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% The execution record is reported evidence, not an independently repeated run.
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\newcommand{\Hil}{\mathcal H}
\newcommand{\BB}{\mathcal B}
\newcommand{\cN}{\mathcal N}
\newcommand{\cC}{\mathcal C}
\newcommand{\cD}{\mathcal D}
\newcommand{\cT}{\mathcal T}
\newcommand{\cP}{\mathcal P}
\newcommand{\cL}{\mathcal L}
\newcommand{\Tr}{\operatorname{Tr}}
\newcommand{\id}{\operatorname{id}}
\newcommand{\Ad}{\operatorname{Ad}}
\newcommand{\ket}[1]{\lvert #1\rangle}
\newcommand{\bra}[1]{\langle #1\rvert}
\newcommand{\kb}[2]{\ket{#1}\bra{#2}}
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\newcommand{\ii}{\mathrm i}
\newcommand{\eps}{\varepsilon}
\newcommand{\vac}{\mathrm{vac}}
\newcommand{\dtr}{d_{\mathrm{tr}}}
\newcommand{\ddiam}{d_\diamond}
\newcommand{\erec}{\varepsilon_{\mathrm{rec}}}
\newcommand{\dleak}{\delta_{\mathrm{leak}}}
\newcommand{\cdim}{c_d}
\newcommand{\Eimp}{\varepsilon_{\mathrm{implemented}}}

\title{\textsf{Hawk02-B}\\[4pt]
Hamiltonian-Derived Emission and Quantitative Recovery\\
in a Finite Evaporation Model}
\author{}
\date{Version 2: diagnostic-record revision\\October 10, 2026}

\begin{document}
\maketitle

\begin{abstract}
We construct an excitation-conserving sequential-emission model
and derive its complete emission isometry from finite-dimensional
Hamiltonians. A $d$-dimensional code occupies a degenerate excited
memory sector and exchanges its excitation with initially vacant
radiation cells through prescribed pulses. After $N$ pulses the
survival probability is $q_N=\prod_{j=1}^N\cos^2\theta_j$.
The optimal single-cell recovery error in half-diamond distance is
$q_N(1-d^{-2})$, and the complete single-cell residual leakage has
the same exact value. For $K$ emission mechanisms carrying an
arbitrarily entangled $K$-cell code, the exact recovery error is
$1-[1-q_N(1-d^{-2})]^K$, hence at most $Kq_N$.
A Duhamel estimate supplies a compatible finite-time implementation
budget.

Emission-time dephasing alone does not destroy the logical code:
time-resolved, label-coherent recovery remains optimal.
Internal-label dephasing, destructive label-blind records,
incomplete collection, and decoder-mode mismatch have different
operational consequences. Entropy turnover is calculated but is
not used to certify recovery.

A supplied execution record reports successful completion of all
32 predicates in the finite-matrix diagnostic script at
$d=2$, $N=3$, $\theta_j=\pi/4$, with a memory-label detuning
of strength $10^{-4}$ and numerical tolerance $2\times10^{-10}$.
The corrected exact targets are $q_N=1/8$, single-cell recovery
and leakage $3/32$, and two-cell recovery $183/1024$.
These reported checks support the selected matrix identities and
Bell witnesses; they do not replace the analytical proofs or
constitute diamond-norm optimization.
The model is an externally scheduled, bare-energy-conserving
evaporation analogue, not gravitational Hawking radiation or an
autonomous microscopic black-hole model.
\end{abstract}

\tableofcontents

\section{Scope, contribution, and evidence}
\label{sec:scope}

\subsection{Dynamics before diagnostics}

The construction follows
\[
\begin{gathered}
\text{code and Hamiltonian}
\longrightarrow
\text{complete emission isometry}\\
\longrightarrow
\text{accessible channel and complement}
\longrightarrow
\text{recovery and perturbation certificate}.
\end{gathered}
The transfer channel is calculated, not selected by matching a
radiation occupation spectrum.

Repeated quantum interactions provide an established Hamiltonian
framework \cite{AttalPautrat}. Here the finite collision sequence
is the specified dynamics itself: no continuous-time limit or
master-equation approximation is used. The collision-model
overview \cite{Ciccarello} supplies background only.
% Reference verification: ([arxiv.org](https://arxiv.org/abs/math-ph/0311002))

After identifying the radiation code, the accessible channel is a
quantum erasure channel. This mechanism is standard
\cite{Erasure}. The contribution is a complete finite
Hamiltonian-to-certificate application, including exact recovery
constants, complete residual accounting, many-cell release,
finite-time robustness, and access controls. No new
erasure-channel mechanism or capacity theorem is claimed.
% Reference verification: ([arxiv.org](https://arxiv.org/abs/quant-ph/9701015))

Hawk02-A records a conditional certification interface rather than
the Hamiltonian instantiated here
\cite[{\texttt{thm:main}, \texttt{thm:sequential}}]{CheungHawk02A}.
Em03-E is a different Hamiltonian-derived finite emission model,
with its own key and environmental accounting
\cite{CheungEm03E}. No early key is generated or assumed here.

\subsection{Source and execution provenance}
\label{sec:provenance}

References attributed to Cheung, N., identify the manuscript
versions and documentary excerpts supplied with the research
programme. They are cited as comparison records, not as
independently verified substitutes for the proofs below.
The supplied corpus locator is
\texttt{d2c1e6b3}, \texttt{retrieval-3}; it is not a public archive
manifest or evidence of a complete corpus search.

The execution evidence is the supplied
\emph{Hawk02-B Optional Diagnostic Script Execution Summary},
filename \texttt{summary (20).txt}, including its transcription of
\texttt{manifest.json} \cite{ExecutionRecord}.
It reports execution on October 10, 2026, at
\texttt{16:28:41.703102+00:00} UTC.

This revision incorporates that report. It does not claim an
independent rerun, direct inspection of the binary
\texttt{matrices.npz}, or verification of the reported source hash.
All mathematical conclusions remain established by the proofs.
The computational report has a narrower, explicitly described role.

\subsection{Corrections and clarification}
\label{sec:debug}

\begin{enumerate}
\item The execution-summary prose briefly suggests $q_N=1/512$
before correcting itself. For its $N=3$ run,
\[
q_N=(\cos^2(\pi/4))^3=1/8.
\]
The value $1/512$ belongs to $N=9$, used separately in the
analytical feasibility example.

\item The statement that the run confirms ``all analytic claims''
is too broad. The script checks selected finite matrices and
witnesses for one configuration. It does not optimize over
decoders, prove arbitrary-input bounds numerically, calculate all
entropy claims, or test general perturbations.

\item Boolean success at a tolerance is not exact-arithmetic
verification. The recorded manifest contains pass/fail predicates,
not their individual residual magnitudes. An additional prose
claim that an isometry residual is exactly zero is not used below.

\item The undefined normalized radiation mode at $q_N=1$ is
avoided by introducing an unnormalized emission operator first.
The retaining comparison uses this operator at all endpoints.

\item The fixed-per-event feasibility criterion is stated with
$\eta>0$ before division by $\eta$; the case $\eta=0$ is treated
separately.

\item Historical ``unexecuted'' wording inside the version-1 source
listing is retained as provenance, not as the evidence status of
this revision. The reported hash belongs to the executed extracted
file, not to this revised \texttt{.tex} file.
\end{enumerate}

\section{Conventions and elementary channel estimates}
\label{sec:conventions}

All Hilbert spaces are finite dimensional. We set $\hbar=1$.
Entropy is in bits; logarithms in pulse-count estimates are natural.
Write
\[
h_2(x)=-x\log_2x-(1-x)\log_2(1-x),
\qquad 0\log_2 0=0.
\]

For identified state and channel spaces, use
\[
\dtr(\rho,\sigma)=\frac12\norm{\rho-\sigma}_1,
\qquad
\ddiam(\Phi,\Psi)=\frac12\norm{\Phi-\Psi}_\diamond.
\]
The diamond norm is the stabilized induced trace norm. For
differences of channels with a $D$-dimensional input it may be
evaluated on density operators with an untouched reference of
dimension $D$. These distances contract under channel
pre- and postcomposition \cite[Chapter 3]{Watrous}.
% Reference verification: ([cs.uwaterloo.ca](https://cs.uwaterloo.ca/~watrous/TQI/))

Define
\[
\Ad_V(X)=VXV^\dagger,\qquad
\cT_\tau(X)=\Tr(X)\tau
\]
for an isometry $V$ and a density operator $\tau$.
For a channel $\cN:C\to A$ and a specified complete complement
$\cC:C\to B$, define
\begin{align}
\erec(\cN)
&=\inf_{\cD:A\to C}\ddiam(\cD\cN,\id_C),
\label{eq:recoverydefinition}\\
\dleak(\cC)
&=\inf_{\tau_B}\ddiam(\cC,\cT_{\tau_B}),
\label{eq:leakdefinition}
\end{align}
where $\cD$ is CPTP. Finite-dimensional compactness gives
attainment of both infima.

The external reference is not available to the decoder.
A half-diamond recovery bound controls every code--reference state,
not merely a selected pure code input.

\begin{lemma}[Identity versus replacement]
\label{lem:replacement}
For a $d$-dimensional system, let
\[
\cdim=1-\frac1{d^2}.
\]
Then
\[
\ddiam(\id,\cT_{I/d})=\cdim,
\qquad
\inf_\sigma\ddiam(\id,\cT_\sigma)=\cdim.
\]
\end{lemma}

\begin{proof}
Temporarily index a basis by $0,\ldots,d-1$. With
$\zeta=e^{2\pi\ii/d}$, define
\[
\mathsf X\ket a=\ket{a+1},\qquad
\mathsf Z\ket a=\zeta^a\ket a,
\]
where addition is modulo $d$. Summation on matrix units gives
\[
\cT_{I/d}
=
\frac1{d^2}\sum_{r,s=0}^{d-1}
\Ad_{\mathsf X^r\mathsf Z^s}.
\]
Separating the identity term and using that the half-diamond
distance between channels is at most one proves
$\ddiam(\id,\cT_{I/d})\le1-d^{-2}$.

