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\hypersetup{
  pdftitle={Hawk03-A: Encoded Partner Algebras and Firewall-Interface Tests},
  pdfsubject={Finite-dimensional compatibility, complete complements,
              and conditional physical certification}
}

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\title{\textbf{Hawk03-A}\\[4pt]
Encoded Partner Algebras, Complete Emission Channels,\\
and Quantitative Firewall-Interface Tests}
\author{}
\date{Version 1}

\begin{document}
\maketitle

\begin{abstract}
We formulate and prove a finite-dimensional interface package for
testing proposals that identify an interior partner with an observable
encoded in radiation. We first give a quantitative independent-factor
AMPS benchmark, an entropy-growth bound under explicit near-pair
hypotheses, and a sharp two-purifier projection inequality. We then
construct one linear unitary finite model with a state-independent
partner algebra, exact reference-preserving message recovery, a complete
constant complement, and exact statistics for a declared pair-vacuum
proxy instrument. The partner algebra is a subalgebra of the radiation
algebra, not an additional commuting copy. A quantitative commutator
obstruction proves why two commuting representations cannot both
intertwine the same noncommutative logical algebra. An explicit
purifier-distillation intervention changes the original pair test,
while preserving the message channel.

All model-specific mathematical statements are proved locally.
Standard entropy and Stinespring results are identified as external
mathematical inputs. A separate stability theorem explains what
additional physical bridges would turn the finite package into a
certificate for named infalling measurements and recovery. No such
gravitational bridge is constructed here. In particular, pair-proxy
regularity is not Hadamard regularity, a message register is not a
spacetime-localized infalling diary, and the construction is not a
Page curve, an evaporation endpoint, or a resolution of the physical
firewall problem.
\end{abstract}

\tableofcontents

\section{Question, contribution, and scope}
\label{sec:scope}

The firewall argument concerns the joint compatibility of microscopic
unitarity, a near-horizon effective description, and an uneventful
horizon for an infalling observer \cite{AMPS,Apologia}. A resolution must
address their interaction in one physical model. Establishing an
outgoing spectrum in one model and information recovery in another
does not establish that compatibility.

The source records of Cheung separate precisely these obligations.
Hawk01-A separates a fixed-background Hawking source from a microscopic
message channel \cite{CHawk01A}. Hawk01-B separates entropy histories
from complete code preservation \cite{CHawk01B}. Hawk02-A requires
geometry, complete emission, comparison errors, and complementary
leakage to be supplied independently \cite{CHawk02A}. Em03-C explicitly
does not construct an interior algebra \cite{CEm03C}. Em03-H separates
microscopic extraction from a physical infalling-diary interpretation
\cite{CEm03H}.

This manuscript completes a \emph{finite-interface mathematical
milestone}. It does not complete the missing gravitational programme.
Its positive construction establishes that an encoded partner,
reference-preserving message recovery, and a specified pair test can
coexist in one finite model when the partner is not an independent
factor of the radiation. Its obstruction results establish that this
observation cannot be upgraded to an independent interior copy, or
to intervention-independent vacuum statistics.

The distinction is essential. Holographic error correction and
entanglement-wedge reconstruction motivate encoded-observable
descriptions \cite{ADH,DHW,Cotler,Penington}. They do not identify the
finite construction below with a gravitational interior.
Replica-wormhole and island calculations supply further external
candidate settings, with their own dynamical and interpretive
assumptions \cite{AMMZ,PSSY}. None is a premise of our finite model.
Conversely, the objections in \cite{Apologia} are not answered merely
by renaming an early-radiation observable as an interior observable.

\subsection{Source status and dependency discipline}

The Cheung references are cited by identifier and supplied version.
Where the supplied material consists of excerpts, the citation does not
represent an independent verification of the full manuscript.
Bibliographic entries do not assign an unprovided publication year,
journal, DOI, or peer-review status.

The mathematical results below use the finite spaces, channels,
operators, and hypotheses explicitly defined here. Results from
different Cheung manuscripts are contextual comparisons, not silently
imported premises. We cite standard quantum-information results where
used and prove the model-specific consequences.

\subsection{Three distinct verdicts}

Throughout, we distinguish:
\begin{enumerate}
\item \textbf{Constructed}: a stated mathematical object and its claimed
properties are established on the declared domain.
\item \textbf{Obstructed}: a stated collection of requirements is proved
incompatible under explicit hypotheses.
\item \textbf{Unresolved}: a required construction or physical
identification is absent.
\end{enumerate}
Failure to construct an object is not a universal nonexistence theorem.

\section{Finite interfaces and normalization}
\label{sec:interfaces}

All Hilbert spaces in Sections \ref{sec:interfaces}--\ref{sec:stability}
are finite dimensional. A continuum or gravitational extension requires
additional hypotheses and is not implicit.

For a Hilbert space $\HH_X$, write $\BB(\HH_X)$ for its matrix algebra
and $\States(\HH_X)$ for its density operators. Channels are linear,
completely positive, trace-preserving maps. For a channel difference,
the diamond norm is stabilized against an arbitrary untouched reference.
We use the conventions
\begin{align}
\td(\rho,\sigma)
  &=\frac12\norm{\rho-\sigma}_1,\\
\dd(\Lambda,\Gamma)
  &=\frac12\norm{\Lambda-\Gamma}_\diamond,\\
\tv(p,q)
  &=\frac12\sum_x\abs{p_x-q_x}.
\end{align}
These distances are at most one for normalized states, channels,
and probability laws. Standard channel facts used here can be found
in \cite{Watrous}.

An external reference is denoted by $Q$. It is not an output environment
and is not traced out in a reference-preservation claim.

\begin{definition}[Complete emission interface]
\label{def:complete}
A complete finite emission interface consists of an input code
$\HH_C$, an accessible output $\HH_R$, an inaccessible output
$\HH_K$, and an isometry
\[
V:\HH_C\longrightarrow\HH_R\otimes\HH_K.
\]
Every initially fixed auxiliary mixed state is purified, and its
purification is assigned to the accessible or inaccessible output.
The interface defines
\begin{align}
\Vchan(X)&=VXV^\dagger,\\
\Nchan(X)&=\Tr_K(VXV^\dagger),\\
\Nchan^c(X)&=\Tr_R(VXV^\dagger).
\end{align}
The superscript $c$ refers to the complement of this specified
dilation.
\end{definition}

This definition is an output-accounting convention. It is not a
theorem that physical inside and outside regions factorize.
Gauge constraints, dressing, and observable algebras can obstruct
naive subsystem identifications \cite{DG}.

\begin{definition}[Recovery and complete leakage]
\label{def:errors}
For the interface in Definition \ref{def:complete}, define
\begin{align}
\rec(\Nchan)
  &=\inf_{\Dchan}\dd(\Dchan\Nchan,\id_C),\\
\leak(\Nchan)
  &=\inf_{\tau\in\States(\HH_K)}
      \dd(\Nchan^c,\Rrep_\tau),\\
\Rrep_\tau(X)&=\Tr(X)\tau.
\end{align}
The infimum defining recovery ranges over deterministic CPTP decoders
$\Dchan:\BB(\HH_R)\to\BB(\HH_C)$.
\end{definition}

The quantifier is over all code inputs, including those entangled
with $Q$. Agreement on a selected preparation or on diagonal matrix
units is insufficient.

\begin{definition}[Complete instrument]
A finite instrument is a family of completely positive maps
$\{\Ichan_x\}_{x\in\mathsf X}$ whose sum is trace preserving.
Its complete classical--quantum output channel is
\[
\Ichan(\rho)=
  \sum_{x\in\mathsf X}\kb{x}{x}\otimes\Ichan_x(\rho).
\]
Failure outcomes are included in $\mathsf X$.
\end{definition}

Comparing only the probabilities $\Tr\Ichan_x(\rho)$ compares records,
not the conditional residual states. This distinction is recorded
in Cheung's En03-A and Psy02-B \cite{CEn03A,CPsy02B}.

\section{Independent-factor AMPS benchmarks}
\label{sec:amps}

Let $A$ denote a proposed partner, $B$ a selected outgoing packet,
and $E$ previously accessible radiation. This section assumes a
genuine tensor product
\[
\HH_A\otimes\HH_B\otimes\HH_E
\]
and one joint density operator on it. This is an explicit hypothesis,
not a consequence of the names ``interior'' and ``radiation.''

All logarithms in entropy expressions are base two. Set
\[
S(X)_\rho=-\Tr(\rho_X\log_2\rho_X),\qquad
S(B|A)_\rho=S(AB)_\rho-S(A)_\rho.
\]
For $0\le x\le1$, define
\[
h_2(x)=-x\log_2x-(1-x)\log_2(1-x),
\]
with the usual zero conventions, and
\begin{equation}
g_d(x)=
2x\log_2d+(1+x)h_2\!\left(\frac{x}{1+x}\right).
\label{eq:gd}
\end{equation}

\begin{lemma}[Conditional-entropy continuity]
\label{lem:continuity}
If $\dim\HH_B=d$ and
$\td(\rho_{AB},\sigma_{AB})\le\eps\le1$, then
\[
\abs{S(B|A)_\rho-S(B|A)_\sigma}\le g_d(\eps).
\]
\end{lemma}

\begin{proof}
We give the finite-dimensional argument corresponding to
Winter's continuity bound \cite{Winter}. Let
$x=\td(\rho,\sigma)$. If $x=0$, the conclusion is immediate.
Otherwise the Jordan decomposition gives states $\rho_+$ and
$\rho_-$ such that
\[
\rho-\sigma=x(\rho_+-\rho_-).
\]
Consequently,
\[
\omega=
\frac{\rho+x\rho_-}{1+x}
=
\frac{\sigma+x\rho_+}{1+x}.
\]

For a binary mixture with weights $1-t,t$, conditional entropy
satisfies
\[
0\le
S(B|A)_{\omega}
-\bigl((1-t)S(B|A)_{\omega_0}
       +tS(B|A)_{\omega_1}\bigr)
\le h_2(t).
\]
Indeed, attaching a classical mixture flag $Z$ identifies the
difference with $I(B:Z|A)$. Its lower bound is strong subadditivity
\cite{LiebRuskai}. Its upper bound follows from
$I(B:Z|A)=S(Z|A)-S(Z|AB)\le H(Z)$, since classical conditional
entropy conditioned on a quantum system is nonnegative.

