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% Corrections incorporated:
%  - The physical background is the prescribed regular conical product.
%  - The plateau is relatively compact and open, not a compact open set.
%  - The regular conical product is not globally hyperbolic.
%  - All wave inverses have fixed Cauchy data on a supported domain.
%  - The scalar Euler operator is density-valued.
%  - Switching is performed before variation, including the mass term.
%  - The quadratic metric vertex carries g, not g^2.
%  - The second metric coefficient transforms with the first.
%  - Functional support is CONTAINED IN D_f; equality is not claimed.
%  - A common observation plateau does not imply response equality.
%  - The classical comparison is unique on the stated solution spaces.
%  - Finite readout encodes evaluations; it does not acquire a field.
%  - Remote control includes a causal classical record wire.
%  - Historical H2, constructed internal I2, and unresolved I3/J2
%    are distinguished.
%=====================================================================

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\hypersetup{
 pdftitle={VII-A-ST: Plateau intertwiners on a prescribed conical spacetime},
 pdfsubject={Supported classical descent, cutoff comparison, and finite Enter readout}
}

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\title{\textbf{VII-A-ST: Plateau intertwiners on a prescribed\\
conical spacetime}\\[5pt]
\large Supported classical descent, cutoff comparison,\\
and finite destructive-clock certificates}
\author{}
\date{}

\begin{document}
\maketitle

\begin{abstract}
We construct a supported local spacetime adaptation of the
plateau intertwiner of Manuscript VII-A. The physical background
is the regular part of I-B's prescribed conical Lorentzian
product, not global Minkowski spacetime. A developing chart
supports a precisely specified switched scalar action,
unswitched finite-time harmonic coordinates, and formal
diffeomorphisms supported inside a switch plateau. We prove
the complete second-order scalar and coordinate intertwiners,
including the density transformation and every switch
derivative, without imposing a field equation on the metric
coefficients. The construction restricts to an explicitly
defined, nonempty Einstein--scalar coupling jet.

The repair is deliberately relative. For general compact gauge
supports we derive the exact first-order residual
\(\mathsf E_m\mathcal Q_{(1-g)K\xi}^{(m)}\phi_0\), construct a
strictly negative smeared witness, and prove that no higher
coupling coefficient or positive \(\hbar\)-order removes the
fixed classical gap. We also prove causal dependence on the
truncated observation past and a unique Cauchy-data-preserving
comparison cocycle between switches. An explicit example shows
why equality on an observation plateau alone is insufficient.

The additional contribution is a finite operational realization
that records both the repaired coefficients and the unrepaired
gap. A sourced massless family supplies a nontrivial evaluation
table; a complete destructive Enter instrument reads its finite
encoding, admits a quantitative complete-channel approximation
bound, and transports the payload and its classical control
record along a prescribed causal route. This separation of
classical covariance, preparation memory, and finite readout
may provide a useful reproducible interface for later analytic
work. It is not field acquisition or a quantum-gravity
measurement construction. The classical plateau gate is closed
on the stated domain; no quantum product, quantum Ward identity,
radiative coefficient, or counterterm separator is constructed.
\end{abstract}

\statementbox{Release boundary}{%
This is a local Grade S construction on supplied geometry.
The classical intertwiner and its larger-support obstruction
are attributed to VII-A and are reproved here.
The spacetime adaptation and finite evaluation interface do not
derive the metric, lapse, deficit, or cone from records.

The regular cone is not globally hyperbolic. All analytical
inverses below belong to a supported developing presentation.
The quantum certificate remains RG-open.}

\tableofcontents

%=====================================================================
\section{Introduction and the Enter interface discipline}
\label{sec:intro}
%=====================================================================

The problem is to keep three statements separate:
\[
 \text{plateau-supported classical covariance},\qquad
 \text{larger-support classical failure},\qquad
 \text{finite operational readout}.
\]
The last statement must neither erase the failure nor supply an
unconstructed quantum field observable.

Manuscript VII-A proves a classical repair for a specified
action-level switch, fixed harmonic Cauchy prescription, and
restricted gauge group \cite{CheungVIIA}. Its repair does not
extend to every compact gauge generator. We give the local
proofs needed to place that construction in the conical
laboratory of I-B \cite{CheungIB}. VI supplies a documentary
comparison for the unrestricted obstruction, not a premise of
our argument \cite{CheungVI}.

Relational observables and their perturbative expansions have
established antecedents \cite{BFHPR,Frob}. In particular, neither
the existence of relational constructions nor previously
computed gauge-invariant quantum correlation functions is
claimed as a discovery of this paper. Our object has its own
finite-time boundary data, switch prescription, and gauge
domain.

\subsection{Background and philosophy of Enter}

Cheung's Enter calculus treats runtime quantum names as access
rights to tensor factors. A destructive readout consumes the
measured names and returns a classical record; unmeasured
factors may remain \cite{CheungEnter,CheungI}. The philosophical
distinction is between an output interface and an unrestricted
claim about the fate of all implementing physical degrees of
freedom. Destructive readout, quantum recovery, a thermodynamic
implementation, and a geometric identification are different
tasks \cite{CheungPhilosophy}.

We adopt the specified finite Born instrument and prove its
complete channel formula. We do not rederive the Enter
characterization from an abbreviated axiom list or promote it
to a field-instrument theorem.

I-B attaches a finite clock, local rods, and causal transport
permissions to a prescribed conical spacetime
\cite{CheungIB}. Consuming that clock does not consume the
geometric time axis. The manifold, worldsheet defect, and future
cone remain input data.

\begin{definition}[Local Grade S]
\label{def:grade}
A local Grade S package specifies and establishes:
\begin{enumerate}[label=\textup{(S\arabic*)}]
\item a prescribed Lorentzian background, local rods, and a
supported analytical laboratory;
\item named classical equations and proved descent identities
on that laboratory;
\item a finite destructive clock with a complete recorded
instrument;
\item an explicit finite encoding of selected classical
evaluations;
\item reference-preserving transport whose complete
dependencies respect the prescribed causal relation.
\end{enumerate}
Grade E would additionally require a construction determining
geometric data from operational records. It is not asserted.
\end{definition}

The supported-presentation pattern is comparable with III-B
and V-B \cite{CheungGradeS}. Their theorems are not substituted
for the arguments below.

%=====================================================================
\section{Prescribed geometry and the supported Cauchy presentation}
\label{sec:geometry}
%=====================================================================

Fix
\[
 c>0,\qquad 0<\mu<2\pi,\qquad
 \alpha=1-\frac{\mu}{2\pi}.
\]
The physical smooth spacetime is
\begin{equation}
 M=\RR_T\times C_\mu^\circ\times\RR_z,\qquad
 \bar g=-c^2\dd T^2+\dd r^2+\alpha^2r^2\dd\phi^2+\dd z^2,
 \label{eq:cone}
\end{equation}
where \(r>0\) and \(\phi\in\RR/(2\pi\mathbb Z)\).
The completion has singular locus
\[
 \mathcal W=\RR_T\times\{o\}\times\RR_z.
\]
It is a worldsheet, not a worldline. No smooth Einstein tensor
is asserted on \(\mathcal W\).

\begin{proposition}[Geometry, rods, and time orientation]
\label{prop:geometry}
The metric \eqref{eq:cone} has signature \((-,+,+,+)\).
Choose the orientation for which \(\partial_T\) is future
directed. A nonzero future-causal vector \(a\partial_T+w\)
satisfies
\[
 a>0,\qquad |w|_{\rm spatial}\le ca.
\]
Static observers have proper-time increment \(\dd\tau=\dd T\).

On a sufficiently small lifted angular interval,
\[
 \operatorname{dev}(T,r,\phi,z)
 =\bigl(T,r\cos(\alpha\phi),r\sin(\alpha\phi),z\bigr)
\]
is a local isometry to Minkowski space. Its three spatial
coordinates are local rods.
\end{proposition}

\begin{proof}
The diagonal metric matrix has one negative and three positive
entries. Its causal inequality is
\(-c^2a^2+|w|_{\rm spatial}^2\le0\).
A nonzero causal vector cannot have \(a=0\), and the chosen
orientation selects \(a>0\). The static proper-time formula
follows by setting \(w=0\).
Finally,
\[
 \dd x^2+\dd y^2=\dd r^2+\alpha^2r^2\dd\phi^2.
\]
\end{proof}

The angular deck transformation changes the developed angle
by \(2\pi-\mu\). In the inverse-deck convention the gluing is
rotation by \(\mu\). Thus local flatness does not remove the
global conical identification.

\begin{proposition}[The regular product is not globally hyperbolic]
\label{prop:nonGH}
The spacetime \(M\) has a noncompact causal diamond.
\end{proposition}

\begin{proof}
Fix \(r_0>0\), one angle, and \(z=0\).
Take events \(p,q\) at that spatial point with times
\(-2r_0/c\) and \(2r_0/c\). For any \(r_n\downarrow0\),
the events \((0,r_n,\phi,0)\) belong to
\(J^+(p)\cap J^-(q)\), by radial causal travel.
They have no convergent subsequence in \(M\), because the
apex is excluded.
\end{proof}

No global Cauchy theorem is inferred from local flatness.
Global Green-operator results have additional hypotheses
\cite{BGP}.

\subsection{The laboratory}

Choose a regular spatial chart
\[
 \chi:\mathcal V\longrightarrow V\subset\RR^3
\]
whose image is convex and contains \(\overline B_R\) in its
interior. Translate the spatial coordinates if necessary, and set
\[
 t=c(T-T_{\rm in}),\qquad
 \eta=\diag(-1,1,1,1).
\]
For \(0<b<R\), use the shrinking domain
\begin{equation}
 \Dom=\{(t,x):0\le t\le b,\ |x|<R-t\},
 \qquad
 \Sigma_0=\{0\}\times B_R.
 \label{eq:domain}
\end{equation}
Its open positive-time part is denoted \(\Dom^\circ_+\).
Fields are smooth up to the initial surface on neighborhoods
of the portions used.

\begin{definition}[Supported presentation]
\label{def:presentation}
The analytical data consist of:
\begin{enumerate}[label=\textup{(\roman*)}]
\item the chart, \(\Dom\), \(\Sigma_0\), and the fixed initial
data prescriptions;
\item compact switches, tests, and gauge generators in
\(\Dom^\circ_+\);
\item a relatively compact open plateau \(U\Subset
\Dom^\circ_+\), with its closure separated from \(\Sigma_0\);
\item compact truncated past cones for all evaluations, with
positive margin inside the developing laboratory.
\end{enumerate}
Solutions are determined from their stated initial data, with no
independent incoming lateral-boundary data.

Supportedness is a dependence condition. It does not require
the metric coefficients themselves to have compact support or
compact Cauchy data.
\end{definition}

The domain \(\Dom\) is backward-cone closed relative to
\(\Sigma_0\): if \((s,z)\) lies between \(0\) and a point
\((t,x)\in\Dom\), with \(|z-x|\le t-s\), then
\[
 |z|\le |x|+t-s<R-s.
\]
This is the property used by all local inverses.

\begin{lemma}[Zero-Cauchy inverses]
\label{lem:Cauchy}
For \(m\ge0\), define
\[
 P_m=-\partial_t^2+\Delta-m^2,\qquad
 \Omega_m=(-\Delta+m^2)^{1/2}.
\]
There is a unique smooth solution of \(P_mu=F\) on \(\Dom\)
with prescribed smooth Cauchy data. The zero-data inverse
\(\Em\) is locally represented by
\begin{equation}
 (\Em F)(t)
 =-\int_0^t
 \frac{\sin((t-s)\Omega_m)}{\Omega_m}F(s)\,\dd s.
 \label{eq:Em}
\end{equation}
It has finite propagation and is continuous in local smooth
topologies on compact truncated cones.

