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\title{\textbf{Hawk05-C}\\[4pt]
Conditional Page-Curve Extrapolations under Enter\\[4pt]
\large Compatible capacities, conserved-sector proxies,
two return types, and cutoff certificates}
\author{Ngo Cheung}
\date{15 October 2026}

\begin{document}
\maketitle

\begin{abstract}
We develop conditional Page-curve extrapolations from a finite
Haar proxy while retaining the access and output distinctions of
the Philosophy of Enter. Nonuniform coherent cell release gives the
Page--Sen mean, whereas destructive readout gives an append-only
classical record with a different exact entropy mean. Their entropy
difference is basis-dependent relative-entropy coherence, not a
whole-code recovery error or a thermodynamic cost.

A compatibility theorem identifies a missing condition in a proposed
clock replacement: a fixed-dimensional tensor-release proxy requires
the sum of its two logarithmic capacities to remain constant.
Consequently, identifying radiation capacity with a coarse entropy
production law larger than black-hole entropy loss is not merely a
reparametrization of Hawk05-B. We give an explicit rounded,
fixed-dimensional realization for the compatible case and an exact
direct-sum calculation for a conserved additive-charge sector.
A positive-mass inverse-square clock yields a conditional capacity
crossover, its cutoff criterion, and a peak-location stability bound.

Physical entropy transfer, ensemble typicality, and quantum recovery
receive separate quantitative allowances. A proposed Petz decoder is
made into a deterministic channel on its entire input space and can
be certified by its decoded Choi discrepancy. No occupation law,
entropy slope, restricted observable algebra, or finite termination
supplies the missing physical comparisons. The result is a proved
finite extrapolation framework and a collection of explicitly
conditional evaporation sketches, not a microscopic gravitational
Page curve, an island construction, or spacetime emergence.
\end{abstract}

\paragraph{Mathematical scope.}
All channel and random-state results below are finite dimensional.
The semiclassical clock is an additional supplied model.
The physical channel, receiver, and typicality allowances remain
unfilled until justified in one compatible microscopic theory.
The manuscript does not replace an unconstructed gravitational
interface by a finite proxy and then declare the identification proved.

\tableofcontents

%=====================================================================
\section{Enter background, provenance, and the corrected extrapolation}
\label{sec:background}
%=====================================================================

\subsection{The Philosophy of Enter}

Cheung's Philosophy of Enter distinguishes destructive readout,
quantum return, physical implementation, and analogy. Enter consumes
a measured quantum name at its interface and returns a classical
record. A decoder instead acts on declared accessible systems and
returns a quantum target; whole-code correctness includes untouched
reference correlations. Consumption is not physical annihilation,
and an input--output channel does not determine a thermal reset
machine \cite{CheungPhilosophy}.

The present paper adopts ordinary finite Born instruments explicitly.
Cheung's operational characterization applies inside its stated
strategy class; it is not used to derive a gravitational field
instrument or enlarge that class \cite{CheungEnter,CheungI}.
The four questions guiding every extrapolation are:
\[
 \boxed{
 \text{What is prepared?}\quad
 \text{What evolves?}\quad
 \text{What is accessible?}\quad
 \text{What is returned?}}
\]

Hawk05-A separates classical accumulation from quantum recovery.
Hawk05-B then constructs a finite coherent Haar history with the
Page--Sen mean and a different exact classical-record mean
\cite{Hawk05A,Hawk05B}. We retain those objects as proxies, not
gravitational identifications.

\subsection{Three levels of claim}

We distinguish:
\begin{enumerate}[label=\textup{(\roman*)}]
\item a finite channel theorem;
\item a conditional identification and its quantitative allowance;
\item a physical conclusion after the identification has been proved.
\end{enumerate}
This paper proves statements at level (i), proves implications at
level (ii), and does not assert that the gravitational hypotheses
required for level (iii) have been discharged.

The Page--Sen average is a random-state theorem
\cite{PageAverage,Sen}. A black-hole time profile requires additional
entropy and luminosity assumptions, as in Page's time-dependent
analysis \cite{PageTime}. Island and entanglement-wedge calculations
have further geometric and dynamical hypotheses
\cite{Penington,IslandReview}. None is instantiated by a finite
capacity plot.

\subsection{Debugging the proposed extrapolations}
\label{sec:debug}

\begingroup
\small
\renewcommand{\arraystretch}{1.12}
\begin{longtable}{@{}p{.36\linewidth}p{.59\linewidth}@{}}
\caption{Repairs required before extrapolating the finite proxy.}
\label{tab:debug}\\
\toprule
Proposed inference & Corrected statement\\
\midrule
\endfirsthead
\toprule
Proposed inference & Corrected statement\\
\midrule
\endhead
\bottomrule
\endfoot
Choose arbitrary continuous \(r(t),b(t)\) by exponentiating
two entropies.
&
Finite dimensions are integers. In a fixed tensor-release proxy,
\(r(t)b(t)=D\); logarithmic capacities must therefore have constant
sum, up to declared rounding error.
\\[5pt]
Clock replacement has only a channel-continuity error.
&
The budget also includes finite Page deficit, capacity rounding,
ensemble mismatch, and temporal compatibility. A collection of
snapshot formulas is not automatically one emission history.
\\[5pt]
A greybody occupation fixes a cell dimension.
&
An occupation is a state statistic. Turning it into a dimension,
probability, cutoff, or entropy-rate weight requires a specified
additional map and truncation convention.
\\[5pt]
The record--quantum entropy gap measures all consumed information.
&
For a specified basis it equals relative-entropy coherence of one
accessible marginal. It is state- and basis-dependent and is not
whole-code recovery loss.
\\[5pt]
Restricting access produces a Page tent.
&
Restriction defines an observable task. A tent additionally requires
a state ensemble and a compatible capacity history. A product state
has zero entropy on every tensor prefix.
\\[5pt]
Two commuting copies of a matrix algebra cannot exist.
&
They can exist on independent tensor factors. What is obstructed is
two commuting faithful representations that both act as the same
unknown logical algebra on the encoded code.
\\[5pt]
A conserved sector remains an ordinary tensor product.
&
A fixed total additive charge generally decomposes into a direct
sum of charge-compatible tensor blocks. Its entropy includes charge
weight entropy; the unrestricted Page formula cannot be applied
with only two aggregate dimensions.
\\[5pt]
A falling radiation entropy implies a constant complement.
&
A pure input's output entropy and the complete complement's
dependence on all code inputs are different questions.
Exact recovery may coexist with a mixed fixed syndrome.
\\[5pt]
A positive-cutoff plot necessarily includes a Page turnover.
&
It includes a turnover only if the declared capacity crossover is
reached before the cutoff. The clock can stop on the increasing branch.
\\[5pt]
A named Petz map establishes physical recovery.
&
It must be a well-defined deterministic channel on the actual
access interface, have a proved task error, and have a certified
implementation. An algebraic target may be narrower than a whole code.
\\
\end{longtable}
\endgroup

The proposal's unspecified ``2026 algebraic reading'' is not treated
as an established universal statement or attributed to an unidentified
recent paper. Section~\ref{sec:algebras} instead states a finite
access theorem and separately identifies an external asymptotic
proposal with its own hypotheses.

%=====================================================================
\section{Complete interfaces and the two return types}
\label{sec:interfaces}
%=====================================================================

Let \(C\) be a variable code with \(\dim C=d\ge2\), and let \(Q\)
be an untouched reference.

\begin{definition}[Complete interface]
A complete interface is an isometry
\[
 V:C\longrightarrow R\otimes B,
\]
with accessible channel and complete complement
\[
 \Nchan(X)=\Tr_B(VXV^\dagger),
 \qquad
 \Cchan(X)=\Tr_R(VXV^\dagger).
\]
Every unreturned purifier, measurement environment, lost mode, and
admitted implementation factor belongs to \(B\).
\end{definition}

Use the normalizations
\[
 \dtr(\rho,\sigma)=\frac12\norm{\rho-\sigma}_1,
 \qquad
 \ddiam(\Phi,\Psi)=\frac12\norm{\Phi-\Psi}_{\diamond},
\]
and define
\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}
The diamond norm includes arbitrary reference extensions
\cite[Chapter 3]{Watrous}.
Set
\[
 a_d=1-d^{-2},\qquad b_d=1-d^{-1}.
\]
All entropies are in bits unless a natural logarithm is printed
explicitly.

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

\begin{proof}
For a fixed state \(\omega_C\), the map
\[
 \mathcal L_V(Y)
 =
 V^\dagger YV+
 \Tr((I-VV^\dagger)Y)\omega_C
\]
is CP, trace preserving, and left-inverts \(\Ad_V\).
An isometric embedding preserves all nonzero singular values, also
after reference extension. Trace-distance contraction on referenced
states gives diamond contraction. Insert an arbitrary decoder in
the triangle inequality, take infima, and interchange the compared
channels to prove stability.
\end{proof}

\begin{theorem}[Exact whole-code recovery]
\label{thm:exact}
An accessible output exactly recovers the whole code if and only if
its complete complement is replacement:
\[
 \Dchan\Nchan=\id_C
 \quad\Longleftrightarrow\quad
 \Cchan=\rep{\tau_B}.
\]
In that case there are a state \(\sigma_K\) 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}
One may take \(\dim K=\rank\tau_B\).
For a pure logical input,
\[
 S(\Nchan(\kb\psi\psi))=S(\sigma_K).
\]
Thus exact recovery does not force purity of the raw accessible
output.
\end{theorem}

\begin{proof}
If the complement is constant, diagonalize
\(\tau_B=\sum_\alpha\lambda_\alpha\kb\alpha\alpha\) on its nonzero
support and write
\[
 V\ket i
 =
 \sum_\alpha\sqrt{\lambda_\alpha}
       \ket{u_{i\alpha}}_R\ket\alpha_B.
\]
The complementary equations on matrix units imply
\(\langle u_{j\beta}\mid u_{i\alpha}\rangle
=\delta_{ij}\delta_{\alpha\beta}\).
They define \(T(\ket i\ket\alpha)=\ket{u_{i\alpha}}\).
Compression by \(T^\dagger\), tracing \(K\), and preparing a fixed
state on the unused accessible subspace gives a deterministic
decoder.

Conversely, dilate an exact decoder by \(W:R\to C G\).
The composite \((W\otimes I_B)V\) dilates the identity channel.
Its Choi operator has rank one, so each Kraus operator is a scalar
multiple of \(I_C\). Hence the composite is
\(\ket\psi\mapsto\ket\psi_C\ket\chi_{GB}\) for fixed \(\chi\).
Tracing \(CG\) proves complement constancy.
The entropy statement follows from \eqref{eq:normalform}.
\end{proof}

This is the standard exact correction criterion
\cite{KnillLaflamme,CheungEm05B}, not a new theorem supplied merely
by the Enter terminology.

\begin{lemma}[Replacement and approximate correction]
\label{lem:correction}
A replacement accessible channel has optimal recovery error \(a_d\);
every channel has recovery error at most \(a_d\).
If
\(\ddiam(\Nchan,\rep{\omega})\le\eta\), then
\[
 \erec(\Nchan)\ge\max\{0,a_d-\eta\}.
\]
For a complete interface, the information--disturbance theorem gives
\begin{equation}
 \erec(\Nchan)\le\min\{a_d,\sqrt{2\dleak(\Cchan)}\}.
 \label{eq:ksw}
\end{equation}
\end{lemma}

\begin{proof}
On a Bell input, a decoded replacement produces
\(I_Q/d\otimes\sigma_C\), with Bell overlap \(d^{-2}\).
Preparing \(I_C/d\) attains the lower bound: depolarization is the
uniform average of the \(d^2\) Weyl conjugations, one of which is
identity. This also gives the universal constant-decoder cap.
The accessible-replacement lower bound follows by contraction and
the reverse triangle inequality.

