% Oai069-A.tex
% Single-file manuscript for pdfLaTeX.
% No external bibliography, graphics, data files, or shell escape are required.
%
% Suggested build:
%   pdflatex -interaction=nonstopmode -halt-on-error Oai069-A.tex
%   pdflatex -interaction=nonstopmode -halt-on-error Oai069-A.tex
%
% These commands have not been executed here.
% Source records are not independently verified theorem premises.
% No statistical dataset, compiler execution, or laboratory realization
% is asserted.

\documentclass[11pt]{article}

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

\hypersetup{
  pdftitle={Oai069-A: Certified Categorical Recovery},
  pdfsubject={Finite-monodromy local systems, access-restricted recovery,
              complete channel certification, and free-field consistency}
}

\setlength{\emergencystretch}{3em}
\setlist[enumerate]{leftmargin=*,itemsep=3pt}
\setlist[itemize]{leftmargin=*,itemsep=3pt}
\newcolumntype{Y}{>{\raggedright\arraybackslash}X}

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

\DeclareMathOperator{\Tr}{Tr}
\DeclareMathOperator{\id}{id}
\DeclareMathOperator{\Ad}{Ad}
\DeclareMathOperator{\rank}{rank}
\DeclareMathOperator{\supp}{supp}
\DeclareMathOperator{\im}{im}
\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\End}{End}
\DeclareMathOperator{\Bun}{Bun}
\DeclareMathOperator{\Rep}{Rep}
\DeclareMathOperator{\Rec}{Rec}
\DeclareMathOperator{\WF}{WF}
\DeclareMathOperator{\diag}{diag}

\newcommand{\bbC}{\mathbb C}
\newcommand{\bbR}{\mathbb R}
\newcommand{\bbZ}{\mathbb Z}
\newcommand{\bbN}{\mathbb N}
\newcommand{\B}{\mathcal B}
\newcommand{\A}{\mathcal A}
\newcommand{\D}{\mathcal D}
\newcommand{\ket}[1]{\lvert #1\rangle}
\newcommand{\bra}[1]{\langle #1\rvert}
\newcommand{\kb}[2]{\lvert #1\rangle\langle #2\rvert}
\newcommand{\norm}[1]{\left\lVert #1\right\rVert}
\newcommand{\abs}[1]{\left\lvert #1\right\rvert}
\newcommand{\pair}[2]{\left\langle #1,#2\right\rangle}
\newcommand{\sid}[1]{\texttt{\detokenize{#1}}}
\newcommand{\ddia}{d_{\diamond}}
\newcommand{\erec}{\varepsilon_{\mathrm{rec}}}
\newcommand{\clip}[1]{\left[#1\right]_{[0,1]}}

\title{\textbf{Oai069-A}\\[4pt]
From Quantum Langlands Equivalence to Certified Categorical Recovery:\\
A Finite-Monodromy Abelian Benchmark and an Independent Free-Field Audit}
\author{}
\date{Version 1 --- 12 October 2026}

\begin{document}
\maketitle

\begin{abstract}
The supplied Family 069-01 source claims an unramified de Rham
quantum geometric Langlands equivalence at irrational shifted levels.
We retain that claim as a documentary record, not as a verified
premise. Its categorical equivalence does not itself specify a
positive state space, accessible registers, a decoder, or a
reference-preserving channel.

We complete a separation-preserving finite benchmark independently
of that source. On the elliptic curve
\(E=\mathbb C/(\mathbb Z+i\mathbb Z)\), we construct the category of
unitary local systems whose monodromy factors through a specified
finite cyclic quotient. Spectral decomposition gives an explicit
dagger equivalence with finite graded Hilbert spaces and a
constructive Fourier realization. A named rank-one local system
carries a qubit multiplicity subsystem and an arbitrary gauge
factor. We construct an access-restricted decoder, prove uniform
recovery with arbitrary references and gauge correlations, and
determine sharp error floors after removing coefficient legs.
The missing-leg interface still recovers a named effect, illustrating
the distinction between observable recovery and complete quantum
return.

For the complete implemented channel family, matrix-unit coverage
and Choi tomography give simultaneous finite-confidence error
intervals, including label corruption, calibration allowances,
and independently justified transfer error. We separately prove
two-probe coverage of a declared homogeneous conserved free-scalar
stress class and an exact finite Ward-defect identity.

The torus benchmark is not a specialization of the connected-simple
Family 069 theorem. No comparison between its categorical functor
and a four-dimensional Weyl net is constructed. The localization--
Whittaker, central-torsor, and collision compatibilities needed for
a genuine Family 069 operational bridge remain explicitly named.
No gravitational or holographic dictionary is inferred.
\end{abstract}

\paragraph{Keywords.}
Quantum geometric Langlands; finite monodromy; local system;
categorical equivalence; subsystem recovery; access restriction;
Choi certification; stress-tensor improvement; Ward identity.

\tableofcontents

\section{Scope and debugging of the proposed programme}
\label{sec:scope}

The purpose of this manuscript is to make a categorical benchmark
operational without confusing three different kinds of conclusion:
\[
 \text{categorical identification},\qquad
 \text{complete quantum-channel recovery},\qquad
 \text{gravitational identification}.
\]
The proposed programme is completed at the finite benchmark level.
A literal operational realization of the full Family 069 functor is
not claimed.

The distinctions between an implemented decoder, an optimal decoder,
a finite algebra task, a subsystem task, and a physical
interpretation are emphasized in the supplied Cheung records
\cite{Cheung280C,Cheung290A,Cheung265B,CheungEn02B,
CheungEn02C,CheungEn05C}. Every finite interface used in the proofs
below is defined and proved locally. Those manuscripts provide
attribution and comparisons, not unchecked theorem premises.

\subsection{Corrections required before a proof is possible}

\begin{enumerate}
\item \textbf{A complex-linear DG equivalence is not automatically
a quantum channel.}
Complete positivity, adjoints, density matrices, trace, and
operational access are additional structures. They are not
specified by presentability and complex linearity alone.

\item \textbf{The abelian example lies outside the printed
connected-simple hypothesis.}
A torus or \(\mathrm{GL}(1)\) is not a connected simple group in
the sense of the supplied Family 069 statement. Replacing \(G\)
by a torus is a change of theorem, not a specialization justified
by that statement. In particular, the printed determinant
normalization involving \(h^\vee\) cannot be substituted unchanged
with \(h^\vee=0\).

\item \textbf{An abelian local-system task is not a nondegenerate
Whittaker task.}
A torus has trivial maximal unipotent subgroup. Our named task
concerns actual finite-monodromy local systems on a fixed curve;
it is not advertised as recovery of a nontrivial Whittaker module.

\item \textbf{The two meanings of free field are not identified.}
A chiral free-field realization of a \(W\)-algebra and a
four-dimensional free massless scalar net are different objects.
No dimension-changing or positive-net realization follows from
the word ``free.''

\item \textbf{Irrational categorical data do not establish
unitarity or strong rationality.}
The source includes non-real levels. It does not provide the
positive Hilbert realization needed by our operational task.
The VOA--net comparison hypotheses discussed in Oai280-A are
not silently imported \cite{Cheung280A}.

\item \textbf{Sector labels do not authorize inter-sector quantum
coherences.}
Our categorical state algebra is a finite direct sum. Arbitrary
quantum coherence is allowed inside each multiplicity space,
including correlations with a gauge factor and an external
reference. Coherences between distinct superselection labels are
not part of the declared task.

\item \textbf{Two stress probes are not process tomography.}
They cover a specified two-parameter conserved stress class.
Complete channel coverage uses all input matrix units, or an
equivalent complete Choi measurement family.

\item \textbf{A failed sufficient certificate is not an
all-decoder exclusion.}
We distinguish exclusion of a tested decoder from bounds proved
uniformly over every CPTP decoder allowed at the stated access.

\item \textbf{Global-phase and categorical coherence data require
their own audit.}
Channel tomography cannot detect a common phase of a unitary
implementation. Equality of implemented density-matrix channels
does not by itself verify all natural isomorphisms, determinant
twists, or central-torsor coherence.

\item \textbf{A marked-point configuration is not a spacetime
boundary.}
We use a specified finite receiver interface. Removing a leg
changes its channel and recovery capability, which is proved.
We do not claim that moving marked points on the curve changes
recovery, or that those points define a continuum entanglement wedge.

\item \textbf{Conserved currents and Ward contacts are distinct.}
The free stress calculation establishes a tensor and its
conformal-Killing current conservation. A Ward-insertion
certificate also needs a specified product and contact
prescription.

\item \textbf{The title denotes a benchmark agenda, not a completed
source-to-physics arrow.}
A positive realization of the Family 069 localization--Whittaker
functor is not constructed. The precise missing bridge is stated
in Section~\ref{sec:bridge}.
\end{enumerate}

\subsection{Related established frameworks}

Kapustin--Witten formulate a gauge-theoretic geometric Langlands
programme using a topologically twisted four-dimensional
\(\mathcal N=4\) theory compactified on a curve
\cite{KapustinWitten}. Gaitsgory's quantum Langlands notes and
Whittaker comparison work concern independently stated categorical
and sheaf-theoretic interfaces
\cite{GaitsgoryQGL,GaitsgoryWhit}. Arakawa--Frenkel prove named
\(W\)-algebra representation dualities \cite{ArakawaFrenkel}.

Abelian Fourier transforms for differential-operator algebras have
their own geometric statements \cite{PolishchukRothstein}.
Our finite Fourier construction below is proved directly; it is
not substituted for those transforms.

The AdS/CFT proposal involves further choices and limits, including
the large-\(N\) setting described by Maldacena
\cite{Maldacena}. The Kapustin--Witten motivation does not identify
the present finite task with that dictionary.

\section{Family 069-01 as a source record}
\label{sec:source}

The referenced catalogue identifies Family 069-01 as
\emph{Global quantum geometric Langlands at irrational level},
dated \textbf{4 October 2026} \cite{OpenAI069Catalogue}.
That concrete date and title govern the present attribution.