Let
\[
\ket{\Phi_d}
=\frac1{\sqrt d}\sum_{a=0}^{d-1}\ket a_C\ket a_Q,
\qquad
\Phi_d=\kb{\Phi_d}{\Phi_d}.
\]
For every density operator $\sigma$,
\[
\Tr\!\left[\Phi_d(\sigma\otimes I_Q/d)\right]=d^{-2}.
\]
The Bell-projector test distinguishes $\Phi_d$ from the replaced
output with probability difference $1-d^{-2}$.
It supplies the matching lower bound for every $\sigma$.
\end{proof}

\begin{lemma}[Sequential channel comparison]
\label{lem:telescoping}
Suppose $\Phi_j,\widehat\Phi_j$ are channels on common retained
spaces and
$\ddiam(\Phi_j,\widehat\Phi_j)\le\eta_j$.
Then
\[
\ddiam(\Phi_m\cdots\Phi_1,
       \widehat\Phi_m\cdots\widehat\Phi_1)
\le\min\left\{1,\sum_{j=1}^m\eta_j\right\}.
\]
\end{lemma}

\begin{proof}
Insert one channel replacement at a time:
\[
\begin{split}
&\Phi_m\cdots\Phi_1-\widehat\Phi_m\cdots\widehat\Phi_1\\
&\quad=
\sum_j\Phi_m\cdots\Phi_{j+1}
(\Phi_j-\widehat\Phi_j)
\widehat\Phi_{j-1}\cdots\widehat\Phi_1.
\end{split}
\]
Contractivity and the triangle inequality prove the bound.
\end{proof}

This is the locally restated comparison architecture of Hawk02-A
and VII-G \cite{CheungHawk02A,CheungVIIG}.
It is not a discretization error of the ideal finite sequence.

\section{Hamiltonian, code, and energy sectors}
\label{sec:hamiltonian}

Fix $d\ge2$, $N\ge1$, and $\omega>0$. Let
\[
\Hil_M=\operatorname{span}\{\ket0,\ket1,\ldots,\ket d\},
\qquad
\Hil_{R_j}\cong\Hil_M.
\]
The code $\Hil_C=\mathbb C^d$ has embedding
\[
J_C\ket a_C=\ket a_M,\qquad 1\le a\le d.
\]
Set
\[
H_M=\omega\sum_{a=1}^d\kb{a}{a}_M,\qquad
H_{R_j}=\omega\sum_{a=1}^d\kb{a}{a}_{R_j},
\qquad
H_0=H_M+\sum_jH_{R_j}.
\]
Each radiation cell begins in $\ket0$. Every code state therefore
has bare energy $\omega$.

Let $I_j$ be chronologically ordered disjoint intervals.
The real integrable pulse $g_j$ is supported on $I_j$ and
prescribed independently of the code and reference.
In the interaction picture, define
\[
H_{\mathrm{int},j}(t)=g_j(t)S_j,
\]
where
\begin{equation}
S_j=\sum_{a=1}^d
\left(
\kb{0}{a}_M\otimes\kb{a}{0}_{R_j}
+
\kb{a}{0}_M\otimes\kb{0}{a}_{R_j}
\right).
\label{eq:S}
\end{equation}
Other cells are acted on by the identity. Let
\[
\theta_j=\int_{I_j}g_j(t)\dd t,
\qquad 0\le\theta_j\le\pi/2.
\]
Previously emitted radiation does not reinteract.

\begin{btheorem}[Hamiltonian-derived, bare-energy-conserving emission]
\label{thm:B1}
The interval propagator is
\begin{equation}
U_j=e^{-\ii\theta_jS_j}
=I+(\cos\theta_j-1)P_j-\ii\sin\theta_jS_j,
\label{eq:Uj}
\end{equation}
where
\[
P_j=S_j^2
=\sum_{a=1}^d
\left(\kb{a,0}{a,0}+\kb{0,a}{0,a}\right)_{MR_j}.
\]
It is unitary on the full space and satisfies
\[
[S_j,H_0]=[U_j,H_0]=0.
\]
In particular,
\begin{align}
U_j\ket a_M\ket0_{R_j}
&=\cos\theta_j\ket a_M\ket0_{R_j}
-\ii\sin\theta_j\ket0_M\ket a_{R_j},
\label{eq:rotation}\\
U_j\ket0_M\ket0_{R_j}&=\ket0_M\ket0_{R_j}.
\nonumber
\end{align}
The sequence acts linearly on arbitrary reference-entangled inputs.
\end{btheorem}

\begin{proof}
The mutually orthogonal spaces
$\operatorname{span}\{\ket{a,0},\ket{0,a}\}$ are invariant.
On each, $S_j$ is the exchange Pauli matrix; it vanishes on their
orthogonal complement. Thus $S_j^2=P_j$ and $S_j^3=S_j$,
which gives \eqref{eq:Uj}. All Hamiltonians within an interval are
scalar multiples of $S_j$, so time ordering reduces to the pulse
area. The expression solves the finite-dimensional Schr\"odinger
equation almost everywhere for integrable pulses.

Both members of each coupled pair have energy $\omega$.
This proves the energy commutator and \eqref{eq:rotation}.
Tensoring all propagators and the preparation with $I_Q$
proves the reference statement.
\end{proof}

\begin{remark}[External drive]
The Schr\"odinger-picture Hamiltonian is
$H(t)=H_0+g_j(t)S_j$ during a pulse.
The conservation claim concerns $H_0$, not a time-independent
total Hamiltonian including a switching apparatus.
No autonomous controller, clock, or work-storage system is
constructed. On the prepared one-excitation sector, free evolution
adds a common phase, so it does not change the derived channels.
The decoder is likewise an allowed CPTP laboratory operation, not
a separately energy-priced apparatus.
\end{remark}

\section{Complete emission isometry and both channels}
\label{sec:isometry}

Write
\[
\ket{\vac}_R=\ket0_{R_1}\cdots\ket0_{R_N},
\]
and let $\ket{a;j}$ denote excitation label $a$ in cell $j$ and
vacuum elsewhere. Define
\begin{equation}
s_N=\prod_j\cos\theta_j,\quad
q_N=s_N^2,\quad
\alpha_j=-\ii\sin\theta_j\prod_{k<j}\cos\theta_k.
\label{eq:amplitudes}
\end{equation}
Introduce the unnormalized emission operator
\begin{equation}
L_N\ket a_C=\sum_j\alpha_j\ket{a;j}_R.
\label{eq:L}
\end{equation}

\begin{btheorem}[Exact complete emission isometry]
\label{thm:B2}
The Hamiltonian sequence induces
\begin{equation}
W_N\ket a_C
=s_N\ket a_M\ket{\vac}_R+\ket0_M L_N\ket a_C.
\label{eq:Wunnormalized}
\end{equation}
Furthermore,
\[
L_N^\dagger L_N=(1-q_N)I_C.
\]
When $q_N<1$, the normalized embedding
\[
J_N=(1-q_N)^{-1/2}L_N
\]
has orthonormal columns and
\begin{equation}
W_N
=\sqrt{q_N}\,J_C\otimes\ket{\vac}
+\sqrt{1-q_N}\,\ket0_M\otimes J_N.
\label{eq:W}
\end{equation}
With complete ideal outputs $A=R_1\cdots R_N$ and $B=M$,
the two channels, for all $q_N$, are
\begin{align}
\cN_N(X)&=L_NXL_N^\dagger
+q_N\Tr(X)\kb{\vac}{\vac},
\label{eq:Nall}\\
\cC_N(X)&=q_NJ_CXJ_C^\dagger
+(1-q_N)\Tr(X)\kb{0}{0}_M.
\label{eq:C}
\end{align}
For $q_N<1$ the accessible formula is equivalently
\begin{equation}
\cN_N(X)
=(1-q_N)J_NXJ_N^\dagger
+q_N\Tr(X)\kb{\vac}{\vac}.
\label{eq:N}
\end{equation}
All formulas are identities on operators and on reference extensions.
\end{btheorem}

\begin{proof}
For one interval, \eqref{eq:Wunnormalized} is
\eqref{eq:rotation}. After emission at an earlier interval the
memory is in $\ket0$, so a later pulse with a fresh vacant cell
acts trivially on that branch. The surviving branch acquires
$\cos\theta_j$, or emits
$-\ii\sin\theta_j\prod_{k<j}\cos\theta_k$.
Induction proves the complete formula.

Let $q_0=1$ and $q_j=q_{j-1}\cos^2\theta_j$.
Then
\[
|\alpha_j|^2=q_{j-1}-q_j,\qquad
\sum_j|\alpha_j|^2=1-q_N.
\]
Orthogonality of the time and label basis vectors proves
$L_N^\dagger L_N=(1-q_N)I_C$.
The memory ground and excited sectors are orthogonal, as are
radiation vacuum and excitation sectors, so $W_N^\dagger W_N=I_C$.

Tracing memory kills the cross terms because $\bra0J_C=0$.
Tracing radiation kills them because $\bra{\vac}L_N=0$.
The respective remaining traces give
\eqref{eq:Nall} and \eqref{eq:C}.
\end{proof}

If $q_N=1$, $L_N=0$, the accessible channel is constant, and
the complement is the memory code embedding.
If $q_N=0$, the final memory is the fixed state $\ket0$ and
radiation is an isometric code.

\begin{definition}[Complete-output enlargement]
\label{def:enlargement}
Every added mixed auxiliary is purified. Its purifier, inaccessible
controller registers, detector environments, discarded radiation,
and other inaccessible dilation outputs belong to the complete
residual unless explicitly made accessible.
The ideal assignment $B=M$ does not justify ignoring new outputs.
\end{definition}

This agrees with the complete-output accounting recorded in
Hawk02-A and Em03-E \cite{CheungHawk02A,CheungEm03E}.

\section{Explicit optimal single-cell recovery}
\label{sec:single-recovery}

For $q_N<1$, let $P_N=J_NJ_N^\dagger$ and define
\begin{equation}
\cD_N(Y)
=J_N^\dagger YJ_N
+\Tr((I_R-P_N)Y)\frac{I_C}{d}.
\label{eq:decoder}
\end{equation}
For $q_N=1$, define $\cD_N=\cT_{I_C/d}$.
The unused-subspace rule is fixed and trace preserving.

The coherent-mode decoder receives all radiation cells, their
internal labels, and knowledge of $J_N$.
It receives neither memory nor reference, uses no early key, and
does not postselect. Section \ref{sec:controls} gives a less demanding
time-resolved decoder with the same optimum.

\begin{btheorem}[Reference-preserving optimal recovery]
\label{thm:B3}
The declared decoder is CPTP, and
\[
\cD_N\cN_N=(1-q_N)\id_C+q_N\cT_{I_C/d}.
\]
For all $q_N\in[0,1]$,
\begin{equation}
\boxed{\erec(\cN_N)=q_N\left(1-\frac1{d^2}\right).}
\label{eq:singleoptimum}
\end{equation}
\end{btheorem}

\begin{proof}
For $q_N<1$, compression and complement measure-and-prepare
are completely positive and their traces sum to $\Tr Y$.
If $\{e_b\}$ is an orthonormal basis of $P_N^\perp$, a Kraus family is
\[
J_N^\dagger,\qquad
d^{-1/2}\ket a_C\bra{e_b}_R,
\quad 1\le a\le d.
\]
The Kraus effects sum to $I_R$. Direct substitution proves the
decoded formula. Lemma \ref{lem:replacement} yields error
$q_N\cdim$. The same conclusion holds for the constant endpoint.

