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% EVIDENCE BOUNDARY
% This manuscript supplies analytical constructions and proofs.
% Compilation, machine verification, numerical execution, calibration,
% and laboratory realization are not asserted.
%
% The field-state prescription is inherited explicitly from the supplied
% Em03-A-v1.tex. The effective apparatus is a declared model, not a local
% Schwarzschild detector.
%
% The joint capture, finite measurement family, statistical certificate,
% and discrimination results are constructed below. Local-field matching,
% collapse-packet convergence, and microscopic evaporation dynamics remain
% separately identified obligations.
%=====================================================================

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\hypersetup{
  pdftitle={Em03-C: Joint Hawking-Packet Correlations and Certified Tests
            of Semiclassical Departures},
  pdfauthor={Ngo Cheung},
  pdfsubject={Joint Gaussian states, shared-port capture,
              informationally complete readout, and discrimination}
}

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\title{\textbf{Em03-C}\\[1mm]
\textbf{Joint Hawking-Packet Correlations and Certified Tests
of Semiclassical Departures}\\[2mm]
\large Finite-mode states, information-sensitive readout,
and local-field matching obligations}

\author{Ngo Cheung\\
Independent Researcher\\
Hong Kong SAR, China\\
\texttt{info@cheungngomedical.com}}

\date{Version 1 --- 8 October 2026}

\begin{document}

\maketitle

\begin{abstract}
We construct a two-packet operational interface for the stationary
free-scalar Hawking model. The inherited outgoing quasifree
prescription determines a normal joint Gaussian density, its joint
count generating function, and its interpacket number covariance.
Orthogonal packet modes are independent precisely when their
off-diagonal occupation covariance vanishes.

The apparatus construction uses two memories coupled to one effective
temporal port. Compact temporal tests are corrected at zero frequency,
jointly orthonormalized, and compared with the desired packet plane.
An explicit matrix-valued passive interaction captures both tests
during the same switching interval. This avoids treating overlapping
windows as independent apparatuses. Joint preparation errors,
memory and register coupling, and correlated assignment noise enter
one complete, reference-stable budget.

After coherent extraction, bounded single-qubit rotations and
destructive pointer readout implement an informationally complete
Pauli measurement family. We prove its effects, its complete-record
implementation bound, an independent-trial statistical estimate, and
a trace-distance reconstruction certificate. Destructive count tails
and coherent state-approximation tails are distinguished.

A bounded-effect discrimination theorem translates a specified
state departure into a certifiable record departure. A separate
input-information theorem proves that an input-independent radiation
channel cannot recover an unknown microscopic label or preserve its
reference entanglement. The results concern a declared effective
apparatus and a fixed-background baseline. They do not construct a
local Schwarzschild detector, a collapse packet limit, a
quantum-gravity correction, or microscopic evaporation dynamics.
\end{abstract}

\tableofcontents

%=====================================================================
\section{Scope, dependencies, and the release boundary}
\label{sec:scope}
%=====================================================================

The progression is
\[
\boxed{
\begin{gathered}
\text{Em03-A: a fixed-background emission observable}\\
\longrightarrow
\text{Em03-B: one-packet effective transfer and readout}\\
\longrightarrow
\text{Em03-C: joint states and information-sensitive tests}.
\end{gathered}
}
\]

The supplied sources \cite{Em03A,Em03B} do not contain an
unknown black-hole microstate as a dynamical input. This manuscript
does not insert one implicitly. A correction state, when discussed,
must be supplied as additional model data.

Three levels remain separate:
\begin{enumerate}[label=\textup{(\roman*)},leftmargin=*]
\item \textbf{Radiation-state interface:}
a specified algebraic field state and its normal finite-mode
restrictions;

\item \textbf{Effective apparatus model:}
a temporal port, memories, register controls, preparations,
and complete readout instruments;

\item \textbf{Physical correspondence:}
a local field--probe coupling, its relation to the effective
port, and a calibrated implementation.
\end{enumerate}
Only the first two levels are constructed here.

Figure~\ref{fig:dependency} records the dependency order used below.
Solid arrows are channels or restrictions defined on their stated
input spaces. Dashed boxes are comparisons that use a named state,
not channels. The third level is drawn only as an open obligation.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[node distance=2.4mm]
\node[qbox,text width=3.15cm] (rad) {Radiation interface\\ $n(\omega)$, $A$};
\node[qbox,text width=3.15cm,right=of rad] (time) {Compact tests\\ $f_1,f_2$};
\node[qbox,text width=3.15cm,right=of time] (port) {Shared port\\ $\lambda,\Omega,\tau$};
\node[qbox,text width=3.15cm,below=of port] (store) {Storage\\ $s$, $\tau_*=\tau s$};
\node[qbox,text width=3.15cm,left=of store] (ext) {Extraction\\ $U_{\boldsymbol a,\boldsymbol K}$};
\node[qbox,text width=3.15cm,left=of ext] (rot) {Settings\\ $U_X,U_Y$};
\node[cbox,text width=3.15cm,below=of rot] (ptr) {Pointer and $W$\\ failures kept};
\node[cbox,text width=3.15cm,right=of ptr] (stat) {Tomography and\\ discrimination};
\node[gbox,text width=3.15cm,right=of stat] (open) {Not constructed:\\ local probe, collapse,\\ microstate channel};
\draw[arr] (rad)--(time);
\draw[arr] (time)--(port);
\draw[arr] (port)--(store);
\draw[arr] (store)--(ext);
\draw[arr] (ext)--(rot);
\draw[arr] (rot)--(ptr);
\draw[arr] (ptr)--(stat);
\node[pbox,text width=9.9cm,anchor=north west] at ([yshift=-4mm]ptr.south west)
  {Retained: spectators $S$ and reference $R$. Discarded: memory after extraction, measured register, reset environment.};
\end{tikzpicture}
\caption{Dependency of Em03-C. The first two columns are the radiation interface and the declared apparatus. The dashed box is not a constructed mechanism. Not a local Schwarzschild detector, and not an admission of a quantum coefficient.}
\label{fig:dependency}
\end{figure}



The finite-mode Gaussian construction is standard in structure
\cite{Weedbrook}. The supplied companion account attributes a
general version to En02-E, at \code{thm:gaussian},
\code{eq:outcov}, and \code{eq:K} \cite{En02EAccount}.
The full En02-E source is not supplied. We therefore give the
needed specialization and its proof explicitly rather than
importing an unseen theorem or claiming novelty for Gaussian
diagonalization.

The remaining constructions are ordinary channel calculations.
Landmark I is optional semantic input for calling finite destructive
readout \(\Enter\); it is not a field-state or apparatus premise.

\paragraph{Evidence boundary.}
The source supplies analytical arguments. No compilation,
machine checking, numerical experiment, statistical calibration,
or physical realization is asserted. The term ``certified'' refers
to explicit mathematical estimates under stated model data.

\paragraph{Conventions.}
We use \(c=\hbar=k_B=1\), \(G,M>0\), and
\[
\dist(\rho,\sigma)=\frac12\|\rho-\sigma\|_1,\qquad
\ddiam(\Phi,\Psi)=\frac12\|\Phi-\Psi\|_\diamond,\qquad
\tv(p,q)=\frac12\sum_y|p(y)-q(y)|.
\]
Channels are normal CPTP maps on trace-class operators.
A finite spectator is denoted \(S\); an arbitrary
finite-dimensional external reference is denoted \(R\).
Their tensor placements are retained throughout. The normal
state identities also hold with separable references.

We write \(\Gamma(U)\) for bosonic second quantization of a
one-particle unitary. This notation is distinct from the
greybody probability \(\Gamma_\ell(\omega)\).

%=====================================================================
\section{C0: the complete two-packet semiclassical baseline}
\label{sec:baseline}
%=====================================================================

\subsection{Inherited state and the covariance convention}

\begin{assumption}[Stationary outgoing radiation input]
\label{ass:radiation}
Fix one real angular harmonic of degree \(\ell\) in the
massless scalar model of Em03-A. The outgoing radiation state
is centered, gauge-invariant, regular, and quasifree, with
\[
\langle a_{\mathrm{out}}(\omega)^\dagger
               a_{\mathrm{out}}(\omega')\rangle
=n(\omega)\delta(\omega-\omega'),\qquad
\langle a_{\mathrm{out}}(\omega)
               a_{\mathrm{out}}(\omega')\rangle=0,
\]
where
\begin{equation}
n(\omega)=\frac{\Gamma_\ell(\omega)}
                    {e^{\betah\omega}-1},
\qquad
\betah=8\pi GM,\qquad
0\le\Gamma_\ell(\omega)\le1.
\label{eq:radiationdensity}
\end{equation}
Annihilation tests are conjugate-linear:
\[
a(h)=\int_0^\infty\overline{h(\omega)}
                         a_{\mathrm{out}}(\omega)\dd\omega.
\]
\end{assumption}

This input uses Em03-A's boundary-state prescription,
\code{lem:horizonoccupation}, \code{eq:quantumscatter},
\code{eq:numberdensity}, and \code{eq:packetlaw}.
Its bulk Hadamard identification remains the declared external
Dappiaggi--Moretti--Pinamonti input \cite{DMP}.
Neither that global proof nor Em03-A's manuscript proofs are
independently verified here.

Choose orthonormal \(h_1,h_2\in C_c^\infty((0,\infty))\),
supported in a common compact band
\[
B=[\omega_-,\omega_+]\subset(0,\infty).
\]
Set \(b_i=a(h_i)\) and \(N_i=b_i^\dagger b_i\).
Then \([b_i,b_j^\dagger]=\delta_{ij}\).
Define
\begin{equation}
\boxed{
A_{ij}:=\langle b_j^\dagger b_i\rangle
=\int_B\overline{h_i(\omega)}h_j(\omega)n(\omega)\dd\omega.
}
\label{eq:A}
\end{equation}
Thus
\[
\langle b_i^\dagger b_j\rangle=A_{ji},\qquad
\nu_i=A_{ii},\qquad
q_i=\frac{\nu_i}{1+\nu_i}.
\]
The transposition is important. Equation~\eqref{eq:A} is not
the formula for \(\langle b_i^\dagger b_j\rangle\) with the same
indices under the predecessor's test convention.

The matrix \(A\) is positive semidefinite and satisfies
\[
0\preceq A\preceq n_{\max}I,\qquad
n_{\max}=(e^{\betah\omega_-}-1)^{-1}.
\]


Figure~\ref{fig:covariance} fixes the index convention that the
following characteristic function, passive transformation, and Wick
formula all use. A complex off-diagonal entry is the control that
exposes an accidental transpose.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[node distance=3mm]
\node[qbox,text width=4.6cm] (test) {Conjugate-linear test\\ $a(h)=\int\overline{h}\,a_{\mathrm{out}}\,d\omega$};
\node[qbox,text width=4.6cm,right=8mm of test] (comm) {$[b_i,b_j^\dagger]=\delta_{ij}$\\ $b_i=a(h_i)$};
\node[cbox,text width=4.6cm,below=of test] (aij) {$A_{ij}=\langle b_j^\dagger b_i\rangle$\\ $\langle b_i^\dagger b_j\rangle=A_{ji}$};
\node[pbox,text width=4.6cm,below=of comm] (use) {Used by $\operatorname{tr}\rho D(z)$,\\ $\Gamma(W)$, and Wick};
\node[gbox,text width=9.7cm,anchor=north west] at ([yshift=-6mm]aij.south west)
  {Synthetic control: $A_{12}=0.2+0.1i$ is not $A_{21}$. Equal diagonals do not license a transpose.};
\draw[arr] (test)--(comm);
\draw[arr] (test)--(aij);
\draw[arr] (comm)--(use);
\draw[arr] (aij)--(use);
\end{tikzpicture}
\caption{Covariance orientation. The diagram is a convention check, not a numerical diagonalization and not a physical calibration.}
\label{fig:covariance}
\end{figure}


\subsection{Normal density and uniqueness}

For \(z\in\mathbb C^2\), define
\[
D(z)=\exp\left(\sum_i z_i b_i^\dagger-\overline{z_i}b_i\right).
\]

\begin{theorem}[Normal two-mode Gaussian restriction]
\label{thm:gaussian}
Under Assumption~\ref{ass:radiation}, the two-packet state is the
unique normal density on
\(\HH_{12}=\FF(\mathbb C^2)\) with characteristic function
\begin{equation}
\tr(\rho_0D(z))
=\exp\left(-\frac12 z^\dagger z-z^\dagger A z\right).
\label{eq:characteristic}
\end{equation}
If
\(A=W\operatorname{diag}(\lambda_1,\lambda_2)W^\dagger\),
then
\begin{equation}
\boxed{
\rho_0=\Gamma(W)
 \left(\bigotimes_{r=1}^2
   \frac1{1+\lambda_r}
   \sum_{m=0}^\infty
   \left(\frac{\lambda_r}{1+\lambda_r}\right)^m
                  \ketbra m m\right)
 \Gamma(W)^\dagger.
}
\label{eq:jointdensity}
\end{equation}
A zero eigenvalue means vacuum in that diagonal mode.
No density operator for the entire stationary continuum is required.
\end{theorem}

\begin{proof}
Restriction and complex-linear scattering preserve the inherited
centered quasifree prescription. Its zero anomalous covariance
and number matrix \eqref{eq:A} give
\eqref{eq:characteristic}.