For the last assertion use the finite information--disturbance
inequality
\[
 \frac14
 \left(\inf_{\Dchan}\norm{\Dchan\Nchan-\id_C}_\diamond\right)^2
 \le
 \inf_\tau\norm{\Cchan-\rep{\tau}}_\diamond
\]
and substitute the half-diamond normalization
\cite{KSW}.
\end{proof}

\subsection{Destructive Enter}

For a basis \(\{\ket y\}\) of a measured factor \(A\), define
\[
 (\Enter_A\otimes\id_K)(X)
 =
 \sum_y\kb y y_Y\otimes
 (\bra y_A\otimes I_K)X(\ket y_A\otimes I_K).
\]
The measured name \(A\) is not returned.
Its complete dilation can be
\[
 \ket y_A\longmapsto\ket y_Y\ket y_L,
\]
with \(L\) unreturned.
A continuation with joint access to \(YL\) would have a different
quantum interface.

\begin{proposition}[Record obstruction and append-only entropy]
\label{prop:classical}
For every purely classical output channel on a \(d\)-dimensional code,
\[
 \erec(\Mchan)\ge b_d.
\]
Computational-basis measurement attains the bound.
For an unedited append-only record,
\[
 H(Y_{\le k+1})
 =H(Y_{\le k})+H(Y_{k+1}\mid Y_{\le k})
 \ge H(Y_{\le k}).
\]
Moreover,
\[
 \Drop_Y\circ\Enter_A=\Tr_A
\]
on all referenced operators.
\end{proposition}

\begin{proof}
Any decoder following a classical-output channel is
measure-and-prepare, so its Bell output is separable
\cite{HorodeckiShorRuskai}.
Every separable state has Bell overlap at most \(1/d\), since a
product state has overlap
\(\Tr(\rho_Q^{\mathsf T}\sigma_C)/d\le1/d\).
The Bell-projector test gives error at least \(b_d\).
Basis-state preparation gives dephasing, the average of \(d\)
phase conjugations, and has matching error \(b_d\).

The entropy assertion is the classical chain rule.
For the last assertion, expand
\(X=\sum_{y,z}X_{yz}\otimes\kb y z_A\).
Readout followed by discard returns \(\sum_yX_{yy}\), the partial
trace. The blocks may include any reference.
\end{proof}

In particular, a record-only endpoint cannot have a constant complete
complement. Terminal residual readout and discard do not improve the
already-returned accessible channel
\cite{CheungEn07A,CheungEn07B}.

%=====================================================================
\section{Nonuniform cells: an exact temporal proxy}
\label{sec:nonuniform}
%=====================================================================

Fix cell dimensions \(s_1,\ldots,s_N\ge2\) and a final core dimension
\(b_*\ge1\). Let
\[
 \HH_{\rm int}
 =
 \left(\bigotimes_{j=1}^N\mathbb C^{s_j}\right)
 \otimes\mathbb C^{b_*},
 \qquad
 D=b_*\prod_{j=1}^Ns_j.
\]
Prepare a fixed isometry \(J:C\to\HH_{\rm int}\), choose one
logical-input-independent unitary \(U\in U(D)\), and then release
the cells in order. Earlier outputs are untouched.

At cut \(k\),
\begin{equation}
 r_k=\prod_{j=1}^ks_j,\qquad
 b_k=b_*\prod_{j=k+1}^Ns_j,\qquad
 r_kb_k=D.
 \label{eq:dimensions}
\end{equation}
Haar statements refer to one Haar sample of \(U\), not independently
sampled states at different cuts. The pure input is fixed independently
of \(U\); \(U\) is known when a decoder is invoked.

A finite conserving implementation may give each occupied memory
and outgoing cell a common gap \(\omega\), with a vacant zero-energy
sector. The preparation occupies all memory cells. Extend \(U\) on
the all-occupied, degenerate internal sector and transfer each cell
by SWAP. These operations commute with the declared sum of
occupied-sector energies. This establishes a register conservation
law, not a gravitational mass law
\cite{Hawk05B,CheungHawk02B}.

For integers \(r,b\ge1\), put
\[
 m=\min(r,b),\quad n=\max(r,b),\quad D=rb,
 \qquad
 \Harm{\ell}=\sum_{j=1}^{\ell}\frac1j,
\]
and define
\begin{align}
 \PageMean(r,b)
 &=
 \frac{\Harm{rb}-\Harm n-(m-1)/(2n)}{\ln2},
 \label{eq:PageMean}\\
 \RecordMean(r,b)
 &=
 \frac{\Harm{rb}-\Harm b}{\ln2}.
 \label{eq:RecordMean}
\end{align}

\begin{lemma}[Page--Sen input]
\label{lem:pagesen}
A Haar pure state on \(\mathbb C^r\otimes\mathbb C^b\) has mean
reduction entropy \(\PageMean(r,b)\).
\end{lemma}

\begin{proof}
The Gaussian--Wishart evaluation in
Appendix~\ref{app:page} gives \eqref{eq:PageMean}.
This is the established Page--Sen formula
\cite{PageAverage,Sen}.
\end{proof}

\begin{theorem}[Two exact means for one cell-release proxy]
\label{thm:two}
For every fixed pure input independent of \(U\), coherent radiation
and immediate destructive cell readout have the exact means
\begin{equation}
 \Expectation_U S(R_k)=\PageMean(r_k,b_k),
 \qquad
 \Expectation_U H(Y_{\le k})=\RecordMean(r_k,b_k).
 \label{eq:twomeans}
\end{equation}
The classical mean increases strictly at each release.
The coherent mean increases on cuts with \(r_k\le b_k\) that remain
on that side, and decreases on cuts with \(r_k\ge b_k\) that remain
on that side. If the capacity crossing is traversed, a maximal cut
is among the two cuts bracketing it.

If \(b_*=1\), the coherent endpoint mean is zero while the record
endpoint mean is
\[
 \frac{\Harm D-1}{\ln2}>0.
\]
If the schedule stops before a crossing, no turnover is guaranteed.
\end{theorem}

\begin{proof}
Haar invariance makes \(UJ\ket\psi\) a Haar unit vector, and
Lemma~\ref{lem:pagesen} applies at every cut.

Write its product-basis components as normalized independent complex
Gaussians. A radiation prefix has \(b_k\) squared Gaussian components.
The \(r_k\) prefix probabilities therefore have a Dirichlet
distribution with all parameters \(b_k\).
Its moment identity is
\[
 \Expectation\sum_y p_y^t
 =
 r_k\frac{\Gamma(b_k+t)\Gamma(D)}
           {\Gamma(b_k)\Gamma(D+t)}.
\]
Differentiate at \(t=1\). Using
\(\psi_0(\ell+1)=\Harm{\ell}-\gamma_{\rm E}\) gives the record mean.
Later SWAPs act on unmeasured factors, so immediate readout has
precisely these prefix probabilities.

For a same-side coherent step, write the smaller and larger
dimensions as \(m,n\), and transfer an integer factor \(s\ge2\)
so that the next early-side dimensions are \(sm,n/s\).
Remaining on that side implies \(n\ge s^2m\).
The Page-mean increment in natural units is
\[
 \sum_{j=n/s+1}^{n}\frac1j
 -
 \frac{(s^2-1)m-(s-1)}{2n}
 \ge
 1-\frac1s-\frac{s^2-1}{2s^2}
 =
 \frac{(s-1)^2}{2s^2}>0.
\]
Exchange the two factors for the late side.
The record increment is
\((\Harm{b_{k-1}}-\Harm{b_k})/\ln2>0\).
The endpoint assertions follow by substitution.
\end{proof}

A large cell can jump over the equal-capacity point. The precise
finite maximum need not be located at a time satisfying
\(r_k=b_k\), because that equality may occur at no cut.

\subsection{Finite deficits and single-history control}

\begin{proposition}[Explicit capacity deficits]
\label{prop:deficits}
Let \(T(r,b)=\log_2m\). Then
\begin{align}
 0\le T(r,b)-\PageMean(r,b)
 &\le u(r,b):=\log_2(1+m/n),
 \label{eq:pagegap}\\
 0\le\log_2r-\RecordMean(r,b)
 &\le\frac1{b\ln2}.
 \label{eq:recordgap}
\end{align}
For a single sampled Haar history and every \(z>0\),
\begin{equation}
 \Probability\left\{
 \max_k\bigl(T(r_k,b_k)-S(R_k)\bigr)>z
 \right\}
 \le
 \min\left\{1,\frac{\sum_ku(r_k,b_k)}z\right\}.
 \label{eq:historymarkov}
\end{equation}
No independence of cuts is assumed.
\end{proposition}

\begin{proof}
Haar purity is
\(\Expectation\Tr\rho_m^2=(m+n)/(mn+1)\), obtained by the two-copy
swap test on
\(\Expectation\kb vv^{\otimes2}=(I+\mathsf F)/(D(D+1))\).
Since \(S(\rho)\ge-\log_2\Tr\rho^2\), Jensen's inequality gives
\[
 \log_2m-\Expectation S(\rho)
 \le
 \log_2\left(\frac{m(m+n)}{mn+1}\right)
 \le u(r,b).
\]
The upper entropy bound is the rank bound.

The harmonic sum obeys
\[
 \ln\frac{D+1}{b+1}
 \le\Harm D-\Harm b
 \le\ln\frac Db.
\]
Its difference from \(\ln r\) is at most
\(\ln(1+1/b)\le1/b\), proving \eqref{eq:recordgap}.
Finally, every capacity deficit is nonnegative, and its expectation
is bounded by \eqref{eq:pagegap}. Apply Markov's inequality to their
sum, which dominates their maximum.
\end{proof}

\subsection{What the entropy gap actually means}

Let \(\Delta_R\) dephase the declared radiation basis. Its diagonal
probabilities equal the destructive prefix record probabilities.

\begin{proposition}[The gap is marginal coherence]
\label{prop:coherence}
For every state \(\rho_R\),
\[
 H(Y)-S(\rho_R)
 =
 S(\Delta_R\rho_R)-S(\rho_R)
 =
 D_2(\rho_R\Vert\Delta_R\rho_R)\ge0.
\]
For the Haar proxy,
\begin{equation}
 \RecordMean(r,b)-\PageMean(r,b)
 =
 \begin{cases}
 \dfrac{r-1}{2b\ln2},&r\le b,\\[6pt]
 \dfrac{\Harm r-\Harm b+(b-1)/(2r)}{\ln2},&r\ge b.
 \end{cases}
 \label{eq:coherencegap}
\end{equation}
This is not a whole-code recovery diagnostic.
\end{proposition}

\begin{proof}
The diagonal logarithm satisfies
\(\Tr(\rho_R\log_2\Delta_R\rho_R)
=\Tr(\Delta_R\rho_R\log_2\Delta_R\rho_R)\).
Positivity ensures the required support containment.
The relative-entropy identity and nonnegativity follow.
Subtract \eqref{eq:PageMean} from \eqref{eq:RecordMean} for the
two cases.