The publicly accessible introduction gives the shifted convention
\begin{equation}
 D_c(\Bun_G(X))
 :=
 D\!\left(L_G^{(c-h^\vee)/(2h^\vee)}\right),
 \qquad
 L_G|_E=\det R\Gamma(X,\mathfrak g_E).
 \label{eq:source-level}
\end{equation}
The complex power describes a differential-operator twisting,
not a positive fractional power of a Hilbert-space operator
\cite{OpenAI069Introduction}.

\begin{record}[Family 069-01]
\label{record:069}
For a connected simple complex algebraic group \(G\), a smooth
projective connected complex curve \(X\), and
\(c\in\mathbb C\setminus\mathbb Q\), the source claims an equivalence
\[
 D_c(\Bun_G(X))
 \simeq
 D_{-1/(rc)}(\Bun_{G^\vee}(X))
\]
of presentable complex-linear DG categories. Here \(r\) is the
lacing number, and the full root datum fixes the global form.

The stated characterization includes localization--Whittaker
comparison, compatibility with families of marked points and
collisions, and interchange of central-torsor actions with the
corresponding transgression local systems
\cite{OpenAI069Introduction}.
\end{record}

The source introduction and supplied excerpts have been consulted
for attribution. No independent proof verification is asserted.
The catalogue explicitly separates this claim from Cheung's
registered series and reports no Lean scope note for this family
\cite{OpenAI069Catalogue}.

The order \(d\) of the finite cyclic group introduced below is
\emph{not} the lacing number \(r\). No shifted level \(c\) is needed
for our finite construction.

\section{Phase 1: fix the information task}
\label{sec:task}

\subsection{Distances and complete comparisons}

For states, write
\[
 d_{\mathrm{tr}}(\rho,\sigma)
 =\frac12\norm{\rho-\sigma}_1.
\]
For finite-dimensional channels, use
\[
 \ddia(\Phi,\Psi)=\frac12\norm{\Phi-\Psi}_\diamond.
\]
Equivalently, for channels this is the supremum trace distance of
their outputs on arbitrary reference-entangled input states.
The finite-dimensional stabilization and channel conventions are
standard \cite{Watrous}. Our bounds below are proved in the
reference-state formulation.

\begin{definition}[Claim specification]
\label{def:claim}
A finite categorical recovery claim specifies:
\begin{enumerate}
\item the curve, monodromy quotient, named objects, and allowed
coefficient spaces;
\item the positive state algebra and its permitted references;
\item the encoding, inaccessible systems, and receiver access;
\item a target channel or named effect;
\item the decoder programme and its allowed elementary operations;
\item the error topology and implementation comparison;
\item a predeclared validation family, calibration allowance,
transfer allowance, and decision rule.
\end{enumerate}
A geometric interpretation is not part of this definition.
\end{definition}

\subsection{The concrete specification used here}

Fix
\[
 E=\bbC/(\bbZ+i\bbZ),\qquad d=2^s,\quad s\ge1,\qquad
 g=2^h,\quad h\ge0,
\]
and put \(\omega=e^{2\pi i/d}\).

The named object is the rank-one local system
\(\mathcal L_1\) constructed in Section~\ref{sec:abelian}.
The finite roster includes all \(\mathcal L_k\),
\(k\in\bbZ/d\bbZ\).

Its quantum multiplicity is
\[
 C=\bbC^2,
\]
and its gauge factor is \(K=\bbC^g\).
The input and target algebras are
\begin{equation}
 \A_{\mathrm{in}}=\bigoplus_{k=0}^{d-1}\B(C\otimes K),
 \qquad
 \A_{\mathrm{tar}}=\bigoplus_{k=0}^{d-1}\B(C).
 \label{eq:task-algebras}
\end{equation}
Their traces are the sums of the ordinary block traces.

The complete target is
\begin{equation}
 \mathcal T((\rho_k)_k)
 =\sum_k\kb{k}{k}_J\otimes\Tr_K\rho_k.
 \label{eq:target}
\end{equation}
Here \(J\) is the returned sector label, represented by an
orthogonal register. Its output is classical in that basis.

Every reference input is of the form
\[
 \rho_{Q,\mathrm{in}}=\bigoplus_k\rho_{QCK,k},
 \qquad
 \rho_{QCK,k}\ge0,\qquad
 \sum_k\Tr\rho_{QCK,k}=1.
\]
Inside each block, \(Q,C,K\) may be arbitrarily correlated.
There is no independent-gauge preparation hypothesis.

\begin{remark}[Named object versus unknown geometric object]
Reading \(k\) identifies an object in the predeclared finite roster.
It does not estimate an arbitrary continuous monodromy parameter,
reconstruct an unknown sheaf from one specimen, or return every
object of a presentable DG category.
\end{remark}

The three tested tasks will be:
\begin{enumerate}
\item return the named local-system label and the qubit multiplicity;
\item return the multiplicity subsystem uniformly over \(K\) and \(Q\);
\item recover the named logical effect
\(P_+=(I+X)/2\) when only one coefficient leg is accessible.
\end{enumerate}

\section{An explicit abelian local-system category}
\label{sec:abelian}

\subsection{Rank-one local systems on the elliptic curve}

Let
\[
 \chi_k(n,m)=\omega^{kn},
 \qquad (n,m)\in\pi_1(E)\cong\bbZ^2.
\]
Define \(\mathcal L_k\) as the quotient of
\(\bbC\times\bbC\) under the action
\[
 (n,m)\cdot(z,v)
 =\bigl(z+n+im,\chi_k(n,m)^{-1}v\bigr).
\]
The trivial connection on the cover descends. With this convention,
parallel transport around the first basic cycle has monodromy
\(\omega^k\), and the second basic cycle has monodromy one.

Equivalently, these local systems arise from the cyclic covering
\[
 \bbC/(d\bbZ+i\bbZ)\longrightarrow E.
\]
This is an actual local-system construction on a fixed curve,
not a category of \(D\)-modules on \(\Bun_{\mathbb G_m}(E)\).

Let \(\mathsf{Loc}_d(E)\) consist of the finite-rank unitary local
systems whose second monodromy is identity and whose first
monodromy \(U\) satisfies \(U^d=I\).
Morphisms are parallel complex-linear maps. Adjoints use the
parallel Hermitian structures.

\begin{lemma}[Orthogonal character projectors]
\label{lem:projectors}
For unitary \(U\) with \(U^d=I\), set
\[
 P_k=\frac1d\sum_{a=0}^{d-1}\omega^{-ka}U^a.
\]
Then
\[
 P_k^*=P_k,\qquad
 P_kP_l=\delta_{kl}P_k,\qquad
 \sum_kP_k=I,\qquad
 UP_k=\omega^kP_k.
\]
\end{lemma}

\begin{proof}
For every \(b\in\bbZ/d\bbZ\),
\[
 \sum_{a=0}^{d-1}\omega^{ab}
 =\begin{cases}d,&b=0,\\0,&b\ne0.\end{cases}
\]
Expanding the products of the finite sums proves orthogonality,
the sum identity, and the eigenvalue identity.
Replacing \(a\) by \(-a\), and using \(U^*=U^{-1}\), proves
self-adjointness.
\end{proof}

\begin{theorem}[Finite-monodromy categorical equivalence]
\label{thm:category}
There is a complex-linear dagger equivalence
\[
 \mathsf{Loc}_d(E)
 \simeq
 \bigoplus_{k=0}^{d-1}\mathsf{Hilb}_{\mathrm{fd}},
 \qquad
 \mathcal V\longmapsto(H_k)_k,\quad H_k=P_k\mathcal V_{z_0}.
\]
Its inverse sends \((H_k)_k\) to
\[
 \bigoplus_k\mathcal L_k\otimes H_k.
\]
The construction extends degreewise to bounded complexes and their
mapping complexes.
\end{theorem}

\begin{proof}
Lemma~\ref{lem:projectors} decomposes each fiber into orthogonal
monodromy eigenspaces. The projectors are parallel, so they
decompose the entire local system.

A parallel map \(A\) commutes with \(U\), hence with every \(P_k\).
It is therefore precisely a family of maps \(A_k:H_k\to H'_k\).
Conversely, any such family defines a parallel map of the direct
sums. Composition, adjoints, and identities are blockwise.

The decomposition and recombination maps are inverse natural
unitaries. For complexes, the differentials and all graded
intertwiners commute with the projectors. The same decomposition
therefore respects differentials in both objects and mapping
complexes.
\end{proof}

The operational use below concerns degree-zero Hilbert coefficient
spaces. A chain complex is not treated as a density matrix.

The character tensor product satisfies
\[
 \mathcal L_k\otimes\mathcal L_l\cong\mathcal L_{k+l}.
\]
This follows by multiplying their monodromies.
It is not a reversible quantum map
\(\ket{k,l}\mapsto\ket{k+l}\) on a full coherent label register.
If a classical fusion channel is desired, it is separately the
complete probability pushforward under addition modulo \(d\).

\subsection{Constructive Fourier realization}

On \(S=\bbC^d\), define
\[
 S_a\ket j=\ket{j+a},\qquad
 Z_b\ket j=\omega^{bj}\ket j,
\]
and
\begin{equation}
 F_d\ket j=\frac1{\sqrt d}\sum_{k=0}^{d-1}\omega^{kj}\ket k,
 \qquad
 v_k=\frac1{\sqrt d}\sum_{j=0}^{d-1}\omega^{-kj}\ket j.
 \label{eq:fourier}
\end{equation}

\begin{proposition}[Fourier identities]
\label{prop:fourier}
The printed matrix \(F_d\) is unitary, and
\[
 F_dv_k=\ket k,\qquad
 S_av_k=\omega^{ak}v_k,
\]
\begin{equation}
 F_dS_aF_d^\dagger=Z_a,\qquad
 F_dZ_bF_d^\dagger=S_{-b}.
 \label{eq:finite-weyl}
\end{equation}
Thus the regular-representation local system is explicitly
identified with its character decomposition.
\end{proposition}

\begin{proof}
The inner products of distinct rows of \(F_d\) vanish by the
finite sum in Lemma~\ref{lem:projectors}; each row has norm one.
The remaining equations follow by substituting the finite sums
and changing the summation index modulo \(d\).
\end{proof}

Equation~\eqref{eq:finite-weyl} is a proved finite
translation--character comparison. It is not the source's
central-torsor/transgression theorem.

On a single charge sector, \(Z_a\) is a scalar.
Consequently its conjugation action on a density matrix is
identity. Density-matrix tomography alone cannot verify that
scalar phase. The linear identity
\(F_dS_a=Z_aF_d\) has been checked algebraically, rather than
inferred from channel tomography.

