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\hypersetup{
 pdftitle={XI-A-ST: A supported Einstein--scalar sheet on the
 prescribed Enter spacetime},
 pdfsubject={Local transfer, switched Einstein--scalar action,
 Noether identities, empty fixed-input jet, and destructive reports}
}

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\theoremstyle{definition}
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\newtheorem{assumption}[theorem]{Assumption}
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\title{\textbf{XI-A-ST: A supported Einstein--scalar sheet\\
on the prescribed Enter spacetime}\\[5pt]
\large Signed action, separated Noether identities,\\
an empty fixed-input Einstein jet, and complete finite reports}
\author{Ngo Cheung}
\date{}

\begin{document}
\maketitle

\begin{abstract}
We give a local Grade S transfer of Cheung's XI-A
Einstein--scalar sheet to the regular I-B conical product.
The host geometry is prescribed, while all analytical equations
remain on an independently declared Minkowski completion.
The transfer retains the unswitched Einstein--Hilbert action,
the minimally switched scalar action, the distinction between
invariant and gauge-fixed Noether identities, and tensor-density
covariance of the invariant metric Euler operator.

The principal obstruction is inherited, not newly claimed:
for the specified free scalar input
\(\phi_m(t)=\sin(mt)/m\), with massless value \(t\), the
fixed-input Einstein jet is empty for every nonzero compact
switch in the frozen class. Its detector is the first
source-compatibility coefficient
\((\partial_\mu g)T_0^{\mu\nu}[\phi_m]\).
A compact bump gives a strictly positive margin, whereas the
vacuum retains a nonempty jet. This distinguishes a
fixed-input incompatibility from a failure of Einstein theory
or of gauge-fixed wave solvability.

The contribution of this sheet is the supported transfer and
its complete reporting interface, not a new gravitational
covariance theorem. We identify two essential locality
qualifications: the switching detector is not a plateau
detector, and global completion-jet emptiness must not be
renamed emptiness of every plateau-local solution space.
A finite supplied catalogue is reported by destructive Enter,
with surviving registers and arbitrary untouched references
retained. An explicit common-interface probability encoding
has an exact diamond-distance formula for rational table
approximations. Finite-probe coverage and an invisible
conserved-perturbation construction prevent a successful
finite table from being mistaken for a distributional identity.

The potential value is an auditable separation of action
identities, solution compatibility, local support, and finite
readout. The fixed positive ramp width is not a worldsheet
thickness. No host reconstruction, field-acquisition map,
gravitational propagator, quantum multiplet, tree, BRST
construction, vacuum-energy calculation, or characteristic
quantum coefficient is produced.
\end{abstract}

\scopebox{Exact release}{%
The classical XI-A sheet is retained on the supported
completion chart. The specified completion Einstein jet is
proved obstructed, with the same compact detector whenever
that detector fits the retained laboratory.

The finite report instrument is constructed for supplied
evaluations. It does not acquire the continuum fields,
source the host, or turn a report of incompatibility into
a metric solution.

X-A-ST's relational intertwiner, scalar covariance, and
edge-zone theorem are cited, not recounted.}

\tableofcontents

%=====================================================================
\section{Purpose, attribution, and the philosophy of Enter}
\label{sec:purpose}
%=====================================================================

\subsection{The scope of the transfer}

XI-A fixes one classical Einstein--scalar action, separates
its invariant and gauge-fixed identities, and identifies
an empty Einstein fiber for a named free scalar input
\cite{CheungXIA}. The present sheet retains those objects.
It changes their supported host interpretation and supplies
a finite Enter report interface.

The action is not obtained from the conical metric.
Conversely, imposing an equation on a formal analysis metric
does not update that conical metric.
These two statements are part of the model specification,
not consequences of a field equation.

The results attributed below to XI-A are supported
restatements of its explicitly named formulas and theorems.
Short variational, sign, and chart checks fix the transfer.
They are not offered as new discoveries of the underlying
Einstein--scalar identities.

The broader distinction between classical gauge symmetry,
gauge fixing, and subsequent perturbative BV quantization
is also explicit in the external literature
\cite{BFR}. No BV construction is imported into this sheet.

\subsection{Enter as an interface discipline}

Cheung's Enter calculus uses a strictly linear treatment of
quantum names and destructive computational-basis readout.
Its operational characterization of measurement is
class-relative; it is not a characterization of every
possible continuum field instrument
\cite{CheungEnter,CheungI}.

The accompanying philosophy essay distinguishes the
classical record returned by destructive readout from a
quantum-return recovery map \cite{CheungPhilosophy}.
The relevant questions are:
\[
 \text{What is supplied?}\qquad
 \text{What is accessible?}\qquad
 \text{What survives?}\qquad
 \text{What is certified?}
\]
Here the supplied objects are finitely many classical
evaluations and their declared encodings.
The output contains a classical code and the actual
unmeasured finite residual.

In particular,
\[
\begin{gathered}
 \text{Euler derivative}\ne\text{Einstein solution},\\
 \text{gauge-fixed solution}\ne\text{invariant Einstein jet},\\
 \text{finite report}\ne\text{continuum acquisition},\\
 \text{prescribed cone}\ne\text{derived spacetime},\\
 \text{clock-register consumption}\ne\text{consumption of }T.
\end{gathered}
\]
The complete finite instrument is proved in
Section~\ref{sec:instrument}; no general measurement
uniqueness theorem is needed for that matrix calculation.

\subsection{What is new, and what is not}

The gravitational action, its signs, its Noether identity,
and the fixed-input obstruction belong to XI-A.
The host and destructive clock interface belong to the
I-B/X-A-ST framework \cite{CheungIB,CheungXAST}.

The present assembly makes explicit:
\begin{enumerate}[label=\textup{(\roman*)}]
\item the local tensor-density transfer with the two
signature conventions kept separate;
\item the distinction between a switching-region detector
and a plateau-local equation problem;
\item a complete finite reporting channel with a specified
rational-approximation metric;
\item the coverage requirements that separate a finite
report from an identity on an unrestricted residual space.
\end{enumerate}
No priority claim is made for the general variational,
linear-algebraic, or channel principles used in this assembly.

%=====================================================================
\section{The retained host and its three ledgers}
\label{sec:host}
%=====================================================================

\begin{assumption}[Prescribed host and supported chart]
\label{ass:host}
Retain the regular conical product
\[
 \mathcal M^\circ
   =\RR_T\times C_{\delta_c}^{\circ}\times\RR_z,
 \qquad
 \alpha_c=1-\frac{\delta_c}{2\pi},
 \qquad 0<\delta_c<2\pi,
\]
with metric
\begin{equation}
 G=-c_{\mathrm h}^{\,2}\dd T^2+\dd r^2
       +\alpha_c^2r^2\dd\varphi^2+\dd z^2,
 \qquad c_{\mathrm h}>0,
 \label{eq:hostmetric}
\end{equation}
and future orientation given by increasing \(T\).
Its completion has the worldsheet
\[
 \mathcal W=\RR_T\times\{o\}\times\RR_z.
\]
The defect is a worldsheet, not a particle worldline.

Retain a regular developing chart
\[
 \iota:V\longrightarrow\mathcal L\subset\mathcal M^\circ
\]
from an open set \(V\) in the declared completion
\[
 \widehat M=\RR^{1+3},
 \qquad
 \eta=\diag(1,-1,-1,-1),
 \qquad
 \Box=\partial_t^2-\Delta_{\mathbf x}.
\]
The signature and time-calibration ledger is
\begin{equation}
 \iota^*G=-\eta,
 \qquad
 t=c_{\mathrm h}(T-T_\Sigma).
 \label{eq:signature}
\end{equation}
Retain the observation region \(\mathcal O\) and the
supported shadow \(D_*\Subset V\) declared for this
laboratory instance.
\end{assumption}

This is the I-B host with the X-A-ST completion convention
\cite{CheungIB,CheungXAST}.
The notation \(V\) makes the chart domain distinct from its
host image \(\mathcal L\).

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.29\linewidth}X@{}}
\toprule
Ledger item & Status in XI-A-ST\\
\midrule
Cone, worldsheet, orientation &
Prescribed I-B data; no reconstruction.\\
Developing chart, observation, shadow &
Retained supported data; no silent enlargement.\\
Analysis metric convention &
\((+,-,-,-)\) on the Minkowski completion.\\
Host metric convention &
\((-,+,+,+)\), with \(\iota^*G=-\eta\).\\
Global inverse on the regular cone &
Not supplied or used. X-A-ST records non-global
hyperbolicity in \loc{prop:nonGH}.\\
Ramp width &
Fixed analytical metadata, not defect thickness.\\
Laboratory storage &
All live clocks, payloads, auxiliaries, and controls charged.\\
Returned storage &
All exported records and retained outputs charged separately.\\
External reference &
Untouched and separately listed; no control wire supplied.\\
\bottomrule
\end{tabularx}
\end{center}

The host declarations and the non-global-transfer boundary are
copied from I-B/X-A-ST rather than reproved
\cite{CheungIB,CheungXAST}.
The three-storage-ledger discipline is the one used in
X-L-ST \cite{CheungXLST}.
It does not import that sheet's action or free normalization.

\begin{remark}[A sign convention is not a coordinate signature change]
Equation~\eqref{eq:signature} does not assert that a real
coordinate transformation changes the signature of a
bilinear form. It specifies a sign-reversed analytical
comparison metric on the developed chart.
All analytical contractions below use \(\eta\), or the
formal metric \(q\), not the host metric \(G\).
\end{remark}

\begin{remark}[Width is not geometry]
The fixed \(\varepsilon>0\) belongs to the ramp prescription.
It neither thickens \(\mathcal W\) nor defines a smooth
core replacing the cone. No width limit is taken.
\end{remark}

\subsection{The local transfer rule}

Let \(x\) be the developed coordinates and \(y\) an
orientation-preserving coordinate system on the host
laboratory. If \(u^{\mu\nu}\) is a contravariant rank-two
density of weight one, its transported components are
\begin{equation}
 \widetilde u^{ab}(y)
 =
 \left|\det\frac{\partial x}{\partial y}\right|
 \frac{\partial y^a}{\partial x^\mu}
 \frac{\partial y^b}{\partial x^\nu}
 u^{\mu\nu}(x(y)).
 \label{eq:passivedensity}
\end{equation}

\begin{lemma}[Supported density transfer]
\label{lem:transfer}
Compactly supported variational pairings, density identities,
and nonzero compact-test pairings are unchanged by the
transport \eqref{eq:passivedensity}, provided all background
tensors and test tensors are transported consistently.

This local rule supplies no global conical Cauchy inverse
and no field equation on the worldsheet.
\end{lemma}

\begin{proof}
A covariant variation transforms with two inverse Jacobian
factors. They cancel the two displayed tensor factors in
\eqref{eq:passivedensity}; the remaining determinant is
exactly the change-of-variables factor in the integral.
Thus the variational pairing is unchanged.

