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\title{\textbf{Hawk05-B}\\[4pt]
A Finite Page-Curve Derivation under the Enter Discipline\\[4pt]
\large Haar-scrambled coherent radiation, destructive records,
and the limits of gravitational identification}
\author{Ngo Cheung}
\date{15 October 2026}

\begin{document}
\maketitle

\begin{abstract}
We derive the finite random-state Page curve within an explicitly
declared Enter-compatible model. A single logical-input-independent
Haar unitary scrambles a prepared finite code; a conserving sequence
then transfers quantum cells into a growing accessible radiation
register. For every fixed pure input chosen independently of the
unitary, the exact mean radiation entropy is
\[
 \overline S_k
 =
 \frac{\mathsf H_{s^N}-\mathsf H_{s^{\max(k,N-k)}}
       -\dfrac{s^{\min(k,N-k)}-1}{2s^{\max(k,N-k)}}}{\ln 2}.
\]
We also prove a simultaneous, all-prefix leading Page law for one
sampled unitary. Destructive Enter defines a different process:
its append-only records have nondecreasing Shannon entropy and
cannot recover an arbitrary unknown quantum code. Their exact mean
entropy is
\[
 \overline H_k
 =
 \frac{\mathsf H_{s^N}-\mathsf H_{s^{N-k}}}{\ln 2},
\]
which does not turn over.

Recovery is addressed separately. We give exact full-code recovery
errors, a conservative Haar subcode-decoupling certificate, corrected
flagged-transfer calibrations, and same-model endpoint and entropy
comparison bounds. The Haar hypothesis, coherent access, and emission
schedule are explicit additional assumptions, not consequences of
Enter. The construction derives a finite Page curve and specifies a
conditional route to a gravitational comparison; it does not construct
a microscopic gravitational evaporation channel, an island formula,
or the unresolved characteristic quantum coefficient of the supplied
analytic programme.
\end{abstract}

\paragraph{Scope of completion.}
The mathematical Page-curve target is completed for the finite
random-state model stated below. A gravitational Page-curve derivation
cannot be obtained from the supplied channel and manuscript
summaries alone: the required gravitational dynamics and
access-preserving comparison are not supplied. That missing
identification is retained as a hypothesis, not replaced by a
finite circuit or claimed as a nonexistence theorem.

\tableofcontents

%=====================================================================
\section{Enter background and the corrected target}
\label{sec:introduction}
%=====================================================================

\subsection{The philosophical discipline}

Cheung's Philosophy of Enter separates destructive readout from
quantum recovery. Destructive computational-basis readout consumes
the measured quantum name and returns classical data; a decoder
instead returns a quantum target through declared access.
Consumption is an interface statement, not physical annihilation.
The philosophy also distinguishes complete energy, erasure cost, and
residual leakage, and distinguishes an analogy from a physical
identification \cite{CheungPhilosophy}.

We implement these distinctions by specifying the code, untouched
reference, complete preparation, accessible outputs, excluded
outputs, and returned type before stating entropy or recovery
claims. Ordinary finite Born instruments are defined explicitly.
Cheung's calculus and Manuscript I provide background; their
measurement-characterization hypotheses are not enlarged by citation
\cite{CheungEnter,CheungI}.

\subsection{Three different statements called a Page curve}

We distinguish:
\begin{enumerate}[label=\textup{(\roman*)}]
\item a designed entropy turnover, such as \(h_2(q)\);
\item the random-state Page law, with entropy approximately the
logarithm of the smaller subsystem dimension;
\item a gravitational radiation-entropy history obtained from a
specified physical theory and radiation interface.
\end{enumerate}
The first does not establish the second, and the second does not
identify the third.

Page proposed the exact random-subsystem average; Sen proved it
\cite{PageAverage,Sen}. Page's radiation discussion supplies a
separate physical interpretation \cite{PageRadiation}.
Existing gravitational entropy analyses have additional dynamical
and geometric content \cite{Penington,AlmheiriEtAl}. They are not
instantiated by naming finite registers ``radiation'' and
``black hole.''

\subsection{Repairs to the supplied Hawk05-A and refined plan}
\label{sec:repairs}

The supplied Hawk05-A is a useful foundation, but its flagged
calibration theorem contains an obsolete reversed-weight equation.
That false equation must be removed from the theorem statement,
rather than retained alongside its correction.

\begin{center}
\small
\renewcommand{\arraystretch}{1.12}
\begin{tabularx}{\linewidth}{@{}p{.34\linewidth}X@{}}
\toprule
Issue & Repair in Hawk05-B\\
\midrule
Residual leakage versus residual recovery &
For occupied residual probability \(q\),
\(\dleak(\Cchan_q)=q(1-d^{-2})\), whereas
\(\erec(\Cchan_q)=(1-q)(1-d^{-2})\).
These are different optimizations.\\[4pt]
Any discrete schedule from \(q=1\) to \(q=0\) has a peak at \(1/2\) &
A discrete schedule can skip \(1/2\), or jump directly from one
to zero. Only a continuous crossing guarantees that point.
A single-excitation turnover is not an extensive Page law.\\[4pt]
Snapshot readout equals a cumulative history &
A refined emission-time record can have the same recovery optimum
as a coarse snapshot while having a different Shannon entropy.
The complete history is computed explicitly below.\\[4pt]
Averaged entropy equals entropy of the averaged state &
The Page quantity is
\(\Expectation_U S(\rho_R^U)\), not
\(S(\Expectation_U\rho_R^U)\). The latter is
\(\log_2\dim R\) in this model.\\[4pt]
A pure-input Page curve certifies the whole code &
Its input is fixed independently of \(U\).
Whole-code recovery quantifies over every referenced input.
The two quantifiers cannot be interchanged.\\[4pt]
A conditional-entropy bound locates maximal leakage &
Without a channel comparison, a conditional-entropy inequality
does not locate a maximum of complete-complement diamond leakage.\\
\bottomrule
\end{tabularx}
\end{center}

The corrections to destructive records, complete complements, and
endpoint identification agree with the distinctions in Cheung's
Hawk and En07 manuscripts
\cite{CheungHawk02B,CheungHawk04D,CheungEn07B,CheungEn07C}.
All elementary mathematical statements used below are proved locally.

%=====================================================================
\section{Finite interfaces, reference preservation, and recovery}
\label{sec:interfaces}
%=====================================================================

All spaces are finite dimensional. Let \(C\) be the variable logical
code, with dimension \(d\ge2\). An external reference \(Q\) is
untouched.

\begin{definition}[Complete interface]
A complete interface is an isometry
\[
 V:\HH_C\longrightarrow\HH_R\otimes\HH_B.
\]
Its two channels are
\[
 \Nchan(X)=\Tr_B(VXV^\dagger),
 \qquad
 \Cchan(X)=\Tr_R(VXV^\dagger).
\]
The complete complement \(B\) includes every unreturned preparation
purifier, lost output, measurement environment, and implementation
factor admitted in the model.
\end{definition}

For states and channels use
\[
 \dtr(\rho,\sigma)=\frac12\norm{\rho-\sigma}_1,
 \qquad
 \ddiam(\Phi,\Psi)=\frac12\norm{\Phi-\Psi}_{\diamond}.
\]
The diamond norm retains arbitrary reference extensions
\cite[Chapter 3]{Watrous}.

Define replacement, recovery error, and complete leakage by
\begin{align}
 \rep{\tau}(X)&=\Tr(X)\tau,\\
 \erec(\Nchan)
 &=\inf_{\Dchan:R\to C\ {\rm CPTP}}
      \ddiam(\Dchan\Nchan,\id_C),\\
 \dleak(\Cchan)
 &=\inf_{\tau_B\in\States(B)}
      \ddiam(\Cchan,\rep{\tau_B}).
\end{align}
We abbreviate
\[
 a_d=1-d^{-2},\qquad b_d=1-d^{-1}.
\]
Compactness ensures that these finite-dimensional infima are
attained. Entropies are in bits:
\[
 S(\rho)=-\Tr(\rho\log_2\rho).
\]

\begin{lemma}[Complete restoration and stability]
\label{lem:restoration}
Every isometric channel \(\Ad_V\) has a CPTP left inverse.
It preserves the trace norm of every referenced operator.
CPTP pre- and postprocessing contract \(\ddiam\), and
\[
 |\erec(\Nchan)-\erec(\Nchan')|
 \le \ddiam(\Nchan,\Nchan').
\]
\end{lemma}

\begin{proof}
For a fixed code state \(\omega\), the channel
\[
 \mathcal L_V(Y)
 =
 V^\dagger YV+\Tr((I-VV^\dagger)Y)\omega
\]
is CP, trace preserving, and satisfies
\(\mathcal L_V\Ad_V=\id_C\).
Isometric embeddings preserve nonzero singular values, also after
tensoring with \(I_Q\). Trace-distance contraction on referenced
states gives diamond-distance contraction.
Insert any decoder into the triangle inequality, take infima, and
interchange the compared channels to obtain the stability bound.
\end{proof}

\begin{theorem}[Exact complete-complement criterion]
\label{thm:exact}
The accessible output exactly recovers the entire code if and only if
\[
 \Cchan=\rep{\tau_B}
\]
for one state \(\tau_B\).
If this holds and \(r=\rank\tau_B\), there are a state
\(\sigma_K\), \(\dim K=r\), and an isometry \(T:C\otimes K\to R\)
such that
\begin{equation}
 \Nchan(X)=T(X\otimes\sigma_K)T^\dagger.
 \label{eq:normalform}
\end{equation}
An exact decoder on all of \(R\) is
\[
 \Dchan(Y)
 =
 \Tr_K(T^\dagger YT)
 +\Tr((I-TT^\dagger)Y)\omega_C.
\]
In particular,
\(\dim R\ge d\,\rank\tau_B\).
Both complementary factors cannot independently recover the whole
code when \(d\ge2\).
\end{theorem}

\begin{proof}
Suppose \(\Cchan=\rep{\tau_B}\), and diagonalize its nonzero support:
\[
 \tau_B=\sum_{\alpha=1}^r\lambda_\alpha\kb{\alpha}{\alpha}.
\]
Write
\[
 V\ket i
 =\sum_\alpha\sqrt{\lambda_\alpha}
       \ket{u_{i\alpha}}_R\ket\alpha_B.
\]
The complementary identities on all matrix units give
\[
 \langle u_{j\beta}\mid u_{i\alpha}\rangle
 =\delta_{ij}\delta_{\alpha\beta}.
\]
Consequently \(T(\ket i\ket\alpha)=\ket{u_{i\alpha}}\) is
isometric and yields \eqref{eq:normalform}. The displayed decoder
is CPTP and recovers every matrix unit.