\subsection{Positive channels require more than the categorical equivalence}

\begin{proposition}[Complex-linear equivalence need not preserve positivity]
\label{prop:nopositive}
An equivalence of complex-linear categories can induce an
endomorphism-algebra isomorphism that is not positive with respect
to a chosen Hilbert adjoint.
\end{proposition}

\begin{proof}
On a one-object complex-linear category with endomorphism algebra
\(M_2(\bbC)\), take
\[
 R=\diag(2,1),\qquad A\longmapsto RAR^{-1}.
\]
This is an algebra automorphism and hence a category equivalence.
It sends the Hermitian matrix \(X\) to
\[
 \begin{pmatrix}0&2\\1/2&0\end{pmatrix},
\]
which is not Hermitian. A positive linear map sends Hermitian
matrices to Hermitian matrices, so this automorphism is not
positive for the fixed Hilbert structure.
\end{proof}

Our equivalence in Theorem~\ref{thm:category} has the additional
dagger and unitary structures needed for the next section.
The general Family 069 record supplies no operational analogue
of those choices.

\section{Phase 2: positive realization and access-restricted recovery}
\label{sec:recovery}

\subsection{Complete channels on the finite direct sums}

For algebras
\[
 A=\bigoplus_k\B(H_k),\qquad B=\bigoplus_l\B(H'_l),
\]
a complete channel can be written
\[
 (\Phi(\rho))_l
 =\sum_{k,a}K_{l|k,a}\rho_kK_{l|k,a}^\dagger,
 \qquad
 \sum_{l,a}K_{l|k,a}^\dagger K_{l|k,a}=I_{H_k}.
 \label{eq:block-channel}
\]

\begin{proposition}[Finite channel realization]
\label{prop:block-channel}
Formula~\eqref{eq:block-channel} is CPTP and preserves every
reference. Conversely, every CPTP map between these finite
direct sums has such a representation.

A unitary spectral realization of
Theorem~\ref{thm:category} transports these channels bijectively
by transporting their Kraus maps.
\end{proposition}

\begin{proof}
After any reference extension, each Kraus expression is positive.
The displayed completeness identity gives trace preservation
separately on each input block.

Conversely, composing a channel with the injection of input block
\(k\) and projection onto output block \(l\) gives a CP map.
Its positive Choi matrix decomposes into rank-one positive
summands, giving the Kraus maps. Trace preservation gives the
completeness identity. This is finite-dimensional Choi theory
\cite{Choi,Watrous}.

Unitary changes of input and output realizations replace
\(K\) by \(U_{\mathrm{out}}KU_{\mathrm{in}}^\dagger\).
They preserve the completeness identity and have inverse
transport. Composition is preserved.
\end{proof}

This reproves the finite direct-sum interface used in the supplied
VII-G and Oai280-C comparisons
\cite{CheungVIIG,Cheung280C}.
It does not identify the category of all categorical morphisms
with the category of all quantum channels.

\subsection{The encoded local-system multiplicity}

Let \(R_1,R_2\) be qubits and \(B\cong K\) an inaccessible gauge
register. Let \(H\), \(X\), and \(Z\) denote the usual qubit
Hadamard and Pauli matrices. Define
\[
 \epsilon_k=k\bmod 2,\qquad
 U_k=\operatorname{CNOT}_{1\to2}
              (H Z^{\epsilon_k}\otimes I).
\]
The encoder is the isometry
\begin{equation}
 V:\ket k\ket\psi_C\ket\xi_K
 \longmapsto
 v_k{}_S\otimes
 U_k(\ket\psi_{R_1}\ket0_{R_2})
 \otimes\ket\xi_B.
 \label{eq:encoding}
\end{equation}
The receiver gets \(S,R_1,R_2\), not \(B\).

The source logical label has monodromy \(\omega^k\).
The encoded monodromy acts as \(S_1\) on \(S\), with identity on
the multiplicity and gauge registers. Thus \(V\) intertwines
the declared local systems.

\begin{lemma}[Encoding isometry and dual realization]
\label{lem:encoding}
The map \(V\) is isometric and preserves monodromy.
Applying \(F_d\) on the received \(S\) gives
\[
 (F_d\otimes I)V:
 \ket k\ket\psi\ket\xi
 \longmapsto
 \ket k_J\otimes U_k(\ket\psi\ket0)\otimes\ket\xi_B.
\]
\end{lemma}

\begin{proof}
The \(v_k\) are orthonormal and each \(U_k\) is unitary.
Therefore inner products are preserved.
The monodromy and Fourier claims are
Proposition~\ref{prop:fourier}.
\end{proof}

The word ``dual'' in this formula means the character-coordinate
side of the explicit finite equivalence. It is not an
unconstructed positive realization of the dual bundle category
in Record~\ref{record:069}.

\subsection{A complete decoder using only the receiver}

Write a received operator, after \(F_d\), as \(Y\). Define
\begin{equation}
 \mathcal D_{12}(Y)
 =\sum_k\kb k k_{J_{\mathrm{out}}}\otimes
 \Tr_{R_2}\!\left[
 U_k^\dagger
 \bigl(\bra k_JY\ket k_J\bigr)
 U_k\right].
 \label{eq:decoder}
\end{equation}
This is defined on the \emph{entire} receiver matrix algebra,
not only on the code range.

The programme is:
\begin{enumerate}
\item apply the printed \(F_d\) to \(S\), unless character
coordinates have already been supplied;
\item read \(J\) in the computational basis, retaining every
outcome \(k\);
\item on that branch apply \(\operatorname{CNOT}_{1\to2}\),
then \(H\) on \(R_1\), then \(Z^{\epsilon_k}\) on \(R_1\);
\item discard \(R_2\), return \(R_1\) as \(C\), and return \(k\)
as the classical label.
\end{enumerate}
The order in step three is \(U_k^\dagger\).

\begin{theorem}[Uniform local-system subsystem recovery]
\label{thm:exact}
Let \(\mathcal N_{12}\) be the accessible channel obtained from
\eqref{eq:encoding} by tracing \(B\), followed by the
character-coordinate transformation.
Then
\begin{equation}
 \mathcal D_{12}\mathcal N_{12}=\mathcal T
 \label{eq:exact}
\end{equation}
on \(\A_{\mathrm{in}}\), including every external reference
and arbitrary correlations of \(C\) with \(K\).

The label \(k=1\) and the full qubit coefficient subsystem of
\(\mathcal L_1\) are recovered with zero ideal error.
\end{theorem}

\begin{proof}
The encoder and partial trace are CPTP.
The label compression and output-tag preparation in
\eqref{eq:decoder} are CP. Unitary conjugation and partial trace
are CP. Summing the label branches preserves trace, so
\(\mathcal D_{12}\) is CPTP.

For each input block \(\rho_k\), conjugation by \(U_k^\dagger\)
restores
\[
 \rho_{R_1B,k}\otimes\kb0 0_{R_2}.
\]
Tracing \(R_2\) and the inaccessible gauge gives precisely
\(\Tr_K\rho_k\), with label \(k\).
The calculation is an equality of linear maps, so it holds
after tensoring with \(\id_Q\).
\end{proof}

\begin{remark}[Subsystem recovery is not full-code recovery]
The complete complement of the encoder can retain the state of
\(K\). It is therefore not constant on \(C\otimes K\) when
\(g>1\). Nevertheless the subsystem \(C\) is exactly recovered.

The constant-complete-complement criterion applies to full-code
identity recovery \cite{KnillLaflamme}. It must not be imposed
as a necessary condition for the present partial-trace target.
\end{remark}

\subsection{Programme soundness and resource accounting}

\begin{proposition}[Complete branch soundness]
\label{prop:programme}
For a finite programme built from isometries, unitaries, complete
instruments, classical controls, and specified discards, let
\(K_\gamma\) be its complete-history Kraus maps.
Then
\[
 \sum_\gamma K_\gamma^\dagger K_\gamma=I.
\]
Its reported selective maps and complete channel preserve all
references. Equality on every input matrix unit proves equality
of the complete channels.
\end{proposition}

\begin{proof}
At the final instruction, instrument completeness gives the
identity sum. Backward induction through each instrument,
unitary, and isometry gives the displayed identity.
The branch maps therefore define a CPTP sum.

Matrix units span the input operator space. Equality on them
gives equality of linear maps and hence of every reference
extension. Normalized branch states are not substituted for
the unnormalized Kraus sums.
\end{proof}

This is the finite execution interface of En02-B/C,
reproved for the stated programme class
\cite{CheungEn02B,CheungEn02C}.

The decoder's quantum peak is at most \(s+2\) qubits:
\(s\) label qubits and two coefficient legs arrive at the
receiver. After destructive label readout, two quantum legs
remain; after discard, one is returned.
Classical label storage is a separate ledger.

The inaccessible supplier holds \(h\) gauge qubits.
A Choi validation experiment additionally uses
\(h+1\) input-reference qubits. Those references are not
available to the decoder and are not omitted from a
whole-experiment ledger.

The exact finite gate library includes the printed \(F_d\).
No laboratory synthesis, optimal gate count, or resource-optimal
compiler is inferred.

\subsection{Heisenberg access certification}

\begin{proposition}[Receiver-only factorization]
\label{prop:access}
A channel from a complete finite system \(R\otimes B\) to an output
can be implemented using only \(R\) if its adjoint factors as
\[
 \Phi^*(O)=K^*(O)\otimes I_B
\]
for a UCP map \(K^*\) on \(R\).
The decoder \(\mathcal D_{12}\) has this factorization when
extended to the complete encoder output.
\end{proposition}

\begin{proof}
The factorization is the adjoint of
\(\Phi=K\circ\Tr_B\), which uses only the \(R\) snapshot.
The decoder's operations and Kraus maps act on \(S,R_1,R_2\)
only; their complete-system adjoints tensor with \(I_B\).
\end{proof}

The following finite tensor-free form also clarifies the scope
of snapshot determination.

\begin{proposition}[Finite snapshot--containment equivalence]
\label{prop:snapshot}
Let \(A\subseteq M\) be a unital finite-dimensional
\(C^*\)-subalgebra, and let \(T:N\to M\) be UCP.
The following are equivalent:
\begin{enumerate}
\item \(T(N)\subseteq A\);
\item states with equal restrictions to \(A\) have equal
expectations on every \(T(b)\), \(b\in N\);
\item \(T\) factors uniquely through a UCP map into \(A\).
\end{enumerate}
\end{proposition}

\begin{proof}
Containment gives state comparison.
For the converse, if a Hermitian \(T(b)\notin A\), its
Hilbert--Schmidt orthogonal component \(h_\perp\) is nonzero,
Hermitian, and traceless because \(I\in A\).
For sufficiently small \(\varepsilon>0\), the density matrices
\[
 \rho_\pm=I/\dim M\pm\varepsilon h_\perp
\]
in a faithful matrix realization are positive.
Their restrictions agree on \(A\), but their expectations on
\(T(b)\) differ by
\(2\varepsilon\|h_\perp\|_2^2\).
This contradicts item two.