For completeness, let \(\beta_1,\beta_2\) be independent circular
complex Gaussian random variables with variances
\(\lambda_1,\lambda_2\), with a zero-variance variable identically
zero, and put \(\alpha=W\beta\).
The positive trace-class mixture
\[
\rho=\mathbb E\ketbra\alpha\alpha
\]
has trace one and characteristic function
\[
\mathbb E\left[
e^{-|z|^2/2+\alpha^\dagger z-z^\dagger\alpha}
\right]
=e^{-|z|^2/2-z^\dagger A z}.
\]
For a single variance \(\lambda>0\), angular and radial
integration give the number probabilities
\[
\frac1{\lambda m!}
\int_0^\infty e^{-(1+\lambda^{-1})t}t^m\dd t
=\frac{\lambda^m}{(1+\lambda)^{m+1}}.
\]
Consequently the mixture is \eqref{eq:jointdensity}.
Equality of Weyl characteristic functions identifies the
abstract restricted state and uniquely determines its normal
density. This proves normality by construction.
\end{proof}


Figure~\ref{fig:gaussian} separates the two constructions of the same
normal density: diagonal thermal modes, and a coherent-state mixture
with covariance $A$. Neither arrow is a density of the whole continuum.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[node distance=3mm]
\node[qbox,text width=4.4cm] (char) {Characteristic function\\ $e^{-|z|^2/2-z^\dagger A z}$};
\node[qbox,text width=4.2cm,anchor=north east] (therm) at ([yshift=-7mm,xshift=-3mm]char.south)
  {$A=W\operatorname{diag}(\lambda_r)W^\dagger$\\ geometric factors};
\node[qbox,text width=4.2cm,anchor=north west] (mix) at ([yshift=-7mm,xshift=3mm]char.south)
  {Circular Gaussians $\beta$\\ $\alpha=W\beta$, $\mathbb E|\alpha\rangle\langle\alpha|$};
\node[cbox,text width=5.2cm,anchor=north] (rho) at ([yshift=-28mm]char.south) {Unique normal $\rho_0$ on $\mathcal F(\mathbb C^2)$};
\draw[arr] (char)--(therm);
\draw[arr] (char)--(mix);
\draw[arr] (therm)--(rho);
\draw[arr] (mix)--(rho);
\end{tikzpicture}
\caption{Two routes to the normal two-mode restriction. A zero eigenvalue is vacuum in that diagonal mode. The figure does not execute a Fock truncation.}
\label{fig:gaussian}
\end{figure}


\subsection{Joint count law and correlations}

Write
\[
p_0(n_1,n_2)=
\bra{n_1,n_2}\rho_0\ket{n_1,n_2}.
\]

\begin{theorem}[Joint count generating function and Wick identities]
\label{thm:counts}
For \(0\le s_1,s_2\le1\), with \(S=\operatorname{diag}(s_1,s_2)\),
\begin{equation}
\boxed{
G(s_1,s_2)
:=\mathbb E_{\rho_0}s_1^{N_1}s_2^{N_2}
=\frac1{\det(I+A(I-S))}.
}
\label{eq:PGF}
\end{equation}
More explicitly,
\[
G(s_1,s_2)=
\frac1{
1+\nu_1(1-s_1)+\nu_2(1-s_2)
+(\nu_1\nu_2-|A_{12}|^2)(1-s_1)(1-s_2)}.
\]
Its Taylor coefficients at \((0,0)\) give the complete joint
count law.

Each marginal is geometric with mean \(\nu_i\), and
\begin{equation}
\boxed{
\Cov(N_1,N_2)=|A_{12}|^2,\qquad
\Var(N_i)=\nu_i(1+\nu_i).
}
\label{eq:numbercov}
\end{equation}
Moreover,
\begin{equation}
\langle b_i^\dagger b_j^\dagger b_k b_l\rangle
=A_{ki}A_{lj}+A_{li}A_{kj}.
\label{eq:wick}
\end{equation}
The packet factors are statistically independent if and only if
\(A_{12}=0\).
\end{theorem}

\begin{proof}
Conditional on a coherent amplitude \(\alpha\), the two counts
are independent Poisson variables of means \(|\alpha_i|^2\).
The coherent-state mixture in Theorem~\ref{thm:gaussian} gives
\[
G=\mathbb E e^{-\alpha^\dagger(I-S)\alpha}.
\]
The circular Gaussian integral equals
\[
\det\bigl(I+A^{1/2}(I-S)A^{1/2}\bigr)^{-1}
=\det(I+A(I-S))^{-1}.
\]
This also covers singular \(A\) by its diagonal Gaussian
construction. Setting either \(s_i=1\) gives its geometric
marginal.

Gaussian contraction gives \eqref{eq:wick}. For different modes,
\(N_1N_2=b_1^\dagger b_2^\dagger b_1b_2\); hence
\(\langle N_1N_2\rangle=\nu_1\nu_2+|A_{12}|^2\).
The same-mode calculation gives the stated variance.
All polynomial moments are finite.

If \(A_{12}=0\), the characteristic function factorizes, so the
state is the product of its marginals. Conversely, a product
centered state has zero cross covariance.
\end{proof}

\begin{example}[Orthogonality without independence]
Let \(u,v\) be normalized tests with disjoint supports and
occupations \(\nu_u\ne\nu_v\). Then
\[
h_1=(u+v)/\sqrt2,\qquad h_2=(u-v)/\sqrt2
\]
are orthonormal, but \(A_{12}=(\nu_u-\nu_v)/2\ne0\).
Disjoint supports of \(h_1,h_2\) themselves would instead force
\(A_{12}=0\).
\end{example}

The joint number law depends on \(A_{12}\) only through its
modulus. A local phase transformation changes its phase without
changing any number-diagonal probability. Number counts therefore
do not determine the complete joint state.


Figure~\ref{fig:counts} records what the generating function determines
and what it does not. Orthogonal packets are independent only when
$A_{12}=0$. Equal moduli with different phases share the number law
and are separated only by a phase-sensitive observable.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[node distance=3mm]
\node[qbox,text width=5.4cm] (g) {$G(s)=1/\det(I+A(I-S))$};
\node[pbox,text width=3.45cm,anchor=north] (cov) at ([yshift=-8mm]g.south)
  {$\operatorname{Cov}=|A_{12}|^2$\\ $\operatorname{Var}=\nu_i(1+\nu_i)$};
\node[pbox,text width=3.35cm,left=3mm of cov] (marg)
  {Marginals\\ geometric, mean $\nu_i$};
\node[cbox,text width=3.45cm,right=3mm of cov] (miss)
  {Phase of $A_{12}$\\ not fixed by counts};
\node[gbox,text width=11.1cm,anchor=north] (note) at ([yshift=-4mm]cov.south)
  {Synthetic split: $h_\pm=(u\pm v)/\sqrt2$ can be orthonormal with $A_{12}\neq0$ when $\nu_u\neq\nu_v$. Disjoint $h_1,h_2$ force $A_{12}=0$.};
\draw[arr] (g)--(marg);
\draw[arr] (g)--(cov);
\draw[arr] (g)--(miss);
\end{tikzpicture}
\caption{Joint count law. The dashed box is a state distinction invisible to number-diagonal records, including the later synthetic pair $\ket{\psi_\pm}$. Not an executed coefficient comparison.}
\label{fig:counts}
\end{figure}


%=====================================================================
\section{C1 interface: joint carriers and referenced records}
\label{sec:carriers}
%=====================================================================

\subsection{Two cutoffs per packet}

Choose reported saturation levels \(N_i^{\mathrm{cut}}\ge1\)
and control-domain cutoffs \(K_i\ge N_i^{\mathrm{cut}}\).
To avoid confusing operators and cutoff values, abbreviate
\[
a_i=N_i^{\mathrm{cut}},\qquad
\boldsymbol a=(a_1,a_2),\qquad
\boldsymbol K=(K_1,K_2).
\]
Let \(P_{i,a_i}=\sum_{n=0}^{a_i}\ketbra n n\).

The one-mode carrier map is
\[
\CC_{a_i}(X)
=V_iP_{i,a_i}XP_{i,a_i}V_i^\dagger
+\tr((I-P_{i,a_i})X)\ketbra{a_i}{a_i}.
\]
Its Kraus maps are \(V_iP_{i,a_i}\) and
\(\ket{a_i}\bra n\), \(n>a_i\), whose effects sum strongly to
the identity.

Set
\[
b_i^{\mathrm{reg}}=\ceil{\log_2(a_i+1)},\qquad
d_i=2^{b_i^{\mathrm{reg}}},\qquad
B_{\mathrm{reg}}=b_1^{\mathrm{reg}}+b_2^{\mathrm{reg}},
\qquad d=d_1d_2.
\]
The binary embeddings \(J_i\) send carrier integer \(k\) to its
fixed binary word. Define
\[
\BB_{\boldsymbol a}
=(J_1\otimes J_2)
(\CC_{a_1}\otimes\CC_{a_2})(\,\cdot\,)
(J_1\otimes J_2)^\dagger.
\]

\subsection{Complete raw alphabet and decoding}

For raw integers \(j_i\in\{0,\ldots,d_i-1\}\), put
\[
c_i(j_i)=\min(j_i,a_i).
\]
An ordinary record contains both raw words, both decoded integers,
and an ``ordinary'' status:
\[
y_{\boldsymbol j}
=(\operatorname{bin}(j_1),\operatorname{bin}(j_2),
                       c_1(j_1),c_2(j_2),\mathrm{ordinary}).
\]
Enlarge this alphabet \(\mathcal Y\) by all declared failure records.
Failure records retain their labels; their decoded pair is
\((0,0)\). A different bounded assignment could be specified, but
must be fixed before using a mean certificate.

The record can be embedded into \(F_1F_2F_{\mathrm{flag}}\).
We denote the whole classical output by \(F\).
For arbitrary record operators, the total decoder is
\begin{equation}
\mathcal G_{\boldsymbol a}(X)=
\sum_{y\in\mathcal Y}\bra yX\ket y
                    \ketbra{c(y)}{c(y)}_{\rec}.
\label{eq:totaldecoder}
\end{equation}
It is CPTP on the full alphabet.

For computational destructive readout, set
\[
M_{\boldsymbol j}=\bra{j_1,j_2}_Q\otimes I_S,\qquad
\PP_Z^S(X)=
\sum_{\boldsymbol j}\ketbra{y_{\boldsymbol j}}{y_{\boldsymbol j}}
                    \otimes M_{\boldsymbol j}XM_{\boldsymbol j}^\dagger.
\]
The measured register is absent from the output.

\subsection{Exact record identity and tail}

For \(\rho_{12SR}\), define the positive blocks
\[
\rho_{SR}^{(\boldsymbol n)}
=\bra{n_1,n_2}\rho_{12SR}\ket{n_1,n_2}.
\]
Let
\[
T_{\boldsymbol a}(\boldsymbol n)
=(\min(n_1,a_1),\min(n_2,a_2)).
\]
The ideal full record is
\begin{equation}
\widetilde\Omega_{\boldsymbol a,SR}(\rho)
=\sum_{\boldsymbol k\le\boldsymbol a}
 \ketbra{y_{\boldsymbol k}}{y_{\boldsymbol k}}
 \otimes
 \sum_{T_{\boldsymbol a}(\boldsymbol n)=\boldsymbol k}
                                      \rho_{SR}^{(\boldsymbol n)}.
\label{eq:jointrecord}
\end{equation}

\begin{theorem}[Joint referenced destructive record]
\label{thm:jointrecord}
For every normal input,
\[
(\PP_Z^S\otimes\id_R)
(\BB_{\boldsymbol a}\otimes\id_{SR})(\rho)
=\widetilde\Omega_{\boldsymbol a,SR}(\rho).
\]
Let \(\Omega_{\boldsymbol a,SR}\) be its decoded version and
\(\Omega_{\infty,SR}\) the unsaturated joint number record.
In the common countable record space,
\begin{equation}
\boxed{
\dist(\Omega_{\boldsymbol a,SR},\Omega_{\infty,SR})
=p_{\boldsymbol a}(\rho)
:=\Pr_\rho(N_1>a_1\ \text{or}\ N_2>a_2).
}
\label{eq:jointtail}
\end{equation}
For the baseline,
\[
p_{\boldsymbol a}(\rho_0)
\le\min\{1,q_1^{a_1+1}+q_2^{a_2+1}\}.
\]
No independence assumption is needed.
\end{theorem}

\begin{proof}
Tensor the two Kraus families. Destructive readout selects the
number-diagonal input blocks and sums exactly the blocks with
the same clipped pair. Positive block traces sum to one, so the
series converge in trace norm.

The two decoded records agree on the rectangular interior.
Outside that rectangle, the unsaturated record has positive
blocks that are absent from the clipped record. The clipped
record adds their positive mass to boundary records inside the
rectangle. These inside and outside differences occupy orthogonal
classical records. Their total trace norm is twice the outside
probability, proving \eqref{eq:jointtail}. The union bound and the
geometric marginal tails give the last assertion.
\end{proof}

The marginal ideal mean deficit is exactly
\[
\nu_i-\mathbb E_{\mathrm{ideal}}\kappa_i=\nu_iq_i^{a_i}.
\]
This statement is unaffected by interpacket correlations.

\subsection{A different tail for coherent comparisons}

Embed the binary register into the oscillator pair by
\[
\iota\ket{j_1,j_2}_Q=\ket{j_1,j_2}_{12},
\]
including unused binary words. Define
\[
P_{\boldsymbol a}=P_{1,a_1}\otimes P_{2,a_2}.
\]

\begin{lemma}[Coherent carrier approximation]
\label{lem:coherenttail}
For any normal packet--reference input,
\begin{equation}
\boxed{
\dist\left(
\rho_{12SR},
(\iota\otimes I_{SR})
(\BB_{\boldsymbol a}\otimes\id_{SR})(\rho)
(\iota^\dagger\otimes I_{SR})
\right)
\le t(p_{\boldsymbol a}),
}
\label{eq:coherenttail}
\end{equation}
where
\[
t(p)=\min\{1,\sqrt p+p/2\}.
\]
\end{lemma}

\begin{proof}
The embedded carrier output is
\[
P_{\boldsymbol a}\rho P_{\boldsymbol a}+T,\qquad
T\succeq0,\quad \tr T=p_{\boldsymbol a}.
\]
This follows by separating the low--low Kraus map from all other
Kraus maps.

Purify \(\rho\) to \(\ket\Psi\), and set
\(\ket\Phi=(P_{\boldsymbol a}\otimes I)\ket\Psi\).
Writing \(p=p_{\boldsymbol a}\), one has
\(\|\Phi\|^2=\langle\Psi,\Phi\rangle=1-p\). The rank-two
calculation gives
\[
\bigl\|\ketbra\Psi\Psi-\ketbra\Phi\Phi\bigr\|_1
=\sqrt{4p-3p^2}\le2\sqrt p.
\]
Partial trace contracts this norm. Adding \(T\) costs at most
\(p\) in trace norm. Divide by two and cap at one.
\end{proof}

The exact probability error \eqref{eq:jointtail} concerns
destructive number records. It is not the coherent state
certificate \eqref{eq:coherenttail}.


Figure~\ref{fig:tails} separates the two cutoff errors. The
destructive distance is exactly the outside probability. The coherent
comparison pays an additional square-root term because the inside and
outside amplitudes need not occupy orthogonal record labels.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[node distance=3mm]
\node[qbox,text width=4.3cm] (box) {Retained rectangle\\ $n_i\le a_i$};
\node[cbox,text width=4.3cm,right=8mm of box] (over) {Overflow\\ either mode, or both};
\node[pbox,text width=4.3cm,below=of box] (rec) {Record distance\\ $=p_{\boldsymbol a}$};
\node[pbox,text width=4.3cm,below=of over] (coh) {Coherent distance\\ $\le t(p)=\min\{1,\sqrt p+p/2\}$};
\draw[arr] (over)--(rec);
\draw[arr] (box)--(rec);
\draw[arr] (over)--(coh);
\node[gbox,text width=9.4cm,anchor=north west] at ([yshift=-5mm]rec.south west)
  {Saturation Kraus: low blocks copied, overflow sent to the corner $\ket{a_i}$. Off-diagonal coherence across the cut is not preserved.};
\end{tikzpicture}
\caption{Destructive record tail against coherent carrier approximation. Independence of the packets is not used. Not a laboratory histogram.}
\label{fig:tails}
\end{figure}


%=====================================================================
\section{Joint temporal modes and participating radiation}
\label{sec:temporal}
%=====================================================================

\subsection{The effective port and its domain}

As in Em03-B, the effective port has one-particle space
\(L^2(\mathbb R,\dd\omega)\). Its positive-frequency sector
carries \eqref{eq:radiationdensity}; its negative-frequency
sector is a new independent auxiliary vacuum.

