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% Scope:
% Finite classical--quantum instruments, observation boundaries,
% stochastic histories, entangled spectators, implementation comparison,
% resource accounting, and channel-level stability.
% A small certified slicing-selection example witnesses consistency.
% No gravitational evolution equation or quantum-field admission is claimed.
%
% Corrections incorporated:
% 1. Hybrid blocks are subnormalized.
% 2. Instruments and controller tables are independent of the unknown state.
% 3. Future-history agreement requires branchwise adjoint closure.
% 4. The direct-sum observation criterion is a specialization of VII-E's
%    block-diagonal criterion, not a new general channel theorem.
% 5. Spectator consistency is proved for every unnormalized history block.
% 6. Informative classical records obstruct exact recovery on all inputs,
%    including recovery with access to the classical record.
% 7. Implementation comparison concerns selective CP maps, not only effects.
% 8. Full-channel and history stability estimates use consistent factors
%    of two and an explicit classical-input diamond norm.
% 9. Resource balance is not identified with physical energy conservation.
% 10. A fixed-cone slicing selection is not gravitational backreaction.
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% The publisher page for the Enter article reports acceptance on
% 28 September 2026 and a forthcoming final formatted version.
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% VIII-B, VIII-C, and En02-C are contextual comparisons; all mathematical
% statements used in the present results are restated and proved below.

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  pdftitle={VII-G: Completely positive observation boundaries for finite matter--geometry feedback},
  pdfsubject={Classical--quantum instruments, stochastic histories, spectators, and certified geometric interfaces}
}

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\title{\textbf{Landmark VII-G}\\[0.45em]
Completely positive observation boundaries\\
for finite matter--geometry feedback}
\author{}
\date{}

\begin{document}
\maketitle

\begin{abstract}
We replace an unrestricted deterministic state-functional coupling by
a finite classical--quantum transition instrument. Classical labels
contain the geometric interface and every controller variable used
to select future operations. Matter occupies a fixed
finite-dimensional Hilbert space, and the unconditional evolution is
linear, completely positive, and trace preserving.

We give an exact observation criterion in the direct-sum state
algebra and strengthen it to agreement of stochastic history laws
and history-conditioned accessible residuals. A finite backward
construction identifies the smallest observable linear spaces needed
for a declared continuation. Arbitrary entangled spectator extensions
preserve every unnormalized geometric-history block after restriction,
provided the complete controller and its certificates are spectator
blind. Implementation equivalence is characterized by selective
adjoint maps and is stable under compatible continuations.

Two limitations are proved independently: a nonconstant deterministic
packet selection cannot be the classical output of an affine channel
on a convex state domain, and an informative classical record is
incompatible with exact recovery of every input state. Resource and
energy balance are stated as operator identities, with resource
accounting kept distinct from physical energy. Choi bounds yield
checkable assisted stability estimates for both terminal states and
complete histories.

A resource-budgeted unsharp qubit instrument selects strictly ordered
horizontal slices of a fixed spinless cone. Every branch has an
explicit certificate; its classical sector description and a
quantitative dephasing approximation are calculated exactly.
This example is matter-controlled slicing, not matter-driven
gravitational dynamics. No quantum-field source, changing deficit,
moving particle, or gravitational field equation is constructed.
\end{abstract}

\tableofcontents

\section{The finite problem and its relation to the series}
\label{sec:intro}

\subsection{The nonlinear benchmark}

The observation interface of VII-E allows a coupling rule of the form
\begin{equation}
 g'=F_v\bigl(g,\omega_\rho|_{\Src_v}\bigr),
 \label{eq:old-coupling}
\end{equation}
while leaving the matter state unchanged. VII-E explicitly notes that
this need not be affine and is not thereby a quantum
channel~\cite[Definition \texttt{def:rules} and
Remark \texttt{rem:nonlinear}]{CheungVIIE}.

A concrete benchmark is the rule called \emph{Option N} in VIII-B:
\begin{equation}
 \mu_\theta(\rho)=\mu_0+\theta\kappa I_f(\rho),
 \qquad
 \rho\longmapsto\Pi_A\bigl(\mu_\theta(\rho)\bigr).
 \label{eq:optionN}
\end{equation}
Its domain requires the selected deficit to stay in the declared
interval. The rule selects one definite geometric packet and does
not acquire a stochastic measurement record. The finite source
letter of VIII-C uses a supplied source token in a prescribed
table-selection command, including the word
\begin{equation}
 W=\mathsf G;\ \mathsf{Stab};\ \mathsf G;\
       \mathsf{E}_{\mathrm{particle}};\ \mathsf G.
 \label{eq:source-word}
\end{equation}
A promised token is not an instrument that extracts an expectation
from unknown matter~\cite{CheungVIIIB,CheungVIIIC}.

We take these as named comparison targets, not as premises providing
a joint completely positive evolution. Proposition~\ref{prop:affine}
below identifies exactly why a nonconstant deterministic selection
cannot simply be promoted to an unconditional affine channel.

\subsection{Established framework and present contribution}

Quantum instruments distinguish outcome probabilities from their
selective residual maps~\cite{DaviesLewis1970}. Finite-dimensional
complete positivity and its matrix certificates are
standard~\cite{Choi1975,Watrous2018}. Linear completely positive
classical--quantum dynamics is also an established framework,
including stochastic trajectories and broader hybrid
generators~\cite{Oppenheim2023,OppenheimTrajectories2023,
OppenheimClasses}.

Accordingly, we do not claim novelty for the direct-sum state space,
the Kraus construction, the exact recovery--complement criterion,
or the general channel observation principle. In particular,
VII-E already proves a composition criterion for block-diagonal
state spaces~\cite[Proposition
\texttt{prop:classical-composition}]{CheungVIIE}.
Our direct-sum criterion is its intrinsic specialization.

The purpose of VII-G is to organize these facts into a complete
finite feedback specification, with:
\begin{enumerate}
 \item a state-independent transition table and certified geometric
       output graph;
 \item branch-sensitive observation boundaries and stochastic histories;
 \item exact consistency with entangled spectators;
 \item selective-map implementation tests;
 \item resource accounting and assisted error certificates;
 \item an explicit finite example satisfying the specification.
\end{enumerate}
All results used in the assembly are proved below.

\subsection{Separation from gravitational dynamics}

A geometric label can describe a slice, a germ, a controller state,
or a spacetime parameter. Updating such a label is not, by itself,
a gravitational evolution law. The distinction is
\begin{equation}
 \boxed{
 \text{matter-controlled slicing}
 \neq
 \text{matter-driven gravitational dynamics}.}
 \label{eq:separation}
\end{equation}

VII-F supplies a demanding fixed-cone incidence certificate, including
full affine holonomy and exact event control, but keeps its spacetime,
deficit, and worldline fixed~\cite{CheungVIIF}.
We use its certificate discipline, not an unproved extension of its
geometry. The example in Section~\ref{sec:example} is simpler:
its slices are horizontal and its particle link does not change.

The field-theoretic manuscripts VII-A--D are not prerequisites of the
finite results. Their relevance to a later source construction is
discussed in Section~\ref{sec:limits}.

\section{Hybrid states, controllers, and transition instruments}
\label{sec:hybrid}

\subsection{Direct-sum conventions}

Fix a nonempty finite label set \(\Geo\) and a Hilbert space
\(\Hil\) of dimension \(d<\infty\). Set
\begin{equation}
 \Alg_{\Geo}=\bigoplus_{g\in\Geo}\Mat(\Hil).
 \label{eq:state-algebra}
\end{equation}
For \(X=(X_g)_g\) and \(B=(B_g)_g\), use
\begin{equation}
 \Tr_{\Geo}X=\sum_g\Tr X_g,
 \qquad
 \langle B,X\rangle=\sum_g\Tr(B_gX_g).
 \label{eq:pairing}
\end{equation}
The identity is \(\one=(I)_g\). The relevant norms are
\begin{equation}
 \norm{X}_{1,\Geo}=\sum_g\norm{X_g}_1,
 \qquad
 \norm{B}_{\infty,\Geo}=\max_g\norm{B_g}_\infty.
 \label{eq:hybrid-norms}
\end{equation}

A hybrid state is a positive element \(\rho=(\rho_g)_g\) with
\(\Tr_{\Geo}\rho=1\). Equivalently,
\begin{equation}
 \sigma_\rho=\sum_g\ketbra g g\otimes\rho_g.
 \label{eq:cq-state}
\end{equation}
The blocks are \emph{subnormalized}:
\begin{equation}
 p_g=\Tr\rho_g,
 \qquad
 \widehat\rho_g=\rho_g/p_g\quad\text{only if }p_g>0.
 \label{eq:conditional}
\end{equation}
Conditional normalization is not used in the unconditional
evolution law.

We use
\[
 \dtr(\rho,\sigma)=\tfrac12\norm{\rho-\sigma}_{1,\Geo},
 \qquad
 \dTV(p,q)=\tfrac12\sum_x\abs{p_x-q_x},
 \qquad
 \ddiam(\Phi,\Xi)=\tfrac12\norm{\Phi-\Xi}_\diamond.
\]
The classical-input meaning of the diamond norm is made precise in
Section~\ref{sec:stability}.

\subsection{A declared controller, not a state-dependent compiler}

The label \(g\) contains every classical variable used to select
subsequent operations. A typical label is
\begin{equation}
 g=(\gamma,\text{clock},\text{permission},
          \text{controller memory},\text{declared records}),
 \label{eq:controller-label}
\end{equation}
where \(\gamma\) denotes geometric data.

An operation table may depend on \(g\), but not on an undeclared
functional of the current unknown density operator. Adaptive
measurement control is represented by recording the outcome in a
classical successor label. If a finer outcome affects future choices,
it must remain in the controller state even when a coarser geometric
report is displayed.

For a fixed finite horizon, any finite instruction tree can be made
Markovian by adjoining its node or relevant history to the label.
This may require exponentially many labels. No bounded-memory
claim for arbitrary horizons is implied.

\begin{definition}[Certified transition graph]
\label{def:graph}
A certified finite interface consists of:
\begin{enumerate}
 \item the finite labels \(\Geo\), each with a declared admissible
       geometric realization and controller interpretation;
 \item a directed edge relation \(\mathcal E\subseteq\Geo\times\Geo\);
 \item a fixed certificate for each edge \((g,h)\in\mathcal E\),
       specifying all affected output data, matching requirements,
       controller updates, and any location transport;
 \item a declared reserve for implementation history and spectators,
       not inspected by geometric control.
\end{enumerate}
The certificates are classical data. The definition does not prove
a gravitational constraint equation.
\end{definition}

\subsection{The channel construction}

For each \(g,h\in\Geo\), let
\[
 \Psi_{h|g}:\Mat(\Hil)\longrightarrow\Mat(\Hil)
\]
be completely positive. Require
\begin{equation}
 \sum_h\Psi_{h|g}^*(I)=I\qquad(g\in\Geo).
 \label{eq:instrument-complete}
\end{equation}
Nonzero branches are permitted only on certified edges.

