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% Scope:
% This revision proves a microscopic construction and certification
% framework for a regulated interacting SYK--scalar-reservoir model.
% It does NOT claim that the required small residual bound,
% gravitational correspondence, or evaporation endpoint has been proved.
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% Evidence:
% No pdfLaTeX execution, numerical execution, formal verification,
% or independent peer review is asserted.
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% Source basis:
% The preceding Hawk02-D text and the excerpts supplied in this
% conversation. References to Cheung identify those supplied versions.
%
% Author metadata must be supplied by the responsible author.

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\hypersetup{
  pdftitle={Hawk02-E: Coherent Diaries and Complete-Residual Certification},
  pdfsubject={A regulated interacting SYK--scalar-reservoir model},
  pdfkeywords={SYK, quantum recovery, complete complement,
               scalar reservoir, channel certification, evaporation}
}

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\title{
Hawk02-E\\[0.3em]
Coherent Diaries and Complete-Residual Certification\\
in a Regulated Interacting SYK--Scalar Model
}
\author{}
\date{}

\begin{document}
\maketitle

\begin{abstract}
We specify a fixed-realization interacting SYK source coupled to
a continuum of free scalar reservoir modes with declared infrared
and ultraviolet cutoffs. A normalized Euclidean-filtered pure
source state is prepared before a logical diary arrives. A single
message-independent, parity-preserving swap loads an even-parity
qubit diary into two source modes. The displaced carrier is retained
in the complete residual, including its correlations with the
remaining source.

We construct the self-adjoint continuum Hamiltonian, determine the
complete code isometry, and derive a reference-preserving finite
approximation for every prescribed finite packet algebra and finite
observation interval. If \(G\) is an explicit interaction bound,
\(m\) is a total reservoir occupation cutoff, and \(\Delta\) is a
frequency-mesh bound, the approximation allowance is
\[
 b_{\mathrm{reg}}(T;m,\Delta)
 \le
 \min\left\{
 1,\,
 2\sum_{r=m+1}^{\infty}\frac{(2GT)^r}{\sqrt{r!}}
 +\frac{Tm\Delta}{2}
 \right\}.
\]
The finite approximation supplies all logical matrix units and both
channel outputs. A complementary Choi-state certificate
\(\widehat e_J\) gives
\[
 \delta_{\mathrm{full}}(t)
 \le d\,\widehat e_J(t)+b_{\mathrm{reg}},
 \qquad
 \varepsilon_{\mathrm{rec}}(t)
 \le \sqrt{2\left(d\,\widehat e_J(t)+b_{\mathrm{reg}}\right)},
\]
with the code dimension factor retained. Independently, a specified
finite CPTP decoder can be certified directly on its normalized
Choi operator. We prove the exact microscopic energy ledger,
an obstruction for commuting source couplings, and continuity
requirements for an information endpoint.

The manuscript establishes a coherent microscopic interface and
a quantitative certification procedure. It does not exhibit a
small dynamical residual certificate for the interacting model,
derive a gravitational geometry from that model, or establish a
gravitational evaporation endpoint. Those missing results are
identified separately rather than assumed inside a purported
evaporation theorem.
\end{abstract}

\begin{center}
\fbox{\begin{minipage}{0.93\linewidth}
\textbf{Claim boundary.}
The proved results concern one regulated interacting microscopic
model and its complete channels. Recovery during controlled
gravitational evaporation requires an additional small dynamical
certificate and a gravitational correspondence on the same time
interval. Complete evaporation requires an additional information
endpoint and a gravitational endpoint. None of these conclusions
is supplied as an assumption disguised as a proof.
\end{minipage}}
\end{center}

\tableofcontents

%=====================================================================
\section{Introduction and contribution}
\label{sec:introduction}

The relevant question is not whether an engineered source can transfer
a quantum register to a reservoir. It is whether one independently
specified microscopic evolution supports all of the following:
\[
\begin{gathered}
\text{coherent physical diary preparation},\\
\text{controlled gravitational energy loss and backreaction},\\
\text{reference-preserving recovery from declared radiation},\\
\text{complete late-time information accounting}.
\end{gathered}
\]
These are separate obligations, and their validity intervals must
overlap.

The supplied Cheung manuscripts maintain the distinctions necessary
for this question. Hawk01-B separates entropy histories from complete
code preservation. Hawk02-A separates a recovery certificate from
a microscopic gravitational transfer law. Hawk02-C and Hawk02-D
provide autonomous benchmarks without identifying their sources
with black holes
\cite{CheungHawk01B,CheungHawk02A,CheungHawk02C,CheungHawk02D}.
Hawk03-A distinguishes an algebra encoded in radiation from an
additional independent interior factor
\cite{CheungHawk03A}.
The operational manuscripts distinguish observable readout,
subsystem recovery, coherent collection, decoder execution, and
complete-record error accounting
\cite{CheungEn02B,CheungEn02C,CheungEn02E,CheungEm03C}.

The present contribution advances the microscopic interface beyond
the quadratic benchmark: the source Hamiltonian is an interacting
quartic SYK Hamiltonian. The reservoir approximation is proved for
its actual unitary dilation, including all logical coherences and
the complete residual. The construction does not turn a quadratic
survival formula into an interacting recovery estimate.

\subsection{Relevant literature and model-selection boundary}

Chen, Qi, and Zhang analyze a SYK system coupled to Majorana chains.
Their preparation is thermofield double, and their retrieval test
uses a classical control bit
\cite{ChenQiZhang}.
That is a relevant architecture, but not the initially pure,
reference-entangled diary task adopted here.

Gaikwad, Kaushal, Mandal, and Wadia study a SYK source coupled to
scalar fields, with a low-energy influence-functional description,
partial-evaporation solutions, and horizonless solutions
\cite{GaikwadEtAl}.
Their bath preparation, disorder averaging, couplings, and
switching prescriptions are substantive parts of that model.
In particular, their Model (b) uses a prescribed late-time reduction
of the coupling to preserve the low-energy description.
Their reconstruction of initial spin labels uses repeated
experiments with different couplings. It is not identified here
with single-run recovery of an arbitrary reference-entangled diary.

We therefore choose a regulated scalar-reservoir architecture,
but do not import the geometry of \cite{GaikwadEtAl}.
The vacuum preparation and finite-band regulator below are explicit
differences. A comparison with that paper would have to control
these differences rather than suppress them in notation.

Pure Euclidean-filtered SYK states motivate the preparation
\cite{KourkoulouMaldacena}.
The SYK low-energy reparametrization sector motivates a prospective
nearly-\(\mathrm{AdS}_2\) description
\cite{MaldacenaStanford}.
Neither result supplies a uniform gravitational dictionary for the
inserted code defined below.

Recovery maps and gravitational reconstruction are already available
in important settings
\cite{NakayamaMiyataUgajin,PSSY}.
The JT-plus-bath analysis of Almheiri, Engelhardt, Marolf, and Maxfield
also supplies an explicit evaporating gravitational setting
\cite{AEMM}.
The missing claim in this manuscript is narrower: those results are
not a certified channel correspondence for the microscopic
preparation, interaction, packet access, and complete residual
specified here.

\subsection{Evidence and source policy}

Cheung references identify the supplied manuscript codes and
versions. Their use here is documentary and comparative.
No proof depends on an unprovided theorem from those manuscripts.

External mathematical inputs are stated explicitly. Model-specific
constructions and estimates are proved below. No exhaustive novelty
claim is made.

The supplied OpenAI catalogue contains proposed mathematical
ingredients, not verified premises of this paper
\cite{SuppliedCatalogue}.
Its spectral-gap, area-law, geometric, and entropy claims are not
used to establish SYK recovery or gravitational evaporation.

%=====================================================================
\section{Task, distances, and complete output spaces}
\label{sec:task}

We use \(\hbar=1\). The code \(C\) has finite dimension \(d\).
The main microscopic example has \(d=2\).

For states and channels, write
\[
\dtr(\rho,\sigma)=\frac12\norm{\rho-\sigma}_1,
\qquad
\ddia(\Phi,\Psi)=\frac12\norm{\Phi-\Psi}_{\diamond}.
\]
Channel comparisons retain arbitrary external references.
Finite-dimensional input permits a reference of dimension \(d\)
for channel discrimination \cite{Watrous}.

For an isometry
\[
V_t:C\longrightarrow R\otimes B,
\]
define the accessible channel and complete complementary channel by
\[
\NN_t(X)=\Tr_B(V_tXV_t^\dagger),
\qquad
\CC_t(X)=\Tr_R(V_tXV_t^\dagger).
\]
The replacement channel for a state \(\sigma_B\) is
\[
\TT_{\sigma_B}(X)=\Tr(X)\sigma_B.
\]
The two operational quantities are
\begin{align}
\erec(t)
&=
\inf_{\DD\ {\rm CPTP}}
\ddia(\DD\NN_t,\id_C),
\label{eq:rec-definition}\\
\dleak(t)
&=
\inf_{\sigma_B}
\ddia(\CC_t,\TT_{\sigma_B}).
\label{eq:leak-definition}
\end{align}

\begin{definition}[Initially pure source diary task]
The source is prepared in a declared pure state before the diary
arrives. The reservoir is initially independent of the diary.
The diary may be entangled with an untouched reference \(Q\).
No early radiation or purification of the source is supplied as a
decoding resource.
\end{definition}

The displaced diary carrier is not discarded. After insertion, it
can be correlated with the source. It remains in \(B\) at every
later time.

\paragraph{Ideal access.}
The first mathematical task permits coherent batch access to a
declared finite packet subsystem. It does not assume an efficient
decoder or a compact local receiver. Any physical receiver requires
its own reference-preserving channel comparison.

%=====================================================================
\section{One fixed interacting microscopic model}
\label{sec:model}

\subsection{Source algebra and frozen couplings}

Let \(N=2n\ge4\), and let \(\chi_1,\ldots,\chi_N\) be Majorana
operators satisfying
\[
\chi_i^\dagger=\chi_i,
\qquad
\{\chi_i,\chi_j\}=2\delta_{ij}I.
\]
They act on
\[
\mathcal H_S=\FF(\C^n),
\qquad
\dim\mathcal H_S=2^n.
\]
This normalization is fixed throughout; conversion to another SYK
normalization must rescale the couplings explicitly.