Apply this mixture inequality to the two presentations of $\omega$
with $t=x/(1+x)$. The difference between their mixture gaps has
absolute value at most $h_2(t)$. Since
$-\log_2d\le S(B|A)\le\log_2d$, we obtain
\[
\abs{S(B|A)_\rho-S(B|A)_\sigma}
\le 2x\log_2d+(1+x)h_2\!\left(\frac{x}{1+x}\right).
\]
The right-hand side is increasing for $0\le x\le1$, so replacing
$x$ by $\eps$ proves the assertion.
\end{proof}

\begin{lemma}[Weak monotonicity on independent factors]
\label{lem:weak}
For every state $\rho_{ABE}$ on the stated tensor product,
\[
S(B|E)_\rho+S(B|A)_\rho\ge0.
\]
\end{lemma}

\begin{proof}
Purify $\rho_{ABE}$ by $F$. Strong subadditivity gives
$S(B|A)\ge S(B|AF)$. Purity of $ABEF$ gives
$S(B|AF)=-S(B|E)$. Combining the two identities proves the claim.
\end{proof}

\begin{theorem}[Quantitative independent-factor benchmark]
\label{thm:amps}
Let $\ket{\beta}_{AB}$ be a pure state, let
\[
s=S(B)_\beta>0,
\]
and suppose
\[
\td(\rho_{AB},\kb{\beta}{\beta})\le\eps.
\]
Under the independent-factor hypothesis,
\begin{equation}
S(B|E)_\rho\ge s-g_d(\eps).
\label{eq:ampsbound}
\end{equation}
If a proposed radiation-purification certificate also requires
\[
S(B|E)_\rho\le-\mu,\qquad \mu\ge0,
\]
then its simultaneous validity requires
\begin{equation}
g_d(\eps)\ge s+\mu.
\label{eq:ampsnecessary}
\end{equation}
\end{theorem}

\begin{proof}
For the pure pair,
$S(B|A)_\beta=-s$. Lemma \ref{lem:continuity} gives
$S(B|A)_\rho\le-s+g_d(\eps)$. Lemma \ref{lem:weak} then gives
\eqref{eq:ampsbound}. Combining this lower bound with the asserted
upper bound $-\mu$ gives \eqref{eq:ampsnecessary}.
\end{proof}

\begin{remark}[What unitarity does not supply]
Microscopic unitarity alone does not imply
$S(B|E)<0$ for every selected late packet. That inequality is a
separate state-dependent purification hypothesis. Likewise,
near-horizon smoothness does not become the trace-distance
hypothesis of Theorem \ref{thm:amps} without a physical derivation
identifying the selected pair and its observables.
\end{remark}

\begin{corollary}[Conditional fresh-pair entropy growth]
\label{cor:pairgrowth}
Suppose radiation grows by independent tensor factors,
$E_t=E_{t-1}B_t$. At each step there is a joint state on
$A_tB_tE_{t-1}$ satisfying the hypotheses of
Theorem \ref{thm:amps}, with pair entropy $s_t$, outgoing dimension
$d_t$, and distance allowance $\eps_t$. Then
\[
S(E_n)-S(E_0)
\ge
\sum_{t=1}^{n}\bigl(s_t-g_{d_t}(\eps_t)\bigr).
\]
\end{corollary}

\begin{proof}
At step $t$,
$S(E_t)-S(E_{t-1})=S(B_t|E_{t-1})$.
Apply \eqref{eq:ampsbound} and sum.
\end{proof}

This is a finite trace-distance version of the entropy obstruction
under explicit pair and independence assumptions. It does not claim
that these assumptions follow from our later model. Mathur's
small-corrections argument and Hawk02-A's pair comparison must be
read with their own hypotheses \cite{Mathur,CHawk02A}.

\subsection{A sharp two-purifier test}

\begin{theorem}[Two independent Bell purifiers]
\label{thm:twopurifiers}
Let $A,B,F$ be independent $d$-dimensional factors, $d\ge2$.
Define
\[
\ket{\Phi_d}=\frac1{\sqrt d}\sum_{j=0}^{d-1}\ket{j}\ket{j},
\]
and, on $ABF$,
\[
P=\kb{\Phi_d}{\Phi_d}_{AB}\otimes I_F,\qquad
Q=I_A\otimes\kb{\Phi_d}{\Phi_d}_{BF}.
\]
For every density operator $\rho_{ABF}$,
\begin{equation}
\Tr(P\rho)+\Tr(Q\rho)\le1+\frac1d.
\label{eq:projectionmonogamy}
\end{equation}
The bound is sharp. Consequently, certificates
$\Tr(P\rho)\ge1-\eta_1$ and
$\Tr(Q\rho)\ge1-\eta_2$ require
\[
\eta_1+\eta_2\ge1-\frac1d.
\]
\end{theorem}

\begin{proof}
Let
\[
T_1x=\ket{\Phi_d}_{AB}\otimes x_F,\qquad
T_2y=y_A\otimes\ket{\Phi_d}_{BF}.
\]
Both are isometries, $P=T_1T_1^\dagger$ and
$Q=T_2T_2^\dagger$, and direct expansion gives
$T_1^\dagger T_2=d^{-1}I$ after identifying $A$ with $F$.
The Gram operator of $T=(T_1\ T_2)$ is
\[
T^\dagger T=
\begin{pmatrix}
I&d^{-1}I\\
d^{-1}I&I
\end{pmatrix}.
\]
Its largest eigenvalue is $1+d^{-1}$. The nonzero eigenvalues
of $TT^\dagger=P+Q$ are the same, proving
$\norm{P+Q}_\infty=1+d^{-1}$ and hence
\eqref{eq:projectionmonogamy}. A normalized top eigenvector attains
the bound. The last statement follows by substituting the two
lower bounds.
\end{proof}

\begin{corollary}[Exterior-only distillation]
\label{cor:exteriordistill}
For a state on independent factors $ABE$, let an exterior-only
channel map $E$ to a $d$-dimensional purifier register $F$.
It cannot produce two Bell-test success probabilities violating
\eqref{eq:projectionmonogamy}.
\end{corollary}

\begin{proof}
Applying the channel on $E$ preserves the $AB$ marginal and produces
a state on the independent factors $ABF$. Apply
Theorem \ref{thm:twopurifiers}.
\end{proof}

The word ``exterior-only'' matters: a decoder that acts on $B$ or on
the partner itself need not preserve the original $AB$ certificate.

\section{Complete complements and reference-preserving recovery}
\label{sec:recovery}

\begin{theorem}[Exact complete-complement criterion]
\label{thm:exact}
For a complete finite interface $V:C\to R\otimes K$, the following
conditions are equivalent:
\begin{enumerate}
\item A deterministic CPTP decoder satisfies
$\Dchan\Nchan=\id_C$.
\item There is a state $\tau_K$ such that
$\Nchan^c(X)=\Tr(X)\tau_K$ for every $X\in\BB(\HH_C)$.
\item In any orthonormal code basis $\{\ket{i}\}$,
\[
\Nchan^c(\kb{i}{j})=\delta_{ij}\tau_K.
\]
\end{enumerate}
The resulting recovery preserves every external reference.
\end{theorem}

\begin{proof}
Conditions 2 and 3 are equivalent because matrix units span
$\BB(\HH_C)$.

Assume condition 3, and diagonalize
$\tau_K=\sum_{a=1}^{r}\lambda_a\kb{a}{a}$ with
$\lambda_a>0$. Write
\[
V\ket{i}=\sum_{a=1}^{r}\sqrt{\lambda_a}\,
             \ket{u_{ia}}_R\ket{a}_K.
\]
The complementary matrix-unit identities imply
\[
\langle u_{jb}|u_{ia}\rangle=\delta_{ij}\delta_{ab}.
\]
Thus the $u_{ia}$ form an orthonormal family. On their span, define
an isometry
\[
W\ket{u_{ia}}=\ket{i}_C\ket{a}_S.
\]
The channel that applies this isometry, traces $S$, and assigns an
arbitrary fixed code state on the orthogonal unused subspace is
CPTP. It maps $\Nchan(X)$ to $X$ for every matrix unit and hence
every $X$.

Conversely, dilate an exact decoder by an isometry
$W_D:R\to C\otimes S$. The composite
\[
T=(W_D\otimes I_K)V
\]
is a dilation of $\id_C$. Purity of the decoded output for
$\ket{i}$ implies
$T\ket{i}=\ket{i}\ket{\chi_i}_{SK}$.
Applying exact recovery to superpositions of any two basis vectors
implies $\langle\chi_j|\chi_i\rangle=1$. Hence all $\chi_i$ are the
same vector $\chi$, and
\[
T\ket{\psi}=\ket{\psi}_C\ket{\chi}_{SK}.
\]
Tracing $C$ and $S$ proves condition 2.

