% Hawk04-D.tex
% Standalone pdfLaTeX source; no external bibliography, figures, or code.
%
% Build:
%   pdflatex -interaction=nonstopmode -halt-on-error Hawk04-D.tex
%   pdflatex -interaction=nonstopmode -halt-on-error Hawk04-D.tex
%
% Evidence boundary:
% This source has not been compiled, numerically executed, independently
% reviewed, or proof-assistant checked in this exchange.
%
% Scope:
% A conditional certification framework is proved.
% No gravitational evaporation channel, positive noncommuting recovery
% window, continuum collapse geometry, or Planck-scale endpoint is constructed.

\documentclass[11pt,letterpaper]{article}

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

\hypersetup{
  pdftitle={Hawk04-D: From Finite Enter Interfaces to Conditional Evaporation Certificates},
  pdfauthor={Ngo Cheung},
  pdfsubject={Controlled limits, mass laws, complete complements, and quantum recovery},
  pdfkeywords={Enter calculus, black-hole information, quantum recovery,
    complementary channel, finite regulator, conditional evaporation}
}

\setlength{\emergencystretch}{3em}
\setlength{\tabcolsep}{5pt}
\allowdisplaybreaks
\setlist[enumerate]{leftmargin=*,itemsep=3pt,topsep=5pt}
\setlist[itemize]{leftmargin=*,itemsep=3pt,topsep=5pt}
\newcolumntype{L}[1]{>{\raggedright\arraybackslash}p{#1}}
\newcolumntype{Y}{>{\raggedright\arraybackslash}X}

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

\numberwithin{equation}{section}

\DeclareMathOperator{\Tr}{Tr}
\DeclareMathOperator{\rank}{rank}
\DeclareMathOperator{\supp}{supp}
\DeclareMathOperator{\Span}{span}
\DeclareMathOperator{\Ad}{Ad}
\DeclareMathOperator{\diag}{diag}

\newcommand{\C}{\mathbb C}
\newcommand{\R}{\mathbb R}
\newcommand{\id}{\mathrm{id}}
\newcommand{\ii}{\mathrm i}
\newcommand{\dd}{\mathrm d}
\newcommand{\Lin}{\mathcal L}
\newcommand{\States}{\mathcal D}
\newcommand{\NN}{\mathcal N}
\newcommand{\EE}{\mathcal E}
\newcommand{\VV}{\mathcal V}
\newcommand{\DD}{\mathcal D}
\newcommand{\TT}{\mathcal T}
\newcommand{\MM}{\mathcal M}
\newcommand{\FS}{\mathcal F_{\mathrm s}}
\newcommand{\Enter}{\mathsf{Enter}}
\newcommand{\Drop}{\mathsf{dropBit}}
\newcommand{\ket}[1]{\lvert #1\rangle}
\newcommand{\bra}[1]{\langle #1\rvert}
\newcommand{\kb}[2]{\lvert #1\rangle\!\langle #2\rvert}
\newcommand{\norm}[1]{\left\lVert #1\right\rVert}
\newcommand{\abs}[1]{\left\lvert #1\right\rvert}
\newcommand{\dtr}{d_{\mathrm{tr}}}
\newcommand{\ddia}{d_{\diamond}}
\newcommand{\erec}{\varepsilon_{\mathrm{rec}}}
\newcommand{\dleak}{\delta_{\mathrm{full}}}
\newcommand{\Jn}{J_{\mathrm n}}
\newcommand{\vac}{\mathrm{vac}}
\newcommand{\CPTP}{\mathrm{CPTP}}

\newenvironment{scope}
  {\par\smallskip\noindent\textbf{Scope boundary.}\ }
  {\par\smallskip}

\title{
\textsf{Hawk04-D}\\[0.4em]
From Finite Enter Interfaces to Conditional Evaporation Certificates:\\
Controlled Limits, Mass Laws, and Quantum Recovery
}
\author{Ngo Cheung}
\date{}

\begin{document}
\maketitle

\begin{abstract}
We develop a proof-based extrapolation framework for the finite Enter
interfaces of Hawk04-C. The operational core consists of a complete
isometry, declared accessible and inaccessible outputs, quantum-return
decoders, destructive readout, and explicit classical discard.
We prove exact calibration formulas for a partial-SWAP schedule and a
flagged sequential-transfer model, including their different information
targets. We establish fixed-code approximation and growing-code
certification statements, reconstruct a finite-band reservoir regulator,
and give an independent thermal-pair truncation certificate.
The regulator is not transferred to a collapsing-shell Bogoliubov model
without a separate comparison theorem.

A supplied luminosity law is treated as an additional physical
hypothesis. Under fixed massless-species and greybody scaling assumptions,
we derive the inverse-square power law and the resulting cubic
cutoff-lifetime formula, together with perturbation and discretization
bounds. These conclusions stop at a positive validity cutoff and do not
determine Planck-scale behavior. Entropy turnover, approximately thermal
observations, complementary decoupling, and full-code recovery are
separated by explicit counterexamples and dimension constraints.
An access-preserving endpoint package yields a fixed-decoder certificate;
terminal residual readout-and-discard leaves the existing radiation
channel unchanged.

The resulting predictions are conditional on independently supplied
dynamics, access, approximation, luminosity, and endpoint hypotheses.
No gravitational evaporation channel, universal Page-time recovery
transition, final-burst spectrum, or absence of physical remnants is
derived. The contribution is a coherent certification architecture and
a set of calibrated tests, not a solution of black-hole evaporation.
\end{abstract}

\begin{center}
\fbox{\begin{minipage}{0.93\linewidth}
\textbf{Central principle.}
A finite channel theorem can certify a physical prediction only after
the physical model supplies a comparison at that channel interface.
Thermal matching, a mass equation, and residual decoupling are separate
hypotheses. None is supplied by the word ``Enter.''
\end{minipage}}
\end{center}

\tableofcontents

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

\subsection{The physical background}

Hawking's particle-creation calculation concerns quantum fields on a
classical collapse background, while his information-loss proposal asks
about a complete final description of the initial quantum state
\cite{Hawking1975,Hawking1976}.
For an uncharged, nonrotating Schwarzschild black hole, the external
semiclassical temperature is
\begin{equation}
T_{\mathrm H}
=\frac{\hbar c^3}{8\pi G M k_{\mathrm B}}.
\label{eq:TH-SI}
\end{equation}
A bosonic mode has a thermal denominator modified by an absorption
factor:
\begin{equation}
\langle n_{\omega\ell m}\rangle
=
\frac{\Gamma_{\omega\ell m}}
{\exp(\hbar\omega/(k_{\mathrm B}T_{\mathrm H}))-1}.
\label{eq:occupation}
\end{equation}
These statements do not specify a complete logical-input-to-radiation
channel. In particular, one occupation number is not a test of every
logical coherence.

Page's random-subsystem analysis and unitary-radiation comparison concern
additional assumptions about states and information in radiation
\cite{PageAverage,PageInformation}.
Random-subsystem recovery also depends on the input message, initial
correlations, and subsystem sizes, rather than on a universal count of
``half the emitted quanta'' \cite{HaydenPreskill}.
Island and replica-wormhole calculations address entropy in their
specified gravitational settings \cite{AlmheiriEtAl}.
They are not replaced here by a finite circuit, and they are not used as
a substitute for a missing complete-channel comparison.

The low-energy scalar-reservoir evaporation model of
Gaikwad, Kaushal, Mandal, and Wadia has its own dynamics and physical
assumptions \cite{GaikwadEtAl}. Its geometry is not assigned to the
finite-band source model below merely because both contain a scalar bath.

\subsection{The Philosophy of Enter}

Cheung's \emph{Philosophy of Enter} treats measurement as a declared
interface: destructive computational-basis readout consumes the
measured quantum name and returns classical data. It does not return the
unknown quantum input with its reference correlations. A decoder has a
different target and must specify its permitted access. The essay also
separates complete energy, physical erasure cost, and residual leakage,
and distinguishes an operational analogy from a physical identification
\cite{CheungPhilosophy}.

We implement that discipline by declaring the input code, its retained
references, the accessible output, every excluded output, and the target
channel before stating a recovery or observation result. Register
consumption is a statement about the returned interface, not a proof
that a physical environment has disappeared.

Cheung's journal article develops a linear quantum lambda calculus and
an operational characterization of destructive measurement within an
explicit strategy class \cite{CheungEnter}. Here the finite Born
instrument is defined directly. We neither reproduce that uniqueness
argument nor enlarge its hypotheses by citation.

\subsection{The analogy and its added obligations}

In the finite analogy:
\begin{itemize}
\item a prepared quantum code plays the role of an information-bearing
source;
\item the split \(R\mid B\) specifies an access restriction;
\item a transfer isometry exposes a new outgoing quantum factor;
\item Enter converts a chosen output into a classical record;
\item a terminal instruction consumes a named interface.
\end{itemize}
These are operational descriptions. A metric, horizon, stress tensor,
radiation flux, and physical endpoint require additional constructions.

Hawk04-C supplies the six-row operational separation, while Hawk04-A
and Hawk04-B supply complete-complement recovery, finite-regulator
certificates, and an interacting-family negative control
\cite{CheungHawk04C,CheungHawk04A,CheungHawk04B}.
The supplied excerpts of Hawk02-A, Hawk02-B, Hawk01-B, and En07-A/B/C
motivate the staged organization used here
\cite{CheungHawk02A,CheungHawk02B,CheungHawk01B,
CheungEn07A,CheungEn07B,CheungEn07C}.
All elementary statements needed below have local proofs.

\paragraph{Corrections to the proposed plan.}
The following repairs are substantive.
\begin{enumerate}
\item The general complementary-Choi recovery allowance is
\(\sqrt{2de}+b\), not \(2de+b\).
A linear bound requires a different, independently justified certificate.
\item A sequence of changing input dimensions does not have a
well-defined diamond comparison until input embeddings and recovery
targets are specified.
\item The finite-band reservoir regulator is model specific. It is not
a regulator theorem for a collapsing-shell Bogoliubov transformation.
\item An arbitrary shrinking core does not imply decoupling of the
complete residual. A whole \(n\)-qubit code and a small injected diary
are different recovery tasks.
\item Temperature matching alone does not determine a luminosity,
emission clock, or lifetime.
\item Entropy turnover does not establish a referenced decoder or fix
its transition time.
\item A fixed information-free residual can have nonzero energy and
many physical degrees of freedom. It is not synonymous with no remnant.
\item A null detector result is not, without a statistical and population
model, proof that remnants do not exist.
\end{enumerate}

The exact channel principles have established antecedents in
Stinespring dilation, quantum error correction, no-hiding, and
information--disturbance theory
\cite{Stinespring,KnillLaflamme,BraunsteinPati,KSW}.
No priority claim is made for those principles.

%=====================================================================
\section{Complete interfaces and the finite operational core}
\label{sec:core}

\subsection{Definitions}

Let \(C\simeq\C^d\), \(d\ge2\), be the entire logical input whose
identity channel is the recovery target.
Output Hilbert spaces may be separable; finite output spaces are stated
explicitly whenever finite-dimensional optimization is used.
Channels act on trace-class operators and are normal CPTP maps.
An external reference \(Q\) is untouched.

\begin{definition}[Complete interface and diagnostics]
\label{def:interface}
A complete interface is an isometry
\[
V:C\longrightarrow R\otimes B,\qquad V^\dagger V=I_C.
\]
Its complete, accessible, and complementary channels are
\[
\VV(X)=VXV^\dagger,\qquad
\NN(X)=\Tr_B\VV(X),\qquad
\EE(X)=\Tr_R\VV(X).
\]
The complete inaccessible output \(B\) includes all unreturned
preparation purifiers, source registers, uncollected modes, apparatus
outputs, work registers, and discarded-record stores in the declared
model. The independent reference \(Q\) is not part of \(B\).

For a state \(\sigma\), write
\[
\TT_\sigma(X)=\Tr(X)\sigma.
\]
Use half-normalized distances
\[
\dtr(\rho,\eta)=\frac12\norm{\rho-\eta}_1,\qquad
\ddia(\Phi,\Psi)=\frac12\norm{\Phi-\Psi}_\diamond,
\]
and define
\begin{align}
\erec(\NN)
&=\inf_{\DD:R\to C\ \CPTP}\ddia(\DD\NN,\id_C),\\
\dleak(\EE)
&=\inf_{\sigma\in\States(B)}\ddia(\EE,\TT_\sigma).
\end{align}
Decoders are deterministic and return a quantum code.
A normalized successful branch alone is not a decoder.
\end{definition}

A reference of dimension \(d\) suffices for differences of channels with
input \(C\), by Schmidt decomposition and convexity.
Consequently all diamond statements below retain arbitrary references.

\begin{theorem}[Complete restoration and exact local recovery]
\label{thm:exact}
For a complete interface, choose any \(\tau_C\in\States(C)\).
The map
\begin{equation}
\mathcal L_V(Y)
=
V^\dagger YV+
\Tr((I-VV^\dagger)Y)\tau_C
\label{eq:restore}
\end{equation}
is CPTP and satisfies \(\mathcal L_V\VV=\id_C\).
Isometric embedding preserves the trace norm of every trace-class
operator, including after tensoring with \(I_Q\).

