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\hypersetup{
  pdftitle={Oai290-A: Typed Boundary Recovery and Modular-Band Witnesses},
  pdfsubject={Complete complements, observable certification, and limits of geometric interpretation}
}

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\title{\textbf{Oai290-A}\\[3pt]
Typed Boundary Recovery, Complete Complements,\\
and Modular-Band Witnesses\\[5pt]
\large A Nonperturbative Certification Framework\\
with Explicit Geometric and Analytic Boundaries}
\author{}
\date{Version 1 --- 12 October 2026}

\begin{document}
\maketitle

\begin{abstract}
We give a typed certification framework for a declared finite
boundary channel and a specified deterministic decoder, without
assuming that quantum corrections are perturbatively small.
The framework distinguishes recovery of an unknown quantum code,
reconstruction of a named bounded observable, and consistency with
a separately declared geometric model.

For a complete finite isometry, we prove the constant-complement
criterion, its residual-rank consequence, observable-specific
certificates, and dimension-explicit approximate recovery bounds.
If a confidence event supplies a normalized decoded Choi-overlap
interval \(F_L\le F_e\le F_U\), then an identified target channel has
half-diamond error in
\[
 \left[
 \max\{0,1-F_U-b\},
 \ \min\{1,d\sqrt{1-F_L}+b\}
 \right],
\]
where \(d\) is the logical dimension and \(b\) is a separately
justified half-diamond transfer allowance. Complete-complement and
accessible-state witnesses give lower bounds applying to every
decoder, whereas the Choi certificate concerns the specified decoder.

A separate operator-algebra module presents, with attribution and
a local proof, the bounded modular-band witness statement of the
supplied OpenAI manuscript dated 23 September 2026. We derive its
one-way consequence for bounded asymptotic defect tests. This
existential modular statement is neither a finite-sample channel
certificate nor a bulk--boundary correspondence.

Two defects in the proposed synthesis are repaired. Nonconstant
complements obstruct exact full-code recovery, but need not obstruct
a named observable. Failure of a sufficient certificate is not an
obstruction and does not imply failure of a geometric description.
The resulting decision procedure has certified, obstructed, and
inconclusive outcomes. The inherited characteristic-map obligations
remain documentary open gates: no joint map, quantum coefficient,
geometric emergence theorem, or gravitational endpoint is claimed.
\end{abstract}

\paragraph{Keywords.}
Quantum recovery; boundary access; complete complement;
observable reconstruction; diamond distance; modular spectrum;
bounded witnesses; geometric identification; source discipline.

\section{Question, contribution, and corrected logical structure}
\label{sec:introduction}

The motivating question is often phrased as ``bulk reconstruction
beyond the semiclassical limit.'' That phrase contains at least
three different mathematical tasks:
\begin{enumerate}
\item recovery of a declared quantum input or a named observable
through a declared accessible channel;
\item determination of whether an exact quantum-error-correcting
description remains valid;
\item justification of a classical geometric interpretation of
the recovered data.
\end{enumerate}
This manuscript answers specified versions of the first two tasks
and formalizes the additional evidence required for the third.
It does not construct a gravitational dictionary.

Exact quantum error correction, information--disturbance relations,
observable correction, and quantum-system certification are
established subjects
\cite{KnillLaflamme,KSW,BenyKempfKribs,KlieschRoth}.
Holographic reconstruction results additionally use their own
bulk--boundary and relative-entropy hypotheses
\cite{JLMS,DongHarlowWall}. Those hypotheses are not supplied merely
by naming a channel ``boundary'' or a logical effect ``bulk.''

Cheung's supplied manuscripts motivate the separation of complete
data, accessible outputs, residual ownership, destructive readout,
and physical correspondence
\cite{CheungI,CheungHawk04A,CheungHawk04B,
CheungEn07A,CheungEn07B,CheungEn07C,CheungXVIIA}.
The finite statements needed here are proved for the objects
defined below. No unavailable model-specific result is an implicit
premise.

\subsection{What is completed}

The written mathematical package consists of:
\begin{enumerate}
\item a complete finite-channel contract;
\item exact and approximate full-code recovery statements;
\item an observable-specific reconstruction criterion and obstruction;
\item deterministic and finite-confidence certificates for a fixed
decoder;
\item a locally proved, separately typed modular bounded-witness
module;
\item a conditional endpoint assembly with four explicitly defined
operational obligations;
\item geometric-consistency and analytic-admission ledgers that
do not convert missing constructions into no-go theorems.
\end{enumerate}

The underlying channel tools and the bounded-witness statement are
not claimed as new foundations. The contribution is their
normalization-consistent assembly, the explicit separation of claim
types, and the repairs needed to make the proposed plan mathematically
valid.

\subsection{Repairs to the proposed plan}

\begin{enumerate}
\item \textbf{Full-code recovery is not one-observable recovery.}
The constant-complement criterion applies to recovery of every
input on the full finite code. A proper observable algebra, or one
effect, can remain reconstructible when the complete complement is
nonconstant.

\item \textbf{A named observable does not itself have a diamond norm.}
Its expectation error is an operator-norm quantity.
A half-diamond statement about its readout concerns a specified
measurement channel, defined in Section~\ref{sec:observable}.

\item \textbf{The decision is not an exhaustive success/no-go dichotomy.}
A sufficient upper bound that exceeds a threshold can be
inconclusive. A missing calibration, spectral hypothesis, or
correspondence map is not a proof that recovery is impossible.

\item \textbf{Modular boundedness is a separate analytic implication.}
The bounded-witness theorem has an invariant-mean hypothesis and
an existential operator-norm conclusion. It supplies no finite
sample count, decoding channel, or gravitational identification.

\item \textbf{Recovery failure does not establish geometric failure.}
A declared geometric model must be tested against its own
identified observables and prediction laws.

\item \textbf{Residual rank is an exact constraint, not a stable
error parameter.}
Small nonzero eigenvalues can change rank without giving a uniform
positive approximate-recovery lower bound.

\item \textbf{Open analytic gates stay open.}
A bounded operator on a standard Hilbert space does not define an
unprovided product of distributions or discharge a characteristic
wavefront theorem.
\end{enumerate}

\section{Objects and the three ledgers}
\label{sec:objects}

All channel results through Section~\ref{sec:examples} are finite
dimensional. The modular module has a different ambient algebra
and is introduced separately.

Let \(C\cong\bbC^d\), \(d\ge2\), be a logical input.
Let \(A\) be the accessible output and \(B\) the complete complement.
A declared complete isometry is
\begin{equation}
 V:C\longrightarrow A\otimes B.
 \label{eq:isometry}
\end{equation}
Define
\begin{equation}
 \Nchan(X)=\Tr_B(VXV^\dagger),
 \qquad
 \Cchan(X)=\Tr_A(VXV^\dagger).
 \label{eq:channels}
\end{equation}
If the physical preparation includes a mixed ancillary state, its
purifier is included in \(B\). Discarded flags, unavailable modes,
and unreturned quantum outputs are included according to the
declared dilation.

A decoder is a completely positive trace-preserving map, abbreviated
CPTP,
\[
 \Dchan:\B(A)\longrightarrow\B(C).
\]
The returned quantum channel is \(\Echan=\Dchan\Nchan\).
A destructive validation analyzer acts afterward and is not included
in the returned quantum task.

\begin{definition}[Three-ledger contract]
\label{def:ledgers}
The following ledgers are separately declared.
\begin{enumerate}
\item \textbf{Complete-data ledger.}
The code \(C\), the preparation class, all reference systems,
and the target quantum channel, effect, or bounded observable.
If a physical or gravitational interpretation is intended, its
identification map and validity class are additional entries.

\item \textbf{Accessible ledger.}
The quantum output \(A\), collection operations, available
instruments, decoder, schedule, and resource restrictions.
A classical record is not silently promoted to a surviving quantum
output.

\item \textbf{Complement ledger.}
The complete \(B\) in the same dilation, including preparation
purifiers and every unreturned output. A selected residual marginal
is not substituted for \(B\).
\end{enumerate}
\end{definition}

The use of these ledgers follows the documentary interfaces of
Cheung's Landmark I, Hawk04-A/B, and En07-A
\cite{CheungI,CheungHawk04A,CheungHawk04B,CheungEn07A}.
Their physical models are not identified with \eqref{eq:isometry}.

\subsection{Metrics}

For states and channels, set
\[
 \dtr(\rho,\sigma)=\frac12\norm{\rho-\sigma}_1,
 \qquad
 \ddia(\Phi,\Psi)=\frac12\norm{\Phi-\Psi}_\diamond.
\]
For a returned logical channel,
\begin{equation}
 e(\Echan)=\ddia(\Echan,\id_C),
 \qquad
 \erec(\Nchan)=
 \inf_{D\ {\rm CPTP}}\ddia(D\Nchan,\id_C).
 \label{eq:errors}
\end{equation}
The first quantity concerns a specified decoder.
The second permits all deterministic CPTP decoders unless a smaller
class is explicitly declared.

Every referenced input state obeys
\[
 \dtr\bigl((\id_Q\otimes\Echan)(\omega_{QC}),\omega_{QC}\bigr)
 \le e(\Echan).
\]
Consequently every joint measurement has total-variation error at
most \(e(\Echan)\).

\subsection{Complete failures}

If the accessible process is an instrument
\(\{\mathcal I_z\}_z\), and branch \(z\) has a declared CPTP output
operation \(D_z\), the deterministic returned channel is
\begin{equation}
 \Echan(X)=\sum_zD_z\mathcal I_z(X).
 \label{eq:completebranches}
\end{equation}
It is CPTP because compositions and sums preserve complete
positivity and \(\sum_z\mathcal I_z\) is trace preserving.

Successful branches alone are not renormalized into a deterministic
certificate. Loss can be completed by a declared fixed output, or
retained as a flag in a different output type. These are different
tasks, not interchangeable conventions
\cite{CheungEn02B,CheungEn02C,CheungEm05B}.

\section{Exact full-code recovery and its obstructions}
\label{sec:exact}

\begin{theorem}[Complete-complement criterion]
\label{thm:exact}
For \eqref{eq:isometry}--\eqref{eq:channels}, the following are
equivalent:
\begin{enumerate}
\item there is a CPTP decoder \(D\) with \(D\Nchan=\id_C\);
\item there is a density operator \(\tau_B\) such that
\begin{equation}
 \Cchan(X)=\Tr(X)\tau_B
 \quad\text{for every }X\in\B(C);
 \label{eq:constantcomplement}
\end{equation}
\item in one, hence every, orthonormal code basis,
\[
 \Cchan(\kb ij)=\delta_{ij}\tau_B;
\]
\item for the Kraus family \(K_a=(I_A\otimes\bra a)V\),
\[
 K_a^\dagger K_b=c_{ab}I_C
\]
for scalar coefficients \(c_{ab}\).
\end{enumerate}
If these conditions hold and \(r=\rank\tau_B\), then
\begin{equation}
 \dim A\ge dr.
 \label{eq:rankcapacity}
\end{equation}
The recovered channel preserves every external reference.
\end{theorem}

\begin{proof}
Suppose \(D\Nchan=\id_C\), and dilate \(D\) by an isometry
\(W:A\to C\otimes G\). Then
\[
 T=(W\otimes I_B)V
\]
dilates the identity channel.
The Choi matrix of identity has rank one. Every Kraus operator
of \(T\), whose vectorizations give positive summands of that
matrix, is therefore proportional to \(I_C\).
Consequently
\[
 T\ket\psi=\ket\psi_C\otimes\ket\chi_{GB}
\]
for a single unit vector \(\chi\). Tracing \(CG\) proves
\eqref{eq:constantcomplement}.

Conversely, diagonalize
\(\tau_B=\sum_{a=1}^r\lambda_a\kb aa\), with \(\lambda_a>0\).
The isometry has no component in the kernel of \(\tau_B\).
Write
\[
 V\ket i=\sum_{a=1}^r\sqrt{\lambda_a}\ket{u_{ia}}_A\ket a_B.
\]
The complementary matrix-unit identities give
\[
 \langle u_{jb},u_{ia}\rangle=\delta_{ij}\delta_{ab}.
\]
Thus \(A\) contains \(dr\) orthonormal vectors, proving
\eqref{eq:rankcapacity}.

Let \(P\) project onto their span, and define the isometry
\(U:P A\to C\otimes\bbC^r\) by \(Uu_{ia}=\ket i\ket a\).
For any fixed code state \(\sigma\),
\[
 D(Y)=
 \Tr_{\bbC^r}(UPYP U^\dagger)
 +\Tr((I_A-P)Y)\sigma
\]
is CPTP and satisfies \(D\Nchan(X)=X\).

