it to \(C\).\\[4pt]

Putting every graviton contraction in \(\DD_C\) &
A remaining graviton slot can range over \(D_f\).\\[4pt]

Equating a cubic kernel with a degree-one free class &
The cubic is a distributional polynomial kernel.
Only its appropriate smooth scalar contractions
pass to \(\Qm\).\\[4pt]

Proving a two-point bound and omitting the cubic incidence &
The diagonal source insertion and its normal
covectors are included explicitly.\\[4pt]

Adding a second \(\kappa^2\) inside a tree coefficient &
The three \(T\)'s are coefficients; the observable
supplies one outer \(\kappa^2\).\\[4pt]

Applying the scalar response again to completed tests &
The X-C-ST scalar pair is retained once.\\[4pt]

Using a scalar equation ideal for a graviton remainder &
Lemma~\ref{lem:gslot} supplies the graviton-slot
membership actually required.\\[4pt]

Inferring equation-ideal membership merely from
free-shell vanishing &
Only the forward implication is used; the needed
preimage is exhibited.\\[4pt]

Using the X-E-ST witness to decide the flat-gauge residual &
They are different functionals. The residual class
remains open here.\\[4pt]

Claiming the one-tree defect obstructs every completion &
The mixed tree cancels that truncation defect on
the background-gauge source sheet.\\[4pt]

Inferring a common Hadamard orientation from \(-4\) &
The orientation of both blocks is declared explicitly.
A matrix sign does not reverse a wavefront.\\[4pt]

Equating a chosen Wick matrix with a physical state &
No positive physical graviton state is selected.\\[4pt]

Promoting smooth multiplication to subtraction &
It does not define missing products or counterterms.\\[4pt]

Promoting a finite evaluation table to field acquisition &
The values or enclosures are supplied inputs.\\[4pt]

Returning measured clock names as quantum outputs &
The complete instrument consumes those names.\\[4pt]

Reading several labels as the actual time of one event &
Distinct event-time calibration permits one positive
label; histories require a separate map.\\[4pt]

Equating scheduled ancestry with the whole metric past &
Only certified causal containment is proved.\\[4pt]

Calling reference-preserving transport generated entanglement &
The route preserves pre-existing correlations.\\[4pt]

Treating a source file as a registry certificate &
No installation, compilation, or tests are asserted.\\

\bottomrule
\end{longtable}

%=====================================================================
\section{Documentary comparisons only}
\label{app:catalogue}
%=====================================================================

The requested OpenAI catalogue families are retained
only at their supplied documentary level
\cite{OpenAICatalogue}:
\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.13\linewidth}p{.39\linewidth}X@{}}
\toprule
Family & Supplied topic & No inference for X-F-ST\\
\midrule
282 &
Scale-to-conformal and free Ward comparisons &
Not this background identity or Einstein dynamics\\
362 &
Relativistic Vlasov--Maxwell smoothness &
Not the metric variation or the three trees\\
376 &
Forced Navier--Stokes analysis &
Not a loop regulator or ultraviolet completion\\
\bottomrule
\end{tabularx}
\end{center}

The catalogue authorship is attributed to OpenAI.
No catalogue proof is verified or used as a premise.
A source description or formalization note is not an
independent proof certificate.

XI-G is not repaired; X-Enter's separate encoding
problem is not solved; and the X-G continuation is not
executed by naming it \cite{ComparisonRecords,XG}.
Every comparison retains its own object, domain,
normalization, and stopping condition.

%=====================================================================
\section{Proposed integration and emergence ledger}
\label{app:registration}
%=====================================================================

This is a proposed integration contract, not an executed
repository change. A maintainer may:
\begin{enumerate}
\item install the source as
\loc{content/manuscripts/X-F-ST.tex};
\item add the actual import, file-list item, and
versioned registry metadata;
\item add bearings for the relational and product problems,
with a refusal of a catalogue-wide chain survey;
\item add separation edges to X-F, X-E-ST, X-D-ST,
I-B, VII-C, the IX controls, and X-G;
\item add the emergence ledger below;
\item update existing sharp-width obligations only where
their actual registered wording still says
``proposed'', without converting the scalar limit
into metric admission;
\item add explicit obligations separating the three trees
from a \(\mathsf T_2\) family, a complete interacting
metric, variational integrability of the frozen
source law, and spacetime emergence;
\item run the repository's actual tests and record their
real outcomes and source hashes.
\end{enumerate}