For a Bell input to a \(d\)-dimensional identity channel, the
accessible marginal is \(I/d\), so the marginal gap is zero.
Nevertheless, destructive basis readout has recovery error \(b_d\).
Thus a vanishing gap does not mean no quantum-recovery loss.
\end{proof}

The quantity is the relative entropy of coherence in the selected
basis \cite{Baumgratz}. It is not a basis-independent count of
information destroyed, a residual leakage, or a Landauer cost.

%=====================================================================
\section{Typicality and physical transfer require different allowances}
\label{sec:transfer}
%=====================================================================

\subsection{A precise ensemble comparison}

Let \(\mu\) be a distribution of fixed unitaries on the same internal
space. Define
\[
 d_{\rm TV}(\mu,\mu_{\rm Haar})
 =
 \sup_E|\mu(E)-\mu_{\rm Haar}(E)|.
\]

\begin{proposition}[Ensemble mismatch]
\label{prop:TV}
If \(d_{\rm TV}(\mu,\mu_{\rm Haar})\le\tau\), then at a fixed cut
\[
 \left|\Expectation_\mu S(R)-\PageMean(r,b)\right|
 \le\tau\log_2m,
\]
\[
 \left|\Expectation_\mu H(Y)-\RecordMean(r,b)\right|
 \le\tau\log_2r.
\]
These statements apply to fixed inputs independent of the sampled
unitary.
\end{proposition}

\begin{proof}
Entropy ranges are \(0\le S(R)\le\log_2m\) for a pure complete
proxy and \(0\le H(Y)\le\log_2r\).
The expectation difference of a function in \([0,L]\) is at most
\(L\,d_{\rm TV}\), by integration of its level sets.
\end{proof}

Total variation is a strong sufficient assumption, not a claim that
physical scrambling achieves it.

A weaker hypothesis suffices for a leading capacity law.
For a fixed pure input, let
\[
 M_2^\mu=\Expectation_\mu\kb vv^{\otimes2},
 \qquad
 \norm{M_2^\mu-\frac{I+\mathsf F}{D(D+1)}}_1\le\chi.
\]
The purity swap witness has operator norm one, so
\[
 \Expectation_\mu\Tr\rho_m^2
 \le\frac{m+n}{mn+1}+\chi.
\]
The proof of Proposition~\ref{prop:deficits} then gives
\begin{equation}
 \Expectation_\mu(\log_2m-S(R))
 \le
 \min\left\{\log_2m,\,
 \log_2\left[m\left(\frac{m+n}{mn+1}+\chi\right)\right]\right\}.
 \label{eq:moment}
\end{equation}
Two-copy information can control this leading deficit; it does not
by itself establish either exact entropy mean.

\begin{remark}[Averaging a device is not knowing its unitary]
If the sampled unitary is forgotten and the accessible channel
itself is Haar-averaged, then
\[
 \Expectation_U\Nchan^U(X)=\Tr(X)\frac{I_R}{r}.
\]
That channel is replacement and has recovery error \(a_d\).
The Page average is \(\Expectation_U S(\rho_R^U)\), not
\(S(\Expectation_U\rho_R^U)\).
Likewise, the expression ``the complement is diamond-close to Haar''
has no meaning until a comparison channel, ensemble, input
quantifiers, and treatment of the random control are specified.
\end{remark}

\subsection{Finite-dimensional entropy continuity}

For an integer \(L\ge1\), define
\[
 c_L(\eta)=
 \begin{cases}
 0,&L=1,\\
 \hb(t)+t\log_2(L-1),
 &L\ge2,\quad t=\min\{\eta,1-1/L\}.
 \end{cases}
\]
If states on an \(L\)-dimensional space have trace distance at most
\(\eta\), then their entropy difference is at most \(c_L(\eta)\)
\cite{Audenaert}.

\begin{definition}[Compared extrapolation at a cut]
\label{def:comparison}
At a declared cut, supply:
\begin{enumerate}[label=\textup{(\roman*)}]
\item a physical accessible channel and a proxy accessible channel
on the same code, after justified access-preserving identifications;
\item a common accessible comparison dimension \(L\), with
\(\ddiam(\Nchan_{\rm phys}^U,\Nchan_{\rm proxy}^U)\le\eta\);
\item a physical detector channel compared with the proxy basis
readout, with channel allowance \(\nu\), and common record dimension
\(K\);
\item a distribution \(\mu\) of proxy unitaries with TV allowance
\(\tau\), if an exact Haar-mean comparison is desired;
\item real target capacities \(A,B\) and integer proxy dimensions
\(r,b\) with
\[
 |\log_2r-A|\le e_R,\qquad
 |\log_2b-B|\le e_B.
\]
\end{enumerate}
The physical and proxy families must be coupled on these interfaces.
If a temporal claim is made, the cuts must belong to compatible
complete histories.
\end{definition}

\begin{theorem}[Two-curve extrapolation budget]
\label{thm:transfer}
Under Definition~\ref{def:comparison}, for the declared fixed pure
input,
\begin{align}
 \left|\Expectation S(R_{\rm phys})-\min(A,B)\right|
 &\le
 c_L(\eta)+\tau\log_2m+u(r,b)+\max(e_R,e_B),
 \label{eq:Qbudget}\\
 \left|\Expectation H(Y_{\rm phys})-A\right|
 &\le
 c_K(\min\{1,\eta+\nu\})
 +\tau\log_2r+\frac1{b\ln2}+e_R.
 \label{eq:Ybudget}
\end{align}
The record-only recovery obstruction remains independent of these
entropy allowances.
\end{theorem}

\begin{proof}
The channel comparison bounds the compared output trace distances,
including references, by \(\eta\).
Detector composition contraction and the triangle inequality give
record distance at most \(\min\{1,\eta+\nu\}\).
Apply the finite entropy continuity bounds and take expectations.
Proposition~\ref{prop:TV} replaces the proxy ensemble mean by its
Haar mean. Proposition~\ref{prop:deficits} compares those means with
\(\log_2m\) and \(\log_2r\).
Finally, \(\min\) is one-Lipschitz in the maximum norm:
\[
 |\min(\log_2r,\log_2b)-\min(A,B)|
 \le\max(e_R,e_B).
\]
This proves both budgets.
The classical recovery lower bound is
Proposition~\ref{prop:classical}.
\end{proof}

The dimensions \(L,K\) are those of the actual common comparison
spaces. If errors populate additional sectors, their dimensions
cannot be deleted from the continuity modulus without a support
estimate.

Compatible preparation, evolution, collection, and implementation
channel errors telescope by replacing one factor at a time.
This yields a sum of channel allowances, capped at one.
A metric discrepancy, power error, or selected occupation error
cannot be inserted into that sum without a conversion theorem
\cite{CheungHawk04D,CheungEn02C}.

%=====================================================================
\section{Replacing the clock: compatibility, crossover, and cutoff}
\label{sec:clock}
%=====================================================================

\subsection{The constant-capacity-sum requirement}

\begin{theorem}[Fixed tensor-release compatibility]
\label{thm:compatibility}
For the proxy of Section~\ref{sec:nonuniform},
\[
 \log_2r_k+\log_2b_k=\log_2D
\]
at every cut. If target capacities \(A(t_k),B(t_k)\) satisfy the
rounding bounds of Definition~\ref{def:comparison}, then
\begin{equation}
 |A(t_k)+B(t_k)-\log_2D|\le e_{R,k}+e_{B,k}.
 \label{eq:sumconstraint}
\end{equation}
In particular, for any two cuts \(j,k\),
\begin{equation}
 |[A+B](t_k)-[A+B](t_j)|
 \le e_{R,k}+e_{B,k}+e_{R,j}+e_{B,j}.
 \label{eq:sumdifference}
\end{equation}
This is a necessary condition for that fixed-dimensional model,
not for every possible evaporation model.
\end{theorem}

\begin{proof}
Equation~\eqref{eq:dimensions} gives \(r_kb_k=D\).
Subtract the two target capacities from the corresponding
logarithms and use the triangle inequality.
\end{proof}

Changing support sectors, adding fresh resources, or admitting
additional inaccessible factors can produce other models.
Such changes require a new complete history and state calculation.
They do not preserve the fixed tensor-release theorem merely by
keeping the word ``Haar.''

\subsection{A supplied semiclassical clock}

Only in this subsection and its clock examples set
\(G=\hbar=c=k_{\mathrm B}=1\).
For a Schwarzschild mass label, the Bekenstein--Hawking entropy in
bits is
\begin{equation}
 B(M)=c_{\rm BH}M^2,\qquad c_{\rm BH}=\frac{4\pi}{\ln2}.
 \label{eq:BH}
\end{equation}
This is a supplied semiclassical entropy law
\cite{Bekenstein,Hawking}. Identifying \(B(M)\) with
\(\log_2\dim B_{\rm proxy}\) is an additional modelling declaration.

Let \(M_0>M_{\rm cut}>0\). Supply a positive continuous power law
\(P(M)\) on \([M_{\rm cut},M_0]\) and define
\[
 \dot M=-P(M),\qquad M(t_0)=M_0.
\]
Its cutoff clock is the inverse of
\[
 t(M)-t_0=\int_M^{M_0}\frac{du}{P(u)}.
\]
Continuity and positivity make this inverse unique.

Supply a positive continuous capacity-production ratio \(\xi(M)\)
and define the \emph{target}, not the fine-grained entropy,
\begin{equation}
 A(M)=\int_M^{M_0}\xi(u)B'(u)\,du.
 \label{eq:A}
\end{equation}
The candidate envelope is
\begin{equation}
 F(t)=\min\{A(M(t)),B(M(t))\}.
 \label{eq:envelope}
\end{equation}

\begin{theorem}[Capacity crossover before a positive cutoff]
\label{thm:crossover}
For the supplied laws above:
\begin{enumerate}[label=\textup{(\roman*)}]
\item \(A(M(t))\) increases strictly and \(B(M(t))\) decreases
strictly;
\item there is exactly one crossover in the retained clock interval
if and only if
\[
 A(M_{\rm cut})\ge B(M_{\rm cut});
\]
\item a strict interior turnover occurs if and only if that inequality
is strict;
\item if \(A(M_{\rm cut})<B(M_{\rm cut})\), the envelope is increasing
throughout the interval.
\end{enumerate}
The crossover is a capacity-envelope time, not automatically a
physical recovery transition or the exact maximum of a finite
Page mean.
\end{theorem}

\begin{proof}
Differentiating gives
\[
 \frac{dA}{dt}=\xi(M)B'(M)P(M)>0,
 \qquad
 \frac{dB}{dt}=-B'(M)P(M)<0.
\]
The difference \(A-B\) starts at \(-B(M_0)\), increases strictly
in time, and is continuous. The intermediate value theorem proves
existence and uniqueness exactly under the endpoint condition.
The minimum selects the increasing branch before the crossing and
the decreasing branch afterward.
\end{proof}

\subsection{An inverse-square example}

Assume
\[
 P(M)=\frac{\lambda}{M^2},\qquad \lambda>0,
 \qquad
 \xi(M)=\xi>0.
\]
The flux coefficient \(\lambda\) is not the surface gravity
\(\kappa_{\rm H}\).