Finally, equality of channels on all matrix units remains equality
after tensoring with $\id_Q$. This establishes reference preservation.
\end{proof}

This is the constant-complement form of exact quantum error
correction \cite{KL,KSW}. Comparable locally stated criteria occur in
Cheung's Hawk01-B, Hawk02-A, and Em03-H
\cite{CHawk01B,CHawk02A,CEm03H}. Their model-specific interfaces are
not identified with ours.

\subsection{Approximate recovery}

We use the following external mathematical input from
Kretschmann, Schlingemann, and Werner \cite{KSW}: two channels can
be represented by common, suitably enlarged Stinespring dilations
whose optimal operator-norm separation $\beta$ satisfies
\[
\beta^2\le\norm{\Lambda-\Gamma}_\diamond\le2\beta.
\]
The dilation spaces may be enlarged; this statement does not prescribe
a physically local decoder.

\begin{theorem}[Information--disturbance in half-diamond normalization]
\label{thm:approx}
For Definition \ref{def:errors},
\begin{align}
\rec(\Nchan)
&\le\min\{1,\sqrt{2\leak(\Nchan)}\},\\
\leak(\Nchan)
&\le\min\{1,\sqrt{2\rec(\Nchan)}\}.
\end{align}
\end{theorem}

\begin{proof}
Fix a replacement channel whose distance from $\Nchan^c$ is
$\delta$. Its dilation is the embedding
$\ket{\psi}\mapsto\ket{\psi}_C\ket{\chi}_{SK}$, where $\chi$
purifies the replacement state. The stated Stinespring comparison
provides, after enlarging $S$, a decoder dilation bringing $V$
within operator norm $\sqrt{2\delta}$ of this embedding.
Tracing its syndrome gives a decoded channel within half-diamond
distance $\sqrt{2\delta}$ of $\id_C$. Minimize over replacement states.

Conversely, for a decoder of error $e$, compare a dilation of
$\Dchan\Nchan$ with a dilation of $\id_C$. Their operator-norm
separation can be made at most $\sqrt{2e}$. Their complementary
channels then differ by half-diamond distance at most $\sqrt{2e}$.
Tracing the decoder syndrome leaves the original complete complement,
while the identity dilation leaves a replacement channel.
Minimize over decoders. The bound by one is automatic.
\end{proof}

\begin{remark}[Diagnostic versus worst-case certificate]
A single Bell-input test is not automatically a diamond-norm
certificate. For input dimension $m$, if
\[
J_\Lambda=(\id_Q\otimes\Lambda)(\kb{\Phi_m}{\Phi_m}),
\]
then
\begin{equation}
\td(J_\Lambda,J_\Gamma)
\le\dd(\Lambda,\Gamma)
\le m\,\td(J_\Lambda,J_\Gamma).
\label{eq:choisandwich}
\end{equation}
For the upper bound, write a purified input as
$(A\otimes I)\ket{\Phi_m}$, with
$\norm{A}_\infty\le\sqrt m$, and apply the trace-norm ideal property.
Convexity and Schmidt compression cover arbitrary inputs and
references. This dimension factor must not be silently discarded.
\end{remark}

\subsection{Why separate residual marginals are insufficient}

\begin{proposition}[Joint residual information with constant marginals]
\label{prop:joint}
Let all registers below be qutrits, with arithmetic modulo three.
The map
\begin{equation}
V\ket{j}=
\frac1{\sqrt3}\sum_{r=0}^{2}
  \ket{r}_R\ket{r+j}_{K_1}\ket{r+2j}_{K_2}
\label{eq:qutritcode}
\end{equation}
is an isometry. Each one-register output channel is replacement
by $I_3/3$. Nevertheless, the joint complement $K_1K_2$ contains
perfectly distinguishable outputs for distinct computational inputs.
In particular, separate constant residual marginals do not certify
recovery from $R$.
\end{proposition}

\begin{proof}
The triples in \eqref{eq:qutritcode} are distinct for distinct
$(r,j)$, proving orthonormality of the encoded basis vectors.

Tracing $K_1K_2$ on $\kb{i}{j}$ forces
$r+i=s+j$ and $r+2i=s+2j$, hence $i=j$ and $r=s$.
The $R$ output is therefore $\delta_{ij}I_3/3$.
Tracing the other two pairs gives the same identity for $K_1$
and $K_2$.

For input $\ket{j}$, the joint complement is supported on
\[
\operatorname{span}\{
  \ket{r+j}\ket{r+2j}:r=0,1,2\}.
\]
These supports are orthogonal for distinct $j$, since the difference
between the two recorded coordinates is $j$.
Thus two joint outputs have trace distance one. By the triangle
inequality, every replacement state is at distance at least $1/2$
from one of them.

The accessible channel $R$ is constant. On a maximally entangled
qutrit input, every decoded output has Bell overlap $1/9$, so its
recovery error is at least $8/9$.
\end{proof}

The encoding is a standard quantum-secret-sharing example
\cite{CGL}. Here it is used only to audit complement completeness,
as required by the corresponding separation in Hawk02-A
\cite{CHawk02A}.

\section{Encoded algebras and the no-independent-copy obstruction}
\label{sec:algebras}

A message code and a structural logical space are different objects.
We now keep that distinction explicit.

\begin{definition}[Structural encoding and prepared slice]
\label{def:structural}
Let
\[
\HH_L=\HH_M\otimes\HH_A\otimes\HH_B.
\]
A structural encoding is an isometry $J:\HH_L\to\HH_{\mathrm{phys}}$.
For a fixed pair vector $\ket{\beta}_{AB}$, its prepared message
slice is the isometry
\[
j_\beta:\HH_M\to\HH_{\mathrm{phys}},\qquad
j_\beta\ket{\psi}=J(\ket{\psi}_M\ket{\beta}_{AB}).
\]
A representation of a logical operator $O$ intertwines if
\[
\widehat OJ=JO.
\]
\end{definition}

The range of $j_\beta$ generally is not invariant under the full
partner algebra. Consequently, compressing partner operators to the
message slice is not a construction of that full algebra.

\begin{lemma}[Exact encoded instrument]
\label{lem:instrument}
Suppose $J$ is unitary from $\HH_L$ onto the physical space and
$\{L_{x,a}\}$ are Kraus operators of a complete logical instrument.
Then
\[
\widehat L_{x,a}=JL_{x,a}J^\dagger
\]
define a complete physical instrument and satisfy
\[
\widehat{\Ichan}_x(J\rho J^\dagger)
   =J\Ichan_x(\rho)J^\dagger.
\]
The identity holds with every untouched reference and under any
finite adaptive composition of such instruments.
\end{lemma}

\begin{proof}
Unitary conjugation preserves
$\sum_{x,a}L_{x,a}^\dagger L_{x,a}=I$ and complete positivity.
The displayed channel identity follows by multiplication.
Tensoring all operators with $I_Q$ proves the reference version.
Induction on unnormalized conditional states proves the adaptive
version, including all failure outcomes.
\end{proof}

\begin{theorem}[Quantitative commuting-copy obstruction]
\label{thm:commuting}
Let $J:\HH_L\to\HH_{\mathrm{phys}}$ be an isometry, and let
$X,Z\in\BB(\HH_L)$ be contractions. Let $\widehat X_1$ and
$\widehat Z_2$ also be contractions, satisfying
\[
\norm{\widehat X_1J-JX}_\infty\le\eps_1,\qquad
\norm{\widehat Z_2J-JZ}_\infty\le\eps_2.
\]
Then
\begin{equation}
\norm{[X,Z]}_\infty
\le
\norm{[\widehat X_1,\widehat Z_2]J}_\infty
+2(\eps_1+\eps_2).
\label{eq:commutatorbound}
\end{equation}
In particular, if the physical representatives commute and
$X,Z$ are logical Pauli operators on a qubit factor, then
\[
\eps_1+\eps_2\ge1.
\]
Two exactly intertwining commuting representations cannot both
represent the same full noncommutative logical factor.
\end{theorem}

\begin{proof}
First,
\begin{align*}
\widehat X_1\widehat Z_2J-JXZ
&=\widehat X_1(\widehat Z_2J-JZ)
  +(\widehat X_1J-JX)Z,\\
\widehat Z_2\widehat X_1J-JZX
&=\widehat Z_2(\widehat X_1J-JX)
  +(\widehat Z_2J-JZ)X.
\end{align*}
Each line has norm at most $\eps_1+\eps_2$.
Subtract, apply the triangle inequality, and use
$\norm{J[X,Z]}_\infty=\norm{[X,Z]}_\infty$.
For logical Pauli operators,
$\norm{[X,Z]}_\infty=2$.
\end{proof}

\begin{remark}[Necessary domain qualification]
The theorem concerns a domain carrying the stated noncommutative
logical action. It must not be applied by assuming that the fixed
pair slice $j_\beta(\HH_M)$ is invariant under arbitrary partner
operations. Definition \ref{def:structural} was introduced precisely
to prevent that mistake.
\end{remark}

The theorem identifies an algebraic obstruction, not a theorem
about gravitational locality. In a physical application, one must
derive both the logical domain and the proposed physical commutators.
Dressing considerations are relevant to that separate task
\cite{DG}.