Matrix units span \(\B(C)\), proving the equivalence of
conditions 2 and 3. Finally,
\[
 \Cchan(X)_{ab}=\Tr(K_b^\dagger K_aX),
\]
so constancy is equivalent to condition 4.
Equality of channels remains equality after tensoring with any
identity map, proving reference preservation.
\end{proof}

This is the constant-complement form of the established
Knill--Laflamme condition \cite{KnillLaflamme}.
Comparable finite criteria are recorded in Cheung's Hawk04-A,
En07-A, Hawk03-A, Hawk01-B, VII-G, and Em05-B
\cite{CheungHawk04A,CheungEn07A,CheungHawk03A,
CheungHawk01B,CheungVIIG,CheungEm05B}.
The proof above does not import their model-specific channels.

\begin{corollary}[Strict loss of exact full-code correctability]
\label{cor:nonconstant}
If \(\Cchan\) is nonconstant, then
\(\erec(\Nchan)>0\).
Also, if
\[
 \dim A<d\,\rank\Cchan(I_C/d),
\]
exact full-code recovery is impossible.
\end{corollary}

\begin{proof}
In finite dimensions the decoder Choi matrices form a compact set:
they are positive, satisfy a fixed partial-trace identity, and have
fixed trace. Channel distance is continuous, so the infimum in
\eqref{eq:errors} is attained.
If the minimum were zero, Theorem~\ref{thm:exact} would force a
constant complement.

For the rank assertion, exact recovery would identify
\(\tau_B=\Cchan(I_C/d)\), and then
\eqref{eq:rankcapacity} would hold.
\end{proof}

The corollary concerns the chosen full code and access interface.
It does not say that every named observable has been lost.

\subsection{Classical accessible records}

\begin{proposition}[Classical readout obstruction]
\label{prop:classical}
If the only accessible output is a finite classical measurement
record, every quantum decoder produces an entanglement-breaking
channel and has full-code recovery error at least
\begin{equation}
 1-\frac1d.
 \label{eq:classicalbound}
\end{equation}
For a replacement accessible channel,
\[
 \Nchan(X)=\Tr(X)\omega_A,
\]
the optimal recovery error is
\begin{equation}
 \erec(\Nchan)=1-\frac1{d^2}.
 \label{eq:replacement}
\end{equation}
\end{proposition}

\begin{proof}
A decoded classical record has the form
\[
 \Lambda(X)=\sum_z\Tr(E_zX)\sigma_z.
\]
On a referenced input its output is a sum of positive reference
operators tensored with \(\sigma_z\), hence is separable.
A product state has overlap at most \(1/d\) with a maximally
entangled state; convexity gives the same bound for separable states.
The Bell-projector effect yields \eqref{eq:classicalbound}.

Every decoded replacement is a replacement with some state \(\tau_C\).
Its normalized Bell output is \(I_Q/d\otimes\tau_C\), whose Bell
overlap is \(d^{-2}\), yielding the lower bound
\(1-d^{-2}\).
Preparing \(I_C/d\) attains it: the completely depolarizing channel
is the uniform average of the \(d^2\) Weyl conjugations. Separating
identity and using the triangle inequality gives half-diamond
distance at most \(1-d^{-2}\).
\end{proof}

This separates destructive verification from quantum return,
as in Cheung's Landmark I and Hawk04-B
\cite{CheungI,CheungHawk04B}. Destruction of an accessible carrier
does not establish physical annihilation of the complete complement.

\section{A named observable is a different task}
\label{sec:observable}

The adjoint \(\Phi^*\) of a channel is defined by
\(\Tr(\Phi(X)Y)=\Tr(X\Phi^*(Y))\).
It is unital and completely positive.

Let \(F\in\B(C)\) be an effect, \(0\preceq F\preceq I_C\).
Its binary measurement channel is
\begin{equation}
 \Mchan_F(X)=
 \Tr(FX)\kb11+\Tr((I_C-F)X)\kb00.
 \label{eq:measurementchannel}
\end{equation}
A boundary effect \(G\in\B(A)\), \(0\preceq G\preceq I_A\),
reconstructs this readout with error
\begin{equation}
 \eta_F(G)=\norm{\Nchan^*(G)-F}_\infty.
 \label{eq:effecterror}
\end{equation}

\begin{theorem}[Effect reconstruction, including references]
\label{thm:effect}
For every boundary effect \(G\),
\begin{equation}
 \ddia(\Mchan_G\Nchan,\Mchan_F)
 =
 \norm{\Nchan^*(G)-F}_\infty.
 \label{eq:effectdiamond}
\end{equation}
Thus exact reconstruction of this binary readout is equivalent to
\(\Nchan^*(G)=F\).

For a specified quantum decoder, \(G=\Dchan^*(F)\) is an effect and
\begin{equation}
 \eta_F(\Dchan^*F)\le e(\Dchan\Nchan).
 \label{eq:effectfromcode}
\end{equation}

For any Hermitian \(O\in\B(C)\), with spectral width
\(w(O)=\lambda_{\max}(O)-\lambda_{\min}(O)\),
\begin{equation}
 \norm{\Nchan^*\Dchan^*(O)-O}_\infty
 \le w(O)e(\Dchan\Nchan).
 \label{eq:observablefromcode}
\end{equation}
\end{theorem}

\begin{proof}
Put \(H=\Nchan^*(G)-F\).
On a referenced state \(\omega_{QC}\), the measurement-channel
difference has two diagonal blocks \(K\) and \(-K\), where
\[
 K=\Tr_C((I_Q\otimes H)\omega_{QC}).
\]
For Hermitian \(L\) with \(\norm L_\infty\le1\),
\[
 |\Tr(LK)|
 =|\Tr((L\otimes H)\omega_{QC})|
 \le\norm H_\infty.
\]
Trace-norm duality gives \(\norm K_1\le\norm H_\infty\).
The output half-trace norm is \(\norm K_1\).
An unreferenced eigenstate attaining the largest absolute
eigenvalue of \(H\) gives equality. The characterization of
channel distance by referenced states then proves
\eqref{eq:effectdiamond}.

Postprocessing contraction proves \eqref{eq:effectfromcode}.
For \eqref{eq:observablefromcode}, subtract the scalar midpoint
\(cI_C\), where
\(c=(\lambda_{\max}(O)+\lambda_{\min}(O))/2\).
Both channel adjoints fix identity. For every input state,
trace duality and the output trace-distance bound give expectation
error at most
\[
 2\norm{O-cI_C}_\infty e=w(O)e.
\]
The difference operator is Hermitian, so its operator norm is the
supremum of these absolute expectation errors.
\end{proof}

Observable-algebra correction has a broader established theory
\cite{BenyKempfKribs}. We use only the explicitly proved
effect and bounded-observable statements above.

\subsection{Observable-specific obstructions}

\begin{proposition}[A target-specific lower witness]
\label{prop:effectlower}
For any two code states \(\rho,\sigma\), every boundary effect \(G\)
satisfies
\begin{equation}
 \eta_F(G)\ge
 \frac12\max\left\{
 0,\,
 |\Tr(F(\rho-\sigma))|
 -\dtr(\Nchan(\rho),\Nchan(\sigma))
 \right\}.
 \label{eq:effectlower}
\end{equation}
For full-code recovery, every decoder satisfies
\begin{equation}
 e(D\Nchan)\ge
 \frac12\max\left\{
 0,\,
 \dtr(\rho,\sigma)
 -\dtr(\Nchan(\rho),\Nchan(\sigma))
 \right\}.
 \label{eq:accesslower}
\end{equation}
\end{proposition}

\begin{proof}
Each target expectation differs from its reconstructed expectation
by at most \(\eta_F(G)\).
The difference between the two boundary-effect expectations is
bounded by the boundary trace distance. The triangle inequality
gives \eqref{eq:effectlower}.

For \eqref{eq:accesslower}, each decoded state is within
\(e(D\Nchan)\) of its ideal input. Apply the triangle inequality
and contraction under \(D\).
\end{proof}

The optimal effect-only error is the finite semidefinite
optimization
\[
 \inf\left\{
 t:\ 0\preceq G\preceq I_A,\quad
 -tI_C\preceq\Nchan^*(G)-F\preceq tI_C
 \right\}.
\]
A verified feasible \(G,t\) gives an upper certificate.
A favorable unvalidated numerical fit is not a certificate.

\subsection{Counterexample to conflating the tasks}

\begin{proposition}[Exact \(Z\)-readout with a nonconstant complement]
\label{prop:dephasing}
Let \(C,A,B\) be qubits and
\(V\ket i=\ket i_A\ket i_B\), \(i=0,1\).
Both \(\Nchan\) and \(\Cchan\) are computational-basis dephasing.
For \(F=\kb00\), boundary effect \(G=\kb00\) gives
\(\eta_F(G)=0\).
Nevertheless exact full-code recovery is impossible, and the
identity decoder has error \(1/2\).
\end{proposition}

\begin{proof}
Dephasing fixes \(F\), so \(\Nchan^*(G)=F\).
The complement distinguishes the two basis inputs and is nonconstant.
For the identity decoder,
\[
 \Nchan=\tfrac12(\id+\Ad_Z).
\]
The triangle inequality gives error at most \(1/2\).
On a Bell input the output is an equal mixture of two orthogonal
Bell states, giving distance \(1/2\) and equality.
\end{proof}

Cheung's En02-B likewise distinguishes a logical readout or
rank-one instrument from subsystem recovery
\cite[\sourceid{prop:observable}]{CheungEn02B}.
The sharper one-live-qubit streaming bound in En02-C has additional
access and scheduling hypotheses
\cite[\sourceid{thm:C3}]{CheungEn02C}; it is not a universal
statement about every one-qubit channel.

\section{Normalized finite-error certificates}
\label{sec:finite}

For logical dimension \(d\), define
\[
 \ket{\Phi_d}=\frac1{\sqrt d}\sum_{i=0}^{d-1}\ket i_Q\ket i_C,
 \quad P_d=\kb{\Phi_d}{\Phi_d},
 \quad J_\Phi=(\id_Q\otimes\Phi)(P_d).
\]
A channel's normalized Choi state has trace one.

\begin{lemma}[Choi and pure-target bounds]
\label{lem:choipure}
For channels with common input dimension \(d\),
\begin{equation}
 \dtr(J_\Phi,J_\Psi)
 \le\ddia(\Phi,\Psi)
 \le\min\{1,d\,\dtr(J_\Phi,J_\Psi)\}.
 \label{eq:choibounds}
\end{equation}
For a state \(\rho\), a rank-one projector \(P\), and
\(F=\Tr(P\rho)\),
\begin{equation}
 1-F\le\dtr(\rho,P)\le\sqrt{1-F}.
 \label{eq:purebounds}
\end{equation}
Consequently,
\begin{equation}
 1-F_e\le e(\Echan)\le\min\{1,d\sqrt{1-F_e}\},
 \qquad F_e=\Tr(P_dJ_\Echan).
 \label{eq:bellbounds}
\end{equation}
\end{lemma}

\begin{proof}
A Bell input proves the first lower bound.
For \(\Delta=\Phi-\Psi\), write unit referenced vectors as
\[
 u=(A\otimes I)\Phi_d,\quad v=(B\otimes I)\Phi_d,
 \quad \Tr(A^\dagger A)=\Tr(B^\dagger B)=d.
\]
Then
\[
 \norm{(\id\otimes\Delta)(\kb uv)}_1
 \le\norm A_\infty\norm{J_\Delta}_1\norm B_\infty
 \le d\norm{J_\Delta}_1.
\]
A singular-value decomposition extends this to all
trace-norm-one inputs and all reference dimensions.
Divide by two and use the channel-distance cap one.

The effect \(P\) proves the pure-target lower bound.
Decompose \(\rho=\sum_kp_k\kb{\psi_k}{\psi_k}\).
Pure-state trace distance is
\(\sqrt{1-|\langle\phi,\psi_k\rangle|^2}\) for
\(P=\kb\phi\phi\).
Trace-norm convexity and square-root concavity give the upper bound.
Combining the two inequalities proves \eqref{eq:bellbounds}.
\end{proof}

These standard normalization-sensitive tools agree with
\cite{FuchsVanDeGraaf,Watrous} and the supplied Oai265-B protocol
\cite{Oai265B}. In particular, the mixed-state upper inequality
in \eqref{eq:purebounds} is not generally an equality.