\begin{corollary}[Explicit clock and crossover]
\label{cor:clock}
Within the positive-cutoff interval,
\begin{align}
 M(t)&=[M_0^3-3\lambda(t-t_0)]^{1/3},\\
 t_{\rm cut}-t_0&=\frac{M_0^3-M_{\rm cut}^3}{3\lambda},\\
 A(M)&=\xi c_{\rm BH}(M_0^2-M^2).
\end{align}
The formal crossover mass and time are
\begin{align}
 M_{\rm P}&=M_0\sqrt{\frac{\xi}{1+\xi}},\\
 t_{\rm P}-t_0
 &=\frac{M_0^3}{3\lambda}
   \left[1-\left(\frac{\xi}{1+\xi}\right)^{3/2}\right],
 \label{eq:tP}\\
 F(t_{\rm P})
 &=\frac{\xi}{1+\xi}B(M_0).
\end{align}
They belong to the retained model only if
\(M_{\rm cut}\le M_{\rm P}\).
No value after \(t_{\rm cut}\) is predicted.
\end{corollary}

\begin{proof}
Integrate \(M^2\,dM=-\lambda\,dt\), then solve \(A(M)=B(M)\).
The cutoff condition is Theorem~\ref{thm:crossover}.
\end{proof}

The dependence on the entropy-production ratio is the same algebra
that appears in Page's semiclassical time profile
\cite{PageTime}. A half-capacity crossing is not generically half
the mass or half the formal lifetime.

\begin{proposition}[The \(\xi\ne1\) compatibility obstruction]
\label{prop:xi}
The inverse-square clock itself can reparametrize a fixed
tensor-release proxy. The assignments
\[
 \log_2r=A(M),\qquad \log_2b=B(M)
\]
cannot both be exact assignments in that proxy over a nontrivial
mass interval when \(\xi\ne1\).
More quantitatively, they must satisfy
\[
 |\xi-1|[B(M_0)-B(M)]
 \le w_0+w_M,
 \qquad
 w=e_R+e_B.
\]
\end{proposition}

\begin{proof}
Here
\[
 A(M)+B(M)
 =\xi B(M_0)+(1-\xi)B(M).
\]
It is constant on a nontrivial interval exactly when \(\xi=1\).
Subtract its initial value and apply
\eqref{eq:sumdifference}.
\end{proof}

Thus taking a coarse entropy-production ratio different from one
does not simply ``replace the clock'' in Hawk05-B. It changes the
capacity model, and potentially its state ensemble.

\subsection{A compatible, rounded two-curve sketch}

Choose \(\xi=1\), assume \(B_0:=B(M_0)\ge2\), and put
\[
 n=\lfloor B_0\rfloor,\qquad
 k(t)=\min\{n,\lfloor A(M(t))\rfloor\}.
\]
Use \(n\) qubit cells and one sampled Haar unitary on their
\(2^n\)-dimensional internal space. Release a cell whenever \(k(t)\)
increments. This gives one nested finite history.

\begin{theorem}[A proved two-curve clocked proxy]
\label{thm:sketch}
For this declared construction,
\[
 r(t)=2^{k(t)},\qquad b(t)=2^{n-k(t)},\qquad D=2^n,
\]
and
\[
 |\log_2r(t)-A(M(t))|\le1,\qquad
 |\log_2b(t)-B(M(t))|\le1.
\]
Its exact coherent and record means are
\[
 Q(t)=\PageMean(2^{k(t)},2^{n-k(t)}),
 \qquad
 L(t)=\RecordMean(2^{k(t)},2^{n-k(t)}).
\]
They obey
\begin{align}
 |Q(t)-F(t)|
 &\le
 1+\log_2\left(1+2^{-|n-2k(t)|}\right),
 \label{eq:sketchQ}\\
 |L(t)-A(M(t))|
 &\le
 1+\frac{2^{-(n-k(t))}}{\ln2}.
 \label{eq:sketchL}
\end{align}
If \(B(M_{\rm cut})\ge1\), at least one residual qubit remains at
the cutoff. No terminal drain is included.
\end{theorem}

\begin{proof}
Before the cap in \(k(t)\), the radiation rounding error is less
than one and
\[
 n-k(t)-B(M(t))
 =
 n-B_0+A(M(t))-\lfloor A(M(t))\rfloor
\]
lies between \(-1\) and \(1\).
After the cap, both target errors are at most one because
\(A+B=B_0\) and \(A\ge n\).
Apply Proposition~\ref{prop:deficits}.
If \(B(M_{\rm cut})\ge1\), then
\(A(M_{\rm cut})\le B_0-1\), so \(k(t_{\rm cut})\le n-1\).
\end{proof}

This is a completed \emph{proxy} calculation combining a clock with
two return types. Its physical extrapolation is exactly
Theorem~\ref{thm:transfer}, with \(\eta(t),\nu(t),\tau\) still
requiring justification.

\subsection{Clock and peak stability}

\begin{proposition}[Power allowance and peak-location allowance]
\label{prop:stability}
Suppose a supplied actual power on the same mass interval satisfies
\[
 (1-\rho)\frac{\lambda}{M^2}
 \le P_{\rm act}(M)
 \le(1+\rho)\frac{\lambda}{M^2},
 \qquad 0\le\rho<1.
\]
Then its time to any mass \(M\) in that interval satisfies
\[
 \frac{M_0^3-M^3}{3\lambda(1+\rho)}
 \le t_{\rm act}(M)-t_0
 \le
 \frac{M_0^3-M^3}{3\lambda(1-\rho)}.
\]

Separately, suppose the inverse-square envelope has an interior
crossover and a function \(f(t)\) satisfies
\(\sup_t|f(t)-F(t)|\le E\).
Every attained global maximum \(\widetilde t\) of \(f\) obeys
\[
 |\widetilde t-t_{\rm P}|
 \le\frac{2E}{v},
 \qquad
 v=\frac{2c_{\rm BH}\lambda\min\{\xi,1\}}{M_0}.
\]
\end{proposition}

\begin{proof}
Integrate the bounds on \(1/P_{\rm act}(M)\).
For the second assertion, the envelope has slope
\(2c_{\rm BH}\lambda\xi/M\) on its early branch and negative slope
of magnitude \(2c_{\rm BH}\lambda/M\) on its late branch.
Both magnitudes are at least \(v\), so
\[
 F(t_{\rm P})-F(t)\ge v|t-t_{\rm P}|.
\]
At a maximizing \(\widetilde t\),
\(f(\widetilde t)\ge f(t_{\rm P})\), which implies
\(F(t_{\rm P})-F(\widetilde t)\le2E\).
\end{proof}

The power allowance controls a clock. The entropy allowance controls
a peak of a specified function. Neither supplies the other.

%=====================================================================
\section{Greybody weighting: what is calculable and what is declared}
\label{sec:greybody}
%=====================================================================

\subsection{Occupation and flux are not dimensions}

In the stationary massless scalar setup, with the additional
time--frequency counting convention, the supplied occupation is
\[
 n_\alpha(\omega;M)
 =
 \frac{\Gamma_\alpha(\omega;M)}
      {e^{8\pi M\omega}-1}.
\]
Its flux is
\begin{equation}
 P(M)
 =
 \sum_\alpha g_\alpha
 \int_0^\infty\frac{d\omega}{2\pi}\,
       \omega n_\alpha(\omega;M).
 \label{eq:power}
\end{equation}
Here \(\alpha\) denotes the declared modes or species and
\(g_\alpha\) their counted multiplicities.
The stationary field derivation and its validity hypotheses are
external physical inputs
\cite{Hawking,PageEmission,CheungEm03A}.
Using them at an evolving \(M(t)\) additionally assumes a
quasistatic approximation.

If
\(\Gamma_\alpha(\omega;M)=\gamma_\alpha(M\omega)\) for a fixed
massless mode set, and the following integrals converge, then
\[
 P(M)=\frac{\lambda}{M^2},\qquad
 \lambda=
 \sum_\alpha g_\alpha
 \int_0^\infty\frac{dx}{2\pi}
 \frac{x\gamma_\alpha(x)}{e^{8\pi x}-1}.
\]
This change of variables explains the inverse-square clock.
It says nothing by itself about \(r(t)\) or \(b(t)\).

\begin{proposition}[Mean occupation underdetermines capacity]
\label{prop:occupation}
For \(0<u<1\), a bosonic number mean \(u\) is compatible with both
\[
 (1-u)\kb0 0+u\kb1 1
\]
and the geometric state
\[
 \rho_u=(1-z)\sum_{n=0}^\infty z^n\kb n n,
 \qquad z=\frac{u}{1+u}.
\]
They have different spectra and support dimensions.
Therefore an occupation mean does not determine a Hilbert-space
dimension or a Haar state ensemble.
\end{proposition}

\begin{proof}
Both states have mean \(u\), by a direct sum of their number
probabilities. The first has rank two and the second has infinite
rank. Equality of one number moment does not identify their states.
\end{proof}

Quasifreeness and covariance data can identify geometric or multimode
Gaussian packet states under the additional assumptions in
Cheung's En02-E and Em03-B records
\cite{CheungEn02E,CheungEm03B}.
Those assumptions are stronger than a selected mean.

For a known geometric state, truncation through level \(K\) has tail
\(z^{K+1}\). The normalized truncated marginal has trace distance
\(z^{K+1}\) from the original marginal. The normalized truncation of
its standard pure purification instead has trace distance
\(\sqrt{z^{K+1}}\). These follow from the orthogonal tail and pure-state
overlap. They are state-specific estimates, not a uniform
whole-code diamond certificate.

\subsection{An additional coarse entropy-rate model}

If one separately assumes independent geometric outgoing modes and
the same counting convention, define
\[
 g_2(u)=(u+1)\log_2(u+1)-u\log_2u,
 \qquad g_2(0)=0,
\]
and a coarse entropy flux
\[
 \dot A_{\rm cg}(M)
 =
 \sum_\alpha g_\alpha
 \int_0^\infty\frac{d\omega}{2\pi}\,
 g_2(n_\alpha(\omega;M)).
\]
The geometric-state entropy is \(g_2(u)\), obtained by summing its
number probabilities. Mode independence makes these entropies
additive. This is a coarse state model, not the fine-grained
entropy of a variable-code evaporation channel.

Under the same massless scaling,
\[
 \dot A_{\rm cg}(M)=\frac{\sigma}{M},
 \qquad
 \sigma=
 \sum_\alpha g_\alpha
 \int_0^\infty\frac{dx}{2\pi}\,
 g_2\!\left(\frac{\gamma_\alpha(x)}{e^{8\pi x}-1}\right).
\]
Since \(-\dot B=8\pi\lambda/(M\ln2)\), its declared ratio is
\begin{equation}
 \xi=\frac{\sigma\ln2}{8\pi\lambda}.
 \label{eq:xi-greybody}
\end{equation}
Equations \eqref{eq:power} and \eqref{eq:xi-greybody} require different
state and counting assumptions. If this ratio is not one, using
\(A_{\rm cg}\) as \(\log_2r\) changes the fixed-dimensional proxy,
by Proposition~\ref{prop:xi}.

A greybody factor suppresses a selected transmitted occupation.
It does not universally specify an ``early-time suppression'' of
a Page curve, nor prove typicality of the source code.