\section{One explicit unitary finite benchmark}
\label{sec:model}

We now construct all positive finite objects in one model.

\subsection{Frozen data}

Fix:
\begin{enumerate}
\item a message dimension $m\ge2$ and pair dimension $d\ge2$;
\item bases for
$\HH_M\simeq\HH_D\simeq\HH_K\simeq\mathbb C^m$
and
$\HH_A\simeq\HH_F\simeq\HH_B\simeq\mathbb C^d$;
\item positive weights $p_j$, $\sum_jp_j=1$, and
\[
\ket{\beta}_{AB}
=\sum_{j=0}^{d-1}\sqrt{p_j}\ket{j}_A\ket{j}_B;
\]
\item a fixed unitary $U$ on
$\HH_E=\HH_F\otimes\HH_D$;
\item the accessible final output
$\HH_R=\HH_E\otimes\HH_B$ and complete residual $\HH_K$.
\end{enumerate}
There are no unassigned auxiliary purifications.

The word ``pair'' denotes the prepared finite vector above.
The labels $F,B,D$ are output names, not spacetime locations.

Let
$J_0:\HH_M\otimes\HH_A\otimes\HH_B\to\HH_F\otimes\HH_D\otimes\HH_B$
be the fixed relabeling and permutation
\[
J_0(\ket{\psi}_M\ket{a}_A\ket{b}_B)
=\ket{a}_F\ket{\psi}_D\ket{b}_B,
\]
and define the structural unitary
\begin{equation}
J=(U_{FD}\otimes I_B)J_0.
\label{eq:structuralJ}
\end{equation}
Set
\begin{equation}
V\ket{\psi}
=J(\ket{\psi}_M\ket{\beta}_{AB})\otimes\ket{0}_K.
\label{eq:modelV}
\end{equation}

\subsection{Specified microscopic finite evolution}

Start with the message in $K$, the ready output $D$ in $\ket0$,
and the independently prepared pair $\ket{\beta}_{FB}$.
Apply $\mathrm{SWAP}_{KD}$, then $U_{FD}$.
The final state is exactly \eqref{eq:modelV}.

These are specified finite unitary dynamics. If durations $t_1,t_2>0$
are desired, the swap is generated by
\[
H_1=\frac{\pi}{2t_1}(I-\mathrm{SWAP}_{KD}),
\qquad e^{-it_1H_1}=\mathrm{SWAP}_{KD}.
\]
Writing
$U=\sum_\alpha e^{-i\theta_\alpha}P_\alpha$ with
$\theta_\alpha\in(-\pi,\pi]$, the second pulse is generated by
\[
H_2=\frac1{t_2}\sum_\alpha\theta_\alpha P_\alpha,
\qquad e^{-it_2H_2}=U.
\]
Both generators are bounded. No spatial support, gravitational
Hamiltonian, passive evaporation law, or black-hole mass balance is
assigned to them.

A possible output schedule releases $F$ before the message-transfer
pulse, releases $D$ after that pulse, and releases $B$ subsequently.
The final unitary $U$ can be regarded as a declared processing
operation on the collected $FD$ registers. This schedule is a finite
interface schedule, not a spacetime embedding.

\begin{theorem}[Complete channels of the benchmark]
\label{thm:modelchannels}
The map $V$ is an isometry. Let
$\tau_F=\sum_jp_j\kb{j}{j}$. The complete accessible channel,
its complement, and the $E$ restriction are
\begin{align}
\Nchan(X)
&=J\bigl(X_M\otimes\kb{\beta}{\beta}_{AB}\bigr)J^\dagger,
\label{eq:modelN}\\
\Nchan^c(X)
&=\Tr(X)\kb{0}{0}_K,
\label{eq:modelNc}\\
\Nchan_E(X)
&=U_{FD}\bigl(\tau_F\otimes X_D\bigr)U_{FD}^\dagger.
\label{eq:modelNE}
\end{align}
The decoder
\begin{equation}
\Dchan_E(Y)
=\Tr_F(U_{FD}^\dagger YU_{FD}),
\label{eq:modeldecoder}
\end{equation}
followed by the fixed relabeling $D\to M$, satisfies
$\Dchan_E\Nchan_E=\id_M$.
Thus recovery from $E$ alone is exact, with every reference.
The complete complement of the $E$ channel is also replacement:
\[
\Nchan_E^c(X)=\Tr(X)\tau_B\otimes\kb0 0_K,
\qquad
\tau_B=\sum_jp_j\kb{j}{j}.
\]
\end{theorem}

\begin{proof}
Since $J$ is unitary and $\beta$ and $\ket0$ are normalized,
$V^\dagger V=I_M$. Taking the specified partial traces proves
\eqref{eq:modelN}--\eqref{eq:modelNE} on every matrix unit, including
off-diagonal units. Applying \eqref{eq:modeldecoder} gives
$\Tr_F(\tau_F\otimes X_D)=X_D$. For the $E$ complement, trace $E$;
the unitary $U$ acts only on the traced output and disappears.
The remaining $BK$ state is the displayed fixed state times $\Tr(X)$.
Every equality is a channel identity, hence is reference stable.
\end{proof}

\begin{proposition}[No message imprint in the declared early output]
\label{prop:early}
Before the swap pulse, the channel from the message in $K$ to the
released register $F$ is
\[
\Nchan_{\mathrm{pre}}(X)=\Tr(X)\tau_F.
\]
It is replacement on all matrix units.
\end{proposition}

\begin{proof}
The initial pair preparation is independent of the message.
Tracing $K,D,B$ gives the stated channel.
\end{proof}

This proposition excludes pre-transfer imprint only for the complete
\emph{declared early interface}. It is not a theorem excluding every
possible exterior imprint in a gravitational setting.
Cheung's exterior-source constructions illustrate why that physical
accounting is a separate obligation \cite{CEm03F,CEm03G}.

\subsection{Partner, outgoing, and message algebras}

Define representations on $R$ by
\begin{align}
\pi_A(O)
&=J(I_M\otimes O_A\otimes I_B)J^\dagger,\\
\pi_B(O)
&=J(I_M\otimes I_A\otimes O_B)J^\dagger
 =I_E\otimes O_B,\\
\pi_M(O)
&=J(O_M\otimes I_A\otimes I_B)J^\dagger.
\end{align}
Set
\[
\mathcal A_{\mathrm{in}}=\pi_A(\BB(\HH_A)),\qquad
\mathcal A_{\mathrm{out}}=\pi_B(\BB(\HH_B)),\qquad
\mathcal A_{\mathrm{rad}}=\BB(\HH_E)\otimes I_B.
\]

\begin{theorem}[State-independent encoded partner]
\label{thm:partner}
The maps $\pi_A,\pi_B,\pi_M$ are unital faithful
$*$-homomorphisms and exactly intertwine the structural unitary $J$.
Their logical-factor images commute pairwise. Moreover,
\[
\mathcal A_{\mathrm{in}}\subset\mathcal A_{\mathrm{rad}}.
\]
The partner does not commute with the whole radiation algebra:
for $d\ge2$ its image is noncommutative.
\end{theorem}

\begin{proof}
Unitary conjugation preserves multiplication, adjoints, units,
and injectivity. Operators acting on different logical factors
commute before conjugation and therefore after conjugation.
Since $\pi_A(O)$ acts only on $FD$, its image lies in the radiation
algebra. Choose two noncommuting matrices on $A$; faithfulness
preserves their nonzero commutator.
\end{proof}

The partner is therefore encoded in radiation in an exact algebraic
sense. It is not an additional independent system.
Theorem \ref{thm:commuting} prevents replacing this overlap by two
commuting copies of the same full logical algebra.

\subsection{The pair-vacuum proxy instrument}

Define
\begin{equation}
\Pi_{\beta}
=J\bigl(I_M\otimes\kb{\beta}{\beta}_{AB}\bigr)J^\dagger.
\label{eq:pairproxy}
\end{equation}
The two Kraus operators $\Pi_\beta$ and $I-\Pi_\beta$ define a
complete projection instrument. We call its outcomes ``proxy pass''
and ``proxy departure.''

\begin{theorem}[One-model finite compatibility]
\label{thm:compatibility}
For the same $J,V,U,\beta$, and for every reference-message state
$\rho_{QM}$:
\begin{enumerate}
\item the emission dynamics are linear and arise from the specified
finite unitaries;
\item recovery from $E$ is exact, with complete residual decoupling;
\item the partner algebra is the state-independent algebra of
Theorem \ref{thm:partner};
\item the pair-proxy instrument passes with probability one and
leaves the emitted state unchanged;
\item every declared logical $AB$ instrument, lifted by $J$, has
exactly the same complete output as its logical counterpart,
including references and adaptive histories.
\end{enumerate}
These statements modify the independent-factor benchmark by
identifying the partner algebra with a subalgebra of radiation.
They do not establish physical horizon smoothness.
\end{theorem}

\begin{proof}
Items 1--3 follow from the specified evolution and
Theorems \ref{thm:modelchannels} and \ref{thm:partner}.
The range of $V$ lies in the range of $\Pi_\beta\otimes I_K$,
so $\Pi_\beta V=V$. This proves item 4, including the selective
residual state. Item 5 is Lemma \ref{lem:instrument}, with the
logical instrument acting as the identity on $M$.
Because $A$ is represented inside $E$, no joint state on independent
factors $A,B,E$ is asserted. The final sentence records the absence
of a spacetime and field-state construction.
\end{proof}

\begin{remark}[Explicit entropy check]
For an unreferenced message state $\rho_M$, let
$s=S(\tau_B)=H(p)$. The final state satisfies
\[
S(EB)=S(\rho_M),\qquad
S(E)=S(\rho_M)+s,\qquad
S(B|E)=-s.
\]
The logical $AB$ state is the pure pair and satisfies
$S(B|A)=-s$. Their sum is $-2s$, but this is not a violation of
Lemma \ref{lem:weak}: $A$ and $E$ are not independent factors.
There are two descriptions of overlapping observables, not two
purifiers on independent systems.
\end{remark}

\begin{remark}[Thermal calibration is not a derivation]
Choosing $p_j$ proportional to $e^{-\beta e_j}$ makes the selected
$B$ marginal a finite Gibbs state. It does not derive a Hawking
spectrum, greybody factors, dimensional emission rates, geometry,
or transfer dynamics. Those separations are explicit in
Hawk01-A, Hawk02-A, and Em03-E
\cite{CHawk01A,CHawk02A,CEm03E}.
\end{remark}

