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\hypersetup{
  pdftitle={En05-C: Tensor-Free Snapshots and Causal Locality for Gauge-Invariant Observable Algebras},
  pdfauthor={Ngo Cheung},
  pdfsubject={State restriction, completely positive factorization, gauge constraints, causal support, and compatible observable refinements}
}

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\title{En05-C: Tensor-Free Snapshots and Causal Locality\\
for Gauge-Invariant Observable Algebras}

\author{Ngo Cheung\\
Independent Researcher, Hong Kong SAR, China\\
\texttt{info@cheungngomedical.com}}

\date{October 2026}

\begin{document}
\maketitle

\begin{abstract}
We reformulate the snapshot--channel--adjoint comparison for unital
local observable subalgebras without a spatial tensor decomposition.
For a unital completely positive Heisenberg channel, equality of
input state restrictions determines the output restriction exactly
when the pulled-back output algebra is contained in the declared
input algebra. The induced local channel is unique, completely
positive, and stable under independent matrix references. The proof
uses separation and Jordan decomposition of functionals, not a
complementary-state preparation channel.

We retain central-sector probabilities, characterize the weaker
conclusion available from restricted preparations, and prove the
normal von Neumann algebra version. Conditional expectations are
optional; their state-preserving existence is not assumed.
An explicit constrained $\mathbb Z_2$ graph supplies centerful regional
algebras, localized complete instruments, successful and failed
containment tests, and an obstruction to independently changing
shared boundary data.

For nets with local commutativity, specified internal Kraus devices
give one-layer containment, algebraic ancestral factorization,
metric support bounds, matched replacement invariance, and
unconditioned no-signalling. A regional channel restriction alone
need not determine its global device extension; this additional
interface is essential. Finally, we construct a bounded-sequence
observable limit and an exact subdivision limit of the finite gauge
model, including compatible states, references, records, and matched
devices. These are observable-level constructions, not a relativistic
field-theory limit. No general gauge-theory transport compilation,
Lorentz covariance, gravitational dynamics, or physical speed
calibration is inferred.
\end{abstract}

\tableofcontents

%=====================================================================
\section{Introduction and precise scope}
\label{sec:intro}
%=====================================================================

The comparison lemmas in Cheung En05-A, En05-B, and En06-A identify
three descriptions of finite hybrid-channel locality:
factorization through a snapshot, determinacy by that snapshot,
and containment of the Heisenberg pullback
\cite{CheungEn05A,CheungEn05B,CheungEn06A}.
Their complementary-state proof uses a spatial tensor interface.
The present paper removes that assumption from the comparison
theorem itself.

Gauge constraints make this distinction useful.
A physical regional observable algebra can have a center and need
not define a tensor factor of the global physical algebra.
Moreover, assigning an observable algebra to a region involves a
boundary convention \cite{CasiniHuertaRosabal}.
Introducing boundary variables or edge modes is a different
construction; it is not required by the theorem proved here
\cite{DonnellyFreidel}.

The elementary operator-algebra ingredients are standard:
Hahn--Banach separation, Jordan decomposition of bounded Hermitian
functionals, inherited matrix positivity of a $C^*$-subalgebra,
and completely positive maps
\cite{Pedersen,Arveson,Stinespring}.
We do not claim a new general separation theorem.
The purpose is to give a complete operational formulation, its
gauge-compatible implementation interface, explicit counterexamples,
and compatible observable-limit constructions.

\subsection{The completed stages}

The supplied plan contains both theorem targets and construction
targets. We address each, with its domain stated before its proof.

\begin{center}
\small
\renewcommand{\arraystretch}{1.12}
\begin{tabularx}{\linewidth}{@{}p{.07\linewidth}p{.33\linewidth}X@{}}
\toprule
Stage & Target & Result and boundary\\
\midrule
A & Tensor-free comparison &
Proved for unital $C^*$-subalgebras and all global states.\\
\addlinespace
B & Centers and preparation classes &
Full center-sensitive snapshots; exact preparation-relative
criterion and counterexamples.\\
\addlinespace
C & References &
Independent matrix amplification; shared-charge references
require a declared joint constrained algebra.\\
\addlinespace
D & Finite gauge demonstrator &
Explicit $\mathbb Z_2$ model, regional algebras, instruments,
and four operational tests.\\
\addlinespace
E & One-layer gauge locality &
Proved for specified internal Kraus devices in a locally
commuting net. Not for arbitrary regional UCP maps.\\
\addlinespace
F & Ancestry and no-signalling &
Proved for that device class, with matched global device
actions and complete records.\\
\addlinespace
G & Normal channels &
Proved for von Neumann subalgebras and all normal states.\\
\addlinespace
H & Observable limits &
A bounded-sequence construction and an exact finite-mode
gauge subdivision limit are constructed. A relativistic
field-theory scaling limit is not asserted.\\
\bottomrule
\end{tabularx}
\end{center}

The last boundary is substantive. Completing an observable-level
refinement construction for a specified model is not the construction
of a continuum Yang--Mills theory, nor the discharge of every
observable-limit or covariance hypothesis in En05-A or En06-A.

\subsection{Two necessary corrections to the proposed route}

First, proving that an operation fixes exterior observables does
not, without an additional argument, place a pulled-back observable
in a chosen regional algebra. It may only place it in a commutant.
Our implementation theorem proves membership by multiplication
inside a declared larger regional algebra.

Second, specifying the restriction of a global device to its own
regional algebra need not specify its action on observables extending
across the boundary. We give an explicit gauge example in
Section~\ref{sec:deviceextension}. Matched replacements therefore
retain complete device actions or specified realizations, not merely
their regional restrictions.

These corrections are needed to turn the plan into a valid
tensor-free causal-network argument.

%=====================================================================
\section{Algebraic interfaces, snapshots, and localization conventions}
\label{sec:interfaces}
%=====================================================================

\subsection{Channels and state restriction}

All $C^*$-algebras are complex and unital. A unital subalgebra has
the same unit as its ambient algebra and is norm closed.

\begin{definition}[Algebraic experiment interface]
An interface consists of global algebras
$\M_{\mathrm{in}},\M_{\mathrm{out}}$, local algebras
$\A_{\mathrm{in}}(I),\A_{\mathrm{out}}(Y)$, unital injective
$*$-homomorphisms
\[
 \iota_I:\A_{\mathrm{in}}(I)\longrightarrow\M_{\mathrm{in}},
 \qquad
 \iota_Y:\A_{\mathrm{out}}(Y)\longrightarrow\M_{\mathrm{out}},
\]
and a Heisenberg channel
\[
 T:\M_{\mathrm{out}}\longrightarrow\M_{\mathrm{in}}
\]
that is unital and completely positive.
\end{definition}

We ordinarily identify the local algebras with their images.
No complementary algebra or tensor decomposition is part of this
definition.

For a state $\omega\in\States(\M_{\mathrm{in}})$, the output state
is
\[
 T_*\omega=\omega\circ T.
\]

\begin{definition}[Tensor-free snapshot]
The input snapshot in $I$ is
\[
 \Snap_I(\omega)=\omega\circ\iota_I.
\]
The output snapshot in $Y$ is
\[
 \Snap_Y(T_*\omega)=\omega\circ T\circ\iota_Y.
\]
\end{definition}

Restriction is affine. It does not require a density operator,
a trace, a partial trace, or a conditional expectation.
For bounded functionals we use the same notation for the dual
restriction map
\[
 r_I:\M_{\mathrm{in}}^*\longrightarrow\A_{\mathrm{in}}(I)^*.
\]

\subsection{Declared local nets}

\begin{definition}[Local observable net]
On a declared collection of regions, a local net assigns unital
subalgebras
\[
 I\longmapsto\A(I)\subseteq\M.
\]
It is isotone if $I\subseteq J$ implies $\A(I)\subseteq\A(J)$.
For a discrete support convention, local commutativity means
\[
 [\A(I),\A(J)]=\{0\}\qquad\text{when }I\cap J=\varnothing.
 \label{eq:localcomm}
\]
A spacetime version may instead impose this condition on spacelike
separated regions.
\end{definition}

Our scheduled implementation theorem uses the discrete condition
\eqref{eq:localcomm}. The abstract comparison theorem does not
require local commutativity.

A gauge-model specification must identify:
\begin{enumerate}[label=(\roman*)]
\item the global physical algebra and its representation;
\item which constraint-preserving observables are assigned to $I$;
\item the treatment of boundary electric flux or other central data;
\item the allowed complete devices and preparations.
\end{enumerate}
These choices precede every locality conclusion.

\begin{remark}[No automatic duality]
Local commutativity gives
\[
 \A(I)\subseteq\A(J)'\cap\M
\]
for disjoint $I,J$. It does not give equality.
We never replace membership in $\A(I)$ by commutation with a
declared exterior algebra unless such an equality has been proved.
\end{remark}

\begin{remark}[No automatic additivity]
We do not assume
\[
 \A(I\cup J)=C^*(\A(I)\cup\A(J)).
\]
A gauge-invariant loop observable can belong to the algebra of a
union without being generated by the algebras of its separate
pieces. Thus equal one-piece snapshots need not determine the
snapshot on the union.
\end{remark}

%=====================================================================
\section{Stage A: the tensor-free comparison theorem}
\label{sec:comparison}
%=====================================================================

We use the standard Jordan decomposition of a bounded Hermitian
functional:
\[
 f=f_+-f_-,
 \qquad f_\pm\ge0,
 \qquad \norm f=f_+(1)+f_-(1).
\]
In a von Neumann algebra, the decomposition of a normal Hermitian
functional is normal. These are background functional-analysis
facts \cite{Pedersen,Sakai}.

\begin{lemma}[Separation by equal-snapshot states]
\label{lem:separation}
Let $\A\subseteq\M$ be a unital $C^*$-subalgebra.
For $h=h^*\in\M$, the following are equivalent:
\begin{enumerate}[label=(\roman*)]
\item $h\in\A$;
\item $\omega(h)=\nu(h)$ whenever $\omega,\nu\in\States(\M)$
have equal restrictions to $\A$.
\end{enumerate}
\end{lemma}

\begin{proof}
The forward implication follows by restriction.

Suppose $h\notin\A$. The real closed linear subspace
$\A_{\sa}$ does not contain $h$.
Real Hahn--Banach separation gives a bounded real functional
$\ell$ on $\M_{\sa}$ such that
\[
 \ell|_{\A_{\sa}}=0,\qquad \ell(h)\ne0.
\]
Its complexification is a bounded Hermitian functional $f$ on
$\M$, vanishing on $\A$.

Write $f=f_+-f_-$.
Because $1\in\A$, we have
\[
 f_+(1)=f_-(1)=m.
\]
The nonzero value $f(h)$ implies $m>0$.
Moreover,
\[
 f_+|_\A=f_-|_\A.
\]
Thus
\[
 \omega=m^{-1}f_+,\qquad \nu=m^{-1}f_-
\]
are states with equal snapshots on $\A$, but
\[
 \omega(h)-\nu(h)=m^{-1}f(h)\ne0.
\]
This contradicts (ii).
\end{proof}

\begin{theorem}[Tensor-free snapshot--channel--adjoint equivalence]
\label{thm:comparison}
For the interface of Section~\ref{sec:interfaces}, the following
are equivalent:
\begin{enumerate}[label=(\roman*)]
\item There exists a UCP map
\[
 K:\A_{\mathrm{out}}(Y)\longrightarrow\A_{\mathrm{in}}(I)
\]
such that
\begin{equation}
 T\circ\iota_Y=\iota_I\circ K.
 \label{eq:algebrafactor}
\end{equation}
\item For every pair of global input states,
\begin{equation}
 \omega\circ\iota_I=\nu\circ\iota_I
 \quad\Longrightarrow\quad
 \omega\circ T\circ\iota_Y
 =\nu\circ T\circ\iota_Y.
 \label{eq:snapshotdet}
\end{equation}
\item The Heisenberg pullback satisfies
\begin{equation}
 T(\iota_Y(\A_{\mathrm{out}}(Y)))
 \subseteq\iota_I(\A_{\mathrm{in}}(I)).
 \label{eq:containment}
\end{equation}
\end{enumerate}
When these conditions hold, $K$ is unique and
\begin{equation}
 r_YT_*=K_*r_I,
 \qquad K_*(\eta)=\eta\circ K,
 \label{eq:dual-factor}
\end{equation}
on bounded functionals, and hence on states.
No spatial complement or state-extension channel is used.
\end{theorem}

\begin{proof}
Equation~\eqref{eq:algebrafactor} gives
\eqref{eq:snapshotdet} by evaluation against states, and also
gives \eqref{eq:containment}.