Moreover, the following are equivalent:
\begin{enumerate}[label=\textup{(\roman*)}]
\item a CPTP decoder satisfies \(\DD\NN=\id_C\);
\item \(\EE=\TT_{\sigma_B}\) for one residual state;
\item \(\EE(\kb{i}{j})=\delta_{ij}\sigma_B\) on all matrix units;
\item
\[
\Jn(\EE)=I_Q/d\otimes\sigma_B,
\qquad
\Jn(\Phi):=(\id_Q\otimes\Phi)(\kb{\Phi_d}{\Phi_d}),
\]
where \(\ket{\Phi_d}=d^{-1/2}\sum_i\ket{i}_Q\ket{i}_C\).
\end{enumerate}
Under these conditions, there are a space \(K\), a state \(\sigma_K\),
and an isometry \(A:C\otimes K\to R\) such that
\begin{equation}
\NN(X)=A(X\otimes\sigma_K)A^\dagger.
\label{eq:normalform}
\end{equation}
An exact decoder on all of \(R\) is
\begin{equation}
\DD(Y)=
\Tr_K(A^\dagger YA)+
\Tr((I-AA^\dagger)Y)\tau_C.
\label{eq:Schmidt-decoder}
\end{equation}
If \(\sigma_B\) has finite rank \(r\), then \(\dim R\ge dr\).
\end{theorem}

\begin{proof}
Compression and positive-effect measurement followed by preparation
are CP. Their traces in \eqref{eq:restore} add to \(\Tr Y\), and direct
substitution gives the left inverse. An isometry preserves all nonzero
singular values, proving trace-norm preservation.

Matrix units span \(\Lin(C)\), and their Bell expansion proves the
equivalence of (ii)--(iv).
Assume (ii), and diagonalize the nonzero residual support:
\[
\sigma_B=\sum_\alpha p_\alpha\kb{b_\alpha}{b_\alpha},
\qquad p_\alpha>0.
\]
The index set is finite or countable. Constancy implies that all code
vectors are supported in \(R\otimes\supp\sigma_B\), so write
\[
V\ket{i}=\sum_\alpha\sqrt{p_\alpha}\,
\ket{u_{i\alpha}}_R\ket{b_\alpha}_B.
\]
The complementary matrix-unit identities give
\[
\langle u_{j\beta},u_{i\alpha}\rangle
=\delta_{ij}\delta_{\alpha\beta}.
\]
Define \(A(\ket{i}\ket{\alpha})=\ket{u_{i\alpha}}\) and
\(\sigma_K=\sum_\alpha p_\alpha\kb{\alpha}{\alpha}\).
These vectors are orthonormal, proving the normal form and the dimension
bound. The two terms in \eqref{eq:Schmidt-decoder} are CP and have traces
summing to \(\Tr Y\); substitution returns \(X\).

Conversely, dilate an exact decoder by a Stinespring isometry
\(L:R\to C\otimes G\) \cite{Stinespring}.
The composite \(Z=(L\otimes I_B)V\) has pure decoded output
\(\ket{i}\) on input \(\ket{i}\), hence
\[
Z\ket{i}=\ket{i}\ket{\xi_i}_{GB}.
\]
Recovery of each off-diagonal matrix unit requires
\(\langle\xi_j,\xi_i\rangle=1\). All \(\xi_i\) are one vector \(\xi\).
Tracing the decoded output and \(G\) gives
\[
\EE(\kb{i}{j})=\delta_{ij}\Tr_G\kb{\xi}{\xi}.
\]
Every equality holds on all operators and therefore on all reference
extensions.
\end{proof}

This is the full-input constant-complement structure of exact correction
\cite{KnillLaflamme,BraunsteinPati}; compare the locally proved
formulations in Hawk04-A/B/C \cite{CheungHawk04A,CheungHawk04B,
CheungHawk04C}.

\begin{corollary}[Destination qualification]
\label{cor:destination}
The residual alone recovers the whole code exactly if and only if the
accessible channel is a replacement channel.
A merely nonconstant complement obstructs exact recovery from \(R\);
it does not establish recovery from \(B\).
For \(d\ge2\), both factors cannot independently recover the whole code.
\end{corollary}

\begin{proof}
Interchange the output factors in Theorem~\ref{thm:exact}.
A replacement channel cannot recover two distinct inputs.
\end{proof}

\subsection{Enter, discard, and finite soundness}

Let \(S\) be a measured finite register and \(T\) the surviving quantum
register. In the computational basis, put
\(M_x=\bra{x}_S\otimes I_T\).
Define the destructive instrument
\begin{equation}
\MM_S(X)=\sum_x\kb{x}{x}_K\otimes M_xXM_x^\dagger.
\label{eq:Enter}
\end{equation}
The output names are \(K,T\), not \(S\); \(K\) carries the diagonal
classical algebra. Define
\[
\Drop_K\left(\sum_x\kb{x}{x}_K\otimes X_x\right)=\sum_xX_x.
\]

\begin{proposition}[Readout, forgetting, and complete accounting]
\label{prop:drop}
The instrument is CPTP and
\begin{equation}
\Drop_K\MM_S=\Tr_S.
\label{eq:drop}
\end{equation}
For any CPTP operation on \(B\), with all its outputs ignored, the
untouched \(QR\) marginal is unchanged.
Thus residual Enter followed by Drop leaves \(\NN\) and
\(\erec(\NN)\) exactly unchanged.

A complete readout dilation is
\[
W_{\mathrm{read}}\ket{\psi}
=\sum_x\ket{x}_K\ket{x}_E\otimes M_x\ket{\psi}.
\]
It is an isometry. Tracing \(E\) gives \eqref{eq:Enter}.
Discarding \(K\) may be represented by moving its carrier to another
inaccessible store. Those outputs remain in a complete dilation.
\end{proposition}

\begin{proof}
The measurement bras satisfy \(\sum_xM_x^\dagger M_x=I\).
Summing the classical blocks gives the basis formula for partial trace.
For a local channel \(\Lambda_B\), its adjoint is unital, so every
untouched observable \(A\) obeys
\[
(\id\otimes\Lambda_B^*)(A\otimes I)=A\otimes I.
\]
All untouched expectations agree, including references.
Orthogonality of the two outcome labels and measurement completeness
give \(W_{\mathrm{read}}^\dagger W_{\mathrm{read}}=I\).
Tracing \(E\) removes the cross-outcome terms.
\end{proof}

Conditional states can change when an outcome is supplied. That is a
different access task. An observer restricted to \(R\) also cannot
execute Enter on \(B\) without an additional permission
\cite{CheungHawk04B,CheungHawk04C,CheungEn07A}.

\begin{proposition}[Finite-program soundness]
\label{prop:soundness}
A finite acyclic instruction tree built from specified pure allocation,
specified unitary constants, destructive Enter, classical control,
classical discard, and a fixed returned hybrid type denotes a CPTP
channel. Every quantum name is returned or explicitly consumed.
There are refined path maps \(K_h\) with
\[
\sum_hK_h^\dagger K_h=I,
\]
and each reported output block is the sum of its unnormalized path maps
\(X\mapsto K_hXK_h^\dagger\).
The denotation retains arbitrary references, and every branch terminates.
\end{proposition}

\begin{proof}
Allocation is isometric, unitaries are trace preserving, and readout
bras are complete. Classical control selects a complete continuation
on each block; discard sums blocks. Backward induction on the finite
tree proves path completeness. Grouping histories proves the block
formula. Tensoring path maps with \(I_Q\) proves the reference statement.
A finite tree has finite maximum branch length.
\end{proof}

The theorem neither compiles the constants over a fixed gate library
nor proves a physical endpoint. It uses the finite ownership discipline
of Hawk04-B/C and the separate sequential-accounting comparison in
En02-C \cite{CheungHawk04B,CheungHawk04C,CheungEn02C}.

\subsection{Two sharp benchmarks}

Set \(a_d=1-d^{-2}\).

\begin{lemma}[Replacement and classical-output benchmarks]
\label{lem:benchmarks}
A replacement accessible channel has optimal recovery error \(a_d\).
A measure-and-prepare channel, followed by any decoder to \(C\), has
error at least \(1-d^{-1}\). Computational readout and basis-state
preparation attain the latter bound.
\end{lemma}

\begin{proof}
A decoded replacement has Bell output \(I_Q/d\otimes\tau_C\) and Bell
overlap \(d^{-2}\). The Bell-projector test gives error at least \(a_d\).
Preparing \(I_C/d\) attains it: the completely depolarizing channel is
the uniform average of the \(d^2\) Weyl unitary channels, and separating
the identity term gives the matching upper bound.

A measure-and-prepare channel has separable Bell output. A normalized
product vector has squared overlap with \(\Phi_d\) at most \(1/d\), by
Cauchy--Schwarz; convexity extends this to separable states.
The Bell test gives \(1-1/d\).
Computational dephasing is the uniform average of the \(d\) phase
unitaries \(Z_d^k\). Its Bell outputs are orthogonal, giving diamond
distance \(1-1/d\) from the identity.
\end{proof}

Measure-and-prepare channels are entanglement breaking
\cite{HorodeckiShorRuskai}. The benchmark does not prohibit classical
assistance when a surviving quantum output is also returned
\cite{CheungHawk04B,CheungHawk04C}.

%=====================================================================
\section{Stage 1: two different finite calibration schedules}
\label{sec:baselines}

\subsection{The whole-code SWAP schedule}

Let \(C=C_A\otimes C_D\), with dimensions \(a,m\), so \(d=am\).
The first \(k\) transferred qubits of an \(n\)-qubit SWAP schedule
correspond to \(a=2^k\), \(m=2^{n-k}\).
Before measurement, the accessible channel is
\[
\NN_k=\Tr_{C_D},
\]
up to fixed output dictionaries and blank factors. The complement
returns \(C_D\).

\begin{theorem}[Exact SWAP and record-only calibration]
\label{thm:swap}
For this whole-code target,
\begin{equation}
\erec(\NN_k)=\dleak(\EE_k)=1-m^{-2}.
\label{eq:swap-quantum}
\end{equation}
If only the computational record on \(C_A\) is returned, the channel is
\[
\mathcal F_k(X)=
\sum_{x=1}^{a}\Tr((\kb{x}{x}\otimes I_m)X)\kb{x}{x},
\]
and its optimal recovery error is
\begin{equation}
\erec(\mathcal F_k)=1-\frac{1}{a m^2}.
\label{eq:swap-record}
\end{equation}
Thus, for \(n\) qubits,
\[
\erec(\NN_k)=1-2^{-2(n-k)},\qquad
\erec(\mathcal F_k)=1-2^{-(2n-k)}.
\]
These are analytic identities, not reported numerical executions.
\end{theorem}

\begin{proof}
The decoder returning \(C_A\) and preparing \(I_{C_D}/m\) has decoded
channel \(\id_{C_A}\otimes\TT_{I/m}\), with error \(1-m^{-2}\).
For the lower bound, use independent Bell states for \(C_A,C_D\).
After \(\Tr_{C_D}\), the reference \(Q_D\) is \(I/m\) and independent
of everything available to the decoder. The decoded full Bell overlap
is at most \(m^{-2}\): tracing \(Q_D\) against its Bell projector gives
\(I_{C_D}/m^2\), and the remaining Bell-projector expectation is at most
one. This proves the recovery lower bound.

The complementary channel is \(\Tr_{C_A}\) with output \(C_D\).
Choosing replacement \(I/m\) gives leakage at most \(1-m^{-2}\).
Fixing \(C_A\) and testing a Bell state on \(C_D\) gives the reverse
bound by Lemma~\ref{lem:benchmarks}.

For the record channel, write
\(F_x=\kb{x}{x}\otimes I_m\) and let \(\sigma_x\) be the decoder output
on record \(x\). Its Bell overlap is
\[
\frac{1}{d^2}\sum_{x=1}^{a}\Tr(F_x\sigma_x)\le\frac{a}{d^2}
=\frac{1}{a m^2}.
\]
For attainability, prepare
\(\kb{x}{x}\otimes I_m/m\).
The decoded channel is
\(\Delta_a\otimes\TT_{I/m}\), the uniform mixture of \(a m^2\)
orthogonal-Bell-output unitary channels, one of which is the identity.
Its distance from the identity is \(1-1/(a m^2)\).
\end{proof}

In particular, emitting half the qubits of a completely unknown
\(n\)-qubit input is not enough to recover that whole input.
Immediate readout makes the task stricter still. This calibrates the
schedule of Hawk04-C without confusing a whole star code with one small
diary \cite{CheungHawk04C}.

\subsection{A single logical cell with an occupied-vacuum flag}

Let a memory cell and each outgoing cell contain
\(\C\ket{\vac}\oplus C\).
At step \(j\), a specified unitary rotates
\[
\ket{i}_M\ket{\vac}_{S_j}
\longmapsto
\cos\theta_j\ket{i}_M\ket{\vac}_{S_j}
+\sin\theta_j\ket{\vac}_M\ket{i}_{S_j}
\]
for all logical labels \(i\), using the complementary rotation on the
opposite occupied-vacuum states and the identity elsewhere.
Choose \(0\le\theta_j\le\pi/2\). Define
\[
q_j=\prod_{\ell=1}^{j}\cos^2\theta_\ell,\qquad q_0=1.
\]
The amplitude of first transfer at step \(s\) is
\(\sin\theta_s\prod_{\ell<s}\cos\theta_\ell\).
Orthogonal transfer locations imply that, when \(q_j<1\), these
amplitudes define an isometry \(A_j:C\to R_j\) after normalization.

\begin{theorem}[Flagged-transfer calibration]
\label{thm:flagged}
After \(j\) pulses, the complete map has the form
\begin{equation}
V_{q_j}\ket{\psi}
=
\sqrt{1-q_j}\,A_j\ket{\psi}\ket{\vac}_B
+\sqrt{q_j}\,\ket{\vac}_R L\ket{\psi},
\label{eq:flagged}
\end{equation}
where the occupied ranges of \(A_j,L\) are orthogonal to their vacuum
flags. Its optimal recovery and leakage are
\begin{equation}
\erec(\NN_j)=\dleak(\EE_j)=q_j a_d.
\label{eq:flagged-errors}
\end{equation}
An optimal decoder compresses the occupied radiation code and prepares
\(I_C/d\) on its orthogonal complement.