\section{Purifier distillation and intervention dependence}
\label{sec:intervention}

A compatibility proposal must specify what happens after an exterior
operation on the encoded partner. Computational difficulty alone is
not an algebraic answer. Harlow--Hayden studies resource restrictions
for particular distillation tasks \cite{HH}; we impose no such
restriction in the finite example.

Append a $d$-dimensional register $X$ in $\ket0_X$. Perform:
\begin{enumerate}
\item undo $U_{FD}$;
\item apply $\mathrm{SWAP}_{FX}$;
\item reapply $U_{FD}$.
\end{enumerate}
This is a specified exterior operation on $E$ and its ancilla.

\begin{proposition}[Distillation changes the original proxy]
\label{prop:distill}
After this operation:
\begin{enumerate}
\item the pair $\ket{\beta}$ is transferred exactly to $XB$;
\item the original decoded partner register $F$ is in $\ket0$;
\item the original proxy-pass probability is
\[
\Pr(\Pi_\beta)=p_0^2;
\]
for a maximally entangled pair it is $d^{-2}$;
\item the message remains exactly recoverable from $E$;
\item any inaccessible outputs added by this operation remain
independent of the message.
\end{enumerate}
\end{proposition}

\begin{proof}
In the decoded coordinates, the operation maps
\[
\rho_{QM,D}\otimes\kb{\beta}{\beta}_{FB}\otimes\kb0 0_X
\quad\longmapsto\quad
\rho_{QM,D}\otimes\kb0 0_F\otimes\kb{\beta}{\beta}_{XB}.
\]
This proves items 1 and 2. The original $FB$ marginal becomes
$\kb0 0_F\otimes\tau_B$. Therefore
\[
\bra{\beta}
   (\kb0 0_F\otimes\tau_B)
 \ket{\beta}=p_0^2.
\]
Undoing the reapplied $U$ still exposes the untouched message in $D$,
proving item 4. All other decoded factors depend only on the fixed
pair preparation, proving item 5.
\end{proof}

\begin{proposition}[A unitary can cause complete proxy departure]
\label{prop:unitarydeparture}
For $d=2$, take
\[
\ket{\beta}
=\sqrt{p_0}\ket{00}+\sqrt{p_1}\ket{11},
\qquad p_0,p_1>0.
\]
The exterior unitary
$U_{FD}(X_F\otimes I_D)U_{FD}^\dagger$
preserves the message channel but changes the original
proxy-pass probability to zero.
\end{proposition}

\begin{proof}
In logical coordinates the unitary acts only as $X_A$.
Its overlap with the original pair is
\[
\bra{\beta}(X_A\otimes I_B)\ket{\beta}
=\Tr(\tau_A X)=0.
\]
It does not act on the logical message, so the message decoder is
unchanged.
\end{proof}

\begin{remark}[No intervention-independent vacuum claim]
Theorem \ref{thm:compatibility} concerns the declared prepared states.
It does not assert that all radiation operations preserve their
pair statistics. Transporting the partner representation to $X$
after Proposition \ref{prop:distill} describes a changed observable
assignment. It does not prove that the original infalling instrument
remains vacuum-like, nor does it create an independent second partner.
\end{remark}

A physical proposal must calculate the actual detector coupling,
trajectory, stress-energy, and backreaction after the distillation
operation. None is determined by the finite permutation above.

\section{Diagnostics that do not determine the required channel}
\label{sec:nonid}

\begin{proposition}[Entropy and occupation underdetermination]
\label{prop:diagnostics}
Let $\HH_C\simeq\mathbb C^m$, $m\ge2$, and fix a density
$\tau_T$ with purification $\ket{\chi}_{TT'}$.
There are complete isometric interfaces whose accessible channels are
\[
\Nchan_G(X)=X_D\otimes\tau_T,\qquad
\Nchan_L(X)=\Tr(X)\kb0 0_D\otimes\tau_T.
\]
For every pure code input, their accessible entropies are equal to
$S(\tau_T)$. Their complete accessible density operators also agree
on the selected input $\ket0$. Every diagnostic on $T$ agrees for
all inputs. Nevertheless,
\[
\rec(\Nchan_G)=0,\qquad
\rec(\Nchan_L)\ge1-\frac1{m^2}.
\]
\end{proposition}

\begin{proof}
Complete isometries are
\[
V_G\ket{i}=\ket{i}_D\ket{\chi}_{TT'}\ket0_K,\qquad
V_L\ket{i}=\ket0_D\ket{\chi}_{TT'}\ket{i}_K.
\]
The accessible outputs have the stated form.
For pure inputs the $D$ state is pure in both models, so the
entropies are $S(\tau_T)$. The remaining diagnostic assertions
follow directly.

Tracing $T$ recovers the good input. The bad accessible channel is
constant. For the maximally entangled code-reference input, every
decoded bad output has the form $I_Q/m\otimes\sigma_C$ and Bell
overlap $m^{-2}$. The Bell projection gives the lower bound.
\end{proof}

If $D$ and $K$ are assigned degenerate bare energies and both
models use the same $TT'$ preparation, the comparison also agrees
on the declared energy diagnostics. This adds no spacetime or
emission-rate theorem.

This elementary separation matches the logic of Cheung's
entropy-history and spectrum non-identification records
\cite{CHawk01B,CHawk02A}. It does not challenge an independently
established reconstruction theorem.

\begin{proposition}[Record equality does not fix the residual instrument]
\label{prop:records}
Two complete one-outcome instruments can have identical record laws
for all inputs while producing different residual states.
\end{proposition}

\begin{proof}
Take the channels $\id_2$ and
$\Delta_Z(\rho)=P_0\rho P_0+P_1\rho P_1$, each with the single
reported outcome $*$. Both report $*$ with probability one.
For $\rho=\kb{+}{+}$, their residual states are
$\kb{+}{+}$ and $I_2/2$, at trace distance $1/2$.
\end{proof}

This is the same type of record--residual separation emphasized
in En03-A \cite{CEn03A}. In particular, a probability comparison
must not be promoted to equality of complete instruments.

\begin{proposition}[Finite records do not bound unobserved energy]
\label{prop:unobserved}
Fix any state $\rho_S$ and any family of measurements acting only
on $S$. Append an unobserved oscillator with Hamiltonian
$H_O=\omega N$, $\omega>0$. The states
\[
\rho_S\otimes\kb{n}{n}_O,\qquad n=0,1,2,\ldots,
\]
give identical records for every such measurement, while their
unobserved energy is $n\omega$.
\end{proposition}

\begin{proof}
The measurements act as the identity on $O$, so their probabilities
depend only on $\rho_S$. The energy follows from
$N\ket n=n\ket n$.
\end{proof}

The proposition does not identify this oscillator with a gravitational
stress tensor. It demonstrates why a finite record certificate alone
cannot supply an omitted energy or ultraviolet bound.

\section{Comparison errors and a conditional physical certificate}
\label{sec:stability}

The finite construction is useful only if any proposed physical
successor proves its comparison interfaces. This section makes the
required stability statement explicit.

\begin{lemma}[Sequential reference-stable comparison]
\label{lem:telescoping}
Let $\Lambda_t,\widetilde\Lambda_t$ be channels on identified
successive interfaces and suppose
\[
\dd(\Lambda_t,\widetilde\Lambda_t)\le e_t.
\]
Then
\[
\dd(\Lambda_n\cdots\Lambda_1,
    \widetilde\Lambda_n\cdots\widetilde\Lambda_1)
\le\min\left\{1,\sum_{t=1}^{n}e_t\right\}.
\]
The same conclusion holds for complete history channels if histories,
conditional residual states, auxiliary outputs, and failures are
retained and each one-step comparison is proved on that augmented
interface.
\end{lemma}

\begin{proof}
Telescope the channel difference by inserting and subtracting one
implemented step at a time. CPTP maps have diamond norm one, so
pre- and post-composition cannot increase the norm of a step
difference. Sum the half-diamond allowances. The proof is unchanged
on an augmented history register.
\end{proof}

The normalization agrees with the sequential stability logic in
Cheung's VII-G \cite{CVIIG}: if full-diamond allowances are used
instead, the resulting state-distance budget includes their factor
of one half.

