\documentclass[11pt]{article}

\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{lmodern}
\usepackage[margin=1in]{geometry}
\usepackage{amsmath,amssymb,amsthm}
\usepackage{array,booktabs,longtable}
\usepackage{enumitem}
\usepackage{microtype}
\usepackage[hidelinks]{hyperref}

\hypersetup{
  pdftitle={XVII-A: One named bulk observable from one declared boundary region},
  pdfauthor={},
  pdfsubject={Exact finite-model operator reconstruction on an assumed background}
}

\numberwithin{equation}{section}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}

\newcommand{\HH}{\mathcal H}
\newcommand{\BB}{\mathcal B}
\newcommand{\AA}{\mathcal A}
\newcommand{\Id}{\mathbf 1}
\newcommand{\Tr}{\operatorname{Tr}}
\newcommand{\Ad}{\operatorname{Ad}}
\newcommand{\dd}{\,\mathrm d}
\newcommand{\ket}[1]{\lvert #1\rangle}
\newcommand{\bra}[1]{\langle #1\rvert}
\newcommand{\kb}[2]{\lvert #1\rangle\langle #2\rvert}
\newcommand{\norm}[1]{\lVert #1\rVert}
\newcommand{\abs}[1]{\lvert #1\rvert}
\newcommand{\Obulk}{\mathcal O_{\mathrm{bulk}}}
\newcommand{\OB}{\mathcal O_B}
\newcommand{\NB}{\mathcal N_B}
\newcommand{\RB}{\mathcal R_B}
\newcommand{\ME}{\mathcal M_{\Obulk}}
\newcommand{\dtr}{d_{\mathrm{tr}}}
\newcommand{\ddiam}{d_\diamond}
\newcommand{\lambdastar}{\lambda_*}

\setlist[enumerate]{label=\textup{(\roman*)},leftmargin=*}
\setlist[itemize]{leftmargin=*}
\setlength{\emergencystretch}{2em}

\title{
  XVII-A --- One boundary reconstruction is not holographic emergence\\[0.4em]
  \large One named bulk observable from one declared boundary region,
  and the maps it does not supply
}
\author{}
\date{}

\begin{document}
\maketitle

\begin{abstract}
We complete an exact finite-model version of the proposed reconstruction
problem. The model is a finite-matrix massless-scalar benchmark on an
assumed global anti-de Sitter background in two dimensions. The boundary
region is a fixed interval on the actual right conformal boundary, not a
collar, a scheduled dependency region, or a packet support. The named
bulk observable is a bounded sine effect at the central bulk point.
An explicit boundary smearing reconstructs its underlying finite-matrix
field, and functional calculus supplies a boundary representative of
the effect. A declared completely positive trace-preserving binary
readout channel then reproduces the bulk measurement channel exactly,
including arbitrary reference systems.

This is a new, independently specified benchmark. It is not an
identification with a VII-A relational scalar, a V represented
response, or an Em03 Schwarzschild packet observable. No connecting
map for those inherited candidates is supplied by the available
excerpts, and none is asserted here. This change of model is explicit.

We also prove a conditional perturbation budget, evaluate one
analytical diagnostic, show that the resulting binary record does not
recover a noncommuting observable, and exhibit nonidentifiability of
the background radius from the declared dimensionless boundary data.
The finite cutoff makes every nonempty boundary time interval
algebraically complete; consequently the result is not an
entanglement-wedge theorem. It derives neither a metric, a causal
structure, nor Einstein dynamics. No numerical execution, compilation,
machine verification, or registration is asserted.
\end{abstract}

\section{Scope, completion choice, and corrected statement}

\subsection{The completion choice}

The proposed XVII-A leaves three essential objects unspecified:
a particular bulk observable, a particular boundary region, and a
connecting map between their models. The supplied manuscript excerpts
do not provide that connecting map for any of the proposed inherited
candidates. A proof cannot manufacture one by identifying similarly
named registers, regions, responses, or observables.

We therefore make the following explicit completion choice:
\begin{quote}
The theorem concerns a newly declared finite-matrix scalar benchmark
on a fixed global $\mathrm{AdS}_2$ background. It does not concern an
inherited VII-A, V, or Em03 observable.
\end{quote}
This fills the open object choices in a modified Work Package A.
It is not a repair of an inherited observable by an unproved
identification. If Work Package A is required to use one of those
specific inherited candidates, that version of the problem remains
unresolved.

The fixed-background smearing calculation is motivated by the
Hamilton--Kabat--Lifschytz--Lowe constructions
\cite{HKLL2006a,HKLL2006b}. Every identity needed below is proved directly
for the finite model. No AdS/CFT duality, boundary conformal field
theory, entanglement-wedge hypothesis, or entropy--area formula is a
premise.

\subsection{What reconstruction means}

A channel acts on states, whereas an observable is an operator.
Accordingly, the schematic expression
\[
 R_B(\text{data on }B)=\Obulk
\]
must be replaced by a statement with specified types.

For the present paper, the exact operator statement will be
\begin{equation}
 U^\dagger \OB U=\Obulk,
 \label{eq:typed-operator-target}
\end{equation}
where $U$ is the declared finite representation correspondence and
$\OB$ belongs to the algebra generated by boundary data on $B$.
The exact operational statement will be
\begin{equation}
 \RB\circ\NB=\ME.
 \label{eq:typed-channel-target}
\end{equation}
Here $\NB$ supplies the declared boundary quantum data, $\RB$ is a
binary record channel, and $\ME$ is the binary measurement channel
associated with $\Obulk$.

Equation~\eqref{eq:typed-channel-target} does not claim recovery of an
unknown bulk quantum state. Nor does it identify the conditional
postmeasurement bulk state with a boundary instrument output.
Observable-algebra recovery and state recovery are distinct questions
\cite{BKK2007,ADH2015}.

\subsection{The nonimplication ledger}

The intended refusal is better expressed as a nonimplication than as
an assertion that the underlying objects can never coincide:
\[
\boxed{
\begin{gathered}
 \text{finite decoder}
 \not\Longrightarrow
 \text{geometrically identified bulk--boundary map},\\
 \text{one bulk--boundary operator map}
 \not\Longrightarrow
 \text{metric reconstruction}
 \not\Longrightarrow
 \text{derivation of Einstein dynamics}.
\end{gathered}}
\]
A decoder can implement a bulk--boundary map after the relevant
identifications have been supplied and proved. It does not supply
those identifications merely by being a decoder. The distinction is
important even for holographic code models \cite{Pastawski2015}.

There are three separate claims:
\begin{enumerate}
\item \emph{Operator reconstruction}: the named boundary data determine
      the named bulk observable in the declared model.
\item \emph{Metric reconstruction}: boundary data determine a bulk
      metric within a specified metric class.
\item \emph{Gravitational dynamics}: microscopic assumptions imply
      specified gravitational field equations.
\end{enumerate}
Only the first is established here. Results concerning
entanglement-wedge reconstruction, entropy--area relations, or
linearized gravitational equations require their own additional
assumptions \cite{DHW2016,RT2006,Faulkner2014}.