An invertible change of coordinates maps zero distributions
to zero distributions and preserves a detecting pairing
after the test is transported. The calculation is confined
to \(V\) and makes no assertion outside the regular chart.
\end{proof}

In particular, no additional minus sign is inserted into
the XI-A action because the host convention is
\((-,+,+,+)\). That action is still the analysis action
with background \(\eta\).

%=====================================================================
\section{Analytical freeze and the already-established baseline}
\label{sec:freeze}
%=====================================================================

Work on
\[
 M_T=[0,T]\times\RR^3,\qquad
 M_T^\circ=(0,T)\times\RR^3,\qquad
 \Sigma=\{t=0\},
\]
as subsets of \(\widehat M\).
Every coefficient field is real and smooth on a
neighborhood of the closed slab.

Fix
\[
 m\ge0,\qquad
 g\in C_c^\infty(M_T^\circ;[0,1]),
 \qquad
 P_g=\operatorname{int}\{g=1\}\ne\varnothing.
\]
Retain \(0<\varepsilon<T\) and
\[
 \sigma_\varepsilon(t)=\chi(t/\varepsilon),
\]
where
\[
 \chi(s)=
 \begin{cases}
  0,&s\le0,\\[2pt]
  \displaystyle
  \frac{e^{-1/s}}{e^{-1/s}+e^{-1/(1-s)}},&0<s<1,\\[7pt]
  1,&s\ge1.
 \end{cases}
\]
Choose the retained plateau domain
\[
 U\Subset P_g\cap\{\varepsilon<t<T\}\cap V.
\]
The admitted parameters are compactly supported in \(U\).

The plateau group \(\mathcal G_P\) consists of the
corresponding orientation-preserving compact flows.
For order bookkeeping, use genuine flows generated by
\(\kappa\xi+\kappa^2\zeta\), with
\(\xi,\zeta\in C_c^\infty(U;TM_T)\).
They preserve \(g\) and \(\sigma_\varepsilon\) and are
the identity near \(\Sigma\).

\begin{definition}[The analysis fields]
\label{def:fields}
The independent classical metric variable is one symmetric
tensor \(\gamma\), with
\[
 q_{\mu\nu}=\eta_{\mu\nu}+\kappa\gamma_{\mu\nu}.
\]
The scalar is \(\phi\), and
\[
 \Pm=\Box+m^2.
\]
The formal metric \(q\) is not \(G\).
Evaluating \(\gamma\) later on a formal coefficient family
does not introduce a second fundamental metric field.
\end{definition}

All expansions are coefficientwise in \(\kappa\).
They are not claims about convergence of a numerical
perturbation series.

\subsection{X-A-ST is cited rather than recounted}

The following remain the relational baseline
\cite[\loc{prop:edge}, \loc{thm:main}]{CheungXAST}:
\begin{enumerate}
\item the complete second-order coordinate intertwiner;
\item off-shell invariance of the label-smeared relational scalar;
\item covariance of the switched scalar solution with its
specified Cauchy data;
\item the strictly negative edge-zone preflight defect,
\[
 -\int_0^\varepsilon
       \sigma_\varepsilon'(s)\,\alpha_{\mathrm e}(s)\,\dd s<0
\]
on its named detecting data.
\end{enumerate}
These results are not proved again here.
The profile \(\alpha_{\mathrm e}\) in this documentary
formula is unrelated to the cone opening \(\alpha_c\).

In particular, contracting the Noether identity below on
a plateau parameter is not presented as a new Route II
theorem for that observable.

X-B, X-C, and X-D remain non-inputs of the classical action
sheet, as in XI-A \cite{CheungXIA}.
The X-E normalization is not substituted for the one below.

%=====================================================================
\section{The signed action and its Euler derivatives}
\label{sec:action}
%=====================================================================

\subsection{One full action before expansion}

Write \(\rho_q=\sqrt{-\det q}\).
Use the curvature convention
\begin{equation}
 R_{\mu\nu}
 =
 \partial_\rho\Gamma^\rho_{\mu\nu}
 -\partial_\nu\Gamma^\rho_{\mu\rho}
 +\Gamma^\rho_{\rho\lambda}\Gamma^\lambda_{\mu\nu}
 -\Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\rho},
 \qquad
 G_{\mu\nu}=R_{\mu\nu}-\frac12q_{\mu\nu}R.
 \label{eq:curvature}
\end{equation}
The retained XI-A normalization is
\begin{align}
 S_{\EH}[q]
   &=-\frac{2}{\kappa^2}
        \int_{M_T}\rho_qR[q]\,\dd^4x,
 \label{eq:EH}\\
 \Lagr_q[\phi]
   &=\frac12\rho_q
       \left(q^{\mu\nu}\partial_\mu\phi\partial_\nu\phi
                                          -m^2\phi^2\right),
 \label{eq:Lq}\\
 \Lagr_\eta[\phi]
   &=\frac12\left(\partial_\mu\phi\partial^\mu\phi
                                          -m^2\phi^2\right),\\
 S_g^{\mathrm{sc}}[q,\phi]
   &=\int_{M_T}
       \left((1-g)\Lagr_\eta[\phi]+g\Lagr_q[\phi]\right)
                                                     \dd^4x,
 \label{eq:scalar-action}\\
 S_{\inv}&=S_{\EH}+S_g^{\mathrm{sc}},
 \qquad
 S_{\tot}=S_{\inv}+S_{\gf}.
 \label{eq:Stot}
\end{align}
There is no cosmological-constant term and no gravitational
switch in \eqref{eq:EH} \cite{CheungXIA}.

The ramp regulates the inherited relational coordinate
prescription. It does not multiply the Einstein--Hilbert
density.

Actions mean variational functionals under compact interior
variations and compact differences.
Their values need not be finite numbers on arbitrary
noncompact smooth configurations.
No boundary variation of the metric is taken.

Define
\[
 \tau=\eta^{\mu\nu}\gamma_{\mu\nu},\qquad
 \bar\gamma_{\mu\nu}
      =\gamma_{\mu\nu}-\frac12\eta_{\mu\nu}\tau,\qquad
 D^\nu(\gamma)=\partial_\mu\bar\gamma^{\mu\nu}.
\]
The single gauge-fixing term is
\begin{equation}
 S_{\gf}[\gamma]=\int_{M_T}D_\nu(\gamma)D^\nu(\gamma)\,\dd^4x.
 \label{eq:gf}
\end{equation}
Its coefficient is frozen. No gauge-parameter family is used.

\subsection{The quadratic normalization}

\begin{restatement}[XI-A quadratic coefficient and signed operator]
\label{rest:quadratic}
The bulk quadratic coefficient of \eqref{eq:EH}, modulo
a total divergence, is
\begin{equation}
\begin{split}
 S_{\FP}[\gamma]=\frac12\int_{M_T}\Big[
 &\partial_\lambda\gamma_{\mu\nu}
                       \partial^\lambda\gamma^{\mu\nu}
 -2(\partial_\mu\gamma^{\mu\nu})
                       (\partial^\rho\gamma_{\rho\nu})\\
 &+2(\partial_\mu\gamma^{\mu\nu})\partial_\nu\tau
 -\partial_\lambda\tau\partial^\lambda\tau
 \Big]\dd^4x.
 \label{eq:FP}
\end{split}
Its Euler derivative is \(2\Glin^{\mu\nu}(\gamma)\), where
\begin{equation}
 \Glin^{\mu\nu}(\gamma)
 =
 -\frac12\Box\bar\gamma^{\mu\nu}
 +\partial^{(\mu}D^{\nu)}
 -\frac12\eta^{\mu\nu}\partial_\rho D^\rho.
 \label{eq:Glin}
\end{equation}
Parentheses include the factor \(1/2\).

The separate gauge-fixing derivative is
\begin{equation}
 e_{\gf}^{\mu\nu}
  =-2\partial^{(\mu}D^{\nu)}
                     +\eta^{\mu\nu}\partial_\rho D^\rho,
 \qquad
 \partial_\mu e_{\gf}^{\mu\nu}=-\Box D^\nu.
 \label{eq:egf}
\end{equation}
Consequently the quadratic gauge-fixed operator is
\begin{equation}
 2\Glin^{\mu\nu}(\gamma)+e_{\gf}^{\mu\nu}(\gamma)
                              =-\Box\bar\gamma^{\mu\nu}.
 \label{eq:quadraticoperator}
\end{equation}
\end{restatement}

\begin{proof}[Normalization and transfer check]
These are XI-A \loc{prop:FP}, \loc{eq:Glin}, and
\loc{eq:egf} \cite{CheungXIA}.
The relevant action variation is
\[
 \delta S_{\EH}
       =\frac{2}{\kappa^2}
           \int\rho_qG^{\mu\nu}\delta q_{\mu\nu}\,\dd^4x.
\]
Since \(\delta q=\kappa\delta\gamma\), its leading
Euler coefficient is \(2\Glin\).
Compact variation of \eqref{eq:gf} gives
\eqref{eq:egf}; addition gives
\eqref{eq:quadraticoperator}.
Lemma~\ref{lem:transfer} transports these variational
pairings without changing the action sign.
\end{proof}

In particular, this handoff is
\(-\Box\Ptr\), not X-E's differently normalized metric
operator. A shared de Donder name does not identify them.

\subsection{The two retained sign diagnostics}

\begin{restatement}[Positive TT and attractive Newtonian signs]
\label{rest:signs}
For
\[
 \gamma_{0\mu}=0,\qquad
 \partial_i\gamma_{ij}=0,\qquad
 \delta^{ij}\gamma_{ij}=0,
\]
the Fierz--Pauli density is
\begin{equation}
 \Lagr_{\FP}^{\TT}
  =\frac12\dot\gamma_{ij}\dot\gamma_{ij}
       -\frac12\partial_k\gamma_{ij}\partial_k\gamma_{ij}.
 \label{eq:TT}
\end{equation}

For the auxiliary static unswitched comparison with
\(T^{00}=\rho\ge0\), \(T^{0i}=T^{ij}=0\), where
\(\rho\) is smooth, compactly supported, and nonzero, put
\(H=q-\eta\). The linear equation has the comparison
\begin{equation}
 H_{00}=2\Phi_N,\qquad
 H_{ij}=2\Phi_N\delta_{ij},\qquad H_{0i}=0,
 \label{eq:Newtonmetric}
\end{equation}
with
\begin{equation}
 \Delta\Phi_N=\frac{\kappa^2}{8}\rho,\qquad
 \Phi_N(\mathbf x)
   =-\frac{\kappa^2}{32\pi}
        \int_{\RR^3}\frac{\rho(\mathbf y)}
                           {|\mathbf x-\mathbf y|}\,\dd^3y<0.
 \label{eq:Newton}
\end{equation}
Thus \(\kappa^2=32\pi G_N\) has the attractive sign.
\end{restatement}

\begin{proof}[Signature check]
These are XI-A \loc{lem:TT} and \loc{lem:Newton}
\cite{CheungXIA}.
Raising both spatial indices in \eqref{eq:TT} contributes
two minus signs, so the time-derivative coefficient remains
positive. In \eqref{eq:Newtonmetric},
\(\bar H_{00}=4\Phi_N\), \(\bar H_{ij}=0\), and \(D(H)=0\).
Equation~\eqref{eq:Glin} gives
\(\Glin^{00}(H)=2\Delta\Phi_N\).
The retained unswitched equation is
\(G^{\mu\nu}=\kappa^2T^{\mu\nu}/4\), giving
\eqref{eq:Newton}.
Neither calculation inserts \(G\) in place of \(q\).
\end{proof}

The Newtonian source is an auxiliary static comparator.
It is not a solution of the compact-time switched system.
Neither sign diagnostic computes vacuum energy.