Freeze a real array \(\mathbf J=(J_{ijkl})_{i<j<k<l}\).
Set
\begin{align}
H_{\mathrm{bare}}
&=
\sum_{i<j<k<l}
J_{ijkl}\chi_i\chi_j\chi_k\chi_l,
\\
B_N&=\sum_{i<j<k<l}|J_{ijkl}|,
\\
H_S&=H_{\mathrm{bare}}+B_NI.
\label{eq:HS}
\end{align}
Products of four distinct Majoranas are self-adjoint and have norm
one. Consequently,
\[
0\le H_S\le M_NI,
\qquad
M_N:=2B_N.
\]
The added constant fixes an energy reference. It is not a
gravitational mass normalization.

No disorder average is taken. Theorems below hold for each frozen
array. They do not imply that every array has a useful recovery
regime or a gravitational interpretation.

Define commuting source spins
\[
S_a=\ii\chi_{2a-1}\chi_{2a},
\qquad a=1,\ldots,n.
\]
For a fixed vector \(s'\in\{-1,1\}^n\), set
\[
O_{s'}=\frac1n\sum_{a=1}^n s'_aS_a.
\]
Then \(O_{s'}\) is even, self-adjoint, and has norm at most one.

Fix real coefficients \(\alpha,\zeta\), and define
\begin{equation}
A=\alpha O_{s'}+\zeta O_{s'}^2.
\label{eq:coupling-A}
\end{equation}
Thus
\[
\norm A\le a_*:=|\alpha|+|\zeta|.
\]
This polynomial coupling is independently specified before channel
certification. It is not selected after seeing a favorable decoder.

For notational flexibility, the analytical estimates also permit
a fixed finite family of even self-adjoint operators \(A_a\).
The model used for the scalar comparison may take all \(A_a=A\).

\subsection{Regulated scalar reservoir}

Fix
\[
0<\mu<\Lambda<\infty,
\qquad
r\ge1.
\]
The reservoir one-particle space and Hamiltonian are
\[
\mathfrak h_{\mu,\Lambda}
=
L^2([\mu,\Lambda],\dd\omega;\C^r),
\qquad
h_Bf(\omega)=\omega f(\omega).
\]
On bosonic Fock space,
\[
\mathcal H_B=\FS(\mathfrak h_{\mu,\Lambda}),
\qquad
H_B=\dd\Gamma(h_B).
\]
The vacuum is denoted \(\Omega_B\).

For species \(a\), define
\[
f_a(\omega)=\frac{\mathbf e_a}{\sqrt{2\omega}},
\]
where \(\mathbf e_a\) is the species basis vector.
Let
\[
\Phi(f)=a(f)+a^\dagger(f),
\]
with \(a(f)\) conjugate-linear in \(f\).
Then
\begin{align}
\norm{f_a}^2
&=\frac12\log\frac{\Lambda}{\mu},
\label{eq:f-norm}\\
\norm{h_B^{-1/2}f_a}^2
&=\frac12\left(\frac1\mu-\frac1\Lambda\right).
\label{eq:f-weighted}
\end{align}

The \(1/\sqrt{2\omega}\) factor specifies a regulated scalar-mode
coupling. Its normalization is a model input, not an inferred
bulk-to-boundary correspondence.

Freeze real \(g_1,\ldots,g_r\), and define
\begin{equation}
V=\sum_{a=1}^r g_a A_a\otimes\Phi(f_a),
\qquad
H=H_S\otimes I+I\otimes H_B+V.
\label{eq:H}
\end{equation}
Introduce the explicit constants
\begin{align}
G
&=
\sum_{a=1}^r |g_a|\,\norm{A_a}\,\norm{f_a},
\label{eq:G}\\
G_\omega
&=
\sum_{a=1}^r |g_a|\,\norm{A_a}\,
\norm{h_B^{-1/2}f_a}.
\label{eq:Gomega}
\end{align}
For the common operator \eqref{eq:coupling-A}, these satisfy
\[
G\le
a_*\left(\sum_a|g_a|\right)
\sqrt{\frac12\log\frac{\Lambda}{\mu}},
\]
and the corresponding bound for \(G_\omega\) follows from
\eqref{eq:f-weighted}.

\begin{remark}[Regulators are not silently removed]
The estimates below take finite observation time and refine a
frequency discretization and a particle cutoff at fixed
\(\mu,\Lambda\). They do not prove the limits
\(\mu\downarrow0\) or \(\Lambda\uparrow\infty\).
Indeed the explicit constants in
\eqref{eq:f-norm}--\eqref{eq:f-weighted} diverge in those limits.
Failure of these estimates to be uniform is not a theorem that every
renormalized or alternative infrared construction is impossible.
\end{remark}

\subsection{Construction of continuum dynamics}

We use the standard Kato--Rellich theorem: a symmetric perturbation
with arbitrarily small relative operator bound preserves
self-adjointness on the original domain.
The same field-operator estimates underlie standard spin--boson
constructions \cite{HaslerHerbst}.

\begin{theorem}[Self-adjoint microscopic Hamiltonian]
\label{thm:selfadjoint}
The operator \eqref{eq:H} is self-adjoint on
\(\mathcal H_S\otimes D(H_B)\), bounded below, and generates
the unitary group \(U(t)=e^{-\ii tH}\). As quadratic forms,
\begin{equation}
H\ge\frac12H_B-2G_\omega^2I.
\label{eq:lower-H}
\end{equation}
\end{theorem}

\begin{proof}
The standard annihilation estimate, proved by Cauchy--Schwarz on
each particle sector, is
\[
\norm{a(f)\psi}
\le
\norm{h_B^{-1/2}f}\,
\norm{H_B^{1/2}\psi}.
\]
The creation identity
\[
\norm{a^\dagger(f)\psi}^2
=
\norm{a(f)\psi}^2+\norm f^2\norm\psi^2
\]
therefore gives
\[
\norm{V\psi}
\le
2G_\omega\norm{H_B^{1/2}\psi}
+G\norm\psi.
\]
For every \(b>0\), the spectral theorem bounds
\(\norm{H_B^{1/2}\psi}\) by
\(b\norm{H_B\psi}+c_b\norm\psi\).
Thus \(V\) has relative operator bound zero.
The bounded source term does not change the domain, so
Kato--Rellich establishes self-adjointness.

For a normalized vector in the form domain,
\[
|\langle V\rangle|
\le
2G_\omega\sqrt{\langle H_B\rangle}
\le
\frac12\langle H_B\rangle+2G_\omega^2.
\]
Since \(H_S\ge0\), this proves \eqref{eq:lower-H}.
The unitary group follows from self-adjointness.
\end{proof}

The carrier introduced next is tensored with the identity during
this evolution. A formal infinite Hamiltonian sum is not needed.

%=====================================================================
\section{Pure preparation and coherent diary loading}
\label{sec:preparation}

\subsection{Source preparation}

Choose a normalized simultaneous spin eigenvector
\[
S_a\ket{B_s}=s_a\ket{B_s},
\qquad s_a\in\{-1,1\},
\]
with a declared total fermion parity.
For \(\beta>0\), define
\begin{align}
Z_s(\beta)
&=
\bra{B_s}e^{-\beta H_S}\ket{B_s},
\\
\ket{\Omega_S}
&=
Z_s(\beta)^{-1/2}e^{-\beta H_S/2}\ket{B_s}.
\label{eq:KM-prep}
\end{align}
This is the Euclidean-filtered preparation motivated by
\cite{KourkoulouMaldacena}. Its exact finite-realization properties
used here are elementary.

\begin{proposition}[Preparation probability and energy distribution]
\label{prop:preparation}
The state \eqref{eq:KM-prep} is well defined and has the same
parity as \(\ket{B_s}\). Its normalization obeys
\[
e^{-\beta M_N}\le Z_s(\beta)\le1.
\]
If \(H_S=\sum_E E P_E\), its energy probabilities are
\[
p_E=
Z_s(\beta)^{-1}e^{-\beta E}
\bra{B_s}P_E\ket{B_s}.
\]
Moreover,
\[
\langle H_S\rangle_{\Omega_S}
=-\partial_\beta\log Z_s(\beta),
\qquad
\operatorname{Var}_{\Omega_S}(H_S)
=\partial_\beta^2\log Z_s(\beta).
\]
\end{proposition}

\begin{proof}
The spectral bounds \(0\le H_S\le M_NI\) give the normalization
inequalities and positivity of \(Z_s\). The quartic Hamiltonian
is even, so its exponential preserves parity.
The spectral decomposition gives \(p_E\).
Differentiating the finite-dimensional spectral sum proves the
mean and variance formulas.
\end{proof}

A preparation instrument may use the contraction
\(K_{\mathrm{succ}}=e^{-\beta H_S/2}\) and the complementary
operator
\[
K_{\mathrm{fail}}=(I-e^{-\beta H_S})^{1/2}.
\]
Its success probability on \(\ket{B_s}\) is \(Z_s(\beta)\).
The preparation occurs before the diary is supplied.
The task may start after a declared successful preparation;
an attempted deterministic protocol must instead retain the
failure flag and include its preparation error.
No input-dependent postselection is used.

No finite-\(N\), fixed-realization thermal or gravitational
correspondence is inferred from
Proposition~\ref{prop:preparation}.

\subsection{Parity-preserving qubit diary}

Introduce a carrier \(D\) containing two fermionic modes.
Encode the logical qubit by
\begin{equation}
J_D\ket0=\ket{00}_D,
\qquad
J_D\ket1=\ket{11}_D.
\label{eq:diary-code}
\end{equation}
Both logical basis vectors have even parity.
The logical reference \(Q\) is an abstract quantum register.

Choose two source modes \(T\) before studying recovery.
Let \(U_{\mathrm{sw}}\) be the second quantization of the
one-particle unitary exchanging those source modes with the
carrier modes. It is number conserving and parity preserving.