If
\[
H_0=\omega\left(\Pi_{\mathrm{occ},M}
+\sum_s\Pi_{\mathrm{occ},S_s}\right),
\]
each pulse conserves \(H_0\). For one initially occupied cell,
\begin{equation}
E_{\mathrm{rad}}(j)=\omega(1-q_j).
\label{eq:flagged-energy}
\end{equation}
For \(K\) independent cells with the same \(q_j\), this expectation is
\(K\omega(1-q_j)\), even for correlated logical labels.
\end{theorem}

\begin{proof}
Induction gives the pulse amplitudes above, and
\[
\sum_{s=1}^{j}\sin^2\theta_s
\prod_{\ell<s}\cos^2\theta_\ell=1-q_j.
\]
This proves \eqref{eq:flagged} and its isometry property.

The accessible channel is
\[
\NN_j(X)=(1-q_j)A_jXA_j^\dagger
+q_j\Tr(X)\kb{\vac}{\vac}.
\]
The stated decoder produces
\((1-q_j)\id_C+q_j\TT_{I/d}\), with error \(q_j a_d\).
Any decoder's Bell overlap is at most
\((1-q_j)+q_j/d^2\), proving optimality.

The complement is
\[
\EE_j(X)=q_j LXL^\dagger
+(1-q_j)\Tr(X)\kb{\vac}{\vac}.
\]
Replacement
\(q_j LI_C L^\dagger/d+(1-q_j)\kb{\vac}{\vac}\)
gives leakage at most \(q_j a_d\).
For an arbitrary replacement state, apply a CPTP pinching and retraction
onto the occupied code block and vacuum flag. Let \(u\) be its occupied
weight. On a Bell input, the occupied Bell projector has expectation
difference \(q_j-u/d^2\). The effect equal to that projector on the
occupied block and identity on vacuum has difference \(u a_d\).
If \(u\le q_j\), the first is at least \(q_j a_d\); if \(u\ge q_j\),
the second is at least \(q_j a_d\). This proves the leakage lower bound.

Each pulse mixes equal-energy one-excitation states and is the identity
on all other sectors. It commutes with \(H_0\).
The transfer probability is \(1-q_j\), proving the energy formula.
Linearity and addition of the independent number observables prove
the \(K\)-cell statement.
\end{proof}

These formulas reconstruct the single-cell benchmark reported in the
supplied Hawk02-B excerpt and the flagged comparison in Hawk04-A
\cite{CheungHawk02B,CheungHawk04A}. They do not describe the whole-code
partial-trace channel of Theorem~\ref{thm:swap}.

\begin{corollary}[An imposed transition can be placed anywhere]
\label{cor:arbitrary-schedule}
Every nonincreasing sequence
\(1=q_0\ge q_1\ge\cdots\ge q_N\ge0\)
is realized by the pulses above. Thus a sharp leakage decrease near a
chosen step is a realizable design choice, not a derived Page-time law.
\end{corollary}

\begin{proof}
When \(q_{j-1}>0\), take
\(\cos^2\theta_j=q_j/q_{j-1}\).
After \(q_j=0\), later pulses can be trivial.
\end{proof}

Bare-energy conservation here is not a complete switching-work or
cyclic controller-reset certificate. Full computational Enter on the
logical labels also destroys their quantum coherence; measuring only an
occupation flag is a different instrument.

%=====================================================================
\section{Stage 2: controlled comparison, fixed codes, and growing codes}
\label{sec:limits}

\subsection{Stability and dimension-explicit conversion}

\begin{lemma}[Channel stability and local dictionaries]
\label{lem:stability}
On common output spaces,
\[
\abs{\erec(\NN_1)-\erec(\NN_2)}
\le\ddia(\NN_1,\NN_2),\qquad
\abs{\dleak(\EE_1)-\dleak(\EE_2)}
\le\ddia(\EE_1,\EE_2).
\]
Separate isometric embeddings of \(R\) and \(B\) preserve both
diagnostics.

For common-output isometries,
\begin{equation}
\ddia(\VV_1,\VV_2)\le\norm{V_1-V_2}_\infty.
\label{eq:isometry-bound}
\end{equation}
For channels with input dimension \(d\),
\begin{equation}
\ddia(\Phi,\Psi)\le
\min\left\{1,\frac d2\norm{\Jn(\Phi)-\Jn(\Psi)}_1\right\}.
\label{eq:Choi-bound}
\end{equation}
\end{lemma}

\begin{proof}
For each decoder, the triangle inequality and CPTP contraction change
its error by at most the channel comparison distance. Take infima in
both directions. Distance to the replacement set is likewise
one-Lipschitz.

An isometry \(J:H\to H'\) has the CPTP retraction
\[
Y\longmapsto J^\dagger YJ+
\Tr((I-JJ^\dagger)Y)\tau_H.
\]
Embedding candidate decoders or replacement states and then retracting
proves invariance.

For referenced input \(\rho\), expand the complete-output difference as
\[
(V_1-V_2)\rho V_1^\dagger
+V_2\rho(V_1^\dagger-V_2^\dagger),
\]
with reference identities understood. Its trace norm is at most
\(2\norm{V_1-V_2}_\infty\). Partial trace is contractive.

Every pure input with a \(d\)-dimensional reference is
\((L\otimes I)\ket{\Phi_d}\), with
\(\Tr(L^\dagger L)=d\) and \(\norm L_\infty^2\le d\).
Its output difference is the Choi difference multiplied on both sides
by \(L,L^\dagger\). Convexity completes the bound.
\end{proof}

These stability statements agree with the supplied Hawk02-A interface
and Hawk04-A/B proofs \cite{CheungHawk02A,CheungHawk04A,
CheungHawk04B}.

\paragraph{The external approximate input.}
For finite-dimensional complementary channels, the KSW
information--disturbance theorem gives
\begin{equation}
\erec(\NN)\le\min\{1,\sqrt{2\dleak(\EE)}\}.
\label{eq:KSW}
\end{equation}
Indeed the full-norm inequality
\[
\frac14\left(\inf_{\DD}
\norm{\DD\NN-\id_C}_\diamond\right)^2
\le
\inf_\sigma\norm{\EE-\TT_\sigma}_\diamond
\]
becomes \eqref{eq:KSW} under the half normalization
\cite[Theorem 3]{KSW}.
We use this approximate theorem only on finite channels.
No infinite-output attainment assumption is needed for the proxy lift.

\begin{theorem}[Finite-proxy certificate and lift]
\label{thm:certificate}
Let \(\widehat V:C\to\widehat R\otimes\widehat B\) be a finite
isometry, embedded factorwise into the output space of \(V\).
Suppose
\[
\ddia(\VV,\widehat\VV)\le b.
\]
For a specified proxy residual state \(\widehat\sigma\), set
\[
e=\frac12\norm{
\Jn(\widehat\EE)-I_Q/d\otimes\widehat\sigma}_1.
\]
Then
\begin{equation}
\dleak(\EE)\le\min\{1,de+b\}.
\label{eq:leak-certificate}
\end{equation}
A finite decoder, extended CPTP to all of \(R\), has error
\begin{equation}
\ddia(\DD\NN,\id_C)\le\min\{1,\sqrt{2de}+b\}.
\label{eq:recovery-certificate}
\end{equation}
Consequently \(\erec(\NN)\) is also at most \(a_d\).

For a specified proxy decoder with
\[
r_{\mathrm{dec}}
=\frac12\norm{
\Jn(\widehat\DD\widehat\NN)-\kb{\Phi_d}{\Phi_d}}_1,
\]
its extension satisfies
\begin{equation}
\ddia(\DD\NN,\id_C)\le\min\{1,d r_{\mathrm{dec}}+b\}.
\label{eq:direct-certificate}
\end{equation}
\end{theorem}

\begin{proof}
The complete comparison gives allowance \(b\) for both reduced channels.
Equation~\eqref{eq:Choi-bound} bounds proxy leakage by \(de\).
The triangle inequality proves \eqref{eq:leak-certificate}.
Finite decoder Choi matrices form a compact set, so an optimal proxy
decoder exists. Apply \eqref{eq:KSW} to the proxy.

Let \(P_R\) project onto the embedded finite accessible subspace.
Extend the decoder by
\[
\DD(Y)=
\widehat\DD(P_RYP_R)+\Tr((I-P_R)Y)\tau_C,
\]
with the compression identified with \(\widehat R\).
This map is CPTP and agrees on proxy outputs.
Postprocessing contraction adds \(b\), outside the square root.
The constant decoder supplies \(a_d\).
The direct certificate follows by applying the Choi bound to the
specified decoded proxy channel, followed by the same lift.
\end{proof}

This is the certificate of Hawk04-A/B with its normalization and
continuum-lift distinction retained
\cite{CheungHawk04A,CheungHawk04B}.
The dimension factor multiplies the normalized Choi discrepancy, not an
allowance already expressed in diamond distance.

\begin{theorem}[Fixed-code limit with an explicit rate]
\label{thm:fixed-limit}
Fix \(C\), common separable output spaces \(R_*,B_*\), and factorwise
embeddings of a sequence \(V_\ell\).
Suppose the embedded isometries \(\overline V_\ell\) converge in operator
norm to \(V_*\), with
\[
\norm{\overline V_\ell-V_*}_\infty\le\beta_\ell,\qquad
\beta_\ell\longrightarrow0.
\]
Then \(V_*\) is an isometry, and
\[
\ddia(\overline\NN_\ell,\NN_*)\le\min\{1,\beta_\ell\},\qquad
\ddia(\overline\EE_\ell,\EE_*)\le\min\{1,\beta_\ell\}.
\]
The recovery and leakage diagnostics differ from their limit values by
at most \(\min\{1,\beta_\ell\}\).
\end{theorem}

\begin{proof}
Inner products pass to the limit, so \(V_*^\dagger V_*=I_C\).
Apply Lemma~\ref{lem:stability}, including invariance under the local
embeddings.
\end{proof}

The theorem requires a complete isometry comparison.
Convergence of selected outgoing occupation numbers is not its premise.

\subsection{Growing input dimensions require a separate contract}

Let \(C_n\simeq\C^{d_n}\), with \(d_n\to\infty\).

\begin{definition}[Two legitimate growing-code comparisons]
\label{def:growing}
A growing-code study may use either:
\begin{enumerate}
\item \textbf{Fixed-subcode convergence.}
For each fixed finite code \(C^{(k)}\), supply embeddings
\(J_{n,k}:C^{(k)}\to C_n\) and compare
\(V_nJ_{n,k}\) on that fixed domain, with compatible output
identifications.
\item \textbf{Whole-code certificates.}
For each \(n\), test the target \(\id_{C_n}\) separately, reporting its
own complete proxy allowance \(b_n\) and Choi discrepancy \(e_n\).
\end{enumerate}
Neither convention silently equates different identity channels.
\end{definition}

\begin{corollary}[Uniform whole-code certification requirement]
\label{cor:growing}
For whole-code finite-proxy certificates,
\[
\erec(\NN_n)\le
\min\{a_{d_n},\sqrt{2d_n e_n}+b_n\}.
\]
Thus \(b_n\to0\) and \(d_n e_n\to0\) are sufficient for these upper
certificates to vanish. The assertion \(e_n\to0\) alone is insufficient
to make this particular certificate vanish.
\end{corollary}

\begin{proof}
Apply Theorem~\ref{thm:certificate} for each \(n\).
\end{proof}

The last sentence concerns the certificate, not a necessary condition
for every successful recovery method. For infinite-dimensional input
spaces, energy-constrained norms provide a different comparison
framework \cite{Shirokov}; no unrestricted infinite-input diamond limit
is inferred here.

%=====================================================================
\section{A model-specific finite-band regulator}
\label{sec:regulator}

This section supplies one concrete realization of the approximation
allowance. It reconstructs the finite-band regulator used in
Hawk04-A/B \cite{CheungHawk04A,CheungHawk04B}.
It is independent of the thermal-pair calibration in the next section.

Let \(H_S\) be a bounded self-adjoint operator on a finite source space.
An optional quartic realization is
\[
H_S=
\sum_{i<j<k<l}J_{ijkl}\chi_i\chi_j\chi_k\chi_l+B_NI,
\qquad
B_N=\sum_{i<j<k<l}|J_{ijkl}|,
\]
where \(\{\chi_i,\chi_j\}=2\delta_{ij}I\). Then
\(0\le H_S\le2B_NI\).