\section{Objects fixed before the reconstruction map}
\label{sec:objects}

\subsection{Assumed geometry and independently declared boundary region}

Fix $\ell>0$. The background is the universal-cover global
$\mathrm{AdS}_2$ strip
\begin{equation}
 M=\mathbb R\times(-\pi/2,\pi/2),\qquad
 g_\ell=\frac{\ell^2}{\cos^2 x}
             \bigl(-\mathrm dt^2+\mathrm dx^2\bigr).
 \label{eq:metric}
\end{equation}
The coordinate $t$ is dimensionless. The conformally rescaled metric
extends to $x=\pm\pi/2$. The right conformal boundary is
\[
 \partial_R M=\{(t,\pi/2):t\in\mathbb R\}.
\]
Before introducing the field or a reconstruction map, declare
\begin{equation}
 I_B=[-\pi/2,\pi/2],\qquad
 B=\{(t,\pi/2):t\in I_B\},\qquad
 p_*=(0,0)\in M.
 \label{eq:region-point}
\end{equation}

Thus $B$ is a boundary \emph{time interval}. It is not a spatial
tensor-factor subregion at one time. Its definition uses the assumed
conformal completion of \eqref{eq:metric}; it is not obtained by
renaming any region from the supplied manuscripts.

The metric, its conformal completion, its coordinates, and the
location $p_*$ are inputs.

\subsection{Finite carrier and finite oscillator matrices}

Fix integers $K\ge1$ and $d\ge2$. Define two separately named copies
\[
 \HH_{\mathrm{bulk}}=(\mathbb C^d)^{\otimes K},
 \qquad
 \HH_{\partial}=(\mathbb C^d)^{\otimes K}.
\]
On $\mathbb C^d$, let
\begin{equation}
 a=\sum_{m=1}^{d-1}\sqrt m\,\kb{m-1}{m},
 \qquad
 N=\sum_{m=0}^{d-1}m\,\kb{m}{m}.
 \label{eq:finite-oscillator}
\end{equation}
Let $a_n,N_n$ denote these matrices on the $n$th bulk factor.
On the boundary copy use the independently named matrices $b_n$
with the same displayed matrix entries. The bulk Hamiltonian is
\[
 H_{\mathrm{bulk}}=\sum_{n=1}^K nN_n.
\]
Direct evaluation on the occupation basis gives
\[
 [N_n,a_n]=-a_n,\qquad
 [a_n,a_n^\dagger]
   =\Id-d\,\kb{d-1}{d-1}_n.
\]
In particular, these are finite oscillator matrices, not an exact
finite-dimensional representation of the canonical commutation
relations. No continuum canonical-field theorem is inferred from
them.

\subsection{Declared bulk field and named observable}

Define the self-adjoint bulk matrix field
\begin{equation}
 \Phi(t,x)=
 \sum_{n=1}^K
 \frac{\sin\bigl(n(x+\pi/2)\bigr)}{\sqrt{\pi n}}
 \left(a_ne^{-int}+a_n^\dagger e^{int}\right).
 \label{eq:bulk-field}
\end{equation}
All sums and all operator domains are finite. Set
\begin{equation}
 Q_{\mathrm{bulk}}=\Phi(p_*)=\Phi(0,0),
 \qquad
 \lambdastar=\frac{\pi^{3/2}}6,
 \qquad
 \Obulk=\frac12
       \left(\Id+\sin(\lambdastar Q_{\mathrm{bulk}})\right).
 \label{eq:named-observable}
\end{equation}
The sine is the ordinary functional calculus of a self-adjoint
finite matrix. Consequently
\[
 0\le\Obulk\le\Id.
\]
This bounded effect is the one named bulk observable attempted in
XVII-A.

\begin{remark}[The observable is not an unproved continuum compression]
\label{rem:compression}
The target in \eqref{eq:named-observable} is defined in the finite
model. If $P$ is an occupation cutoff in an infinite oscillator
representation, then in general
\[
 \sin(PQP)\ne P\sin(Q)P
\]
when both sides are interpreted on the cutoff space. We do not
identify our finite-matrix sine effect with the compression of an
unregulated field effect. No cutoff-removal theorem or error to a
continuum observable is asserted.
\end{remark}

\begin{proposition}[Fixed-background field equation]
\label{prop:wave}
The matrix field \eqref{eq:bulk-field} satisfies
\[
 \Box_{g_\ell}\Phi=0
\]
and has zero leading boundary value at both ends of the strip.
\end{proposition}

\begin{proof}
In two dimensions,
\[
 \Box_{g_\ell}
  =\frac{\cos^2x}{\ell^2}
          \left(-\partial_t^2+\partial_x^2\right).
\]
Each term of \eqref{eq:bulk-field} is annihilated by the operator in
parentheses because its temporal and spatial frequencies are both
$n$. The sine spatial profile vanishes at $x=-\pi/2$ and $x=\pi/2$.
Finite summation preserves both assertions.
\end{proof}

This proposition is an equation satisfied by a declared
fixed-background benchmark. It is not a derivation of the metric,
a proof of microcausality, or a canonical quantization theorem.

\section{Boundary data and the declared representation correspondence}
\label{sec:boundary}

\subsection{Boundary operator family}

On the boundary copy define
\begin{equation}
 \beta(t)=
 \sum_{n=1}^K c_n
       \left(b_ne^{-int}+b_n^\dagger e^{int}\right),
 \qquad
 c_n=(-1)^{n+1}\sqrt{\frac n\pi}.
 \label{eq:boundary-field}
\end{equation}
Let $\AA(B)$ be the unital $*$-algebra generated by
\begin{equation}
 \beta(f)=\int_{I_B}f(t)\beta(t)\dd t,
 \qquad
 f\in C_c^\infty(I_B^\circ;\mathbb R),
 \label{eq:boundary-algebra}
\end{equation}
inside $\BB(\HH_\partial)$. In finite dimension this algebra is
norm closed.

\begin{definition}[Declared data on the boundary region]
\label{def:data}
Boundary data mean a quantum state on $\AA(B)$, with its extension to
$\AA(B)\otimes\BB(\HH_R)$ when a reference system $R$ is retained.
They do not mean only the first moment of $\beta$, an occupation
count, or one finite sample of a classical measurement record.
\end{definition}

The data assumption is substantial. The paper constructs a
mathematical readout channel from these data; it does not construct
a spacetime-local detector or a sequential acquisition protocol for
the entire algebra.

\subsection{The correspondence is declared, not inferred}

Define a unitary $U:\HH_{\mathrm{bulk}}\to\HH_\partial$ by
\begin{equation}
 U\ket{m_1,\ldots,m_K}_{\mathrm{bulk}}
   =\ket{m_1,\ldots,m_K}_\partial.
 \label{eq:U}
\end{equation}
Then $Ua_nU^\dagger=b_n$. This occupation-basis correspondence is
an additional model input. It is not obtained from any decoder in
En02, any packet state in Em03, or any finite readout semantics in I.

The boundary-data interface is
\begin{equation}
 \NB(\rho)
   =\left.(U\rho U^\dagger)\right|_{\AA(B)}.
 \label{eq:NB-restriction}
\end{equation}
The following proposition verifies that the independently displayed
boundary family is the normalizable boundary limit of the displayed
bulk family under this declared correspondence.

\begin{proposition}[Actual boundary limit]
\label{prop:boundary-limit}
For every $t$,
\begin{equation}
 \lim_{r\downarrow0}
       r^{-1}\Phi(t,\pi/2-r)
       =U^\dagger\beta(t)U,
 \label{eq:boundary-limit}
\end{equation}
in operator norm. The convergence is uniform on compact $t$ sets.
\end{proposition}

\begin{proof}
For each $n$,
\[
 \sin\bigl(n(\pi-r)\bigr)
       =(-1)^{n+1}\sin(nr),
 \qquad
 r^{-1}\sin(nr)\longrightarrow n.
\]
Multiplication by $1/\sqrt{\pi n}$ gives precisely $c_n$.
The sum is finite, the oscillator matrices are bounded, and the
temporal phases have modulus one. These facts give norm convergence
and the stated uniformity.
\end{proof}

The normalization in \eqref{eq:boundary-limit} is part of the
declared dictionary. The proposition does not construct an
independent interacting boundary conformal field theory.