Define
\begin{equation}
 (\F\rho)_h=\sum_g\Psi_{h|g}(\rho_g).
 \label{eq:hybrid-map}
\end{equation}
Here and below the adjoint is characterized by
\(\Tr(B\Psi(X))=\Tr(\Psi^*(B)X)\).

\begin{theorem}[Finite hybrid channel structure]
\label{thm:structure}
The map \eqref{eq:hybrid-map} is completely positive and trace
preserving on \(\Alg_{\Geo}\). It preserves the classical--quantum
state space and has adjoint
\begin{equation}
 (\F^*B)_g=\sum_h\Psi_{h|g}^*(B_h).
 \label{eq:hybrid-adjoint}
\end{equation}

Conversely, every completely positive trace-preserving map between
two finite direct sums has this branch form. If
\[
 \Psi_{h|g}(X)=\sum_\alpha
 K_{h|g,\alpha}XK_{h|g,\alpha}^\dagger,
\]
then
\begin{equation}
 \sum_{h,\alpha}K_{h|g,\alpha}^\dagger K_{h|g,\alpha}=I.
 \label{eq:kraus-complete}
\end{equation}
A canonical extension to the full matrix space on
\(\C^{\Geo}\otimes\Hil\) has Kraus operators
\begin{equation}
 \ketbra h g\otimes K_{h|g,\alpha}.
 \label{eq:lift-kraus}
\end{equation}
This extension discards off-diagonal input label blocks and has
block-diagonal output.
\end{theorem}

\begin{proof}
After adjoining any matrix ancilla, a positive direct-sum input has
positive blocks. Each amplified branch map is positive, and sums
of its outputs are positive. Thus \(\F\) is completely positive.
Furthermore,
\[
 \Tr_{\Geo}\F(X)
 =\sum_g\Tr\left(
      \left[\sum_h\Psi_{h|g}^*(I)\right]X_g
    \right)
 =\Tr_{\Geo}X.
\]
The block formula proves preservation of classicality, and pairing
with \(B\) gives \eqref{eq:hybrid-adjoint}.

For the converse, compose the given map with the completely positive
injection into input block \(g\) and projection onto output block
\(h\). These compositions define completely positive
\(\Psi_{h|g}\). Linearity gives the branch formula, and trace
preservation on each input block gives
\eqref{eq:instrument-complete}.

Finite-dimensional completely positive maps have Kraus
representations: the positive Choi matrix can be decomposed into
rank-one positive terms and each vector unvectorized into a Kraus
operator~\cite{Choi1975}. Completeness gives
\eqref{eq:kraus-complete}. The operators
\eqref{eq:lift-kraus} have sum of squared moduli equal to the full
identity. Their action reads only the diagonal input blocks and
writes only diagonal output blocks, proving the final assertion.
\end{proof}

\begin{proposition}[Composition]
\label{prop:composition-channel}
If a second transition has branches \(\Lambda_{k|h}\), the composed
channel has branches
\begin{equation}
 \Omega_{k|g}=\sum_h\Lambda_{k|h}\circ\Psi_{h|g}.
 \label{eq:branch-composition}
\end{equation}
Its certificates are concatenations of the declared edge certificates.
\end{proposition}

\begin{proof}
Substitution into the two block formulas gives
\eqref{eq:branch-composition}. Sums and compositions preserve
complete positivity. Moreover,
\[
 \sum_k\Omega_{k|g}^*(I)
 =\sum_h\Psi_{h|g}^*
       \left(\sum_k\Lambda_{k|h}^*(I)\right)
 =\sum_h\Psi_{h|g}^*(I)=I.
\]
The certificate assertion means concatenation of already declared
certificates; it introduces no new geometric existence claim.
\end{proof}

\section{Observation boundaries and stochastic histories}
\label{sec:observations}

\subsection{Observable linear spaces}

\begin{definition}[Observation space]
An observation space is a complex linear, adjoint-closed subspace
\(\Obs\subseteq\Alg_{\Geo}\) containing \(\one\). Thus it is a
finite-dimensional operator system. It need not be closed under
multiplication.

Two states agree on \(\Obs\) if
\[
 \langle B,\rho\rangle=\langle B,\sigma\rangle
 \qquad(B\in\Obs).
\]
A frequently used choice is
\begin{equation}
 \Obs_{\Src}=\bigoplus_g\Src_g,
 \label{eq:direct-observation}
\end{equation}
where each \(\Src_g\subseteq\Mat(\Hil)\) is a unital
\(^{*}\)-subalgebra. This choice observes the complete classical
label, because it contains each central block projection.
\end{definition}

Use of linear spaces is important: the smallest future-sensitive
observation space constructed below need not be an algebra.

\begin{theorem}[Intrinsic hybrid observation criterion]
\label{thm:observation}
Let \(\F:\Alg_{\mathrm{in}}\to\Alg_{\mathrm{out}}\) be a channel
and let \(\Obs_{\mathrm{in}},\Obs_{\mathrm{out}}\) be observation
spaces. The following are equivalent:
\begin{enumerate}[label=\textup{(\roman*)}]
 \item for all hybrid states,
 \[
 \rho|_{\Obs_{\mathrm{in}}}=\sigma|_{\Obs_{\mathrm{in}}}
 \ \Longrightarrow\
 \F(\rho)|_{\Obs_{\mathrm{out}}}
       =\F(\sigma)|_{\Obs_{\mathrm{out}}};
 \]
 \item
 \begin{equation}
 \F^*(\Obs_{\mathrm{out}})\subseteq\Obs_{\mathrm{in}}.
 \label{eq:observation-inclusion}
 \end{equation}
\end{enumerate}
For the direct-sum spaces
\(\Obs_{\mathrm{in}}=\bigoplus_g\Src_g\) and
\(\Obs_{\mathrm{out}}=\bigoplus_h\mathcal B_h\), this is equivalent to
\begin{equation}
 \Psi_{h|g}^*(\mathcal B_h)\subseteq\Src_g
 \qquad(g,h).
 \label{eq:branch-inclusion}
\end{equation}
\end{theorem}

\begin{proof}
The inclusion implies agreement by pulling each output observable
back to the input space.

Conversely, let \(H=H^\dagger\in\Alg_{\mathrm{in}}\) annihilate
the Hermitian part of \(\Obs_{\mathrm{in}}\) under the trace pairing.
Because \(\one\in\Obs_{\mathrm{in}}\), \(\Tr H=0\).
Choose a faithful hybrid state \(\omega\); a multiple of the
block identity suffices. For sufficiently small \(a>0\),
\(\omega\pm aH\) are states agreeing on \(\Obs_{\mathrm{in}}\).
The assumed output agreement implies
\[
 \langle \F^*B,H\rangle=0
 \qquad(B=B^\dagger\in\Obs_{\mathrm{out}}).
\]
Finite-dimensional double orthogonal complementation places
\(\F^*B\) in the Hermitian part of
\(\Obs_{\mathrm{in}}\). Complex linearity gives
\eqref{eq:observation-inclusion}.

If the spaces are direct sums, an output observable may have one
nonzero block \(B_h\). Formula \eqref{eq:hybrid-adjoint} then makes
\eqref{eq:observation-inclusion} imply every
\eqref{eq:branch-inclusion}. The reverse implication follows by
summing those branch images.
\end{proof}

This is the direct-sum specialization of VII-E's composition
principle and block-diagonal refinement, proved here without an
ambient pinching notation~\cite[Theorem \texttt{thm:composition}
and Proposition \texttt{prop:classical-composition}]{CheungVIIE}.

\subsection{One-step geometric effects}

Define
\begin{equation}
 E_{h|g}=\Psi_{h|g}^*(I).
 \label{eq:effects}
\end{equation}
For a supported input block,
\begin{equation}
 p(h\mid g,\widehat\rho_g)
       =\Tr(E_{h|g}\widehat\rho_g).
 \label{eq:transition-probability}
\end{equation}

\begin{corollary}[Exact source restriction]
\label{cor:source}
For a fixed \(g\), all successor probabilities depend only on the
state functional on \(\Src_g\) if and only if
\[
 E_{h|g}\in\Src_g\qquad(h\in\Geo).
\]
\end{corollary}

\begin{proof}
Sufficiency is \eqref{eq:transition-probability}. For necessity,
apply the faithful-state perturbation proof of
Theorem~\ref{thm:observation} to each effect.
\end{proof}

This is the effect-pullback observation of VII-E with the successor
geometry as the classical outcome~\cite[Theorem
\texttt{thm:pullback}]{CheungVIIE}. It says nothing by itself about
the residual matter state.

\subsection{History instruments}

Allow a finite sequence of channels with branches
\(\Psi^{(t)}_{h|g}\), \(t=1,\ldots,T\). All tables are fixed;
the current label already contains the controller memory.

