% Hawk01-B.tex
% Standalone manuscript for pdfLaTeX.
% Build:
%   pdflatex Hawk01-B.tex
%   pdflatex Hawk01-B.tex
%
% No external bibliography, figures, or source files are required.
%
% Preparation notes:
% - The construction and its exact calculations are given in full.
% - The approximate recovery constants are derived from the cited
%   Kretschmann--Schlingemann--Werner theorem, with normalization explicit.
% - Cheung manuscript references use the identifiers and versions supplied
%   in the brief; publication metadata not supplied are not invented.
% - This source has not been compiled or machine-checked in this interface.
% - Author and affiliation information should be inserted by the submitter.

\documentclass[11pt]{article}

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

\hypersetup{
  colorlinks=true,
  linkcolor=blue!45!black,
  citecolor=blue!45!black,
  urlcolor=blue!45!black,
  pdftitle={Hawk01-B: Complete Quantum-Code Preservation in a Sequential Evaporation Model},
  pdfsubject={Reference-preserving recovery, residual leakage, conserving radiation channels, and entropy-history separation}
}

\setlength{\emergencystretch}{3em}
\setlist[enumerate]{leftmargin=*,itemsep=3pt}
\setlist[itemize]{leftmargin=*,itemsep=3pt}
\numberwithin{equation}{section}

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

\DeclareMathOperator{\Tr}{Tr}
\DeclareMathOperator{\rank}{rank}
\DeclareMathOperator{\supp}{supp}
\DeclareMathOperator{\id}{id}
\DeclareMathOperator{\Ad}{Ad}
\DeclareMathOperator{\TV}{TV}

\newcommand{\ii}{\mathrm{i}}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\HH}{\mathcal H}
\newcommand{\BB}{\mathcal B}
\newcommand{\NN}{\mathcal N}
\newcommand{\CC}{\mathcal C}
\newcommand{\DD}{\mathcal D}
\newcommand{\TT}{\mathcal T}
\newcommand{\RR}{\mathcal R}
\newcommand{\I}{I}
\newcommand{\ket}[1]{\lvert #1\rangle}
\newcommand{\bra}[1]{\langle #1\rvert}
\newcommand{\kb}[2]{\lvert #1\rangle\langle #2\rvert}
\newcommand{\norm}[1]{\left\lVert #1\right\rVert}
\newcommand{\abs}[1]{\left\lvert #1\right\rvert}
\newcommand{\Dtr}{D_{\mathrm{tr}}}
\newcommand{\Ddia}{D_{\diamond}}
\newcommand{\erec}{\epsilon_{\mathrm{rec}}}
\newcommand{\dleak}{\delta_{\mathrm{leak}}}
\newcommand{\vac}{\mathrm{vac}}
\newcommand{\pend}{\mathrm{f}}
\newcommand{\TH}{T_{\mathrm H}}
\newcommand{\PhiD}{\Phi_d}
\newcommand{\cdim}{c_d}
\newcommand{\hbin}{h_2}

\title{\textbf{Hawk01-B}\\[0.4em]
Complete Quantum-Code Preservation in a Sequential Evaporation Model:\\
Radiation Correlations, Residual Leakage, and the
Page-Curve--Recovery Gap}
\author{}
\date{}

\begin{document}
\maketitle

\begin{abstract}
We study preservation of an arbitrary finite-dimensional quantum code,
including all external reference correlations, in a declared sequential
emission process. We translate the
Kretschmann--Schlingemann--Werner information--disturbance theorem into
half-diamond normalization and separate its worst-case guarantee from
single-input entanglement and entropy diagnostics. We then construct an
energy-conserving sequence that generates its own key resource, transfers
an encrypted logical system through occupied radiation sectors, retains
vacuum and unsuccessful-transfer branches, and releases every residual
degree of freedom through an explicitly separate endpoint map. At each
pre-endpoint prefix, the optimal recovery error and optimal
constant-channel leakage distance are both
\(s_t(1-d^{-2})\), where \(s_t\) is the probability that transfer remains
pending. Individual emitted blocks are message-independent, whereas the
joint radiation carries the logical coherence. Reference mutual
information and its conditional increments are calculated exactly.
Thermal occupation comparisons include explicit truncation errors and
compatibility obstructions; no agreement with Hawking pair-production
dynamics is inferred from local statistics. A second conserving evolution
has the same radiation density operators for one specified preparation,
and the same radiation entropy histories for every pure code
preparation, but retains the code in an inaccessible memory. The
counterexample discloses its different endpoint and does not challenge
island or entanglement-wedge reconstruction results. Coherent access,
destructive number measurement, complete-output erasure, and regulated
implementation errors have separate certificates. The construction is a
finite field-sector completion, not a microscopic theory of realistic
black-hole evaporation.
\end{abstract}

\section{Problem, scope, and contributions}
\label{sec:introduction}

Hawk01-A distinguished delayed-query retrieval from source spectra,
observer access, and an assumed information-transfer mechanism
\cite{CheungHawkA}. Here the logical task is stronger:
\begin{quote}
Under what declared dynamical, conservation, endpoint, and access
assumptions does radiation preserve every state of a specified quantum
code, together with every external reference correlation?
\end{quote}
Distinguishing selected preparations is not substituted for this
arbitrary-input requirement.

We establish three connected results.
First, residual leakage and arbitrary-input recoverability are related
by the established channel information--disturbance framework, with
all normalization factors displayed
\cite{KSW2006,BenyOreshkov2010}.
Second, a consistent conserving sequence realizes complete code
preservation using identifiable joint radiation correlations.
Third, an explicit entropy-history separation shows why scalar
radiation entropies, detached from their dynamical and reconstruction
setting, do not determine that channel property.

The exact recovery interfaces in Hawk01-A and Em05-B, the key-transfer
separation in Em03-E, and the readout/recovery/resource distinctions in
En02-B--E provide useful source precedents
\cite{CheungHawkA,CheungEm05B,CheungEm03E,CheungEn02B,
CheungEn02C,CheungEn02D,CheungEn02E}.
Their models are not identified with the model below, and their
conclusions are not transferred as black-hole theorems.

\paragraph{Established versus assumed.}
The isometries, conservation identities, channels, decoders, entropy
formulas, and recovery optima below are mathematical results within the
specified finite model. The degenerate radiation sectors, their
couplings, transfer schedule, and endpoint map are model assumptions.
They are not derived from the Schwarzschild geometry or the Hawking
spectrum.

\paragraph{No unpriced assistance.}
The construction starts from one logical input and fixed pure
ancillary preparations. It does not require pre-collected radiation or
a supplied correlated early key. It generates its early resource in
the first emission and includes its purifier in subsequent residual
and endpoint accounting.

\paragraph{No implicit terminal extrapolation.}
The pre-endpoint construction is stopped above a declared
\(M_{\mathrm{cut}}\). A separate endpoint rule is then imposed.
Setting that rule to complete evaporation is an additional assumption,
not an extrapolation of fixed-background quantum field theory.

\section{Preservation criteria and the complete complement}
\label{sec:criteria}

All Hilbert spaces in the channel theorems are finite-dimensional.
Logarithms and entropies use base two.
For states and channels, respectively, define
\begin{equation}
\Dtr(\rho,\sigma)=\frac12\norm{\rho-\sigma}_1,
\qquad
\Ddia(\Phi,\Psi)=\frac12\norm{\Phi-\Psi}_{\diamond}.
\label{eq:normalization}
\end{equation}
The diamond norm is the stabilized trace norm in the
Schr\"odinger picture. Its dual is the completely bounded operator norm
used in the Heisenberg-picture formulation of
Kretschmann--Schlingemann--Werner \cite{KSW2006}.

Let \(A\) be a code of dimension \(d\ge2\), contained in one declared
energy and charge sector. Let \(S\cong A\) be an external reference,
and write
\begin{equation}
\ket{\PhiD}^{SA}
=\frac1{\sqrt d}\sum_{i=0}^{d-1}\ket i_S\ket i_A.
\label{eq:Phi}
\end{equation}

\begin{definition}[Complete channel interface]
\label{def:interface}
A prefix isometry is
\[
W_t:\HH_A\longrightarrow\HH_{Z_t}\otimes\HH_{F_t}.
\]
Its accessible and inaccessible channels are
\begin{equation}
\NN_t(X)=\Tr_{F_t}(W_tXW_t^\dagger),
\qquad
\CC_t(X)=\Tr_{Z_t}(W_tXW_t^\dagger).
\label{eq:channels}
\end{equation}
The system \(Z_t\) contains every accessible quantum system and
classical record. The system \(F_t\) contains the residual source,
excluded radiation, detector environments, stored-resource purifiers,
and every other inaccessible output.

Fixed mixed preparations are purified explicitly before defining
\(W_t\); their purifiers are assigned to \(Z_t\) or \(F_t\), not omitted.
\end{definition}

\begin{definition}[Recovery and leakage]
\label{def:errors}
The unrestricted deterministic recovery error and inaccessible
leakage distance are
\begin{align}
\erec(t)
&=\inf_{\DD_t:Z_t\to A}
\Ddia(\DD_t\NN_t,\id_A),
\label{eq:recovery}\\
\dleak(t)
&=\inf_{\tau_{F_t}}
\Ddia(\CC_t,\TT_{\tau_{F_t}}),
\qquad
\TT_\tau(X)=\Tr(X)\tau.
\label{eq:leakage}
\end{align}
The decoder optimization is over CPTP maps.
A resource-restricted decoder class defines a different optimum.
\end{definition}

\begin{proposition}[Operational content of the recovery metric]
\label{prop:operational}
If a decoder satisfies
\(\Ddia(\DD_t\NN_t,\id_A)\le\epsilon\), then, for every
reference \(S'\) and every joint state \(\rho^{S'A}\),
\begin{equation}
\Dtr\!\left[
(\id_{S'}\otimes\DD_t\NN_t)(\rho),
\rho
\right]\le\epsilon.
\label{eq:arbitraryreference}
\end{equation}
Conversely, it is sufficient to verify this uniform state bound for a
reference of dimension \(d\).
\end{proposition}

\begin{proof}
Apply the definition of the stabilized trace norm to the channel
difference. For a Hermiticity-preserving difference of channels,
the diamond norm is attained on a state with reference dimension
equal to the input dimension. Larger references do not increase the
supremum \cite{Watrous2018}.
\end{proof}

\paragraph{Two distinct initial-state scopes.}
A \emph{formation-code} statement preserves all variable states admitted
by a declared initial code subspace of the regulated source.
An \emph{injected-message} statement preserves a message introduced
into a separately specified source and early-radiation preparation.
The construction below is a formation-code construction with fixed
ancillaries. It does not claim preservation of arbitrary states of its
entire enlarged ambient source Hilbert space.

\section{Exact and approximate residual-leakage certificates}
\label{sec:certificates}

\subsection{Exact arbitrary-code criterion}

\begin{theorem}[Exact recovery and constant complement]
\label{thm:exact}
For a finite-dimensional isometry
\(W:A\to Z\otimes F\), the following are equivalent:
\begin{enumerate}
\item there is a CPTP map \(\DD:Z\to A\) with
\(\DD\NN=\id_A\);
\item there is a density operator \(\tau_F\) such that
\begin{equation}
\CC(X)=\Tr(X)\tau_F
\quad\text{for every }X\in\BB(\HH_A);
\label{eq:constantcomplement}
\end{equation}
\item for a Kraus family \(\{K_\alpha\}\) of \(\NN\),
\begin{equation}
K_\alpha^\dagger K_\beta
=c_{\alpha\beta}I_A
\label{eq:KL}
\end{equation}
with scalar coefficients \(c_{\alpha\beta}\).
\end{enumerate}
If these conditions hold and \(\rank\tau_F=r\), then
\begin{equation}
\dim Z\ge dr.
\label{eq:rankcapacity}
\end{equation}
\end{theorem}

\begin{proof}
Dilate an exact decoder by
\(T:Z\to A\otimes G\).
The composite \((T\otimes I_F)W\) dilates the identity channel.
The Choi matrix of the identity has rank one, so every Kraus operator
of that composite is proportional to \(I_A\).
Consequently
\[
(T\otimes I_F)W\ket\psi
=\ket\psi_A\otimes\ket\chi_{GF}
\]
for one fixed normalized vector \(\chi\).
Tracing \(AG\) proves \eqref{eq:constantcomplement}.