For an arbitrary CPTP decoder, use a Bell input $\Phi_d$.
The success branch contributes at most $1-q_N$ to decoded Bell
overlap. The vacuum branch is independent of the input and has
reference marginal $I_Q/d$; its contribution is exactly $q_N/d^2$.
The Bell test therefore gives error at least $q_N(1-d^{-2})$.
\end{proof}

The argument restates the Bell-witness architecture reported in
Hawk01-B without importing its constants
\cite{CheungHawk01B}. At complete discharge it agrees with the
constant-complement criterion recorded in Em05-B
\cite[{\texttt{thm:constantComplement}}]{CheungEm05B}.

\section{Complete residual leakage}
\label{sec:leakage}

\begin{btheorem}[Exact single-cell residual decoupling]
\label{thm:B4}
For the complete ideal complement,
\begin{equation}
\boxed{\dleak(\cC_N)=q_N\left(1-\frac1{d^2}\right).}
\label{eq:exactleak}
\end{equation}
An optimal comparison state is
\[
\tau_{M,N}
=(1-q_N)\kb{0}{0}
+q_NJ_C(I_C/d)J_C^\dagger.
\]
For every $\rho_{CQ}$,
\begin{equation}
\dtr\!\left(
(\cC_N\otimes\id_Q)(\rho_{CQ}),
\tau_{M,N}\otimes\rho_Q
\right)\le q_N\cdim.
\label{eq:uniformdecoupling}
\end{equation}
Comparison with the discharged state alone has distance $q_N$.
\end{btheorem}

\begin{proof}
For the stated comparison,
\[
\cC_N-\cT_{\tau_{M,N}}
=q_N\Ad_{J_C}(\id-\cT_{I/d}).
\]
Isometric output embedding preserves the diamond norm.
Lemma \ref{lem:replacement} proves the upper bound.

For any alternative $\tau_M$, set
$P_{\mathrm{ex}}=J_CJ_C^\dagger$ and
$t=\Tr(P_{\mathrm{ex}}\tau_M)$.
On a Bell input the actual complement output is
\[
q_N\Phi_{J_C}+(1-q_N)\kb{0}{0}\otimes I_Q/d.
\]
Pinch the output difference by the projections
\[
\Phi_{J_C},\quad
P_{\mathrm{ex}}\otimes I_Q-\Phi_{J_C},\quad
\kb{0}{0}\otimes I_Q.
\]
Their block trace norms sum to
\[
\left|q_N-\frac{t}{d^2}\right|
+t(1-d^{-2})+|t-q_N|.
\]
For $t\le q_N$, half this is
$q_N-t/d^2\ge q_N\cdim$.
For $t\ge q_N$, using
$|q_N-t/d^2|\ge q_N-t/d^2$ gives at least
$t\cdim\ge q_N\cdim$.
Pinching contracts trace norm, so this is a lower bound on the
original distance for every $\tau_M$.

The reference estimate follows from the diamond bound.
For comparison with $\kb{0}{0}$, the difference is $q_N$
times two channels with orthogonal output supports.
Their half-diamond distance is one.
\end{proof}

General information--disturbance results relate recovery and
complementary decoupling \cite{KSW}. Here the constants are proved
directly in the stated half-diamond convention.
% Reference verification: ([arxiv.org](https://arxiv.org/abs/quant-ph/0605009))

Because the residual memory space is fixed,
\eqref{eq:uniformdecoupling} is uniform as $q_N\to0$.
It is not an assertion that arbitrarily long implemented evolution
remains accurate.

\section{Many-cell recovery and extensive energy release}
\label{sec:many}

Take $K$ memory cells with distinct radiation cells $R_{\ell,j}$.
The code is
\[
\Hil_{C^{(K)}}=(\mathbb C^d)^{\otimes K}.
\]
Allow arbitrary entanglement between cells and with a reference.

Ideal interactions act on distinct cell mechanisms.
Their complete map factorizes, up to output ordering:
\[
W^{(K)}=\bigotimes_\ell W_\ell,\quad
\cN^{(K)}=\bigotimes_\ell\cN_\ell,\quad
\cC^{(K)}=\bigotimes_\ell\cC_\ell.
\]
This factorization of maps does not assume a product input.
Write $q_\ell$ for the individual survival probabilities.

\begin{bcorollary}[Exact many-cell recovery and energy release]
\label{thm:B5}
The product of the declared single-cell decoders is optimal:
\begin{equation}
\boxed{
\erec(\cN^{(K)})
=1-\prod_{\ell=1}^K(1-\cdim q_\ell).
}
\label{eq:manyexact}
\end{equation}
For identical mechanisms,
\begin{align}
\erec(\cN_N^{\otimes K})
&=1-(1-\cdim q_N)^K,
\label{eq:manycommon}\\
\cdim q_N
&\le\erec(\cN_N^{\otimes K})
\le1-(1-q_N)^K\le Kq_N.
\nonumber
\end{align}
The complete residual has the sufficient bound
\begin{equation}
\dleak(\cC^{(K)})
\le1-\prod_\ell(1-\cdim q_\ell).
\label{eq:manyleak}
\end{equation}
No exact joint leakage optimization is required or asserted.

For identical mechanisms, the initial bare energy is $K\omega$,
the probability of full discharge is $(1-q_N)^K$, and
\[
\mathbb E E_{\mathrm{rad}}=K\omega(1-q_N),\qquad
\mathbb E E_{\mathrm{memory}}=K\omega q_N.
\]
The emitted-excitation count is binomial with parameters
$K$ and $1-q_N$, for every input code state.
\end{bcorollary}

\begin{proof}
The decoded map is
\[
\bigotimes_\ell
\left[(1-q_\ell)\id+q_\ell\cT_{I/d}\right].
\]
Each factor is a Weyl mixture with identity weight
$1-\cdim q_\ell$. The product mixture has identity weight
$\prod_\ell(1-\cdim q_\ell)$.
Separating that term gives the upper bound
\eqref{eq:manyexact}.

For the lower bound, use a product of $K$ Bell states between code
cells and reference cells. Let $F$ be the erased-cell subset.
Its probability is
\[
p_F=\prod_{\ell\in F}q_\ell
\prod_{\ell\notin F}(1-q_\ell).
\]
On that branch the erased references are independent and
maximally mixed. After any decoder the normalized state has form
\[
\sigma_{C^{(K)}Q_{F^c}}\otimes I_{Q_F}/d^{|F|}.
\]
Its overlap with the full Bell projector is
\[
d^{-2|F|}
\Tr\!\left[
\left(I_{C_F}\otimes\Phi_{C_{F^c}Q_{F^c}}\right)
\sigma_{C^{(K)}Q_{F^c}}
\right]
\le d^{-2|F|}.
\]
The total overlap is at most
\[
\sum_Fp_Fd^{-2|F|}
=\prod_\ell(1-\cdim q_\ell).
\]
The Bell test proves the matching unrestricted-decoder lower bound.

The conservative inequalities follow from the union bound.

For leakage, compare with the product of the single-cell
comparison states. The orthogonal memory sector $F$ has weight
$p_F$ and contains an embedded copy of $\Tr_{F^c}X$.
Its distance from the corresponding replacement is at most
$p_F(1-d^{-2|F|})$, by Lemma \ref{lem:replacement} in dimension
$d^{|F|}$ and partial-trace contractivity.
Summing gives \eqref{eq:manyleak}.
The distance to this particular product comparison is attained by
the product Bell input, but alternative joint comparison states
are not optimized here.

Finally, each emitted/vacuum flag effect is a scalar multiple of
the cell-code identity. The product flag effects therefore have
probabilities $p_F$ for every input, including entangled inputs.
The branch emits $K-|F|$ excitations, giving the count and energy laws.
\end{proof}

The all-discharged-event decoder supplies a valid
$1-(1-q_N)^K$ upper bound. Partial-discharge branches also retain
useful quantum information; their inclusion explains the
smaller exact optimum \eqref{eq:manycommon}.

\section{Hamiltonian perturbations and finite-time certification}
\label{sec:robustness}

\subsection{A direct unitary comparison}

For each event $e$, on common finite retained spaces, let
\[
\widetilde H_e(t)=H_e(t)+E_e(t),
\qquad
h_e=\int_{I_e}\norm{E_e(t)}_\infty\dd t,
\]
where $E_e(t)$ is Hermitian and integrable.

\begin{lemma}[Duhamel allowance]
\label{lem:duhamel}
The event propagators satisfy
\[
\norm{\widetilde U_e-U_e}_\infty\le h_e,\qquad
\ddiam(\Ad_{\widetilde U_e},\Ad_{U_e})
\le\min\{1,h_e\}.
\]
The allowance also holds after common preparation isometries,
with arbitrary references.
\end{lemma}

\begin{proof}
Duhamel's formula is
\[
\widetilde U_e(t_1,t_0)-U_e(t_1,t_0)
=-\ii\int_{t_0}^{t_1}
\widetilde U_e(t_1,s)E_e(s)U_e(s,t_0)\dd s.
\]
Unitarity bounds the integral by $h_e$.

For every amplified operator $Z$, insert one unitary replacement
on each side. The trace norm of the output difference is at most
\[
2\norm{\widetilde U_e-U_e}_\infty\norm{Z}_1.
\]
Take the stabilized supremum, divide by two, and apply preparation
contractivity.
\end{proof}

With $\hbar$ restored, $h_e$ has an additional $\hbar^{-1}$.
This channel allowance is not Em06-B's classical vector-stability
estimate \cite[{\texttt{thm:input-stability}}]{CheungEm06B}.

\subsection{Implementation comparisons}

Let the ideal preparation embed the code, append all vacant radiation,
and append any declared fixed auxiliary states. For complete-output
claims, auxiliaries are purified and retained.

\begin{assumption}[Common-space implementation comparisons]
\label{ass:implementation}
The implementation specifies:
\begin{enumerate}[label=\textup{(I\arabic*)}]
\item ideal and actual complete preparation maps
$\cP,\widetilde\cP$ with
$\ddiam(\widetilde\cP,\cP)\le e_{\mathrm{prep}}$;
\item event Hamiltonian deviations with integrals $h_e$;
\item a collected-register channel $\widetilde{\cL}:R\to R$
with
$\ddiam(\widetilde{\cL},\id_R)\le e_{\mathrm{collection}}$;
\item a decoder $\widetilde\cD:R\to C^{(K)}$ with
$\ddiam(\widetilde\cD,\cD^{(K)})\le e_{\mathrm{decoder}}$.
\end{enumerate}
All allowances are nonnegative. Space identifications and padding
are fixed before comparison. Every inaccessible dilation output
is included for complete residual claims.
\end{assumption}

A preparation estimate for one selected input is not (I1).
For an otherwise identical preparation using different fixed
auxiliary states, their trace distance gives an accessible
preparation-channel allowance. Comparing chosen purifications
may require a separate complete allowance.