For a history
\(\gamma=(g_0,\ldots,g_n)\), define
\begin{equation}
 \rho_\gamma
 =
 \Psi^{(n)}_{g_n|g_{n-1}}\circ\cdots\circ
 \Psi^{(1)}_{g_1|g_0}(\rho_{g_0}).
 \label{eq:path-block}
\end{equation}
Its probability is \(p_\gamma=\Tr\rho_\gamma\). The complete
history-retaining output is
\begin{equation}
 \mathscr H_n(\rho)=
       \bigoplus_{\gamma\in\Geo^{n+1}}\rho_\gamma.
 \label{eq:history-channel}
\end{equation}
Forbidden paths have zero blocks.

\begin{lemma}[Completeness of history retention]
\label{lem:history-channel}
The map \(\mathscr H_n\) is a channel. Summing its blocks with a
common terminal label gives the ordinary terminal hybrid state.
\end{lemma}

\begin{proof}
Each path map is completely positive. Summing probabilities over
the final label uses \eqref{eq:instrument-complete}; repeating
this sum backwards over the path leaves
\(\sum_{g_0}\Tr\rho_{g_0}=1\). This proves trace preservation.
The terminal summation is precisely repeated substitution in
\eqref{eq:hybrid-map}.
\end{proof}

\begin{theorem}[History-sensitive observation principle]
\label{thm:histories}
Suppose that, for each \(t,g,h\),
\begin{equation}
 \Psi^{(t)*}_{h|g}(\Obs_{t,h})
        \subseteq\Obs_{t-1,g},
 \label{eq:time-branch-inclusion}
\end{equation}
where each \(\Obs_{t,g}\subseteq\Mat(\Hil)\) contains \(I\).
If two initial hybrid states agree on
\(\bigoplus_g\Obs_{0,g}\), then:
\begin{enumerate}[label=\textup{(\roman*)}]
 \item all geometric history probabilities through time \(T\) agree;
 \item for every history ending at \(g_n\), the corresponding
       unnormalized blocks agree on \(\Obs_{n,g_n}\);
 \item on every positive-probability history, the normalized
       conditional states agree on the same space.
\end{enumerate}
\end{theorem}

\begin{proof}
For \(B\in\Obs_{n,g_n}\), its history-conditioned expectation is
\[
 \Tr(B\rho_\gamma)
 =
 \Tr\left(
   \left[
    \Psi^{(1)*}_{g_1|g_0}\circ\cdots\circ
    \Psi^{(n)*}_{g_n|g_{n-1}}(B)
   \right]\rho_{g_0}
 \right).
\]
Successive applications of
\eqref{eq:time-branch-inclusion} put the bracket in
\(\Obs_{0,g_0}\). Initial agreement therefore proves (ii).
Taking \(B=I\) proves (i). Equal positive probabilities permit
division by the same scalar, proving (iii).
\end{proof}

The theorem compares correlated stochastic histories, not merely
one-step probabilities or terminal marginals.

\subsection{The smallest boundary for a fixed continuation}

\begin{proposition}[Finite backward observation construction]
\label{prop:minimal}
Fix terminal observation spaces \(\Obs_{T,g}\). Define backwards
\begin{equation}
 \Obs_{t-1,g}
 =
 \spanC\left(
 \{I\}\cup
 \bigcup_h\Psi^{(t)*}_{h|g}(\Obs_{t,h})
 \right).
 \label{eq:backward-space}
\end{equation}
These are observation spaces. They are the smallest family containing
the terminal spaces and satisfying
\eqref{eq:time-branch-inclusion}.

Agreement on the initial direct sum is equivalent to agreement of
all complete length-\(T\) history-conditioned terminal expectations.
It consequently implies agreement of all shorter history laws.
\end{proposition}

\begin{proof}
Adjoints of completely positive maps preserve adjoints. Hence the
displayed span is adjoint closed and contains \(I\). Any family
satisfying the inclusions must contain the indicated span at each
backward step, proving minimality by induction.

The pullbacks of all length-\(T\) history-conditioned terminal
observables span the recursively constructed spaces. The inserted
identity adds no independent expectation: at each step it equals
\(\sum_h\Psi^{(t)*}_{h|g}(I)\). Thus agreement of all those
expectations is exactly agreement on the initial spaces.
Summing extensions of a shorter history gives its probability.
\end{proof}

For a time-homogeneous table and geometry-only terminal observations,
one can also construct the boundary for every finite history.

\begin{corollary}[Finite stabilization of the all-history boundary]
\label{cor:stabilization}
Set \(\Obs_g^{(0)}=\C I\) and
\begin{equation}
 \Obs_g^{(n+1)}
 =
 \spanC\left(
   \Obs_g^{(n)}\cup
   \bigcup_h\Psi_{h|g}^*(\Obs_h^{(n)})
 \right).
 \label{eq:iterated-boundary}
\end{equation}
The family stabilizes after at most
\(\abs{\Geo}(d^2-1)\) strict enlargements. Its stable value is the
smallest branch-invariant observation family containing the classical
effects. Agreement on its direct sum is equivalent to agreement of
all finite geometric history laws.
\end{corollary}

\begin{proof}
The dimensions are nondecreasing and their sum is at most
\(\abs{\Geo}d^2\), starting at \(\abs{\Geo}\). If one iteration
does not enlarge the family, all subsequent iterations are identical.
This proves the bound and branch invariance.

The recursively generated elements are spans of finite path effects.
Conversely, every finite path effect appears at some iteration.
The equivalence follows from the trace pairing with those effects.
\end{proof}

No algebra closure is asserted for these minimal spaces. Replacing
them by generated \(^{*}\)-algebras is permissible but may declare
more information accessible than the histories actually require.

\subsection{The negative observation test}

\begin{proposition}[Indistinguishability obstruction]
\label{prop:obs-obstruction}
If two admissible inputs agree on the declared observation space
but a target requires different accessible output expectations,
no channel satisfying the corresponding adjoint inclusion realizes
that target on both inputs.
\end{proposition}

\begin{proof}
Theorem~\ref{thm:observation} forces equality of the output
expectations.
\end{proof}

This is the linear-channel form of the observation-factorization
obstruction in VII-F~\cite[Proposition
\texttt{prop:obs-obstruction}]{CheungVIIF}. A capacity check or
rounded report is an additional observation, not a consequence of
a declared matter source algebra.

\section{Entangled spectators and complete controller blindness}
\label{sec:spectators}

Let \(N\) be a finite untouched spectator. An initial extension
satisfies
\begin{equation}
 \Tr_N\widetilde\rho_g=\rho_g
 \qquad(g\in\Geo).
 \label{eq:block-extension}
\end{equation}
It may be entangled within every block.

\begin{assumption}[Blindness of the complete transition specification]
\label{ass:blind}
Adding the spectator does not change:
\begin{enumerate}
 \item the label dictionary or any input-label interpretation;
 \item the selected branch maps, except for tensoring with
       \(\id_N\);
 \item guard results, successor labels, certificates, or transport
       of original names;
 \item controller updates or the declared resource reserve.
\end{enumerate}
Spectator storage and any spectator location history are separately
reserved. Spectator outcomes are not read by the controller.
\end{assumption}

The assumption concerns the whole transition specification.
Equality of one displayed source functional is not sufficient.

\begin{lemma}[Restriction of a selective map]
\label{lem:partial-trace}
For every linear map \(\Psi\) on the original matter,
\begin{equation}
 \Tr_N[(\Psi\otimes\id_N)(X)]
       =\Psi(\Tr_NX).
 \label{eq:selective-restriction}
\end{equation}
No trace-preservation or product-state assumption is needed.
\end{lemma}

\begin{proof}
Pair both sides with an arbitrary output operator \(B\). Their
pairings equal
\[
 \Tr((\Psi^*(B)\otimes I_N)X).
\]
Equality against all \(B\) proves the statement.
\end{proof}

\begin{theorem}[Spectator consistency for stochastic histories]
\label{thm:spectators}
Under Assumption~\ref{ass:blind} and
\eqref{eq:block-extension}, extend every branch by
\(\Psi^{(t)}_{h|g}\otimes\id_N\). Then, for every finite history,
\begin{equation}
 \Tr_N\widetilde\rho_\gamma=\rho_\gamma.
 \label{eq:path-spectator}
\end{equation}
Consequently:
\begin{enumerate}[label=\textup{(\roman*)}]
 \item the complete geometric history distributions agree;
 \item the reduced history--matter outputs agree;
 \item on every supported history, the conditional original-matter
       reductions agree;
 \item the nonselective spectator marginal is unchanged.
\end{enumerate}
\end{theorem}

\begin{proof}
Apply Lemma~\ref{lem:partial-trace} to each map in
\eqref{eq:path-block}. This proves \eqref{eq:path-spectator}
without normalizing any branch. Taking traces proves (i);
taking the direct sum proves (ii); division by a common positive
trace proves (iii).

For (iv), the sum of all branches of a one-step transition is
trace preserving on each input label. Pair the spectator reduction
with \(B_N\). Its expectation after the step is
\[
 \sum_g\Tr\left(
  \left[\sum_h\Psi_{h|g}^*(I)\right]\otimes B_N
  \ \widetilde\rho_g
 \right)
 =
 \sum_g\Tr((I\otimes B_N)\widetilde\rho_g).
\]
Iteration proves nonselective invariance.
\end{proof}

This extends the partial-trace mechanism of VII-E to selective
branches and their stochastic correlations~\cite[Lemma
\texttt{lem:partial-trace}, Theorem \texttt{thm:spectators},
and Assumption \texttt{ass:blind}]{CheungVIIE}.
It does not identify two different spectator extensions as joint
outputs.

\begin{example}[A capacity leak]
\label{ex:capacity}
Suppose an update is allowed only when the total number of matter
names is at most \(K\). An original interface with \(K\) names passes.
Adding one product spectator makes the same geometric update fail,
even if every source algebra is \(\C I\) and no quantum operation
touches the spectator.
\end{example}

This violates Assumption~\ref{ass:blind}. It is the capacity leak
identified in VII-E~\cite[Example \texttt{ex:capacity}]{CheungVIIE}.

\begin{remark}[Selective spectator states]
For an entangled pair
\((\ket{00}+\ket{11})/\sqrt2\), local projective readout leaves the
remote marginal \(I/2\) nonselectively, but its conditional states
are \(\ketbra00\) and \(\ketbra11\). Thus (iv) is not a claim that
the spectator state is unchanged on each outcome. Nor may a
subsequent controller inspect an otherwise excluded spectator
record without changing the hypothesis.
\end{remark}

\section{Implementation equivalence and represented readout}
\label{sec:implementation}

\subsection{Selective adjoint comparison}

Let \(\Psi_{h|g}\) and \(\widetilde\Psi_{h|g}\) be two implementations
with the same input and output label dictionaries.

\begin{proposition}[Exact implementation criterion]
\label{prop:implementation}
The implementations agree on
\(\Obs_{\mathrm{out}}=\bigoplus_h\mathcal B_h\) for every hybrid
input if and only if
\begin{equation}
 \Psi_{h|g}^*(B)
 =\widetilde\Psi_{h|g}^*(B)
 \qquad(B\in\mathcal B_h,\ g,h).
 \label{eq:implementation-adjoint}
\end{equation}
The equality also holds on
\(\Obs_{\mathrm{out}}\otimes\Mat(N)\) for every untouched reference
\(N\), using the same extended input in both implementations.
\end{proposition}

\begin{proof}
Choose an input supported in block \(g\) and an output observable
supported in block \(h\). Equality of all state expectations is
equality of the Hermitian adjoint images. Complex linearity gives
\eqref{eq:implementation-adjoint}. Conversely, summing these
identities proves output agreement.