\subsection{A finite-cutoff localization caveat}

\begin{lemma}[Every open time interval is algebraically complete]
\label{lem:timeband}
For the finite model,
\begin{equation}
 \AA(B)=\BB(\HH_\partial).
 \label{eq:full-algebra}
\end{equation}
The same conclusion holds when $I_B^\circ$ is replaced by any
nonempty open time interval.
\end{lemma}

\begin{proof}
Write
\[
 C_n=b_n+b_n^\dagger,\qquad
 T_n=i(b_n^\dagger-b_n).
\]
Then
\[
 \beta(t)=
 \sum_{n=1}^K c_n
          \bigl(C_n\cos(nt)+T_n\sin(nt)\bigr).
\]
The $2K$ real functions
\[
 u_\alpha(t)\in
 \{\cos(nt),\sin(nt):1\le n\le K\}
\]
are linearly independent on every nonempty open interval.
Indeed, a trigonometric polynomial vanishing there is identically
zero, and its distinct Fourier coefficients then vanish.

Choose $w\in C_c^\infty(I_B^\circ)$, $w\ge0$, positive on some
nonempty open subinterval. The real Gram matrix
\[
 G_{\alpha\gamma}
    =\int w(t)u_\alpha(t)u_\gamma(t)\dd t
\]
is positive definite. The test functions
\[
 f_\alpha(t)
    =w(t)\sum_\gamma
                  (G^{-1})_{\alpha\gamma}u_\gamma(t)
\]
satisfy
\[
 \int f_\alpha(t)u_\eta(t)\dd t=\delta_{\alpha\eta}.
\]
The smearings $\beta(f_\alpha)$ therefore isolate $c_nC_n$ and
$c_nT_n$. Since $c_n\ne0$, the algebra contains
\[
 b_n=\frac12(C_n+iT_n)
 \quad\hbox{and}\quad b_n^\dagger.
\]

On one factor, $b_n^\dagger b_n$ has the distinct eigenvalues
$0,\ldots,d-1$. Polynomial functional calculus supplies all its
one-dimensional spectral projections $P_m$. Moreover,
\[
 P_{m-1}b_nP_m=\sqrt m\,\kb{m-1}{m}.
\]
Taking adjoints and products supplies every matrix unit on that
factor. The factor algebras together generate
$\BB((\mathbb C^d)^{\otimes K})$, proving the assertion.
\end{proof}

\begin{remark}[No entanglement-wedge inference]
\label{rem:no-wedge}
Lemma~\ref{lem:timeband} is a finite-cutoff consequence of finitely
many analytic frequencies and finite matrices. It neither identifies
a boundary tensor factor with a spacetime subregion nor proves
locality of the regulated bulk field. Its Gram-matrix inversion also
does not provide a resource bound or a conditioning estimate.

In particular, the present time-band result is not the spatial
subregion reconstruction theorem considered in
\cite{ADH2015,DHW2016}, and it is not a holographic tensor-network code
of the kind constructed in \cite{Pastawski2015}.
\end{remark}

By Lemma~\ref{lem:timeband}, the restricted state in
\eqref{eq:NB-restriction} has the unique density-matrix representative
$U\rho U^\dagger$. Henceforth we write
\begin{equation}
 \NB=\Ad_U,\qquad \Ad_U(\rho)=U\rho U^\dagger,
 \label{eq:NB}
\end{equation}
while retaining Definition~\ref{def:data} as the meaning of access
to $B$.

\section{Exact reconstruction of the named observable}
\label{sec:reconstruction}

\subsection{The boundary smearing}

After fixing the objects above, define
\begin{equation}
 Q_B=\frac12\int_{-\pi/2}^{\pi/2}\beta(t)\dd t,
 \qquad
 \OB=\frac12
      \left(\Id+\sin(\lambdastar Q_B)\right).
 \label{eq:boundary-representative}
\end{equation}
The integral is an operator-norm integral of a continuous
finite-matrix function.

It belongs to $\AA(B)$ even though its displayed scalar weight
has endpoints at $\partial I_B$. To see this, choose smooth
$\chi_j$ compactly supported in $I_B^\circ$ with
$\chi_j\to1$ in $L^1(I_B)$. Then
\[
 \left\|
 \frac12\int_{I_B}(1-\chi_j(t))\beta(t)\dd t
 \right\|_\infty
 \le
 \frac12\sup_{t\in I_B}\norm{\beta(t)}_\infty
              \norm{1-\chi_j}_{L^1(I_B)}
 \longrightarrow0.
\]
Norm closure and functional calculus give $Q_B,\OB\in\AA(B)$.

\begin{lemma}[Central smearing identity]
\label{lem:smearing}
The boundary smearing satisfies
\begin{equation}
 Q_B=UQ_{\mathrm{bulk}}U^\dagger.
 \label{eq:Q-identity}
\end{equation}
\end{lemma}

\begin{proof}
For every positive integer $n$,
\[
 \frac12\int_{-\pi/2}^{\pi/2}e^{\pm int}\dd t
       =\frac{\sin(n\pi/2)}n.
\]
Substituting \eqref{eq:boundary-field} into
\eqref{eq:boundary-representative} gives
\[
 Q_B=
 \sum_{n=1}^K
 \frac{(-1)^{n+1}\sin(n\pi/2)}{\sqrt{\pi n}}
                   (b_n+b_n^\dagger).
\]
If $n$ is even, the sine coefficient vanishes. If $n$ is odd,
$(-1)^{n+1}=1$. Thus
\[
 Q_B=
 \sum_{n=1}^K
 \frac{\sin(n\pi/2)}{\sqrt{\pi n}}
                   (b_n+b_n^\dagger)
       =U\Phi(0,0)U^\dagger.
\]
\end{proof}

This is the finite-model central-point version of a
fixed-background smearing construction. Its relation to the
$\Delta=1$ global $\mathrm{AdS}_2$ smearing calculation is a
comparison with \cite{HKLL2006a}, not an imported duality premise.

\subsection{The allowed reconstruction channel}

Let $\HH_C=\mathbb C^2$ be a classical output register in the
basis $\{\ket0,\ket1\}$. Declare
\begin{align}
 F_1&=\OB,\qquad F_0=\Id-\OB, \label{eq:F}\\
 \RB(\sigma)
   &=\sum_{r=0}^1\Tr(F_r\sigma)\kb r r, \label{eq:RB}\\
 \ME(\rho)
   &=\Tr((\Id-\Obulk)\rho)\kb0 0
          +\Tr(\Obulk\rho)\kb1 1. \label{eq:ME}
\end{align}
These are complete, deterministic binary record channels:
both outcomes are retained, and no postselection is used.