Conversely, dilate an exact decoder by \(W:R\to C G\).
The composite \((W\otimes I_B)V\) dilates the identity channel.
The Choi operator of the identity has rank one. Every Kraus
operator of that composite is therefore proportional to \(I_C\),
so the composite has the form
\[
 \ket\psi\longmapsto\ket\psi_C\ket\chi_{GB}
\]
with fixed \(\chi\). Its \(B\) channel is replacement.
All identities extend to references by linearity.
If both factors recovered, the complement of either recoverable
factor would be both replacement and recoverable, which is
impossible on two distinct inputs.
\end{proof}

This is the standard exact-error-correction criterion, with the
complete complement specified
\cite{KnillLaflamme,CheungEm05B}. It is not a new error-correction
principle.

\begin{lemma}[Replacement and Choi benchmarks]
\label{lem:benchmarks}
A replacement accessible channel has recovery error \(a_d\);
every accessible channel has recovery error at most \(a_d\).
For
\[
 \Phi_d=\kb{\Phi_d}{\Phi_d},
 \qquad
 \ket{\Phi_d}=\frac1{\sqrt d}\sum_i\ket i_Q\ket i_C,
\]
put \(\Jnorm(\Phi)=(\id_Q\otimes\Phi)(\Phi_d)\).
Then
\begin{equation}
 \frac12\norm{\Jnorm(\Phi)-\Jnorm(\Psi)}_1
 \le\ddiam(\Phi,\Psi)
 \le\frac d2\norm{\Jnorm(\Phi)-\Jnorm(\Psi)}_1.
 \label{eq:choi}
\end{equation}
If \(\ddiam(\Nchan,\rep{\tau})\le\eta\), then
\[
 \erec(\Nchan)\ge\max\{0,a_d-\eta\}.
\]
\end{lemma}

\begin{proof}
A decoded replacement has Bell output \(I_Q/d\otimes\omega_C\)
and Bell overlap \(d^{-2}\), giving the lower bound \(a_d\).
Preparing \(I_C/d\) attains it. Indeed, depolarization is the
uniform average of the \(d^2\) Weyl conjugations; separating its
identity term gives diamond distance \(1-d^{-2}\).

The first Choi inequality uses the Bell input.
For a pure referenced input, write
\(\ket\psi=(L\otimes I)\ket{\Phi_d}\), where
\(\Tr(L^\dagger L)=d\).
Then \(\norm L_\infty^2\le d\). Its output difference is obtained
from the normalized Choi difference by multiplication with
\(L\otimes I\) on both sides, giving the second inequality.
Convexity handles mixed inputs. The final assertion follows by
postprocessing contraction and the triangle inequality.
\end{proof}

\begin{proposition}[Approximate correction input]
\label{prop:ksw}
The finite information--disturbance theorem gives
\begin{equation}
 \erec(\Nchan)\le\min\{a_d,\sqrt{2\dleak(\Cchan)}\},
 \qquad
 \dleak(\Cchan)\le\min\{1,\sqrt{2\erec(\Nchan)}\}.
 \label{eq:ksw}
\end{equation}
\end{proposition}

\begin{proof}
Use the Stinespring continuity theorem and its
information--disturbance consequence in raw diamond normalization
\cite[Theorems 1 and 3]{KSW}:
\[
 \frac14\epsilon_{\rm raw}^2\le\delta_{\rm raw},
 \qquad
 \delta_{\rm raw}\le2\sqrt{\epsilon_{\rm raw}}.
\]
Here the replacement state is optimized, and the converse requires
existence of an appropriate replacement state, not every state.
Substitute
\(\epsilon_{\rm raw}=2\erec\) and
\(\delta_{\rm raw}=2\dleak\).
The independent constant-decoder cap is Lemma~\ref{lem:benchmarks}.
\end{proof}

Adding fixed pure spectators or access-preserving isometric output
embeddings changes none of the optimized distances above: compression
with a CPTP off-support completion gives the reverse comparison.
This observation permits us to suppress fixed vacant factors in the
finite model.

%=====================================================================
\section{One coherent emission history with a Haar hypothesis}
\label{sec:model}
%=====================================================================

Fix integers \(s\ge2\) and \(N\ge2\). Let
\[
 \HH_{\rm int}=(\mathbb C^s)^{\otimes N},
 \qquad
 D=s^N,
 \qquad
 J:C\to\HH_{\rm int}
\]
be a fixed isometry, with \(2\le d\le D\).
Choose a single unitary \(U\in U(D)\).
For ensemble statements only, \(U\) is distributed according to
normalized Haar measure.

\begin{definition}[Meaning of the random-unitary hypothesis]
\label{def:haar}
Haar randomness is an additional ensemble assumption about a family
of devices. In each device, \(U\) is fixed, independent of the
variable logical input, and known to the decoder.
It is not resampled between emission steps.

If \(U\) is instead an unknown discarded random control and the
accessible channel is averaged over it, that is a different,
replacement-channel task.
\end{definition}

This extra typicality hypothesis is not derived from the Enter
philosophy, energy conservation, or Hawking occupations.

\subsection{A nontrivial conserving realization}

For each memory cell \(M_j\) and radiation cell \(R_j\), use
\[
 \HH_{M_j}=\mathbb C\ket{\varnothing}\oplus\mathbb C^s,
 \qquad
 \HH_{R_j}=\mathbb C\ket{\varnothing}\oplus\mathbb C^s.
\]
The preparation occupies all memory cells with the internal state
\(J\ket\psi\) and leaves every radiation cell vacant.

Let \(P_X\) project onto the occupied sector of cell \(X\).
For an energy gap \(\omega>0\), declare
\begin{equation}
 H_0=\omega\sum_{j=1}^N(P_{M_j}+P_{R_j}).
 \label{eq:H0}
\end{equation}
Extend \(U\) to act on the all-occupied memory sector and to be
identity elsewhere. Emit cell \(j\) by the full SWAP between
\(M_j\) and \(R_j\).

\begin{proposition}[Complete conserving implementation]
\label{prop:implementation}
The extended \(U\) and all emission SWAPs commute with \(H_0\).
At cut \(k\), the effective quantum output is
\[
 R_k=(\mathbb C^s)^{\otimes k},
 \qquad
 B_k=(\mathbb C^s)^{\otimes(N-k)},
\]
with
\[
 r_k=s^k,\qquad b_k=s^{N-k},\qquad r_kb_k=D.
\]
Up to fixed vacant factors and tensor permutations, its complete
isometry is
\begin{equation}
 V_k^U=F_kUJ:C\to R_k\otimes B_k,
 \label{eq:Vk}
\end{equation}
where \(F_k\) is the canonical tensor regrouping.
Earlier emitted cells are never acted on again.
At \(k=N\), the residual is a fixed pure vacant state.
\end{proposition}

\begin{proof}
The all-occupied memory sector has one \(H_0\) eigenvalue,
\(N\omega\); thus arbitrary \(U\) on that sector commutes with
\(H_0\). SWAP preserves \(P_{M_j}+P_{R_j}\).
After \(k\) swaps, the first \(k\) internal tensor factors are in
radiation, while the remaining factors are in memory.
All other factors are fixed vacancies. This is
\eqref{eq:Vk}. At the last cut no internal memory factor remains.
\end{proof}

The energy distribution of \(H_0\) is conserved. The ideal mean
radiation and memory energies are \(k\omega\) and \((N-k)\omega\).
This register Hamiltonian is not a black-hole mass law, and
implementation work and control reset are not priced by it.

\subsection{Quantifiers and input types}

Fix a pure logical input \(\ket\psi\) independently of \(U\).
Then \(UJ\ket\psi\) is a Haar-distributed unit vector in
\(\mathbb C^D\). Define
\begin{equation}
 \rho_{R_k}^{U,\psi}
 =
 \Tr_{B_k}\kb{F_kUJ\psi}{F_kUJ\psi},
 \qquad
 S_k(U,\psi)=S(\rho_{R_k}^{U,\psi}).
 \label{eq:Sk}
\end{equation}
All cuts in \eqref{eq:Sk} belong to one state and one sampled \(U\).
We do not replace this history by independently sampled states at
different cuts.

The fixed-pure-input entropy problem is distinct from recovery of
every state of \(C\), including reference-entangled states.

%=====================================================================
\section{Derivation of the finite Page curve}
\label{sec:page}
%=====================================================================

For a positive integer \(\ell\), let
\[
 \Harm{\ell}=\sum_{j=1}^{\ell}\frac1j.
\]
Put
\[
 m_k=\min(r_k,b_k),\qquad n_k=\max(r_k,b_k).
\]

\begin{theorem}[Exact finite Page curve]
\label{thm:page}
Under Definition~\ref{def:haar}, for every fixed pure input
independent of \(U\),
\begin{equation}
 \boxed{
 \overline S_k:=\Expectation_U S_k(U,\psi)
 =
 \frac{\Harm{D}-\Harm{n_k}-(m_k-1)/(2n_k)}{\ln2}.
 }
 \label{eq:page}
\end{equation}
The value is independent of that fixed input and of the code
embedding \(J\).
It satisfies
\[
 \overline S_0=\overline S_N=0,
 \qquad
 \overline S_k=\overline S_{N-k}.
\]
It increases strictly up to the middle and decreases strictly
afterward. For odd \(N\), the two central cuts have equal maximal
mean entropy. For even \(N\), the unique maximal mean occurs at
\(k=N/2\).
\end{theorem}

\begin{proof}
Haar invariance makes \(UJ\ket\psi\) a uniform unit vector.
The two reductions of a bipartite pure state have the same
nonzero spectrum. The calculation in Appendix~\ref{app:page}
therefore applies with \(m=m_k\), \(n=n_k\), and gives
\eqref{eq:page}. This is the Page--Sen average, rederived there
with normalization explicit \cite{PageAverage,Sen}.

