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\title{One declared classical-limit package for a relational observable,
and the stops it does not close\\[4pt]
\large Em06-A: a scalar-only coherent limit of the first plateau coefficient}
\author{}
\date{}

\begin{document}
\maketitle

\begin{abstract}
We construct a controlled classical limit for one precisely weakened
relational observable: the first plateau-compatible scalar coefficient
with a prescribed classical metric coefficient and only the free scalar
quantized. The classical functional, its restricted gauge domain, and
its dependence support are specified and proved before quantization.
An exact compact scalar test represents the functional. For a declared
family of scaled canonical-commutation-relation representations and
coherent states, its mean equals the named classical value for every
positive classicality parameter, while its variance is exactly that
parameter times a finite one-particle norm. Its spectral law converges
to the classical point mass in the quadratic Wasserstein metric.
A nonzero example and a preparation-response diagnostic are evaluated.

The outcome is a repaired, scalar-only classical-limit theorem, not
quantization of the metric-dependent insertion as a joint gravitational
functional. The metric jet, Minkowski causal structure, scalar equation,
switch, initial surface, and preparation are inputs. No metric
reconstruction, Einstein dynamics, interacting gravitational product,
holographic map, or inherited quantum-response coefficient is obtained.
The separate product and completion stops remain untouched.
\end{abstract}

\section{Question, outcome, and evidence boundary}
\label{sec:scope}

The object studied here is the first-order plateau relational
coefficient associated with the construction described in
Cheung's Manuscript VII-A, specifically its
\texttt{thm:first} \cite{CheungVIIA}. We retain its first-order
architecture: a compactly switched scalar response, unswitched
zero-Cauchy harmonic coordinates, and gauge parameters supported
inside the switch plateau and away from the initial surface.
All definitions and the required classical identity are reproduced
and proved below.

The quantum question is deliberately narrower. The metric coefficient
is fixed as an external classical parameter. Only the scalar is
quantized, and only the first relational coefficient is represented.
This restriction is substantial: the joint metric--scalar insertion
problem of VII-B is not the scalar-only problem constructed here
\cite{CheungVIIB}. The distinction between external-metric,
finite-mode metric, and composed stress-response objects is also
explicit in Manuscript V \cite{CheungV}.

\paragraph{Declared outcome.}
In the terminology of the Em06 prospectus \cite{CheungDossier},
the result is a \emph{repaired limit}. The exact change from the
larger proposed quantum target is:
\begin{enumerate}
\item the metric coefficient is classical, not a quantum field;
\item the observable is the first plateau coefficient, not the full
nonlinear relational scalar or its second-order completion;
\item the quantum carrier is a free scalar Weyl representation,
not an interacting gravitational polynomial algebra;
\item the limit varies the effective CCR parameter, not a switch,
boundary regulator, or ultraviolet cutoff.
\end{enumerate}
These restrictions are part of the theorem, not conclusions about
the unrestricted problem.

The coherent-state concentration mechanism is standard
\cite{Glauber,Hepp}. The contribution here is its explicit application
to a named first-order relational coefficient, including the density
operator, sign audit, compact test, restricted gauge identification,
nontrivial example, and separate fluctuation estimate.

\paragraph{Three different claims.}
The construction attempts only operator--symbol correspondence in
this scalar-only model:
\[
\begin{array}{ll}
\text{Operator reconstruction:}
 & \text{proved for the named scalar-only coefficient;}\\
\text{Metric reconstruction:}
 & \text{not attempted;}\\
\text{Derivation of Einstein dynamics:}
 & \text{not attempted.}
\end{array}
\]
Here ``operator reconstruction'' does not mean boundary-to-bulk
reconstruction. Holographic entropy and entanglement-wedge
reconstruction require additional, different structures
\cite{RT,DHW}; none is an input of this proof.

\paragraph{Source discipline.}
Cheung references are identified by the versions and labels supplied
with the problem. No conclusion about an unexamined later version
is asserted. Objects described only in the supplied prospectus,
including the VII-H comparison, are cited as documentary comparison
material, not used as theorem premises. The mathematical proof below
does not assume registration, compilation, machine verification,
numerical execution, calibration, or experimental realization.

\section{The classical object before the limit}
\label{sec:classical}

\subsection{Background, scalar equation, switch, and gauge domain}

\begin{specification}[Fixed classical data]
\label{spec:data}
Fix
\[
M=\R^{1+3},\qquad
\eta=\operatorname{diag}(1,-1,-1,-1),\qquad
\Sigma_0=\{t=0\},
\]
and a mass \(m>0\). Set
\[
\Box=\partial_t^2-\Delta,\qquad P_m=\Box+m^2.
\]
Fix \(T>0\), a real switch
\[
\chi\in C_c^\infty((0,T)\times\R^3),\qquad 0\leq\chi\leq1,
\]
and an open set \(U\) with compact closure in
\((0,T)\times\R^3\), such that \(\chi=1\) on a neighborhood of
\(\overline U\).

The admitted gauge parameters are exactly
\[
\mathfrak g_U=C_c^\infty(U;TM).
\]
For \(\xi\in\mathfrak g_U\), define
\[
(K\xi)_{\mu\nu}=\partial_\mu\xi_\nu+\partial_\nu\xi_\mu,
\qquad X_\xi=\xi^\mu\partial_\mu.
\]
The metric coefficient \(h\) is an arbitrary real smooth symmetric
covariant tensor on \(M\). The scalar input \(\phi_0\) is a real smooth
solution of \(P_m\phi_0=0\). The observation density is
\[
f\in C_c^\infty((0,T)\times\R^3;\R).
\]
All these data are independent of the classicality parameter introduced
later.
\end{specification}

Indices are raised and lowered with \(\eta\). Write
\[
\tau(h)=\eta^{\mu\nu}h_{\mu\nu},\qquad
a^{\mu\nu}(h)=\frac12\tau(h)\eta^{\mu\nu}-h^{\mu\nu},
\qquad b(h)=\frac12\tau(h),
\]
and
\begin{equation}
C(h)^\nu=\partial_\mu h^{\mu\nu}
              -\frac12\partial^\nu\tau(h).
\label{eq:C}
\end{equation}

The scalar density perturbation is the differential operator
\begin{equation}
Q_h v=\partial_\mu\bigl(a^{\mu\nu}(h)\partial_\nu v\bigr)
                         +m^2b(h)v.
\label{eq:Q}
\end{equation}
In particular, \(Q_{\chi h}\), not \(\chi Q_h\), is the switched
operator. Derivatives of the switch are retained.

For orientation, this is the coefficient in the Euler equation of the
declared first-order scalar action
\begin{equation}
S_\chi[\phi;h]
=
\frac12\int_M
  \bigl(\eta^{\mu\nu}\partial_\mu\phi\partial_\nu\phi-m^2\phi^2\bigr)
  \dd^4x
+
\frac{\kappa}{2}\int_M\chi
 \bigl(a^{\mu\nu}(h)\partial_\mu\phi\partial_\nu\phi-m^2b(h)\phi^2\bigr)
 \dd^4x.
\label{eq:action}
\end{equation}
Its equation is \(P_m\phi+\kappa Q_{\chi h}\phi=0\).
The parameter \(\kappa\) labels the classical metric jet; it is not the
classicality parameter.