Assume (ii). For a self-adjoint
$b\in\A_{\mathrm{out}}(Y)$, complete positivity makes
$T(\iota_Y(b))$ self-adjoint.
Its expectation is equal on every pair of states with equal
input snapshots. Lemma~\ref{lem:separation} places it in
$\iota_I(\A_{\mathrm{in}}(I))$.
Every output element is a complex linear combination of two
self-adjoint elements, proving (iii).

Assume (iii). Define
\[
 K=\iota_I^{-1}\circ T\circ\iota_Y.
\]
It is linear and unital.
For every matrix size $r$, the inclusion $\iota_I^{(r)}$
is a complete order embedding.
If $[b_{ij}]\ge0$ in $M_r(\A_{\mathrm{out}}(Y))$, then
\[
 [T(\iota_Y(b_{ij}))]\ge0
\]
in $M_r(\M_{\mathrm{in}})$.
Its entries lie in the input local algebra, where the inherited
positive cone is the same. Therefore $K$ is completely positive.

Injectivity of $\iota_I$ proves uniqueness.
Taking dual maps in \eqref{eq:algebrafactor} gives
\eqref{eq:dual-factor}.
\end{proof}

\begin{remark}[The local channel is not a homomorphism]
Nothing in Theorem~\ref{thm:comparison} makes $K$
multiplicative. For example, dephasing on $M_2$ sends the
Pauli operator $X$ to zero and $X^2=1$ to $1$.
\end{remark}

\begin{corollary}[Recovery of the supplied hybrid comparison]
\label{cor:tensorcase}
For finite tensor-product hybrid interfaces, use the local
inclusions by identity extension and let $T=\Phi^*$.
Then \eqref{eq:dual-factor} is precisely
\[
 R_Y\Phi=K_*R_I.
\]
Thus Theorem~\ref{thm:comparison} recovers
\Source{lem:comparison} in En05-A, En05-B, and En06-A
without their complementary-state proof.
\end{corollary}

\begin{proof}
Under finite-dimensional trace duality, the dual of identity
extension is partial trace together with classical marginalization.
The dual of a UCP map is a complete trace-preserving channel.
Apply Theorem~\ref{thm:comparison}.
\end{proof}

\begin{remark}[Restriction versus extension]
A state on a unital $C^*$-subalgebra has a state extension to the
ambient algebra by the usual positive extension theorem.
This does not provide a canonical extension, an affine
extension channel, or independently compatible extensions for
several regions. None is required in the proof above.
\end{remark}

%=====================================================================
\section{Stage B: centers and restricted preparations}
\label{sec:centers}
%=====================================================================

\subsection{Center-sensitive affine snapshots}

A finite-dimensional regional algebra has an intrinsic decomposition
\[
 \A(I)\cong\bigoplus_{\alpha=1}^m M_{d_\alpha}.
\]
Let $e_\alpha$ be its minimal central projections.

\begin{proposition}[Finite center-sensitive snapshot]
\label{prop:centers}
Every state on $\A(I)$ is uniquely represented by positive matrices
\[
 \omega_I=\bigoplus_{\alpha=1}^m\omega_\alpha,
 \qquad
 \sum_\alpha\Tr\omega_\alpha=1.
\]
Its sector probabilities and conditional states are
\[
 p_\alpha=\omega_I(e_\alpha)=\Tr\omega_\alpha,
 \qquad
 \rho_\alpha=\omega_\alpha/p_\alpha\quad(p_\alpha>0).
\]
The complete array is affine in the state.
The collection of normalized conditional states alone is not
a complete snapshot.
\end{proposition}

\begin{proof}
Trace duality on each matrix summand gives the unique positive
matrix $\omega_\alpha$. Evaluation on the unit gives the
normalization, and evaluation on $e_\alpha$ gives $p_\alpha$.
Conditional normalization introduces division by $p_\alpha$.
Without the probabilities, distinct arrays can have the same
normalized matrices.
\end{proof}

This is the direct-sum snapshot convention already retained in
En05-A and En06-A \cite{CheungEn05A,CheungEn06A}.
Gauge-theory discussions of local operational access similarly
require care with superselection sectors
\cite{VanAcoleyenEtAl}.

\begin{example}[Missing sector probabilities]
\label{ex:missingcenter}
Let $\A=M_2\oplus M_2$, and fix a density matrix $\rho$.
The states
\[
 \eta_p=p\rho\oplus(1-p)\rho,\qquad 0<p<1,
\]
have the same normalized state $\rho$ in both sectors.
Nevertheless the central projection $e_+=(1,0)$ has
\[
 \eta_p(e_+)=p.
\]
Thus a complete central readout distinguishes $p=1/4$ from $p=3/4$.
The finite gauge example below realizes these states as restrictions
of physical global states.
\end{example}

\subsection{The exact preparation-relative criterion}

\begin{definition}[Equal-snapshot difference space]
For an admitted set
$\mathcal S_{\mathrm{adm}}\subseteq\States(\M)$, define
\[
 \mathcal D_I=
 \spanlin_{\CC}
 \{\omega-\nu:
   \omega,\nu\in\mathcal S_{\mathrm{adm}},
   \ \omega|_{\A(I)}=\nu|_{\A(I)}\}.
\]
Its annihilator is
\[
 \mathcal D_I^\perp
 =
 \{a\in\M:f(a)=0\text{ for every }f\in\mathcal D_I\}.
\]
\end{definition}

\begin{proposition}[Preparation-relative snapshot determinacy]
\label{prop:relative}
Snapshot determinacy tested only on $\mathcal S_{\mathrm{adm}}$
is equivalent to
\[
 T(\A_{\mathrm{out}}(Y))\subseteq\mathcal D_I^\perp.
\]
It implies literal regional containment if
\[
 \mathcal D_I^\perp=\A_{\mathrm{in}}(I).
 \label{eq:fiberseparation}
\]
For the full global state space, \eqref{eq:fiberseparation} holds.
\end{proposition}

\begin{proof}
Equality of the two output restrictions is exactly
\[
 (\omega-\nu)(T(b))=0
 \qquad(b\in\A_{\mathrm{out}}(Y))
\]
for the generating pairs in the definition.
Taking their linear span gives the first assertion.
The second is immediate.
For the full state space, local elements belong to the annihilator,
and Lemma~\ref{lem:separation} excludes every self-adjoint element
outside the local algebra. Complexification gives equality.
\end{proof}

\begin{remark}[No automatic algebra on the annihilator]
For a restricted state class, $\mathcal D_I^\perp$ is a closed
linear space containing the local algebra. It need not be closed
under multiplication. Nor need the globally undetectable space
\[
 \{a:\omega(a)=0\text{ for all }\omega\in\mathcal S_{\mathrm{adm}}\}
\]
be an ideal. An operational quotient is not automatically a
$C^*$-algebra quotient.
\end{remark}

\begin{example}[Vacuous determinacy]
\label{ex:restricted}
Take $\M=M_2$, $\A=\CC1$, and $\mathcal S_{\mathrm{adm}}=\{\omega_0\}$.
The identity channel passes every equal-snapshot test on this
preparation class, but
\[
 \id(M_2)\not\subseteq\CC1.
\]
All-state quantification, or an explicit fiber-separation condition,
is therefore essential.
\end{example}

\begin{remark}[Fixed sectors]
If preparations are confined to a superselection sector, the
appropriate comparison theorem is formulated in that sector's
represented ambient algebra and its represented regional
subalgebras. The sector unit replaces the original unit.
The output channel must preserve the stated sector interface.
A full-algebra conclusion cannot be recovered from untested sectors.
\end{remark}

%=====================================================================
\section{Stage C: independent references and complete instruments}
\label{sec:references}
%=====================================================================

\begin{corollary}[Reference-stable comparison]
\label{cor:reference}
Under Theorem~\ref{thm:comparison}, for every $r\ge1$,
\[
 (T\circ\iota_Y)\otimes\id_{M_r}
 =
 (\iota_I\otimes\id_{M_r})\circ(K\otimes\id_{M_r}).
 \label{eq:referencefactor}
\]
Consequently, equal joint restrictions of two states on
$\M_{\mathrm{in}}\otimes M_r$ to
$\A_{\mathrm{in}}(I)\otimes M_r$ give equal joint output
restrictions on $\A_{\mathrm{out}}(Y)\otimes M_r$.
\end{corollary}

\begin{proof}
Amplify the linear identity \eqref{eq:algebrafactor}.
Complete positivity ensures positivity at every matrix size.
Evaluation against the joint states gives the conclusion.
\end{proof}

This tensor product represents an independent external reference,
not a spatial complement of the gauge system.
It is the matrix-level version of the reference clause in the supplied
comparison and soundness interfaces
\cite{CheungEn05A,CheungEn05B,CheungEn06A,CheungEn02C}.
The operator-system framework distinguishes matrix-level positivity
from mere scalar positivity \cite{Arveson}.

\begin{remark}[The joint-snapshot caveat]
Equal restrictions on $\A(I)$ alone do not imply equal restrictions
on $\A(I)\otimes M_r$.
Two different Bell states have the same system marginal but different
joint system--reference states.
Theorem~\ref{thm:comparison} does not erase those correlations.
\end{remark}

\begin{remark}[References carrying shared gauge charge]
If the reference shares a Gauss constraint with the system, an
unrestricted tensor product is not its declared physical interface.
Specify the joint constrained global and local algebras and apply
Theorem~\ref{thm:comparison} to those inclusions.
No claim of independent reference preparation then follows.
\end{remark}

\begin{definition}[Complete algebraic recorded instrument]
A finite instrument with outcomes $y$ consists of CP maps
\[
 T_y:\M_{\mathrm{out}}\longrightarrow\M_{\mathrm{in}},
 \qquad
 \sum_yT_y(1)=1.
\]
Its recorded Heisenberg channel is
\[
 \widehat T:\bigoplus_y\M_{\mathrm{out}}\longrightarrow\M_{\mathrm{in}},
 \qquad
 \widehat T((a_y)_y)=\sum_yT_y(a_y).
\]
\end{definition}

\begin{proposition}[Complete record accounting]
\label{prop:records}
The map $\widehat T$ is UCP.
The outcome probability in state $\omega$ is
$\omega(T_y(1))$.
Discarding the record gives the nonselective channel
\[
 T_{\mathrm{ns}}=\sum_yT_y.
\]
These constructions preserve all independent matrix references.
\end{proposition}

\begin{proof}
At every matrix level, positivity in a direct sum is blockwise.
Each summand is mapped positively, so their sum is completely
positive. The stated completeness identity is unitality.
The central unit of outcome block $y$ pulls back to $T_y(1)$,
giving its probability.
Embedding an unrecorded observable diagonally gives
$T_{\mathrm{ns}}$. Amplification preserves the identities.
\end{proof}

Selective normalized states are not substituted for this complete
channel. This is the same complete-history distinction used in the
finite program and readout accounting of Landmark I, En04-A,
En02-C, and Hawk04-B
\cite{CheungI,CheungEn04A,CheungEn02C,CheungHawk04B}.