The endpoint and symmetry statements follow immediately.
To verify monotonicity, consider consecutive cuts lying on the
early half, so \(m'=sm\), \(n'=n/s\), with \(m'\le n'\).
Their difference, in natural entropy units, is
\[
 \sum_{j=n/s+1}^{n}\frac1j
 -
 \frac{(s^2-1)m-(s-1)}{2n}.
\]
The first term is at least \(1-1/s\).
Since \(n\ge s^2m\), the difference is at least
\[
 1-\frac1s-\frac{s^2-1}{2s^2}
 =\frac{(s-1)^2}{2s^2}>0.
\]
Symmetry proves the decreasing half and the central statements.
\end{proof}

\begin{remark}[Meaning of the midpoint]
The midpoint arises from the equal-dimension equation
\(r_k=b_k\), the equal-size emission rule, and the Haar ensemble.
It is not a universal ``half the emitted quanta'' law.
Different cell capacities, conservation sectors, or dynamical
ensembles require a new calculation.
\end{remark}

\subsection{A simultaneous leading Page law without resampling}

Define the capacity tent
\begin{equation}
 T_k=\min(k,N-k)\log_2s=\log_2m_k.
 \label{eq:tent}
\end{equation}

\begin{lemma}[Haar purity]
\label{lem:purity}
For a Haar unit vector in \(\mathbb C^m\otimes\mathbb C^n\),
\[
 \Expectation\Tr\rho_m^2=\frac{m+n}{mn+1}.
\]
\end{lemma}

\begin{proof}
For \(D=mn\), unitary invariance and symmetry give
\[
 \Expectation\bigl(\kb v v^{\otimes2}\bigr)
 =\frac{I+\mathsf F}{D(D+1)},
\]
where \(\mathsf F\) swaps the two complete copies.
The normalization follows by taking the trace.
The purity swap test uses
\(\mathsf F_m\otimes I_{n^2}\).
Its traces against \(I\) and \(\mathsf F_m\otimes\mathsf F_n\)
are \(mn^2\) and \(m^2n\), respectively.
\end{proof}

\begin{theorem}[All-prefix capacity approximation]
\label{thm:uniform}
For one sampled \(U\), set
\[
 \Delta_k(U,\psi)=T_k-S_k(U,\psi)\ge0.
\]
Then
\begin{equation}
 0\le\Expectation\Delta_k
 \le\log_2(1+m_k/n_k)
 \le\frac{s^{-|N-2k|}}{\ln2}.
 \label{eq:gap}
\end{equation}
Consequently, with
\[
 C_s=\frac{s^2+1}{(s^2-1)\ln2},
\]
for every \(z>0\),
\begin{equation}
 \Probability\left\{\max_{0\le k\le N}\Delta_k>z\right\}
 \le\min\{1,C_s/z\}.
 \label{eq:allprefix}
\end{equation}
In particular, for fixed \(s\) and inputs \(\psi_N\) independent
of \(U_N\),
\begin{equation}
 \max_k
 \left|
 \frac{S_k(U_N,\psi_N)}{N\log_2s}
 -\min\left(\frac{k}{N},1-\frac{k}{N}\right)
 \right|
 \longrightarrow0
 \quad\hbox{in probability}.
 \label{eq:leadingpage}
\end{equation}
\end{theorem}

\begin{proof}
A reduction has rank at most \(m_k\), so its entropy is at most
\(\log_2m_k\).
The elementary Rényi bound
\(S(\rho)\ge-\log_2\Tr\rho^2\) follows by Jensen's inequality
with weights equal to the eigenvalues.
Thus
\[
 \Expectation\Delta_k
 \le\log_2\left(
 m_k\frac{m_k+n_k}{m_kn_k+1}\right)
 \le\log_2(1+m_k/n_k).
\]
This proves \eqref{eq:gap}.

For even \(N\), the infinite extension of the sum of
\(s^{-|N-2k|}\) is
\(1+2/(s^2-1)\).
For odd \(N\), it is \(2s/(s^2-1)\), which is no larger.
Therefore
\[
 \Expectation\sum_k\Delta_k\le C_s.
\]
Markov's inequality and
\(\max_k\Delta_k\le\sum_k\Delta_k\) prove
\eqref{eq:allprefix}. Take \(z=\epsilon N\log_2s\) to obtain
\eqref{eq:leadingpage}.
\end{proof}

No independence between cuts was used. This is a temporal statement
about one nested sequence, although the exact finite statement is
an ensemble mean and an individual realization can fluctuate.

\subsection{Two averages that must not be confused}

\begin{proposition}[Entropy after ensemble forgetting]
\label{prop:averaging}
For the fixed pure input,
\[
 \Expectation_U\rho_{R_k}^{U,\psi}=\frac{I_{R_k}}{r_k},
 \qquad
 S(\Expectation_U\rho_{R_k}^{U,\psi})=\log_2r_k.
\]
If the unitary is not known and the accessible channel itself is
averaged over Haar measure, then
\[
 \Expectation_U\Nchan_k^U(X)
 =\Tr(X)\frac{I_{R_k}}{r_k}.
\]
Its recovery error is \(a_d\) at every cut, including \(k=N\).
\end{proposition}

\begin{proof}
The Haar twirl of \(JXJ^\dagger\) is
\(\Tr(X)I_D/D\).
Take the residual partial trace. The recovery conclusion is
Lemma~\ref{lem:benchmarks}.
\end{proof}

An unknown discarded scrambling control cannot be omitted from
complete output accounting and then treated as known by a decoder.

%=====================================================================
\section{Destructive Enter gives a different entropy history}
\label{sec:records}
%=====================================================================

For a register \(A\) with basis \(\{\ket y\}\), define destructive
readout by
\[
 (\Enter_A\otimes\id_K)(X)
 =
 \sum_y\kb y y_Y\otimes
 (\bra y_A\otimes I_K)X(\ket y_A\otimes I_K).
\]
The returned quantum names are those of \(K\), not \(A\).
A complete dilation is
\[
 W_{\rm read}\ket y_A=\ket y_Y\ket y_L,
\]
where \(L\) is unreturned.
The complete evolution remains isometric; the returned classical
record alone is dephased.

Finite adaptive compositions are CPTP: composing the unnormalized
path maps \(K_h\) gives
\(\sum_hK_h^\dagger K_h=I\).
This is the ordinary finite instrument setting also used in
Cheung's hybrid and sequential interfaces
\cite{CheungVIIG,CheungEn02C}.

\subsection{Immediate cell readout}

Apply Enter in the occupied computational basis immediately after
each emission of Section~\ref{sec:model}. Let
\(Y_{\le k}=(Y_1,\ldots,Y_k)\) be the complete unedited record.

\begin{theorem}[Exact record law and mean entropy]
\label{thm:classicalpage}
For a fixed pure input and fixed \(U\),
\begin{equation}
 p_U(y_{\le k})
 =
 \sum_{z\in\{0,\ldots,s-1\}^{N-k}}
 \left|\bra{y_{\le k},z}UJ\ket\psi\right|^2.
 \label{eq:recordlaw}
\end{equation}
The averaged cumulative record entropy is
\begin{equation}
 \boxed{
 \overline H_k
 =
 \Expectation_U H(Y_{\le k})
 =
 \frac{\Harm{D}-\Harm{b_k}}{\ln2}.
 }
 \label{eq:recordmean}
\end{equation}
It increases strictly with \(k\).
At the endpoint,
\[
 \overline H_N=\frac{\Harm{D}-1}{\ln2}>0,
 \qquad
 \overline S_N=0.
\]
For every fixed \(U,\psi\), not merely on average,
\[
 H(Y_{\le k+1})\ge H(Y_{\le k}).
\]
\end{theorem}

\begin{proof}
Later operations are SWAPs on other cells. The projectors defining
the full outcome history therefore correspond to the commuting
tensor-basis projectors on the original scrambled state.
Their Born law is \eqref{eq:recordlaw}; no postselection is used.

Represent a Haar vector by independent standard complex Gaussians
divided by their Euclidean norm.
The sums of squared moduli over the \(b_k\) entries associated with
each prefix are independent Gamma variables of shape \(b_k\).
After normalization, the \(r_k\) prefix probabilities have the
Dirichlet distribution with all parameters \(b_k\).
For real \(t\) near one,
\[
 \Expectation\sum_y p_y^t
 =
 r_k\frac{\Gamma(b_k+t)\Gamma(D)}
           {\Gamma(b_k)\Gamma(D+t)}.
\]
Differentiation at \(t=1\) yields
\[
 \Expectation\sum_y p_y\ln p_y
 =\psi_0(b_k+1)-\psi_0(D+1)
 =\Harm{b_k}-\Harm{D},
\]
where \(\psi_0=(\ln\Gamma)'\).
This proves \eqref{eq:recordmean}.
Its increment is
\((\Harm{b_k}-\Harm{b_{k+1}})/\ln2>0\).

For a fixed process, Shannon's chain rule gives
\[
 H(Y_{\le k+1})
 =H(Y_{\le k})+H(Y_{k+1}\mid Y_{\le k}),
\]
and the conditional entropy is nonnegative.
\end{proof}

\begin{proposition}[Permanent classical recovery obstruction]
\label{prop:classical}
Every purely classical channel with code input dimension \(d\ge2\)
satisfies
\[
 \erec(\Mchan)\ge b_d.
\]
The bound is sharp for computational-basis measurement.
Its complete complement is never constant.
\end{proposition}

\begin{proof}
Following a classical channel by any quantum decoder gives a
measure-and-prepare channel. Its Bell output is separable
\cite{HorodeckiShorRuskai}.
For product states,
\[
 \Tr\bigl(\Phi_d(\rho_Q\otimes\sigma_C)\bigr)
 =\frac1d\Tr(\rho_Q^{\mathsf T}\sigma_C)\le\frac1d.
\]
Convexity extends the bound to separable states.
The Bell-projector test gives error at least \(1-1/d\).