Conversely, diagonalize
\(\tau_F=\sum_{j=1}^{r}\lambda_j\kb jj\), with
\(\lambda_j>0\), and write
\[
W\ket i=\sum_{j=1}^{r}\sqrt{\lambda_j}
\ket{\phi_{ij}}_Z\ket j_F.
\]
The constant-complement identities on the matrix units
\(\kb i{i'}\) give
\[
\langle\phi_{i'j'}\mid\phi_{ij}\rangle
=\delta_{ii'}\delta_{jj'}.
\]
Thus there are \(dr\) orthonormal vectors in \(Z\), proving
\eqref{eq:rankcapacity}.
On their span \(L\), define
\(U\ket{\phi_{ij}}=\ket i_A\ket j_J\).
For a fixed state \(\sigma_A\), the map
\[
\DD(Y)
=
\Tr_J(UP_LYP_LU^\dagger)
+\Tr((I-P_L)Y)\sigma_A
\]
is CPTP and satisfies \(\DD\NN(X)=X\).

Finally, in the environment basis associated with the Kraus family,
\[
\CC(X)_{\alpha\beta}
=\Tr(K_\beta^\dagger K_\alpha X).
\]
Constancy on every operator is equivalent to
\eqref{eq:KL}.
\end{proof}

This is the arbitrary-code constant-complement form of exact
Knill--Laflamme correctability
\cite{KnillLaflamme1997,CheungEm05B,CheungHawkA}.
Testing only a small preparation ensemble does not establish its
matrix-unit requirement.

\subsection{The half-diamond normalization audit}

We use the following published continuity theorem, rather than
introducing a new general information--disturbance theorem.

\begin{lemma}[KSW continuity input]
\label{lem:KSW}
After padding their dilation spaces when necessary, channels
\(\Phi,\Psi\) admit Stinespring isometries with optimized operator-norm
distance \(\beta\) satisfying
\begin{equation}
\beta^2
\le\norm{\Phi-\Psi}_{\diamond}
\le2\beta.
\label{eq:KSWraw}
\end{equation}
This is Theorem~1 of Kretschmann--Schlingemann--Werner, expressed
through the Schr\"odinger dual of its completely bounded norm.
\end{lemma}

\begin{theorem}[Recovery--leakage bounds with explicit constants]
\label{thm:approximate}
For the complete complementary channels of
Definition~\ref{def:interface},
\begin{equation}
\boxed{
\erec\le\sqrt{2\dleak},
\qquad
\dleak\le\sqrt{2\erec}.
}
\label{eq:approximatepair}
\end{equation}
Equivalently,
\begin{equation}
\boxed{
\frac{\dleak^2}{2}
\le\erec
\le\sqrt{2\dleak}.
}
\label{eq:approximatebounds}
\end{equation}
The corresponding symmetric bounds follow by exchanging
\(\erec\) and \(\dleak\).
In particular, \(\erec=0\) if and only if \(\dleak=0\).
\end{theorem}

\begin{proof}
Fix a state \(\tau_F\).
A dilation of its constant channel is
\[
W_\tau\ket\psi
=\ket\psi_A\otimes\ket{\chi_\tau}_{GF},
\]
where \(\chi_\tau\) purifies \(\tau_F\).
Applying Lemma~\ref{lem:KSW} to
\(\CC\) and \(\TT_\tau\) supplies, after padding, an isometry
\(T:Z\to A\otimes G\) such that
\[
\norm{(T\otimes I_F)W-W_\tau}_\infty
\le
\sqrt{\norm{\CC-\TT_\tau}_{\diamond}}.
\]
Tracing \(GF\), and using the right inequality of
\eqref{eq:KSWraw}, gives
\[
\Ddia(\DD\NN,\id_A)
\le
\sqrt{\norm{\CC-\TT_\tau}_{\diamond}}
=
\sqrt{2\Ddia(\CC,\TT_\tau)}.
\]
Optimize over \(\tau\).

Conversely, fix a decoder with error \(\epsilon\) and dilate it.
The resulting channel is within raw diamond distance \(2\epsilon\)
of the identity. Lemma~\ref{lem:KSW} aligns its dilation with
\(\ket\psi\mapsto\ket\psi\otimes\ket\chi\), with operator distance
at most \(\sqrt{2\epsilon}\).
Tracing the recovered logical output and the decoder environment
therefore bounds the original complement by a constant channel
with half-diamond error at most \(\sqrt{2\epsilon}\).
Optimize over the decoder.

This reproduces the information--disturbance argument of
KSW Theorem~3 with the two infima explicit.
The factors of two arise solely from
\eqref{eq:normalization}; no square-root constant is guessed.
\end{proof}

\begin{remark}[What is, and is not, optimized]
The state \(\tau_F\) depends on the channel and may depend on the
decoder in the converse proof. The theorem does not state that
every chosen constant output state gives the converse bound.
An executable decoder and its resource cost remain separate from
the unrestricted existence statement.
Approximate-QEC complementary-channel duality offers additional
fidelity-based and near-optimal recovery formulations
\cite{BenyOreshkov2010,BenyOreshkov2011}.
\end{remark}

\subsection{Single-reference diagnostics}

For a channel \(\Lambda:A\to A\), define its normalized Choi state and
Bell-input entanglement fidelity by
\begin{equation}
J(\Lambda)
=(\id_S\otimes\Lambda)(\kb{\PhiD}{\PhiD}),
\qquad
F_e(\Lambda)
=\bra{\PhiD}J(\Lambda)\ket{\PhiD}.
\label{eq:Choi}
\end{equation}

\begin{proposition}[Bell tests and dimension-dependent conversion]
\label{prop:Bell}
For every \(\Lambda:A\to A\),
\begin{equation}
1-F_e(\Lambda)
\le
\Ddia(\Lambda,\id_A)
\le
\min\{1,d\sqrt{1-F_e(\Lambda)}\}.
\label{eq:Bellconversion}
\end{equation}
For any two channels with input dimension \(d\),
\begin{equation}
\Dtr(J(\Phi),J(\Psi))
\le
\Ddia(\Phi,\Psi)
\le
d\,\Dtr(J(\Phi),J(\Psi)).
\label{eq:Choiconversion}
\end{equation}
\end{proposition}

\begin{proof}
The Bell input and Bell-projector effect give the lower bound
in \eqref{eq:Bellconversion}.
The Choi trace-norm comparison gives
\eqref{eq:Choiconversion}; equivalently,
the raw diamond norm is at most the trace norm of the
unnormalized Choi matrix, which is \(dJ\)
\cite{Watrous2018}.
Finally, the trace distance to a pure state is at most the square
root of one minus its overlap. Combine these statements.
\end{proof}

Thus \(F_e=1\) proves exact channel identity, but approximate
Bell fidelity alone does not give a dimension-independent
worst-case guarantee.

For the complementary Bell state
\[
\omega^{SF}
=(\id_S\otimes\CC)(\kb{\PhiD}{\PhiD}),
\]
put
\[
\mu=\Dtr\left(\omega^{SF},
\frac{I_S}{d}\otimes\omega^F\right).
\]
Then \eqref{eq:Choiconversion} gives
\begin{equation}
\dleak\le\min\{1,d\mu\}.
\label{eq:Bellleakage}
\end{equation}
If only its mutual information is known, quantum Pinsker's inequality
in bit units gives
\begin{equation}
\mu\le
\sqrt{\frac{\ln2}{2}\,I(S:F)},
\qquad
\dleak\le
\min\left\{1,
d\sqrt{\frac{\ln2}{2}\,I(S:F)}
\right\}.
\label{eq:MIconversion}
\end{equation}
These are diagnostic-to-channel conversions, not replacements
for the diamond metric.

\begin{proposition}[Two-preparation leakage witness]
\label{prop:pairwitness}
For any orthonormal pair \(\ket0,\ket1\) in the code, define
\[
\ell=\Dtr\left(
\CC(\kb00),\CC(\kb11)
\right).
\]
Every decoder satisfies
\begin{equation}
\Ddia(\DD\NN,\id_A)\ge\frac{\ell^2}{4}.
\label{eq:pairwitness}
\end{equation}
\end{proposition}

\begin{proof}
Use a maximally entangled input on that two-dimensional subcode.
If the decoder error is \(\epsilon\), its overlap with the
corresponding Bell vector is at least \(1-\epsilon\).
Dilate the decoder and project the global pure state onto that
Bell vector. The normalized projected state is a Bell vector
times an environmental vector, and its distance from the original
pure state is at most \(\sqrt{\epsilon}\).
Partial trace gives
\[
\Dtr\left(\rho^{SF},
\frac{I_S}{2}\otimes\tau_F\right)
\le\sqrt{\epsilon}.
\]
Pinching the two-dimensional reference in the pair basis yields
\[
\frac12\left[
\Dtr(\CC(\kb00),\tau_F)
+
\Dtr(\CC(\kb11),\tau_F)
\right]\le\sqrt{\epsilon}.
\]
The triangle inequality bounds the left side below by \(\ell/2\).
\end{proof}

This reproduces the Bell-projection witness reported in Em05-B,
with its normalization and proof explicit
\cite{CheungEm05B}.

\section{Sequential conserving interface}
\label{sec:sequential}

A consistent sequence begins with one fixed preparation and acts on
the actual residual system:
\begin{equation}
V_t:B_{t-1}\longrightarrow B_t\otimes R_t,
\qquad
W_t=(V_t\otimes I_{R_{<t}})W_{t-1}.
\label{eq:sequence}
\end{equation}
No internal register is re-prepared between steps.

For each declared conserved quantity \(Q\), conservation means
\begin{equation}
(Q_{B_t}+Q_{R_t})V_t=V_tQ_{B_{t-1}},
\label{eq:chargeintertwining}
\end{equation}
including the analogous Hamiltonian equation.
Energy sectors and their dimensions are part of the model data.

\begin{assumption}[Regulated radiation sectors]
\label{ass:sectors}
We use orthogonal outgoing energy-eigenmode sectors:
\begin{enumerate}
\item a key block \(R_1\) with \(d^2\) orthogonal one-particle
states \(\ket r\), all of energy \(e_{\mathrm K}>0\);
\item each transfer block \(R_t\), \(2\le t\le N\), with vacuum
of energy zero and \(d\) orthogonal one-particle states
\(J_t\ket i\), all of energy \(e>0\);
\item an endpoint radiation space with the energy-labelled
orthogonal states specified in Section~\ref{sec:endpoint}.
\end{enumerate}
All these states have the same declared charge and angular-momentum
labels within their respective occupation sectors; in the explicit
neutral implementation these labels vanish.

Sufficient degeneracy is assumed in the residual source sectors.
No Bekenstein--Hawking state-count formula is assigned to them.
\end{assumption}

One finite field realization uses distinct neutral scalar species of
equal dispersion in the same regulated s-wave energy mode.
No additional conserved flavor number is imposed, and the specified
couplings may mix the degenerate species. This is an artificial but
explicit field-sector choice, not an identification of known
astrophysical particles.

Exact energy conservation below concerns this energy-eigenmode
regulator. A finite-time wavepacket is not an exact energy eigenmode.
Replacing these sectors by finite-time collected packets requires
a separate spectral-width and channel-error certificate.

\paragraph{Cutoff and energy budget.}
The initial source energy is \(M\), and
\begin{equation}
M-e_{\mathrm K}-e\ge M_{\mathrm{cut}}>0.
\label{eq:cutoffenergy}
\end{equation}
The code is degenerate within that initial sector.
Its logical Hamiltonian is zero after subtraction of the common
sector energy.

\paragraph{Dimensions.}
At prefix \(t\ge1\), the construction uses
\begin{equation}
\HH_{B_t}
=
\left[
\bigoplus_{j=2}^{t}\mathbb C^{d^2}
\right]
\oplus
\left[
\bigoplus_{r=1}^{d^2}\mathbb C^d
\right],
\qquad
\dim B_t=d^2(t-1+d).
\label{eq:residualdimensions}
\end{equation}
The first summands record completed transfer time and key.
The last summands contain a pending encrypted logical system.
The growth of this particular ledger is disclosed; no claim that it
models a decreasing black-hole microstate dimension is made.