Fix \(0<\mu<\Lambda<\infty\), and set
\[
\mathfrak h=L^2([\mu,\Lambda],\dd\omega;\C^r),\qquad
(hf)(\omega)=\omega f(\omega).
\]
On \(\FS(\mathfrak h)\), let \(H_B=\dd\Gamma(h)\).
For fixed self-adjoint source operators \(A_a\), real \(g_a\), and
\(f_a\in\mathfrak h\), define
\[
H=H_S+H_B+\sum_a g_aA_a\otimes\Phi(f_a),
\qquad
\Phi(f)=a(f)+a^\dagger(f).
\]
Annihilation is conjugate-linear in \(f\). Put
\[
G=\sum_a|g_a|\norm{A_a}\norm{f_a},\qquad
G_\omega=\sum_a|g_a|\norm{A_a}\norm{h^{-1/2}f_a}.
\]
The bath is initially vacuum, and all finite source/carrier preparation
is given by one fixed code isometry.
Choose a finite collected packet space
\(\mathfrak r=\Span\{p_1,\ldots,p_q\}\).

\begin{lemma}[Complete dynamics exists]
\label{lem:H}
The Hamiltonian is self-adjoint on the free bath domain and generates
a complete unitary group.
If \(H_S\ge0\), it has the form lower bound
\[
H\ge\frac12H_B-2G_\omega^2I.
\]
\end{lemma}

\begin{proof}
The field estimates give
\[
\norm{a(f)\psi}\le
\norm{h^{-1/2}f}\norm{H_B^{1/2}\psi},
\]
and
\[
\norm{a^\dagger(f)\psi}^2
=\norm{a(f)\psi}^2+\norm f^2\norm\psi^2.
\]
Hence the interaction obeys
\[
\norm{V_{\mathrm{int}}\psi}
\le2G_\omega\norm{H_B^{1/2}\psi}+G\norm\psi.
\]
The square-root term has arbitrarily small relative operator bound
with respect to \(H_B\). Kato--Rellich therefore gives self-adjointness
\cite{Kato}; these are standard field-operator estimates
\cite{HaslerHerbst}.
For normalized form-domain vectors,
\[
|\langle V_{\mathrm{int}}\rangle|
\le2G_\omega\sqrt{\langle H_B\rangle}
\le\tfrac12\langle H_B\rangle+2G_\omega^2.
\]
The spectral theorem supplies the unitary group.
\end{proof}

No ground-state or scattering theorem is imported from the field
references.

Let \(P_m\) project onto bath occupation at most \(m\), and put
\[
Z_m(x)=\sum_{\ell=m+1}^{\infty}\frac{x^\ell}{\sqrt{\ell!}}.
\]

\begin{theorem}[Finite-time complete regulator]
\label{thm:regulator}
For fixed cutoffs, packets, and \(T<\infty\), there is a finite prepared
proxy isometry \(\widehat V_t\), embedded factorwise into the complete
output, such that uniformly for \(0\le t\le T\),
\begin{equation}
\ddia(\VV_t,\widehat\VV_t)\le
b_{\mathrm{reg}}(T;m,\Delta)
:=
\min\left\{1,2Z_m(2GT)+\frac{Tm\Delta}{2}\right\}.
\label{eq:breg}
\end{equation}
The same bound applies to both reduced channels and their diagnostic
differences.

For \(T>0\) and \(0<b<1\), sufficient choices are
\begin{equation}
m\ge1,\quad
m+1\ge4\mathrm e(2GT)^2,\quad
m\ge2+\log_2(1/b),\quad
\Delta\le\frac{b}{Tm}.
\label{eq:regulator-selection}
\end{equation}
They give \(b_{\mathrm{reg}}\le b\).
\end{theorem}

\begin{proof}
In the interaction picture,
\[
V_I(s)=\sum_a g_a A_a(s)\otimes\Phi(e^{\ii sh}f_a).
\]
On sectors with occupation at most \(k\),
\[
\norm{V_I(s)\psi}\le2G\sqrt{k+1}\norm\psi.
\]
An \(\ell\)-fold product from vacuum has norm at most
\((2G)^\ell\sqrt{\ell!}\).
After integration over the ordered simplex, its Dyson term is bounded
by \((2Gt)^\ell/\sqrt{\ell!}\).
The series converges also after multiplication by any fixed polynomial
in occupation. On these prepared vectors it solves the Schr\"odinger
equation in the required domains and equals the self-adjoint evolution.

The compressed Hamiltonian \(P_mHP_m\) has identical Dyson terms through
order \(m\). Inserted projections are contractions, so the two prepared
isometries differ by at most \(2Z_m(2GT)\).

Partition the frequency band into bins of width at most \(\Delta\).
Replacing frequency multiplication by bin midpoints gives
\(\norm{h-\widetilde h}\le\Delta/2\).
On the occupation-cutoff space,
\[
\norm{P_m\dd\Gamma(h-\widetilde h)P_m}\le m\Delta/2.
\]
Both compressed Hamiltonians are bounded, so Duhamel's formula adds
at most \(Tm\Delta/2\).

The active one-particle space
\[
\mathfrak f=
\Span\{\mathbf1_{I_b}f_a,\mathbf1_{I_b}p_j:a,b,j\}
\]
is finite dimensional, invariant under \(\widetilde h\), and contains
all couplings and collected packets. Its orthogonal complement remains
vacuum. The total-occupation-cutoff space is finite dimensional and
embeds into
\[
\FS_{\le m}(\mathfrak r)\otimes
\FS_{\le m}(\mathfrak f\cap\mathfrak r^\perp).
\]
This is an embedding, not an equality of the total cutoff with a tensor
product. Source and carrier stay in the inaccessible factor.
Lemma~\ref{lem:stability} converts the isometry allowance to
\eqref{eq:breg}.

Finally let \(x=2GT\). The factorial estimate yields
\[
\frac{x^{m+1}}{\sqrt{(m+1)!}}
\le\left(\frac{\mathrm e x^2}{m+1}\right)^{(m+1)/2}
\le2^{-(m+1)}.
\]
Later term ratios are at most \(1/2\), hence \(Z_m(x)\le2^{-m}\).
The occupation and mesh terms are each at most \(b/2\) under
\eqref{eq:regulator-selection}.
\end{proof}

\begin{scope}
The theorem refines occupation and frequency proxies on a bounded time
interval. It does not remove \(\mu,\Lambda\), and its constants are not
uniform under their removal. It proves no collapse geometry, no local
field-to-memory receiver, and no asymptotic completeness theorem.
A different Hamiltonian requires a different analytical comparison.
\end{scope}

%=====================================================================
\section{Thermal calibration is not a complete emission channel}
\label{sec:thermal}

\subsection{An independent thermal-pair cutoff}

Use \(\hbar=k_{\mathrm B}=1\) in this subsection.
For one oscillator frequency \(\omega>0\), temperature \(T>0\), and
\(\lambda=e^{-\omega/T}\), define
\[
\ket{\Theta_\lambda}
=\sqrt{1-\lambda}\sum_{r=0}^{\infty}\lambda^{r/2}\ket r_T\ket r_P.
\]
Its \(T\)-marginal is
\[
\tau_\lambda=(1-\lambda)\sum_{r=0}^{\infty}\lambda^r\kb r r,
\qquad
\Tr(N\tau_\lambda)=\frac{\lambda}{1-\lambda}.
\]
The normalized cutoff preparation is
\[
\ket{\Theta_{\lambda,m}}
=
\sqrt{\frac{1-\lambda}{1-\lambda^{m+1}}}
\sum_{r=0}^{m}\lambda^{r/2}\ket r_T\ket r_P.
\]

\begin{proposition}[Thermal-pair state and moment certificate]
\label{prop:thermal-cutoff}
These preparations satisfy
\begin{align}
\dtr(\kb{\Theta_\lambda}{\Theta_\lambda},
      \kb{\Theta_{\lambda,m}}{\Theta_{\lambda,m}})
&=\lambda^{(m+1)/2},\\
\dtr(\tau_\lambda,\tau_{\lambda,m})
&=\lambda^{m+1},\\
\Tr(N\tau_\lambda)-\Tr(N\tau_{\lambda,m})
&=\frac{(m+1)\lambda^{m+1}}{1-\lambda^{m+1}}.
\label{eq:thermal-moment}
\end{align}
For \(L\) independent pairs with individual cutoffs \(m_\ell\), the
complete pure-state distance is
\[
\sqrt{1-\prod_{\ell=1}^{L}
(1-\lambda_\ell^{m_\ell+1})}
\le
\sqrt{\sum_{\ell=1}^{L}\lambda_\ell^{m_\ell+1}}.
\]
Appending these fixed preparations to the same logical isometry gives
the same complete-channel allowance, independently of the code dimension.
\end{proposition}

\begin{proof}
The squared overlap of the complete pair preparations is
\(1-\lambda^{m+1}\), giving the pure-state distance.
The discarded geometric tail has weight \(\lambda^{m+1}\);
the normalized retained distribution gives the marginal distance.
Differentiating the finite geometric sum gives
\[
\Tr(N\tau_{\lambda,m})
=\frac{\lambda}{1-\lambda}
-\frac{(m+1)\lambda^{m+1}}{1-\lambda^{m+1}}.
\]
Product overlaps multiply, and
\(1-\prod_\ell(1-t_\ell)\le\sum_\ell t_\ell\).
For any referenced logical input, the preparation difference is a tensor
product with a trace-one logical output, so its trace norm is precisely
the preparation trace norm.
\end{proof}

This is a state-preparation certificate. It does not establish that a
finite matrix is an exact truncation of a Bogoliubov propagator.

\subsection{The Bogoliubov bridge has separate hypotheses}

A proposed free-field bridge must provide:
\begin{enumerate}
\item the spacetime and state preparation, field algebra, mode
normalizations, and incoming and outgoing access;
\item a canonical transformation with its complete partner and
purifier accounting;
\item an implementability statement in the chosen representation;
\item a fixed-code complete-channel comparison to its finite proxy;
\item moment or flux control for the unbounded observables used in
energy predictions.
\end{enumerate}
Unitary implementability of bosonic canonical transformations is a
nontrivial issue in infinite dimensions \cite{BruneauDerezinski}.
It is not supplied by choosing a lattice.

Nor can finite matrices satisfy the exact oscillator relation
\([a,a^\dagger]=I\), because the trace of a finite commutator is zero.
Finite truncation therefore requires a stated approximation and cannot
be described as an exact finite representation of the full CCR.

Even trace-distance convergence does not alone control an unbounded
number observable. For example,
\[
\rho_n=(1-n^{-1})\kb0 0+n^{-1}\kb{n^2}{n^2}
\]
converges in trace distance to vacuum, but \(\Tr(N\rho_n)=n\).
Equation~\eqref{eq:thermal-moment} supplies moment control for its
particular geometric preparation; a collapse model must supply its own.

\subsection{Equal thermal and entropy diagnostics, opposite recovery}

\begin{proposition}[Spectral underdetermination]
\label{prop:spectral-separation}
Choose a finite state \(\sigma_K\), a purification
\(\ket{\chi}_{KP}\), and \(L\simeq C\). Let
\(R=C_R\otimes K\), \(B=P\otimes L\), and define
\[
V_{\mathrm{return}}\ket\psi
=\ket\psi_{C_R}\ket\chi_{KP}\ket0_L,\qquad
V_{\mathrm{retain}}\ket\psi
=\ket0_{C_R}\ket\chi_{KP}\ket\psi_L.
\]
Then
\[
\NN_{\mathrm{return}}(X)=X\otimes\sigma_K,\qquad
\NN_{\mathrm{retain}}(X)=\Tr(X)\kb0 0\otimes\sigma_K.
\]
For every pure code input, their accessible spectra and entropies
coincide. For \(H_R=I_{C_R}\otimes H_K\), all accessible energy-measurement
distributions coincide for every input. Nevertheless,
\[
\erec(\NN_{\mathrm{return}})=0,\qquad
\erec(\NN_{\mathrm{retain}})=a_d.
\]
Taking \(\sigma_K\) Gibbs makes these selected energy statistics thermal.
\end{proposition}

\begin{proof}
Both maps preserve inner products, and partial traces give the channels.
For pure inputs, both outputs have the nonzero spectrum of \(\sigma_K\).
Every spectral effect of \(H_R\) acts identically on the logical factor.
Tracing \(K\) decodes the return map; the retain map is a replacement,
so Lemma~\ref{lem:benchmarks} applies.
\end{proof}

The same separation is reported in the supplied En07-C excerpts and
Hawk04-C \cite{CheungEn07C,CheungHawk04C}.
A complete channel, once specified, fixes its complementary information
up to the standard environmental equivalence. The missing information
here is the complete channel specification, not a failure of dilation
theory \cite{Stinespring}.

%=====================================================================
\section{Stage 3: energy, erasure cost, and a conditional mass law}
\label{sec:mass}

\subsection{Three independent ledgers}

The ledgers have different defining data:
\begin{enumerate}
\item \textbf{Complete energy.}
Supply the Hamiltonians and complete dynamics.
Conservation means \(U^\dagger H_{\mathrm{tot}}U=H_{\mathrm{tot}}\),
or, for an energy-intertwining isometry,
\(H_{\mathrm{out}}V=VH_{\mathrm{in}}\).
Unitarity alone does not imply either statement.
\item \textbf{Physical erasure/reset cost.}
Supply the memory, input class, reservoirs, work storage, controls,
initial correlations, heat/work conventions, and resource-restoration
conditions.
\item \textbf{Residual leakage.}
Compute the dimensionless complete-channel diagnostic \(\dleak\).
It is not an energy, stress tensor, area, or bit count.
\end{enumerate}
This separation is central to En07-A/B/C
\cite{CheungEn07A,CheungEn07B,CheungEn07C}.

\begin{proposition}[No unique implementation cost from a visible channel]
\label{prop:no-cost}
Equality of a selected visible channel does not determine a unique
heat or work ledger, even after a conserving spectator Hamiltonian is
fixed.
\end{proposition}

\begin{proof}
Append independent qubits \(W,L\) with
\(H_W=H_L=\epsilon\kb1 1\), \(\epsilon>0\).
Prepare \(W\) excited and \(L\) in
\(\gamma_L=\diag(1-p,p)\), \(0<p<1\).
Doing nothing and applying SWAP both conserve \(H_W+H_L\) and leave
the selected visible channel unchanged.
The first has designated reservoir energy change zero, while the second
has \(Q_L=\epsilon(1-p)>0\), supplied by the decrease of \(W\)'s energy.
The consumed resource differs. These are not cyclic erasure protocols
unless the spectator resources are restored.
\end{proof}

\begin{proposition}[A specified finite thermal balance]
\label{prop:thermal-balance}
Let a memory and finite reservoir start in
\(\rho_M\otimes\gamma_L\), with
\(\gamma_L=e^{-\beta H_L}/\Tr e^{-\beta H_L}\), \(\beta>0\).
After a specified unitary, define
\[
Q_L=\Tr(H_L(\rho_L'-\gamma_L)),\qquad
\Delta S_M=S(\rho_M)-S(\rho_M').
\]
Using natural logarithms,
\begin{equation}
\beta Q_L
=\Delta S_M+I(M:L)_{\rho'}+
D(\rho_L'\Vert\gamma_L)\ge\Delta S_M.
\label{eq:Landauer}
\end{equation}
\end{proposition}

\begin{proof}
Unitary entropy preservation and the initial product state give
\[
S(\rho_L')-S(\gamma_L)=\Delta S_M+I(M:L)_{\rho'}.
\]
Using \(\log\gamma_L=-\beta H_L-\log Z\) gives
\[
D(\rho_L'\Vert\gamma_L)
=\beta Q_L-[S(\rho_L')-S(\gamma_L)].
\]
Combine the identities and use nonnegativity.
\end{proof}

Equation~\eqref{eq:Landauer} is the thermal entropy-balance identity
associated with the additional implementation assumptions, not a cost
derived from \(\Drop\circ\Enter=\Tr\)
\cite{Landauer,ReebWolf}.