Let $\widetilde{\cN}^{(K)}$ be the actual precollection radiation
channel. Define
\[
\Eimp
=
\ddiam\!\left(
\widetilde\cD\,\widetilde{\cL}\,
\widetilde{\cN}^{(K)},\id_{C^{(K)}}
\right).
\]

\begin{btheorem}[Robust finite-time recovery]
\label{thm:B6}
Under Assumption \ref{ass:implementation}, put
\[
e_{\mathrm{comp}}
=\min\{1,e_{\mathrm{prep}}+\sum_eh_e\},
\]
and
\[
e_{\mathrm{tot}}
=\min\{1,e_{\mathrm{prep}}+\sum_eh_e+
e_{\mathrm{collection}}+e_{\mathrm{decoder}}\}.
\]
Then
\begin{equation}
\boxed{
\Eimp\le
\min\left\{1,\,
1-\prod_\ell(1-\cdim q_\ell)+e_{\mathrm{tot}}
\right\}.
}
\label{eq:robustexact}
\end{equation}
For identical mechanisms,
\begin{equation}
\boxed{\Eimp\le\min\{1,Kq_N+e_{\mathrm{tot}}\}.}
\label{eq:robustrequested}
\end{equation}
For one cell the ideal term is $q_N\cdim$.

If ideal and actual complete outputs are identified before
collection, their residual channels satisfy
\[
\dleak(\widetilde{\cC}^{(K)})
\le
\min\left\{1,\,
1-\prod_\ell(1-\cdim q_\ell)+e_{\mathrm{comp}}
\right\}.
\]
An accessible collection allowance alone does not compare particular
complete collection dilations with the same linear error.
\end{btheorem}

\begin{proof}
Lemmas \ref{lem:duhamel} and \ref{lem:telescoping} bound the
complete dynamics by $e_{\mathrm{comp}}$.
Partial trace yields the same accessible comparison.
Insert collection and decoder replacements; contractivity gives
\[
\ddiam(
\widetilde\cD\,\widetilde{\cL}\,
\widetilde{\cN}^{(K)},
\cD^{(K)}\cN^{(K)})
\le e_{\mathrm{tot}}.
\]
Add Corollary \ref{thm:B5}.

The residual comparison follows by partial trace of the complete
dynamical comparison, followed by comparison with the ideal
product replacement state.

A distance between accessible collection channels does not,
by itself, compare their chosen isometries in operator norm.
That requires a specified complete comparison or an appropriate
Stinespring continuity estimate \cite{KSW}.
\end{proof}

\begin{remark}[Perturbed energy]
Recovery permits arbitrary integrable $E_e$.
Actual bare-energy conservation additionally requires
$[E_e(t),H_0]=0$ and an appropriate initial sector.
It is not implied by a small norm error.

For $0\le H\le H_{\max}I$, a trace-distance comparison of size
$e_{\mathrm{comp}}$ bounds the expectation discrepancy by
$H_{\max}e_{\mathrm{comp}}$.
The full $KN$-cell radiation Hamiltonian has maximum energy
$KN\omega$, not $K\omega$, unless conservation of the initial
sector is separately established.
\end{remark}

\subsection{A compatible finite-time regime}

For a constant $0<\theta<\pi/2$, set
\[
r=\cos^2\theta,\qquad q_N=r^N.
\]

\begin{corollary}[Positive finite-time perturbation budget]
\label{cor:feasibility}
Fix $K\ge1$ and $0<\varepsilon<1$. Let
\[
N_*=
\left\lceil
\frac{\log(2K/\varepsilon)}{-\log r}
\right\rceil.
\]
Use $KN_*$ serial events of duration $\tau>0$. If
\[
e_{\mathrm{prep}}+e_{\mathrm{collection}}
+e_{\mathrm{decoder}}\le\varepsilon/4,
\qquad
h_e\le\frac{\varepsilon}{4KN_*},
\]
then $\Eimp\le\varepsilon$.

The allowed set contains nonzero bare-energy-conserving
perturbations, for example constant event perturbations
\[
E_e(t)=\lambda_e S_e,\qquad
0<|\lambda_e|\tau\le\frac{\varepsilon}{4KN_*}.
\]
\end{corollary}

\begin{proof}
$Kr^{N_*}\le\varepsilon/2$, and the accumulated implementation
allowance is at most $\varepsilon/2$.
Apply \eqref{eq:robustrequested}.
Since $\norm{S_e}_\infty=1$ and $[S_e,H_0]=0$,
the displayed perturbations have the claimed norm and conservation.
\end{proof}

\begin{corollary}[Fixed event accuracy]
\label{cor:fixedaccuracy}
Suppose $h_e\le\eta$ for each of $KN$ events and the remaining
allowances sum to $e_0<\varepsilon/2$.
For $\eta>0$, the sufficient conditions
\[
Kr^N\le\varepsilon/2,\qquad
KN\eta+e_0\le\varepsilon/2
\]
are simultaneously feasible if
\[
N_*\le
\left\lfloor\frac{\varepsilon/2-e_0}{K\eta}\right\rfloor.
\]
For $\eta=0$, $N=N_*$ is sufficient.
\end{corollary}

\begin{proof}
The first condition holds for integers $N\ge N_*$.
For $\eta>0$, the second holds up to the displayed integer upper
bound. For $\eta=0$ it is independent of $N$.
\end{proof}

This supplies a finite accuracy regime, not a claim that residual
information can be reduced indefinitely at fixed positive event error.

\paragraph{Analytical $N=9$ example; not the reported run.}
For $d=2$, $K=2$, $\theta=\pi/4$, and $\varepsilon=1/100$,
take $N_*=9$. Then $q_N=1/512$ and $Kq_N=1/256$.
Fixed allowances at most $1/400$ and $18$ event allowances each
at most $1/7200$ give $e_{\mathrm{tot}}\le1/200$.
Thus
\[
\frac1{256}+\frac1{200}<\frac1{100}.
\]
The ideal exact error is
\[
1-\left(1-\frac3{2048}\right)^2
=\frac{12279}{4194304}.
\]
These parameter values are analytical and were not the configuration
executed in the supplied report.

\section{Entropy and the Hawking-pair assumption audit}
\label{sec:entropy}

\begin{bproposition}[Entropy history and reference interpretation]
\label{thm:B7}
For a single-cell code state $\rho_C$,
\[
S(R)=h_2(q_N)+(1-q_N)S(\rho_C),\qquad
S(M)=h_2(q_N)+q_NS(\rho_C).
\]
For a pure unreferenced input, $S(R)=h_2(q_N)$.
For a reference-entangled pure input, radiation need not become
pure at discharge.

For identical $K$-cell mechanisms and arbitrary reduced code state
$\rho_{C^{(K)}}$,
\begin{equation}
S(R^{(K)})
=
Kh_2(q_N)
+\sum_{S\subseteq\{1,\ldots,K\}}
(1-q_N)^{|S|}q_N^{K-|S|}S(\rho_{C_S}),
\label{eq:manyentropy}
\end{equation}
where the empty subsystem has entropy zero.
For independent pure cell states this reduces to $Kh_2(q_N)$.
\end{bproposition}

\begin{proof}
The accessible output has orthogonal vacuum and code blocks.
Their eigenvalues are $q_N$ and $(1-q_N)$ times the eigenvalues
of $\rho_C$. The complement has the reversed block weights.
The entropy formulas follow, including endpoints by continuity.

For many cells, the emitted subset $S$ is an orthogonal flag with
the displayed probability. Its conditional state is an isometric
copy of $\rho_{C_S}$.
The flag entropy is $Kh_2(q_N)$.
The entropy of orthogonal blocks gives \eqref{eq:manyentropy}.
Tracing an external reference first gives the same formulas for
the reduced code state.
\end{proof}

As $q$ decreases from one to $1/2$, $h_2(q)$ increases; as $q$
decreases from $1/2$ to zero, it decreases.
A discrete schedule may skip its continuous maximum.
These are finite-model entropy histories, not gravitational
Page-curve calculations.

\subsection{Absent pair hypotheses}

Small-corrections results constrain evolution under stated
near-horizon pair and locality hypotheses \cite{Mathur}.
Hawk02-A records a corresponding pair-entropy interface
\cite[{\texttt{thm:pairentropy}}]{CheungHawk02A}.
% Reference verification: ([arxiv.org](https://arxiv.org/abs/0909.1038))

This model:
\begin{enumerate}
\item transfers an existing excitation instead of creating a fresh
approximately Bell-entangled outgoing/interior pair at every step;
\item reuses one finite memory instead of introducing independently
initialized interior partners;
\item supplies no smooth horizon, interior field algebra,
gravitational dressing, or controlled near-horizon approximation;
\item retains emitted radiation without reinteraction;
\item does not stipulate a thermal outgoing state.
\end{enumerate}
The absence of the pair-production premises explains why entropy
can decrease. It is not an evasion or contradiction of a
small-corrections theorem.

\subsection{A retaining comparison}

\begin{proposition}[Equal entropy histories without recovery]
\label{prop:retainer}
A second bare-energy-conserving sequential model has the same
radiation entropy $h_2(q_N)$ for every pure code input and every
prefix, but constant radiation and recovery error $\cdim$.
\end{proposition}

\begin{proof}
Store the variable code in a zero-energy degenerate spectator $V$.
Prepare another source memory $D$ in the fixed excited state
$\ket1_D$, of energy $\omega$, and apply the same exchange pulses
to $D$ and radiation. All Hamiltonians act as identity on $V$.

The radiation channel is
\[
\cN_N^{\mathrm{ret}}(X)
=\Tr(X)
\left[
q_N\kb{\vac}{\vac}
+(L_N\ket1)(\bra1L_N^\dagger)
\right].
\]
The unnormalized expression is valid even when $q_N=1$.
Its two nonzero block eigenvalues are $q_N$ and $1-q_N$,
so its entropy is $h_2(q_N)$.
Any decoder produces a fixed state; Lemma \ref{lem:replacement}
gives optimal error $\cdim$.
The complete residual retains the original code in $V$.
\end{proof}

The comparison explicitly changes the source preparation and
enlarges the residual ledger. It does not give two channels for one
fixed Hamiltonian and preparation.
It illustrates the entropy-history distinction in Hawk01-B
\cite[{\texttt{thm:entropyseparation}}]{CheungHawk01B}.