\subsection{Green operators and the harmonic coordinate coefficient}

Let \(G_m^{\mathrm R}\) and \(G_m^{\mathrm A}\) be the retarded and
advanced Green operators of \(P_m\). For compact smooth sources,
\[
P_mG_m^{\mathrm R}v=P_mG_m^{\mathrm A}v=v,
\qquad
(G_m^{\mathrm R})^*=G_m^{\mathrm A}.
\]
With \(D_m=\sqrt{-\Delta+m^2}\), the chosen signs are
\begin{align}
(G_m^{\mathrm R}v)(t)
 &=\int_{-\infty}^{t}
       \frac{\sin((t-s)D_m)}{D_m}v(s)\dd s,
\label{eq:retarded}\\
(G_m^{\mathrm A}v)(t)
 &=\int_t^\infty
       \frac{\sin((s-t)D_m)}{D_m}v(s)\dd s.
\label{eq:advanced}
\end{align}
These formulas give the Green identities and finite propagation.
They also fix the signs independently of differently normalized
Green operators in comparison manuscripts. Standard general
Cauchy and Green-operator results may be found in \cite{BGP}.

Define \(w[h]\), for \(t\geq0\), by the componentwise massless Cauchy
problem
\begin{equation}
\Box w[h]^\mu=C(h)^\mu,\qquad
w[h]^\mu|_{\Sigma_0}=0,\qquad
\partial_t w[h]^\mu|_{\Sigma_0}=0.
\label{eq:w}
\end{equation}
This problem has a unique smooth solution on every finite time slab.
For arbitrary smooth \(h\), it is understood locally through finite
propagation; no spatial Fourier transform of an arbitrary
nondecaying \(h\) is required.

The formal metric is \(g_\kappa=\eta+\kappa h\). Indeed,
\[
\left.\partial_\kappa\Box_{g_\kappa}v\right|_{\kappa=0}
=-h^{\mu\nu}\partial_\mu\partial_\nu v-C(h)^\nu\partial_\nu v.
\]
Consequently, \(X_\kappa^\mu=x^\mu+\kappa w[h]^\mu\) obeys the
harmonic coordinate equation through first order. Its inverse has
first coefficient \(-w[h]\).

The switched first scalar coefficient is
\begin{equation}
\phi_{1,\chi}[h;\phi_0]
=-G_m^{\mathrm R}Q_{\chi h}\phi_0.
\label{eq:phi1}
\end{equation}
Its source is compact and lies strictly to the future of \(\Sigma_0\),
so it has zero Cauchy data there.

\begin{definition}[The named observable]
\label{def:observable}
The observable of Em06-A is exactly
\begin{equation}
\boxed{
\Obs_f[h;\phi_0]
=
\int_M f(x)
 \left(\phi_{1,\chi}[h;\phi_0](x)
          -w[h]^\mu(x)\partial_\mu\phi_0(x)\right)\dd^4x.
}
\label{eq:observable}
\end{equation}
It is the first coefficient of the relational scalar obtained by
composing the switched scalar solution with the inverse harmonic
coordinate map and integrating against the fixed density in
relational-coordinate space.
\end{definition}

This definition names a coefficient, not an approximation whose
uncontrolled remainder is silently discarded in the classical limit.
The theorem below concerns \eqref{eq:observable} itself. It asserts
no \(O(\kappa^2)\) estimate for a full finite-\(\kappa\) observable.

\begin{remark}[What the background notation means]
\label{rem:background}
Later \(g_{\cl}\) denotes the prescribed metric one-jet
\[
g_{\cl}=(\eta,h_{\cl}).
\]
The scalar preparation must also be specified: a relational scalar
coefficient is not a functional of the metric alone. We therefore
write the classical target honestly as
\[
\Obs_f[g_{\cl};\phi_{\cl}]
:=\Obs_f[h_{\cl};\phi_{\cl}].
\]
If \(\Obs_f[g_{\cl}]\) is used as shorthand, the fixed scalar
preparation, switch, and initial surface are suppressed inputs.
None is reconstructed from quantum data.
\end{remark}

\section{Classical gauge legitimacy and support}
\label{sec:gauge}

\begin{lemma}[Density and pure-gauge identities]
\label{lem:gaugeidentities}
For every smooth \(\xi\),
\begin{align}
C(K\xi)&=\Box\xi,
\label{eq:CK}\\
Q_{K\xi}&=(\partial_\mu\xi^\mu)P_m-[P_m,X_\xi].
\label{eq:QK}
\end{align}
Thus, on a free scalar solution,
\[
Q_{K\xi}\phi_0=-P_m(X_\xi\phi_0).
\]
Moreover, \(Q_h\) is formally self-adjoint for real \(h\).
\end{lemma}

\begin{proof}
The trace of \(K\xi\) is \(2\partial_\mu\xi^\mu\). Differentiating
\eqref{eq:C} gives \eqref{eq:CK}.

Expanding \eqref{eq:Q} gives
\[
Q_h=-h^{\mu\nu}\partial_\mu\partial_\nu
          -C(h)^\nu\partial_\nu+\frac12\tau(h)P_m.
\]
Also,
\[
[P_m,X_\xi]
=(\Box\xi^\nu)\partial_\nu
       +2(\partial^\mu\xi^\nu)\partial_\mu\partial_\nu.
\]
Substitution proves \eqref{eq:QK}. The on-shell identity follows
from \(P_m\phi_0=0\). Formal self-adjointness follows by integration
by parts in the divergence-form expression \eqref{eq:Q}.
\end{proof}

\begin{proposition}[Exact first-order plateau invariance]
\label{prop:plateau}
For every datum in Specification~\ref{spec:data} and every
\(\xi\in\mathfrak g_U\),
\begin{equation}
\Obs_f[h+K\xi;\phi_0]=\Obs_f[h;\phi_0].
\label{eq:plateau}
\end{equation}
This is invariance under the declared additive first-order gauge
action, not unrestricted diffeomorphism invariance.
\end{proposition}

\begin{proof}
Since all derivatives of \(\xi\) are supported where \(\chi=1\),
\(\chi K\xi=K\xi\). Lemma~\ref{lem:gaugeidentities} gives
\[
\phi_{1,\chi}[h+K\xi;\phi_0]-\phi_{1,\chi}[h;\phi_0]
=
G_m^{\mathrm R}P_m(X_\xi\phi_0)
=X_\xi\phi_0.
\]
The last equality holds because \(X_\xi\phi_0\) is compactly
supported and vanishes near \(\Sigma_0\).

Likewise, \eqref{eq:CK}, the vanishing initial data of \(\xi\),
and uniqueness of \eqref{eq:w} imply
\[
w[h+K\xi]-w[h]=\xi.
\]
The two terms in \eqref{eq:observable} therefore cancel exactly.
\end{proof}

This reproduces the needed first-order content of VII-A's
\texttt{thm:first} within the explicitly printed conventions
\cite{CheungVIIA}. No second-order source theorem is imported.

\begin{proposition}[Dependence support]
\label{prop:support}
Let \(K_f=\supp f\), and let
\[
\mathcal D_f=J^-_\eta(K_f)\cap\{0\leq t\leq T\}.
\]
This is compact. The functional \(\Obs_f\), regarded as a functional
of smooth \(h,\phi_0\) before restricting \(\phi_0\) to the free
shell, has dependence support contained in \(\mathcal D_f\).
\end{proposition}

\begin{proof}
The retarded scalar response observed in \(K_f\) uses sources only
in \(J^-_\eta(K_f)\). Its source is restricted by \(\chi\) to
\(0<t<T\). The coordinate response in \(K_f\) uses \(C(h)\) only
between \(\Sigma_0\) and \(K_f\), again inside \(J^-_\eta(K_f)\).
The direct scalar factor in the coordinate term is evaluated in
\(K_f\). Differentiation does not enlarge these supports.