\subsection{The luminosity and clock are additional inputs}

Use \(G=\hbar=c=k_{\mathrm B}=1\) only for the physical matching formulas
in this subsection and the mass estimates below.
Thus \(T_{\mathrm H}(M)=1/(8\pi M)\).

\begin{assumption}[Semiclassical luminosity matching on a validity domain]
\label{ass:luminosity}
On \(M\in[M_{\mathrm c},M_0]\), with \(M_{\mathrm c}>0\), a proposed
model supplies a time normalization, outgoing flux observable, and
power law
\begin{equation}
P(M)=
\sum_\alpha\int_0^\infty\frac{\dd\omega}{2\pi}
\frac{\omega\,\Gamma_\alpha(M,\omega)}
{\exp(\omega/T_{\mathrm H}(M))-\eta_\alpha}.
\label{eq:power}
\end{equation}
Here \(\eta_\alpha=1\) for bosons and \(-1\) for fermions; degeneracies
are included in \(\alpha\).
The integrals and sum are finite.
For the cubic-scaling specialization, the species are massless and
fixed, and
\[
\Gamma_\alpha(M,\omega)=\gamma_\alpha(M\omega).
\]
The mass balance is additionally supplied as \(\dot M=-P(M)\), or with
a declared forcing error.
\end{assumption}

The flux normalization and absorption factors are part of a physical
emission calculation, as in Hawking and Page
\cite{Hawking1975,PageEmission}.
A temperature-labelled finite state does not supply them.

\begin{theorem}[Conditional inverse-square power and cutoff lifetime]
\label{thm:lifetime}
Under Assumption~\ref{ass:luminosity} in its fixed massless-species
specialization,
\[
P(M)=\frac{\kappa}{M^2},
\]
where
\begin{equation}
\kappa=
\sum_\alpha\int_0^\infty\frac{\dd x}{2\pi}
\frac{x\,\gamma_\alpha(x)}{\exp(8\pi x)-\eta_\alpha}.
\label{eq:kappa}
\end{equation}
If \(\kappa>0\) and \(\dot M=-\kappa/M^2\), then until the validity cutoff,
\begin{equation}
M(t)^3=M_0^3-3\kappa t,\qquad
t_{\mathrm c}=\frac{M_0^3-M_{\mathrm c}^3}{3\kappa}.
\label{eq:lifetime}
\end{equation}
For fixed \(M_{\mathrm c}\) and \(\kappa\), this cutoff lifetime has cubic
large-\(M_0\) scaling.
\end{theorem}

\begin{proof}
Set \(x=M\omega\) in \eqref{eq:power}.
The measure times the energy factor contributes \(M^{-2}\), and the
thermal denominator becomes \(\exp(8\pi x)-\eta_\alpha\).
Then \(\dd(M^3)/\dd t=3M^2\dot M=-3\kappa\).
Integrate and solve for the first crossing of \(M_{\mathrm c}\).
\end{proof}

Massive thresholds, changing species, rotation, charge, incoming
radiation, or time-dependent validity assumptions can change this
specialization. No coefficient or Planck-scale extrapolation is
predicted by the finite Enter interface.

Moreover, temperature matching without a clock is insufficient even
abstractly: traversing the same sequence of thermal states with a
rescaled time parameter preserves those state observations while
changing its duration. A lifetime requires a flux-and-time law.

\begin{scope}
The theorem predicts time to a positive cutoff under its explicit
luminosity assumptions. It does not predict the remaining lifetime,
final burst, or remnant behavior below that cutoff.
Finite termination of an imposed schedule supplies none of those
missing physical conclusions.
\end{scope}

\subsection{Stability, adiabaticity, and finite-step errors}

\begin{theorem}[Mass-law robustness]
\label{thm:mass-stability}
Suppose an absolutely continuous trajectory remains in
\([M_{\mathrm c},M_0]\) until its first cutoff crossing and satisfies
\[
(1-\rho)\frac{\kappa}{M^2}
\le-\dot M\le
(1+\rho)\frac{\kappa}{M^2},
\qquad 0\le\rho<1.
\]
Then it reaches the cutoff and
\begin{equation}
\frac{M_0^3-M_{\mathrm c}^3}{3\kappa(1+\rho)}
\le t_{\mathrm c}\le
\frac{M_0^3-M_{\mathrm c}^3}{3\kappa(1-\rho)}.
\label{eq:lifetime-bracket}
\end{equation}

More generally, if
\[
\dot M=-P(M),\qquad
\dot{\widetilde M}=-P(\widetilde M)+r(t),
\]
and \(P\) is Lipschitz with constant \(L\) on a common validity interval,
then, while both trajectories stay there,
\begin{equation}
|\widetilde M(t)-M(t)|
\le e^{Lt}|\widetilde M(0)-M(0)|
+\int_0^t e^{L(t-s)}|r(s)|\,\dd s.
\label{eq:Gronwall}
\end{equation}
\end{theorem}

\begin{proof}
Multiply the first differential inequalities by \(3M^2\).
The derivative of \(M^3\) lies between
\(-3\kappa(1+\rho)\) and \(-3\kappa(1-\rho)\).
Integration gives finite cutoff crossing and the time bracket.
For the second claim, the difference satisfies the integral inequality
\[
|z(t)|\le |z(0)|+\int_0^t L|z(s)|\,\dd s+\int_0^t|r(s)|\,\dd s.
\]
Gronwall's inequality gives \eqref{eq:Gronwall}.
\end{proof}

This locally proves the type of classical forcing stability requested
in the Hawk02-A plan, without importing an unprovided numerical forcing
estimate \cite{CheungHawk02A}.

\begin{proposition}[Euler mass schedule and frozen-step channel comparison]
\label{prop:discrete}
Let \(f(M)=-\kappa/M^2\), with \(|f|\le K\) and Lipschitz constant \(L\)
on a common positive-mass domain.
For step size \(h\), let
\(M_{j+1}=M_j+h f(M_j)\).
If the exact solution, numerical iterates, and connecting intervals
remain in that domain, then
\[
|M_j-M(jh)|\le\frac{Kh}{2}(e^{Ljh}-1).
\]
One may take \(K=\kappa/M_{\mathrm c}^2\) and
\(L=2\kappa/M_{\mathrm c}^3\) on \([M_{\mathrm c},M_0]\).

Separately, suppose a finite proxy Hamiltonian obeys
\[
\norm{H(M)-H(M')} \le K_H|M-M'|,
\qquad |\dot M|\le v.
\]
Replacing the actual mass path over one step by its initial value changes
the complete unitary channel by at most \(K_Hvh^2/2\) in half-diamond
distance. Multiple compatible step allowances add by telescoping.
\end{proposition}

\begin{proof}
The exact solution's local Euler residual is at most
\[
\int_0^h L|M(t_j+s)-M(t_j)|\,\dd s
\le LK h^2/2.
\]
Thus the global error obeys
\(e_{j+1}\le(1+Lh)e_j+LK h^2/2\).
Sum the geometric series and use \((1+Lh)^j\le e^{Ljh}\).

Duhamel's formula bounds the propagator difference by
\[
\int_0^h\norm{H(M(t_j+s))-H(M(t_j))}\,\dd s
\le K_Hvh^2/2.
\]
Apply the isometry-to-channel estimate and CPTP contraction at each
matched interface.
\end{proof}

For \(T(M)\propto M^{-1}\), a dimensionless slow-variation indicator is
\[
\chi_{\mathrm{ad}}=\tau_{\mathrm{corr}}
\frac{|\dot T|}{T}
=\tau_{\mathrm{corr}}\frac{|\dot M|}{M}.
\]
Small \(\chi_{\mathrm{ad}}\) is not a channel certificate by itself.
The final statement of Proposition~\ref{prop:discrete} requires a
Hamiltonian comparison constant, which may depend on the regulator.

\subsection{Feedback and logical-input dependence}

A prescribed external \(M(t)\) gives an input-independent family of
quantum channels. A measured record controlling later emissions gives
a different adaptive hybrid channel, which is CPTP when the complete
instruction tree is specified.

A rule that computes \(M\) from an unknown input state's expectation and
then selects a state-dependent propagator is not automatically a linear
quantum channel. It must be replaced by an explicit quantum or classical
control implementation, or restricted to an input-independent energy
sector. Neither fresh inputs nor input-dependent preparations may be
silently inserted while testing one fixed code.

Mass updates must use specified energy increments, not the number of
classical bits produced by Enter. An apparatus can generate a record
without that record count being an outgoing gravitational energy flux.

%=====================================================================
\section{Stage 4: information flow, entropy tests, and Page-time limits}
\label{sec:information}

\subsection{The two directions of approximate constancy}

\begin{proposition}[Featureless accessible output is a lower certificate]
\label{prop:featureless}
If \(\ddia(\NN,\TT_\omega)\le\eta\), then
\begin{equation}
\erec(\NN)\ge\max\{0,a_d-\eta\}.
\label{eq:featureless}
\end{equation}
Near constancy of the complete complement instead gives a recovery
upper certificate through \eqref{eq:KSW} or its finite-proxy lift.
\end{proposition}

\begin{proof}
Every decoded replacement is still a replacement.
Postprocessing contraction and the triangle inequality give
\[
\ddia(\DD\NN,\id_C)
\ge\ddia(\DD\TT_\omega,\id_C)-\eta
\ge a_d-\eta.
\]
Take the decoder infimum.
\end{proof}

This is the directional distinction retained in En07-B/C and
Hawk04-A/B/C \cite{CheungEn07B,CheungEn07C,CheungHawk04A,
CheungHawk04B,CheungHawk04C}.
A radiation state near maximally mixed for one input is not the uniform
referenced channel hypothesis in \eqref{eq:featureless}.

\begin{lemma}[A residual-response witness]
\label{lem:response}
If
\(x=\dtr(\EE(\rho_0),\EE(\rho_1))\),
then \(\dleak(\EE)\ge x/2\).
\end{lemma}

\begin{proof}
For every replacement state \(\sigma\), the triangle inequality gives
\[
x\le\dtr(\EE(\rho_0),\sigma)
+\dtr(\sigma,\EE(\rho_1))
\le2\ddia(\EE,\TT_\sigma).
\]
Take the infimum.
\end{proof}

\subsection{A whole-code dimension constraint}

\begin{proposition}[Accessible dimension bounds whole-code recovery]
\label{prop:dimension}
For a finite accessible output of dimension \(r\le d\),
\begin{equation}
\erec(\NN)\ge1-\frac rd.
\label{eq:dimension-lower}
\end{equation}
Exact recovery with constant residual rank \(s\) additionally requires
\(r\ge ds\).
\end{proposition}

\begin{proof}
The Bell-input state on \(Q\otimes R\) is a mixture of pure vectors of
Schmidt rank at most \(r\). Applying each decoder Kraus operator cannot
increase that rank. A normalized vector of Schmidt rank at most \(r\)
has squared Bell overlap at most \(r/d\), by Cauchy--Schwarz on its
Schmidt coefficients. Convexity and the Bell-projector test prove
\eqref{eq:dimension-lower}.
The exact statement is Theorem~\ref{thm:exact}.
\end{proof}

If \(C\) is the entire unknown \(n\)-qubit input and only \(k\) qubits
are accessible, this gives
\(\erec\ge1-2^{k-n}\).
A claim of accurate whole-code recovery after half the qubits must
therefore change its input target, output dimension, or access
assumptions. A small diary inserted into a previously correlated
system is not this whole-code task \cite{HaydenPreskill}.

\subsection{Entropy turnover is not a decoder transition}

For a pure input in the flagged model, the radiation state has
orthogonal occupied and vacuum blocks, so
\begin{equation}
S(R_j)=h_2(q_j),\qquad
h_2(q)=-q\log q-(1-q)\log(1-q).
\label{eq:flagged-entropy}
\end{equation}
Its entropy is maximal at \(q_j=1/2\), where the optimal recovery error
is \(a_d/2\), not zero. Also \(h_2(q)=h_2(1-q)\), although the respective
recovery errors \(qa_d\) and \((1-q)a_d\) differ.