Under containment, restricting the codomain gives the unique
factor. Matrix-level positivity is reflected by the subalgebra
inclusion, so the factor is UCP.
\end{proof}

The proposition is the finite-algebra instance of the comparison
in En05-C \cite{CheungEn05C}. It compares already specified
algebras and maps; it does not construct a relativistic net.

\section{Access dependence, named effects, and sharp exclusions}
\label{sec:loss}

Let \(\mathcal N_1\) retain \(J,R_1\) but discard \(R_2,B\).
For a fixed sector,
\[
 \mathcal N_{1,k}(\rho)
 =\Delta_Z\!\left(HZ^{\epsilon_k}
             (\Tr_K\rho)Z^{\epsilon_k}H\right),
\]
where
\[
 \Delta_P(\sigma)=\frac12(\sigma+P\sigma P).
\]
Applying \(Z^{\epsilon_k}H\) to \(R_1\) gives the complete
decoded channel
\begin{equation}
 \mathcal D_1\mathcal N_1
 =\text{label return}\ \otimes\ \Delta_X\circ\Tr_K.
 \label{eq:one-leg}
\end{equation}
The same description holds for \(R_2\) alone, with the appropriate
identification of that receiver leg.

\begin{theorem}[Sharp coefficient-access verdicts]
\label{thm:loss}
For each sector, uniformly over gauge states and references:
\begin{enumerate}
\item both coefficient legs admit zero-error subsystem recovery;
\item one coefficient leg has optimal half-diamond error \(1/2\)
for complete qubit return;
\item no coefficient leg has optimal half-diamond error \(3/4\);
\item one coefficient leg still exactly recovers the named effect
\(P_+=(I+X)/2\).
\end{enumerate}
Items two and three are bounds over \emph{all} receiver-only
CPTP decoders, not merely over the printed programmes.
\end{theorem}

\begin{proof}
Item one is Theorem~\ref{thm:exact}.

The channel retaining one leg is measure-and-prepare after
discarding \(K\), hence entanglement breaking on \(C\).
Fixing \(K\) to a pure state and using a Bell input on \(QC\),
every receiver-only decoded output is separable across
\(Q\) and the returned qubit. Its Bell overlap is at most \(1/2\):
for a pure product vector this is Cauchy--Schwarz, and convexity
gives the mixed separable case.
The Bell projector therefore gives error at least \(1/2\).

Equation~\eqref{eq:one-leg} attains the bound.
Indeed \(\Delta_X=(\id+\Ad_X)/2\), so the triangle inequality
gives error at most \(1/2\), while the Bell output has
orthogonal Bell components of weights \(1/2,1/2\).
Composition with \(\Tr_K\) gives the same uniform upper bound.

Without coefficient legs, the accessible channel is independent
of the qubit. For a fixed sector every decoded qubit is a
replacement state \(\tau\). Its Bell-input output has overlap
\(1/4\), so the error is at least \(3/4\).
Returning \(I/2\) attains \(3/4\), because qubit depolarization
is the equal mixture of the four Pauli conjugations.
Again composition with \(\Tr_K\) gives a uniform bound.

Finally, \(\Delta_X^*(P_+)=P_+\).
Before the one-leg inverse, the corresponding received effect is
\[
 \frac12\bigl(I+(-1)^{\epsilon_k}Z_{R_1}\bigr).
\]
Its pullback is \(P_+\otimes I_K\).
\end{proof}

The optimal subsystem errors are therefore
\[
 \boxed{\quad
 0,\qquad\frac12,\qquad\frac34
 \quad}
\]
for two, one, and zero coefficient legs.

\begin{remark}[A demonstrated finite access change]
Removing the \(R_2\) receiver edge changes the channel from an
exactly recoverable one to an entanglement-breaking coefficient
channel. This is a proved dependence on the finite access graph.
It is not a dependence on a derived metric or a marked-point
collision on \(E\).
\end{remark}

A sector-label measurement followed by classical output cannot
replace complete qubit recovery. In particular, measuring a
finite list of module tags does not establish recovery of their
quantum multiplicity spaces.

\section{Complete coverage and simultaneous finite-confidence certification}
\label{sec:certification}

\subsection{All matrix units, not a few selected observables}

For an input block \(C\otimes K\), put
\[
 n=2g.
\]
Let \(O=J_{\mathrm{out}}\otimes C_{\mathrm{out}}\), of dimension
\[
 o=2d.
\]
Allow a tested complete channel
\[
 \Phi_{\nu,k}:\B(C\otimes K)\to\B(O)
\]
for every predeclared decoder/access task \(\nu\) and input sector
\(k\). The implementation may corrupt its returned label.
That possibility is not removed from the certificate.

The ideal target on block \(k\) is
\[
 \mathcal T_k(\rho)=\kb k k\otimes\Tr_K\rho.
\]
Its normalized Choi state is
\begin{equation}
 J(\mathcal T_k)
 =
 P_{\mathrm{Bell},Q_CC_{\mathrm{out}}}
 \otimes\frac{I_{Q_K}}g\otimes\kb k k_{J_{\mathrm{out}}},
 \label{eq:targetchoi}
\end{equation}
with the indicated fixed rearrangement into
\(Q_C,Q_K,J_{\mathrm{out}},C_{\mathrm{out}}\) order.

\begin{proposition}[Complete finite coverage]
\label{prop:channelcoverage}
Values of \(\Phi_{\nu,k}\) on every matrix unit of
\(\B(C\otimes K)\) determine the entire block channel.
Testing every sector determines the complete channel on
\(\A_{\mathrm{in}}\), including every reference and every
gauge correlation.

Equivalently, its normalized Choi state determines the channel.
\end{proposition}

\begin{proof}
Matrix units span the input operator space.
Their coefficients are precisely the blocks of
\[
 J(\Phi_{\nu,k})
 =\frac1n\sum_{i,j}\kb i j\otimes\Phi_{\nu,k}(\kb i j).
\]
Thus either set of data gives the other and determines the
linear map. Direct-sum decomposition and reference extension
give the stated complete scope.
\end{proof}

This covers the subsystem task without reducing it to a
fixed gauge preparation. A Choi experiment that entangles only
\(C\), while fixing \(K\), would not cover the declared target.

\subsection{A dimension-explicit Choi bound}

Let
\[
 \delta(\Phi,\Psi)=\frac12\|J(\Phi)-J(\Psi)\|_1.
\]

\begin{lemma}[Choi and complete channel distances]
\label{lem:choi}
For channels with input dimension \(n\),
\begin{equation}
 \delta(\Phi,\Psi)
 \le \ddia(\Phi,\Psi)
 \le \min\{1,n\,\delta(\Phi,\Psi)\}.
 \label{eq:choibound}
\end{equation}
\end{lemma}

\begin{proof}
The normalized Bell input gives the lower bound.

For the upper bound, purify any referenced input.
By its Schmidt decomposition it can be written, up to an
isometry on the reference, as
\[
 \ket\psi=(A\otimes I)\ket{\Phi_n},
 \qquad
 \|A\|_{\mathrm{op}}^2\le n.
\]
The output difference is the corresponding multiplication of
\(J(\Phi)-J(\Psi)\) by \(A\) and \(A^\dagger\).
The trace-norm ideal property bounds its half trace norm by
\(n\delta\). Partial trace of a purification cannot increase
trace distance. The distance between channels is at most one.
\end{proof}

The dimension factor is retained. A high average Choi overlap
does not become a dimension-free worst-case recovery assertion.

If a direct-sum implementation is specified by its block channels,
then
\begin{equation}
 \ddia_{\oplus}(\Phi,\mathcal T)
 =\max_k\ddia(\Phi_k,\mathcal T_k).
 \label{eq:blockerror}
\end{equation}
The upper bound follows by summing block trace distances weighted
by block trace. The reverse bound is obtained by choosing one
input block.

\subsection{Tomographic setting and measurement contract}

Because \(d,g\) are powers of two, the Choi system has
\[
 D=no=4dg
\]
dimensions and \(h+s+2\) qubits.
Let \(\mathcal P\) be its Hermitian Pauli-string basis:
\[
 \Tr(PP')=D\delta_{P,P'},\qquad P^2=I.
\]
For a Choi state,
\[
 J=\frac1D\sum_{P\in\mathcal P}t_P P,
 \qquad t_P=\Tr(PJ),\qquad t_I=1.
\]

Fix finitely many task labels \(\nu\), all sectors, and every
nonidentity Pauli setting. Let \(M\) be the total number of these
settings.

\begin{assumption}[Complete validation contract]
\label{ass:validation}
For each setting \((\nu,k,P)\):
\begin{enumerate}
\item conditional on a declared calibration record \(\mathcal Z\),
the positive sample count \(N_{\nu kP}\) is fixed before validation;
\item the scores \(Z_{\nu kP,i}\in[-1,1]\) are conditionally
independent with a common mean \(m_{\nu kP}\);
\item all settings for \((\nu,k)\) test the same complete channel;
\item on an event \(G\), with \(\Pr(G^c)\le\beta\),
\[
 |m_{\nu kP}-t_{\nu kP}|\le b_{\nu kP};
\]
\item on that event, the deployment channel obeys
\[
 \ddia(\Phi^\star_{\nu,k},\Phi_{\nu,k})
 \le b^{\mathrm{tr}}_{\nu,k}.
\]
\end{enumerate}
Preparation, decoding, losses, failure branches, and score
assignment are specified completely. No success-postselected
channel is substituted for the deterministic channel being claimed.
\end{assumption}

A failure flag can be converted to a predeclared replacement output
before scoring. If failures are instead retained as extra output
states, the output dimension and tomography must be enlarged.
Unsupported calibration or transfer hypotheses make the deployment
verdict inconclusive.

\subsection{The simultaneous certificate}

For \(0<\alpha_{\mathrm s}<1\), define
\begin{align}
 \widehat t_{\nu kP}
 &=\frac1{N_{\nu kP}}\sum_iZ_{\nu kP,i},\\
 r_{\nu kP}
 &=\sqrt{\frac2{N_{\nu kP}}
                  \log\frac{2M}{\alpha_{\mathrm s}}},\\
 a_{\nu kP}&=r_{\nu kP}+b_{\nu kP}.
\end{align}
Set \(\widehat t_{\nu kI}=1\) exactly and form
\[
 \widehat J_{\nu,k}
 =\frac1D\sum_P\widehat t_{\nu kP}P.
\]
It need not be a physical density matrix. No physicality
projection is assumed.

