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  pdftitle={Observation boundaries for destructive quantum readout and external geometry},
  pdfsubject={Typed observation boundaries, quantum instruments, and a certified slicing exchange}
}

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\title{Observation boundaries for destructive quantum readout\\
and external geometry}
\author{}
\date{}

\usepackage{tikz}
\begin{document}

\maketitle

\begin{abstract}
We specify an observation boundary for a finite linearly typed quantum
interface with destructive computational-basis readout and primitive
classical geometric labels. Geometric rules inspect declared labels;
coupling rules inspect declared matter observables; control and resource
checks are part of the interface. Under spectator-blind control, arbitrary
extensions with the same accessible reduced state preserve geometric
traces and accessible matter states. A restricted coherent implementation
of destructive readout agrees with immediate readout on their common
record--matter output.

For a channel \(T\), coarse effects \(E_w\), and aggregated fine effects
\(\Pi_w\), record compatibility is equivalent to
\(T^*(E_w)=\Pi_w\). The exact unassisted worst-case total-variation error
is a maximum of operator norms over outcome events. Exact record
compatibility extends to untouched entangled references, but does not
determine residual instruments. For sharp measurements and a coarse
L\"uders instrument, full compatibility is characterized exactly by
the removal of fine coherence within each coarse outcome. A compositional
criterion identifies when continuations preserve agreement on a declared
observable algebra.

One complete four-face adjacency exchange illustrates the interface.
It is induced by a prescribed slicing of a fixed conical spacetime and
has explicit charts, edge laws, a compact collar, a transition stratum,
an exact guard, location transport, and a decoder-correct finite reversible
implementation. The general interface results are independent of this
example. A separate dictionary proves three linear observation rules
for distributional comparisons without assuming any particular
renormalized composite.
\end{abstract}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (a) at (0,0) {Destructive readout};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Coherent record};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {External geometry};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {These interfaces are not identified with one another.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{Three observation interfaces stay distinct. The three boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:scope-VII-E}
\end{figure}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (a) at (0,0) {Destructive readout};
\node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {External geometry};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {These interfaces are not identified with one another.};
\draw[->] (a) -- (s);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{Three observation interfaces stay distinct. The boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:split-VII-E}
\end{figure}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (a) at (0,0) {Destructive readout};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Coherent record};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {External geometry};
\node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {These interfaces are not identified with one another.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (c) -- (s);
\end{tikzpicture}
\caption{Reading downward is not a transfer. The boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:chain-VII-E}
\end{figure}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=4.2cm,font=\small,inner sep=3pt] (a) at (0,0) {Destructive readout};
\node[align=center,font=\small] (m) at (5.2,0) {not the same object};
\node[draw,rounded corners,align=center,text width=4.2cm,font=\small,inner sep=3pt] (b) at (10.4,0) {External geometry};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Coherent record. These interfaces are not identified with one another.};
\draw (a) -- (m);
\draw (m) -- (b);
\draw[->] (m) -- (s);
\end{tikzpicture}
\caption{Destructive readout is not External geometry. The boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:pair-VII-E}
\end{figure}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (a) at (0,0) {Named: Destructive readout};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Coherent record};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: External geometry};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {These interfaces are not identified with one another.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{Three statuses stay separate. The boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:gate-VII-E}
\end{figure}

Figure~\ref{fig:split-VII-E} records one view of the same separation. Figure~\ref{fig:chain-VII-E} records one view of the same separation. Figure~\ref{fig:pair-VII-E} records one view of the same separation. Figure~\ref{fig:gate-VII-E} records one view of the same separation. The drawings add no hypothesis.


Figure~\ref{fig:scope-VII-E} records that separation.
The drawing adds no hypothesis.



\section{The closed observation problem}
\label{sec:introduction}

Destructive measurement, a coherent record carrier, and a measurement
instrument with a surviving quantum output are different interfaces.
A geometric extension of a quantum language must say which of these
interfaces it can observe. Linear ownership of quantum names alone
does not answer that question.

We take destructive computational-basis readout as a given primitive,
with type
\begin{equation}
  \Enter:\Qtype^{\otimes n}\longrightarrow\Bit^{\otimes n}.
  \label{eq:enter-type}
\end{equation}
Its measured quantum interface is consumed. We use its Born semantics,
as in the corresponding finite circuit fragment of the Enter
calculus~\cite{Cheung2026Enter}. No characterization or uniqueness proof
for the measurement law is repeated here.