More generally, Proposition~\ref{prop:spectral-separation} can be applied
at successive times with the same prepared ancilla spectrum
\(\sigma_{K,t}\). It gives identical selected entropy histories while
one logical channel returns the code and the other retains it.
Producing a turnover in \(\sigma_{K,t}\) does not repair the hidden-code
model. This is the entropy/recovery separation motivating the supplied
Hawk01-B comparison \cite{CheungHawk01B}.

If radiation is cumulatively retained and never reinteracts, more access
cannot worsen unrestricted recovery: whenever
\(\NN_s=P_{t\to s}\NN_t\) with \(P_{t\to s}\) CPTP,
\[
\erec(\NN_t)\le\erec(\NN_s).
\]
Indeed a decoder at \(s\) can be composed with the restriction.
This availability statement is not a streaming-memory or efficient
execution theorem \cite{CheungEn02C}.

\subsection{A joint-pair consistency test}

\begin{theorem}[Persistent near-Bell pairs constrain entropy turnover]
\label{thm:pair}
Let \(R\) be earlier radiation and \(b,c\) a new outgoing qubit and
partner. If
\[
S(bc)\le s,\qquad S(c)\ge\log2-u,
\]
then
\begin{equation}
S(Rb)-S(R)\ge\log2-s-u.
\label{eq:pair-growth}
\end{equation}
If
\(\dtr(\rho_{bc},\kb{\Phi^+}{\Phi^+})\le e\le1/2\), then
\begin{equation}
S(Rb)-S(R)\ge
\log2-2h_2(e)-e\log3.
\label{eq:pair-trace}
\end{equation}
If successive steps have no reinteraction of earlier radiation and
each right-hand side is at least \(\gamma>0\), then
\(S(R_N)\ge S(R_0)+N\gamma\).
\end{theorem}

\begin{proof}
Weak monotonicity, a consequence of strong subadditivity
\cite{LiebRuskai}, gives
\[
S(Rb)+S(bc)\ge S(R)+S(c).
\]
For the trace-distance specialization, finite-dimensional entropy
continuity gives
\(S(bc)\le h_2(e)+e\log3\), and partial-trace contraction gives
\(S(c)\ge\log2-h_2(e)\) \cite{Audenaert}.
The bounds are monotone on the stated range.
The final assertion is telescoping under the no-reinteraction premise.
\end{proof}

This reconstructs the supplied Hawk02-A pair test
\cite{CheungHawk02A}. It is consistent with the role of specified joint
pair structure in small-corrections arguments \cite{Mathur}.
Its trace-distance parameter is not silently identified with another
paper's amplitude parameter. A thermal-looking outgoing marginal does
not establish its joint-pair premise.

\subsection{A nonzero-coupling negative control}

The following check uses the commuting four-Majorana member of the
family in Hawk04-A/B \cite{CheungHawk04A,CheungHawk04B}.
Take
\[
H_S=J(I-S_1S_2),\qquad A=(S_1+S_2)/2,\qquad J>0,\quad g\ne0,
\]
where \(S_j=2c_j^\dagger c_j-I\).
Insert the diary into the source states
\(s_0=\ket{00}\), \(s_1=\ket{11}\).
They have source energy zero and \(A\)-eigenvalues \(-1,+1\).
Couple to one finite-band scalar field with
\(f(\omega)=1/\sqrt{2\omega}\).
Define
\[
\eta_i(t)=ga_i\frac{e^{-\ii\omega t}-1}{\omega}f(\omega),
\qquad a_0=-1,\quad a_1=1.
\]
For a collected packet projection \(P_{\mathfrak r}\), let
\[
c_R(t)=
\exp\left[-\frac12
\norm{P_{\mathfrak r}(\eta_0-\eta_1)}^2\right],
\qquad z_R=\sqrt{1-c_R^2}.
\]

\begin{proposition}[Commuting dynamical sanity check]
\label{prop:negative}
The radiation channel is
\[
\NN_t(\rho)=
\rho_{00}\kb{\eta_{0,R}}{\eta_{0,R}}
+\rho_{11}\kb{\eta_{1,R}}{\eta_{1,R}}.
\]
It has
\[
\erec(\NN_t)=\frac{3-z_R(t)}4\ge\frac12,\qquad
\dleak(\EE_t)\ge\frac12,
\]
although the complete output is exactly recoverable.
Its energy ledger is
\[
E_S(t)=0,\qquad
E_B(t)=g^2\int_\mu^\Lambda
\frac{1-\cos(\omega t)}{\omega^2}\,\dd\omega,\qquad
E_I(t)=-E_B(t).
\]
\end{proposition}

\begin{proof}
For a fixed source eigenvalue, Weyl displacement completes the square in
\(H_B+ga_i\Phi(f)\). Vacuum evolves, up to a common phase, to the coherent
amplitude \(\eta_i\). Tracing the orthogonal source labels gives the
displayed measure-and-prepare channel.

The two collected pure states have trace distance \(z_R\).
Any decoder's Bell overlap is at most
\((1+z_R)/4\): its two logical diagonal adjoint effects define a binary
discrimination measurement, whose equal-prior success probability is at
most \((1+z_R)/2\), obtained from the positive part of the state
difference. The Bell test gives error at least \((3-z_R)/4\).
Parity interchanges the two states, so an optimal discrimination
measurement may be symmetrized to have equal conditional errors.
Measuring it and preparing the inferred basis state yields
\[
z_R\Delta_Z+(1-z_R)\TT_{I_2/2}.
\]
Its Pauli identity weight is \((1+z_R)/4\); orthogonal Pauli Bell outputs
give the matching error.

The residual retains orthogonal source labels on the two basis inputs,
so Lemma~\ref{lem:response} gives leakage at least \(1/2\).
Complete preservation follows from Lemma~\ref{lem:H} and
Theorem~\ref{thm:exact}.
The coherent bath energy is
\(\norm{h^{1/2}\eta_i}^2\), giving the integral.
The initial total energy is zero, and conservation gives \(E_I=-E_B\).
\end{proof}

This is a negative control, not a generic scrambling claim.
Bath energy here is a dressing response of an initially undressed
preparation, not a derived gravitational mass loss.

%=====================================================================
\section{Stage 5: endpoint obligations and order of limits}
\label{sec:endpoint}

\begin{definition}[Four same-model endpoint obligations]
\label{def:endpoint}
For a fixed finite code \(C\), an endpoint package supplies:
\begin{enumerate}[label=\textbf{H\arabic*.}]
\item A complete pure preparation \(P_0:C\to H_{\mathrm{tot}}\),
including all fixed purifiers.
\item Input-independent complete unitary evolution \(U_t\), with a
justified time domain.
\item Output partitions \(R_t\otimes B_t\), factorwise isometries
\(j_t:R_t\to R_*\), \(k_t:B_t\to B_*\), and an endpoint map \(V_*\)
such that
\[
\overline V_t=(j_t\otimes k_t)V_t,\qquad
\norm{\overline V_t-V_*}_\infty\le\nu_t,\qquad \nu_t\to0.
\]
The embeddings preserve access roles.
\item Either exact constancy of
\(\EE_*=\Tr_{R_*}\Ad_{V_*}\), or an explicit approximate endpoint
certificate:
\begin{itemize}
\item finite endpoint outputs with
\(\ddia(\EE_*,\TT_{\sigma_*})\le\zeta\); or
\item a finite endpoint proxy with complete allowance \(b_*\) and
residual Choi discrepancy \(e_*\).
\end{itemize}
\end{enumerate}
For a physical application, these data must be justified in one
compatible model, with a receiver realizing the declared access.
\end{definition}

This is a sufficient contract, not a necessary characterization of all
possible endpoints. It is the four-obligation distinction in the
supplied En07-B/C excerpts, with the finite-proxy option added to avoid
an unproved infinite-output approximate-decoder assertion
\cite{CheungEn07B,CheungEn07C}.

\begin{theorem}[Fixed-decoder endpoint certificate]
\label{thm:endpoint}
Under Definition~\ref{def:endpoint}, \(V_*\) is an isometry and the
embedded complete and reduced channels are within
\(\min\{1,\nu_t\}\) of their endpoint channels.

If \(\EE_*\) is exactly constant, one fixed endpoint decoder
\(\DD_*:R_*\to C\) gives local finite-time decoders
\(\DD_t=\DD_*\Ad_{j_t}\) with
\[
\ddia(\DD_t\NN_t,\id_C)\le\min\{1,\nu_t\}.
\]
For a finite endpoint with allowance \(\zeta\), the same construction
has error at most
\[
\min\{1,\nu_t+\sqrt{2\zeta}\}.
\]
For the finite-proxy endpoint option, a fixed extended proxy decoder
has error at most
\[
\min\{1,\nu_t+b_*+\sqrt{2d e_*}\}.
\]
A specified endpoint proxy decoder instead gives
\(\min\{1,\nu_t+b_*+d r_{\mathrm{dec},*}\}\).

If the exact complete residual is pure, the endpoint radiation is an
isometric encoding. If it is mixed, the syndrome normal form applies.
\end{theorem}

\begin{proof}
Operator-norm convergence preserves \(V_*^\dagger V_*=I_C\).
Lemma~\ref{lem:stability} gives channel convergence.
For exact constancy, Theorem~\ref{thm:exact} supplies \(\DD_*\).
For a finite approximate endpoint, \eqref{eq:KSW} and finite attainment
supply an endpoint decoder. For the proxy option, use
Theorem~\ref{thm:certificate}.
In each case, postprocessing contraction adds at most \(\nu_t\).
The decoder \(\DD_t\) acts only on \(R_t\).
Rank one makes the syndrome space one dimensional.
\end{proof}

The endpoint statement continues the access-preserving construction in
Hawk04-A/B/C \cite{CheungHawk04A,CheungHawk04B,CheungHawk04C}.
A comparison unitary that swaps a still-inaccessible logical factor into
the radiation output is not an access-preserving identification.

\begin{proposition}[Zero endpoint leakage is exact constancy]
\label{prop:zero-leakage}
For finite input and separable normal outputs, \(\dleak(\EE)=0\)
implies that \(\EE\) is a replacement channel.
\end{proposition}

\begin{proof}
Choose states \(\sigma_n\) with
\(\ddia(\EE,\TT_{\sigma_n})\to0\).
Evaluation on \(I_C/d\) gives
\(\sigma_n\to\EE(I_C/d)\) in trace distance.
Replacement-channel distance equals the corresponding state distance.
The triangle inequality gives
\(\EE=\TT_{\EE(I_C/d)}\).
\end{proof}

\subsection{Information-free is not physically absent}

A constant complement contains no logical input dependence in the
declared full-code task. It need not be one dimensional, pure, empty,
or zero energy. Conversely, a zero-energy source can have a degenerate
memory retaining the entire code:
\[
V\ket\psi=\ket0_R\ket{\vac}_S\ket\psi_M,\qquad H_S\ket{\vac}=0.
\]
Radiation is constant, while the complete residual recovers the input.

The explicit endpoint maps in the supplied Hawk01-B excerpts transfer
all pending code and records to radiation and leave a fixed residual
\cite{CheungHawk01B}. That rule is an additional construction.
Calling a source ``evaporated'' does not supply it.

A complete classical-output channel on \(d\ge2\) cannot have a
one-dimensional pure-dilation environment: otherwise it would itself
be an isometry and have an exact decoder, contradicting
Lemma~\ref{lem:benchmarks}. Thus an empty list of live program names does
not prove an empty complete physical complement.

\subsection{Recurrence and a diagonal limiting schedule}

\begin{proposition}[A fixed finite autonomous proxy does not prove permanence]
\label{prop:recurrence}
For a finite-dimensional autonomous Hamiltonian, its complete prepared
channel returns arbitrarily close to its initial channel at arbitrarily
late times.
If accessible radiation is initially fixed vacuum, then
\[
\limsup_{t\to\infty}\erec(\widehat\NN_t)=a_d.
\]
\end{proposition}

\begin{proof}
The finitely many spectral phases have simultaneous approximate returns
to one. Equivalently, compactness of the phase torus and the group
property produce arbitrarily late returns; a periodic orbit uses
multiples of its period.
The propagator approaches the identity along such a sequence.
Lemma~\ref{lem:stability} gives channel recurrence and recovery stability.
The initial channel is a replacement with error \(a_d\), and every
channel has recovery error at most \(a_d\) using the constant decoder.
\end{proof}

This is the fixed-proxy obstruction in Hawk04-A/B/C
\cite{CheungHawk04A,CheungHawk04B,CheungHawk04C}.

If a continuum endpoint is approximated at increasing times, one must
choose a diagonal regulator schedule satisfying, for example,
\[
b_{\mathrm{reg}}(T_\ell;m_\ell,\Delta_\ell)\to0
\]
on the actual increasing windows \(T_\ell\), together with the endpoint
access comparison and decoupling estimates.
For the sufficient selection in \eqref{eq:regulator-selection},
occupation grows at least quadratically with \(GT_\ell\), and the mesh
must shrink accordingly.
A finite-time theorem does not justify interchanging the regulator and
late-time limits.