%=====================================================================
\section{Conditional expectations are optional}
\label{sec:expectations}
%=====================================================================

\begin{definition}[Conditional expectation]
A conditional expectation onto $\A\subseteq\M$ is a UCP map
$E:\M\to\A$ that fixes $\A$ pointwise and satisfies
\[
 E(a_1xa_2)=a_1E(x)a_2
 \qquad(a_1,a_2\in\A,\ x\in\M).
\]
When convenient, its inclusion into $\M$ is understood.
\end{definition}

\begin{proposition}[Expectation-based certificate]
\label{prop:expectation}
If such an $E$ exists, then the conditions of
Theorem~\ref{thm:comparison} are equivalent to
\[
 E\,T(b)=T(b)\qquad(b\in\A_{\mathrm{out}}(Y)).
\]
In that case $K=E\,T|_{\A_{\mathrm{out}}(Y)}$.
Also, every local state $\eta$ has the common affine extension
$\eta\circ E$.
\end{proposition}

\begin{proof}
The fixed-point range of $E$ is exactly $\A$.
Thus the displayed equality is equivalent to containment.
Composition with $E$ then gives the unique local channel.
Since $E|_\A=\id$, $\eta\circ E$ restricts to $\eta$.
\end{proof}

This is additional infrastructure, not a premise of
Theorem~\ref{thm:comparison}.

\begin{proposition}[Finite-dimensional expectation]
\label{prop:finiteexpect}
Suppose $\M=\mathcal B(\Hil)$ and
\[
 \Hil=\bigoplus_\alpha(\CC^{d_\alpha}\otimes\CC^{m_\alpha}),
 \qquad
 \A=\bigoplus_\alpha(M_{d_\alpha}\otimes1_{m_\alpha}).
\]
Then
\[
 E(x)=
 \bigoplus_\alpha
 \left[
  (\id\otimes\tau_{m_\alpha})(e_\alpha x e_\alpha)
       \otimes1_{m_\alpha}
 \right],
 \qquad
 \tau_m=\Tr/m,
 \label{eq:finiteexpect}
\]
is a trace-preserving conditional expectation.
\end{proposition}

\begin{proof}
Central-block pinching, partial normalized trace, and identity
extension are CP. The formula is unital and fixes $\A$.
Moving the left and right matrix factors through the partial trace
gives bimodularity. The ordinary trace of the resulting block equals
that of the original diagonal block, and off-diagonal blocks have
zero trace.
\end{proof}

\begin{remark}[State-preserving expectations]
For a von Neumann subalgebra and a faithful normal state $\varphi$,
the existence of a normal $\varphi$-preserving conditional
expectation is characterized by invariance of the subalgebra under
the modular automorphism group of $\varphi$.
We use this only as a stated external criterion
\cite{Takesaki}.
Neither a regional declaration nor gauge averaging automatically
verifies it.
\end{remark}

The finite example in Section~\ref{sec:finiteexpectexample} explicitly
distinguishes gauge averaging, regional expectation, and
state-preserving expectation.

%=====================================================================
\section{Stage D: a finite constrained \texorpdfstring{$\mathbb Z_2$}{Z2} model}
\label{sec:gauge}
%=====================================================================

\subsection{Constraint-preserving regional algebras}

Let $\Gamma=(V,E)$ be a finite connected graph without self-loops;
parallel links are allowed. Associate a qubit to each link:
\[
 \Hil_{\kin}=\bigotimes_{e\in E}\CC^2.
\]
Let $X_e,Z_e$ be its Pauli operators.
The Gauss operators are
\[
 G_v=\prod_{e\ni v}X_e.
 \label{eq:gauss}
\]
They commute, and $\prod_vG_v=1$.

Define
\[
 \Hil_{\phys}=\{\psi:G_v\psi=\psi\text{ for every }v\},
 \qquad
 P=\text{orthogonal projection onto }\Hil_{\phys}.
\]
The tensor product above is kinematic.
We do not assume that the constrained physical Hilbert space factors
across a link-region partition.

This finite pure-gauge choice is a specified discrete model.
The broader Hamiltonian lattice-gauge setting is reviewed in the
original formulation of Kogut and Susskind \cite{KogutSusskind};
no Hamiltonian from that work is a premise here.

For $S\subseteq E$, let
\[
 \F(S)=
 \{a\in\mathcal B(\Hil_{\kin}):
   a\text{ acts identically outside }S,\ [a,G_v]=0\ \forall v\}.
\]
Its physical image is
\[
 \A(S)=\qphys(\F(S)),
 \qquad
 \qphys(a)=a|_{\Hil_{\phys}},
 \qquad
 \M=\mathcal B(\Hil_{\phys}).
 \label{eq:gaugenet}
\]

\begin{proposition}[A physical gauge net]
\label{prop:gaugenet}
The map $\qphys$ is a unital $*$-homomorphism on the global
constraint-preserving algebra.
Each $\A(S)$ is a unital $C^*$-subalgebra of $\M$.
The net is isotone and locally commuting on disjoint link sets.
Its global regional algebra satisfies $\A(E)=\M$.
Every state on $\M$ is an admitted physical state in this model.
\end{proposition}

\begin{proof}
A constraint-preserving observable commutes with $P$.
Hence its restriction preserves products and adjoints:
\[
 PabP|_{\Hil_{\phys}}
 =
 (PaP)(PbP)|_{\Hil_{\phys}}.
\]
The image is a finite-dimensional unital $*$-algebra, therefore
norm closed.
Isotony follows from kinematic support inclusion.
Disjoint kinematic supports commute, and their physical images
continue to commute because restriction is multiplicative.

For $c\in\mathcal B(\Hil_{\phys})$, extend it by zero on
$\Hil_{\phys}^\perp$. Every $G_v$ is the identity on
$\Hil_{\phys}$ and preserves its orthogonal complement, so this
extension commutes with every $G_v$. Its restriction is $c$,
proving $\A(E)=\M$.
\end{proof}

\begin{remark}[Compression is not generally multiplicative]
For arbitrary kinematic operators that do not preserve the physical
subspace, $a\mapsto PaP$ need not preserve products.
The proof above uses $[a,P]=0$ explicitly.
\end{remark}

\subsection{Pauli description and boundary convention}

Write, up to a scalar phase,
\[
 X^\alpha Z^\beta
 =
 \prod_{e\in E}X_e^{\alpha_e}Z_e^{\beta_e},
 \qquad
 \alpha,\beta\in\Ztwo^E.
\]
Let $\partial:\Ztwo^E\to\Ztwo^V$ be the graph incidence map.

\begin{lemma}[Invariant Pauli basis]
\label{lem:paulibasis}
The algebra $\F(S)$ is spanned by Pauli monomials supported in $S$
whose $Z$-exponent satisfies
\[
 \partial\beta=0.
 \label{eq:cyclecondition}
\]
For connected $\Gamma$,
\[
 \dim\Hil_{\phys}=2^{|E|-|V|+1}.
\]
\end{lemma}

\begin{proof}
Conjugation by $G_v$ multiplies a Pauli monomial by
$(-1)^{(\partial\beta)_v}$.
Since Pauli monomials form a basis, the invariant subspace is exactly
the stated span.

In the joint $X$-eigenbasis, a physical basis vector has eigenvalue
$(-1)^{f_e}$ on link $e$, with $\partial f=0$.
The incidence map of a connected graph has rank $|V|-1$ over
$\Ztwo$, so its kernel has dimension $|E|-|V|+1$.
\end{proof}

Thus all supported electric operators $X_e$ are retained.
Supported closed $Z$-strings are retained when their full cycles
lie in the region. Open charged strings are not inserted.
This is the boundary convention for the present paper.

\subsection{The theta graph}

Take two vertices joined by three distinct links $1,2,3$.
Both vertex constraints are
\[
 G=X_1X_2X_3.
\]
The physical Hilbert space has dimension four.

Use the $X$-eigenbasis with eigenvalues $a,b,c\in\{+1,-1\}$.
The constraint is $abc=1$.
Define an orthonormal physical basis
\[
 \ket{f,q}_{\phys}
 =
 \ket{q}_{X,1}\ket{fq}_{X,2}\ket f_{X,3},
 \qquad f,q\in\{+1,-1\}.
 \label{eq:theta-basis}
\]
This identifies the vector space with a sector label and a logical
qubit. It is not a spatial factorization into the region
$S=\{1,2\}$ and its complement $\{3\}$.

On $\Hil_{\phys}$ set
\[
 F=X_3=X_1X_2,\qquad
 Q=X_1,\qquad
 R=Z_1Z_2.
\]
Then
\[
 F\ket{f,q}=f\ket{f,q},\quad
 Q\ket{f,q}=q\ket{f,q},\quad
 R\ket{f,q}=\ket{f,-q}.
 \label{eq:logical-paulis}
\]
Let $e_f=(1+fF)/2$.

\begin{theorem}[Regional algebras and their shared center]
\label{thm:thetaalgebras}
For the theta graph,
\[
 \A(\{1,2\})
 =
 C^*(F,Q,R)
 =
 e_+M_4e_+\oplus e_-M_4e_-
 \cong M_2\oplus M_2,
 \label{eq:A12}
\]
and
\[
 \A(\{3\})=C^*(F)\cong\CC^2.
 \label{eq:A3}
\]
Consequently
\[
 Z(\A(\{1,2\}))=\A(\{3\}),
 \qquad
 \A(\{1,2\})\cap\A(\{3\})=C^*(F).
\]
No spatial tensor-factor decomposition of $\M$ realizes these two
regional inclusions as complementary tensor factors.
\end{theorem}

\begin{proof}
An invariant Pauli monomial supported on links $1,2$ can have
$\beta=(0,0,0)$ or $\beta=(1,1,0)$.
Hence it is generated by $X_1,X_2,Z_1Z_2$.
On the physical space $X_2=FQ$.
Within each $f$ sector, $Q,R$ are a Pauli pair and generate $M_2$.
The projection $e_f$ selects the sector, proving \eqref{eq:A12}.

On link $3$ alone, invariance permits only $1,X_3$, proving
\eqref{eq:A3}.
The center and intersection follow.
Complementary tensor factors with common unit intersect only in
scalars; here the intersection contains the non-scalar $F$.
\end{proof}

In this example the commutant identity
$\A(\{3\})'\cap M_4=\A(\{1,2\})$ happens to hold.
The general theory does not assume it.

\subsection{Four operational tests}

\begin{example}[Containment with genuine local dependence]
\label{ex:localread}
Set $P_\pm=(1\pm Q)/2$.
The supported Kraus operators $P_\pm$ commute with the Gauss
constraint. The complete recorded instrument is
\[
 \widehat T(a_+,a_-)
 =
 P_+a_+P_++P_-a_-P_-.
\]
On the local recorded output
$\A(\{1,2\})\oplus\A(\{1,2\})$, its range is in
$\A(\{1,2\})$.
The scalar report channel has effects $P_+,P_-$ and distinguishes
the physical inputs $\ket{f,+}$ and $\ket{f,-}$ with gap one.
Thus containment does not imply a constant output.

The nonselective channel acts on the local basis by
\[
 T_{\mathrm{ns}}(e_f)=e_f,\qquad
 T_{\mathrm{ns}}(e_fQ)=e_fQ,\qquad
 T_{\mathrm{ns}}(e_fR)=0,\qquad
 T_{\mathrm{ns}}(e_fQR)=0.
\]
These formulas test containment on a complete spanning set.
\end{example}

\begin{example}[Containment with a constant report]
\label{ex:constant}
The channel from a two-outcome report algebra is
\[
 T_0(\lambda_+,\lambda_-)=\lambda_+1.
\]
It has range in every unital regional algebra and always reports $+$.
A neutral local pointer initialized in $+$ implements this report
without resetting the gauge system.
Declared ancestry need not be influential.
\end{example}

\begin{example}[Omitted central data]
\label{ex:gaugecenter}
In the coordinates \eqref{eq:theta-basis}, take
\[
 \rho_p=
 \begin{pmatrix}p&0\\0&1-p\end{pmatrix}_{f}
 \otimes\rho_q.
\]
Its full local snapshot is
\[
 p\rho_q\oplus(1-p)\rho_q.
\]
The normalized states within both sectors are $\rho_q$, but the
local central readout $e_+,e_-$ has law $(p,1-p)$.
A smaller algebra $1_f\otimes M_2$ also fails to distinguish these
inputs. It is not a replacement for the declared full regional
algebra \eqref{eq:A12}.
\end{example}

\begin{example}[A concrete failed containment test]
\label{ex:failure}
Let
\[
 W=Z_1Z_3|_{\Hil_{\phys}}.
\]
It is gauge invariant but anticommutes with $F$.
Thus $W\notin\A(\{1,2\})$.
Define a binary readout by
\[
 T_W(\lambda_+,\lambda_-)
 =
 \lambda_+\frac{1+W}{2}
 +\lambda_-\frac{1-W}{2}.
\]
The global density matrices
\[
 \rho_\pm=\frac{1\pm W}{4}
\]
are positive and normalized.
Every element of $\A(\{1,2\})$ is block diagonal in $f$,
whereas $W$ is off diagonal, so $\rho_+$ and $\rho_-$ have
equal regional snapshots. Their output laws under $T_W$
are respectively $\delta_+$ and $\delta_-$.
This is an exact witness for the failure of containment.
\end{example}

Restricting Example~\ref{ex:failure} to the singleton preparation
class $\{1/4\}$ would conceal the failure, providing the fourth,
preparation-class test.

