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% VERSION 2
% The mathematical interface is supplemented by:
%   - a specified passive capture model;
%   - a proved finite-switching packet approximation;
%   - explicit coherent oscillator--register extraction;
%   - bounded-generator control certificates;
%   - a complete noisy readout instrument, including failure records;
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% EVIDENCE BOUNDARY
% This source supplies analytical constructions and proofs.
% Compilation, formal verification, numerical execution, experimental
% calibration, and a local Schwarzschild detector are not asserted.
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\hypersetup{
  pdftitle={Em03-B: A Certified Apparatus-Model Readout of One Hawking Packet},
  pdfauthor={Ngo Cheung},
  pdfsubject={Passive packet capture, coherent finite extraction,
              destructive instruments, and reference-stable errors}
}

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\title{\textbf{Em03-B}\\[1mm]
\textbf{A Certified Apparatus-Model Readout of One Hawking Packet}\\[2mm]
\large Passive capture, coherent finite extraction,
and complete destructive records}

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

\date{Version 2 --- 8 October 2026}

\begin{document}

\maketitle

\begin{abstract}
We construct and analyze a specified apparatus-model realization of
finite destructive number readout for one outgoing Hawking packet.
The stationary field prescription is an explicit predecessor input.
Its centered quasifree restriction gives a normal geometric
one-mode density, proved here without assuming a density operator
for the entire stationary radiation field.

The apparatus consists of a passive temporal-mode capture port, a
memory oscillator, a binary register, and a complete readout
instrument. A smooth, finitely switched capture interaction has an
explicit transfer kernel. A zero-integral temporal-window
construction controls packet mismatch and prevents an infrared
integrability shortcut. Unselected positive-frequency radiation is
not replaced by vacuum: its occupation and correlations enter a
reference-stable mode-comparison bound. Vacuum auxiliary ports and
independent apparatus preparations are separately declared model
data.

Coherent oscillator--register extraction is generated by explicit
number-selective swap pulses. Its finite control-domain error is
exactly the probability above a second cutoff when the stated global
block structure is present. Bounded control perturbations and a
distinct support-only certificate are treated separately. A complete
noisy readout instrument includes assignment errors, quantum
readout errors, spectator coupling, imperfect preparation, and
visible failure outcomes.

The resulting transfer and implementation budgets are derived from
model parameters and substituted into finite-count, mean-bias, and
record--spectator--reference bounds. A constructive tolerance
schedule and an apparatus resource ledger are supplied. These are
model-level results. A local Schwarzschild-field coupling, a justified
laboratory effective model, and experimental implementation remain
separate obligations.
\end{abstract}

\tableofcontents

%=====================================================================
\section{Scope, evidence, and the three levels of the claim}
\label{sec:scope}
%=====================================================================

The architecture is
\[
\boxed{
\text{radiation input}
\ \xrightarrow{\text{passive capture}}\
c
\ \xrightarrow{\text{coherent extraction}}\
Q
\ \xrightarrow{\text{complete readout}}\
F .
}
\]
Here \(c\) is a memory oscillator, \(Q\) is a finite binary register,
and \(F\) contains raw words, decoded values, and declared failures.

Three levels are kept separate.

\begin{center}
\renewcommand{\arraystretch}{1.15}
\begin{tabular}{@{}p{0.22\textwidth}p{0.38\textwidth}p{0.30\textwidth}@{}}
\toprule
Level & Evidence supplied & Permitted conclusion\\
\midrule
Mathematical interface
& Channels, normal finite-mode states, adjoints, and record identities
& The B0--B3 interface\\
Specified apparatus model
& Capture dynamics, preparations, extraction pulses, noise
instruments, and derived budgets
& Model-level B4--B5 realization\\
Physical realization
& Local coupling, effective-model justification, calibrated
parameters, and implementation evidence
& Not established here\\
\bottomrule
\end{tabular}
\end{center}

The capture port is a one-dimensional Markov input--output
\emph{model}. It is not identified with a local detector coupled
to the massless scalar on Schwarzschild. In particular, its
two-sided temporal envelope includes auxiliary degrees of freedom
that are not negative-energy modes of that scalar field.

Our model uses explicitly available number-selective control
Hamiltonians. This is a declared control set, not a claim that a
dispersive oscillator--qubit laboratory automatically implements
those Hamiltonians. General oscillator-control constructions
\cite{Krastanov} do not supply a compiler or a leakage certificate
for the pulses constructed below.

Local system--probe measurement theory requires a separate
spacetime coupling and an induced-observable identification
\cite{FewsterVerch}. No theorem below supplies that missing
correspondence.

\paragraph{Evidence boundary.}
The arguments are analytical. Compilation, formal verification,
numerical pulse optimization, parameter calibration, and laboratory
execution are not asserted. ``Certified'' means that the
displayed implications have stated hypotheses and mathematical
proofs.

\paragraph{Units and distances.}
We use \(c=\hbar=k_B=1\), \(G>0\), and
\[
\dist(\rho,\sigma)=\frac12\norm{\rho-\sigma}_1,\qquad
\ddiam(\Phi,\Psi)=\frac12\norm{\Phi-\Psi}_\diamond,\qquad
\tv(p,q)=\frac12\sum_j|p(j)-q(j)|.
\]
A channel on trace-class operators is understood as the predual
of a normal completely positive unital map when its input is
infinite dimensional. All operational comparisons retain finite
spectators and arbitrary finite-dimensional external references.
The mathematical identities also hold for normal extensions to
separable reference spaces.

\paragraph{Predecessor status.}
Em03-A and Landmark I are used through the supplied text
\cite{Em03A,LandmarkI}. Their proofs and registry status are not
independently verified here. Other internal manuscript names are
not formal premises of the apparatus construction.

%=====================================================================
\section{Audited field input and the packet marginal}
\label{sec:field}
%=====================================================================

\subsection{The inherited data}

Fix one angular harmonic of degree \(\ell\), a mass parameter
\(M>0\), and
\[
B=[\omega_-,\omega_+]\subset(0,\infty),\qquad
h\in C_c^\infty((0,\infty)),\qquad
\supp h\subset B,\qquad
\int_B|h(\omega)|^2\dd\omega=1.
\]
The normalized Killing time is the predecessor's time, and
\(\betah=8\pi GM\).

\begin{assumption}[Inherited radiation prescription]
\label{ass:field}
The horizon and past-infinity input states are independent,
centered, regular, and quasifree. Their positive-Killing-frequency
covariances are
\begin{align}
\langle a_{\mathrm H}(\omega)^\dagger
                   a_{\mathrm H}(\omega')\rangle
 &=\nh(\omega)\delta(\omega-\omega'),
&
\nh(\omega)&=(e^{\betah\omega}-1)^{-1},\\
\langle a_{\mathrm H}(\omega)a_{\mathrm H}(\omega')\rangle&=0,
&
\langle a_{\mathrm I}(\omega)^\dagger
                   a_{\mathrm I}(\omega')\rangle&=0.
\end{align}
The infinity state is its positive-frequency vacuum, hence is
centered with zero anomalous covariance. Cross-channel
covariances vanish.

Stationary scattering is complex linear:
\[
a_{\mathrm{out}}(\omega)
=\tau_\ell(\omega)a_{\mathrm H}(\omega)
+\rho_{\mathrm I,\ell}(\omega)a_{\mathrm I}(\omega),
\qquad
|\tau_\ell|^2+|\rho_{\mathrm I,\ell}|^2=1.
\]
Write
\begin{equation}
n(\omega)=
\frac{\Gamma_\ell(\omega)}{e^{\betah\omega}-1},
\qquad
\Gamma_\ell=|\tau_\ell|^2,\qquad 0\le\Gamma_\ell\le1.
\label{eq:nomega}
\end{equation}
The outgoing state is therefore centered, gauge-invariant, and
quasifree, with number covariance \(n(\omega)\delta(\omega-\omega')\).
\end{assumption}

The components of this assumption have distinct source roles.

\begin{center}
\small
\renewcommand{\arraystretch}{1.15}
\begin{tabular}{@{}p{0.38\textwidth}p{0.53\textwidth}@{}}
\toprule
Component & Supplied Em03-A locator or status\\
\midrule
Product boundary prescription and quasifreeness
& State subsection and declared bulk-to-boundary input\\
Horizon number and anomalous covariance
& \code{lem:horizonoccupation}\\
Infinity vacuum and independence
& State subsection and \code{lem:horizonoccupation}\\
Complex-linear stationary scattering
& \code{eq:quantumscatter}\\
Outgoing compact-band number covariance
& \code{prop:packet}, \code{eq:numberdensity}\\
Packet occupation integral
& \code{eq:packetlaw}\\
Bulk existence and Hadamard identification
& Declared Dappiaggi--Moretti--Pinamonti input \cite{DMP}\\
Weighted radiation-test completion used below
& Explicit mathematical completion in
Section~\ref{subsec:completion}; not a new bulk theorem\\
\bottomrule
\end{tabular}
\end{center}

Hadamard regularity alone does not select these covariances.

Define
\[
\bop=a_{\mathrm{out}}(h),\qquad
[\bop,\bop^\dagger]=1,\qquad
\Nop=\bop^\dagger\bop,
\]
where annihilation operators are conjugate-linear in their test:
\(a(h)=\int\overline{h(\omega)}a(\omega)\dd\omega\).
Set
\begin{equation}
\nuh=\int_B|h(\omega)|^2 n(\omega)\dd\omega,\qquad
\qh=\frac{\nuh}{1+\nuh}.
\label{eq:nu}
\end{equation}
Then
\[
0\le\nuh\le n_{\max}:=(e^{\betah\omega_-}-1)^{-1}<\infty.
\]