%=====================================================================
\section{The combined conditional certification theorem}
\label{sec:combined}

\begin{definition}[Physical acceptance package]
\label{def:acceptance}
A proposed extrapolation supplies:
\begin{enumerate}[label=\textup{(P\arabic*)}]
\item A fixed logical target, complete preparation, dynamics, field
content, regulators, validity domain, and all inaccessible outputs.
\item A justified accessible radiation interface and receiver, with
compatible complete comparisons for preparation, evolution, collection,
and implementation.
\item A finite proxy and a rigorously feasible residual state or
quantum-return decoder. The transfer law is labelled
\emph{derived} or \emph{additionally assumed}.
\item For a mass prediction, a specified flux, time normalization,
mass balance, cutoff, and quantitative power-error control.
\item For an endpoint prediction, the same-model package in
Definition~\ref{def:endpoint}.
\item For an experimental claim, a detector instrument, input class,
efficiency, background model, and statistical confidence convention.
\end{enumerate}
No scalar entropy, metric, or energy error is added to a diamond budget
without a conversion theorem.
\end{definition}

For compatible complete-channel comparisons let
\[
b_{\mathrm{tot}}=
\min\left\{1,\,
b_{\mathrm{prep}}+\sum_j b_j+
b_{\mathrm{reg}}+b_{\mathrm{collection}}+b_{\mathrm{implementation}}
\right\}.
\]
Every allowance must refer to the same complete output accounting after
specified identifications; omitted sectors already counted in the
complement are not charged twice.

\begin{theorem}[Hawk04-D conditional extrapolation certificate]
\label{thm:main}
Under the applicable parts of Definition~\ref{def:acceptance}:
\begin{enumerate}[label=\textup{(\roman*)}]
\item Compatible stepwise complete comparisons give actual-to-proxy
complete-channel distance at most \(b_{\mathrm{tot}}\).
\item A proxy residual Choi discrepancy \(e\) gives
\[
\dleak(\EE)\le\min\{1,de+b_{\mathrm{tot}}\},\qquad
\erec(\NN)\le
\min\{a_d,\sqrt{2de}+b_{\mathrm{tot}}\}.
\]
A specified decoder gives error at most
\(\min\{1,d r_{\mathrm{dec}}+b_{\mathrm{tot}}\}\).
\item An independently proved accessible replacement comparison gives
the lower bound \(\max\{0,a_d-\eta\}\).
\item A matched inverse-square luminosity with relative power error
\(\rho<1\) gives the cutoff-lifetime bracket
\eqref{eq:lifetime-bracket}. It does not determine below-cutoff behavior.
\item The endpoint package gives the fixed-decoder bounds of
Theorem~\ref{thm:endpoint}.
\item Any prescribed subsequent detector channel inherits the decoded
channel allowance by CPTP contraction.
\item Residual Enter followed by Drop leaves the already supplied
accessible channel unchanged and adds no final-burst conclusion.
\end{enumerate}
The channel, mass, and endpoint conclusions follow from separate
hypotheses. None supplies a missing hypothesis of another.
\end{theorem}

\begin{proof}
Telescoping matched CPTP compositions and contraction prove (i).
Theorem~\ref{thm:certificate} proves (ii);
Proposition~\ref{prop:featureless} proves (iii).
Theorems~\ref{thm:lifetime} and~\ref{thm:mass-stability} prove (iv).
Theorem~\ref{thm:endpoint} proves (v).
Postprocessing contraction proves (vi), and
Proposition~\ref{prop:drop} proves (vii).
The hypotheses are explicitly partitioned by their mathematical
objects, so the final scope assertion makes no additional physical
identification.
\end{proof}

The theorem is a conditional certification architecture in the spirit
of the supplied Hawk02-A acceptance interface
\cite{CheungHawk02A}. It is not a report that its gravitational
hypotheses have been discharged.

\subsection{What the numerical and experimental tests can establish}

For a specified decoded channel, let its Bell overlap be
\[
F=\bra{\Phi_d}
\Jn(\widehat\DD\widehat\NN)
\ket{\Phi_d}.
\]
A Bell-projector test always gives the error lower witness \(1-F\).
A validated lower bound \(F\ge F_0\) also gives
\[
r_{\mathrm{dec}}\le\sqrt{1-F_0},
\]
by the pure-target fidelity--trace-distance inequality
\cite{FuchsVanDeGraaf}. Hence the dimension-explicit upper certificate is
\[
\ddia(\DD\NN,\id_C)
\le\min\{1,d\sqrt{1-F_0}+b_{\mathrm{tot}}\}.
\]
High Bell overlap is therefore useful, but its conversion to a uniform
diamond certificate must retain the dimension factor or use a sharper
independently proved channel estimate.

Repeated calibrated preparations and measurements can estimate such
quantities. They are not one-run tomography of an unknown diary.
Statistical bounds must state their confidence level; numerical
feasibility and physical preparation/receiver errors remain separate.

\begin{proposition}[Observation-level testability]
\label{prop:testability}
If two predicted channels are within \(b\) in half-diamond distance,
then for every allowed referenced preparation and subsequent detector
instrument, their classical output distributions have total variation
distance at most \(b\).
\end{proposition}

\begin{proof}
Compose the preparation, channels, and detector.
CPTP contraction bounds the final trace distance by \(b\).
For diagonal classical states this is total variation distance.
\end{proof}

A failure of a matched empirical prediction challenges at least one
member of the physical, approximation, receiver, or statistical package.
It does not falsify the identity \(\Drop\circ\Enter=\Tr\).
Conversely, agreement with selected counts does not establish a
complete-channel certificate.

%=====================================================================
\section{Prediction targets and a reproducible next-calculation plan}
\label{sec:targets}

\subsection{Corrected targets}

\begingroup
\small
\renewcommand{\arraystretch}{1.12}
\begin{longtable}{@{}L{.17\textwidth}L{.26\textwidth}
L{.27\textwidth}L{.23\textwidth}@{}}
\caption{Conditional prediction targets. An experimental signature is a
test of the specified combined model, not a consequence of typing alone.}
\label{tab:targets}\\
\toprule
Target & Proved finite ingredient & Additional input & Defensible test\\
\midrule
\endfirsthead
\toprule
Target & Proved finite ingredient & Additional input & Defensible test\\
\midrule
\endhead
\bottomrule
\endfoot

Cutoff lifetime &
Sequential energy calibration; ODE and perturbation bounds &
Flux/time normalization, greybody scaling, species, validity cutoff &
Luminosity or lifetime data compared with the stated bracket, including
detector and population assumptions\\[4pt]

Entropy turnover &
Selected entropy examples and the joint-pair consistency test &
A specified state history and microscopic entropy calculation &
Entropy measurements in a calibrated analogue; not automatically a
decoder test\\[4pt]

Late quantum recovery &
Constant-complement criterion and finite Choi/decoder certificates &
Complete residual audit, coherent collection, feasible quantum decoder &
Referenced quantum-return tests with a dimension-explicit confidence
certificate\\[4pt]

Terminal energetic emission &
A terminal instruction is well typed; readout-and-drop leaves the old
radiation channel unchanged &
Below-cutoff dynamics, charged energy budget, spectral and detector model &
A specified energetic distribution or correlation; no burst follows
from Enter rate\\[4pt]

Information-free endpoint &
Exact or approximate endpoint complement certificate &
Same-model preparation, dynamics, access, and endpoint limit &
Recovery or complete residual-independence tests in a controlled model\\[4pt]

Absence of physical remnants &
No finite channel identity proves it &
Microscopic endpoint ontology, stable-sector audit, production and
detection model &
Statistical exclusion of a stated remnant class, not universal absence
from a null observation\\
\end{longtable}
\endgroup

\subsection{Six calculations, with completion criteria}

\begin{enumerate}[label=\textbf{C\arabic*.}]
\item \textbf{Analytic baseline before simulation.}
Implement the \(n=8\)--\(12\) SWAP schedule only after specifying whether
the target is the whole \(n\)-qubit input or a smaller diary.
Compare against Theorem~\ref{thm:swap}.
Independently implement the occupied-vacuum pulse model and compare
against \(q_j a_d\).
These are different channels and must not share an unqualified formula.

\item \textbf{A finite-band microscopic proxy.}
Archive \(H_S,A_a,g_a,f_a\), cutoffs, code preparation, packets,
\(T,m,\Delta\), and the exact finite active space.
Choose regulators before evaluating the recovery certificate.
Report the proved \(b_{\mathrm{reg}}\), not merely a matrix size.

\item \textbf{A separate field bridge.}
If studying a collapsing shell, archive the geometry, canonical
transformation, representation, partner modes, and complete fixed-code
comparison theorem.
Do not substitute \eqref{eq:breg} from a different Hamiltonian.
Thermal-pair preparation may first be checked using
Proposition~\ref{prop:thermal-cutoff}; it is not the collapse calculation.

\item \textbf{Mass matching and stability.}
Declare whether the mass path is imposed or derived.
Supply the outgoing energy observable and emission clock.
Validate the power law on a positive-mass domain, report the cutoff
lifetime bracket, and retain all forcing, discretization, switching, and
interaction-energy terms.

\item \textbf{Information and negative controls.}
Compute every logical matrix unit of both proxy channels or exploit a
proved symmetry reduction.
Validate a residual state or CPTP decoder.
Include the commuting control of Proposition~\ref{prop:negative},
the spectral separation, and a record-only test.
A designed \(q_j\) drop is labelled designed, not discovered.

\item \textbf{Release the full certificate package.}
Archive channel data, feasible Choi matrices, validated positivity,
trace-preservation and trace-norm bounds, arithmetic allowances,
quantum-return programs, separate terminal-readout programs, complete
apparatus outputs, and all physical matching assumptions.
For endpoint claims, include access-preserving identifications and the
order of limits.
\end{enumerate}

No numerical run is reported in this manuscript.
For a whole \(n\)-qubit input, there are \(d^2=4^n\) logical matrix units.
Generic complete-channel certification is therefore not automatically
tractable at \(n=12\). Analytic symmetries, smaller fixed diaries, or
validated structured representations must be justified rather than
assumed.

For a finite residual Choi state \(J\), a feasible trace-norm upper
certificate can be produced by
\[
\begin{aligned}
\text{minimize}\quad&
\tfrac12\Tr(X_++X_-),\\
\text{subject to}\quad&
J-I_Q/d\otimes\sigma=X_+-X_-,\\
&X_\pm\succeq0,\quad \sigma\succeq0,\quad\Tr\sigma=1.
\end{aligned}
\]
A feasible tuple certifies an upper bound on \(e\), not the exact diamond
norm. A finite decoder's unnormalized Choi operator must satisfy
\[
J_D\succeq0,\qquad \Tr_CJ_D=I_{\widehat R}.
\]
Floating-point objective values alone certify neither condition.

%=====================================================================
\section{Standing constraints and conclusion}
\label{sec:conclusion}

The completed plan yields local channel theorems, explicit calibration
models, controlled finite-time proxy comparisons, and conditional mass
and endpoint implications. Its conclusions retain the following limits.

\begin{enumerate}[label=\textup{(S\arabic*)}]
\item No metric, event horizon, area theorem, stress-energy solution,
backreaction geometry, or gravitational field-to-memory receiver is
constructed.
\item Hawking temperature and greybody matching are external physical
data. A selected Gibbs marginal is not a complete logical channel.
\item The cubic formula is a time-to-cutoff consequence of an imposed or
independently derived inverse-square luminosity. It is not a
Planck-scale endpoint theorem.
\item No universal Page-time transition is obtained. Whole-code recovery,
small-diary recovery, entropy turnover, and reference decoupling have
different hypotheses.
\item Readout-and-drop is operational forgetting, not annihilation.
Complete apparatus and discarded-record outputs remain accounted for.
\item An information-free residual need not be physically absent.
Absence of detectable objects requires a specified empirical exclusion
model.
\item A mathematical decoder and an abstract qubit dictionary are not
an efficient implementation, a finite-gate compilation certificate, or
a coherent physical receiver.
\item The finite-band regulator is not transferred to another
Hamiltonian or to cutoff removal by analogy.
Fixed finite autonomous proxies recur.
\item No catalogue attributed to OpenAI is used as a premise.
The separate documentary statuses and analytic-admission programmes
mentioned by the companion manuscripts are not updated by this paper.
\end{enumerate}

The mathematical extrapolation is therefore:
\[
\boxed{
\begin{gathered}
\text{complete physical comparison}
+\text{finite feasible certificate}
\Longrightarrow
\text{referenced channel bound},\\
\text{specified luminosity and mass balance}
\Longrightarrow
\text{conditional cutoff-lifetime bound},\\
\text{access-preserving endpoint and decoupling}
\Longrightarrow
\text{endpoint quantum recovery}.
\end{gathered}}
\]
None of these arrows supplies the missing premise of another.

The Enter philosophy is useful precisely because it keeps the
additional obligations visible. The resulting programme can generate
quantitative conditional predictions and calibrated analogue tests.
A claim about actual black-hole evaporation requires a same-model
physical construction, not a renaming of the finite outputs.

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

\paragraph{Verification and reproducibility.}
The evidence consists of the analytical arguments printed here.
Compilation, numerical optimization, proof-assistant checking,
independent technical review, and acceptance of this manuscript are
not asserted. No experimental or simulated data are reported.

\paragraph{Attribution.}
Established dilation, correction, no-hiding, entropy, and
information--disturbance results are attributed to their external
literature. Cheung companion manuscripts are version-qualified
comparisons. Local proofs do not establish priority for the underlying
standard principles.

\paragraph{Authorship and submission metadata.}
The author field follows the supplied Hawk04-A/B/C sources.
The responsible submitter should confirm authorship, affiliations,
journal-specific declarations, and archival availability of unpublished
comparisons before submission.