\subsection{Expectations in the demonstrator}
\label{sec:finiteexpectexample}

\begin{proposition}[Regional expectation versus gauge averaging]
\label{prop:thetaexpect}
The regional conditional expectation onto $\A(\{1,2\})$ is
\[
 E_{12}(a)=e_+ae_++e_-ae_-=\frac12(a+FaF).
 \label{eq:thetaexpect}
\]
It preserves the ordinary trace.
Gauge averaging by $G$ acts as the identity on the physical algebra
and is therefore not $E_{12}$.

There are faithful physical states for which no state-preserving
conditional expectation onto $\A(\{1,2\})$ exists.
\end{proposition}

\begin{proof}
Equation~\eqref{eq:thetaexpect} is block pinching onto the two full
matrix blocks, hence a conditional expectation by
Proposition~\ref{prop:finiteexpect}.
The constraint $G$ is the identity on $\Hil_{\phys}$.

Any conditional expectation onto these full blocks is forced by
bimodularity to kill every off-diagonal block $e_+ae_-$ and
$e_-ae_+$, and to fix every diagonal block.
It must therefore equal $E_{12}$.

Choose
\[
 \rho_\varepsilon=\frac14\,1+\varepsilon W,
 \qquad 0<\varepsilon<\frac14.
\]
This state is faithful, and
\[
 \Tr(\rho_\varepsilon W)=4\varepsilon,\qquad
 E_{12}(W)=0.
\]
The unique expectation does not preserve it.
\end{proof}

%=====================================================================
\section{The complete device extension and boundary compatibility}
\label{sec:deviceextension}
%=====================================================================

\begin{proposition}[A regional channel does not determine its device]
\label{prop:extensiongap}
There are two gauge-invariant devices supported on $\{1,2\}$
that agree on $\A(\{1,2\})$ but disagree on $\M$.
\end{proposition}

\begin{proof}
The kinematic unitary $U=X_1X_2$ is supported on $\{1,2\}$ and
commutes with every Gauss operator.
Its physical image is $F$.
The channels
\[
 T_{\id}(a)=a,\qquad T_U(a)=FaF
\]
agree on $\A(\{1,2\})$, because $F$ is central there.
But for the cross-boundary observable $W=Z_1Z_3$,
\[
 T_U(W)=-W\ne W.
\]
\end{proof}

The factor $F$ is not central in the global algebra.
This example prevents a proof of matched replacement invariance
from agreement of regional restrictions alone.

\begin{definition}[Specified localized device]
A specified device includes its complete global UCP action.
A localized Kraus realization is one sufficient way to specify it:
\[
 T_b(a)=\sum_{\mu=1}^{m_b}V_{b,\mu}^*aV_{b,\mu},
 \qquad
 V_{b,\mu}\in\A(b),
 \qquad
 \sum_\mu V_{b,\mu}^*V_{b,\mu}=1.
 \label{eq:internaldevice}
\]
A chosen Kraus family is realization data, not a canonical
observable associated to the channel.
Different Kraus families may specify the same global channel.
\end{definition}

This distinction preserves En06-A's warning that arbitrary Kraus
representatives are not intrinsic observables
\cite{CheungEn06A}.

\begin{proposition}[Shared boundary data cannot be independently changed]
\label{prop:sharedcenter}
Suppose $I,J$ are disjoint regions and
$z\in\A(I)\cap\A(J)$.
Every internal Kraus device localized in $I$ fixes $z$.
In the theta graph, no device that fixes $\A(\{3\})$
can induce the sector-swap automorphism
\[
 (a_+,a_-)\longmapsto(a_-,a_+)
\]
on $\A(\{1,2\})$.
\end{proposition}

\begin{proof}
Local commutativity implies that each Kraus operator in $\A(I)$
commutes with $z\in\A(J)$.
Hence
\[
 T_I(z)=
 z\sum_\mu V_\mu^*V_\mu=z.
\]
For the theta graph, the proposed swap sends $F$ to $-F$,
whereas exterior fixation requires $F$ to remain $F$.
\end{proof}

The sector-swap map is a valid abstract UCP automorphism of
$M_2\oplus M_2$. It is not thereby an admissible exterior-fixing
physical device. This is an explicit failure of arbitrary independent
regional operation closure.

Likewise, two separately assigned snapshots on $\A(\{1,2\})$
and $\A(\{3\})$ cannot be restrictions of one global state if they
assign different probabilities to their common projection $e_+$.

%=====================================================================
\section{Stage E: one-layer locality without product channels}
\label{sec:onelayer}
%=====================================================================

\begin{assumption}[Complete local control]
\label{ass:control}
Every controller, classical record, seed, resource, and quantum
observable used by a device belongs to its declared localization
interface. The device action is fixed before the unknown input
state is supplied.

If an auxiliary neutral apparatus is used, its local algebra,
preparation, record outputs, and downstream control paths are
included in the experiment.
No unknown-state description or undeclared remote scheduler input
is admitted.
\end{assumption}

This is the operational support discipline compared with VII-G
and the supplied En05 series
\cite{CheungVIIG,CheungEn05A,CheungEn05B,CheungEn06A}.

\begin{lemma}[Localized internal-device support]
\label{lem:devicesupport}
Let $\A$ be an isotone locally commuting net.
For the device \eqref{eq:internaldevice},
\[
 T_b(\A(Y))\subseteq\A(Y\cup b).
 \label{eq:enlargement}
\]
If $b\cap Y=\varnothing$, then
\[
 T_b|_{\A(Y)}=\id.
 \label{eq:exteriorfix}
\]
Devices with disjoint localization regions commute as global
superoperators.
\end{lemma}

\begin{proof}
For $a\in\A(Y)$, isotony places $a$ and all $V_{b,\mu}$ in
$\A(Y\cup b)$. This algebra is closed under products and sums,
proving \eqref{eq:enlargement}.

For disjoint $b,Y$, local commutativity gives
\[
 T_b(a)=a\sum_\mu V_{b,\mu}^*V_{b,\mu}=a.
\]
For disjoint devices $b,c$, their Kraus operators and adjoints
commute. Expanding both compositions and reordering those
operators proves $T_bT_c=T_cT_b$.
\end{proof}

Membership in \eqref{eq:enlargement} follows from multiplication
inside the declared algebra. It is not inferred solely from
\eqref{eq:exteriorfix}.

\begin{proposition}[Constraint-preserving realization of the device class]
\label{prop:gaugeddevices}
In the finite gauge net \eqref{eq:gaugenet}, choose kinematic
Kraus operators $v_{b,\mu}\in\F(b)$ satisfying
\[
 \sum_\mu v_{b,\mu}^*v_{b,\mu}=1_{\kin}.
\]
Their physical restrictions satisfy \eqref{eq:internaldevice}
and all conclusions of Lemma~\ref{lem:devicesupport}.
Complete selected outcomes may be grouped and recorded as in
Proposition~\ref{prop:records}.
\end{proposition}

\begin{proof}
Constraint preservation makes $\qphys$ a $*$-homomorphism, so
the kinematic completeness identity restricts to the physical
completeness identity.
The physical Kraus operators lie in $\A(b)$ by definition.
Apply Lemma~\ref{lem:devicesupport} and
Proposition~\ref{prop:records}.
\end{proof}

\subsection{Partitioned layers}

Let a finite site set $X$ have a partition $\mathcal P_s$ at tick $s$.
The sites may be links of the gauge graph.
Each block $b$ carries a specified internal Kraus device.
Define the global Heisenberg layer by composition:
\[
 T_s=\prod_{b\in\mathcal P_s}T_{s,b}.
 \label{eq:layercomposition}
\]
The order is irrelevant by Lemma~\ref{lem:devicesupport}.
Equation~\eqref{eq:layercomposition} is not a spatial tensor product.

For $Y\subseteq X$, put
\[
 U_s(Y)=\bigcup\{b\in\mathcal P_s:b\cap Y\ne\varnothing\}.
\]

\begin{theorem}[Tensor-free one-layer support]
\label{thm:onelayer}
For the layer \eqref{eq:layercomposition},
\[
 T_s(\A(Y))\subseteq\A(U_s(Y)).
 \label{eq:onelayer}
\]
Its restriction to $\A(Y)$ depends only on the specified global
devices in the blocks meeting $Y$.
\end{theorem}

\begin{proof}
Let $\mathcal P_s(Y)$ be the meeting blocks.
By commutation, compose them first.
Repeated use of \eqref{eq:enlargement} places the result in
$\A(U_s(Y))$.
Every remaining block is disjoint from $U_s(Y)$, so its device
fixes that entire algebra by \eqref{eq:exteriorfix}.
It makes no contribution.
\end{proof}

\begin{remark}[Countable locally finite substrates]
If finite-region algebras generate a quasi-local global algebra and
every finite region meets only finitely many blocks, the same formula
defines a layer on the local union. Enlarging a region adds only
devices that fix the smaller calculation, so the definition is
consistent. The finite compositions are UCP contractions at every
matrix level; norm completion gives a UCP global map.
Finite horizons still have finite backward regions under the usual
finite-ball and finite-block hypotheses.
No infinite-past network semantics is inferred.
\end{remark}

\subsection{Records and local apparatuses}

For a finite experiment, gauge-neutral pointer and controller
registers can be reserved locally.
Tensoring the constrained system with these declared neutral
registers does not impose a spatial tensor decomposition on the
gauge degrees of freedom.

At finite matrix interfaces, a local instrument can be recorded
using a local pointer and the isometry
\[
 \psi\longmapsto
 \sum_{y,\mu}V_{y,\mu}\psi\otimes\ket{y,\mu}.
\]
Completeness gives the isometry identity.
Recording or dephasing the pointer yields the direct-sum instrument
of Proposition~\ref{prop:records}.
For finite-dimensional local algebras, this local isometry can be
completed blockwise to a unitary on a sufficiently large reserved
neutral apparatus. All such apparatus factors must be retained
in the locality ledger \cite{Stinespring,Watrous}.

Consequently Theorem~\ref{thm:onelayer} applies to the complete
record-retaining implementation, including later locally controlled
continuations. It is not a theorem about an isolated selectively
normalized branch.

%=====================================================================
\section{Stage F: ancestry, matched replacements, and metric bounds}
\label{sec:ancestry}
%=====================================================================

\subsection{Algebraic backward recursion}

For terminal region $Y$ and terminal tick $T$, define
\[
 N_T(Y)=Y,\qquad N_s(Y)=U_s(N_{s+1}(Y)).
 \label{eq:backwardsets}
\]
Let
\[
 T_{s:T}=T_s\circ T_{s+1}\circ\cdots\circ T_{T-1}.
\]

\begin{theorem}[Tensor-free ancestral factorization]
\label{thm:ancestry}
For the specified layers,
\[
 T_{s:T}(\A(Y))\subseteq\A(N_s(Y)).
 \label{eq:ancestrycontainment}
\]
There is a unique UCP channel
\[
 K_{s,Y}:\A(Y)\longrightarrow\A(N_s(Y))
\]
whose inclusion equals the displayed restriction of $T_{s:T}$.
The terminal snapshot therefore depends only on the initial
state restriction to $\A(N_s(Y))$, including joint restrictions
with any independent matrix reference.
\end{theorem}

\begin{proof}
Apply Theorem~\ref{thm:onelayer} successively in reverse tick order.
The set recursion is exactly \eqref{eq:backwardsets}.
Theorem~\ref{thm:comparison} gives the local channel and snapshot
factorization, and Corollary~\ref{cor:reference} gives amplification.
\end{proof}

\begin{theorem}[Matched replacement invariance]
\label{thm:matched}
Consider two runs on the same net, partitions, and record interfaces.
Suppose:
\begin{enumerate}[label=(\roman*)]
\item their initial states agree on $\A(N_{t_0}(Y))$;
\item at every tick $s$, their specified global block devices agree
on every block meeting $N_{s+1}(Y)$.
\end{enumerate}
Then their terminal snapshots on $\A(Y)$ agree.
Devices outside the backward recursion may be replaced by arbitrary
devices from the stated localized class.