The required finite isometries can be embedded in a larger fixed
energy-sector Hilbert space and extended to block-energy-conserving
unitaries on prepared source-plus-vacuum ports. Unused orthogonal
states are included in that embedding. This finite-dimensional
extension does not supply a local gravitational Hamiltonian.

\section{Internally generated key and sequential logical transfer}
\label{sec:construction}

\subsection{Weyl operators and the first emission}

Let \(\zeta=e^{2\pi\ii/d}\), and define
\begin{equation}
X\ket i=\ket{i+1\bmod d},
\qquad
Z\ket i=\zeta^i\ket i,
\qquad
U_{a,b}=X^aZ^b.
\label{eq:Weyl}
\end{equation}
Write \(r=(a,b)\in\{0,\ldots,d-1\}^2\).

\begin{lemma}[Weyl twirl]
\label{lem:twirl}
For every \(Y\in\BB(\mathbb C^d)\),
\begin{equation}
\frac1{d^2}\sum_rU_rYU_r^\dagger
=\Tr(Y)\frac{I_d}{d}.
\label{eq:twirl}
\end{equation}
\end{lemma}

\begin{proof}
Averaging over \(b\) removes every off-diagonal matrix unit.
Averaging over \(a\) distributes each surviving diagonal entry
uniformly over the computational basis.
\end{proof}

The first emission is
\begin{equation}
V_1\ket\psi_A
=
\frac1d\sum_r
\ket{\pend,r;U_r\psi}_{B_1}
\otimes\ket r_{R_1}.
\label{eq:firstemission}
\end{equation}
Here \(\ket{\pend,r;\psi}\) is a linear isometric embedding of the
logical vector into the pending residual summand labelled by \(r\).
All pending residual states have energy \(M-e_{\mathrm K}\).

Equation~\eqref{eq:firstemission} creates the key correlation inside
the evolution. Its coefficient and the \(d^2\) terms give
\(V_1^\dagger V_1=I_A\).
There is no externally supplied early register or classical random
key. The first outgoing marginal is
\begin{equation}
\NN_{R_1}(Y)=\Tr(Y)\frac{I_{d^2}}{d^2}.
\label{eq:earlymarginal}
\end{equation}
The complementary output retains the encrypted code and the key.

\subsection{Transfer attempts with vacuum branches}

Choose fixed probabilities
\(\lambda_t\in[0,1]\), \(2\le t\le N\).
For a pending residual input define
\begin{equation}
\begin{split}
V_t\ket{\pend,r;\psi}
={}&
\sqrt{\lambda_t}\ket{t,r}_{B_t}\otimes J_t\ket\psi_{R_t}\\
&+
\sqrt{1-\lambda_t}
\ket{\pend,r;\psi}_{B_t}\otimes\ket{\vac}_{R_t}.
\end{split}
\label{eq:attemptmap}
\end{equation}
For an already completed branch \(j<t\),
\begin{equation}
V_t\ket{j,r}_{B_{t-1}}
=
\ket{j,r}_{B_t}\otimes\ket{\vac}_{R_t}.
\label{eq:completedmap}
\end{equation}
Completed residual states have energy
\(M-e_{\mathrm K}-e\).

Define
\begin{equation}
s_1=1,\qquad
s_t=\prod_{k=2}^{t}(1-\lambda_k),\qquad
w_j=s_{j-1}\lambda_j,\qquad
p_t=1-s_t=\sum_{j=2}^{t}w_j.
\label{eq:weights}
\end{equation}
The label \(s_t\) denotes pending-transfer probability.
A zero-probability branch may be retained in the ambient ledger.

\begin{theorem}[Sequential physicality and energy accounting]
\label{thm:physicality}
Equations~\eqref{eq:firstemission}--\eqref{eq:completedmap}
define a consistent sequence of isometries on the declared residual
spaces. They conserve energy and the specified charges, retain every
vacuum and unsuccessful branch, and perform no reset.

For \(t\ge1\), the accumulated isometry is
\begin{equation}
\begin{split}
W_t\ket\psi
=\frac1d\sum_r\ket r_{R_1}\otimes
\Bigg[
&\sum_{j=2}^{t}\sqrt{w_j}\,
\ket{j,r}_{B_t}\otimes
J_jU_r\ket\psi_{R_j}
\otimes\ket{\vac}_{R_{[2,t]\setminus\{j\}}}\\
&+
\sqrt{s_t}\,
\ket{\pend,r;U_r\psi}_{B_t}
\otimes\ket{\vac}_{R_{[2,t]}}
\Bigg].
\end{split}
\label{eq:accumulated}
\end{equation}
Every branch has total energy \(M\).
The mean pre-endpoint residual energy is
\begin{equation}
\langle H_{B_t}\rangle
=M-e_{\mathrm K}-p_te.
\label{eq:meanmass}
\end{equation}
\end{theorem}

\begin{proof}
For pending inputs, the two terms in
\eqref{eq:attemptmap} are orthogonal, and their squared
norms sum to the input norm. For distinct key labels the residual
summands are orthogonal. A newly completed branch has a different
residual label from every previously completed branch.
Thus the maps preserve all input inner products.

A pending successful term changes residual energy by \(e\) and emits
energy \(e\). An unsuccessful or previously completed term emits
vacuum and retains its residual energy. The first emission changes
residual energy by \(e_{\mathrm K}\) and emits that amount.
The same reasoning applies to the declared charges.

Induction on \(t\) proves \eqref{eq:accumulated}:
a pending amplitude is multiplied by
\(\sqrt{1-\lambda_t}\), and its newly completed amplitude is
multiplied by \(\sqrt{\lambda_t}\).
The earlier amplitudes are unchanged.
Their probabilities are \eqref{eq:weights} and sum to one.
The energy average follows.
\end{proof}

The model emits at most one logical transfer quantum across the
attempt blocks. It is not a forecast that a physical black hole ceases
ordinary emission once a message has been transferred.

\subsection{Complete prefix channels}

For concision, identify each occupied block with its logical
\(d\)-dimensional subspace and denote the associated
all-other-vacuum embeddings by \(J_j\).

\begin{theorem}[Accessible and complementary channels]
\label{thm:prefixchannels}
For \(t\ge1\), with all emitted blocks accessible,
\begin{equation}
\begin{split}
\NN_t(Y)
=\frac1{d^2}\sum_r\kb rr_{R_1}\otimes
\left[
\sum_{j=2}^{t}w_jJ_jU_rYU_r^\dagger J_j^\dagger
+s_t\Tr(Y)\kb{\vac}{\vac}
\right].
\end{split}
\label{eq:accessibleprefix}
\end{equation}
The complete residual complement is
\begin{equation}
\begin{split}
\CC_t(Y)
=\frac1{d^2}\sum_r
\left[
\sum_{j=2}^{t}w_j\Tr(Y)\kb{j,r}{j,r}
+s_t\,I_{\pend,r}U_rYU_r^\dagger I_{\pend,r}^\dagger
\right],
\end{split}
\label{eq:complementprefix}
\end{equation}
where \(I_{\pend,r}\) embeds the pending logical system into its
residual summand.
Every individual emitted block is message-independent:
\begin{align}
\NN_{R_1}(Y)
&=\Tr(Y)\frac{I_{d^2}}{d^2},
\label{eq:localkey}\\
\NN_{R_j}(Y)
&=\Tr(Y)\left[
(1-w_j)\kb{\vac}{\vac}
+w_j\frac{I_d^{(1)}}{d}
\right],
\qquad j\ge2.
\label{eq:localdata}
\end{align}
\end{theorem}

\begin{proof}
Trace the orthogonal residual summands in
\eqref{eq:accumulated}. Different key labels also become diagonal
because their residual supports are orthogonal. This gives
\eqref{eq:accessibleprefix}.

Conversely, tracing \(R_1\) removes cross terms in the key.
Tracing occupied/vacuum radiation removes cross terms between
different transfer times and between transfer and pending branches.
The resulting residual state is \eqref{eq:complementprefix}.

The local formulas follow by tracing all other radiation blocks and
applying Lemma~\ref{lem:twirl}.
\end{proof}

The logical information is therefore not inferred from one-block
distinguishability. It is present in the correlations between the
generated key and an occupied transfer block.

\section{Exact recovery, leakage, and reference correlations}
\label{sec:exactoptima}

Set
\begin{equation}
\cdim=1-\frac1{d^2}.
\label{eq:cd}
\end{equation}

\begin{lemma}[Identity versus complete depolarization]
\label{lem:depolarization}
For \(\TT_d(Y)=\Tr(Y)I_d/d\),
\begin{equation}
\Ddia(\id_d,\TT_d)=\cdim,
\qquad
\inf_\sigma\Ddia(\id_d,\TT_\sigma)=\cdim.
\label{eq:depolarization}
\end{equation}
\end{lemma}

\begin{proof}
The Weyl twirl represents
\(\TT_d=d^{-2}\sum_r\Ad_{U_r}\).
The identity term has weight \(d^{-2}\).
The triangle inequality and
\(\Ddia(\Ad_U,\id)\le1\) give the upper bound \(\cdim\).
On the Bell input, the \(d^2\) Weyl outputs are orthogonal Bell states,
so the matching trace distance is \(\cdim\).

For any constant channel \(\TT_\sigma\), the Bell output is
\(I_S/d\otimes\sigma\). Its Bell-projector expectation is \(d^{-2}\).
The effect bound gives distance at least \(\cdim\).
\end{proof}

\begin{theorem}[Exact arbitrary-code recovery and residual leakage]
\label{thm:optima}
For the complete pre-endpoint prefix,
\begin{equation}
\boxed{
\erec(t)=s_t\cdim,
\qquad
\dleak(t)=s_t\cdim.
}
\label{eq:exacterrors}
\end{equation}
The maximum Bell-input entanglement fidelity is
\begin{equation}
\sup_{\DD_t}F_e(\DD_t\NN_t)
=p_t+\frac{s_t}{d^2}
=1-s_t\cdim.
\label{eq:exactfidelity}
\end{equation}
An explicit optimal decoder reads \(r\), locates the occupied transfer
block without reading its logical basis, applies \(U_r^\dagger\)
to that block, and returns it. On the all-vacuum pending branch it
prepares \(I_d/d\).
\end{theorem}

\begin{proof}
The stated decoder has complete decoded channel
\begin{equation}
\Lambda_t=p_t\id_d+s_t\TT_d.
\label{eq:decodedprefix}
\end{equation}
Lemma~\ref{lem:depolarization} gives error \(s_t\cdim\).

For any decoder, a completed branch contributes Bell overlap at most
one. The accessible pending branch is input-independent.
Its reference state remains \(I_S/d\), so after any local decoding its
Bell overlap is \(d^{-2}\). Therefore
\[
F_e\le p_t+s_td^{-2}.
\]
The Bell-projector lower bound proves that no decoder has error
smaller than \(s_t\cdim\), and the stated decoder attains both optima.

For leakage, choose the constant state
\(\tau=\CC_t(I_d/d)\).
The completed summands cancel in
\(\CC_t-\TT_\tau\).
On the pending summands the difference is \(s_t\) times a reversible
flagged Weyl encoding of \(\id_d-\TT_d\).
Contractivity, followed by its left inverse, gives the upper bound
\(s_t\cdim\).

To prove optimality against an arbitrary constant state, apply a
CPTP map to the residual that retains a pending/completed flag,
corrects \(U_r\) and returns the logical qudit on pending branches,
and returns a fixed state on completed branches.
On the Bell input, the pending block of the actual state is
\(s_t\kb{\PhiD}{\PhiD}\).
Let \(a\) be the pending weight of the mapped candidate constant
state. Its pending Bell expectation is \(a/d^2\).