If \(u\) has zero Cauchy data, then
\[
 \Em P_mu=u.
 \label{eq:leftinverse}
\]
For general \(u\),
\begin{equation}
 \Em P_mu=u-H_m[u],\qquad
 H_m[u](t)=
 \cos(t\Omega_m)u|_0+
 \frac{\sin(t\Omega_m)}{\Omega_m}\dot u|_0.
 \label{eq:homogeneous-subtraction}
\end{equation}
\end{lemma}

\begin{proof}
For compact spatial data and sources, spatial Fourier
transformation reduces the equation to forced oscillators.
Solving them gives \eqref{eq:Em} and the displayed homogeneous
solution.

Finite propagation and smooth uniqueness follow from the
energy identity on a truncated backward cone: the outward
lateral flux is nonnegative, and zero data and zero source
give zero energy. Differentiated energy estimates and local
Sobolev estimates give smooth dependence.

For a compact target in \(\Dom\), cut off and extend the
required data and source from its truncated past to compact
spatial data. Different extensions agree on the target by
the preceding uniqueness argument. Exhaustion by compact
targets defines the smooth local solution. Subtracting
\(H_m[u]\) proves \eqref{eq:leftinverse} and
\eqref{eq:homogeneous-subtraction}.
\end{proof}

The flat Fourier formula is an auxiliary realization of the
specified local problem, not a declaration that all of
\(\RR^{1,3}\) is physical spacetime. After construction the
result pulls back to the regular conical laboratory.
No intrinsic global conical inverse is selected.

%=====================================================================
\section{Switching before variation}
\label{sec:scalar}
%=====================================================================

Work in \(C^\infty(\Dom)[[\kappa]]/(\kappa^3)\), without
factorials in the coefficient expansion:
\begin{equation}
 \gamma=\eta+\kappa h+\kappa^2k\formal.
 \label{eq:metricjet}
\end{equation}
The prescribed background \(\bar g\) and the formal unknown
\(\gamma\) have different roles. The former supplies the
laboratory and apparatus cone. The latter is an input to the
coefficient equations. No record-dependent metric update is
introduced.

Raise coefficient indices with \(\eta\). Define
\begin{align}
 a(h)&=\frac12\tr_\eta h,&
 b(h)&=\frac18(\tr_\eta h)^2-\frac14h_{\mu\nu}h^{\mu\nu},
 \nonumber\\
 A^{\mu\nu}(h)&=a(h)\eta^{\mu\nu}-h^{\mu\nu},&
 B^{\mu\nu}(h)&=
 h^\mu{}_\rho h^{\rho\nu}
 -a(h)h^{\mu\nu}+b(h)\eta^{\mu\nu}.
 \label{eq:AB}
\end{align}
Writing
\[
 \rho_\gamma=\sqrt{-\det\gamma},\qquad
 \mathcal A_\gamma^{\mu\nu}=\rho_\gamma\gamma^{\mu\nu},
\]
matrix inversion and determinant expansion give
\begin{align}
 \rho_\gamma
 &=1+\kappa a(h)+\kappa^2(a(k)+b(h))\formal,
 \nonumber\\
 \mathcal A_\gamma^{\mu\nu}
 &=\eta^{\mu\nu}
 +\kappa A^{\mu\nu}(h)
 +\kappa^2(A^{\mu\nu}(k)+B^{\mu\nu}(h))\formal.
 \label{eq:density-expansion}
\end{align}
Indeed,
\[
 \gamma^{-1}
 =\eta^{-1}-\kappa h^\sharp+
 \kappa^2(h^\sharp h^\sharp-k^\sharp),
\]
and
\(\rho_\gamma=\exp(\frac12\tr\log(\eta^{-1}\gamma))\);
multiplication proves \eqref{eq:density-expansion}.

\begin{definition}[Plateau and restricted gauge data]
\label{def:plateau}
Fix
\[
 g\in C_c^\infty(\Dom^\circ_+),\qquad 0\le g\le1,
 \qquad
 \overline U\subset\operatorname{int}\{g=1\}.
\]
The test is real \(f\in C_c^\infty(U)\).
Let
\[
 \mathfrak g_U=C_c^\infty(U;T\Dom).
\]
The admitted formal transformations are
\begin{equation}
 F_\kappa^*
 =\exp(\kappa\LL_\xi+\kappa^2\LL_\zeta)\formal,
 \qquad \xi,\zeta\in\mathfrak g_U.
 \label{eq:formal-group}
\end{equation}
They are formal jets of orientation-preserving diffeomorphisms
generated inside \(U\). Their group is denoted \(\GG_U\).
\end{definition}

The Lie bracket preserves \(\mathfrak g_U\), and the truncated
Baker--Campbell--Hausdorff formula preserves the class
\eqref{eq:formal-group}. Including \(\zeta\) is necessary for
this convenient composition-stable parametrization.
Every admitted actual flow is the identity outside a compact
subset of \(U\), preserves \(g\), and is the identity near
\(\Sigma_0\).

Define the switched coefficients
\[
 \mathcal A_g[\gamma]=(1-g)\mathcal A_\eta+g\mathcal A_\gamma,
 \qquad
 \rho_g[\gamma]=(1-g)+g\rho_\gamma.
\]
Our scalar action and Euler operator are
\begin{align}
 S_g^{\rm sc}[\gamma,\phi]
 &=-\frac12\int
 \left(
 \mathcal A_g[\gamma]^{\mu\nu}
             \partial_\mu\phi\partial_\nu\phi
 +m^2\rho_g[\gamma]\phi^2
 \right)\dd^4x,
 \label{eq:action}\\
 \Dg_g[\gamma]\phi
 &=\partial_\mu\left(
 \mathcal A_g[\gamma]^{\mu\nu}\partial_\nu\phi
 \right)-m^2\rho_g[\gamma]\phi.
 \label{eq:Euler}
\end{align}
The action is variationally defined with compact variations;
its numerical value need not be finite on every smooth input.
Its Euler output represents a scalar density relative to
\(\dd^4x\).

For a symmetric tensor \(H\), put
\begin{align}
 \Qg_H^{(m)}u
 &=\partial_\mu(A^{\mu\nu}(H)\partial_\nu u)-m^2a(H)u,
 \nonumber\\
 \Tg_h^{(m)}u
 &=\partial_\mu(B^{\mu\nu}(h)\partial_\nu u)-m^2b(h)u.
 \label{eq:QT}
\end{align}
Then
\begin{equation}
 \Dg_g[\gamma]
 =P_m+\kappa V_g[h]+\kappa^2W_g[h,k]\formal,
 \label{eq:DW}
\end{equation}
where
\begin{align}
 V_g[h]&=\Qg_{gh}^{(m)},\nonumber\\
 W_g[h,k]u
 &=\partial_\mu\left(
 g(A^{\mu\nu}(k)+B^{\mu\nu}(h))\partial_\nu u
 \right)-m^2g(a(k)+b(h))u.
 \label{eq:VW}
\end{align}
In particular,
\begin{align}
 V_g[h]u
 &=g\Qg_h^{(m)}u
     +(\partial_\mu g)A^{\mu\nu}(h)\partial_\nu u,
 \nonumber\\
 W_g[h,k]u
 &=g(\Qg_k^{(m)}+\Tg_h^{(m)})u
 +(\partial_\mu g)(A^{\mu\nu}(k)+B^{\mu\nu}(h))
                                           \partial_\nu u.
 \label{eq:switch-derivatives}
\end{align}
The quadratic metric vertex has one factor \(g\), not \(g^2\).
The iterated first-order response below contains two switch
insertions for a different reason.

\subsection{The scalar assignment}

Fix a real smooth free scalar \(P_m\phi_0=0\) with declared
Cauchy data. Define
\begin{align}
 \phi_g[\gamma;\phi_0]
 &=\phi_0+\kappa u_{1,g}+\kappa^2u_{2,g}\formal,
 \nonumber\\
 u_{1,g}&=-\Em V_g[h]\phi_0,\nonumber\\
 u_{2,g}&=
 \Em V_g[h]\Em V_g[h]\phi_0-\Em W_g[h,k]\phi_0.
 \label{eq:scalar-coeff}
\end{align}
Both positive-order coefficients have zero Cauchy data.

\begin{proposition}[Existence and uniqueness of the scalar assignment]
\label{prop:scalar}
The expression \eqref{eq:scalar-coeff} uniquely solves
\[
 \Dg_g[\gamma]\phi_g=0\formal
\]
with the stated leading data and zero positive-order data.
It is defined for arbitrary smooth \(h,k\).
\end{proposition}

\begin{proof}
The two coefficient equations are
\[
 P_mu_{1,g}=-V_g[h]\phi_0,\qquad
 P_mu_{2,g}=-V_g[h]u_{1,g}-W_g[h,k]\phi_0.
\]
Apply Lemma~\ref{lem:Cauchy} successively. Substituting the
first solution gives \eqref{eq:scalar-coeff}.
\end{proof}

%=====================================================================
\section{Unswitched harmonic labels and the relational coefficients}
\label{sec:coordinates}
%=====================================================================

Define
\[
 \mathcal C^\mu[h]
 =\partial_\nu h^{\nu\mu}
       -\frac12\partial^\mu\tr_\eta h
 =-\partial_\nu A^{\nu\mu}(h).
\]
The harmonic labels are four scalar functions
\[
 X_\gamma^\mu=x^\mu+\kappa w^\mu+\kappa^2v^\mu\formal
\]
satisfying the \emph{unswitched} equation
\[
 \partial_\alpha(
 \mathcal A_\gamma^{\alpha\beta}
                    \partial_\beta X_\gamma^\mu)=0\formal
\]
and data \(X_\gamma^\mu|_0=x^\mu|_0\),
\(\partial_tX_\gamma^\mu|_0=\partial_tx^\mu|_0\).

\begin{proposition}[Harmonic Cauchy construction]
\label{prop:coordinates}
The unique coefficients are
\begin{align}
 w^\mu[h]&=\Ezero\mathcal C^\mu[h],\nonumber\\
 v^\mu[h,k]&=
 \Ezero\left(
 \mathcal C^\mu[k]-\partial_\alpha B^{\alpha\mu}(h)
                         -\Qg_h^{(0)}w^\mu[h]
 \right).
 \label{eq:coordinates}
\end{align}
\end{proposition}

\begin{proof}
At first order the equation is
\(\Box_\eta w^\mu=\mathcal C^\mu[h]\).
At second order it is
\[
 \Box_\eta v^\mu=
 \mathcal C^\mu[k]-\partial_\alpha B^{\alpha\mu}(h)
                         -\Qg_h^{(0)}w^\mu.
\]
Both have zero data, so Lemma~\ref{lem:Cauchy} applies.
\end{proof}

The index \(\mu\) labels the four scalar functions.
It does not make the labels the components of a globally
defined conical vector field.

Their formal inverse is
\begin{equation}
 Y_\gamma^\mu(y)
 =y^\mu-\kappa w^\mu(y)
 +\kappa^2\left(
 w^\nu\partial_\nu w^\mu-v^\mu
 \right)(y)\formal.
 \label{eq:inverse}
\end{equation}
Substitution in \(X_\gamma(Y_\gamma(y))\) proves the formula
and uniqueness order by order.

Define
\begin{equation}
 \Phi_g[\gamma;\phi_0]=\phi_g[\gamma;\phi_0]\circ Y_\gamma,
 \qquad
 \OO_f[g](\gamma,\phi_0)=
           \int f(y)\Phi_g[\gamma;\phi_0](y)\,\dd^4y.
 \label{eq:observable}
\end{equation}
The smearing uses the fixed label density, not
\(\rho_\gamma\dd^4y\). No global numerical invertibility of
\(X_\gamma\) is required.