\subsection{Probability-weighted transfer is a different exact proxy}

Let \(C=\bigotimes_{j=1}^K C_j\), with \(\dim C_j=d_j\), and supply
probabilities \(q_j\in[0,1]\).
On each factor use the flagged transfer isometry
\[
 \ket\psi_{C_j}
 \longmapsto
 \sqrt{1-q_j}\ket\psi_{R_j}\ket{\varnothing}_{B_j}
 +
 \sqrt{q_j}\ket{\varnothing}_{R_j}\ket\psi_{B_j}.
\]
The occupied logical sector and vacancy are orthogonal.
Let \(S\subseteq\{1,\ldots,K\}\) denote the surviving accessible
factors and put
\[
 w_S=\prod_{j\in S}(1-q_j)\prod_{j\notin S}q_j.
\]

\begin{theorem}[Nonuniform product-transfer calibration]
\label{thm:product}
For arbitrary input \(\rho_C\),
\begin{equation}
 S(R)=\sum_{j=1}^K\hb(q_j)+\sum_Sw_SS(\rho_{C_S}).
 \label{eq:productentropy}
\end{equation}
Its exact whole-product-code recovery error is
\begin{equation}
 \erec(\Nchan)
 =
 1-\prod_{j=1}^K\left[1-q_j(1-d_j^{-2})\right].
 \label{eq:productrecovery}
\end{equation}
These are product-erasure formulas, not Haar Page formulas.
\end{theorem}

\begin{proof}
The accessible output is an orthogonal direct sum over flags \(S\),
with weights \(w_S\) and states \(\rho_{C_S}\).
The flag entropy is the sum of the independent binary entropies,
which proves \eqref{eq:productentropy}.

Compress every occupied block and append \(I_{C_j}/d_j\) on
vacancy. The decoded product channel is
\[
 \bigotimes_j
 \left[(1-q_j)\id_{C_j}+q_j\rep{I_{C_j}/d_j}\right].
\]
Write each replacement as a Weyl average. The identity-conjugation
coefficient is
\(\prod_j[1-q_j(1-d_j^{-2})]\); separating that term gives the
upper bound \eqref{eq:productrecovery}.

For the lower bound use the product Bell input.
On flag \(S\), the missing references are independent maximally
mixed states and no decoder can attain full Bell overlap larger
than \(\prod_{j\notin S}d_j^{-2}\).
Average over flags. The resulting overlap bound is the same product
coefficient, proving optimality.
\end{proof}

This extends the supplied multi-cell calibration in Hawk02-B
\cite{CheungHawk02B}. A proposed greybody-to-\(q_j\) map must still
be stated and justified. At \(q_j=0\) for every \(j\), the code is
released, not retained. Entropy turnover alone does not prove that
a particular process retains the code; a retaining comparison must
be constructed separately.

%=====================================================================
\section{Conserved-sector scrambling: a direct-sum Page calculation}
\label{sec:sectors}
%=====================================================================

A conserved total additive charge need not leave a tensor-product
sector. At a cut, suppose the justified sector is
\begin{equation}
 \HH_{\mathsf q}
 =
 \bigoplus_{\lambda}
 \left(R_\lambda\otimes B_{\mathsf q-\lambda}\right),
 \label{eq:charge-sector}
\end{equation}
embedded in the physical tensor product. Distinct \(\lambda\)
have orthogonal radiation charge sectors and orthogonal residual
charge sectors. Retain only nonzero blocks, and set
\[
 r_\lambda=\dim R_\lambda,\quad
 b_\lambda=\dim B_{\mathsf q-\lambda},\quad
 D_\lambda=r_\lambda b_\lambda,\quad
 D_{\mathsf q}=\sum_\lambda D_\lambda,
\]
\[
 w_\lambda=\frac{D_\lambda}{D_{\mathsf q}},
 \qquad
 m_\lambda=\min(r_\lambda,b_\lambda),\quad
 n_\lambda=\max(r_\lambda,b_\lambda).
\]

\begin{theorem}[Sector Page and sector record means]
\label{thm:sector}
For a Haar pure vector in \(\HH_{\mathsf q}\), the mean radiation
entropy is
\begin{equation}
 \boxed{
 \Expectation S(R)
 =
 \frac{
 \Harm{D_{\mathsf q}}
 -\sum_\lambda w_\lambda\Harm{n_\lambda}
 -\sum_\lambda w_\lambda(m_\lambda-1)/(2n_\lambda)}
 {\ln2}.
 }
 \label{eq:sectorpage}
\end{equation}
Destructive basis readout resolving each radiation charge sector has
mean entropy
\begin{equation}
 \boxed{
 \Expectation H(Y)
 =
 \frac{\Harm{D_{\mathsf q}}
       -\sum_\lambda w_\lambda\Harm{b_\lambda}}{\ln2}.
 }
 \label{eq:sectorrecord}
\end{equation}
The sector geometry and weights cannot be replaced by two aggregate
dimensions in the unrestricted Page formula.
\end{theorem}

\begin{proof}
Use independent Gaussian coordinates in all sector blocks.
Their squared block norms \(p_\lambda\) have the Dirichlet law
with parameters \(D_\lambda\). Conditional normalized directions
are independent Haar vectors in the individual blocks and
independent of the weights.

The residual trace kills cross-charge coherences, so
\[
 \rho_R=\bigoplus_\lambda p_\lambda\rho_{R_\lambda},
 \qquad
 S(\rho_R)=H(p)+\sum_\lambda p_\lambda S(\rho_{R_\lambda}).
\]
Dirichlet differentiation gives
\[
 \Expectation H(p)
 =
 \frac{\Harm{D_{\mathsf q}}
       -\sum_\lambda w_\lambda\Harm{D_\lambda}}{\ln2}.
\]
Add \(\sum_\lambda w_\lambda\PageMean(r_\lambda,b_\lambda)\);
the \(\Harm{D_\lambda}\) terms cancel, yielding
\eqref{eq:sectorpage}.

Every individual radiation basis outcome in sector \(\lambda\)
collects \(b_\lambda\) Gaussian coordinates.
Its probability has a Dirichlet parameter \(b_\lambda\),
repeated \(r_\lambda\) times.
The same moment differentiation gives
\eqref{eq:sectorrecord}.
\end{proof}

This direct-sum structure is the standard symmetry-resolved
random-state setting \cite{SymmetryPage}. Its derivation is included
to make the missing tensor-product hypothesis visible.

If the sector probabilities are instead fixed at specified
\(p_\lambda\), with independent conditional Haar directions, the
means become
\[
 H(p)+\sum_\lambda p_\lambda\PageMean(r_\lambda,b_\lambda)
\]
and
\[
 H(p)+\sum_\lambda p_\lambda\RecordMean(r_\lambda,b_\lambda).
\]
Externally assigned sector weights define a different ensemble from
Haar measure on the whole conserved sector.

\begin{example}[One excitation does not give an extensive tent]
\label{ex:onecharge}
For \(N\) qubits with total excitation number one and a cut after
\(k\) qubits, the sector has dimension \(N\).
For \(0<k<N\), its two blocks have dimensions
\[
 (r_0,b_0)=(1,N-k),\qquad
 (r_1,b_1)=(k,1).
\]
The coherent reduction has rank at most two and entropy at most
one bit. The exact mean is
\[
 \Expectation S(R_k)
 =
 \frac{
 \Harm N-\frac{N-k}{N}\Harm{N-k}
          -\frac{k}{N}\Harm k}{\ln2}.
\]
The record mean is
\[
 \Expectation H(Y_{\le k})
 =
 \frac{
 \Harm N-\frac{N-k}{N}\Harm{N-k}
          -\frac{k}{N}\Harm1}{\ln2}.
\]
The endpoint coherent entropy is zero, but the curve is not
\(\min(k,N-k)\) bits.
\end{example}

More generally, for fixed excitation number \(q\),
\[
 r_\lambda=\binom{k}{\lambda},\qquad
 b_\lambda=\binom{N-k}{q-\lambda},\qquad
 D_q=\binom Nq.
\]
Vandermonde's identity verifies the total sector dimension.
A single Haar unitary within that fixed sector and passive cell
release give compatible cuts to which Theorem~\ref{thm:sector}
applies. A mass window, non-Abelian angular momentum sector, or
interacting energy shell needs its own decomposition; it is not
automatically \eqref{eq:charge-sector}.

Sector projection is also not free postselection. The code must be
prepared in the stated sector, or the selective operation and its
failure branch must belong to the task.

%=====================================================================
\section{Access algebras and independently certified recovery}
\label{sec:algebras}
%=====================================================================

\subsection{Access is not automatically a spatial tensor factor}

In a finite physical output space, declare a unital observable algebra
\(\mathcal A\subseteq\BB(\HH_{\rm phys})\).
For a tensor factor this can be
\(\BB(R)\otimes I_B\).
For a general finite algebra, a decomposition is
\[
 \HH_{\rm phys}
 =
 \bigoplus_\alpha
 \left(\mathbb C^{n_\alpha}\otimes\mathbb C^{m_\alpha}\right),
 \qquad
 \mathcal A
 =
 \bigoplus_\alpha
 \left(M_{n_\alpha}\otimes I_{m_\alpha}\right).
\]
Its restricted state can be described by weights \(p_\alpha\) and
normalized accessible matrices \(\rho_\alpha\).
With the explicitly chosen ordinary matrix trace on each accessible
block, one may define
\[
 S_{\mathcal A}
 =
 H(p)+\sum_\alpha p_\alpha S(\rho_\alpha).
\]
This entropy convention excludes the multiplicity factors
\(m_\alpha\); it is not automatically the entropy of an ambient
density operator.

Cheung's En05-C compares state restrictions with Heisenberg
containment without assuming a spatial complement
\cite{CheungEn05C}.
A continuum gravitational algebra requires its own representation,
trace or entropy prescription, and comparison theorem.

\begin{proposition}[Nested access and entropy]
\label{prop:nested}
If a later accessible channel admits the earlier one as CPTP
postprocessing, then its unrestricted recovery error cannot increase.
No corresponding monotonicity holds for quantum subsystem entropy.
Restriction of access alone does not imply a Page curve.
\end{proposition}

\begin{proof}
A later decoder can first apply the postprocessing and then an
earlier decoder. Quantum entropy can decrease on adding a factor:
the first half of a Bell pair has positive entropy, whereas the
complete pair is pure.
Conversely, a pure product state has zero entropy on every tensor
prefix despite the same sequence of increasing access dimensions.
\end{proof}

Laddha, Prabhu, Raju, and Shrivastava argue, under their asymptotically
flat quantum-gravity assumptions, for information availability near
early cuts of null infinity and discuss an entropy behavior unlike
the usual Page curve \cite{NullInfinity}.
That is an external proposal, not a proof that the present finite
proxy is an algebra at infinity.
Removing an observable such as the Hamiltonian also requires defining
a remaining observable algebra; arbitrary deletion need not leave an
algebra closed under multiplication.

\subsection{The precise obstruction to duplicate logical algebras}

\begin{proposition}[No commuting duplicate action on one code]
\label{prop:duplicates}
Let \(V:C\to H\) be an isometry.
There cannot be two commuting unital star representations
\(\pi_1,\pi_2:\BB(C)\to\BB(H)\) such that
\[
 \pi_1(X)V=VX,\qquad \pi_2(X)V=VX
\]
for every \(X\in\BB(C)\), when \(d\ge2\).
\end{proposition}

\begin{proof}
For all \(X,Y\),
\[
 VXY=\pi_1(X)\pi_2(Y)V
     =\pi_2(Y)\pi_1(X)V
     =VYX.
\]
Injectivity of \(V\) would imply \(XY=YX\) for every pair of
logical matrices, a contradiction.
\end{proof}

Two independent commuting matrix algebras can certainly exist.
The obstruction concerns their representing the \emph{same}
logical input. The partner comparisons in Hawk03-A must retain
that distinction \cite{CheungHawk03A}.
An island formula is not a theorem of this finite algebraic
proposition.