Use first the effect equal to the Bell projector on pending and
zero on completed. Its expectation difference is
\(s_t-a/d^2\).
Use next the effect equal to that Bell projector on pending and
identity on completed. Its expectation difference is
\(a(1-d^{-2})\).
If \(a\le s_t\), the first is at least \(s_t\cdim\);
if \(a\ge s_t\), the second is at least \(s_t\cdim\).
The trace-distance effect bound, and hence the diamond norm, gives
the required lower bound for every constant state.
\end{proof}

\begin{remark}[No deletion of the key]
The residual key is not ignored in the leakage calculation.
Once transfer has succeeded, the remaining key and transfer-time
labels are constant channels on the logical input, even though
they remain correlated with accessible radiation.
Constancy means absence of logical-input dependence, not absence
of all physical correlations.
\end{remark}

\subsection{Joint logical observables}

For any logical operator \(O\in\BB(\mathbb C^d)\), define on the
completed accessible sectors
\begin{equation}
\widehat O_t
=
\sum_r\kb rr_{R_1}\otimes
\sum_{j=2}^{t}
J_jU_rOU_r^\dagger J_j^\dagger.
\label{eq:jointobservable}
\end{equation}
Then
\begin{equation}
\Tr[\NN_t(\rho)\widehat O_t]
=p_t\Tr(\rho O).
\label{eq:logicalobservable}
\end{equation}
Thus Weyl \(X\) and \(Z\) coherence is accessed by key-conditioned
joint observables, not by local number statistics.
On a successful flag, the same expression normalized by that
branch weight gives the exact logical expectation.
The primary channel remains complete and unconditioned.

The decoder needs one surviving \(d\)-dimensional quantum logical
carrier and classical key/time records. The key can be measured in
its label basis and stored classically without destroying the
recoverable logical coherence. This is different from measuring
the occupied logical radiation in its occupation basis.

\subsection{Entropy and mutual information}

Define the branch distribution
\[
\bm b_t=(w_2,\ldots,w_t,s_t),
\qquad
H(\bm b_t)=-\sum_k b_{t,k}\log b_{t,k},
\]
with \(0\log0=0\), and put \(L_d=\log d\).

\begin{theorem}[Exact entropy and reference-information ledger]
\label{thm:information}
For an arbitrary code state \(\rho\) and \(t\ge1\),
\begin{align}
H(R_{\le t})
&=H(\bm b_t)+2L_d+p_tH(\rho),
\label{eq:radiationentropy}\\
H(B_t)
&=H(\bm b_t)+2L_d+s_tH(\rho).
\label{eq:residualentropy}
\end{align}
For the input \(\ket{\PhiD}^{SA}\),
\begin{align}
I(S:R_{\le t})&=2p_tL_d,
\label{eq:MIradiation}\\
I(S:B_t)&=2s_tL_d,
\label{eq:MIresidual}\\
I(S:R_t\mid R_{<t})&=2w_tL_d,
\qquad t\ge2.
\label{eq:CMIincrement}
\end{align}
The key emission has \(I(S:R_1)=0\).
\end{theorem}

\begin{proof}
In \eqref{eq:accessibleprefix}, completed blocks have conditional
entropy \(H(\rho)\), pending blocks have entropy zero, and the
classical key has entropy \(2L_d\).
The orthogonal branch labels contribute \(H(\bm b_t)\).
This proves \eqref{eq:radiationentropy}.
The complementary channel has the opposite placement of the
logical state, proving \eqref{eq:residualentropy}.

For the Bell input, \(H(\rho)=L_d\) and the global state \(SRB_t\)
is pure. Therefore
\[
I(S:R)=L_d+H(R)-H(B_t)=2p_tL_d,
\]
and the complementary identity gives \(I(S:B_t)=2s_tL_d\).
The conditional mutual information is the difference of successive
reference--radiation mutual informations. Since
\(p_t-p_{t-1}=w_t\), \eqref{eq:CMIincrement} follows.
\end{proof}

The mutual-information identity
\[
I(S:R_{\le t})+I(S:B_t)=2\log d
\]
is an exact information ledger for this pure complete interface.
It is not an independent argument for recoverability in another
model. Here the diamond errors are separately calculated in
Theorem~\ref{thm:optima}.

The construction is consistent with no-hiding statements:
when a selected marginal is constant, its complete complement
contains the corresponding logical information
\cite{BraunsteinPati2007}. It does not claim that arbitrary quantum
information can be stored solely in correlations of two inaccessible
marginals while disappearing from their complete complement.

\section{Endpoint completion and the no-remnant condition}
\label{sec:endpoint}

Choose a fixed final residual energy \(m_*\) satisfying
\begin{equation}
0\le m_*<M-e_{\mathrm K}-e.
\label{eq:endpointenergy}
\end{equation}
The information-free endpoint residual is one fixed state
\(\ket v\) of energy \(m_*\).
For completed residual basis states, introduce endpoint radiation
states \(\ket{j,r}_*\) of energy
\[
E_*^{\mathrm s}=M-e_{\mathrm K}-e-m_*.
\]
For pending residual basis states, introduce
\(\ket{\pend,r,i}_*\) of energy
\[
E_*^{\mathrm f}=M-e_{\mathrm K}-m_*.
\]
All these radiation states are orthogonal, with the specified
conserved labels.

Define
\begin{align}
V_*\ket{j,r}_{B_N}
&=\ket v_{B_*}\otimes\ket{j,r}_*,
\label{eq:endpointsuccess}\\
V_*\ket{\pend,r;\psi}_{B_N}
&=\ket v_{B_*}\otimes
\sum_i\psi_i\ket{\pend,r,i}_*.
\label{eq:endpointfailure}
\end{align}

\begin{theorem}[Complete endpoint preservation]
\label{thm:endpoint}
The endpoint map is an energy-conserving isometry and transfers every
residual key, transfer-time record, and pending logical system into
radiation. Its complement is
\begin{equation}
\CC_*(Y)=\Tr(Y)\kb vv.
\label{eq:endpointcomplement}
\end{equation}
Hence
\begin{equation}
\boxed{\erec(*)=\dleak(*)=0.}
\label{eq:endpointerrors}
\end{equation}
For input \(\rho\),
\begin{equation}
H(R_{\mathrm{all}})=H(\rho),
\label{eq:endpointentropy}
\end{equation}
and for the Bell input,
\begin{equation}
I(S:R_{\mathrm{all}})=2\log d,
\qquad
I(S:B_*)=0.
\label{eq:endpointMI}
\end{equation}
The endpoint conditional information increment is
\(2s_N\log d\).
\end{theorem}

\begin{proof}
The maps send an orthonormal residual basis to an orthonormal
endpoint radiation basis with one fixed source factor.
Every term preserves residual-plus-radiation energy.
There is no residual record or purifier outside that fixed factor.

On completed branches, the earlier occupied block is decoded using
\(r\). On pending branches, the endpoint occupied subspace contains
the encrypted logical system and its key label; apply \(U_r^\dagger\)
there. The branch labels are orthogonal.
This is an explicit exact decoder on the emitted state.

Alternatively, \eqref{eq:endpointcomplement} and
Theorem~\ref{thm:exact} establish exact recovery.
The full radiation map is an isometric encoding, so it preserves
the nonzero spectrum of \(\rho\), proving
\eqref{eq:endpointentropy}.
The reference statements follow from purity, or from exact decoding.
Their increment over \eqref{eq:MIradiation} is \(2s_N\log d\).
\end{proof}

For \(m_*=0\) and a unique fixed source vacuum, this is a declared
complete-evaporation endpoint. For \(m_*>0\), it is an
information-free remnant endpoint.
Neither option is derived from semiclassical physics below
\(M_{\mathrm{cut}}\).

\begin{proposition}[Endpoint sufficiency and its missing-premise test]
\label{prop:endpointtest}
If a pure-preparation isometry has all non-source outputs included in
accessible radiation and its final source space is one-dimensional,
then the radiation preserves the entire input code exactly.

A final mass label, or vanishing final source energy, does not by
itself imply this one-dimensional condition.
\end{proposition}

\begin{proof}
A one-dimensional source factor makes its complementary channel
constant, so Theorem~\ref{thm:exact} applies.
Conversely, an energy eigenspace can have dimension greater than one.
A code can remain in that degeneracy without changing the energy
label.
\end{proof}

The endpoint rule is therefore a substantive part of the preservation
claim. An inaccessible message-bearing auxiliary cannot be erased
from the account by calling the visible source ``evaporated.''

\section{Hawking-mode calibration and compatibility limits}
\label{sec:thermal}

The source spectrum does not determine the transfer schedule
\(\lambda_t\). We make only a declared comparison with selected
outgoing occupation states.

For a neutral scalar stationary energy mode, the fixed-background
outgoing occupation is
\begin{equation}
\overline n(\omega;M)
=
\frac{\Gamma_0(\omega;M)}
{e^{\omega/\TH}-1},
\qquad
q=\frac{\overline n}{1+\overline n}.
\label{eq:Hawkoccupation}
\end{equation}
This is the greybody-modified source interface developed in
Hawk01-A and Em03-A
\cite{CheungHawkA,CheungEm03A,Hawking1975}.
We do not repeat its field-theory derivation.

For \(g\) identical degenerate scalar modes, define the product thermal
state
\begin{equation}
\sigma_g(q)
=(1-q)^g
\sum_{\bm n\in\mathbb N^g}
q^{|\bm n|}\kb{\bm n}{\bm n},
\qquad 0\le q<1.
\label{eq:thermalstate}
\end{equation}
Its normalized restriction to total occupation zero or one is
\begin{equation}
\chi_g(q)
=
\frac{
\kb{\vac}{\vac}
+q I_g^{(1)}
}{1+gq},
\qquad
a_g(q)=\frac{gq}{1+gq}.
\label{eq:thermal01}
\end{equation}
The discarded thermal probability is
\begin{equation}
\theta_g(q)
=
1-(1-q)^g(1+gq).
\label{eq:thermaltail}
\end{equation}

\begin{proposition}[Local deviation certificate]
\label{prop:thermalcomparison}
For
\[
\pi_g(w)
=(1-w)\kb{\vac}{\vac}+wI_g^{(1)}/g,
\]
one has
\begin{align}
\Dtr(\pi_g(w),\chi_g(q))
&=\abs{w-a_g(q)},
\label{eq:Bernoullideviation}\\
\Dtr(\chi_g(q),\sigma_g(q))
&=\theta_g(q),
\label{eq:thermalcutdistance}\\
\Dtr(\pi_g(w),\sigma_g(q))
&\le
\min\{1,\abs{w-a_g(q)}+\theta_g(q)\}.
\label{eq:fullthermaldeviation}
\end{align}
Thus the transfer block \(R_j\) has the displayed bound with
\(g=d\) and \(w=w_j\).
\end{proposition}

\begin{proof}
The vacuum and uniform one-particle sectors are orthogonal, so their
probability difference gives \eqref{eq:Bernoullideviation}.
The conditional low-occupation state and the removed tail occupy
orthogonal sectors; renormalization gives trace distance equal to
the removed probability. Apply the triangle inequality.
\end{proof}

The conditional thermal state in \eqref{eq:thermal01} is a calibration
object, not a heralding rule for the model data.
All model attempts remain in the primary channel.

\begin{proposition}[Schedule compatibility obstruction]
\label{prop:schedulecompatibility}
Prescribed unconditional one-particle weights \(a_j\ge0\) can be
realized by the transfer attempts exactly when
\begin{equation}
\sum_{j=2}^{N}a_j\le1.
\label{eq:sumcompatibility}
\end{equation}
Before exhaustion, a realizing schedule is
\begin{equation}
\lambda_j
=
\frac{a_j}{1-\sum_{k=2}^{j-1}a_k}.
\label{eq:calibratedschedule}
\end{equation}
After exhaustion all later \(a_j\) must vanish, and later
\(\lambda_j\) are immaterial on the actual state.
\end{proposition}

\begin{proof}
The model emits at most one transfer excitation, and
\(\sum_jw_j=p_N\le1\), proving necessity.
For sufficiency, use \eqref{eq:calibratedschedule} and induction:
the pending weight before attempt \(j\) is
\(1-\sum_{k<j}a_k\), so \(w_j=a_j\).
\end{proof}

Consequently arbitrary exact local thermal restrictions cannot be
promised independently at every step. A target
\(a_j=a_d(q_j)\) must satisfy
\eqref{eq:sumcompatibility}, and the parameters \(q_j\) must first be
obtained from an actual declared source calculation if an
astrophysical calibration is intended.

The key block is deliberately not Hawking-like in a small-tail regime.
For \(g=d^2\), its state is \(I_g^{(1)}/g\), and
\begin{equation}
\Dtr\left(I_g^{(1)}/g,\sigma_g(q)\right)
=
1-gq(1-q)^g
\ge
1-\left(\frac{g}{g+1}\right)^{g+1}.
\label{eq:keythermalobstruction}
\end{equation}
The minimum follows by maximizing the one-particle probability
at \(q=1/(g+1)\). Thus this construction does not claim uniformly
small thermal deviations for all emitted blocks.