\subsection{Fixed decoder and transfer}

Suppose the tested returned channel is \(\Echan_0\), and the
identified target channel is \(\Echan_\star\), on the same
logical interfaces. A proved transfer allowance is
\begin{equation}
 \ddia(\Echan_\star,\Echan_0)\le b.
 \label{eq:transfer}
\end{equation}

\begin{theorem}[Typed finite-confidence recovery certificate]
\label{thm:certificate}
Suppose an event of probability at least \(1-\alpha\) jointly gives
\[
 F_L\le\Tr(P_dJ_{\Echan_0})\le F_U,
 \qquad
 \ddia(\Echan_\star,\Echan_0)\le b,
 \qquad 0\le F_L\le F_U\le1.
\]
Define
\begin{equation}
 L=\max\{0,1-F_U-b\},
 \qquad
 U=\min\{1,d\sqrt{1-F_L}+b\}.
 \label{eq:LU}
\end{equation}
On that event,
\[
 L\le e(\Echan_\star)\le U.
\]
For every target-code effect \(F\),
\[
 \ddia(\Mchan_F\Echan_\star,\Mchan_F)\le U.
\]
For every target-code Hermitian \(O\),
\[
 \norm{\Echan_\star^*(O)-O}_\infty\le w(O)U.
\]

If \(\Echan_\star=D_\star\Nchan_\star\) is the specified physical
factorization, then also
\[
 \erec(\Nchan_\star)\le
 \min\{1-d^{-2},U\}.
\]
This last bound does not establish optimality of \(D_\star\).
\end{theorem}

\begin{proof}
Lemma~\ref{lem:choipure} bounds \(e(\Echan_0)\).
The reverse triangle inequality gives
\[
 |e(\Echan_\star)-e(\Echan_0)|\le b.
\]
Theorem~\ref{thm:effect} gives the observable conclusions.
Optimal recovery is no worse than the specified decoder or the
decoder that ignores its input and prepares \(I_C/d\);
Proposition~\ref{prop:classical} evaluates the latter.
\end{proof}

The transfer allowance is in half-diamond units and is added
outside the square root. It is not multiplied by \(d\)
\cite[\sourceid{thm:certificate}]{CheungHawk04B}.

If
\[
 \ddia(\Nchan_\star,\Nchan_0)\le a,\qquad
 \ddia(D_\star,D_0)\le k
\]
on common identified interfaces, contraction gives \(b\le a+k\).
Neither comparison follows from a match of selected spectra.

\subsection{Nonperturbative meaning}

No proof above expands in a gravitational coupling or assumes that
\(a,k,b\) are small. Large allowances remain legitimate but can make
the certificate vacuous. Thus ``beyond a small-correction input''
means that the inequalities remain valid for the declared channels;
it does not mean that an unprovided quantum-gravitational channel
has been constructed.

If a physical bounded observable is represented on the code by
\(O\), and a separately proved dictionary allowance satisfies
\[
 \left|\langle O_{\rm physical}\rangle_\rho-\Tr(O\rho_C)\right|
 \le a_O
\]
for the named preparation class, the complete expectation allowance
is
\[
 a_O+w(O)U+a_{\rm readout},
\]
where \(a_{\rm readout}\) is a distinct detector allowance.
The dictionary and detector estimates are inputs to this statement.
Unbounded observables require an additional domain and
energy-constrained analysis absent from this finite contract.

\subsection{A checkable upper matrix witness}

\begin{lemma}[Feasible diamond upper witness]
\label{lem:dual}
Let \(\Delta\) be Hermiticity preserving and
\(K_\Delta=dJ_\Delta\).
If a Hermitian \(Z\) and \(\nu\ge0\) satisfy
\[
 Z\succeq K_\Delta,\qquad Z\succeq-K_\Delta,\qquad
 \Tr_{\rm out}Z\preceq\nu I_{\rm in},
\]
then
\[
 \frac12\norm{\Delta}_\diamond\le\frac{\nu}{2}.
\]
\end{lemma}

\begin{proof}
For a pure referenced input
\(\psi=(A\otimes I)\Phi_d\), \(\Tr(A^\dagger A)=d\),
its Hermitian output difference \(H\) satisfies
\(-W\preceq H\preceq W\), where
\[
 W=d^{-1}(A\otimes I)Z(A^\dagger\otimes I).
\]
The first two matrix inequalities imply \(Z\succeq0\).
Testing against the positive and negative spectral projections
of \(H\) gives
\[
 \norm H_1\le\Tr W
 \le d^{-1}\nu\Tr(A^\dagger A)=\nu.
\]
For arbitrary input operators, Hermitian block dilation preserves
input and output trace norms for a Hermiticity-preserving map.
Jordan decomposition reduces Hermitian inputs to states; convexity
reduces states to pure states. This proves the diamond bound.
\end{proof}

Semidefinite formulations are standard \cite{Watrous}.
Numerical feasibility, positivity, trace preservation, and rounding
must be validated before this lemma is used as a certificate.

\section{Complete-complement quantitative bounds}
\label{sec:leakage}

Define the complete leakage distance
\begin{equation}
 \lambda(\Cchan)=
 \inf_{\tau_B}\ddia(\Cchan,\Trep{\tau_B}),
 \qquad
 \Trep{\tau_B}(X)=\Tr(X)\tau_B.
 \label{eq:leakage}
\end{equation}
The infimum is attained because the finite state space is compact.

We use two standard finite-dimensional fidelity facts in the
sufficiency direction below:
\begin{enumerate}
\item root fidelity
\(f(\rho,\sigma)=\norm{\sqrt\rho\sqrt\sigma}_1\) satisfies
\(f(\rho,\sigma)\ge1-\dtr(\rho,\sigma)\)
\cite{FuchsVanDeGraaf};
\item purifications can be aligned, after enlarging the purifying
space, with overlap \(f(\rho,\sigma)\)
\cite{Uhlmann}.
\end{enumerate}
These are explicitly identified standard inputs, not consequences
of the supplied modular manuscript.

\begin{theorem}[Dimension-explicit recovery--leakage bounds]
\label{thm:leakage}
For the complete finite interface,
\begin{equation}
 \frac{\lambda(\Cchan)^2}{d^2}
 \le \erec(\Nchan)
 \le
 \min\left\{1-d^{-2},\,d\sqrt{2\lambda(\Cchan)}\right\}.
 \label{eq:leakagebounds}
\end{equation}
The upper bound is an information-theoretic existence statement.
It is not a resource certificate for an implemented decoder.
\end{theorem}

\begin{proof}
Fix any decoder \(D\) and dilate it to \(W:A\to C\otimes G\).
Apply \((W\otimes I_B)V\) to one half of a Bell vector, producing
a pure vector \(\Psi\) on \(Q,C,G,B\).
Let \(F_e\) be the decoded Bell overlap.

For \(F_e>0\), project \(\Psi\) onto \(\Phi_d\) in \(QC\)
and normalize the environmental projection to \(\chi\).
The pure-state distance from \(\Phi_d\otimes\chi\) is
\(\sqrt{1-F_e}\). Tracing \(C,G\) gives some \(\tau_B\) with
\[
 \dtr(J_\Cchan,I_Q/d\otimes\tau_B)
 \le\sqrt{1-F_e}.
\]
By Lemma~\ref{lem:choipure},
\[
 \lambda(\Cchan)\le d\sqrt{1-F_e}
 \le d\sqrt{e(D\Nchan)}.
\]
If \(F_e=0\), the same final inequality is trivial because
\(e(D\Nchan)=1\) and \(\lambda\le1\).
Taking the infimum over \(D\) proves the lower bound.

For the upper bound, choose a minimizing \(\tau_B\) and put
\(\delta=\lambda(\Cchan)\).
The Bell-input comparison gives
\[
 \dtr(J_\Cchan,I_Q/d\otimes\tau_B)\le\delta.
\]
The second state has a purification
\(\Phi_d{}_{QC}\otimes\chi_{BG}\).
The first has the purification obtained from \(V\), with purifying
system \(A\).
After enlarging \(G\), purification alignment supplies an isometry
\(U:A\to C\otimes G\) whose aligned overlap with this product
purification is at least \(1-\delta\).

Define \(D(Y)=\Tr_G(UYU^\dagger)\).
Its Bell overlap is at least \((1-\delta)^2\), because projection
onto \(\Phi_d\) includes the particular environmental vector
\(\chi\). Therefore
\[
 e(D\Nchan)
 \le d\sqrt{1-(1-\delta)^2}
 \le d\sqrt{2\delta}.
\]
The constant-decoder alternative gives the other cap.
\end{proof}

Sharper dimension-independent continuity theorems are available
in the information--disturbance literature \cite{KSW}.
The weaker bounds \eqref{eq:leakagebounds} suffice here and keep
the local derivation explicit.

\begin{corollary}[An explicit complementary witness]
\label{cor:complementwitness}
Suppose states \(\rho,\sigma\) and a complementary effect \(H_B\)
give
\[
 q=
 |\Tr(H_B(\Cchan(\rho)-\Cchan(\sigma)))|>0.
\]
Then every deterministic decoder obeys
\begin{equation}
 e(D\Nchan)\ge\frac{q^2}{4d^2}.
 \label{eq:complementlower}
\end{equation}
A proved lower confidence bound \(q_L\) can replace \(q\)
on its confidence event.
\end{corollary}

\begin{proof}
The complementary output trace distance is at least \(q\).
For every \(\tau_B\), its two distances from the corresponding
outputs are each at most
\(\ddia(\Cchan,\Trep{\tau_B})\). Thus \(q\le2\lambda(\Cchan)\).
Apply Theorem~\ref{thm:leakage}.
\end{proof}

A nonconstant finite complement always has some such positive
witness: if it agreed on all states, linearity and the spanning
property of states would make it a replacement channel.
This establishes exact nonrecoverability, but a small positive
\(q\) need not obstruct a prescribed positive error threshold.

\begin{proposition}[Accessible-dimension lower bound]
\label{prop:dimension}
If \(\dim A=m<d\), every decoder has
\[
 e(D\Nchan)\ge1-\frac md.
\]
\end{proposition}

\begin{proof}
A state on \(Q\otimes A\) has a pure-state decomposition with Schmidt
rank at most \(m\). Local Kraus maps on \(A\) cannot increase that
rank. Every decoded Bell-output state therefore has Schmidt number
at most \(m\).
A unit vector of Schmidt rank at most \(m\) has squared Bell
overlap at most \(m/d\), by Cauchy--Schwarz. Convexity extends this
bound to the decoded state.
The Bell-projector test proves the claim.
\end{proof}

\section{A complete-record qubit confidence construction}
\label{sec:statistics}

This section supplies one way to instantiate
Theorem~\ref{thm:certificate}; it does not report an experiment.
The construction follows the same declared-channel discipline as
Oai265-B and Cheung's bounded complete-record interfaces
\cite{Oai265B,CheungEn01C}.

For a qubit,
\[
 P_2=\frac{I+X\otimes X-Y\otimes Y+Z\otimes Z}{4}.
\]
Let \(s_x=1,s_y=-1,s_z=1\).
Every attempted trial in setting \(j\) has a predeclared score
\(Z_{ji}\in[-1,1]\), including failure records.

Conditional on training and calibration data, assume independent
scores within each setting, fixed counts \(n_j\), means \(m_j\),
and a common pre-analyzer channel \(\Echan_0\).
Set
\[
 F_{\rm obs}=\frac{1+m_x-m_y+m_z}{4}.
\]
On a calibration event \(G\), assume
\[
 |F_{\rm obs}-F_e|\le b_{\rm cal},
 \qquad
 \ddia(\Echan_\star,\Echan_0)\le b,
 \qquad \Pr(G)\ge1-\beta.
\]

\begin{lemma}[A concrete calibration conversion]
\label{lem:calibration}
Suppose the actual common pre-analyzer state \(\sigma\) obeys
\(\dtr(\sigma,J_{\Echan_0})\le\delta\).
Let the complete score observable \(B_j\) satisfy
\[
 \norm{B_j-A_j}_\infty\le\mu_j,
 \quad
 A_x=X\otimes X,\ A_y=Y\otimes Y,\ A_z=Z\otimes Z.
\]
Then a valid bias allowance is
\[
 b_{\rm cal}=\delta+\frac{\mu_x+\mu_y+\mu_z}{4}.
\]
\end{lemma}

\begin{proof}
Replacing \(B_j\) by \(A_j\) changes the fidelity combination
by at most \(\sum_j\mu_j/4\).
The ideal combination on \(\sigma\) is \(\Tr(P_2\sigma)\).
Since \(P_2\) is an effect, its difference from
\(\Tr(P_2J_{\Echan_0})\) is at most \(\delta\).
\end{proof}

For example, independently proved preparation, reference, and
joint cross-talk half-distance bounds can give
\[
 \delta\le
 \min\{1,\delta_{\rm prep}+\delta_{\rm ref}+\delta_{\rm joint}\}.
\]
This requires a fixed environment preparation and an identified
channel. A reduced preparation estimate alone does not justify an
input-correlated system--environment assignment.

\begin{theorem}[Bounded-score confidence instance]
\label{thm:statistics}
Let \(\alpha_{\rm s}>0\) and
\(\alpha_{\rm s}+\beta\le\alpha<1\). Define
\[
 \widehat c_j=\frac1{n_j}\sum_i Z_{ji},
 \qquad
 r_j=\sqrt{\frac2{n_j}\log\frac6{\alpha_{\rm s}}},
 \qquad
 R_{\rm s}=\frac{r_x+r_y+r_z}{4},
\]
and
\[
 \widehat F=\frac{1+\widehat c_x-\widehat c_y+\widehat c_z}{4}.
\]
With \(\clipone u=\max\{0,\min\{1,u\}\}\), set
\[
 F_L=\clipone{\widehat F-R_{\rm s}-b_{\rm cal}},
 \qquad
 F_U=\clipone{\widehat F+R_{\rm s}+b_{\rm cal}}.
\]
These endpoints instantiate Theorem~\ref{thm:certificate}
with coverage at least \(1-\alpha_{\rm s}-\beta\).