\paragraph{Proposed routing note.}
X-F-ST is a supported local Grade S realization of
three fixed-width retarded source coefficients and
their direct background identity through
\(\kappa^2\hbar^0\). It preserves the inherited host,
free carrier, scalar-completion counting, and matter
normalization. Its response is equation-level; it
does not supply a variational switched-action
M{\o}ller map, a loop coefficient, or a
time-ordered hierarchy.

\begin{verbatim}
identifier: X-F-ST
sourceStatus: proposed
relationshipRole: supported-transfer-and-audit
primarySourceRecord: X-F@1
predecessorSources:
  - X-D-ST supplied source
  - X-E-ST supplied source
host: unchanged X-A-ST / I-B instance
widthStatus: fixed-positive-width
releasedOrder: kappa^2 hbar^0
responseStatus: equation-level retarded solution
variationalMollerIdentification: not asserted
allTreeBackgroundIdentity: proved modulo equation ideal
flatGaugeResidualClass: not decided
timeOrderedHierarchy: not constructed
completeInteractingQuantumMetric: not constructed
sharpMetricAdmission: not constructed
emergenceStatus: not constructed
registryInstallationExecuted: false
repositoryTestsExecuted: false
compilationAsserted: false
\end{verbatim}

If later installed, ``registered transfer'' describes
a repository status only. It does not change
\loc{emergenceStatus}, prove a field detector, or
upgrade the analytical release.

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

\paragraph{Authorship and attribution.}
The author name follows the supplied X-D-ST and
X-E-ST source metadata. The action and three-tree
mechanism are attributed separately to Cheung's
X-E and X-F. No priority claim is made for those
parent results.

\paragraph{Source status.}
The supplied adaptations, rendered parent records,
and documentary comparisons are distinguished.
No publication date, DOI, registration, or external
proof verification is invented for an internal
manuscript. The X-G variational caution is used at
the level of the supplied record; the local
formal-adjoint calculation is proved here.

\paragraph{Analytical scope.}
The construction is a finite-order, fixed-width
polynomial response calculation on the completion.
Classical backgrounds remain off shell.
There is no all-order, loop, sharp joint-metric,
or characteristic cross-line admission theorem.

\paragraph{Operational scope.}
A finite dictionary encodes supplied evaluations.
The clock consumes finite names, not geometric time.
Transport uses declared regular finite-operation
permissions and preserves existing reference
correlations.

\paragraph{Verification.}
No successful \loc{pdflatex} execution,
proof-assistant verification, numerical evaluation,
symbolic-program execution, hardware synthesis,
repository installation, independent execution,
experiment, independent referee review, or journal
acceptance is asserted. The mathematical release
consists of the explicit definitions and written
proofs in this source.

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

\bibitem{XF}
N.~Cheung,
\emph{Landmark X-F: The second-order quantum metric
as the retarded M{\o}ller response, and the closure
of the \(\kappa^2\) background identity}.
\loc{X-F.tex}, version 1.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/X-F}.
Especially the typed covariance identity,
background-gauge commutator, three source coefficients,
sum-zero calculus, graviton-slot ideal, direct
identity, and flat-gauge comparison.
The stronger variational terminology is qualified
in the present paper.

\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}.
Action normalization, Palatini variation, free
operator, matter response, and one-tree defect.

\bibitem{XEST}
N.~Cheung,
\emph{X-E-ST: Einstein--Hilbert normalization and
one matter-induced tree}.
Supplied \loc{X-E-ST.tex}.
Fixed-width supported predecessor, including
corrected scalar support, three-slot matter admission,
and the truncation audit.
No external registration is inferred.

\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 topology, closed scalar equation
subspace, reduction identities, and the separate
classical-background scalar sharp limit.

\bibitem{XG}
N.~Cheung,
\emph{Landmark X-G}.
\loc{X-G.tex}, version 1.
Version-specific supplied excerpts and correction record,
Enter Calculus Companion.
\url{https://enter.grok.me/browse/X-G}.
Retarded-solution versus variational-M{\o}ller distinction,
completed support, and ordering-data audit.
No successor loop construction is imported.

\bibitem{XAST}
N.~Cheung,
\emph{X-A-ST: Supported classical covariance of a
fixed-ramp relational scalar}.
Version-specific predecessor source record.
Inherited host instance, supported chart, completion,
signature ledger, coordinate prescription, and
plateau algebra.