\section{Access controls and exact operational effects}
\label{sec:controls}

Write
\[
q=q_N,\quad p=1-q,\quad p_j=|\alpha_j|^2.
\]
Let $J_j\ket a=\ket{a;j}$.
When $p>0$, set $v_j=\alpha_j/\sqrt p$.

\subsection{Incomplete collection}

\begin{proposition}[Discarded radiation]
\label{prop:collection}
Collect only cells in $\mathcal S\subseteq\{1,\ldots,N\}$.
Set
\[
p_{\mathcal S}=\sum_{j\in\mathcal S}p_j,\qquad
q_{\mathcal S}=1-p_{\mathcal S}.
\]
The collected channel is an isometric erasure channel with
\[
\erec(\cN_{\mathcal S})=q_{\mathcal S}\cdim.
\]
The complete residual includes memory and uncollected radiation.
\end{proposition}

\begin{proof}
In \eqref{eq:Wunnormalized}, all collected emission amplitudes share
a vacuum residual. The remaining amplitudes share collected
vacuum and have residual support orthogonal to that vacuum.
Tracing the residual gives the erasure-channel form with success
weight $p_{\mathcal S}$. Apply Theorem \ref{thm:B3}.
If $p_{\mathcal S}=0$, the channel is constant.
\end{proof}

\subsection{Time dephasing preserves the code}

Pinch radiation into vacuum, the label-preserving time sectors
$J_j\Hil_C$, and the unused sector.

\begin{proposition}[Time-resolved optimal recovery]
\label{prop:time}
Time dephasing produces
\[
(\Delta_{\mathrm{time}}\cN_N)(X)
=q\Tr(X)\kb{\vac}{\vac}
+\sum_jp_jJ_jXJ_j^\dagger.
\]
Its optimal recovery error remains $q\cdim$.
A schedule-independent CPTP decoder is
\begin{equation}
\cD_{\mathrm{time}}(Y)
=\sum_jJ_j^\dagger YJ_j
+\Tr\!\left[
\left(I_R-\sum_jJ_jJ_j^\dagger\right)Y
\right]\frac{I_C}{d}.
\label{eq:timedecoder}
\end{equation}
It requires internal-label coherence within a time sector, not
coherence between different time sectors.
\end{proposition}

\begin{proof}
Pinching removes only inter-time cross terms.
The orthogonal sector embeddings and complement replacement
make \eqref{eq:timedecoder} CPTP.
Its composition is $p\id+q\cT_{I/d}$.
The Bell lower bound of Theorem \ref{thm:B3} still applies.
\end{proof}

The measurement environment must be included in a complete
postmeasurement ledger. Nevertheless, measuring time alone does
not make the quantum code unrecoverable.

For $p>0$, applying the \emph{original} coherent-mode decoder after
time dephasing instead gives
\[
a\id+(1-a)\cT_{I/d},\qquad
a=\frac{\sum_jp_j^2}{p}.
\]
Indeed $J_N^\dagger J_j=\overline v_j I_C$.
Its error is $(1-a)\cdim$.
This is decoder-mode failure, not an optimal-recovery obstruction.
At $p=0$, use a constant decoder without defining $v$ or $a$.

\subsection{Incorrect radiation mode}

\begin{proposition}[Mode mismatch]
\label{prop:mismatch}
For $p>0$, let $\widetilde v$ be normalized and
$\widetilde J\ket a=\sum_j\widetilde v_j\ket{a;j}$.
The coherent-mode decoder using $\widetilde J$ instead of $J_N$
has exact error
\[
\left[1-p|\langle\widetilde v,v\rangle|^2\right]\cdim.
\]
At $p=0$ its error is $\cdim$.
\end{proposition}

\begin{proof}
$\widetilde J^\dagger J_N
=\langle\widetilde v,v\rangle I_C$.
Compression transfers the stated weight and replaces its complement
by $I_C/d$. Apply Lemma \ref{lem:replacement}.
\end{proof}

\subsection{Logical-label dephasing and destructive records}

Let
\[
\Delta_C(X)=\sum_a\kb{a}{a}X\kb{a}{a}.
\]
Internal-label dephasing in radiation gives $\cN_N\Delta_C$,
whether or not time is additionally dephased.

\begin{proposition}[Two distinct destructive controls]
\label{prop:destructive}
After internal-label dephasing,
\[
\erec(\cN_N\Delta_C)
=1-\frac{1-q}{d}-\frac{q}{d^2}.
\]
A label-blind occupation measurement that retains only the record
``vacuum'' or ``occupied cell $j$'' and discards all quantum carriers
gives a constant channel and recovery error $\cdim$.
\end{proposition}

\begin{proof}
The coherent-mode decoder gives
$(1-q)\Delta_C+q\cT_{I/d}$.
Since $\Delta_C$ is the uniform mixture of the $d$ phase unitaries,
this is a Weyl mixture with identity weight
$(1-q)/d+q/d^2$. Separating that term gives the upper bound.

On a Bell input, the successful label-dephased branch is separable
between radiation and reference. Decoding preserves separability.
A separable $d$-by-$d$ state has Bell overlap at most $1/d$:
Cauchy--Schwarz proves this for pure product vectors, and mixtures
preserve it. The vacuum contribution is $q/d^2$.
The Bell test yields the matching lower bound.

The label-blind record probabilities $q,p_1,\ldots,p_N$ depend on
the input only through its trace. Discarding every carrier therefore
leaves a constant channel. Apply Lemma \ref{lem:replacement}.
\end{proof}

A destructive record-only measurement is not a nondestructive
number measurement leaving logical carriers available.
Likewise, Psy04-A's $3/4$ score bounds a different two-bit
delayed-query task with a two-state classical memory
\cite[{\texttt{thm:classical}}]{CheungPsy04A};
it is not a half-diamond theorem for the present task.

\subsection{Permanently incomplete release}

\begin{proposition}[Persistent weak coupling]
\label{prop:weak}
If one cell obeys
\[
\sum_{j=1}^\infty\sin^2\theta_j\le b<1,
\]
then every prefix has $q_N\ge1-b$.
Any full $K$-cell code containing that cell has ideal recovery
error at least $\cdim(1-b)$.
\end{proposition}

\begin{proof}
For $x_j\in[0,1]$,
$\prod_j(1-x_j)\ge1-\sum_jx_j$ on each finite prefix.
Use $x_j=\sin^2\theta_j$ and then \eqref{eq:manyexact}.
\end{proof}

For example, $\theta_j=\theta_0\,2^{-j}$, $0<\theta_0<1$, gives
$b\le\theta_0^2/3<1$.
More pulses of this schedule do not yield arbitrarily small error.

\section{Endpoint and operational accounting}
\label{sec:endpoint}

At finite $N$, memory remains inaccessible.
Operationally discarding it does not remove it from a complete
complementary-channel calculation.

\begin{proposition}[A separately declared discharge endpoint]
\label{prop:endpoint}
Append one fresh radiation cell and a pulse of area $\pi/2$ to
each memory mechanism. The resulting survival probability is zero,
the final memory is $\ket0^{\otimes K}$, and radiation recovery is exact.
\end{proposition}

\begin{proof}
Each appended survival factor is $\cos^2(\pi/2)=0$.
Apply Theorem \ref{thm:B2} and Corollary \ref{thm:B5}.
\end{proof}

The extra pulse is specified engineered dynamics, not a free
memory reset or a gravitational endpoint. Its implementation error
must be included when it is used.

Finite CPTP decoders preserve the declared reference semantics.
Hardware storage, ancillas, collection, and discarded systems require
a separate resource ledger. En02-B and En02-C supply comparison
interfaces for such accounting
\cite{CheungEn02B,CheungEn02C}, not a hardware or gravitational
memory-cost theorem for this decoder.

\section{Reported Python diagnostics}
\label{sec:numerics}

\subsection{Status and provenance}

The supplied execution record \cite{ExecutionRecord} reports:
\begin{center}
\begin{tabularx}{\linewidth}{@{}lX@{}}
\toprule
Field & Reported value\\
\midrule
Execution date & October 10, 2026\\
Manifest timestamp &
\texttt{2026-10-10T16:28:41.703102+00:00}\\
Status & \texttt{OPTIONAL\_DIAGNOSTICS\_COMPLETED}\\
Python & \texttt{3.10.21}, GCC \texttt{12.2.0}\\
NumPy & \texttt{2.2.6}\\
Configuration & $d=2$, $N=3$, $\omega=1$, $\theta_j=\pi/4$\\
Event duration & $1$\\
Detuning strength & $10^{-4}$\\
Tolerance & $2\times10^{-10}$\\
Diagnostic predicates & $32$, all reported \texttt{true}\\
\bottomrule
\end{tabularx}
\end{center}

The reported SHA256 of the extracted executed script is:
\begin{center}
\footnotesize
\texttt{0cd724f6588ab5b87dbefc4a7b5a0d11221}\\
\texttt{ace68518b0c3e9167b460d691e6dc}
\end{center}
The two lines concatenate without a separator.
This is a reported identifier, not a recomputed hash.

The reported output directory is
\begin{quote}
\small\path{/workspace/artifacts/hawk02b_diagnostics_output/}
\end{quote}
with \texttt{manifest.json} and \texttt{matrices.npz}.
The path is a local execution locator, not a public repository.

The manifest explicitly records
\[
\begin{gathered}
\texttt{diamond\_optimization\_executed=false},\\
\texttt{proof\_machine\_verified=false},\\
\texttt{latex\_compilation\_verified=false}.
\end{gathered}
\]

\subsection{Correct analytical targets}

For this run,
\[
q=\frac18,\qquad p=\frac78,\qquad \cdim=\frac34,
\]
and
\[
\alpha_1=-\frac{\ii}{\sqrt2},\qquad
\alpha_2=-\frac{\ii}{2},\qquad
\alpha_3=-\frac{\ii}{2\sqrt2}.
\]
Thus $(p_1,p_2,p_3)=(1/2,1/4,1/8)$ and
$\sum_jp_j=7/8$.

\begin{center}
\begin{tabular}{@{}lll@{}}
\toprule
Quantity & Exact value & Decimal value\\
\midrule
Survival $q$ & $1/8$ & $0.125$\\
Single-cell recovery & $3/32$ & $0.09375$\\
Single-cell leakage & $3/32$ & $0.09375$\\
Two-cell recovery & $183/1024$ & $0.1787109375$\\
Time-resolved recovery & $3/32$ & $0.09375$\\
Fixed-mode error after time pinching & $15/32$ & $0.46875$\\
Internal-label-dephasing error & $17/32$ & $0.53125$\\
First-cell-only collection error & $3/8$ & $0.375$\\
Detuning Duhamel allowance & $3/10000$ & $0.0003$\\
\bottomrule
\end{tabular}
\end{center}

The fixed-mode time-pinching coefficient is
\[
a=\frac{(1/2)^2+(1/4)^2+(1/8)^2}{7/8}=\frac38.
\]
The table's control values follow from the analytical propositions.
The script checks the corresponding matrix identities or Bell
witnesses; its manifest does not separately print all these values.