Declaring ``certified'' only when \(U\le\varepsilon\) gives
\[
 \Pr\{\text{certified and }e(\Echan_\star)>\varepsilon\}
 \le\alpha_{\rm s}+\beta.
\]
\end{theorem}

\begin{proof}
Hoeffding's inequality \cite{Hoeffding} gives, conditional on
the calibration record,
\[
 \Pr\{|\widehat c_j-m_j|>r_j\}
 \le2e^{-n_jr_j^2/2}=\alpha_{\rm s}/3.
\]
A union bound gives simultaneous score-mean coverage.
On that event,
\[
 |\widehat F-F_e|\le R_{\rm s}+b_{\rm cal}
\]
when \(G\) also holds. Clipping preserves coverage.
Integrating the conditional statistical bound and adding
\(\Pr(G^c)\le\beta\) proves the result.
An invalid acceptance lies in the upper-certificate failure event.
\end{proof}

Exact binomial alternatives are available for iid binary pass
records, with their own complete-record contract
\cite{Oai265B}. No new execution of that predecessor's script,
and no transfer of its simulation results to modular or
gravitational data, is claimed.

Fixed-sample formulas must not be applied after unaccounted
optional stopping. Calibration, decoder selection, missing
readout, reference disturbance, and setting-dependent drift are
separate parts of the probability model.

\section{A separately typed modular bounded-witness module}
\label{sec:modular}

The supplied manuscript \emph{Bounded recovery for modular spectral
averages}, dated 23 September 2026, is attributed to OpenAI
\cite{OpenAI29002}. The supplied dossier labels it
\sourceid{290-02} in Family 290 \cite{Dossier}.
The following theorem reproduces its bounded-witness content with
a local argument following the supplied proof. This is not a
verification of the paper's bicentralizer application, a machine
proof, or authentication of its catalogue metadata.

No theorem in Sections~\ref{sec:exact}--\ref{sec:statistics}
uses Family 290 as a premise. Conversely, no finite decoder theorem
is a premise of the modular argument.

\subsection{Modular objects and the fixed mean}

Let \(M\) be a von Neumann algebra with faithful normal state
\(\varphi\). In standard form, put
\[
 H=L^2(M),\qquad \xi=\varphi^{1/2},\qquad
 D=\log\Delta_\varphi,\qquad U_t=e^{itD}.
\]
Write \(\sigma_t=\sigma_t^\varphi\) and let \(E_D\) be the spectral
resolution of \(D\).
We use the standard modular identities
\begin{equation}
 U_t(a\xi)=\sigma_t(a)\xi,
 \qquad e^{D/2}a\xi=\xi a,
 \qquad L(a)h=ah,\quad R(a)h=ha.
 \label{eq:modularidentities}
\end{equation}
These are standard-form modular-theory inputs
\cite{Takesaki}. Inner products in this section are linear in the
first variable.

The scalar-centralizer assumption is
\begin{equation}
 M_\varphi=\{a:\sigma_t(a)=a\ \forall t\}=\bbC1.
 \label{eq:scalarcentralizer}
\end{equation}
It is not assumed for finite logical codes.

Let \(\mu_k\) be uniform probability measure on \([-k,k]\).
Fix once and for all a free ultrafilter \(\mathcal U\) on \(\bbN\).
For bounded continuous scalar \(f\), define
\[
 m_t f(t)=\lim_{k\to\mathcal U}\int f(t)\,d\mu_k(t).
\]
Boundary estimates make this mean translation invariant and
reflection invariant. It is positive and normalized.
For bounded continuous \(H\)-valued \(F\), define
\[
 \norm F_{\rm av}=
 \bigl(m_t\norm{F(t)}^2\bigr)^{1/2}.
\]
This is a seminorm by the finite-interval triangle inequality and
passage to the ultrafilter limit.

\begin{lemma}[Fourier localization and predual control]
\label{lem:modularlocal}
If \(f(u)=\int g(t)e^{itu}\,dt\), with
\(f\in C_c^\infty(\bbR)\), then
\[
 \mathcal F_f(a)=\int g(t)\sigma_t(a)\,dt\in M,
 \qquad
 \norm{\mathcal F_f(a)}\le\norm g_1\norm a,
 \qquad
 \mathcal F_f(a)\xi=f(D)a\xi.
\]
The integral is weak, defined against normal functionals.

With predual conventions
\[
 (w\varphi)(y)=\varphi(yw),\qquad
 (\varphi w)(y)=\varphi(wy),
\]
one has
\begin{equation}
 \norm{\varphi w-e^s w\varphi}_{M_*}
 \le(1+e^{s/2})
 \norm{(e^{D/2}-e^{s/2})w\xi}.
 \label{eq:predualband}
\end{equation}
Consequently uniformly bounded \(w_n\) whose vectors are supported
in bands shrinking to \(s\) satisfy
\(\norm{\varphi w_n-e^s w_n\varphi}_{M_*}\to0\).
\end{lemma}

\begin{proof}
The inverse Fourier transform of \(f\) is integrable.
The weak integral has the asserted operator-norm bound.
Applying it to \(\xi\) and using the spectral theorem proves the
vector identity.

For the second claim, put \(c=e^{s/2}\) and
\(\zeta=\xi w-cw\xi\).
Modular conjugation gives
\(J\zeta=w^*\xi-c\xi w^*\).
For \(\norm y\le1\), write
\[
 \varphi(wy)=\langle y\xi,w^*\xi\rangle.
\]
Replacing \(w^*\xi\) by \(c\xi w^*\) costs at most
\(\norm\zeta\).
Commutation of left and right actions gives
\(\langle y\xi,\xi w^*\rangle=\langle y(\xi w),\xi\rangle\).
Replacing \(\xi w\) by \(cw\xi\) costs at most
\(c\norm\zeta\). The resulting expression is
\(c^2\varphi(yw)\), proving \eqref{eq:predualband}.

On a band \([s-\delta_n,s+\delta_n]\), functional calculus bounds
the last norm by
\[
 \sup_{|u-s|\le\delta_n}|e^{u/2}-e^{s/2}|\,
 \norm{w_n\xi},
\]
which tends to zero.
\end{proof}

\begin{lemma}[Two ordinary ergodic limits]
\label{lem:ergodic}
Under \eqref{eq:scalarcentralizer}, for every \(a\in M\) and
\(\psi\in M_*\),
\begin{align}
 \int\sigma_t(a)\xi\,d\mu_k(t)&\longrightarrow\varphi(a)\xi
 &&\text{in }H, \label{eq:Hilbertmean}\\
 \int\psi\circ\sigma_{-t}\,d\mu_k(t)&\longrightarrow\psi(1)\varphi
 &&\text{in }M_* . \label{eq:predualmean}
\end{align}
\end{lemma}

\begin{proof}
The unitary mean ergodic theorem gives a Hilbert-norm limit in
\eqref{eq:Hilbertmean}.
The operator averages are bounded by \(\norm a\).
Translation by fixed \(u\) changes them in norm by at most
\(|u|\norm a/k\) for large \(k\).
Every weak-star cluster point is therefore fixed by \(\sigma\)
and hence scalar. Evaluation by \(\varphi\) identifies it as
\(\varphi(a)1\), which also identifies the vector limit.

The modular action on \(M_*\) is norm continuous on each
functional: positive functionals are standard-form vector
functionals, and general functionals are linear combinations
of positive ones. Thus the averages are Bochner integrals
and contractions.

Let \(N\) be the norm-closed span of
\(\chi\circ\sigma_u-\chi\), \(\chi\in M_*,u\in\bbR\).
Its annihilator in \(M\) is the scalar centralizer.
Hahn--Banach consequently identifies
\(N=\{\psi:\psi(1)=0\}\).
Interval boundary estimates make the averages tend to zero
on the generators of \(N\), hence on \(N\) by contractivity.
They fix \(\varphi\), proving \eqref{eq:predualmean}.
\end{proof}

\begin{lemma}[Averaged quadratic form]
\label{lem:averaging}
For \(S\in\B(H)\), define \(P(S)\) by
\[
 \langle P(S)h,k\rangle
 =m_t\langle U_t^*SU_th,k\rangle.
\]
Then \(P\) is a positive unital contraction, \(P(S)\) commutes
with every spectral projection of \(D\), and
\[
 P(L(a))=\varphi(a)I_H.
\]
In particular,
\begin{equation}
 m_t\norm{TU_th}^2=\langle P(T^*T)h,h\rangle.
 \label{eq:averagedform}
\end{equation}
No normality of \(P\) is asserted.
\end{lemma}

\begin{proof}
The bounded sesquilinear form defines \(P(S)\) with norm at most
\(\norm S\). Positivity and unitality follow from the mean.
Translation invariance gives \(U_u^*P(S)U_u=P(S)\), hence spectral
commutation.

For \(L(a)\), the conjugated operators belong to the weakly closed
algebra \(L(M)\). Their averaged weak limit belongs to \(L(M)\)
and commutes with the modular group, so is scalar.
Evaluation at \(\xi\) gives \(\varphi(a)I_H\).
Equation~\eqref{eq:averagedform} is the definition for \(T^*T\).
\end{proof}

\subsection{The attributed bounded-witness theorem}

\begin{theorem}[Bounded modular-band witnesses]
\label{thm:bounded}
Assume \eqref{eq:scalarcentralizer}.
Fix \(T\in\B(H)\), \(s\in\bbR\), and the mean above.
Let \(\delta_n>0\) tend to zero, and let unit vectors \(h_n\)
have \(D\)-support in \([s-\delta_n,s+\delta_n]\).
If
\begin{equation}
 \limsup_n m_t\norm{TU_th_n}^2>0,
 \label{eq:positiveaverage}
\end{equation}
then a subsequence has \(v_j\in M\) and constants
\(C_*<\infty,\eta>0\) such that
\[
 \norm{v_j}\le C_*,
 \qquad \norm{Tv_j\xi}\ge\eta,
 \qquad
 v_j\xi\in E_D([s-4\delta_{n_j},s+4\delta_{n_j}])H.
\]
This is the bounded-recovery statement of
\cite[Theorem 4.1]{OpenAI29002}; the proof below follows its
supplied construction.
\end{theorem}

\begin{proof}
Choose \(f_0\in C_c^\infty(\bbR)\) with
\(0\le f_0\le1\), \(f_0=1\) on \([-1,1]\), and support in
\([-2,2]\). Write \(f_0(u)=\int g_0(t)e^{itu}\,dt\).
For \(\rho>0\), set
\[
 f_\rho(u)=f_0((u-s)/\rho),\qquad
 g_\rho(t)=\rho e^{-ist}g_0(\rho t),\qquad
 C_1=\norm{g_0}_1.
\]
Then \(\norm{g_\rho}_1=C_1\), and
\[
 \mathcal F_\rho(a)\xi=f_\rho(D)a\xi,\qquad
 \norm{\mathcal F_\rho(a)}\le C_1\norm a.
\]

Pass to a subsequence on which the averages in
\eqref{eq:positiveaverage} are at least \(4\gamma\) for
some \(\gamma>0\).
Density of \(M\xi\), filtering by \(f_{\delta_n}(D)\), and
normalization give \(y_n\in M\) satisfying
\[
 \norm{y_n\xi}=1,\quad
 \supp_D(y_n\xi)\subset[s-d_n,s+d_n],\quad d_n=2\delta_n,
\]
and, by sufficiently close approximation,
\[
 m_t\norm{TU_ty_n\xi}^2\ge2\gamma.
\]
Here the averaged-seminorm difference is bounded by
\(\norm T\norm{y_n\xi-h_n}\).
Discard finitely many terms so that \(d_n\le1\).
The modular identity yields
\begin{equation}
 \varphi(y_n^*y_n)=1,\qquad
 \varphi(y_ny_n^*)=\norm{e^{D/2}y_n\xi}^2\le e^{s+1}.
 \label{eq:twomasses}
\end{equation}
There is no uniform bound on \(\norm{y_n}\).