For a reference observable \(B\otimes C\), the two adjoint images
are equal by tensoring \eqref{eq:implementation-adjoint} with \(C\).
Such tensors span the reference-extended observation space.
\end{proof}

\begin{corollary}[Implementation agreement through histories]
\label{cor:implementation-histories}
Suppose both implementations satisfy the same branchwise inclusions
\eqref{eq:time-branch-inclusion}, and their branch adjoints agree on
each \(\Obs_{t,h}\). Starting from the same input, their complete
histories have equal probabilities and equal accessible
history-conditioned residual expectations.
\end{corollary}

\begin{proof}
In the backward expectation formula from
Theorem~\ref{thm:histories}, replace the final adjoint by the other
implementation's adjoint. The inclusion keeps its image in the
previous observation space, where the next replacement is valid.
Continue backwards to the initial state.
\end{proof}

Equality of effects is only the case \(B=I\).
It cannot replace the selective-map criterion.

\subsection{An exact dilation and its resource meaning}

Fix an input label \(g\). Choose a Kraus representation for every
nonzero branch and write \(a=(h,\alpha)\). Define
\begin{equation}
 V_g\psi
 =\sum_{h,\alpha}
       \ket{h,\alpha}_A\otimes K_{h|g,\alpha}\psi.
 \label{eq:dilation}
\end{equation}
Completeness implies \(V_g^\dagger V_g=I\).

Destructive computational readout of \(A\), followed by reporting
only \(h\), gives
\begin{equation}
 \sum_h\ketbra h h_R\otimes
        \sum_\alpha K_{h|g,\alpha}XK_{h|g,\alpha}^\dagger.
 \label{eq:dilated-instrument}
\end{equation}
The matter register is returned. The carrier is consumed.
This is a represented instrument, not destructive readout of the
matter source itself.

The Born readout interface is the specified semantic primitive of
the Enter calculus~\cite{CheungEnter}. No measurement-strategy
uniqueness theorem is used as a premise here.

\begin{proposition}[One-shot environment ledger]
\label{prop:environment}
Let
\[
 r_{h|g}=\rank J(\Psi_{h|g}),
 \qquad
 R_g=\sum_h r_{h|g},
\]
where \(J\) is the unnormalized Choi matrix. Among dilations with
orthogonal pointer sectors for the reported outcomes \(h\), the
minimum environment dimension is \(R_g\).

A common pointer reserve of
\begin{equation}
 e=\left\lceil\log_2\max_g R_g\right\rceil
 \label{eq:pointer-bits}
\end{equation}
qubits suffices. If matter is encoded in \(s\) qubits, a unitary
extension of \eqref{eq:dilation} uses those \(s\) matter qubits and
\(e\) fresh pointer qubits. Full fine readout produces at most \(e\)
classical bits per step.
\end{proposition}

\begin{proof}
A Choi matrix of rank \(r_{h|g}\) has a decomposition into precisely
that many rank-one positive terms, giving a Kraus family of that
size. Conversely, a family with \(q\) operators gives a Choi sum
of \(q\) rank-one terms, so \(q\geq r_{h|g}\).

An environment sector of dimension \(q_h\) gives at most \(q_h\)
Kraus operators when expanded in a sector basis. Thus
\(q_h\geq r_{h|g}\), and summing sectors gives the lower bound
\(R_g\). Minimal Kraus families in \eqref{eq:dilation} attain it.

Encode the pointer in \(e\) qubits. The isometry on the subspace
with pointer input \(\ket{0}^{\otimes e}\) extends to a unitary
by completing orthonormal bases. Destructive readout consumes
the pointer and returns a finite classical fine index.
\end{proof}

This is an existence statement for a unitary dilation. It is not
an exact-synthesis claim for arbitrary matrix entries in a fixed
discrete gate library. After readout, any retained fine index is
classical implementation history. It must either be explicitly
available to the controller or remain outside its observation
boundary.

For a horizon \(T\), retaining all fine indices requires at most
\(Te\) bits. Storing all geometry labels additionally requires
at most \((T+1)\lceil\log_2\abs{\Geo}\rceil\) bits.
These are sufficient reserves, not optimal adaptive-memory bounds.
No spectator is charged against them.

\subsection{Immediate and coherent implementations that agree}

Suppose a source instrument has Kraus operators \(M_a\), with
\(\sum_aM_a^\dagger M_a=I\), and its outcome selects a successor
\(\Gamma_a(g)\) and unitary \(U_{\Gamma_a(g)}\).

The immediate implementation reads out the pointer of
\[
 V_g^{\mathrm{src}}\psi=\sum_a\ket a_A\otimes M_a\psi
\]
and then applies \(U_{\Gamma_a(g)}\).
The coherent implementation first applies
\begin{equation}
 C_g=\sum_a\ketbra a a_A\otimes U_{\Gamma_a(g)}
 \label{eq:controlled-feedback}
\end{equation}
and then reads out \(A\).

\begin{proposition}[Exact source--feedback readout equivalence]
\label{prop:deferred}
Both implementations have selective maps
\begin{equation}
 X\longmapsto
 U_{\Gamma_a(g)}M_aXM_a^\dagger U_{\Gamma_a(g)}^\dagger.
 \label{eq:source-feedback-map}
\end{equation}
They therefore agree on the full classical-record--matter output,
including every entangled reference.
\end{proposition}

\begin{proof}
Projection onto pointer outcome \(a\) gives
\[
 (\bra a_A\otimes I)C_gV_g^{\mathrm{src}}
       =U_{\Gamma_a(g)}M_a.
\]
This is the same branch operator as immediate readout followed by
feedback. Equality of branch maps, and then tensoring with a
reference, proves the conclusion.
\end{proof}

Before readout, the coherent pointer is quantum, not an already
existing classical geometry. The proposition compares the completed
outputs. It does not equate arbitrary intermediate geometric traces
or authorize an intervening controller to inspect pointer
coherences. This is consistent with the restricted deferred-readout
boundary of VII-E and with precise circuit formulations of deferred
measurement~\cite[Theorem \texttt{thm:deferred}]{CheungVIIE,
GurevichBlass2022}.

\subsection{A counterexample using the same effects}

\begin{example}[Equal records, different future geometry]
\label{ex:residual}
For a qubit, consider two fair two-outcome instruments:
\[
 \Psi_a(X)=\tfrac12X,
 \qquad
 \widetilde\Psi_a(X)=\tfrac12\Delta_Z(X),
 \qquad a\in\{0,1\},
\]
where
\[
 \Delta_Z(X)=P_0XP_0+P_1XP_1.
\]
Both have effects \(I/2\). On input \(\ketbra++\), their normalized
residuals are respectively \(\ketbra++\) and \(I/2\), at trace
distance \(1/2\).

If the next geometric output records an \(X\)-basis measurement,
the first implementation gives
\[
 p(a,+)=\tfrac12,\qquad p(a,-)=0,
\]
whereas the second gives \(p(a,\pm)=1/4\).
The complete two-step geometric laws have total-variation distance
\(1/2\).
\end{example}

The continuation observes a residual direction not determined by the
first record effects. Its permission must be reflected in the
backward observation space. This is the same record--residual
separation emphasized by VII-E~\cite[Theorem
\texttt{thm:residual-obstruction} and Theorem
\texttt{thm:luders}]{CheungVIIE}.
Identical accessible geometry does not certify quantum recovery.

\section{Two limitations of informative classical feedback}
\label{sec:obstructions}

\subsection{Deterministic packet selection and affinity}

\begin{proposition}[Affine obstruction to nonconstant definite packets]
\label{prop:affine}
Let \(\mathcal S\) be a convex set of quantum states. Suppose an
affine map assigns to every \(\rho\in\mathcal S\) a probability
distribution concentrated at one classical label:
\[
 T(\rho)=\delta_{\gamma(\rho)}.
\]
Then \(\gamma\) is constant on \(\mathcal S\).

The same conclusion holds for packets in any classical outcome
space in which distinct packets can be separated by a measurable
event.
\end{proposition}

\begin{proof}
If \(\gamma(\rho)\neq\gamma(\sigma)\), then for \(0<t<1\),
affinity gives
\[
 T(t\rho+(1-t)\sigma)
 =t\delta_{\gamma(\rho)}
   +(1-t)\delta_{\gamma(\sigma)}.
\]
This distribution is not concentrated at one label. The mixture
belongs to \(\mathcal S\), giving a contradiction. In a general
outcome space, an event separating the two packets has probability
strictly between zero and one on that mixture, with the same
contradiction.
\end{proof}

Applied to \eqref{eq:optionN}, this obstruction requires that the
selected packet genuinely vary: \(\theta\kappa\neq0\), the source
functional is nonconstant on the convex domain, and the packet's
declared deficit distinguishes the outputs. It does not prohibit
deterministic evolution of an already classical label.

Stochastic output, a nonconvex promised input class, or a controlled
approximation is a different target. A nonlinear rule may also
be retained explicitly outside the present channel programme.

\subsection{Identity decompositions}

\begin{lemma}[Completely positive branches of the identity]
\label{lem:identity-branches}
If completely positive maps satisfy
\[
 \sum_a\Psi_a=\id_{\Mat(\Hil)},
\]
then
\[
 \Psi_a=p_a\id,
 \qquad p_a\geq0,\quad\sum_ap_a=1.
\]
In particular, all record probabilities are state independent.
\end{lemma}

\begin{proof}
The Choi matrix of the identity is
\[
 J(\id)=\ketbra{\Omega}{\Omega},
 \qquad
 \ket{\Omega}=\sum_{i=1}^d\ket i\ket i,
\]
and has rank one. Each positive \(J(\Psi_a)\) lies below
\(J(\id)\), so its range is contained in the one-dimensional
range of \(J(\id)\). Thus
\(J(\Psi_a)=p_aJ(\id)\). Choi injectivity gives the result.
\end{proof}

Thus an informative instrument cannot retain the matter-unchanged
property of \eqref{eq:old-coupling} on every quantum state.