Taylor expansion gives
\begin{align}
 \Phi_g&=\phi_0+\kappa A_1^{(g)}
                       +\kappa^2A_2^{(g)}\formal,\nonumber\\
 A_1^{(g)}
 &=u_{1,g}-w^\mu\partial_\mu\phi_0,\nonumber\\
 A_2^{(g)}
 &=u_{2,g}-w^\mu\partial_\mu u_{1,g}
 +(w^\nu\partial_\nu w^\mu-v^\mu)\partial_\mu\phi_0
 +\frac12w^\mu w^\nu\partial_\mu\partial_\nu\phi_0.
 \label{eq:A12}
\end{align}
Write
\[
 C_r[g](h,k;\phi_0,f)=\int f A_r^{(g)}\,\dd^4y,
 \qquad r=1,2.
\]
In particular, the complete second coefficient is
\begin{align}
 A_2^{(g)}
 ={}&\Em V_g\Em V_g\phi_0-\Em W_g\phi_0
      +X_w\Em V_g\phi_0\nonumber\\
 &+(w^\nu\partial_\nu w^\mu-v^\mu)\partial_\mu\phi_0
 +\frac12w^\mu w^\nu\partial_\mu\partial_\nu\phi_0.
 \label{eq:fullA2}
\end{align}
All five displayed contributions are retained.

%=====================================================================
\section{First-order repair and the larger-support residual}
\label{sec:first}
%=====================================================================

For a vector field \(\xi\), let
\[
 (K\xi)_{\mu\nu}
 =\partial_\mu\xi_\nu+\partial_\nu\xi_\mu,\qquad
 X_\xi=\xi^\mu\partial_\mu,\qquad
 D_\xi=X_\xi+\partial_\mu\xi^\mu.
\]
Here \(D_\xi\) is the Lie derivative on scalar densities
represented in the fixed chart.

\begin{lemma}[Two linear identities]
\label{lem:linear}
For smooth \(\xi,u\),
\begin{align}
 \mathcal C[K\xi]&=\Box_\eta\xi,\nonumber\\
 \Qg_{K\xi}^{(m)}u&=D_\xi P_mu-P_mX_\xi u.
 \label{eq:linear}
\end{align}
If \(\xi\) vanishes near \(\Sigma_0\), then
\(w[K\xi]=\xi\). On \(P_m\phi_0=0\),
\[
 \Qg_{K\xi}^{(m)}\phi_0=-P_m(X_\xi\phi_0).
 \label{eq:QK-shell}
\]
\end{lemma}

\begin{proof}
The trace of \(K\xi\) is \(2\partial_\rho\xi^\rho\);
the mixed derivative cancels in \(\mathcal C[K\xi]\).
The zero-data left-inverse identity then gives \(w[K\xi]=\xi\).

For the second formula,
\[
 A^{\mu\nu}(K\xi)
 =(\partial_\rho\xi^\rho)\eta^{\mu\nu}
       -\partial^\mu\xi^\nu-\partial^\nu\xi^\mu,
 \qquad a(K\xi)=\partial_\rho\xi^\rho.
\]
Its divergence is \(-\Box_\eta\xi^\nu\). Thus
\[
 \Qg_{K\xi}^{(m)}u
 =(\partial_\rho\xi^\rho)P_mu
 -(\Box_\eta\xi^\nu)\partial_\nu u
 -2(\partial^\mu\xi^\nu)\partial_\mu\partial_\nu u.
\]
This is \(D_\xi P_mu-P_mX_\xi u\).
\end{proof}

\begin{theorem}[Exact first-order plateau cancellation]
\label{thm:first}
For every smooth \(h\), every free \(\phi_0\), and
\(\xi\in\mathfrak g_U\),
\[
 gK\xi=K\xi,\qquad
 u_{1,g}[h+K\xi]-u_{1,g}[h]=X_\xi\phi_0,\qquad
 w[h+K\xi]-w[h]=\xi.
 \label{eq:first-three}
\]
Consequently \(A_1^{(g)}\), and hence \(C_1[g]\), is invariant.
No metric field equation is required.
\end{theorem}

\begin{proof}
All derivatives of \(\xi\) have support in the plateau.
Linearity and \eqref{eq:QK-shell} give
\[
 \delta_\xi u_{1,g}
 =-\Em\Qg_{K\xi}^{(m)}\phi_0
 =\Em P_m(X_\xi\phi_0)=X_\xi\phi_0.
\]
The last equality uses the vanishing near \(\Sigma_0\), not an
unrestricted inverse identity. The coordinate identity follows
from Lemma~\ref{lem:linear}. Substitution in \eqref{eq:A12}
cancels the two variations.
\end{proof}

\begin{proposition}[Exact defect for the larger compact class]
\label{prop:defect}
Let
\(\xi\in C_c^\infty(\Dom^\circ_+;T\Dom)\), without the plateau
support restriction. Under the unchanged-free-input comparison
\((h,\phi_0)\mapsto(h+K\xi,\phi_0)\),
\begin{equation}
 A_1^{(g)}[h+K\xi;\phi_0]-A_1^{(g)}[h;\phi_0]
 =\Em\Qg_{(1-g)K\xi}^{(m)}\phi_0.
 \label{eq:defect}
\end{equation}
\end{proposition}

\begin{proof}
The difference is
\[
 -\Em\Qg_{gK\xi}^{(m)}\phi_0-X_\xi\phi_0.
\]
The zero-data identity and \eqref{eq:QK-shell} give
\(X_\xi\phi_0=-\Em\Qg_{K\xi}^{(m)}\phi_0\).
Linearity in the tensor argument proves \eqref{eq:defect}.
\end{proof}

For comparison with an expanded retarded formula, write
\(\overline H^{\mu\nu}
 =H^{\mu\nu}-\frac12\eta^{\mu\nu}\tr_\eta H\).
Then
\begin{align}
 \delta_\xi A_1^{(g)}
 ={}&(g-1)X_\xi\phi_0\nonumber\\
 &+\Em\left(
 (\partial_\mu g)\overline{K\xi}^{\mu\nu}\partial_\nu\phi_0
                  -[P_m,g]X_\xi\phi_0
 \right),\nonumber\\
 [P_m,g]u&=(\Box_\eta g)u+
                         2(\partial^\mu g)\partial_\mu u.
 \label{eq:expandeddefect}
\end{align}
Indeed, expand \(\Qg_{gK\xi}\), and use
\(\Em P_m(gX_\xi\phi_0)=gX_\xi\phi_0\).
This independently reproduces the classical mechanism compared
with VI \cite{CheungVI}.

\begin{proposition}[A normalized negative witness]
\label{prop:witness}
For every \(m\ge0\), an admitted packet exists with
\(f\ge0\), \(\int f=1\), and a compact larger-support generator
\(\xi\), constant near \(\supp g\), such that
\[
 g=1\text{ near }\supp f,\qquad gK\xi=0,
\]
but
\begin{equation}
 \Delta=
 C_1[g](h+K\xi;\phi_0,f)-C_1[g](h;\phi_0,f)
 \le-\frac12.
 \label{eq:negativegap}
\end{equation}
The gap is independent of the baseline \(h\).
\end{proposition}

\begin{proof}
Choose an interior event \((t_*,0)\) and free scalar data at
\(t=t_*\) equal to \(x^1\) near zero, with zero time derivative,
and smoothly compactly supported in space.
The resulting free solution satisfies
\(\partial_1\phi_0(t_*,0)=1\).
Choose a small plateau neighborhood \(U\) where
\(\partial_1\phi_0\ge1/2\), and a unit-integral nonnegative
test \(f\) there. Choose \(g=1\) near \(\overline U\).

A compact smooth \(\chi_\xi\), supported after \(0\), can be
chosen equal to one near the entire \(\supp g\).
Set \(\xi=\chi_\xi\partial_1\).
All derivatives of \(\xi\) vanish near \(\supp g\), so
\(gK\xi=0\). The scalar response has zero variation, while
the coordinate response changes by \(\xi\). Therefore
\[
 \Delta=-\int f\,\partial_1\phi_0\,\dd^4x\le-\frac12.
\]
The cutoff of \(\xi\) occurs outside the plateau containing
the observation. Its support is not contained in \(U\).
\end{proof}

\begin{corollary}[No higher-order repair of the fixed residual]
\label{cor:nohigherrepair}
With \(A_1^{(g)}\), the free input, and the larger-support
identification fixed, no regular addition beginning at
\(\kappa^2\) makes the observable invariant under all those
compact generators. Terms of positive \(\hbar\)-degree cannot
cancel the nonzero \(\kappa\hbar^0\) residual either.
\end{corollary}

\begin{proof}
Equality of formal series with nonnegative powers is
coefficientwise. Extract the coefficient of \(\kappa\hbar^0\)
and use \eqref{eq:negativegap}.
\end{proof}

This is an existential obstruction to extending the theorem
to the full compact class. It is not a claim that every
generator outside the chosen \(U\) produces a nonzero residual.
For example, additional generators supported in another part
of \(\{g=1\}^{\circ}\) may also satisfy the cancellation.

The defect is classical, not a loop anomaly.
Restricting the gauge domain does not refute the unrestricted
obstruction.

%=====================================================================
\section{Density covariance and the second-order intertwiner}
\label{sec:second}
%=====================================================================

Expansion of the metric pullback gives
\begin{align}
 h'&=h+K\xi,\nonumber\\
 k'&=k+\LL_\xi h+K\zeta+\frac12\LL_\xi^2\eta.
 \label{eq:metric-transform}
\end{align}
Holding \(k\) fixed while changing \(h\) by \(K\xi\) is
generally not this transformation.

\begin{lemma}[Plateau covariance of the density operator]
\label{lem:plateaucov}
For an admitted actual diffeomorphism \(F\), and hence
coefficientwise for \eqref{eq:formal-group},
\begin{equation}
 \Dg_g[F^*\gamma](F^*u)
       =F^*_{\rm dens}(\Dg_g[\gamma]u).
 \label{eq:plateaucov}
\end{equation}
The density pullback is
\(F^*_{\rm dens}r=(\det DF)(r\circ F)\).
\end{lemma}

\begin{proof}
The inverse-metric density \(\mathcal A_\gamma\) is a
contravariant tensor density, and \(\rho_\gamma\) is a scalar
density. Their coordinate transformation laws follow by
matrix inversion and the determinant chain rule.

Because \(F^*g=g\), the discrepancy between
\(F^*\mathcal A_g[\gamma]\) and
\(\mathcal A_g[F^*\gamma]\) is
\[
 (1-g)(F^*\mathcal A_\eta-\mathcal A_\eta).
\]
Inside \(U\), \(1-g=0\). Outside \(U\), \(F\) and its derivative
are the identity. The discrepancy therefore vanishes.
The same argument applies to \(\rho_g\).

For completeness, divergence of a vector density is natural:
pair it against a compact scalar test, integrate by parts,
and change variables by \(F\). The transformed integral is
exactly the integral of the pulled-back divergence.
Applying this fact to
\(\mathcal A_g[\gamma]^{\mu\nu}\partial_\nu u\)
and adding the mass density proves \eqref{eq:plateaucov}.
Formal expansion gives the coefficient identity.
\end{proof}

This lemma concerns the full divergence-form operator.
It does not delete the terms in
\eqref{eq:switch-derivatives}, and by itself it is not the
scalar intertwiner.

\begin{proposition}[Scalar and coordinate Cauchy intertwiners]
\label{prop:intertwiners}
For \(\xi,\zeta\in\mathfrak g_U\),
\begin{align}
 \phi_g[F_\kappa^*\gamma;\phi_0]
 &=F_\kappa^*\phi_g[\gamma;\phi_0]\formal,
 \nonumber\\
 X_{F_\kappa^*\gamma}
 &=X_\gamma\circ F_\kappa\formal.
 \label{eq:intertwiners}
\end{align}
In coefficient form,
\begin{align}
 u'_{1,g}&=u_{1,g}+X_\xi\phi_0,\nonumber\\
 u'_{2,g}&=u_{2,g}+X_\xi u_{1,g}
                 +X_\zeta\phi_0+\frac12X_\xi^2\phi_0,
 \nonumber\\
 w'^\mu&=w^\mu+\xi^\mu,\nonumber\\
 v'^\mu&=v^\mu+\xi^\nu\partial_\nu w^\mu
                 +\zeta^\mu+\frac12\xi^\nu\partial_\nu\xi^\mu.
 \label{eq:intertwiner-coeff}
\end{align}
The free input \(\phi_0\) is unchanged.
\end{proposition}

\begin{proof}
Lemma~\ref{lem:plateaucov} shows that the pulled-back scalar
solves the transformed equation. The pullback is the identity
near \(\Sigma_0\), so all its prescribed Cauchy data agree with
those of the transformed scalar assignment.
Proposition~\ref{prop:scalar} proves equality.