A finite-time Minkowski causal past of a compact set has bounded
spatial extent on the closed slab \(0\leq t\leq T\), proving
compactness. The conclusion follows from finite propagation.
\end{proof}

\begin{remark}[The plateau is not optional]
For a compact \(\xi\) supported to the future of \(\Sigma_0\), but
not necessarily in the plateau, the same calculation yields
\begin{equation}
\Obs_f[h+K\xi;\phi_0]-\Obs_f[h;\phi_0]
=
-\int_M f\,G_m^{\mathrm R}Q_{(\chi-1)K\xi}\phi_0\dd^4x.
\label{eq:outsideplateau}
\end{equation}
There is no reason for this expression to vanish. No step below
enlarges \(\mathfrak g_U\).
\end{remark}

\section{An exact compact scalar test}
\label{sec:test}

Fix \(h\) as a classical parameter. Put
\[
u_f=G_m^{\mathrm A}f
\]
and define
\begin{equation}
q_{f,h}
=
-Q_{\chi h}u_f+\partial_\mu\bigl(fw[h]^\mu\bigr).
\label{eq:q}
\end{equation}

\begin{proposition}[Scalar-only representation]
\label{prop:test}
The function \(q_{f,h}\) is real, smooth, and compactly supported
strictly to the future of \(\Sigma_0\). For every smooth free solution,
\begin{equation}
\boxed{
\Obs_f[h;\phi_0]=\int_M q_{f,h}\phi_0\dd^4x.
}
\label{eq:representation}
\end{equation}
More precisely,
\begin{equation}
\supp q_{f,h}
\subseteq
K_f\ \cup\
 \bigl(\supp\chi\cap J^-_\eta(K_f)\bigr).
\label{eq:q-support}
\end{equation}
For fixed \(h,\chi\) and fixed compact observation support \(K\),
the map \(f\mapsto q_{f,h}\) is continuous from
\(C_K^\infty\) into \(C_{K\cup\supp\chi}^\infty\).
\end{proposition}

\begin{proof}
Transpose the retarded Green operator and then the density operator:
\[
\int f\phi_{1,\chi}\dd^4x
=
-\int (G_m^{\mathrm A}f)Q_{\chi h}\phi_0\dd^4x
=
-\int (Q_{\chi h}u_f)\phi_0\dd^4x.
\]
The integrations by parts have no boundary contribution because
\(\chi h\) is compactly supported away from \(\Sigma_0\).
For the coordinate term,
\[
-\int f w[h]^\mu\partial_\mu\phi_0\dd^4x
=
\int\partial_\mu(fw[h]^\mu)\phi_0\dd^4x.
\]
This proves \eqref{eq:representation} with the sign in
\eqref{eq:q}.

Smoothness follows from smooth Green evolution and
\eqref{eq:w}. The first term has support in
\(\supp\chi\cap J^-_\eta(K_f)\); the second has support in \(K_f\).
Both sets are compact and separated from \(t=0\).

For continuity, on the finite time interval containing the source
and observation supports, \eqref{eq:advanced} gives continuous
dependence of every derivative of \(u_f\) on finitely many
derivatives of \(f\). This can be seen by commuting spatial
derivatives through the sine propagator and applying Sobolev
estimates on an enlarged compact spatial set; time derivatives
follow from \(P_mu_f=f\). Multiplication by the fixed smooth
coefficients and application of the displayed finite differential
operators then give the asserted fixed-support continuity.
\end{proof}

There is also a useful exact gauge identity at the level of tests.

\begin{proposition}[Gauge equality in the free scalar quotient]
\label{prop:testgauge}
For \(\xi\in\mathfrak g_U\),
\begin{equation}
q_{f,h+K\xi}-q_{f,h}
=P_m\bigl(X_\xi u_f\bigr),
\qquad X_\xi u_f\in C_c^\infty(M).
\label{eq:q-gauge}
\end{equation}
Thus the tests define the same class modulo
\(P_mC_c^\infty(M)\).
\end{proposition}

\begin{proof}
Use \(\chi K\xi=K\xi\), \(w[K\xi]=\xi\), and \(P_mu_f=f\):
\[
q_{f,h+K\xi}-q_{f,h}
=-Q_{K\xi}u_f+\partial_\mu(f\xi^\mu).
\]
Writing \(d_\xi=\partial_\mu\xi^\mu\), \eqref{eq:QK} gives
\[
-Q_{K\xi}u_f
=-d_\xi f+P_m(X_\xi u_f)-X_\xi f.
\]
The remaining divergence is \(d_\xi f+X_\xi f\), proving
\eqref{eq:q-gauge}. Compactness follows from the compact support
of \(\xi\).
\end{proof}

\begin{remark}[Why this does not repair joint admission]
\label{rem:notjoint}
The compact test \(q_{f,h}\) is obtained \emph{after fixing \(h\)}.
It does not establish joint microcausal admission of the original
metric--scalar kernel. In particular, the metric dependence through
the zero-Cauchy map \(w[h]\) can retain the initial-surface structure
which is central to VII-B's obstruction \cite{CheungVIIB}.
A smooth scalar test with a classical parameter is a different
typed object from a distributional kernel with a quantized metric
slot.
\end{remark}

\section{The quantum family is constructed, not inferred}
\label{sec:quantum}

\subsection{One-particle map and scalar preparation}

Let
\[
\Hil=L^2(\R^3,\dd^3p),\qquad
\langle v,z\rangle=\int_{\R^3}\overline{v(p)}z(p)\dd^3p,
\qquad \omega_p=\sqrt{|p|^2+m^2}.
\]
The inner product is linear in its second argument. For a real
compact smooth scalar test \(F\), define
\begin{equation}
k_F(p)=
\frac{1}{(2\pi)^{3/2}\sqrt{2\omega_p}}
 \int_M F(t,x)e^{\ii\omega_pt-\ii p\cdot x}\dd t\dd^3x.
\label{eq:k}
\end{equation}

\begin{lemma}[Finite norm and shell descent]
\label{lem:k}
The map \(F\mapsto k_F\) is continuous from each fixed-support
\(C^\infty\) test space into \(\Hil\). Also
\[
k_{P_m v}=0\qquad(v\in C_c^\infty(M)).
\]
\end{lemma}

\begin{proof}
Integrating by parts spatially gives, for every integer \(n\geq1\),
\[
|k_F(p)|
\leq
\frac{C}{\sqrt{\omega_p}(1+|p|^2)^n}
 \bigl\|(1-\Delta_x)^nF\bigr\|_{L^1(M)}.
\]
Since \(m>0\), already a suitable fixed \(n\) gives an integrable
square on \(\R^3\). On a fixed compact support, the \(L^1\) norm
is controlled by finitely many \(C^\infty\) seminorms. This proves
the norm estimate and continuity.