\subsection{Minimal metric coupling and the actual scalar switch}

Let \(p_\mu=\partial_\mu\phi\), and put
\[
 \ell_q=\frac12(q^{\alpha\beta}p_\alpha p_\beta-m^2\phi^2),
 \qquad
 T_q^{\mu\nu}
   =q^{\mu\alpha}q^{\nu\beta}p_\alpha p_\beta
                                         -q^{\mu\nu}\ell_q.
\]
Write \(T_0=T_\eta\).

The covariant metric variation is
\begin{equation}
 \delta_q\Lagr_q=-\frac12\rho_qT_q^{\mu\nu}\delta q_{\mu\nu}.
 \label{eq:stressvariation}
\end{equation}
Set
\[
 \rho_q=1+\kappa b_1+\kappa^2b_2+O(\kappa^3),
 \qquad
 \rho_qq^{\mu\nu}
      =\eta^{\mu\nu}+\kappa a_1^{\mu\nu}
                           +\kappa^2a_2^{\mu\nu}+O(\kappa^3),
\]
where
\begin{align}
 b_1&=\frac12\tau,
 &
 b_2&=\frac18\tau^2-\frac14\gamma_{\alpha\beta}\gamma^{\alpha\beta},
 \label{eq:bs}\\
 a_1^{\mu\nu}&=\frac12\tau\eta^{\mu\nu}-\gamma^{\mu\nu},
 &
 a_2^{\mu\nu}
   &=\gamma^\mu{}_\rho\gamma^{\rho\nu}
       -\frac12\tau\gamma^{\mu\nu}+b_2\eta^{\mu\nu}.
 \label{eq:as}
\end{align}
The matter vertices are therefore
\begin{align}
 V_{1,g}
  &=\frac12\int g
       (a_1^{\mu\nu}p_\mu p_\nu-m^2b_1\phi^2)\,\dd^4x
    =-\frac12\int g\gamma_{\mu\nu}T_0^{\mu\nu}\,\dd^4x,
 \label{eq:vertex}\\
 V_{2,g}
  &=\frac12\int g
       (a_2^{\mu\nu}p_\mu p_\nu-m^2b_2\phi^2)\,\dd^4x.
 \label{eq:secondvertex}
\end{align}
There is one factor \(g\), not \(g^2\), in the second
coefficient.

Define \(H_n^{\mu\nu}\) by
\begin{equation}
 \rho_qG^{\mu\nu}(q)
   =\kappa H_1^{\mu\nu}
       +\kappa^2H_2^{\mu\nu}
       +\kappa^3H_3^{\mu\nu}+O(\kappa^4).
 \label{eq:Hdefinition}
\end{equation}
Thus \(H_1=\Glin\).
Appendix~\ref{app:coefficients} gives a finite
coefficient prescription.

Also,
\[
 \rho_qT_q^{\mu\nu}
       =T_0^{\mu\nu}+\kappa J_1^{\mu\nu}+O(\kappa^2),
\]
with
\begin{equation}
\begin{split}
 J_1^{\mu\nu}={}&
 b_1T_0^{\mu\nu}
 -\gamma^{\mu\alpha}p_\alpha p^\nu
 -\gamma^{\nu\alpha}p_\alpha p^\mu\\
 &+\gamma^{\mu\nu}\Lagr_\eta
 +\frac12\eta^{\mu\nu}\gamma^{\alpha\beta}p_\alpha p_\beta.
 \label{eq:J1}
\end{split}
\end{equation}

\begin{restatement}[XI-A Euler derivative through order two]
\label{rest:euler}
The invariant metric Euler density is
\begin{align}
 e^{\mu\nu}
  :=\frac{\delta S_{\inv}}{\delta\gamma_{\mu\nu}}
  &=\frac{2}{\kappa}\rho_qG^{\mu\nu}(q)
                   -\frac{\kappa}{2}g\rho_qT_q^{\mu\nu},
 \label{eq:eexact}\\
  &=2H_1^{\mu\nu}
     +\kappa\left(2H_2^{\mu\nu}-\frac12gT_0^{\mu\nu}\right)
 \nonumber\\
  &\quad
     +\kappa^2\left(2H_3^{\mu\nu}-\frac12gJ_1^{\mu\nu}\right)
     +O(\kappa^3).
 \label{eq:eexp}
\end{align}
It contains no derivatives of \(g\).

By contrast, the scalar Euler density is
\begin{equation}
 E_\phi
   =-\partial_\mu\left[
       ((1-g)\eta^{\mu\nu}+g\rho_qq^{\mu\nu})
                                      \partial_\nu\phi\right]
     -m^2((1-g)+g\rho_q)\phi.
 \label{eq:Ephiexact}
\end{equation}
Equivalently,
\[
 E_\phi
   =-(\Pm+\kappa\mathsf V_{1,g}
                     +\kappa^2\mathsf V_{2,g})\phi+O(\kappa^3),
\]
where
\begin{equation}
 \mathsf V_{j,g}\phi
   =g\left[\partial_\mu(a_j^{\mu\nu}\partial_\nu\phi)
                                      +m^2b_j\phi\right]
       +(\partial_\mu g)a_j^{\mu\nu}\partial_\nu\phi,
 \qquad j=1,2.
 \label{eq:scalar-switch}
\end{equation}
Finally,
\begin{equation}
 \partial_\mu(gT_0^{\mu\nu})
   =g(\Pm\phi)\partial^\nu\phi
                              +(\partial_\mu g)T_0^{\mu\nu}.
 \label{eq:sourcedivergence}
\end{equation}
\end{restatement}

\begin{proof}[Local variational check]
The formulas are XI-A \loc{thm:euler} and
\loc{eq:scalar-switch} \cite{CheungXIA}.
Combining \eqref{eq:stressvariation} with
\(\delta q=\kappa\delta\gamma\) gives
\eqref{eq:eexact}; no metric integration by parts in
the matter action is involved.
Scalar integration by parts instead gives
\eqref{eq:Ephiexact}, and hence
\eqref{eq:scalar-switch}.
The identity
\(\partial_\mu T_0^{\mu\nu}
       =(\Pm\phi)\partial^\nu\phi\)
gives \eqref{eq:sourcedivergence}.
All operations are completion-chart operations and
transport by Lemma~\ref{lem:transfer}.
\end{proof}

\begin{remark}[Order bookkeeping]
The action is fixed before its derivative is projected.
The classical cubic and quartic Einstein coefficients
supply \(2H_2\) and \(2H_3\).
Because \eqref{eq:eexact} contains \(1/\kappa\),
a covariance check through order \(\kappa^2\) must
retain the required order-three metric curvature data.
A premature order-two truncation of the pulled-back
metric is not sufficient.
\end{remark}

%=====================================================================
\section{Noether identities and tensor-density covariance}
\label{sec:noether}
%=====================================================================

\subsection{The correct adjoint operator}

For a symmetric contravariant density \(u^{\mu\nu}\), define
\begin{equation}
 \Ncal_\nu(u)
   =2\partial_\mu(u^{\mu\rho}q_{\rho\nu})
                         -u^{\mu\rho}\partial_\nu q_{\mu\rho}.
 \label{eq:N}
\end{equation}
The scaled active variations are
\[
 \delta_\xi q=\kappa\Lie_\xi q,\qquad
 \delta_\xi\gamma=\Lie_\xi q=K\xi+\kappa\Lie_\xi\gamma,\qquad
 \delta_\xi\phi=\kappa\xi^\rho\partial_\rho\phi,
\]
where
\[
 (K\xi)_{\mu\nu}=\partial_\mu\xi_\nu+\partial_\nu\xi_\mu.
\]
The backgrounds \(\eta\), \(g\), and
\(\sigma_\varepsilon\) are held fixed in this active
transformation.

Define
\begin{equation}
 \Bcal_\nu
   =(\partial_\mu g)T_0^\mu{}_\nu
       -(1-g)(\Pm\phi)\partial_\nu\phi
       +(\partial_\nu g)(\Lagr_\eta-\Lagr_q).
 \label{eq:B}
\end{equation}

\begin{restatement}[Invariant and gauge-fixed identities]
\label{rest:noether}
The exact invariant identity is
\begin{equation}
 \boxed{\Ncal_\nu(e)
          =\kappa(E_\phi\partial_\nu\phi-\Bcal_\nu).}
 \label{eq:noether}
\end{equation}
The gauge-fixed identity is instead
\begin{equation}
 \Ncal_\nu(e+e_{\gf})
   =\kappa(E_\phi\partial_\nu\phi-\Bcal_\nu)
                                  +\Ncal_\nu(e_{\gf}).
 \label{eq:noethergf}
\end{equation}
On \(U\), \(\Bcal_\nu=0\), but the gauge-fixing
contribution is not removed.
\end{restatement}

\begin{proof}[Adjoint and chart check]
These are XI-A \loc{thm:noether},
\loc{eq:noether}, and \loc{eq:noether-gf}
\cite{CheungXIA}.
For a compact interior parameter, direct integration by
parts gives
\[
 \delta_\xi S_{\inv}
 =\int\xi^\nu
       [-\Ncal_\nu(e)+\kappa E_\phi\partial_\nu\phi]\,\dd^4x.
\]
The inherited fixed-background action variation has
coefficient \(\kappa\int\xi^\nu\Bcal_\nu\dd^4x\).
Comparison gives \eqref{eq:noether}.
The variation of the independent term
\eqref{eq:gf} gives \eqref{eq:noethergf}.
The pairing transfers by Lemma~\ref{lem:transfer}.
No new proof of the relational Route II theorem is asserted.
\end{proof}

\begin{remark}[A factor of two must not be lost]
The raw divergence and the adjoint Noether operator are
different:
\[
 \partial_\mu e_{\gf}^{\mu\nu}=-\Box D^\nu,
 \qquad
 \Ncal_\nu(e_{\gf})\big|_{\kappa=0}=-2\Box D_\nu.
\]
The factor two comes from the leading flat part of
\eqref{eq:N}.
\end{remark}