Fix \(n\), write \(y=y_n\), and define
\[
 A_k=\int\sigma_t(y^*y)\,d\mu_k(t),\qquad
 B_k=\int\sigma_t(yy^*)\,d\mu_k(t),
\]
\[
 \psi(x)=\varphi(y^*xy),\qquad
 \psi_k=\int\psi\circ\sigma_{-t}\,d\mu_k(t).
\]
The functional \(\psi\) is a normal state.
By Lemma~\ref{lem:ergodic},
\[
 A_k\xi\to\xi,\qquad \norm{\psi_k-\varphi}_{M_*}\to0.
\]
Although \(B_k\) varies with \(k\),
\[
 \norm{B_k}\le\norm y^2,\qquad
 \varphi(B_k)\le e^{s+1}.
\]
Thus, with this \(n\) fixed, all sufficiently large \(k\) satisfy
\[
 \varphi(A_k^2)\le2,\qquad
 |\psi_k(B_k)-\varphi(B_k)|\le1,
\]
and hence
\begin{equation}
 \varphi(A_k^2)+\psi_k(B_k)\le3+e^{s+1}.
 \label{eq:continuousmoments}
\end{equation}

The positive ultrafilter average ensures that the set of \(k\)
with
\[
 \int\norm{T\sigma_t(y)\xi}^2\,d\mu_k(t)>\tfrac32\gamma
\]
belongs to \(\mathcal U\). It meets the cofinite set on which
\eqref{eq:continuousmoments} holds.

For one such \(k\), approximate the three continuous scalar
integrands
\[
 \norm{T\sigma_t(y)\xi}^2,\quad
 \varphi(z_t^*z_tz_u^*z_u),\quad
 \varphi(z_t^*z_uz_u^*z_t),
 \qquad z_t=\sigma_t(y),
\]
on the compact interval or square by one common finite partition.
Strong-star continuity on bounded sets justifies these
approximations. This gives \(z_i=\sigma_{t_i}(y)\) and weights
\(p_i\ge0\), \(\sum_i p_i=1\), with
\[
 A=\sum_i p_i z_i^*z_i,\qquad B=\sum_i p_i z_iz_i^*,
\]
such that
\begin{equation}
 \sum_i p_i\norm{Tz_i\xi}^2\ge\gamma,\qquad
 \varphi(A^2)+\sum_i p_i\varphi(z_i^*Bz_i)\le C,
 \quad C=4+e^{s+1}.
 \label{eq:finitemoments}
\end{equation}
The times and weights may depend on \(n\); \(C,\gamma\) do not.

Let \(\epsilon_i\) be independent uniform circle phases and put
\(z=\sum_i\sqrt{p_i}\epsilon_i z_i\).
Its vector remains in the band of radius \(d_n\), and
\[
 \mathbb E\norm{Tz\xi}^2\ge\gamma.
\]
Expanding the fourth moment, the two surviving phase pairings give
\[
 \mathbb E[(z^*z)^2]
 =A^2+\sum_i p_i z_i^*Bz_i
       -\sum_i p_i^2(z_i^*z_i)^2.
\]
The subtracted terms are positive, so
\(\mathbb E\varphi(|z|^4)\le C\).
This is the circle-phase fourth-moment device underlying the
Haagerup--Musat construction \cite{HaagerupMusat}.

For polar decomposition \(z=u|z|\), truncate at a fixed \(K>0\):
\[
 z^{[K]}=u\min\{|z|,K\}.
\]
Then
\[
 \norm{z^{[K]}}\le K,\qquad
 \mathbb E\norm{(z-z^{[K]})\xi}^2
 \le K^{-2}\mathbb E\varphi(|z|^4)\le C/K^2.
\]
Set \(v=\mathcal F_{d_n}(z^{[K]})\).
Every realization satisfies
\[
 \norm v\le C_1K,\qquad
 \supp_D(v\xi)\subset[s-2d_n,s+2d_n].
\]
Since \(f_{d_n}(D)\) fixes \(z\xi\),
\[
 \mathbb E\norm{v\xi-z\xi}^2\le C/K^2.
\]
Therefore
\[
 \bigl(\mathbb E\norm{Tv\xi}^2\bigr)^{1/2}
 \ge\sqrt\gamma-\norm T\sqrt C/K.
\]
Choose once and for all
\(K\ge2\norm T\sqrt{C/\gamma}\).
The resulting average is at least \(\sqrt\gamma/2\).

The phase torus is compact. The truncation is norm continuous,
because
\[
 z^{[K]}=zq_K(z^*z),\qquad
 q_K(r)=\min\{1,K/\sqrt r\},\quad q_K(0)=1.
\]
Filtering is bounded linear, so \(\norm{Tv\xi}^2\) attains
a maximum at least its expectation.
Choose one maximizing realization for each \(n\).
It gives \(C_*=C_1K\), \(\eta=\sqrt\gamma/2\), and band radius
\(2d_n=4\delta_n\).
\end{proof}

\subsection{What follows, and only in which direction}

Put \(C_T=P(T^*T)\), and define
\begin{equation}
 \Lambda_s(T)=
 \lim_{\delta\downarrow0}
 \norm{E_D([s-\delta,s+\delta])C_T
                   E_D([s-\delta,s+\delta])}_\infty.
 \label{eq:localdefect}
\end{equation}
The limit exists by monotonicity; \(C_T\) commutes with the
spectral projections. Empty bands contribute zero.

\begin{corollary}[Bounded-defect implication]
\label{cor:boundeddefect}
If \(\Lambda_s(T)>0\), there is a uniformly bounded sequence
\(v_n\in M\) with spectral support shrinking to \(s\) and
\(\inf_n\norm{Tv_n\xi}>0\).

Consequently, if
\begin{equation}
 \norm{Tw_n\xi}\longrightarrow0
 \quad\text{for every uniformly bounded sequence }w_n
 \text{ supported in bands shrinking to }s,
 \label{eq:boundedvanishing}
\end{equation}
then \(\Lambda_s(T)=0\).
\end{corollary}

\begin{proof}
For radii \(1/n\), choose unit vectors whose quadratic-form
values in \eqref{eq:averagedform} stay above a fixed positive
fraction of \(\Lambda_s(T)\). Apply
Theorem~\ref{thm:bounded}. The second claim is its contrapositive.
\end{proof}

No converse from zero invariant mean to pointwise defect vanishing
is asserted. The theorem also gives no rate for the averaging
times, which were chosen after fixing representatives whose
operator norms may grow.

\begin{corollary}[A named asymptotic defect contract]
\label{cor:intertwining}
Suppose a fixed \(T\in\B(H)\) obeys the additional contract
\[
 \norm{\varphi w_n-e^s w_n\varphi}_{M_*}\to0,\quad
 \sup_n\norm{w_n}<\infty
 \quad\Longrightarrow\quad
 \norm{Tw_n\xi}\to0.
\]
Then \(\Lambda_s(T)=0\).
\end{corollary}

\begin{proof}
Lemma~\ref{lem:modularlocal} supplies the premise for every
bounded shrinking-band sequence. Apply
Corollary~\ref{cor:boundeddefect}.
\end{proof}

A typical defect operator can be
\(T=R(a^\flat)-L(k)\), when a named compactly localized
\(a\) satisfies \(a\xi=\xi a^\flat\).
The required asymptotic identity is still a separate contract;
bounded recovery does not invent it.

\subsection{No finite-code identification}

\begin{proposition}[Scalar centralizer is not a finite-code assumption]
\label{prop:finitecentralizer}
A finite-dimensional von Neumann algebra with a faithful state and
scalar modular centralizer is \(\bbC1\).
\end{proposition}

\begin{proof}
The center is fixed by every modular automorphism. A scalar
centralizer therefore forces the finite algebra to be a matrix
factor \(M_n\).
Write the faithful state with positive density matrix \(\rho\).
Its modular group is \(\Ad_{\rho^{it}}\).
If \(n\ge2\), the commutant of \(\rho\) contains nontrivial spectral
projections, or all of \(M_n\) if \(\rho\) is scalar.
The centralizer cannot be scalar. Thus \(n=1\).
\end{proof}

A type-\(\mathrm{III}\) algebra is not replaced by a finite tensor
factor with a global density matrix. A proposed bridge to the finite
channel module must specify:
\begin{enumerate}
\item the logical algebra and its actual inclusion or encoding;
\item normal preparation, access, and recovery maps;
\item the complete dilation or a separately justified complement;
\item a reference-stable error topology and comparison allowances;
\item the relation of the modular defect \(T\) to the chosen task.
\end{enumerate}
No such bridge is supplied merely by
Theorem~\ref{thm:bounded}.
The separate existence results for ergodic states on
type-\(\mathrm{III}_1\) factors retain their own hypotheses
\cite{MarrakchiVaes}.

\section{Four same-model endpoint obligations}
\label{sec:endpoint}

The supplied summary of En07-B reports an endpoint theorem with
four same-model obligations, but does not provide their complete
verbatim clauses \cite[\sourceid{thm:endpoint}]{CheungEn07B}.
We therefore do not manufacture those clauses or claim to instantiate
that theorem. The following four obligations are an independently
defined operational refinement for this manuscript.

\begin{definition}[Endpoint certificate inputs]
\label{def:endpoint}
A conditional endpoint recovery claim supplies:
\begin{enumerate}[label=\textbf{E\arabic*.}]
\item \textbf{Identified complete interface.}
A named code, physical preparation class, accessible output,
complete complement, and endpoint or outgoing representation.
Any gravitational dictionary is separately justified.

\item \textbf{Model complementary certificate.}
A complete finite model isometry \(V_0\), with complement
\(\Cchan_0\), and a state \(\tau\) satisfying
\[
 \ddia(\Cchan_0,\Trep{\tau})\le\delta.
\]
Selected thermal modes or retained source factors do not replace
the complete complement.

\item \textbf{Controlled physical transfer.}
On common identified finite interfaces,
\[
 \ddia(\Nchan_{\rm physical},\Nchan_0)\le a,
\]
with regulator, truncation, collection, and validity errors
accounted for once.

\item \textbf{Deterministic executable return.}
A model decoder \(D_0\) with a proved recovery bound and an actual
implementation \(D_{\rm imp}\), including every failure output,
satisfy
\[
 \ddia(D_{\rm imp},D_0)\le k.
\]
Its quantum access, schedule, workspace, and output type are declared.
Information-theoretic decoder existence alone does not prove this
implementation obligation.
\end{enumerate}
\end{definition}

\begin{theorem}[Conditional outgoing recovery]
\label{thm:endpoint}
Under Definition~\ref{def:endpoint}, if \(D_0\) is chosen with the
information-theoretic bound furnished by
Theorem~\ref{thm:leakage}, then
\begin{equation}
 e(D_{\rm imp}\Nchan_{\rm physical})
 \le
 \min\{1,d\sqrt{2\delta}+a+k\}.
 \label{eq:endpointbudget}
\end{equation}
For a named effect or bounded observable, the corresponding bounds
are those of Theorem~\ref{thm:effect}.
If the inputs hold on a joint confidence event, the conclusions
hold on that event.
\end{theorem}

\begin{proof}
The complete complementary bound gives a model decoder with
error at most \(\min\{1,d\sqrt{2\delta}\}\).
Contraction and the triangle inequality give
\[
 e(D_{\rm imp}\Nchan_{\rm physical})
 \le e(D_0\Nchan_0)+a+k.
\]
Apply the channel cap one and
Theorem~\ref{thm:effect}.
\end{proof}

An approximate replacement \emph{accessible} channel has the opposite
direction. If
\(\ddia(\Nchan,\Trep{\omega})\le a\), then every decoder obeys
\[
 e(D\Nchan)\ge\max\{0,1-d^{-2}-a\}.
\]
This follows from postprocessing contraction, the reverse triangle
inequality, and \eqref{eq:replacement}.
It must not be used as a high-accuracy upper bound.

The obligations above are not constructed here for a
quantum-gravitational evaporation model.
Cheung's Hawk02-A/E likewise separate channel transfer, physical
identification, and endpoint dynamics
\cite{CheungHawk02A,CheungHawk02E}.

\begin{remark}[An asymptotic representation has a direction]
The detailed supplied Hawk02-D endpoint statement uses
\[
 \Omega_+:\mathfrak h_B\to\mathfrak h,\qquad
 J_S:\bbC^n\to\mathfrak h,\qquad
 F_{\rm out}=\Omega_+^\dagger J_S.
\]
The adjoint is required by the domains. The shortened expression
\(\Omega_+J_S\) in part of the supplied summary is not type correct
on these interfaces. This correction concerns that documentary
formula, not a new scattering proof or a gravitational endpoint
\cite[\sourceid{thm:endpoint}]{CheungHawk02D}.
\end{remark}

\section{Analytic examples and separation tests}
\label{sec:examples}

\subsection{Rank is not a stable approximate error parameter}

For \(p\in[0,1]\), let
\(A=C\oplus\bbC\ket e\) and
\(B=\bbC\ket v\oplus C\), and define the flagged-erasure isometry
\[
 V_p\ket\psi=
 \sqrt{1-p}\ket\psi_A\ket v_B
 +\sqrt p\ket e_A\ket\psi_B.
\]
Then
\[
 \Nchan_p(X)=(1-p)X\oplus p\Tr(X)\kb ee,
\]
\[
 \Cchan_p(X)=(1-p)\Tr(X)\kb vv\oplus pX.
\]

\begin{proposition}[Exact erasure benchmark]
\label{prop:erasure}
For this channel,
\[
 \erec(\Nchan_p)=p(1-d^{-2}).
\]
For every \(0<p<1\),
\[
 \rank\Cchan_p(I_C/d)=d+1,\qquad \dim A=d+1,
\]
so the exact rank-capacity condition fails, although optimal error
tends to zero as \(p\downarrow0\).
\end{proposition}

\begin{proof}
Return the transmitted code state on success and prepare \(I_C/d\)
on erasure. The complete decoded channel is
\((1-p)\id+p\Trep{I_C/d}\), with error
\(p(1-d^{-2})\).