Put
\begin{align}
 R_{\nu,k}
 &=\left(\sum_{P\ne I}a_{\nu kP}^2\right)^{1/2},\\
 \widehat\delta_{\nu,k}
 &=\frac12\|\widehat J_{\nu,k}-J(\mathcal T_k)\|_1,\\
 L_{\nu,k}
 &=\clip{\widehat\delta_{\nu,k}
                  -\tfrac12R_{\nu,k}-b^{\mathrm{tr}}_{\nu,k}},\\
 U_{\nu,k}
 &=\min\left\{1,\,
 n\bigl(\widehat\delta_{\nu,k}+\tfrac12R_{\nu,k}\bigr)
                         +b^{\mathrm{tr}}_{\nu,k}\right\}.
 \label{eq:intervals}
\end{align}
Here \(\clip{x}=\max\{0,\min\{1,x\}\}\).
For a complete direct-sum task set
\[
 L_\nu=\max_kL_{\nu,k},\qquad
 U_\nu=\max_kU_{\nu,k}.
\]

\begin{theorem}[Simultaneous reference-preserving certification]
\label{thm:simultaneous}
Under Assumption~\ref{ass:validation},
\[
 \Pr\left\{
 L_{\nu,k}\le
 \ddia(\Phi^\star_{\nu,k},\mathcal T_k)
 \le U_{\nu,k}
 \text{ for every }\nu,k
 \right\}
 \ge1-\alpha_{\mathrm s}-\beta.
\]
The same event gives
\[
 L_\nu\le
 \ddia_{\oplus}(\Phi^\star_\nu,\mathcal T)
 \le U_\nu
\]
for every task.

If certification is declared only when \(U_\nu\le\varepsilon_\nu\),
then
\[
 \Pr\{\text{some certified task has actual error }
                 >\varepsilon_\nu\}
 \le\alpha_{\mathrm s}+\beta.
\]
\end{theorem}

\begin{proof}
Conditional Hoeffding bounds give
\[
 \Pr\{|\widehat t_{\nu kP}-m_{\nu kP}|>r_{\nu kP}
                       \mid\mathcal Z\}
 \le\frac{\alpha_{\mathrm s}}M
 \quad\cite{Hoeffding}.
\]
A union bound gives simultaneous score coverage.
On its intersection with \(G\), orthogonality of the Pauli basis
implies
\[
 \|\widehat J_{\nu,k}-J(\Phi_{\nu,k})\|_2^2
 \le\frac1D\sum_{P\ne I}a_{\nu kP}^2.
\]
Since \(\|A\|_1\le\sqrt D\|A\|_2\),
\[
 \|\widehat J_{\nu,k}-J(\Phi_{\nu,k})\|_1
 \le R_{\nu,k}.
\]
The reverse triangle inequality gives
\[
 |\widehat\delta_{\nu,k}
                 -\delta(\Phi_{\nu,k},\mathcal T_k)|
 \le\tfrac12R_{\nu,k}.
\]
Lemma~\ref{lem:choi} and the deployment triangle inequality give
\eqref{eq:intervals}. Equation~\eqref{eq:blockerror} gives the
direct-sum intervals.

Integrating the conditional statistical failure probability and
adding \(\Pr(G^c)\) proves the probability bound.
Every invalid acceptance is outside the simultaneous coverage event.
\end{proof}

The proof re-establishes the finite-confidence and complete-accounting
interfaces compared in Oai265-B, Oai280-C, and Oai290-A
\cite{Cheung265B,Cheung280C,Cheung290A}.
No data are reported as collected.

\begin{corollary}[Finite sample sufficiency for the ideal benchmark]
\label{cor:samples}
Suppose the tested and deployed two-leg channels are ideal,
calibration bias is zero, and all setting counts equal \(N\).
For a common tolerance \(\varepsilon>0\), the sufficient condition
\[
 N\ge
 \frac{2n^2(D^2-1)}{\varepsilon^2}
                  \log\frac{2M}{\alpha_{\mathrm s}}
\]
ensures \(U_\nu\le\varepsilon\) on the simultaneous score-coverage
event.
\end{corollary}

\begin{proof}
For the ideal channel,
\(J(\Phi_{\nu,k})=J(\mathcal T_k)\).
Score coverage gives
\(\widehat\delta_{\nu,k}\le R_{\nu,k}/2\).
Thus \(U_{\nu,k}\le nR_{\nu,k}\).
Here
\[
 R_{\nu,k}
 =\sqrt{D^2-1}\sqrt{\frac2N
                         \log\frac{2M}{\alpha_{\mathrm s}}}.
\]
The displayed count makes this upper bound at most \(\varepsilon\).
\end{proof}

This is a sufficient finite count, not sample-optimality.
The statistical guarantee is simultaneous coverage and
false-acceptance control, not a posterior probability conditional
on acceptance.

\subsection{Decision language}

For each tested programme:
\begin{enumerate}
\item \(U_\nu\le\varepsilon_\nu\): certified at the declared tolerance;
\item \(L_\nu>\varepsilon_\nu\): that implemented programme is excluded;
\item otherwise: inconclusive.
\end{enumerate}
An all-decoder exclusion is available only from an additional
uniform argument, such as Theorem~\ref{thm:loss}.
A failed upper certificate alone supplies no such exclusion.

\section{Phase 3: an independent free-scalar consistency audit}
\label{sec:free}

\subsection{Field input and its limited role}

Independently supply the four-dimensional vacuum free massless
scalar representation on Minkowski space,
\[
 \eta=\diag(1,-1,-1,-1),\qquad \Box\phi=0,
\]
its normally ordered Wick distributions, and their common
polynomial domain. Whenever bounded-net affiliation is invoked,
a compatible free-field Wick realization is explicitly part
of this separate input.

No theorem identifies this representation with
\(\mathsf{Loc}_d(E)\), a \(W\)-algebra category, or either category
in Record~\ref{record:069}.
We prove its algebraic improvement and finite coverage directly.

Set
\[
 Q=:\!\phi^2\!:,\qquad
 K_0=:\!\partial_\rho\phi\,\partial^\rho\phi\!:,
\]
\[
 A_{\mu\nu}=:\!\partial_\mu\phi\,\partial_\nu\phi\!:,
 \qquad
 T^{\mathrm{can}}_{\mu\nu}
 =A_{\mu\nu}-\frac12\eta_{\mu\nu}K_0,
\]
\[
 C_{\mu\nu}
 =(\partial_\mu\partial_\nu-\eta_{\mu\nu}\Box)Q.
\]

\begin{proposition}[Free-scalar improvement and currents]
\label{prop:improvement}
Distributionally on the supplied Wick domain,
\[
 \Box Q=2K_0,\qquad
 (T^{\mathrm{can}})^\mu{}_\mu=-K_0,\qquad
 \partial^\mu C_{\mu\nu}=0.
\]
Hence
\[
 L=-\frac12Q,\qquad
 t_{\mu\nu}=T^{\mathrm{can}}_{\mu\nu}-\frac16C_{\mu\nu}
\]
satisfy
\[
 (T^{\mathrm{can}})^\mu{}_\mu=\Box L,\qquad
 \partial^\mu t_{\mu\nu}=0,\qquad t^\mu{}_\mu=0.
\]
For a conformal Killing field \(X\), the current
\(X^\nu t_{\mu\nu}\) is conserved.
\end{proposition}

\begin{proof}
The Wick derivative rule and \(\Box\phi=0\) give
\(\Box Q=2K_0\).
Contraction of \(T^{\mathrm{can}}\) in four dimensions gives
\(-K_0\). Commuting distributional derivatives gives
\(\partial^\mu C_{\mu\nu}=0\), and
\[
 C^\mu{}_\mu=-3\Box Q=-6K_0.
\]
Canonical conservation follows from the free equation.

For
\[
 \partial_\mu X_\nu+\partial_\nu X_\mu
 =2\sigma_X\eta_{\mu\nu},
 \qquad \sigma_X=\tfrac14\partial_\rho X^\rho,
\]
symmetry gives
\[
 \partial^\mu(X^\nu t_{\mu\nu})
 =X^\nu\partial^\mu t_{\mu\nu}+\sigma_Xt^\mu{}_\mu=0.
\]
\end{proof}

This fixes the standard scalar improvement normalization
\cite{CCJ}. The physical admission of a newly reconstructed
scalar under the general Family 282 hypotheses is not proved
by this calculation.

\subsection{A precisely declared homogeneous class}

Let \(\mathscr W_4\) consist of symmetric rank-two Wick
polynomials:
\begin{enumerate}
\item homogeneous of scaling dimension four under \([\phi]=1\);
\item built from \(\phi\) and finitely many derivatives with
constant real Poincar\'e-covariant coefficients;
\item containing no dimensionful parameter, external tensor,
or new field;
\item identified modulo the free field equation and its derivatives.
\end{enumerate}
Only tensor identities inside this class are asserted.

\begin{lemma}[Conserved stress class]
\label{lem:stressclass}
Every conserved member of \(\mathscr W_4\) has the form
\begin{equation}
 T^{a,b}_{\mu\nu}
 =aT^{\mathrm{can}}_{\mu\nu}+bC_{\mu\nu},
 \qquad a,b\in\bbR.
 \label{eq:stressclass}
\end{equation}
\end{lemma}

\begin{proof}
Dimension and tensor rank restrict a general member to the span
of
\[
 A_{\mu\nu},\quad \eta_{\mu\nu}K_0,\quad
 \partial_\mu\partial_\nu Q,\quad
 \eta_{\mu\nu}:\!\phi^4\!:.
\]
Indeed the bilinear terms have two derivatives;
\(:\!\phi\partial_\mu\partial_\nu\phi\!:\) is
\(\tfrac12\partial_\mu\partial_\nu Q-A_{\mu\nu}\).
The scalar \(:\!\phi\Box\phi\!:\) vanishes on the free equation,
and \(\Box Q=2K_0\).
Odd numbers of remaining tensor indices cannot be contracted
to a symmetric rank-two tensor using invariant tensors.
Any possible antisymmetric epsilon contraction at this
derivative order vanishes after symmetrization.