Basis measurement followed by basis-state preparation is dephasing.
Dephasing is the average of the \(d\) phase conjugations, so separating
its identity term gives matching error \(1-1/d\).
A constant complete complement would imply exact recovery by
Theorem~\ref{thm:exact}, a contradiction.
\end{proof}

The bound concerns the entire adaptive record, not just one cell.
A quantum-output decoder cannot repair entanglement breaking merely
by preparing a quantum state from classical data.

\begin{proposition}[Hybrid output and terminal discard]
\label{prop:hybrid-drop}
For any returned block-diagonal output
\(\rho_{YK}=\sum_yp_y\kb y y\otimes\rho_{K|y}\),
\[
 S(YK)=H(Y)+\sum_yp_yS(\rho_{K|y})\ge H(Y).
\]
Thus an unchanged positive-entropy record prevents purity of the raw
hybrid output.

Furthermore, for all referenced operators,
\[
 (\id\otimes\Drop_Y)(\id\otimes\Enter_A)
 =\Tr_A.
\]
Terminal residual Enter followed by drop leaves the already-returned
accessible channel unchanged.
\end{proposition}

\begin{proof}
The spectrum of a diagonal block is \(p_y\) times the spectrum of
\(\rho_{K|y}\), proving the entropy formula.
Writing an operator as
\(\sum_{y,z}X_{yz}\otimes\kb y z_A\), readout followed by discard
returns \(\sum_yX_{yy}\), precisely the partial trace.
The blocks may contain an arbitrary reference.
\end{proof}

This identity is an interface equality, not physical annihilation
or a reset-cost theorem
\cite{CheungEn07A,CheungEn07B,CheungEn07C}.

%=====================================================================
\section{The Page curve and whole-code recovery are different tasks}
\label{sec:recovery}
%=====================================================================

\subsection{An exact full-code calibration in the same model}

Take \(d=D\) and identify \(J\) with a unitary input dictionary.

\begin{theorem}[Whole ambient code: exact errors]
\label{thm:fullcode}
For every fixed \(U\), at a cut with \(r=r_k\), \(b=b_k\),
the coherent channels satisfy
\begin{equation}
 \erec(\Nchan_k^U)=\dleak(\Cchan_k^U)=1-b^{-2}.
 \label{eq:fullcode}
\end{equation}
If the returned radiation is destructively read out, its classical
channel satisfies
\begin{equation}
 \erec(\Mchan_k^U)=1-\frac1{rb^2}.
 \label{eq:fullrecord}
\end{equation}
In particular, a midpoint mean-entropy peak does not imply accurate
recovery of the whole \(D\)-dimensional input.
\end{theorem}

\begin{proof}
Conjugating the logical target by \(U\) leaves the optimization
unchanged. The coherent channel becomes \(\Tr_B\).
The decoder appending \(I_B/b\) has error \(1-b^{-2}\), by the
Weyl twirl on \(B\).
For the reverse bound, use the product Bell input
\(\Phi_r\otimes\Phi_b\).
After \(B\) is lost, its reference is independent of every decoder
input. The decoded full Bell overlap is at most \(b^{-2}\).
This proves the recovery formula.

The complementary channel is \(\Tr_R\).
Replacement by \(I_B/b\) has distance at most \(1-b^{-2}\).
Fixing the input on \(R\) and testing a Bell input on \(B\) gives
the matching lower bound for every replacement.

For record-only access, the measurement effects on the transformed
code are
\[
 F_y=\kb y y_R\otimes I_B,\qquad y=1,\ldots,r.
\]
If \(\sigma_y\) is any decoder output, its full Bell overlap is
\[
 \frac1{D^2}\sum_y\Tr(F_y\sigma_y)
 \le\frac r{D^2}=\frac1{rb^2}.
\]
The decoder preparing \(\kb y y_R\otimes I_B/b\) attains the bound:
its channel is \(\Delta_R\otimes\rep{I_B/b}\), the average of
\(rb^2\) phase/Weyl conjugations, one of which is the identity.
Separating that term proves the matching upper bound.
\end{proof}

Theorem~\ref{thm:fullcode} is the whole-code SWAP calibration
documented in Cheung's Hawk04-D, here applied to the same
Haar-prepared entropy model \cite{CheungHawk04D}.
It is not a Hayden--Preskill small-diary claim
\cite{HaydenPreskill}.

\begin{example}[The reference changes the entropy history]
For the full-code Bell input, the marginal logical state is \(I_D/D\).
For every \(U\),
\[
 \rho_{R_k}=I_{R_k}/r_k,
 \qquad
 S(R_k)=k\log_2s.
\]
This referenced-input marginal entropy is monotone and ends at
\(\log_2D\), not zero. Its final reference entanglement is correctly
preserved by coherent recovery. The pure-unreferenced Page theorem
does not apply to that marginal.
\end{example}

\begin{remark}[No uniform pure-input Page theorem on the full code]
For every \(U\) on the full code, the input
\(U^\dagger\ket{0,\ldots,0}\) produces zero entropy at every cut.
It depends on \(U\), so it violates the fixed-input hypothesis.
This explicitly prevents promotion of Theorem~\ref{thm:uniform}
to a uniform entropy assertion over all inputs.
\end{remark}

\subsection{A smaller code: a conservative complete-channel certificate}

For a Haar embedding \(V:C\to\mathbb C^r\otimes\mathbb C^b\),
\(D=rb\), define
\[
 \omega_{QB}=(\id_Q\otimes\Cchan)(\Phi_d),
 \qquad
 \sigma_B=\Cchan(I_C/d).
\]
Put
\begin{equation}
 \gamma_B
 =
 \frac{(1-d^{-2})\,r(b^2-1)}{D^2-1},
 \qquad
 \Lambda_B=\frac d2\sqrt{db\,\gamma_B}.
 \label{eq:gamma}
\end{equation}

\begin{theorem}[Haar subcode decoupling]
\label{thm:decoupling}
For the fixed code dimension \(d\),
\begin{equation}
 \Expectation_U
 \norm{\omega_{QB}-I_Q/d\otimes\sigma_B}_2^2
 =\gamma_B.
 \label{eq:HS}
\end{equation}
Consequently,
\[
 \Expectation_U\dleak(\Cchan)\le\min\{1,\Lambda_B\}.
\]
For \(0<\alpha<1\), with probability at least \(1-\alpha\),
\begin{equation}
 \dleak(\Cchan)\le\min\{1,\Lambda_B/\alpha\},
 \qquad
 \erec(\Nchan)
 \le\min\{a_d,\sqrt{2\Lambda_B/\alpha}\}.
 \label{eq:prob-decouple}
\end{equation}
These are whole-code, reference-preserving bounds.

For fixed \(d\) and \(D\to\infty\),
\(\Lambda_B\to0\) whenever \(b/r\to0\).
More generally, the sufficient regime is \(d^3b/r\to0\).
This last dimension dependence is a conservative certificate, not
an optimal random-code threshold.
\end{theorem}

\begin{proof}
Appendix~\ref{app:fourth} proves the fourth-moment identity
\eqref{eq:HS}.
The operator acts on a space of dimension \(db\), so
\[
 \frac12\norm{\omega_{QB}-I_Q/d\otimes\sigma_B}_1
 \le\frac12\sqrt{db}\,
       \norm{\omega_{QB}-I_Q/d\otimes\sigma_B}_2.
\]
Cauchy--Schwarz in expectation, followed by
Lemma~\ref{lem:benchmarks}, gives
\[
 \Expectation\dleak(\Cchan)\le\Lambda_B.
\]
Markov's inequality gives the leakage statement, and
Proposition~\ref{prop:ksw} gives recovery.

Since
\[
 \Lambda_B^2
 =\frac{d^3b(1-d^{-2})r(b^2-1)}
        {4(r^2b^2-1)},
\]
the stated sufficient regimes follow.
At \(b=1\), the expression vanishes exactly.
\end{proof}

This theorem supplies a channel estimate not inferred from the Page
entropy slope. It deliberately retains the input-dimension factor in
the Choi-to-diamond conversion.

An early-access comparison follows by interchanging \(r\) and \(b\):
\[
 \gamma_R
 =
 \frac{(1-d^{-2})\,b(r^2-1)}{D^2-1},
 \qquad
 \Lambda_R=\frac d2\sqrt{dr\,\gamma_R}.
\]
With probability at least \(1-\alpha\),
\[
 \ddiam(\Nchan,\rep{\Nchan(I_C/d)})
 \le\min\{1,\Lambda_R/\alpha\},
\]
and therefore
\[
 \erec(\Nchan)\ge\max\{0,a_d-\Lambda_R/\alpha\}.
\]
Early replacement and late recovery are thus separate proved
comparisons in the Haar subcode model.

\begin{remark}[Partners and extra commuting copies]
The normal form \eqref{eq:normalform} represents the recovered
logical algebra on the accessible support as
\(T(\BB(C)\otimes I_K)T^\dagger\).
Its commutant on that support is
\(T(I_C\otimes\BB(K))T^\dagger\), as follows by commuting with all
logical matrix units.
The syndrome algebra is not a second recoverable copy of the same
unknown logical state. An abstract algebra of the same dimension
may exist there without carrying that input.

This is the distinction relevant to the supplied partner
comparisons in Hawk03-A \cite{CheungHawk03A}.
No independent-factor entropy bound is used to estimate complete
leakage or identify its maximum.
\end{remark}

%=====================================================================
\section{Explicit entropy--recovery separations}
\label{sec:separation}
%=====================================================================

\subsection{Same entropy history, inaccessible logical memory}

\begin{theorem}[An extensive Page history without code release]
\label{thm:retain}
There are two conserving sequential complete interfaces with the same
accessible and residual spaces such that:
\begin{enumerate}[label=\textup{(\roman*)}]
\item for one fixed pure code input, their accessible states agree
at every prefix;
\item the two ensembles have the same exact Page mean
\eqref{eq:page} for every fixed pure code input;
\item the endpoint recovery errors are respectively \(0\) and
\(a_d\), and endpoint complete leakages are respectively \(0\) and
\(a_d\).
\end{enumerate}
Thus even the extensive Page mean does not determine endpoint
whole-code recovery.
\end{theorem}

\begin{proof}
Append a degenerate inaccessible register \(L\simeq C\).
In the returning model, run Section~\ref{sec:model} on the variable
input and fix \(L\) in \(\ket0\).
In the retaining model, store the variable input in \(L\) and run
the same radiation schedule on the fixed dummy input \(\ket0_C\).