The semantic additions are fixed throughout: an external classical
geometric configuration, locations for active names, declared coupling
algebras, and an explicit control and resource interface. The question
is then precise:
\begin{quote}
Which spectator extensions, deferred implementations, and scale/readout
comparisons preserve the outputs that the interface declares accessible?
\end{quote}
Our answers concern equality of observable outputs, not a mechanism
that constructs geometry from quantum syntax.

\subsection{Results and dependencies}

The paper proves six interface statements.

\begin{enumerate}
\item On a prescribed substrate, block channels have an exact finite-step
backwards support bound. Retiming a fixed labelled gate partial order
preserves the total channel.

\item Under spectator-blind control and resources, any admissible
spectator extension with the same accessible reduction preserves the
geometric trace and that reduction at every step.

\item Immediate destructive readout agrees with a restricted
coherent-then-readout implementation after both outputs are restricted
to the same record--matter interface.

\item Record laws across evolution and label aggregation agree exactly
when their effects obey a pullback identity. Their unassisted
worst-case total-variation error has an exact event-norm formula.

\item Record agreement does not imply residual-instrument agreement.
For sharp measurements with a coarse L\"uders instrument, an exact
criterion isolates the remaining intra-outcome coherence.

\item Agreement on an input observable algebra is preserved as agreement
on an output observable algebra exactly when the adjoint continuation
maps the latter into the former. A corresponding statement is given
for classical block-diagonal state spaces.
\end{enumerate}

Finite-dimensional channel conventions follow
Watrous~\cite{Watrous2018}. The circuit principle of deferred measurement
has a precise treatment in Gurevich and Blass~\cite{GurevichBlass2022}.
Our emphasis is the additional observation boundary: geometry, control,
resource tests, and accessible residual observables are all named.

The geometric certificate uses a fixed conical spacetime and the
polygonal-slicing setting of
\cite{DeserJackiwHooft1984,tHooft1992,tHooft1993,Kadar2005}.
The proofs of the interface theorems do not depend on that certificate.
The certificate shows that one nontrivial change of cell incidence can
meet the declared interface; it does not supply the assumptions of the
interface theorems.

\subsection{Operational ambition and limits}

Within the proposed model, a future gravitational update may inspect
only its declared observable interface. Destroyed names, inaccessible
carriers, and resource bookkeeping are not undeclared geometric source
variables. This is an operational requirement for a later
quantum--geometric composite, not a derivation of its dynamics.

In particular, the paper proves no gravitational field equation,
continuum limit, curvature-convergence theorem, physical causal
dominance, or interacting-product theorem. It does not evaluate a
singular triangle graph or establish a distributional finite part for
one. The short distributional dictionary in \cref{sec:dictionary}
contains conditional linear inference rules only.

The distinction is structural:
\[
 \text{names are not mass},\qquad
 \text{capacity is not curvature},\qquad
 \text{readout is not coarse-graining}.
\]
Likewise, a scheduled circuit past is not a physical light cone.
A computational implementation requires an additional physical model
before such identifications can be tested.

\section{Types, states, and external geometry}
\label{sec:semantics}

\subsection{Finite-dimensional conventions}

For a finite-dimensional Hilbert space \(\cH\), write \(L(\cH)\) for
its operators and \(D(\cH)\) for its density operators. A channel is a
completely positive, trace-preserving linear map. Its adjoint satisfies
\begin{equation}
 \Tr(A\Phi(X))=\Tr(\Phi^*(A)X).
 \label{eq:adjoint}
\end{equation}
The adjoint of a channel is unital and completely positive. A Kraus
representation is
\[
 \Phi(X)=\sum_\alpha K_\alpha X K_\alpha^\dagger,
 \qquad
 \sum_\alpha K_\alpha^\dagger K_\alpha=I.
\]
These are the standard finite-dimensional
conventions~\cite{Watrous2018}.

A classical register has a distinguished orthonormal basis.
Admissible joint states are block diagonal in that basis, including
their correlations with other registers. A diagonal classical marginal
alone is insufficient. Operations declared to preserve classicality
preserve this joint block-diagonal state space.

A unital observable algebra means a unital \(^{*}\)-subalgebra
\(\cA\subseteq L(\cH)\). Its accessible state functional is
\begin{equation}
 \omega_\rho|_{\cA}:A\longmapsto\Tr(\rho A).
 \label{eq:functional}
\end{equation}
Agreement on \(\cA\) means equality of these functionals.