The unswitched massless density equation is natural by the same
change-of-variables argument, without the switch discrepancy.
Each pulled-back harmonic label has the prescribed initial
data. Proposition~\ref{prop:coordinates} therefore identifies it
with the transformed coordinate solution.
Expansion proves \eqref{eq:intertwiner-coeff}.
\end{proof}

\subsection{An operator audit of the density terms}

Let \(V,W\) and \(V',W'\) be the operators before and after
\eqref{eq:metric-transform}. Put
\[
 X=X_\xi,\quad
 Y=X_\zeta+\frac12X_\xi^2,\quad
 D=D_\xi,\quad
 Z=D_\zeta+\frac12D_\xi^2.
\]
Expanding \eqref{eq:plateaucov} gives the off-shell identities
\begin{align}
 V'+P_mX&=V+DP_m,\nonumber\\
 W'+V'X+P_mY&=W+DV+ZP_m.
 \label{eq:operator-audit}
\end{align}
These retain the density divergence in \(D\) and the terms
proportional to \(P_m\).

One can verify the second scalar law without appealing further
to a covariance slogan. Set \(a=u_{1,g}\), \(b=u_{2,g}\), and
\(b_{\rm cand}=b+Xa+Y\phi_0\).
Using \(P_ma=-V\phi_0\), \(P_mb=-Va-W\phi_0\), and
\eqref{eq:operator-audit},
\[
 P_mb_{\rm cand}
 =-V'(a+X\phi_0)-W'\phi_0.
\]
The candidate has zero Cauchy data because the gauge generators
vanish near \(\Sigma_0\). Uniqueness gives the second scalar law.

\begin{theorem}[Second-order plateau descent]
\label{thm:second}
For arbitrary smooth \(h,k\), the pointwise relational scalar
and its fixed-label smearing satisfy
\[
 \Phi_g[F_\kappa^*\gamma;\phi_0]
       =\Phi_g[\gamma;\phi_0]\formal,\qquad
 \OO_f[g](F_\kappa^*\gamma,\phi_0)
       =\OO_f[g](\gamma,\phi_0)\formal
\]
under \(\GG_U\). In particular, \(C_1[g]\) and \(C_2[g]\)
descend under this restricted formal action.
\end{theorem}

\begin{proof}
The coordinate intertwiner implies
\[
 Y_{F_\kappa^*\gamma}
       =F_\kappa^{-1}\circ Y_\gamma\formal.
\]
Composing with the scalar intertwiner gives
\[
 \phi_g[F_\kappa^*\gamma;\phi_0]\circ
              Y_{F_\kappa^*\gamma}
 =(\phi_g[\gamma;\phi_0]\circ F_\kappa)
               \circ F_\kappa^{-1}\circ Y_\gamma
 =\Phi_g[\gamma;\phi_0]\formal.
\]
The fixed label density gives the smeared identity.
Appendix~\ref{app:direct} checks the second coefficient directly.
\end{proof}

For a pure-gauge pullback of the flat metric,
\[
 h=K\xi,\qquad
 k=K\zeta+\frac12\LL_\xi^2\eta,
\]
the first and second relational corrections vanish.
The metric family \(\eta+\kappa K\xi\) with \(k=0\) does not
generally have this property: it omits part of the pullback.

%=====================================================================
\section{The named plateau Einstein--scalar coupling jet}
\label{sec:jet}
%=====================================================================

The off-shell intertwiner does not require gravitational
equations. To state its intended on-shell restriction, fix the
normalization
\begin{equation}
 G_{\mu\nu}[\gamma]
       =\kappa^2T_{\mu\nu}[\gamma,\phi].
 \label{eq:Einstein-normalization}
\end{equation}
This is VII-A's normalization. A coefficient such as the
\(1/4\) used in another action convention must not be transferred
without rescaling the coupling.

Define
\[
 G[\eta+\kappa h+\kappa^2k]
 =\kappa\Glin[h]
  +\kappa^2\bigl(\Glin[k]+G^{(2)}[h,h]\bigr)\formal.
\]
The differential definition is given in
Appendix~\ref{app:curvature}. The leading stress tensor is
\[
 T^{(0)}_{\mu\nu}[\phi_0]
 =\partial_\mu\phi_0\partial_\nu\phi_0
 -\frac12\eta_{\mu\nu}
 \left(\partial^\rho\phi_0\partial_\rho\phi_0
                              +m^2\phi_0^2\right).
\]

\begin{definition}[Plateau coupling jet]
\label{def:jet}
Let \(\JJ_g^2\) consist of smooth tuples
\[
 (h,k,\phi_0,u_{1,g},u_{2,g})
\]
with \(P_m\phi_0=0\), the scalar assignment
\eqref{eq:scalar-coeff}, and
\begin{equation}
 \Glin[h]=0,\qquad
 \Glin[k]+G^{(2)}[h,h]=T^{(0)}[\phi_0].
 \label{eq:metric-equations}
\end{equation}
No zero-data condition is imposed on \(h\) or \(k\).
Their gravitational constraints are not replaced by
zero-data wave evolution.
\end{definition}

The metric equations are unswitched at this order, whereas the
scalar equation is switched. On the plateau the scalar operator
equals the unswitched curved density operator, so these are the
Einstein--scalar coefficient equations there.
Outside it, no nonlinear completion or switched gauge-fixed
gravitational action is claimed.

A nonzero \(\phi_0\) is a coupling-expansion coefficient.
It is not an assertion that \((\eta,\phi_0)\) is an exact
nonzero-scalar Minkowski solution at nonzero coupling.

\begin{proposition}[Stability of the named jet]
\label{prop:jetstable}
The transformations \eqref{eq:metric-transform} and
\eqref{eq:intertwiner-coeff} preserve \(\JJ_g^2\).
The relational coefficients therefore descend on its
\(\GG_U\)-orbit set.
\end{proposition}

\begin{proof}
The scalar equations and data are preserved by
Proposition~\ref{prop:intertwiners}.

For the metric, pullback preserves the Levi--Civita connection
because it preserves torsion freedom and metric compatibility.
It therefore preserves the curvature and Einstein tensor.
Taking coefficients gives
\[
 \Glin[h']=\Glin[h],
\]
\[
 \Glin[k']+G^{(2)}[h',h']
 =\Glin[k]+G^{(2)}[h,h]+\LL_\xi\Glin[h].
\]
The last term vanishes on \eqref{eq:metric-equations}.
The leading scalar remains \(\phi_0\), so its order-two source
is unchanged. The final assertion follows from
Theorem~\ref{thm:second}.
\end{proof}

\begin{proposition}[A nonempty sourced massless subclass]
\label{prop:nonempty}
For \(m=0\), let \(a_\mu\ne0\) be constant and put
\[
 \phi_0=a_\mu x^\mu,\qquad h=0,\qquad
 T_{\mu\nu}=a_\mu a_\nu
       -\frac12\eta_{\mu\nu}a_\rho a^\rho,\qquad
 T=\eta^{\mu\nu}T_{\mu\nu}.
\]
Define
\begin{equation}
 \sigma(x)=
 \left(-\frac14T_{\alpha\beta}
             +\frac1{12}\eta_{\alpha\beta}T\right)x^\alpha x^\beta,
 \qquad k_{\mu\nu}=2\sigma\eta_{\mu\nu}.
 \label{eq:sourced-example}
\end{equation}
Together with the assigned switched scalar coefficients,
this is a nonzero-source member of \(\JJ_g^2\).
\end{proposition}

\begin{proof}
The leading scalar is free and its stress is the displayed
constant nonzero tensor. Direct linearized curvature gives
\[
 \Glin[2\sigma\eta]_{\mu\nu}
       =-2\partial_\mu\partial_\nu\sigma
                              +2\eta_{\mu\nu}\Box_\eta\sigma.
\]
Here
\[
 \partial_\mu\partial_\nu\sigma
 =-\frac12T_{\mu\nu}+\frac16\eta_{\mu\nu}T,\qquad
 \Box_\eta\sigma=\frac16T.
\]
Thus \(\Glin[k]=T\). The scalar coefficients exist by
Proposition~\ref{prop:scalar}; \(u_{1,g}=0\) and
\(u_{2,g}=-\Ezero\Qg_{gk}^{(0)}\phi_0\).
\end{proof}

This polynomial local example imposes neither asymptotic
flatness nor compact metric Cauchy data. It proves nonvacuity
of the stated equations, not Einstein dynamics derived from
records.

%=====================================================================
\section{Causal support, switch equality, and classical comparison}
\label{sec:comparison}
%=====================================================================

For a nonzero test \(f\), define its truncated flat past
\begin{equation}
 \Dom_f=J^-_\eta(\supp f)\cap\{t\ge0\}.
 \label{eq:Df}
\end{equation}
Because \(\supp f\Subset\Dom^\circ_+\), this set is compact,
lies inside the supported laboratory, and may meet \(\Sigma_0\).

\begin{proposition}[Coefficientwise causal dependence]
\label{prop:support}
For fixed free input \(\phi_0\), the coefficients of
\(\OO_f[g]\) through order two depend on \(h,k,g\) only through
their restrictions to every neighborhood of \(\Dom_f\).

In particular, their functional support in the metric
variables is contained in \(\Dom_f\). Equality with all of
\(\Dom_f\) is not asserted.
\end{proposition}

\begin{proof}
First-order scalar and coordinate solutions at \(y\) use only
their sources in the truncated past of \(y\).
All coefficient operators are local differential operators.

A nested second-order term has an intermediate point \(z\)
in the past of \(y\). Every further point affecting that term
is in the past of \(z\), hence in the past of \(y\).
The same reasoning applies to \(v\), including
\(\Qg_h^{(0)}w\).
Taking the union over \(y\in\supp f\) proves the statement.
Agreement on a neighborhood includes the derivatives occurring
in these formulas.
\end{proof}

The developing map pulls each causal segment in this argument
back to a \(\bar g\)-causal segment. Consequently the local
metric dependence is contained in the prescribed conical past.
This does not identify functional dependence with operational
signalling, or supply the full global conical past.

\begin{proposition}[Equality under causal switch agreement]
\label{prop:cutoffequal}
Fix \(h,k,\phi_0\) and the harmonic Cauchy prescription.
If \(g_1=g_2\) on a neighborhood of \(\Dom_f\), then
\[
 \OO_f[g_1]=\OO_f[g_2]\formal.
\]
\end{proposition}

\begin{proof}
The scalar coefficient operators agree there, including their
switch derivatives. Proposition~\ref{prop:support} gives
identical scalar evaluations on \(\supp f\).
The coordinates do not depend on the switch.
\end{proof}

Agreement only near \(\supp f\) is insufficient. At first order,
\begin{equation}
 A_1^{(g_2)}-A_1^{(g_1)}
       =-\Em\Qg_{(g_2-g_1)h}^{(m)}\phi_0.
 \label{eq:switchdifference}
\end{equation}
The following is an explicit example, not merely a support
warning.

\begin{proposition}[Preparation memory despite a common plateau]
\label{prop:memory}
There are two admitted switches sharing an observation plateau,
the same metric coefficients, the same free scalar, and the
same harmonic prescription, for which \(C_1[g_1]\ne C_1[g_2]\).
The example can be chosen in the massless coupling-jet class.
\end{proposition}

\begin{proof}
Take \(m=0\), \(\phi_0=t\), and \(h=2\eta\).
Then \(A(h)=2\eta\), \(\mathcal C[h]=0\), and \(w=0\).
Choose a compact spatial cutoff equal to one near the spatial
projection of \(\Dom_f\). On that causal region let
\(g_i(t,x)=q_i(t)\), with \(q_i\) compactly supported after \(0\)
and equal to one near all observation times.