For the reference version, the initial restrictions must agree
jointly on $\A(N_{t_0}(Y))\otimes M_r$.
\end{theorem}

\begin{proof}
Pull an arbitrary terminal observable backward.
At each layer only the meeting block devices contribute by
Theorem~\ref{thm:onelayer}, and these are common.
Thus the two terminal pullbacks are equal and belong to
$\A(N_{t_0}(Y))$. Their expectations agree by (i).
The amplified calculation is identical.
\end{proof}

Agreement of chosen complete kinematic block realizations is a
sufficient certificate for condition (ii).
Agreement only on $\A(b)$ is not sufficient, by
Proposition~\ref{prop:extensiongap}.
A weaker sufficient condition is agreement of the layer
restrictions on every algebra $\A(N_{s+1}(Y))$ used in the
backward proof.

The theorem is the gauge-net counterpart of the prescribed-support
comparisons in En03-A, VII-E, VII-J, Psy04-B, and En06-A
\cite{CheungEn03A,CheungVIIE,CheungVIIJ,CheungPsy04B,CheungEn06A}.
Their model-specific tensor wiring is not imported.

\subsection{A general finite event sequence}

\begin{corollary}[Finite event ancestry]
\label{cor:eventancestry}
For a finite sequence of localized internal devices with envelopes
$b_1,\ldots,b_m$, start from $S_m=Y$ and put
\[
 S_{j-1}=
 \begin{cases}
 S_j,&b_j\cap S_j=\varnothing,\\
 S_j\cup b_j,&b_j\cap S_j\ne\varnothing.
 \end{cases}
\]
The complete pullback of $\A(Y)$ lies in $\A(S_0)$.
Replacing excluded devices does not change the terminal snapshot
when the initial restriction and the included complete device
actions agree.
\end{corollary}

\begin{proof}
Apply Lemma~\ref{lem:devicesupport} at each backward step.
Then apply Theorem~\ref{thm:comparison}.
\end{proof}

This formulation uses declared algebras at the successive cuts.
It does not assert that every arbitrary graph of regional UCP maps
has a uniquely defined global wiring operation.

\subsection{Metric envelope}

Let $d$ be an integer-valued metric on $X$, and suppose every
partition block has diameter at most $r$.

\begin{corollary}[Metric support and signalling upper bound]
\label{cor:metric}
For finite $Y$,
\[
 N_s(Y)\subseteq\{x:d(x,Y)\le r(T-s)\}.
 \label{eq:metricenvelope}
\]
Equal initial state restrictions on the algebra of that metric
neighborhood, and common included complete device actions, give
equal terminal snapshots.

Any admitted matched local signalling pair whose changes are
excluded from this backward region has zero unconditioned law gap.
For ticks of physical duration $\Delta t$, the resulting scalar
metric propagation upper bound is $r/\Delta t$.
No lower attainment statement follows.
\end{corollary}

\begin{proof}
If $x$ is added through a block meeting $N_{s+1}(Y)$ at $y$, then
\[
 d(x,Y)\le d(x,y)+d(y,Y)\le r+r(T-s-1).
\]
Backward induction proves \eqref{eq:metricenvelope}.
Isotony permits restriction from the larger metric-neighborhood
algebra to the ancestor algebra.
Theorem~\ref{thm:matched} gives the snapshot statement.
The speed assertion is the same upper-bound comparison in physical
units.
\end{proof}

Gauss constraints can prevent an independently adjustable
single-site perturbation from existing.
A declared metric envelope is therefore not an attainment theorem.
Nor is algebraic localization necessarily unique: two distinct
regions can share the same central observable.
The admissible intervention, not a label alone, determines whether
a local variation is possible.

\subsection{Legal reorderings}

\begin{proposition}[Ordering invariance for specified devices]
\label{prop:ordering}
Let a finite dependency partial order compare every pair of devices
that are not declared independent.
Suppose incomparable devices commute as complete global maps;
in particular, disjoint internal Kraus devices satisfy this condition.
Every linear extension gives the same complete composite channel.
\end{proposition}

\begin{proof}
Any two linear extensions are connected by adjacent swaps of
incomparable events. Each swap leaves the composition unchanged.
\end{proof}

With complete local records and their dependencies included, this
also preserves the joint labelled record law.
It does not assert boost covariance or unrestricted inter-frame
device composition.

%=====================================================================
\section{Operational no-signalling and approximate certificates}
\label{sec:nosignalling}
%=====================================================================

\begin{definition}[Exterior-fixing intervention]
Relative to a protected input algebra $\A(I)\subseteq\M$, an
exterior-fixing intervention is a UCP map $L:\M\to\M$ with
\[
 L(a)=a\qquad(a\in\A(I)).
\]
\end{definition}

\begin{theorem}[Exterior-fixing no-signalling]
\label{thm:nosignalling}
If $T(\A_{\mathrm{out}}(Y))\subseteq\A_{\mathrm{in}}(I)$ and
$L$ is exterior fixing, then
\[
 L\circ T|_{\A_{\mathrm{out}}(Y)}
 =
 T|_{\A_{\mathrm{out}}(Y)}.
\]
Thus $\omega$ and $\omega\circ L$ give the same unconditioned
output snapshot for every global input state.
The conclusion holds jointly with every independent untouched
matrix reference.
\end{theorem}

\begin{proof}
Every $T(b)$ belongs to the fixed algebra of $L$.
Amplify that pointwise identity for the reference conclusion.
\end{proof}

\begin{corollary}[Influence is contained in scheduled ancestry]
\label{cor:influence}
For the device class of Section~\ref{sec:onelayer}, an intervention
localized in a region disjoint from the initial ancestor region
cannot change the terminal unconditioned law. Hence
\[
 \Inf^-(Y)\subseteq\Sch^-(Y)
\]
for the declared operational interventions.
The inclusion can be strict.
\end{corollary}

\begin{proof}
Lemma~\ref{lem:devicesupport} makes the exterior device fix
the ancestor algebra, and Theorem~\ref{thm:ancestry} places the
terminal pullback there. Apply Theorem~\ref{thm:nosignalling}.
The constant report in Example~\ref{ex:constant} supplies strictness.
\end{proof}

\begin{remark}[No-signalling is not independence]
The result neither requires nor implies product states,
independent outcomes, or arbitrary independent preparation of
regional snapshots. Proposition~\ref{prop:sharedcenter} exhibits
a compatibility constraint even before correlations are considered.
Stronger subsystem-independence statements, such as split-property
results in AQFT, require additional assumptions
\cite{Fewster}.
\end{remark}

\begin{remark}[No automatic converse from a small intervention class]
Absence of signalling under a restricted set of exterior devices
does not imply all-state snapshot determinacy.
A device class containing only the identity tests nothing.
A converse needs a sufficiently rich intervention or
state-separation interface.
\end{remark}

\subsection{Approximate comparison}

For states on any unital $C^*$-algebra use
\[
 d(\omega,\nu)=\frac12\norm{\omega-\nu}.
\]
For finite matrix or hybrid algebras this is trace distance.
For classical finite algebras it is total variation.

\begin{proposition}[Tensor-free approximate cone certificate]
\label{prop:approx}
Let $\widetilde T:\B\to\M$ be UCP and let
$T_0=\iota_IK:\B\to\M$ be a UCP map factoring through $\A(I)$.
If
\[
 \frac12\norm{\widetilde T-T_0}_{\cb}\le e,
\]
then states with equal restrictions on $\A(I)$ have output
distance at most
\[
 \min\{1,2e\}.
\]
The same estimate holds at every independent reference size when
the corresponding joint input restrictions agree.
\end{proposition}

\begin{proof}
The ideal output states agree.
Each implemented output differs from its ideal output by at most
$e$, because composition of a norm-one state with the difference
map has norm at most $2e$.
The triangle inequality gives $2e$.
State distance is at most one.
At matrix size $r$, the difference-map norm is bounded by its
completely bounded norm.
\end{proof}

\begin{proposition}[Finite composition error ledger]
\label{prop:telescoping}
For compatible UCP maps $T_j,\widetilde T_j$,
\[
 \norm{\widetilde T_1\cdots\widetilde T_m-
              T_1\cdots T_m}_{\cb}
 \le
 \sum_{j=1}^m\norm{\widetilde T_j-T_j}_{\cb}.
\]
\end{proposition}

\begin{proof}
Replace one factor at a time.
Each UCP factor is a complete contraction, so every term in the
telescoping sum is bounded by the corresponding local error.
\end{proof}

On finite interfaces, this is the Heisenberg dual of the
half-diamond accounting used in the supplied channel-certificate
comparisons \cite{Watrous,CheungEn05A,CheungEn05B,CheungPsy04B}.
A small observational defect alone is not asserted to supply a
nearby UCP regional factorization.

%=====================================================================
\section{Stage G: normal von Neumann channels}
\label{sec:normal}
%=====================================================================

\begin{lemma}[Normal equal-snapshot separation]
\label{lem:normalseparation}
Let $\A\subseteq\M$ be a unital von Neumann subalgebra.
If $h=h^*\notin\A$, there exist normal states $\omega,\nu$ on
$\M$ with equal restrictions to $\A$ and
$\omega(h)\ne\nu(h)$.
\end{lemma}

\begin{proof}
The self-adjoint part of $\A$ is ultraweakly closed.
Separation in the dual pairing $(\M,\M_*)$ gives a nonzero
normal Hermitian functional $f$ vanishing on $\A$ with
$f(h)\ne0$.
Its Jordan components $f_\pm$ are normal.
As in Lemma~\ref{lem:separation},
\[
 f_+(1)=f_-(1)=m>0,\qquad f_+|_\A=f_-|_\A.
\]
Normalize them to obtain the stated normal states.
\end{proof}

\begin{theorem}[Normal tensor-free comparison]
\label{thm:normal}
Suppose the ambient algebras and local algebras are von Neumann
algebras, their inclusions are normal, and $T$ is normal UCP.
The three conditions of Theorem~\ref{thm:comparison} remain
equivalent when snapshot determinacy is tested on all normal
global input states.
The unique induced local map $K$ is normal UCP.
The factorization holds on preduals and is stable under normal
independent matrix amplification.
\end{theorem}

\begin{proof}
Use Lemma~\ref{lem:normalseparation} in place of
Lemma~\ref{lem:separation}.
The matrix-positivity proof of complete positivity is unchanged.

To prove normality, let $b_\lambda$ be a bounded increasing net
of positive local output elements, with supremum $b$.
Normality of $T$ gives
\[
 T(b_\lambda)\uparrow T(b)
\]
in the ambient input algebra.
These elements belong to the input von Neumann subalgebra,
where the same supremum is computed.
Thus $K(b_\lambda)\uparrow K(b)$, proving normality.
Dualization gives the predual identity.
\end{proof}

\begin{remark}[The closure hypothesis matters]
For a merely norm-closed subalgebra represented inside a von
Neumann algebra, testing only normal states need not recover
that norm-closed algebra.
The normal theorem requires the regional algebra to be a
von Neumann subalgebra.
\end{remark}

For example, the algebra of compact operators plus scalars on an
infinite-dimensional Hilbert space is ultraweakly dense in
$\mathcal B(\Hil)$.
Two normal states agreeing there agree everywhere, so normal-state
determinacy cannot detect membership in that proper norm-closed
subalgebra.

No normal conditional expectation has been added to
Theorem~\ref{thm:normal}. Its optional state-preserving existence
continues to require a separate criterion, as discussed in
Section~\ref{sec:expectations}.

%=====================================================================
\section{Stage H: a constructed bounded-sequence observable limit}
\label{sec:sequencelimit}
%=====================================================================

This section gives an actual observable algebra, not a conclusion
from commuting superoperators or convergent outcome numbers.