This two-sided Markov envelope is model data. It is not a
physical negative-energy sector of the Schwarzschild scalar.
Unselected positive-frequency radiation is not replaced by vacuum.

Use
\[
p_i(t)=\frac1{\sqrt{2\pi}}\int h_i(\omega)e^{-\ii\omega t}\dd\omega,
\qquad
\widehat f(\omega)=\frac1{\sqrt{2\pi}}
                   \int f(t)e^{\ii\omega t}\dd t.
\]
The input annihilator \(a(\widehat f)\) is the temporal smearing
\(\int\overline{f(t)}b_{\mathrm{in}}(t)\dd t\).

Finite restrictions are defined on the weighted domain
\[
\int_{\mathbb R}(1+n_{\mathrm{port}}(\omega))|v(\omega)|^2
                                                    \dd\omega<\infty,
\]
where \(n_{\mathrm{port}}=n\) on positive frequencies and zero
on negative frequencies. Compact positive-frequency tests are
dense in the positive-frequency weighted space. Covariances and
Gaussian Weyl positivity extend by continuity. Every finite
canonical subspace with finite occupation matrix has a normal
density by the same diagonalization proof as
Theorem~\ref{thm:gaussian}.

This extension is a radiation-test construction, not a new local
bulk characteristic-restriction theorem.

\subsection{Joint compact-window construction}

Choose real
\(\chi\in C_c^\infty((-2,2))\), equal to one on \([-1,1]\),
and choose \(\psi\in C_c^\infty((-2,2))\) with
\(\int\psi=1\). For \(L>0\), put
\[
v_{i,L}(t)=\chi(t/L)p_i(t)-m_{i,L}L^{-1}\psi(t/L),
\qquad
m_{i,L}=\int\chi(t/L)p_i(t)\dd t.
\]
Let \(V_L:\mathbb C^2\to L^2(\mathbb R,\dd t)\) have columns
\(v_{i,L}\). For sufficiently large \(L\),
\(G_L=V_L^\dagger V_L\) is positive definite. First
orthonormalize:
\[
\widetilde F_L=\widehat{V_LG_L^{-1/2}}.
\]
Let \(H:\mathbb C^2\to L^2(\mathbb R,\dd\omega)\) have columns
\(h_1,h_2\). Write the left polar decomposition
\[
H^\dagger\widetilde F_L=B_LU_L,\qquad B_L\succ0,
\]
and define
\[
F_L=\widetilde F_LU_L^\dagger,\qquad
H^\dagger F_L=B_L.
\]
The columns of \(F_L\) are denoted \(\widehat f_{1,L},
\widehat f_{2,L}\).

\begin{lemma}[Orthonormal finite windows with infrared control]
\label{lem:windows}
For sufficiently large \(L\):
\begin{enumerate}[label=\textup{(\roman*)},leftmargin=*]
\item \(f_{1,L},f_{2,L}\) are orthonormal smooth tests supported
in \((-2L,2L)\), and each has zero integral;

\item their Fourier transforms converge columnwise to
\(h_1,h_2\) in the Schwartz topology and in the weighted
occupation norm;

\item the participating plane
\[
\mathfrak v_L=\operatorname{ran}H+\operatorname{ran}F_L
\]
has dimension at most four and a finite occupation matrix.
\end{enumerate}
\end{lemma}

\begin{proof}
Each \(p_i\) is Schwartz, and \(\int p_i=0\) because \(h_i\)
vanishes near zero. Hence
\[
m_{i,L}=-\int(1-\chi(t/L))p_i(t)\dd t
\]
decays faster than every inverse power of \(L\).
The windowed error and the correcting bump tend to zero in
every Schwartz seminorm. Thus \(G_L\to I\); the inverse square
root and the polar factors tend to \(I\). Their finite matrix
operations preserve supports and zero integrals.

For \(\omega>0\),
\[
n(\omega)\le(\betah\omega)^{-1}.
\]
A Fourier test \(w\) with \(w(0)=0\) satisfies
\[
\int_0^1n(\omega)|w(\omega)|^2\dd\omega
\le\frac{\sup_{[0,1]}|w'|^2}{2\betah}.
\]
On \([1,\infty)\), \(n\) is bounded. Schwartz convergence
therefore implies weighted convergence. It also proves finite
occupation for every vector in \(\mathfrak v_L\).
\end{proof}

Ordinary temporal windowing alone would not establish
integrability at zero using only \(\Gamma_\ell\le1\).
The zero-integral correction avoids importing a low-frequency
greybody asymptotic that has not been proved here.


Figure~\ref{fig:windows} is the temporal-mode interface used before
capture. The zero-integral correction is what makes the Fourier test
vanish at $\omega=0$. Ordinary compact windowing, without that
correction, does not by itself give finite occupation from
$n(\omega)\le 1/(\beta_{\mathrm H}\omega)$.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[node distance=2.8mm]
\node[qbox,text width=3.5cm] (h) {Band-limited\\ targets $h_i$};
\node[qbox,text width=3.5cm,right=of h] (win) {Window on $(-2L,2L)$\\ plus zero-mean correction};
\node[qbox,text width=3.5cm,right=of win] (on) {Joint Gram step\\ then polar alignment};
\node[cbox,text width=3.5cm,below=of on] (f) {Orthonormal $f_i$\\ $\int f_i=0$};
\node[pbox,text width=3.5cm,left=of f] (ir) {$\widehat f_i(0)=0$\\ weighted $n$ integrable};
\node[gbox,text width=3.5cm,left=of ir] (bad) {Uncorrected window\\ need not have finite $\nu$};
\draw[arr] (h)--(win);
\draw[arr] (win)--(on);
\draw[arr] (on)--(f);
\draw[arr] (f)--(ir);
\draw[arr] (win)--(bad);
\end{tikzpicture}
\caption{Compact temporal tests and infrared control. Convergence as $L$ increases is the analytical Schwartz statement of the window lemma. The figure is not a numerical quadrature.}
\label{fig:windows}
\end{figure}


\subsection{Principal-plane comparison}

Diagonalize
\[
B_L=R_L\operatorname{diag}(\cos\theta_1,\cos\theta_2)R_L^\dagger,
\qquad 0\le\theta_r<\pi/2.
\]
The principal target modes \(h'_r=(HR_L)_r\) and captured modes
\(f'_r=(F_LR_L)_r\) obey
\[
f'_r=\cos\theta_r\,h'_r+\sin\theta_r\,g_r.
\]
For nonzero \(\theta_r\), the \(g_r\) are orthonormal and
orthogonal to the target plane. Zero-angle terms are omitted.

Let \(v_L\) be the passive one-particle rotation sending
\(h'_r\) to \(f'_r\) in each principal plane and acting identically
on its orthogonal complement. Let \(\mathcal I_H\) select the
target pair and discard the other participating modes. Then the
captured-pair selector is
\[
\mathcal S_F(\rho)
=\mathcal I_H\bigl(\Gamma(v_L)^\dagger
                           \rho\Gamma(v_L)\bigr),
\]
with both selected pairs identified with the same two memory
coordinates. The operational input is the normal joint state
\(\rho_{\mathfrak v_L SR}\), not merely its target marginal.

\begin{lemma}[Joint Gaussian mode-selection error]
\label{lem:mode}
For the inherited Gaussian marginal on \(\mathfrak v_L\), and
any spectator--reference extension,
\begin{equation}
\dist(\mathcal S_F(\rho),\mathcal I_H(\rho))
\le
\mu_L:=
\min\left\{1,
\sum_{r:\theta_r>0}
\theta_r\sqrt{E_{h'_r}+E_{g_r}+6E_{h'_r}E_{g_r}}
\right\}.
\label{eq:jointmode}
\end{equation}
Here each occupation is computed from the actual port covariance.
Moreover, \(\mu_L\to0\).
\end{lemma}

\begin{proof}
The rotation generator in plane \(r\) is
\[
J_r=\ii(a(g_r)^\dagger a(h'_r)
                  -a(h'_r)^\dagger a(g_r)).
\]
These generators are self-adjoint on their total-number-sector
direct sums. Purification, the spectral inequality
\(|e^{\ii x}-1|\le|x|\), and the vector triangle inequality
give the bound
\[
\dist\le
\sum_r\theta_r\sqrt{\langle J_r^2\rangle}.
\]
For a centered gauge-invariant Gaussian two-mode marginal,
put \(m=\langle a(g_r)^\dagger a(h'_r)\rangle\).
Wick contraction gives
\[
\langle J_r^2\rangle
=E_{h'_r}+E_{g_r}
 +2E_{h'_r}E_{g_r}+2|m|^2-4\Repart(m^2)
\le E_{h'_r}+E_{g_r}+6E_{h'_r}E_{g_r},
\]
using \(|m|^2\le E_{h'_r}E_{g_r}\).
This depends only on the participating marginal, so arbitrary
reference extensions obey the same estimate.

The principal target occupations are bounded by \(n_{\max}\).
The residual vectors
\(f'_r-\cos\theta_r h'_r\) have weighted occupation tending
uniformly to zero by Lemma~\ref{lem:windows}. Their occupations
are \(\sin^2\theta_r E_{g_r}\). Therefore
\(\theta_r^2 E_{g_r}\to0\), proving the last assertion.
\end{proof}

For a non-Gaussian input the same purification proof gives
\[
\mu(\rho)=
\min\left\{1,\sum_r\theta_r
                      \sqrt{\tr(\rho J_r^2)}\right\},
\]
provided these moments are finite. Otherwise the trivial bound
one remains valid. A proposed correction must justify its own
participating-mode moments; the baseline's Wick estimate cannot
be transferred to it.

%=====================================================================

Figure~\ref{fig:plane} is the comparison between the desired packet
plane and the plane spanned by the compact tests. A small weighted
residual can still be a normalized mode of large occupation. The
bound uses the residual occupations, not the norm of the residual alone.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[node distance=3mm]
\node[qbox,text width=3.6cm] (want) {Target plane\\ $\operatorname{span}\{h_1,h_2\}$};
\node[qbox,text width=3.6cm,right=of want] (have) {Realized plane\\ $\operatorname{span}\{f_1,f_2\}$};
\node[cbox,text width=3.6cm,right=of have] (res) {Residual modes\\ principal angles};
\node[pbox,text width=7.6cm,below=6mm of have] (pass) {Passive $W$ and $\Gamma(W)$\\ Gaussian error from residual occupations};
\draw[arr] (want)--(have);
\draw[arr] (have)--(res);
\draw[arr] (res)--(pass);
\draw[arr] (have)--(pass);
\end{tikzpicture}
\caption{Principal-plane selection. The passive rotation is one-particle data; its second quantization acts on the joint Gaussian state. Not a collapse-packet limit.}
\label{fig:plane}
\end{figure}


\section{A shared-port two-memory capture construction}
\label{sec:capture}
%=====================================================================

This section constructs a combined apparatus. It does not add
two one-packet capture budgets.

\subsection{Matrix synthesis}

Fix the orthonormal temporal tests \(f_1,f_2\) just constructed,
supported in \((a,b)=(-2L,2L)\), and \(0<\tau<1\). Set
\begin{equation}
\begin{aligned}
q(t)&=(\overline{f_1(t)},\overline{f_2(t)})^{\mathsf T},\\
D(t)&=(1-\tau)I+\tau\int_a^t q(s)q(s)^\dagger\dd s,\\
G(t)&=D(t)^{1/2},\\
\lambda(t)&=\sqrt\tau\,G(t)^{-1}q(t),\\
C(t)&=-G(t)^{-1}\dot G(t),\\
\Omega(t)&=\frac{\ii}{2}(C(t)-C(t)^\dagger).
\end{aligned}
\label{eq:matrixcapture}
\end{equation}
Orthonormality gives \(D(a)=(1-\tau)I\), \(D(b)=I\).

\begin{specification}[Shared effective capture port]
\label{spec:jointcapture}
Two memory amplitudes \(z(t)\in\mathbb C^2\) and one incoming
port amplitude \(u(t)\) evolve by
\begin{equation}
\dot z(t)=C(t)z(t)+\lambda(t)u(t),\qquad
u_{\mathrm{out}}(t)=u(t)-\lambda(t)^\dagger z(t).
\label{eq:vectorODE}
\end{equation}
The quantum evolution is the second quantization of this
one-particle map.

The initial memories have a named joint state \(\sigma_c\),
independent of radiation, spectators, and references. Write
\(p_c=\dist(\sigma_c,\ketbra{0,0}{0,0})\).
An explicit allowed preparation is
\[
\sigma_c=(1-p_c)\ketbra{0,0}{0,0}+p_c\sigma_c^\perp,
\qquad \sigma_c^\perp\ket{0,0}=0.
\]
It may have internal correlations.

A subsequent common storage interval has vacuum transmissivity
\(s=e^{-\kappa_sT_s}\) on each memory, with independent vacuum
loss ports and no additional unpriced interaction. Define
\(\tau_*=\tau s\).
The clock is the predecessor's normalized asymptotic time.
\end{specification}

This is a specified linear input--output model, in the setting
of passive memory models \cite{GardinerCollett,YamamotoJames}.
The following synthesis and comparison are proved directly.

\begin{theorem}[Simultaneous capture from one port]
\label{thm:sharedcapture}
Equation~\eqref{eq:vectorODE} is unitary from
\(\mathbb C^2\oplus L^2([a,b])\) to itself. Its final memory
annihilators obey
\begin{equation}
\boxed{
c_{i,b}=\sqrt{1-\tau}\,c_{i,a}
                         +\sqrt\tau\,a(\widehat f_i),
\qquad i=1,2.
}
\label{eq:capturepair}
\end{equation}
The interaction is smooth and finitely switched, and
\[
C+C^\dagger=-\lambda\lambda^\dagger,\qquad
C=-\tfrac12\lambda\lambda^\dagger-\ii\Omega.
\]
Its resources satisfy
\begin{equation}
\begin{aligned}
\int_a^b\|\lambda(t)\|^2\dd t&=-2\log(1-\tau),\\
\|\lambda(t)\|^2&\le
                    \frac{\tau}{1-\tau}\|q(t)\|^2,\\
\|\Omega(t)\|&\le
                    \frac{\tau}{2(1-\tau)}\|q(t)\|^2.
\end{aligned}
\label{eq:capturebounds}
\end{equation}
\end{theorem}

\begin{proof}
Differentiating \(D=G^2\) gives
\(\dot D=\dot G G+G\dot G=\tau qq^\dagger\).
It follows that
\[
C+C^\dagger
=-G^{-1}\dot D G^{-1}=-\lambda\lambda^\dagger.
\]
Consequently
\[
\frac{\dd}{\dd t}\|z(t)\|^2
=|u(t)|^2-|u_{\mathrm{out}}(t)|^2.
\]
Integration proves isometry. Substituting
\(u=u_{\mathrm{out}}+\lambda^\dagger z\) gives a bounded
backward linear equation with a unique solution from the final
memory and outgoing amplitude. The isometry is therefore onto.