Since
\[
 V_{g_i}[h]t=-2q_i'(t),
\]
the zero-data solution is
\[
 u_{1,g_i}(t,x)=-2\int_0^tq_i(s)\,\dd s
 \quad\text{on }\Dom_f.
\]
Choose \(q_2=q_1+\varepsilon p\), where \(p\ge0\) is a nonzero
smooth pulse strictly before the observation plateau and
supported where \(q_1=0\). Choose \(\varepsilon>0\) small enough
to keep \(0\le q_2\le1\).
For \(\int f=1\),
\[
 C_1[g_2]-C_1[g_1]
       =-2\varepsilon\int p(s)\,\dd s\ne0.
\]

To impose the metric equations, choose \(k\) from
\eqref{eq:sourced-example} for \(\phi_0=t\).
The constant \(h\) has vanishing linear and quadratic Einstein
coefficients, so the same \(k\) works for both switches.
\end{proof}

Thus late agreement of sources or local operators is not
agreement of their responses. The comparison with XVI-B is
methodological only; its prepared-source witness is not used
here \cite{CheungPrepared}.

\subsection{A unique comparison, rather than false equality}

Fix a metric jet and two switches. Write
\[
 \Dg_i=P_m+\kappa V_i+\kappa^2W_i.
\]
Let \(\operatorname{Sol}_i\) be its formal scalar solution
space through order two, with arbitrary smooth formal Cauchy
data. Define
\begin{equation}
 \mathsf G_i=
 \Em-\kappa\Em V_i\Em
 +\kappa^2\left(
 \Em V_i\Em V_i\Em-\Em W_i\Em
 \right)\formal.
 \label{eq:formalGreen}
\end{equation}
Multiplication gives \(\Dg_i\mathsf G_i=\id\formal\) on smooth
sources. Every coefficient output has zero Cauchy data.

\begin{proposition}[Classical Cauchy comparison and cocycle]
\label{prop:classicalcomparison}
The map
\begin{equation}
 r_{21}[\gamma]\psi
 =\psi-\mathsf G_2(\Dg_2-\Dg_1)\psi\formal
 \label{eq:r21}
\end{equation}
takes \(\operatorname{Sol}_1\) to \(\operatorname{Sol}_2\)
and preserves formal Cauchy data. For each source solution
it is the unique target solution with those data. Hence it
is the unique map between these solution spaces with those
properties, and
\[
 r_{32}r_{21}=r_{31},\qquad r_{12}r_{21}=\id\formal.
\]

If a gauge transformation is supported in a common plateau,
\[
 r_{21}[F_\kappa^*\gamma](F_\kappa^*\psi)
       =F_\kappa^*(r_{21}[\gamma]\psi)\formal.
 \label{eq:comparison-equivariance}
\]
\end{proposition}

\begin{proof}
For \(\Dg_1\psi=0\formal\),
\[
 \Dg_2r_{21}\psi
 =\Dg_2\psi-(\Dg_2-\Dg_1)\psi
 =\Dg_1\psi=0\formal.
\]
The correction has zero data. Coefficientwise uniqueness
follows successively from Lemma~\ref{lem:Cauchy}.
The composites in the cocycle assertions solve the same final
equation with the same initial data, so uniqueness proves them.

For \eqref{eq:comparison-equivariance}, both sides solve the
transformed target equation by plateau covariance and have the
same Cauchy data, since the transformation is the identity
near \(\Sigma_0\). Apply uniqueness again.
\end{proof}

The comparison also transports the corresponding
\(\JJ_{g_1}^2\) to \(\JJ_{g_2}^2\), since it changes neither
the metric coefficients nor the leading scalar source.

For an observable \(B_1\) on \(\operatorname{Sol}_1\), its
transport is
\[
 \alpha_{21}^{\rm cl}B_1=B_1\circ r_{12}.
\]
This need not be the unchanged evaluation formula on
\(\operatorname{Sol}_2\). Proposition~\ref{prop:memory}
already detects that distinction.

The map is classical. It does not construct the quantum
algebraic cutoff comparison, which requires an admitted
product domain and additional locality hypotheses
\cite{BFR,DF}.

%=====================================================================
\section{An explicit finite table from actual coupling jets}
\label{sec:table}
%=====================================================================

We now make the finite encoding nonvacuous.
Specialize to \(m=0\), a unit-integral \(f\ge0\) supported in a
small cylindrical plateau \(U\), and
\[
 g(t,x)=q(t)\chi(x),
\]
where \(\chi=1\) near the spatial projection of \(\Dom_f\)
and \(q=1\) near the observation times.
All cutoffs are compactly supported within the laboratory.

Let \(\phi_0=t\), let \(k=2\sigma\eta\) be the tensor of
\eqref{eq:sourced-example}, and for real \(\lambda\) put
\[
 h_\lambda=2\lambda\eta.
\]
Here
\[
 T_{00}=\frac12,\qquad
 T_{ij}=\frac12\delta_{ij},\qquad
 \sigma=-\frac{5t^2+|x|^2}{24}.
\]
The metric coefficients satisfy
\eqref{eq:metric-equations} for every \(\lambda\):
\(h_\lambda\) is constant, and \(\Glin[k]=T^{(0)}[t]\).

Set
\[
 Q(t)=\int_0^tq(s)\,\dd s,\qquad
 J(t)=\int_0^tq(s)^2\,\dd s.
\]
For constant conformal \(h_\lambda\), direct substitution in
\eqref{eq:AB} gives
\[
 A(h_\lambda)=2\lambda\eta,\qquad B(h_\lambda)=0,
 \qquad w=0,\qquad v=\Ezero\mathcal C[k].
\]
On \(\Dom_f\), zero-data uniqueness gives
\[
 u_{1,g}=-2\lambda Q(t),\qquad
 u_{2,g}=4\lambda^2J(t)-\Ezero\Qg_{gk}^{(0)}t.
\]
For example, in the temporally homogeneous portion,
\(V_g[h_\lambda]u=-2\lambda(q\dot u)'\);
inserting \(u_{1,g}\) proves the \(4\lambda^2J\) term.
The cutoff outside \(\Dom_f\) cannot affect these evaluations.

Define the finite real calibration constants
\begin{align}
 b_1&=-2\int fQ(t)\,\dd^4x<0,\nonumber\\
 b_2&=4\int fJ(t)\,\dd^4x>0,\nonumber\\
 \beta&=\int f\left(
 -\Ezero\Qg_{gk}^{(0)}t-v^0
 \right)\dd^4x.
 \label{eq:calibrationconstants}
\end{align}
Thus
\begin{equation}
 C_1[g](h_\lambda,k;t,f)=\lambda b_1,\qquad
 C_2[g](h_\lambda,k;t,f)=\beta+\lambda^2b_2.
 \label{eq:pilotvalues}
\end{equation}
These are evaluations of actual members of the named coupling
jet, not merely off-shell placeholders.

For a larger-support generator
\(\xi_{\rm out}=\chi_{\rm out}\partial_t\), with
\(\chi_{\rm out}=1\) near \(\supp g\),
\[
 gK\xi_{\rm out}=0,\qquad
 \Delta_{\rm out}=-\int fX_{\xi_{\rm out}}t\,\dd^4x=-1.
 \label{eq:pilotgap}
\]
Choose also any nonzero \(\xi_{\rm in}\in\mathfrak g_U\).
The second-order covariance residual of its complete metric
pullback is exactly zero.

Use four samples \(\lambda_a=a\), \(a=0,1,2,3\), and four
settings \(s=0,1,2,3\):
\begin{equation}
 b_{sa}=
 \begin{cases}
 a\,b_1,&s=0 \quad\text{(first plateau coefficient)},\\
 \beta+a^2b_2,&s=1 \quad\text{(second plateau coefficient)},\\
 -1,&s=2 \quad\text{(larger-support first-order gap)},\\
 0,&s=3 \quad\text{(restricted second-order covariance residual)}.
 \end{cases}
 \label{eq:finite-table}
\end{equation}
Rows \(0,1\) are descended evaluations.
Row \(2\) records the failure of a different, larger-support
identification; it is not relabelled as a descended observable.
Row \(3\) records an identity already proved for the restricted
group.

\subsection{Finite sample logic}

If defined functions \(q,n_1,\ldots,n_L\) are sampled at
finitely many points, yielding a vector \(q_S\) and a column
matrix \(N_S\), then exactly one alternative holds:
\[
 q_S=N_Sc\text{ for some }c,
 \quad\text{or}\quad
 z^{\mathsf T}N_S=0,\quad z^{\mathsf T}q_S\ne0
 \text{ for some }z.
\]
To prove this, project \(q_S\) orthogonally onto \(\im N_S\).
If the residual is nonzero, use it as \(z\).

A finite fit is not a global identity. For example, \(q(x)=x\)
and \(n_1(x)=1\) fit at \(x=0\) but not globally.
Injective evaluation on the stated finite-dimensional function
span is sufficient to promote a fit to an identity, because it
annihilates no nonzero difference. It is not necessary:
\(q=1,n_1=1,n_2=x\) has a global fit even though evaluation
at \(0\) is noninjective.

These are finite-table distinctions comparable with II and II-B
\cite{CheungCertificates}. They do not provide a complete
quantum counterterm catalogue or a renormalization separator.

%=====================================================================
\section{Finite encoding and the destructive-clock instrument}
\label{sec:instrument}
%=====================================================================

Let the sample payload be two qubits,
\[
 \HH_X=\CC^4,\qquad
 \mathsf A_s=\sum_{a=0}^3b_{sa}\proj a_X.
\]
The sample basis labels classical entries of
\eqref{eq:finite-table}; it is not a one-particle basis of a
scalar or gravitational field.

Choose a declared real calibration
\[
 \Lambda>2\max_{s,a}|b_{sa}|.
\]
Define a binary table probe by
\[
 E_{s,y}=\frac12(I+y\mathsf A_s/\Lambda),\qquad
 M_{s,y}=E_{s,y}^{1/2},\qquad y=\pm1.
 \label{eq:effects}
\]
These satisfy
\begin{equation}
 \sum_yM_{s,y}^\dagger M_{s,y}=I,\qquad
 \sum_y\Lambda y E_{s,y}=\mathsf A_s.
 \label{eq:scoreidentity}
\end{equation}
On input \(\proj a\) at setting \(s\), the score
\(\Lambda y\) has mean \(b_{sa}\).

The finite constants are prescribed real metadata. Their
appearance in a finite model does not assert that arbitrary
real calibration data have been obtained exactly in finite
physical precision.

\subsection{Clock and complete recorded map}

Let the clock consist of two qubits,
\[
 \HH_C=\CC^4,\qquad
 \{\ket{\tau_s}\}_{s=0}^3\text{ orthonormal},\qquad
 P_s=\proj{\tau_s}.
\]
Apply a fixed basis change
\(B_C\ket{\tau_s}=\ket s\), then destructive computational
readout of both clock names. Put
\[
 D_s=\bra{\tau_s}_C\otimes I_X.
\]
The clock record is \(R\), and the score record is \(Y\).

\begin{theorem}[Complete destructive readout]
\label{thm:instrument}
The clock channel is
\begin{align}
 \Mclock(Z)
 &=\sum_s\proj s_R\otimes D_sZD_s^\dagger\nonumber\\
 &=\sum_s\proj s_R\otimes
 \Tr_C\bigl[(P_s\otimes I_X)Z(P_s\otimes I_X)\bigr].
 \label{eq:clock}
\end{align}
The clock-selected table instrument is
\begin{equation}
 \mathcal I(Z)=
 \sum_{s,y}\proj s_R\otimes\proj y_Y\otimes
 M_{s,y}D_sZD_s^\dagger M_{s,y}^\dagger.
 \label{eq:instrument}
\end{equation}
Both maps are completely positive and trace preserving,
including every untouched reference extension.