\subsection{Exact recovery and a constant complement}

The next criterion is standard. It is also recorded in
En02-C, but is restated and proved for the present use
rather than imported with that manuscript's radiation
benchmark~\cite[Lemma \texttt{lem:decouple}]{CheungEn02C}.
Channel-level continuity versions belong to the
information--disturbance literature~\cite{KSW2008}.

\begin{theorem}[Exact recovery criterion]
\label{thm:decouple}
A finite-dimensional channel \(\Nchan\) has a CPTP left inverse if
and only if a complementary channel has the form
\begin{equation}
 \Nchan^c(X)=\Tr(X)\tau
 \label{eq:constant-complement}
\end{equation}
for a fixed density operator \(\tau\).
\end{theorem}

\begin{proof}
Let \(V:\Hil_{\mathrm{in}}\to\Hil_{\mathrm{out}}\otimes E\)
be a Stinespring isometry.

Suppose a recovery exists and let
\(W:\Hil_{\mathrm{out}}\to\Hil_{\mathrm{in}}\otimes F\)
dilate it. Then \((W\otimes I_E)V\) dilates the identity channel.
Every Kraus operator of the identity is proportional to the
identity by the rank-one Choi argument. Consequently this
composite isometry has the form
\[
 \psi\longmapsto\psi\otimes\chi_{FE}
\]
for one fixed unit vector. Tracing over the recovered input and
\(F\) shows that the original complementary output is constant.

Conversely, diagonalize
\(\tau=\sum_j\lambda_j\ketbra j j\), omitting zero eigenvalues.
Write \(V=\sum_jV_j^{\mathrm{raw}}\otimes\ket j\).
The complementary identity implies
\[
 (V_k^{\mathrm{raw}})^\dagger V_j^{\mathrm{raw}}
       =\delta_{jk}\lambda_j I.
\]
Hence \(V_j^{\mathrm{raw}}=\sqrt{\lambda_j}V_j\), where the
\(V_j\) are isometries with mutually orthogonal ranges.
Let \(P=\sum_jV_jV_j^\dagger\), and choose a density operator
\(\omega_*\) on the input. Define
\[
 \Rchan(Y)=
     \sum_jV_j^\dagger YV_j
     +\Tr((I-P)Y)\omega_*.
\]
This is completely positive and trace preserving. Since
\(\Nchan(X)=\sum_j\lambda_jV_jXV_j^\dagger\),
orthogonality gives \(\Rchan\Nchan(X)=X\).
\end{proof}

\begin{corollary}[An informative record prevents universal recovery]
\label{cor:informative}
Let \(\{\Psi_a\}\) be an instrument. If at least one effect
\(\Psi_a^*(I)\) is not a scalar multiple of \(I\), then:
\begin{enumerate}[label=\textup{(\roman*)}]
 \item its nonselective matter channel \(\sum_a\Psi_a\) has no
       CPTP left inverse;
 \item its complete classical-record--matter channel
       \(\Ichan(X)=\bigoplus_a\Psi_a(X)\) also has no CPTP left
       inverse.
\end{enumerate}
Both assertions concern recovery of every input state.
\end{corollary}

\begin{proof}
For (i), use the dilation with environment basis \(\ket{a,\alpha}\).
Measuring its environment gives the instrument probabilities.
A constant complement would make every such probability
independent of the input, contradicting the nonscalar effect.
Apply Theorem~\ref{thm:decouple}.

For (ii), suppose \(\Rchan\Ichan=\id\). Restrict the recovery to
each classical output block, obtaining a CPTP map \(\Rchan_a\).
The completely positive maps \(\Rchan_a\Psi_a\) sum to the
identity, so Lemma~\ref{lem:identity-branches} gives
\(\Rchan_a\Psi_a=p_a\id\).
Taking traces and using trace preservation of \(\Rchan_a\)
implies
\(\Psi_a^*(I)=p_aI\), again a contradiction.
\end{proof}

A nondemolition measurement of one observable is compatible with this
corollary: its eigenstates may be preserved while coherence between
eigenspaces is disturbed. Recovery on a proper code subspace is
also a different task.

\begin{proposition}[Constant-qubit calibration]
\label{prop:constant-qubit}
For a constant qubit channel \(\Nchan(X)=\Tr(X)\tau\),
\[
 \inf_{\Rchan\ \mathrm{CPTP}}
 \ddiam(\Rchan\Nchan,\id_2)=\frac34.
\]
\end{proposition}

\begin{proof}
Every decoded channel is constant, with some output \(\sigma\).
On a maximally entangled input its output is \(I/2\otimes\sigma\),
whose Bell overlap is \(1/4\). Testing the Bell projector gives
half-diamond error at least \(3/4\).

Preparing \(I/2\) attains the bound. Indeed, the completely
depolarizing qubit channel equals the uniform mixture of the
four Pauli conjugations. Subtracting the identity and using the
triangle inequality gives half-diamond distance at most \(3/4\).
\end{proof}

Only this constant-channel calibration is used. No streaming-memory
or radiation error from a different channel is transferred.

\section{Balance laws and admissibility without postselection}
\label{sec:balance}

\subsection{Operator tests for energy}

Let a declared Hermitian hybrid observable be
\begin{equation}
 \mathcal E_g=
 E_{\mathrm{geom}}(g)I+
 H_{\mathrm m}(g)+E_{\mathrm{res}}(g)I.
 \label{eq:energy}
\end{equation}
The reservoir term, if used, is part of the label specification.
The evolution parameter and physical interpretation must be declared
independently of a circuit tick or slicing parameter.

\begin{proposition}[Exact expectation balance]
\label{prop:energy}
Expected energy is conserved for every hybrid input if and only if
\begin{equation}
 \sum_h\Psi_{h|g}^*(\mathcal E_h)=\mathcal E_g
 \qquad(g\in\Geo).
 \label{eq:energy-identity}
\end{equation}
The branchwise relations
\begin{equation}
 \mathcal E_hK_{h|g,\alpha}
       =K_{h|g,\alpha}\mathcal E_g
 \label{eq:branch-energy}
\end{equation}
are sufficient.

If the norm of the defect in \eqref{eq:energy-identity} is at most
\(\delta\) for every \(g\), the expected one-step error is at most
\(\delta\), and the accumulated error after \(n\) steps is at most
\(n\delta\) for a common observable and uniform defect.
\end{proposition}

\begin{proof}
Expected conservation is
\(\langle\mathcal E,\F\rho\rangle
 =\langle\mathcal E,\rho\rangle\).
Testing all states supported in one input block gives the operator
identity.

Under \eqref{eq:branch-energy},
\[
 \sum_{h,\alpha}K_{h|g,\alpha}^\dagger
      \mathcal E_hK_{h|g,\alpha}
 =
 \sum_{h,\alpha}K_{h|g,\alpha}^\dagger
      K_{h|g,\alpha}\mathcal E_g
 =\mathcal E_g.
\]
For a defect \(D_g\),
\[
 \abs{\sum_g\Tr(D_g\rho_g)}
 \leq\max_g\norm{D_g}_\infty\sum_g\Tr\rho_g.
\]
Iteration and the triangle inequality give the accumulated bound.
\end{proof}

Condition \eqref{eq:branch-energy} is spectral intertwining. It is not
the assertion that postselecting an energy-correlated outcome preserves
the unconditional input mean on that branch.

\subsection{A separate resource ledger}

Let \(r(g)\geq0\) be a resource budget and let \(c_{h|g}\geq0\)
be a declared edge cost satisfying
\begin{equation}
 r(h)+c_{h|g}=r(g)
 \quad\text{on every nonzero branch}.
 \label{eq:path-resource}
\end{equation}

\begin{proposition}[Resource balance]
\label{prop:resource}
On every supported history,
\[
 r(g_n)+\sum_{t=1}^n c_{g_t|g_{t-1}}=r(g_0).
\]
Its expectation satisfies
\begin{equation}
 (\F^*r)_g
 =
 r(g)I-\sum_h c_{h|g}E_{h|g}.
 \label{eq:expected-resource}
\end{equation}
\end{proposition}

\begin{proof}
The history identity telescopes
\eqref{eq:path-resource}. For the expected identity,
\[
 (\F^*r)_g=\sum_hr(h)E_{h|g}
 =\sum_h(r(g)-c_{h|g})E_{h|g},
\]
and the effects sum to \(I\).
\end{proof}

These costs may count permissions, attempts, records, or other declared
finite resources. They are not automatically physical energies.

\subsection{Complete fallback design}

\begin{proposition}[Admissible rerouting]
\label{prop:fallback}
Suppose \(\{\Phi_{a|g}\}_a\) is a complete instrument and
\(a\mapsto\Gamma_a(g)\) is a fixed routing table whose outputs all
have valid certificates, including any declared no-update output.
Then
\begin{equation}
 \Psi_{h|g}=\sum_{a:\Gamma_a(g)=h}\Phi_{a|g}
 \label{eq:routing}
\end{equation}
is a complete hybrid transition.

Observation compatibility holds if each routed map satisfies
\[
 \Phi_{a|g}^*(\Obs_{\Gamma_a(g)})
        \subseteq\Obs_g.
\]
\end{proposition}

\begin{proof}
Grouping completely positive maps preserves complete positivity,
and summing all groups gives the original trace-preserving
instrument. Summing the indicated adjoint inclusions gives the
branch inclusions.
\end{proof}

Rerouting does not undo the matter disturbance of the rejected
proposal. Nor does rerouting automatically preserve an observation
boundary established for a different output algebra.

\begin{example}[Deleting a branch is not a channel repair]
\label{ex:postselection}
If the only retained branch is
\(\Phi(X)=P_0XP_0\), its trace is \(\Tr(P_0X)\), not \(\Tr X\).
The normalized successful output
\[
 \rho\longmapsto
 \frac{P_0\rho P_0}{\Tr(P_0\rho)}
\]
is not a trace-preserving linear map on the full input domain.
A complete model must retain an admissible failure branch or another
complete fallback.
\end{example}

\section{Operational locality with explicit controller support}
\label{sec:locality}

\subsection{Nonselective remote invariance}

\begin{proposition}[Untouched remote reduction]
\label{prop:remote}
For a local instrument \(\{\Psi_a^R\}\) on \(R\), with
\(\sum_a\Psi_a^R\) trace preserving,
\[
 \Tr_R\left[
   \sum_a(\Psi_a^R\otimes\id_{R^c})(\rho)
 \right]
 =\Tr_R\rho.
\]
This holds for entangled inputs. It does not assert equality of
conditional remote states.
\end{proposition}

\begin{proof}
Pair the remote output with \(B\). Its expectation is
\[
 \Tr\left(
   \left[\sum_a\Psi_a^{R*}(I_R)\right]\otimes B\ \rho
 \right)
 =\Tr((I_R\otimes B)\rho).
\]
\end{proof}

\subsection{Feedback compiled into a local hybrid network}

Let a finite site set \(X\) carry local algebras
\[
 \Alg_x=\bigoplus_{c_x}\Mat(\Hil_x).
\]
The local classical variables include all controller records read
at that site. The global algebra is
\(\bigotimes_{x\in X}\Alg_x\), equivalently a direct sum over
classical configurations.