Our norm and distance conventions are
\begin{equation}
 \norm{X}_1=\Tr\sqrt{X^\dagger X},
 \qquad
 d_{\mathrm{tr}}(\rho,\sigma)=\frac12\norm{\rho-\sigma}_1,
 \qquad
 d_{\mathrm{TV}}(p,q)=\frac12\sum_w\abs{p(w)-q(w)}.
 \label{eq:distance-conventions}
\end{equation}
The operator norm is \(\norm{\cdot}_\infty\). No ancillary reference
is included in an optimization unless explicitly stated.

\subsection{A minimal linear interface}

Wire types are generated by
\[
 A::=I\mid\Qtype\mid\Bit\mid A\otimes A.
\]
A judgement \(f:A\to B\) consumes the complete input interface and
returns the declared output interface. Composition and tensoring obey
\[
 \frac{f:A\to B\quad g:B\to C}{g\circ f:A\to C},
 \qquad
 \frac{f:A\to B\quad g:C\to D}
      {f\otimes g:A\otimes C\to B\otimes D}.
\]
Identities and wire permutations are available. Circuit constants used
below include quantum unitaries, preparation of \(\ket0\), and
destructive computational-basis readout. The finite classical-code
implementation also uses reversible NOT, CNOT, and Toffoli operations
on classical interfaces. Their quantum-carrier realizations are the
corresponding permutation unitaries.

Linearity is an ownership condition. It permits a sequence of gates
to consume and return the same physical tensor factor. It provides
neither duplication of an arbitrary quantum wire nor an implicit
discard operation.

The semantic matter-rule class below permits declared channels.
We do not claim that every such channel has an exact implementation
in the displayed finite gate library. Exact gate realizability is proved
only for the finite reversible event code in \cref{sec:event-code}.

\subsection{Three measurement interfaces}

Let \(S\) have computational basis
\(\{\ket z:z\in Z\}\), and let \(B\) be its unmeasured complement.
Put
\[
 M_z=\bra z_S\otimes I_B,
 \qquad
 P_z=\ket z\bra z_S\otimes I_B.
\]

\begin{definition}[Destructive readout]
\label{def:destructive}
The interpretation of \(\Enter_S\) is the channel
\begin{equation}
 \cM_S(\rho)
 =\sum_z\ket z\bra z_R\otimes M_z\rho M_z^\dagger,
 \qquad
 \cM_S:L(\cH_{SB})\longrightarrow L(\cH_{RB}).
 \label{eq:destructive}
\end{equation}
The record \(R\) is classical. The factor \(S\) is absent from the
output.
\end{definition}

\begin{definition}[Coherent recording]
\label{def:coherent}
A fresh quantum carrier \(A\), with basis indexed by \(Z\), is introduced
by the isometry
\begin{equation}
 V(\ket z_S\otimes\ket b_B)
 =\ket z_S\otimes\ket z_A\otimes\ket b_B.
 \label{eq:record-isometry}
\end{equation}
The corresponding channel is \(\Copy_S(\rho)=V\rho V^\dagger\).
It returns both \(S\) and \(A\).
\end{definition}

\begin{definition}[Represented nondestructive instrument]
\label{def:represented}
The represented computational-basis instrument is
\begin{equation}
 \cI_S^{\mathrm{nd}}(\rho)
 =\sum_z\ket z\bra z_R\otimes P_z\rho P_z.
 \label{eq:nondestructive}
\end{equation}
Its residual output is on \(S\otimes B\). It is not the destructive
primitive.
\end{definition}

\begin{center}
\begin{tabularx}{\textwidth}{@{}lXXX@{}}
\toprule
Operation & Measured factor & Record interface & Status\\
\midrule
\(\Enter_S\) & \(S\) consumed & Classical \(R\) & Primitive\\
\(\Copy_S\) & \(S\) retained & Quantum carrier \(A\) & Isometry\\
\(\cI_S^{\mathrm{nd}}\) & \(S\) retained & Classical \(R\) & Represented instrument\\
\bottomrule
\end{tabularx}
\end{center}

Coherent recording is not cloning. For a qubit,
\[
 V\ket{+}=\frac{\ket{00}+\ket{11}}{\sqrt2}
 \neq \ket{+}\otimes\ket{+}.
\]
The three interfaces must therefore remain distinct both in theorem
statements and in geometric couplings.