For \(\sigma_s=D_s\rho D_s^\dagger\) and
\(p_s=\Tr\sigma_s>0\),
\[
 \mathbb E[\Lambda y\mid s]
       =\frac{\Tr(\mathsf A_s\sigma_s)}{p_s}.
\]
The measured clock names are consumed. No quantum clock factor
is returned.
\end{theorem}

\begin{proof}
The clock Kraus operators are
\(\ket s_R\otimes D_s\), and their squared moduli sum to
\(\sum_sP_s\otimes I_X=I_{CX}\).
Expanding \(Z\) in the \(\tau\)-basis proves the partial-trace
expression.

For the full instrument use
\[
 L_{s,y}=\ket s_R\otimes\ket y_Y\otimes M_{s,y}D_s.
\]
Equation~\eqref{eq:scoreidentity} gives
\(\sum_{s,y}L_{s,y}^\dagger L_{s,y}=I_{CX}\).
These are operator identities, so tensoring with any reference
identity proves the complete statements.
The score formula follows from \eqref{eq:scoreidentity}.
The output interface contains \(R,Y,X\), not quantum \(C\).
\end{proof}

A finite probe realization is explicit. Allocate one fresh
qubit \(A\), apply the isometry
\[
 \psi\longmapsto
 \sum_{y=\pm1}\ket y_A\otimes M_{s,y}\psi,
\]
extend it to a unitary on the finite input space with the
specified blank ancilla, then destructively read \(A\).
Completeness proves that the displayed map is isometric.
Its unitary extension follows by completing orthonormal bases.

This is an abstract finite unitary permission, not a compiled
hardware circuit or a compact field--probe coupling.

\begin{proposition}[Clock limits and observation boundaries]
\label{prop:clocklimits}
The following hold:
\begin{enumerate}[label=\textup{(\roman*)}]
\item \(\Mclock\) has no universal left inverse.
\item Its record marginal alone does not certify the residual.
\item Discarding its clock record gives
\(\Tr_R\Mclock(Z)=\Tr_CZ\), including references.
\item No finite matrices obey \([T_C,H_C]=iI_C\).
\item A record-only channel followed by a preparation decoder
cannot recover every input of a quantum code of dimension
at least two.
\end{enumerate}
\end{proposition}

\begin{proof}
With the same payload, the two clock states
\[
 (\ket{\tau_0}+\ket{\tau_1})/\sqrt2,\qquad
 (\ket{\tau_0}-\ket{\tau_1})/\sqrt2
\]
have identical destructive outputs, proving (i).
Inputs \(P_0\otimes\rho_X\) and
\(P_0\otimes\rho_X'\) have the same deterministic record
and different residuals, proving (ii).
Summing diagonal clock blocks proves (iii).
Taking traces of the proposed commutator proves (iv).

For (v), a channel factoring through a classical register
has measure-and-prepare form
\[
 \rho\longmapsto\sum_j\Tr(E_j\rho)\tau_j.
\]
On one half of a maximally entangled code-reference state,
its output is separable. A separable state has overlap at most
\(1/d<1\) with that maximally entangled state: on a product
state the overlap is
\(\Tr(\sigma^{\mathsf T}\tau)/d\le1/d\), and the inequality
is preserved by convex combinations.
It cannot be the identity channel.
\end{proof}

Record agreement is weaker than residual-instrument agreement.
For example, identity and dephasing have the same one-outcome
record law but send \(\proj+\) to different states.
The complete instrument, not just its effects, is our
comparison target. These distinctions parallel VII-E, VII-G,
and En09-A \cite{CheungOperational,CheungEn09}.
No stationarity or clock-covariance hypothesis is imposed here;
En09-A's additional relational-evolution interpretation is not
claimed.

\subsection{Three declared dictionary maps}

Choose distinct geometric history times
\[
 \theta_0<\theta_1<\theta_2<\theta_3
\]
and a sender port \(x_A\in\mathcal V\). The dictionary is:
\begin{enumerate}[label=\textup{(\roman*)}]
\item \emph{Clock map:}
\(s\mapsto(\theta_s,\text{row }s\text{ of }\eqref{eq:finite-table})\),
with history event \(\kappa(s)=(\theta_s,x_A)\).
The readout event is specified independently.
\item \emph{Rod map:} each live carrier has an assigned
spatial port \(x(a)\), and rod coordinates \(\chi(x(a))\).
\item \emph{Causal map:} every complete scheduled dependency
edge, including classical control, is assigned future-causal
input and output ports in \((M,\bar g)\).
\end{enumerate}
These maps are supplied, not inferred from probabilities.

\begin{proposition}[Calibration and geometric nonuniqueness]
\label{prop:dictionary}
At a fixed event \(e\), exact event calibration
\[
 p_s>0\Longrightarrow\theta_s=T(e)
\]
with distinct labels forces one deterministic outcome, or
admits no normalized input if \(T(e)\) is not a label.

The abstract instrument does not uniquely determine its
history times, rod frame, or the global conical deficit.
\end{proposition}

\begin{proof}
Distinctness permits at most one supported index, and total
probability is one. If \(p_s=1\), positivity implies support
in \(P_s\otimes I_X\).

For nonuniqueness, retain every finite matrix and reassign
the history labels, or rotate and translate the local rod
frame with corresponding port coordinates.
Different prescribed deficits also admit isometric small
regular laboratories carrying the same finite instrument.
The finite record law is unchanged.
\end{proof}

A clock-basis change generally changes the clock observable;
it is not automatically a spacetime coordinate transformation.
Likewise, consuming the clock does not remove \(T\) or
\(\mathcal W\).

\subsection{A controlled finite-compression limit}

\begin{proposition}[Fixed-table approximation]
\label{prop:compressionlimit}
Suppose finite diagonal tables \(\mathsf A_s^{(n)}\) satisfy
\[
 \epsilon_n=\max_s
       \norm{\mathsf A_s^{(n)}-\mathsf A_s}\longrightarrow0.
\]
Use a common \(\Lambda\) such that every effect considered has
spectrum in \([1/4,3/4]\). The corresponding complete
instruments satisfy
\begin{equation}
 d_\diamond(\mathcal I_n,\mathcal I)
 \le\min\{1,4\epsilon_n/\Lambda\}\longrightarrow0,
 \qquad
 d_\diamond=\frac12\norm{\cdot}_\diamond.
 \label{eq:compression}
\end{equation}
Smooth convergence of the classical inputs on common compact
dependency neighborhoods gives such entrywise table convergence.
\end{proposition}

\begin{proof}
For diagonal positive matrices with spectra in
\([1/4,3/4]\), the scalar square-root identity gives
\[
 \norm{M_{s,y}^{(n)}-M_{s,y}}
       \le\frac{\epsilon_n}{2\Lambda}.
\]
For contraction Kraus operators,
\[
 \norm{\Ad_L-\Ad_{\widetilde L}}_\diamond
       \le2\norm{L-\widetilde L},
\]
by inserting and subtracting
\(LZ\widetilde L^\dagger\), also after a reference extension.
Sum over the eight recorded branches and divide by two to
obtain \eqref{eq:compression}.

The classical continuity assertion follows from
Lemma~\ref{lem:Cauchy}, finite differentiation, multiplication,
and compact integration in the explicit coefficient formulas.
\end{proof}

The norm and channel language are standard
\cite{Watrous}; the needed estimate has been proved here.
The limit concerns one fixed finite catalogue.
It is not a finite-dimensional one-particle space, a
finite-qubit field instrument, or a continuum-field limit.

No acquisition map from an unknown scalar--metric configuration
to \(\HH_X\) has been supplied.
The returned finite word does not determine an arbitrary real
coefficient exactly in one shot. Its scored mean realizes a
declared finite evaluation.
Local field measurements require a separately specified
coupling and dynamical comparison \cite{FV}.

%=====================================================================
\section{Schedule locality and causal certificate transport}
\label{sec:transport}
%=====================================================================

A complete finite schedule includes every quantum input,
classical control, record, guard, and routing decision actually
used. At each tick the complete channel factors over disjoint
declared blocks.

For terminal region \(Y\), define backward block sets by
\[
 N_n(Y)=Y,\qquad
 N_s(Y)=\bigcup\{B:
  B\text{ is a tick-}s\text{ block and }
                        B\cap N_{s+1}(Y)\ne\varnothing\}.
\]

\begin{theorem}[Complete schedule locality]
\label{thm:locality}
The schedule channel obeys
\[
 \Phi^*(\mathcal A(Y))\subseteq\mathcal A(N_0(Y)).
\]
Hence its output on \(Y\), together with any untouched
reference \(E\), depends only on the input reduction to
\(N_0(Y)E\).

If every scheduled dependency edge has future-causal ports in
\((M,\bar g)\), scheduled ancestors lie in the output's metric
past. Operational intervention influence is contained in
scheduled ancestry, but need not equal it.
\end{theorem}

\begin{proof}
On a block disjoint from the current backward support, the
observable is the identity. A trace-preserving channel's
adjoint fixes that identity. A block that meets the support
can enlarge it only within its declared union.
Backward induction proves the algebra inclusion.

Equal input reductions therefore give equal expectations for
every output observable, including referenced observables.
Causal transitivity gives metric containment along each
dependency path.

Replacing an operation outside the backward region acts only
on identity factors and cannot change the output.
Strictness is possible: a scheduled input to a replacement
channel is an ancestor even though that channel ignores it.
\end{proof}

Thus scheduled reachability, intervention influence, and the
supplied metric relation are three different objects. This is
the split compared with VII-J, En03-A, and En05-A
\cite{CheungLocality}. It is independent of the scalar
intertwiner proof.

\subsection{A target-adapted route}

Choose sender and receiver \(x_A,x_B\in\mathcal V\) with
\[
 L=\abs{\chi(x_B)-\chi(x_A)}>0,
\]
and a straight developed segment inside the regular convex
laboratory. Install sites
\[
 x_j=\chi^{-1}\left(
 \chi(x_A)+\frac jn(\chi(x_B)-\chi(x_A))\right),
 \qquad j=0,\ldots,n.
\]
Each site has a two-qubit payload register \(X_j\).
Initialize \(X_1,\ldots,X_n\) in a fixed pure blank \(\omega\).

At hop \(j\), transmit the contents of \(X_{j-1}\) to \(X_j\)
and the blank contents back to \(X_{j-1}\), giving the endpoint
SWAP channel. Allocate duration
\[
 \delta T=\frac{L}{nc}
\]
to that hop. Both directions have causal ports; equivalently,
this is an admitted finite communication block with all
input-output dependencies causally certified.

\begin{theorem}[Complete transport on the prescribed route]
\label{thm:transport}
After hop \(j\), every payload-reference input has state
\[
 \rho_{X_jE}\otimes
 \bigotimes_{\substack{0\le i\le n\\i\ne j}}\omega_{X_i}.
\]
The logical channel from \(X_0\) to \(X_n\) is the identity,
including all references. Arrival is at \(T_{\rm d}+L/c\).
All communication dependencies remain in the prescribed cone.
\end{theorem}

\begin{proof}
SWAP transfers the complete input factor and every one of
its reference correlations into the next independent blank.
Induction proves the displayed state identity.

Each hop has length \(L/n=c\delta T\).
The developed connecting paths are null, and convexity keeps
them inside the chart. Their pullbacks are therefore causal
in \((M,\bar g)\).
\end{proof}

Read the clock before departure. The transported payload may
then be correlated with both its classical code \(R\) and an
untouched reference \(E\).
The theorem applies with \(RE\) as the reference system.