Write a general member as
\[
 \alpha A_{\mu\nu}+\beta\eta_{\mu\nu}K_0
        +\gamma C_{\mu\nu}
        +\zeta\eta_{\mu\nu}:\!\phi^4\!:.
\]
Its divergence is
\[
 (\alpha/2+\beta)\partial_\nu K_0
                  +\zeta\partial_\nu:\!\phi^4\!:.
\]
Different Wick degrees are independent. Equivalently,
coherent displacements scaled by a real parameter distinguish
the quadratic and quartic powers. Since
\(\partial_\nu:\!\phi^4\!:\) is not identically zero,
conservation forces \(\zeta=0\).
Also \(\partial_\nu K_0\) is not identically zero, as may be
checked on the classical free solution
\(\phi=(x^0)^2+(x^1)^2\).
Therefore \(\beta=-\alpha/2\), proving \eqref{eq:stressclass}.
\end{proof}

\subsection{Two probes with complete coverage inside that class}

Choose a nonnegative \(f\in C_c^\infty(M)\), with
\(\int f=1\), supported in a small double cone.
Choose a smooth free solution \(u\) equal to \(x^0\) on a
neighborhood of that cone, with smooth compactly supported
Cauchy data.

Such a solution is obtained by taking zero initial field and a
smooth velocity cutoff equal to one on a sufficiently large ball.
Finite propagation speed makes it equal to \(x^0\) on the desired
cone. Its data have a finite massless one-particle norm, so its
Weyl displacement defines a coherent state \(\omega_u\).

For vacuum-normal-ordered fields, coherent displacement gives
\[
 \omega_u\bigl(:\!\partial^{\alpha_1}\phi\cdots
                     \partial^{\alpha_m}\phi\!:\bigr)
 =\prod_j\partial^{\alpha_j}u.
\]
This follows by shifting the creation and annihilation operators
by their displacement constants.

Define
\[
 p_1(T)=\omega_u(T_{00}(f)),\qquad
 p_2(T)=\omega_u(T^\mu{}_\mu(f)).
\]

\begin{theorem}[Two-probe homogeneous stress coverage]
\label{thm:stresscoverage}
For \eqref{eq:stressclass},
\[
 p_1(T^{a,b})=\frac a2,\qquad
 p_2(T^{a,b})=-a-6b.
\]
The two probes determine \(a,b\) uniquely:
\[
 a=2p_1,\qquad
 b=-\frac{p_2+2p_1}{6}.
\]
Therefore agreement of the two probes implies equality of every
member of the conserved class \(\mathscr W_4\).

The normalized traceless choice is uniquely
\[
 a=1,\qquad b=-\frac16,\qquad L=-\frac12Q.
\]
\end{theorem}

\begin{proof}
On the probe neighborhood, \(u=x^0\), so
\[
 K_0(u)=1,\qquad
 T^{\mathrm{can}}_{00}(u)=1/2,\qquad
 C_{00}(u)=0,\qquad C^\mu{}_\mu(u)=-6.
\]
The unit-integral smear gives the displayed values.
The coefficient matrix has nonzero determinant \(-3\).
The inverse formulas prove coverage.

The first probe fixes \(a=1\) under the declared normalization
\(p_1=1/2\). Zero trace then fixes \(b=-1/6\).
Within the homogeneous scalar dimension-two Wick class,
\(L=\lambda Q\), and
\(\Box L=2\lambda K_0=-K_0\) gives \(\lambda=-1/2\).
\end{proof}

If two matched probes have discrepancies
\(|\Delta p_1|\le e_1\), \(|\Delta p_2|\le e_2\), then
\[
 |\Delta a|\le2e_1,\qquad
 |\Delta b|\le\frac{2e_1+e_2}{6}.
\]
Consequently every further declared matrix-element test \(F\)
inside the class obeys
\[
 |\Delta T(F)|
 \le2e_1|T^{\mathrm{can}}(F)|
        +\frac{2e_1+e_2}{6}|C(F)|.
\]
A bound on those fixed matrix elements is still required for a
numerical tolerance.

This reproduces the finite-class coverage discipline discussed
in Oai282-B \cite{Cheung282B}, without treating its supplied
theorem as a proof premise.

\subsection{A finite Ward-contact consistency certificate}

Supply matrix-valued joint insertion distributions
\[
 \mathscr T_{\mu\nu},\qquad \mathscr L,\qquad
 \mathscr P_\nu,\qquad \mathscr S,\qquad\mathscr C_X
\]
on a declared joint-test domain. All mixed products and
derivative conventions are specified. Define
\[
 \mathscr t_{\mu\nu}
 =\mathscr T_{\mu\nu}
       +\frac13(\partial_\mu\partial_\nu-\eta_{\mu\nu}\Box)
                                                        \mathscr L,
\]
\[
 E_\nu=\partial^\mu\mathscr T_{\mu\nu}-\mathscr P_\nu,\qquad
 E_{\mathrm{tr}}=\mathscr T^\mu{}_\mu-\Box\mathscr L-\mathscr S,
\]
\[
 K_X=X^\nu\mathscr P_\nu+\sigma_X\mathscr S-\mathscr C_X.
\]

\begin{proposition}[Finite Ward-defect identity]
\label{prop:ward}
For a conformal Killing field \(X\) and a compact joint test \(F\),
\begin{equation}
 \bigl[\partial^\mu(X^\nu\mathscr t_{\mu\nu})
                            -\mathscr C_X\bigr](F)
 =
 \sum_\nu E_\nu(X^\nu F)
     +E_{\mathrm{tr}}(\sigma_XF)+K_X(F).
 \label{eq:ward}
\end{equation}
Thus equality of the named insertion and contact data on both
declared sides gives equality of the named finite Ward residuals.
\end{proposition}

\begin{proof}
The improvement has zero divergence and trace
\(-\Box\mathscr L\).
Contracting the conformal Killing equation with the symmetric
insertion gives
\[
 \partial^\mu(X^\nu\mathscr t_{\mu\nu})
 =X^\nu\partial^\mu\mathscr T_{\mu\nu}
       +\sigma_X(\mathscr T^\mu{}_\mu-\Box\mathscr L).
\]
Substitution and evaluation on \(F\) prove \eqref{eq:ward}.
\end{proof}

This is the finite residual interface of Oai282-A, reproved here
\cite{Cheung282A}. It certifies the named family, including the
multiplied tests \(X^\nu F\) and \(\sigma_XF\).

A conserved Wightman current distribution does not acquire
time-ordered contact terms merely by calling its divergence a
Ward identity. Conversely, derivatives of time-ordered products
need not equal time ordering of field derivatives.
The contact prescription must resolve that distinction
\cite{HollandsWald}.

No stress-current realization of a source categorical functor is
deduced by \eqref{eq:ward}. No bounded-score measurement bridge
for these unbounded field insertions is constructed.

\subsection{Why the stress probes do not certify channel recovery}

\begin{proposition}[Independent coverage requirements]
\label{prop:independentcoverage}
Two-probe equality of the stress family does not determine a
multiplicity-register channel.
\end{proposition}

\begin{proof}
Append the same free scalar representation to the finite register
in two completions. Use identity on the coefficient register in
one completion and \(\Ad_Z\) in the other, while keeping the scalar
representation and both stress probes fixed.
The scalar data agree exactly. The two qubit channels have
half-diamond distance one, detected by a Bell input.
\end{proof}

This is a completion-level separation, not a statement that
field observables can never constrain an independently specified
coupling. No such coupling is part of the present construction.

\section{The exact remaining source-to-operation bridge}
\label{sec:bridge}

\begin{definition}[A Family 069 operational bridge]
\label{def:bridge}
A genuine application of the source correspondence to an
operational recovery claim must supply:
\begin{enumerate}
\item a verified instance of the categorical functor, or an
independently constructed finite restriction with its precise
domain and inverse;
\item named objects and coefficient-extraction maps on both sides;
\item positive finite realizations, traces, preparation maps,
and complete reference conventions;
\item implemented CPTP maps and an access-restricted decoder
whose complete channel represents the specified task;
\item comparison diagrams with the localization--Whittaker maps
and their actual shifts and twists;
\item the relevant central-torsor/transgression compatibility,
including the natural-isomorphism coherence that channels
alone do not detect;
\item the required moving-point and collision compatibilities;
\item separately justified errors between the categorical,
finite-model, tested, and deployed tasks.
\end{enumerate}
\end{definition}

\begin{proposition}[Conditional transfer, with a named norm]
\label{prop:transfer}
Suppose a bridge provides channels on the same declared finite
input and output algebras, and
\[
 \ddia(\Phi_{\mathrm{source}},\Phi_{\mathrm{benchmark}})\le b_1,
 \qquad
 \ddia(\Phi_{\mathrm{benchmark}},\mathcal T)\le b_2.
\]
Then
\[
 \ddia(\Phi_{\mathrm{source}},\mathcal T)\le\min\{1,b_1+b_2\}.
\]
All reference comparisons are retained.
\end{proposition}

\begin{proof}
Apply the triangle inequality for the complete channel norm.
\end{proof}

There is no diamond norm between abstract DG objects without a
positive operational realization. Proposition~\ref{prop:transfer}
therefore does not assign a numerical \(b_1\) to Record~\ref{record:069}.

The finite construction proves the benchmark side of this
interface. It supplies no identification
\[
 D_c(\Bun_G(E))
 \longrightarrow
 \mathsf{Loc}_d(E)
\]
representing the source functor, and no proof of the source
localization--Whittaker characterization.

Likewise, our cyclic character identities are not evidence of
all central-torsor compatibilities, and finite character fusion
is not a proof of compatibility on every Ran collision stratum.

\begin{proposition}[Equivalence alone does not determine accessible recovery]
\label{prop:source-separation}
Fix the same finite categorical equivalence and named label.
Without a fixed positive encoding/access bridge, independent
operational completions can have zero or nonzero full qubit
recovery error.
\end{proposition}

\begin{proof}
For a fixed \(k\), append equal-dimensional accessible and
inaccessible coefficient spaces. One isometry sends
\[
 \psi\longmapsto\psi_R\otimes\ket0_B,
\]
and another sends
\[
 \psi\longmapsto\ket0_R\otimes\psi_B.
\]
Keep the local-system monodromy and categorical character
decomposition unchanged. The first accessible channel is identity;
the second is replacement. Their optimal qubit recovery errors
are respectively zero and \(3/4\), by the Bell argument in
Theorem~\ref{thm:loss}.
\end{proof}

The completions are not asserted to realize the source functor.
They identify precisely the missing quantifier: a category
record without an operational bridge does not select the channel.