The exponential in \eqref{eq:k} solves \(P_m v=0\).
Integration by parts against a compact \(v\) proves
\(k_{P_mv}=0\).
\end{proof}

Choose once and for all \(z_{\cl}\in\Sch(\R^3;\C)\). The corresponding
real smooth free scalar is
\begin{equation}
\phi_{\cl}(t,x)
=
\frac{1}{(2\pi)^{3/2}}
 \int_{\R^3}
 \frac{
 z_{\cl}(p)e^{-\ii\omega_pt+\ii p\cdot x}
 +\overline{z_{\cl}(p)}e^{\ii\omega_pt-\ii p\cdot x}
 }{\sqrt{2\omega_p}}\dd^3p.
\label{eq:classicalfield}
\end{equation}
It satisfies
\begin{equation}
\int_M F\phi_{\cl}\dd^4x
=2\Rea\langle k_F,z_{\cl}\rangle.
\label{eq:classicalpairing}
\end{equation}
Real compact smooth Cauchy data are included in this preparation
class. With the unitary spatial Fourier transform, their amplitude is
\[
z_{\cl}(p)
=\sqrt{\frac{\omega_p}{2}}\,\widehat{\phi_{\cl}(0)}(p)
 +\frac{\ii}{\sqrt{2\omega_p}}\,
       \widehat{\partial_t\phi_{\cl}(0)}(p).
\]

\subsection{CCR scaling and the observable section}

On \(\mathcal H=\Fock(\Hil)\), let \(a(v)\) be antilinear in \(v\),
with
\[
[a(v),a^\dagger(z)]=\langle v,z\rangle\One.
\]
Define
\[
\Phi(v)=a(v)+a^\dagger(v).
\]
For \(0<\epsilon\leq1\), the scaled scalar field is
\[
\phi_\epsilon(F)=\sqrt{\epsilon}\,\Phi(k_F).
\]
Equivalently, on the underlying real space of \(\Hil\), set
\[
W_\epsilon(v)=e^{\ii\sqrt{\epsilon}\Phi(v)}.
\]
Its Weyl relation is
\begin{equation}
W_\epsilon(v)W_\epsilon(z)
=e^{-\ii\epsilon\Ima\langle v,z\rangle}W_\epsilon(v+z).
\label{eq:weyl}
\end{equation}
Thus \(\epsilon\) is a dimensionless effective Planck parameter,
with a reference unit of action fixed. It is not \(\kappa\),
not an edge width, and not a cutoff.

Fix the prescribed \(h_{\cl}\), and abbreviate
\[
v_f=k_{q_{f,h_{\cl}}}.
\]
The quantum representative of the named observable is the section
\begin{equation}
\widehat{\Obs}_{f,\epsilon}
=\sqrt{\epsilon}\,\Phi(v_f).
\label{eq:quantumobservable}
\end{equation}
The classical symbol \(\Obs_f[h_{\cl};\,\cdot\,]\) is fixed while
its CCR representation varies. This is the states-or-representations
interpretation of the Em06 acceptance pair.

\begin{lemma}[Domains and self-adjointness]
\label{lem:domains}
For every \(v\in\Hil\), \(\Phi(v)\) is essentially self-adjoint
on the finite-particle domain \(\Dfin\). Coherent vectors have
finite moments of every order of its self-adjoint closure.
\end{lemma}

\begin{proof}
On vectors with at most \(r\) particles, the creation and
annihilation estimates give
\[
\|\Phi(v)^n\psi\|
\leq (2\|v\|)^n
 \sqrt{\frac{(r+n)!}{r!}}\|\psi\|.
\]
The analytic-vector series converges for every fixed sufficiently
small positive argument, indeed for every argument. The dense
finite-particle domain therefore consists of analytic vectors.
The analytic-vector criterion gives essential self-adjointness
\cite{Nelson}.

For completeness, if
\[
D(z)=\exp\bigl(a^\dagger(z)-a(z)\bigr),
\]
the exponential-vector expansion yields the unitary displacement
identity
\begin{equation}
D(z)^*\Phi(v)D(z)
=\Phi(v)+2\Rea\langle v,z\rangle\One.
\label{eq:displacement}
\end{equation}
The vacuum has finite moments of every order, by the same
finite-particle estimates. Applying \eqref{eq:displacement} to
the vacuum proves the coherent-vector assertion.
\end{proof}

All field powers and expectations below refer to these closures
and their finite coherent-vector moments. No formal expectation
of an undefined unbounded insertion is used.

\subsection{The state family}

Define
\begin{equation}
\Psi_\epsilon=D(z_{\cl}/\sqrt{\epsilon})\Omega,\qquad
\omega_\epsilon(A)=\langle\Psi_\epsilon,A\Psi_\epsilon\rangle.
\label{eq:states}
\end{equation}
Each \(\Psi_\epsilon\) is a normalized vector. Its restriction to
the represented Weyl algebra is consequently a positive normalized
state. Explicitly,
\begin{equation}
\omega_\epsilon(W_\epsilon(v))
=
\exp\left(
 \ii\,2\Rea\langle v,z_{\cl}\rangle
       -\frac{\epsilon}{2}\|v\|^2
\right).
\label{eq:characteristic}
\end{equation}

To verify the formula, normal ordering gives
\[
e^{\ii t\Phi(v)}
=e^{-t^2\|v\|^2/2}
 e^{\ii t a^\dagger(v)}e^{\ii t a(v)},
\]
and therefore
\(\langle\Omega,e^{\ii t\Phi(v)}\Omega\rangle
=e^{-t^2\|v\|^2/2}\).
Conjugating by the displacement in \eqref{eq:states} then gives
\eqref{eq:characteristic}. This construction is the familiar
coherent-state mechanism \cite{Glauber,Hepp}, with its normalization
printed here.

\section{The acceptance pair and its topology}
\label{sec:limit}

\begin{theorem}[Em06-A scalar-only classical-limit theorem]
\label{thm:main}
Fix Specification~\ref{spec:data}, \(h=h_{\cl}\), and the preparation
\(z_{\cl}\) of \eqref{eq:classicalfield}. For the observable
\eqref{eq:observable}, the quantum section
\eqref{eq:quantumobservable}, and the states \eqref{eq:states},
the following statements hold.

\begin{enumerate}[label=\textup{(\roman*)}]
\item \textbf{Mean.} For every \(0<\epsilon\leq1\),
\begin{equation}
\boxed{
\omega_\epsilon(\widehat{\Obs}_{f,\epsilon})
=\Obs_f[g_{\cl};\phi_{\cl}].
}
\label{eq:mean}
\end{equation}
In particular, the mean converges in the ordinary real topology
to the named classical value.

\item \textbf{Variance.} For every \(0<\epsilon\leq1\),
\begin{equation}
\boxed{
\Var_{\omega_\epsilon}(\widehat{\Obs}_{f,\epsilon})
=\epsilon\,\|k_{q_{f,h_{\cl}}}\|^2
\longrightarrow0.
}
\label{eq:variance}
\end{equation}

\item \textbf{Spectral topology.} Let \(\mu_{f,\epsilon}\) be the
spectral probability law of \(\widehat{\Obs}_{f,\epsilon}\) in
\(\Psi_\epsilon\), and put
\(o_f=\Obs_f[g_{\cl};\phi_{\cl}]\). Then
\[
\mu_{f,\epsilon}
=\mathcal N(o_f,\epsilon\|v_f\|^2),
\]
with the point-mass interpretation if \(v_f=0\), and
\begin{equation}
W_2(\mu_{f,\epsilon},\delta_{o_f})
=\sqrt{\epsilon}\,\|v_f\|.
\label{eq:W2}
\end{equation}
Thus the spectral laws converge in the quadratic Wasserstein
metric, not merely by a cutoff estimate.

\item \textbf{Restricted gauge independence.} Replacing
\(h_{\cl}\) by \(h_{\cl}+K\xi\), \(\xi\in\mathfrak g_U\),
leaves the represented observable, its classical value,
and its spectral law unchanged.