\begin{lemma}[Isometric full-output comparison]
\label{lem:isometrystability}
For isometries $V,\widetilde V:C\to R\otimes K$,
\[
\dd(\Vchan,\widetilde\Vchan)
\le\norm{V-\widetilde V}_\infty.
\]
Consequently both accessible channels and their specified complements
are within the same half-diamond allowance.
\end{lemma}

\begin{proof}
For a purified input, also including a reference, the output vectors
differ in norm by at most $\norm{V-\widetilde V}_\infty$.
The trace distance of their pure-state densities is no greater than
that vector difference. Convexity covers mixed inputs. Partial trace
is a channel and contracts trace distance and diamond distance.
\end{proof}

\begin{lemma}[Finite POVM comparison]
\label{lem:povm}
Let $\{F_x\}$ be a physical POVM, $\{G_x\}$ a logical POVM,
and $J$ an isometry. If
\[
\norm{J^\dagger F_xJ-G_x}_\infty\le a_x
\]
for every outcome, then for every logical state $\rho$,
\[
\tv\bigl(P_F^{J\rho J^\dagger},P_G^\rho\bigr)
\le\min\left\{1,\frac12\sum_xa_x\right\}.
\]
If the prepared physical state is within trace distance $e_{\rm prep}$
of $J\rho J^\dagger$, add $e_{\rm prep}$ to the allowance.
\end{lemma}

\begin{proof}
Each outcome probability differs by at most $a_x$.
Sum their absolute differences and divide by two. Measurement
contracts preparation trace distance, so the second assertion follows
by the triangle inequality.
\end{proof}

\begin{remark}[Effect and instrument budgets remain different]
Lemma \ref{lem:povm} controls classical probabilities. A guarantee
about residual states, untouched-reference outputs, or subsequent
adaptive experiments requires a complete instrument comparison.
Neither a one-particle seminorm bound nor a list of effect
expectations automatically supplies it. This retains the boundaries
of Em06-B and Em03-D \cite{CEm06B,CEm03D}.
\end{remark}

\begin{assumption}[One-model physical bridge package]
\label{ass:physical}
For each declared time $t$ up to a stopping criterion, a candidate
physical model supplies the following data.
\begin{enumerate}[label=\textup{(H\arabic*)}]
\item A stated microscopic action or evolution, field content,
initial data, physical observable algebra, regulators,
renormalization prescription, and validity domain.
\item A complete physical isometric emission interface, with a fixed
identification of every accessible and inaccessible output used in
the comparison. Any gravitational factorization or algebra-to-factor
bridge is separately justified.
\item A reference-uniform comparison on those identified outputs,
\[
\dd(\Vchan_t^{\rm phys},\Vchan_t^0)\le e_{\rm em}(t),
\]
where $\Vchan_t^0$ is a complete comparison interface with proved
recovery error $r_0(t)$ and complete leakage $\delta_0(t)$.
The comparison map is linear and its transfer law is labeled
derived or additionally assumed.
\item For each admitted infalling experiment, complete
physical and comparison instruments on the identified output
interface satisfying
\[
\dd(\Ichan_t^{\rm phys},\Ichan_t^0)\le e_{\rm inst}(t).
\]
Their comparison record laws are within
$e_{\rm reg}(t)$ of a stated regular reference experiment with
matched excitations and detector preparation.
\item Independently proved state and geometry certificates:
appropriate field-state regularity; renormalized stress-energy and
conservation; curvature and constraint estimates; total energy
balance; and controlled backreaction through the stopping criterion.
Any use of those estimates to bound $e_{\rm reg}$ or
$e_{\rm inst}$ is proved in a declared topology.
\item A physical partner algebra and an operational assignment for
joint radiation and infalling interventions, on the same admissible
states. Reconstruction domains, commutators, state-preparation
quantifiers, and post-intervention validity domains are stated.
\end{enumerate}
\end{assumption}

The hypotheses are obligations, not results of this manuscript.
In particular, item (H5) is not derived from (H3) or from a mass ledger.

\begin{theorem}[Conditional one-model certification]
\label{thm:physical}
Under Assumption \ref{ass:physical},
\begin{align}
\rec(\Nchan_t^{\rm phys})
&\le\min\{1,r_0(t)+e_{\rm em}(t)\},
\label{eq:physicalrec}\\
\leak(\Nchan_t^{\rm phys})
&\le\min\{1,\delta_0(t)+e_{\rm em}(t)\},
\label{eq:physicalleak}\\
\tv(P_{\rm fall}^{\rm phys},P_{\rm regular})
&\le
\min\{1,e_{\rm em}(t)+e_{\rm inst}(t)+e_{\rm reg}(t)\}.
\label{eq:physicalfall}
\end{align}
The channel bounds preserve arbitrary external references.
If complete history comparisons are used, the corresponding
history-record bound holds with their sequential budgets.
For the exact benchmark of Section \ref{sec:model},
$r_0=\delta_0=0$.

These conclusions certify only the declared recovery and infalling
experiments. Physical horizon regularity additionally depends on
the independently proved content of (H5)--(H6).
\end{theorem}

\begin{proof}
Trace the full-output comparison over $K$ and apply the same
comparison decoder. Contractivity and the triangle inequality give
\eqref{eq:physicalrec}, taking an arbitrarily near-optimal comparison
decoder if necessary.

Trace instead over $R$ and compare to an arbitrarily near-optimal
replacement channel for the comparison complement. This gives
\eqref{eq:physicalleak}.

For the instrument outputs, insert
$\Ichan_t^{\rm phys}\Vchan_t^0$ between
$\Ichan_t^{\rm phys}\Vchan_t^{\rm phys}$ and
$\Ichan_t^0\Vchan_t^0$. Channel contractivity and the instrument
comparison bound the first two differences by $e_{\rm em}$ and
$e_{\rm inst}$. Reading only the classical record cannot enlarge
the distance. Add the stated regular-reference allowance
$e_{\rm reg}$.

All comparisons were in diamond norm, so the proof holds with
$\id_Q$ tensored throughout. The history assertion follows from
Lemma \ref{lem:telescoping}. The exact benchmark values follow
from Theorem \ref{thm:modelchannels}. Nothing in this argument
constructs the independent physical hypotheses (H5)--(H6).
\end{proof}

\begin{corollary}[Leakage-only recovery certificate]
Under the same output identifications, if only the comparison
leakage $\delta_0$ is available, then
\[
\rec(\Nchan_t^{\rm phys})
\le
\min\left\{
1,\sqrt{2\bigl(\delta_0(t)+e_{\rm em}(t)\bigr)}
\right\}.
\]
\end{corollary}

\begin{proof}
Combine \eqref{eq:physicalleak} with Theorem \ref{thm:approx}.
\end{proof}

\subsection{Cutoffs and stopping criteria}

A physical cutoff certificate must specify a channel on all admitted
inputs, including references. A tail probability for one preparation
is not such a certificate.

If truncation, preparation, and implementation comparisons are
proved on the same complete interface, their half-diamond allowances
can be combined by Lemma \ref{lem:telescoping}. They cannot be added
merely because they share the word ``error.''

A stopping mass $\Mcut$ limits the claimed physical domain.
A recovery bound valid at or above $\Mcut$ does not determine the
subsequent evaporation endpoint.

\section{The remaining gravitational and diary obligations}
\label{sec:physicalobligations}

The finite partner representation is not a relationally localized
gravitational observable. The pair projector is not a freely falling
detector coupling. Neither has a continuum short-distance structure.

Hadamard regularity, renormalized stress-energy, and detector response
are separate objects in curved-spacetime quantum field theory
\cite{Wald,SahlmannVerch,LoukoSatz}. A state can satisfy named finite
tests without those tests supplying a uniform bound on all unobserved
degrees of freedom. Proposition \ref{prop:unobserved} gives the
elementary energy version of this warning.

\begin{requirement}[Physical horizon certificate]
\label{req:horizon}
A claimed smooth-horizon successor must supply:
\begin{enumerate}
\item a metric/state solution, or a stated controlled approximation,
with a named crossing region and validity interval;
\item physically represented infalling observables and complete
measurement couplings;
\item the appropriate short-distance state condition across that
region;
\item stress-energy renormalization and backreaction estimates;
\item a theorem relating these physical data to the finite
comparison instruments and their error topology;
\item a post-distillation calculation for every intervention class
included in the claim.
\end{enumerate}
Passing a finite pair test alone does not discharge this requirement.
\end{requirement}

\begin{requirement}[Spacetime-localized diary certificate]
\label{req:diary}
A diary claim additionally fixes:
\begin{enumerate}
\item a physical diary subsystem $D_{\rm diary}$, initially entangled
with an untouched reference $Q$;
\item its preparation, trajectory, crossing event, interactions,
and code embedding in the same physical model;
\item the complete pre-crossing exterior interface, including retained
records and auxiliary outputs, with a bound on any diary imprint;
\item the post-crossing emission channel and its complete complement;
\item a decoder preserving the diary-reference correlations;
\item the joint infalling and recovery experiment, with detector
excitations matched in the regular reference.
\end{enumerate}
A logical register transfer without these data is not an
infalling-diary theorem.
\end{requirement}

Hayden--Preskill gives a recovery precedent under specified unitary
and mixing assumptions \cite{HP}. It does not derive those assumptions
for a new gravitational model. Likewise,
Proposition \ref{prop:early} is a statement about a finite early port,
not a theorem about a diary's causal history.

The external-source caution is particularly important:
information-sensitive radiation records can arise from strictly
exterior interactions. The Cheung Em03-F and Em03-G models
are therefore comparisons for pre-crossing accounting, not
post-horizon-transfer premises \cite{CEm03F,CEm03G}.