Its printed targets are
\[
q=0.12500000000000006,\qquad
\text{single recovery}=\text{single leakage}
=0.09375000000000004,
\]
\[
\text{two-cell recovery}=0.1787109375,\qquad
\text{Duhamel allowance}=0.00030000000000000003.
\]
Their small departures from the rational targets are floating-point
representations, not new physical effects.

\subsection{What the 32 predicates test}

\begin{center}
\begin{tabularx}{\linewidth}{@{}rX@{}}
\toprule
Count & Predicate group\\
\midrule
3 & Ideal isometry, Hamiltonian-to-isometry agreement,
and amplitude normalization\\
1 & Decoder Kraus completeness\\
12 & Radiation, memory, and decoded identities on the four
$d=2$ matrix units\\
2 & Single-cell recovery and leakage Bell-state witnesses\\
3 & Ideal prepared energy sector, perturbed isometry, and
Duhamel operator comparison\\
8 & Time-resolved and fixed-mode-after-pinching identities
on the four matrix units\\
3 & Label-dephasing, incomplete-collection, and two-cell
product-decoder Bell witnesses\\
\midrule
32 & All predicates reported \texttt{true}\\
\bottomrule
\end{tabularx}
\end{center}

Most predicates compare an operator residual or scalar discrepancy
with the stated tolerance.
The Duhamel predicate instead compares
$\norm{W_{\mathrm{actual}}-W_{\mathrm{ideal}}}_\infty$
with $3\times10^{-4}$ plus that tolerance.
It should not be described as a residual smaller than the tolerance.

The two-cell check uses the tensor product of the decoded
single-cell channels and a dimension-four Bell state.
It is not an independently simulated two-memory Hamiltonian
experiment or a decoder optimization.

\subsection{The tested perturbation and its theorem-level consequence}

On the ordered one-excitation sector, the script uses a detuning
operator $D_M$ with memory-label eigenvalues $-1,+1$ and zero
radiation-sector entries. The unit-duration event propagator is
\[
\widetilde U_j
=\exp[-\ii(\theta_jS_j+10^{-4}D_M)].
\]
Since $\norm{D_M}_\infty=1$, each event has $h_j=10^{-4}$.
This perturbation stays in the initial energy sector; extending
$D_M$ by zero on other sectors gives a bare-energy-conserving
finite-space perturbation.

With exact preparation, collection, and the ideal decoder in this
specified matrix model, Theorem \ref{thm:B6} gives
\begin{equation}
\Eimp\le\frac3{32}+\frac3{10000}
=\frac{1881}{20000}=0.09405.
\label{eq:diagnosticrobust}
\end{equation}
Equation \eqref{eq:diagnosticrobust} is an analytical consequence
of the specified detuning and norm budget.
The script does not compute the actual perturbed recovery optimum
or print the perturbed decoded diamond distance.

\subsection{Reported saved matrices}

The execution summary reports the following keys and shapes.
The binary arrays were not independently inspected for this revision.

\begin{center}
\small
\begin{tabular}{@{}lll@{}}
\toprule
Key & Shape & Reported dtype\\
\midrule
\texttt{S\_sector} & $(3,8,8)$ & \texttt{complex128}\\
\texttt{W\_ideal} & $(21,2)$ & \texttt{complex128}\\
\texttt{W\_actual} & $(21,2)$ & \texttt{complex128}\\
\texttt{J\_radiation} & $(7,2)$ & \texttt{complex128}\\
\texttt{J\_memory} & $(3,2)$ & \texttt{complex128}\\
\texttt{radiation\_Kraus} & $(3,7,2)$ & \texttt{complex128}\\
\texttt{memory\_Kraus} & $(7,3,2)$ & \texttt{complex128}\\
\texttt{decoder\_Kraus} & $(11,2,7)$ & \texttt{complex128}\\
\texttt{tau\_memory} & $(3,3)$ & \texttt{complex128}\\
\texttt{alpha} & $(3,)$ & \texttt{complex128}\\
\texttt{theta} & $(3,)$ & \texttt{float64}\\
\texttt{H\_memory} & $(3,3)$ & \texttt{float64}\\
\texttt{H\_radiation\_compressed} & $(7,7)$ & \texttt{float64}\\
\bottomrule
\end{tabular}
\end{center}

These dimensions match the analytical sector representation:
$d(N+1)=8$, $1+Nd=7$, memory dimension $3$, and tensor-output
dimension $3\cdot7=21$.
The decoder has one compression Kraus operator and
$d[(1+Nd)-d]=10$ complement Kraus operators.

\subsection{Limits of the computational evidence}

The recorded execution supports the selected formulas at the
reported configuration and floating-point tolerance.
It does not establish:
\begin{itemize}
\item exact arithmetic or universal correctness for other dimensions;
\item a numerical upper bound on the diamond norm merely from a Bell test;
\item unrestricted decoder optimality independently of the proofs;
\item general environmental perturbation or collection robustness;
\item entropy histories, arbitrary mode-mismatch instances, or
the infinite weak-coupling example;
\item hardware implementation, calibration, or black-hole identification.
\end{itemize}
No plots or numerical optimization results are reported.
Future diagnostic revisions should log residual magnitudes and
hashes of output artifacts, but that additional logging is not
claimed for the reported run.

\section{Records, physical bridges, and final release}
\label{sec:release}

\subsection{Records are not complete mechanisms}

The channel identities on matrix units determine linear maps,
including reference extensions. Occupation records alone do not.
En03-A records a residual-instrument obstruction under equal coarse
records \cite[{\texttt{thm:obstruction}}]{CheungEn03A}.
Psy02-B records equality of adaptive record laws under equality
of complete accessible process data
\cite[{\texttt{thm:process}}]{CheungPsy02B}.
Neither replaces the Hamiltonian and partial-trace derivation here.

\subsection{Outstanding physical obligations}

A gravitational application would have to identify:
\begin{enumerate}
\item physical memory and radiation degrees of freedom and the code;
\item dynamics producing the exchange or a replacement mechanism
from a specified gravitational theory;
\item a gravitational energy, source, geometry, and backreaction ledger;
\item physically justified radiation-access and collection assumptions;
\item complete endpoint dynamics;
\item a controlled comparison with the actual near-horizon pair structure.
\end{enumerate}

Em03-A and Hawk01-A remain fixed-background/source comparisons
\cite{CheungEm03A,CheungHawk01A}.
The spectrum-nonidentification examples in Hawk01-A and Hawk02-A
do not select this Hamiltonian \cite{CheungHawk01A,CheungHawk02A}.
Em03-F's exterior source-to-field channel is a different mechanism
and is not black-hole mass loss
\cite[{\texttt{thm:fieldchannel}}]{CheungEm03F}.

\subsection{Non-import boundaries}

Em06-B's classical seminorm stability is not imported as a diamond
allowance. En02-E's discarded-probability cutoff equality uses its
commuting-state hypothesis and is not imported for arbitrary
coherent code--reference inputs
\cite{CheungEm06B,CheungEn02E}.

The unrelated characteristic-admission programme is not a premise
of this finite-matrix construction.
The supplied XIV-I/XIV-J records leave the full joint characteristic
map unresolved at I3/J2 \cite{CheungXIVI,CheungXIVJ}.
This paper does not open XV-B's downstream admission exit,
discharge XVI-N, or close XVI-O's bulk O.1
\cite{CheungXVIN,CheungXVIO}.
No inherited quantum coefficient is admitted.

\subsection{Exact release statement}

For identical mechanisms and the stated implementation comparisons,
\[
\boxed{
\begin{aligned}
\Eimp
&\le
\min\left\{1,\,
1-\left[1-(1-d^{-2})(\cos^2\theta)^N\right]^K
+e_{\mathrm{tot}}
\right\}\\
&\le
\min\{1,K(\cos^2\theta)^N+e_{\mathrm{tot}}\}.
\end{aligned}
}
\]
Under Corollary \ref{cor:feasibility}, the error is at most
$\varepsilon$, uniformly over arbitrary quantum inputs and their
external references.

The emission channel and complete residual are derived from the
Hamiltonian; single-cell recovery and leakage and many-cell
recovery are calculated exactly.
The supplied diagnostic report adds a concrete executed
finite-matrix check of selected identities and witnesses,
without replacing their proofs.

No gravitational Hawking-emission mechanism, autonomous evaporator,
microscopic black-hole Hamiltonian, gravitational Page curve,
spacetime-emergence theorem, or ultraviolet completion is asserted.
No independent rerun, diamond-norm optimization, machine proof
verification, or compilation of this revision is asserted.

\appendix

\section{Exact invariant-sector matrix representation}
\label{app:matrices}

The full physical radiation space is
$(\mathbb C^{d+1})^{\otimes N}$.
The ideal prepared trajectory occupies exactly
\[
\Hil_{R,\le1}
=\operatorname{span}\{\ket{\vac},\ket{a;j}\},
\]
of dimension $1+Nd$.

The total one-excitation sector has ordered basis
\[
\ket{m,a}=\ket a_M\ket{\vac}_R,\qquad
\ket{r,j,a}=\ket0_M\ket{a;j}_R.
\]
Its dimension is $d(N+1)$, and
\[
S_j=\sum_a
\left(\kb{m,a}{r,j,a}+\kb{r,j,a}{m,a}\right)
\]
on that sector. There $H_0=\omega I$.
This is an exact invariant-sector representation, not a
probabilistic cutoff.

Arbitrary perturbations leaving this sector require the common
full-space comparison in Theorem \ref{thm:B6}.
The reported detuning was sector preserving.

\section{Recorded diagnostic source listing}
\label{app:implementation}

The following is the version-1 source content associated with the
reported execution. Its historical docstring describes its status
when originally supplied; Section \ref{sec:numerics} gives the
updated reported execution status.