\item \textbf{Fixed-support uniformity.} For any bounded subset
\(\mathcal B\subset C_K^\infty((0,T)\times\R^3;\R)\),
\[
\sup_{f\in\mathcal B}
\Var_{\omega_\epsilon}(\widehat{\Obs}_{f,\epsilon})
\leq \epsilon C_{\mathcal B},
\qquad
\sup_{f\in\mathcal B}
W_2(\mu_{f,\epsilon},\delta_{o_f})
\leq\sqrt{\epsilon C_{\mathcal B}},
\]
where \(C_{\mathcal B}<\infty\) is independent of \(\epsilon\).
\end{enumerate}

The metric jet and causal structure are prescribed inputs.
The theorem asserts neither metric reconstruction nor a derivation
of Einstein dynamics.
\end{theorem}

\begin{proof}
By \eqref{eq:displacement},
\begin{equation}
D(z_{\cl}/\sqrt{\epsilon})^*
 \widehat{\Obs}_{f,\epsilon}
D(z_{\cl}/\sqrt{\epsilon})
=
2\Rea\langle v_f,z_{\cl}\rangle\One
          +\sqrt{\epsilon}\Phi(v_f).
\label{eq:shiftedoperator}
\end{equation}
The vacuum mean of \(\Phi(v_f)\) is zero. Equations
\eqref{eq:classicalpairing} and \eqref{eq:representation} identify
the constant in \eqref{eq:shiftedoperator} with \(o_f\).
This proves the mean statement separately.

For the variance, center \eqref{eq:shiftedoperator}. Since
\(\Phi(v_f)\Omega=a^\dagger(v_f)\Omega\),
\[
\begin{split}
\Var_{\omega_\epsilon}(\widehat{\Obs}_{f,\epsilon})
&=
\|(\widehat{\Obs}_{f,\epsilon}-o_f)\Psi_\epsilon\|^2\\
&=\epsilon\|\Phi(v_f)\Omega\|^2
=\epsilon\|v_f\|^2.
\end{split}
\]
Lemma~\ref{lem:k} proves that the norm is finite. This proves
the vanishing-variance statement without inferring it from
the mean.

The characteristic function of the spectral law, obtained from
\eqref{eq:characteristic} with \(v=t v_f\), is
\[
\int_\R e^{\ii t x}\dd\mu_{f,\epsilon}(x)
=
e^{\ii t o_f-\epsilon t^2\|v_f\|^2/2}.
\]
This identifies the stated Gaussian law. Every coupling to the
point mass \(\delta_{o_f}\) has second coordinate \(o_f\), so
\[
W_2(\mu_{f,\epsilon},\delta_{o_f})^2
=\int_\R |x-o_f|^2\dd\mu_{f,\epsilon}(x)
=\epsilon\|v_f\|^2.
\]

For gauge independence, Proposition~\ref{prop:testgauge} and
Lemma~\ref{lem:k} give
\[
k_{q_{f,h_{\cl}+K\xi}}=k_{q_{f,h_{\cl}}}.
\]
Hence the quantum representatives agree as operators. Classical
agreement is Proposition~\ref{prop:plateau}.

Finally, the continuous linear maps of
Proposition~\ref{prop:test} and Lemma~\ref{lem:k} send a bounded
fixed-support test set to a bounded subset of \(\Hil\).
Taking
\[
C_{\mathcal B}
=\sup_{f\in\mathcal B}\|k_{q_{f,h_{\cl}}}\|^2
\]
proves both uniform estimates.
\end{proof}

\begin{corollary}[State-relative \(L^2\) convergence]
\label{cor:L2}
For every named \(f\),
\[
\|(\widehat{\Obs}_{f,\epsilon}
       -\Obs_f[g_{\cl};\phi_{\cl}])\Psi_\epsilon\|
=\sqrt{\epsilon}\,\|v_f\|\longrightarrow0.
\]
\end{corollary}

\begin{proof}
This is the centered norm evaluated in the proof of
Theorem~\ref{thm:main}.
\end{proof}

\begin{remark}[Not operator-norm convergence]
The convergence is state-relative and spectral. It is not
operator-norm convergence of unbounded fields, and it is not
uniform over all states. Even for the bounded Weyl functions,
if \(v\ne0\), the spectrum of \(\Phi(v)\) is \(\R\), and
\[
\|e^{\ii\sqrt{\epsilon}\Phi(v)}-\One\|=2
\qquad(\epsilon>0).
\]
The theorem therefore makes no norm-continuous identification
of the entire quantum algebra with a classical algebra.
\end{remark}

\section{A nonzero example and one evaluated response}
\label{sec:diagnostic}

The acceptance pair should not be confused with concentration of
the zero operator. The following admissible data give a nonzero
observable and an explicit classical target.

\begin{proposition}[Nontrivial plateau coefficient]
\label{prop:witness}
Choose
\[
0<t_1<t_2<t_3<t_4<T<\frac{\pi}{2m}.
\]
Let
\[
a\in C_c^\infty((t_1,t_2)),\qquad
F\in C_c^\infty((t_3,t_4))
\]
be nonnegative and nonzero, and let
\(\rho\in C_c^\infty(\R^3)\) be nonnegative with
\(\int\rho\dd^3x=1\). Set
\[
f(t,x)=F(t)\rho(x),\qquad
h_{00}=h_{0i}=0,\qquad h_{ij}=-a(t)\delta_{ij}.
\]
Choose \(\chi\) equal to one on a neighborhood of
\[
[t_1,t_4]\times
 \{\text{spatial projection of }\mathcal D_f\}.
\]
Such a compact switch separated from \(t=0\) exists.

Choose compact smooth scalar Cauchy data equal to
\((1,0)\) on a ball containing the initial causal dependence
region of \(\mathcal D_f\). Then their amplitude belongs to
\(\Sch(\R^3)\), and
\begin{equation}
\boxed{
\Obs_f[h;\phi_{\cl}]
=
-3m\int_\R F(t)
       \int_0^t a(s)\cos(ms)\sin(m(t-s))\dd s\dd t
<0.
}
\label{eq:witness}
\end{equation}
In particular \(v_f\ne0\).
\end{proposition}

\begin{proof}
Finite propagation and uniqueness imply
\(\phi_{\cl}(t,x)=\cos(mt)\) throughout \(\mathcal D_f\).
For the stated metric coefficient,
\[
\tau(h)=3a(t),\qquad C(h)^0=-\frac32a'(t),\qquad C(h)^i=0.
\]
The unswitched coordinate solution is consequently
\[
w[h]^0(t)=-\frac32\int_0^t a(s)\dd s,\qquad w[h]^i=0.
\]
On the observation dependence region, the switch is one whenever
\(h\) is nonzero. The scalar density source on the free shell is
\[
Q_h\phi_{\cl}
=-C(h)^0\partial_t\phi_{\cl}
=-\frac{3m}{2}a'(t)\sin(mt).
\]
The retarded solution therefore gives
\[
\phi_{1,\chi}(t,x)
=\frac32\int_0^t
       \sin(m(t-s))a'(s)\sin(ms)\dd s
\]
in the observation region. Subtracting the coordinate term yields
\[
\phi_{1,\chi}-w[h]^0\partial_t\phi_{\cl}
=
\frac32\int_0^t\sin(m(t-s))a'(s)\sin(ms)\dd s
-\frac{3m}{2}\sin(mt)\int_0^t a(s)\dd s.
\]
Integrate the first term by parts. Its endpoint terms vanish.
The trigonometric identity
\[
\sin(2ms-mt)-\sin(mt)
=-2\cos(ms)\sin(m(t-s))
\]
gives
\[
\phi_{1,\chi}-w[h]^0\partial_t\phi_{\cl}
=-3m\int_0^t a(s)\cos(ms)\sin(m(t-s))\dd s.
\]
Integrating against \(f\) proves \eqref{eq:witness}.