\subsection{A deterministic Petz candidate and its certificate}

Let \(\widehat{\Nchan}:C\to R\) be a supplied finite proxy channel
and choose a full-rank code state \(\sigma_C\).
Set
\[
 s_R=\widehat{\Nchan}(\sigma_C),
 \qquad P=\supp s_R,
\]
and use the inverse of \(s_R\) only on its support.
For a fixed code state \(\omega_C\), define
\begin{equation}
 \mathcal R_{\sigma,\widehat{\Nchan}}(Y)
 =
 \sigma_C^{1/2}\widehat{\Nchan}^{\,*}
 \left(s_R^{-1/2}PYPs_R^{-1/2}\right)\sigma_C^{1/2}
 +
 \Tr((I-P)Y)\omega_C.
 \label{eq:petz}
\end{equation}

\begin{lemma}[Full-domain Petz completion]
\label{lem:petz}
Equation~\eqref{eq:petz} is CPTP on the entire accessible input
space. It is a candidate decoder, not an automatic recovery theorem.
\end{lemma}

\begin{proof}
Both terms are CP.
The trace of the first is
\[
 \Tr\left(
 s_Rs_R^{-1/2}PYPs_R^{-1/2}
 \right)=\Tr(PY).
\]
The second contributes \(\Tr((I-P)Y)\), proving trace preservation.
\end{proof}

Petz and universal recovery channels have important uses in
entanglement-wedge reconstruction under additional hypotheses
\cite{ChenPetz,CotlerRecovery}. Those geometric conclusions are
not imported here.

Let
\[
 \ket{\Phi_d}=\frac1{\sqrt d}\sum_i\ket i_Q\ket i_C,
 \qquad
 \Phi_d=\kb{\Phi_d}{\Phi_d},
 \qquad
 \Jnorm(\Phi)=(\id_Q\otimes\Phi)(\Phi_d).
\]

\begin{lemma}[Normalized Choi comparison]
\label{lem:choi}
For channels with input dimension \(d\),
\[
 \frac12\norm{\Jnorm(\Phi)-\Jnorm(\Psi)}_1
 \le\ddiam(\Phi,\Psi)
 \le\frac d2\norm{\Jnorm(\Phi)-\Jnorm(\Psi)}_1.
\]
\end{lemma}

\begin{proof}
The first inequality uses the Bell input.
Any pure referenced input can be written
\((L\otimes I)\ket{\Phi_d}\), with
\(\Tr(L^\dagger L)=d\) and \(\norm L_\infty^2\le d\).
Multiplying the Choi difference on both sides by \(L\otimes I\)
bounds its trace norm by \(d\) times the Choi trace norm.
Convexity handles mixed inputs
\cite[Chapter 3]{Watrous}.
\end{proof}

\begin{theorem}[Independent recovery extrapolation]
\label{thm:recovery}
Suppose actual and proxy complete channels are compared under
access-preserving identifications with distance at most \(\epsilon\).
If
\[
 \dleak(\widehat{\Cchan})\le\zeta,
\]
then
\begin{align}
 \dleak(\Cchan_{\rm act})&\le\min\{1,\zeta+\epsilon\},\\
 \erec(\Nchan_{\rm act})
 &\le\min\{a_d,\epsilon+\sqrt{2\zeta}\}.
 \label{eq:recoverybudget}
\end{align}
This is an information-theoretic decoder-existence conclusion.

Alternatively, let a specified proxy decoder \(\widehat{\Dchan}\),
possibly \eqref{eq:petz}, have discrepancy
\[
 r_{\rm dec}
 =
 \frac12\norm{
 \Jnorm(\widehat{\Dchan}\widehat{\Nchan})-\Phi_d}_1.
\]
If its actual implementation has
\(\ddiam(\Dchan_{\rm imp},\widehat{\Dchan})\le e_{\rm imp}\),
then that fixed implementation has error at most
\begin{equation}
 \min\{1,\,d r_{\rm dec}+\epsilon+e_{\rm imp}\}.
 \label{eq:specifiedbudget}
\end{equation}
\end{theorem}

\begin{proof}
Partial traces contract the complete-channel comparison.
The replacement triangle inequality proves leakage.
Apply a proxy-optimal decoder to the actual accessible channel
and use Lemma~\ref{lem:correction}.
The independent constant decoder gives the \(a_d\) cap.
For the specified decoder, use Lemma~\ref{lem:choi} and telescope
the channel and implementation differences.
\end{proof}

No entropy or greybody statistic was substituted for \(\zeta\),
\(\epsilon\), or \(r_{\rm dec}\).
A recovery claim additionally needs the same-model preparation,
complete evolution, access identification, and complement
certificate emphasized in En07-B and Hawk04-D
\cite{CheungEn07B,CheungHawk04D}.

Whole-code recovery is also stronger than one-effect recovery.
For example, qubit dephasing preserves the basis effect
\(\kb0 0\) exactly but has whole-code recovery error \(1/2\).
This is the kind of target separation documented in Oai290-A
\cite{CheungOai290A}.
Finite streaming workspace obstructions belong to their specified
benchmark and resource models, not to every possible decoder
\cite{CheungEn02B,CheungEn02C}.

\subsection{An entropy-history control with opposite recovery}

For completeness, append a degenerate inaccessible register
\(L\simeq C\) to a \(b_*=1\) proxy.
The returning model scrambles and releases the variable input while
fixing \(L\).
The retaining model stores the variable input in \(L\) and runs the
same proxy on a fixed dummy input \(\ket0_C\).

For \(\ket0_C\), the accessible histories agree exactly for the same
unitary. For every fixed pure input, both ensembles have the same
Page and record means. At the coherent endpoint, recovery errors
are respectively \(0\) and \(a_d\).
The retaining complement carries the code isometrically.
This proves that the entropy means and the supplied clock do not
determine endpoint recovery
\cite{CheungHawk01B,Hawk05B}.
It does not compare two models under an identical requirement of
complete outgoing access and a unique empty residual.

%=====================================================================
\section{Cutoffs, recurrence, three ledgers, and analogies}
\label{sec:ledgers}
%=====================================================================

\subsection{The cutoff is not an endpoint certificate}

A positive \(M_{\rm cut}\) specifies the boundary of the supplied
clock's validity. It does not establish complement constancy at that
boundary, or determine later mass loss, a burst, a remnant, or
outgoing quantum access.

Conversely, zero leakage does not force zero energy.
The isometry
\[
 V\ket\psi=\ket\psi_R\ket1_B,\qquad
 H_B=\varepsilon\kb1 1,\quad \varepsilon>0,
\]
has zero leakage and positive residual energy.
Likewise, a zero-energy residual sector may have a degeneracy
that retains the entire code.
Energy labels do not count all inaccessible logical degrees
of freedom \cite{CheungHawk01B,CheungEn07B}.

For a pure, unreferenced complete state, radiation purity is
equivalent to purity of its residual marginal.
It is not equivalent to channel constancy of that residual on
every logical input. The latter is the recovery criterion.

\begin{proposition}[A fixed autonomous finite proxy is recurrent]
\label{prop:recurrence}
For a fixed finite-dimensional Hamiltonian \(H\), fixed preparation
\(V_0\), and fixed access partition, the complete channel of
\(e^{-itH}V_0\) returns arbitrarily close to its initial channel at
arbitrarily late times.
If the initial accessible channel is replacement, then
\[
 \limsup_{t\to\infty}\erec(\Nchan_t)=a_d.
\]
\end{proposition}

\begin{proof}
There are finitely many eigenvalue phases.
Simultaneous recurrence on their compact torus gives arbitrarily
late times when \(e^{-itH}\) is arbitrarily close to identity in
operator norm.
For isometries,
\(\ddiam(\Ad_V,\Ad_W)\le\norm{V-W}_\infty\), by referenced
pure-state comparison and convexity.
Partial trace and Lemma~\ref{lem:basic} give recurrence of the
accessible channel and its recovery error.
The initial error and universal upper cap both equal \(a_d\).
\end{proof}

This concerns a fixed closed autonomous finite model
\cite{CheungHawk04D}. It does not establish recurrence of an
outgoing continuum radiation interface or an externally terminated
cell-release schedule. A regulator limit taken together with late
times requires a controlled order of limits.

\subsection{Three panels, not three interchangeable numbers}

An extrapolation report should keep separate:
\[
 \boxed{
 \begin{array}{c}
 M(t),\ P(t),\ \text{declared energy conservation}\\
 \text{heat, work, consumed resources, and cyclic restoration}\\
 S(R_t),\ H(Y_{\le t}),\ \dleak(\Cchan_t),\ \erec(\Nchan_t)
 \end{array}}
\]
Even the information panel contains different quantities.
A decreasing mass is not a decreasing fine-grained radiation
entropy, and a decreasing entropy is not a recovery certificate.

\begin{proposition}[No reset cost from channel equality]
A reduced reset channel or partial trace does not determine a unique
heat, work, or reusable-memory reset cost.
\end{proposition}

\begin{proof}
SWAP a memory with an independent blank qubit.
The memory is reset and its referenced input is transferred to the
other qubit.
For equal gaps
\(H_M=H_L=\varepsilon\kb1 1\), SWAP conserves their sum and
transfers energy \(\varepsilon p\), where \(p\) is the initial
excitation probability.
The same reset channel is obtained for every positive
\(\varepsilon\).
Independent fuel, bath, or actuator processes may also change the
implementation ledger without changing that channel.
The consumed blank and fuel must be restored before a cyclic cost
is claimed.
\end{proof}

This underdetermination is the En07-A ledger distinction
\cite{CheungEn07A}.
A thermal erasure bound belongs to an additionally specified
memory--reservoir setup, not to
\(\Drop\circ\Enter=\Tr\)
\cite{ReebWolf}.

\subsection{Processor, film, album, and library analogies}

\paragraph{Processor and coherent film.}
A processor scrambles a pure prepared code and releases unequal
quantum cells. The receiver keeps those cells coherently.
The Haar hypothesis supplies the finite Page mean.
Replacing the equal-cell clock by a compatible release schedule
changes the horizontal axis, not the output type.

\paragraph{Click album.}
Installing Enter on every emitted cell produces an append-only
classical record. Its exact proxy entropy mean is
\eqref{eq:RecordMean}, not \eqref{eq:PageMean}.
Later clicks may improve answers to classical questions, but they
cannot restore an arbitrary unknown quantum state from that record.
Postprocessing an entanglement-breaking channel remains
entanglement breaking
\cite{HorodeckiShorRuskai,CheungEm03F}.

\paragraph{Torn strip.}
For a classical message \(L\), more photographs can decrease
\(H(L\mid Y_{\le k})\) while the full album entropy increases.
For a quantum strip, photographs retain outcomes rather than
coherent fragments. Successful quantum recovery needs coherent
fragments or another explicitly accessible quantum carrier.

\paragraph{Library cards.}
Photocopies can reconstruct known-alphabet classical text.
A classical index can also select a decoder acting on a genuine
quantum output. It is not itself the unknown quantum book.
Emptying a named drawer does not erase unreturned purifiers or
measurement environments.