\paragraph{Conservation and symmetry scope.}
The logical operations act inside a degenerate fixed-charge code.
No claim is made about preserving coherent superpositions of arbitrary
different charge sectors with the same rules.
Conservation can impose additional recovery constraints in scrambling
models and symmetry-covariant codes
\cite{NakataWakakuwaKoashi2023,FaistEtAl2020}.
Those constraints are not removed by assuming sufficient degeneracy
without disclosing it.

\section{Small corrections: the changed assumption}
\label{sec:smallcorrections}

Outgoing number statistics, a one-block marginal, a near-horizon pair
state, and local semiclassical dynamics are four different tests.
The construction proves only its explicitly stated marginal and
channel results.

A useful entropy form of the pair-production obstruction is elementary.

\begin{proposition}[Pair-entropy obstruction]
\label{prop:pairentropy}
Let \(R\) be previously emitted radiation, \(b\) the next outgoing
system, and \(c\) its declared inaccessible partner.
Suppose the actual post-step state satisfies
\begin{equation}
H(bc)\le\eta,
\qquad
H(c)\ge h-\eta.
\label{eq:pairhypotheses}
\end{equation}
Then
\begin{equation}
H(Rb)-H(R)\ge h-2\eta.
\label{eq:pairgrowth}
\end{equation}
In particular, if \(h>2\eta\), radiation entropy cannot decrease
at that step.
\end{proposition}

\begin{proof}
Weak monotonicity, equivalent to strong subadditivity after
purification, gives
\[
H(Rb)+H(bc)\ge H(R)+H(c).
\]
Insert \eqref{eq:pairhypotheses}.
\end{proof}

This proposition states explicit entropy hypotheses.
Its parameter \(\eta\) is not identified with Mathur's
pair-amplitude correction parameter.
Mathur's small-corrections theorem establishes positive entropy growth
from a more specific near-Hawking pair assumption
\cite{Mathur2009,MathurPlumberg2011}.
Thermal-looking outgoing marginals alone do not provide those
hypotheses.

\begin{table}[ht]
\small
\caption{The four distinct comparisons for the constructed process.}
\label{tab:paircomparison}
\begin{tabularx}{\linewidth}{@{}p{0.29\linewidth}X@{}}
\toprule
Comparison & Result or explicit boundary\\
\midrule
Outgoing number statistics
& Vacuum/one-particle probabilities are calculated exactly.
Selected thermal comparisons have
\eqref{eq:fullthermaldeviation}, not an assumed zero error.\\
Outgoing one-block marginal
& Every key or attempt block is exactly message-independent.
This does not make their joint state independent.\\
Near-horizon \(bc\) pair state
& No near-horizon pair identification or norm-closeness theorem
is supplied. The logical-transfer map replaces a fresh-pair
production rule.\\
Local semiclassical evolution
& No local curved-spacetime interaction deriving
\eqref{eq:attemptmap} is supplied.
Its coupling of residual logical state to an outgoing sector
is a phenomenological transfer assumption.\\
Endpoint
& The residual is transferred by a separate completion.
It is not evolved through the terminal regime using the
fixed-background source law.\\
\bottomrule
\end{tabularx}
\end{table}

The model therefore does not evade the small-corrections theorem while
retaining its assumptions. It changes the emission rule and does not
assert small corrections to the full Hawking pair structure.
At an entropy-decreasing endpoint, the positive-growth hypotheses of
Proposition~\ref{prop:pairentropy} cannot all hold.

\section{Entropy-history separation with residual memory disclosed}
\label{sec:entropyseparation}

We now construct the counterexample rather than inferring it from
similarity of spectra.

Let the preserving evolution above be denoted \(G\).
Fix a pure code preparation \(\ket0\).
Construct a second evolution \(L\) as follows:
\begin{enumerate}
\item retain the variable input \(\ket\psi_A\) in a separate
inaccessible degenerate memory \(F\);
\item run the entire \(G\) sequence on an internal dummy code fixed
initially to \(\ket0\);
\item emit the dummy sequence's residual through the same endpoint
radiation embedding;
\item retain the variable memory \(F\) at the endpoint.
\end{enumerate}
The dummy code and all other ancillaries are fixed pure preparations,
not additional variable inputs.
Their common energies are included in the initial mass ledger.

Choose the endpoint memory's energy sector to have energy \(m_*\)
and degeneracy at least \(d\).
The preserving process occupies one fixed vector of this sector;
the retaining process occupies the logical input in it.
Thus the two processes can have the same macroscopic energy ledger
while having different complementary channels.

\begin{theorem}[Entropy histories do not certify code preservation]
\label{thm:entropyseparation}
There are conserving sequential evolutions \(G,L\) with the following
properties.
For every prefix including the endpoint,
\begin{equation}
\NN_t^{L}(Y)
=
\Tr(Y)\NN_t^{G}(\kb00).
\label{eq:retainingchannel}
\end{equation}
Consequently their full accessible radiation density operators agree
for input \(\kb00\).

Their scalar radiation entropy histories agree for every pure
code input, although their radiation density operators need not agree
for those other inputs.
At the endpoint,
\begin{equation}
\erec_G(*)=0,
\qquad
\erec_L(*)=\cdim,
\qquad
\dleak_L(*)=\cdim.
\label{eq:separationerrors}
\end{equation}
The nonpreserving accessible channel retains the input in an
inaccessible \(d\)-dimensional memory; it does not destroy it globally.
\end{theorem}

\begin{proof}
The retaining evolution acts as identity on its variable memory and
as the preserving sequence on its fixed dummy preparation.
Tracing the memory gives \eqref{eq:retainingchannel}.
All maps remain isometries, and the variable code labels have equal
energy, so the declared energy accounting is unchanged.

For the preserving sequence, Theorem~\ref{thm:information} shows that
the radiation entropy depends on the input only through \(H(\rho)\).
Every pure code input has \(H(\rho)=0\), so its pre-endpoint entropy is
\[
H(\bm b_t)+2\log d.
\]
The endpoint radiation is an isometric encoding and is pure for
every pure input. Its entropy is therefore zero.
The retaining sequence emits the fixed dummy history, giving exactly
the same scalar values.

At the endpoint, the retaining radiation is constant and its
complement contains the original input isometrically.
A decoder on constant radiation can only prepare a fixed state.
Lemma~\ref{lem:depolarization} gives optimal error \(\cdim\).
The same lemma and a left inverse for the retained input give
leakage distance \(\cdim\).
The preserving endpoint result is
Theorem~\ref{thm:endpoint}.
\end{proof}

\begin{remark}[Endpoint distinction is essential]
For \(m_*>0\), \(L\) is an information-bearing remnant model.
For \(m_*=0\), it postulates a degenerate zero-energy inaccessible
memory, rather than a unique vacuum.
If such memory is forbidden and all outputs are accessible,
the retaining process is excluded by
Proposition~\ref{prop:endpointtest}.
The counterexample does not assert different recoverability under
identical complete-evaporation and complete-access assumptions.
\end{remark}

On the Bell input, the models are immediately distinguishable at the
channel level: the preserving process has
\(I(S:R_{\mathrm{all}})=2\log d\), whereas the retaining process has
\(I(S:R_{\mathrm{all}})=0\) and \(I(S:F)=2\log d\).
Mixtures and reference-entangled inputs contain information absent
from a list of pure-input scalar entropies.

Equal entropy histories are also weaker than equal complete record
laws, and equal selected records are weaker than equal residual
instruments. These distinctions parallel the complete-process and
record interfaces in Psy02-B and En03-A
\cite{CheungPsy02B,CheungEn03A}.

\section{Controlled island comparison without a reconstruction overclaim}
\label{sec:islands}

The preceding finite counterexample concerns scalar entropy data
without additional structure. It is not a counterexample to the island
prescription.

Replica-wormhole calculations supply additional gravitational saddles
and derive island entropy formulas in controlled settings.
Quantum-extremal-surface and entanglement-wedge analyses additionally
address reconstruction and its code-space dependence
\cite{AlmheiriEtAl2020,Penington2020,EngelhardtWall2015}.
The replica-wormhole work of Penington, Shenker, Stanford, and Yang
also gives a direct gravitational reconstruction argument using a
Petz-map formula \cite{PeningtonEtAl2022}.
These advances are more than a fit to one scalar entropy curve.

For a concrete comparison, select the evaporating JT-plus-bath setting
of Almheiri, Engelhardt, Marolf, and Maxfield
\cite{AlmheiriEngelhardtMarolfMaxfield2019}.
Table~\ref{tab:islandaudit} records what must be extracted or added
before making the particular finite-code diamond-norm statement
of this paper. ``Not derived here'' is a boundary of this comparison,
not a claim that the cited literature lacks reconstruction results.

\begin{table}[ht]
\small
\caption{Operational audit for the selected controlled-gravity comparison.}
\label{tab:islandaudit}
\begin{tabularx}{\linewidth}{@{}p{0.30\linewidth}X@{}}
\toprule
Required operational datum & Treatment in this comparison\\
\midrule
Logical code and allowed inputs
& A finite code, conserved-sector restriction, and fixed ancillary
preparation must be selected. They are not identified with the
\(d\)-dimensional code of the constructed model.\\
Radiation algebra and access
& Specify the bath region, regulator, and observer algebra
in the JT-plus-bath setting. Do not replace them by an arbitrary
tensor factor without justification.\\
Reference correlations
& Introduce a reference to that selected code and determine the
corresponding channel or complementary leakage.
No such calculation is supplied by the finite example here.\\
QES and entanglement wedge
& Use the gravitational model's stated entropy and wedge results.
Their geometric and holographic assumptions remain explicit.\\
Uniform reconstruction
& Determine whether one recovery map works on every allowed code
state, and what uniform error the reconstruction argument supplies.\\
Decoder and diamond error
& A specific channel or reconstruction map must be supplied and
its finite-code error bounded. Not derived here.\\
Four-dimensional transfer
& No identification with neutral Schwarzschild evaporation,
its endpoint, or its microscopic interaction law is made.\\
\bottomrule
\end{tabularx}
\end{table}

For example, once a channel \(\NN:A\to Z\) and a full-rank code state
\(\sigma\) are supplied, a Petz candidate can be written as
\begin{equation}
\begin{split}
\DD_{\sigma,\NN}(Y)
={}&
\sigma^{1/2}
\NN^\dagger\!\left[
\NN(\sigma)^{-1/2}Y\NN(\sigma)^{-1/2}
\right]\sigma^{1/2}\\
&+\Tr((I-P)Y)\sigma,
\end{split}
\label{eq:Petz}
\end{equation}
where inverse powers are on
\(P=\supp\NN(\sigma)\).
The support-completion term makes this a CPTP map on the whole output
space: the trace of the first term is \(\Tr(PY)\).

Writing \eqref{eq:Petz} does not prove that it corrects a given
evaporation code. One must evaluate or bound
\(\Ddia(\DD_{\sigma,\NN}\NN,\id_A)\).
A scalar entropy history supplies neither the channel on every matrix
unit nor its adjoint, and therefore does not by itself specify this
calculation.

The operational gap addressed here is consequently narrow:
\emph{an entropy history alone is not a code-wide channel certificate}.
It does not deny reconstruction conclusions derived from entropy
together with additional gravitational, algebraic, and code-space
structure. Page-curve expectations likewise remain valuable
unitary-evaporation diagnostics without being substituted for an
arbitrary-input decoder error \cite{Page2013}.

\section{Observer access: coherence, destructive measurement, and loss}
\label{sec:access}

For a complete collection and storage channel \(\mathcal A_t\),
\begin{equation}
\NN_t^{\mathrm{obs}}=\mathcal A_t\NN_t.
\label{eq:observerchannel}
\end{equation}
Its complete complement includes the collection environment and every
excluded mode.

\begin{proposition}[Access monotonicity]
\label{prop:accessmonotonicity}
Message-independent CPTP restriction of the accessible output cannot
improve unrestricted recovery:
\begin{equation}
\erec(\mathcal A_t\NN_t)\ge\erec(\NN_t).
\label{eq:accessmonotonicity}
\end{equation}
For consistent coherent radiation prefixes,
\(\erec(t+1)\le\erec(t)\).
\end{proposition}

\begin{proof}
Every decoder after \(\mathcal A_t\) is a decoder for the original
output by composition. For prefixes, a later decoder may discard the
new block and use an earlier optimal decoder.
\end{proof}

Storage degradation can violate the prefix premise.
Information-theoretic access and executable coherent recovery are
separate resource questions, as in En02-B--D
\cite{CheungEn02B,CheungEn02C,CheungEn02D}.