\subsection{B0: complete one-mode identification}

\begin{theorem}[Normal geometric packet marginal]
\label{thm:marginal}
Under Assumption~\ref{ass:field}, the selected packet state is
represented normally on
\(\HH_h=\FF(\mathbb C)\cong\ell^2(\mathbb N_0)\) by
\begin{equation}
\boxed{
\rho_h=(1-\qh)\sum_{n=0}^\infty\qh^n\ketbra n n.
}
\label{eq:rhoh}
\end{equation}
For \(\nuh=0\), it is the vacuum. Its count law has mean
\(\nuh\), variance \(\nuh(1+\nuh)\), and generating function
\[
\mathbb E s^{N_h}=\frac{1-\qh}{1-\qh s},
\qquad |s|<\qh^{-1}
\]
when \(\qh>0\).
\end{theorem}

\begin{proof}
Complex-linear scattering and restriction preserve centered
quasifreeness. The packet has
\(\langle b_h^2\rangle=0\) and
\(\langle b_h^\dagger b_h\rangle=\nuh\). Thus its displacement
characteristic function is
\[
\langle e^{zb_h^\dagger-\overline z b_h}\rangle
=e^{-(\nuh+1/2)|z|^2}.
\]
For \(\nuh>0\), construct the trace-class coherent-state mixture
\[
\sigma=
\int_{\mathbb C}\frac{e^{-|\alpha|^2/\nuh}}{\pi\nuh}
                   \ketbra\alpha\alpha\,\dd^2\alpha.
\]
The measure has total mass one. Its characteristic function is
the displayed Gaussian, since
\[
\bra\alpha e^{zb_h^\dagger-\overline z b_h}\ket\alpha
=e^{-|z|^2/2+z\overline\alpha-\overline z\alpha}.
\]
Angular and radial integration give
\[
\bra n\sigma\ket m
=\delta_{nm}\frac1{\nuh n!}
       \int_0^\infty e^{-(1+\nuh^{-1})t}t^n\dd t
=\delta_{nm}\frac{\nuh^n}{(1+\nuh)^{n+1}}.
\]
This is \eqref{eq:rhoh}. Equality of characteristic functions
identifies the states on the Weyl algebra, because its dense
linear span consists of Weyl operators. The vacuum case follows
directly. Summing the geometric series proves the moment and
generating-function statements.
\end{proof}

This is a finite-mode identification, consistent with the Gaussian
framework \cite{Weedbrook}. It neither constructs nor requires a
trace-class density operator for the entire stationary continuum.

\begin{example}[The mean-only stop]
The density
\(\frac12\ketbra00+\frac12\ketbra22\)
is centered and gauge-invariant, has
\(\langle b_h^2\rangle=0\), and has mean one. It is not the
geometric state with mean one. Quasifreeness is essential to
Theorem~\ref{thm:marginal}.
\end{example}

\subsection{A weighted test completion, not a global Fock density}
\label{subsec:completion}

For positive-frequency tests, use the norm
\[
\norm{v}_{n}^2
=\int_0^\infty(1+n(\omega))|v(\omega)|^2\dd\omega.
\]
The compact positive-frequency covariance extends by continuity
to its completion. The resulting Gaussian characteristic
functional remains positive: each finite matrix and each Weyl
positivity condition is a limit of the corresponding compact-test
condition.

For any finite-dimensional canonical subspace with finite
occupation matrix \(A\succeq0\), the restricted state is normal.
Indeed, diagonalize \(A\) by a passive finite-mode unitary.
The characteristic function becomes a product of one-mode
characteristic functions; Theorem~\ref{thm:marginal} constructs
their normal product density. Transforming it back constructs the
normal finite-mode restriction.

This completion provides the radiation-mode calculations used
below. It does not assert a new characteristic restriction theorem
for local bulk fields.

%=====================================================================
\section{The B1--B3 mathematical interface}
\label{sec:interface}
%=====================================================================

\subsection{The saturating carrier}

Fix integers \(N\ge1\) and, later, \(K\ge N\). The reported
saturation level \(N\) and the control-domain cutoff \(K\) are
different parameters.

Let
\[
\KK_N=\Span\{\ket0_C,\ldots,\ket N_C\},\qquad
P_N=\sum_{n=0}^N\ketbra n n,\qquad Q_N=I-P_N.
\]
Let \(V_N\) identify the number bases on \(P_N\HH_h\) and
\(\KK_N\). Define
\begin{equation}
\CC_N(X)=V_NP_NXP_NV_N^\dagger
               +\tr(Q_NX)\ketbra N N_C.
\label{eq:carrier}
\end{equation}

\begin{proposition}[Channel and observable identification]
\label{prop:carrier}
The map \(\CC_N\) is CPTP. Its adjoint is
\[
\CC_N^*(A)=P_NV_N^\dagger A V_NP_N
                   +\bra N A\ket N_C Q_N.
\]
For \(D_N=\sum_{k=0}^N k\ketbra k k_C\),
\[
\CC_N^*(D_N)=\min(N_h,N).
\]
The readout effects pulled back to the oscillator are
\[
\CC_N^*(\ketbra k k_C)=
\begin{cases}
\ketbra k k,&k<N,\\
\displaystyle\sum_{n=N}^\infty\ketbra n n,&k=N.
\end{cases}
\]
\end{proposition}

\begin{proof}
The Kraus maps are
\(K_0=V_NP_N\) and \(K_n=\ket N_C\bra n\), \(n>N\).
Their effects sum strongly to \(I\). The positive output series
converges in trace norm on positive trace-class inputs. This
proves the normal CPTP extension. The adjoint and its
specializations follow by substitution.
\end{proof}

Low-block coherences are preserved before readout. Saturation is
not postselection.

\subsection{Binary register and a total decoder}

Set
\[
b_N=\ceil{\log_2(N+1)},\qquad d_N=2^{b_N},\qquad
\QQ_N=(\mathbb C^2)^{\otimes b_N}.
\]
Identify carrier basis \(k\) with binary integer \(k\) in \(Q\);
write the isometry as \(J_N\), and set
\[
\BB_N(X)=J_N\CC_N(X)J_N^\dagger.
\]

For every \(0\le j<d_N\), including unused words, define
\[
c_N(j)=\min(j,N),\qquad
y_j=(\operatorname{bin}_{b_N}(j),c_N(j)).
\]
Let \(\mathcal Y\) contain all distinct \(y_j\) and any declared
failure symbols. The full record space is
\(\HH_F=\mathbb C^{\mathcal Y}\). Ideal readout has zero weight on
failures.

For a spectator \(S\), define
\[
M_j=\bra j_Q\otimes I_S,\qquad
E_j=M_j^\dagger M_j.
\]
The selective map is \(X\mapsto M_jXM_j^\dagger\), whereas its
effect is \(E_j\). Define
\begin{equation}
\PP_N^S(X)=
\sum_{j=0}^{d_N-1}\ketbra{y_j}{y_j}_F
                       \otimes M_jXM_j^\dagger.
\label{eq:idealread}
\end{equation}

Assign every failure symbol a fixed decoded value in
\(\{0,\ldots,N\}\); below all failure values are zero. Write this
total decoder as \(d(y)\), with \(d(y_j)=c_N(j)\). Its channel on
\emph{arbitrary} record operators is
\begin{equation}
\mathcal G_N(X)=
\sum_{y\in\mathcal Y}\bra yX\ket y\,
                  \ketbra{d(y)}{d(y)}_{\rec}.
\label{eq:decoder}
\end{equation}
This is CPTP on the full record space, not merely a prescription
on its populated diagonal.

\subsection{Complete referenced record identity}

For a normal state \(\rho_{hA}\), where \(A=SR\) collects
unmeasured systems, put
\[
\rho_A^{(n)}=\bra n\rho_{hA}\ket n.
\]
Define
\begin{align}
\widetilde\Omega_{NA}(\rho)
&=\sum_{k<N}\ketbra{y_k}{y_k}\otimes\rho_A^{(k)}
 +\ketbra{y_N}{y_N}\otimes\sum_{n\ge N}\rho_A^{(n)},
\label{eq:fullomega}\\
\Omega_{NA}(\rho)
&=\sum_{k<N}\ketbra k k\otimes\rho_A^{(k)}
 +\ketbra N N\otimes\sum_{n\ge N}\rho_A^{(n)}.
\label{eq:omega}
\end{align}

\begin{theorem}[Complete destructive record]
\label{thm:record}
For every normal packet--spectator--reference state,
\[
(\PP_N^S\otimes\id_R)
(\BB_N\otimes\id_{SR})(\rho)
=\widetilde\Omega_{N,SR}(\rho).
\]
Applying \(\mathcal G_N\) gives \(\Omega_{N,SR}(\rho)\).
All positive diagonal block sums converge in trace norm.
\end{theorem}

\begin{proof}
The low-block Kraus map contributes the diagonal blocks at
\(k\le N\). Every Kraus index \(n>N\) contributes its reference
block to the same carrier state \(N\). Readout adds those
positive blocks to the \(n=N\) block. Their traces sum to at most
one, proving convergence. The total decoder gives the second
identity.
\end{proof}

Spectators undergo the identity operation in this ideal channel.
Their \emph{conditional} state can change when they are correlated
with the carrier. No nondestructive residual field instrument is
identified by this destructive record.

\subsection{Exact tails and mean bias}

For an arbitrary normal oscillator state of finite mean \(\mu\),
\begin{equation}
\mu-\mathbb E_{\mathrm{ideal}}\kappa
=B_N(\rho):=\tr\bigl(\rho(N_h-N)_+\bigr)
=\sum_{j=N+1}^\infty\Pr_\rho(N_h\ge j).
\label{eq:generalbias}
\end{equation}
Let
\[
\Omega_{\infty A}(\rho)
=\sum_{n=0}^\infty\ketbra n n\otimes\rho_A^{(n)}.
\]
The finite decoded record is embedded into this countable record
space by assigning zero weight above \(N\).

\begin{proposition}[Exact referenced record tail]
\label{prop:recordtail}
For every normal input,
\[
\dist(\Omega_{NA}(\rho),\Omega_{\infty A}(\rho))
=\Pr_\rho(N_h>N).
\]
Consequently, for the geometric marginal,
\begin{equation}
\boxed{
\begin{aligned}
p_N(k)&=(1-\qh)\qh^k &&(k<N),&
p_N(N)&=\qh^N,\\
\mathbb E_{\mathrm{ideal}}\kappa&=\nuh(1-\qh^N),&
B_N(\rho_h)&=\nuh\qh^N,\\
\dist(\Omega_{NA},\Omega_{\infty A})
&=\tv(p_N,p_h)=\qh^{N+1}.
\end{aligned}
}
\label{eq:idealtails}
\end{equation}
\end{proposition}

\begin{proof}
At record \(N\), the difference is the positive block
\(\sum_{n>N}\rho_A^{(n)}\); at each larger record it is
\(-\rho_A^{(n)}\). Orthogonal record blocks have additive trace
norm. The half trace norm is therefore the discarded probability.
Equation~\eqref{eq:generalbias} follows from
\(N_h-\min(N_h,N)=(N_h-N)_+\). Geometric summation gives
\(\Pr(N_h\ge j)=\qh^j\) and
\[
B_N=\frac{\qh^{N+1}}{1-\qh}=\nuh\qh^N.
\]
\end{proof}

For \(E>0\),
\[
\sup_{\rho:\tr(\rho N_h)\le E}B_N(\rho)=E.
\]
Indeed,
\((1-E/L)\ketbra00+(E/L)\ketbra L L\), with
\(L>\max(E,N)\), has bias \(E(1-N/L)\to E\).
A mean-only constraint gives a probability bound
\(\Pr(N_h>N)\le E/(N+1)\), not a uniformly vanishing mean bias.