Let $\M_n$ be unital $C^*$-algebras. Define
\[
 \prod\nolimits_b\M_n
 =
 \{(a_n):\sup_n\norm{a_n}<\infty\},
 \qquad
 \bigoplus\nolimits_0\M_n
 =
 \{(a_n):\norm{a_n}\to0\}.
\]
The latter is a closed two-sided ideal. Put
\[
 \M_\infty=
 \left(\prod\nolimits_b\M_n\right)
 \big/
 \left(\bigoplus\nolimits_0\M_n\right).
 \label{eq:sequencequotient}
\]
Write $[a_n]$ for a class. Its norm is
\[
 \norm{[a_n]}=\limsup_n\norm{a_n}.
 \label{eq:quotientnorm}
\]

For nets $\A_n(O)\subseteq\M_n$, define
\[
 \A_\infty(O)=
 \{[a_n]:a_n\in\A_n(O),\ \sup_n\norm{a_n}<\infty\}.
 \label{eq:limitlocalalgebra}
\]

\begin{theorem}[Observable sequence-limit construction]
\label{thm:sequence}
The following statements hold.
\begin{enumerate}[label=(\roman*)]
\item Each $\A_\infty(O)$ is a unital $C^*$-subalgebra of
$\M_\infty$, and isotony passes to the limit.
\item If the two scale algebras commute for all sufficiently
large $n$, their limit local algebras commute.
More generally, the same conclusion follows if every pair of
bounded local representative sequences has commutator norm
tending to zero.
\item UCP maps $T_n:\M_{\mathrm{out},n}\to\M_{\mathrm{in},n}$
induce a UCP map
\[
 T_\infty([a_n])=[T_n(a_n)].
\]
\item If
\[
 T_n(\A_{\mathrm{out},n}(Y))
 \subseteq\A_{\mathrm{in},n}(I)
\]
for all sufficiently large $n$, the induced map has the same
containment and the tensor-free comparison theorem applies.
\item Complete recorded instruments with fixed finite alphabets,
independent matrix references, and matched equalities of device
actions pass to this quotient.
\end{enumerate}
\end{theorem}

\begin{proof}
The local bounded product is a $C^*$-algebra.
Its intersection with the ambient null ideal is its own
norm-null ideal. Formula~\eqref{eq:quotientnorm} shows that its
quotient embeds isometrically into the ambient quotient.
Its image is therefore closed, proving (i).

For (ii),
\[
 [[a_n],[b_n]]=[[a_n,b_n]],
\]
and the stated commutator sequences are norm null.

UCP maps are contractions at every matrix size.
They send bounded sequences to bounded sequences and norm-null
sequences to norm-null sequences.
The coordinatewise product map is UCP, and quotienting gives a
UCP map, proving (iii).

For (iv), discard finitely many terms and use the indicated local
representatives. Equivalently, use their local UCP restrictions
to construct the induced factorization.
The resulting local image is in \eqref{eq:limitlocalalgebra}.

Finite direct sums and finite matrix amplifications commute with
this construction. Equality on the specified local sequence
algebra passes term by term or after a norm-null error, proving (v).
\end{proof}

\subsection{Compatible states and effects}

Choose a state $\lambda$ on $\ell^\infty(\NN)$ extending the ordinary
limit on convergent scalar sequences.
Such a state follows from the positive state-extension theorem.
It vanishes on $c_0$.

\begin{proposition}[State--effect compatibility]
\label{prop:limitstates}
For states $\omega_n$ on $\M_n$,
\[
 \omega_\infty([a_n])
 =
 \lambda\bigl((\omega_n(a_n))_n\bigr)
 \label{eq:limitstate}
\]
is a state on $\M_\infty$.
If $\omega_n(a_n)\to p$, then
$\omega_\infty([a_n])=p$.
For a complete finite readout with convergent outcome laws, the
limit state reproduces the entire limiting law.
\end{proposition}

\begin{proof}
Norm-null representatives have scalar evaluations tending to zero,
so \eqref{eq:limitstate} is well defined.
Positivity and unit normalization hold coordinatewise and then
under the positive functional $\lambda$.
On convergent sequences, $\lambda$ is the ordinary limit.
Apply that statement to every effect of the finite readout.
\end{proof}

\begin{proposition}[Coherent central sectors]
\label{prop:limitcenters}
Suppose $e_{\alpha,n}$ are a fixed finite family of central projections
of $\A_n(O)$, pairwise orthogonal and summing to one.
Then $[e_{\alpha,n}]$ are central projections of $\A_\infty(O)$.
If $\omega_n(e_{\alpha,n})\to p_\alpha$, the limit snapshot retains
the weights $p_\alpha$.
A sector projection is nonzero in the abstract quotient whenever it
is nonzero at infinitely many scales.
\end{proposition}

\begin{proof}
All projection, orthogonality, completeness, and commutation
identities pass coordinatewise.
The norm of a nonzero projection is one, so
\eqref{eq:quotientnorm} gives the last assertion.
Proposition~\ref{prop:limitstates} gives the weights.
\end{proof}

A subsequent state representation can kill a zero-weight sector.
We do not identify such a representation quotient with the full
abstract limit algebra.

\begin{remark}[What this limit does and does not select]
The quotient is generally large, and the state $\lambda$ need not
be unique. It supplies a rigorous product-preserving observable
interface, not a preferred vacuum, a renormalization-group limit,
or a unique physical continuum theory.
Scaling algebras in AQFT provide a more specialized framework with
additional spacetime and continuity structure
\cite{BuchholzVerch}. We do not identify
\eqref{eq:sequencequotient} with that construction.
\end{remark}

%=====================================================================
\section{An exact gauge subdivision limit}
\label{sec:subdivision}
%=====================================================================

We now provide a second, concrete realization of Stage H.
Its physical algebra is finite dimensional, but its geometric
mesh tends to zero and its regional assignments are defined on
open subsets of a continuous graph.

\subsection{Subdivided constraints and physical identifications}

Realize the theta graph as three closed arcs joining two vertices.
At scale $n$, divide each arc into $m_n=2^n$ links.
Use the Gauss constraints \eqref{eq:gauss} at the original vertices
and all new degree-two vertices.

At a degree-two vertex, the physical $X$-eigenvalues of its two
incident links must agree. Hence all sublinks of one original
arc have a common eigenvalue.
The two original endpoint constraints require that the product
of the three arc eigenvalues be one.

\begin{lemma}[Exact physical refinement]
\label{lem:physicalrefinement}
The physical Hilbert space $\Hil_{\phys,n}$ has dimension four.
The map
\[
 V_n:\Hil_{\phys,0}\longrightarrow\Hil_{\phys,n}
\]
that replaces each arc's $X$-eigenvalue by the same value on all
its sublinks is unitary between the physical spaces.
Define
\[
 J_n(a)=V_naV_n^*.
\]
Then $J_n:M_4\to\mathcal B(\Hil_{\phys,n})$ is a unital
$*$-isomorphism and a complete isometry.
\end{lemma}

\begin{proof}
The constraints give exactly the four configurations
\eqref{eq:theta-basis}, now constant along each arc.
They form an orthonormal basis.
The stated map sends one physical basis bijectively to the other.
The conclusions follow from unitary conjugation.
\end{proof}

This is a comparison identification of constrained physical
spaces. It is not a device that clones an unknown qubit into
independent sublink systems. Sublink snapshots are constrained and
are not independently adjustable quantum outputs.

Under $J_n$:
\[
 X_j\longmapsto X_{j,k}|_{\Hil_{\phys,n}}
 \quad\text{for any sublink }k,
\]
whereas
\[
 Z_1Z_2
 \longmapsto
 \left(\prod_{k=1}^{m_n}Z_{1,k}\right)
 \left(\prod_{k=1}^{m_n}Z_{2,k}\right)
 \bigg|_{\Hil_{\phys,n}}.
 \label{eq:refinedwilson}
\]
The second expression is a complete closed cycle.
A single sublink $Z_{j,k}$ does not preserve the Gauss constraints.

\subsection{Regional observable convergence}

For an open subset $O$ of the continuous theta graph, let $E_n(O)$
be the sublinks whose closed geometric arcs lie in $O$.
Define $\A_n(O)$ by the same supported-invariant prescription
as in \eqref{eq:gaugenet}, and set
\[
 \widetilde\A_n(O)=J_n^{-1}(\A_n(O))\subseteq M_4.
\]

\begin{lemma}[Nested regional comparison algebras]
\label{lem:nestedregions}
For every open $O$,
\[
 \widetilde\A_n(O)\subseteq\widetilde\A_{n+1}(O).
\]
For disjoint open $O_1,O_2$, the two algebras commute at every scale.
\end{lemma}

\begin{proof}
By Lemma~\ref{lem:paulibasis}, a supported invariant algebra is
spanned by electric monomials and cycle-type $Z$ monomials.
Under one subdivision, represent an old electric operator on one
of its two child links and replace every old cycle $Z$ factor by
the product on both children.
The cycle condition remains valid at the new degree-two vertices.
On the physical basis this realizes exactly the same operator.

If an old closed link lies in $O$, both children lie in $O$,
so support is preserved. This proves the inclusion.
Disjoint $O_1,O_2$ have disjoint retained link sets, and
Proposition~\ref{prop:gaugenet} proves commutation.
\end{proof}

\begin{theorem}[Constructed spatial gauge refinement net]
\label{thm:refinementnet}
The assignment
\[
 \A_{\mathrm{ref}}(O)
 =
 \overline{\bigcup_n\widetilde\A_n(O)}
 \subseteq M_4
 \label{eq:refinementnet}
\]
is an isotone locally commuting spatial net.
For each fixed $O$, the increasing algebras
$\widetilde\A_n(O)$ eventually stabilize.
Admitted bounded observable sequences whose $J_n^{-1}$ images
converge in norm have well-defined product- and adjoint-preserving
limits.

The full graph algebra is $M_4$, and centerful proper regional
algebras persist.
\end{theorem}

\begin{proof}
Isotony holds at every scale.
Lemma~\ref{lem:nestedregions} gives an increasing family.
Its union is a $*$-algebra, and norm closure gives a
$C^*$-algebra. Local commutativity passes to the closure.

The dimensions of the increasing subspaces are integers bounded
by $\dim M_4=16$. Only finitely many strict increases are possible.
Hence each fixed regional sequence eventually stabilizes.

If $a_n,b_n$ have norm-convergent comparison images, those images
are uniformly bounded and
\[
 J_n^{-1}(a_nb_n)
 =
 J_n^{-1}(a_n)J_n^{-1}(b_n)
 \longrightarrow ab.
\]
Adjoints behave similarly.

The global physical algebra is $M_4$ by
Proposition~\ref{prop:gaugenet}.
Take $O_{12}$ containing arcs $1,2$ completely but not arc $3$
completely. Its algebra is $M_2\oplus M_2$.
Small portions of arc $3$ add only the already present electric
observable $F$.
An open interval $O_3$ in the interior of arc $3$ has algebra
$C^*(F)$ for sufficiently fine scales.
Thus the shared center persists.
\end{proof}

Choose $O_3$ away from the endpoint neighborhoods in $O_{12}$.
Then $O_{12}$ and $O_3$ are disjoint despite their shared physical
central observable. Their snapshots must still agree on that
observable.

\subsection{States, instruments, and matched devices}

\begin{theorem}[Exact compatible operational refinement]
\label{thm:operationalrefinement}
Let $T_0:M_4\to M_4$ be a specified complete device, or a fixed
finite complete recorded instrument, built from the gauge-preserving
realizations above. Put
\[
 T_n=J_nT_0J_n^{-1},
 \qquad
 \omega_n=\omega\circ J_n^{-1}.
 \label{eq:exactrefinement}
\]
Then:
\begin{enumerate}[label=(\roman*)]
\item every $T_n$ is UCP and every $\omega_n$ is a state;
\item all complete outcome laws are identical at every scale;
\item the logical channel equality is exact at every independent
reference size;
\item compatible regional containments are preserved;
\item matched pairs retain the same unchanged device actions,
central data, resources, and report designations;
\item finite compositions and complete recorded continuations
are preserved exactly.
\end{enumerate}
The electric readout in Example~\ref{ex:localread} and closed-cycle
unitaries generated by $R$ give nonconstant instances.
\end{theorem}

\begin{proof}
Conjugation by a $*$-isomorphism preserves complete positivity,
unitality, and states.
For every output observable,
\[
 \omega_n(T_n(J_n(a)))=\omega(T_0(a)).
\]
The same identity holds componentwise for recorded instruments
and after matrix amplification, proving (ii)--(iii).