Define the insertion isometry
\begin{equation}
W\ket\psi
=
U_{\mathrm{sw}}
\left(J_D\ket\psi\otimes\ket{\Omega_S}\right),
\qquad
W:C\longrightarrow\mathcal H_S\otimes\mathcal H_D.
\label{eq:W}
\end{equation}
Fix the fermionic occupation ordering once and for all.

\begin{proposition}[Coherent insertion and displaced carrier]
\label{prop:insertion}
The map \(W\) is an isometry implemented by one
message-independent unitary. It preserves every logical matrix
unit and arbitrary reference entanglement.

After insertion, the carrier marginal is independent of the diary,
but the carrier can be correlated with the remaining source.
It must therefore remain in the complete residual.

If the carrier Hamiltonian is degenerate and preparation
interactions are absent before and after the swap, the change
in mean source energy satisfies
\[
\left|
\Tr(H_S W\rho W^\dagger)
-
\bra{\Omega_S}H_S\ket{\Omega_S}
\right|
\le M_N
\]
for every logical density operator \(\rho\).
\end{proposition}

\begin{proof}
Composition of the isometry \(J_D\), a fixed pure ancillary
preparation, and a unitary proves \(W^\dagger W=I_C\).
Linearity gives the action on all matrix units; tensoring the
unitary with \(I_Q\) proves the reference statement.

Exchanging the entire two-mode subsystems transfers the old source
two-mode marginal to the carrier. Its correlations with the
remaining source are also transferred, rather than erased.
Fermionic permutation signs are fixed by the mode ordering and the
even logical code; they do not create a message-dependent
operation.

Both source energy expectations lie in \([0,M_N]\), proving
the final inequality.
\end{proof}

\begin{remark}[A repaired use of the benchmark swap]
In Hawk02-D the destination source was vacuum, and the carrier
became a fixed vacuum. That conclusion does not apply to the
Euclidean-filtered source here.
The carrier is generally not vacuum and is not removable from the
residual ledger. Initial carrier independence alone does not
establish its later joint independence.
\end{remark}

The energy bound is deliberately crude. It does not prove that
the insertion remains within a gravitational low-energy window.
An infalling interpretation and sharper backreaction control
remain additional obligations.

%=====================================================================
\section{Complete channels and exact microscopic accounting}
\label{sec:complete}

\subsection{Packet access}

Fix an orthonormal packet family \(p_1,\ldots,p_q\) in
\(\mathfrak h_{\mu,\Lambda}\), independently of any recovery
optimization. Put
\[
\mathfrak r=\Span\{p_1,\ldots,p_q\}.
\]
The bosonic Fock factorization is
\[
\FS(\mathfrak h_{\mu,\Lambda})
\simeq
\FS(\mathfrak r)\otimes\FS(\mathfrak r^\perp).
\]
Define
\begin{align}
R&=\FS(\mathfrak r),\\
B&=
\mathcal H_S\otimes\mathcal H_D
\otimes\FS(\mathfrak r^\perp).
\label{eq:residual-ledger}
\end{align}
The complete code isometry is
\begin{equation}
V_t\ket\psi
=
\mathscr R\,
(U(t)\otimes I_D)
\left(W\ket\psi\otimes\Omega_B\right),
\label{eq:Vt}
\end{equation}
where \(\mathscr R\) is the fixed regrouping into \(R\otimes B\).

The vacuum reservoir has no inaccessible preparation purifier.
If a mixed reservoir preparation is substituted, its purification
must be added to \(B\); that is a different input specification.

For \(\Xi_i(t)=V_t\ket i\),
\begin{equation}
\NN_t(E_{ij})
=\Tr_B\kb{\Xi_i(t)}{\Xi_j(t)},
\qquad
\CC_t(E_{ij})
=\Tr_R\kb{\Xi_i(t)}{\Xi_j(t)}.
\label{eq:matrix-units}
\end{equation}
These formulas determine every reference-entangled input.

\subsection{Energy ledger}

Let \(E_0(\rho)=\Tr(H_S W\rho W^\dagger)\).
The initial interaction mean is zero because the reservoir is vacuum.

\begin{theorem}[Exact microscopic conservation and bounded reservoir energy]
\label{thm:energy}
For every logical input, define
\[
E_S(t)=\langle H_S\rangle_t,
\qquad
E_B(t)=\langle H_B\rangle_t,
\qquad
E_I(t)=\langle V\rangle_t.
\]
Then
\begin{equation}
E_S(t)+E_B(t)+E_I(t)=E_0(\rho).
\label{eq:energy-conservation}
\end{equation}
Uniformly in the logical input and time,
\begin{equation}
0\le E_B(t)\le2M_N+4G_\omega^2.
\label{eq:bath-energy-bound}
\end{equation}
The source current, with positive sign for loss from the source,
is
\begin{equation}
\mathcal J_S(t)
=-\frac{\dd E_S}{\dd t}
=-\ii\langle[V,H_S]\rangle_t.
\label{eq:source-current}
\end{equation}
Consequently,
\begin{equation}
\frac{\dd E_B}{\dd t}
=
\mathcal J_S(t)-\frac{\dd E_I}{\dd t}.
\label{eq:interaction-energy}
\end{equation}
No sign is asserted for either current at every time.
\end{theorem}

\begin{proof}
The initially prepared vectors belong to \(D(H)\).
Autonomous unitary evolution preserves the total Hamiltonian
expectation. The initial bath and interaction means vanish,
giving \eqref{eq:energy-conservation}.

The form inequality \eqref{eq:lower-H} gives
\[
\frac12E_B(t)-2G_\omega^2\le E_0(\rho)\le M_N,
\]
which proves \eqref{eq:bath-energy-bound}.

The particle-sector bounds established in
Section~\ref{sec:regulator} also show that these prepared evolved
vectors have finite moments of every fixed particle-number order
on bounded time intervals. Since \(\omega\) is bounded on the
one-particle band, the commutator expectations can be obtained by
finite-particle approximation. Differentiation gives
\(\dot E_S=\ii\langle[H,H_S]\rangle
=\ii\langle[V,H_S]\rangle\).
Differentiating \eqref{eq:energy-conservation} proves
\eqref{eq:interaction-energy}.
\end{proof}

\begin{remark}[Interaction energy cannot be omitted]
Source energy loss and reservoir energy gain differ by the
interaction-energy derivative. A source-only mass-loss formula
requires an additional approximation or a declared dressing
convention. Equation~\eqref{eq:energy-conservation} supplies neither
a positive outward stress tensor nor a gravitational metric.
\end{remark}

For a prescribed differentiable switching schedule \(g_a(t)\),
a separate time-dependent model has the work term
\[
\frac{\dd}{\dd t}\langle H(t)\rangle
=
\sum_a\dot g_a(t)
\langle A_a\otimes\Phi(f_a)\rangle.
\]
A switch used in a gravitational comparison must retain this work.
It is not an autonomous endpoint operation.

%=====================================================================
\section{Reference-preserving finite approximation}
\label{sec:regulator}

This section proves the finite-channel approximation for the
interacting model itself. It does not assume source relaxation,
Markovian reduced dynamics, or a small complementary channel.

Let \(N_B=\dd\Gamma(I)\), and let \(P_m\) project onto reservoir
sectors with at most \(m\) particles.

\subsection{A convergent vacuum Dyson expansion}

In the interaction picture with respect to \(H_S+H_B\), write
\[
V_I(s)=
\sum_a g_a A_a(s)\otimes\Phi(f_{a,s}),
\]
where
\[
A_a(s)=e^{\ii sH_S}A_ae^{-\ii sH_S},
\qquad
f_{a,s}=e^{\ii s h_B}f_a.
\]
These transformations preserve the norms in \eqref{eq:G}.

For \(x\ge0\), define
\begin{equation}
Z_m(x)=
\sum_{\ell=m+1}^{\infty}
\frac{x^\ell}{\sqrt{\ell!}}.
\label{eq:Z-tail}
\end{equation}
This series is finite for every finite \(x\).

\begin{lemma}[Vacuum Dyson and occupation-tail estimates]
\label{lem:dyson}
For an initial vector consisting of an arbitrary normalized
source--carrier state and the reservoir vacuum, the interaction
picture Dyson term of order \(\ell\) has norm at most
\[
\frac{(2Gt)^\ell}{\sqrt{\ell!}}.
\]
The expansion converges in norm, uniformly on bounded time
intervals and uniformly over normalized code-reference inputs.

For the complete prepared isometry,
\begin{equation}
\norm{(I-P_m)V_t}
\le Z_m(2Gt),
\label{eq:occupation-tail}
\end{equation}
where \(P_m\) acts on the reservoir before regrouping.
\end{lemma}

\begin{proof}
On sectors containing at most \(k\) particles,
\[
\norm{V_I(s)\psi}
\le
G(\sqrt{k}+\sqrt{k+1})\norm\psi
\le
2G\sqrt{k+1}\norm\psi.
\]
An \(\ell\)-fold product acting on vacuum is therefore bounded
by \((2G)^\ell\sqrt{\ell!}\).
The ordered integration simplex has volume \(t^\ell/\ell!\),
giving the claimed term bound.

The ratio of consecutive scalar bounds is
\(2Gt/\sqrt{\ell+1}\), so the series converges.
The same proof holds with an untouched reference and carrier.
Each term is supported in sectors with at most \(\ell\) particles.

For completeness, multiplication of the scalar term bound by
any fixed power of \(\ell\) still gives a convergent series.
Thus the Dyson series converges in every fixed particle-number
graph norm. With the finite frequency band and bounded source
operators, the integral equation and its derivative hold on the
prepared vectors. The resulting Schr\"odinger-picture solution
is the unique solution generated by the self-adjoint Hamiltonian
of Theorem~\ref{thm:selfadjoint}. This identifies the expansion
with the actual evolution.

Terms of order \(\ell\le m\) vanish under \(I-P_m\).
Summing the remaining bounds proves
\eqref{eq:occupation-tail}.
\end{proof}

\subsection{Dynamics truncated at total occupation}

On \(P_m(\mathcal H_S\otimes\mathcal H_B)\), set
\[
H^{(m)}=P_mHP_m.
\]
It is bounded and self-adjoint, since the reservoir energy is
at most \(m\Lambda\) there and
\(\norm{P_mVP_m}\le2G\sqrt{m+1}\).
Let \(V_t^{(m)}\) be its complete prepared isometry, embedded
into the original output space.

\begin{lemma}[Occupation-truncated dynamics]
\label{lem:occupation-dynamics}
For \(0\le t\le T\),
\begin{equation}
\norm{V_t-V_t^{(m)}}
\le2Z_m(2GT).
\label{eq:occupation-dynamics}
\end{equation}
\end{lemma}

\begin{proof}
The free evolution preserves particle number. In interaction
picture, the exact and truncated Dyson terms acting on vacuum
coincide through order \(m\): no intermediate state at those
orders can exceed \(m\) particles.