%=====================================================================
\section{The apparatus input: radiation plus participating modes}
\label{sec:port}
%=====================================================================

\subsection{An effective temporal port}

Let
\[
\mathfrak h_{\mathrm{port}}=L^2(\mathbb R,\dd\omega).
\]
Its positive-frequency part carries the outgoing radiation
covariance \eqref{eq:nomega}. Its negative-frequency part is a
new, independent auxiliary vacuum in the \emph{apparatus model}.
Define
\[
n_{\mathrm{port}}(\omega)=
\begin{cases}
n(\omega),&\omega>0,\\
0,&\omega<0.
\end{cases}
\]
The Fourier convention is
\[
p(t)=\frac1{\sqrt{2\pi}}\int_{\mathbb R}
                h(\omega)e^{-\ii\omega t}\dd\omega,\qquad
\widehat f(\omega)=\frac1{\sqrt{2\pi}}\int_{\mathbb R}
                f(t)e^{\ii\omega t}\dd t,
\]
with \(h\) extended by zero outside its band.

Annihilators on this port satisfy
\[
a(\widehat f)=\int_{\mathbb R}\overline{f(t)}
                                      b_{\mathrm{in}}(t)\dd t,
\qquad
[b_{\mathrm{in}}(t),b_{\mathrm{in}}(s)^\dagger]=\delta(t-s),
\]
as smeared identities. The two-sided envelope makes the Markov
port canonical. It is not a claim that the physical scalar field
contains independent negative-Killing-frequency annihilators.

All positive-frequency components outside the target packet retain
their actual prescribed radiation covariance. In particular, no
vacuum substitution is made for unselected Hawking radiation.

\subsection{Finite participating reductions}

For a chosen capture envelope \(f\), the memory output depends
only on \(a(\widehat f)\). Together with the target \(a(h)\), it
therefore requires at most two canonical input modes. Define their
plane by
\[
\mathfrak v_f=\Span\{h,\widehat f\}.
\]
The normal state on its Fock space is justified by
Section~\ref{subsec:completion} whenever its occupation matrix is
finite.

The declared operational input is a normal state
\[
\rho_{h g SR}
\quad\hbox{on}\quad
\FF(\mathfrak v_f)\otimes\HH_S\otimes\HH_R.
\]
Here \(g\) will be the orthogonal participating mode. The ideal
capture target identifies \(h\) with \(c\) and traces out \(g\):
\begin{equation}
\mathcal I_h(\rho_{h g SR})
=\rho_{cSR}\equiv\Tr_g\rho_{h g SR}.
\label{eq:idealcapture}
\end{equation}
It does not assume \(\rho_{h g SR}\) factorizes.

The Hawking input class used for the model theorem consists of
all such normal extensions with the specified two-mode Gaussian
marginal. External finite references are unrestricted. A general
occupation-constrained class is treated separately below.

No normal density representation of the entire continuum is
assumed. The capture model defines its reduced channel through its
output Weyl operators on these finite participating modes. A
physical continuum interaction with further participating
degrees of freedom would require a larger input interface.

%=====================================================================
\section{Finite switching and nonvacuum mode mismatch}
\label{sec:window}
%=====================================================================

A compactly switched temporal capture cannot exactly select the
nonzero band-limited packet \(p\). We construct an explicit
approximation, including a low-frequency integrability repair.

Choose real \(\chi\in C_c^\infty((-2,2))\), with
\(0\le\chi\le1\) and \(\chi=1\) on \([-1,1]\).
Choose \(\psi\in C_c^\infty((-2,2))\) with
\(\int\psi(t)\dd t=1\). For \(L>0\), set
\begin{equation}
\begin{aligned}
\chi_L(t)&=\chi(t/L),&
\psi_L(t)&=L^{-1}\psi(t/L),\\
m_L&=\int\chi_L(t)p(t)\dd t,&
v_L(t)&=\chi_L(t)p(t)-m_L\psi_L(t).
\end{aligned}
\label{eq:windowconstruction}
\end{equation}
For sufficiently large \(L\), \(v_L\ne0\).
Normalize \(v_L\) and choose its constant phase so that
\[
f_L=e^{\ii\varphi_L}\frac{v_L}{\norm{v_L}_2},
\qquad
\alpha_L=\langle h,\widehat f_L\rangle\in(0,1].
\]
Write
\begin{equation}
\alpha_L=\cos\theta_L,\qquad
\widehat f_L=\cos\theta_L\,h+\sin\theta_L\,g_L,
\qquad 0\le\theta_L<\pi/2.
\label{eq:modeplane}
\end{equation}
When \(\theta_L=0\), omit \(g_L\) and all mismatch terms.

\begin{lemma}[Window, zero moment, and finite occupations]
\label{lem:window}
For sufficiently large \(L\):
\begin{enumerate}[label=\textup{(\roman*)},leftmargin=*]
\item \(f_L\in C_c^\infty((-2L,2L))\), \(\norm{f_L}_2=1\),
and \(\int f_L=0\).

\item \(f_L\to p\) in the Schwartz topology and
\(\widehat f_L\to h\) in that topology. In particular,
\(\theta_L\to0\).

\item The occupations
\[
E_f=\int_0^\infty n(\omega)|\widehat f_L(\omega)|^2\dd\omega,
\qquad
E_g=\int_0^\infty n(\omega)|g_L(\omega)|^2\dd\omega
\]
are finite for every nondegenerate participating plane.

\item With
\[
w_L=\widehat f_L-\alpha_L h,
\qquad
\mathcal N(w)=\int_0^\infty n(\omega)|w(\omega)|^2\dd\omega,
\]
one has \(\mathcal N(w_L)\to0\), \(E_f\to\nuh\), and
\[
\theta_L^2E_g\longrightarrow0.
\]
\end{enumerate}
\end{lemma}

\begin{proof}
The support and zero integral follow from
\eqref{eq:windowconstruction}. The inverse Fourier transform \(p\)
is Schwartz. Also \(\int p=0\), because \(h\) vanishes near zero.
Hence
\[
m_L=-\int(1-\chi_L)p,
\]
which decays faster than every inverse power of \(L\).
Differentiating the window shows that
\((1-\chi_L)p\to0\) in every Schwartz seminorm. The scaled bump
\(m_L\psi_L\) has the same property. Normalization and the
phase alignment therefore preserve Schwartz convergence.

For positive frequencies,
\[
n(\omega)\le(e^{\betah\omega}-1)^{-1}
\le(\betah\omega)^{-1}.
\]
The zero integral gives \(\widehat f_L(0)=0\). Thus
\(\widehat f_L(\omega)=O(\omega)\) at zero; the same holds for
\(w_L\) and, for fixed \(L\), \(g_L\). Their weighted integrals
are finite near zero. At high frequencies, the occupation is
bounded and the tests are square integrable.

More quantitatively, for any differentiable \(w\) with \(w(0)=0\),
\[
\int_0^1n(\omega)|w(\omega)|^2\dd\omega
\le\frac{\sup_{[0,1]}|w'|^2}{2\betah}.
\]
On \([1,\infty)\), \(n\) is bounded. Schwartz convergence proves
\(\mathcal N(w_L)\to0\) and \(E_f\to\nuh\).
Finally,
\[
\theta_L^2E_g
=\frac{\theta_L^2}{\sin^2\theta_L}\mathcal N(w_L)\to0.
\]
\end{proof}

\begin{remark}[Why ordinary windowing was insufficient]
Without the subtraction in \eqref{eq:windowconstruction}, a
compact temporal window generally has nonzero Fourier value at
zero. The predecessor bound \(\Gamma_\ell\le1\) alone would then
not prove integrability against \(n(\omega)\lesssim1/\omega\).
We do not import an unproved low-frequency greybody asymptotic
to bypass this issue.
\end{remark}

\subsection{Reference-stable mode comparison with a nonvacuum background}

For simplicity write \(\theta=\theta_L\), \(g=g_L\).
Let
\[
J=\ii(a(g)^\dagger a(h)-a(h)^\dagger a(g)).
\]
Up to the choice of rotation sign,
\(V_\theta=e^{-\ii\theta J}\) identifies mode \(h\) with
\(\cos\theta\,h+\sin\theta\,g\).
The generator is self-adjoint as the direct sum of finite
total-number sectors.

\begin{lemma}[Gaussian mode-mismatch certificate]
\label{lem:modeerror}
Suppose the \(h,g\) marginal is centered, gauge-invariant, and
quasifree, with occupations \(E_h,E_g\). For every normal
spectator--reference extension,
\begin{equation}
\dist\bigl(\hbox{mode-\(\widehat f_L\) reduction},
                  \hbox{mode-\(h\) reduction}\bigr)
\le
\mu_L:=
\min\{1,\theta_L\sqrt{E_h+E_g+6E_hE_g}\}.
\label{eq:modeerror}
\end{equation}
For the radiation marginal, \(E_h=\nuh\), and \(\mu_L\to0\).
No independence between the two modes is required.
\end{lemma}

\begin{proof}
Purify the joint input, retaining its external systems. The
unitary comparison and partial-trace contractivity give
\[
\dist\le\norm{(V_\theta-I)\ket\Psi}
\le|\theta|\sqrt{\langle J^2\rangle}.
\]
The last inequality follows from the scalar spectral inequality
\(|e^{-\ii\theta x}-1|\le|\theta x|\).

Put \(m=\langle a(g)^\dagger a(h)\rangle\).
Wick's rule gives
\[
\langle J^2\rangle
=E_h+E_g+2E_hE_g+2|m|^2-4\Re(m^2).
\]
Cauchy--Schwarz gives \(|m|^2\le E_hE_g\), hence
\(\langle J^2\rangle\le E_h+E_g+6E_hE_g\).
All these moments depend only on the two-mode marginal, so the
bound is unchanged by its reference extension.