The final-time fundamental solution is \(G(t)\), since
\(\dot G=-GC\) and \(G(b)=I\). Thus
\[
z(b)=G(a)z(a)+\int_a^bG(t)\lambda(t)u(t)\dd t
=\sqrt{1-\tau}\,z(a)+\sqrt\tau\int_a^bq(t)u(t)\dd t.
\]
Second quantization proves \eqref{eq:capturepair}.

The integrated coupling follows from
\[
\tr(D^{-1}\dot D)=\|\lambda\|^2
                   =\frac{\dd}{\dd t}\log\det D.
\]
The peak bound uses \(D\succeq(1-\tau)I\).
The Sylvester equation for \(\dot G\) has the integral solution
\[
\dot G=\int_0^\infty e^{-rG}\dot D e^{-rG}\dd r.
\]
Hence
\(\|\dot G\|\le\|\dot D\|/(2\sqrt{1-\tau})\), and
\(\|C\|\le\tau\|q\|^2/(2(1-\tau))\).
The same bounds \(\|\Omega\|\).
Smoothness and finite switching follow from the supports of
\(f_i\) and the positive lower bound on \(D\).
\end{proof}

The off-diagonal memory Hamiltonian \(\Omega(t)\) is part of
the construction, not an omitted cross-coupling error.
The underlying bosonic evolution is not treated as a bounded
full-oscillator Hamiltonian.


Figure~\ref{fig:port} is the shared-port synthesis. One scalar input
$u(t)$ drives both memories because $\lambda(t)$ is a vector. The
generator $C(t)$ contains the memory mixing required by passivity.
Two separate window approximations are not this joint port.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[node distance=3mm]
\node[qbox,text width=3.2cm] (u) {Incoming port\\ $u(t)$};
\node[cbox,text width=3.4cm,right=of u] (lam) {$\lambda(t)=\sqrt\tau\,G^{-1}q$\\ $q=(\overline f_1,\overline f_2)^{\mathsf T}$};
\node[qbox,text width=3.2cm,right=of lam] (z) {Memories $z\in\mathbb C^2$\\ $\dot z=Cz+\lambda u$};
\node[pbox,text width=3.2cm,below=of z] (out) {Final amplitudes\\ $\sqrt{1-\tau}\,c_a+\sqrt\tau\,a(\widehat f_i)$};
\node[gbox,text width=3.4cm,below=of lam] (mix) {$\Omega=(C-C^\dagger)/(2i)$\\ off-diagonal mixing kept};
\draw[arr] (u)--(lam);
\draw[arr] (lam)--(z);
\draw[arr] (z)--(out);
\draw[arr] (lam)--(mix);
\end{tikzpicture}
\caption{One temporal port, two memories. Passivity is $C+C^\dagger=-\lambda\lambda^\dagger$. The drawing is the proved one-particle synthesis, not a laboratory capture trace.}
\label{fig:port}
\end{figure}


\subsection{Preparation, loss, and joint capture error}

\begin{lemma}[Referenced multimode vacuum-loss comparison]
\label{lem:jointloss}
Vacuum pure loss of common transmissivity \(t\) on a finite
mode family satisfies
\[
\dist((\mathcal L_t\otimes\id)(\rho),\rho)
\le
\ell(E,t):=\min\{1,\sqrt{2E(1-\sqrt t)}\}
\]
when \(E=\tr(\rho N_{\mathrm{tot}})<\infty\).
\end{lemma}

\begin{proof}
Purify the input and append vacuum loss modes. The overlap of
the initial and beam-splitter-evolved purifications is
\[
a=\tr(\rho\,t^{N_{\mathrm{tot}}/2}).
\]
For \(0\le r\le1\),
\(1-r^n\le n(1-r)\), so
\(1-a\le E(1-\sqrt t)\).
The pure-state distance is \(\sqrt{1-a^2}\le\sqrt{2(1-a)}\).
Partial trace proves the assertion.
\end{proof}

Let
\[
E_F=\sum_{i=1}^2\int_0^\infty
                         n(\omega)|\widehat f_i(\omega)|^2\dd\omega.
\]

\begin{theorem}[Combined capture certificate]
\label{thm:capturebudget}
For the baseline participating marginal and every admitted
normal spectator--reference extension, the capture-plus-storage
channel obeys
\begin{equation}
\boxed{
\dist(\MM(\rho),\mathcal I_H(\rho))
\le \xi_{\mathrm{cap}}
:=\min\{1,p_c+\mu_L+\ell(E_F,\tau_*)\}.
}
\label{eq:capbudget}
\end{equation}
\end{theorem}

\begin{proof}
Replace the independent memory preparation by vacuum at cost
\(p_c\). By \eqref{eq:capturepair}, the resulting reduced
capture and storage are joint vacuum pure loss on the orthonormal
\(F\)-pair. Lemma~\ref{lem:jointloss} costs
\(\ell(E_F,\tau_*)\). Finally compare the \(F\)-selector and
target selector using Lemma~\ref{lem:mode}. All comparisons
retain \(SR\), and contractivity gives the result.
\end{proof}

This prices a common port, overlapping windows, and the
nonvacuum background in one comparison. It does not assume
that the two target packets or their environmental modes are
independent.

For a non-Gaussian input, replace \(\mu_L\) and \(E_F\) by its
justified participating-mode values. The same interaction and
loss proof then apply.

%=====================================================================

Figure~\ref{fig:capbudget} prices preparation and storage once, on the
joint memories. Unselected positive-frequency radiation stays in the
input; it is not replaced by vacuum. The composite transmissivity is
$\tau_*=\tau s$.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[node distance=3mm]
\node[cbox,text width=3.4cm] (prep) {Joint preparation\\ distance $p_c$};
\node[qbox,text width=3.4cm,right=of prep] (loss) {Vacuum storage ports\\ transmissivity $s$};
\node[qbox,text width=3.4cm,right=of loss] (cap) {Capture $\tau$\\ shared port};
\node[gbox,text width=3.4cm,below=of cap] (no) {Unselected modes\\ not reset to vacuum};
\path (prep.south west |- no.south) ++(0,-3.5mm) coordinate (xipos);
\node[pbox,text width=10.8cm,anchor=north west] (xi) at (xipos)
  {$\tau_*=\tau s$\quad $\xi_{\mathrm{cap}}=\min\{1,\,p_c+\mu_L+\ell(E_F,\tau_*)\}$};
\draw[arr] (prep)--(loss);
\draw[arr] (loss)--(cap);
\draw[arr] (cap)--(no);
\draw[arr] (loss.south)--(loss.south |- xi.north);
\draw[arr] (no.south)--(no.south |- xi.north);
\end{tikzpicture}
\caption{Joint capture budget. Correlated preparation is allowed and is not two independent one-packet errors added after the fact.}
\label{fig:capbudget}
\end{figure}


\section{Joint extraction and complete correlated readout}
\label{sec:implementation}
%=====================================================================

\subsection{Exact extraction and global control error}

Initialize two binary blocks in their joint zero state.
For each memory \(i\) and \(1\le n\le K_i\), swap the disjoint
basis pairs
\[
\ket n_{c_i}\ket0_{Q_i}
\longleftrightarrow
\begin{cases}
\ket0_{c_i}\ket n_{Q_i},&n\le a_i,\\
\ket n_{c_i}\ket{a_i}_{Q_i},&a_i<n\le K_i.
\end{cases}
\]
All other states are fixed. Denote the resulting product
unitary by \(U_{\boldsymbol a,\boldsymbol K}\).

For each pair of normalized vectors \(A_{i,n},B_{i,n}\), set
\[
R_{i,n}=\tfrac12
          \ket{A_{i,n}-B_{i,n}}\bra{A_{i,n}-B_{i,n}}.
\]
An exact interaction-picture pulse is
\[
H_E(t)=\sum_{i,n}u_i(t)R_{i,n},\qquad
u_i(t)=\frac{2\pi}{T_E}\sin^2(\pi t/T_E),
\quad 0\le t\le T_E.
\]
Within each block the projections are orthogonal; between
blocks they commute. Since each pulse has area \(\pi\),
its evolution is the stated swap with no relative swap phase.

These number-selective matrix elements are a declared
effective control set. They are not inferred from a generic
oscillator controllability result.

Let \(\EE_{\boldsymbol a,\boldsymbol K}\) be the register
channel obtained by discarding the memories.

\begin{proposition}[Joint extraction-domain tail]
\label{prop:extraction}
For every normal referenced input,
\begin{equation}
\dist(\EE_{\boldsymbol a,\boldsymbol K}(\rho),
                     \BB_{\boldsymbol a}(\rho))
\le
p_{\boldsymbol K}(\rho)
:=\Pr_\rho(N_1>K_1\ \text{or}\ N_2>K_2).
\label{eq:Ktail}
\end{equation}
For the baseline,
\[
p_{\boldsymbol K}(\rho_0)
\le q_1^{K_1+1}+q_2^{K_2+1}.
\]
\end{proposition}

\begin{proof}
For one memory, the extraction channel preserves the number
block through \(a_i\), prepares \(a_i\) on
\(a_i<n\le K_i\), and prepares zero on \(n>K_i\).
This follows by tracing the distinct memory labels after
the swaps.

Both joint channels annihilate cross terms between
\(P_{1,K_1}\otimes P_{2,K_2}\) and its complement, and agree
on the first block. On the complementary positive input block
they are CP and have the same trace \(p_{\boldsymbol K}\).
The half trace norm of the difference of two positive operators
of that trace is at most \(p_{\boldsymbol K}\).
The union bound gives the final assertion.
\end{proof}

The joint bound need not be an equality: different tail branches
can overlap after finite preparation.

Allow a global bounded perturbation
\[
\widehat H_E(t)=H_E(t)+\Delta H_E(t)
\]
on memories, registers, and \(S\), with identity on \(R\).
It may include cross-memory and cross-register coupling.
A permitted finite leakage model has
\[
\Delta H_E(t)
=(P_{1,J_1}\otimes P_{2,J_2}\otimes I_{QS})
 \Delta H_E(t)
(P_{1,J_1}\otimes P_{2,J_2}\otimes I_{QS}),
\qquad J_i\ge K_i.
\]
Define
\[
\zeta_E=\min\left\{1,\int_0^{T_E}\|\Delta H_E(t)\|\dd t\right\}.
\]

\begin{lemma}[Bounded propagator and composition estimates]
\label{lem:stability}
For bounded self-adjoint generators,
\[
\|\widehat U(T)-U(T)\|
\le\int_0^T\|\widehat H(t)-H(t)\|\dd t.
\]
Their channels, including ancillas and partial traces, have
half-diamond error bounded by the same quantity, capped at one.

Compatible channel compositions telescope in half-diamond
distance. Their error is at most the sum of the component
errors, capped at one.
\end{lemma}

\begin{proof}
Variation of constants gives
\[
\widehat U(T)-U(T)
=-\ii\int_0^T\widehat U(T,t)
       (\widehat H(t)-H(t))U(t,0)\dd t.
\]
Unitary factors have norm one. The purified-output vector
comparison bounds trace distance, including references.
Alternatively the complete induced trace norm of the channel
difference is at most twice \(\|\widehat U-U\|\).
The composition statement follows by inserting one changed
factor at a time and using channel contraction.
\end{proof}

No unbounded oscillator drive is bounded by
Lemma~\ref{lem:stability} without an additional domain theorem.

Let the joint register preparation be
\[
\sigma_Q=(1-p_Q)\ketbra{0,0}{0,0}+p_Q\sigma_Q^\perp,
\qquad \sigma_Q^\perp\ket{0,0}=0,
\]
independent of the participating radiation and \(SR\).
It may have internal register correlations.

\begin{theorem}[Derived joint transfer budget]
\label{thm:transfer}
Capture, storage, joint preparation, implemented extraction,
and memory discard have referenced transfer error at most
\begin{equation}
\boxed{
\eta=
\min\{1,\xi_{\mathrm{cap}}+p_Q+\zeta_E
                              +p_{\boldsymbol K}(\rho_0)\}
}
\label{eq:transferbudget}
\end{equation}
against
\((\BB_{\boldsymbol a}\otimes\id_{SR})\mathcal I_H\).
\end{theorem}

\begin{proof}
Apply the same implemented extraction first to the actual
captured state and to the ideal target-in-memory state.
Their difference costs \(\xi_{\mathrm{cap}}\).
Replace the independent register preparation at cost \(p_Q\).
The remaining ideal memory input is the baseline packet pair,
so Lemma~\ref{lem:stability} and
Proposition~\ref{prop:extraction} apply with its own tail.
This ordering does not assume a geometric tail for an
imperfectly captured state.
\end{proof}

If controls are certified only on the initial
\(P_{1,K_1}\otimes P_{2,K_2}\) domain, with arbitrary behavior
outside it, the valid replacement for
\(\zeta_E+p_{\boldsymbol K}\) is
\[
\zeta_{\mathrm{low}}+2\sqrt{p_{\boldsymbol K}}.
\]
Indeed, compare each channel through the normalized projected
input, whose referenced distance from the original is at most
\(\sqrt{p_{\boldsymbol K}}\). The sharp probability tail requires
the global block structure just proved.