At every higher order, both terms satisfy the bound of
Lemma~\ref{lem:dyson}; inserted projections are contractions.
The difference is therefore bounded by twice the tail series.
Returning to Schr\"odinger picture does not change the norm.
\end{proof}

\subsection{Frequency discretization with exact packet identification}

Partition \([\mu,\Lambda]\) into finitely many intervals \(I_b\)
of width at most \(\Delta\). Let \(\omega_b\) be their midpoints,
and define
\[
\widetilde h_B f(\omega)=\omega_b f(\omega)
\quad(\omega\in I_b).
\]
Then
\[
\norm{h_B-\widetilde h_B}\le\Delta/2.
\]

Let \(\mathfrak f\) be the finite-dimensional space spanned by
\[
\mathbf1_{I_b}f_a,
\qquad
\mathbf1_{I_b}p_j,
\]
for every bin, species, and declared packet.
Zero vectors are omitted and the remaining vectors are
orthonormalized.

This space contains the coupling form factors and the entire
packet space \(\mathfrak r\). It is invariant under
\(\widetilde h_B\).
All modes in \(\mathfrak f^\perp\) remain vacuum in the
discretized evolution. Thus restriction to
\(\FS(\mathfrak f)\) is exact for that discretized model.

Let
\[
\widehat H^{(m,\Delta)}
=
P_m\left(H_S+\dd\Gamma(\widetilde h_B)+V\right)P_m
\]
on the active finite-mode space. Its Hilbert space has finite
dimension. If \(p=\dim\mathfrak f\), the reservoir dimension is
\[
\sum_{k=0}^m\binom{p+k-1}{k}=\binom{p+m}{m}.
\]
The source and carrier dimensions are included separately.

\begin{lemma}[Frequency-mesh error]
\label{lem:frequency}
For the occupation-truncated prepared isometries,
\[
\norm{
V_t^{(m)}-\widehat V_t^{(m,\Delta)}
}
\le tm\Delta/2.
\]
\end{lemma}

\begin{proof}
On sectors with at most \(m\) particles,
\[
\norm{
\dd\Gamma(h_B)-\dd\Gamma(\widetilde h_B)
}
\le m\Delta/2.
\]
The source and interaction terms are unchanged.
Duhamel's formula for bounded self-adjoint operators gives
the stated unitary difference.
The invariant active-mode space realizes the discretized
prepared evolution exactly. Restriction to the code preserves
the bound.
\end{proof}

\begin{lemma}[Isometry-to-channel conversion]
\label{lem:isometry}
If \(V_1,V_2:C\to R\otimes B\) are isometries into a common
output space, then
\[
\ddia(\NN_{V_1},\NN_{V_2})\le\norm{V_1-V_2},
\qquad
\ddia(\CC_{V_1},\CC_{V_2})\le\norm{V_1-V_2}.
\]
The same bound holds for the complete output channels.
\end{lemma}

\begin{proof}
For a pure code-reference input, the two output vectors differ
in norm by at most \(\norm{V_1-V_2}\).
Their pure-state trace distance is no greater than that norm.
Convexity handles mixed inputs, and partial trace is contractive.
Taking the reference-input supremum proves the result.
\end{proof}

\begin{theorem}[Complete finite-regulator approximation]
\label{thm:regulator}
At fixed \(\mu,\Lambda\), for the preparation, interaction, and
packet access specified above,
\begin{equation}
b_{\mathrm{reg}}(T;m,\Delta)
=
\min\left\{
1,\,
2Z_m(2GT)+\frac{Tm\Delta}{2}
\right\}
\label{eq:breg}
\end{equation}
bounds the half-diamond distance between exact and finite
complete output channels uniformly for \(0\le t\le T\).
It also bounds the distances between their accessible channels
and between their complete complementary channels, after the
specified common embeddings.

For every \(T<\infty\) and \(b>0\), finite \(m\) and a finite
mesh can be selected so that \(b_{\mathrm{reg}}\le b\).
\end{theorem}

\begin{proof}
Combine Lemmas~\ref{lem:occupation-dynamics},
\ref{lem:frequency}, and \ref{lem:isometry}.
The finite output has the same packet space as the continuum
output. The displaced carrier and all uncollected active modes
are retained; unused orthogonal modes are fixed vacuum.
Thus both partial traces refer to the same declared partition.

Convergence follows because \(Z_m(2GT)\to0\) as \(m\to\infty\),
followed by \(\Delta\to0\).
\end{proof}

\begin{corollary}[Explicit regulator selection]
\label{cor:regulator-selection}
Suppose \(T>0\), \(0<b<1\), and \(x=2GT\).
Choose an integer \(m\ge1\) such that
\[
m+1\ge4ex^2,
\qquad
m\ge2+\log_2(1/b),
\]
and choose the mesh so that
\[
\Delta\le\frac{b}{Tm}.
\]
Then \(b_{\mathrm{reg}}(T;m,\Delta)\le b\).
\end{corollary}

\begin{proof}
For \(\ell\ge m+1\), consecutive terms of
\eqref{eq:Z-tail} have ratio at most \(1/2\).
Also,
\[
\frac{x^{m+1}}{\sqrt{(m+1)!}}
\le
\left(\frac{ex^2}{m+1}\right)^{(m+1)/2}
\le2^{-(m+1)}.
\]
Thus \(Z_m(x)\le2^{-m}\), and \(2Z_m(x)\le b/2\).
The mesh term is also at most \(b/2\).
\end{proof}

\begin{remark}[What the finite approximation is]
The frequency bins are a spectral discretization, not finite
spatial boxes. The theorem makes no locality or detector-support
claim. It proves a reference-preserving channel approximation
for the declared spectral packets.

The occupation cutoff is an analytical approximation to dynamics,
not a physical process that destroys excess particles.
\end{remark}

%=====================================================================
\section{Complete-residual certificates and decoder certificates}
\label{sec:certificates}

Let
\[
\ket{\Phi_d}
=
d^{-1/2}\sum_{i=1}^d\ket i_Q\ket i_C,
\qquad
\tau_Q=I_Q/d.
\]
For a channel \(\Phi\), define the normalized Choi operator
\[
J_{\mathrm n}(\Phi)
=
(\id_Q\otimes\Phi)(\kb{\Phi_d}{\Phi_d}).
\]

\subsection{The dimension factor}

\begin{lemma}[Normalized Choi-to-diamond bound]
\label{lem:choi}
For channels \(\Phi,\Psi\) with input dimension \(d\),
\[
\ddia(\Phi,\Psi)
\le
\frac d2
\norm{J_{\mathrm n}(\Phi)-J_{\mathrm n}(\Psi)}_1.
\]
The output may be infinite dimensional provided the channel
outputs are normal states.
\end{lemma}

\begin{proof}
A pure input with a \(d\)-dimensional reference can be written as
\[
\ket\psi=(L\otimes I)\ket{\Phi_d},
\qquad
\Tr(L^\dagger L)=d.
\]
Its channel-output difference is
\[
(L\otimes I)
\left[J_{\mathrm n}(\Phi)-J_{\mathrm n}(\Psi)\right]
(L^\dagger\otimes I).
\]
Its trace norm is at most \(d\) times the Choi trace norm because
\(\norm L^2\le d\).
Convexity and the finite-reference characterization of channel
diamond distance complete the proof.
\end{proof}

The information--disturbance theorem supplies
\begin{equation}
\erec\le\sqrt{2\dleak}
\label{eq:KSW}
\end{equation}
in half-diamond normalization \cite{KSW}.
This is an external channel theorem. It does not estimate
\(\dleak\) from the interacting Hamiltonian.

\begin{theorem}[Certified residual and recovery bounds]
\label{thm:certificate}
Fix \(t\le T\), and compute the finite complementary channel
\(\widehat{\CC}_t\) from
\(\widehat V_t^{(m,\Delta)}\).
Let \(\widehat\sigma_B\) be any rigorously specified finite
residual density operator and put
\[
\widehat e_J(t)
=
\frac12
\norm{
J_{\mathrm n}(\widehat{\CC}_t)
-\tau_Q\otimes\widehat\sigma_B
}_1.
\]
Embed \(\widehat\sigma_B\) into the complete residual space,
with the unused modes in vacuum. Then
\begin{align}
\dleak(t)
&\le
\min\{1,d\,\widehat e_J(t)+b_{\mathrm{reg}}\},
\label{eq:residual-certificate}\\
\erec(t)
&\le
\min\left\{
1,\sqrt{2\left[d\,\widehat e_J(t)+b_{\mathrm{reg}}\right]}
\right\}.
\label{eq:recovery-certificate}
\end{align}
If a validated calculation provides
\(\widehat e_J(t)\le e_{\mathrm{cert}}(t)\), that certified
upper bound may replace \(\widehat e_J(t)\).
\end{theorem}

\begin{proof}
Theorem~\ref{thm:regulator} gives
\[
\ddia(\CC_t,\widehat{\CC}_t)\le b_{\mathrm{reg}}.
\]
The triangle inequality and Lemma~\ref{lem:choi} give
\[
\ddia(\CC_t,\TT_{\widehat\sigma_B})
\le b_{\mathrm{reg}}+d\,\widehat e_J(t).
\]
Taking the replacement-state infimum proves
\eqref{eq:residual-certificate}.
Equation~\eqref{eq:KSW} proves
\eqref{eq:recovery-certificate}.
\end{proof}

\begin{remark}[Why the regulator error is not multiplied by \(d\)]
The regulator theorem already bounds the complementary channel
in diamond distance. Its allowance is therefore added after
the Choi conversion. Replacing it by a Choi-state estimate and
multiplying it by \(d\) would be an unnecessary weakening.
\end{remark}

\begin{remark}[Certificate versus successful recovery]
Theorem~\ref{thm:certificate} is a proved implication with a
specified finite computation. It does not exhibit a time at
which \(e_{\mathrm{cert}}\) is small. No such numerical or
analytical certificate has been supplied for this interacting
model. The manuscript consequently does not claim an actual
interacting recovery window.
\end{remark}