The reported hash applies to the extracted file used by the
execution environment. It has not been recomputed from a new
extraction, whose whitespace or line endings could differ.
To rerun, extract the listing as
\texttt{hawk02b\_diagnostics.py} and use a new output directory.

\begin{lstlisting}[language=Python]
#!/usr/bin/env python3
"""Hawk02-B: optional diagnostic artifact; unexecuted in manuscript."""
import datetime
import hashlib
import itertools
import json
from pathlib import Path
import sys
import numpy as np

D, N = 2, 3
OMEGA = 1.0
THETA = np.full(N, np.pi / 4)
PERTURBATION = 1e-4
TOL = 2e-10

def adj(A):
    return A.conj().T

def opnorm(A):
    return float(np.linalg.norm(A, 2))

def td(A, B):
    X = (A - B + adj(A - B)) / 2
    return float(np.sum(np.abs(np.linalg.eigvalsh(X))) / 2)

def exp_hermitian(H):
    w, V = np.linalg.eigh(H)
    return (V * np.exp(-1j * w)) @ adj(V)

def channel(Ks, X):
    out = np.zeros((Ks[0].shape[0], Ks[0].shape[0]), complex)
    for A in Ks:
        out += A @ X @ adj(A)
    return out

def matrix_units(d):
    for a in range(d):
        for b in range(d):
            X = np.zeros((d, d), complex)
            X[a, b] = 1
            yield a, b, X

def referenced(chan, d):
    # Normalized Choi state: channel applied to half of Phi_d.
    sample = chan(np.eye(d) / d)
    out = np.zeros((sample.shape[0] * d,) * 2, complex)
    for a, b, X in matrix_units(d):
        out += np.kron(chan(X), X) / d
    return out

def bell(d):
    v = np.eye(d).reshape(-1) / np.sqrt(d)
    return np.outer(v, v.conj())

def check(name, condition, manifest):
    passed = bool(condition)
    manifest["checks"][name] = passed
    if not passed:
        raise AssertionError(name)

def diagnostics(root, manifest):
    d, n = D, N
    r, m, z = 1 + n * d, d + 1, d * (n + 1)
    Id = np.eye(d)
    sigma = Id / d

    # One-excitation sector ordering: memory, then radiation bins.
    S = []
    U = np.eye(z, dtype=complex)
    Ua = np.eye(z, dtype=complex)
    detuning = np.zeros((z, z), complex)
    detuning[:d, :d] = np.diag(np.linspace(-1, 1, d))

    for j, theta in enumerate(THETA):
        A = np.zeros((z, z), complex)
        for a in range(d):
            b = d + j * d + a
            A[a, b] = A[b, a] = 1
        S.append(A)
        P = A @ A
        step = np.eye(z) + (np.cos(theta) - 1) * P
        step = step - 1j * np.sin(theta) * A
        U = step @ U
        actual = exp_hermitian(theta * A +
                               PERTURBATION * detuning)
        Ua = actual @ Ua

    E = np.zeros((z, d), complex)
    E[:d] = Id
    T = np.zeros((m * r, z), complex)
    for a in range(d):
        T[(a + 1) * r, a] = 1
    for j in range(n):
        for a in range(d):
            T[1 + j * d + a, d + j * d + a] = 1
    W, Wa = T @ U @ E, T @ Ua @ E

    survive = 1.0
    alpha = []
    for theta in THETA:
        alpha.append(-1j * np.sin(theta) * survive)
        survive *= np.cos(theta)
    alpha = np.asarray(alpha)
    q, p = float(survive**2), float(1 - survive**2)
    cd = 1 - 1 / d**2

    JC = np.zeros((m, d), complex)
    JC[1:] = Id
    vac = np.zeros((r, 1), complex)
    vac[0, 0] = 1
    ground = np.zeros((m, 1), complex)
    ground[0, 0] = 1
    J = np.zeros((r, d), complex)
    for j in range(n):
        J[1 + j*d:1 + (j+1)*d] = alpha[j] * Id / np.sqrt(p)
    Wan = np.sqrt(q) * np.kron(JC, vac)
    Wan += np.sqrt(p) * np.kron(ground, J)

    check("isometry", opnorm(adj(W) @ W - Id) < TOL, manifest)
    check("Hamiltonian_to_isometry", opnorm(W - Wan) < TOL,
          manifest)
    check("amplitude_normalization",
          abs(np.vdot(alpha, alpha).real - p) < TOL, manifest)

    # Accessible and residual Kraus operators.
    tensor = W.reshape(m, r, d)
    Kr = [tensor[a, :, :] for a in range(m)]
    Km = [tensor[:, b, :] for b in range(r)]

    PJ = J @ adj(J)
    def decode(Y):
        return adj(J) @ Y @ J + np.trace((np.eye(r)-PJ) @ Y)*sigma

    # Explicit decoder Kraus matrices.
    vals, vecs = np.linalg.eigh(np.eye(r) - PJ)
    Kd = [adj(J)]
    for k in np.flatnonzero(vals > 0.5):
        for a in range(d):
            B = np.zeros((d, r), complex)
            B[a, :] = vecs[:, k].conj() / np.sqrt(d)
            Kd.append(B)

    check("decoder_Kraus_complete",
          opnorm(sum(adj(A) @ A for A in Kd)-np.eye(r)) < TOL,
          manifest)

    for a, b, X in matrix_units(d):
        rad = channel(Kr, X)
        mem = channel(Km, X)
        target_r = p * J @ X @ adj(J)
        target_r += q * np.trace(X) * (vac @ adj(vac))
        target_m = q * JC @ X @ adj(JC)
        target_m += p * np.trace(X) * (ground @ adj(ground))
        check(f"radiation_unit_{a}_{b}",
              opnorm(rad-target_r) < TOL, manifest)
        check(f"residual_unit_{a}_{b}",
              opnorm(mem-target_m) < TOL, manifest)
        check(f"decoded_unit_{a}_{b}",
              opnorm(decode(rad)-(p*X+q*np.trace(X)*sigma)) < TOL,
              manifest)

    tau = p * ground @ adj(ground) + q * JC @ sigma @ adj(JC)
    rho_dec = referenced(lambda X: decode(channel(Kr, X)), d)
    rho_mem = referenced(lambda X: channel(Km, X), d)
    check("recovery_Bell_witness",
          abs(td(rho_dec, bell(d))-q*cd) < TOL, manifest)
    check("leakage_Bell_witness",
          abs(td(rho_mem, np.kron(tau, sigma))-q*cd) < TOL,
          manifest)

    Hm = np.diag([0] + [OMEGA]*d)
    Hr = np.diag([0] + [OMEGA]*(n*d))
    Htotal = np.kron(Hm, np.eye(r)) + np.kron(np.eye(m), Hr)
    check("prepared_energy_sector",
          opnorm(adj(W) @ Htotal @ W - OMEGA*Id) < TOL,
          manifest)
    check("perturbed_isometry",
          opnorm(adj(Wa) @ Wa - Id) < TOL, manifest)
    check("Duhamel_operator_allowance",
          opnorm(Wa-W) <= n*abs(PERTURBATION)+TOL, manifest)

    # Time pinching: preserve the internal labels in each bin.
    blocks = [slice(0, 1)]
    blocks += [slice(1+j*d, 1+(j+1)*d) for j in range(n)]
    def time_pinch(Y):
        out = np.zeros_like(Y)
        for block in blocks:
            out[block, block] = Y[block, block]
        return out

    def time_decode(Y):
        out = Y[0, 0] * sigma
        for j in range(n):
            block = slice(1+j*d, 1+(j+1)*d)
            out = out + Y[block, block]
        return out

    coherent_weight = float(np.sum(np.abs(alpha)**4) / p)
    for a, b, X in matrix_units(d):
        Y = time_pinch(channel(Kr, X))
        check(f"time_preserves_code_{a}_{b}",
              opnorm(time_decode(Y) -
                     (p*X+q*np.trace(X)*sigma)) < TOL, manifest)
        check(f"fixed_mode_after_time_pinch_{a}_{b}",
              opnorm(decode(Y) -
                     (coherent_weight*X +
                      (1-coherent_weight)*np.trace(X)*sigma))
              < TOL, manifest)

    # Internal-label dephasing.
    label_chan = lambda X: decode(channel(Kr, np.diag(np.diag(X))))
    label_error = 1 - p/d - q/d**2
    check("label_dephasing_Bell_witness",
          abs(td(referenced(label_chan, d), bell(d))-label_error)
          < TOL, manifest)

    # Keep only the first emitted cell: CP loss map on compressed R.
    keep = np.zeros((r, r), complex)
    keep[:1+d, :1+d] = np.eye(1+d)
    Kloss = [keep]
    for b in range(1+d, r):
        B = np.zeros((r, r), complex)
        B[0, b] = 1
        Kloss.append(B)
    Jfirst = np.zeros((r, d), complex)
    Jfirst[1:1+d] = Id
    def decode_first(Y):
        Pfirst = Jfirst @ adj(Jfirst)
        return (adj(Jfirst) @ Y @ Jfirst +
                np.trace((np.eye(r)-Pfirst) @ Y)*sigma)
    collected = lambda X: decode_first(
        channel(Kloss, channel(Kr, X)))
    pfirst = float(abs(alpha[0])**2)
    check("incomplete_collection_Bell_witness",
          abs(td(referenced(collected, d), bell(d)) -
              (1-pfirst)*cd) < TOL, manifest)

    # Product decoder, K=2: witness for the analytic many-cell optimum.
    phase = np.diag(np.exp(2j*np.pi*np.arange(d)/d))
    shift = np.roll(np.eye(d), 1, axis=0)
    single = [np.sqrt(p)*Id]
    for a in range(d):
        for b in range(d):
            single.append(np.sqrt(q)/d *
                          np.linalg.matrix_power(shift, a) @
                          np.linalg.matrix_power(phase, b))
    product = [np.kron(A, B) for A, B in
               itertools.product(single, repeat=2)]
    target_many = 1 - (1-cd*q)**2
    check("two_cell_Bell_witness",
          abs(td(referenced(lambda X: channel(product, X), d*d),
                 bell(d*d))-target_many) < TOL, manifest)

    arrays = {
        "S_sector": np.stack(S), "W_ideal": W, "W_actual": Wa,
        "J_radiation": J, "J_memory": JC,
        "radiation_Kraus": np.stack(Kr),
        "memory_Kraus": np.stack(Km),
        "decoder_Kraus": np.stack(Kd), "tau_memory": tau,
        "alpha": alpha, "theta": THETA,
        "H_memory": Hm, "H_radiation_compressed": Hr
    }
    np.savez(root/"matrices.npz", **arrays)
    manifest["analytic_targets"] = {
        "q": q, "single_recovery": q*cd,
        "single_leakage": q*cd, "two_cell_recovery": target_many,
        "detuning_Duhamel_allowance": n*abs(PERTURBATION)
    }

def main():
    if len(sys.argv) != 2:
        raise SystemExit("usage: script.py NEW_OUTPUT_DIRECTORY")
    root = Path(sys.argv[1])
    root.mkdir(parents=True, exist_ok=False)
    manifest = {
        "status": "RUNNING",
        "time_utc": datetime.datetime.now(
            datetime.timezone.utc).isoformat(),
        "python": sys.version,
        "numpy": np.__version__,
        "parameters": {
            "d": D, "N": N, "omega": OMEGA,
            "angles": THETA.tolist(), "event_duration": 1,
            "perturbation": PERTURBATION, "tolerance": TOL
        },
        "checks": {},
        "diamond_optimization_executed": False,
        "proof_machine_verified": False,
        "latex_compilation_verified": False
    }
    source = Path(__file__).read_bytes()
    manifest["script_sha256"] = hashlib.sha256(source).hexdigest()
    try:
        diagnostics(root, manifest)
        manifest["status"] = "OPTIONAL_DIAGNOSTICS_COMPLETED"
    except BaseException as exc:
        manifest["status"] = "FAILED"
        manifest["failure"] = repr(exc)
        raise
    finally:
        (root/"manifest.json").write_text(
            json.dumps(manifest, indent=2, allow_nan=False))
    print(json.dumps(manifest, indent=2, allow_nan=False))

if __name__ == "__main__":
    main()
\end{lstlisting}

\section{Supplied manifest transcription}
\label{app:manifest}

The following reproduces the manifest transcription in the supplied
execution summary. It is not a newly generated manifest.