For the radiation input,
\[
\theta_L^2(\nuh+E_g+6\nuh E_g)\to0
\]
by Lemma~\ref{lem:window}. This proves convergence.
\end{proof}

The background contribution in \eqref{eq:modeerror} is computed
from \(n(\omega)\) and the actual orthogonal mode \(g_L\). It
includes its possible correlation \(m\) with \(h\), through a
uniform Wick bound. There is no separate ``background vacuum''
assumption or overlapping additive background-error label.

\begin{proposition}[A non-Gaussian occupation-constrained alternative]
\label{prop:generalmodemismatch}
If only
\(\langle N_h+N_g\rangle\le E_{\mathrm{pl}}\) is known, then
for every integer \(D\ge0\) the same mode comparison obeys
\[
\dist\le
\min\left\{1,\,
D|\theta|+2\sqrt{\min\{1,E_{\mathrm{pl}}/(D+1)\}}\right\}.
\]
\end{proposition}

\begin{proof}
Project onto total occupation at most \(D\). Its discarded
probability is at most \(E_{\mathrm{pl}}/(D+1)\).
The distance to the normalized projected state is at most the
square root of that probability, as follows by purification.
On this invariant sector, \(\norm J\le D\), so the unitary
comparison is at most \(D|\theta|\). Two projection comparisons
and contractivity give the result.
\end{proof}

The constraint is packet-plane occupation, not a separately
identified packet Hamiltonian. Occupation-constrained comparisons
are a specialization of infinite-dimensional constrained-channel
methods \cite{Shirokov}.

%=====================================================================
\section{B4 capture: an explicit passive input--output interaction}
\label{sec:capture}
%=====================================================================

\subsection{Dynamics, switching, and the unitary model}

Fix a normalized \(f=f_L\), supported in \((a,b)=(-2L,2L)\),
and a capture transmissivity \(0<\tau<1\). Set
\begin{equation}
D(t)=1-\tau+\tau\int_a^t|f(s)|^2\dd s,\qquad
\lambda(t)=\frac{\sqrt{\tau}\,\overline{f(t)}}{\sqrt{D(t)}},
\qquad
\kappa(t)=|\lambda(t)|^2.
\label{eq:captcontrol}
\end{equation}
Outside \([a,b]\), the coupling is zero.

\begin{specification}[Passive capture port]
\label{spec:capture}
The effective interaction is defined first on one-particle
amplitudes \(z(t)\) of the memory and incoming amplitudes \(u(t)\):
\begin{equation}
\dot z(t)=-\frac{\kappa(t)}2z(t)+\lambda(t)u(t),
\qquad
u_{\mathrm{out}}(t)=u(t)-\overline{\lambda(t)}z(t).
\label{eq:passiveODE}
\end{equation}
Its quantum evolution is the bosonic second quantization of this
one-particle input--output map.

The memory is initially independent of the radiation, spectators,
and references. Its specified preparation is the geometric state
of mean \(m_0\ge0\). A subsequent storage interval of duration
\(T_s\) has vacuum attenuation rate \(\kappa_s\ge0\), implemented
by an independent auxiliary loss port. Define
\[
\tau_*=\tau e^{-\kappa_sT_s}.
\]
All phases are defined in a fixed interaction picture. Its clock is
the predecessor's asymptotic retarded-time coordinate, up to a fixed
translation.
\end{specification}

This is a declared passive model in the input--output tradition
\cite{GardinerCollett,YamamotoJames}. We give its transfer
calculation directly, rather than assuming an exact pulse-capture
claim from those references.

\begin{theorem}[Unitary passive evolution and transfer kernel]
\label{thm:capturekernel}
Equation~\eqref{eq:passiveODE} defines a unitary map
\[
\mathbb C\oplus L^2([a,b])
\longrightarrow
\mathbb C\oplus L^2([a,b])
\]
from initial memory and incoming port to final memory and outgoing
port. Its memory annihilator is
\begin{equation}
c_b=\sqrt{1-\tau}\,c_a+\sqrt{\tau}\,a(\widehat f).
\label{eq:captureann}
\end{equation}
The switching is smooth and finite, and
\begin{equation}
\int_a^b\kappa(t)\dd t=-\log(1-\tau),\qquad
\|\kappa\|_\infty\le
\frac{\tau}{1-\tau}\|f\|_\infty^2.
\label{eq:capturecost}
\end{equation}
\end{theorem}

\begin{proof}
The energy identity
\[
\frac{\dd}{\dd t}|z(t)|^2
=|u(t)|^2-|u_{\mathrm{out}}(t)|^2
\]
follows by substitution. Integration proves that the map is
isometric. Given final memory and outgoing amplitude, solve
backwards the equation obtained by substituting
\(u=u_{\mathrm{out}}+\overline{\lambda}z\); bounded coefficients
give a unique initial memory and incoming amplitude. Thus the
isometry is onto.

The memory solution is
\[
z(b)=e^{-\frac12\int_a^b\kappa}z(a)
+\int_a^b\lambda(t)
           e^{-\frac12\int_t^b\kappa(s)\dd s}u(t)\dd t.
\]
Since \(D'=\kappa D\), \(D(a)=1-\tau\), and \(D(b)=1\),
\[
e^{-\frac12\int_t^b\kappa}=\sqrt{D(t)}.
\]
The transfer kernel is therefore
\(\sqrt{\tau}\,\overline{f(t)}\).
Second quantization gives \eqref{eq:captureann}. The integral and
peak bound follow from the same identities. Because \(f\) is
smooth and supported away from the endpoints and \(D\ge1-\tau>0\),
the switched coefficient \(\lambda\) is smooth.
\end{proof}

The formal white-noise interaction has not been treated as a
bounded Hamiltonian. Its well-defined unitary model is the
one-particle map just proved and its second quantization.
Accordingly, no finite-dimensional Hamiltonian norm estimate is
applied to this unbounded bosonic capture evolution.

The storage attenuation is an ordinary vacuum beam-splitter
channel with transmissivity \(e^{-\kappa_sT_s}\); it can be
defined by the same passive dilation. With the initial memory
replaced by vacuum, capture and storage together are vacuum
pure loss of transmissivity \(\tau_*\) on the selected port mode.

\subsection{A direct pure-loss benchmark}

\begin{lemma}[Referenced pure-loss bound]
\label{lem:loss}
Let \(\LL_t\) be the vacuum pure-loss channel of transmissivity
\(0\le t\le1\). If \(\tr(\rho N)\le E\), then, including any
normal reference,
\begin{equation}
\dist\bigl((\LL_t\otimes\id)(\rho),\rho\bigr)
\le\ell(E,t):=
\min\{1,\sqrt{2E(1-\sqrt t)}\}.
\label{eq:lossbound}
\end{equation}
\end{lemma}

\begin{proof}
Purify \(\rho\), append a vacuum loss oscillator, and apply the
beam splitter. The overlap with the initial purified state is
\[
A=\tr(\rho\,t^{N/2})
=\sum_np(n)(\sqrt t)^n.
\]
This is nonnegative. The inequality
\(1-s^n\le n(1-s)\), \(0\le s\le1\), gives
\(1-A\le E(1-\sqrt t)\). The pure-state trace distance is
\(\sqrt{1-A^2}\le\sqrt{2(1-A)}\). Partial trace proves
\eqref{eq:lossbound}.
\end{proof}

This lemma is used only after the independent vacuum-environment
reduction has been established by Specification~\ref{spec:capture}.
It is not applied directly to unselected radiation.

\subsection{Derived capture budget}

The initial memory's distance from vacuum is exactly
\[
p_c=\frac{m_0}{1+m_0}.
\]
This preparation error is priced once, before the common capture
evolution.

\begin{theorem}[Capture certificate on the stated joint input]
\label{thm:captureerror}
For the radiation two-mode marginal and every admitted
spectator--reference extension,
\begin{equation}
\boxed{
\dist\bigl((\MM\otimes\id_{SR})(\rho_{h g SR}),
                     \mathcal I_h(\rho_{h g SR})\bigr)
\le
\xi_{\mathrm{cap}}
:=\min\{1,p_c+\mu_L+\ell(E_f,\tau_*)\}.
}
\label{eq:capbudget}
\end{equation}
Here \(\MM\) is the capture-plus-storage channel obtained from
Specification~\ref{spec:capture}.
\end{theorem}

\begin{proof}
Replacing the independently prepared memory by vacuum costs
\(p_c\), including references, by tensor-product equality and
contractivity. For that preparation, capture and storage reduce
to \(\LL_{\tau_*}\) on the \(f\)-mode, so
Lemma~\ref{lem:loss} costs \(\ell(E_f,\tau_*)\).
The comparison between the \(f\)-mode and the target \(h\)-mode
costs \(\mu_L\), by Lemma~\ref{lem:modeerror}.
The ideal target traces out the other participating mode; no
independence between that mode and \(hSR\) is used. Apply the
triangle inequality.
\end{proof}

The three terms have different comparisons:
preparation, mode selection including the nonvacuum background,
and vacuum loss. There is no additional overlapping
``window'' or ``background'' term.

For this Gaussian input, the actual memory marginal is geometric
with mean
\begin{equation}
\nu_c=\tau_*E_f
+e^{-\kappa_sT_s}(1-\tau)m_0.
\label{eq:memorymean}
\end{equation}
This follows directly from \eqref{eq:captureann}, independence of
the memory preparation, and gauge-invariant quasifreeness.
It is an apparatus-model prediction, not a replacement for the
target value \(\nuh\).

%=====================================================================
\section{B4 extraction: explicit swaps and control certificates}
\label{sec:extraction}
%=====================================================================

\subsection{The exact finite-domain interaction}

Use the binary register \(Q\) itself, initially in \(\ket0_Q\).
For \(1\le n\le K\), define the disjoint pairs
\[
A_n=\ket n_c\ket0_Q,\qquad
B_n=
\begin{cases}
\ket0_c\ket n_Q,&n\le N,\\
\ket n_c\ket N_Q,&N<n\le K.
\end{cases}
\]
These pairs are mutually disjoint. Let \(U_{N,K}\) swap each
\(A_n,B_n\) and act as the identity on the remaining basis states.

\begin{theorem}[Exact coherent extraction and sharp tail]
\label{thm:extraction}
Tracing out \(c\) after \(U_{N,K}\), with initial register
\(\ket0_Q\), induces
\begin{equation}
\begin{split}
\EE_{N,K}(X)={}&
J_NV_NP_NXP_NV_N^\dagger J_N^\dagger\\
&+\sum_{n=N+1}^K\bra nX\ket n\,\ketbra N N_Q\\
&+\sum_{n>K}\bra nX\ket n\,\ketbra00_Q.
\end{split}
\label{eq:extractionchannel}
\end{equation}
For every normal memory--reference state,
\begin{equation}
\boxed{
\dist\bigl((\EE_{N,K}\otimes\id)(\rho),
                   (\BB_N\otimes\id)(\rho)\bigr)
=\Pr_\rho(N_c>K).
}
\label{eq:sharpK}
\end{equation}
For the ideal target packet this error is \(\qh^{K+1}\).
\end{theorem}

\begin{proof}
For \(0\le n\le N\), the output memory is the same vacuum.
Consequently all low-block coherences survive in \(Q\).
For \(N<n\le K\), memory state \(n\) labels the environment and
the register is \(N\). For \(n>K\), the untouched input has memory
state \(n\) and register zero. Distinct such memory states are
orthogonal, proving \eqref{eq:extractionchannel}.