\subsection{All qubit coherences}

For \(d=2\), use the logical Pauli matrices \(X,Y,Z\).

\begin{proposition}[Qubit complete-residual test]
\label{prop:pauli}
Choose \(\sigma_B=\CC_t(I_C/2)\). Then
\[
e_J(t)
\le
\frac14
\left[
\norm{\CC_t(X)}_1+
\norm{\CC_t(Y)}_1+
\norm{\CC_t(Z)}_1
\right].
\]
Hence
\[
\dleak(t)
\le
\frac12
\left[
\norm{\CC_t(X)}_1+
\norm{\CC_t(Y)}_1+
\norm{\CC_t(Z)}_1
\right].
\]
The same statement holds for the finite approximation.
\end{proposition}

\begin{proof}
The Bell projector has the expansion
\[
\kb{\Phi_2}{\Phi_2}
=
\frac14
(I\otimes I+X\otimes X-Y\otimes Y+Z\otimes Z).
\]
Subtracting
\((I_Q/2)\otimes\CC_t(I_C/2)\)
removes the identity term.
Each Pauli matrix has trace norm two.
The triangle inequality gives the first estimate, and
Lemma~\ref{lem:choi} gives the second.
\end{proof}

Both \(X\) and \(Y\) involve off-diagonal logical matrix units.
A calculation limited to two diagonal preparations does not
verify this test.

\subsection{Finite semidefinite certificates}

The finite complementary Choi operator can be used in the
optimization
\begin{equation}
\begin{aligned}
\text{minimize}\quad&
\frac12\Tr(X_++X_-),\\
\text{subject to}\quad&
J_{\mathrm n}(\widehat{\CC}_t)
-\tau_Q\otimes\sigma_B=X_+-X_-,\\
&X_\pm\succeq0,\qquad
\sigma_B\succeq0,\qquad
\Tr\sigma_B=1.
\end{aligned}
\label{eq:complement-SDP}
\end{equation}
This is a finite semidefinite program. Its value is the optimized
normalized Choi trace distance, not automatically the optimized
diamond distance.

Any rigorously feasible tuple supplies an upper certificate.
An exploratory floating-point optimum, an unverified positive
semidefiniteness test, or a solver residual is not a proof of
feasibility.

A direct decoder certificate is also available.

\begin{proposition}[A specified decoder and complete extension]
\label{prop:decoder}
Let \(\widehat{\DD}\) be a rigorously CPTP decoder on the finite
packet output space, and define
\[
\widehat r(t)
=
\frac12
\norm{
J_{\mathrm n}(\widehat{\DD}\widehat{\NN}_t)
-\kb{\Phi_d}{\Phi_d}
}_1.
\]
Complete \(\widehat{\DD}\) to the full packet Fock space by
\[
\DD(Y)
=
\widehat{\DD}(P_RYP_R)
+
\Tr((I-P_R)Y)\frac{I_C}{d},
\]
where \(P_R\) projects onto the finite packet occupation space.
Then
\[
\ddia(\DD\NN_t,\id_C)
\le
\min\{1,d\,\widehat r(t)+b_{\mathrm{reg}}\}.
\]
\end{proposition}

\begin{proof}
Compression by \(P_R\) is completely positive. The second term
is a positive-effect measurement followed by state preparation.
Their traces add to \(\Tr Y\), so the extension is CPTP.

The approximate channel is supported on \(P_R\).
Lemma~\ref{lem:choi} gives its decoded error
\(d\,\widehat r(t)\).
Postprocessing contraction and
Theorem~\ref{thm:regulator} give the additional
\(b_{\mathrm{reg}}\).
\end{proof}

For decoder search, its unnormalized Choi operator \(J_D\)
must obey
\[
J_D\succeq0,
\qquad
\Tr_CJ_D=I_{\widehat R}.
\]
Composition with the known finite channel is linear in \(J_D\);
the trace-norm objective can be expressed using positive and
negative parts as in \eqref{eq:complement-SDP}.
A complete feasible decoder, not only a successful branch,
is the required output.

\paragraph{Petz candidate.}
For a finite channel \(\widehat{\NN}\), a full-rank code state
\(\tau\), and \(P=\supp\widehat{\NN}(\tau)\), a candidate is
\[
\begin{split}
\DD_{\tau,\widehat{\NN}}(Y)
={}&
\tau^{1/2}
\widehat{\NN}^{\dagger}\!
\left[
\widehat{\NN}(\tau)^{-1/2}
Y
\widehat{\NN}(\tau)^{-1/2}
\right]\tau^{1/2}\\
&+\Tr((I-P)Y)\tau.
\end{split}
\]
Inverse powers are taken on \(P\).
The first term is completely positive and has trace
\(\Tr(PY)\); the completion therefore makes the map CPTP.
Writing it down does not prove a small recovery error.
Proposition~\ref{prop:decoder} supplies a direct test.
This agrees with the distinction emphasized in the supplied
Hawk01-B comparison \cite{CheungHawk01B}.

\subsection{End-to-end errors}

\begin{proposition}[Named-interface error composition]
\label{prop:composition}
Suppose an implemented channel and decoder satisfy the declared
comparisons
\[
b_{\mathrm{prep}},\quad
b_{\mathrm{collection}},\quad
b_{\mathrm{implementation}}
\]
in half-diamond distance at their respective interfaces.
Then a finite direct decoder certificate gives
\[
\begin{split}
\varepsilon_{\mathrm{tot}}
\le{}&
d\,\widehat r(t)+b_{\mathrm{reg}}
+b_{\mathrm{prep}}\\
&+b_{\mathrm{collection}}
+b_{\mathrm{implementation}},
\end{split}
\]
capped at one.
A decoder supplied by the residual theorem instead replaces
\(d\,\widehat r(t)+b_{\mathrm{reg}}\) by the bound in
\eqref{eq:recovery-certificate}.
\end{proposition}

\begin{proof}
Insert the intermediate channels one at a time.
Use the triangle inequality and contraction of diamond
distance under CPTP pre- and postprocessing.
\end{proof}

Loss already included in a complete complementary channel is
not added again as an independent collection allowance.
A separate term is appropriate only for a separate proved
comparison. This avoids the double-counting problem noted in
Hawk02-C \cite{CheungHawk02C}.

%=====================================================================
\section{Gaussian influence and a recovery obstruction}
\label{sec:influence}

The reservoir covariance is exactly
\begin{equation}
K_{ab}(s,u)
=
\langle\Omega_B,\Phi(f_{a,s})\Phi(f_{b,u})\Omega_B\rangle
=
\delta_{ab}\int_\mu^\Lambda
\frac{e^{-\ii\omega(s-u)}}{2\omega}\dd\omega.
\label{eq:covariance}
\end{equation}
This follows directly from the canonical commutation relations.
It supplies a microscopic Gaussian influence kernel, in the
sense of the influence-functional framework
\cite{FeynmanVernon}.
It is not yet a gravitational effective action.

\begin{proposition}[Exact prescribed-history bath overlap]
\label{prop:history}
For real integrable histories \(j_a(s)\) on a finite interval,
consider the interaction-picture bath Hamiltonian
\[
H_j(s)=\sum_a j_a(s)\Phi(f_{a,s}).
\]
Its propagator has the form
\[
U_j(t)=e^{\ii\theta_j(t)}\mathcal W(\eta_j(t)),
\qquad
\eta_j(t)=-\ii\sum_a\int_0^t j_a(s)f_{a,s}\dd s,
\]
where
\[
\mathcal W(\eta)=
\exp(a^\dagger(\eta)-a(\eta))
\]
is the Weyl unitary and \(\theta_j(t)\) is real.
For two histories,
\[
\left|
\langle\Omega_B,U_k(t)^\dagger U_j(t)\Omega_B\rangle
\right|
=
\exp\left[-\frac12\norm{\eta_j(t)-\eta_k(t)}^2\right].
\]
\end{proposition}

\begin{proof}
The commutator of \(H_j(s)\) and \(H_j(u)\) is a scalar
multiple of the identity:
\[
[\Phi(f),\Phi(g)]
=2\ii\operatorname{Im}\langle f,g\rangle I.
\]
Consequently chronological Weyl multiplication yields one
Weyl displacement and a real common phase.
The displacement follows by integrating the linear
generator. The vacuum characteristic function is
\[
\langle\Omega_B,\mathcal W(\eta)\Omega_B\rangle
=e^{-\norm\eta^2/2}.
\]
The Weyl product phase has unit modulus, proving the overlap
formula.
\end{proof}

Replacing the quantum source operators \(A_a(s)\) by prescribed
classical histories is an additional approximation. When they
do not commute, their time ordering and both logical branches
must be retained. A scalar-history saddle or a diagonal
reduced density operator is not the complete channel
\eqref{eq:matrix-units}.

\begin{theorem}[Commuting-coupling obstruction]
\label{thm:QND}
Suppose
\[
[H_S,A_a]=0,
\qquad
[A_a,A_b]=0
\]
for every pair of couplings.
For any insertion isometry \(W\), the accessible reservoir
channel is entanglement breaking.
Every deterministic decoder therefore satisfies
\[
\ddia(\DD\NN_t,\id_C)\ge1-d^{-1}.
\]
For a qubit diary, the lower bound is \(1/2\).
\end{theorem}

\begin{proof}
Choose a common source eigenbasis, with spectral projectors
\(P_\lambda\). The complete propagator is block diagonal:
\[
U(t)=\sum_\lambda
P_\lambda\otimes
e^{-\ii t(E_\lambda+H_B+\sum_a g_a a_{a,\lambda}\Phi(f_a))}.
\]
Tracing the source removes terms between different projectors.
Tracing the carrier does not restore them.
The accessible channel therefore has measure-and-prepare form
\[
\NN_t(\rho)
=
\sum_\lambda
\Tr(E_\lambda^{\mathrm{code}}\rho)\,
\sigma_{\lambda,t},
\]
where
\[
E_\lambda^{\mathrm{code}}
=
W^\dagger(P_\lambda\otimes I_D)W
\]
are positive effects summing to \(I_C\).