Supported invariant realizations refine by the construction in
Lemma~\ref{lem:nestedregions}. Their physical channels are
\eqref{eq:exactrefinement}, so containment passes under the same
identifications.
Choose the refinements once for a matched pair; unchanged
realizations and resources remain common.
Unitary conjugation respects composition and finite direct sums,
giving (v)--(vi).
The specified examples depend nontrivially on physical inputs.
\end{proof}

\begin{remark}[The spacetime boundary of this construction]
The mesh of the spatial graph tends to zero, but the number of
physical logical modes remains two.
A refined closed-cycle gate such as \eqref{eq:refinedwilson} has
support around the whole cycle. It has not been compiled into
shrinking finite-range blocks under a relativistic clock.

Accordingly, Theorems~\ref{thm:refinementnet} and
\ref{thm:operationalrefinement} construct a finite-mode spatial
observable refinement limit. They do not establish a
Lorentz-covariant dynamical spacetime net, a continuum gauge-field
Hamiltonian, or an invariant propagation speed.
\end{remark}

This explicit construction closes the observable-compatibility
target for its stated refinement class, rather than leaving Stage H
as a name. It does not enlarge the scope of the continuum hypotheses
in the supplied manuscripts.

%=====================================================================
\section{Continuum transfer under a specified spacetime realization}
\label{sec:continuumtransfer}
%=====================================================================

To connect the algebraic results to a separate spacetime scaling
family, the geometric and operational interfaces must still be
provided. The following transfer statement isolates exactly what
is needed.

\begin{assumption}[Matched gauge spacetime realization]
\label{ass:continuum}
A family has:
\begin{enumerate}[label=(\roman*)]
\item local physical observable nets and complete localized devices
satisfying the one-layer support theorem;
\item an embedding of their complete device and report footprints
into a common spacetime, with shrinking footprints;
\item a uniform physical cone upper bound on every dependency path;
\item matched realizing families for local modifications, retaining
the same unchanged complete device actions and joint initial
restrictions;
\item convergent complete report laws on fixed finite alphabets.
\end{enumerate}
For a referenced conclusion, the corresponding joint reference
interfaces and restrictions are included.
\end{assumption}

\begin{theorem}[Gauge-algebra continuum domain-of-dependence transfer]
\label{thm:continuum}
Under Assumption~\ref{ass:continuum}, suppose a compact
modification support is at positive distance from the limiting
past of the common finite report tuple.
Then the two limiting joint report laws agree.
The conclusion does not require spatial tensor factorization.
\end{theorem}

\begin{proof}
The positive margin, shrinking footprints, and uniform path bound
place every changed device outside the report ancestors for all
sufficiently fine scales.
The matched background and included complete device actions agree.
Theorem~\ref{thm:matched}, applied to the full record-retaining
implementation, gives equality of the discrete joint report laws.
Their limits are therefore equal.
\end{proof}

If a product-preserving observable limit of those spacetime nets
is supplied, the commutation transfer of
Theorem~\ref{thm:sequence} also applies.
Operational no-signalling and observable commutation remain
different statements.

En05-B supplies a declared tensor-interface transport construction,
whereas En06-A separates its outcome-law theory from a matching
dynamical observable realization
\cite{CheungEn05B,CheungEn06A}.
Neither construction is silently converted into a constrained
gauge transport implementation here.
In particular, arbitrary swaps of charged or constraint-linked
slots require a separate gauge-compatible realization.

%=====================================================================
\section{Finite certificates and complete release}
\label{sec:certificates}
%=====================================================================

\subsection{A constructive finite separator}

\begin{proposition}[Finite matrix separation certificate]
\label{prop:finiteseparator}
Let $\A\subseteq M_d$ be a unital $*$-subalgebra and
$h=h^*\notin\A$.
Let $E_{\mathrm{HS}}$ be the Hilbert--Schmidt orthogonal projection
onto $\A$, and put
\[
 H=h-E_{\mathrm{HS}}h.
\]
For sufficiently small $t>0$,
\[
 \rho_\pm=\frac1d1\pm tH
\]
are states with equal restrictions to $\A$, but different
expectations of $h$.
\end{proposition}

\begin{proof}
The projection preserves adjoints because the subspace is
adjoint closed and the Hilbert--Schmidt inner product is compatible
with adjoints.
Thus $H$ is Hermitian.
Orthogonality to the unit gives $\Tr H=0$.
For $t\norm H<1/d$, both matrices are positive and normalized.

For every $a\in\A$, orthogonality gives $\Tr(Ha)=0$,
so the restrictions agree. Finally,
\[
 \Tr((\rho_+-\rho_-)h)
 =
 2t\Tr(Hh)
 =
 2t\Tr(H^2)>0.
\]
\end{proof}

For a finite channel, test containment on a complete output
algebra basis. If a Hermitian pullback has a nonzero component
outside the input algebra, Proposition~\ref{prop:finiteseparator}
provides an exact equal-snapshot witness.
Kraus completeness, Gauss commutation, regional algebra generation,
and equality of complete device actions are separate checks.

A finite numerical agreement is not substituted for an exact
containment identity. If approximate arithmetic is used, its
validated allowance must enter Proposition~\ref{prop:approx}.

\subsection{Combined mathematical conclusion}

\begin{theorem}[En05-C release]
\label{thm:release}
On the domains specified in this manuscript:
\begin{enumerate}[label=(\roman*)]
\item snapshot determinacy for all global states, UCP local
factorization, and Heisenberg regional containment are equivalent;
\item the factorization is unique and stable under independent
matrix references;
\item center probabilities are part of the affine snapshot, and
restricted preparation tests have the exact annihilator criterion;
\item the normal von Neumann comparison holds for all normal
states;
\item the finite $\mathbb Z_2$ net, its centerful regional algebras,
complete instruments, and successful and failed locality tests
are explicitly constructed;
\item specified internal Kraus devices give one-layer containment,
ancestral factorization, matched replacement invariance, metric
upper bounds, and exterior-fixing no-signalling;
\item regional channel restrictions alone do not specify complete
devices, and shared central data obstruct unrestricted independent
regional changes;
\item a product-preserving bounded-sequence observable limit and
an exact finite-mode spatial gauge refinement limit are constructed;
\item continuum domain of dependence transfers under the displayed
matched spacetime-realization assumptions.
\end{enumerate}
No general relativistic gauge-field scaling limit or physical
frame-covariance conclusion is part of this release.
\end{theorem}

\begin{proof}
Items (i)--(ii) are Theorem~\ref{thm:comparison} and
Corollary~\ref{cor:reference}.
Item (iii) is Proposition~\ref{prop:centers} and
Proposition~\ref{prop:relative}.
Item (iv) is Theorem~\ref{thm:normal}.
Item (v) is Proposition~\ref{prop:gaugenet},
Theorem~\ref{thm:thetaalgebras}, and the explicit examples.
Item (vi) is Theorems~\ref{thm:onelayer},
\ref{thm:ancestry}, \ref{thm:matched},
\ref{thm:nosignalling}, and Corollary~\ref{cor:metric}.
Item (vii) is Propositions~\ref{prop:extensiongap}
and \ref{prop:sharedcenter}.
Item (viii) is Theorems~\ref{thm:sequence},
\ref{thm:refinementnet}, and
\ref{thm:operationalrefinement}.
Item (ix) is Theorem~\ref{thm:continuum}.
\end{proof}

\subsection{Conclusion}

The tensor-free comparison is
\[
 \boxed{
 \begin{gathered}
 \text{equal state restrictions determine the output}\\
 \Longleftrightarrow\
 \text{UCP factorization through the declared local algebra}\\
 \Longleftrightarrow\
 \text{containment of the Heisenberg pullback}.
 \end{gathered}}
\]

Its proof requires a unital closed local algebra and a sufficiently
rich state class, but no spatial complement.
Its causal application additionally requires complete localized
devices whose support action is proved.
The finite gauge example shows both why the theorem applies and why
a regional restriction alone is not a complete device interface.

The constructed observable limits retain products, sector data,
state--effect pairings, and matched actions on their stated classes.
They do not turn a finite constrained graph into a relativistic
field theory by a change of terminology.

%=====================================================================
\appendix
\section{Debugging ledger}
\label{app:debug}
%=====================================================================

The following corrections are required for a proof-ready
implementation of the supplied plan.

\begin{enumerate}[label=(\arabic*)]
\item \textbf{A complementary preparation is unnecessary.}
The proof uses separation and Jordan decomposition.
No map $\rho_I\mapsto\rho_I\otimes\sigma$ is assumed.

\item \textbf{The local unit must be the ambient unit.}
The equality $f_+(1)=f_-(1)$ in the separation proof depends on it.
Corners and fixed sectors must be reformulated with their own
ambient unit.

\item \textbf{State richness is not optional.}
Restricted preparations establish membership in
$\mathcal D_I^\perp$, not automatically in $\A(I)$.
Even scalar-separating preparations need not supply all
equal-snapshot fibers required by the theorem.

\item \textbf{Centers must not be discarded.}
Equal normalized sector states do not determine the sector
probabilities. Complete central effects distinguish them.

\item \textbf{A block decomposition is not spatial factorization.}
The decomposition $M_2\oplus M_2$ describes a regional algebra.
It does not identify complementary physical tensor factors.

\item \textbf{Exterior fixation is not algebra containment.}
Commutant membership and declared regional membership are
different. The internal-device proof establishes membership
by algebra multiplication and isotony.

\item \textbf{Global device extensions must be retained.}
The identity and $\Ad_F$ in
Proposition~\ref{prop:extensiongap} have the same regional
restriction but different cross-boundary actions.

\item \textbf{Arbitrary regional UCP maps need not be admissible.}
The sector-swap automorphism is incompatible with fixation of
the shared exterior flux.

\item \textbf{Constraint compression needs invariant inputs.}
Multiplicativity of restriction was proved only for observables
preserving the physical subspace.

\item \textbf{Complete records are required.}
The unital map is the complete recorded or nonselective channel.
Selective normalization is not substituted for it.

\item \textbf{Joint outputs require a joint algebra.}
Separate one-region or one-record tests are not automatically
a test of their full joint output.

\item \textbf{References are joint interfaces.}
An independent matrix reference is not a spatial complement.
A charge-linked reference requires a joint constrained algebra.

\item \textbf{Expectations are extra structure.}
Gauge averaging is not automatically regional expectation.
Trace-preserving expectation need not preserve a chosen state.

\item \textbf{Normal tests need normal closure.}
The normal theorem uses von Neumann subalgebras, not arbitrary
norm-closed subalgebras tested only on normal states.

\item \textbf{Convergent numbers are not observable convergence.}
The bounded-sequence quotient and the exact refinement maps
provide products and state pairings explicitly.
An arbitrary convergent terminal law supplies neither.

\item \textbf{The refinement limit has a stated size.}
The subdivision family retains four physical dimensions.
It is not an extensive field-theory scaling construction.

\item \textbf{Long loop operations are not microscopic blocks.}
Refining their support does not compile them into a finite-speed
shrinking-block implementation.