For a state $\omega_B$ on the boundary algebra, the same definition
is
\[
 \RB(\omega_B)=
  \omega_B(\Id-\OB)\kb0 0+\omega_B(\OB)\kb1 1.
\]
The map class is fixed: exact observable reconstruction with an
explicit completely positive trace-preserving record channel.
It is not an unrestricted quantum-state recovery problem.

\begin{theorem}[One named observable from one declared region]
\label{thm:main}
For the model of Sections~\ref{sec:objects}--\ref{sec:boundary},
the independently declared region $B$, the named observable
$\Obulk$ in \eqref{eq:named-observable}, and the map
\eqref{eq:RB}, the following statements hold.
\begin{enumerate}
\item The boundary representative belongs to the declared algebra:
      $\OB\in\AA(B)$.
\item Exact operator reconstruction holds:
      \[
       \OB U=U\Obulk,
       \qquad U^\dagger\OB U=\Obulk.
      \]
\item $\RB$ is completely positive and trace preserving.
\item Exact operational reconstruction holds:
      \[
       \RB\circ\NB=\ME.
      \]
      Equivalently, in the Heisenberg picture,
      \[
       \NB^*\!\left(\RB^*(\kb1 1)\right)=\Obulk.
      \]
\item For every finite reference system $R$ and every state
      $\rho_{\mathrm{bulk},R}$,
      \begin{equation}
       (\RB\otimes\operatorname{id}_R)
       \bigl((U\otimes\Id_R)\rho_{\mathrm{bulk},R}
                       (U^\dagger\otimes\Id_R)\bigr)
       =
       (\ME\otimes\operatorname{id}_R)
                         (\rho_{\mathrm{bulk},R}).
       \label{eq:reference-exact}
      \end{equation}
      In particular,
      \[
       \ddiam(\RB\circ\NB,\ME)=0.
      \]
\end{enumerate}
The metric and the bulk--boundary representation correspondence are
inputs. Metric reconstruction and derivation of Einstein dynamics
are not conclusions of this theorem.
\end{theorem}

\begin{proof}
Membership was proved after
\eqref{eq:boundary-representative}. Lemma~\ref{lem:smearing} and
unitary covariance of finite-matrix functional calculus give
\[
 \sin(\lambdastar Q_B)
        =U\sin(\lambdastar Q_{\mathrm{bulk}})U^\dagger.
\]
This proves the operator identities.

Since $Q_B$ is self-adjoint, $\OB$ is an effect; hence $F_0,F_1$
are positive and sum to $\Id$. A Stinespring isometry for
\eqref{eq:RB} is
\[
 W:\HH_\partial\longrightarrow
       \HH_C\otimes\mathbb C^2\otimes\HH_\partial,
 \qquad
 Wv=\sum_{r=0}^1
             \ket r_C\otimes\ket r_E\otimes\sqrt{F_r}\,v.
\]
Indeed, $W^\dagger W=F_0+F_1=\Id$. Tracing out the last two
factors yields \eqref{eq:RB}. Thus $\RB$ is CPTP.

The operator identity gives, for each outcome,
\[
 \Tr(F_rU\rho U^\dagger)
       =\Tr(U^\dagger F_rU\rho),
\]
with $U^\dagger F_1U=\Obulk$ and
$U^\dagger F_0U=\Id-\Obulk$. This proves the channel identity.

With a reference retained, the unnormalized reference block
associated with outcome $r$ is
\[
 \Tr_{\partial}
       \left[(F_r\otimes\Id_R)
           (U\otimes\Id_R)\rho_{\mathrm{bulk},R}
                    (U^\dagger\otimes\Id_R)\right].
\]
Unitary transport of the partial trace gives the corresponding
bulk block with effect $U^\dagger F_rU$. Equality of both
classical blocks proves \eqref{eq:reference-exact}. Equality of
channels gives zero diamond distance.
\end{proof}

\begin{remark}[Operator determination is not a single-shot estimate]
Knowing a state on $\AA(B)$ determines
$\omega_B(\OB)$ mathematically. One attempted measurement produces
a random binary outcome, not its exact probability. No finite-sample
tomography or statistical confidence claim is included in
Theorem~\ref{thm:main}.
\end{remark}

\begin{remark}[Functional calculus is not applied to a first moment]
For a general input state,
\[
 \omega_B\bigl(\sin(\lambdastar Q_B)\bigr)
 \ne
 \sin\bigl(\lambdastar\omega_B(Q_B)\bigr).
\]
The reconstruction therefore uses the declared observable algebra,
not merely a mean boundary waveform.
\end{remark}

\section{A conditional, reference-stable error budget}
\label{sec:stability}

For states and channels use
\[
 \dtr(\rho,\sigma)=\frac12\norm{\rho-\sigma}_1,
 \qquad
 \ddiam(\mathcal C,\mathcal D)
       =\frac12\norm{\mathcal C-\mathcal D}_\diamond.
\]
These conventions retain arbitrary reference systems in channel
comparisons.

\begin{lemma}[Exact distance between binary record channels]
\label{lem:binary-diamond}
Let $E,F$ be effects on the same finite-dimensional Hilbert space.
For their binary record channels,
\begin{equation}
 \ddiam(\mathcal M_E,\mathcal M_F)
             =\norm{E-F}_\infty.
 \label{eq:binary-diamond}
\end{equation}
\end{lemma}

\begin{proof}
Set $D=E-F$. With any reference, the output difference has blocks
$-Y,Y$, where
\[
 Y=\Tr_{\HH}\bigl[(D\otimes\Id)X\bigr]
\]
for an input trace-class operator $X$. Trace-norm duality gives
\[
 \norm{Y}_1
  =\sup_{\norm{J}_\infty\le1}
          \abs{\Tr((D\otimes J)X)}
  \le\norm{D}_\infty\norm{X}_1.
\]
The two output blocks have total trace norm $2\norm{Y}_1$.
This proves the upper bound on the diamond norm. A pure eigenstate
of the self-adjoint matrix $D$ corresponding to an eigenvalue of
largest absolute value attains that bound without a reference.
\end{proof}

\begin{lemma}[Sine perturbation bound]
\label{lem:sine}
For self-adjoint finite matrices $A,B$,
\[
 \norm{\sin A-\sin B}_\infty\le\norm{A-B}_\infty.
\]
\end{lemma}

\begin{proof}
The Duhamel identity
\[
 e^{iA}-e^{iB}
   =i\int_0^1 e^{i(1-s)A}(A-B)e^{isB}\dd s
\]
gives $\norm{e^{iA}-e^{iB}}_\infty\le\norm{A-B}_\infty$.
Apply the same argument to $-A,-B$ and use
$\sin A=(e^{iA}-e^{-iA})/(2i)$.
\end{proof}

\begin{proposition}[Named conditional perturbation allowance]
\label{prop:stability}
Suppose a self-adjoint boundary matrix family
$\widetilde\beta(t)$ satisfies
\[
 \sup_{t\in I_B}
       \norm{\widetilde\beta(t)-\beta(t)}_\infty
       \le\delta.
\]
Define $\widetilde Q_B,\widetilde\OB,\widetilde\RB$ by the same
integral, functional calculus, and complete record prescription as
before. Then
\begin{equation}
 \ddiam(\widetilde\RB\circ\NB,\ME)
       \le
 \min\left\{1,\frac{\abs{\lambdastar}\pi}{4}\delta\right\}.
 \label{eq:channel-perturbation}
\end{equation}
More generally, if an actual CPTP boundary interface
$\widetilde{\mathcal N}$ obeys
$\ddiam(\widetilde{\mathcal N},\NB)\le\eta$, then
\begin{equation}
 \ddiam(\widetilde\RB\circ\widetilde{\mathcal N},\ME)
       \le
 \min\left\{1,\eta+
                \frac{\abs{\lambdastar}\pi}{4}\delta\right\}.
 \label{eq:complete-budget}
\end{equation}
\end{proposition}