\begin{lstlisting}[language={}]
{
  "status": "OPTIONAL_DIAGNOSTICS_COMPLETED",
  "time_utc": "2026-10-10T16:28:41.703102+00:00",
  "python": "3.10.21 (main, Sep 19 2026, 01:08:03) [GCC 12.2.0]",
  "numpy": "2.2.6",
  "parameters": {
    "d": 2,
    "N": 3,
    "omega": 1.0,
    "angles": [
      0.7853981633974483,
      0.7853981633974483,
      0.7853981633974483
    ],
    "event_duration": 1,
    "perturbation": 0.0001,
    "tolerance": 2e-10
  },
  "checks": {
    "isometry": true,
    "Hamiltonian_to_isometry": true,
    "amplitude_normalization": true,
    "decoder_Kraus_complete": true,
    "radiation_unit_0_0": true,
    "residual_unit_0_0": true,
    "decoded_unit_0_0": true,
    "radiation_unit_0_1": true,
    "residual_unit_0_1": true,
    "decoded_unit_0_1": true,
    "radiation_unit_1_0": true,
    "residual_unit_1_0": true,
    "decoded_unit_1_0": true,
    "radiation_unit_1_1": true,
    "residual_unit_1_1": true,
    "decoded_unit_1_1": true,
    "recovery_Bell_witness": true,
    "leakage_Bell_witness": true,
    "prepared_energy_sector": true,
    "perturbed_isometry": true,
    "Duhamel_operator_allowance": true,
    "time_preserves_code_0_0": true,
    "fixed_mode_after_time_pinch_0_0": true,
    "time_preserves_code_0_1": true,
    "fixed_mode_after_time_pinch_0_1": true,
    "time_preserves_code_1_0": true,
    "fixed_mode_after_time_pinch_1_0": true,
    "time_preserves_code_1_1": true,
    "fixed_mode_after_time_pinch_1_1": true,
    "label_dephasing_Bell_witness": true,
    "incomplete_collection_Bell_witness": true,
    "two_cell_Bell_witness": true
  },
  "diamond_optimization_executed": false,
  "proof_machine_verified": false,
  "latex_compilation_verified": false,
  "script_sha256": "0cd724f6588ab5b87dbefc4a7b5a0d11221ace68518b0c3e9167b460d691e6dc",
  "analytic_targets": {
    "q": 0.12500000000000006,
    "single_recovery": 0.09375000000000004,
    "single_leakage": 0.09375000000000004,
    "two_cell_recovery": 0.1787109375,
    "detuning_Duhamel_allowance": 0.00030000000000000003
  }
}
\end{lstlisting}

\section{Internal source locators}
\label{app:sources}

The principal Cheung comparisons are:
\begin{itemize}
\item Hawk02-A, version 1:
\texttt{def:completeoutput}, \texttt{thm:sequential},
\texttt{thm:constant}, \texttt{thm:stability},
\texttt{thm:pairentropy}, \texttt{thm:main}.
\item Hawk01-B, version 1:
reported recovery and endpoint interfaces and supplied
\texttt{thm:entropyseparation}.
\item Em03-E, version 1:
Hamiltonian-derived finite dynamics and complete auxiliary accounting.
\item Em05-B, version 1:
\texttt{thm:constantComplement}.
\item VII-G, version 1:
\texttt{thm:stability}.
\item Em06-B, version 1:
\texttt{thm:input-stability}, not a channel allowance here.
\item En02-B and En02-C, version 1:
finite-programme soundness and storage accounting.
\item En02-E, version 2:
\texttt{thm:cutoff}, not imported for arbitrary coherent inputs.
\item Psy04-A, version 1:
\texttt{thm:classical}, a different task and score.
\item En03-A, version 1:
\texttt{thm:obstruction}.
\item Psy02-B, version 2:
\texttt{thm:process}.
\end{itemize}

These locators identify the supplied documentary comparisons.
All model-specific proofs are local to this manuscript.

\begin{thebibliography}{99}

\bibitem{AttalPautrat}
S. Attal and Y. Pautrat,
``From repeated to continuous quantum interactions,''
\emph{Annales Henri Poincar\'e}
\textbf{7}, 59--104 (2006).
arXiv:math-ph/0311002.

\bibitem{Ciccarello}
F. Ciccarello, S. Lorenzo, V. Giovannetti, and G. M. Palma,
``Quantum collision models: Open system dynamics from repeated
interactions,''
\emph{Physics Reports} \textbf{954}, 1 (2022).
arXiv:2106.11974.
Background overview; not a premise of the model-specific proofs.

\bibitem{Erasure}
C. H. Bennett, D. P. DiVincenzo, and J. A. Smolin,
``Capacities of quantum erasure channels,''
\emph{Physical Review Letters}
\textbf{78}, 3217--3220 (1997).
arXiv:quant-ph/9701015.

\bibitem{Watrous}
J. Watrous,
\emph{The Theory of Quantum Information},
Cambridge University Press (2018).

\bibitem{KSW}
D. Kretschmann, D. Schlingemann, and R. F. Werner,
``The information-disturbance tradeoff and the continuity of
Stinespring's representation,''
\emph{IEEE Transactions on Information Theory}
\textbf{54}, 1708--1717 (2008).
arXiv:quant-ph/0605009.

\bibitem{Mathur}
S. D. Mathur,
``The information paradox: A pedagogical introduction,''
\emph{Classical and Quantum Gravity}
\textbf{26}, 224001 (2009).
arXiv:0909.1038.

\bibitem{ExecutionRecord}
\emph{Hawk02-B Optional Diagnostic Script Execution Summary},
supplied execution record, October 10, 2026.
Filename: \texttt{summary (20).txt}.
Includes a transcription of \texttt{manifest.json} and reported
matrix shapes. Reported source SHA256:
\texttt{0cd724f6588ab5b87dbefc4a7b5a0d11221}\\
\texttt{ace68518b0c3e9167b460d691e6dc}.
The two hash lines concatenate without a separator.

\bibitem{CheungHawk02A}
N. Cheung,
\emph{Hawk02-A}, version 1, manuscript, n.d.
Supplied certification-interface and complete-output source records.

\bibitem{CheungHawk01B}
N. Cheung,
\emph{Hawk01-B}, version 1, manuscript, n.d.
Supplied recovery, residual-decoupling, endpoint, and
\texttt{thm:entropyseparation} records.

\bibitem{CheungHawk01A}
N. Cheung,
\emph{Hawk01-A}, version 1, manuscript, n.d.
Supplied source and conserving-completion comparisons, including
\texttt{thm:source} and \texttt{thm:completion}.

\bibitem{CheungEm03E}
N. Cheung,
\emph{Em03-E}, version 1, manuscript, n.d.
Supplied Hamiltonian-derived finite emission and complete-complement
records.

\bibitem{CheungEm05B}
N. Cheung,
\emph{Em05-B}, version 1, manuscript, n.d.
Supplied \texttt{thm:constantComplement}.

\bibitem{CheungEm06B}
N. Cheung,
\emph{Em06-B}, version 1, manuscript, n.d.
Supplied \texttt{thm:input-stability}; its classical vector estimate
is not a channel allowance in this paper.

\bibitem{CheungVIIG}
N. Cheung,
\emph{VII-G}, version 1, manuscript, n.d.
Supplied \texttt{thm:stability}, finite-execution comparison.

\bibitem{CheungEn02B}
N. Cheung,
\emph{En02-B}, version 1, manuscript, n.d.
Supplied \texttt{thm:B1}, resource-certified decoder soundness.

\bibitem{CheungEn02C}
N. Cheung,
\emph{En02-C}, version 1, manuscript, n.d.
Supplied \texttt{thm:C1}, reference-preserving sequential soundness.

\bibitem{CheungEn02E}
N. Cheung,
\emph{En02-E}, version 2, manuscript, n.d.
Supplied \texttt{thm:cutoff}, commuting Gaussian-state flagged cutoff.

\bibitem{CheungPsy04A}
N. Cheung,
\emph{Psy04-A}, version 1, manuscript, n.d.
Supplied \texttt{thm:classical}, bounded classical-memory
delayed-query benchmark.

\bibitem{CheungEn03A}
N. Cheung,
\emph{En03-A}, version 1, manuscript, n.d.
Supplied \texttt{thm:obstruction}, record/residual separation.

\bibitem{CheungPsy02B}
N. Cheung,
\emph{Psy02-B}, version 2, manuscript, n.d.
Supplied \texttt{thm:process}, complete-process equality and testing.

\bibitem{CheungEm03A}
N. Cheung,
\emph{Em03-A}, version 1, manuscript, n.d.
Supplied \texttt{thm:main}, fixed-background scalar emission law.

\bibitem{CheungEm03F}
N. Cheung,
\emph{Em03-F}, version 1, manuscript, n.d.
Supplied \texttt{thm:fieldchannel}, exterior source-to-field channel.

\bibitem{CheungXIVI}
N. Cheung,
\emph{XIV-I}, version 1, manuscript, n.d.
Supplied \texttt{thm:I2}, \texttt{thm:I3}, and release boundaries.
Not a premise of this finite model.

\bibitem{CheungXIVJ}
N. Cheung,
\emph{XIV-J}, version 1, manuscript, n.d.
Supplied \texttt{thm:J2} and \texttt{cor:noadmission}.
Not a premise of this finite model.

\bibitem{CheungXVIN}
N. Cheung,
\emph{XVI-N}, version r12, manuscript, n.d.
Supplied \texttt{thm:massless-obstruction}.
Its separate hypotheses and boundary are unchanged.

\bibitem{CheungXVIO}
N. Cheung,
\emph{XVI-O}, version o70-sync, manuscript, n.d.
Supplied kernel-subsector/bulk verdict distinction.
Bulk O.1 remains \texttt{NOT\_CLOSED}.

\end{thebibliography}

\end{document}