\item \textbf{No verification claim replaces a mathematical premise.}
Compilation and machine checking can audit a supplied proof;
they cannot supply an absent localization, extension, or
spacetime-realization theorem.
\end{enumerate}

%=====================================================================
\section{Source alignment and non-transfers}
\label{app:alignment}
%=====================================================================

\begin{center}
\small
\renewcommand{\arraystretch}{1.12}
\begin{tabularx}{\linewidth}{@{}p{.28\linewidth}p{.27\linewidth}X@{}}
\toprule
Present result & Cheung comparison & Exact boundary\\
\midrule
Tensor-free comparison &
En05-A, En05-B, En06-A \Source{lem:comparison} &
Recovers their tensor-interface special case without
a complementary-state channel.\\
\addlinespace
Center-sensitive snapshot &
En05-A and En06-A affine hybrid arrays &
Sector weights retained; direct sums do not imply
spatial tensor factors.\\
\addlinespace
Controller support &
VII-G \Source{thm:locality} &
All control data belong to the localization interface;
no hidden global scheduler.\\
\addlinespace
Ancestral support &
En03-A, VII-E, VII-J, Psy04-B &
An upper support region, not equality with actual influence.\\
\addlinespace
References and complete histories &
En02-C \Source{thm:C1}; Landmark I &
Joint input restrictions required; arbitrary gauge-device
compilation is not supplied.\\
\addlinespace
Complete readout and forgetting &
En04-A; Hawk04-B \Source{thm:drop} &
Destructive readout and operational discard are not
unknown-state transport or physical annihilation.\\
\addlinespace
Observable comparison &
En06-A C6$^{\mathrm{obs}}$ &
Products and state--effect compatibility supplied for the
present quotient and refinement classes only.\\
\bottomrule
\end{tabularx}
\end{center}

\paragraph{Residuals.}
Restriction to a subalgebra does not, by itself, choose a tensor
environment or a complementary channel.
Such an object needs a specified dilation and access interface.
The residual-response distinctions in En04-A and Hawk04-B remain
comparisons in their own models
\cite{CheungEn04A,CheungHawk04B}.
A nontrivial center, an inaccessible observable, or nonconstant
residual response is not geometric backreaction.

\paragraph{Enter.}
In a declared finite measured-register interface, Enter is
destructive Born--L\"uders readout returning a complete classical
record. It is not a decoder, deletion axiom, or unitary.
The tensor-free comparison neither changes that primitive nor
constructs a gauge-field measurement uniqueness theorem
\cite{CheungI,CheungEn05A}.

\paragraph{Unrelated analytic programmes.}
The supplied XIV-H--XIV-J records concern a different
distributional construction.
The later internal-amplitude result at I2 is not the full joint
characteristic map absent at I3 and J2
\cite{CheungXIVH,CheungXIVI,CheungXIVJ}.
No such map or inherited quantum coefficient is constructed here.
The downstream XV-B admission exit is not opened.
The supplied O.1 \Source{NOT\_CLOSED} marker and the separate
XVI-N prepared-insertion obligations are unchanged.
These are documentary non-transfers, not premises of any
operator-algebra theorem in this manuscript.

\paragraph{Documentary catalogue.}
The supplied OpenAI-labelled catalogue remains documentary.
No catalogue entry, family number, factoring claim, circuit
complexity statement, capacity result, or metric-curvature label
is used as a gauge-net, compilation, or scaling-limit theorem.
No independent catalogue verification is asserted.

\paragraph{Physical scope.}
No metric is derived from entanglement.
No Einstein equation, spacetime emergence, Page curve, island,
horizon, holographic reconstruction, gravitationally generated
entanglement test, or invariant physical speed is derived.
No Lorentz covariance or unrestricted device resynchronization
closure is reinstated.

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

\paragraph{Mathematical status.}
The claims of this manuscript are the displayed theorems on their
specified domains. Standard functional-analysis inputs are cited.
No unprovided result from a companion manuscript is a theorem premise.

\paragraph{Resources.}
The finite examples and refinements are existence constructions.
No hardware efficiency, energy-cost, bounded-degree continuum
architecture, uniform block-cardinality bound, or arbitrary
gate-library compilation theorem is claimed.

\paragraph{Verification.}
No successful \Source{pdflatex} execution, proof-assistant
verification, executed numerical check, independent referee
certification, journal acceptance, or laboratory implementation is
asserted. The source has an internal bibliography and requires no
external figures, bibliography file, or shell escape.

\paragraph{Funding.}
No external funding is reported for this manuscript.

%=====================================================================
\begin{thebibliography}{99}
\addcontentsline{toc}{section}{References}
%=====================================================================

\bibitem{CheungEn05A}
N.~Cheung,
\emph{Exact Causal Cones for Partitioned Enter Dynamics:
Snapshot--Channel Comparison, Sharp Relay Speed,
and Conditional Continuum Limits}.
En05-A, version 1, supplied manuscript
\Source{En05-A-v1.tex}, October 2026.
Especially \Source{lem:comparison}, \Source{thm:cone},
and the C1--C7 interfaces.
Cited as a supplied source, not as an independently certified
publication.

\bibitem{CheungEn05B}
N.~Cheung,
\emph{En05-B: An Isotropic Matched Operational Continuum Limit
for Partitioned Quantum--Classical Networks}.
Version 1, supplied manuscript \Source{En05-B-v1.tex},
October 2026.
Especially \Source{lem:comparison}, \Source{thm:matched},
and the fixed-skeleton completion interface.

\bibitem{CheungEn06A}
N.~Cheung,
\emph{En06-A: Causal Cones, Increasing Stencils,
and Operational Transport:
A Corrected Layer Account of Partitioned Quantum Dynamics}.
Version 1, supplied manuscript \Source{En06-A-v1.tex},
October 2026.
Especially \Source{lem:comparison}, \Source{thm:order},
and C6$^{\mathrm{obs}}$.

\bibitem{CheungEn03A}
N.~Cheung,
\emph{En03-A}.
Version 1, supplied companion excerpts.
Especially \Source{lem:one-layer}, \Source{thm:locality},
and the prescribed schedule support comparison.

\bibitem{CheungVIIE}
N.~Cheung,
\emph{Landmark VII-E}.
Version 1, supplied companion excerpts.
Prescribed schedule locality and fixed-labelled-channel
ordering invariance.

\bibitem{CheungVIIG}
N.~Cheung,
\emph{Landmark VII-G}.
Version 1, supplied companion excerpts.
Operational locality with explicit hybrid-controller support;
especially \Source{thm:locality}.

\bibitem{CheungVIIJ}
N.~Cheung,
\emph{Landmark VII-J}.
Version 1, supplied companion excerpts.
Complete-channel no-signalling, operational influence,
and prescribed causal-support interfaces.

\bibitem{CheungPsy04B}
N.~Cheung,
\emph{Psy04-B}.
Version 1, supplied companion excerpts from
\Source{Psy04-B-v1.tex}.
Complete finite networks, scheduled ancestry, and approximate
no-signalling certificates.

\bibitem{CheungEn02C}
N.~Cheung,
\emph{En02-C}.
Version 1, supplied companion excerpts from
\Source{En02-C-v1.tex}.
Reference-preserving sequential-program soundness;
especially \Source{thm:C1}.

\bibitem{CheungI}
N.~Cheung,
\emph{Landmark I}.
Version 2, supplied companion source record and excerpts.
Finite complete instruments and the specified destructive
Enter interface. No gauge-field implementation theorem is
imported.

\bibitem{CheungEn04A}
N.~Cheung,
\emph{En04-A}.
Version 1, supplied companion excerpts.
Complete readout accounting and the distinction between
accessible recovery, complete outputs, and residual response.

\bibitem{CheungHawk04B}
N.~Cheung,
\emph{Hawk04-B}.
Version 2, supplied companion excerpts.
Especially \Source{thm:drop} and the classical-record
reconstruction interface.

\bibitem{CheungXIVH}
N.~Cheung,
\emph{Landmark XIV-H:
Internal Kernel Relations on the Operator Route}.
Version 1, supplied companion excerpts.
Historical H2 internal-amplitude boundary.
Documentary non-transfer only.

\bibitem{CheungXIVI}
N.~Cheung,
\emph{Landmark XIV-I}.
Version 1, supplied companion excerpts.
Internal amplitude constructed at I2; full joint characteristic
map unresolved at I3.
Documentary non-transfer only.

\bibitem{CheungXIVJ}
N.~Cheung,
\emph{Landmark XIV-J}.
Version 1, supplied companion excerpts.
Reconstruction descent, supported-sector off-characteristic
reference, and the unresolved J2 full-map boundary.
Documentary non-transfer only.

\bibitem{CasiniHuertaRosabal}
H.~Casini, M.~Huerta, and J.~A.~Rosabal,
``Remarks on entanglement entropy for gauge fields,''
\emph{Physical Review D} \textbf{89} (2014), 085012.
\href{https://doi.org/10.1103/PhysRevD.89.085012}
{doi:10.1103/PhysRevD.89.085012}.
arXiv:\href{https://arxiv.org/abs/1312.1183}{1312.1183}.

\bibitem{DonnellyFreidel}
W.~Donnelly and L.~Freidel,
``Local subsystems in gauge theory and gravity,''
\emph{Journal of High Energy Physics}
\textbf{09} (2016), 102.
\href{https://doi.org/10.1007/JHEP09(2016)102}
{doi:10.1007/JHEP09(2016)102}.
arXiv:\href{https://arxiv.org/abs/1601.04744}{1601.04744}.

\bibitem{VanAcoleyenEtAl}
K.~Van Acoleyen, N.~Bultinck, J.~Haegeman, M.~Marien,
V.~B.~Scholz, and F.~Verstraete,
``Entanglement of distillation for lattice gauge theories,''
\emph{Physical Review Letters}
\textbf{117} (2016), 131602.
\href{https://doi.org/10.1103/PhysRevLett.117.131602}
{doi:10.1103/PhysRevLett.117.131602}.
arXiv:\href{https://arxiv.org/abs/1511.04369}{1511.04369}.

\bibitem{Arveson}
W.~B.~Arveson,
``Subalgebras of $C^*$-algebras,''
\emph{Acta Mathematica}
\textbf{123} (1969), 141--224.
\href{https://doi.org/10.1007/BF02392388}
{doi:10.1007/BF02392388}.
Cited for the matrix-level completely positive framework,
not as a required extension theorem in the comparison proof.

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

\bibitem{Pedersen}
G.~K.~Pedersen,
\emph{$C^*$-Algebras and Their Automorphism Groups}.
London Mathematical Society Monographs 14,
Academic Press, 1979.
Background on states, Hermitian functionals,
Jordan decomposition, and positive extension.

\bibitem{Sakai}
S.~Sakai,
\emph{$C^*$-Algebras and $W^*$-Algebras}.
Springer, 1971.
Background on von Neumann preduals and normal functionals.

\bibitem{Takesaki}
M.~Takesaki,
``Conditional expectations in von Neumann algebras,''
\emph{Journal of Functional Analysis}
\textbf{9} (1972), 306--321.
\href{https://doi.org/10.1016/0022-1236(72)90004-3}
{doi:10.1016/0022-1236(72)90004-3}.

\bibitem{KogutSusskind}
J.~Kogut and L.~Susskind,
``Hamiltonian formulation of Wilson's lattice gauge theories,''
\emph{Physical Review D}
\textbf{11} (1975), 395--408.
\href{https://doi.org/10.1103/PhysRevD.11.395}
{doi:10.1103/PhysRevD.11.395}.
The finite $\mathbb Z_2$ demonstrator in this paper is proved
directly.

\bibitem{Fewster}
C.~J.~Fewster,
``The split property for locally covariant quantum field
theories in curved spacetime,''
\emph{Letters in Mathematical Physics}
\textbf{105} (2015), 1633--1661.
\href{https://doi.org/10.1007/s11005-015-0798-2}
{doi:10.1007/s11005-015-0798-2}.
arXiv:\href{https://arxiv.org/abs/1501.02682}{1501.02682}.
Cited as a comparison for stronger independence assumptions.

\bibitem{BuchholzVerch}
D.~Buchholz and R.~Verch,
``Scaling algebras and renormalization group in algebraic
quantum field theory,''
\emph{Reviews in Mathematical Physics}
\textbf{7} (1995), 1195--1240.
\href{https://doi.org/10.1142/S0129055X9500044X}
{doi:10.1142/S0129055X9500044X}.
arXiv:\href{https://arxiv.org/abs/hep-th/9501063}
{hep-th/9501063}.
Cited as a specialized AQFT comparison, not identified with the
bounded-sequence quotient constructed here.

\bibitem{Watrous}
J.~Watrous,
\emph{The Theory of Quantum Information}.
Cambridge University Press, 2018.
\href{https://doi.org/10.1017/9781316848142}
{doi:10.1017/9781316848142}.
Finite complete instruments and reference-stable channel norms.

\end{thebibliography}

\end{document}