\section{Source-boundary audit and release}
\label{sec:release}

\subsection{What changes, and what does not}

\begin{longtable}{@{}p{0.20\textwidth}p{0.35\textwidth}p{0.38\textwidth}@{}}
\toprule
Object & Verdict here & Retained boundary\\
\midrule
\endfirsthead
\toprule
Object & Verdict here & Retained boundary\\
\midrule
\endhead
Independent-factor AMPS benchmark
&
Quantitative obstruction proved
&
The factorization and pair hypotheses remain explicit.
\\[4pt]
Complete channel and complement
&
Constructed for the finite benchmark
&
No physical inside/outside factorization is derived.
\\[4pt]
Encoded partner algebra
&
Constructed on the full structural logical space
&
Not a gravitational interior dictionary.
\\[4pt]
Pair-proxy instrument
&
Exact prepared-state statistics and residuals
&
Not Hadamard regularity or a local infalling detector.
\\[4pt]
Purifier distillation
&
Exact finite intervention calculated
&
The original proxy changes; no gravitational work or backreaction
calculation is supplied.
\\[4pt]
Message recovery
&
Exact and reference preserving
&
A message register is not a physical infalling diary.
\\[4pt]
Physical comparison theorem
&
Proved conditionally
&
Its model-specific bridge hypotheses are not constructed.
\\[4pt]
Evaporation endpoint
&
Unresolved
&
No endpoint dynamics or Page curve is asserted.
\\
\bottomrule
\end{longtable}

\subsection{Non-opening boundaries}

The supplied XIV-I excerpt distinguishes its constructed I2 internal
retarded-amplitude module from the unresolved I3 joint
characteristic map \cite{CXIVI}. Nothing here acts on the missing
grouped quantum coefficient or proves its admission.

The following documentary boundaries are retained:
\begin{itemize}
\item No full characteristic map at I3 or J2 is supplied.
\item No downstream quantum-admission exit is opened by this paper.
The label ``XV-B'' used for that exit in supplied records must not
be conflated with an assertion that a separately titled technical
XV-B manuscript contains no mathematical results.
\item XVI-N is not discharged, and XVI-O's O.1 is not closed.
These are retained source-record statuses, not newly proved
theorems about those programmes.
\item No inherited quantum coefficient, quantum Ward identity,
spacetime emergence, holographic identification, or physical
compilation is asserted.
\end{itemize}

The finite unitary products in Section \ref{sec:model} are explicitly
defined matrix operations. They are not an admission of the
unresolved characteristic coefficient and not a claim of
laboratory implementation.

\subsection{Corrections to tempting but invalid shortcuts}

The following repairs are part of the mathematical formulation:
\begin{enumerate}
\item Negative conditional entropy of a selected late packet is
stated separately; it is not inferred from global unitarity alone.
\item The structural logical space is distinguished from the fixed-pair
message slice. Full partner operators are not claimed to act
invariantly on that slice.
\item Partner reconstruction is one linear prescription on the whole
declared structural space, not unrelated state-by-state rules.
\item A reconstructed partner is not treated as an additional
commuting quantum copy.
\item Complete complements retain joint residual outputs and fixed
preparation purifications.
\item Entropy, spectrum, record, Choi, and diamond-norm certificates
are not identified.
\item The distilled-purifier calculation uses the original proxy
before considering a transported observable assignment.
\item Geometry, state regularity, detector realization, and
backreaction remain independent physical obligations.
\end{enumerate}

\begin{theorem}[Hawk03-A mathematical release]
\label{thm:release}
On the finite domains specified in this manuscript:
\begin{enumerate}
\item the independent-factor entropy benchmark and conditional
fresh-pair growth bound are proved;
\item the sharp two-purifier projection bound is proved;
\item the exact complete-complement recovery criterion and
half-diamond information--disturbance consequences are established;
\item the fixed unitary benchmark has a complete emission interface,
a state-independent encoded partner algebra, exact message recovery,
and exact prepared-state pair-proxy instrument statistics;
\item two commuting copies cannot exactly intertwine the same
noncommutative logical factor, with the quantitative bound
\eqref{eq:commutatorbound};
\item exterior purifier distillation and a proxy-destroying unitary
are calculated explicitly;
\item comparison, instrument, and complete-output errors propagate
according to Theorem \ref{thm:physical}.
\end{enumerate}
The finite mathematical ending is \textbf{Constructed}.
The independent-copy requirement is \textbf{Obstructed} under
Theorem \ref{thm:commuting}'s hypotheses.
The physical firewall and infalling-diary targets are
\textbf{Unresolved}: Requirements \ref{req:horizon} and
\ref{req:diary} are not discharged.
\end{theorem}

\begin{proof}
Items 1 and 2 are Theorem \ref{thm:amps},
Corollary \ref{cor:pairgrowth}, and
Theorem \ref{thm:twopurifiers}.
Item 3 is Theorems \ref{thm:exact} and \ref{thm:approx}.
Item 4 is Theorems \ref{thm:modelchannels},
\ref{thm:partner}, and \ref{thm:compatibility}.
Item 5 is Theorem \ref{thm:commuting}.
Item 6 is Propositions \ref{prop:distill} and
\ref{prop:unitarydeparture}.
Item 7 is the comparison lemmas and
Theorem \ref{thm:physical}.

Every constructed object is finite dimensional and has the explicit
domain stated above. No metric, continuum field-state construction,
physical crossing trajectory, or controlled gravitational
backreaction is supplied. Accordingly the physical requirements
remain unresolved.
\end{proof}

\section{Conclusion}

The finite feasibility question has an exact answer: reference-preserving
message recovery and a specified encoded pair test can coexist in one
linear unitary model when the partner is part of the radiation algebra.
There is no independent second partner. Moreover, operations on that
radiation can move the purifier or change the original pair statistics.

This answers an interface question, not the physical firewall question.
The next indispensable result is a physical bridge connecting a
constructed infalling algebra and complete detector instrument to a
complete emission channel, with state regularity and backreaction
controlled in the same model. The conditional theorem above specifies
how such bridges would be used without substituting a spectrum, entropy
curve, or finite register identity for them.

\appendix
\section{Elementary proof checks}
\label{app:checks}

The following identities provide reproducible algebraic checks.
They are not reports of executed numerical tests.

\begin{enumerate}
\item For $d=2$ and a Bell pair, the two-purifier Gram matrix has
eigenvalues $3/2$ and $1/2$. Thus two unit-success Bell tests on
independent purifiers exceed the permitted sum by $1/2$.
\item The benchmark matrix-unit complement is
\[
\Nchan^c(\kb{i}{j})=\delta_{ij}\kb0 0_K.
\]
Checking only diagonal units would omit the coherence condition.
\item On the prepared slice,
$\Pi_\beta V=V$ and
$(I-\Pi_\beta)V=0$.
\item The decoded $E$ channel is
$\tau_F\otimes X_D$, so tracing $F$ returns every matrix unit
$\kb{i}{j}_D$.
\item For a Bell pair, exterior swap distillation changes the original
proxy-pass probability from one to $1/4$, while preserving the
message-reference state.
\item For qubit Pauli matrices, $[X,Z]=-2iY$ and
$\norm{[X,Z]}_\infty=2$. The sign does not affect the commutator
obstruction.
\item The qutrit code in \eqref{eq:qutritcode} has constant
single-register channels but orthogonal joint residual supports for
distinct computational inputs.
\item Full-diamond allowances $\eps_t$ correspond to half-diamond
allowances $e_t=\eps_t/2$. Mixing the conventions changes the
stated constants.
\end{enumerate}

\section{Reference and artifact status}

The bibliography contains two kinds of sources:
\begin{enumerate}
\item external published or arXiv mathematical and physics references;
\item Cheung manuscript records, cited by identifier and version, with
the supplied excerpt labels where relevant.
\end{enumerate}
The latter are not assigned invented publication metadata.
The current document is not claimed to have been deposited, registered,
peer reviewed, compiled, or machine checked. Its author and affiliation
metadata require completion by the actual submitting author.

\begin{thebibliography}{99}

\bibitem{AMPS}
A.~Almheiri, D.~Marolf, J.~Polchinski, and J.~Sully,
\emph{Black Holes: Complementarity or Firewalls?},
arXiv:1207.3123 (2012).
\url{https://arxiv.org/abs/1207.3123}.

\bibitem{Apologia}
A.~Almheiri, D.~Marolf, J.~Polchinski, D.~Stanford, and J.~Sully,
\emph{An Apologia for Firewalls},
arXiv:1304.6483 (2013).
\url{https://arxiv.org/abs/1304.6483}.

\bibitem{Mathur}
S.~D.~Mathur,
\emph{The information paradox: A pedagogical introduction},
arXiv:0909.1038 (2009).
\url{https://arxiv.org/abs/0909.1038}.

\bibitem{Winter}
A.~Winter,
\emph{Tight uniform continuity bounds for quantum entropies:
conditional entropy, relative entropy distance and energy constraints},
arXiv:1507.07775 (2015), especially Lemma 2.
\url{https://arxiv.org/abs/1507.07775}.

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

\bibitem{KL}
E.~Knill and R.~Laflamme,
\emph{A Theory of Quantum Error-Correcting Codes},
arXiv:quant-ph/9604034 (1996).
\url{https://arxiv.org/abs/quant-ph/9604034}.