For \(s\in(t_1,t_2)\), \(t\in(t_3,t_4)\), both
\(\cos(ms)\) and \(\sin(m(t-s))\) are strictly positive.
The integral is therefore strictly negative. If \(v_f=0\),
\eqref{eq:classicalpairing} and \eqref{eq:representation} would
make the same value zero, a contradiction.
\end{proof}

\subsection{The diagnostic is the response of the named observable}

A response diagnostic must specify what is varied. Here it is a
classical scalar Cauchy amplitude, not a gravitational source law.

For \(\zeta\in\Sch(\R^3;\C)\), let
\[
z_\alpha=z_{\cl}+\alpha\zeta,\qquad
\Psi_{\epsilon,\alpha}=D(z_\alpha/\sqrt{\epsilon})\Omega,
\qquad \alpha\in\R.
\]
Write \(\phi_\zeta\) for the real free scalar given by
\eqref{eq:classicalfield} with amplitude \(\zeta\).

\begin{theorem}[Preparation-response diagnostic]
\label{thm:diagnostic}
For the same observable as in Theorem~\ref{thm:main},
\begin{equation}
\left.\frac{\dd}{\dd\alpha}\right|_{\alpha=0}
 \omega_{\epsilon,\alpha}(\widehat{\Obs}_{f,\epsilon})
=
2\Rea\langle v_f,\zeta\rangle
=
\Obs_f[h_{\cl};\phi_\zeta].
\label{eq:diagnostic}
\end{equation}
The equality holds for every \(\epsilon>0\); hence the response
has the stated classical limit.

For the data of Proposition~\ref{prop:witness}, take
\(\zeta=z_{\cl}\). The response is precisely the strictly negative
number in \eqref{eq:witness}.
\end{theorem}

\begin{proof}
The displacement calculation gives the exact affine formula
\[
\omega_{\epsilon,\alpha}(\widehat{\Obs}_{f,\epsilon})
=
2\Rea\langle v_f,z_{\cl}\rangle
+\alpha\,2\Rea\langle v_f,\zeta\rangle.
\]
Differentiate this identity. The second equality in
\eqref{eq:diagnostic} follows from
\eqref{eq:classicalpairing} and \eqref{eq:representation}.
There is no interchange of an uncontrolled derivative and limit.
The final assertion follows from \(\phi_\zeta=\phi_{\cl}\) and
Proposition~\ref{prop:witness}.
\end{proof}

This diagnostic evaluates the named relational coefficient.
It is not VII-G's instrument, an entanglement or recovery statistic,
En01-B's sphere contrast, or En01-C's total-diffusion certificate
\cite{CheungDossier}. It does not identify a physical gravitational
mechanism or supply a detector preparation and residual-error ledger.

\section{What the limit does not repair}
\label{sec:stops}

\subsection{The joint product obstruction remains a separate statement}

For the frozen data and F1 hypotheses of Cheung's XI-G
\texttt{thm:obstruction}, the supplied result is
\begin{equation}
[u]_{\mathrm{sol}}\cap\mathscr F_{\mc,\mathrm{pol}}
=\varnothing.
\label{eq:XIG}
\end{equation}
Here \([u]_{\mathrm{sol}}\) is the class of compact cubic kernels
with two metric slots and one scalar slot, identified by equality
on the declared free solutions; \(\mathscr F_{\mc,\mathrm{pol}}\)
is the predecessor's admissible microcausal polynomial target
\cite{CheungXIG}. We neither enlarge that target nor change the
frozen class.

\begin{proposition}[No transfer from concentration to joint admission]
\label{prop:noadmission}
Theorem~\ref{thm:main} constructs no representative of
\([u]_{\mathrm{sol}}\) in
\(\mathscr F_{\mc,\mathrm{pol}}\), and no interacting gravitational
product. It therefore provides no repair of \eqref{eq:XIG}.
In particular, any additional claim assigning such a representative
under the unchanged XI-G hypotheses would still contradict
\eqref{eq:XIG}, independently of a classical-limit estimate.
\end{proposition}

\begin{proof}
The constructed map is
\[
(f,h_{\cl})\longmapsto q_{f,h_{\cl}}
\longmapsto k_{q_{f,h_{\cl}}}
\longmapsto\sqrt{\epsilon}\Phi(k_{q_{f,h_{\cl}}}).
\]
The metric argument has already been fixed before the first
quantum map. Its output is a scalar-linear field in a free scalar
representation. No step has an output with the two quantum metric
slots and scalar slot of \(u\); no map between those kernel classes
is defined. Products of the resulting free scalar operators, when
defined on their domains, are not the missing gravitational
retarded products.

If a further construction produced a representative in the stated
intersection, its existence would contradict \eqref{eq:XIG}.
An estimate on expectation values of a different, already-defined
operator does not alter that set-theoretic contradiction.
\end{proof}

The same typing distinction prevents transferring this theorem to
the sharp joint insertion of VII-B \cite{CheungVIIB}, or to
X-G's separate variational and one-loop questions
\cite{CheungDossier}. The limit here is not an interacting quantum
product.

\subsection{VII-H is a comparison, not a premise}

The supplied prospectus describes VII-H as a free massless
Minkowski Wick-square susceptibility, with contact ambiguity
\(\R\hbar\delta_0\) and a prescription-independent pairing
\[
-\frac{\hbar t_0}{2\pi^2}
\]
under that construction's hypotheses \cite{CheungDossier}.
This is documentary comparison material only. Its full
construction is not assumed here.

The present coefficient is a massive scalar-linear relational
response, and its variance coefficient is
\[
\|k_{q_{f,h_{\cl}}}\|^2.
\]
It is not VII-H's pairing and is not an inherited gravitational
characteristic coefficient. Even if both quantities scale with
an effective Planck parameter, that common scaling does not
identify their domains, kernels, states, or observables.

No counterterm is introduced in this proof. More generally, a
local distributional correction and its propagated image are
different objects: for a retarded inverse, propagation can enlarge
support. Nothing here replaces the propagated image by the local
contact itself.

\subsection{Non-premise and non-transfer ledger}

The following is a scope ledger, not a list of assumptions used
in the proof.

\small
\begin{longtable}{@{}p{0.23\textwidth}p{0.71\textwidth}@{}}
\toprule
Construction & Boundary retained by Em06-A\\
\midrule
\endfirsthead
\toprule
Construction & Boundary retained by Em06-A\\
\midrule
\endhead
\bottomrule
\endfoot

III and VI
&
Classical quotient and solution-jet constructions do not supply
the state family or the CCR scaling. Their interpretation may be
used after a limit has been constructed; they are not premises
here. Equations included in an Einstein--scalar jet are not
equations derived by this theorem
\cite{CheungIII,CheungVI}.\\[4pt]

IV Model 3 and V
&
A prescribed Newton--wave stress response, a represented form,
and a coherent comparison are different objects. No inverse
prescription from those models is inserted into the present
proof. Their Einstein operator, current domains, and signs are
not transferred
\cite{CheungIV,CheungV}.\\[4pt]

VIII-B
&
An expectation selecting a member of a preconstructed conical
family is not reconstruction of that family from quantum data.
The conical dictionary is a non-premise
\cite{CheungVIIIB}.\\[4pt]

VIII-C and VII-E
&
Support of prescribed block channels and scheduled directional
dependency are not derived spacetime causal cones. The proof
uses the already declared Minkowski causal structure
\cite{CheungVIIIC,CheungDossier}.\\[4pt]