For a \(\kappa\)-independent scalar, write
\(e=e_0+\kappa e_1+\kappa^2e_2+\cdots\), and put
\[
 \Qcal_\nu(u)
 =2\partial_\mu(u^{\mu\rho}\gamma_{\rho\nu})
                              -u^{\mu\rho}\partial_\nu\gamma_{\mu\rho}.
\]
The retained coefficient audit is
\begin{align}
 2\partial_\mu(e_0^{\mu\rho}\eta_{\rho\nu})&=0,
 \label{eq:N0}\\
 2\partial_\mu(e_1^{\mu\rho}\eta_{\rho\nu})
       +\Qcal_\nu(e_0)
  &=-g(\Pm\phi)\partial_\nu\phi
                      -(\partial_\mu g)T_0^\mu{}_\nu,
 \label{eq:N1}\\
 2\partial_\mu(e_2^{\mu\rho}\eta_{\rho\nu})
       +\Qcal_\nu(e_1)
  &=-(\mathsf V_{1,g}\phi)\partial_\nu\phi
                     +(\partial_\nu g)\Lagr_{(1)},
 \label{eq:N2}
\end{align}
where
\[
 \Lagr_{(1)}
    =\frac12(a_1^{\alpha\beta}p_\alpha p_\beta-m^2b_1\phi^2).
\]
These formulas retain the nonlinear adjoint contribution;
a bare partial divergence is not the complete identity.

\subsection{The two rejected replacements}

\begin{restatement}[XI-A's quadratic-only and unbroken-gauge-fixed
alternatives remain false]
\label{rest:rejections}
Replacing the full Einstein action by the Fierz--Pauli
coefficient does not preserve the nonlinear Noether identity.
The fixed gauge-fixing term is not invariant under
\(\mathcal G_P\).
\end{restatement}

\begin{proof}[Retained diagnostic checks]
For the first assertion, XI-A's order-one vacuum identity
contains
\[
 2\partial_\mu(2H_2^{\mu\rho}\eta_{\rho\nu})
                          +\Qcal_\nu(2H_1)=0.
\]
For a tensor equal to \(\psi(t)\eta\) near a point,
\[
 \Qcal_0(2\Glin(\psi\eta))=-6\psi'\psi''.
\]
Deleting \(H_2\) therefore leaves a generally nonzero
term.

For the second assertion, take the same compact spatial
parameter used in XI-A,
\(\xi=f\partial_1\), \(0\ne f\in C_c^\infty(U)\).
Then
\[
 D(K\xi)=\Box\xi,\qquad
 S_{\gf}[K\xi]=-\int(\Box f)^2\,\dd^4x<0.
\]
A nonzero spacetime-compact homogeneous wave is excluded
by Cauchy uniqueness, so \(\Box f\not\equiv0\).
These are the existing XI-A checks, now inside the retained
regular chart \cite{CheungXIA}.
\end{proof}

No gauge algebra is recovered by imposing both compact
spacetime support and \(\Box\xi=0\): that restriction
would leave only the zero parameter.

\subsection{Active tensor-density covariance}

For an orientation-preserving diffeomorphism \(F\), define
\begin{equation}
 (F^*_{(2,0),\dens}u)^{\mu\nu}(x)
 =
 \det DF(x)\,
 (DF(x)^{-1})^\mu{}_\alpha
 (DF(x)^{-1})^\nu{}_\beta
 u^{\alpha\beta}(F(x)).
 \label{eq:tensorpull}
\end{equation}

\begin{restatement}[XI-A invariant Euler covariance]
\label{rest:covariance}
For \(F\in\mathcal G_P\), put
\(q'=F^*q\), \(\phi'=F^*\phi\), and
\(\gamma'=(q'-\eta)/\kappa\).
Then
\begin{equation}
 e[q',\phi']=F^*_{(2,0),\dens}e[q,\phi].
 \label{eq:covariance}
\end{equation}
The formal identity retains its coefficientwise meaning
through order \(\kappa^2\) when projection is performed
after the required coefficients have been retained.
It is not valid in general with \(e+e_{\gf}\) in place
of \(e\).

For a compact near-identity flow not required to preserve
\(g\), its metric covariance defect is
\begin{equation}
 e[q',\phi']-F^*_{(2,0),\dens}e[q,\phi]
 =
 -\frac{\kappa}{2}(g-F^*g)\rho_{q'}T_{q'}[\phi'].
 \label{eq:covariance-defect}
\end{equation}
\end{restatement}

\begin{proof}[Naturality and support check]
This is XI-A \loc{thm:covariance} and
\loc{eq:covariance-defect} \cite{CheungXIA}.
The two tensor densities in \eqref{eq:eexact} are
natural. Their switched matter terms differ only
by \(g-F^*g\), giving
\eqref{eq:covariance-defect}.
The plateau support gives \(F^*g=g\).
The explicit gauge-fixing operator
\eqref{eq:egf} retains the fixed background and fails
the corresponding assertion, as already diagnosed
in Restatement~\ref{rest:rejections}.
Lemma~\ref{lem:transfer} supplies the local host reading.
\end{proof}

The infinitesimal covariance defect involves
\(\xi^\mu\partial_\mu g\).
The source-divergence defect involves
\((\partial_\mu g)T_0^{\mu\nu}\).
They are related background-switch effects, not
identical formulas.

%=====================================================================
\section{The completion Einstein jet and its compact detector}
\label{sec:jet}
%=====================================================================

\subsection{The equation space being tested}

\begin{definition}[Fixed-input completion Einstein jet]
\label{def:jet}
For a smooth free scalar \(\phi_0\), let
\(\Jcal_g(\phi_0)\) consist of smooth coefficient families
\[
 \gamma(\kappa)=\gamma_0+\kappa\gamma_1+\kappa^2\gamma_2,
 \qquad
 \phi(\kappa)=\phi_0+\kappa\phi_1+\kappa^2\phi_2,
\]
such that the coefficients through order \(\kappa^2\)
of
\[
 e=0,\qquad E_\phi=0
\]
vanish on the completion slab.
The scalar satisfies \(\Pm\phi_0=0\), and
\(\phi_1,\phi_2\) have the inherited zero Cauchy data
at \(\Sigma\).
Metric Cauchy data are unrestricted.

This is the invariant Einstein equation, not the
gauge-fixed Euler equation.
\end{definition}

For clarity, the scalar coefficient equations include
\begin{align}
 \Pm\phi_1
   &=-\mathsf V_{1,g}[\gamma_0]\phi_0,\\
 \Pm\phi_2
   &=-\mathsf V_{1,g}[\gamma_0]\phi_1
     -\bigl(\mathsf V_{1,g}[\gamma_1]
               +\mathsf V_{2,g}[\gamma_0]\bigr)\phi_0.
 \label{eq:scalarjets}
\end{align}
These equations specify the inherited classical
Cauchy problem. No inverse is constructed here.

The metric coefficients include
\begin{align}
 e^{[0]}={}&2H_1[\gamma_0],\\
 e^{[1]}={}&2H_1[\gamma_1]+2H_2[\gamma_0]
                                      -\frac12gT_0[\phi_0],\\
 e^{[2]}={}&2H_1[\gamma_2]
       +2\,\mathrm DH_2[\gamma_0]\gamma_1+2H_3[\gamma_0]
 \nonumber\\
 &-\frac12g\bigl(J_1[\gamma_0,\phi_0]
                         +\dot T_0[\phi_0;\phi_1]\bigr),
 \label{eq:metricjets}
\end{align}
where
\[
 \dot T_0^{\mu\nu}[\phi_0;\phi_1]
 =p_0^\mu p_1^\nu+p_1^\mu p_0^\nu
       -\eta^{\mu\nu}
            (p_{0,\alpha}p_1^\alpha-m^2\phi_0\phi_1).
\]
Thus the scalar solution correction is not confused
with the metric correction to the stress density.

\begin{restatement}[Necessary source compatibility]
\label{rest:compatibility}
Nonemptiness of \(\Jcal_g(\phi_0)\) requires
\begin{equation}
 (\partial_\mu g)T_0^{\mu\nu}[\phi_0]=0.
 \label{eq:compatibility}
\end{equation}
In the subclass \(\gamma_0=0\), this is already
necessary for
\begin{equation}
 \Glin^{\mu\nu}(\gamma_1)=\frac14gT_0^{\mu\nu}[\phi_0].
 \label{eq:sourcejet}
\end{equation}
\end{restatement}

\begin{proof}[Retained coefficient check]
This is XI-A \loc{lem:compatibility}
\cite{CheungXIA}.
For \(\gamma_0=0\), take the divergence of
\eqref{eq:sourcejet} and use
\(\partial_\mu\Glin^{\mu\nu}=0\) and
\(\partial_\mu T_0^{\mu\nu}[\phi_0]=0\).
For arbitrary \(\gamma_0\), use the order-one
coefficient of \eqref{eq:noether}.
On a purported jet its left side vanishes;
\(E_\phi^{[0]}=-\Pm\phi_0=0\), while
\(\Bcal_\nu^{[0]}=(\partial_\mu g)T_0^\mu{}_\nu[\phi_0]\).
The same condition follows without restricting the
leading vacuum metric data.
\end{proof}

\subsection{The inherited empty fiber}

Define
\begin{equation}
 \phi_m(t,\mathbf x)=
 \begin{cases}
  \sin(mt)/m,&m>0,\\
  t,&m=0.
 \end{cases}
 \label{eq:input}
\end{equation}

\begin{restatement}[XI-A's empty fixed-input Einstein jet]
\label{rest:empty}
For every nonzero compact switch of the frozen class
and every fixed \(m\ge0\),
\begin{equation}
 \boxed{\Jcal_g(\phi_m)=\varnothing.}
 \label{eq:empty}
\end{equation}
The vacuum provides a nonempty fiber, so universal
emptiness is false.
\end{restatement}

\begin{proof}[Stress and locality check]
The inherited proof is XI-A \loc{thm:empty}
\cite{CheungXIA}.
For \eqref{eq:input},
\[
 T_0^{00}=\frac12,\qquad T_0^{0i}=0,
 \qquad
 T_0^{ij}=
 \begin{cases}
  \frac12\cos(2mt)\delta^{ij},&m>0,\\
  \frac12\delta^{ij},&m=0.
 \end{cases}
\]
The time component of \eqref{eq:compatibility}
would require \(\partial_tg/2=0\).
A nonzero switch compactly supported in the slab
interior cannot be independent of time.
Hence compatibility fails.