\begin{proof}
The interval has length $\pi$, so
\[
 \norm{\widetilde Q_B-Q_B}_\infty
   \le\frac{\pi}{2}\delta.
\]
Lemma~\ref{lem:sine} yields
\[
 \norm{\widetilde\OB-\OB}_\infty
   \le\frac{\abs{\lambdastar}}2
                         \norm{\widetilde Q_B-Q_B}_\infty
   \le\frac{\abs{\lambdastar}\pi}{4}\delta.
\]
Lemma~\ref{lem:binary-diamond}, unitary precomposition, and
Theorem~\ref{thm:main} prove
\eqref{eq:channel-perturbation}. CPTP postprocessing contracts
diamond distance, so the triangle inequality gives
\eqref{eq:complete-budget}. Half-diamond distance between channels
is at most one.
\end{proof}

This is a conditional mathematical allowance. It does not establish
an implemented apparatus, a bound on resources needed to access
$\AA(B)$, or convergence as $K,d\to\infty$. No value of $\delta$
or $\eta$ is reported as experimentally or numerically certified.

\section{One analytical diagnostic}
\label{sec:diagnostic}

Specialize the declared model to $K=1$, $d=2$. Let
\[
 X=\begin{pmatrix}0&1\\1&0\end{pmatrix},
 \qquad
 Y=\begin{pmatrix}0&-i\\i&0\end{pmatrix},
 \qquad
 Z=\begin{pmatrix}1&0\\0&-1\end{pmatrix}.
\]
Then
\begin{equation}
 Q_{\mathrm{bulk}}=\frac{X}{\sqrt\pi},
 \qquad
 \beta(t)=\frac{X\cos t+Y\sin t}{\sqrt\pi},
 \qquad
 Q_B=\frac{X}{\sqrt\pi}.
 \label{eq:diagnostic-fields}
\end{equation}
Since $\lambdastar/\sqrt\pi=\pi/6$ and $X^2=\Id$,
\begin{equation}
 \Obulk=\OB
   =\frac12\Id+\frac14X.
 \label{eq:diagnostic-effect}
\end{equation}

Choose the explicitly prepared state
\[
 \ket\psi=\frac{\sqrt3}{2}\ket0+\frac12\ket1,
 \qquad \rho_\psi=\kb\psi\psi.
\]
Its Pauli expectations are
\[
 \Tr(X\rho_\psi)=\frac{\sqrt3}{2},
 \qquad
 \Tr(Y\rho_\psi)=0.
\]
Thus the boundary mean waveform is
\begin{equation}
 \omega_B(\beta(t))
       =\frac{\sqrt3}{2\sqrt\pi}\cos t.
 \label{eq:diagnostic-waveform}
\end{equation}
For this particular two-dimensional effect, the boundary algebra
identity \eqref{eq:diagnostic-effect} also gives
\[
 \omega_B(\OB)
   =\frac12+\frac{\sqrt\pi}{4}\omega_B(\beta(0))
   =\frac12+\frac{\sqrt3}{8}.
\]
The direct bulk expression agrees:
\begin{equation}
 \Tr(\Obulk\rho_\psi)=\frac12+\frac{\sqrt3}{8}.
 \label{eq:diagnostic-agreement}
\end{equation}
The complete reconstructed record is therefore
\[
 \RB(\NB(\rho_\psi))
  =
 \left(\frac12-\frac{\sqrt3}{8}\right)\kb0 0
 +
 \left(\frac12+\frac{\sqrt3}{8}\right)\kb1 1.
\]

This is an exact analytical evaluation, not a numerical execution.
The special first-moment formula used here follows from
\eqref{eq:diagnostic-effect}; it is not an assertion that arbitrary
nonlinear field effects are determined by a first moment.
VII-G's instrument, VII-H's susceptibility, and Em04-A's concurrence
are not this diagnostic \cite{CheungVII,CheungEm04}.

\section{The reconstructed record does not recover a second observable}
\label{sec:record-obstruction}

The input quantum algebra on $B$ must not be confused with the
classical record returned by $\RB$. Lemma~\ref{lem:timeband} concerns
the former. The following obstruction concerns only the latter.

\begin{lemma}[Indistinguishable-input obstruction]
\label{lem:indistinguishable}
Let $\mathcal C$ be a channel and let $J$ be an effect. If
\[
 \mathcal C(\rho_0)=\mathcal C(\rho_1),
 \qquad
 \abs{\Tr(J\rho_0)-\Tr(J\rho_1)}=\Delta,
\]
then every scalar estimator $h$ depending only on the output of
$\mathcal C$ has worst-case expectation error at least $\Delta/2$:
\[
 \sup_\rho
  \abs{h(\mathcal C(\rho))-\Tr(J\rho)}
       \ge\frac{\Delta}{2}.
\]
The assertion does not require $h$ to be linear.
\end{lemma}

\begin{proof}
The estimator assigns the same value $s$ to the two equal outputs.
For the two target values $j_0,j_1$,
\[
 \Delta=\abs{j_0-j_1}
          \le\abs{j_0-s}+\abs{s-j_1}.
\]
At least one error is at least $\Delta/2$.
\end{proof}

\begin{proposition}[Sharp obstruction after the named binary readout]
\label{prop:record-obstruction}
In the specialization $K=1,d=2$, define the second bulk effect
\[
 J=\kb0 0=\frac12(\Id+Z).
\]
For the already fixed record channel
$\mathcal C=\RB\circ\NB$, no processing of its record determines
$\Tr(J\rho)$ for every bulk state. The optimal worst-case scalar
expectation error is exactly $1/2$.

Moreover, every CPTP decoder $\mathcal D$ from that record to a bulk
qubit satisfies
\[
 \ddiam(\mathcal D\circ\mathcal C,
                     \operatorname{id}_{\HH_{\mathrm{bulk}}})
       \ge\frac12.
\]
No optimality claim for this quantum-channel lower bound is made.
\end{proposition}

\begin{proof}
By \eqref{eq:diagnostic-effect},
\[
 \mathcal C(\kb0 0)
       =\mathcal C(\kb1 1)
       =\frac12\kb0 0_C+\frac12\kb1 1_C.
\]
The corresponding target probabilities for $J$ are $1$ and $0$.
Lemma~\ref{lem:indistinguishable} gives the lower bound $1/2$.
The constant estimator $h=1/2$ attains it because every effect
probability lies in $[0,1]$.

For a CPTP decoder, both inputs give the same decoded state $\tau$.
Since $\dtr(\kb0 0,\kb1 1)=1$, the trace-distance triangle inequality
implies
\[
 \max\{\dtr(\tau,\kb0 0),\dtr(\tau,\kb1 1)\}\ge\frac12.
\]
Worst-case input-state distance is a lower bound on half-diamond
distance, proving the channel assertion.
\end{proof}

This is a proved obstruction for a stated processing class and
a stated output interface. It is not an obstruction to access to
the full boundary quantum algebra. The same distinction underlies
observable-algebra formulations of recovery \cite{BKK2007}.