VII-G, VII-H, Em01-A, Em02-E
&
The finite instrument, massless susceptibility, wave-shell
comparison, and inflationary lapse are separate constructions.
None supplies the observable, quantum state, or acceptance
pair proved here. In particular, no quantum regulator-removal
audit for the lapse is completed
\cite{CheungDossier,CheungEm02E}.\\[4pt]

Em04-A, Em05-A, Em05-B
&
The XY-chain reduction, signed extractor, leakage estimates,
and recovery results are non-premises. No identification with
their Hilbert spaces or dynamics is made
\cite{CheungDossier}.\\[4pt]

En01-B and En01-C
&
The sphere contrast and total flat-stage diffusion certificate
are not the preparation-response diagnostic
\eqref{eq:diagnostic}. No detector or source-identification
ledger is inferred from those statistics
\cite{CheungDossier}.\\[4pt]

En02 and Em03 examples
&
Finite decoders, perfect-tensor codes, packet channels, and
fixed-background emission laws do not supply a bulk--boundary
encoding or a metric reconstruction map. All are non-premises
\cite{CheungDossier}.\\[4pt]

XI-G
&
The frozen free-shell class obstruction \eqref{eq:XIG} is
untouched. The scalar-only restriction is not admission of
that class \cite{CheungXIG}.\\[4pt]

XVI-N
&
The conditional obstruction of its regular seed-preserving
antifield-linear completion is not negated. No BV complex is
constructed. Its separately named Route B is not discharged
\cite{CheungXVIN}.\\[4pt]

XVI-O
&
A kernel counterterm is not bulk BV restoration. The supplied
O.1 status remains \texttt{NOT\_CLOSED}; no inherited insertion
completion is obtained \cite{CheungXVIO}.\\[4pt]

XIV-H, XIV-I, XIV-J; XV-B exit
&
No change to the recorded H2, I3, or J2 gates is claimed.
In particular, a constructed internal amplitude module is
not the missing full joint characteristic map. No quantum
coefficient is admitted, and XV-B is not opened
\cite{CheungXIVH,CheungXIVI,CheungXIVJ}.\\

\end{longtable}
\normalsize

\subsection{Why this is not spacetime emergence}

The data flow is
\[
\boxed{
\begin{gathered}
\text{prescribed Minkowski background and metric jet}\\
+\ \text{declared scalar equation, switch, and harmonic prescription}\\
+\ \text{declared scalar preparation and CCR scaling}
\end{gathered}
\longrightarrow
\begin{gathered}
\text{one represented relational coefficient}\\
\text{with classical mean and vanishing variance}.
\end{gathered}
}
\]
There is no reverse map reconstructing the background.

The construction neither uses nor derives an entropy--area
dictionary. It supplies no boundary region, code subspace,
bulk--boundary encoding, entanglement wedge, or reconstruction
theorem of the type studied in \cite{RT,DHW}. It also does not
derive gravitational field equations from entanglement constraints;
results of that different kind retain their own holographic
hypotheses \cite{Faulkner}. No Einstein equation is imposed or
deduced for \(h_{\cl}\).

Consequently the logical separation is retained:
\[
\boxed{
\begin{gathered}
\text{classical quotient}\ne\text{prescribed response},\\
\text{prescribed response}\ne\text{kinematic dictionary},\\
\text{none of these alone is emergence of spacetime}.
\end{gathered}
}
\]

\section{Repairs to the proof architecture and final release}
\label{sec:release}

\subsection{Three errors excluded by the construction}

\begin{proposition}[Fixed-CCR coherent displacement does not suffice]
\label{prop:fixedCCR}
Let \(v_f\ne0\). In the unscaled representation, every coherent
state \(D(z)\Omega\) has
\[
\Var_{D(z)\Omega}(\Phi(v_f))=\|v_f\|^2>0.
\]
Thus coherent displacement alone, at fixed CCR and for the
unscaled observable, does not prove the absolute
vanishing-variance acceptance condition.
\end{proposition}

\begin{proof}
Apply \eqref{eq:displacement}. Displacement changes the mean
by a constant and leaves the centered vacuum variance unchanged.
\end{proof}

This is why \(\epsilon\) and its representation are specified
explicitly. The theorem is not a claim that large occupation
alone makes the unscaled absolute fluctuations vanish.

\begin{remark}[The preparation has a resource cost]
The chosen coherent states obey
\[
\langle\Psi_\epsilon,N\Psi_\epsilon\rangle
=\frac{\|z_{\cl}\|^2}{\epsilon}.
\]
For the smooth preparation,
\[
\left\langle\Psi_\epsilon,
 \epsilon\,\mathrm d\Gamma(\omega_p)\Psi_\epsilon\right\rangle
=\int_{\R^3}\omega_p|z_{\cl}(p)|^2\dd^3p.
\]
Thus the scaled classical energy is finite while the expected
unscaled particle number grows. No bounded-resource preparation
or laboratory implementation is inferred.
\end{remark}

\begin{remark}[Other repairs]
Two further substitutions have been excluded:
\begin{enumerate}
\item The correct scalar density perturbation is
\(Q_{\chi h}\). Replacing it by \(\chi Q_h\), or omitting its
trace--mass term, would change the declared action and the
off-shell test calculation.
\item The notation \(\Obs_f[g_{\cl}]\) cannot erase the scalar
preparation. The exact theorem names both \(g_{\cl}\) and
\(\phi_{\cl}\), and concerns the first coefficient only.
\end{enumerate}
\end{remark}

\subsection{Exact release}

\begin{theorem}[Em06-A analytical release]
\label{thm:release}
For the fixed data printed in this manuscript:
\begin{enumerate}[label=\textup{(\roman*)}]
\item the classical functional \eqref{eq:observable}, its dependence
support, and its plateau gauge domain are defined and proved;
\item its scalar-only compact-test representation
\eqref{eq:q}--\eqref{eq:representation} is exact;
\item the quantum representative, its self-adjoint domain, and
the positive normalized state family are constructed;
\item the mean and variance acceptance conditions are proved
separately by \eqref{eq:mean} and \eqref{eq:variance};
\item spectral convergence holds in \(W_2\), with the exact rate
\eqref{eq:W2} and fixed-support uniformity;
\item a nonzero classical target and the preparation-response
diagnostic are evaluated by
\eqref{eq:witness} and \eqref{eq:diagnostic}.
\end{enumerate}
The outcome is a repaired scalar-only limit for the first plateau
relational coefficient.

No quantized metric insertion, full nonlinear relational limit,
interacting gravitational product, unrestricted gauge theorem,
Einstein dynamics, metric reconstruction, or holographic map is
released. The XI-G class obstruction is untouched. No inherited
quantum-response coefficient is admitted; XVI-N's separate Route B
is not discharged; O.1 is not closed; H2, I3, and J2 are not resolved
by this manuscript; XV-B is not opened.
\end{theorem}

\begin{proof}
Propositions~\ref{prop:plateau} and \ref{prop:support} prove (i).
Propositions~\ref{prop:test} and \ref{prop:testgauge} prove (ii).
Lemma~\ref{lem:domains} and the explicit vector construction
\eqref{eq:states} prove (iii).
Theorem~\ref{thm:main} proves (iv) and (v).
Proposition~\ref{prop:witness} and
Theorem~\ref{thm:diagnostic} prove (vi).