For \(\phi_0=0\), the choice
\[
 (\gamma_0,\gamma_1,\gamma_2)=(0,0,0),
 \qquad
 (\phi_0,\phi_1,\phi_2)=(0,0,0)
\]
solves the coefficient equations.
A higher-order correction cannot cancel a nonzero
order-one coefficient.
\end{proof}

\subsection{The quantitative bump and its actual support}

There is a relatively compact open region \(B\) and
a constant \(c_{\mathrm b}>0\) with
\[
 \partial_tg\ge c_{\mathrm b}\quad\text{on }B.
\]
Choose \(f_{\mathrm b}\ge0\) with
\[
 f_{\mathrm b}\in C_c^\infty(B),\qquad
 \int f_{\mathrm b}\,\dd^4x=1.
\]
The retained XI-A detector is
\begin{equation}
 a_{\mathrm b}
   :=\int f_{\mathrm b}
          (\partial_\mu g)T_0^{\mu0}[\phi_m]\,\dd^4x
     =\frac12\int f_{\mathrm b}\partial_tg\,\dd^4x
     \ge\frac{c_{\mathrm b}}2>0.
 \label{eq:bump}
\end{equation}

\begin{proposition}[What the local transfer of the detector requires]
\label{prop:localdetector}
If \(\supp f_{\mathrm b}\subset D_*\), the nonzero pairing
\eqref{eq:bump} transfers to the retained laboratory
with the same value.

The detector is a switching-region detector:
its support is not contained in the plateau \(U\).
If it does not fit the retained chart or shadow,
the completion obstruction remains valid, but a report
of its supplied value is not a locally acquired detector
in that laboratory. Enlarging the shadow is not authorized.
\end{proposition}

\begin{proof}
Lemma~\ref{lem:transfer} preserves the compact pairing.
On \(U\), \(g=1\) on a neighborhood and \(\partial_tg=0\);
thus a test supported there cannot have the strictly
positive pairing \eqref{eq:bump}.
The remaining statements distinguish the domain of
the analytical calculation from the location and
access contract of its report.
\end{proof}

\begin{remark}[Completion emptiness is not every local emptiness statement]
Definition~\ref{def:jet} requires the equations on the
completion slab.
It is not the space obtained by requiring equations
only on a plateau neighborhood.
The obstruction proves no universal nonexistence
theorem for plateau-local solutions.

Where the detecting switching region is in the chart,
the same local compatibility identity excludes a jet
satisfying the invariant equations on a neighborhood
of that detector.
This is the precise local consequence.
\end{remark}

For the massless input, the existing XI-A conformal check is
\[
 \Glin[\psi\eta]_{\mu\nu}
       =-\partial_\mu\partial_\nu\psi+\eta_{\mu\nu}\Box\psi.
\]
Selecting \(\gamma=\psi\eta\) does not change
\eqref{eq:bump}, which depends only on \(g,\phi_m,f_{\mathrm b}\).
It therefore does not repair the sourced equation.

The other retained diagnostic is
\begin{equation}
 \Glin[K\xi]=0,\qquad D(K\xi)=\Box\xi.
 \label{eq:puregauge}
\end{equation}
It checks the linearization, not existence of a sourced jet.

The optional compact-scalar-data version of the witness uses
a spatial cutoff equal to one on a ball containing the initial
causal shadows of \(\supp g\).
Finite propagation makes its scalar solution agree with
\eqref{eq:input} near \(\supp g\), preserving the detector
\cite{CheungXIA,BGP}.
This does not construct a gravitational propagator.

%=====================================================================
\section{Why the gauge-fixed wave equation is different}
\label{sec:different}
%=====================================================================

For the source-response subclass, the quadratic gauge-fixed
equation would be
\begin{equation}
 -\Box\bar\gamma_1^{\mu\nu}
                      =\frac12gT_0^{\mu\nu}[\phi_0].
 \label{eq:wave}
\end{equation}
Taking its divergence on the free scalar shell yields
\begin{equation}
 \Box D^\nu(\gamma_1)
       =-\frac12(\partial_\mu g)T_0^{\mu\nu}[\phi_0].
 \label{eq:constraintbreaking}
\end{equation}
For \(\phi_m\), the right side at \(\nu=0\) is
\(-\partial_tg/4\), so \(D\gamma_1\) cannot vanish identically.

Componentwise smooth Cauchy theory can solve
\eqref{eq:wave} without a conserved source.
That fact does not solve \eqref{eq:sourcejet}.
This is XI-A's distinction, later used in XI-B
\cite{CheungXIA,CheungXIB}.

\scopebox{Operator only}{%
Equation~\eqref{eq:quadraticoperator} specifies the signed
quadratic operator.
This paper supplies no gravitational retarded or advanced
inverse, no two-point function, no ghost or auxiliary
multiplet, no antifield, no tree, and no state.

XI-B is a downstream comparison, not a premise used to
construct those objects here.}

The source obstruction is not a statement that Einstein theory
fails. It is an incompatibility between the specified compact
switch, the named free input, and the invariant equation space.

%=====================================================================
\section{Finite coverage and the limits of a report catalogue}
\label{sec:coverage}
%=====================================================================

The proposed catalogue records finitely many evaluations.
It cannot replace an identity on an unrestricted smooth
or distributional space.
The following local statements give the coverage and
invisibility discipline compared with Oai282-A
\cite{CheungOai282}.

\begin{proposition}[Finite-probe coverage]
\label{prop:coverage}
Let \(Z\) be a proved finite-dimensional residual space,
and let
\[
 P:Z\longrightarrow\RR^N,\qquad
 P(r)=(\ell_1(r),\ldots,\ell_N(r))
\]
be a specified linear probe map.
The conditions \(\ell_j(r)=0\) for every \(j\) force
\(r=0\) for every \(r\in Z\) if and only if \(P\) is
injective.

For any fixed norm on \(Z\), injectivity implies a
finite constant \(C_Z\) with
\[
 \norm r_Z\le C_Z\norm{P(r)}_\infty.
\]
Neither the subspace membership \(r\in Z\) nor injectivity
follows from having printed a finite table.
\end{proposition}

\begin{proof}
The exact assertion is \(\ker P=\{0\}\).
For the quantitative assertion, the continuous function
\(r\mapsto\norm{P(r)}_\infty\) has a positive minimum
on the unit sphere of a nonzero finite-dimensional \(Z\).
Its reciprocal gives \(C_Z\).
The zero space is immediate.
\end{proof}

\begin{proposition}[A finite family can miss a conserved symmetric tensor]
\label{prop:invisible}
Let \(O\subset V\) be any nonempty open set and
\(\ell_1,\ldots,\ell_N\) any finite family of linear
evaluations on smooth compactly supported symmetric tensors.
There is a nonzero
\[
 S^{\mu\nu}\in C_c^\infty(O;\operatorname{Sym}^2)
\]
such that
\[
 \partial_\mu S^{\mu\nu}=0,\qquad
 \ell_j(S)=0\quad(1\le j\le N).
\]
No assertion of a zero trace is required.
\end{proposition}

\begin{proof}
For \(\psi\in C_c^\infty(O)\), define
\[
 S_\psi^{\mu\nu}
      =\partial^\mu\partial^\nu\psi-\eta^{\mu\nu}\Box\psi.
\]
It is symmetric and divergence free.
If \(S_\psi=0\), taking its trace gives
\(-3\Box\psi=0\).
A spacetime-compact homogeneous wave is zero by
Cauchy uniqueness, so \(\psi=0\).

Choose \(N+1\) nonzero bump functions with mutually
disjoint compact supports in \(O\).
Their tensors \(S_{\psi_a}\) are nonzero and linearly
independent.
The \(N\) evaluations on their span have a nonzero
kernel. A nonzero tensor in that kernel has all the
required properties.
\end{proof}

This establishes the finite-probe limitation locally.
It does not append a new stress source to the action,
construct an Einstein solution, or verify OpenAI Family 282.

The independent non-implication from record conservation
to Einstein dynamics is the existing I-B
\loc{prop:not-source} and X-A-ST \loc{prop:noeinstein}
boundary \cite{CheungIB,CheungXAST}.
No third geometric counterexample is needed here.

%=====================================================================
\section{The complete Enter report interface}
\label{sec:instrument}
%=====================================================================

\subsection{The finite supplied catalogue}

Before execution, freeze all evaluation configurations,
tests, units, source data, and approximation enclosures.
A possible catalogue consists of the following entries.

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.20\linewidth}p{.40\linewidth}X@{}}
\toprule
Entry & Supplied classical value & Interpretation\\
\midrule
TT sign &
The coefficient \(+1/2\) in \eqref{eq:TT} &
A linear kinetic-sign diagnostic.\\[3pt]
Newtonian sign &
A fixed value \(\Phi_N(\mathbf x_N)<0\) from
\eqref{eq:Newton} &
An auxiliary static sign comparison.\\[3pt]
Compatibility &
\((\partial_\mu g)T_0^{\mu0}[\phi_m](x_{\mathrm b})\) &
A specified smooth evaluation, not all sources.\\[3pt]
Compact detector &
\(a_{\mathrm b}\) in \eqref{eq:bump} &
A positive conservation obstruction.\\[3pt]
Vacuum &
Selected zero coefficients of the zero metric and
zero scalar jet &
An analytic vacuum witness, not a measured field state.\\[3pt]
Pure gauge &
A specified compact pairing of
\(\Glin[K\xi]=0\) &
A linearized identity evaluated on supplied data.\\[3pt]
Gauge-fixing divergence &
A specified compact pairing of \(-\Box D^\nu(\gamma)\) &
The breaking term, not an Einstein source.\\
\bottomrule
\end{tabularx}
\end{center}

Vector entries are flattened into finitely many real
components. Every entry is an ordinary real number.
A field, tensor density, equation space, and truth value
are not identified with one another by this flattening.

An approximation enclosure must be justified by its own
mathematical calculation or supplied certificate.
A channel-distance bound does not validate that enclosure.