The effect $J$ is a finite-mode population effect. It is not called
a black-hole interior observable. Neither the proposition nor the
reference system constructs an interior partner.

\section{The maps not supplied}
\label{sec:missing-maps}

\subsection{A concrete metric nonidentifiability statement}

The background radius was fixed as an input in
\eqref{eq:metric}. Nevertheless, one can ask whether the particular
boundary data declared here would identify it if it were unknown.

\begin{proposition}[Radius is not determined by the declared data]
\label{prop:radius}
Consider the family of models obtained by replacing $\ell$ in
\eqref{eq:metric} by any positive radius, while keeping the
dimensionless coordinates, $K,d$, the displayed mode normalization,
$U$, and the corresponding preparation fixed.

All boundary operator families, boundary states, and reconstructed
records declared in this paper are identical throughout this
family. Therefore no deterministic rule using only these data can
return the correct radius for every member of the family.
The corresponding metrics have different scalar curvature when
their radii differ.
\end{proposition}

\begin{proof}
The finite matrices \eqref{eq:finite-oscillator}, the bulk expansion
\eqref{eq:bulk-field}, the boundary family
\eqref{eq:boundary-field}, and the correspondence \eqref{eq:U}
contain no $\ell$. Consequently the declared boundary states and
the record channel are also independent of $\ell$. A deterministic
rule applied to identical data must return the same answer.

For completeness, write
\[
 g_\ell=e^{2\sigma}(-\mathrm dt^2+\mathrm dx^2),
 \qquad \sigma=\log\ell-\log\cos x.
\]
The two-dimensional conformal curvature formula gives
\[
 \operatorname{Scal}(g_\ell)
   =-2e^{-2\sigma}\partial_x^2\sigma
   =-\frac2{\ell^2}.
\]
Distinct positive radii therefore give distinct scalar curvatures.
\end{proof}

The proposition concerns exactly the dimensionless and
normalization-fixed data specified here. Adding an independently
calibrated proper-time scale, a stress-tensor normalization, or
other radius-sensitive information changes the input and is not
covered by the obstruction.

\subsection{No derivation of dynamics or causal structure}

Proposition~\ref{prop:wave} uses an already specified metric.
There is no dynamical metric variable, gravitational action,
gravitational constraint algebra, or backreaction equation in the
finite model. Thus the paper proves no gravitational field equation.
This is not a nonexistence theorem for every possible gravitational
completion; it is the exact boundary of the present result.

The lightlike directions of \eqref{eq:metric} are likewise assumed.
Neither Lemma~\ref{lem:timeband} nor the record channel derives a
causal structure. The finite-matrix regulator is not promoted to a
local relativistic quantum field.

The distinction from entropy-based gravitational arguments is
substantive: no Ryu--Takayanagi relation or entanglement-first-law
hypothesis appears in the proof \cite{RT2006,Faulkner2014}.

\subsection{No interior, entropy--area, or evaporation conclusion}

The model supplies none of the following:
\begin{itemize}
\item an interior-partner operator or a black-hole interior algebra;
\item an entropy--area relation, a geometric area observable, or an
      entanglement wedge;
\item a Page curve, an island prescription, Hawking evaporation,
      backreaction, or an evaporation endpoint;
\item microscopic gravitational dynamics or a quantum coefficient
      of an inherited gravitational response.
\end{itemize}

A formal complementary output is not an interior partner. A finite
Hilbert-space dimension is not an area. A scalar field equation on
a chosen background is not Einstein dynamics. These distinctions
are compatible with the broader reconstruction frameworks
\cite{ADH2015,DHW2016}; those frameworks are not assumptions that
can silently enlarge Theorem~\ref{thm:main}.

\section{Source-alignment and non-premise ledger}
\label{sec:ledger}

\subsection{Status of the supplied Cheung manuscripts}

The references to Cheung below identify the supplied manuscript
versions and their reported scopes. They are comparisons and
non-premise records, not an independent verification of the full
manuscripts. No theorem in Sections~\ref{sec:objects}--%
\ref{sec:missing-maps} relies on an unprovided result from them.

In particular, an emission-law calculation on an assumed metric is
not already operator reconstruction from a declared boundary region.
A reconstruction theorem additionally needs the boundary objects,
the connecting map, and its proof.

{\small
\renewcommand{\arraystretch}{1.15}
\begin{longtable}{@{}p{0.17\textwidth}p{0.38\textwidth}p{0.39\textwidth}@{}}
\toprule
Source & Reported supply & Status in XVII-A\\
\midrule
\endfirsthead
\toprule
Source & Reported supply & Status in XVII-A\\
\midrule
\endhead
\bottomrule
\endfoot

I \cite{CheungI}
&
Finite instruments, records, ownership, and budgeted readout
stability.
&
Not an encoding or a bulk--boundary map. Not a premise.
\\[0.4em]

V; VII-A \cite{CheungV,CheungVII}
&
Represented responses and a relational-scalar plateau result
in their separately declared models.
&
Candidate bulk-side comparisons only. No identification with
\eqref{eq:named-observable} is asserted. Not premises.
\\[0.4em]

VII-E \cite{CheungVII}
&
Scheduled dependency and a directional velocity set.
&
Not a derived light cone or the region $B$. Not a premise.
\\[0.4em]

VII-G; VII-H \cite{CheungVII}
&
A finite hybrid instrument on a fixed cone; a free massless
Minkowski susceptibility.
&
Neither supplies the present boundary family, effect, or
readout. Not premises.
\\[0.4em]

VIII-A--C \cite{CheungVIII}
&
A source-free conical dictionary, a variable-deficit family,
and prescribed-schedule support bounds.
&
The zero-cosmological-constant conical data are not the AdS
background \eqref{eq:metric}. No collar is renamed a boundary.
Not premises.
\\[0.4em]

En02-A \cite{CheungEn02}
&
Cut conservation, register cost $N_0$, and conditional Born
rigidity.
&
Resource counters are not geometry. The X-Enter occupancy
class is not $N_0$. Not a premise.
\\[0.4em]

En02-B \cite{CheungEn02}
&
An exact two-live-qubit perfect-code decoder and a
module-class tree bound.
&
Not the correspondence $U$, an entanglement wedge, or
unrestricted decoder optimality. Not a premise.
\\[0.4em]

En02-C \cite{CheungEn02}
&
Exact finite-prefix recovery and a specified one-live-qubit
streaming error.
&
Not the record obstruction of
Proposition~\ref{prop:record-obstruction}, despite possible
numerical coincidence of a bound. Not a premise.
\\[0.4em]

En02-D \cite{CheungEn02}
&
A transmission-dependent memory gap and a prepared entropy
sequence; an unexecuted computational supplement.
&
Not a Page curve or Hawking dynamics. No supplement execution
is imported. Not a premise.
\\[0.4em]

En02-E \cite{CheungEn02}
&
A finite Gaussian packet state and flagged cutoff; a benchmark
calibrated by nonvacuum probability.
&
Not a metric reconstruction or the present finite-matrix
effect. Not a premise.
\\[0.4em]

Em03-A \cite{CheungEm03}
&
A fixed-background free-scalar emission law, retaining an
external bulk Unruh-state theorem.
&
No Unruh state or Schwarzschild emission law is used here.
An emission law alone is not the required connecting map.
Not a premise.
\\[0.4em]