For any decoder, the success contribution to Bell overlap is at
most \(1-p\). The erasure contribution has a product Bell output
and overlap exactly \(p/d^2\).
The Bell-projector effect therefore gives the matching lower bound.
The rank calculation is immediate from the displayed complement.
\end{proof}

This also demonstrates why a failure branch cannot be removed:
conditional success is perfect, while deterministic recovery has
positive error.

\subsection{Identical selected thermal data with opposite recovery}

\begin{proposition}[Selected marginal nonidentification]
\label{prop:thermal}
For any finite state \(\tau_S\), there are complete isometries
with the same selected accessible marginal \(\tau_S\) for every
input, but with full accessible optimal recovery errors respectively
\(0\) and \(1-d^{-2}\).
\end{proposition}

\begin{proof}
Choose a purification \(\chi_{SB_0}\) of \(\tau_S\).
Let the full accessible output be \(A=S\otimes A_1\), with
\(A_1\cong C\), and take \(B=B_0\otimes B_1\),
\(B_1\cong C\).
Define
\[
 V_{\rm good}\psi=
 \chi_{SB_0}\otimes\psi_{A_1}\otimes\ket0_{B_1},
\]
\[
 V_{\rm bad}\psi=
 \chi_{SB_0}\otimes\ket0_{A_1}\otimes\psi_{B_1}.
\]
Both are isometries and have selected \(S\) marginal \(\tau_S\).
The good channel returns the entire input in \(A_1\).
The bad full accessible channel is replacement.
Apply Theorem~\ref{thm:exact} and
Proposition~\ref{prop:classical}.
\end{proof}

The proposition does not construct a Hawking state.
It proves that even a complete selected density, and hence its
occupation data, does not determine the full code-transfer channel.
En07-C treats thermal occupation as an external input, and
Cheung's field and packet manuscripts distinguish source laws,
finite marginals, and coherent reception
\cite{CheungEn07C,CheungEm03A,CheungEm03B,CheungEm03C,
CheungEm03F,CheungEm03G}.

\subsection{Two commuting logical copies}

\begin{proposition}[Exact duplicate noncommutative representations]
\label{prop:copies}
Suppose unital representations of \(\B(C)\) on \(A\) and \(B\)
have operators \(X_A\) and \(X_B\) satisfying
\[
 (X_A\otimes I_B)V=VX,\qquad
 (I_A\otimes X_B)V=VX
\]
for every logical \(X\).
Then all logical operators commute. Hence such two commuting
exact copies do not exist for \(d\ge2\).
\end{proposition}

\begin{proof}
For logical \(X,Y\), the two physical operators commute because
they act on separate factors. Their intertwining identities give
\[
 VXY=(X_A\otimes I)(I\otimes Y_B)V
     =(I\otimes Y_B)(X_A\otimes I)V=VYX.
\]
Injectivity of \(V\) gives \(XY=YX\), contradicted by matrix units
in \(\B(C)\) when \(d\ge2\).
\end{proof}

This is the finite obstruction relevant to the encoded-partner
discussion of Hawk03-A \cite{CheungHawk03A}. It does not establish
a smooth horizon or settle a firewall question.

\subsection{Modular success and finite recovery are not identified}

At modular energy \(s=0\), the fixed vector \(\xi\) lies in every
shrinking band. With \(T=I_H\), its averaged squared norm is one
and \(v=1\) is a bounded witness.
With \(T=0\), the positive-average hypothesis fails.

Either modular choice can be placed alongside an identity finite
channel or a replacement finite channel in the absence of a
connecting map. Thus the modular witness by itself does not
distinguish those finite recovery tasks.
Failure of the positive-average hypothesis likewise does not imply
finite nonrecoverability.

\section{Geometry requires an additional certificate}
\label{sec:geometry}

A channel theorem has a mathematical input and output.
To call either one gravitational requires a further correspondence,
not a change of notation.

\begin{definition}[Declared geometric prediction model]
\label{def:geometry}
A geometric interpretation specifies:
\begin{enumerate}
\item a class \(\mathcal G\) of geometries, gauge equivalences,
and admissible parameters;
\item a named family of preparations and bounded operational
observables;
\item predictions \(p(g)\), \(g\in\mathcal G\), for their complete
record laws or named expectations;
\item a proof or explicit assumption linking those predictions
to the physical channels, including approximation, fluctuation,
and nuisance allowances;
\item the statistical acceptance or exclusion rule.
\end{enumerate}
\end{definition}

This is distinct from the three recovery ledgers.
It follows the separation of an accessible state functional from
a prescribed geometric admissibility dictionary in Cheung's
VIII-A/B/C and the explicit non-emergence boundaries of Em06-A/B
\cite{CheungVIIIA,CheungVIIIB,CheungVIIIC,
CheungEm06A,CheungEm06B}.

\begin{proposition}[Geometry-class exclusion is its own test]
\label{prop:geometrytest}
Suppose a confidence region \(\mathcal K\) covers the true complete
operational law with probability at least \(1-\alpha\).
Let \(\mathcal P_{\mathcal G}\) include all laws permitted by the
declared geometric model and its joint nuisance allowances.
Reject that model only if
\[
 \mathcal K\cap\mathcal P_{\mathcal G}=\varnothing.
\]
The probability of rejecting a true member of that declared class
is at most \(\alpha\).
Nonempty intersection establishes consistency, not uniqueness or
spacetime emergence.
\end{proposition}

\begin{proof}
If the model is true, its actual law belongs to
\(\mathcal P_{\mathcal G}\). On the confidence-coverage event it
also belongs to \(\mathcal K\), preventing rejection.
\end{proof}

There is no universal implication
\[
 \text{nonconstant complement}
 \Longrightarrow
 \text{failure of classical geometry}.
\]
Proposition~\ref{prop:dephasing} already shows that an exactly
available observable can coexist with failure of full-code
recovery. Nor does perfect quantum recovery establish classical
concentration: identity recovery of a qubit in \(\ket+\) has zero
channel error, while the variance of \(Z\) is one.

Without Definition~\ref{def:geometry}, the finite and modular
data impose no mathematical equations identifying a metric.
This is a relative nonidentification statement, not a theorem
that arbitrary geometries realize those data.

\subsection{External mathematical labels do not provide the dictionary}

The supplied dossier records Family 265 as a lattice area-law
entry, Family 260 as a Penrose-inequality entry, and Family 272 as
an entanglement/secret-key entry \cite{Dossier}.
Their source proofs and catalogue provenance are not verified here.
No theorem from those entries is used.

A ground-state spatial entanglement bound does not specify
\(\Nchan\) or \(\Cchan\). A spacetime inequality with its own
asymptotic and energy-condition hypotheses does not supply a
decoder. A protocol-specific distinction between entanglement and
secret key does not identify a gravitational mediator.
These missing interfaces are also emphasized in Cheung's
Hawk02-E audit \cite{CheungHawk02E}.

XVII-A's supplied finite-matrix sine-effect reconstruction is
a distinct declared benchmark \cite{CheungXVIIA}.
It is not identified with the present code effect, an entanglement
wedge, or a reconstructed metric.

\section{The corrected assembly and decision theorem}
\label{sec:assembly}

\begin{theorem}[Typed recovery decision]
\label{thm:assembly}
Fix the three-ledger finite contract, a named effect \(F\) or
bounded Hermitian \(O\), a specified decoder, and threshold
\(0\le\varepsilon<1\).
Suppose a jointly valid event supplies the inputs of
Theorem~\ref{thm:certificate} and any separately announced lower
witnesses.

Then the following decisions are justified on that event:
\begin{enumerate}
\item \textbf{Specified quantum recovery certified.}
If \(U\le\varepsilon\), every input and reference has returned
quantum error at most \(\varepsilon\).
Named effect and bounded-observable errors follow from
Theorem~\ref{thm:effect}.

\item \textbf{Specified decoder excluded at the threshold.}
If \(L>\varepsilon\), that implemented target channel has error
larger than \(\varepsilon\).
This does not exclude a different decoder.

\item \textbf{All deterministic decoders excluded.}
If a lower bound from
\eqref{eq:accesslower}, \eqref{eq:complementlower}, or
Proposition~\ref{prop:dimension} exceeds \(\varepsilon\),
every decoder on that same access interface has error larger than
\(\varepsilon\).

\item \textbf{Effect-only reconstruction certified or excluded.}
A proved upper bound on \(\eta_F(G)\) certifies the measurement
task at that bound. A lower bound from
\eqref{eq:effectlower} applies to every boundary effect.
Neither conclusion is automatically a full-code verdict.

\item \textbf{Exact correctability decided.}
A constant complete complement gives exact recovery and
\eqref{eq:rankcapacity}.
A nonconstant complete complement excludes exact full-code
recovery, but not every positive-error target.

\item \textbf{Otherwise inconclusive.}
If no applicable upper or lower bound separates the target,
the result is inconclusive. Unverified contract inputs do not
become certified by observing favorable data.
\end{enumerate}

If the separate scalar-centralizer modular contract is also supplied,
Theorem~\ref{thm:bounded} and
Corollaries~\ref{cor:boundeddefect}--\ref{cor:intertwining} give their
bounded-witness conclusions. This conjunction does not identify
the modular witnesses with the finite reconstructed observables.

A geometric conclusion requires Definition~\ref{def:geometry}.
An endpoint conclusion requires Definition~\ref{def:endpoint}.
\end{theorem}

\begin{proof}
Items 1 and 2 are Theorem~\ref{thm:certificate}.
Item 3 uses the lower bounds that are uniform over all decoders.
Item 4 is Theorem~\ref{thm:effect} and
Proposition~\ref{prop:effectlower}.
Item 5 is Theorem~\ref{thm:exact} and
Corollary~\ref{cor:nonconstant}.
Item 6 follows because none of those theorems supplies a converse
to an unsuccessful sufficient certificate.

The modular conclusions have their own printed hypotheses and
proofs. No connecting map is among the assembly inputs.
The geometric and endpoint definitions supply additional objects,
so they cannot be inferred by omitting those inputs.
\end{proof}

The conjunction architecture is comparable to the source-qualified
conditional certificates in Cheung's series: valid modules can
be assembled without identifying their carriers or interpretation
maps \cite{CheungHawk02A,CheungEn07B,CheungEm03G}.
Theorem~\ref{thm:assembly} does not turn that methodological
comparison into a gravitational theorem.

\section{Source alignment, release, and verification boundary}
\label{sec:release}

\subsection{Source roles}

{\small
\renewcommand{\arraystretch}{1.12}
\begin{longtable}{@{}p{0.20\linewidth}p{0.36\linewidth}
                       p{0.38\linewidth}@{}}
\toprule
Source & Supplied interface & Role here\\
\midrule
\endfirsthead
\toprule
Source & Supplied interface & Role here\\
\midrule
\endhead
\bottomrule
\endfoot

OpenAI \sourceid{290-02} &
Bounded modular-band witnesses; separate rigidity and
bicentralizer application. &
Bounded-witness statement attributed and locally argued.
The other applications are not proved or used here.\\[4pt]

OpenAI \sourceid{290-01} &
Relative-bicentralizer and expected-subalgebra package reported
in the dossier. &
Different preprint; full source not supplied.
No theorem imported.\\[4pt]

Hawk04-A; En07-A; Em05-B &
Finite exact recovery and complete complements. &
Comparisons for Theorem~\ref{thm:exact}; all finite proofs local.\\[4pt]

Hawk04-B; Landmark I &
Classical readout is not coherent recovery. &
Comparisons for Proposition~\ref{prop:classical}.\\[4pt]

En07-B &
Endpoint theorem conditional on four same-model obligations. &
Reported scope only; our four operational inputs are separately
defined, not presented as verbatim source clauses.\\[4pt]

En07-C; Em03-A--C &
Thermal/source input, finite marginals, and apparatus interfaces. &
No field state, spectrum, or detector is derived here.\\[4pt]

En02-B/C &
Logical readout versus subsystem recovery; a restricted streaming
obstruction. &
Access-class comparison only; no streaming number is transferred
to a different channel.\\[4pt]

Hawk03-A &
Encoded partner and commuting-copy obstruction. &
Finite algebra comparison; no horizon identification.\\[4pt]

XVII-A &
One finite-matrix effect from a declared boundary region. &
Distinct benchmark, not the present gravitational dictionary.\\[4pt]

Hawk02-A/D/E &
Transfer, asymptotic endpoint, and gravitational-source audits. &
Conditional-interface comparison; wave-operator direction checked
at the documentary level.\\[4pt]

XIV-H/I/J &
Internal transition, internal amplitude, and partial reference;
full joint map unresolved. &
Appendix~\ref{app:analytic} preserves the stated open gates.
No quantum admission.\\[4pt]

Families 265/260/272 &
External documentary labels. &
No proof or catalogue/formal-verification status authenticated.
No premise.\\
\end{longtable}
}

The dossier says that a Family 290 Lean note names
\sourceid{290-02}, not \sourceid{290-01}; it separately names
\sourceid{260-13} and \sourceid{272-01} for other entries
\cite{Dossier}. These are reported identifiers.
No Lean source, manifest, compilation, or coverage of the cited
theorems is established by this manuscript.
A family identifier is not a proof-assistant verification.