To perform the code-selected score probe remotely, reserve
a two-bit classical register at each site and forward the code
along the same causal hops. Apply \(M_{s,y}\) only after both
payload and code arrive. Copying a diagonal classical code is
not cloning an unknown quantum state.

This composition equals \eqref{eq:instrument}, with the
payload relocated. It does not assume instantaneous remote
record control.

\begin{proposition}[No zero-duration remote SWAP]
\label{prop:instant}
A complete identity transmission or SWAP between distinct
spacelike ports cannot be an admitted zero-duration remote
operation in the causal contract.
\end{proposition}

\begin{proof}
Changing a sender input between two orthogonal states changes
the receiver output under identity transmission or SWAP.
There is operational influence. The causal contract requires
a future-causal dependency for that influence, whereas
distinct simultaneous spatial ports are spacelike.
\end{proof}

For \(n\) hops the laboratory resource ledger is
\[
\begin{array}{l|l}
\text{item}&\text{charge}\\
\hline
\text{payload storage}&n+1\text{ two-qubit registers}\\
\text{clock}&2\text{ qubits, then consumed}\\
\text{table probe}&1\text{ fresh qubit after clock consumption}\\
\text{payload communication}&n\text{ endpoint SWAPs}\\
\text{classical route storage}&2(n+1)\text{ bits}\\
\text{score record}&1\text{ bit}
\end{array}
\]
With all route sites reserved in advance,
\[
 Q_{\rm lab}\le2(n+1)+2,\qquad
 C_{\rm lab}\le2(n+1)+1.
\]
External reference storage is charged separately.

The exact theorem uses a target-adapted installed chain, not
exact null attainment on every endpoint of one fixed lattice.
It supplies no continuous relativistic Hamiltonian for a
spatially extended SWAP. Its model is a finite network of
admitted causal communication channels, not a continuum field
simulation.

For implemented compatible channels, replacing one factor at
a time and using complete trace-norm contraction gives
\[
 d_\diamond(\widetilde\Phi,\Phi)
 \le\min\left\{
 1,\epsilon_{\rm enc}+\sum_j\epsilon_j+\epsilon_{\rm dec}
 \right\}.
\]
This is a conditional error ledger, not reported experimental
performance or an executed compilation.

%=====================================================================
\section{Release and limits of the construction}
\label{sec:release}
%=====================================================================

\begin{theorem}[VII-A-ST local Grade S release]
\label{thm:release}
On the supported presentation and finite interface above:
\begin{enumerate}[label=\textup{(\roman*)}]
\item the specified action-level switch and unswitched harmonic
Cauchy prescription define unique scalar and coordinate
coefficients through second order;
\item the first-order plateau cancellation and second-order
intertwiner hold under \(\GG_U\), with the full transformation
of \(k\);
\item the larger compact class retains the exact defect
\eqref{eq:defect}, a negative witness, and the coefficientwise
no-higher-repair obstruction;
\item the named plateau coupling jet is stable and has an
explicit nonempty sourced subclass;
\item metric and switch dependence obey the truncated-past
bound; causal switch agreement gives equality, while a common
observation plateau alone does not;
\item the Cauchy-data-preserving classical comparison map is
unique and satisfies its cocycle and common-plateau equivariance;
\item the finite evaluation table has a complete destructive
instrument, declared dictionaries, a controlled compression
limit, and reference-preserving causal transport;
\item no quantum product, quantum Ward identity, radiative
coefficient, or counterterm-separation certificate is supplied.
\end{enumerate}
Items (i)--(vii) satisfy Definition~\ref{def:grade}.
The repair remains plateau-restricted.
\end{theorem}

\begin{proof}
Propositions~\ref{prop:scalar} and \ref{prop:coordinates}
give (i). Theorem~\ref{thm:first},
Lemma~\ref{lem:plateaucov}, and
Theorem~\ref{thm:second} give (ii).

Propositions~\ref{prop:defect}, \ref{prop:witness}, and
Corollary~\ref{cor:nohigherrepair} give (iii).
Propositions~\ref{prop:jetstable} and \ref{prop:nonempty}
give (iv).

Propositions~\ref{prop:support}, \ref{prop:cutoffequal},
and \ref{prop:memory} prove (v).
Proposition~\ref{prop:classicalcomparison} proves (vi).
Sections~\ref{sec:table}--\ref{sec:transport} prove (vii).

Every field-theoretic construction used in these proofs is a
classical smooth coefficient construction. The additional
quantum objects named in (viii) have not been defined or
evaluated here. This is a release limit, not a theorem excluding
all future constructions.
\end{proof}

\subsection{What is added, and what is not a novelty claim}

The classical plateau intertwiner and larger-support defect
are those of VII-A \cite{CheungVIIA}.
The additional contribution is their supported conical
realization, together with:
\begin{itemize}
\item a domain on which no global conical Cauchy premise is
needed;
\item an explicit separation of the apparatus metric from
the formal metric input;
\item a sourced finite table with nontrivial first- and
second-order entries;
\item a complete finite readout and a quantitative
instrument-level approximation;
\item a causal route carrying both the payload and its
classical control record.
\end{itemize}
The possible usefulness is that a correct restricted identity
and a surviving unrestricted failure can be carried through
one operational interface without confusing either with a
quantum anomaly or a field measurement.

\subsection{Quantum and microlocal boundaries}

A classical density identity is not a quantum Master Ward
identity. It neither supplies time-ordered products nor
classifies their renormalization freedom. The general
retarded-product and locally covariant frameworks retain their
own admission hypotheses \cite{DF,BFR}.

For a smooth multiplier \(g\),
\[
 \WF(gu)\subseteq\WF(u),
\]
with local equality where \(g\ne0\), by multiplication with
the smooth reciprocal there. That elementary observation
does not construct the retarded quantum coefficient or remove
the finite-initial-surface singularities discussed in the
comparison records \cite{CheungPrepared,CheungProductTracks}.

In particular:
\begin{itemize}
\item VII-B, VII-C, VII-D, and VII-I concern separate sharp
admission, product, finite-part, and finite-width questions.
The present intertwiner does not select their product domains,
remove a logarithmic triple, or turn a finite part into an
algebra.
\item VII-H's free susceptibility is not \(C_2[g]\) and is not
the inherited characteristic coefficient.
\item IX-A through IX-E, X-G, and XI-G address different
carriers, variational questions, or free-shell classes.
No missing nonlocal one-loop product is defined here.
\end{itemize}

The quantum certificate therefore remains \textbf{RG-open}.
The finite table in Section~\ref{sec:table} is not an evaluated
quantum response matrix. Finite linear algebra cannot define a
missing distributional product.

\subsection{Named non-identifications}

The following documentary separations are retained
\cite{CheungComparisons}:
\begin{itemize}
\item III's four equation spaces and its unconstructed sourced
successor are not identified with \(\JJ_g^2\).
The retained metric equations here are unswitched.
\item VII-F's three-to-six particle slicing event is not this
plateau and is not a dynamical source. The defect of our
background is the worldsheet \(\mathcal W\).
\item VIII-A through VIII-C do not select our prescribed
deficit. Symmetry, conservation, and zero trace are not a
derivation of Einstein dynamics.
\item Em02-E, XX-A, and XIX-A keep a relational lapse,
harmonic-coordinate coefficients, clock prescriptions, and
event-time labels distinct.
\item En08-A's scheduler consistency does not select the cone.
En08-B's null-boundary approximation is not the exact installed
chain proved above.
\item IV and IV-B's represented readouts have their own
completion and apparatus permissions. A timestamp is not a
compact causal realization of a global apparatus, and a Ramsey
contrast is not automatically a field measurement.
\item Oai282-A's finite local Ward residual uses its own
improvement data. It is not this classical intertwiner and is
not a gravitational quantum Ward identity.
\item The OpenAI catalogue is not a premise or a verification
of this construction. No family supplies the cone, plateau,
or clock. A static area law is not this intertwiner or a
Page curve.
\item The Hawk, Psy, and Em packet sheets named in the supplied
dossier are not identified with this plateau scalar.
\end{itemize}

\subsection{Analytical stops and standing refusals}

We do not alter the historical H2 verdict of XIV-H.
The narrower internal amplitude constructed at I2 in XIV-I
must nevertheless be acknowledged: it is not withdrawn by
calling the full characteristic programme unresolved.
That internal amplitude is not the missing full joint map
at I3/J2 \cite{CheungAnalytic}.

No result here closes that joint map, its remainder and
extension obligations, or its quantum coefficient admission.
``XV-B is not opened'' means that the downstream quantum
admission exit is not activated; it does not deny the existence
of a technical manuscript with that identifier.

XVI-N is not discharged. Within the XVI-O prescription and the
scope of this release,
\[
 \mathrm{O.1}=\loc{NOT_CLOSED}.
\]
Separately scoped subsector reports with enlarged prescriptions
do not update that statement or admit the inherited insertion
\cite{CheungStops}.

The construction is not spacetime emergence, Grade E, a global
conical Cauchy theorem, an intrinsic cone field theory,
Einstein dynamics derived from records, quantized relational
time without external scheduling, a Page curve, an island, an
interior, an ultraviolet completion, or an interacting
Hilbert-space construction.

The clock encoding is not the geometric time axis.
A complete channel does not specify work, heat, a bath, reset
cost, or the physical fate of all implementation degrees of
freedom. No quantum field coefficient is admitted and no
compilation is asserted.

%=====================================================================
\appendix
\section{Direct second-order cancellation}
\label{app:direct}
%=====================================================================

Write
\[
 d^\mu=w^\nu\partial_\nu w^\mu-v^\mu.
\]
From \eqref{eq:intertwiner-coeff},
\[
 d'^\mu=d^\mu+w^\nu\partial_\nu\xi^\mu
                 +\frac12\xi^\nu\partial_\nu\xi^\mu-\zeta^\mu.
\]
Substitute this, \(w'=w+\xi\), and the scalar transformations
in \eqref{eq:A12}.

Terms differentiating \(u_{1,g}\) cancel between
\(X_\xi u_{1,g}\) and the changed transport term.
Terms proportional to \(\partial_\mu\phi_0\) have coefficient
\[
 \zeta^\mu+\frac12\xi^\nu\partial_\nu\xi^\mu
 -(w^\nu+\xi^\nu)\partial_\nu\xi^\mu
 +w^\nu\partial_\nu\xi^\mu
 +\frac12\xi^\nu\partial_\nu\xi^\mu-\zeta^\mu=0.
\]
The remaining Hessian coefficient is the symmetric contraction
of
\[
 \frac12\xi^\mu\xi^\nu
 -(w^\mu+\xi^\mu)\xi^\nu
 +\frac12\left(
 w^\mu\xi^\nu+\xi^\mu w^\nu+\xi^\mu\xi^\nu
 \right),
\]
which vanishes against
\(\partial_\mu\partial_\nu\phi_0\).
Thus \(A_2^{(g)\prime}=A_2^{(g)}\) directly.

This check requires both the inverse-coordinate displacement
\(w\cdot\partial w-v\) and the quadratic Taylor term.
Omitting either would invalidate the cancellation.