The channel difference on a referenced input is exactly
\[
\bigl(\ketbra00_Q-\ketbra N N_Q\bigr)
 \otimes H,\qquad H=\sum_{n>K}\bra n\rho\ket n\succeq0.
\]
Because \(N\ge1\), the two register states are orthogonal.
Its half trace norm is \(\tr H\).
\end{proof}

This is coherent extraction, not copying all number labels while
leaving a distinguishable low-number environment. Finite output
does not imply a finite environment on the whole oscillator:
an isometry from an infinite-dimensional input cannot have
finite-dimensional output and environment simultaneously.

\subsection{A declared control set and an exact pulse sequence}

Define the orthogonal rank-one projections
\[
R_n=\frac12(A_n-B_n)(A_n^\dagger-B_n^\dagger).
\]
The available extraction controls in our effective model are
\[
H_{\mathrm E}(t)=\sum_{n=1}^K u_n(t)R_n.
\]
These are the specified number-selective matrix elements; their
availability is part of the model.

For a common duration \(T_E>0\), choose
\begin{equation}
u_n(t)=\frac{2\pi}{T_E}
                 \sin^2\left(\frac{\pi t}{T_E}\right),
\qquad 0\le t\le T_E.
\label{eq:swapPulse}
\end{equation}
The pulses are zero outside this interval. They have area \(\pi\),
maximum \(2\pi/T_E\), and derivative bounded by
\(2\pi^2/T_E^2\).

\begin{proposition}[Exact pulse compilation in the declared control set]
\label{prop:compile}
The \(K\) simultaneous pulses \eqref{eq:swapPulse} generate
\(U_{N,K}\) exactly, with no relative swap phases.
If the envelope limits are \(\Lambda_E>0\) and \(B_E>0\), it
suffices to take
\[
T_E\ge
\max\left\{\frac{2\pi}{\Lambda_E},
           \sqrt{\frac{2\pi^2}{B_E}}\right\}.
\]
For input supported at occupation at most \(K\), the ideal
sequence never exceeds occupation \(K\).
\end{proposition}

\begin{proof}
The projections commute and are mutually orthogonal.
On each pair,
\[
e^{-\ii\pi R_n}=I-2R_n
\]
is the swap matrix, including its phase. Outside the pairs the
evolution is the identity. The bounds follow by differentiating
\eqref{eq:swapPulse}. Its invariant subspace contains only
memory occupations \(0,\ldots,K\).
\end{proof}

The constant-pulse version is the Hamiltonian
\[
H_{AB}=\frac{\pi}{2T_E}
 \bigl(\ketbra A A+\ketbra B B-\ketbra A B-\ketbra B A\bigr)
\]
on each pair, as a special case.

\paragraph{Free-energy convention.}
A memory Hamiltonian \(\Omega_cN_c\), \(\Omega_c>0\), may be
specified, with degenerate computational register levels.
The displayed controls are in its interaction picture.
Their laboratory-frame version contains the corresponding
number-dependent phase modulation. The required frequency range
grows with \(K\Omega_c\); it is not hidden in the envelope derivative
bound. No field packet Hamiltonian is identified with
\(\Omega_cN_h\), and extraction is not asserted to conserve the
memory's free energy without work exchange.

\subsection{Global bounded perturbations and finite leakage domains}

Suppose the implemented interaction-picture generator is
\[
\widehat H_E(t)
=H_E(t)+\sum_{n=1}^K\Delta u_n(t)R_n+G_E(t).
\]
Here \(G_E(t)\) is explicitly bounded and self-adjoint. One
permitted leakage model takes it supported on
\[
P_J^c\otimes I_Q,\qquad J\ge K,
\]
with finite-dimensional matrix elements. This allows temporary
occupation up to \(J\), rather than silently identifying the
initial cutoff \(K\) with the maximum populated level.

Put
\begin{equation}
I_E=\int_0^{T_E}
 \left(\max_{1\le n\le K}|\Delta u_n(t)|
                    +\|G_E(t)\|\right)\dd t,\qquad
\zeta_{\mathrm{glob}}=\min\{1,I_E\}.
\label{eq:globalcontrol}
\end{equation}

\begin{lemma}[Bounded propagator comparison]
\label{lem:Duhamel}
For bounded self-adjoint generators \(H,\widehat H\),
\[
\|\widehat U(T)-U(T)\|
\le\int_0^T\|\widehat H(t)-H(t)\|\dd t.
\]
The induced channels, including references and partial traces,
have half-diamond distance at most the same bound, capped at one.
\end{lemma}

\begin{proof}
Variation of constants gives
\[
\widehat U(T)-U(T)
=-\ii\int_0^T
 \widehat U(T,t)(\widehat H(t)-H(t))U(t,0)\dd t.
\]
The unitary factors have norm one. For a purified input, output
trace distance is at most the norm of the difference of output
vectors; then take the supremum and use contractivity.
\end{proof}

Because the \(R_n\) are orthogonal,
\(\|\sum_n\Delta u_nR_n\|=\max_n|\Delta u_n|\).
Thus \eqref{eq:globalcontrol} is a derived global certificate.
It is not an estimate for an unbounded full-oscillator drive.

\begin{proposition}[Two distinct extraction-error regimes]
\label{prop:controlregimes}
For an ideal target input with geometric marginal:
\begin{enumerate}[label=\textup{(\roman*)},leftmargin=*]
\item Under the global bounded model above, implemented extraction
with a correctly prepared register differs from \(\BB_N\) by at most
\[
t_{N,K}^{\mathrm{glob}}
=\min\{1,\zeta_{\mathrm{glob}}+\qh^{K+1}\}.
\]

\item If instead an implementation is certified only on inputs
supported in \(P_K\), with referenced error \(\zeta_{\mathrm{low}}\),
and its behavior outside that domain is arbitrary, then a valid
bound is
\[
t_{N,K}^{\mathrm{low}}
=\min\{1,\zeta_{\mathrm{low}}+2\sqrt{\qh^{K+1}}\}.
\]
\end{enumerate}
The sharp probability tail is not available merely from a
support-only control certificate.
\end{proposition}

\begin{proof}
For (i), compare the implemented channel with \(\EE_{N,K}\)
using Lemma~\ref{lem:Duhamel}, then use
Theorem~\ref{thm:extraction}.

For (ii), project and normalize the referenced input on \(P_K\).
Its distance from the original is at most the square root of the
discarded probability, by purification. The implementation and
\(\BB_N\) each contract this distance. On the projected input,
\(\EE_{N,K}=\BB_N\), and the support certificate costs
\(\zeta_{\mathrm{low}}\). Add the two projection costs.
\end{proof}

\subsection{Register preparation and the derived transfer budget}

The extraction register preparation is independent and specified as
\[
\sigma_Q=(1-p_Q)\ketbra00+p_Q\sigma_Q^\perp,\qquad
\sigma_Q^\perp\ket0=0,\qquad 0\le p_Q\le1.
\]
Its distance from \(\ketbra00\) is exactly \(p_Q\).

The actual model transfer is capture, storage, register preparation,
implemented extraction, and memory discard. Denote it by
\(\TT_{N,K}^{\mathrm{model}}\).
Its ideal target is
\[
(\BB_N\otimes\id_{SR})\mathcal I_h(\rho_{h g SR}).
\]

\begin{theorem}[Derived B4 transfer budget]
\label{thm:transfer}
In the global bounded-control model,
\begin{equation}
\boxed{
\eta_{N,K}^{\mathrm{model}}
=\min\{1,\xi_{\mathrm{cap}}+p_Q
                +\zeta_{\mathrm{glob}}+\qh^{K+1}\}
}
\label{eq:transferbudget}
\end{equation}
bounds the transfer trace distance, including all admitted
spectators and references.
For a support-only implementation, replace the last two terms
by \(\zeta_{\mathrm{low}}+2\sqrt{\qh^{K+1}}\).
\end{theorem}

\begin{proof}
First apply the same implemented extraction to the actual captured
state and the ideal packet-in-memory state. Contractivity and
Theorem~\ref{thm:captureerror} cost \(\xi_{\mathrm{cap}}\).
Replacing the independent register preparation costs \(p_Q\).
Now the input memory is the ideal packet, so
Proposition~\ref{prop:controlregimes} supplies the appropriate
control and tail terms. This ordering avoids making an
unjustified geometric-tail claim about an imperfectly captured
state.
\end{proof}

%=====================================================================
\section{B5: a complete readout apparatus and its errors}
\label{sec:readoutmodel}
%=====================================================================

\subsection{Pointer interaction and complete instrument}

Let \(A\) consist of \(b_N\) pointer qubits. With ideal preparation
\(\ket{0^{b_N}}_A\), the readout interaction is
\[
U_{\mathrm R}=\prod_{j=1}^{b_N}\operatorname{CNOT}_{Q_j\to A_j}.
\]
One explicit bounded generator for duration \(T_R\) is
\[
H_R=\sum_{j=1}^{b_N}
 \frac{\pi}{2T_R}\ketbra11_{Q_j}\otimes(I-X_{A_j}),
\]
with identity placement on the other factors. Its summands commute,
and its time-\(T_R\) evolution is \(U_R\).

The pointer is read in its computational basis, both \(Q\) and
\(A\) are consumed, and \(S\) is returned. With no noise this
induces exactly \eqref{eq:idealread}, including correlated
references.

For a stochastic assignment matrix \(W(y\mid j)\), \(y\in\mathcal Y\),
define
\begin{equation}
\PP_W^S(X)=
\sum_{j,y}W(y\mid j)\ketbra y y_F
                              \otimes M_jXM_j^\dagger.
\label{eq:assignmentinstrument}
\end{equation}
This is a complete instrument, not a fitted probability rule for
one input.