A coherent packet decoder in a declared source-field model is a
useful comparison, not a local Schwarzschild receiver
\cite{CheungEm03G}. The analogy's usefulness is preserved by
stating that missing identification.

%=====================================================================
\section{A disciplined extrapolation programme}
\label{sec:programme}
%=====================================================================

\begin{center}
\small
\renewcommand{\arraystretch}{1.12}
\begin{tabularx}{\linewidth}{@{}p{.24\linewidth}X@{}}
\toprule
Possibility & What must additionally be supplied\\
\midrule
Clocked fixed-capacity proxy &
Integer compatible dimensions, one nested history, and
\eqref{eq:sumconstraint}; a power law alone is insufficient.\\[4pt]
Greybody-weighted capacities &
An explicit occupation-to-capacity rule, cutoff and tail,
and a typicality hypothesis for that rule.\\[4pt]
Two return-type curves &
One complete process with two declared receiver/readout
continuations; publish both entropy definitions.\\[4pt]
Restricted exterior access &
A genuine observable algebra or factor, its state restriction,
and an entropy convention; no emergence follows from restriction.\\[4pt]
Positive-mass wall &
The validity interval and the crossover-before-cutoff test;
do not attach an unproved zero-entropy endpoint.\\[4pt]
Petz or island decoder comparison &
A deterministic task channel, correct supports, complete
comparison allowance, and executable implementation budget.\\[4pt]
Conserved-sector scrambling &
The actual sector decomposition and its ensemble weights;
use \eqref{eq:sectorpage}, not an unrestricted tensor formula.\\[4pt]
Three-ledger report &
Energy, reset resources, entropy, and recovery estimates
must retain their different hypotheses and units.\\[4pt]
Gravitational conclusion &
One microscopic preparation, dynamics, access dictionary,
receiver, typicality theorem, and endpoint comparison.\\
\bottomrule
\end{tabularx}
\end{center}

The following are specifically not imported:
\begin{itemize}
\item a static area-law or catalogue record as a temporal channel
history \cite{CheungOai265A};
\item a prepared entropy sequence as Hawking emission
\cite{CheungEn02D};
\item an island or replica-wormhole reference as a complete
evaporation isometry \cite{IslandReview};
\item an autonomous finite source-field Hamiltonian as a
gravitational collapse Hamiltonian \cite{CheungHawk02C};
\item any soft, lost, purifier, or other admitted output factor
removed from complete accounting merely by a final mass label.
\end{itemize}

If an actual theory includes such factors, the complete interface
must account for them or prove why they are irrelevant to the
declared finite code. This paper does not assume their existence,
absence, or specific dynamics.

%=====================================================================
\section{Hawk05-C release theorem}
\label{sec:release}
%=====================================================================

\begin{theorem}[Conditional extrapolation certificate]
\label{thm:main}
Within the stated finite interface discipline:
\begin{enumerate}[label=\textup{(\roman*)}]
\item nonuniform nested tensor release has the exact coherent and
record means \eqref{eq:twomeans};
\item its fixed-dimensional capacities obey
\eqref{eq:sumconstraint}; a coarse entropy-production ratio
\(\xi\ne1\) cannot be inserted as a mere clock change;
\item the compatible inverse-square example has the explicit
rounded two-curve sketch
\eqref{eq:sketchQ}--\eqref{eq:sketchL}, while the general supplied
capacity envelope has the crossover and cutoff conditions of
Theorem~\ref{thm:crossover};
\item additive-charge-sector Haar scrambling has the exact
direct-sum means
\eqref{eq:sectorpage}--\eqref{eq:sectorrecord};
\item physical two-curve transfer is conditional on the separate
capacity, ensemble, channel, and detector allowances in
\eqref{eq:Qbudget}--\eqref{eq:Ybudget};
\item whole-code recovery is a separate complete-complement or
decoded-channel certificate, with bounds
\eqref{eq:recoverybudget} and \eqref{eq:specifiedbudget};
\item every record-only endpoint has the permanent error lower
bound \(1-1/d\), regardless of the clock or capacity plot.
\end{enumerate}
These results establish finite extrapolation possibilities and their
compatibility tests. They do not construct a gravitational Page curve,
an island, spacetime emergence, or the inherited characteristic
quantum coefficient.
\end{theorem}

\begin{proof}
Items (i)--(iv) are Theorems~\ref{thm:two},
\ref{thm:compatibility}, \ref{thm:crossover}, \ref{thm:sketch},
and \ref{thm:sector}, with Proposition~\ref{prop:xi}.
Item (v) is Theorem~\ref{thm:transfer}.
Item (vi) is Theorem~\ref{thm:recovery} and the exact criterion
of Theorem~\ref{thm:exact}.
Item (vii) is Proposition~\ref{prop:classical}.
No physical comparison theorem supplying the excluded
identifications is among the constructed objects.
\end{proof}

\paragraph{Final conclusion.}
The clean next calculation is a compatible capacity clock with two
declared return types and explicit, unfilled physical comparison
slots. The coherent proxy may turn over; the unedited classical
album cannot. A crossover of capacities, a maximum of a mean entropy,
a decrease of complete leakage, and an executable recovery threshold
remain four different events until a same-model theorem relates them.

%=====================================================================
\appendix
\section{The Page--Sen calculation used in the proxy}
\label{app:page}
%=====================================================================

We record the finite calculation to make the imported random-state
input explicit. The result is due to Page and Sen
\cite{PageAverage,Sen}; no priority is claimed.

Let \(m\le n\), \(D=mn\), and let \(G\) be an \(m\times n\)
standard complex Gaussian matrix. Put
\[
 W=GG^\dagger,\qquad T=\Tr W,\qquad \rho=W/T.
\]
Gaussian radius and direction are independent, \(T\) is Gamma
with shape \(D\), and \(\rho\) is the Haar reduction.
Hence, in natural entropy units,
\begin{equation}
 \Expectation S_{\rm nat}(\rho)
 =
 \psi_0(D+1)-D^{-1}\Expectation\Tr(W\ln W),
 \label{eq:gamma-page}
\end{equation}
because
\(\Tr(W\ln W)=T\ln T+T\Tr(\rho\ln\rho)\) and
\(\Expectation(T\ln T)=D\psi_0(D+1)\).

Write \(\alpha=n-m\).
The singular-value Jacobian gives the eigenvalue density proportional
to
\[
 e^{-\sum_i x_i}\prod_i x_i^\alpha
 \prod_{i<j}(x_i-x_j)^2,\qquad x_i>0.
\]
Orthogonalizing the Vandermonde with generalized Laguerre
polynomials gives the one-point density
\[
 R_1(x)
 =
 e^{-x}x^\alpha
 \sum_{j=0}^{m-1}\frac{[L_j^\alpha(x)]^2}{h_j},
 \qquad
 h_j=\frac{\Gamma(j+\alpha+1)}{j!}.
\]
These statements follow by the singular-value change of variables
and expansion of the squared orthogonal-polynomial determinant.

Rodrigues's formula and integration by parts give the norms and
orthogonality. The identities
\[
 xL_j^\alpha
 =
 (2j+\alpha+1)L_j^\alpha
 -(j+1)L_{j+1}^\alpha-(j+\alpha)L_{j-1}^\alpha,
\]
\[
 \partial_\alpha L_j^\alpha
 =
 \sum_{\ell=0}^{j-1}\frac{L_\ell^\alpha}{j-\ell}
\]
follow respectively from coefficient comparison and differentiation
of the generating function
\[
 \sum_{j\ge0}L_j^\alpha(x)t^j
 =
 (1-t)^{-\alpha-1}\exp\!\left(-\frac{xt}{1-t}\right).
\]
Consequently
\[
 \frac1{h_j}\int_0^\infty
 x^{\alpha+1}e^{-x}\ln x[L_j^\alpha(x)]^2\,dx
 =
 (2j+\alpha+1)\psi_0(j+\alpha+1)+1+2j.
\]
Indeed, differentiate
\[
 \int_0^\infty x^{\alpha+1}e^{-x}[L_j^\alpha(x)]^2\,dx
 =(2j+\alpha+1)h_j;
\]
the polynomial derivative cross term is \(-jh_j\).
Differentiation under the integral is justified locally for
\(\alpha>-1\) by the exponential weight and integrability at zero.

Summing the displayed logarithmic integrals gives
\[
 \Expectation\Tr(W\ln W)
 =
 mn\psi_0(n+1)+\frac{m(m-1)}2.
\]
For example, substitute
\(\psi_0(r)=\psi_0(n+1)-\sum_{\ell=r}^{n}\ell^{-1}\) and use
\[
 \sum_{r=\alpha+1}^{\ell}(2r-\alpha-1)
 =\ell(\ell-\alpha)
\]
to evaluate the finite double sum.
Equation~\eqref{eq:gamma-page} becomes
\[
 \Expectation S_{\rm nat}(\rho)
 =
 \psi_0(mn+1)-\psi_0(n+1)-\frac{m-1}{2n}.
\]
Using \(\psi_0(\ell+1)=\Harm{\ell}-\gamma_{\rm E}\) and dividing
by \(\ln2\) proves Lemma~\ref{lem:pagesen}, including \(m=1\).

%=====================================================================
\section{The independent characteristic-map boundary}
\label{app:analytic}
%=====================================================================

The supplied XIV-I, XIV-J, and XV-B passages concern a different
distributional programme
\cite{CheungXIVI,CheungXIVJ,CheungXVB}.
The finite capacity and entropy results above do not act on its
six-branch characteristic target.

For a surjective reconstruction map
\(\Rec:\mathscr P\to\mathscr K\), a presentation-level linear map
\(\widetilde M\) descends uniquely exactly when
\[
 \ker\Rec\subseteq\ker\widetilde M.
\]
This follows directly because equal reconstructions differ by a
reconstruction-null presentation.
The identity
\[
 q\delta'(q)+\delta(q)=0
\]
likewise prevents a representative-dependent highest degree from
automatically defining an intrinsic reconstructed symbol:
pairing \(q\delta'\) with a test function gives minus its value at zero.

These compatibility facts do not construct the full joint
characteristic map. According to the supplied versions, the named
internal amplitudes are constructed at I2, while the full map remains
unresolved at I3/J2 and in the graph-specific XV-B continuation.
The required section, complete remainder, transverse estimates,
contact compatibility, output wavefront theorem, and normalized
quantum admission are not supplied here.

The standing supplied boundaries remain unchanged:
\(O.1=\texttt{NOT\_CLOSED}\); the downstream quantum-admission exit is
not opened; XVI-N's universal obligation is not discharged; and no
inherited characteristic quantum coefficient is admitted.
The conditional massless obstruction in the supplied XVI-N instance
is not contradicted or enlarged \cite{CheungXVIN}.
These are scoped provenance statements, not universal
nonexistence theorems.

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

\paragraph{Attribution.}
The Page average, finite entropy-continuity theorem, exact correction
criterion, entanglement-breaking structure, and information--disturbance
theorem are established results cited at their use.
The direct-sum sector calculation is consistent with the
symmetry-resolved random-state literature.
The contribution is their assembly into a corrected extrapolation
framework, the capacity compatibility test, the two-return-type
clocked sketch, and explicit separation of the required allowances.

\paragraph{Source provenance.}
Cheung's internal manuscripts are cited by the supplied identifiers
and versions. Documentary comparison is not independent verification
of the entire corpus and is not an identification of their models.
The supplied Hawk05-A and Hawk05-B author fields are empty;
authorship of those files is not inferred.
Cheung's journal article is distinguished from the internal
manuscripts.