\bibitem{XBST}
\emph{X-B-ST: One scalar retarded map on the X-A-ST host}.
Version-specific predecessor adaptation record.
Scalar map, off-shell insertion convention, and
one-contraction boundary; no general
time-ordered hierarchy is imported.

\bibitem{XCST}
N.~Cheung,
\emph{X-C-ST: Second-order scalar completion on the
inherited X-A-ST/X-B-ST host}.
Version-specific predecessor source record.
Completed representatives, affine slot, scalar-pair
counting, and background identity.

\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{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 and worldsheet, finite clock,
calibration, locality, and regular transport.

\bibitem{EnterI}
N.~Cheung,
\emph{Manuscript I: Finite operational characterization
of destructive measurement in the Enter calculus}.
\loc{I.tex}, version 2.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/I}.
Characterization within its full stated class;
contextual attribution, not a premise of the tree proof.

\bibitem{EnterPhilosophy}
N.~Cheung,
\emph{The Philosophy of Enter: Destructive Readout,
Quantum Recovery, and the Refusal of Identification}.
Introductory essay, Enter Calculus Companion.
\url{https://enter.grok.me/enter}.
Interpretive background, not a gravitational
construction theorem.

\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 causal extensions; 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
dynamical equation relations.
No renormalized interaction 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, distributional kernels, wavefront
products, 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{Landmark VII-C: The mixed-Wick obstruction
for the sharp Cauchy sector}.
\loc{VII-C.tex}, version 1.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/VII-C}.
Comparison only, retaining its exact product,
topology, and mandatory-word hypotheses.

\bibitem{VIIE}
N.~Cheung,
\emph{VII-E}.
Version 1, Enter Calculus Companion.
\url{https://enter.grok.me/browse/VII-E}.
Complete scheduled-support and observation
interfaces.

\bibitem{En08A}
N.~Cheung,
\emph{En08-A}.
Version 1, Enter Calculus Companion.
\url{https://enter.grok.me/browse/En08-A}.
Scheduler consistency versus Lorentz-cone
reconstruction.

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

\bibitem{En09}
N.~Cheung,
\emph{En09-A: Realization of Relational Time by
Destructive Enter on a Clock Register}.
\loc{En09-A.tex}, version 1.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/En09-A}.
Complete instrument identity, untouched references,
conditioning, and separate dynamical hypotheses.

\bibitem{En10A}
N.~Cheung,
\emph{En10-A}.
Version 1, Enter Calculus Companion.
\url{https://enter.grok.me/browse/En10-A}.
Separate finite source-weight estimation under a
declared access contract; not supplied by an installed
tree-evaluation table.

\bibitem{Watrous}
J.~Watrous,
\emph{The Theory of Quantum Information}.
Cambridge University Press, 2018.
Complete finite channels, instruments, and
diamond-norm comparisons.

\bibitem{XVIA}
N.~Cheung,
\emph{XVI-A}.
\loc{XVI-A.tex}, version 1.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/XVI-A}.
Conditional retarded-slot wavefront interface,
not a constructed subtraction or a replacement
for a missing coefficient.

\bibitem{ComparisonRecords}
N.~Cheung,
\emph{VII-I}, version 1;
\emph{IX-A}, version 1;
\emph{IX-D}, version 3;
\emph{IX-E}, version 1;
\emph{XI-G}, version 1;
and \emph{X-Enter}, version 1.
Also the version-specific \emph{IX-A-ST} and
\emph{IX-D-ST} adaptation records.
Enter Calculus Companion.
\url{https://enter.grok.me/browse}.
Documentary separation controls, not analytical
inputs to the three-tree proof.

\bibitem{EmRecords}
N.~Cheung,
\emph{Em01-B}, version 3;
\emph{Em01-C}, version 3;
and \emph{Em02-D}, version 4.
Version-specific supplied excerpts,
Enter Calculus Companion.
\url{https://enter.grok.me/browse}.
Separate polynomial-growth, aligned-null diagnostic,
and raw-family asymptotic records.

\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 H2, the narrower internal amplitude,
missing full I3/J2 maps, and downstream
admission boundaries remain distinct.

\bibitem{OpenAICatalogue}
OpenAI,
\emph{Mathematics catalogue}.
Supplied documentary descriptions of families
282, 362, and 376 in the Enter Calculus Companion.
Comparison-only attribution.
No catalogue proof is independently verified or used
as a premise.

\end{thebibliography}

\end{document}