\subsection{Exact release}

The manuscript releases the written finite proofs and the
attributed local modular argument. It supplies a coherent conditional
certificate, not an instantiated microscopic gravitational model.

No Page curve, island, quantum extremal surface, replica wormhole,
bulk metric, smooth horizon, gravitationally generated entanglement,
or derivation of Einstein dynamics is constructed.
The supplied semiclassical spectrum is treated as an external
model input, not a complete channel or a constant-complement verdict.

The documentary inherited status is unchanged:
\[
 \boxed{
 \begin{gathered}
 \sourceid{O.1}:\ \sourceid{NOT_CLOSED},\\
 \text{XV-B downstream admission exit: not opened},\\
 Q_HR_1^{(1)}\in\mathscr E_H:\ \text{not asserted}.
 \end{gathered}}
\]
XV-B's existence as a technical continuation is not denied.
Its downstream admission requirements are not discharged.

No successful pdfLaTeX build, numerical execution, laboratory
calibration, proof-assistant check, independent verification of
the supplied manuscripts, referee approval, or acceptance is
asserted.

\appendix

\section{Why modular boundedness does not close characteristic admission}
\label{app:analytic}

The later portion of the supplied plan concerns a distinct
distributional construction. Its stages cannot be completed by
renaming \(T\), \(v\xi\), or a finite decoder.
We record the precise generic compatibility statements that can be
proved without inventing the absent joint map.

\subsection{Descent through reconstructed presentations}

Let \(\mathscr P\) be a linear presentation space,
\[
 \Rec:\mathscr P\longrightarrow\mathcal D'(W),
 \qquad \mathscr K=\im\Rec.
\]
For a linear candidate
\(\widetilde{\mathsf M}:\mathscr P\to\mathcal D'(W)\), a map on
actual reconstructed kernels must be independent of presentation.

\begin{proposition}[Reconstruction descent]
\label{prop:descent}
There is a unique linear map
\(\mathsf M:\mathscr K\to\mathcal D'(W)\) satisfying
\(\widetilde{\mathsf M}=\mathsf M\Rec\) if and only if
\[
 \ker\Rec\subseteq\ker\widetilde{\mathsf M}.
\]
Agreement after restriction to an open \(U\subset W\) proves this
criterion only after that restriction.
\end{proposition}

\begin{proof}
A factorization annihilates every null presentation.
Conversely, if two presentations have the same reconstruction,
their difference is in \(\ker\Rec\); the criterion makes their
images equal. This defines the unique induced map.

Restriction to \(U\) does not detect distributions supported
in \(W\setminus U\). Thus restricted annihilation need not imply
global annihilation.
\end{proof}

This is the compatibility condition explicitly recorded in
XIV-J \cite[\sourceid{prop:descent}]{CheungXIVJ}.

\subsection{The retarded pole is an existing line, not a new product}

For \(\Box=\partial_t^2-\Delta_{\mathbf x}\), the fixed retarded
kernel
\[
 R(t,\mathbf x)=\frac1{2\pi}\Theta(t)\delta(t^2-|\mathbf x|^2)
\]
has spatial Fourier transform
\(\Theta(t)\sin(\rho t)/\rho\), \(\rho=|\xi|\).
With transform \(e^{-i(t\omega+\mathbf x\cdot\xi)}\), positive
damping gives
\[
 \int_0^\infty e^{-(\epsilon+i\omega)t}
        \frac{\sin(\rho t)}{\rho}\,dt
 =-\frac1{(\omega-i\epsilon)^2-\rho^2}.
\]
Thus, on \(\rho>0\),
\[
 M_R=-\frac1{2\rho}
 \left[(\omega-\rho-i0)^{-1}
       -(\omega+\rho-i0)^{-1}\right].
\]
Near sheet \(s\in\{1,-1\}\),
\[
 M_R=
 -\frac{s}{2\rho}(\omega-s\rho-i0)^{-1}
 +\frac{s}{2\rho(\omega+s\rho)}.
\]
The pole boundary value is fixed by the original retarded kernel.
This computation agrees with the supplied XIV-I internal
normalization \cite[\sourceid{prop:pole}]{CheungXIVI}.
It computes one existing distribution, not its product with
unprovided quantum cross-lines.

If an internal reconstruction convention is
\[
 \Rec\bigl((i\theta)^jM_R(\kappa)\bigr)
 =\delta^{(j)}(q)R(r),
\]
then
\begin{equation}
 \Rec\bigl((q\,i\theta+1)M_R(\kappa)\bigr)=0.
 \label{eq:nullpresentation}
\end{equation}
Indeed \(q\delta'(q)=-\delta(q)\).
The two summands are separately nonzero on a nontrivial chart.
Therefore polynomial degree of a presentation is not automatically
an intrinsic symbol filtration on the reconstructed object
\cite[\sourceid{lem:null}]{CheungXIVJ}.

The identity does not invalidate internal estimates.
It does require any later coefficient-level construction to
respect reconstruction descent.

\subsection{A conditional normal-jet extension is not a prescription}

Let \(U=X\times\bbR^c\), with normal coordinate \(q\), and let
\(S=X\times\{0\}\). Fix a closed \(F\subset X\).
For coefficient distributions \(a_\alpha\in\mathcal D'(X)\), define
\[
 \langle E_\alpha a_\alpha,\psi\rangle
 =
 (-1)^{|\alpha|}
 \langle a_\alpha,\partial_q^\alpha\psi(x,0)\rangle.
\]
A supported finite-normal-order distribution, in this appendix,
means a distribution of the explicit form
\[
 t=\sum_{|\alpha|\le k}E_\alpha a_\alpha.
\]
This definition includes the support condition; normal order alone
must not be used to hide an arbitrary off-\(S\) distribution.

\begin{proposition}[Conditional coefficient extension]
\label{prop:normalextension}
Suppose
\[
 t^\circ=\sum_{|\alpha|\le k}E_\alpha a_\alpha^\circ
\]
is given away from \(F\times\{0\}\), with
\(a_\alpha^\circ\in\mathcal D'(X\setminus F)\).
A supported finite-normal-order extension \(t\) exists if and only if
every \(a_\alpha^\circ\) has an extension
\(a_\alpha\in\mathcal D'(X)\).
The coefficient extensions are unique for a given \(t\), but are
not necessarily unique from \(t^\circ\).
\end{proposition}

\begin{proof}
Choose a smooth cutoff \(\chi(q)=1\) near zero.
Testing against
\(\chi(q)q^\beta\phi(x)/\beta!\) isolates \(a_\beta\), up to
the known sign, since its normal derivative at zero is
\(\delta_{\alpha\beta}\phi(x)\).
Thus a supported extension \(t\) uniquely determines coefficient
extensions.

Conversely, given the coefficient extensions, their finite
inserted sum defines \(t\) and has the required restriction.
Two coefficient extensions can differ by distributions supported
on \(F\), producing different contact terms.
\end{proof}

This is the generic content of the conditional supported interface
in XIV-J \cite[\sourceid{thm:conditionalextension}]{CheungXIVJ}.
It asserts neither existence of the needed coefficient extensions
nor their characteristic normalization, growth, or wavefront
bounds.

\subsection{The inherited six-gate ledger}

The following is a source-qualified status ledger, not a new proof
of the inherited field-theory construction. Its definitions,
tensor coefficients, gauge group, cutoffs, and target spaces remain
those of the cited supplied excerpts
\cite{CheungXIVH,CheungXIVI,CheungXIVJ}.

{\small
\renewcommand{\arraystretch}{1.15}
\begin{longtable}{@{}p{0.08\linewidth}p{0.36\linewidth}
                       p{0.50\linewidth}@{}}
\toprule
Gate & Required object & Status retained here\\
\midrule
\endfirsthead
\toprule
Gate & Required object & Status retained here\\
\midrule
\endhead
\bottomrule
\endfoot

1 &
Full joint characteristic map on tests meeting the locus. &
XIV-I reports an internal amplitude construction at I2, with the
joint map unresolved at I3. XIV-J's full-map stop remains J2.
A supported-sector reference for \(K_6+K_8+K_9\) away from the
locus is not the complete six-branch map.\\[4pt]

2 &
Joint section and complete remainder for that map. &
Not supplied for the absent full map. Internal reconstruction
and a finite internal remainder do not supply it.\\[4pt]

3 &
Uniform transverse estimates. &
Not supplied for the full joint section. Frequency homogeneity
of one retarded pole is not a transverse degree of the complete
graph.\\[4pt]

4 &
Contact compatibility and singular-spectator gluing. &
Not supplied for that section. A finite contact fit or an
off-locus coefficient interface is not this compatibility
theorem.\\[4pt]

5 &
Output wavefront theorem after the actual operations and proper
pushforward. &
Not supplied for the absent complete joint distribution.
An internal wavefront bound does not define a missing composition.\\[4pt]

6 &
The stated quantum-output admission. &
\(Q_HR_1^{(1)}\in\mathscr E_H\) is not asserted.
The downstream XV-B admission exit is not opened.\\
\end{longtable}
}

H\"ormander's product and pullback calculus has hypotheses on the
actual distributions and their covectors \cite{Hormander}.
Theorem~\ref{thm:bounded} supplies no estimate of those covectors,
no distributional multiplication map, and no renormalization
normalization. The six gates therefore do not follow from modular
boundedness.

The historical H2 stop of XIV-H and the later I2 construction of
XIV-I are not conflated. The latter advances an internal
representation gate, while the full joint map remains absent.
The supplied XIV-J no-admission conclusion is retained at precisely
that scope \cite[\sourceid{cor:noadmission}]{CheungXIVJ}.

The separate XVI prepared-source and O.1 programmes are not
discharged by this appendix
\cite{CheungXVIA,CheungXVIN}.
Missing constructions are not thereby proved nonexistent.

\section*{Conclusion}

The corrected mathematical statement is not bulk reconstruction:
\[
 \boxed{
 \begin{gathered}
 \text{declared finite channel and complete complement}\\
 +\ \text{specified deterministic decoder}\\
 +\ \text{proved calibration and transfer inputs}\\
 \Longrightarrow\\
 L\le e(\Echan_\star)\le U,\qquad
 \eta_F\le U,\qquad
 \Delta_O\le w(O)U .
 \end{gathered}}
\]
Exact full-code recovery is equivalent to a constant complete
complement. One observable can remain exactly reconstructible
outside that condition. Complementary and accessible witnesses
can prove obstructions; an unsuccessful sufficient certificate
can instead be inconclusive.

The modular theorem independently converts positive averaged
spectral defects into bounded operator witnesses under a
scalar-centralizer hypothesis. It supplies no decoder, no
finite-sample confidence guarantee, and no gravitational dictionary.

Geometry, endpoint dynamics, and characteristic quantum admission
each require their own identified objects and estimates.
Their missing interfaces remain missing rather than being filled
by analogy.

\begin{thebibliography}{99}

\bibitem{KnillLaflamme}
E.~Knill and R.~Laflamme.
Theory of quantum error-correcting codes.
\emph{Physical Review A} \textbf{55} (1997), 900--911.
doi:10.1103/PhysRevA.55.900.
arXiv:quant-ph/9604034.

\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.
arXiv:quant-ph/0605009.

\bibitem{BenyKempfKribs}
C.~B\'eny, A.~Kempf, and D.~W.~Kribs.
Quantum error correction of observables.
\emph{Physical Review A} \textbf{76} (2007), 042303.
arXiv:0705.1574.

\bibitem{KlieschRoth}
M.~Kliesch and I.~Roth.
Theory of quantum system certification---a tutorial.
\emph{PRX Quantum} \textbf{2} (2021), 010201.
arXiv:2010.05925.

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

\bibitem{Uhlmann}
A.~Uhlmann.
The ``transition probability'' in the state space of a
\(^{*}\)-algebra.
\emph{Reports on Mathematical Physics}
\textbf{9} (1976), 273--279.
doi:10.1016/0034-4877(76)90060-4.