\subsection{Destructive clock readout}

Choose
\[
 \HH_C\cong\CC^{D},\qquad D=2^q,\quad q\ge1,
\]
with orthonormal calibration basis
\(\{\ket{\tau_j}:0\le j<D\}\).
Let \(S\) contain every unmeasured finite register.
Set
\[
 P_j=\proj{\tau_j},\qquad
 D_j=\bra{\tau_j}_C\otimes I_S.
\]
Unused clock codes have a predeclared no-task or failure
interpretation; no branch is omitted.

\begin{theorem}[Complete destructive report clock]
\label{thm:instrument}
The channel
\begin{equation}
 \Mclock(X)
   =\sum_{j=0}^{D-1}\proj j_R\otimes D_jXD_j^\dagger
 \label{eq:clock}
\end{equation}
is CPTP and equals
\[
 \sum_j\proj j_R\otimes
       \Tr_C[(P_j\otimes I_S)X(P_j\otimes I_S)].
\]
For every untouched reference \(E\),
\begin{equation}
 \Tr_R[(\Mclock\otimes\id_E)(X)]=\Tr_CX.
 \label{eq:discard}
\end{equation}
The returned quantum interface contains \(S\), not \(C\).
\end{theorem}

\begin{proof}
The recorded Kraus maps are
\(\ket j_R\otimes D_j\), with squared moduli summing to
\[
 \sum_jP_j\otimes I_S=I_{CS}.
\]
Expansion
\[
 X=\sum_{j,k}\kb{\tau_j}{\tau_k}\otimes X_{jk}
\]
shows that the \(j\)-block is \(X_{jj}\) in both formulas.
Summing gives \eqref{eq:discard}.
The same calculation includes \(E\).
The codomain has no quantum \(C\)-factor.
\end{proof}

This is precisely the I-B/X-A-ST clock interface,
also compared with En09-A
\cite{CheungIB,CheungXAST,CheungEn09}.
For \(p_j=\Tr(D_j\rho D_j^\dagger)>0\), the normalized
residual is \(D_j\rho D_j^\dagger/p_j\).
No normalized residual is assigned to a zero-probability
branch.

Conditional reference steering is compatible with
\eqref{eq:discard}, which concerns the unconditional
reference marginal.

\subsection{A common-interface rational table encoding}

A deterministic replacement of one printed rational word
by another generally changes an orthogonal classical label.
Its channel distance need not be small when the numerical
difference is small.
A quantitative comparison must therefore specify the
common encoding interface.

For each real catalogue entry \(a_j\), choose a positive
rational scale \(\Lambda_j\) and a supplied rational
approximation \(q_j\) such that
\[
 |a_j|\le\Lambda_j,\qquad |q_j|\le\Lambda_j,\qquad
 |a_j-q_j|\le\delta_j.
\]
Define
\begin{equation}
 p_j(a_j)=\frac12\left(1+\frac{a_j}{\Lambda_j}\right).
 \label{eq:probencode}
\end{equation}
After clock outcome \(j\), prepare one fresh score qubit
in
\[
 \sqrt{1-p_j}\ket0+\sqrt{p_j}\ket1
\]
using the declared finite preparation permission, and
destructively read it.
The score qubit is consumed. The surviving register \(S\)
is not measured.

The complete common-interface channel is
\begin{equation}
 \mathcal I_a(X)
  =\sum_{j,y\in\{0,1\}}
       w_{j,y}(a_j)\,
       \proj j_R\otimes\proj y_Y\otimes D_jXD_j^\dagger,
 \label{eq:tablechannel}
\end{equation}
where
\[
 w_{j,1}(a_j)=p_j(a_j),\qquad
 w_{j,0}(a_j)=1-p_j(a_j).
\]
In \eqref{eq:tablechannel}, the index \(j\) ranges over
\(0,\ldots,D-1\), and \(y\) ranges over \(\{0,1\}\).

\begin{theorem}[Exact complete-channel table comparison]
\label{thm:table}
The maps \(\mathcal I_a\) and \(\mathcal I_q\) are CPTP,
including arbitrary untouched references, and
\begin{equation}
 \ddia(\mathcal I_a,\mathcal I_q)
   =\max_j\frac{|a_j-q_j|}{2\Lambda_j}
   \le\max_j\frac{\delta_j}{2\Lambda_j},
 \qquad
 \ddia(\Phi,\Psi)=\frac12\norm{\Phi-\Psi}_\diamond.
 \label{eq:tablebound}
\end{equation}
Both maps return the complete conditional \(S\)-blocks.
Discarding \(R,Y\) gives \(\Tr_CX\).

The bound concerns the declared encoded table.
It is not a field-acquisition allowance or a certificate
of an Einstein jet.
\end{theorem}

\begin{proof}
Kraus maps for \eqref{eq:tablechannel} are
\[
 L_{j,y}
   =\sqrt{w_{j,y}(a_j)}\,
      \ket j_R\otimes\ket y_Y\otimes D_j.
\]
Their squared moduli sum to \(I_{CS}\).

For any input--reference state, write its positive
diagonal clock blocks as \(\rho_{jj}\).
The output difference is block diagonal in \(j,y\).
Its half trace norm is
\[
 \sum_j
   |p_j(a_j)-p_j(q_j)|\,\Tr\rho_{jj}
 \le \max_j|p_j(a_j)-p_j(q_j)|.
\]
The standard state characterization of diamond distance
for a difference of channels gives the upper bound
\cite{Watrous}.
A deterministic clock input \(P_j\), with \(j\) maximizing
the difference and a fixed surviving state, attains it.
Equation~\eqref{eq:probencode} gives
\eqref{eq:tablebound}.
Summing the score probabilities and the clock blocks
proves the discard statement.
\end{proof}

This is an explicit probability encoding implementing the
fixed-table comparison discipline of X-A-ST
\loc{prop:table} \cite{CheungXAST}.
Its constants refer to
\eqref{eq:probencode}, not to an unspecified encoding.
Oai283-A is a finite-interface comparator only
\cite{CheungOai283}.

A deterministic table lookup
\[
 j\longmapsto(j,q_j,\delta_j,\text{task metadata})
\]
may also be returned.
If its numerical word is changed, that change is a
separate classical-output comparison, not silently
covered by \eqref{eq:tablebound}.

\begin{corollary}[Robust mathematical sign report]
\label{cor:signreport}
For the bump entry, a valid enclosure
\(|a_{\mathrm b}-q_{\mathrm b}|\le\delta_{\mathrm b}\)
with
\[
 q_{\mathrm b}-\delta_{\mathrm b}>0
\]
certifies that the named compatibility pairing is nonzero.
It certifies no universal identity for other stresses.

A single score outcome \(Y\) does not certify this sign
or determine \(a_{\mathrm b}\).
\end{corollary}

\begin{proof}
The enclosure gives
\(a_{\mathrm b}\ge q_{\mathrm b}-\delta_{\mathrm b}>0\).
The conclusion concerns the named number only.
The score law in \eqref{eq:probencode} is Bernoulli;
one random outcome does not identify its probability.
\end{proof}

No finite-shot confidence theorem is asserted without a
separately declared repetition, calibration, and
attempted-run model.

\subsection{Clock limits and the acquisition boundary}

\begin{proposition}[Retained clock limitations]
\label{prop:clocklimits}
The clock interface has the following limitations.
\begin{enumerate}[label=\textup{(\roman*)}]
\item If distinct labels \(t_j\) must equal the time coordinate
of one fixed readout event whenever they occur, at most
one label can have positive probability.
\item Nontrivial history readout requires a separate event
dictionary, not the assertion that all outcomes occurred
at different times at one event.
\item The channel has no universal left inverse on its
full clock input space.
\item The record marginal does not determine the surviving
quantum residual.
\item No finite matrices satisfy
\([T_C,H_C]=i\hbar I_C\) for \(\hbar\ne0\).
\end{enumerate}
\end{proposition}

\begin{proof}
Distinct labels prove (i).
For (ii), a history dictionary assigns a code to a
separately specified event or preparation; it does not
change the actual reading event.

The two clock states
\[
 \frac{\ket{\tau_0}+\ket{\tau_1}}{\sqrt2},
 \qquad
 \frac{\ket{\tau_0}-\ket{\tau_1}}{\sqrt2}
\]
with identical surviving inputs have identical destructive
outputs, proving (iii).
Inputs \(P_0\otimes\rho_S\) and
\(P_0\otimes\rho'_S\) have the same deterministic record
and arbitrary different residuals, proving (iv).
Taking traces proves (v).
These are the I-B/X-A-ST limitations
\cite{CheungIB,CheungXAST}.
\end{proof}

No stationarity or covariant-clock hypothesis is imposed.
Consequently En09-A's additional relational-evolution
interpretation is not claimed \cite{CheungEn09}.

A basis change acts on finite registers.
It is not a coordinate change on \(V\), and it does not
transform \(G\).
Scheduler consistency likewise does not select the
prescribed Lorentz cone; En08-A's countermodels retain
that distinction \cite{CheungEn08}.

Finally, the report program has no continuum field input
port. A numerical value obtained by a prior calculation
is supplied preparation data.
To identify this with a field measurement would require
a separately defined acquisition map, complete apparatus,
and comparison theorem.
None is constructed.

%=====================================================================
\section{Resource fit and fixed-host discipline}
\label{sec:resources}
%=====================================================================

A local realization supplies:
\begin{enumerate}
\item the clock and surviving-register dimensions;
\item all installed rational words, scales, and enclosures;
\item finite preparation and basis-change permissions;
\item one regular readout location and actual duration bounds;
\item complete failure outcomes and returned storage;
\item all classical control wires.
\end{enumerate}

For \(p\) surviving qubits, clock dimension \(2^q\), and
a score qubit allocated only after clock consumption,
the active quantum occupancy of this readout pattern is
\[
 Q_{\mathrm{active}}=\max\{q+p,p+1\},
\]
before adding any other declared live workspace.
The output record requires \(q+1\) bits, in addition to
the installed table and any returned metadata.
Returned quantum storage and external references have
their own ledgers.

The actual inequalities
\[
 Q_{\mathrm{peak}}\le Q_{\mathrm{reserve}},\qquad
 B_{\mathrm{peak}}\le B_{\mathrm{reserve}},\qquad
 \Delta T_{\mathrm{program}}\le\Delta T_{\mathrm{reserve}}
\]
must hold on the supplied schedule.
Finiteness alone is not a fit certificate.

All conditional score operations may be co-located with
clock readout.
If a remote controller is used, its record dependency
must be delivered on a declared causal wire.
No instantaneous distant use of a classical record
is implicit.

\begin{proposition}[Fixed-host release]
\label{prop:fixedhost}
A realization passing the stated fit checks does not
alter the host, worldsheet, calibration map, or external
time coordinate. It does not change \(g\),
\(\sigma_\varepsilon\), \(\varepsilon\), or the action
when a table entry is read or approximated.
\end{proposition}

\begin{proof}
The command types operate on finite registers, records,
and declared controls. They contain no metric-update,
worldsheet-update, regulator-update, or action-update
output. The resource inequalities refer to the retained
reserve rather than creating new space.
\end{proof}

This is the fixed-host discipline of I-B, X-A-ST, and
the X-L-ST ledger comparison
\cite{CheungIB,CheungXAST,CheungXLST}.
It supplies neither a thermodynamic reset cost nor an
Einstein source law.