An explicit assignment dilation maps pointer word \(j\) to
\[
\sum_y\sqrt{W(y\mid j)}\,\ket y_F\ket{j,y}_E.
\]
The orthonormal environment labels make this an isometry and
produce \eqref{eq:assignmentinstrument} after the discarded
systems are traced out. Failure records remain in \(\mathcal Y\).

\begin{theorem}[Exact assignment-noise distance]
\label{thm:assignment}
With the ideal output alphabet enlarged by zero-weight failure
records,
\begin{equation}
\boxed{
\ddiam(\PP_W^S,\PP_N^S)
=\delta_{\mathrm{as}}
:=\max_j[1-W(y_j\mid j)].
}
\label{eq:assignmenterror}
\end{equation}
\end{theorem}

\begin{proof}
For an arbitrary operator on the input and a reference, first
pinch into the \(Q\)-basis blocks \(X_{jj}\).
Pinching is trace-norm contractive and
\(\sum_j\|X_{jj}\|_1\le\|X\|_1\).
The difference of the two output maps has norm at most
\[
\sum_j\left(\sum_y|W(y\mid j)-\delta_{y,y_j}|\right)
                                      \|X_{jj}\|_1
\le2\delta_{\mathrm{as}}\|X\|_1.
\]
This gives the complete upper bound. A basis input attaining the
maximum has classical output distance
\(1-W(y_j\mid j)\), proving equality.
\end{proof}

Independent reported-bit flips with probabilities \(r_j\) give
the special case
\[
\delta_{\mathrm{as}}
=1-\prod_j(1-r_j)\le\sum_jr_j,
\]
provided raw strings are retained. Coarsening to decoded counts
can merge erroneous strings and need not preserve this equality.

\subsection{Specified quantum errors before assignment}

The following is one complete model, not an assertion that all
real detector imperfections have this form.

\begin{specification}[Readout noise and preparation]
\label{spec:readnoise}
All apparatus baths and preparation records are fresh and
independent of \(QSR\).
\begin{enumerate}[label=\textup{(\arabic*)},leftmargin=*]
\item With probability \(r_{\mathrm F}\), a preparation protocol
reports the visible symbol \(\mathrm{reset\!-\!failed}\), consumes
\(Q\), and returns \(\Tr_Q X\) on \(S\). Its decoded value is zero.

\item Conditional on readiness, pointer preparation is
\[
\sigma_A=\bigotimes_{j=1}^{b_N}
 \bigl((1-p_{A,j})\ketbra00+p_{A,j}\ketbra11\bigr).
\]
Its distance from the all-zero pointer is
\[
p_A=1-\prod_j(1-p_{A,j}).
\]

\item Before pointer copying, register qubit \(j\) undergoes
amplitude damping with
\[
r_{D,j}=1-e^{-\gamma_j T_D},
\qquad \gamma_j\ge0.
\]
Its Kraus maps are
\[
L_{0,j}=\ketbra00+\sqrt{1-r_{D,j}}\ketbra11,\qquad
L_{1,j}=\sqrt{r_{D,j}}\ket0\bra1.
\]
A known residual basis rotation is
\[
V_B=\exp\left(-\frac{\ii}{2}
                    \sum_j\alpha_jY_{Q_j}\right).
\]

\item A separate spectator-coupling interval has a bounded
self-adjoint generator \(G_X(t)\) on \(QS\). Its ideal comparison
is the identity.

\item The pointer-copy generator is \(H_R+\Delta H_R(t)\),
where \(\Delta H_R\) is bounded on the finite \(QAS\) space.
It includes the errors assigned to this interval, but not the
separate interval already priced by \(G_X\).

\item Computational pointer readout is followed by the fixed
matrix \(W(y\mid j)\), including any timeout or other declared
assignment failures. The raw record or failure label and its
decoded value are always retained.
\end{enumerate}
\end{specification}

Amplitude damping is explicitly a bath model; each displayed
Kraus pair has a unitary system--bath dilation. The pointer
assignment isometry was given above. Thus these are complete
channels and instruments, rather than unsigned additive
``measurement errors.''

For clarity, write \(\mathcal Z\) for the composition of damping,
basis rotation, and spectator evolution. On a ready branch, the
selective record map is
\begin{equation}
\begin{split}
\widehat\Phi_y(X)
=\sum_jW(y\mid j)\,
\Tr_{QA}\bigl[
 \Pi_j\widehat U_R
       (\mathcal Z(X)\otimes\sigma_A)
 \widehat U_R^\dagger\Pi_j
\bigr],
\end{split}
\label{eq:noisyselective}
\end{equation}
where \(\Pi_j=I_{QS}\otimes\ketbra j j_A\), with tensor factors
placed in the stated order. Summing
\(\ketbra y y\otimes\widehat\Phi_y(X)\) gives the ready
record channel. The full channel is its mixture with the
reset-failure branch.

No failure is deleted and no surviving branch is renormalized.

\subsection{Derived quantum and preparation bounds}

Define
\[
a(r)=\min\{1,\,1-\sqrt{1-r}+r/2\},
\]
and
\begin{equation}
\begin{aligned}
\delta_D&=\min\{1,\sum_ja(r_{D,j})\},\\
\delta_B&=\min\{1,\tfrac12\sum_j|\alpha_j|\},\\
\delta_X&=\min\{1,\int\|G_X(t)\|\dd t\},\\
\delta_R&=\min\{1,\int_0^{T_R}\|\Delta H_R(t)\|\dd t\}.
\end{aligned}
\label{eq:readcomponents}
\end{equation}

\begin{lemma}[Complete error certificates for the readout components]
\label{lem:readcomponents}
The quantities in \eqref{eq:readcomponents} bound the respective
half-diamond distances from ideal damping-free, basis-correct,
spectator-identity, and exact-pointer-copy operations.
\end{lemma}

\begin{proof}
For one damping channel, the no-jump operator \(L_0\) is a
contraction with \(\|L_0-I\|=1-\sqrt{1-r}\).
On any input operator, including a reference,
\[
\|L_0XL_0^\dagger-X\|_1
\le2(1-\sqrt{1-r})\|X\|_1.
\]
The jump map has complete induced trace norm \(r\), since
\(\|L_1\|^2=r\). Divide the sum by two.
Tensor-product telescoping proves the \(\delta_D\) bound.

For \(V_B\),
\(\|V_B-I\|\le\frac12\sum_j|\alpha_j|\) by integrating its
bounded generator. Lemma~\ref{lem:Duhamel} gives the two
remaining bounds.
\end{proof}

\begin{lemma}[Complete-channel telescoping]
\label{lem:telescoping}
If \(\Lambda_m\cdots\Lambda_1\) and
\(\widehat\Lambda_m\cdots\widehat\Lambda_1\) are compatible
compositions of channels and
\(\ddiam(\widehat\Lambda_j,\Lambda_j)\le e_j\), then
their half-diamond distance is at most
\(\min\{1,\sum_je_j\}\).
Every step interface must include retained quantum systems and
classical histories.
\end{lemma}

\begin{proof}
Insert and subtract compositions differing at one factor.
Pre- and postcomposition by channels contract the complete
trace norm. Apply the triangle inequality.
\end{proof}

\begin{theorem}[Derived B5 implementation budget]
\label{thm:readbudget}
The complete readout channel of
Specification~\ref{spec:readnoise} obeys
\begin{equation}
\boxed{
\ddiam(\widehat\PP_N^S,\PP_N^S)
\le
\delta_N^{\mathrm{model}}
:=
\min\left\{1,\,
r_{\mathrm F}
+(1-r_{\mathrm F})
 (p_A+\delta_D+\delta_B+\delta_X+\delta_R+\delta_{\mathrm{as}})
\right\}.
}
\label{eq:readbudget}
\end{equation}
\end{theorem}

\begin{proof}
On a ready branch, replace the independent pointer preparation
by the all-zero state at cost \(p_A\). Telescope the quantum
operations using Lemma~\ref{lem:readcomponents}. Then apply
Theorem~\ref{thm:assignment}. This gives the ready-branch bound.

The failure branch has the same output alphabet and differs from
the ideal channel by half-diamond distance at most one. Convexity
therefore costs \(r_F\), while the ready budget has weight
\(1-r_F\).
\end{proof}

The displayed budget separates actual comparisons. For example,
a spectator error during copying belongs in \(\Delta H_R\);
the same interaction must not also be charged to \(G_X\).
A correlated assignment matrix is allowed, but hidden apparatus
memory correlated with the message is not covered by this fresh
preparation model.

%=====================================================================
\section{Finite Enter interpretation}
\label{sec:enter}
%=====================================================================

\begin{assumption}[Supplied Landmark I characterization]
\label{ass:I}
The supplied theorem \code{thm:main} of Landmark I is admitted
with its \code{ass:strategy-class} and
\code{def:requirements}. The global strategy, on every finite
qubit configuration, satisfies the stated structural class and
\(\Cstar,\NSstar,\Nstar,\Kstar,\Rstar\).

These are global composability, no-signaling, nullity,
continuity, and record requirements, with the source's
continuation restriction. They are not established by testing
one packet or the program in this paper.
\end{assumption}

Its conclusion identifies destructive finite-register readout
with the Born selective maps and the normalized residual on
positive-probability pure branches. No residual ray is specified
on a zero-probability branch.

\begin{proposition}[Semantic scope]
Under Assumption~\ref{ass:I}, the ideal finite program reads the
\(b_N\) carrier qubits destructively, retains their raw tuple,
applies the total decoder, and returns untested spectators with
the channel \eqref{eq:idealread}. The specified finite pointer
instrument has the corresponding ordinary Born interpretation.
\end{proposition}

\begin{proof}
For pure finite-register inputs the characterization gives the
displayed measurement maps. Sequential reads compose to
\(\bra j_Q\). A finite mixed carrier--spectator state has a finite
purification; embed its purifying space into qubits, apply the
pure-state statement, and trace out the ancillary factors.
The result is \(M_jXM_j^\dagger\), independent of the chosen
purification. Pointer allocation, finite interactions, readout,
and classical assignment compose the same way.
\end{proof}

This semantic statement begins after the finite registers exist.
It is not a derivation of the radiation state, the Markov capture
port, or the oscillator's control Hamiltonians.

%=====================================================================
\section{End-to-end theorem and constructive tolerances}
\label{sec:endtoend}
%=====================================================================

\subsection{The general interface theorem}

The following statement remains useful for any independently
certified transfer and implementation.

\begin{theorem}[General finite-record interface]
\label{thm:interface}
Suppose a transfer on a named joint input class differs from the
ideal \((\BB_N\otimes\id_S)\mathcal I_h\) by at most \(\eta\)
in half trace norm, including finite references. Suppose a
complete readout channel with the common alphabet \(\mathcal Y\)
has half-diamond error at most \(\delta\).
Set \(e=\min\{1,\eta+\delta\}\).