\section{Phases 4 and 5: assembly, outcomes, and separation ledger}
\label{sec:assembly}

\begin{theorem}[Oai069-A finite recovery and consistency release]
\label{thm:main}
For the explicitly defined benchmark:
\begin{enumerate}
\item the finite-monodromy local systems on \(E\) have the
constructive dagger equivalence of
Theorem~\ref{thm:category};
\item the Fourier matrix realizes its regular-representation
character comparison and the finite identities
\eqref{eq:finite-weyl};
\item the receiver-only programme
\eqref{eq:decoder} exactly returns the named local-system label
and qubit multiplicity subsystem, uniformly over references
and gauge correlations;
\item removal of coefficient legs has the sharp all-decoder
errors and exact named-effect recovery of
Theorem~\ref{thm:loss};
\item every predeclared complete implemented task has the
matrix-unit coverage and simultaneous finite-confidence
certificate of Section~\ref{sec:certification};
\item the independently supplied free scalar has the unique
normalized traceless stress in the declared homogeneous
conserved class, with two-probe coverage and the finite
Ward-defect comparison of Section~\ref{sec:free}.
\end{enumerate}

No item uses the truth of Record~\ref{record:069}.
No item identifies the finite abelian functor or the free scalar
with a specialization of its connected-simple quantum Langlands
equivalence.
\end{theorem}

\begin{proof}
Items one and two are
Theorem~\ref{thm:category} and Proposition~\ref{prop:fourier}.
Items three and four are
Theorems~\ref{thm:exact} and \ref{thm:loss}.
Item five combines
Proposition~\ref{prop:channelcoverage},
Lemma~\ref{lem:choi}, and Theorem~\ref{thm:simultaneous}.
Item six is
Proposition~\ref{prop:improvement},
Theorem~\ref{thm:stresscoverage}, and
Proposition~\ref{prop:ward}.
Their assumptions and proofs are independent of the source
equivalence.
\end{proof}

\subsection{Completion table for the five phases}

{\small
\renewcommand{\arraystretch}{1.14}
\begin{longtable}{@{}p{0.09\linewidth}p{0.40\linewidth}
                         p{0.45\linewidth}@{}}
\toprule
Phase & Delivered result & Exact boundary\\
\midrule
\endfirsthead
\toprule
Phase & Delivered result & Exact boundary\\
\midrule
\endhead
\bottomrule
\endfoot
1 &
Fixed curve, finite monodromy, named local systems, multiplicity
subsystem, gauge factor, effect, access, error, and validation. &
No unknown arbitrary DG object or continuous monodromy is inferred
from finite label data.\\[4pt]
2 &
Constructive category equivalence, complete channel transport,
receiver-only decoder, programme soundness, access factorization,
sharp reduced-access verdicts, and simultaneous certification. &
Not an operational realization of the source bundle-stack functor.
No laboratory execution is asserted.\\[4pt]
3 &
Independent free-scalar improvement, complete two-probe coverage
inside a specified conserved class, and finite Ward-defect identity. &
No chiral-to-four-dimensional bridge, new general improvement
scalar, or source Ward-current implementation is constructed.\\[4pt]
4 &
Explicit nonidentification and source-to-operation bridge contract. &
No gravitational dictionary, Einstein dynamics, spacetime
emergence, or admission of the inherited graph coefficient.\\[4pt]
5 &
A genuine finite local-system task with constructive decoder and
reference-preserving ideal and finite-confidence certificates. &
The minimum finite benchmark is achieved; the literal Family 069
operational bridge remains a separate construction.\\
\end{longtable}
}

\subsection{Not holography or gravity}

The curve \(E\), Minkowski metric \(\eta\), receiver graph,
Hilbert spaces, and code are inputs.
Their role is not inverted by the proof.

A gravitational extension would need a same-model gauge theory,
states, observables, a boundary/bulk identification, a justified
regime or limit, dynamics, and error control.
The source's gauge-theory motivation does not supply those data.

In particular:
\begin{itemize}
\item a DG equivalence is not an AdS dual;
\item a receiver leg is not a continuum entanglement wedge;
\item a cyclic character is not a metric coefficient;
\item a traceless matter tensor is not an Einstein equation;
\item a finite recovery error is not a Page curve;
\item an abstract inverse functor is not a laboratory decoder.
\end{itemize}

These are scope separations, not universal nonexistence theorems.
Additional bridge theorems are not excluded.

\subsection{The standing series ledger}

As recorded in the supplied dossier \cite{Dossier},
\[
 \boxed{
 \begin{gathered}
 \sid{DeclaredPath}_{\mathrm{gravity}}=\sid{null},\\
 \sid{O.1}=\sid{NOT_CLOSED},\\
 \text{XV-B downstream quantum-admission exit: not opened},\\
 \text{XVI-N and spatially compact Route B: not discharged},\\
 \text{XI-G frozen-class obstruction: not repaired},\\
 Q_HR_1^{(1)}\in\mathscr E_H:\ \text{not asserted}.
 \end{gathered}}
\]
The finite model changes none of the equation spaces, frozen
kernels, cutoffs, or gauge groups of those records.

The supplied III, VI, and Em02-E constructions concern their
declared classical quotients, jets, and relational quantities
\cite{CheungIII,CheungVI,CheungEm02E}.
Those quantities are not local-system multiplicities and are not
derived from our Fourier matrix.

The inherited characteristic-product programme has a different
target. Its internal amplitude and partial supported reference
are not its full joint characteristic map
\cite{CheungXIVI,CheungXIVJ}.
The technical XV-B version 2 proves specified descent and extension
interfaces but does not admit that complete coefficient in the
supplied account \cite{CheungXVB}.

The conditional XVI-N and XI-G results retain their own
quantifiers \cite{CheungXVIN,CheungXIG}.
The word ``free'' does not identify their prepared observable
with the independent scalar audit in this paper.

\subsection{A compact operational report}

A report for an application of this benchmark must contain:
\[
 \begin{array}{ll}
 \text{Object:}&\mathcal L_k\otimes C\text{, with its gauge convention};\\
 \text{Access:}&J\text{ and the declared coefficient legs};\\
 \text{Target:}&\mathcal T\text{ or the named effect};\\
 \text{Programme:}&\mathcal D_{12},\mathcal D_1,\text{ or another tested map};\\
 \text{Resources:}&\text{receiver, supplier, reference, and classical ledgers};\\
 \text{Certificate:}&[L_\nu,U_\nu]\text{ under the complete contract};\\
 \text{Decision:}&\text{certified, tested-programme excluded, or inconclusive};\\
 \text{Uniform exclusion:}&\text{only if a separately proved all-decoder bound applies}.
 \end{array}
\]
Unsupported source transfer or calibration is explicitly
``inconclusive,'' not a small error allowance chosen by fiat.

\appendix

\section{The unrelated six-step characteristic programme}
\label{app:characteristic}

The dossier appends a six-step analytic plan for a different
quantum coefficient. It is not completed by categorical recovery.

For a reconstruction map
\[
 \Rec:\mathscr P\to\mathscr K,
\]
a linear presentation-level candidate \(\widetilde M\) descends
to reconstructed objects exactly when
\[
 \ker\Rec\subseteq\ker\widetilde M.
\]
Indeed, equal reconstructions have a difference in \(\ker\Rec\),
and the inclusion is precisely what makes their images equal.

This elementary condition does not define a missing
distributional product. The supplied relation
\[
 q\delta'(q)+\delta(q)=0
\]
shows why a highest-degree presentation term is not automatically
an intrinsic reconstructed symbol \cite{CheungXIVJ,CheungXVB}.

The remaining analytic chain retains all six separate obligations:
\begin{enumerate}
\item construct the full joint characteristic map on tests meeting
the characteristic locus;
\item construct its joint section and complete controlled remainder;
\item prove the actual uniform transverse estimates;
\item prove compatibility with the permitted contacts and singular
spectators;
\item prove the full output wavefront theorem after composition
and proper pushforward;
\item identify and admit the normalized coefficient in its named
output space.
\end{enumerate}

The supplied internal I2 advancement is not undone by repeating
the historical H2 stop. The missing full target at I3/J2 remains
a different operation \cite{CheungXIVH,CheungXIVI,CheungXIVJ}.

Neither our finite Kraus matrices nor the Ward-defect evaluation
defines that graph. Failure to construct it is not a proof that
every permitted prescription is impossible.

\section{Evidence and verification declarations}
\label{app:evidence}

\paragraph{Source consultation.}
The Family 069 catalogue and public introduction were consulted
for date, normalization, and theorem attribution
\cite{OpenAI069Catalogue,OpenAI069Introduction}.
The supplied long dossier gives further source excerpts and
comparative manuscript labels \cite{Dossier}.
This is not an exhaustive scan or independent proof audit.

\paragraph{Cheung attribution.}
Cheung manuscripts are cited by their supplied identifiers and
versions. Unknown publication metadata are not invented.
Their citations are documentary comparisons.
The locally used finite channel, access, coverage, statistical,
and Ward-defect interfaces have explicit proofs in this manuscript.

\paragraph{Free-field input.}
The independent free representation, Wick products, and compatible
domains are specified inputs. This paper constructs no new field
from the full Family 282 framework and proves no new affiliation
theorem for an unspecified reconstructed operator.

\paragraph{No execution claim.}
No successful pdfLaTeX execution, proof-assistant checking,
statistical validation run, symbolic-code execution, compiler
execution, pulse optimization, calibration, or laboratory
implementation is asserted.

\paragraph{Meaning of certified.}
Theorems provide exact mathematical implications and finite-data
coverage under explicit contracts. An absent dataset is not
replaced by a claimed numerical interval. A certificate for a
complete channel is not evidence for a missing geometric bridge.

\section*{Conclusion}

The proved operational chain is
\[
 \boxed{
 \begin{gathered}
 \text{explicit finite-monodromy local-system equivalence}\\
 +\text{positive coefficient realization}
 +\text{declared receiver access}\\
 \Longrightarrow
 \text{constructive subsystem decoder}\\
 \Longrightarrow
 \text{complete reference-preserving finite certificate}.
 \end{gathered}}
\]
An independent free-scalar audit supplies a separate finite-class
stress and Ward consistency test.

The chain from the full Family 069 functor to this realization has
not been manufactured by replacing its objects or hypotheses.
Its required localization--Whittaker, central-torsor, positivity,
and collision comparisons remain explicit bridge obligations.