%=====================================================================
\appendix

\section{Debugging and provenance ledger}
\label{app:debug}

\subsection{Mathematical repairs incorporated into the paper}

\begin{longtable}{@{}L{.31\textwidth}L{.62\textwidth}@{}}
\toprule
Draft shortcut & Replacement used here\\
\midrule
\endfirsthead
\toprule
Draft shortcut & Replacement used here\\
\midrule
\endhead
\bottomrule
\endfoot

Decoder allowance \(2de+b\) &
The general complementary-Choi allowance is
\(\sqrt{2de}+b\). A specified decoded Choi state instead gives
\(d r_{\mathrm{dec}}+b\).\\[4pt]

Diamond convergence with changing \(d_n\) &
Fixed-subcode comparisons or separate whole-code certificates are
required. Whole-code upper certificates retain \(d_ne_n\).\\[4pt]

One regulator for scalar reservoir and collapse Bogoliubov map &
The finite-band theorem is proved for its printed Hamiltonian.
A field bridge has separate representation, implementability, access,
and approximation obligations.\\[4pt]

Exact finite oscillator algebra &
Finite matrices cannot obey the full CCR.
Normalized finite preparations and finite-channel comparisons are used
instead.\\[4pt]

Bits emitted imply mass decrease &
Energy and mass updates use declared flux observables and a clock.
Record count alone is not an energy ledger.\\[4pt]

\(T\propto M^{-1}\) implies \(M^3\) lifetime &
Fixed species, greybody scaling, flux normalization, and mass balance
are also required; the theorem stops at \(M_{\mathrm c}>0\).\\[4pt]

Half the quanta imply recovery of the whole code &
The target and accessible dimension are fixed first.
The SWAP calibration and dimension lower bound obstruct that universal
claim.\\[4pt]

Entropy turnover implies small complete leakage &
Explicit equal-entropy/unequal-recovery channels disprove this inference.
A separate complete-complement certificate is required.\\[4pt]

Final Enter creates an energetic burst &
Readout-and-drop leaves prior radiation unchanged.
Any burst requires a new energetic dynamics and detector model.\\[4pt]

Thermalization makes the environment disappear &
Input independence, thermal state preparation, and physical absence are
different claims. Purifiers remain in complete accounting.\\[4pt]

Null remnant detection proves no remnants &
Only a stated class can be statistically excluded under a production,
efficiency, and background model.\\[4pt]

Finite termination proves irreversible evaporation &
Finite syntax terminates; finite autonomous proxies recur.
An endpoint needs its own limit and access construction.\\
\end{longtable}

\subsection{Source boundary}

The complete Hawk04-A version~1, Hawk04-B version~2, and Hawk04-C
version~1 sources were supplied for this development.
The Hawk01-B, Hawk02-A/B, and En07-A/B/C comparisons additionally use the
version-qualified excerpts supplied with the request.
Their full unseen proof packages are not asserted to have been inspected.

The Philosophy essay was consulted at its stated companion address.
Cheung's journal article has an independently identifiable publisher
record. That article's status does not assign publication dates or
journal acceptance to the companion manuscripts.

The retrieval labels, duplicated catalogue extracts, and automated
``further development'' paragraphs in the supplied material are not
mathematical premises. OpenAI catalogue material is excluded.

\begin{longtable}{@{}L{.22\textwidth}L{.70\textwidth}@{}}
\toprule
Source & Comparison retained\\
\midrule
\endfirsthead
\toprule
Source & Comparison retained\\
\midrule
\endhead
\bottomrule
\endfoot

Hawk04-A@1 &
Complete restoration, Schmidt decoder, finite-band regulator,
dimension-explicit certificate, negative control, and access-preserving
endpoint.\\[4pt]

Hawk04-B@2 &
Typed destructive readout, quantum return, apparatus accounting,
finite proxies, and recurrence.\\[4pt]

Hawk04-C@1 &
Six-row operational analogy, SWAP schedule, complete-complement
criterion, three ledgers, and standing non-identifications.\\[4pt]

Hawk02-A@1 &
Supplied stability, pair-entropy, mass-forcing, and conditional physical
acceptance interfaces. Corresponding statements used here are proved
locally.\\[4pt]

Hawk02-B@2 &
Supplied single-cell pulse survival and optimal flagged recovery
benchmark. The model and formulas are reconstructed locally.\\[4pt]

Hawk01-B@1 &
Supplied entropy/recovery separation and explicit information-free
endpoint transfer. No endpoint is inferred without that added rule.\\[4pt]

En07-A/B/C@1 &
Supplied three-ledger separation, endpoint obligations, KSW direction,
and selected spectral/entropy counterexamples.\\[4pt]

En02-C@1 &
Sequential reference accounting and the distinction between available
information and executable recovery under a resource budget.\\
\end{longtable}

\section{A compact archival checklist}
\label{app:archive}

A submission reporting a future positive physical or numerical result
should additionally archive:
\begin{enumerate}
\item the exact code target and reference convention;
\item complete preparation, all purifiers, source/carrier ordering,
and every inaccessible factor;
\item the frozen Hamiltonian or canonical transformation and its
existence/implementability proof;
\item the accessible algebra and physical receiver comparison;
\item regulator choices and complete-channel allowances on the actual
time domain;
\item all logical matrix-unit data or a proved equivalent reduction;
\item feasible residual and decoder Choi operators with validated
positivity, normalization, and arithmetic bounds;
\item the quantum-return program and any distinct terminal observation;
\item full energy, interaction, switching, heat/work, and resource-return
ledgers;
\item luminosity matching, clock, mass cutoff, and perturbation bounds;
\item endpoint embeddings, complete residual audit, and order of limits;
\item detector likelihoods, efficiencies, backgrounds, and confidence
statements.
\end{enumerate}
No successful branch, selected population test, disorder-averaged
channel, or entropy plot substitutes for this package.

%=====================================================================
\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}.
Introductory essay, EntPhil, version~2, n.d.
Enter Calculus Companion.
\url{https://enter.grok.me/enter}.
Consulted October~14, 2026.
Introductory source, not a registered gravitational theorem.

\bibitem{CheungEnter}
N.~Cheung,
``The Enter Calculus: A Linear Quantum Lambda Calculus with an
Operational Characterization of Measurement,''
\emph{Frontiers in Physics} \textbf{14} (2026).
DOI: \url{https://doi.org/10.3389/fphy.2026.1954943}.
Publisher record consulted October~14, 2026:
accepted September~28, 2026; final formatted version indicated as
forthcoming.

\bibitem{CheungHawk04A}
N.~Cheung,
\emph{Hawk04-A: Unitary Information Preservation and Radiation
Recovery: A Complete-Complement Formulation of the Black-Hole
Information Problem}.
Version~1, supplied unpublished manuscript, n.d.
Source supplied as \texttt{Hawk04-A-v1 (1).tex}.
Comparison record; channel and regulator arguments used here are
printed locally.

\bibitem{CheungHawk04B}
N.~Cheung,
\emph{Hawk04-B: Black-Hole Information as an Access Problem:
Enter, Quantum Recovery, and Complete Residuals}.
Version~2, supplied unpublished manuscript, n.d.
Source supplied as \texttt{Hawk04-B-v2 (1).tex}.

\bibitem{CheungHawk04C}
N.~Cheung,
\emph{Hawk04-C: Finite Enter Analogues of the Hawking Narrative:
Access, Complete Complements, and Destructive Readout}.
Version~1, supplied unpublished manuscript, n.d.
Source supplied as \texttt{Hawk04-C-v1.tex}.

\bibitem{CheungHawk02A}
N.~Cheung,
\emph{Hawk02-A}.
Version~1, supplied unpublished manuscript excerpts, n.d.
Comparisons: \texttt{thm:stability}, \texttt{prop:adiabatic},
\texttt{prop:massstability}, \texttt{thm:pairentropy}, and the
conditional physical acceptance package.
No unprovided numerical estimate is imported.

\bibitem{CheungHawk02B}
N.~Cheung,
\emph{Hawk02-B}.
Version~2, supplied unpublished manuscript excerpts, n.d.
Comparisons: sequential occupied-vacuum pulse model,
\texttt{thm:B3}, and \texttt{eq:singleoptimum}.
The pulse model and optimal error used here are reconstructed locally.

\bibitem{CheungHawk01B}
N.~Cheung,
\emph{Hawk01-B}.
Version~1, supplied unpublished manuscript excerpts, n.d.
Comparisons: full-code recovery criterion, entropy/recovery separation,
and the explicit complete endpoint transfer.
The endpoint rule is not inferred from mass or entropy alone.

\bibitem{CheungEn07A}
N.~Cheung,
\emph{En07-A}.
Version~1, supplied unpublished manuscript excerpts, n.d.
Comparisons: complete restoration, destructive readout, discard,
apparatus accounting, and thermodynamic underdetermination.

\bibitem{CheungEn07B}
N.~Cheung,
\emph{Access-preserving endpoint certificates under destructive Enter:
Exact recovery, approximate certification, and the limits of
channel-only closure}.
En07-B, version~1, supplied unpublished manuscript excerpts, n.d.
In particular, \texttt{def:four} and \texttt{thm:endpoint}.

\bibitem{CheungEn07C}
N.~Cheung,
\emph{Hawking Radiation under Destructive Enter}.
En07-C, version~1, supplied unpublished manuscript excerpts, n.d.
Comparisons: external spectrum typing, selected entropy and energy
statistics, three ledgers, and same-model endpoint obligations.

\bibitem{CheungEn02C}
N.~Cheung,
\emph{Resource-Limited Black-Hole Information Recovery:
Radiation Access, Decoder Memory, and Operational Recovery Thresholds}.
En02-C, version~1, supplied unpublished comparison excerpts, n.d.
Sequential reference accounting and availability-versus-execution
distinctions; no model-specific optimum is transferred here.

\bibitem{Hawking1975}
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{Hawking1976}
S.~W.~Hawking,
``Breakdown of predictability in gravitational collapse,''
\emph{Physical Review D} \textbf{14} (1976), 2460--2473.
DOI: \url{https://doi.org/10.1103/PhysRevD.14.2460}.

\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}.
Cited for the physical emission framework, not for a modern
species-independent numerical lifetime coefficient.

\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}.
arXiv:gr-qc/9305007.

\bibitem{PageInformation}
D.~N.~Page,
``Information in black hole radiation,''
\emph{Physical Review Letters} \textbf{71} (1993), 3743--3746.
DOI: \url{https://doi.org/10.1103/PhysRevLett.71.3743}.
arXiv:hep-th/9306083.

\bibitem{HaydenPreskill}
P.~Hayden and J.~Preskill,
``Black holes as mirrors: Quantum information in random subsystems,''
\emph{Journal of High Energy Physics} 09 (2007), 120.
DOI: \url{https://doi.org/10.1088/1126-6708/2007/09/120}.
arXiv:0708.4025.

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

\bibitem{GaikwadEtAl}
A.~Gaikwad, A.~Kaushal, G.~Mandal, and S.~R.~Wadia,
``A microscopic model of black hole evaporation in two dimensions,''
\emph{Journal of High Energy Physics} 08 (2023), 171.
DOI: \url{https://doi.org/10.1007/JHEP08(2023)171}.
arXiv:2210.15579.

\bibitem{Stinespring}
W.~F.~Stinespring,
``Positive functions on \(C^*\)-algebras,''
\emph{Proceedings of the American Mathematical Society}
\textbf{6} (1955), 211--216.
DOI: \url{https://doi.org/10.1090/S0002-9939-1955-0069403-4}.

\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}.
arXiv:quant-ph/9604034.

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

\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}.
arXiv: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}.
arXiv:quant-ph/0302031.

\bibitem{Shirokov}
M.~E.~Shirokov,
``On the energy-constrained diamond norm and its application in
quantum information theory,''
\emph{Problems of Information Transmission}
\textbf{54} (2018), 20--33.
DOI: \url{https://doi.org/10.1134/S0032946018010027}.
Preprint: arXiv:1706.00361.

\bibitem{BruneauDerezinski}
L.~Bruneau and J.~Derezi\'nski,
\emph{Bogoliubov Hamiltonians and one parameter groups of
Bogoliubov transformations}.
arXiv:math-ph/0511069.
Cited for implementability and generator obligations, not as a
collapse-channel construction.

\bibitem{Kato}
T.~Kato,
\emph{Perturbation Theory for Linear Operators}.
Classics in Mathematics, Springer, 1995.
Reprint of the second edition.
DOI: \url{https://doi.org/10.1007/978-3-642-66282-9}.

\bibitem{HaslerHerbst}
D.~Hasler and I.~Herbst,
\emph{Ground States in the Spin Boson Model}.
arXiv:1003.5923.
Field-operator context only; no ground-state or scattering theorem
is imported into the present finite-band many-level model.

\bibitem{Landauer}
R.~Landauer,
``Irreversibility and heat generation in the computing process,''
\emph{IBM Journal of Research and Development}
\textbf{5} (1961), 183--191.
DOI: \url{https://doi.org/10.1147/rd.53.0183}.

\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}.
arXiv:1306.4352.

\bibitem{LiebRuskai}
E.~H.~Lieb and M.~B.~Ruskai,
``Proof of the strong subadditivity of quantum-mechanical entropy,''
\emph{Journal of Mathematical Physics}
\textbf{14} (1973), 1938--1941.
DOI: \url{https://doi.org/10.1063/1.1666274}.

\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}.
arXiv:quant-ph/0610146.

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

\bibitem{FuchsVanDeGraaf}
C.~A.~Fuchs and J.~van de Graaf,
``Cryptographic distinguishability measures for quantum-mechanical
states,''
\emph{IEEE Transactions on Information Theory}
\textbf{45} (1999), 1216--1227.
DOI: \url{https://doi.org/10.1109/18.761271}.
arXiv:quant-ph/9712042.

\end{thebibliography}

\end{document}