Figure~\ref{fig:extraction} is the selective swap on one memory.
Levels $1\le n\le a_i$ move into the register. Levels $a_i<n\le K_i$
are clipped to the corner. States outside the control domain are not
given the same exact tail; an implementation certified only on the low
domain pays the square-root comparison.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[node distance=2.6mm]
\node[qbox,text width=3.3cm] (low) {$1\le n\le a_i$\\ swap into $Q_i$};
\node[cbox,text width=3.3cm,right=of low] (mid) {$a_i<n\le K_i$\\ clip to $\ket{a_i}$};
\node[gbox,text width=3.3cm,right=of mid] (out) {$n>K_i$\\ not in the exact tail};
\node[pbox,text width=10.4cm,below=5mm of mid] (u)
  {$U_{\boldsymbol a,\boldsymbol K}$: orthogonal pulses, area $\pi$, no relative swap phase.};
\draw[arr] (low)--(u);
\draw[arr] (mid)--(u);
\draw[arr] (out)--(u);
\end{tikzpicture}
\caption{Extraction-domain unitary for the synthetic cutoffs used as a schematic, for instance $a_i=1$ and $K_i=3$. Not an executed pulse sequence.}
\label{fig:extraction}
\end{figure}


\subsection{One complete joint pointer model}

Let \(A_{\mathrm{ptr}}\) contain \(B_{\mathrm{reg}}\) pointer
qubits. Its joint state is independently prepared as
\[
\sigma_A=(1-p_A)\ketbra{0^B}{0^B}+p_A\sigma_A^\perp,
\qquad \sigma_A^\perp\ket{0^B}=0.
\]
Internal pointer correlations are allowed.

The ideal copying interaction is
\[
U_R=\prod_{j=1}^{B_{\mathrm{reg}}}
                       \operatorname{CNOT}_{Q_j\to A_j}.
\]
For duration \(T_R\), the bounded generator
\[
H_R=\sum_{j=1}^{B_{\mathrm{reg}}}
 \frac{\pi}{2T_R}\ketbra11_{Q_j}\otimes(I-X_{A_j})
\]
generates it exactly.

Append a fresh finite bath \(D\) in its stated state, independent
of \(QASR\). The actual pointer interaction is
\[
H_R\otimes I_{SD}+G_s(t),\qquad
G_s(t)=\sum_r\epsilon_{s,r}(t)L_r,
\]
where \(L_r=L_r^\dagger\), \(\|L_r\|\le1\), are declared finite
matrices on \(QASD\). They may couple both registers, pointers,
spectators, and bath. This is one joint bounded noise model;
its terms need not factor by packet.

Define
\[
I_{R,s}=\int_0^{T_R}\sum_r|\epsilon_{s,r}(t)|\dd t,\qquad
\delta_{R,s}=\min\{1,I_{R,s}\}.
\]
This includes only errors during that interval. It does not
purport to include an unspecified unbounded leakage process.

The pointer is destructively read in its computational basis.
All of \(Q,A,D\) are then discarded; \(S\) is returned.
Let \(W_s(y\mid\boldsymbol j)\) be a fixed stochastic matrix
from the ideal raw pair to the reported joint record.
It may include correlated assignment errors, timeout, and other
declared flags.

A fresh readiness variable has failure probability \(r_F\).
On that branch the apparatus reports
\(\mathrm{reset\!-\!failed}\), consumes \(Q\), and returns
\(\Tr_QX\) on \(S\). No trial is deleted.

\begin{theorem}[Exact joint assignment error and readout budget]
\label{thm:readout}
For ideal destructive readout followed only by \(W_s\),
\begin{equation}
\boxed{
\ddiam(\PP_{W_s}^S,\PP_Z^S)
=\delta_{\mathrm{as},s}
:=\max_{\boldsymbol j}
                  [1-W_s(y_{\boldsymbol j}\mid\boldsymbol j)].
}
\label{eq:assignment}
\end{equation}
For the complete pointer model above, its error against the
corresponding ideal readout is at most
\begin{equation}
\boxed{
\delta_s=
\min\{1,\,
r_F+(1-r_F)(p_A+\delta_{R,s}+\delta_{\mathrm{as},s})\}.
}
\label{eq:readbudget}
\end{equation}
\end{theorem}

\begin{proof}
The assignment instrument is
\[
\PP_{W_s}^S(X)=
\sum_{\boldsymbol j,y}W_s(y\mid\boldsymbol j)
 \ketbra y y\otimes
                  M_{\boldsymbol j}XM_{\boldsymbol j}^\dagger.
\]
An isometry realizing its assignment sends
\(\ket{\boldsymbol j}\) to
\(\sum_y\sqrt{W_s(y\mid\boldsymbol j)}
              \ket y\ket{\boldsymbol j,y}_D\),
with orthonormal environmental labels.

For any input operator with a reference, pinching in the
computational basis is trace-norm contractive. Its diagonal
blocks \(X_{\boldsymbol j\boldsymbol j}\) obey
\(\sum_{\boldsymbol j}\|X_{\boldsymbol j\boldsymbol j}\|_1
\le\|X\|_1\).
The complete norm of the assignment difference is therefore at
most
\[
\max_{\boldsymbol j}
\sum_y|W_s(y\mid\boldsymbol j)-\delta_{y,y_{\boldsymbol j}}|
=2\delta_{\mathrm{as},s}.
\]
A basis input attaining the maximum gives equality.

On a ready branch, replacing the independent pointer preparation
costs \(p_A\). Lemma~\ref{lem:stability} gives
\(\delta_{R,s}\) for the actual joint pointer evolution.
Assignment then costs \(\delta_{\mathrm{as},s}\).
Convexity prices the visible failure branch by \(r_F\), proving
\eqref{eq:readbudget}.
\end{proof}

This construction includes common preparation and correlated
assignment noise explicitly. It assumes fresh apparatus state,
not unknown memory carried between trials.
Finite-channel and instrument terminology follows the standard
framework \cite{Watrous}; the particular equalities above have
their own proofs.

%=====================================================================

Figure~\ref{fig:pointer} is one complete instrument. The measured
register is absent from the output. Assignment noise may be correlated
across the two words. Failure labels stay in the alphabet; a reset
failure is a visible record, not a discarded trial.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[node distance=2.8mm]
\node[qbox,text width=3.2cm] (q) {Register $Q$\\ both words};
\node[cbox,text width=3.2cm,right=of q] (p) {Pointers\\ correlated preparation};
\node[qbox,text width=3.2cm,right=of p] (b) {Finite bath\\ joint generator};
\node[pbox,text width=3.2cm,below=of b] (w) {Assignment $W$\\ plus reset failure};
\node[gbox,text width=3.2cm,below=of p] (y) {Output $y$\\ raw, decoded, flags};
\node[qbox,text width=3.2cm,below=of q] (gone) {Register removed\\ after the copy};
\draw[arr] (q)--(p);
\draw[arr] (p)--(b);
\draw[arr] (b)--(w);
\draw[arr] (w)--(y);
\draw[arr] (q)--(gone);
\end{tikzpicture}
\caption{Joint pointer model. The half-diamond assignment error is an exact function of $W$ on the declared alphabet. Not a calibrated laboratory instrument.}
\label{fig:pointer}
\end{figure}


\section{C2: information-sensitive finite-register measurements}
\label{sec:tomography}
%=====================================================================

\subsection{A declared rotation family}

Let \(\mathscr P_B=\{I,X,Y,Z\}^{\otimes B_{\mathrm{reg}}}\).
For each nonidentity Pauli string \(P\), choose a product basis
rotation \(R_P\) using
\[
U_X=e^{+\ii\pi Y/4},\qquad
U_Y=e^{-\ii\pi X/4},\qquad U_Z=I,
\]
and use \(I\) on positions where \(P\) is the identity.
These obey
\[
U_X^\dagger ZU_X=X,\qquad U_Y^\dagger ZU_Y=Y.
\]
The rotations are generated by bounded single-qubit Hamiltonians
with pulse area of magnitude \(\pi/2\). Their phase convention
is fixed by the apparatus reference.

The ideal selective maps are
\[
M_{P,\boldsymbol j}
=(\bra{\boldsymbol j}_QR_P)\otimes I_S.
\]
Their effects on \(Q\) are
\[
E_{P,\boldsymbol j}
=R_P^\dagger\ketbra{\boldsymbol j}{\boldsymbol j}R_P.
\]
Thus
\[
\PP_P^S(X)=
\sum_{\boldsymbol j}\ketbra{y_{\boldsymbol j}}{y_{\boldsymbol j}}
 \otimes M_{P,\boldsymbol j}XM_{P,\boldsymbol j}^\dagger
\]
is complete, including unused raw inputs. The setting label is
also retained.

For a selected setting \(P\), let its implemented rotation have
bounded generator deviation \(\Delta H_P(t)\) on \(QS\), and set
\[
\gamma_P=\min\left\{1,\int\|\Delta H_P(t)\|\dd t\right\}.
\]
The pointer budget \(\delta_P\) is
\eqref{eq:readbudget} for that setting.

\begin{theorem}[Complete-record comparison in every setting]
\label{thm:settings}
Let \(\rho_{12SR}=\mathcal I_H(\rho_{\mathfrak v_LSR})\).
For each number or Pauli setting \(s\), the complete model record
obeys
\begin{equation}
\boxed{
\dist\left(\Xi_{s,FSR},
(\PP_s^S\otimes\id_R)
       (\BB_{\boldsymbol a}\otimes\id_{SR})(\rho_{12SR})\right)
\le e_s:=\min\{1,\eta+\gamma_s+\delta_s\}.
}
\label{eq:settingbudget}
\end{equation}
For the number setting, \(\gamma_Z=0\) and the ideal target is
\eqref{eq:jointrecord}.
\end{theorem}

\begin{proof}
Telescope through ideal transfer, ideal setting rotation, and
ideal pointer readout. Their errors are respectively
Theorem~\ref{thm:transfer},
Lemma~\ref{lem:stability}, and
Theorem~\ref{thm:readout}. Every intermediate interface includes
\(S\) and the appended reference \(R\).
\end{proof}

If settings are randomly chosen independently with probabilities
\(\pi_s\), the full setting-labelled state has error at most
\(\sum_s\pi_se_s\). The record does not conceal which setting
was used.

In non-number settings, the raw-word decoder remains a declared
classical label. It is not interpreted as measuring the original
packet number after a basis rotation.

\subsection{Informational completeness}

For ordinary raw output \(\boldsymbol j\), let
\[
v_P(y_{\boldsymbol j})
=\prod_{\text{nonidentity positions of }P}(-1)^{j_{\text{position}}}.
\]
Assign score zero to every failure record.
Then \(v_P\in[-1,1]\).

\begin{proposition}[Informationally complete measurement family]
\label{prop:IC}
For an ideal register density \(\tau\),
\[
\mathbb E_{\PP_P(\tau)}v_P=\tr(P\tau).
\]
The family reconstructs every density on the full binary
register:
\begin{equation}
\tau=\frac1d\left(I+
       \sum_{P\in\mathscr P_B\setminus\{I\}}\tr(P\tau)P\right).
\label{eq:PauliExpansion}
\end{equation}
It is therefore informationally complete on the encoded
retained subspace as well.
\end{proposition}

\begin{proof}
The diagonal sign observable before rotation is the tensor product
of \(Z\)'s on the nonidentity positions. Conjugation by \(R_P\)
gives \(P\). Pauli matrices obey
\(\tr(PP')=d\delta_{P,P'}\) and form a basis for register
operators, proving \eqref{eq:PauliExpansion}.
\end{proof}


Figure~\ref{fig:pauli} is the informationally complete family on the
encoded register, including unused binary words. Scores of failure
records are zero. A rotation error enters the complete-record budget
before any statistical estimate.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[node distance=2.8mm]
\node[qbox,text width=2.6cm] (z) {$Z$ basis\\ raw words};
\node[qbox,text width=2.6cm,right=of z] (x) {$U_X$ then $Z$\\ score of $X$};
\node[qbox,text width=2.6cm,right=of x] (y) {$U_Y$ then $Z$\\ score of $Y$};
\node[cbox,text width=2.6cm,right=of y] (id) {Identity setting\\ normalization};
\node[pbox,text width=11.6cm,anchor=north west] at ([yshift=-5mm]z.south west)
  {$\mathbb E v_P=\operatorname{tr}(P\tau)$ on an ideal register density. Reconstruction is a repeated-preparation statement, not one-shot recovery of $R$.};
\draw[arr] (z)--(x);
\draw[arr] (x)--(y);
\draw[arr] (y)--(id);
\end{tikzpicture}
\caption{Pauli measurement family. Signs of $U_X$ and $U_Y$ are part of the declared convention. The figure does not report an executed tomography.}
\label{fig:pauli}
\end{figure}


\subsection{Statistical and reconstruction certificates}

Let
\[
\tau_{\boldsymbol a}=\BB_{\boldsymbol a}(\rho_{12})
\]
be the ideal finite register state. Put \(m=d^2-1\) and
\(e_*=\max_P e_P\).

\begin{assumption}[Independent repeated preparations]
\label{ass:trials}
For statistical conclusions, each setting uses \(n\) independent
trials of the same participating radiation preparation and the
specified fresh apparatus model. Scores in a setting are identically
distributed. The input and apparatus parameters do not drift.
\end{assumption}

This is an additional repeated-experiment assumption. The
one-shot referenced certificate alone does not imply it.

\begin{lemma}[Bounded-score concentration]
\label{lem:concentration}
For independent \(Z_r\in[-1,1]\),
\[
\Pr\left(\left|\frac1n\sum_rZ_r-\mathbb E\frac1n\sum_rZ_r\right|
>u\right)\le2e^{-nu^2/2}.
\]
For independent Bernoulli variables the exponent improves to
\(2e^{-2nu^2}\).
\end{lemma}

\begin{proof}
For a bounded variable on an interval of length \(b-a\), let
\(f(t)=\log\mathbb E e^{t(Z-\mathbb EZ)}\).
Under exponential tilting,
\(f''(t)\) is a variance and is at most \((b-a)^2/4\);
this follows by centering at the interval midpoint.
Since \(f(0)=f'(0)=0\),
\(f(t)\le t^2(b-a)^2/8\).
Multiply the moment bounds for independent variables, apply
exponential Markov inequality, and optimize over \(t\).
Apply the same argument to the lower tail.
\end{proof}

Let \(\widehat t_P\) be the empirical score in setting \(P\).
Define the Hermitian trace-one estimate
\[
\tau_{\mathrm{lin}}
=\frac1d\left(I+\sum_{P\ne I}\widehat t_PP\right).
\]
Let \(\widehat\tau\) be its nearest density matrix in
Hilbert--Schmidt norm. This finite convex minimization has a
unique solution.