\subsection{Destructive occupation measurements}

Let all emitted blocks be measured destructively in their fixed
occupation/rail bases. The complete record retains key, transfer time,
vacuum, and saturation or loss flags. No selected branch is omitted.

Define logical dephasing
\begin{equation}
\Delta_Z(Y)=\sum_{i=0}^{d-1}\kb iiY\kb ii.
\label{eq:dephasing}
\end{equation}

\begin{theorem}[Exact number-record recovery obstruction]
\label{thm:number}
For the pre-endpoint construction with only those complete classical
number records accessible,
\begin{align}
\sup_{\DD}F_e(\DD\NN_t^{\mathrm{num}})
&=\frac{p_t}{d}+\frac{s_t}{d^2},
\label{eq:numberfidelity}\\
\boxed{
\erec^{\mathrm{num}}(t)
=
1-\frac{p_t}{d}-\frac{s_t}{d^2}.
}
\label{eq:numbererror}
\end{align}
After the endpoint, with all endpoint rails also measured,
\begin{equation}
\boxed{\erec^{\mathrm{num}}(*)=1-\frac1d.}
\label{eq:numberendpoint}
\end{equation}
Thus complete evaporation in the model does not make an ordinary
destructive detector a quantum-code decoder.
\end{theorem}

\begin{proof}
On a completed branch, the key identifies the cyclic shift in
\(U_r=X^aZ^b\). Correcting that shift in the classical number outcome
leaves a measurement of the original logical computational basis.
Its channel is \(\Delta_Z\).
Pending branches remain constant.
Preparing the measured logical basis state and preparing \(I_d/d\)
on pending gives
\begin{equation}
\Lambda_t^{\mathrm{num}}
=p_t\Delta_Z+s_t\TT_d.
\label{eq:numberdecoded}
\end{equation}

Every completed branch is entanglement breaking.
A separable state of a \(d\)-dimensional reference and logical output
has Bell overlap at most \(1/d\): the bound holds for pure product
states by Cauchy--Schwarz and then by convexity.
A constant pending branch has Bell overlap \(1/d^2\).
This proves the upper bound in \eqref{eq:numberfidelity}.

Now
\[
\Delta_Z=\frac1d\sum_{b=0}^{d-1}\Ad_{Z^b},
\qquad
\TT_d=\frac1{d^2}\sum_r\Ad_{U_r}.
\]
The identity Weyl weight in
\eqref{eq:numberdecoded} is \(p_t/d+s_t/d^2\).
Orthogonal Bell outputs give error at least one minus that weight.
The triangle inequality over the remaining unitary channels gives
the matching upper bound. This proves optimality.

At the endpoint, pending encrypted logical systems are also measured
in a known-key computational basis. Every branch therefore gives
\(\Delta_Z\), proving \eqref{eq:numberendpoint}.
\end{proof}

This is a channel-level version of the distinction between destructive
records and coherent extraction in Em05-A/B and En03-A
\cite{CheungEm05A,CheungEm05B,CheungEn03A}.
Counting all emitted particles does not restore the lost phase
information.

\subsection{A declared whole-output erasure model}

Suppose the entire accessible prefix is transmitted intact with
message-independent probability \(\eta\), otherwise replaced by a
flagged input-independent state:
\begin{equation}
\mathcal E_\eta(Y)
=
\eta Y\oplus(1-\eta)\Tr(Y)\kb{\perp}{\perp}.
\label{eq:wholeerasure}
\end{equation}

\begin{proposition}[Exact erasure optimum]
\label{prop:erasure}
For the constructed prefix,
\begin{equation}
\erec(\mathcal E_\eta\NN_t)
=(1-\eta p_t)\cdim.
\label{eq:erasureoptimum}
\end{equation}
At the endpoint this becomes
\((1-\eta)\cdim\).
\end{proposition}

\begin{proof}
An intact completed branch has total weight \(\eta p_t\) and permits
perfect recovery. Every other accessible branch is constant.
The branchwise Bell bound and decoder of
Theorem~\ref{thm:optima} apply with success probability \(\eta p_t\).
\end{proof}

Equation~\eqref{eq:wholeerasure} is not a general aperture, bosonic
loss, or detector-efficiency model.
Mode-dependent propagation, thermal background, partial collection,
and storage noise require their actual multimode channels.
The corresponding loss environments must remain in the complement.

\section{Regulators, implementation errors, and certificates}
\label{sec:robustness}

\begin{proposition}[Lipschitz error propagation]
\label{prop:robustness}
For channels on common interfaces,
\begin{align}
\abs{\erec(\NN)-\erec(\widetilde\NN)}
&\le\Ddia(\NN,\widetilde\NN),
\label{eq:recLipschitz}\\
\abs{\dleak(\CC)-\dleak(\widetilde\CC)}
&\le\Ddia(\CC,\widetilde\CC).
\label{eq:leakLipschitz}
\end{align}
For composable CPTP maps,
\begin{equation}
\Ddia(\Phi_n\cdots\Phi_1,
\widetilde\Phi_n\cdots\widetilde\Phi_1)
\le
\sum_{k=1}^{n}\Ddia(\Phi_k,\widetilde\Phi_k).
\label{eq:telescoping}
\end{equation}
\end{proposition}

\begin{proof}
For any decoder, apply the triangle inequality and contractivity
under decoder composition, then optimize and exchange the channels.
For leakage use the same triangle inequality before optimizing the
constant state.
Expand the sequential difference as a telescoping sum and use
contractivity of the diamond norm under CPTP pre- and
post-composition.
\end{proof}

Preparation, mode restriction, occupation cutoff, collection, storage,
and decoder errors must be stated on common channels or referenced
states. A classical metric or source-input norm is not automatically
a diamond allowance.

If \(W,\widetilde W\) are complete aligned isometries and
\(\norm{W-\widetilde W}_\infty\le a\), then both accessible and
complementary half-diamond errors are at most \(a\).
This follows from the right inequality of
\eqref{eq:KSWraw}.
A bound on the accessible channel alone does not justify reusing that
same numerical bound for an arbitrarily chosen complementary
dilation.

\subsection{Complete occupation cutoffs}

For a finite collection of bosonic modes, let \(P_L\) project onto
total occupation at most \(L\), and define the trace-preserving
overflow regulator
\begin{equation}
\RR_L(\rho)
=
P_L\rho P_L+
\Tr((I-P_L)\rho)\kb{\perp}{\perp}.
\label{eq:cutoff}
\end{equation}
With an untouched reference, the overflow branch retains its
corresponding reference reduction.

\begin{proposition}[Uniform referenced cutoff allowance]
\label{prop:cutoff}
If a complete output family satisfies
\[
\sup_\rho\Tr((I-P_L)\NN(\rho))\le\nu,
\]
then, after embedding the original output in the enlarged flag space,
\begin{equation}
\Ddia(\NN,\RR_L\NN)
\le
\min\{1,\sqrt{\nu}+\nu/2\}.
\label{eq:cutoffdiamond}
\end{equation}
A uniform total-number bound
\(\sup_\rho\langle N_{\mathrm{tot}}\rangle\le\mu\)
implies
\begin{equation}
\nu\le\min\{1,\mu/(L+1)\}.
\label{eq:cutoffMarkov}
\end{equation}
For number-block-diagonal output states, the state trace-distance
cutoff error equals their discarded probability.
\end{proposition}

\begin{proof}
For a purification with discarded probability \(\delta\), direct
two-component diagonalization gives
\[
\norm{\kb\psi\psi-P_L\kb\psi\psi P_L}_1
=\sqrt{4\delta-3\delta^2}\le2\sqrt{\delta}.
\]
Partial trace is contractive. Adding the overflow block contributes
trace norm \(\delta\), proving the referenced bound.
The same discarded probability is determined by the input marginal,
so the uniform assumption holds for reference-entangled inputs.
The channel supremum therefore gives
\eqref{eq:cutoffdiamond}.

The operator inequality
\(N_{\mathrm{tot}}\ge(L+1)(I-P_L)\) proves
\eqref{eq:cutoffMarkov}.
For block-diagonal states, removal and overflow insertion occur in
orthogonal supports of equal weight, giving exact distance \(\delta\).
\end{proof}

This is the flagged regulator interface used in Hawk01-A and
En02-E \cite{CheungHawkA,CheungEn02E}.
Its quantum square-root allowance is not replaced by a classical
tail identity unless commutation is established.
Occupation bounds also do not certify discarded angular or frequency
sectors.

In the explicit finite construction, key, transfer, and endpoint
radiation contain at most three particles in total. Its stated
finite occupation regulator is therefore exact once \(L\ge3\).
The required degeneracies and endpoint energies, rather than an
unbounded occupation tail, carry its finite-dimensional resource cost.

\subsection{Using leakage to certify a perturbed observer}

For an approximate complete observer model, suppose a chosen constant
state has a certified complementary distance at most \(a\), and the
difference between the modeled and physical complete complement is
at most \(b\). Then
\begin{equation}
\erec^{\mathrm{physical}}
\le
\min\{1,\sqrt{2(a+b)}\}.
\label{eq:perturbedleakcertificate}
\end{equation}
If instead an explicit implemented decoder has modeled error \(e\)
and the complete accessible-channel allowance is \(r\), then
\begin{equation}
\Ddia(\DD_{\mathrm{impl}}\NN_{\mathrm{physical}},\id_A)
\le e+r,
\label{eq:implementedcertificate}
\end{equation}
with any additional decoder implementation error included in \(e\)
or \(r\).
These certify different resources: the first proves unrestricted
existence, while the second certifies a specified implementation.

\section{An independently checkable small example}
\label{sec:example}

Choose
\begin{equation}
d=2,\quad N=3,\quad
\lambda_2=\lambda_3=\frac12,\quad
e_{\mathrm K}=e=1,\quad M=5,\quad
M_{\mathrm{cut}}=3,\quad m_*=0.
\label{eq:exampleparameters}
\end{equation}
The units here are model energy units, not astrophysical mass or
temperature assignments. Then
\[
w_2=\frac12,\qquad
w_3=\frac14,\qquad
s_3=\frac14,\qquad
\cdim=\frac34.
\]
The endpoint radiation has eight completed-state rails of energy
three and eight pending-state rails of energy four.
The residual source becomes one vacuum state.

\begin{table}[ht]
\centering
\small
\caption{Exact model outputs. Entropies and mutual informations are in
bits. \(H_{\mathrm{pure}}\) uses a pure code input;
\(H_\Phi\) uses the maximally entangled reference input.
At \(t=0\), \(R\) is empty.}
\label{tab:exactexample}
\begin{tabular}{@{}cccccccc@{}}
\toprule
Step & \(p_t\) & \(H_{\mathrm{pure}}(R)\) &
\(H_\Phi(R)\) & \(I(S:R)\) & \(I(S:B)\) &
\(\erec\) & \(\dleak\)\\
\midrule
\(0\) & \(0\) & \(0\) & \(0\) & \(0\) & \(2\) &
\(3/4\) & \(3/4\)\\
\(1\) & \(0\) & \(2\) & \(2\) & \(0\) & \(2\) &
\(3/4\) & \(3/4\)\\
\(2\) & \(1/2\) & \(3\) & \(7/2\) & \(1\) & \(1\) &
\(3/8\) & \(3/8\)\\
\(3\) & \(3/4\) & \(7/2\) & \(17/4\) & \(3/2\) & \(1/2\) &
\(3/16\) & \(3/16\)\\
Endpoint & \(1\) & \(0\) & \(1\) & \(2\) & \(0\) &
\(0\) & \(0\)\\
\bottomrule
\end{tabular}
\end{table}

The Bell-input optimal fidelities are, in order,
\[
\frac14,\quad\frac14,\quad\frac58,\quad\frac{13}{16},
\quad1.
\]
The conditional reference-information increments are zero for the key,
one bit for attempt two, one-half bit for attempt three, and one-half
bit for the endpoint.