Such a channel breaks entanglement with every reference.
A decoder preserves that property.
A separable state on \(Q\otimes C\) has overlap at most \(1/d\)
with \(\ket{\Phi_d}\): this is immediate for pure product
states by Cauchy--Schwarz and extends by convexity.
Testing the Bell projector gives trace distance at least
\(1-1/d\), hence the stated diamond lower bound.
\end{proof}

Noncommutativity removes this particular obstruction; it does
not prove recovery. The theorem generalizes the negative-control
role of the effective single-observable scalar source discussed
in the supplied Em03-F \cite{CheungEm03F}.
The noncommuting exterior construction in Em03-G is a different
model and is not used as a decoder for the present Hamiltonian
\cite{CheungEm03G}.

%=====================================================================
\section{Disorder, omitted residuals, and cutoff discipline}
\label{sec:debug}

\subsection{Averaging channels is not fixed-realization decoupling}

\begin{proposition}[Disorder-averaging counterexample]
\label{prop:average}
An averaged channel can be constant while every realization
retains distinguishable logical information.
\end{proposition}

\begin{proof}
For a qubit, let
\[
\CC_0(\rho)=\diag(\rho),
\qquad
\CC_1(\rho)=X\diag(\rho)X.
\]
Their equal mixture is the replacement channel
\(\Tr(\rho)I/2\).
For each realization, the outputs of \(\ket0\) and \(\ket1\)
are orthogonal.
By the triangle inequality, its distance from any replacement
channel is at least \(1/2\).
Thus averaging has erased information in the description,
not proved its absence in each realization.
\end{proof}

\begin{proposition}[A legitimate probabilistic certificate]
\label{prop:probability}
Suppose a declared disorder distribution and a measurable family
of residual states satisfy
\[
\mathbb E_{\mathbf J}
\widehat e_{J,\mathbf J}(t)\le a(t).
\]
For \(0<p<1\),
\[
\Pr\left\{
\widehat e_{J,\mathbf J}(t)>a(t)/p
\right\}\le p.
\]
On the complementary event,
\[
\dleak_{\mathbf J}(t)
\le d\,a(t)/p+b_{\mathrm{reg},\mathbf J}.
\]
Any decoder dependence on \(\mathbf J\), and any realization
information needed to implement it, must be declared.
\end{proposition}

\begin{proof}
Apply Markov's inequality to the nonnegative random trace
distance, and then apply Theorem~\ref{thm:certificate}
realization by realization.
\end{proof}

This proposition uses an expectation of distances. It does not
use the distance of an averaged channel.
A result uniform on a time interval requires a separate uniform
estimate or a proved discretization and continuity argument.

\subsection{Source relaxation is not complete residual relaxation}

\begin{proposition}[Omitted-output counterexample]
\label{prop:omitted}
A fixed source marginal does not imply complete residual
decoupling.
\end{proposition}

\begin{proof}
The isometry
\[
V\ket\psi=\ket0_S\ket0_R\ket\psi_E
\]
has a constant source and a constant accessible output, but
the inaccessible system \(E\) retains the code.
The complete residual distinguishes orthogonal inputs
perfectly, so its distance from every replacement channel
is at least \(1/2\) for a nontrivial code.
\end{proof}

In the present model, the displaced carrier is one concrete
reason that a source-only reduced-dynamics calculation is
insufficient.

\subsection{Discarded probability is not a coherent cutoff error}

\begin{proposition}[Flagged-cutoff check]
\label{prop:cutoff-check}
Let
\[
\ket+=\frac{\ket0+\ket1}{\sqrt2}.
\]
Replacing the rejected \(\ket1\) component by a distinct
failure flag gives
\[
\sigma=\frac12\kb00+\frac12\kb{\perp}{\perp}.
\]
In the common three-dimensional output space,
\[
\dtr(\kb++ ,\sigma)=\frac{\sqrt5+1}{4}>\frac12.
\]
\end{proposition}

\begin{proof}
The nonzero two-dimensional block of
\(\kb++-\sigma\) has eigenvalues
\((1+\sqrt5)/4\) and \((1-\sqrt5)/4\).
The failure block has eigenvalue \(-1/2\).
Taking half the sum of their absolute values proves the result.
\end{proof}

The exact discarded-probability identity reported in En02-E
uses its commuting state-cutoff hypotheses
\cite{CheungEn02E}. It is not a generic channel theorem.
The regulator estimate in this paper is instead proved from
the actual complete isometries.

\subsection{A decoder's existence does not determine its resources}

The finite optimization does not price a streaming schedule,
coherent collection hardware, or computational complexity.
The decoder resource distinctions in En02-B and En02-C remain
separate \cite{CheungEn02B,CheungEn02C}.

Likewise, a local bounded-effect approximation does not
automatically supply a coherent packet-to-memory channel.
The supplied Em03-C and Em03-D maintain that boundary
\cite{CheungEm03C,CheungEm03D}.

%=====================================================================
\section{Gravitational correspondence and a common validity interval}
\label{sec:gravity}

This section records the gravitational obligations in a form
that can be tested. It does not assert their completion.

\subsection{Required correspondence}

A gravitational successor must provide, for this same microscopic
specification:

\begin{enumerate}
\item
A state and code correspondence covering the preparation
\eqref{eq:KM-prep} and insertion \eqref{eq:W}, including
superpositions and reference-entangled inputs.

\item
A derivation of the low-energy boundary variables, the relevant
metric and dilaton, and the effective bath coupling, with
fixed-realization or explicitly probabilistic remainder bounds.

\item
An identification of the accessible packet algebra with the
outgoing radiation available to the observer.

\item
A sourced energy balance and backreaction calculation, including
interaction energy and any externally prescribed switching work.

\item
A validity interval with explicit cutoff, finite-\(N\),
low-energy, preparation, and detector conditions.

\item
A channel calculation controlling the reference--complete-residual
state, not only a radiation entropy or a diagonal influence saddle.
\end{enumerate}

The correspondence must be derived on the declared code.
Agreement on selected semiclassical observables is not replaced
by diamond closeness to an information-losing semiclassical
channel.

\begin{proposition}[Replacement-channel incompatibility]
\label{prop:replacement}
If \(\NN_{\mathrm{rep}}\) is constant and
\(\ddia(\NN,\NN_{\mathrm{rep}})\le b\), then
\[
\erec(\NN)\ge1-d^{-2}-b.
\]
\end{proposition}

\begin{proof}
For every decoder,
\[
\ddia(\DD\NN,\id_C)
\ge
\ddia(\DD\NN_{\mathrm{rep}},\id_C)-b.
\]
The decoded replacement channel is constant.
For a Bell input, every constant channel has target Bell overlap
\(d^{-2}\), so its error is at least \(1-d^{-2}\).
The maximally mixed replacement attains that bound, as follows
from its Weyl-unitary twirl and the triangle inequality.
Taking the decoder infimum proves the claim.
\end{proof}

Small-corrections and firewall arguments have specific pair-state,
factorization, and interior-partner hypotheses
\cite{Mathur,AMPS}. No horizon pair state or physical interior
algebra is constructed here. The finite encoded-partner results
of Hawk03-A are comparisons, not a substitute for that construction
\cite{CheungHawk03A}.

\subsection{Integrated energy comparison is safer than an unsupported derivative}

\begin{proposition}[Integrated gravitational energy allowance]
\label{prop:integrated-energy}
Suppose a future correspondence proves, on \([t_0,t_1]\),
\[
|E_{\mathrm{grav}}(t)-E_{\mathrm{mic}}(t)|
\le\eta_E(t),
\]
and the microscopic balance satisfies
\[
E_{\mathrm{mic}}(t_1)-E_{\mathrm{mic}}(t_0)
=
-\int_{t_0}^{t_1}F_{\mathrm{mic}}(s)\dd s.
\]
If it also proves
\[
\left|
\int_{t_0}^{t_1}
(F_{\mathrm{out}}-F_{\mathrm{mic}})(s)\dd s
\right|
\le\eta_F,
\]
then
\[
\begin{split}
\left|
E_{\mathrm{grav}}(t_1)-E_{\mathrm{grav}}(t_0)
+\int_{t_0}^{t_1}F_{\mathrm{out}}(s)\dd s
\right|
\le
\eta_E(t_0)+\eta_E(t_1)+\eta_F.
\end{split}
\]
\end{proposition}

\begin{proof}
Insert \(E_{\mathrm{mic}}\) at both endpoints and insert
\(F_{\mathrm{mic}}\) in the integral.
Use the exact microscopic balance and the triangle inequality.
\end{proof}

A bound on energy values alone does not bound their derivatives:
an error \(\eta\sin(\Omega t)\) has amplitude \(\eta\) but
derivative amplitude \(\eta\Omega\).
Thus a claimed pointwise remainder
\(\mathcal R_E(t)\) needs derivative control.
For the autonomous model, a natural exact stored-energy convention
is \(E_{\mathrm{mic}}=E_S+E_I\), whose loss equals the full
reservoir energy gain. Identifying that gain with collected
outgoing physical flux is a separate comparison.

\subsection{Common validity interval}

Let \(I_{\mathrm{geom}}\) be an interval with a proved gravitational
correspondence, \(I_{\mathrm{access}}\) an interval with a proved
radiation receiver comparison, and \(I_{\mathrm{cert}}\) an
interval on which an actual microscopic certificate is small.
The required success set is
\[
I_{\mathrm{success}}
=
I_{\mathrm{geom}}\cap I_{\mathrm{access}}\cap I_{\mathrm{cert}}.
\]

A recovery-during-evaporation result must exhibit parameters and
a nonempty interval in this intersection. The existence of finite
approximations in Theorem~\ref{thm:regulator} proves neither
smallness of the residual nor nonemptiness of that intersection.

The gravitational approximation budget remains distinct from
the channel budget. A metric, dilaton, or stress-tensor error is
not added to a diamond-norm error without a proved conversion.