\begin{theorem}[Finite-state reconstruction and coherent cutoff error]
\label{thm:reconstruction}
Under Assumption~\ref{ass:trials}, for \(0<\beta<1\), define
\[
u=\sqrt{\frac{2\log(2m/\beta)}{n}}.
\]
With probability at least \(1-\beta\),
\begin{equation}
\boxed{
\dist(\widehat\tau,\tau_{\boldsymbol a})
\le
r_{\mathrm{tom}}
:=\min\{1,\tfrac12\sqrt m\,(2e_*+u)\}.
}
\label{eq:tomography}
\end{equation}
In the oscillator embedding,
\begin{equation}
\boxed{
\dist(\iota\widehat\tau\iota^\dagger,\rho_{12})
\le\min\{1,t(p_{\boldsymbol a}(\rho_{12}))
                                  +r_{\mathrm{tom}}\}.
}
\label{eq:fullreconstruction}
\end{equation}
For the baseline,
\(p_{\boldsymbol a}\le q_1^{a_1+1}+q_2^{a_2+1}\).
\end{theorem}

\begin{proof}
A score in \([-1,1]\) has expectation error at most \(2e_P\)
under a record trace-distance error \(e_P\).
Lemma~\ref{lem:concentration} and a union bound over \(m\)
settings therefore imply, with the stated probability,
\[
|\widehat t_P-\tr(P\tau_{\boldsymbol a})|\le2e_*+u
\quad\text{for every }P.
\]
Pauli orthogonality gives
\[
\|\tau_{\mathrm{lin}}-\tau_{\boldsymbol a}\|_2
\le\sqrt{m/d}\,(2e_*+u).
\]
Euclidean projection onto the convex set of densities cannot
increase distance from a member of that set. To see this,
the minimizing projection satisfies
\(\langle\tau_{\mathrm{lin}}-\widehat\tau,
\sigma-\widehat\tau\rangle_{\mathrm{HS}}\le0\)
for every density \(\sigma\); expanding squares proves
the asserted distance property. Thus
\[
\frac12\|\widehat\tau-\tau_{\boldsymbol a}\|_1
\le\frac12\sqrt d\,
           \|\widehat\tau-\tau_{\boldsymbol a}\|_2
\le\tfrac12\sqrt m\,(2e_*+u).
\]
The second conclusion is Lemma~\ref{lem:coherenttail} and the
triangle inequality.
\end{proof}

The theorem reconstructs the \emph{reduced packet state}.
Its measurement comparisons are reference-stable, but observations
of the packets alone do not reconstruct an inaccessible
packet--reference density.

\begin{corollary}[A finite tomography tolerance schedule]
For desired reduced-state trace-distance tolerance \(\varepsilon>0\),
it suffices that
\[
t(p_{\boldsymbol a})\le\varepsilon/2,\qquad
e_*\le\varepsilon/(4\sqrt m),\qquad
n\ge\frac{8m}{\varepsilon^2}\log(2m/\beta).
\]
Then \eqref{eq:fullreconstruction} is at most \(\varepsilon\)
with probability at least \(1-\beta\).
\end{corollary}

This schedule is usable only when the apparatus can meet its
systematic-error allowance. The dimension and number of settings
increase with the cutoffs. A fixed error floor is not removed
by increasing the number of trials.

%=====================================================================

Figure~\ref{fig:ledger} is the order in which errors are composed.
The ideal baseline tail is applied only after the apparatus has been
compared with the ideal memory input. Adding two one-packet histogram
errors is not this ledger. Trials reduce the statistical term only;
they do not remove a fixed systematic floor.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[node distance=2.4mm]
\node[cbox,text width=2.5cm] (xi) {$\xi_{\mathrm{cap}}$};
\node[cbox,text width=2.3cm,right=of xi] (eta) {transfer $\eta$};
\node[cbox,text width=2.3cm,right=of eta] (rot) {rotations};
\node[cbox,text width=2.3cm,right=of rot] (d) {pointer $\delta_s$};
\node[pbox,text width=2.4cm,right=of d] (e) {record $e_s$};
\node[gbox,text width=12.4cm,below=5mm of rot]
  {Caps at $1$ stay visible. State-specific tails are not replaced by the baseline tail before the ideal-input comparison.};
\draw[arr] (xi)--(eta);
\draw[arr] (eta)--(rot);
\draw[arr] (rot)--(d);
\draw[arr] (d)--(e);
\end{tikzpicture}
\caption{Joint error ledger. Synthetic numerical values are not supplied: the manuscript prices symbols, and this sheet does not report an executed evaluation.}
\label{fig:ledger}
\end{figure}


\section{Joint count certificates and bounded diagnostics}
\label{sec:diagnostics}
%=====================================================================

\subsection{Operational joint counts}

Let \(e_Z\) be the number-setting budget of
Theorem~\ref{thm:settings}. Then
\begin{equation}
\begin{aligned}
|\mathbb E_{\mathrm{model}}\kappa_i-\nu_i|
&\le\nu_iq_i^{a_i}+a_ie_Z,\\
\tv(p_{\mathrm{model}},p_0)
&\le\min\{1,p_{\boldsymbol a}(\rho_0)+e_Z\}.
\end{aligned}
\label{eq:countcertificates}
\end{equation}
The second line embeds the decoded actual law into
\(\mathbb N_0^2\). The same bound compares decoded records
with retained \(SR\).

\begin{proof}
The report \(\kappa_i\) is an observable between zero and \(a_iI\).
Its mean changes by at most \(a_ie_Z\).
Use the exact marginal ideal deficit and
Theorem~\ref{thm:jointrecord}.
\end{proof}

The clipped covariance, unlike the true unbounded covariance,
is automatically controlled:
\[
\left|\Cov_{\mathrm{model}}(\kappa_1,\kappa_2)
-\Cov_{\mathrm{ideal}}(\kappa_1,\kappa_2)\right|
\le3a_1a_2e_Z.
\]
The product expectation costs \(a_1a_2e_Z\), and the product of
means costs at most \(2a_1a_2e_Z\).
This does not assert that the clipped ideal covariance equals
\(|A_{12}|^2\).

\subsection{Information that number records miss}

\begin{example}[Same number law, different coherences]
The states
\[
\ket{\psi_\pm}
=\frac{\ket{1,0}\pm\ket{0,1}}{\sqrt2}
\]
have identical joint number laws. They are orthogonal.
For cutoffs \(a_i\ge1\), their exact finite encodings have a
two-qubit \(X\otimes X\) expectation of \(+1\) and \(-1\),
respectively, on the two active qubits; unused qubits are zero.
A Pauli setting distinguishes them, whereas any number-diagonal
record does not.

These are synthetic controls, not states claimed to be produced
by the inherited Hawking model.
\end{example}


Figure~\ref{fig:phase} is that example. The two states share every
number-diagonal probability. On the two active qubits the correlator
$X\otimes X$ has opposite signs. A number record cannot see the sign.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[node distance=3mm]
\node[qbox,text width=4.4cm] (p) {$\ket{\psi_+}=(\ket{10}+\ket{01})/\sqrt2$\\ $X\otimes X=+1$};
\node[qbox,text width=4.4cm,right=10mm of p] (m) {$\ket{\psi_-}=(\ket{10}-\ket{01})/\sqrt2$\\ $X\otimes X=-1$};
\node[font=\scriptsize,text=sgray,above=1pt] at ($(p.east)!0.5!(m.west)$) {same $N$};
\node[cbox,text width=9.6cm,anchor=north west] at ([yshift=-5mm]p.south west)
  {Same joint number law. Unused encoded qubits are zero. The distinguishing setting is a Pauli string, not a count.};
\draw[sgray,dashed] (p)--(m);
\end{tikzpicture}
\caption{A phase-sensitive distinction missed by number records. The states are synthetic controls.}
\label{fig:phase}
\end{figure}


\subsection{A hierarchy of bounded tests}

The baseline gives the unbounded connected expression
\[
K_{ijkl}
=\langle b_i^\dagger b_j^\dagger b_kb_l\rangle
-A_{ki}A_{lj}-A_{li}A_{kj}=0.
\]
A general apparatus conclusion about this unbounded expression
requires uniform moment and tail control.

A safe finite alternative is to choose a finite matrix observable
\(O_{\boldsymbol a}=O_{\boldsymbol a}^\dagger\) on the retained
subspace, compute its baseline expectation from
\eqref{eq:jointdensity}, and compare the corresponding bounded
register observable. Examples include real and imaginary parts of
compressed \(b_1^\dagger b_2\), compressed \(b_1b_2\), or a
compressed fourth-order polynomial, each rescaled by its actual
finite norm.

If \(O\) is measured indirectly through tomography, the certificate
is
\[
|\tr(O(\widehat\tau-\tau_{\boldsymbol a}))|
\le2\|O\|\,r_{\mathrm{tom}}.
\]
Compression and saturation need not preserve Gaussian Wick
identities. Their finite baseline value must be calculated;
it must not be set to zero merely because the uncompressed
Gaussian connected expression vanishes.

The diagnostic hierarchy is:
\begin{gather*}
\text{changed means}\\
\longrightarrow\;\text{changed cross covariance}\\
\longrightarrow\;\text{anomalous covariance}\\
\longrightarrow\;\text{non-Gaussian bounded tests}\\
\longrightarrow\;\text{dependence on a named microscopic input.}
\end{gather*}
These are distinct questions, not interchangeable conclusions.

%=====================================================================

Figure~\ref{fig:diagnostics} keeps bounded diagnostics separate from
field moments. A saturated output is not Gaussian, so a truncated
Wick identity is not an automatic baseline. The same record error that
controls a bounded observable does not control an unbounded moment.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[node distance=3mm]
\node[pbox,text width=4.2cm] (ok) {Bounded $O_{\boldsymbol a}$\\ cross coherence,\\ clipped covariance};
\node[gbox,text width=4.2cm,right=8mm of ok] (no) {Unbounded $K_{ijkl}$\\ needs its own tails};
\node[qbox,text width=9cm,anchor=north west] at ([yshift=-5mm]ok.south west)
  {Baseline values are computed on the saturated state, not imported from the unsaturated Wick formula.};
\draw[sgray,dashed] (ok)--(no);
\end{tikzpicture}
\caption{Diagnostic hierarchy. Count means and clipped covariances are operational. They are not a mechanism identification.}
\label{fig:diagnostics}
\end{figure}


\section{C3: certified discrimination of specified departures}
\label{sec:discrimination}
%=====================================================================

\subsection{Bounded-effect theorem}

Let \(\rho_{12}^{(0)}=\rho_0\) and let
\(\rho_{12}^{(\theta)}\) be a named alternative normal state.
For the effective capture model, an alternative must also specify
its participating environmental state and provide the needed
mode moments and occupations. A two-packet marginal alone does
not determine its capture certificate.

Let \(F\) be a finite-register effect implemented by one of the
constructed settings and classical postprocessing:
\[
F=\sum_{\boldsymbol j}r_{\boldsymbol j}
                 E_{s,\boldsymbol j},
\qquad 0\le r_{\boldsymbol j}\le1.
\]
Assign a bounded score in \([0,1]\) to each failure record as part
of the test definition. Define the exact field-interface effect
\[
E_{\boldsymbol a}=\BB_{\boldsymbol a}^*(F).
\]
It is a bounded effect on the packet pair.

\begin{theorem}[Operational departure certificate]
\label{thm:discrimination}
Suppose the two participating preparations have complete
record errors \(e_\theta,e_0\) for the same declared test.
Then
\begin{equation}
\boxed{
|p_{\mathrm{actual}}^{(\theta)}-p_{\mathrm{actual}}^{(0)}|
\ge
\left|\tr E_{\boldsymbol a}
       (\rho_{12}^{(\theta)}-\rho_{12}^{(0)})\right|
-e_\theta-e_0.
}
\label{eq:discrimination}
\end{equation}
The right-hand side may be replaced by its positive part.

More generally, if an independently specified bounded field
effect \(E\) is compared with the compressed register effect
\[
F_E=J P_{\boldsymbol a} E P_{\boldsymbol a}J^\dagger
\]
and that finite effect is implemented with the same type of
certificate, then
\begin{equation}
\begin{split}
|p_{\mathrm{actual}}^{(\theta)}-p_{\mathrm{actual}}^{(0)}|
\ge{}&
|\tr E(\rho_{12}^{(\theta)}-\rho_{12}^{(0)})|\\
&-t(p_{\boldsymbol a}(\rho_{12}^{(\theta)}))
 -t(p_{\boldsymbol a}(\rho_{12}^{(0)}))
 -e_\theta-e_0.
\end{split}
\label{eq:fielddiscrimination}
\end{equation}
Here \(J\) is the finite low-number encoding.
The second assertion does not claim that every \(F_E\) is a
single setting of the constructed product-basis family.
\end{theorem}

\begin{proof}
An effect expectation changes by at most the record trace
distance. Apply this once for each preparation and use the
reverse triangle inequality. The ideal probabilities equal
the expectations of \(E_{\boldsymbol a}\), proving
\eqref{eq:discrimination}.

For the second assertion, the embedded carrier is within
\(t(p_{\boldsymbol a})\) of the original state by
Lemma~\ref{lem:coherenttail}. Its expectation of \(E\) equals
the register expectation of \(F_E\), since the ideal carrier
is supported in the low-number box. Add both coherent errors
before the record comparisons.
\end{proof}

The primary release is the first assertion for the explicitly
constructed Pauli settings and their postprocessings.
An arbitrary spectral measurement of \(F_E\) would require its
own finite control construction or would instead be estimated
tomographically.

\begin{corollary}[Some constructed setting detects every finite departure]
\label{cor:Pauliwitness}
Let
\(\tau_\theta=\BB_{\boldsymbol a}(\rho_{12}^{(\theta)})\),
\(\tau_0=\BB_{\boldsymbol a}(\rho_{12}^{(0)})\), and
\(D=\dist(\tau_\theta,\tau_0)\).
There is a nonidentity Pauli string \(P\) for which the effect
\((I+P)/2\) has ideal probability gap at least
\[
D/\sqrt{d^2-1}.
\]
Its actual gap is at least this value minus its two complete
implementation errors.
\end{corollary}

\begin{proof}
Expand the traceless difference in the Pauli basis.
Writing \(\Delta t_P=\tr(P(\tau_\theta-\tau_0))\),
\[
\|\tau_\theta-\tau_0\|_1
\le\sqrt{\sum_{P\ne I}|\Delta t_P|^2}
\le\sqrt{d^2-1}\max_P|\Delta t_P|.
\]
The effect gap is \(|\Delta t_P|/2\). The constructed setting
measures its sign event, and
Theorem~\ref{thm:discrimination} gives the actual bound.
\end{proof}

Finite informational completeness does not guarantee a useful
gap on a fixed apparatus: dimension factors, cutoff distortion,
and error floors remain in the bound.

\subsection{Finite-trial discrimination}

For two independent ensembles with \(n_\theta,n_0\) trials,
consider the binary postprocessed event in a fixed setting.
It can be realized by independent classical randomization when
the scores \(r_{\boldsymbol j}\) are not zero or one.