The exact mean residual energies are
\[
5,\quad4,\quad\frac72,\quad\frac{13}{4},\quad0,
\]
with the corresponding emitted energy giving total five in every
branch, not merely on average.

Destructive number-record recovery gives
\[
\erec^{\mathrm{num}}(1)=\frac34,\qquad
\erec^{\mathrm{num}}(2)=\frac58,\qquad
\erec^{\mathrm{num}}(3)=\frac9{16},\qquad
\erec^{\mathrm{num}}(*)=\frac12.
\]
These values differ from the coherent-access column in
Table~\ref{tab:exactexample}.

\paragraph{Formal thermal comparison.}
To match the vacuum/one-particle restrictions of the two attempt
marginals, solve \(a_2(q)=w_j\). This gives
\[
q_2=\frac12,\qquad q_3=\frac16.
\]
The omitted product-thermal probabilities are
\[
\theta_2(q_2)=\frac12,\qquad
\theta_2(q_3)=\frac2{27}.
\]
Thus even this simple calibration exposes a large thermal discrepancy
at the first attempt. These \(q\) values are formal comparison
parameters, not claims that a Schwarzschild greybody calculation
produces them.
For the four-rail key block,
\eqref{eq:keythermalobstruction} gives minimal product-thermal distance
\[
1-\left(\frac45\right)^5
=\frac{2101}{3125}.
\]

\subsection{Exact matrix-entry recipe}

The transition maps themselves are the archive needed to reproduce
the example. For general \(d\),
\begin{equation}
(U_{a,b})_{ki}
=
\zeta^{bi}\mathbf 1\{k=i+a\bmod d\}.
\label{eq:matrixentries}
\end{equation}
At prefix \(t\), nonzero entries of \(W_t\) are:
\begin{align}
\bra{j,r}_{B_t}\bra{r,j,k}_{R}
W_t\ket i_A
&=\frac{\sqrt{w_j}}{d}(U_r)_{ki},
\label{eq:successentries}\\
\bra{\pend,r,k}_{B_t}\bra{r,\vac}_{R}
W_t\ket i_A
&=\frac{\sqrt{s_t}}{d}(U_r)_{ki}.
\label{eq:failureentries}
\end{align}
Every omitted entry is zero. The radiation labels in these equations
include the all-other-vacuum pattern.
The endpoint relocates the residual basis index into the endpoint
radiation factor exactly as in
\eqref{eq:endpointsuccess}--\eqref{eq:endpointfailure}.

An independent implementation can verify:
\begin{enumerate}
\item \(W_t^\dagger W_t=I_d\);
\item
\((H_{B_t}+H_{R_{\le t}})W_t=MW_t\);
\item the corresponding charge-intertwining identities;
\item the two partial traces
\eqref{eq:accessibleprefix} and \eqref{eq:complementprefix};
\item the decoder identity
\eqref{eq:decodedprefix};
\item the entropy eigenvalues used in
Theorem~\ref{thm:information};
\item the endpoint constant complement.
\end{enumerate}
All table values are consequences of these exact formulas, not
unreported numerical experiments.

\section{Assessment and reproducibility boundary}
\label{sec:assessment}

A laboratory or simulated verification of a specified decoder can
use independent preparations of the Bell input and the success
variable
\[
Y_i=\mathbf 1\{\text{Bell projector accepts on attempt }i\}.
\]
With \(n\) independent complete attempts,
\(\widehat F_e=n^{-1}\sum_iY_i\), Hoeffding's inequality gives
\begin{equation}
\Pr\left(\abs{\widehat F_e-F_e}>a\right)
\le2e^{-2na^2}.
\label{eq:BellSampling}
\end{equation}
This certifies a Bell-input diagnostic.
A worst-case channel claim additionally uses
\eqref{eq:Bellconversion}, full process data, or another
uniform channel certificate.

For complete-record tomography with a finite alphabet of size \(K\),
coordinatewise Hoeffding and a union bound give, with probability
at least \(1-\alpha\),
\begin{equation}
\TV(\widehat P,P)
\le
\min\left\{
1,\frac K2
\sqrt{\frac{\log(2K/\alpha)}{2n}}
\right\}.
\label{eq:recordsampling}
\end{equation}
Complete-record laws and their residual instruments remain distinct,
as emphasized in En03-A and Psy02-B
\cite{CheungEn03A,CheungPsy02B}.
Repeated records inside one correlated attempt do not increase \(n\).

If normalized Choi states have a certified trace-distance discrepancy
at most \(a_J\), then
\eqref{eq:Choiconversion} supplies channel discrepancy at most
\(da_J\), and Proposition~\ref{prop:robustness} propagates that
allowance to the recovery optimum.

Repeated injections into one evolving source are not automatically
independent. Shared residual memory, changing energy, and overlapping
radiation require a history-conditional assessment model.
Simulation of independent initial preparations is not observation
of an astrophysical black hole.

Passive source observation also does not ordinarily supply a prepared
logical input, its reference, or a controlled channel-tomography
interface. Replacing a classical observation device by a quantum one
does not create those absent access channels
\cite{CheungPsy03D}.
The task-specific numerical bounds of that manuscript are not
imported here.

A reproducibility record for any extension of the model should include:
\begin{enumerate}
\item code dimension, conserved sector, and allowed input states;
\item all initially fixed states and their purifiers;
\item every transition map, Hamiltonian, and charge operator;
\item radiation mode and occupation regulators;
\item pending, completed, vacuum, loss, and endpoint branches;
\item the complete observer channel and its excluded environment;
\item decoder and quantum/classical storage requirements;
\item exact or certified numerical norm and entropy errors;
\item the endpoint resource ledger;
\item sampling independence and validation-selection assumptions.
\end{enumerate}

\section{Conclusions and mathematical release}
\label{sec:conclusion}

\begin{theorem}[Hawk01-B release]
\label{thm:release}
Within the declared conserving model:
\begin{enumerate}
\item arbitrary-input, reference-preserving recovery is exactly
equivalent to a constant complete complement;
\item the KSW framework gives
\(\erec\le\sqrt{2\dleak}\) and
\(\dleak\le\sqrt{2\erec}\) in half-diamond normalization;
\item the maps \(V_1,\ldots,V_N\) form a conserving sequential
process with internally generated key, no reset, and explicit
vacuum and pending branches;
\item every individual pre-endpoint emitted block is
message-independent, but the joint radiation supports optimal
recovery error \(s_t(1-d^{-2})\);
\item the complete residual leakage distance is exactly the same
quantity, with reference-information transfer
\(I(S:R_{\le t})=2p_t\log d\);
\item the declared information-free endpoint releases every residual
degree of freedom and permits exact recovery;
\item complete destructive occupation access instead has the
obstruction and exact optima of
Theorem~\ref{thm:number};
\item a second conserving process has identical radiation density
operators for one chosen preparation, and identical radiation
entropy histories for every pure preparation, but retains the
code in a disclosed inaccessible endpoint memory.
\end{enumerate}
None of these statements derives the microscopic information-transfer
dynamics or terminal evolution of a realistic black hole.
\end{theorem}

\begin{proof}
Items (i)--(ii) are Theorems~\ref{thm:exact} and
\ref{thm:approximate}.
Item (iii) is Theorem~\ref{thm:physicality}.
Items (iv)--(v) follow from
Theorems~\ref{thm:prefixchannels}, \ref{thm:optima},
and \ref{thm:information}.
Item (vi) is Theorem~\ref{thm:endpoint}.
Item (vii) is Theorem~\ref{thm:number}.
Item (viii) is Theorem~\ref{thm:entropyseparation}.
The coupling, degeneracies, and endpoint remain explicit model
inputs, so no microscopic gravitational derivation follows.
\end{proof}

The preservation certificate is residual decoupling on the entire
logical operator space. A spectrum, local marginal, entropy turnover,
or delayed-query advantage is not substituted for that certificate.

The construction answers where information is located at each stage:
initially in the pending residual with its generated key; subsequently
in key-conditioned joint radiation on completed branches; and, under
the specified endpoint, entirely in radiation for every branch.
It also identifies precisely what must change to obtain a
nonpreserving accessible channel: a message-bearing residual or
excluded output must remain.

The entropy-history counterexample is therefore a separation of data
and channel claims, not a rejection of islands.
Likewise the thermal comparison is a calibration and compatibility
audit, not a claim to reproduce the Hawking pair state or local
semiclassical dynamics.

The defensible physical conclusion is:
\begin{quote}
Within the declared conserving model, the identified joint radiation
correlations and complete residual-decoupling conditions suffice for
preservation of an arbitrary quantum code. Spectrum or specified
entropy-history data alone do not certify that preservation.
The model does not derive the microscopic dynamics of realistic
black-hole evaporation.
\end{quote}

\appendix

\section{Source provenance and non-transfer restrictions}
\label{app:provenance}

The Cheung references below are identified by the supplied manuscript
codes and versions. Their publication dates and canonical article
titles were not supplied and are not invented.
They should be made available to reviewers in their cited versions.

\begin{table}[ht]
\small
\caption{Source interfaces used for comparison, not transferred
black-hole premises.}
\label{tab:provenance}
\begin{tabularx}{\linewidth}{@{}p{0.24\linewidth}X@{}}
\toprule
Source & Restricted role\\
\midrule
Hawk01-A; Em05-B
& Exact complement criterion, reference-preserving metrics,
and spectrum/recovery nonidentifiability.\\
Em03-E
& Sequential key/transfer and complete complementary-output
bookkeeping. Its finite source is not Hawking dynamics.\\
Em05-A/B
& Occupation transfer, logical coherence, reference information,
and recovery are separate diagnostics.\\
En02-B--D
& Information-theoretic recovery, executable decoder resources,
and observable readout are separate tasks.\\
En02-E
& Finite-packet covariance and flagged occupation cutoff.
A regulator is not an exact gravitational tensor decomposition.\\
En03-A; Psy02-B
& Complete records, residual instruments, process equality,
and independent-attempt sampling.\\
Psy03-D
& Missing observation channels are not supplied merely by
changing the carrier. Its separate numerical task is not this one.\\
Em03-A
& Fixed-background source normalization, not the logical
transfer schedule or endpoint.\\
\bottomrule
\end{tabularx}
\end{table}

Hawk01-B does not identify its finite channels with the separate
characteristic-distribution programme described in the supplied
XIV-I/J excerpts. Those excerpts retain an unresolved full joint
characteristic map and do not admit the specified quantum coefficient
\cite{CheungXIVI,CheungXIVJ}.
Finite isometries, complementary channels, and decoder identities here
do not construct that missing operation or open its downstream
admission gate.

Similarly, a quotient or relative-lifting criterion only constrains a
candidate map after that map and its domain have been defined.
It is not a prescription for an unresolved distributional product
\cite{CheungXVB}.
No interacting quantum Ward identity, characteristic coefficient
admission, or completion of those separate analytic obligations is
asserted by this paper.

\section{Normalization and debugging audit}
\label{app:debug}

The following safeguards are incorporated into the statements and proofs.

\begin{enumerate}
\item \textbf{Half-diamond constants.}
KSW uses a raw stabilized norm.
The substitution
\(\norm{\CC-\TT}_{\diamond}=2\dleak\)
and the output half-norm give \(\sqrt{2\dleak}\), not an
unjustified different square-root coefficient.

\item \textbf{Complete complement.}
Key purifiers, pending code, excluded radiation, and detector
environments are included.
Visible-source mass alone does not characterize \(F_t\).

\item \textbf{Internal resource generation.}
The first isometry creates the key correlation from a fixed
preparation. No early-radiation assistance is silently supplied.

\item \textbf{Sequential consistency.}
Every \(V_t\) acts on the actual previous residual.
Previously completed branches emit vacuum; pending branches
retain their state when transfer fails.

\item \textbf{Energy and dimensions.}
Energy conservation is branchwise.
The degeneracy and source-dimension ledger are stated explicitly.
They are not inferred from an astrophysical entropy formula.

\item \textbf{Endpoint accounting.}
The main endpoint emits all residual basis information.
The retaining counterexample explicitly keeps a logical memory
and therefore has a different endpoint condition.

\item \textbf{Entropy-input distinction.}
Pure-input radiation entropy histories can coincide while the
channels differ.
Reference and mixed-input data distinguish the examples.