\bibitem{KSW}
D.~Kretschmann, D.~Schlingemann, and R.~F.~Werner,
\emph{The Information-Disturbance Tradeoff and the Continuity
of Stinespring's Representation},
arXiv:quant-ph/0605009 (2006);
IEEE Transactions on Information Theory \textbf{54} (2008),
1708--1717.
\url{https://arxiv.org/abs/quant-ph/0605009}.

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

\bibitem{CGL}
R.~Cleve, D.~Gottesman, and H.-K.~Lo,
\emph{How to share a quantum secret},
arXiv:quant-ph/9901025 (1999).
\url{https://arxiv.org/abs/quant-ph/9901025}.

\bibitem{ADH}
A.~Almheiri, X.~Dong, and D.~Harlow,
\emph{Bulk Locality and Quantum Error Correction in AdS/CFT},
arXiv:1411.7041 (2014).
\url{https://arxiv.org/abs/1411.7041}.

\bibitem{DHW}
X.~Dong, D.~Harlow, and A.~C.~Wall,
\emph{Reconstruction of Bulk Operators within the Entanglement Wedge
in Gauge-Gravity Duality},
arXiv:1601.05416 (2016).
\url{https://arxiv.org/abs/1601.05416}.

\bibitem{Cotler}
J.~Cotler, P.~Hayden, G.~Penington, G.~Salton, B.~Swingle,
and M.~Walter,
\emph{Entanglement Wedge Reconstruction via Universal Recovery Channels},
arXiv:1704.05839 (2017).
\url{https://arxiv.org/abs/1704.05839}.

\bibitem{Penington}
G.~Penington,
\emph{Entanglement Wedge Reconstruction and the Information Paradox},
arXiv:1905.08255 (2019).
\url{https://arxiv.org/abs/1905.08255}.

\bibitem{AMMZ}
A.~Almheiri, R.~Mahajan, J.~Maldacena, and Y.~Zhao,
\emph{The Page curve of Hawking radiation from semiclassical geometry},
arXiv:1908.10996 (2019).
\url{https://arxiv.org/abs/1908.10996}.

\bibitem{PSSY}
G.~Penington, S.~H.~Shenker, D.~Stanford, and Z.~Yang,
\emph{Replica wormholes and the black hole interior},
arXiv:1911.11977 (2019).
\url{https://arxiv.org/abs/1911.11977}.

\bibitem{DG}
W.~Donnelly and S.~B.~Giddings,
\emph{Observables, gravitational dressing, and obstructions
to locality and subsystems},
arXiv:1607.01025 (2016).
\url{https://arxiv.org/abs/1607.01025}.

\bibitem{Wald}
R.~M.~Wald,
\emph{Quantum Field Theory in Curved Spacetime},
arXiv:gr-qc/9509057 (1995).
\url{https://arxiv.org/abs/gr-qc/9509057}.

\bibitem{SahlmannVerch}
H.~Sahlmann and R.~Verch,
\emph{Microlocal spectrum condition and Hadamard form
for vector-valued quantum fields in curved spacetime},
arXiv:math-ph/0008029 (2000).
\url{https://arxiv.org/abs/math-ph/0008029}.

\bibitem{LoukoSatz}
J.~Louko and A.~Satz,
\emph{Transition rate of the Unruh-DeWitt detector in curved spacetime},
arXiv:0710.5671 (2007).
\url{https://arxiv.org/abs/0710.5671}.

\bibitem{HP}
P.~Hayden and J.~Preskill,
\emph{Black holes as mirrors: quantum information in random subsystems},
arXiv:0708.4025 (2007).
\url{https://arxiv.org/abs/0708.4025}.

\bibitem{HH}
D.~Harlow and P.~Hayden,
\emph{Quantum Computation vs. Firewalls},
arXiv:1301.4504 (2013).
\url{https://arxiv.org/abs/1301.4504}.

\bibitem{CHawk01A}
N.~Cheung,
\emph{Hawk01-A: Delayed-Query Information Retrieval from Astrophysical
Hawking Radiation: Fixed-Background Baselines, Conserving Completions,
and Observer-Access Bounds},
version 1, manuscript, n.d.
Supplied source labels include
\texttt{thm:source}, \texttt{thm:completion}, and
\texttt{ass:unruh}.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/Hawk01-A}.

\bibitem{CHawk01B}
N.~Cheung,
\emph{Hawk01-B: Complete Quantum-Code Preservation in a Sequential
Evaporation Model: Radiation Correlations, Residual Leakage,
and the Page-Curve--Recovery Gap},
version 1, manuscript, n.d.
Supplied source labels include
\texttt{thm:exact} and \texttt{thm:entropyseparation}.
Enter Calculus Companion catalogue:
\url{https://enter.grok.me/browse}.

\bibitem{CHawk02A}
N.~Cheung,
\emph{Hawk02-A: Controlled Interfaces for Black-Hole Evaporation:
Backreaction, Pair Structure, and Residual-Leakage Certification},
version 1, manuscript, n.d.
Supplied source labels include
\texttt{ass:physical}, \texttt{def:completeoutput},
\texttt{thm:pairentropy}, \texttt{thm:constant},
\texttt{thm:stability}, \texttt{cor:spectrum},
and \texttt{prop:jointresidual}.
Enter Calculus Companion catalogue:
\url{https://enter.grok.me/browse}.

\bibitem{CEm03C}
N.~Cheung,
\emph{Em03-C: Joint Hawking-Packet Correlations and Certified Tests
of Semiclassical Departures},
version 1, manuscript, n.d.
Supplied section:
``Open physical bridges: not additional release claims'';
supplied labels include
\texttt{thm:discrimination} and \texttt{thm:nolabel}.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/Em03-C}.

\bibitem{CEm03D}
N.~Cheung,
\emph{Em03-D: A Local System--Probe Realization
of a Hawking-Packet Observable},
version 1, manuscript, n.d.
Supplied source labels include
\texttt{thm:main}, \texttt{cor:departure},
and \texttt{thm:release}.
Enter Calculus Companion catalogue:
\url{https://enter.grok.me/browse}.

\bibitem{CEm03E}
N.~Cheung,
\emph{Em03-E: Microscopic Input Dependence and Certified Recovery
in a Dynamical Radiation Model},
version 1, manuscript, n.d.
Supplied source labels include
\texttt{thm:recovery}, \texttt{prop:distinguish},
and \texttt{thm:entropy}.
Enter Calculus Companion catalogue:
\url{https://enter.grok.me/browse}.

\bibitem{CEm03F}
N.~Cheung,
\emph{Em03-F},
version 1, manuscript, n.d.
Supplied source sections:
``The source-to-field channel and retained-source back-action''
and ``Quantum-transfer obstruction and exact recovery optimum'';
labels include
\texttt{thm:fieldchannel}, \texttt{thm:EB},
and \texttt{thm:counts}.
Enter Calculus Companion catalogue:
\url{https://enter.grok.me/browse}.

\bibitem{CEm03G}
N.~Cheung,
\emph{Em03-G},
version 1, manuscript, n.d.
Supplied source labels include
\texttt{thm:compacttransmission},
\texttt{prop:Fobstruction}, and \texttt{thm:lower}.
Enter Calculus Companion catalogue:
\url{https://enter.grok.me/browse}.

\bibitem{CEm03H}
N.~Cheung,
\emph{Em03-H-v8: Exact Fermionic Extraction, Quench Diagnostics,
and Stored-Packet Excitation-Channel Recovery},
version 8, manuscript, n.d.
Supplied source labels include
\texttt{thm:complementary}, \texttt{thm:specified},
and \texttt{thm:microrecovery}.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/Em03-H}.

\bibitem{CEm06B}
N.~Cheung,
\emph{Em06-B},
version 1, manuscript, n.d.
Supplied source section:
``Quantitative stability of the classical inputs'';
label \texttt{thm:input-stability}.
Enter Calculus Companion catalogue:
\url{https://enter.grok.me/browse}.

\bibitem{CEn03A}
N.~Cheung,
\emph{En03-A},
version 1, manuscript, n.d.
Supplied source sections:
``Accessible output algebras'' and ``Residual instruments'';
labels include
\texttt{def:accessible} and \texttt{thm:obstruction}.
Enter Calculus Companion catalogue:
\url{https://enter.grok.me/browse}.

\bibitem{CPsy02B}
N.~Cheung,
\emph{Psy02-B},
version 2, manuscript, n.d.
Supplied source section:
``Track A5: complete-process identification'';
label \texttt{thm:process}.
Enter Calculus Companion catalogue:
\url{https://enter.grok.me/browse}.

\bibitem{CVIIG}
N.~Cheung,
\emph{VII-G},
version 1, manuscript, n.d.
Supplied source section:
``Assisted stability and finite Choi certificates'';
label \texttt{thm:stability}.
Enter Calculus Companion catalogue:
\url{https://enter.grok.me/browse}.

\bibitem{CXIVI}
N.~Cheung,
\emph{XIV-I: Uniform internal retarded amplitudes
at the diagonal--flow-out transition},
version 1, manuscript, n.d.
Supplied source labels include
\texttt{thm:I2}, \texttt{thm:I3},
\texttt{prop:pole}, and \texttt{thm:release}.
The supplied release distinguishes the constructed internal amplitude
from the unresolved joint characteristic map.
Enter Calculus Companion catalogue:
\url{https://enter.grok.me/browse}.

\end{thebibliography}

\end{document}