\subsection{Audit of the proposed auxiliary mathematics}

\begin{longtable}{@{}p{0.22\linewidth}p{0.70\linewidth}@{}}
\toprule
Proposed tool & Boundary of a valid application\\
\midrule
\endfirsthead
\toprule
Proposed tool & Boundary of a valid application\\
\midrule
\endhead
\bottomrule
\endfoot

Classical SK gap &
A classical Glauber or heat-bath estimate is not a quantum SYK
dilation estimate. Matching temperature names does not prove
a reduction between the processes.\\[0.35em]

Ground-state area law &
A unique gapped ground-state theorem does not automatically
apply to an inserted nonequilibrium code. Spatial entanglement
bounds do not imply complete residual independence.\\[0.35em]

Lieb--Thirring inequality &
A bound on spectral moments is not a propagator, influence
functional, or encoded channel estimate without a connecting
argument.\\[0.35em]

Penrose or Kerr result &
A classical asymptotically flat or near-Kerr theorem requires
its own dimension, constraint, energy-condition, and boundary
hypotheses. It is not automatically a statement about JT
dilaton gravity or the present reservoir.\\[0.35em]

Bosonic entropy inequality &
Its precise theorem and hypotheses must be checked. An
entropy inequality does not construct a decoder or prove
a constant complete complement.\\[0.35em]

Residual entanglement diagnostic &
Entanglement, distillable key, and code-wide complementary
independence are different properties. None can be substituted
for \eqref{eq:leak-definition}.\\
\end{longtable}

No result from the supplied numbered catalogue is used as a
premise of any theorem above. Incomplete catalogue excerpts
cannot discharge the missing interacting or gravitational
calculations \cite{SuppliedCatalogue}.

%=====================================================================
\section{Information endpoint and gravitational endpoint}
\label{sec:endpoint}

An asymptotic endpoint is not inferred from small source energy,
a horizonless effective saddle, or the existence of an infinite
reservoir.

\subsection{A correctly formulated information endpoint}

\begin{definition}[Common-output asymptotic construction]
An information endpoint requires identifications \(S_t\) into
one fixed complete output space
\(\mathcal H_{R_{\mathrm{out}}}\otimes\mathcal H_{B_\infty}\)
such that
\[
S_tV_t\longrightarrow V_{\mathrm{out}}
\]
on every code vector.
The identifications must specify outgoing access and every
inaccessible output. They may express a derived scattering
representation; they may not be chosen as new operations that
release the residual code.
\end{definition}

\begin{theorem}[Finite-code endpoint continuity]
\label{thm:endpoint-continuity}
If the preceding strong convergence holds for a finite code, it
holds in operator norm, and \(V_{\mathrm{out}}\) is an isometry.
Writing
\[
a_t=\norm{S_tV_t-V_{\mathrm{out}}},
\]
the complete channel and either reduced channel differ from
their endpoint channels by half-diamond distance at most \(a_t\).

In particular, on a common residual space,
\[
\left|
\delta_{\mathrm{full}}(t)
-\delta_{\mathrm{end}}
\right|\le a_t.
\]
Thus a proved sequence
\(\delta_{\mathrm{full}}(t)\to0\), together with the declared
asymptotic construction, implies
\(\delta_{\mathrm{end}}=0\).
\end{theorem}

\begin{proof}
Choose an orthonormal code basis.
Convergence on its finitely many vectors bounds the operator
norm difference by the square root of the sum of their squared
differences. Preservation of inner products passes to the limit,
so \(V_{\mathrm{out}}\) is an isometry.

Lemma~\ref{lem:isometry} gives channel convergence.
For every fixed replacement state, the two distances differ
by at most the channel comparison allowance.
Taking infima in both directions proves the last estimate.
\end{proof}

The theorem does not construct wave operators or prove asymptotic
completeness for the interacting Hamiltonian \eqref{eq:H}.
Those are additional analytical problems.

\subsection{A full-residual energy criterion and its limitations}

\begin{proposition}[Unique gapped full-residual vacuum criterion]
\label{prop:residual-energy}
Suppose the \emph{entire} residual has a specified nonnegative
operator \(K_B\) satisfying
\[
K_B\ge\gamma
\left(I_B-\kb{\Omega_B}{\Omega_B}\right),
\qquad \gamma>0,
\]
with one normalized vacuum \(\Omega_B\).
If a complete code isometry \(V\) obeys
\[
\sup_{\rho_C}
\Tr(K_B\CC(\rho_C))\le u<\gamma,
\]
then
\[
\dleak,\ \erec
\le
\min\left\{
1,\sqrt{2\left(1-\sqrt{1-u/\gamma}\right)}
\right\}.
\]
\end{proposition}

\begin{proof}
Set \(P=I_R\otimes\kb{\Omega_B}{\Omega_B}\).
The energy bound gives
\[
V^\dagger(I-P)V\le(u/\gamma)I_C.
\]
Thus \(M=V^\dagger PV\) is positive and invertible.
The polar-normalized map
\[
V_0=PV\,M^{-1/2}
\]
is an isometry of the form \(A\otimes\ket{\Omega_B}\).
Its complementary channel is constant and its accessible
channel is exactly recoverable.

Moreover \(V^\dagger V_0=M^{1/2}\), so
\[
\norm{V-V_0}^2
=
\norm{2I-2M^{1/2}}
\le
2\left(1-\sqrt{1-u/\gamma}\right).
\]
Apply Lemma~\ref{lem:isometry} and decode the isometry \(A\)
on its range with a complete CPTP extension.
\end{proof}

This criterion requires energy control of the full residual,
including the carrier and all uncollected modes.
The present scalar bath and displaced carrier do not provide
that hypothesis automatically. A gapless residual or a
degenerate zero-energy sector defeats this particular argument.

\begin{proposition}[Zero source energy can coexist with perfect residual storage]
\label{prop:zero-energy}
Vanishing source energy does not prove an information endpoint.
\end{proposition}

\begin{proof}
Take a source with vacuum \(\ket0_S\) of energy zero, an
inaccessible degenerate memory \(E\), and the isometry
\[
V\ket\psi
=
\ket0_S\ket0_R\ket\psi_E.
\]
The source energy is zero, but \(E\) stores the code.
\end{proof}

Bound states, dark sectors, conserved labels, persistent
source--bath dressing, and inaccessible radiation channels
must therefore be tested directly in the proposed asymptotic
representation.

\subsection{Finite-proxy recurrence}

\begin{proposition}[Recurrence of a fixed finite proxy]
\label{prop:recurrence}
At fixed occupation cutoff and fixed finite frequency mesh,
the approximate total system is finite dimensional.
Its prepared channel returns arbitrarily closely to the initial
channel at arbitrarily large times.
Consequently the finite proxy cannot, by itself, establish
permanent information release.
\end{proposition}

\begin{proof}
A finite-dimensional Hamiltonian has finitely many eigenvalues.
Simultaneous recurrence of their phases gives arbitrarily large
times at which its unitary is arbitrarily close to the identity.
Lemma~\ref{lem:isometry} then gives return of the prepared channel.

Initially the reservoir is vacuum and its accessible channel
is constant. These recurrences are incompatible with permanent
high-accuracy recovery from that closed proxy.
\end{proof}

The regulator theorem is explicitly finite time.
Taking \(t\to\infty\) at fixed proxy parameters is not a
substitute for a controlled continuum endpoint.

\subsection{The separate gravitational endpoint}

A gravitational endpoint additionally requires the late-time metric,
dilaton, horizon diagnostics, and the regime in which they remain
controlled. Neither Theorem~\ref{thm:endpoint-continuity} nor
Proposition~\ref{prop:residual-energy} supplies these objects.

A surviving code-independent microscopic dressing can be compatible
with information release. A surviving input-dependent sector cannot
be omitted. Conversely, absence of an effective horizon does not
establish complete microscopic release.

%=====================================================================
\section{Release statement}
\label{sec:release}

\begin{theorem}[Mathematical release]
\label{thm:release}
For the explicitly regulated fixed-realization model:
\begin{enumerate}
\item
The interacting source, scalar reservoir, coupling, and
self-adjoint complete dynamics are specified.

\item
A pure source preparation and message-independent parity-preserving
diary insertion are constructed. Their energy and apparatus
boundaries are retained.

\item
Every logical matrix unit determines an accessible output and a
complete complementary output, including the displaced carrier.

\item
An explicit finite-mode, finite-occupation approximation controls
the complete prepared channel uniformly on every finite
observation interval.

\item
Finite complementary Choi certificates and specified CPTP decoder
certificates give the reference-preserving bounds in
Section~\ref{sec:certificates}, with all code dimension factors and
regulator allowances explicit.

\item
The microscopic energy ledger, commuting-coupling recovery
obstruction, disorder-averaging counterexample, coherent-cutoff
check, and finite-proxy recurrence statement are proved.

\item
A common-output endpoint construction has the continuity
consequences of Theorem~\ref{thm:endpoint-continuity}.
A unique gapped full-residual vacuum has the separate sufficient
criterion of Proposition~\ref{prop:residual-energy}.
\end{enumerate}
No item establishes a small dynamical complete-residual bound
for the interacting model, a controlled gravitational
correspondence, or a gravitational evaporation endpoint.
\end{theorem}

\begin{proof}
The items follow respectively from
Theorem~\ref{thm:selfadjoint},
Propositions~\ref{prop:preparation} and \ref{prop:insertion},
equations~\eqref{eq:Vt}--\eqref{eq:matrix-units},
Theorem~\ref{thm:regulator},
Theorem~\ref{thm:certificate} and
Proposition~\ref{prop:decoder},
the results of Sections~\ref{sec:complete},
\ref{sec:influence}, \ref{sec:debug}, and \ref{sec:endpoint},
and the two named endpoint results.