%=====================================================================
\section{Explicit Einstein-tensor coefficient convention}
\label{app:curvature}
%=====================================================================

For a symmetric \(h\), set
\[
 H_{\mu\nu\sigma}
 =\partial_\mu h_{\nu\sigma}
  +\partial_\nu h_{\mu\sigma}
  -\partial_\sigma h_{\mu\nu},
\]
\[
 \Gamma_1^\rho{}_{\mu\nu}
       =\frac12\eta^{\rho\sigma}H_{\mu\nu\sigma},
 \qquad
 \Gamma_2^\rho{}_{\mu\nu}
       =-\frac12h^{\rho\sigma}H_{\mu\nu\sigma}.
\]
Define
\[
 R_{1,\mu\nu}
 =\partial_\rho\Gamma_1^\rho{}_{\mu\nu}
            -\partial_\nu\Gamma_1^\rho{}_{\mu\rho},
\]
\begin{align*}
 R_{2,\mu\nu}
 ={}&\partial_\rho\Gamma_2^\rho{}_{\mu\nu}
       -\partial_\nu\Gamma_2^\rho{}_{\mu\rho}\\
 &+\Gamma_1^\rho{}_{\rho\lambda}
                     \Gamma_1^\lambda{}_{\mu\nu}
       -\Gamma_1^\rho{}_{\nu\lambda}
                     \Gamma_1^\lambda{}_{\mu\rho}.
\end{align*}
Then
\[
 \Glin[h]_{\mu\nu}
 =R_{1,\mu\nu}
             -\frac12\eta_{\mu\nu}\eta^{ab}R_{1,ab},
\]
\begin{align*}
 G^{(2)}[h,h]_{\mu\nu}
 ={}&R_{2,\mu\nu}
 -\frac12\eta_{\mu\nu}
        \left(\eta^{ab}R_{2,ab}-h^{ab}R_{1,ab}\right)\\
 &-\frac12h_{\mu\nu}\eta^{ab}R_{1,ab}.
\end{align*}
These are obtained by expanding the Christoffel and Ricci
formulas with the displayed Ricci convention.
Terms linear in \(k\) give \(\Glin[k]\).

As a direct check,
\[
 \Glin[2\sigma\eta]_{\mu\nu}
       =-2\partial_\mu\partial_\nu\sigma
                               +2\eta_{\mu\nu}\Box_\eta\sigma.
\]
Also,
\[
 \partial^\mu T^{(0)}_{\mu\nu}[\phi_0]
       =(P_m\phi_0)\partial_\nu\phi_0=0.
\]
This necessary conservation law does not imply that arbitrary
zero metric Cauchy data satisfy the gravitational constraints.

%=====================================================================
\section{Prescription and scope audit}
%=====================================================================

\paragraph{Initial data.}
Every use of \(\Em P_m=\id\) in the proof concerns a function
with zero Cauchy data, usually one vanishing near \(\Sigma_0\).
For unspecified data the correct identity is
\eqref{eq:homogeneous-subtraction}.

\paragraph{Subtraction.}
A chartwise retarded subtraction modifies the specified
prescription. It does not become a tensorial operation on the
whole cone by being written in a developing chart. In
particular,
\[
 [P_m,X_\xi]\ne0
\]
for general compact \(\xi\); a fixed inverse need not commute
with an arbitrary pullback. The plateau proof supplies a
specific intertwining construction, not an unrestricted
commutation rule.

\paragraph{Support.}
The proven bound is containment in \(\Dom_f\), not equality
of functional support with \(\Dom_f\). Special inputs and
cancellations can shrink the actual support.
Nor can this bound generally be replaced by \(\supp f\), as
Proposition~\ref{prop:memory} demonstrates.

\paragraph{Gauge domain.}
The data defining \(g,U,\Sigma_0\), and the harmonic
prescription are fixed. Restricting to \(\GG_U\) changes the
admitted identification; it does not repair the old larger
identification without changing its domain.

\paragraph{Classical comparison.}
The comparison cocycle is unique on the specified solution
spaces because the Cauchy problem is unique. It is not a
statement that every off-shell extension of that map is unique,
or that unchanged observable formulas agree at different
switches.

\paragraph{Finite encoding.}
The encoding is defined after the finite classical evaluations
are supplied. A coherent payload state is not declared to be a
superposition of continuum solution jets.
No source-acquisition channel is hidden in its preparation.

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

\paragraph{Authorship.}
The author field is intentionally empty for anonymous review.
Authorized author and affiliation metadata are required for a
signed submission. Registry attribution to N.~Cheung and
citation of Cheung's articles do not assign authorship of this
adaptation.

\paragraph{Provenance.}
VII-A, version 1, supplies the source classical construction.
I-B supplies the documentary spacetime-interface pattern.
The mathematical assertions released here are supported by
the explicit local proofs above. Companion references are
comparisons and source records, not unprinted theorem premises.

\paragraph{Verification and implementation.}
No successful \texttt{pdflatex} execution, proof-assistant
verification, independently executed symbolic or numerical
calculation, physical circuit compilation, experiment,
independent referee certification, or journal acceptance of
this adaptation is asserted.

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

\bibitem{CheungVIIA}
N.~Cheung,
\emph{Plateau-compatible relational scalars:
a second-order classical intertwiner. Landmark VII-A}.
\loc{VII-A.tex}, version 1, Enter Calculus Companion,
unpublished manuscript.
The registered source has an empty author field; attribution
here follows the companion record.
Relevant locators include \loc{thm:first}, \loc{prop:defect},
\loc{prop:witness}, \loc{lem:plateaucov}, \loc{thm:second},
\loc{def:jet}, \loc{prop:support},
and \loc{prop:classicalcomparison}.
\url{https://enter.grok.me/browse/VII-A}.

\bibitem{CheungIB}
N.~Cheung,
\emph{I-B-v1: A Finite Enter Spacetime:
Calibrated Destructive Time Readout on a Prescribed
Conical Product}.
\loc{I-B-v1.tex}, version 1, Enter Calculus Companion,
unpublished manuscript.
\url{https://enter.grok.me/browse/I-B}.

\bibitem{CheungVI}
N.~Cheung,
\emph{VI}, \loc{VI.tex}, version 1.
Enter Calculus Companion, supplied unpublished manuscript.
Second-order classical descent and the compact-switch
obstruction are separate conclusions.
\url{https://enter.grok.me/browse/VI}.

\bibitem{CheungEnter}
N.~Cheung,
``The Enter Calculus: A Linear Quantum Lambda Calculus with
an Operational Characterization of Measurement,''
\emph{Frontiers in Physics} 14 (2026), article 1954943.
Accepted 28 September 2026; the consulted publisher record
identifies final formatting as forthcoming.
DOI: \doi{10.3389/fphy.2026.1954943}.

\bibitem{CheungI}
N.~Cheung,
\emph{I}, \loc{I.tex}, version 2.
Enter Calculus Companion, supplied unpublished manuscript.
Finite measurement characterization inside its declared class.
\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 essay, Enter Calculus Companion.
\url{https://enter.grok.me/enter}.

\bibitem{BFHPR}
R.~Brunetti, K.~Fredenhagen, T.-P.~Hack, N.~Pinamonti,
and K.~Rejzner,
``Cosmological perturbation theory and quantum gravity,''
\emph{Journal of High Energy Physics} 08 (2016), 032.
DOI: \doi{10.1007/JHEP08(2016)032}.
\url{https://arxiv.org/abs/1605.02573}.

\bibitem{Frob}
M.~B.~Fr\"ob,
``Gauge-invariant quantum gravitational corrections to
correlation functions,''
\emph{Classical and Quantum Gravity} 35 (2018), 055006.
DOI: \doi{10.1088/1361-6382/aaa74c}.
\url{https://arxiv.org/abs/1710.00839}.

\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}.

\bibitem{BFR}
R.~Brunetti, K.~Fredenhagen, and K.~Rejzner,
``Quantum gravity from the point of view of locally covariant
quantum field theory,''
\emph{Communications in Mathematical Physics} 345 (2016),
741--779.
DOI: \doi{10.1007/s00220-016-2676-x}.
\url{https://arxiv.org/abs/1306.1058}.

\bibitem{DF}
M.~D\"utsch and K.~Fredenhagen,
``Causal perturbation theory in terms of retarded products,
and a proof of the Action Ward Identity,''
\emph{Reviews in Mathematical Physics} 16 (2004), 1291--1348.
DOI: \doi{10.1142/S0129055X04002266}.
\url{https://arxiv.org/abs/hep-th/0403213}.

\bibitem{FV}
C.~J.~Fewster and R.~Verch,
``Quantum fields and local measurements,''
\emph{Communications in Mathematical Physics} 378 (2020),
851--889.
DOI: \doi{10.1007/s00220-020-03800-6}.
\url{https://arxiv.org/abs/1810.06512}.

\bibitem{Watrous}
J.~Watrous,
\emph{The Theory of Quantum Information}.
Cambridge University Press, 2018.
Author's book interface:
\url{https://cs.uwaterloo.ca/~watrous/TQI/}.

\bibitem{CheungGradeS}
N.~Cheung,
\emph{III-B}, \loc{III-B-v1.tex}, version 1; and
\emph{V-B}, \loc{V-B-v1.tex}, version 1.
Enter Calculus Companion, supplied unpublished source records.
Supported-presentation and local Grade S comparisons.

\bibitem{CheungCertificates}
N.~Cheung,
\emph{II}, \loc{II.tex}, version 2; and
\emph{II-B}, \loc{II-B-v1.tex}, version 1.
Enter Calculus Companion, supplied unpublished manuscripts.
Finite certificate logic; no response catalogue is transferred.

\bibitem{CheungEn09}
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, supplied unpublished manuscript.
Complete conditioning and separately stated dynamical
covariance hypotheses.
\url{https://enter.grok.me/browse/En09-A}.

\bibitem{CheungOperational}
N.~Cheung,
\emph{VII-E}, \emph{VII-G}, and
\emph{VIII-A}, version 3.
Enter Calculus Companion, supplied unpublished source records.
Record, residual-instrument, and observation-boundary
comparisons only.

\bibitem{CheungLocality}
N.~Cheung,
\emph{VII-J}, \loc{VII-J.tex}, version 1;
\emph{En03-A}, \loc{En03-A.tex}, version 1; and
\emph{En05-A}, \loc{En05-A.tex}, version 1.
Enter Calculus Companion, supplied unpublished source records.
Schedule locality, physical containment, and complete-channel
transport contracts.

\bibitem{CheungPrepared}
N.~Cheung,
\emph{XVI-A}, \loc{XVI-A.tex}, version 1; and
\emph{XVI-B}, \loc{XVI-B.tex}, version 1.
Enter Calculus Companion, supplied unpublished manuscripts.
Conditional wavefront interfaces and prepared-source memory;
their witnesses and retarded targets are not imported.

\bibitem{CheungProductTracks}
N.~Cheung,
\emph{VII-B}, \emph{VII-C}, \emph{VII-D},
\emph{VII-H}, \emph{VII-I};
\emph{IX-A} through \emph{IX-E};
\emph{X-G}; and \emph{XI-G}.
Enter Calculus Companion, version-specific unpublished
source records supplied with the problem.
These are separately scoped admission, product, susceptibility,
variational, and free-shell questions.

\bibitem{CheungComparisons}
N.~Cheung,
\emph{III}, version 2;
\emph{IV} and \emph{IV-B};
\emph{VII-F};
\emph{VIII-A} through \emph{VIII-C}, version 3;
\emph{Em02-E}, \emph{XX-A}, \emph{XIX-A};
\emph{En08-A}, \emph{En08-B};
\emph{Oai282-A}, and the named Hawk, Psy, and Em packet sheets.
Enter Calculus Companion, supplied unpublished documentary
records. Grouping is bibliographic only and asserts no
common model or transferred theorem.

The separately supplied OpenAI catalogue is attributed to
OpenAI, not to N.~Cheung. It is not a premise and no family
proof or verification status is imported.

\bibitem{CheungAnalytic}
N.~Cheung,
\emph{XIV-H}, version 1;
\emph{XIV-I}, version 1;
\emph{XIV-J}, version 1; and technical
\emph{XV-B}, version 2.
Enter Calculus Companion, supplied unpublished manuscripts.
Historical H2, constructed internal I2, unresolved full
I3/J2, and the relative-lifting interfaces are distinct records.

\bibitem{CheungStops}
N.~Cheung,
\emph{XVI-N}, revision r12;
\emph{XVI-O}, revision o70-sync; and
\emph{XVI-Q}, release Q11.
Enter Calculus Companion, supplied unpublished source records.
The broader standing insertion and XVI-O boundaries are not
updated by separately scoped subsector statements.

\end{thebibliography}

\end{document}