Em03-B--D \cite{CheungEm03}
&
An apparatus-model packet readout, a joint-packet interface,
and a compact local probe for a bounded s-wave effect.
&
No detector, packet-transfer, or local-probe identification
with $\RB$ is asserted. Exterior correlations are not
interior partners. Not premises.
\\[0.4em]

Em03-E--F \cite{CheungEm03}
&
A finite message-dependent emission model and an exterior
source channel with an entanglement-breaking packet output.
&
Neither supplies $U$ or $\NB$. Classical source dependence
is not quantum recovery. Not premises.
\\[0.4em]

Em03-G--H \cite{CheungEm03}
&
An ordered two-pulse source and a version-8 quartic bath
channel.
&
Not a holographic connecting map. Reported P8 numbers are
not validated here. Not premises.
\\[0.4em]

Em04-A; Em05-A--B
\cite{CheungEm04,CheungEm05}
&
An XY reduction, signed extraction, and complementary
leakage results.
&
Neither Einstein dynamics nor the present boundary
reconstruction. Not premises.
\\[0.4em]

XIV-H--J \cite{CheungXIV}
&
An internal-symbol stop, an internal amplitude construction,
and a supported-sector reference away from the characteristic
locus.
&
No full joint characteristic map or inherited quantum
coefficient is constructed here. Not premises.
\\[0.4em]

XV-B \cite{CheungXV}
&
A supplied technical interface for descent and supported
extensions.
&
Not used here and not opened as a downstream admission exit.
No coefficient admission is inferred. Not a premise.
\\[0.4em]

XVI-N; XVI-O \cite{CheungXVI}
&
A conditional prepared-insertion obstruction and a
kernel counterterm under a stated prescription.
&
Not a bulk BV restoration or a holographic map. XVI-N's
Route B is not discharged; O.1 remains
\texttt{NOT\_CLOSED}. Not premises.
\\

\end{longtable}
}

\subsection{What is, and is not, inherited}

All formal premises of the reconstruction theorem are displayed in
this paper:
\begin{enumerate}
\item the fixed metric and conformal boundary;
\item the finite Hilbert spaces and matrices;
\item the finite bulk and boundary mode expansions;
\item the declared occupation-basis correspondence $U$;
\item quantum access to the stated boundary algebra.
\end{enumerate}
These are not claimed to follow from the supplied manuscript ledger.

Cheung's XIV-J distinguishes an absent full map from a universal
obstruction \cite{CheungXIV}. The same logical caution applies here:
the lack of an inherited connecting map is not a theorem that
every such map is impossible. The inherited candidate versions of
XVII-A remain unconstructed, rather than universally obstructed.

The elementary obstruction in
Proposition~\ref{prop:record-obstruction} has a different object,
domain, and map class from H2, I3, or J2. Similar language does not
identify these problems.

\section{Decision gates and exact release}
\label{sec:release}

\subsection{Decision gates}

{\small
\renewcommand{\arraystretch}{1.15}
\begin{tabular}{@{}p{0.22\textwidth}p{0.72\textwidth}@{}}
\toprule
Gate & Outcome in this paper\\
\midrule
Objects named
&
$B$ and $p_*$ are fixed in \eqref{eq:region-point};
$\Obulk$ is fixed in \eqref{eq:named-observable}, before
the boundary smearing and record map.
\\[0.4em]

Map class fixed
&
Exact finite operator reconstruction with the CPTP
binary record channel \eqref{eq:RB}.
\\[0.4em]

Route fixed
&
The metric, coordinates, boundary completion, and finite
dictionary are inputs. No metric or Einstein derivation.
\\[0.4em]

Non-premises marked
&
The supplied manuscripts are explicitly separated in
Section~\ref{sec:ledger}.
\\[0.4em]

Geometry check
&
$B$ lies on the assumed conformal boundary of a newly
declared AdS model. No inherited collar, schedule
region, or packet support is renamed.
\\[0.4em]

Interior check
&
No interior partner, wedge, island, or evaporation
channel is asserted.
\\[0.4em]

Diagnostic
&
One exact analytical comparison is evaluated in
Section~\ref{sec:diagnostic}; no numerical execution.
\\[0.4em]

Cutoff check
&
The target is a finite-matrix sine effect.
Remark~\ref{rem:compression} excludes an unproved
continuum compression or cutoff-removal claim.
\\
\bottomrule
\end{tabular}
}

\subsection{Release theorem}

\begin{theorem}[XVII-A finite-model release]
\label{thm:release}
For the newly declared finite-matrix benchmark:
\begin{enumerate}
\item the named bulk effect has an exact representative in the
      algebra of the independently declared conformal-boundary
      region $B$;
\item the declared complete binary record channel reproduces its
      bulk measurement channel exactly, including reference
      systems;
\item the conditional perturbation allowance is
      \eqref{eq:complete-budget};
\item the diagnostic probability is exactly
      $1/2+\sqrt3/8$;
\item the named record does not reconstruct the noncommuting
      population effect $J$, with optimal scalar error $1/2$;
\item the declared dimensionless boundary data do not identify
      the background radius within the family of
      Proposition~\ref{prop:radius}.
\end{enumerate}

The ending is \textbf{Constructed: exact finite-model operator
reconstruction and complete binary readout} for this benchmark.

No identification with an inherited VII-A, V, or Em03 observable,
continuum field, boundary CFT, entanglement wedge, or gravitational
microscopic model is released. Those connecting problems are not
discharged by this theorem.
\end{theorem}

\begin{proof}
Theorem~\ref{thm:main} proves the operator and referenced channel
identities. Proposition~\ref{prop:stability} proves the conditional
budget. Equation~\eqref{eq:diagnostic-agreement} evaluates the
diagnostic. Proposition~\ref{prop:record-obstruction} proves the
stated processing obstruction.
Proposition~\ref{prop:radius} proves the metric nonidentifiability
statement. Each assertion retains its displayed model and input
class.
\end{proof}

\subsection{Retained programme boundaries}

This holographic-reconstruction comparison is not XVI-N's Route B.
No result here discharges that route. The paper makes no new
assertion resolving H2, I3, or J2, does not open XV-B as an
admission exit, and does not close O.1. No inherited quantum
coefficient is admitted \cite{CheungXIV,CheungXV,CheungXVI}.

The distinction between the classical-limit project associated
with Em06 in the supplied plan and the present boundary
reconstruction project is retained. Neither is enlarged into a
derivation of classical spacetime by the present finite theorem.

\paragraph{Evidence status.}
All results released here are analytical finite-matrix results.
The text does not assert executed numerical checks, successful
pdfLaTeX compilation, machine verification, a laboratory
realization, an operational resource optimum, or registration of
XVII-A. The source is self-contained and requires no external
bibliography file.

\paragraph{Conclusion.}
One operator has been reconstructed from one declared boundary
region in one explicit finite model. The representation
correspondence and geometry were not inferred from the
reconstruction. The complete record is weaker than quantum-state
recovery, and the declared data do not identify even the background
radius in the displayed normalization-fixed family. Consequently
the completed finite theorem is not holographic emergence of
classical spacetime.

\begin{thebibliography}{99}

\bibitem{HKLL2006a}
A.~Hamilton, D.~Kabat, G.~Lifschytz, and D.~A.~Lowe,
\emph{Local bulk operators in AdS/CFT: A boundary view of horizons
and locality},
Phys.\ Rev.\ D \textbf{73}, 086003 (2006).
arXiv:hep-th/0506118.
\url{https://arxiv.org/abs/hep-th/0506118}.