The excluded claims require additional dynamical and physical
objects not constructed by these results.
\end{proof}

\begin{center}
\small
\renewcommand{\arraystretch}{1.15}
\begin{tabularx}{\linewidth}{@{}p{0.31\linewidth}X@{}}
\toprule
Milestone & Status and permissible claim\\
\midrule
Microscopic E1 interface &
Constructed at fixed \(\mu,\Lambda\): interacting Hamiltonian,
pure preparation, coherent diary, and complete output ledger.\\

Gravitational diary interpretation &
Not constructed. The insertion is a specified microscopic
loading operation, not a proved infalling trajectory.\\

Controlled gravitational evaporation E2 &
Not established for this microscopic specification.\\

Quantum recovery during evaporation E3 &
Not established. A small complete-residual certificate and a
common gravitational/access interval are missing.\\

Information endpoint &
Continuity criteria proved; interacting asymptotic channel and
its decoupling remain unproved.\\

Complete gravitational endpoint E4 &
Not established.\\

Four-dimensional physical solution &
Not claimed.\\
\bottomrule
\end{tabularx}
\end{center}

\paragraph{Conclusion.}
The revision replaces an unspecified interacting channel by a
well-defined one and replaces informal regulator removal by a
reference-preserving finite-time theorem. It also repairs the
carrier ledger, the scalar regulator assumptions, the interaction
energy accounting, and the distinction between averaged and
fixed-realization decoupling.

The decisive next result is an actual quantitative estimate for
the complete residual of this interacting model. Its derivation
must preserve the diary's off-diagonal matrix units and must be
valid in the same regime as a derived gravitational description
and a declared radiation receiver. Without that result, the
correct release is a microscopic construction and certification
framework, not a solution of evaporation.

\paragraph{Submission and reproducibility boundary.}
The manuscript is a self-contained analytical methods draft.
No numerical certificate, solver execution, pdfLaTeX build,
formal verification, or independent review is asserted.
A responsible submission should supply author metadata,
archival copies of any unpublished references relied upon,
a successful build, and independent checking of the proofs.
A submission claiming E3 or E4 would additionally need the
missing dynamical and gravitational results, not stronger wording.

%=====================================================================
\appendix
\section{Minimal certificate packet for an interacting recovery claim}
\label{app:certificate}

A completed microscopic recovery result should archive:

\begin{enumerate}
\item
The exact \(N,\mathbf J,\beta,s,s'\), source normalization,
coupling polynomial, couplings, cutoffs, and packet functions.

\item
The mode and occupation basis dictionaries, including source and
carrier fermionic ordering.

\item
The insertion isometry and the complete residual inventory.

\item
The finite Hermitian Hamiltonian and the exact definition of its
propagator. A floating-point array is not the exact specification
unless a proved error enclosure accompanies it.

\item
All accessible and complementary logical matrix-unit outputs.

\item
A rigorously feasible replacement state or complete decoder,
together with certified positivity and trace-preservation
conditions.

\item
Validated trace-norm upper bounds and any arithmetic allowances.

\item
The regulator parameters selected from
Corollary~\ref{cor:regulator-selection}, or sharper proved bounds.

\item
A declared time or a uniform time interval, with a proved
continuity allowance when finite sampling in time is used.

\item
For a gravitational claim, the separately derived state,
observable, energy, geometry, and radiation-access comparisons
on the same interval.

\item
For an endpoint claim, the asymptotic identification, bound-sector
audit, complete residual channel, order of limits, and separate
gravitational endpoint.
\end{enumerate}

The finite-channel interface in the supplied Em03-H is a useful
documentary comparison for items involving complete purifications
and matrix units \cite{CheungEm03H}. Its eigenvector-selected
code and finite path are not substituted for the physical loading
and continuum evolution specified here.

\section{Relation to the autonomous benchmarks}

The supplied Hawk02-C and Hawk02-D report exact one-particle
benchmarks of the form
\[
\erec=\dleak=(1-d^{-2})s,
\]
with \(s\) containing survival and uncollected weight
\cite{CheungHawk02C,CheungHawk02D}.
Those formulas are valuable normalization checks, but they use
their own excitation-sector structure.

The present quartic Hamiltonian generally lacks that structure.
No scalar survival factor is asserted for its code, and no
quadratic wave-operator completeness theorem is transferred
to it. The regulator estimate derived here applies without
assuming those simplifications.

The resource and access distinctions in En02-B, En02-C, En02-E,
and Em03-C likewise remain attached to their own channels
\cite{CheungEn02B,CheungEn02C,CheungEn02E,CheungEm03C}.
Their explicit decoders or packet-state approximations are not
automatically implementations of the present decoder.

%=====================================================================
\begin{thebibliography}{99}

\bibitem{CheungHawk01B}
N.~Cheung,
\emph{Hawk01-B}, version 1,
unpublished manuscript, n.d.
Supplied excerpts:
``Controlled island comparison without a reconstruction overclaim''
and ``Conclusions and mathematical release,''
including \texttt{eq:Petz} and \texttt{thm:release}.
Version-qualified documentary citation; no independent verification
of the full manuscript is asserted.

\bibitem{CheungHawk02A}
N.~Cheung,
\emph{Hawk02-A}, version 1,
unpublished manuscript, n.d.
Supplied excerpt:
``Decision ledger and exact ending.''

\bibitem{CheungHawk02C}
N.~Cheung,
\emph{Hawk02-C}, version 2,
unpublished manuscript, n.d.
Supplied abstract, gravitational-obligation ledger, and
``Discarded probability is not a generic coherent error.''

\bibitem{CheungHawk02D}
N.~Cheung,
\emph{Hawk02-D}, version 1,
supplied manuscript text, n.d.
Autonomous quadratic fermionic benchmark; supplied abstract
and mathematical release.
Author attribution follows the supplied source record.

\bibitem{CheungHawk03A}
N.~Cheung,
\emph{Hawk03-A}, version 1,
unpublished manuscript, n.d.
Supplied abstract and
``Source-boundary audit and release.''

\bibitem{CheungEn02B}
N.~Cheung,
\emph{En02-B}, version 1,
unpublished manuscript, n.d.
Supplied abstract and finite programme-certificate excerpts.
Batch and streaming decoder statements refer to its declared
perfect-code model.

\bibitem{CheungEn02C}
N.~Cheung,
\emph{En02-C}, version 1,
unpublished manuscript, n.d.
Supplied excerpts:
``Information availability versus executable recovery''
and \texttt{thm:C1}, sequential reference-preserving soundness.

\bibitem{CheungEn02E}
N.~Cheung,
\emph{En02-E}, version 2,
unpublished manuscript, n.d.
Supplied abstract and cutoff/interface excerpts.
Selected Gaussian-state cutoff identities retain their stated
commutation and preparation hypotheses.

\bibitem{CheungEm03C}
N.~Cheung,
\emph{Em03-C}, version 1,
unpublished manuscript, n.d.
Supplied excerpts:
``Source repairs and logical checks'' and
``One controlled correction and microscopic dynamics.''

\bibitem{CheungEm03D}
N.~Cheung,
\emph{Em03-D}, version 1,
unpublished manuscript, n.d.
Supplied excerpts:
``Repairs to the proposed proof architecture''
and ``Access and effort.''

\bibitem{CheungEm03F}
N.~Cheung,
\emph{Em03-F}, version 1,
unpublished manuscript, n.d.
Supplied logical-repair and exterior scalar-source release excerpts.
Effective exterior source model; no gravitational identification
is imported.

\bibitem{CheungEm03G}
N.~Cheung,
\emph{Em03-G}, version 1,
unpublished manuscript, n.d.
Supplied resource ledger and ordered exterior-source release.
Its packet decoder is not a coherent local receiver for the
present model.

\bibitem{CheungEm03H}
N.~Cheung,
\emph{Em03-H}, version 8,
unpublished manuscript, n.d.
Supplied \texttt{thm:channels},
``Complete finite channel interface.''

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

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

\bibitem{HaslerHerbst}
D.~Hasler and I.~Herbst,
``Ground states in the spin boson model,''
arXiv:1003.5923.
Used as context for standard field-operator and
spin--boson Hamiltonian constructions; no ground-state or
endpoint theorem from this reference is assumed for the present
many-level source.

\bibitem{FeynmanVernon}
R.~P.~Feynman and F.~L.~Vernon, Jr.,
``The theory of a general quantum system interacting with
a linear dissipative system,''
\emph{Annals of Physics}
\textbf{24} (1963), 118--173.
doi: \texttt{10.1016/0003-4916(63)90068-X}.

\bibitem{KourkoulouMaldacena}
I.~Kourkoulou and J.~Maldacena,
``Pure states in the SYK model and nearly-\(\mathrm{AdS}_2\)
gravity,''
arXiv:1707.02325.

\bibitem{MaldacenaStanford}
J.~Maldacena and D.~Stanford,
``Comments on the Sachdev--Ye--Kitaev model,''
\emph{Physical Review D}
\textbf{94} (2016), 106002.
doi: \texttt{10.1103/PhysRevD.94.106002}.
arXiv:1604.07818.

\bibitem{ChenQiZhang}
Y.~Chen, X.-L.~Qi, and P.~Zhang,
``Replica wormhole and information retrieval in the SYK model
coupled to Majorana chains,''
\emph{Journal of High Energy Physics}
06 (2020), 121.
doi: \texttt{10.1007/JHEP06(2020)121}.
arXiv:2003.13147.

\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: \texttt{10.1007/JHEP08(2023)171}.
arXiv:2210.15579.

\bibitem{NakayamaMiyataUgajin}
Y.~Nakayama, A.~Miyata, and T.~Ugajin,
``The Petz (lite) recovery map for the scrambling channel,''
\emph{Progress of Theoretical and Experimental Physics}
2023 (2023), 123B04.
doi: \texttt{10.1093/ptep/ptad147}.
arXiv:2310.18991.

\bibitem{PSSY}
G.~Penington, S.~H.~Shenker, D.~Stanford, and Z.~Yang,
``Replica wormholes and the black hole interior,''
arXiv:1911.11977.

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

\bibitem{Mathur}
S.~D.~Mathur,
``The information paradox: A pedagogical introduction,''
arXiv:0909.1038.

\bibitem{AMPS}
A.~Almheiri, D.~Marolf, J.~Polchinski, and J.~Sully,
``Black holes: Complementarity or firewalls?''
arXiv:1207.3123.

\bibitem{SuppliedCatalogue}
OpenAI,
\emph{External mathematics catalogue},
supplied manuscript records and excerpts,
including families 227, 260, 262, 264, 265, 272, and 273.
The supplied record identifies a catalogue dated
7 October 2026.
Documentary citation only: the catalogue's claimed proofs are
not independently verified here and none is a premise of the
microscopic, recovery, or gravitational statements in this paper.

\end{thebibliography}

\end{document}