%=====================================================================
\section{Decision, handoff, and unchanged stops}
\label{sec:release}
%=====================================================================

\subsection{Decision table}

\begin{longtable}{@{}p{.45\linewidth}p{.49\linewidth}@{}}
\toprule
Gate & XI-A-ST result\\
\midrule
\endfirsthead
\toprule
Gate & XI-A-ST result\\
\midrule
\endhead

Host, worldsheet, future orientation, signature &
Prescribed and retained; not reconstructed.\\[4pt]

Unswitched Einstein--Hilbert density &
Retained with coefficient \(-2/\kappa^2\).\\[4pt]

Minimal metric Euler derivative &
Contains \(g\), not \(\partial g\).\\[4pt]

Scalar equation and source divergence &
Switch-derivative terms retained.\\[4pt]

TT and Newtonian sign diagnostics &
Retained; no vacuum-energy conclusion.\\[4pt]

Invariant Noether identity &
Retained with the full adjoint operator.\\[4pt]

Unbroken gauge-fixed Noether identity &
False; its breaking term remains explicit.\\[4pt]

Fierz--Pauli-only nonlinear replacement &
Insufficient; higher classical Einstein coefficients remain.\\[4pt]

Invariant metric Euler covariance &
Retained as rank-two, weight-one density covariance
on \(\mathcal G_P\).\\[4pt]

X-A-ST relational intertwiner and edge defect &
Cited; not recounted or counted as new theorems.\\[4pt]

Completion fiber \(\Jcal_g(\phi_m)\) &
Empty; the vacuum remains a counterexample to
universal emptiness.\\[4pt]

Switching detector on the host &
Transferred when its support fits the retained chart
and shadow; not a plateau detector.\\[4pt]

Gauge-fixed wave equation &
A different equation; not an Einstein-jet repair.\\[4pt]

Complete Enter instrument &
Constructed for supplied finite evaluations and residuals.\\[4pt]

Finite report coverage &
Requires an independent residual-space coverage theorem.\\[4pt]

Field acquisition, metric update, propagator,
multiplet, tree, BRST &
Not constructed.\\
\bottomrule
\end{longtable}

\subsection{The six-object handoff}

A later XI-B-ST, if constructed, receives:
\begin{enumerate}
\item the signed quadratic operator
\[
 2\Glin+e_{\gf}=-\Box\Ptr;
\]
\item the de Donder term
\(S_{\gf}=\int D_\nu D^\nu\), with its divergence
\(-\Box D^\nu\) and adjoint Noether contribution;
\item the stress vertex
\(-\frac12\int g\gamma_{\mu\nu}T_0^{\mu\nu}\);
\item the full unswitched Einstein action and the
classical coefficient bookkeeping needed through
the specified coupling order;
\item the invariant Noether identity with its exact
fixed-background switching defect;
\item the empty fixed-input completion Einstein fiber,
its compact detector, and its distinction from a
gauge-fixed Cauchy problem.
\end{enumerate}
These are tagged with the host/signature ledger and
the complete supplied-report catalogue.

They are the same six types of object handed from
XI-A to XI-B \cite{CheungXIA}.
No propagator, ghost, antifield, tree, or state is
included in this handoff.

\subsection{Problem reading for this sheet alone}

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.11\linewidth}p{.19\linewidth}X@{}}
\toprule
Problem & Status & Boundary\\
\midrule
1 & Unfinished &
A classical action on a prescribed host is not a
quantized gravitational observable.\\[3pt]
2 & Unfinished &
An Euler derivative and finite report do not define
a renormalized interacting product.\\[3pt]
6 & Excluded &
The host and action are inputs; emergence is not attempted.\\[3pt]
8 & Unfinished &
External events, preparation, calibration, and time remain.\\[3pt]
10 & Not reached &
The Newtonian sign comparison is not a vacuum-energy
calculation.\\[3pt]
12 & Unresolved &
No aligned-null contraction or characteristic quantum
coefficient is produced.\\
\bottomrule
\end{tabularx}
\end{center}

The other numbered problems receive no additional
construction from this sheet.

\subsection{Standing stops}

The convolution gap of X-A-conv-ST is not repaired.
No width-removal, adiabatic-limit, or new subtraction
prescription is introduced.

XV-B is not opened as a downstream quantum-admission exit.
XVI-N and Route B are not discharged.
No O.1 status is changed:
\[
 \loc{O.1 = NOT_CLOSED}
 \quad\text{outside any independently earned scope},
 \qquad
 \loc{bulk_symmetry_restored = false}.
\]

This sheet does not discharge the characteristic obligations
identified by the historical H2, I3, and J2 records.
That statement does not erase XIV-I's version-specific
constructed I2 internal-amplitude result.
An internal kernel or its amplitude is not its full
joint quantum composition
\cite{CheungCharacteristic}.

No Einstein equation is derived from a record.
No Page curve, de Sitter state, cosmological bounce,
gravity-generated-entanglement test, ultraviolet
completion, or holographic identification is supplied.

\begin{definition}[Local Grade S in this sheet]
A local Grade S package consists of the retained host
and chart, the exact action and equation conventions,
the supported evaluations being reported, the finite
complete instrument, and actual resource and causal
fit data.

A report of a supplied completion calculation does not
become local acquisition merely by assigning the
report register a laboratory location.
\end{definition}

\begin{theorem}[XI-A-ST release]
\label{thm:release}
For the frozen instance and supported interfaces stated
in this paper:
\begin{enumerate}[label=\textup{(\roman*)}]
\item XI-A's signed action, quadratic operator, Euler
coefficients, Noether split, and invariant tensor-density
covariance have the retained local-chart interpretation;
\item the fixed-input completion jet
\(\Jcal_g(\phi_m)\) is empty, while the vacuum fiber
is nonempty;
\item the detector \eqref{eq:bump} has the same nonzero
value after supported chart transfer whenever its support
fits \(D_*\);
\item the complete destructive report instrument retains
the actual unmeasured residual and arbitrary untouched
references;
\item the common-interface rational probability table
obeys \eqref{eq:tablebound}, and the finite-probe limitations
of Propositions~\ref{prop:coverage} and \ref{prop:invisible}
remain explicit;
\item no operation reconstructs or sources the host,
and no additional quantum-gravitational gate is passed.
\end{enumerate}
\end{theorem}

\begin{proof}
Lemma~\ref{lem:transfer} and the supported restatements
in Sections~\ref{sec:action} and \ref{sec:noether}
give (i).
Restatement~\ref{rest:empty} gives (ii).
Proposition~\ref{prop:localdetector} gives (iii).
Theorem~\ref{thm:instrument} gives (iv).
Theorem~\ref{thm:table} and
Section~\ref{sec:coverage} give (v).
Proposition~\ref{prop:fixedhost}, the declared analytical
domain, and the absence of any quantum product or
metric-update construction give (vi).
\end{proof}

%=====================================================================
\appendix
\section{Finite specification of the retained gravitational coefficients}
\label{app:coefficients}
%=====================================================================

This appendix records XI-A's finite coefficient prescription
for \eqref{eq:Hdefinition}; it does not select or calculate
a graph \cite{CheungXIA}.

Set
\[
 u_0^{\mu\nu}=\eta^{\mu\nu},\qquad
 u_1^{\mu\nu}=-\gamma^{\mu\nu},\qquad
 u_2^{\mu\nu}=\gamma^\mu{}_\rho\gamma^{\rho\nu}.
\]
For \(n=1,2,3\), put
\[
 \Gamma_n{}^\rho_{\mu\nu}
  =\frac12u_{n-1}^{\rho\sigma}
       (\partial_\mu\gamma_{\nu\sigma}
        +\partial_\nu\gamma_{\mu\sigma}
        -\partial_\sigma\gamma_{\mu\nu}).
\]
Define
\begin{align}
 R_{n,\mu\nu}
  ={}&\partial_\rho\Gamma_n{}^\rho_{\mu\nu}
      -\partial_\nu\Gamma_n{}^\rho_{\mu\rho}
 \nonumber\\
   &+\sum_{\substack{a+b=n\\a,b\ge1}}
      \left(
       \Gamma_a{}^\rho_{\rho\lambda}
                           \Gamma_b{}^\lambda_{\mu\nu}
       -\Gamma_a{}^\rho_{\nu\lambda}
                           \Gamma_b{}^\lambda_{\mu\rho}
      \right),\\
 r_n
   ={}&\sum_{\substack{i+j=n\\i\ge0,\ j\ge1}}
                           u_i^{\mu\nu}R_{j,\mu\nu},
 \qquad r_0=0,\\
 C_{n,\mu\nu}
   ={}&R_{n,\mu\nu}-\frac12\eta_{\mu\nu}r_n
                                -\frac12\gamma_{\mu\nu}r_{n-1}.
\end{align}
With \(b_0=1\) and \(b_1,b_2\) from \eqref{eq:bs},
\begin{equation}
 H_n^{\mu\nu}
 =
 \sum_{\substack{i+j+\ell+s=n\\i,j,\ell\ge0,\ s\ge1}}
    b_i u_j^{\mu\alpha}u_\ell^{\nu\beta}C_{s,\alpha\beta},
 \qquad 1\le n\le3.
 \label{eq:Hfinite}
\end{equation}
Only indices through two occur on \(u\) and \(b\)
in the required sums.

These formulas are the coefficient expansion of the
displayed Christoffel, curvature, inverse-metric,
and density expressions.
They fix \(H_1,H_2,H_3\) without leaving an unknown
classical vertex in \eqref{eq:eexp}.
They do not construct a quantum product involving
those coefficients.

%=====================================================================
\section{Documentary comparisons and non-inputs}
\label{app:comparisons}
%=====================================================================

The following are role comparisons, not premise transfers.

\begin{longtable}{@{}p{.28\linewidth}p{.66\linewidth}@{}}
\toprule
Record & Role and non-identification\\
\midrule
\endfirsthead
\toprule
Record & Role and non-identification\\
\midrule
\endhead

I-B; X-A-ST &
Prescribed regular host, signature ledger, destructive
clock, and the existing refusal to infer Einstein
dynamics from records.\\[4pt]

X-L-ST &
Storage and supported-release discipline only.
Its action and free operator are not this action.\\[4pt]

VIII-A, VIII-B, VIII-C &
Conical certificates, persistent packet, and finite
composition comparisons. No Einstein--scalar action
is imported \cite{CheungVIII}.\\[4pt]

Oai282-A &
Finite Ward, coverage, invisibility, and identification
boundaries. No verification of OpenAI Family 282.\\[4pt]

Oai283-A &
A finite supplied-instance comparator, not the
Einstein jet, tree, or field-acquisition map.\\[4pt]

En08-A; En09-A &
Scheduler/geometry separation and finite clock readout.
Neither supplies this action or a free ghost sector.\\[4pt]

III; VI; VII-A &
Different equation spaces, signatures, or metric
prescriptions. They do not replace XI-A's second-order
classical sheet.\\[4pt]

VII-C; VII-H; IX-A &
Separate extension, background, and admission questions.
No theorem is applied or contradicted by the present
finite table.\\[4pt]

XX-A; Em02-E &
Their own lapse backgrounds, not this conical host.
No lapse attachment is needed for the release.\\[4pt]

XII-C &
A later changed cubic weighting and changed Noether
identity, not an input to \eqref{eq:EH}.\\[4pt]

XI-B &
Receives the six-object classical handoff.
Its propagator and multiplet are not imported.\\[4pt]

X-B, X-C, X-D; X-E &
Non-inputs of this classical transfer; no normalization,
sharp limit, quantum map, or tree is borrowed.\\
\bottomrule
\end{longtable}

Unspecified source versions and unretrieved full proofs
are not certified by their appearance in this table.