\item \textbf{Marginal compatibility.}
One transfer excitation across the attempt sequence imposes
\(\sum_jw_j\le1\).
Arbitrary independent exact thermal weights are not promised.

\item \textbf{Thermal versus near-horizon state.}
The local trace-distance calibration is not a proof of
near-Hawking pair creation or local semiclassical coupling.

\item \textbf{Coherent versus destructive access.}
Reading the key or the occupied-sector flag need not measure the
logical basis. Reading every occupation rail does, and has a
different exact optimum.

\item \textbf{Single input versus worst case.}
Approximate Bell fidelity and mutual information use explicit
dimension-dependent conversions before any diamond claim.

\item \textbf{Regulator completeness.}
Overflow is flagged, not postselected.
A general quantum cutoff uses its referenced projection allowance.

\item \textbf{Reproducibility.}
The small example is specified by exact matrix entries and rational
parameters. No unperformed numerical solver or compilation run is
claimed.

\item \textbf{Scope.}
Neither the sequential model nor the entropy counterexample
disproves island reconstruction or solves realistic evaporation.
\end{enumerate}

\begin{thebibliography}{99}

\bibitem{KSW2006}
D.~Kretschmann, D.~Schlingemann, and R.~F.~Werner,
``The Information-Disturbance Tradeoff and the Continuity of
Stinespring's Representation'' (2006).
arXiv:\href{https://arxiv.org/abs/quant-ph/0605009}
{quant-ph/0605009}.
Theorems 1 and 3, with their completely bounded norm conventions.

\bibitem{BenyOreshkov2010}
C.~B\'eny and O.~Oreshkov,
``General conditions for approximate quantum error correction
and near-optimal recovery channels,''
\emph{Physical Review Letters} \textbf{104}, 120501 (2010).
doi:\href{https://doi.org/10.1103/PhysRevLett.104.120501}
{10.1103/PhysRevLett.104.120501}.
arXiv:\href{https://arxiv.org/abs/0907.5391}{0907.5391}.

\bibitem{BenyOreshkov2011}
C.~B\'eny and O.~Oreshkov,
``Approximate simulation of quantum channels'' (2011).
arXiv:\href{https://arxiv.org/abs/1103.0649}{1103.0649}.

\bibitem{KnillLaflamme1997}
E.~Knill and R.~Laflamme,
``Theory of quantum error-correcting codes,''
\emph{Physical Review A} \textbf{55}, 900--911 (1997).
doi:\href{https://doi.org/10.1103/PhysRevA.55.900}
{10.1103/PhysRevA.55.900}.
Preprint:
\href{https://arxiv.org/abs/quant-ph/9604034}
{arXiv:quant-ph/9604034}.

\bibitem{Watrous2018}
J.~Watrous,
\emph{The Theory of Quantum Information}
(Cambridge University Press, 2018).
Author-hosted manuscript and errata:
\url{https://cs.uwaterloo.ca/~watrous/TQI/}.

\bibitem{Hawking1975}
S.~W.~Hawking,
``Particle creation by black holes,''
\emph{Communications in Mathematical Physics}
\textbf{43}, 199--220 (1975).
doi:\href{https://doi.org/10.1007/BF02345020}
{10.1007/BF02345020}.

\bibitem{BraunsteinPati2007}
S.~L.~Braunstein and A.~K.~Pati,
``Quantum information cannot be completely hidden in correlations:
implications for the black-hole information paradox,''
\emph{Physical Review Letters} \textbf{98}, 080502 (2007).
doi:\href{https://doi.org/10.1103/PhysRevLett.98.080502}
{10.1103/PhysRevLett.98.080502}.
arXiv:\href{https://arxiv.org/abs/gr-qc/0603046}
{gr-qc/0603046}.

\bibitem{NakataWakakuwaKoashi2023}
Y.~Nakata, E.~Wakakuwa, and M.~Koashi,
``Black holes as clouded mirrors: the Hayden--Preskill protocol
with symmetry,''
\emph{Quantum} \textbf{7}, 928 (2023).
doi:\href{https://doi.org/10.22331/q-2023-02-21-928}
{10.22331/q-2023-02-21-928}.
arXiv:\href{https://arxiv.org/abs/2007.00895}{2007.00895}.

\bibitem{FaistEtAl2020}
P.~Faist, S.~Nezami, V.~V.~Albert, G.~Salton,
F.~Pastawski, P.~Hayden, and J.~Preskill,
``Continuous symmetries and approximate quantum error correction,''
\emph{Physical Review X} \textbf{10}, 041018 (2020).
doi:\href{https://doi.org/10.1103/PhysRevX.10.041018}
{10.1103/PhysRevX.10.041018}.
arXiv:\href{https://arxiv.org/abs/1902.07714}{1902.07714}.

\bibitem{Mathur2009}
S.~D.~Mathur,
``The information paradox: a pedagogical introduction,''
\emph{Classical and Quantum Gravity}
\textbf{26}, 224001 (2009).
doi:\href{https://doi.org/10.1088/0264-9381/26/22/224001}
{10.1088/0264-9381/26/22/224001}.
arXiv:\href{https://arxiv.org/abs/0909.1038}{0909.1038}.

\bibitem{MathurPlumberg2011}
S.~D.~Mathur and C.~Plumberg,
``Correlations in Hawking radiation and the infall problem'' (2011).
arXiv:\href{https://arxiv.org/abs/1101.4899}{1101.4899}.

\bibitem{AlmheiriEtAl2020}
A.~Almheiri, T.~Hartman, J.~Maldacena,
E.~Shaghoulian, and A.~Tajdini,
``Replica wormholes and the entropy of Hawking radiation,''
\emph{Journal of High Energy Physics} 05 (2020), 013.
doi:\href{https://doi.org/10.1007/JHEP05(2020)013}
{10.1007/JHEP05(2020)013}.
arXiv:\href{https://arxiv.org/abs/1911.12333}{1911.12333}.

\bibitem{Penington2020}
G.~Penington,
``Entanglement Wedge Reconstruction and the Information Paradox.''
arXiv:\href{https://arxiv.org/abs/1905.08255}{1905.08255},
version 3 (2020).

\bibitem{EngelhardtWall2015}
N.~Engelhardt and A.~C.~Wall,
``Quantum extremal surfaces: holographic entanglement entropy beyond
the classical regime,''
\emph{Journal of High Energy Physics} 01 (2015), 073.
doi:\href{https://doi.org/10.1007/JHEP01(2015)073}
{10.1007/JHEP01(2015)073}.
arXiv:\href{https://arxiv.org/abs/1408.3203}{1408.3203}.

\bibitem{PeningtonEtAl2022}
G.~Penington, S.~H.~Shenker, D.~Stanford, and Z.~Yang,
``Replica wormholes and the black hole interior.''
arXiv:\href{https://arxiv.org/abs/1911.11977}{1911.11977}.
The cited preprint includes the Petz-map reconstruction discussion.

\bibitem{AlmheiriEngelhardtMarolfMaxfield2019}
A.~Almheiri, N.~Engelhardt, D.~Marolf, and H.~Maxfield,
``The entropy of bulk quantum fields and the entanglement wedge
of an evaporating black hole,''
\emph{Journal of High Energy Physics} 12 (2019), 063.
doi:\href{https://doi.org/10.1007/JHEP12(2019)063}
{10.1007/JHEP12(2019)063}.
arXiv:\href{https://arxiv.org/abs/1905.08762}{1905.08762}.

\bibitem{Page2013}
D.~N.~Page,
``Time Dependence of Hawking Radiation Entropy,''
\emph{Journal of Cosmology and Astroparticle Physics}
09 (2013), 028.
arXiv:\href{https://arxiv.org/abs/1301.4995}{1301.4995}.

\bibitem{CheungHawkA}
N.~Cheung,
\emph{Hawk01-A}, version 1,
unpublished manuscript, n.d.
Supplied source: exact constant-complement criterion,
conserving completions, delayed-query optima, and complete
observer/regulator interfaces.
Labels include
\texttt{thm:complement},
\texttt{thm:completion},
and \texttt{thm:completionoptima}.
Source collection: Enter Calculus Companion.

\bibitem{CheungEm03A}
N.~Cheung,
\emph{Em03-A}, version 1,
unpublished manuscript, n.d.
Supplied source: fixed-background scalar emission law,
\texttt{thm:main}, and finite-packet source normalization.
Source collection: Enter Calculus Companion.

\bibitem{CheungEm03E}
N.~Cheung,
\emph{Em03-E}, version 1,
unpublished manuscript, n.d.
Supplied source descriptions: sequential key/transfer dynamics,
complete complement, leakage, and referenced recovery.
Supplied theorem excerpt: \texttt{thm:recovery}.
Source collection: Enter Calculus Companion.

\bibitem{CheungEm05A}
N.~Cheung,
\emph{Em05-A}, version 1,
unpublished manuscript, n.d.
Supplied source: signed extraction isometry and complete
complement, \texttt{thm:channels}; information and diagnostic
separation interfaces.
Source collection: Enter Calculus Companion.

\bibitem{CheungEm05B}
N.~Cheung,
\emph{Em05-B}, version 1,
unpublished manuscript, n.d.
Supplied source: exact constant-complement criterion,
\texttt{thm:constantComplement}, and Bell-projection
leakage witness, \texttt{lem:leakageBell}.
Source collection: Enter Calculus Companion.

\bibitem{CheungEn02B}
N.~Cheung,
\emph{En02-B}, version 1,
unpublished manuscript, n.d.
Supplied source: resource-certified decoder soundness,
\texttt{thm:B1}, and separation of observable, subsystem,
and instrument tasks.
Source collection: Enter Calculus Companion.

\bibitem{CheungEn02C}
N.~Cheung,
\emph{En02-C}, version 1,
unpublished manuscript, n.d.
Supplied source: finite sequential radiation recovery,
\texttt{thm:C2}, and executable-access distinctions.
Source collection: Enter Calculus Companion.

\bibitem{CheungEn02D}
N.~Cheung,
\emph{En02-D}, version 4,
unpublished manuscript, n.d.
Supplied source: energy-labelled unrestricted and
capacity-limited recovery optima,
\texttt{thm:benchmark}.
Source collection: Enter Calculus Companion.

\bibitem{CheungEn02E}
N.~Cheung,
\emph{En02-E}, version 2,
unpublished manuscript, n.d.
Supplied source descriptions: finite Gaussian packets,
complete occupation cutoffs, and calibrated recovery.
Supplied theorem excerpt: \texttt{thm:recovery}.
Source collection: Enter Calculus Companion.

\bibitem{CheungEn03A}
N.~Cheung,
\emph{En03-A}, version 1,
unpublished manuscript, n.d.
Supplied source: exact record compatibility,
\texttt{thm:pullback}; residual-instrument, stability,
and sampling interfaces.
Source collection: Enter Calculus Companion.

\bibitem{CheungPsy02B}
N.~Cheung,
\emph{Psy02-B}, version 2,
unpublished manuscript, n.d.
Supplied source: complete-process equality and testing
obstruction, \texttt{thm:process}.
Source collection: Enter Calculus Companion.

\bibitem{CheungPsy03D}
N.~Cheung,
\emph{Psy03-D}, version 1,
unpublished manuscript, n.d.
Supplied source: carrier-independent uncued observation
obstruction, \texttt{cor:qobstruction}.
Source collection: Enter Calculus Companion.

\bibitem{CheungXIVI}
N.~Cheung,
\emph{XIV-I}, version 1,
unpublished manuscript, n.d.
Supplied source: constructed internal amplitude interface and
unresolved full joint characteristic map,
\texttt{thm:I2}, \texttt{thm:I3}.
Source collection: Enter Calculus Companion.

\bibitem{CheungXIVJ}
N.~Cheung,
\emph{XIV-J}, version 1,
unpublished manuscript, n.d.
Supplied source: reconstruction descent, restricted supported-sector
reference, unresolved full characteristic map, and no quantum
admission,
\texttt{prop:descent}, \texttt{thm:reference},
\texttt{thm:J2}, \texttt{cor:noadmission}.
Source collection: Enter Calculus Companion.

\bibitem{CheungXVB}
N.~Cheung,
\emph{XV-B}, version 1,
unpublished manuscript, n.d.
Supplied source: conditional relative-lifting criterion,
\texttt{thm:lifting}.
Source collection: Enter Calculus Companion.

\end{thebibliography}

\end{document}