\bibitem{HKLL2006b}
A.~Hamilton, D.~Kabat, G.~Lifschytz, and D.~A.~Lowe,
\emph{Holographic representation of local bulk operators},
Phys.\ Rev.\ D \textbf{74}, 066009 (2006).
arXiv:hep-th/0606141.
\url{https://arxiv.org/abs/hep-th/0606141}.

\bibitem{BKK2007}
C.~B\'eny, A.~Kempf, and D.~W.~Kribs,
\emph{Generalization of quantum error correction via the
Heisenberg picture},
Phys.\ Rev.\ Lett.\ \textbf{98}, 100502 (2007).
arXiv:quant-ph/0608071.
\url{https://arxiv.org/abs/quant-ph/0608071}.

\bibitem{ADH2015}
A.~Almheiri, X.~Dong, and D.~Harlow,
\emph{Bulk locality and quantum error correction in AdS/CFT},
JHEP \textbf{04} (2015), 163.
arXiv:1411.7041.
\url{https://arxiv.org/abs/1411.7041}.

\bibitem{Pastawski2015}
F.~Pastawski, B.~Yoshida, D.~Harlow, and J.~Preskill,
\emph{Holographic quantum error-correcting codes: Toy models for
the bulk/boundary correspondence},
JHEP \textbf{06} (2015), 149.
arXiv:1503.06237.
\url{https://arxiv.org/abs/1503.06237}.

\bibitem{DHW2016}
X.~Dong, D.~Harlow, and A.~C.~Wall,
\emph{Reconstruction of bulk operators within the entanglement
wedge in gauge-gravity duality},
Phys.\ Rev.\ Lett.\ \textbf{117}, 021601 (2016).
arXiv:1601.05416.
\url{https://arxiv.org/abs/1601.05416}.

\bibitem{RT2006}
S.~Ryu and T.~Takayanagi,
\emph{Holographic derivation of entanglement entropy from AdS/CFT},
Phys.\ Rev.\ Lett.\ \textbf{96}, 181602 (2006).
arXiv:hep-th/0603001.
\url{https://arxiv.org/abs/hep-th/0603001}.

\bibitem{Faulkner2014}
T.~Faulkner, M.~Guica, T.~Hartman, R.~C.~Myers, and
M.~Van Raamsdonk,
\emph{Gravitation from entanglement in holographic CFTs},
JHEP \textbf{03} (2014), 051.
arXiv:1312.7856.
\url{https://arxiv.org/abs/1312.7856}.

\bibitem{CheungI}
N.~Cheung,
\emph{I}, version 1, manuscript, n.d.
Enter Calculus Companion; supplied excerpts.
Relevant supplied locator:
\texttt{thm:stability}.
Publication date and independent verification are not supplied.
Archive description:
\url{https://enter.grok.me}.

\bibitem{CheungV}
N.~Cheung,
\emph{V}, version 1, manuscript, n.d.
Enter Calculus Companion; supplied excerpts.
Relevant supplied locators:
scope and non-goals;
\texttt{eq:mixedcore};
\texttt{eq:nonlocalcomm}.
This citation identifies the supplied manuscript, not an
independent holographic identification.

\bibitem{CheungVII}
N.~Cheung,
\emph{VII-A, VII-E, VII-G, and VII-H},
version 1 manuscripts, n.d.
Enter Calculus Companion; supplied excerpts.
Relevant supplied locators:
VII-A \texttt{thm:first};
VII-E \texttt{prop:direction};
VII-G \texttt{thm:locality} and the reported fixed-cone
finite-model interface;
VII-H \texttt{thm:release}.
The four manuscripts have distinct models and are grouped here
only for the non-premise ledger.

\bibitem{CheungVIII}
N.~Cheung,
\emph{VIII-A, VIII-B, and VIII-C},
version 2 manuscripts, n.d.
Enter Calculus Companion; supplied excerpts.
Relevant supplied locators:
VIII-A \texttt{A:thm:A1} and
\texttt{A:cor:spectators};
VIII-B \texttt{thm:dictionary} and
\texttt{thm:spectators};
VIII-C \texttt{thm:C5}.
These conical and schedule interfaces are not AdS data.

\bibitem{CheungEn02}
N.~Cheung,
\emph{En02-A, En02-B, En02-C, En02-D, and En02-E},
manuscripts, n.d.
Supplied versions:
A--C, version 1; D, version 4; E, version 2.
Enter Calculus Companion; supplied excerpts.
Relevant supplied locators:
A \texttt{thm:E} and the register/Born ledger;
B \texttt{thm:tree} and the reported
\texttt{thm:B5};
C the reported \texttt{thm:C2}, \texttt{thm:C3},
and supplied \texttt{thm:C5};
D abstract and evidence limitations;
E scope, dependency, and evidence statements.
No execution of their computational supplements is inferred.

\bibitem{CheungEm03}
N.~Cheung,
\emph{Em03-A through Em03-H},
manuscripts, n.d.
Supplied versions:
A and C--G, version 1; B, version 2; H, version 8.
Enter Calculus Companion; supplied excerpts.
Relevant supplied locators:
A--G \texttt{thm:release};
H \texttt{thm:channels};
the supplied packet, source, probe, and evidence limitations.
The separate manuscripts are not identified with the finite
AdS benchmark of this paper.

\bibitem{CheungEm04}
N.~Cheung,
\emph{Em04-A}, version 2, manuscript, n.d.
Enter Calculus Companion; supplied excerpts.
Relevant supplied locator:
\texttt{thm:release}.
Source alignment, numerical execution, and compilation are not
inferred from that analytical release.

\bibitem{CheungEm05}
N.~Cheung,
\emph{Em05-A and Em05-B},
version 1 manuscripts, n.d.
Enter Calculus Companion; supplied excerpts.
Relevant supplied locators:
Em05-A \texttt{thm:release};
Em05-B revision audit and remaining verification work.
Complementary leakage is not an interior-partner construction.

\bibitem{CheungXIV}
N.~Cheung,
\emph{XIV-H, XIV-I, and XIV-J},
version 1 manuscripts, n.d.
Enter Calculus Companion; supplied excerpts.
Relevant supplied locators:
XIV-H \texttt{thm:H2};
XIV-I \texttt{thm:I2}, \texttt{thm:I3},
and \texttt{thm:release};
XIV-J \texttt{prop:descent},
\texttt{thm:reference},
\texttt{thm:conditionalextension},
\texttt{thm:J2},
\texttt{cor:noadmission},
and \texttt{thm:release}.
Their supplied partial constructions do not constitute the
full inherited characteristic map.

\bibitem{CheungXV}
N.~Cheung,
\emph{XV-B}, version 1, manuscript, n.d.
Enter Calculus Companion; supplied excerpts.
Relevant supplied locator:
\texttt{thm:release}.
The supplied technical interface does not construct or admit
the inherited six-branch quantum coefficient.

\bibitem{CheungXVI}
N.~Cheung,
\emph{XVI-N and XVI-O},
manuscripts, n.d.
Supplied versions:
XVI-N, r12; XVI-O, o70-sync.
Enter Calculus Companion; supplied excerpts.
Relevant supplied locators:
XVI-N \texttt{thm:massless-obstruction};
XVI-O abstract and the reported kernel-prescription boundaries.
XVI-N's Route B is not discharged here, and bulk O.1 remains
\texttt{NOT\_CLOSED}.

\end{thebibliography}

\end{document}