At instruction \(t\), suppose a fixed envelope \(R_t\subseteq X\)
contains every quantum and classical register read or written by
that instruction, and the complete instruction is a channel
\[
 \Phi_t^{R_t}\otimes\id_{X\setminus R_t}.
\]
Within \(R_t\), the channel may select different operations from
its classical controller blocks. The envelope must nevertheless
include those controls.

For \(Y\subseteq X\), define backwards
\begin{equation}
 S_T=Y,\qquad
 S_{t-1}=
 \begin{cases}
 S_t\cup R_t,&R_t\cap S_t\neq\varnothing,\\
 S_t,&R_t\cap S_t=\varnothing.
 \end{cases}
 \label{eq:local-backwards}
\end{equation}

\begin{theorem}[Hybrid-controller support bound]
\label{thm:locality}
Every output observable supported on \(Y\) pulls back to an input
observable supported on \(S_0\). Equal input reductions on \(S_0\)
therefore give equal output reductions on \(Y\).

If \(X\) has a metric and \(\diam R_t\leq\Delta\), then
\[
 S_0\subseteq\{x:d(x,Y)\leq T\Delta\}.
\]
\end{theorem}

\begin{proof}
If \(R_t\) is disjoint from the current support, the observable
is the identity on that entire envelope. The channel adjoint is
unital, so the envelope contributes no support.
If they meet, support can enlarge only to their union.
Induction proves \eqref{eq:local-backwards}.

A new point lies within distance \(\Delta\) of a point in the
previous backwards set. The triangle inequality gives the metric
bound. Finally, equality of input reductions gives equality of
every pulled-back output expectation.
\end{proof}

This theorem does not inherit locality from a controller that reads
a global geometry label and then selects a remote support. Such a
controller must itself be included in the envelopes.

The distinction is the one already present in VII-E's prescribed
schedule theorem and VIII-C's prescribed block-channel
bound~\cite[Theorem \texttt{thm:locality} and Proposition
\texttt{prop:direction}]{CheungVIIE},
\cite[Theorem \texttt{thm:C5}]{CheungVIIIC}.
The present extension permits feedback inside a declared local
hybrid channel. It is still a dependency theorem, not a spacetime
light-cone theorem. No physical directional causal dominance is
deduced.

\section{Assisted stability and finite Choi certificates}
\label{sec:stability}

\subsection{A classical-input diamond norm}

For a hybrid linear map \(D\), define its diamond norm through the
canonical full-matrix extension that first pinches the input label
and then applies the direct-sum map. Equivalently, one may amplify
the direct-sum map by arbitrary matrix references and use the trace
norm \eqref{eq:hybrid-norms}.

For two transitions, set
\[
 D_{h|g}=\Psi_{h|g}-\widetilde\Psi_{h|g},
 \qquad
 D_g(X)=\bigoplus_hD_{h|g}(X).
\]

\begin{proposition}[Exact column reduction]
\label{prop:diamond-column}
The assisted hybrid norm satisfies
\begin{equation}
 \norm{\F-\widetilde\F}_\diamond
       =\max_g\norm{D_g}_\diamond.
 \label{eq:column-diamond}
\end{equation}
Moreover,
\[
 \norm{D_g}_\diamond
 \leq\sum_h\norm{D_{h|g}}_\diamond.
\]
\end{proposition}

\begin{proof}
For an amplified block input \(X=(X_g)_g\),
\[
 \norm{[(\F-\widetilde\F)\otimes\id](X)}_1
 \leq\sum_g\norm{(D_g\otimes\id)(X_g)}_1
 \leq\max_g\norm{D_g}_\diamond\sum_g\norm{X_g}_1.
\]
For the full-matrix extension, pinching is a trace-norm contraction
also after amplification. This follows either from duality with
a unital completely positive adjoint or by averaging diagonal
phase unitaries. Hence the same upper bound holds there.

An input supported in one maximizing block gives the reverse
inequality. The branch bound is the triangle inequality in the
output direct sum.
\end{proof}

\subsection{Choi upper and lower bounds}

For an arbitrary linear map \(D:\Mat(\C^d)\to\Mat(K)\), write
\begin{equation}
 J(D)=\sum_{i,j=1}^d\ketbra i j\otimes D(\ketbra i j),
 \qquad
 \chi(D)=J(D)/d.
 \label{eq:choi-convention}
\end{equation}

\begin{lemma}[Finite Choi certificate]
\label{lem:choi}
For every such map,
\begin{equation}
 \norm{\chi(D)}_1
 \leq\norm{D}_\diamond
 \leq d\norm{\chi(D)}_1.
 \label{eq:choi-bounds}
\end{equation}
For a column map \(D_g\), its Choi matrix is the direct sum of
the branch Choi matrices in the output label.
\end{lemma}

\begin{proof}
Apply \(D\) to one half of the normalized maximally entangled
state. Its output is \(\chi(D)\), giving the lower bound.

For the upper bound, adjoin an arbitrary reference space \(N\).
Any unit vector in \(N\otimes\C^d\) can be written
\[
 u=(A\otimes I)\ket{\Omega},
 \qquad \Tr(A^\dagger A)=1,
\]
with \(A:\C^d\to N\), and therefore \(\norm A_\infty\leq1\).
For two unit vectors \(u,v\), represented by \(A,B\),
\[
 (\id_N\otimes D)(\ketbra u v)
       =(A\otimes I)J(D)(B^\dagger\otimes I).
\]
Its trace norm is at most \(\norm{J(D)}_1\).
A singular-value decomposition of an arbitrary input operator
extends the bound to
\[
 \norm{(\id_N\otimes D)(X)}_1
       \leq\norm{J(D)}_1\norm X_1.
\]
Taking the supremum proves the upper bound. The direct-sum
assertion follows from linearity of the Choi construction.
\end{proof}

These are certification bounds, not numerical results inherited
from another finite channel. They use the normalization
\eqref{eq:choi-convention} explicitly.

\subsection{Accumulation for terminal states and full histories}

\begin{theorem}[Finite-execution stability]
\label{thm:stability}
Suppose two executions use the same label dictionaries and satisfy
\[
 \norm{\F_t-\widetilde\F_t}_\diamond\leq\eps_t.
\]
Their terminal channels obey
\begin{equation}
 \norm{\F_T\cdots\F_1-
       \widetilde\F_T\cdots\widetilde\F_1}_\diamond
 \leq\sum_{t=1}^T\eps_t.
 \label{eq:telescoping}
\end{equation}
The same bound holds for their complete geometry-history channels.

For a common input, including an arbitrary reference, the trace
distance of the corresponding outputs, and the total-variation
distance of their geometric history laws, are at most
\begin{equation}
 \min\left\{1,\frac12\sum_{t=1}^T\eps_t\right\}.
 \label{eq:state-history-error}
\end{equation}
\end{theorem}

\begin{proof}
Insert and subtract one implemented step at a time. Each term has
the difference of one pair of steps sandwiched between channels.
Channels have diamond norm one, and the diamond norm is
submultiplicative~\cite{Watrous2018}. The triangle inequality
gives \eqref{eq:telescoping}.

For history retention, a current history is a classical input label.
Its next-step column is \(D_{g_t}\), with the preceding history
copied into the output label. Copying that fixed classical prefix
does not change the norm of the column. Proposition
\ref{prop:diamond-column} therefore gives the same one-step
bound, now on the history algebra. Apply the same telescoping
argument.

A common normalized input gives trace-norm difference at most the
channel diamond bound. Trace distance contributes the factor
\(1/2\). Reading only the geometric record is a channel, so it
cannot increase trace distance; on diagonal states that distance
is total variation. Distances between states are at most one.
\end{proof}

\subsection{Restricted unassisted comparison}

For an observation space \(\Obs\), define
\[
 d_{\Obs}(\F,\widetilde\F)
 =
 \frac12\sup_{\rho}
 \sup_{\substack{B=B^\dagger\in\Obs\\
                 \norm B_{\infty,\Geo}\leq1}}
 \abs{\langle B,(\F-\widetilde\F)(\rho)\rangle}.
\]
Then
\begin{equation}
 d_{\Obs}(\F,\widetilde\F)
 =
 \frac12
 \sup_{\substack{B=B^\dagger\in\Obs\\
                 \norm B_{\infty,\Geo}\leq1}}
 \norm{(\F^*-\widetilde\F^*)(B)}_{\infty,\Geo}.
 \label{eq:restricted-distance}
\end{equation}
Indeed, interchange the suprema and optimize the Hermitian
expectation by an eigenstate in a maximizing block.
This is the hybrid form of VII-E's accessible-output
distance~\cite[Proposition
\texttt{prop:accessible}]{CheungVIIE}.
It is unassisted unless a reference is included explicitly.

\subsection{Guard stability is a separate obligation}

An exact clock guard \(j=0\) is stable under an exact finite
representation, but not under arbitrary perturbation of a real
parameter. Similarly, if a length is \(c|s|\), a floor report at
resolution \(h\) is zero whenever \(0<c|s|<h\). It cannot serve as
an exact event detector.

Theorem~\ref{thm:stability} applies to already specified channels on
the same finite dictionaries. It does not convert a discontinuous
geometric guard into a uniformly continuous map of real geometric
data. Such a claim requires a separate margin or event-resolution
theorem, as emphasized in VII-E and VII-F~\cite{CheungVIIE,
CheungVIIF}.

\section{An explicit resource-budgeted feedback witness}
\label{sec:example}

This section witnesses the consistency of the finite specification.
It is not a replacement for VII-F's particle-link incidence
construction.

\subsection{A finite certified geometric family}

Fix an integer budget \(B\geq1\), a tick duration \(\tau>0\),
and \(0<\delta<\tau\). Set
\begin{equation}
 \Geo_B=
 \{(k,q):0\leq k\leq B,\ q\in\Z,\ |q|\leq k,\
                       q\equiv k\pmod2\}.
 \label{eq:example-labels}
\end{equation}
The resource is \(r(k,q)=B-k\). The controller tick is \(k\tau\).
The selected slice height is
\begin{equation}
 T(k,q)=k\tau+\delta q.
 \label{eq:slice-height}
\end{equation}
These two quantities are distinct.