The finite benchmark is complete. Family 069 remains an external
source record. No gravitational path is declared, and
\sid{O.1} remains \sid{NOT_CLOSED}.

\begin{thebibliography}{99}

\bibitem{OpenAI069Catalogue}
OpenAI, as attributed in the catalogue.
\emph{Global quantum geometric Langlands at irrational level}.
Family 069-01, 4 October 2026.
Enter Calculus Companion:
\url{https://enter.grok.me/openai/069-01}.
Documentary source record; not independently verified here.

\bibitem{OpenAI069Introduction}
OpenAI, as attributed in the public preprint source.
\emph{Global quantum geometric Langlands at irrational level}.
Introduction, including the shifted-level convention and
\sid{thm:main}, 4 October 2026.
Public source:
\url{https://github.com/openai/math/blob/main/preprints/Global-quantum-geometric-Langlands-at-irrational-level-October-4-2026/build/sections/01-introduction.tex}.
Source consultation is not proof verification.

\bibitem{KapustinWitten}
A.~Kapustin and E.~Witten.
Electric-magnetic duality and the geometric Langlands program.
\emph{Communications in Number Theory and Physics}
\textbf{1} (2007), 1--236.
arXiv:hep-th/0604151.
\url{https://arxiv.org/abs/hep-th/0604151}.
Gauge-theoretic context, not a gravitational premise of this paper.

\bibitem{GaitsgoryQGL}
D.~Gaitsgory.
\emph{Quantum Langlands Correspondence}.
Informal notes written in 2007, posted in 2016.
arXiv:1601.05279.
\url{https://arxiv.org/abs/1601.05279}.

\bibitem{GaitsgoryWhit}
D.~Gaitsgory.
\emph{The local and global versions of the Whittaker category}.
arXiv:1811.02468.
\url{https://arxiv.org/abs/1811.02468}.
The independently stated local/global Whittaker framework is not
replaced by the finite cyclic example.

\bibitem{ArakawaFrenkel}
T.~Arakawa and E.~Frenkel.
Quantum Langlands duality of representations of \(W\)-algebras.
\emph{Compositio Mathematica}
\textbf{155} (2019), 2235--2262.
doi:10.1112/S0010437X19007553.
arXiv:1807.01536.
\url{https://arxiv.org/abs/1807.01536}.

\bibitem{PolishchukRothstein}
A.~Polishchuk and M.~Rothstein.
\emph{Fourier transform for \(D\)-algebras}.
arXiv:math/9901009.
\url{https://arxiv.org/abs/math/9901009}.
Abelian differential-operator transforms have their own geometric
hypotheses and are not the finite matrix transform proved here.

\bibitem{Maldacena}
J.~M.~Maldacena.
The large \(N\) limit of superconformal field theories and supergravity.
\emph{Advances in Theoretical and Mathematical Physics}
\textbf{2} (1998), 231--252.
arXiv:hep-th/9711200.
\url{https://arxiv.org/abs/hep-th/9711200}.
External AdS/CFT context only.

\bibitem{Choi}
M.-D.~Choi.
Completely positive linear maps on complex matrices.
\emph{Linear Algebra and its Applications}
\textbf{10} (1975), 285--290.
doi:10.1016/0024-3795(75)90075-0.

\bibitem{Watrous}
J.~Watrous.
\emph{The Theory of Quantum Information}.
Cambridge University Press, 2018.
Author's book materials:
\url{https://cs.uwaterloo.ca/~watrous/TQI/}.
Channel, Choi, trace-norm, and reference conventions.

\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.
\url{https://arxiv.org/abs/quant-ph/9604034}.

\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{CCJ}
C.~G.~Callan, Jr., S.~Coleman, and R.~Jackiw.
A new improved energy-momentum tensor.
\emph{Annals of Physics}
\textbf{59} (1970), 42--73.
doi:10.1016/0003-4916(70)90394-5.

\bibitem{HollandsWald}
S.~Hollands and R.~M.~Wald.
Conservation of the stress tensor in perturbative interacting
quantum field theory in curved spacetimes.
arXiv:gr-qc/0404074.
\url{https://arxiv.org/abs/gr-qc/0404074}.
Derivative, product, and conservation normalization distinctions.

\bibitem{Dossier}
\emph{User-supplied Oai069-A development plan and comparative
manuscript dossier}.
Includes source statements, retrieved passages, and status records.
Retrieval label \sid{0d99297f}.
No exhaustive corpus scan or independent manuscript-proof
verification is asserted.

\bibitem{Cheung280C}
N.~Cheung.
\emph{Oai280-C}, version 1.
Supplied \sid{Oai280-C.tex} excerpts.
Especially \sid{thm:simultaneous} and the operational-map contract.
Logical-code, subsystem, and finite-algebra certification.
Enter Calculus Companion, \url{https://enter.grok.me}.

\bibitem{Cheung280A}
N.~Cheung.
\emph{Oai280-A}, version 1.
Supplied \sid{Oai280-A.tex} excerpts.
Sector-label certificates and VOA--net separation.
The strong-rationality, positivity, and locality boundaries are retained.
Enter Calculus Companion.

\bibitem{Cheung280B}
N.~Cheung.
\emph{Oai280-B}, version 1.
Supplied \sid{Oai280-B.tex} record.
Finite perfect-tensor and five-qubit decoder benchmarks.
No five-qubit theorem is imported into the different encoder
of this paper.
Enter Calculus Companion.

\bibitem{Cheung290A}
N.~Cheung.
\emph{Oai290-A}, version 1.
Supplied \sid{Oai290-A.tex} excerpts.
Implemented-decoder versus optimal-decoder decisions and
modular/operational separation.
Enter Calculus Companion.

\bibitem{Cheung265B}
N.~Cheung.
\emph{Oai265-B}, version 2.
Supplied \sid{Oai265-B-v2.tex} excerpts.
Especially \sid{thm:main}.
Complete finite-confidence recovery accounting.
Enter Calculus Companion.

\bibitem{CheungEn02B}
N.~Cheung.
\emph{En02-B}, version 1.
Supplied \sid{En02-B-v1.tex} excerpts.
Especially \sid{thm:B1}.
Complete decoder-programme soundness and resource ledgers.
Enter Calculus Companion.

\bibitem{CheungEn02C}
N.~Cheung.
\emph{En02-C}, version 1.
Supplied \sid{En02-C-v1.tex} excerpts.
Especially \sid{thm:C1}.
Reference-preserving sequential execution and actual-storage accounting.
Enter Calculus Companion.

\bibitem{CheungEn05C}
N.~Cheung.
\emph{En05-C}, version 1.
Supplied \sid{En05-C.tex} excerpts.
Especially \sid{thm:comparison}.
Tensor-free snapshot--channel--adjoint equivalence.
Only its finite-algebra interface is reproved and used here.
Enter Calculus Companion.

\bibitem{CheungVIIG}
N.~Cheung.
\emph{VII-G}, version 1.
Supplied \sid{VII-G.tex} excerpts.
Especially \sid{thm:structure}.
Finite direct-sum hybrid channels and complete branch maps.
Enter Calculus Companion.

\bibitem{Cheung282A}
N.~Cheung, as attributed in the supplied dossier.
\emph{Oai282-A}, version 1.
Supplied predecessor text and excerpts.
Especially \sid{thm:ward}.
Declared improvement data, finite Ward-defect certificates,
and local/global separation.
Enter Calculus Companion.

\bibitem{Cheung282B}
N.~Cheung.
\emph{Oai282-B}, version 1.
Supplied \sid{Oai282-B.tex} excerpts and description.
Especially \sid{thm:coverage}.
Free-scalar homogeneous stress-class coverage and its stated
physical-field boundaries.
The full manuscript is not an unchecked proof premise here.
Enter Calculus Companion.

\bibitem{CheungIII}
N.~Cheung.
\emph{Landmark III}, version 2.
Supplied \sid{III.tex} excerpts.
Classical equation-space quotient and jet distinctions.
Enter Calculus Companion.

\bibitem{CheungVI}
N.~Cheung.
\emph{Landmark VI}, version 1.
Supplied \sid{VI.tex} excerpts.
Especially \sid{thm:descent}.
Finite-time second-order classical relational descent.
Enter Calculus Companion.

\bibitem{CheungEm02E}
N.~Cheung.
\emph{Em02-E}, version 1.
Supplied \sid{Em02-E-v1.tex} excerpts.
Especially \sid{thm:lapse}.
Relational lapse on a declared Einstein--scalar background;
not the improvement scalar or a local-system coefficient.
Enter Calculus Companion.

\bibitem{CheungXIVH}
N.~Cheung.
\emph{Landmark XIV-H: Internal kernel relations on the operator route},
version 1.
Supplied \sid{XIV-H.tex} excerpts.
Historical H2 boundary and exact internal kernel maps.
Enter Calculus Companion.

\bibitem{CheungXIVI}
N.~Cheung.
\emph{Landmark XIV-I}, version 1.
Supplied \sid{XIV-I.tex} excerpts.
Especially \sid{thm:I2}, \sid{thm:I3}, and \sid{thm:release}.
Internal amplitude construction and the missing full characteristic map.
Enter Calculus Companion.

\bibitem{CheungXIVJ}
N.~Cheung.
\emph{Landmark XIV-J}, version 1.
Supplied \sid{XIV-J.tex} excerpts.
Especially \sid{prop:descent}, \sid{thm:J2},
and \sid{cor:noadmission}.
Supported-sector references do not admit the complete coefficient.
Enter Calculus Companion.

\bibitem{CheungXVB}
N.~Cheung.
\emph{XV-B}, version 2.
Supplied \sid{XV-B.tex} excerpts.
Relative lifting and normal-jet extension interfaces.
The technical manuscript is distinct from the downstream
quantum-admission exit.
Enter Calculus Companion.

\bibitem{CheungXVIN}
N.~Cheung.
\emph{XVI-N}, version r12.
Supplied \sid{XVI-N.tex} excerpts.
Especially \sid{thm:massless-obstruction}.
Conditional prepared-insertion obstruction and the retained
Route B limitation.
Enter Calculus Companion.

\bibitem{CheungXIG}
N.~Cheung.
\emph{XI-G}, version 1.
Supplied \sid{XI-G.tex} excerpts.
Frozen-class microlocal separation on its stated quantifiers;
not a universal obstruction to every bimodule.
Enter Calculus Companion.

\end{thebibliography}

\end{document}