\paragraph{Physical scope.}
No gravitational microscopic Hamiltonian, collapse channel,
backreaction law, local coherent receiver, island surface,
replica-wormhole calculation, or spacetime-emergence theorem is
constructed. The supplied clock and black-hole entropy law do not
remove those obligations.

\paragraph{Build and verification.}
This is a standalone source written for \texttt{pdflatex}.
No successful compilation, proof-assistant verification, numerical
execution, independent referee approval, or journal acceptance of
Hawk05-C 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 exposition, 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.''
\emph{Frontiers in Physics}, volume 14 (2026).
Publisher abstract record: accepted 28 September 2026;
final formatted version described there as forthcoming.
DOI \url{https://doi.org/10.3389/fphy.2026.1954943}.

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

\bibitem{Hawk05A}
\emph{Hawk05-A: Enter, Page-Type Entropy Turnover, and the
Classical-Record Obstruction}.
Supplied \texttt{Hawk05-A.tex}, dated 15 October 2026.
Author field unspecified.
The obsolete reversed-weight flagged-leakage equation is not adopted.

\bibitem{Hawk05B}
\emph{Hawk05-B: A Finite Page-Curve Derivation under the Enter
Discipline}.
Supplied \texttt{Hawk05-B.tex}, version 1.
Author field unspecified.
Finite Haar, record, and channel-separation proxy.

\bibitem{CheungHawk01B}
N.~Cheung,
\emph{Hawk01-B}, version 1.
Enter Calculus Companion.
Supplied endpoint and entropy-history separation records.
\url{https://enter.grok.me/browse/Hawk01-B}.

\bibitem{CheungHawk02B}
N.~Cheung,
\emph{Hawk02-B: Hamiltonian-Derived Emission and Quantitative
Recovery in a Finite Evaporation Model}, version 2.
Enter Calculus Companion.
Supplied single- and multi-cell calibration records.
\url{https://enter.grok.me/browse/Hawk02-B}.

\bibitem{CheungHawk02C}
N.~Cheung,
\emph{Hawk02-C}, version 2.
Enter Calculus Companion.
Supplied autonomous source-field comparison; not a gravitational
Hamiltonian identification.
\url{https://enter.grok.me/browse/Hawk02-C}.

\bibitem{CheungHawk03A}
N.~Cheung,
\emph{Hawk03-A}, version 1.
Enter Calculus Companion.
Supplied complete-complement, independent-factor, and partner
comparisons.
\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.
Supplied cutoff clock, complete-channel budgets, endpoint obligations,
and finite recurrence.
\url{https://enter.grok.me/browse/Hawk04-D}.

\bibitem{CheungEn07A}
N.~Cheung,
\emph{En07-A}, version 1.
Enter Calculus Companion.
Supplied readout-and-drop 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 access-preserving endpoint certificate.
\url{https://enter.grok.me/browse/En07-B}.

\bibitem{CheungEn05C}
N.~Cheung,
\emph{En05-C}, version 1.
Enter Calculus Companion.
Supplied tensor-free snapshot--channel--adjoint comparison.
\url{https://enter.grok.me/browse/En05-C}.

\bibitem{CheungEn02B}
N.~Cheung,
\emph{En02-B}, version 1.
Enter Calculus Companion.
Supplied finite resource-certified decoder soundness and
benchmark-specific workspace statements.
\url{https://enter.grok.me/browse/En02-B}.

\bibitem{CheungEn02C}
N.~Cheung,
\emph{En02-C}, version 1.
Enter Calculus Companion.
Supplied sequential soundness and streaming benchmark.
\url{https://enter.grok.me/browse/En02-C}.

\bibitem{CheungEn02D}
N.~Cheung,
\emph{En02-D}, version 4.
Enter Calculus Companion.
Supplied prepared turnover and transmission-dependent recovery
comparisons.
\url{https://enter.grok.me/browse/En02-D}.

\bibitem{CheungEn02E}
N.~Cheung,
\emph{En02-E}, version 2.
Enter Calculus Companion.
Supplied finite-mode Gaussian restriction and cutoff records.
\url{https://enter.grok.me/browse/En02-E}.

\bibitem{CheungEm03A}
N.~Cheung,
\emph{Em03-A}, version 1.
Enter Calculus Companion.
Supplied fixed-background scalar emission law,
\texttt{thm:main}.
\url{https://enter.grok.me/browse/Em03-A}.

\bibitem{CheungEm03B}
N.~Cheung,
\emph{Em03-B}, version 2.
Enter Calculus Companion.
Supplied geometric packet marginal and apparatus-model transfer;
not a local Schwarzschild receiver.
\url{https://enter.grok.me/browse/Em03-B}.

\bibitem{CheungEm03F}
N.~Cheung,
\emph{Em03-F}, version 1.
Enter Calculus Companion.
Supplied entanglement-breaking source-to-packet comparison.
\url{https://enter.grok.me/browse/Em03-F}.

\bibitem{CheungEm03G}
N.~Cheung,
\emph{Em03-G}, version 1.
Enter Calculus Companion.
Supplied ordered source and coherent-decoder comparisons.
\url{https://enter.grok.me/browse/Em03-G}.

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

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

\bibitem{CheungOai290A}
N.~Cheung,
\emph{Oai290-A}, version 1.
Enter Calculus Companion.
Supplied whole-code, named-effect, and implemented-channel
certificate distinctions.
\url{https://enter.grok.me/browse/Oai290-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-sector 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{CheungXVIN}
N.~Cheung,
\emph{XVI-N}, version r12.
Enter Calculus Companion.
Supplied conditional massless special-family obstruction and
undischarged universal obligation.
\url{https://enter.grok.me/browse/XVI-N}.

\bibitem{PageAverage}
D.~N.~Page,
``Average entropy of a subsystem.''
\emph{Physical Review Letters} \textbf{71} (1993), 1291--1294.
DOI \url{https://doi.org/10.1103/PhysRevLett.71.1291}.
\url{https://arxiv.org/abs/gr-qc/9305007}.

\bibitem{Sen}
S.~Sen,
``Average entropy of a subsystem.''
\emph{Physical Review Letters} \textbf{77} (1996), 1--3.
DOI \url{https://doi.org/10.1103/PhysRevLett.77.1}.
\url{https://arxiv.org/abs/hep-th/9601132}.

\bibitem{PageTime}
D.~N.~Page,
``Time dependence of Hawking radiation entropy.''
\emph{Journal of Cosmology and Astroparticle Physics}
\textbf{2013}(09), 028.
DOI \url{https://doi.org/10.1088/1475-7516/2013/09/028}.
\url{https://arxiv.org/abs/1301.4995}.

\bibitem{PageEmission}
D.~N.~Page,
``Particle emission rates from a black hole:
Massless particles from an uncharged, nonrotating hole.''
\emph{Physical Review D} \textbf{13} (1976), 198--206.
DOI \url{https://doi.org/10.1103/PhysRevD.13.198}.

\bibitem{Bekenstein}
J.~D.~Bekenstein,
``Black holes and entropy.''
\emph{Physical Review D} \textbf{7} (1973), 2333--2346.
DOI \url{https://doi.org/10.1103/PhysRevD.7.2333}.

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

\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.
DOI \url{https://doi.org/10.1103/PhysRevA.55.900}.

\bibitem{KSW}
D.~Kretschmann, D.~Schlingemann, and R.~F.~Werner,
``The information-disturbance tradeoff and the continuity of
Stinespring's representation.''
\emph{IEEE Transactions on Information Theory}
\textbf{54} (2008), 1708--1717.
DOI \url{https://doi.org/10.1109/TIT.2008.917696}.
\url{https://arxiv.org/abs/quant-ph/0605009}.

\bibitem{HorodeckiShorRuskai}
M.~Horodecki, P.~W.~Shor, and M.~B.~Ruskai,
``Entanglement breaking channels.''
\emph{Reviews in Mathematical Physics}
\textbf{15} (2003), 629--641.
DOI \url{https://doi.org/10.1142/S0129055X03001709}.
\url{https://arxiv.org/abs/quant-ph/0302031}.

\bibitem{Audenaert}
K.~M.~R.~Audenaert,
``A sharp continuity estimate for the von Neumann entropy.''
\emph{Journal of Physics A: Mathematical and Theoretical}
\textbf{40} (2007), 8127--8136.
DOI \url{https://doi.org/10.1088/1751-8113/40/28/S18}.
\url{https://arxiv.org/abs/quant-ph/0610146}.

\bibitem{Baumgratz}
T.~Baumgratz, M.~Cramer, and M.~B.~Plenio,
``Quantifying coherence.''
\emph{Physical Review Letters} \textbf{113} (2014), 140401.
DOI \url{https://doi.org/10.1103/PhysRevLett.113.140401}.
\url{https://arxiv.org/abs/1311.0275}.

\bibitem{SymmetryPage}
S.~Murciano, P.~Calabrese, and L.~Piroli,
``Symmetry-resolved Page curves.''
\emph{Physical Review D} \textbf{106} (2022), 046015.
DOI \url{https://doi.org/10.1103/PhysRevD.106.046015}.
\url{https://arxiv.org/abs/2206.05083}.

\bibitem{ChenPetz}
C.-F.~Chen, G.~Penington, and G.~Salton,
``Entanglement wedge reconstruction using the Petz map.''
\emph{Journal of High Energy Physics} \textbf{2020}(01), 168.
DOI \url{https://doi.org/10.1007/JHEP01(2020)168}.
\url{https://arxiv.org/abs/1902.02844}.

\bibitem{CotlerRecovery}
J.~Cotler, P.~Hayden, G.~Penington, G.~Salton, B.~Swingle,
and M.~Walter,
``Entanglement wedge reconstruction via universal recovery channels.''
\emph{Physical Review X} \textbf{9} (2019), 031011.
DOI \url{https://doi.org/10.1103/PhysRevX.9.031011}.

\bibitem{NullInfinity}
A.~Laddha, S.~G.~Prabhu, S.~Raju, and P.~Shrivastava,
``The holographic nature of null infinity.''
\emph{SciPost Physics} \textbf{10} (2021), 041.
DOI \url{https://doi.org/10.21468/SciPostPhys.10.2.041}.
\url{https://arxiv.org/abs/2002.02448}.

\bibitem{Penington}
G.~Penington,
\emph{Entanglement Wedge Reconstruction and the Information Paradox}.
arXiv:1905.08255.
External gravitational comparison, not a premise identifying the proxy.
\url{https://arxiv.org/abs/1905.08255}.

\bibitem{IslandReview}
A.~Almheiri, T.~Hartman, J.~Maldacena, E.~Shaghoulian,
and A.~Tajdini,
``The entropy of Hawking radiation.''
\emph{Reviews of Modern Physics} \textbf{93} (2021), 035002.
DOI \url{https://doi.org/10.1103/RevModPhys.93.035002}.
\url{https://arxiv.org/abs/2006.06872}.

\bibitem{ReebWolf}
D.~Reeb and M.~M.~Wolf,
``An improved Landauer principle with finite-size corrections.''
\emph{New Journal of Physics} \textbf{16} (2014), 103011.
DOI \url{https://doi.org/10.1088/1367-2630/16/10/103011}.
\url{https://arxiv.org/abs/1306.4352}.

\end{thebibliography}

\end{document}