Use the fixed ultrastatic spinless cone
\[
 M=\R_T\times C_{\pi/2},
 \qquad
 ds^2=-dT^2+dr^2+r^2d\theta^2,
 \qquad 0\leq\theta\leq3\pi/2,
\]
with its wedge boundary rays identified. Each label denotes the
horizontal slice \(T=T(k,q)\), with the same fixed sector
cellulation, stationary axis, and any fixed annular collar.

For \(k<B\), define
\begin{equation}
 \Gamma_a(k,q)=(k+1,q+a),\qquad a\in\{-1,+1\}.
 \label{eq:example-successors}
\end{equation}
At \(k=B\), the label is absorbing.

\begin{lemma}[Complete branch certificate]
\label{lem:example-certificate}
Every successor in \eqref{eq:example-successors} belongs to
\(\Geo_B\). Its slice is embedded and spacelike, and advances
in spacetime time by \(\tau+a\delta>0\).

The complete affine peripheral word is
\[
 J=(R(\pi/2),0)
\]
on every label, with the same marked stationary axis.
Location transport is the identity on the persistent cell labels.
The resource decreases by exactly one on every active edge.
\end{lemma}

\begin{proof}
The inequalities and parity in \eqref{eq:example-labels} are
preserved by \(k\mapsto k+1\), \(q\mapsto q+a\).
A horizontal graph has induced metric equal to the spatial cone
metric and is embedded. Formula \eqref{eq:slice-height} gives
the time increment.

The upper seam maps to the lower seam by the rotation
\(R(\pi/2)\), which fixes the time axis pointwise.
A positively oriented primitive meridian crosses that seam once;
its angular lift advances by \(3\pi/2\). Its affine word is
therefore \(J\). Moving the adapted horizontal chart by
\(-T(k,q)e_0\) commutes with \(J\), so there is no induced affine
time translation. The cellulation and collar are unchanged
spatially, giving identity cell-label transport.
Finally, \(B-(k+1)=B-k-1\).
\end{proof}

The cone interpretation and the need to retain full affine data are
consistent with the classical point-particle description and the
fixed-cone certificate of VII-F~\cite{DJH1984,CheungVIIF}.
Here no deficit or worldline changes.

\subsection{The unsharp source instrument}

Let \(\Hil=\C^2\), \(P_j=\ketbra j j\), and
\[
 Q=P_1-P_0.
\]
Choose \(0\leq\eta\leq1\) and define
\begin{equation}
 E_a=\tfrac12(I+a\eta Q),
 \qquad
 M_a=\sqrt{E_a}.
 \label{eq:unsharp}
\end{equation}
Let
\begin{equation}
 U_{k,q}=\exp(-i\vartheta_{k,q}Q),
 \qquad
 \vartheta_{k,q}=\pi q/8.
 \label{eq:example-unitaries}
\end{equation}
For an active input \(g=(k,q)\), set
\begin{equation}
 \Psi_{\Gamma_a(g)|g}(X)
 =
 U_{\Gamma_a(g)}M_aXM_a
       U_{\Gamma_a(g)}^\dagger,
 \label{eq:example-instrument}
\end{equation}
with all other branches zero. At \(k=B\), use
\(\Psi_{g|g}=\id\).

This is an operational measurement--feedback prescription.
Newtonian measurement-feedback models provide related motivation,
not a gravitational validation of this finite
construction~\cite{TilloyDiosi2016}.

\begin{proposition}[Exact source and disturbance formulas]
\label{prop:unsharp}
The example is a complete hybrid channel. For a normalized active
matter state \(\rho\),
\begin{align}
 p_a&=\tfrac12(1+a\eta\Tr(Q\rho)), \label{eq:unsharp-p}\\
 p_+-p_-&=\eta\Tr(Q\rho). \label{eq:unsharp-signal}
\end{align}
Before the geometry-dependent unitary, its nonselective source map
\[
 \mathcal M_\eta(X)=M_+XM_++M_-XM_-
\]
preserves diagonal blocks and satisfies
\begin{equation}
 P_0\mathcal M_\eta(X)P_1
   =\sqrt{1-\eta^2}\,P_0XP_1.
 \label{eq:unsharp-coherence}
\end{equation}

Writing \(x=\delta q\), the active geometric increment has
\begin{align}
 \mathbb E[\Delta x\mid\rho]
     &=\delta\eta\Tr(Q\rho), \label{eq:drift}\\
 \operatorname{Var}(\Delta x\mid\rho)
     &=\delta^2\bigl(1-\eta^2\Tr(Q\rho)^2\bigr).
 \label{eq:variance}
\end{align}
\end{proposition}

\begin{proof}
The positive effects sum to \(I\), so
\(\sum_aM_a^\dagger M_a=I\). Unitary feedback does not change
branch traces, proving \eqref{eq:unsharp-p} and completeness.

On \(P_0\), the two square-root eigenvalues are
\(\sqrt{(1-\eta)/2}\) and \(\sqrt{(1+\eta)/2}\), in opposite
order to their values on \(P_1\). Their two cross products sum
to \(\sqrt{1-\eta^2}\); their squared values sum to one on
each sector. This proves \eqref{eq:unsharp-coherence}.
Finally, \(\Delta x=\delta a\), so its mean follows from
\eqref{eq:unsharp-signal} and its second moment is \(\delta^2\).
\end{proof}

These are model-specific formulas, not continuous
decoherence--diffusion inequalities. The latter require their
own generator hypotheses~\cite{OppenheimDecoherence2023}.

\subsection{Observation, feedback, and implementation}

Choose the source observation algebra
\begin{equation}
 \Src=\spanC\{P_0,P_1\}.
 \label{eq:example-algebra}
\end{equation}
Both \(M_a\) and \(U_{k,q}\) commute with this algebra. Therefore
\[
 \Psi_{h|g}^*(\Src)\subseteq\Src
\]
on every branch, including absorbing ones. Theorems
\ref{thm:histories} and \ref{thm:spectators} apply to every input
and every entangled spectator extension.

For \(0<\eta<1\), the two feedback unitaries at the successors of
one active label act differently on surviving off-diagonal matter
blocks. Thus geometry affects the subsequent matter channel.
That phase dependence is not observable on \(\Src\), which records
only populations. A continuation measuring coherence must declare
an enlarged observation boundary; it is not licensed to treat
coherence as both inaccessible and later observable.

The source dilation
\[
 \psi\longmapsto\ket+_AM_+\psi+\ket-_AM_-\psi
\]
uses one fresh pointer qubit. Immediate readout followed by
\eqref{eq:example-unitaries}, and coherent controlled feedback
followed by readout, agree as full transition instruments by
Proposition~\ref{prop:deferred}. Their complete residual outputs,
not merely their geometric probabilities, coincide.

The classical label update \eqref{eq:example-successors} is a finite
table and has a reversible completion with retained input history:
an injective map \(z\mapsto(f(z),z)\) can be extended to a
permutation on a finite enlarged code space by exchanging disjoint
input--output pairs. This does not claim discrete-gate exact
synthesis of the unsharp quantum rotations for arbitrary \(\eta\).

\subsection{Resource accounting}

Every active branch consumes one permission, regardless of the
outcome. No measurement is attempted at \(k=B\).
Consequently,
\begin{equation}
 (\F^*r)_{k,q}
 =
 \begin{cases}
 (B-k-1)I,&k<B,\\
 0,&k=B.
 \end{cases}
 \label{eq:example-resource}
\end{equation}
On an execution initially at \(k_0\), the number of source attempts
after \(n\) ticks is exactly \(\min\{n,B-k_0\}\).

One fresh pointer qubit suffices per active attempt. Retaining its
readout consumes one additional classical history bit. The matter
qubit is returned; the pointer is destructively consumed. Spectator
storage is separate. These are resource statements, not an energy
model for the measurement apparatus.

\subsection{An exact classical sector description}

Let \(\lambda_j=2j-1\), so \(Q P_j=\lambda_jP_j\).
The branches satisfy
\begin{equation}
 \Psi_{\Gamma_a(g)|g}(P_j)
      =p_a(j)P_j,
 \qquad
 p_a(j)=\tfrac12(1+a\eta\lambda_j).
 \label{eq:classical-sector}
\end{equation}
Thus, on states diagonal in the source basis, the full evolution is
exactly the classical Markov chain on \((g,j)\) with fixed \(j\)
and transition probabilities \(p_a(j)\).

For an arbitrary initial qubit state at one label, its geometric
history law is the mixture of those two classical sector laws
with weights \(\rho_{00},\rho_{11}\). Off-diagonal blocks have
zero trace at every history and do not contribute. Outcomes are
independent conditional on \(j\), not generally independent after
mixing the sectors.

\begin{proposition}[Quantitative approximation to the diagonal model]
\label{prop:classical-approx}
Start at one label \(g_0=(k_0,q_0)\). Let
\(\Delta_Z\rho=P_0\rho P_0+P_1\rho P_1\), and compare the complete
history--matter outputs from \(\rho\) and \(\Delta_Z\rho\).
After \(n\leq B-k_0\) active steps,
\begin{equation}
 \dtr\bigl(\mathscr H_n(\rho),
             \mathscr H_n(\Delta_Z\rho)\bigr)
 =
 \abs{\rho_{01}}(1-\eta^2)^{n/2}.
 \label{eq:classical-approximation}
\end{equation}
Their geometric history laws agree exactly. Forgetting histories
can only decrease the distance in
\eqref{eq:classical-approximation}.
\end{proposition}

\begin{proof}
On a fixed history, the difference has only a pair of conjugate
off-diagonal entries. Its trace distance contribution is the
absolute value of one entry. The diagonal feedback unitaries
change only its phase.

For either outcome,
\[
 \langle0|M_a|0\rangle
 \langle1|M_a|1\rangle
       =\tfrac12\sqrt{1-\eta^2}.
\]
Each length-\(n\) history therefore contributes
\(\abs{\rho_{01}}(\sqrt{1-\eta^2}/2)^n\).
There are \(2^n\) histories, so their direct-sum contributions give
\eqref{eq:classical-approximation}. Equal diagonals give equal
history probabilities by \eqref{eq:classical-sector}.
Marginalization over histories is a channel.
\end{proof}

This is a controlled approximation to a named finite classical model.
It is not an \(\hbar\to0\) limit, a continuum limit, or an Einstein
limit. At \(\eta=0\), the source learns nothing and no dephasing
occurs. At \(\eta=1\), one active measurement removes all source
coherence, while its sector eigenstates remain unchanged.