For an input whose packet marginal is \(\rho_h\), its actual
full record satisfies
\[
\dist(\Xi_{FSR},\widetilde\Omega_{N,SR}(\rho_{hSR}))\le e,
\]
where \(\rho_{hSR}=\Tr_g\rho_{h g SR}\). Its decoded mean and
law satisfy
\begin{equation}
\boxed{
\begin{aligned}
|\mathbb E_{\mathrm{actual}}\kappa-\nuh|
 &\le\nuh\qh^N+Ne,\\
\tv(p_{\mathrm{actual}},p_h)
 &\le\min\{1,\qh^{N+1}+e\}.
\end{aligned}
}
\label{eq:outerbounds}
\end{equation}
The decoded record--spectator--reference state has the same
second bound against \(\Omega_{\infty,SR}\).
\end{theorem}

\begin{proof}
Telescope through the ideal readout applied to the actual
transferred state. Transfer contractivity and implementation
complete-channel error give the full-record bound.

The reported observable
\[
A_F=\sum_{y\in\mathcal Y}d(y)\ketbra y y
\]
satisfies \(0\le A_F\le NI\). For two states and such an
observable,
\[
|\tr(A_F(\rho-\sigma))|\le N\dist(\rho,\sigma),
\]
because the positive and negative parts of \(\rho-\sigma\) each
have trace \(\dist(\rho,\sigma)\). Combine this with the exact
ideal bias \(\nuh\qh^N\).

Classical decoding and reference discard are contractive.
A triangle inequality through the saturated ideal record, followed
by Proposition~\ref{prop:recordtail}, costs exactly
\(\qh^{N+1}\) for the unsaturated comparison.
\end{proof}

For an arbitrary normal packet state of finite mean \(\mu\),
replace the two geometric errors by \(B_N(\rho_h)\) and
\(\Pr(N_h>N)\), respectively. A mean alone does not supply
the geometric law.

\subsection{The model-level theorem}

Define
\[
e_{N,K}^{\mathrm{model}}
=\min\{1,\eta_{N,K}^{\mathrm{model}}
                    +\delta_N^{\mathrm{model}}\},
\]
using the derived budgets
\eqref{eq:transferbudget} and \eqref{eq:readbudget}.

\begin{theorem}[Apparatus-model realization of the finite Hawking readout]
\label{thm:model}
Retain Assumption~\ref{ass:field}, the weighted test completion,
the finite-window construction, and the model specifications for
capture, controls, independent preparations, and readout.
The input has the specified participating Gaussian marginal and
any admitted normal spectator--reference extension.

The complete model record obeys
\begin{equation}
\boxed{
\dist(\Xi_{FSR}^{\mathrm{model}},
      \widetilde\Omega_{N,SR}(\Tr_g\rho))
\le e_{N,K}^{\mathrm{model}}.
}
\label{eq:modelrecord}
\end{equation}
Its decoded statistics obey
\begin{equation}
\boxed{
\begin{aligned}
|\mathbb E_{\mathrm{model}}\kappa-\nuh|
&\le\nuh\qh^N+N e_{N,K}^{\mathrm{model}},\\
\tv(p_{\mathrm{model}},p_h)
&\le\min\{1,\qh^{N+1}+e_{N,K}^{\mathrm{model}}\}.
\end{aligned}
}
\label{eq:modelbounds}
\end{equation}
The decoded complete-record comparison with the unsaturated
number record retains spectators and references and has the
same bound as the second line.

For the finite Enter terminology, also retain
Assumption~\ref{ass:I}.
\end{theorem}

\begin{proof}
Theorem~\ref{thm:transfer} supplies the transfer certificate
from the capture kernel, preparation, pulses, and control domain.
Theorem~\ref{thm:readbudget} supplies the complete-instrument
certificate from its specified components. Substitute both into
Theorem~\ref{thm:interface}.
\end{proof}

This theorem replaces unspecified \(\eta_N,\delta_N\) by explicit
model quantities. It does not identify the model with a local
Schwarzschild detector.

\subsection{An exact noise-free model control}

With exact extraction, correctly prepared register, and ideal
readout, let \(q_c=\nu_c/(1+\nu_c)\), where \(\nu_c\) is
\eqref{eq:memorymean}. Then the actual model's decoded law is
\begin{equation}
\begin{aligned}
p(0)&=1-q_c+q_c^{K+1},\\
p(k)&=(1-q_c)q_c^k,\qquad 1\le k<N,\\
p(N)&=q_c^N-q_c^{K+1}.
\end{aligned}
\label{eq:exactmodelcounts}
\end{equation}
Its mean is
\[
\nu_c(1-q_c^N)-Nq_c^{K+1}.
\]
These formulas follow directly from
\eqref{eq:extractionchannel}; they also hold for \(N=1\), with
the middle line empty. Classical assignment noise subsequently
gives
\[
\Pr(y)=\sum_{j=0}^N W(y\mid j)p(j).
\]
This is an analytical model control, not an experimentally
measured Hawking histogram.

\subsection{Constructive accuracy and its limitations}

\begin{theorem}[Tolerance schedule]
\label{thm:tolerance}
Fix a packet with finite \(\nuh\) and a requested mean tolerance
\(\varepsilon>0\). Any admissible model parameters satisfying
\[
\nuh\qh^N\le\varepsilon/2,\qquad
\eta_{N,K}^{\mathrm{model}}\le\frac{\varepsilon}{4N},\qquad
\delta_N^{\mathrm{model}}\le\frac{\varepsilon}{4N}
\]
give
\[
|\mathbb E_{\mathrm{model}}\kappa-\nuh|\le\varepsilon.
\]

In the family of idealized models allowing arbitrary finite
windows, capture transmissivity approaching one, arbitrarily
accurate independent preparation, no fixed readout or control
error floor, and the stated selective controls for every finite
\(K\), such parameters exist.
\end{theorem}

\begin{proof}
The first assertion is \eqref{eq:modelbounds}.

For existence in the stated model family, choose \(N\) by geometric
decay; if \(\nuh=0\), the ideal bias already vanishes.
Then choose finite \(K\ge N\) with \(\qh^{K+1}\) sufficiently
small. Choose \(L\) large enough that
\(\mu_L\) is sufficiently small, using
Lemmas~\ref{lem:window}--\ref{lem:modeerror}.
For this finite \(L\), \(E_f<\infty\). Choose
\(\tau_*<1\) sufficiently close to one that
\(\ell(E_f,\tau_*)\) is small. The corresponding capture
coupling is smooth with finite peak for each chosen parameter.

Choose \(m_0,p_Q\) and the bounded pulse deviations sufficiently
small; exact pulses are included in the family.
Choose the readiness, pointer-preparation, damping, basis,
cross-talk, pointer-control, and assignment parameters so that
\eqref{eq:readbudget} meets the final allowance.
There are finitely many error terms, so these choices meet the
two required budgets.
\end{proof}

For \(\nuh>0\), a sufficient bias choice is
\[
N\ge\max\left\{1,
\ceil{\frac{\log(2\nuh/\varepsilon)}{-\log\qh}}\right\}.
\]
The conservative band estimate
\[
\qh\le e^{-\betah\omega_-}
\]
can be used without a numerical transmission table.

The existence assertion is not arbitrary accuracy on fixed
hardware. A maximum capture rate, restricted bandwidth, fixed
temperature, fixed storage loss, or irreducible assignment error
may prevent the stated parameter choices. A nonzero lower bound
on the \emph{certificate budget} prevents that certificate from
proving arbitrary accuracy; it is not, by itself, a lower bound
on the actual statistical error, which can exhibit cancellations.

For convergence of the displayed mean certificates as \(N\) grows,
one needs \(N e_{N,K}^{\mathrm{model}}\to0\), not merely a small
fixed record error.

%=====================================================================
\section{Resource ledger, reset model, and verification requirements}
\label{sec:resources}
%=====================================================================

\subsection{What the model counts}

The capture memory is an oscillator, not one qubit. The
extraction input control-domain dimension is \(K+1\).
With the finite leakage model it is \(J+1\), and these dimensions
must not be confused.

The capture state itself need not have finite occupation support.
Its geometric tail is priced by the extraction comparison.
The control Hamiltonian is defined on the full oscillator and
acts identically outside its specified finite domain, except for
the explicitly bounded perturbation when included.

Let \(s\) be the number of laboratory spectator qubits.
The construction uses \(b_N\) register qubits and \(b_N\) pointer
qubits. Allocating both blocks before readout gives peak
laboratory qubit occupancy
\[
2b_N+s.
\]
External references are retained in correctness specifications
but are not laboratory storage.

\begin{center}
\small
\renewcommand{\arraystretch}{1.15}
\begin{longtable}{@{}p{0.19\textwidth}p{0.69\textwidth}@{}}
\toprule
Component & Explicit resource or boundary\\
\midrule
\endfirsthead
\toprule
Component & Explicit resource or boundary\\
\midrule
\endhead
\bottomrule
\endfoot
Capture &
Window duration \(4L\); coefficient \(\lambda(t)\);
integrated coupling \(-\log(1-\tau)\);
peak rate and derivatives obtained from
\eqref{eq:captcontrol}. A hardware bandwidth restriction must be
checked against this actual coefficient.\\
Memory &
One oscillator; initial mean \(m_0\); storage time \(T_s\);
storage rate \(\kappa_s\); control input dimension \(K+1\);
maximum modeled leakage domain \(J+1\).\\
Extraction &
\(K\) simultaneous number-selective pulse components;
duration \(T_E\); amplitude and envelope-derivative limits;
frequency modulation through \(K\Omega_c\);
deviation integral \(I_E\).
No oscillator drive or ancillary control qubit is concealed.\\
Register &
\(b_N\) qubits, independently initialized with error \(p_Q\).
The Hamiltonian operates directly on their binary basis; a
separate unpriced encoding unitary is unnecessary.\\
Pointer readout &
\(b_N\) pointer qubits; copy duration \(T_R\);
computational readout; damping time \(T_D\);
assignment matrix \(W\); pointer preparation \(p_A\).
The aggregate ideal copy uses \(b_N\) controlled-NOT operations.\\
Spectators &
Storage through the capture, storage, extraction, and readout
intervals. The only modeled error is the stated
spectator-coupling budget; additional decoherence must be added
as a separate complete channel.\\
Classical output &
Raw words and decoded values, together with all failure labels.
Classical storage is not counted as quantum slots.\\
Preparation and reset &
Fresh bath systems, their specified states, reset duration,
and readiness probability. Independence across shots must be
justified before invoking this one-shot model repeatedly.\\
Energy and work &
If memory energy is \(\Omega_cN_c\), its pre-extraction mean is
\(\Omega_c\nu_c\). Control norm and duration are specified.
A thermodynamic work or reset-cost theorem is not asserted.\\
\end{longtable}
\end{center}

\subsection{A specified reset realization}

The preparations can be implemented within the effective model by
replacement through swaps with fresh independent bath systems:
one oscillator in the geometric state of mean \(m_0\), a register
bath block in \(\sigma_Q\), and pointer bath qubits in their
specified diagonal states. Discard the old apparatus states into
those baths. Finite-dimensional swaps have bounded generators;
the oscillator swap is a passive two-mode rotation with a known
phase correction.