All transfer operations conserve \eqref{eq:H0}; \(L\)'s logical
labels are degenerate. Both are complete isometries into the same
spaces. For the input \(\ket0_C\), the radiation histories agree
exactly for the same \(U\).
Haar invariance proves that the returning model's mean entropy for
any fixed pure input equals the dummy model's mean, namely
\eqref{eq:page}.

At the endpoint the returning radiation is \(UJ\ket\psi\), with
fixed complete residual. The retaining radiation is fixed at
\(UJ\ket0\), while the residual contains \(\ket\psi_L\).
The endpoint recovery errors follow from exact correction and the
replacement benchmark. The retained logical copy has leakage
\(a_d\): a CPTP left inverse gives the identity-channel lower bound,
and an embedded maximally mixed replacement gives the upper bound.
\end{proof}

This is an ensemble Page-history extension of the disclosed-memory
separation in Cheung's Hawk01-B \cite{CheungHawk01B}.
The comparison does not impose identical assumptions of complete
outgoing access and a unique empty physical endpoint.

\begin{proposition}[Selected spectra and thermal marginals]
\label{prop:thermal}
Let \(\sigma_K\) have purification \(\ket\chi_{KP}\), and put
\(R=C_RK\), \(B=PL\), with \(L\simeq C\).
The isometries
\[
 V_{\rm good}\ket\psi
 =\ket\psi_{C_R}\ket\chi_{KP}\ket0_L,
 \qquad
 V_{\rm bad}\ket\psi
 =\ket0_{C_R}\ket\chi_{KP}\ket\psi_L
\]
have identical accessible spectra for every pure input, but
recovery errors \(0\) and \(a_d\).
For \(H_R=I_{C_R}\otimes H_K\), their selected energy laws agree.
One may take \(\sigma_K\) to be Gibbs.
\end{proposition}

\begin{proof}
The accessible channels are \(X\mapsto X\otimes\sigma_K\) and
\(X\mapsto\Tr(X)\kb0 0\otimes\sigma_K\).
The spectrum, recovery, and energy conclusions follow directly.
\end{proof}

The supplied En07-C and Oai265-A records contain related separations
\cite{CheungEn07C,CheungOai265A}. Oai265-A is used only as a
documentary comparison; no catalogue family is a premise.

\subsection{The corrected single-excitation calibration}

For comparison with Hawk05-A and Hawk02-B, let each memory and
emission cell have space
\(\mathbb C\ket{\varnothing}\oplus C\).
At opportunity \(j\), rotate
\[
 \ket{i,\varnothing}
 \longmapsto
 \cos\theta_j\ket{i,\varnothing}
 +\sin\theta_j\ket{\varnothing,i},
\]
with the complementary rotation on
\(\ket{\varnothing,i}\).
The occupied labels are degenerate, so the rotation conserves
the sum of occupied-sector energies.

Set
\[
 q_n=\prod_{j=1}^n\cos^2\theta_j,
 \qquad
 p_j=q_{j-1}-q_j.
\]
The complete isometry has the form
\[
 V_n\ket\psi
 =
 \sqrt{1-q_n}\,J_n\ket\psi_R\ket{\varnothing}_M
 +\sqrt{q_n}\,\ket{\varnothing}_R\ket\psi_M,
\]
where the normalized emission amplitudes define an isometry \(J_n\).

\begin{theorem}[Clean flagged formulas and the complete record]
\label{thm:flagged}
For \(q=q_n\),
\begin{align}
 \erec(\Nchan_q)&=q a_d,
 &
 \dleak(\Cchan_q)&=q a_d,
 &
 \erec(\Cchan_q)&=(1-q)a_d,
 \label{eq:flaggederrors}\\
 S(\Nchan_q(\rho))
 &=\hb(q)+(1-q)S(\rho),
 &
 S(\Cchan_q(\rho))
 &=\hb(q)+qS(\rho).
 \label{eq:flaggedentropy}
\end{align}
Immediate cell readout returns histories consisting of either
no emission or an emission time \(j\) and basis label \(i\).
For \(u_i=\bra i\rho\ket i\), their probabilities are
\[
 \Probability(\mathrm e)=q_n,
 \qquad
 \Probability(j,i)=p_j u_i.
\]
Its recovery error and complete-record entropy are
\begin{align}
 \erec(\Mchan_n)&=q_n a_d+(1-q_n)b_d,
 \label{eq:flagrecorderror}\\
 H_n
 &=H(p_1,\ldots,p_n,q_n)+(1-q_n)H(u).
 \label{eq:flagrecordentropy}
\end{align}
If \(q_{n-1}>0\) and \(t_n=q_n/q_{n-1}\), then
\[
 H_n-H_{n-1}
 =
 q_{n-1}\hb(t_n)+(q_{n-1}-q_n)H(u)\ge0.
\]
If \(q_{n-1}=0\), the increment is zero.
\end{theorem}

\begin{proof}
The channels are orthogonal-block channels
\[
 \Nchan_q(X)
 =(1-q)J_nXJ_n^\dagger
 +q\Tr(X)\kb{\varnothing}{\varnothing},
\]
\[
 \Cchan_q(X)
 =qX+(1-q)\Tr(X)\kb{\varnothing}{\varnothing}.
\]
The decoder compressing the occupied block and preparing \(I_C/d\)
on vacancy gives the radiation upper bound \(qa_d\).
On a Bell input, no decoder has overlap greater than
\(1-q+q/d^2\), proving optimality.
Interchanging occupied probabilities gives the residual recovery
error. The exact residual leakage is proved in
Appendix~\ref{app:flag}; it is \(qa_d\), not residual recovery.
The entropy formulas follow from the orthogonal blocks.

Successive local readouts may be commuted past later rotations on
other cells for the purpose of computing the complete outcome law.
Their probabilities are therefore \(p_ju_i\) and \(q_n\).

Preparing \(\ket i\) on a reported label and \(I_C/d\) on no emission
gives decoded channel
\((1-q_n)\Delta+q_n\rep{I_C/d}\).
Its error is at most \((1-q_n)b_d+q_na_d\).
The Bell test gives the same lower bound, irrespective of emission
time. The history entropy follows by splitting the no-emission
branch and then its label.
At the next opportunity only the former no-emission outcome is
refined, giving the displayed increment.
\end{proof}

The recovery formula agrees with the supplied Hawk02-B calibration
\cite{CheungHawk02B}. Erasing the emission-time label gives a coarse
snapshot record with entropy
\(\hb(q_n)+(1-q_n)H(u)\), but does not change the recovery optimum:
time can be resampled with input-independent conditional
probabilities \(p_j/(1-q_n)\).
The refined and coarse records have different entropy ledgers.

For a pure unreferenced input the coherent entropy is \(\hb(q)\),
bounded by one bit regardless of the number of cells.
Taking \(\theta_1=\pi/4\), \(\theta_2=\pi/2\) gives coherent
entropies \((0,1,0)\) for a basis input and cumulative classical
entropies \((0,1,1)\).
Other prepared turnover sequences, including the supplied
En02-D example, are comparisons rather than premises of the Haar
Page derivation \cite{CheungEn02D}.

%=====================================================================
\section{Endpoint certification and controlled physical comparison}
\label{sec:bridge}
%=====================================================================

\subsection{Four obligations, supplied in one model}

A physical endpoint package must provide:
\begin{enumerate}[label=\textup{(E\arabic*)}]
\item a justified finite code and complete prepared isometry,
including preparation purifiers;
\item complete logical-input-independent unitary evolution on a
justified domain;
\item access-preserving identifications into common comparison
spaces, including a justified quantum receiver if physical radiation
is to be collected;
\item a proved complete-complement replacement allowance.
\end{enumerate}
These are the same-model obligations in Cheung's En07 endpoint
formulations \cite{CheungEn07B,CheungEn07C}.
They are not consequences of selected thermal occupations.

\begin{theorem}[Endpoint channel certificate]
\label{thm:endpoint}
After the declared identifications, suppose actual and proxy
complete channels satisfy
\[
 \ddiam(\Vchan,\widehat{\Vchan})\le\epsilon.
\]
If the proxy complement obeys
\(\dleak(\widehat{\Cchan})\le\zeta\), then
\[
 \dleak(\Cchan)\le\min\{1,\zeta+\epsilon\},
\]
\[
 \erec(\Nchan)
 \le\min\{a_d,\epsilon+\sqrt{2\zeta}\}.
\]
A fixed proxy-optimal decoder has actual error at most
\[
 \min\{1,\epsilon+\erec(\widehat{\Nchan})\}.
\]
If a feasible state \(\tau_B\) instead supplies the normalized
Choi discrepancy
\[
 e=\frac12\norm{
 \Jnorm(\widehat{\Cchan})-I_Q/d\otimes\tau_B}_1,
\]
one may take \(\zeta=\min\{1,de\}\).
\end{theorem}

\begin{proof}
Partial traces contract the complete-channel comparison.
The triangle inequality proves leakage.
Apply a fixed proxy-optimal decoder to the actual accessible channel,
use contraction, and then use \eqref{eq:ksw}.
The constant decoder gives the independent \(a_d\) cap.
Equation \eqref{eq:choi} gives the Choi conversion.
\end{proof}

The finite model of Section~\ref{sec:model} meets these obligations
mathematically: \(J\) fixes preparation, its unitary extension and
SWAPs fix evolution, tensor regrouping fixes access, and the terminal
residual is a fixed pure vacancy.
These are not gravitationally justified objects.