The maps in the proof retain a classical metric parameter and a
free scalar carrier throughout. Proposition~\ref{prop:noadmission}
establishes the non-transfer to the frozen cubic admission problem.
No map or dynamical object needed for any of the excluded claims
is constructed. Those claims are therefore not additional parts
of this release.
\end{proof}

\paragraph{Evidence boundary.}
All conclusions are analytical statements under the printed
hypotheses. Registration, compilation, formal verification,
numerical execution, calibration, and physical realization are
not asserted.

\begin{thebibliography}{99}

\bibitem{CheungVIIA}
N.~Cheung,
\emph{Plateau-compatible relational scalars:
a second-order classical intertwiner. Landmark VII-A}.
Manuscript VII-A, version~1, n.d.
Relevant supplied statement:
\texttt{thm:first}, lines 973--1012.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/VII-A}.
The first-order construction needed here is reproduced explicitly.

\bibitem{CheungVIIB}
N.~Cheung,
\emph{Landmark VII-B: Cauchy harmonic insertions and second-order
retarded products: a sharp-boundary obstruction in an off-shell
polynomial domain}.
Manuscript VII-B, version~1, n.d.
Relevant supplied statement:
\texttt{thm:main}, lines 924--985.
\url{https://enter.grok.me/browse/VII-B}.

\bibitem{CheungV}
N.~Cheung,
\emph{Admissible currents and first retarded corrections
for a fixed-time scalar response}.
Manuscript V, version~1, n.d.
External-metric, one-oscillator, and Newton--wave objects are
separate constructions.
\url{https://enter.grok.me/browse/V}.

\bibitem{CheungIV}
N.~Cheung,
\emph{Represented records for three named scalar-field models:
finite-plane corrections, dense-TT corrections, and sourced
curvature pairings}.
Manuscript IV, version~1, n.d.
Relevant supplied material: Model~3 and \texttt{thm:model3}.
\url{https://enter.grok.me/browse/IV}.

\bibitem{CheungIII}
N.~Cheung,
\emph{Descent of first-order relational scalars,
and the free-product obstruction}.
Manuscript III, version~1, n.d.
Classical quotient and gauge constructions; comparison only.
\url{https://enter.grok.me/browse/III}.

\bibitem{CheungVI}
N.~Cheung,
\emph{Relational Einstein--scalar observables at second order:
classical descent and a compact-switch obstruction}.
Manuscript VI, version~1, n.d.
Classical descent and supplied quantum-status audit; comparison only.
\url{https://enter.grok.me/browse/VI}.

\bibitem{CheungVIIIB}
N.~Cheung,
\emph{A sourced spinless conical packet with a persistent
three-to-six polyhedral slicing event. Landmark VIII-B}.
Manuscript VIII-B, version~2, n.d.
Relevant supplied statement:
\texttt{thm:dictionary}, lines 451--469.
\url{https://enter.grok.me/browse/VIII-B}.

\bibitem{CheungVIIIC}
N.~Cheung,
\emph{Landmark VIII-C: Certified Fixed-Packet Flow,
Guarded Events, and Finite Reversible Composition}.
Manuscript VIII-C, version~2, n.d.
Relevant supplied statement:
\texttt{thm:C5}, lines 1512--1566.
\url{https://enter.grok.me/browse/VIII-C}.

\bibitem{CheungXIG}
N.~Cheung,
Manuscript XI-G, version~1, n.d.
Relevant supplied statements:
\texttt{def:class}, lines 576--589;
\texttt{thm:criterion}, lines 1036--1069;
\texttt{thm:obstruction}, lines 1189--1216.
\url{https://enter.grok.me/browse/XI-G}.

\bibitem{CheungEm02E}
N.~Cheung,
Manuscript Em02-E, version~1, n.d.
Relevant supplied statements:
\texttt{thm:lapse}, lines 758--796;
\texttt{thm:release}, lines 1795--1846.
Classical relational lapse on a declared background.
\url{https://enter.grok.me/browse/Em02-E}.

\bibitem{CheungXVIN}
N.~Cheung,
Manuscript XVI-N, revision r12, n.d.
Relevant supplied statement:
Theorem~11, \texttt{thm:massless-obstruction},
lines 306--312.
Conditional obstruction in its declared completion class.
\url{https://enter.grok.me/browse/XVI-N}.

\bibitem{CheungXVIO}
N.~Cheung,
Manuscript XVI-O, revision o70-sync, n.d.
Relevant supplied material:
\texttt{rem:three-verdicts}, lines 74--97;
status ledger, lines 413--446.
Kernel subsector and retained \texttt{NOT\_CLOSED} status.
\url{https://enter.grok.me/browse/XVI-O}.

\bibitem{CheungXIVH}
N.~Cheung,
Manuscript XIV-H, version~1, n.d.
Relevant supplied statements:
\texttt{thm:H2}, lines 1621--1664;
\texttt{thm:release}, lines 1929--1977.
\url{https://enter.grok.me/browse/XIV-H}.

\bibitem{CheungXIVI}
N.~Cheung,
Manuscript XIV-I, version~1, n.d.
Relevant supplied statements:
\texttt{thm:I2}, lines 1370--1460;
\texttt{thm:I3}, lines 1541--1630;
\texttt{thm:release}, lines 1811--1896.
\url{https://enter.grok.me/browse/XIV-I}.

\bibitem{CheungXIVJ}
N.~Cheung,
Manuscript XIV-J, version~1, n.d.
Relevant supplied statements:
\texttt{thm:J2}, lines 1690--1731;
\texttt{cor:noadmission}, lines 1890--1904;
\texttt{thm:release}, lines 1976--2021.
\url{https://enter.grok.me/browse/XIV-J}.

\bibitem{CheungDossier}
N.~Cheung,
\emph{Em06 prospectus and accompanying manuscript-comparison dossier}.
Unpublished material supplied with the problem, n.d.
Documentary descriptions of VII-G, VII-H, Em04-A, Em05-A,
Em05-B, and the En01/En02/Em03 comparison families are used
only to state non-transfer boundaries.
This entry does not assert independent publication, registration,
or verification of those described sources.

\bibitem{Glauber}
R.~J.~Glauber,
Coherent and incoherent states of the radiation field,
\emph{Physical Review} \textbf{131} (1963), 2766--2788.
doi:10.1103/PhysRev.131.2766.

\bibitem{Hepp}
K.~Hepp,
The classical limit for quantum mechanical correlation functions,
\emph{Communications in Mathematical Physics}
\textbf{35} (1974), 265--277.
doi:10.1007/BF01646348.

\bibitem{Nelson}
E.~Nelson,
Analytic vectors,
\emph{Annals of Mathematics} (2)
\textbf{70} (1959), 572--615.
doi:10.2307/1970331.

\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.
doi:10.4171/037.
arXiv:0806.1036.

\bibitem{RT}
S.~Ryu and T.~Takayanagi,
Holographic derivation of entanglement entropy from AdS/CFT,
\emph{Physical Review Letters}
\textbf{96} (2006), 181602.
doi:10.1103/PhysRevLett.96.181602.
arXiv:hep-th/0603001.
External comparison, not a premise.

\bibitem{DHW}
X.~Dong, D.~Harlow, and A.~C.~Wall,
Reconstruction of bulk operators within the entanglement wedge
in gauge-gravity duality,
\emph{Physical Review Letters}
\textbf{117} (2016), 021601.
doi:10.1103/PhysRevLett.117.021601.
arXiv:1601.05416.
External comparison, not a premise.

\bibitem{Faulkner}
T.~Faulkner, M.~Guica, T.~Hartman, R.~C.~Myers,
and M.~Van Raamsdonk,
Gravitation from entanglement in holographic CFTs,
\emph{Journal of High Energy Physics}
\textbf{03} (2014), 051.
doi:10.1007/JHEP03(2014)051.
arXiv:1312.7856.
External comparison, not a premise.

\end{thebibliography}

\end{document}