%=====================================================================
\section{Debugging and acceptance audit}
\label{app:debug}
%=====================================================================

\begin{longtable}{@{}p{.40\linewidth}p{.54\linewidth}@{}}
\toprule
Potential error & Required correction\\
\midrule
\endfirsthead
\toprule
Potential error & Required correction\\
\midrule
\endhead

Identifying \(q\) with the conical host \(G\) &
\(q\) is an analysis field around \(\eta\);
the host is unchanged metadata.\\[4pt]

Changing the action sign to match host notation &
Retain \(\iota^*G=-\eta\) and the XI-A action
normalization.\\[4pt]

Using X-E or X-L-ST's free operator &
The XI-A quadratic operator is
\(-\Box\Ptr\) with \(S_{\gf}=\int D^2\).\\[4pt]

Switching the Einstein density by the ramp &
The Einstein action remains unswitched.\\[4pt]

Putting \(\partial g\) in the minimal metric derivative &
Such derivatives belong to the scalar equation and
source divergence, not \eqref{eq:eexact}.\\[4pt]

Dropping cubic or quartic classical coefficients &
Retain the full generating action before projection.\\[4pt]

Replacing the Noether adjoint by a raw divergence &
Use \eqref{eq:N}; retain its factor two and nonlinear term.\\[4pt]

Removing gauge-fixing breaking on the plateau &
Only \(\Bcal_\nu\) vanishes there.\\[4pt]

Calling \eqref{eq:covariance} a new ramp theorem &
It is XI-A's Euler-density covariance; X-A-ST's
relational theorem is only cited.\\[4pt]

Making the positive bump a plateau test &
Its support is in the switching region, where
\(\partial_tg>0\).\\[4pt]

Renaming completion-jet emptiness as all local emptiness &
Keep Definition~\ref{def:jet} and
Proposition~\ref{prop:localdetector} separate.\\[4pt]

Repairing the source with a conformal or pure-gauge ansatz &
Neither changes the nonzero compatibility pairing.\\[4pt]

Equating a wave solution with an Einstein jet &
Equation~\eqref{eq:constraintbreaking} retains its
nonzero constraint source.\\[4pt]

Inferring all residuals vanish from a finite table &
Prove residual-space coverage and probe injectivity.\\[4pt]

Inferring a small channel error from nearby printed labels &
Use the common probability encoding and the actual
bound \eqref{eq:tablebound}.\\[4pt]

Treating a supplied value as acquired field data &
An acquisition interface is additional and absent.\\[4pt]

Reading one Bernoulli code as an exact real number &
The code is one random outcome; no finite-shot
inference is asserted.\\[4pt]

Consuming the time axis with the clock &
Only measured quantum names are removed.\\[4pt]

Calling a finite schedule a resource fit &
Actual storage, location, timing, and permission
inequalities must pass.\\
\bottomrule
\end{longtable}

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

\paragraph{Authorship.}
The author field is left blank for submission preparation.
The bibliography attributes the source manuscripts to
N.~Cheung; this does not assign authorship of the present draft.

\paragraph{Provenance and attribution.}
XI-A's action, signs, Noether identities, covariance, and
fixed-input obstruction are retained results, not new
priority claims.
The online XI-A and I-B records were consulted.
Specific X-A-ST, X-L-ST, and comparator attributions also
use the supplied version-qualified passages.
An attribution is not an independent verification of an
entire manuscript.

\paragraph{Novelty scope.}
The claim is a supported transfer with an explicit complete
report interface, support qualifications, and finite
coverage/error discipline.
No new Einstein equation, general measurement theorem,
or quantum-gravity completion is claimed.

\paragraph{Physical scope.}
The action, host, time orientation, chart, speed calibration,
finite preparations, and permitted controls are inputs.
No microscopic detector-field coupling or host-update
mechanism is constructed.

\paragraph{Verification.}
No successful \loc{pdflatex} execution, proof-assistant
verification, numerical computation, compiled laboratory
circuit, repository installation, independent rerun,
independent review, or journal acceptance of this draft
is asserted.
The submission source contains the stated definitions,
supported restatements, and written local proofs.

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

\bibitem{CheungXIA}
N.~Cheung,
\emph{Landmark XI-A: A switched Einstein--scalar action
for the ramp relational scalar: sign conventions,
Noether identities, and an empty fixed-input Einstein jet}.
\loc{XI-A.tex}, version 1.
Enter Calculus Companion.
Especially \loc{prop:FP}, \loc{lem:TT},
\loc{lem:Newton}, \loc{thm:euler},
\loc{thm:noether}, \loc{prop:gfbad},
\loc{thm:covariance}, \loc{lem:compatibility},
\loc{thm:empty}, and \loc{eq:bump}.
\url{https://enter.grok.me/browse/XI-A}.

\bibitem{CheungIB}
N.~Cheung,
\emph{I-B}.
\loc{I-B-v1.tex}, version 1.
Enter Calculus Companion.
Especially the prescribed regular conical product,
\loc{thm:instrument}, \loc{prop:fixed-event},
\loc{prop:not-source}, and \loc{cor:manipulation}.
\url{https://enter.grok.me/browse/I-B}.

\bibitem{CheungXAST}
N.~Cheung,
\emph{X-A-ST}.
Version 1, supplied manuscript passages.
Host chart and completion ledger, \loc{prop:nonGH},
\loc{prop:no-global-transfer}, \loc{prop:edge},
\loc{thm:main}, \loc{prop:table},
\loc{thm:instrument}, \loc{prop:clocklimits},
\loc{prop:noeinstein}, and \loc{prop:fixedhost}.
The relational results are documentary baseline results,
not reproved here.
\url{https://enter.grok.me/browse/X-A-ST}.

\bibitem{CheungXLST}
N.~Cheung,
\emph{X-L-ST: Restricted classical BRST domains,
exact free recovery, and complete residual reports}.
Version 1 in the supplied record.
Host, supported-release, and three-storage-ledger
comparison only.
Its classical action and free normalization are not
premises of XI-A-ST.
\url{https://enter.grok.me/browse/X-L-ST}.

\bibitem{CheungEnter}
N.~Cheung,
``The Enter Calculus: A Linear Quantum Lambda Calculus
with an Operational Characterization of Measurement,''
\emph{Frontiers in Physics} \textbf{14} (2026),
article 1954943.
doi:10.3389/fphy.2026.1954943.
Publisher record: accepted September 28, 2026;
final formatted version forthcoming in the consulted record.
\url{https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2026.1954943/abstract}.

\bibitem{CheungI}
N.~Cheung,
\emph{Manuscript I}.
\loc{I.tex}, version 2, supplied manuscript passages.
Class-relative operational characterization of destructive
measurement and positive-branch projective residuals.
\url{https://enter.grok.me/browse/I}.

\bibitem{CheungPhilosophy}
N.~Cheung,
\emph{The Philosophy of Enter: Destructive Readout,
Quantum Recovery, and the Refusal of Identification}.
\loc{EntPhil-v2.tex}, version 2.
Introductory interpretive essay, not a theorem premise.
\url{https://enter.grok.me/enter}.

\bibitem{CheungEn09}
N.~Cheung,
\emph{En09-A}.
Version 1, supplied manuscript passages.
Complete destructive clock extraction, residual and reference
semantics, and the additional hypotheses required for a
relational-evolution interpretation.
\url{https://enter.grok.me/browse/En09-A}.

\bibitem{CheungEn08}
N.~Cheung,
\emph{En08-A}.
Version 1, especially \loc{thm:main}.
Scheduler-consistency countermodels and the distinction
between operational order and additional cone-selection
hypotheses.
\url{https://enter.grok.me/browse/En08-A}.

\bibitem{CheungOai282}
N.~Cheung,
\emph{Oai282-A}.
Version 1, supplied manuscript passages.
Finite Ward audit, \loc{prop:coverage},
\loc{prop:invisible}, and the boundary between
finite certificates and identification.
No verification of OpenAI Family 282 is imported.
\url{https://enter.grok.me/browse/Oai282-A}.

\bibitem{CheungOai283}
N.~Cheung,
\emph{Oai283-A}.
Version 1 in the supplied documentary comparison.
Finite supplied-instance and resource-interface comparison
only; not the Einstein jet or a field-acquisition theorem.
\url{https://enter.grok.me/browse/Oai283-A}.

\bibitem{CheungXIB}
N.~Cheung,
\emph{XI-B}.
\loc{XI-B.tex}, version 1, supplied manuscript passages.
Downstream free-multiplet construction and the switched-source
divergence diagnostic.
The construction is not imported into the present handoff.
\url{https://enter.grok.me/browse/XI-B}.

\bibitem{CheungVIII}
N.~Cheung,
\emph{VIII-A}, \emph{VIII-B}, and \emph{VIII-C}.
Version-qualified supplied conical-geometry records.
Used only for documentary comparisons of fixed geometric
packets, constraints, and finite composition, not for
the action or dynamics in XI-A-ST.

\bibitem{CheungCharacteristic}
N.~Cheung,
\emph{XIV-H}, \emph{XIV-I}, and \emph{XIV-J}.
Version 1 passages supplied with the problem.
Historical H2 boundary, constructed I2 internal amplitude,
and unresolved full joint characteristic constructions
at I3 and J2.
No characteristic coefficient or downstream quantum admission
is imported.

\bibitem{BFR}
R.~Brunetti, K.~Fredenhagen, and K.~Rejzner,
\emph{Quantum gravity from the point of view of locally
covariant quantum field theory}.
arXiv:1306.1058, version 5 (2015).
External perturbative-gravity and renormalized-BV framework.
It does not supply a missing construction in this manuscript.
\url{https://arxiv.org/abs/1306.1058}.

\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.
arXiv:0806.1036.
Classical Cauchy uniqueness and finite-propagation background;
no global regular-cone inverse is inferred.
\url{https://arxiv.org/abs/0806.1036}.

\bibitem{Watrous}
J.~Watrous,
\emph{The Theory of Quantum Information}.
Cambridge University Press, 2018.
Especially the finite-channel and completely bounded
trace-norm framework.
Author manuscript and errata:
\url{https://cs.uwaterloo.ca/~watrous/TQI/}.

\end{thebibliography}

\end{document}