\bibitem{Watrous}
J.~Watrous.
Simpler semidefinite programs for completely bounded norms.
arXiv:1207.5726, 2012.

\bibitem{Hoeffding}
W.~Hoeffding.
Probability inequalities for sums of bounded random variables.
\emph{Journal of the American Statistical Association}
\textbf{58} (1963), 13--30.
doi:10.1080/01621459.1963.10500830.

\bibitem{Takesaki}
M.~Takesaki.
\emph{Theory of Operator Algebras II}.
Encyclopaedia of Mathematical Sciences, vol.~125.
Springer, 2003.
doi:10.1007/978-3-662-10451-4.

\bibitem{HaagerupMusat}
U.~Haagerup and M.~Musat.
The Effros--Ruan conjecture for bilinear forms on \(C^*\)-algebras.
\emph{Inventiones Mathematicae}
\textbf{174} (2008), 139--163.
arXiv:0711.1851.

\bibitem{MarrakchiVaes}
A.~Marrakchi and S.~Vaes.
\emph{Ergodic states on type \(\mathrm{III}_1\) factors and ergodic
actions}.
arXiv:2305.14217.
Cited for the separate ergodic-state context, not as a gravitational
or finite-decoder theorem.

\bibitem{JLMS}
D.~L.~Jafferis, A.~Lewkowycz, J.~Maldacena, and S.~J.~Suh.
Relative entropy equals bulk relative entropy.
\emph{Journal of High Energy Physics}
2016, issue 6, article 004.
arXiv:1512.06431.

\bibitem{DongHarlowWall}
X.~Dong, D.~Harlow, and A.~C.~Wall.
Reconstruction of bulk operators within the entanglement wedge
in gauge-gravity duality.
\emph{Physical Review Letters}
\textbf{117} (2016), 021601.
arXiv:1601.05416.

\bibitem{Hormander}
L.~H\"ormander.
\emph{The Analysis of Linear Partial Differential Operators I:
Distribution Theory and Fourier Analysis}.
Second edition, Springer, 1990.
Especially the wavefront, product, and pullback calculus.

\bibitem{OpenAI29002}
OpenAI, as attributed in the supplied document.
\emph{Bounded recovery for modular spectral averages}.
23 September 2026.
User-supplied document
\sourceid{Bounded-recovery-for-modular-spectral-averages-September-23-2026.pdf}.
The dossier identifies it as Family 290, preprint
\sourceid{290-02}. The bounded-witness statement is Theorem 4.1.
No independent verification of the complete paper, catalogue
metadata, or formalization is asserted here.

\bibitem{Dossier}
\emph{User-supplied development plan and comparative source dossier}.
Undated material supplied with this manuscript request.
Includes corpus label \sourceid{6f617598},
retrieval label \sourceid{retrieval-3}, Family 290 identifiers,
other catalogue summaries, and selected Cheung excerpts.
Retrieval and documentary status are not proof verification.
The full source of \sourceid{290-01} and formalization manifests
were not supplied.

\bibitem{Oai265B}
\emph{Oai265-B: Reference-Preserving Quantum-Memory Certification
from Measured Correlations}.
Supplied companion source and version-2 revision,
12 October 2026.
Finite-confidence and complete-record comparison.
No new execution is imported into Oai290-A.

\bibitem{CheungI}
N.~Cheung.
\emph{Landmark I}, version 2.
Undated manuscript; supplied \sourceid{I.tex} excerpts.
Finite instruments, destructive access, records, and decoder
boundaries.
Enter Calculus Companion.
Cited at the supplied-source level.

\bibitem{CheungHawk04A}
N.~Cheung.
\emph{Hawk04-A}, version 1.
Undated manuscript; supplied statement summaries.
Especially \sourceid{thm:global} and \sourceid{thm:exact}:
global code preservation versus accessible recovery.
Enter Calculus Companion.
Full source not independently verified.

\bibitem{CheungHawk04B}
N.~Cheung.
\emph{Hawk04-B: Black-Hole Information as an Access Problem:
Enter, Quantum Recovery, and Complete Residuals}.
Version 2, undated manuscript.
Supplied summaries and excerpts:
\sourceid{thm:classical}, \sourceid{thm:certificate},
and decoder/error-normalization interfaces.
Enter Calculus Companion.

\bibitem{CheungEn07A}
N.~Cheung.
\emph{En07-A}, version 1.
Undated manuscript; supplied summaries of
\sourceid{thm:exact}, \sourceid{cor:dimension}, and
\sourceid{thm:release}.
Complete restoration and the separation from thermodynamic cost.
Enter Calculus Companion.

\bibitem{CheungEn07B}
N.~Cheung.
\emph{En07-B}, version 1.
Undated manuscript; supplied summary of
\sourceid{thm:endpoint}.
Four same-model obligations are reported, but their complete
verbatim clauses are not reconstructed here.
Enter Calculus Companion.

\bibitem{CheungEn07C}
N.~Cheung.
\emph{En07-C}, version 1.
Undated manuscript; supplied thermal-input and
complete-channel scope records, including
\sourceid{thm:release}.
Enter Calculus Companion.

\bibitem{CheungEn01C}
N.~Cheung.
\emph{En01-C}, version 1.
Undated manuscript; supplied complete-record statistical excerpts
and calibration-risk interface.
Enter Calculus Companion.

\bibitem{CheungEn02B}
N.~Cheung.
\emph{En02-B}, version 1.
Supplied \sourceid{En02-B-v1.tex} excerpts.
Especially \sourceid{prop:observable} and the declared streaming
and decoder-module interfaces.
Enter Calculus Companion.

\bibitem{CheungEn02C}
N.~Cheung.
\emph{En02-C}, version 1.
Supplied \sourceid{En02-C-v1.tex} excerpts.
Especially \sourceid{thm:C2}, \sourceid{thm:C3},
and \sourceid{lem:firstmeasurement}.
Exact information-theoretic recovery and a restricted
one-live-qubit streaming obstruction.
Enter Calculus Companion.

\bibitem{CheungEm05B}
N.~Cheung.
\emph{Em05-B}, version 1.
Supplied \sourceid{Em05-B-v1.tex} excerpts.
Especially \sourceid{thm:constantComplement} and the
flagged-instrument interface.
Enter Calculus Companion.

\bibitem{CheungHawk03A}
N.~Cheung.
\emph{Hawk03-A}, version 1.
Supplied \sourceid{Hawk03-A.tex} excerpts and summaries.
Especially \sourceid{thm:exact}, \sourceid{thm:partner},
and the exact commuting-copy obstruction.
Enter Calculus Companion.

\bibitem{CheungHawk01B}
N.~Cheung.
\emph{Hawk01-B}, version 1.
Supplied \sourceid{Hawk01-B.tex} excerpts.
Especially \sourceid{thm:exact} and
\sourceid{eq:rankcapacity}.
Enter Calculus Companion.

\bibitem{CheungVIIG}
N.~Cheung.
\emph{VII-G}, version 1.
Supplied \sourceid{VII-G.tex} excerpts.
Especially \sourceid{thm:decouple} and the constant-channel
and informative-record distinctions.
Enter Calculus Companion.

\bibitem{CheungEm03A}
N.~Cheung.
\emph{Em03-A}, version 1.
Supplied fixed-background emission and documentary-audit excerpts.
The physical source hypotheses are not imported here.
Enter Calculus Companion.

\bibitem{CheungEm03B}
N.~Cheung.
\emph{Em03-B}, version 2.
Supplied \sourceid{Em03-B-v2.tex} excerpts on finite packet
marginals, apparatus access, complete failures, and non-transfer.
Enter Calculus Companion.

\bibitem{CheungEm03C}
N.~Cheung.
\emph{Em03-C}, version 1.
Supplied \sourceid{Em03-C-v1.tex} excerpts on joint packet
interfaces, reference stability, and mechanism nonidentification.
Enter Calculus Companion.

\bibitem{CheungEm03F}
N.~Cheung.
\emph{Em03-F}, version 1.
Supplied \sourceid{Em03-F-v1.tex} excerpts.
Source-dependent exterior field signals and an
entanglement-breaking accessible channel are distinct from
quantum recovery.
Enter Calculus Companion.

\bibitem{CheungEm03G}
N.~Cheung.
\emph{Em03-G}, version 1.
Supplied \sourceid{Em03-G-v1.tex} excerpts.
Information-theoretic packet transmission and local coherent
reception are separate release gates.
Enter Calculus Companion.

\bibitem{CheungXVIIA}
N.~Cheung.
\emph{XVII-A}, version 1.
Supplied \sourceid{XVII-A.tex} excerpts and summaries.
One finite-matrix sine effect from a declared boundary region;
source-alignment and non-premise ledger.
Enter Calculus Companion.

\bibitem{CheungHawk02A}
N.~Cheung.
\emph{Hawk02-A}, version 1.
Supplied \sourceid{Hawk02-A.tex} excerpts.
Complete transfer interfaces, residual leakage, and the
outstanding gravitational and analytic obligations.
Enter Calculus Companion.

\bibitem{CheungHawk02D}
N.~Cheung.
\emph{Hawk02-D}, version 1.
Supplied \sourceid{Hawk02-D.tex} excerpts.
Especially the detailed \sourceid{thm:endpoint} statement with
\(F_{\rm out}=\Omega_+^\dagger J_S\).
The finite reservoir endpoint is not a gravitational endpoint.
Enter Calculus Companion.

\bibitem{CheungHawk02E}
N.~Cheung.
\emph{Hawk02-E}, version 1.
Supplied \sourceid{Hawk02-E.tex} excerpts.
Gravitational correspondence, common validity intervals,
and the auxiliary-mathematics non-transfer audit.
Enter Calculus Companion.

\bibitem{CheungVIIIA}
N.~Cheung.
\emph{VIII-A}, version 2.
Supplied \sourceid{VIII-A-v2.tex} excerpts.
Accessible state functionals and independently specified
conical geometric admissibility.
Enter Calculus Companion.

\bibitem{CheungVIIIB}
N.~Cheung.
\emph{VIII-B}, version 2.
Supplied \sourceid{VIII-B-v2.tex} excerpts.
Especially \sourceid{thm:spectators} and
\sourceid{thm:source-map}.
A prescribed source dictionary is not derived geometry.
Enter Calculus Companion.

\bibitem{CheungVIIIC}
N.~Cheung.
\emph{VIII-C}, version 2.
Supplied \sourceid{VIII-C-v2.tex} excerpts.
Especially the token-boundary and destructive-record distinctions.
Enter Calculus Companion.

\bibitem{CheungEm06A}
N.~Cheung.
\emph{Em06-A}, version 1.
Supplied \sourceid{Em06-A-v1.tex} excerpts.
A scalar-only coherent limit on a prescribed background,
with explicit non-emergence boundaries.
Enter Calculus Companion.

\bibitem{CheungEm06B}
N.~Cheung.
\emph{Em06-B}, version 1.
Supplied \sourceid{Em06-B-v1.tex} excerpts.
Declared scalar-family concentration and residual fluctuations;
the metric coefficient remains an input.
Enter Calculus Companion.

\bibitem{CheungXIVH}
N.~Cheung.
\emph{Landmark XIV-H: Internal kernel relations on the operator route}.
Version 1, supplied \sourceid{XIV-H.tex} excerpts.
Especially \sourceid{thm:H2} and \sourceid{prop:K5exact}.
Historical internal-amplitude stop, not a current full-map release.
Enter Calculus Companion.

\bibitem{CheungXIVI}
N.~Cheung.
\emph{Landmark XIV-I}.
Version 1, supplied \sourceid{XIV-I.tex} excerpts.
Especially \sourceid{prop:pole}, \sourceid{thm:I2},
\sourceid{thm:I3}, and \sourceid{thm:release}.
Internal amplitudes are constructed at I2; the joint
characteristic map remains unresolved at I3.
Enter Calculus Companion.

\bibitem{CheungXIVJ}
N.~Cheung.
\emph{Landmark XIV-J}.
Version 1, supplied \sourceid{XIV-J.tex} excerpts.
Especially \sourceid{prop:descent}, \sourceid{lem:null},
\sourceid{thm:reference}, \sourceid{thm:conditionalextension},
\sourceid{thm:J2}, and \sourceid{cor:noadmission}.
The supported-sector reference is not the full joint map.
Enter Calculus Companion.

\bibitem{CheungXVIA}
N.~Cheung.
\emph{XVI-A}, version 1.
Supplied \sourceid{XVI-A.tex} excerpts, especially the
separate prepared-source obligations 4--6.
Enter Calculus Companion.

\bibitem{CheungXVIN}
N.~Cheung.
\emph{XVI-N}, version r12.
Supplied \sourceid{XVI-N.tex} package-boundary excerpt and
documentary status summary.
No coefficient admission or O.1 closure is inferred.
Enter Calculus Companion.

\end{thebibliography}

\end{document}