For \(0<\beta<1\), put
\[
u_x=\sqrt{\frac{\log(4/\beta)}{2n_x}},
\qquad x\in\{0,\theta\}.
\]
With probability at least \(1-\beta\),
\[
|\widehat p_\theta-\widehat p_0|
\ge
\Delta_{\boldsymbol a}-e_\theta-e_0-u_\theta-u_0,
\]
where
\(\Delta_{\boldsymbol a}
=|\tr E_{\boldsymbol a}(\rho_\theta-\rho_0)|\).
This follows from the Bernoulli form of
Lemma~\ref{lem:concentration} and the two probability comparisons.

Conversely, an observed gap exceeding
\(e_\theta+e_0+u_\theta+u_0\) excludes equality of the two ideal
test probabilities at that confidence level, under the stated
preparation and apparatus assumptions.

\subsection{Interpretation and externally supplied corrections}

A departure from \(\rho_0\) can arise from altered preparation,
ordinary matter interactions, changed scattering, or a changed
background. It does not by itself identify quantum gravity or
information recovery.

If a separate theory supplies
\[
p_\lambda(E)=p_0(E)+\lambda p_1(E)+R_\lambda(E),
\qquad |R_\lambda(E)|\le r(\lambda),
\]
then the certified lower bound is obtained by subtracting
\(r(\lambda)\), both apparatus errors, and any proved cutoff
comparison from \(|\lambda p_1(E)|\).
This is a substitution rule, not an evaluated effective-field-theory
correction. No effective action, loop coefficient, regulator removal,
or microscopic horizon model is introduced by that formula.

%=====================================================================

Figure~\ref{fig:discrimination} is the operational gap. A specified
ideal effect gives a probability difference. Both implementation errors
and, when trials are used, the finite-sample allowance are subtracted.
A positive lower bound certifies a record departure. It does not name
the physical cause.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[node distance=2.6mm]
\node[qbox,text width=3.6cm] (gap) {Ideal gap\\ $|\operatorname{tr} E(\rho_\theta-\rho_0)|$};
\node[cbox,text width=3.4cm,right=of gap] (err) {Subtract $e_\theta+e_0$\\ and trial allowance};
\node[pbox,text width=3.4cm,right=of err] (yes) {Certified record\\ departure, or not};
\node[gbox,text width=10.8cm,below=5mm of err]
  {The alternative state must name its participating preparation and moment bounds. A detectable gap is not quantum gravity, recovery, or a firewall resolution.};
\draw[arr] (gap)--(err);
\draw[arr] (err)--(yes);
\end{tikzpicture}
\caption{Discrimination certificate. The figure does not exhibit a numerical gap for a named alternative.}
\label{fig:discrimination}
\end{figure}


\section{C4: input information and reference limitations}
\label{sec:information}
%=====================================================================

\begin{theorem}[No label recovery from identical accessible inputs]
\label{thm:nolabel}
Let \(x\) have prior distribution \(\pi_x\).
If every \(x\) gives the same density on \emph{all degrees of freedom
participating in the apparatus}, and the apparatus channel and
independent preparations do not depend on \(x\), then all output
record--spectator states are identical.

Any classical estimator based on those outputs has success
probability at most \(\max_x\pi_x\).
For equal priors on \(L\) labels the bound is \(1/L\).
\end{theorem}

\begin{proof}
A fixed linear channel maps identical inputs to identical outputs.
Every subsequent measurement or postprocessing therefore has an
\(x\)-independent output law \(q(z)\).
For an estimator \(g(z)\),
\[
\Pr(g(z)=x)
=\sum_zq(z)\pi_{g(z)}
\le\max_x\pi_x.
\]
The same conclusion holds if the returned spectator is measured
as part of the output.
\end{proof}

\begin{remark}[Necessary accessible-input correction]
Equality of the target two-packet marginals alone is insufficient
for this theorem if the apparatus also receives other modes or
spectators. Those systems could carry \(x\), or their correlations
with the packets could depend on \(x\).
The shared capture acts on \(\mathfrak v_L\), not only on
\(\operatorname{ran}H\). A label dependence in the orthogonal
participating modes can change captured records even when
\(\rho_{12}\) is fixed.

For a radiation-only ideal channel acting exclusively on the
target pair, equality of its two-packet inputs does suffice.
This distinction prevents a side-channel signal from being
renamed microscopic radiation recovery.
\end{remark}

\begin{proposition}[Approximate accessible-input limitation]
If all accessible inputs obey
\(\dist(\rho_x,\rho_*)\le r\), and the apparatus is fixed, then
every estimator has success probability at most
\(\min\{1,\max_x\pi_x+r\}\).
\end{proposition}

\begin{proof}
Contractivity gives output law distance at most \(r\) from the
law for \(\rho_*\). The probability of the estimator's correct
decision changes by at most \(r\) for each \(x\).
Average over the priors and apply
Theorem~\ref{thm:nolabel}.
\end{proof}

\begin{theorem}[Constant radiation channel cannot recover entanglement]
\label{thm:noreference}
Let a finite microscopic message space have dimension \(D_M\).
If a channel \(\mathcal N\) maps every message density to the same
radiation density \(\rho_{\mathrm{rad}}\), then it is a replacement
channel:
\[
\mathcal N(X)=\tr(X)\rho_{\mathrm{rad}}.
\]
For a maximally entangled message--reference input, every decoder
acting only on radiation has entanglement fidelity exactly
\(D_M^{-2}\), and its output has trace distance at least
\(1-D_M^{-2}\) from the maximally entangled target.
\end{theorem}

\begin{proof}
Linearity and decomposition of Hermitian matrices into scalar
and traceless parts show that a channel constant on all densities
vanishes on traceless matrices. Complex linearity proves the
replacement formula.

The referenced radiation output is
\(I_R/D_M\otimes\rho_{\mathrm{rad}}\).
A decoder produces
\(I_R/D_M\otimes\sigma_M\).
Its overlap with the maximally entangled projector is
\[
\bra{\Phi}\left(I_R/D_M\otimes\sigma_M\right)\ket{\Phi}
=\frac{\tr\sigma_M}{D_M^2}=D_M^{-2}.
\]
Testing that projector gives the trace-distance lower bound.
\end{proof}

This theorem requires constancy on \emph{all} message inputs,
not merely on a chosen set of classical basis states.

The A--B stationary preparation is a fixed state, not a
microstate-dependent evaporation channel. A successor claiming
microscopic recovery must specify a message space, a
preparation or encoding map, evolution, radiation access,
and a referenced recovery criterion. Adding tomography or a
decoder cannot supply those dynamics.


Figure~\ref{fig:information} is the information limitation. Identical
accessible inputs cannot reveal a label, even if some unaccessed
environment still carries it. A channel that ignores the message is a
replacement channel, and a radiation-only decoder then has entanglement
fidelity $D_M^{-2}$.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[node distance=2.8mm]
\node[qbox,text width=3.6cm] (lab) {Label or message\\ dimension $D_M$};
\node[cbox,text width=3.6cm,right=of lab] (ch) {Input-independent\\ radiation channel};
\node[pbox,text width=3.6cm,right=of ch] (out) {Same $\rho_{\mathrm{rad}}$\\ fidelity $D_M^{-2}$};
\node[gbox,text width=3.8cm,below=of lab] (hid) {Counterexample:\\ label only in an\\ unaccessed mode};
\draw[arr] (lab)--(ch);
\draw[arr] (ch)--(out);
\draw[arr] (lab)--(hid);
\end{tikzpicture}
\caption{No label recovery and no preserved reference entanglement from a constant radiation channel. Equal target marginals are not equal records of every environmental mode.}
\label{fig:information}
\end{figure}


%=====================================================================
\section{Resources, attainable tolerances, and semantic scope}
\label{sec:resources}
%=====================================================================

\subsection{A joint apparatus ledger}

The construction uses two oscillator memories, not two qubits.
Initial extraction-domain dimensions are \(K_i+1\); the modeled
maximum leakage-domain dimensions are \(J_i+1\).
Capture itself is not restricted to finite occupation support.

With \(s_{\mathrm{lab}}\) spectator qubits, the retained register
and pointer occupancy is
\(2B_{\mathrm{reg}}+s_{\mathrm{lab}}\).
If the finite joint noise bath \(D\) is stored in the laboratory,
its qubit embedding must also be counted. Fresh reset registers
and their peak overlap with retained registers are additional
preparation resources, not silently included in this number.

\begin{center}
\small
\renewcommand{\arraystretch}{1.12}
\begin{longtable}{@{}p{0.22\textwidth}p{0.68\textwidth}@{}}
\toprule
Resource & Declared quantity\\
\midrule
\endfirsthead
\toprule
Resource & Declared quantity\\
\midrule
\endhead
\bottomrule
\endfoot
Shared capture &
One effective temporal port; window \(4L\);
two compact orthonormal tests; vector \(\lambda(t)\);
memory mixing \(\Omega(t)\);
bounds \eqref{eq:capturebounds}.\\
Memories &
Two oscillators; joint preparation \(\sigma_c\);
storage duration \(T_s\), rate \(\kappa_s\);
initial cutoffs \(K_i\); maximum controlled leakage levels \(J_i\).\\
Extraction &
\(K_1+K_2\) selective pulse components;
duration \(T_E\); finite-rank exact generator;
joint bounded deviation \(\Delta H_E\);
register preparation \(\sigma_Q\).\\
Phase-sensitive controls &
At most \(B_{\mathrm{reg}}\) single-qubit rotations per setting;
fixed phase convention;
rotation error integrals \(\gamma_s\).\\
Readout &
\(B_{\mathrm{reg}}\) pointer qubits;
\(B_{\mathrm{reg}}\) ideal controlled-NOT copies;
duration \(T_R\);
joint finite bath and generator \(G_s\);
full matrix \(W_s\).\\
Reset and readiness &
Fresh independent preparation systems;
joint errors \(p_c,p_Q,p_A\);
readiness failure probability \(r_F\);
visible failure output.\\
Repeated tests &
A sufficient, not optimized, schedule has
\(n(d^2-1)\) preparations for tomography.
Different Pauli strings can share bases, but such reuse is not
needed by the proved schedule.\\
Classical records &
Setting, both raw words, both declared decoded values,
all failure flags, and trial identifier.
No failed trial is removed.\\
Spectators &
Stored throughout the sequence; all modeled coupling errors
are included in the named joint generators. Additional
decoherence requires an additional complete comparison.\\
References &
Retained in correctness statements; not laboratory memory.\\
Energy and work &
Oscillator free energies, control phase references, cooling,
bath delivery, and reset work require separately supplied
physical models. No thermodynamic optimum is claimed.\\
\end{longtable}
\end{center}

The preparations can be represented by replacement through swaps
with fresh systems in the stated states, discarding old apparatus
states into the preparation environment. This supplies complete
preparation channels. It does not prove cooling costs or
independence in a reused experimental device.

\subsection{Accuracy within the declared model family}

For a fixed finite retained box, the model budgets can approach
zero if the allowed family permits:
\begin{enumerate}[label=\textup{(\arabic*)},leftmargin=*]
\item arbitrarily long finite windows;
\item capture and storage transmissivity approaching one;
\item arbitrarily accurate joint independent preparations;
\item the stated selective controls for each finite \(K_i\);
\item bounded control deviations approaching zero;
\item joint pointer, assignment, and readiness errors approaching zero.
\end{enumerate}
Indeed, Lemma~\ref{lem:mode} controls the first item,
Lemma~\ref{lem:jointloss} controls the second, geometric tails
control finite \(K_i\), and the displayed perturbation integrals
control the remaining dynamics.

These are tunability properties of a model family, not a
theorem of arbitrary accuracy for fixed hardware.
In particular, \eqref{eq:capturebounds} shows the increasing
capture resources as \(\tau\to1\), and tomography's statistical
and systematic requirements scale with \(d\).

\subsection{Optional Enter interpretation}

\begin{assumption}[Finite semantic characterization]
\label{ass:Enter}
The supplied Landmark I characterization \cite{LandmarkI} is
admitted with \code{ass:strategy-class},
\code{def:requirements}, and \code{thm:main}.
The global finite-qubit strategy satisfies
\(\Cstar,\NSstar,\Nstar,\Kstar,\Rstar\) with the source's
continuation restrictions.
\end{assumption}

Under this assumption, finite allocation, rotations, pointer
couplings, destructive Born readout, and classical postprocessing
have the displayed Enter interpretation. Mixed finite inputs
are treated by finite purification. Values of residual rays on
zero-probability branches are not required.

Without this assumption, all channel theorems remain ordinary
quantum-channel statements with specified Born instruments.
Landmark I neither selects the Unruh state nor constructs the
effective port or its local field counterpart.

%=====================================================================

Figure~\ref{fig:resources} separates parameters that can be driven down
inside the declared family from resources that grow with that demand,
and from error floors that extra trials do not remove. No thermodynamic
optimum is claimed.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[node distance=2.6mm]
\node[qbox,text width=3.5cm] (tun) {Tunable in the family:\\ $L$, $\tau$, $p_c$, $K_i$, $\Delta H$};
\node[cbox,text width=3.7cm,right=of tun] (grow) {Grows with the demand:\\ window, $\int\|\lambda\|^2$,\\ $n(d^2-1)$ trials};
\node[gbox,text width=3.5cm,right=of grow] (floor) {Not removed by trials:\\ fixed hardware floors,\\ open local coupling};
\draw[arr] (tun)--(grow);
\draw[arr] (grow)--(floor);
\end{tikzpicture}
\caption{Resource reading of the joint ledger. Attainability is a property of the model family, not a theorem for fixed hardware, and not a reported optimization.}
\label{fig:resources}
\end{figure}


\section{Open physical bridges: not additional release claims}
\label{sec:open}
%=====================================================================


Figure~\ref{fig:open} is the boundary of the release. Each column is
an obligation list, not a mechanism constructed in this manuscript.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[node distance=2.4mm]
\node[gbox,text width=3.6cm] (loc) {Local probe\\ coupling region,\\ scattering map,\\ packet observable,\\ derived error};
\node[gbox,text width=3.6cm,right=of loc] (col) {Collapse packet\\ named geometry,\\ propagated profile,\\ local, tail, and\\ propagation errors};
\node[gbox,text width=3.6cm,right=of col] (mic) {Microscopic channel\\ one theory, action,\\ state, regulator,\\ message space};
\node[cbox,text width=11.2cm,below=5mm of col]
  {Em03-C supplies the joint baseline and the effective tests. It does not cross any of these three bridges.};
\end{tikzpicture}
\caption{Open physical bridges. A certified departure from the semiclassical baseline is not identification of a quantum-gravity mechanism. O.1 stays open. No quantum coefficient is admitted.}
\label{fig:open}
\end{figure}


\subsection{A local Schwarzschild system--probe correspondence}

The effective port has not been identified with a local
Schwarzschild coupling. A possible successor begins with a
specified probe field \(\psi\) and compact bilinear interaction
\[
S_{\mathrm{int}}
=-\int\lambda(x)\phi(x)\psi(x)\dd\mathrm{vol}_g,
\qquad \lambda\in C_c^\infty(\mathcal M;\mathbb R).
\]
The Fewster--Verch framework supplies a setting for local
system--probe scattering and induced observables
\cite{FewsterVerch}. Asymptotic local-observable measurement
constructions give additional comparison tools \cite{FewsterJubbRuep}.
Neither result is an identification of this manuscript's
matrix capture with a local asymptotic packet-number instrument.