This specifies a replacement channel and the destination of the
discarded information. It does not claim unitary erasure in an
isolated finite environment. The readiness branch is generated
by a fresh classical variable with probability \(r_F\); its
failure output remains visible.

The bath preparations, cooling work, and delivery of fresh bath
systems are external resources of this model. Their
thermodynamic costs are not fixed by channel notation.
Post-measurement reset does not retroactively change a completed
record. Reuse in a later trial requires the newly specified
preparation and independence, rather than an unreported apparatus
memory carried between shots.

\subsection{Verification required for a physical successor}

A physical implementation must supply:
\begin{enumerate}[label=\textup{(\arabic*)},leftmargin=*]
\item a justified effective port and its relation to the scalar
field species, positive-frequency convention, and selected
partial wave;

\item a coupling region and switching in spacetime, with a
proved induced-observable or transfer identification;

\item the clock and redshift conversion. Our clock is asymptotic
time. If a static proper-time clock at radius \(r_0\) is used,
the Schwarzschild relation
\(\dd\tau_{\mathrm{proper}}
=\sqrt{1-2GM/r_0}\,\dd t\) must be carried through all pulse
parameters. This relation alone does not construct the detector;

\item the actual participating field degrees of freedom and their
joint state. The auxiliary vacuum port and nonvacuum radiation
occupations cannot be inferred from a mean histogram;

\item availability or a certified compilation of the
number-selective control set, including detunings, amplitude,
frequency range, temporary occupation, and leakage;

\item complete readout-process characterization, not only a
single-state assignment histogram. Spectator effects, apparatus
memory, reset failures, and correlated errors must be included;

\item independently justified values or bounds for every
parameter substituted into the analytical certificate.
Statistical calibration uncertainty must be propagated into
those bounds.
\end{enumerate}

A compilation log, archive manifest, or numerical optimization
objective is not a replacement for any of these proofs.

%=====================================================================
\section{Debugging and retained non-transfer boundaries}
\label{sec:debug}
%=====================================================================

\begin{enumerate}[label=\textbf{D\arabic*.},leftmargin=*]
\item \textbf{Source repairs.}
The Fock macro \(\FF\) is defined.
The decoder \eqref{eq:decoder} acts on arbitrary record
operators. Inherited radiation inputs and new apparatus
preparations are separated explicitly.

\item \textbf{A mean does not determine the packet density.}
The geometric law is proved using quasifreeness.
The mean-only stop remains valid.

\item \textbf{A finite marginal is not a global radiation density.}
Only finite participating reductions are represented normally.
The weighted radiation-test extension does not reprove the bulk
Hadamard construction.

\item \textbf{A finite temporal switch is not exact band-limited
capture.}
The constructed \(f_L\) differs from \(p\).
Its mismatch is proved, not silently ignored.

\item \textbf{Unselected radiation is not vacuum.}
The participating \(g_L\) occupation and its correlation with \(h\)
are retained. Only newly declared auxiliary apparatus ports are
prepared in vacuum.

\item \textbf{The Markov envelope is an effective-model extension.}
Its negative-frequency auxiliary sector is not a new physical
negative-energy scalar sector. Correspondence to a local
Schwarzschild interaction is unproved.

\item \textbf{Coherent transfer is not number copying.}
Low occupations leave the same vacuum memory state and preserve
register coherences. High occupations are saturating environment
branches.

\item \textbf{The two cutoffs and three tails differ.}
Reported saturation begins at \(N\), giving probability \(\qh^N\).
The ideal count-law error is \(\qh^{N+1}\).
The global extraction-domain tail is \(\qh^{K+1}\).
A support-only certificate instead uses a square-root tail.

\item \textbf{A bounded finite-domain estimate is not an
unbounded-drive estimate.}
Global control certification here uses explicitly bounded
generators. Other oscillator drives require their own invariant
domain or leakage theorem.

\item \textbf{Complete instruments include failures.}
Raw words, decoder values, spectators, and failure records share
one output interface. No trial is postselected away.

\item \textbf{Spectator identity is an operation statement.}
Conditioning on a correlated readout can change the spectator
state even when the ideal operation on its tensor factor is the
identity.

\item \textbf{Landmark I starts with finite qubit kinematics.}
Its global starred requirements do not supply field preparation,
capture, or oscillator control.

\item \textbf{Packet occupation is not an emission rate.}
The inherited \(\nuh\) is a normalized packet mean.
The predecessor's rate uses a separate temporal-window
normalization and observation schedule.

\item \textbf{The result is not a recovery or evaporation model.}
The parameter \(\qh\) is not an energy-labelled transmission
probability from another channel. No Page curve, microscopic
message recovery, backreaction, or mass-loss theorem is inferred.

\item \textbf{Unrelated admission gates are untouched.}
No construction concerns O.1, XVI-N, XV-B, or an inherited
characteristic quantum coefficient. Their objects are not
identified with this apparatus model.
\end{enumerate}

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

\begin{theorem}[Em03-B version 2 release]
\label{thm:release}
Under the explicitly stated radiation input and apparatus model:
\begin{enumerate}[label=\textup{(\roman*)},leftmargin=*]
\item the normal geometric packet marginal and finite
carrier--record interface are constructed;

\item a smooth finite-switching passive capture interaction has
the exact kernel \eqref{eq:captureann}, and its referenced
certificate \eqref{eq:capbudget} includes nonvacuum mode mismatch,
preparation, and justified vacuum loss;

\item coherent oscillator--register extraction is implemented by
the explicit swap pulses, with exact sharp-tail comparison
\eqref{eq:sharpK};

\item global bounded perturbations and support-only control
certificates yield their distinct proved tail regimes;

\item a complete pointer instrument with quantum noise,
assignment noise, spectator coupling, preparation imperfections,
and visible failures has the derived budget
\eqref{eq:readbudget};

\item the derived transfer and readout budgets give the complete
referenced record comparison and the mean and count-law bounds
of Theorem~\ref{thm:model};

\item the tolerance schedule is constructive within its stated
parameter family, with its resource growth and possible error
floors retained.
\end{enumerate}

These statements establish a specified apparatus-model realization,
not a local Schwarzschild detector or a laboratory realization.
No global radiation Fock density, collapse packet limit,
nondestructive field instrument, evaporation mechanism, or
quantum-gravitational correction is established.
\end{theorem}

\begin{proof}
Theorem~\ref{thm:marginal},
Proposition~\ref{prop:carrier}, and
Theorem~\ref{thm:record} prove (i).
Theorem~\ref{thm:capturekernel},
Lemmas~\ref{lem:window}--\ref{lem:modeerror}, and
Theorem~\ref{thm:captureerror} prove (ii).
Theorem~\ref{thm:extraction} and
Proposition~\ref{prop:compile} prove (iii).
Lemma~\ref{lem:Duhamel} and
Proposition~\ref{prop:controlregimes} prove (iv).
Theorems~\ref{thm:assignment} and
\ref{thm:readbudget} prove (v).
Theorems~\ref{thm:transfer} and
\ref{thm:model} prove (vi).
Theorem~\ref{thm:tolerance} and
Section~\ref{sec:resources} prove (vii).

The specifications include no local Schwarzschild system--probe
coupling or experimental correspondence theorem. None of the
excluded conclusions is an additional implication of the
constructed finite interfaces.
\end{proof}

\paragraph{Final scope statement.}
Em03-B version 2 constructs and analyzes an effective
apparatus-model realization of finite packet transfer and destructive
readout. Its reference-stable budgets are derived from declared
preparations, interactions, control pulses, and complete instruments.
Its correspondence to a local Schwarzschild-field detector and its
laboratory realization remain separately identified obligations.

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

\paragraph{Data and verification availability.}
The analytical constructions are contained in this source.
No executed numerical dataset, compilation record, formal proof
certificate, experimental calibration, or apparatus demonstration
is supplied or asserted.

\paragraph{Scope of attribution.}
The inherited field prescription and supplied Landmark I theorem
are identified as predecessor inputs. The capture, extraction,
instrument calculations, and error-budget substitutions are
mathematical constructions of this manuscript. Bibliographic
comparisons are not transfers of all results from the cited works.

%=====================================================================
\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}.
Used through its stated boundary prescription,
\code{lem:horizonoccupation},
\code{eq:quantumscatter},
\code{eq:numberdensity}, and \code{eq:packetlaw}.
Its manuscript proofs are not independently verified here.

\bibitem{LandmarkI}
N.~Cheung,
\emph{I}, version 1.
Supplied characterization statement and excerpts:
\code{ass:strategy-class},
\code{def:requirements}, and \code{thm:main}.
The characterization is an admitted semantic premise, not
reproved in this paper.

\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 bulk-state construction is inherited through Em03-A.

\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 Gaussian-state context; the finite-mode identifications
used here are proved explicitly.

\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.
Pulse-selective linear-memory context, not a proof of the
particular Hawking capture model.

\bibitem{Shirokov}
M.~E.~Shirokov,
\emph{Energy-constrained diamond norms and their use in quantum
information theory},
Problems of Information Transmission
\textbf{54} (2018), 20--33.
arXiv:1706.00361.
doi:10.1134/S0032946018010027.
The constraints used in this paper are explicitly oscillator
occupation constraints.

\bibitem{Krastanov}
S.~Krastanov, V.~V.~Albert, C.~Shen, C.-L.~Zou,
R.~W.~Heeres, B.~Vlastakis, R.~J.~Schoelkopf, and L.~Jiang,
\emph{Universal Control of an Oscillator with Dispersive
Coupling to a Qubit},
Physical Review A \textbf{92} (2015), 040303(R).
arXiv:1502.08015.
doi:10.1103/PhysRevA.92.040303.
A comparison for a different control set, not a compiler or
leakage certificate for the number-selective primitives here.

\bibitem{FewsterVerch}
C.~J.~Fewster and R.~Verch,
\emph{Quantum fields and local measurements},
arXiv:1810.06512, published version.
Used for the separate local system--probe measurement obligation,
not to identify the present Markov port with a local scalar-field
apparatus.

\bibitem{Watrous}
J.~Watrous,
\emph{The Theory of Quantum Information},
Cambridge University Press, 2018.
Chapters 2 and 3: channels, instruments, trace distances, and the
completely bounded trace norm.
The specific channel identities and distance formulas in this
manuscript are proved directly.

\end{thebibliography}

\end{document}