\subsection{An explicit interior parameter bound}

\begin{proposition}[Pilot implementation stability]
\label{prop:pilot-stability}
Compare two active instruments with parameters
\(\eta,\widetilde\eta\in[0,\eta_*]\), where \(\eta_*<1\), and
feedback phases \(\vartheta_h,\widetilde\vartheta_h\).
For the same routing table,
\begin{equation}
 \norm{\F-\widetilde\F}_\diamond
 \leq
 4\left[
 \frac{\abs{\eta-\widetilde\eta}}
      {2\sqrt{2(1-\eta_*)}}
 +\max_h\abs{\vartheta_h-\widetilde\vartheta_h}
 \right].
 \label{eq:pilot-stability}
\end{equation}
The bound may be capped by \(2\).
\end{proposition}

\begin{proof}
The derivatives of the scalar entries of \(M_a\) have magnitude
at most \(1/(2\sqrt{2(1-\eta_*)})\). Hence the first term bounds
\(\norm{M_a-\widetilde M_a}_\infty\).
Also
\[
 \norm{e^{-i\vartheta Q}
          -e^{-i\widetilde\vartheta Q}}_\infty
 \leq\abs{\vartheta-\widetilde\vartheta}.
\]
The branch operators \(K_a=U_aM_a\) are contractions, and
\[
 \norm{\Ad_{K_a}-\Ad_{\widetilde K_a}}_\diamond
 \leq(\norm{K_a}_\infty+\norm{\widetilde K_a}_\infty)
          \norm{K_a-\widetilde K_a}_\infty.
\]
This follows by expanding
\(K_aXK_a^\dagger-\widetilde K_aX\widetilde K_a^\dagger\)
into two terms, also after amplification.
Summing the two branch bounds and using
Proposition~\ref{prop:diamond-column} gives
\eqref{eq:pilot-stability}. Two channels have diamond distance
at most \(2\).
\end{proof}

A uniform Lipschitz estimate in the parameter \(\eta\) is not asserted
at \(\eta=1\): the square-root coherence factor itself is not
Lipschitz there. Channel-norm certification remains meaningful at
that endpoint.

\begin{theorem}[Finite observation-compatible feedback model]
\label{thm:finite-model}
The model \eqref{eq:example-labels}--\eqref{eq:example-instrument}
has:
\begin{enumerate}[label=\textup{(\roman*)}]
 \item a finite set of completely specified, admissible geometric
       and controller labels;
 \item a fixed qubit matter space and state-independent completely
       positive trace-preserving hybrid evolution;
 \item matter-dependent successor probabilities determined by the
       declared source algebra;
 \item an admissibility certificate for every nonzero branch,
       without postselection;
 \item branchwise compositional observation compatibility;
 \item exact spectator consistency for entangled extensions and
       complete histories;
 \item two full-instrument-equivalent readout implementations;
 \item an exact permission balance and finite pointer/history
       resource reserve;
 \item assisted implementation stability and a quantitative
       approximation to the declared diagonal classical model.
\end{enumerate}
For \(0<\eta<1\), both source sensitivity and geometry-dependent
matter phase operations are nontrivial. The ambient cone, particle
worldline, deficit, and peripheral affine holonomy remain fixed.
\end{theorem}

\begin{proof}
Lemma~\ref{lem:example-certificate} proves (i) and (iv).
Theorem~\ref{thm:structure} and
Proposition~\ref{prop:unsharp} prove (ii) and (iii).
Commutation with \eqref{eq:example-algebra} gives the branch
inclusions, proving (v) by Theorem~\ref{thm:histories}.
The specified controller and separate reserves satisfy
Assumption~\ref{ass:blind}; Theorem~\ref{thm:spectators} gives (vi).

Proposition~\ref{prop:deferred} gives (vii).
Equation~\eqref{eq:example-resource} and
Proposition~\ref{prop:environment} give (viii).
Theorem~\ref{thm:stability}, Lemma~\ref{lem:choi},
Proposition~\ref{prop:pilot-stability}, and
Proposition~\ref{prop:classical-approx} give (ix).
The final statements follow from the explicit source effects,
distinct successor phases, and the fixed-cone certificate.
\end{proof}

\section{Geometric and field-theoretic boundaries}
\label{sec:limits}

\subsection{How a VII-F event may enter a later instrument}

A later transition may first produce a classical source outcome and
then select a certified geometric edge. The source instrument and the
geometry-only event remain distinct operations.

For the VII-F incidence event, the exact guard is still \(j=0\),
with incoming/outgoing germ and permission data included in the
classical label. A source outcome may prepare a declared readiness
register, but it may not replace the exact guard by a rounded length
test. Applying Proposition~\ref{prop:fallback} requires an admissible
fallback for every outcome, not deletion of an invalid event
branch~\cite[Contract \texttt{contract:G}, Proposition
\texttt{prop:blind}, and Theorem
\texttt{thm:implementation}]{CheungVIIF}.

Such an integration would change neither the deficit nor the
worldline merely by invoking the finite channel theorem.
VIII-B's supplied packet family and VIII-C's promised-token
composition are comparison constructions, not a derived dynamical
law~\cite{CheungVIIIB,CheungVIIIC}.

\subsection{Two geometric negative tests}

\begin{example}[A stale deficit identification]
Suppose a cone realization has deficit \(\mu\), but a proposed output
packet reports \(\nu\neq\mu\) while retaining its seam word
\((R(\mu),0)\). That output does not match the reported deficit in
the fixed framing. For planar rotations,
\[
 \norm{R(\mu)-R(\nu)}_2
       =2\abs{\sin((\mu-\nu)/2)}.
\]
The mismatch is not repaired by changing a packet label.
\end{example}

\begin{example}[Correct Lorentz part, wrong affine word]
For \(\epsilon\neq0\),
\[
 J_\epsilon(z)=R(\mu)z+\epsilon e_0
\]
has the same Lorentz part as the spinless identification, but no
fixed point: its time coordinate increases by \(\epsilon\).
In particular, it does not fix the stationary worldline pointwise.
This is why a certificate must check the full affine peripheral
word, not only its Lorentz factor~\cite{CheungVIIF}.
\end{example}

Changing a deficit, moving an axis, or transferring physical energy
between matter and geometry requires new matching and evolution
proofs. The finite operator test
\eqref{eq:energy-identity} is necessary for a declared energy
balance but does not supply those geometric proofs.

\subsection{Why VII-A--D do not provide a source instrument}

The supplied manuscripts establish distinct results:
\begin{itemize}
 \item VII-A gives a second-order classical plateau intertwiner,
       not a quantum source algebra~\cite{CheungVIIA}.
 \item VII-B excludes the unchanged sharp-Cauchy insertion from its
       declared off-shell microcausal polynomial domain; its linear
       on-shell repair is not an interacting source
       instrument~\cite[Theorem \texttt{thm:main}]{CheungVIIB}.
 \item VII-C excludes one normal-topology Wick extension under its
       stated continuity and old-product agreement requirements,
       not every possible boundary theory~\cite[Theorem
       \texttt{thm:coefficient} and Corollary
       \texttt{cor:noalgebra}]{CheungVIIC}.
 \item VII-D gives one strong numerical finite part and a complete
       regulator conversion, not an interacting observable algebra
       or stress tensor~\cite[Theorems \texttt{thm:finitepart} and
       \texttt{thm:conversion}]{CheungVIID}.
\end{itemize}
Consequently,
\[
 \text{finite numerical graph coefficient}
 \not\Longrightarrow
 \text{quantum source instrument}.
\]

A field-theoretic successor must independently specify a source
domain, state, renormalization, smearing, and completely positive
instrument. Only then may normalization changes be tested against
the geometric observation boundary. VII-E's descent and
counterterm-blind tests provide a comparison dictionary, not a
product construction~\cite[Propositions \texttt{prop:descent},
\texttt{prop:blind-tests}, and
\texttt{prop:presentation}]{CheungVIIE}.

Nothing in the present finite argument constructs a missing
characteristic graph product, a quantum Ward identity, or an admitted
interacting coefficient.

\subsection{What is left to a genuinely gravitational successor}

A gravitational extension must declare:
\begin{enumerate}
 \item the dynamical geometric variable and its physical clock;
 \item the gravitational evolution and constraints;
 \item complete affine transitions, axes, collars, and matching
       for every branch;
 \item total energy and any apparatus or reservoir;
 \item directional physical propagation, not merely scheduled support;
 \item a controlled approximation to a specified gravitational model.
\end{enumerate}
A continuous-time or continuum limit is not obtained by the finite
proofs above. Existing classical--quantum limiting constructions
provide comparison frameworks, whose scaling assumptions must be
matched rather than imported~\cite{LaytonOppenheim2024}.

\section{Conclusion}

A finite matter--geometry feedback step can be a genuine channel only
after its classical controller, quantum instrument, geometric
certificates, and observation boundary have been declared together.

The unconditional law uses subnormalized blocks and fixed completely
positive transition maps. Exact observation compatibility is an
adjoint inclusion. For stochastic histories, it must hold branchwise;
the smallest future-sensitive observation spaces can be constructed
backwards and, in the homogeneous finite case, stabilize after
finitely many enlargements. Entangled spectators preserve every
unnormalized history block after restriction when the entire
controller is blind to them.

Implementation independence is conditional and testable: selective
maps must agree on the declared continuation observables. Equal
effects alone are insufficient. Informative classical records also
cannot coexist with exact preservation, or exact recovery, of every
quantum input.

The explicit qubit model proves that these requirements, an exact
resource ledger, and quantitative stability can be satisfied
simultaneously. Its geometric branches are strictly ordered slices of
one fixed cone. That is a closed finite operational result, not a
theory of semiclassical gravity.

\[
 \boxed{
 \begin{gathered}
 \text{fixed CP transition instrument}\\
 +\ \text{complete classical controller}\\
 +\ \text{branchwise observation closure}\\
 +\ \text{declared geometric certificates}\\
 \Longrightarrow\
 \text{consistent finite feedback and history comparison}.
 \end{gathered}}
\]

The gravitational dynamics and field-theoretic source stages remain
separate constructions.

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\end{document}