The successor must specify:
\begin{enumerate}[label=\textup{(\arabic*)},leftmargin=*]
\item probe state, coupling region, and switching;
\item coupled classical evolution and quantum scattering map;
\item a selected outgoing probe observable;
\item its relation to the asymptotic packet algebra;
\item unwanted modes, creation-operator mixing, probe noise,
angular resolution, and finite-radius propagation;
\item a common comparison topology and a derived error.
\end{enumerate}
A passive pure-loss reduction is a theorem target, not a default
property of a bilinear local coupling.
A first induced smeared-field observable is a narrower target
than exact packet-number detection.

\subsection{A collapse packet theorem}

Em03-A's \code{thm:peeling} concerns compact time tests under
explicit peeling and affine-regularity assumptions. The Fourier
packet profiles here are noncompact. A collapse successor needs
a named collapse geometry and initial state, characteristic
restriction, propagated packet identification, and uniform tails.

A possible error decomposition is
\[
\epsilon_{\mathrm{collapse}}
\le\epsilon_{\mathrm{local}}
 +\epsilon_{\mathrm{tail}}
 +\epsilon_{\mathrm{propagation}}.
\]
No term in this decomposition is proved for a collapse spacetime
by the present stationary apparatus analysis. The stationary
Unruh construction \cite{DMP} does not discharge those additional
obligations.

\subsection{One controlled correction and microscopic dynamics}

A correction successor must choose one theory and specify its
action, expansion parameter, state, renormalization, observable,
regulator removal, and remainder. Altered classical geometry,
interacting matter, graviton loops, and quantum horizon dynamics
are different models.

The broader information problem motivates joint-state rather
than mean-spectrum questions. Fine-grained radiation entropy
calculations \cite{EntropyReview} are not, by themselves, a
complete set of radiation amplitudes or the message channel
required in Section~\ref{sec:information}.
The horizon-smoothness and unitarity tension \cite{AMPS} likewise
requires hypotheses about interior observables and their relation
to exterior radiation. No interior algebra is constructed here.

\begin{center}
\small
\renewcommand{\arraystretch}{1.12}
\begin{tabular}{@{}p{0.29\textwidth}p{0.60\textwidth}@{}}
\toprule
Question & Boundary of this construction\\
\midrule
Microscopic recovery &
We supply distinguishability tests; a microscopic
input-dependent evaporation channel is absent.\\
Smooth horizon and unitarity &
Exterior finite-mode tests do not construct interior partners
or decide which locality or factorization hypothesis changes.\\
Gravitational path integrals &
No path integral, ensemble identification, or factorization
theorem is introduced.\\
Four-dimensional physical correspondence &
The field baseline is Schwarzschild scalar radiation;
the apparatus-to-local-field bridge remains open.\\
Ultraviolet robustness &
No universality class of ultraviolet completions is specified
or compared.\\
Evaporation endpoint &
The fixed-background model has no Planck-regime endpoint
dynamics.\\
\bottomrule
\end{tabular}
\end{center}

No result here closes O.1, changes XVI-N's hypotheses, discharges
the XIV--XV characteristic-product chain, or admits its separate
quantum coefficient. The supplied documentary excerpts about
those programmes are not apparatus premises.

%=====================================================================
\section{Source repairs and logical checks}
\label{sec:checks}
%=====================================================================

\begin{enumerate}[label=\textbf{D\arabic*.},leftmargin=*]
\item \textbf{Covariance orientation.}
With conjugate-linear annihilation tests,
\(A_{ij}=\langle b_j^\dagger b_i\rangle\), while
\(\langle b_i^\dagger b_j\rangle=A_{ji}\).
The characteristic function, passive transformation, and Wick
formula use this convention consistently.

\item \textbf{Orthogonality is not independence.}
Joint normality comes from quasifreeness and the full occupation
matrix, not from two means.

\item \textbf{Two separate window approximations are not a joint port.}
The compact tests are jointly orthonormalized. A shared-port
matrix interaction captures them simultaneously and explicitly
contains the required memory mixing.

\item \textbf{Background modes are retained.}
The participating input has dimension at most four in its
one-particle restriction and includes nonvacuum background
covariance. Auxiliary vacuum ports are new model preparations.

\item \textbf{Errors are derived at named interfaces.}
Joint preparation and assignment are priced once. Shared
Hamiltonian perturbations are not silently replaced by the
sum of two one-packet histogram errors.

\item \textbf{A global control bound is not a support-only bound.}
The sharp extraction tail uses the defined global block behavior.
An arbitrary outside-domain implementation requires the
separately stated square-root estimate.

\item \textbf{Count clipping and coherent approximation differ.}
Equation~\eqref{eq:jointtail} is an exact record probability.
Equation~\eqref{eq:coherenttail} controls a coherent state
comparison, including references.

\item \textbf{Failure labels stay visible.}
Raw words, decoder values, setting, and failure records remain
in the comparison. Statistical samples do not discard failures.

\item \textbf{Tomography is a repeated-preparation statement.}
Reference-stable one-shot correctness does not prove independent
trials or reconstruct an inaccessible reference state.

\item \textbf{Compressed Wick tests require their own baseline.}
The finite saturated output is generally not Gaussian.
A truncated cumulant is not automatically zero.

\item \textbf{Equal target marginals need not imply equal actual records.}
The no-information theorem concerns all apparatus-accessible
inputs, not an omitted environment or spectator.

\item \textbf{Departure is not mechanism identification.}
A certified departure from this specified semiclassical state
does not establish quantum gravity, recovery, evaporation,
or a firewall resolution.
\end{enumerate}

%=====================================================================
\section{Final analytical release}
\label{sec:release}
%=====================================================================

\begin{theorem}[Em03-C joint operational release]
\label{thm:release}
Under the stated stationary radiation input and effective
apparatus specification:
\begin{enumerate}[label=\textup{(\roman*)},leftmargin=*]
\item the complete normal two-packet Gaussian state, joint count
law, independence criterion, and Wick correlations are constructed;

\item the joint finite carrier and complete destructive record
retain spectators and references, with exact joint count-tail
error and a separately proved coherent-tail comparison;

\item compact orthonormal temporal modes and a shared-port
two-memory interaction are constructed, with a derived joint
capture certificate;

\item joint preparations, coherent extraction, global bounded
controls, correlated assignment noise, and complete pointer
readout give the derived setting budgets
\eqref{eq:transferbudget}--\eqref{eq:settingbudget};

\item a phase-sensitive informationally complete finite
measurement family has the statistical and trace-distance
reconstruction bounds of Theorem~\ref{thm:reconstruction};

\item specified bounded state departures have the operational
discrimination certificate of
Theorem~\ref{thm:discrimination};

\item identical accessible inputs cannot reveal an unknown
label, and a constant message-to-radiation channel cannot
recover reference entanglement.
\end{enumerate}

Statistical conclusions retain
Assumption~\ref{ass:trials}; Enter terminology retains
Assumption~\ref{ass:Enter}.
Alternative-state certificates require their own named joint
participating preparation and moment bounds.

No local Schwarzschild detector, collapse-packet limit,
quantum-gravity correction, microstate-dependent evaporation
channel, Page curve, island construction, restored unitarity,
firewall resolution, or evaporation endpoint is established.
\end{theorem}

\begin{proof}
Theorems~\ref{thm:gaussian} and \ref{thm:counts} prove (i).
Theorem~\ref{thm:jointrecord} and
Lemma~\ref{lem:coherenttail} prove (ii).
Lemmas~\ref{lem:windows}--\ref{lem:mode} and
Theorems~\ref{thm:sharedcapture}--\ref{thm:capturebudget}
prove (iii).
Proposition~\ref{prop:extraction} and
Theorems~\ref{thm:transfer}, \ref{thm:readout}, and
\ref{thm:settings} prove (iv).
Proposition~\ref{prop:IC} and
Theorem~\ref{thm:reconstruction} prove (v).
Theorem~\ref{thm:discrimination} proves (vi).
Theorems~\ref{thm:nolabel} and
\ref{thm:noreference} prove (vii).

The constructed model contains none of the local-coupling,
collapse, or microscopic dynamics required for the excluded
physical conclusions. Their outstanding interfaces are stated
in Section~\ref{sec:open}.
\end{proof}

\paragraph{Final scope statement.}
Em03-C constructs a joint-packet semiclassical baseline and
certified, information-sensitive measurement interfaces.
It identifies which specified departures can be distinguished
operationally, while retaining explicit local-coupling, collapse,
and microscopic-dynamics obligations. It does not derive a Page
curve, restore unitarity, resolve the firewall paradox, or determine
the endpoint of evaporation.

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

\paragraph{Data and verification availability.}
All analytical constructions used in the release are included.
No executed packet dataset, control optimization, experimental
calibration, formal proof certificate, or compilation record is
supplied or asserted.

\paragraph{Attribution and predecessor status.}
The radiation prescription is inherited from the supplied
Em03-A source. Em03-B supplies the one-packet construction
being extended. The En02-E comparison is identified as a
supplied companion account, not independently checked full text.
The finite-mode Gaussian specialization is proved here for
self-containment and is not presented as a new Gaussian-state
classification theorem.

%=====================================================================
\begin{thebibliography}{99}

\bibitem{Em03A}
N.~Cheung,
\emph{Em03-A: A Fixed-Background Hawking Emission Theorem},
version 1, 8 October 2026.
Supplied source \code{Em03-A-v1.tex}.
Inputs: boundary-state prescription,
\code{lem:horizonoccupation},
\code{eq:quantumscatter},
\code{eq:numberdensity}, and \code{eq:packetlaw}.
The compact-test collapse boundary is
\code{thm:peeling}.

\bibitem{Em03B}
N.~Cheung,
\emph{Em03-B: A Certified Apparatus-Model Readout of One Hawking Packet},
version 2, 8 October 2026.
Supplied source \code{Em03-B-v2.tex}.
One-mode normality, coherent extraction, effective capture,
complete instruments, and the distinction between destructive
record tails and coherent comparisons.

\bibitem{En02EAccount}
\emph{Supplied companion account of En02-E and related manuscripts}.
Documentary source reproduced with the revision programme,
including corpus identifier \code{48702295} and retrieval
identifier \code{retrieval-3}.
Reported locators: \code{thm:gaussian},
\code{eq:outcov}, \code{eq:K}, \code{cor:stats},
\code{thm:cutoff}, and \code{thm:ancilla}.
The full En02-E manuscript was not supplied; this entry is not
represented as full-text verification or a formal premise.

\bibitem{LandmarkI}
N.~Cheung,
\emph{I}, version 1.
Supplied characterization theorem and excerpts:
\code{ass:strategy-class},
\code{def:requirements}, and \code{thm:main}.
Used only for the optional finite-qubit semantic interpretation.

\bibitem{DMP}
C.~Dappiaggi, V.~Moretti, and N.~Pinamonti,
\emph{Rigorous construction and Hadamard property of the Unruh
state in Schwarzschild spacetime},
Advances in Theoretical and Mathematical Physics
\textbf{15} (2011), 355--447.
arXiv:0907.1034.
The global state construction is inherited through Em03-A,
not reproved here.

\bibitem{Weedbrook}
C.~Weedbrook, S.~Pirandola, R.~Garc\'ia-Patr\'on,
N.~J.~Cerf, T.~C.~Ralph, J.~H.~Shapiro, and S.~Lloyd,
\emph{Gaussian Quantum Information},
Reviews of Modern Physics \textbf{84} (2012), 621--669.
arXiv:1110.3234.
doi:10.1103/RevModPhys.84.621.
General context; the required finite-mode construction and
statistics are proved explicitly in this manuscript.

\bibitem{GardinerCollett}
C.~W.~Gardiner and M.~J.~Collett,
\emph{Input and output in damped quantum systems:
Quantum stochastic differential equations and the master equation},
Physical Review A \textbf{31} (1985), 3761--3774.
doi:10.1103/PhysRevA.31.3761.

\bibitem{YamamotoJames}
N.~Yamamoto and M.~R.~James,
\emph{Zero-dynamics principle for perfect quantum memory in
linear networks},
New Journal of Physics \textbf{16} (2014), 073032.
arXiv:1403.1698.
doi:10.1088/1367-2630/16/7/073032.
Linear-memory context, not a local Hawking-detector
correspondence or a substitute for the matrix synthesis proved here.

\bibitem{Watrous}
J.~Watrous,
\emph{The Theory of Quantum Information},
Cambridge University Press, 2018.
Chapters 2 and 3: channels, instruments, state distinguishability,
and completely bounded trace norms.

\bibitem{FewsterVerch}
C.~J.~Fewster and R.~Verch,
\emph{Quantum fields and local measurements},
arXiv:1810.06512v3 (2020).
Local system--probe scattering, induced observables, and
measurement instruments.
No identification of the present effective port with a local
Schwarzschild coupling is imported.

\bibitem{FewsterJubbRuep}
C.~J.~Fewster, I.~Jubb, and M.~H.~Ruep,
\emph{Asymptotic measurement schemes for every observable of
a quantum field theory},
arXiv:2203.09529.
doi:10.1007/s00023-022-01239-0.
A comparison framework for the open local-observable programme,
not an exact finite packet-number apparatus certificate.

\bibitem{EntropyReview}
A.~Almheiri, T.~Hartman, J.~Maldacena, E.~Shaghoulian,
and A.~Tajdini,
\emph{The entropy of Hawking radiation},
Reviews of Modern Physics \textbf{93} (2021), 035002.
arXiv:2006.06872.
doi:10.1103/RevModPhys.93.035002.
Information-problem context only; not a premise for a radiation
encoding or recovery channel in this manuscript.

\bibitem{AMPS}
A.~Almheiri, D.~Marolf, J.~Polchinski, and J.~Sully,
\emph{Black Holes: Complementarity or Firewalls?},
Journal of High Energy Physics \textbf{02} (2013), 062.
arXiv:1207.3123.
doi:10.1007/JHEP02(2013)062.
Interpretation boundary only; exterior tomography here does not
decide the compatibility assumptions in that argument.

\end{thebibliography}

\end{document}