\subsection{Entropy needs its own comparison allowance}

For an output dimension \(L\), define the monotone entropy modulus
\[
 c_L(\eta)=
 \begin{cases}
 0,&L=1,\\[2pt]
 \hb(t)+t\log_2(L-1),
 &L\ge2,\quad t=\min\{\eta,1-1/L\}.
 \end{cases}
\]

\begin{theorem}[Conditional Page-history transfer]
\label{thm:transfer}
Suppose one physical model and one receiver supply complete
access-preserving comparisons to a fixed sampled finite proxy
\(U\), at every declared cut \(k\).
Let the common accessible comparison dimension be \(L_k\), including
any necessary padding, and suppose
\[
 \ddiam(\Nchan_k^{\rm phys},\Nchan_k^U)\le\eta_k.
\]
For every fixed compared input,
\begin{equation}
 |S(R_k^{\rm phys})-S(R_k^U)|\le c_{L_k}(\eta_k).
 \label{eq:entropytransfer}
\end{equation}
If the physical model itself has an ensemble coupled to the Haar
proxy with these allowances, then
\[
 |\Expectation S(R_k^{\rm phys})-\overline S_k|
 \le c_{L_k}(\eta_k).
\]
For coupled single histories, the all-prefix deviation obeys
\[
 \max_k|S(R_k^{\rm phys})-T_k|
 \le
 \max_k\Delta_k+
 \max_k c_{L_k}(\eta_k).
\]
Hence the normalized leading Page law transfers whenever
\[
 \frac{\max_k c_{L_k}(\eta_k)}{N\log_2s}\longrightarrow0
\]
and the other hypotheses of Theorem~\ref{thm:uniform} hold.
\end{theorem}

\begin{proof}
The channel comparison bounds output trace distance by \(\eta_k\).
Apply the sharp finite-dimensional entropy continuity estimate
\cite{Audenaert}. Maximizing its right-hand side over trace distances
at most \(\eta_k\) gives \(c_{L_k}(\eta_k)\).
Take expectations or the maximum and use the triangle inequality.
\end{proof}

The dimension \(L_k\) is the actual common comparison dimension,
not automatically the ideal smaller Schmidt rank.
If errors populate additional sectors, those sectors cannot be
removed from the continuity allowance without a separate support
estimate.

Compatible complete preparation, evolution, collection, and
implementation errors can be accumulated by replacing one channel
factor at a time:
\[
 \epsilon_k
 \le
 \epsilon_{\rm prep}+\sum_{j\le k}\epsilon_j
 +\epsilon_{\rm collection}+\epsilon_{\rm implementation}.
\]
CPTP contraction proves this telescoping bound.
It does not convert a scalar occupation or metric error into a
diamond allowance. This partition of hypotheses is emphasized in
Cheung's Hawk04-D \cite{CheungHawk04D}.

\subsection{What remains missing for gravity}

For a gravitational application, one must still supply:
\begin{enumerate}
\item a microscopic complete state and dynamics, with justified
radiation and residual algebras or finite encodings;
\item a physical derivation or controlled comparison establishing the
required random-state typicality in the relevant conserved sectors;
\item a justified emission clock and radiation-capacity history;
\item a coherent radiation-to-memory receiver with a complete-channel
allowance, if recovery is claimed;
\item a same-model endpoint comparison and complete-complement
allowance.
\end{enumerate}

Hawking's semiclassical emission calculation is an external
field-theoretic result \cite{Hawking}.
The supplied fixed-background emission and En07-C records do not
provide these microscopic identifications
\cite{CheungEm03A,CheungEn07C}.
A physical mass balance, even if separately supplied, does not
construct them.

No finite-band regulator, conditional luminosity law, or independently
defined characteristic coefficient is transferred into this model.
Existing gravitational Page-curve calculations remain external
comparisons, not premises or objects constructed here
\cite{Penington,AlmheiriEtAl}.

%=====================================================================
\section{Three ledgers and the Enter analogies}
\label{sec:analogies}
%=====================================================================

\subsection{Energy, reset resources, and information}

The three ledgers remain distinct:
\[
 \boxed{
 \text{declared complete energy}
 \quad;\quad
 \text{heat/work and cyclic reset resources}
 \quad;\quad
 \text{complete residual leakage}.}
\]

\begin{proposition}[No thermal cost from a channel identity alone]
A reduced channel does not determine its heat, work, or reusable-memory
reset cost.
\end{proposition}

\begin{proof}
SWAP with an independent blank qubit implements reset of the observed
memory while transferring its referenced input to the blank.
With equal Hamiltonians
\(H_M=H_L=\epsilon\kb1 1\), it conserves their sum and transfers
energy \(\epsilon p\) for excitation probability \(p\).
The same reset channel is obtained for every positive \(\epsilon\).
Independent unobserved fuel or reservoir processes can also be appended
without changing the observed channel. The blank and fuel resources
must be restored before a cyclic cost is claimed.
\end{proof}

This is the channel-versus-implementation separation in
Cheung's En07-A/B \cite{CheungEn07A,CheungEn07B}.
In a separately specified initially uncorrelated memory--Gibbs
reservoir setup undergoing a complete unitary,
\[
 \beta Q_{\rm bath}
 =
 \ln2\left[
 S(\rho_M)-S(\rho_M')
 +I(M:L)_{\rho'}+
 D_2(\rho_L'\Vert\gamma_L)
 \right].
\]
Thus thermal erasure has additional hypotheses
\cite{ReebWolf}. No such thermal machine is constructed by the
readout-and-drop identity.

An information-free residual can retain energy and degrees of
freedom: \(V\ket\psi=\ket\psi_R\ket1_B\) has zero leakage but
positive residual energy for \(H_B=\epsilon\kb1 1\).

\subsection{Processor analogy}

An \(N\)-cell processor coherently scrambles a prepared pure input
with a fixed unitary and releases its cells one by one.
The receiver stores all cells as quantum systems.
Under the additional Haar ensemble, the exact mean entropy is the
Page curve. Globally the computation remains reversible.

Installing Enter on every released cell changes the returned type
to classical strings. The same apparatus now has the monotone record
entropy \eqref{eq:recordmean} and the permanent quantum-recovery
obstruction. Retaining coherent measurement environments would
declare a different, larger quantum access interface.

\subsection{Torn strip and photograph album}

For a classical message \(L\), more photographs can reduce
\(H(L\mid Y_{\le k})\), even though the complete album entropy
\(H(Y_{\le k})\) cannot decrease.

For an unknown quantum strip, photographs are measurement outcomes,
not quantum fragments. A Page turnover belongs to the coherently
retained quantum fragments in the stated model. It is not a
turnover of the append-only album, and the album does not acquire
the original reference entanglement after the last photograph.

\subsection{Library cards and quantum books}

Classical photocopies can identify classical text.
If a book denotes an arbitrary unknown quantum state, a classical
card catalogue cannot return every amplitude and reference
correlation from one copy. A quantum carrier must remain accessible
for whole-code recovery.

A classical index can legitimately select a quantum branch decoder
when an appropriate quantum output survives. It must not be renamed
the quantum carrier itself. Emptying the named locked stack is an
interface event, not proof of physical annihilation or a final
energetic burst.

%=====================================================================
\section{Hawk05-B mathematical release}
\label{sec:release}
%=====================================================================

\begin{theorem}[Page derivation with typed access]
\label{thm:main}
For the explicitly prepared, Haar-scrambled, conserving finite
emission model:
\begin{enumerate}[label=\textup{(\roman*)}]
\item coherent quantum radiation has the exact finite Page mean
\eqref{eq:page} for every fixed pure input independent of \(U\);
\item one sampled unitary supplies an all-prefix normalized leading
Page history as in \eqref{eq:leadingpage};
\item destructive Enter has the exact cumulative-record mean
\eqref{eq:recordmean}, not a Page turnover, and record-only whole-code
recovery error is at least \(1-1/d\);
\item the same coherent Page ensemble can describe a full code whose
exact radiation recovery error is \(1-b_k^{-2}\), or a smaller code
with the separate decoupling certificate
\eqref{eq:prob-decouple};
\item the corrected single-excitation calibration is
\eqref{eq:flaggederrors}, with its complete classical history given by
\eqref{eq:flagrecordentropy};
\item entropy history, selected thermal data, complete-complement
constancy, and executable quantum recovery remain separate
certification targets;
\item a physical Page-history or endpoint conclusion follows only
after its same-model comparison hypotheses are supplied.
\end{enumerate}
The finite random-state Page curve is constructed.
A microscopic gravitational evaporation channel and the inherited
characteristic quantum coefficient are not constructed.
\end{theorem}

\begin{proof}
Items (i)--(ii) are Theorems~\ref{thm:page} and \ref{thm:uniform}.
Item (iii) is Theorem~\ref{thm:classicalpage} and
Proposition~\ref{prop:classical}.
Item (iv) is Theorems~\ref{thm:fullcode} and \ref{thm:decoupling}.
Item (v) is Theorem~\ref{thm:flagged}.
Item (vi) follows from the explicit separations and the recovery
criterion. Item (vii) is Theorems~\ref{thm:endpoint} and
\ref{thm:transfer}.
No gravitational comparison or characteristic product is supplied
by the objects in these proofs.
\end{proof}

\paragraph{Conclusion.}
Enter is a discipline for keeping the returned output and access
contract explicit. With additional coherent-state and typicality
hypotheses, it accommodates a genuine finite Page-law derivation.
With destructive record-only access, it excludes the classical-album
version of that curve. Neither conclusion supplies a missing
gravitational dynamics.

%=====================================================================
\appendix
\section{Evaluation of the exact Page average}
\label{app:page}
%=====================================================================

This appendix evaluates the standard random-state average. The result
is due to Page and Sen; no priority is claimed
\cite{PageAverage,Sen}. Natural logarithms are used until the final
conversion to bits.

Let \(1\le m\le n\), \(D=mn\), and let \(G\) be an \(m\times n\)
matrix of independent standard complex Gaussians. Set
\[
 W=GG^\dagger,\qquad T=\Tr W,\qquad \rho=W/T.
\]
The normalized Gaussian vector is Haar on the unit sphere and has
reduction \(\rho\). Gaussian radius and direction are independent,
so \(T\) and \(\rho\) are independent. Moreover,
\(T\) has Gamma density with shape \(D\):
\[
 \Expectation T=D,\qquad
 \Expectation(T\ln T)=D\psi_0(D+1).
\]
Since
\[
 \Tr(W\ln W)=T\ln T+T\Tr(\rho\ln\rho),
\]
the desired natural-log entropy satisfies
\begin{equation}
 \Expectation S_{\rm nat}(\rho)
 =
 \psi_0(D+1)-\frac1D\Expectation\Tr(W\ln W).
 \label{eq:pagegamma}
\end{equation}

\subsection{The eigenvalue density}

Write \(\alpha=n-m\).
The squared singular values \(x_1,\ldots,x_m\) of \(G\) have joint
density proportional to
\begin{equation}
 e^{-\sum_i x_i}
 \prod_i x_i^\alpha
 \prod_{i<j}(x_i-x_j)^2,
 \qquad x_i>0.
 \label{eq:wishart}
\end{equation}
For completeness, the singular-value change of variables gives one
factor \(x_i^{n-m}\) from the \(n-m\) unused column directions for
each singular value. The two complex angular directions associated
with a pair of distinct singular values give
\((x_i-x_j)^2\). The radial coordinates contribute \(dx_i\).
Integrating the angular variables leaves a constant independent of
the \(x_i\). Repeated singular values form a set of Gaussian measure
zero, so this change of variables determines the density.

Define generalized Laguerre polynomials by Rodrigues's formula
\[
 L_j^\alpha(x)
 =
 \frac{x^{-\alpha}e^x}{j!}
 \frac{d^j}{dx^j}\left(e^{-x}x^{j+\alpha}\right).
\]
Integration by parts gives orthogonality with weight
\(w_\alpha(x)=e^{-x}x^\alpha\), and norm
\[
 h_j
 =
 \int_0^\infty w_\alpha(x)[L_j^\alpha(x)]^2\,dx
 =
 \frac{\Gamma(j+\alpha+1)}{j!}.
\]
Indeed, \(L_j^\alpha\) has leading coefficient
\((-1)^j/j!\); it is orthogonal to every lower-degree polynomial,
and integrating it against \(x^j\) gives
\((-1)^j\Gamma(j+\alpha+1)\).

Replacing the monomials in the Vandermonde determinant by these
polynomials changes only a nonzero constant. Expanding the squared
determinant and integrating with orthogonality shows that the
one-point eigenvalue density is
\begin{equation}
 R_1(x)
 =
 w_\alpha(x)\sum_{j=0}^{m-1}\frac{[L_j^\alpha(x)]^2}{h_j}.
 \label{eq:onepoint}
\end{equation}
Its integral is \(m\). Therefore
\[
 \Expectation\Tr(W\ln W)
 =
 \sum_{j=0}^{m-1}
 \frac1{h_j}
 \int_0^\infty x^{\alpha+1}e^{-x}\ln x
           [L_j^\alpha(x)]^2\,dx.
\]

\subsection{The logarithmic integral}

The finite-polynomial or generating-function identities give
\begin{align}
 xL_j^\alpha
 &=(2j+\alpha+1)L_j^\alpha
   -(j+1)L_{j+1}^\alpha-(j+\alpha)L_{j-1}^\alpha,
 \label{eq:lagrec}\\
 \partial_\alpha L_j^\alpha
 &=\sum_{\ell=0}^{j-1}\frac{L_\ell^\alpha}{j-\ell}.
 \label{eq:lagder}
\end{align}
The first follows by comparing coefficients. For the second, use
\[
 \sum_{j\ge0}L_j^\alpha(x)t^j
 =(1-t)^{-\alpha-1}
       \exp\left(-\frac{xt}{1-t}\right)
\]
and differentiate in \(\alpha\).
Terms with index \(-1\) are zero.

Orthogonality and \eqref{eq:lagrec} give
\[
 A_j:=
 \int_0^\infty x^{\alpha+1}e^{-x}[L_j^\alpha(x)]^2\,dx
 =(2j+\alpha+1)h_j.
\]
The cross term in the derivative of \(A_j\) is
\[
 \int_0^\infty x^{\alpha+1}e^{-x}
 L_j^\alpha\partial_\alpha L_j^\alpha\,dx
 =-j h_j.
\]
Only the \(j-1\) term of \eqref{eq:lagder} contributes.
Differentiation under the integral is justified for
\(\alpha>-1\), locally uniformly in \(\alpha\), by the exponential
weight and integrability of \(x^\alpha\ln x\).
Thus
\begin{equation}
 \frac1{h_j}
 \int_0^\infty x^{\alpha+1}e^{-x}\ln x
       [L_j^\alpha(x)]^2\,dx
 =
 (2j+\alpha+1)\psi_0(j+\alpha+1)+1+2j.
 \label{eq:logint}
\end{equation}

To sum \eqref{eq:logint}, put \(r=j+\alpha+1\) and use
\[
 \psi_0(r)
 =
 \psi_0(n+1)-\sum_{\ell=r}^{n}\frac1\ell.
\]
The coefficient sum is
\[
 \sum_{j=0}^{m-1}(2j+\alpha+1)=mn.
\]
The double sum simplifies because
\[
 \sum_{r=\alpha+1}^{\ell}(2r-\alpha-1)
 =\ell(\ell-\alpha).
\]
Consequently
\[
 \sum_{j=0}^{m-1}
 (2j+\alpha+1)\psi_0(j+\alpha+1)
 =
 mn\psi_0(n+1)-\frac{m(m+1)}2.
\]
Since \(\sum_{j=0}^{m-1}(1+2j)=m^2\), we obtain
\[
 \Expectation\Tr(W\ln W)
 =
 mn\psi_0(n+1)+\frac{m(m-1)}2.
\]
Substitute this expression into \eqref{eq:pagegamma}:
\[
 \Expectation S_{\rm nat}(\rho)
 =
 \psi_0(mn+1)-\psi_0(n+1)-\frac{m-1}{2n}.
\]
Finally
\(\psi_0(\ell+1)=\Harm{\ell}-\gamma_{\rm Euler}\).
Division by \(\ln2\) proves \eqref{eq:page}, including \(m=1\).

%=====================================================================
\section{The Haar fourth-moment calculation for code decoupling}
\label{app:fourth}
%=====================================================================

Haar invariance permits us to take \(J\ket i\) to be the first \(d\)
coordinate vectors. Write
\(V_{ab,i}=U_{(a,b),i}\), with \(a=1,\ldots,r\),
\(b=1,\ldots,b\).

For \(D\ge2\), the required Haar fourth moment is
\begin{align}
 &\Expectation[
 U_{xi}U_{zk}\overline{U_{yj}}\overline{U_{w\ell}}]
 \nonumber\\
 &\quad=
 \frac{
 \delta_{xy}\delta_{zw}\delta_{ij}\delta_{k\ell}
 +\delta_{xw}\delta_{zy}\delta_{i\ell}\delta_{kj}}
 {D^2-1}
 \nonumber\\
 &\qquad-
 \frac{
 \delta_{xy}\delta_{zw}\delta_{i\ell}\delta_{kj}
 +\delta_{xw}\delta_{zy}\delta_{ij}\delta_{k\ell}}
 {D(D^2-1)}.
 \label{eq:fourth}
\end{align}
This is the order-two Haar integration identity
\cite{CollinsSniady}.
It can also be obtained directly: left and right unitary invariance
restrict the tensor to the two row pairings and two column pairings;
symmetry under exchange gives two coefficients. Contracting with
\(\sum_xU_{xi}\overline{U_{xj}}=\delta_{ij}\) gives the two linear
equations whose solution is
\(1/(D^2-1)\) and \(-1/[D(D^2-1)]\).

For the normalized complementary Choi state,
\[
 (\omega_{QB})_{ib,jb'}
 =
 \frac1d\sum_a V_{ab,i}\overline{V_{ab',j}}.
\]
Therefore
\[
 \Tr\omega_{QB}^2
 =
 \frac1{d^2}
 \sum_{i,j,a,a',b,b'}
 V_{ab,i}\overline{V_{ab',j}}
 V_{a'b',j}\overline{V_{a'b,i}}.
\]
Substitution of \eqref{eq:fourth} gives four index counts:
\[
 dr^2b,\quad d^2rb^2,\quad d^2r^2b,\quad drb^2.
\]
The last two carry the negative coefficient divided by \(D\).
Thus
\begin{equation}
 \Expectation\Tr\omega_{QB}^2
 =
 \frac{r(D/d-1)+b(D-1/d)}{D^2-1}.
 \label{eq:omegapurity}
\end{equation}

Similarly,
\[
 (\sigma_B)_{b,b'}
 =
 \frac1d\sum_{a,i}V_{ab,i}\overline{V_{ab',i}},
\]
and the corresponding counts are
\[
 d^2r^2b,\quad drb^2,\quad dr^2b,\quad d^2rb^2.
\]
Hence
\begin{equation}
 \Expectation\Tr\sigma_B^2
 =
 \frac{r(D-1/d)+b(D/d-1)}{D^2-1}.
 \label{eq:sigmapurity}
\end{equation}

Because \(\Tr_Q\omega_{QB}=\sigma_B\),
\[
 \norm{\omega_{QB}-I_Q/d\otimes\sigma_B}_2^2
 =
 \Tr\omega_{QB}^2-\frac1d\Tr\sigma_B^2.
\]
Subtracting \eqref{eq:sigmapurity} from
\eqref{eq:omegapurity} with this factor gives
\[
 \frac{(1-d^{-2})(bD-r)}{D^2-1}
 =
 \frac{(1-d^{-2})r(b^2-1)}{D^2-1},
\]
which proves \eqref{eq:HS}.
The expression vanishes for a one-dimensional residual, as it must.

%=====================================================================
\section{Correct residual leakage in the flagged model}
\label{app:flag}
%=====================================================================

Consider
\[
 \Cchan_q(X)=qX\oplus(1-q)\Tr(X)
\]
on occupied code plus vacancy.
Replacement by
\(\tau_q=qI_C/d\oplus(1-q)\) gives distance \(qa_d\).

For the lower bound, pinching any candidate replacement state into
occupied and vacant blocks cannot increase its distance from this
channel. Covariance under logical unitary conjugations, convexity,
and averaging then reduce the optimization to states
\[
 \tau_s=sI_C/d\oplus(1-s),\qquad 0\le s\le1.
\]
On the Bell input, the occupied block difference is
\(q\Phi_d-sI_{QC}/d^2\), and the vacant block has trace norm
\(|s-q|\). Thus twice the tested trace distance is
\[
 \left|q-\frac{s}{d^2}\right|
 +(d^2-1)\frac{s}{d^2}+|s-q|.
\]
For \(s\le q\), this is
\(2q-2s/d^2\ge2qa_d\).
For \(q\le s\le qd^2\), it is
\(2sa_d\ge2qa_d\).
For \(s\ge qd^2\), it is
\(2(s-q)\ge2qa_d\).
Therefore
\[
 \boxed{\dleak(\Cchan_q)=qa_d.}
\]

The reversed weight \((1-q)a_d\) is the residual recovery error,
not its leakage. No contradictory equation is retained as a theorem
statement in this manuscript.

%=====================================================================
\section{The separate characteristic-map programme is not completed}
\label{app:boundary}
%=====================================================================

The supplied XIV-I, XIV-J, and XV-B excerpts concern a distinct
distributional construction, not the finite Haar model
\cite{CheungXIVI,CheungXIVJ,CheungXVB}.
Their supplied version-specific boundary is:
internal amplitudes are constructed at I2; the full grouped
characteristic map remains unresolved at I3/J2, with the
graph-specific continuation unresolved in XV-B version 2.

For a reconstruction map \(\Rec:\mathscr P\to\mathscr K\), a linear
presentation-level candidate \(\widetilde M\) descends precisely when
\[
 \ker\Rec\subseteq\ker\widetilde M.
\]
Indeed, this is exactly the condition that equal reconstructed
presentations have equal candidate outputs.
The identity
\[
 q\delta'(q)+\delta(q)=0
\]
also shows why highest presentation degree is not automatically an
intrinsic reconstructed symbol.

Neither these elementary compatibility statements nor the Page
integrals above construct the full six-branch characteristic
composition. The remaining graph-specific map, section, remainder,
transverse estimates, contact compatibility, output wavefront,
and normalized quantum admission are not supplied here.

The standing supplied statuses are unchanged:
\(O.1=\texttt{NOT\_CLOSED}\); the downstream quantum-admission
exit is not opened; XVI-N is not discharged; no inherited
characteristic quantum coefficient is admitted.
The existence of a technical XV-B manuscript must not be confused
with passage through the earlier admission exit.
These are scoped provenance statements, not universal
nonexistence theorems.

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

\paragraph{Attribution and novelty.}
The exact random-state average is the Page--Sen result.
Exact correction, entanglement breaking, and Stinespring continuity
are established theories.
The contribution is a complete finite emission realization, a
simultaneous leading-history estimate, an exact destructive-record
comparison, and a typed separation of entropy and recovery targets.
No priority is claimed for the standard theorems.

\paragraph{Source provenance.}
Cheung's internal manuscripts are cited by identifier and supplied
version. Their documentary comparison does not import their physical
models or discharge their unrelated open gates.
Hawk05-A is cited as the supplied source file, whose author field is
empty; authorship is not inferred.
The journal entry for Cheung's Enter article is attributed to the
companion's bibliographic record, not represented as an independently
verified publisher-status report.

\paragraph{Physical and computational scope.}
No gravitational Hamiltonian, backreaction equation, physical
radiation receiver, island, quantum extremal surface, or realistic
microscopic evaporation endpoint is constructed.
The prescribed Haar unitary is not compiled over a fixed gate
library, and its implementation complexity and cyclic thermodynamic
cost are not claimed.

\paragraph{Verification.}
This source is standalone and written for \texttt{pdflatex}.
No successful compiler execution, numerical experiment,
proof-assistant verification, independent referee approval, or
journal acceptance is asserted.

%=====================================================================
\begin{thebibliography}{99}
\addcontentsline{toc}{section}{References}
%=====================================================================

\bibitem{CheungPhilosophy}
N.~Cheung,
\emph{The Philosophy of Enter: Destructive Readout, Quantum Recovery,
and the Refusal of Identification},
EntPhil, version 2.
Enter Calculus Companion.
Introductory essay, not a theorem premise.
\url{https://enter.grok.me/enter}.

\bibitem{CheungEnter}
N.~Cheung,
``The Enter Calculus: A linear quantum lambda calculus with an
operational characterization of measurement.''
Bibliographic entry supplied in EntPhil version 2:
\emph{Frontiers in Physics}, volume 14 (2026),
DOI \url{https://doi.org/10.3389/fphy.2026.1954943}.
The companion describes an accepted article with final formatted
version forthcoming; that publisher status is not independently
certified by this manuscript.

\bibitem{CheungI}
N.~Cheung,
\emph{Manuscript I}, version 2.
Enter Calculus Companion.
Supplied finite destructive-measurement characterization,
including \texttt{thm:main}.
\url{https://enter.grok.me/browse/I}.

\bibitem{Hawk05A}
\emph{Hawk05-A: Enter, Page-Type Entropy Turnover, and the
Classical-Record Obstruction}.
Supplied source \texttt{Hawk05-A.tex}, dated 15 October 2026.
Author field unspecified.
The reversed-weight equation in its flagged theorem is corrected
in Section~\ref{sec:repairs} and Appendix~\ref{app:flag}.

\bibitem{CheungHawk02B}
N.~Cheung,
\emph{Hawk02-B: Hamiltonian-Derived Emission and Quantitative
Recovery in a Finite Evaporation Model}, version 2.
Enter Calculus Companion.
Supplied labels \texttt{thm:B3}, \texttt{thm:B7}.
\url{https://enter.grok.me/browse/Hawk02-B}.

\bibitem{CheungHawk01B}
N.~Cheung,
\emph{Hawk01-B: Complete Quantum-Code Preservation in a Sequential
Evaporation Model: Radiation Correlations, Residual Leakage, and
the Page-Curve--Recovery Gap}, version 1.
Enter Calculus Companion.
Supplied \texttt{thm:entropyseparation}.
\url{https://enter.grok.me/browse/Hawk01-B}.

\bibitem{CheungHawk03A}
N.~Cheung,
\emph{Hawk03-A}, version 1.
Enter Calculus Companion.
Supplied complete-complement, independent-factor, and partner
comparisons. Their model-specific hypotheses are not imported.
\url{https://enter.grok.me/browse/Hawk03-A}.

\bibitem{CheungHawk04D}
N.~Cheung,
\emph{Hawk04-D: From Finite Enter Interfaces to Conditional
Evaporation Certificates: Controlled Limits, Mass Laws, and
Quantum Recovery}, version 1.
Enter Calculus Companion.
Relevant supplied labels:
\texttt{thm:swap}, \texttt{thm:flagged}, \texttt{thm:endpoint}.
\url{https://enter.grok.me/browse/Hawk04-D}.

\bibitem{CheungEn07A}
N.~Cheung,
\emph{En07-A}, version 1.
Enter Calculus Companion.
Supplied readout/discard and
\texttt{prop:no-landauer-from-channel} records.
\url{https://enter.grok.me/browse/En07-A}.

\bibitem{CheungEn07B}
N.~Cheung,
\emph{En07-B}, version 1.
Enter Calculus Companion.
Supplied endpoint and separate-ledger records.
\url{https://enter.grok.me/browse/En07-B}.

\bibitem{CheungEn07C}
N.~Cheung,
\emph{En07-C: Hawking Radiation under Destructive Enter}, version 1.
Enter Calculus Companion.
Supplied \texttt{def:four}, \texttt{thm:endpoint},
and spectral/entropy separation records.
\url{https://enter.grok.me/browse/En07-C}.

\bibitem{CheungEn02C}
N.~Cheung,
\emph{En02-C}, version 1.
Enter Calculus Companion.
Supplied reference-preserving sequential soundness,
\texttt{thm:C1}.
\url{https://enter.grok.me/browse/En02-C}.

\bibitem{CheungEn02D}
N.~Cheung,
\emph{En02-D}, version 4.
Enter Calculus Companion.
Supplied prepared-state turnover,
\texttt{thm:turnover}.
\url{https://enter.grok.me/browse/En02-D}.

\bibitem{CheungVIIG}
N.~Cheung,
\emph{VII-G}, version 1.
Enter Calculus Companion.
Supplied finite hybrid channel and history-instrument structure.
\url{https://enter.grok.me/browse/VII-G}.

\bibitem{CheungEm05B}
N.~Cheung,
\emph{Em05-B}, version 1.
Enter Calculus Companion.
Supplied \texttt{thm:constantComplement}.
\url{https://enter.grok.me/browse/Em05-B}.

\bibitem{CheungEm03A}
N.~Cheung,
\emph{Em03-A}, version 1.
Enter Calculus Companion.
Supplied fixed-background scalar emission law,
\texttt{thm:main}; not a complete microscopic evaporation channel.
\url{https://enter.grok.me/browse/Em03-A}.

\bibitem{CheungOai265A}
N.~Cheung,
\emph{Oai265-A}, version 2.
Enter Calculus Companion.
Supplied record/spectrum/recovery separation,
\texttt{thm:separation}.
Documentary comparison only.
\url{https://enter.grok.me/browse/Oai265-A}.

\bibitem{CheungXIVI}
N.~Cheung,
\emph{XIV-I}, version 1.
Enter Calculus Companion.
Supplied I2 construction and I3 boundary.
\url{https://enter.grok.me/browse/XIV-I}.

\bibitem{CheungXIVJ}
N.~Cheung,
\emph{XIV-J}, version 1.
Enter Calculus Companion.
Supplied reconstruction descent, supported reference, and J2 boundary.
\url{https://enter.grok.me/browse/XIV-J}.

\bibitem{CheungXVB}
N.~Cheung,
\emph{XV-B}, version 2.
Enter Calculus Companion.
Supplied relative-lifting and extension interfaces;
the inherited graph-specific map remains unresolved.
\url{https://enter.grok.me/browse/XV-B}.

\bibitem{Watrous}
J.~Watrous,
\emph{The Theory of Quantum Information}.
Cambridge University Press, 2018.
Author's manuscript:
\url{https://cs.uwaterloo.ca/~watrous/TQI/}.

\bibitem{KnillLaflamme}
E.~Knill and R.~Laflamme,
``Theory of quantum error-correcting codes.''
\emph{Physical Review A} \textbf{55} (1997), 900--911.
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\end{thebibliography}

\end{document}