\subsection{Geometric configurations and admissibility}

\begin{definition}[External geometric interface]
\label{def:geometry}
An external geometric interface supplies:
\begin{enumerate}
\item a set \(\cQ\) of classical geometric and declared resource labels;
\item an admissible subset
\[
 \cQ_{\mathrm{adm}}=\cQ_{\mathrm{reg}}\cup\cQ_{\mathrm{tr}};
\]
\item a finite cell-label set \(\Cell(g)\) for each admissible \(g\);
\item support, boundary, and resource certificates for permitted updates;
\item an explicitly supplied gauge equivalence, when one is used.
\end{enumerate}
Regular configurations satisfy the model's nondegeneracy conditions.
A transition configuration is admitted only with an event certificate
specifying its realization and its permitted one-sided continuations.
\end{definition}

In the example, positive-length edges belong to regular cellulations.
At the event, a vanishing edge slot and coincident endpoint labels may
remain in a germ description. Their realization is a single junction;
they are not regular embedded one-cells.

\begin{definition}[Configuration]
\label{def:configuration}
A configuration is
\begin{equation}
 C=(g;\Gamma;\rho;\loc;h),
 \qquad
 \loc:\Gamma\longrightarrow\Cell(g),
 \label{eq:configuration}
\end{equation}
where \(g\in\cQ_{\mathrm{adm}}\), \(\Gamma\) is a finite labelled
matter interface, \(\rho\in D(\cH_\Gamma)\), and \(h\) is a finite
control state. Classical names satisfy the joint classicality convention.
The geometric projection is \(\pi_g(C)=g\).
\end{definition}

A location is an abstract cell label, not an undeclared developed
position or a stress-energy assignment. Input and output matter
interfaces may change. For example, destructive readout replaces
measured names with fresh classical record names.

\subsection{The three rule kinds}

\begin{definition}[Certified rules]
\label{def:rules}
A finite execution uses three kinds of certified rule.
\begin{enumerate}
\item A \emph{geometric rule}, or \(G\)-rule, changes supported geometric
labels as a specified function of geometric and declared resource
labels. It does not inspect the matter density operator or matter-name
cardinality. It returns a complete admissible configuration of geometric
labels and a location transport map.

A transition rule includes its common event realization and both
one-sided continuations.

\item A \emph{matter rule} applies a declared channel
\begin{equation}
 \Phi_g:L(\cH_{\Gamma_0})\longrightarrow L(\cH_{\Gamma'_0})
 \label{eq:matter-rule}
\end{equation}
on a declared interface, extended by the identity on unacted-on names.
It may depend on the current geometric labels but does not change them.
Its output-name locations are declared.

\item A \emph{coupling rule} declares an accessible unital algebra
\(\cA_v\) and updates geometry by
\begin{equation}
 g'=F_v\bigl(g,\omega_\rho|_{\cA_v}\bigr).
 \label{eq:coupling}
\end{equation}
It leaves the matter density operator unchanged and supplies a complete
admissibility certificate and location transport.
\end{enumerate}
\end{definition}

A coupling update must return all affected labels needed for
admissibility. Changing a particle parameter while silently freezing
incompatible gluing data is not a certified update.

The control state, guards, and resource checks determine which rule
is selected. Our executions are nonselective density-operator
executions. A selectively normalized measurement branch is not silently
treated as a trace-preserving matter rule. Classical control can instead
be represented inside a channel that retains its records.

\begin{assumption}[Spectator-blind control and resources]
\label{ass:blind}
For the compared executions, equal geometric labels, equal finite
control states, and equal declared accessible state functionals select
the same:
\begin{enumerate}
\item rule and matter channel;
\item applicability and capacity-check result;
\item support and resource certificate;
\item transport of original names and successor control state.
\end{enumerate}
Adding inaccessible spectator names does not change these selections.
Spectator storage and required transport history are reserved separately
from the geometric-update budget. Random external control is compared
under a common seed or an explicitly coupled control law.
\end{assumption}

This is a hypothesis about the complete controller, not merely about
the written coupling formula.

\begin{remark}[Nonlinear coupling]
\label{rem:nonlinear}
The map in \cref{eq:coupling} need not be affine in \(\rho\).
Consequently the complete coupled input--output map need not be a
quantum channel. The proofs below compare equal accessible inputs
step by step. They do not establish general no-signalling or a physical
consistency theorem for arbitrary nonlinear backreaction.
\end{remark}

\subsection{Gauge and observable geometry}

A gauge equivalence is an explicitly supplied equivalence relation
on descriptions. An admissible geometric observable is constant on
the specified classes. Chart changes and retriangulations may be
declared gauge when they represent the same metric and marked slice.

Equality of an ambient spacetime does not imply equality of every
observable of different marked slices. Likewise, equality of total
channels under a prescribed circuit retiming is not automatically
a gauge symmetry of coupled geometric evolution.

\section{Prescribed kinematics and schedule dependence}
\label{sec:schedule}

All geometry, partitions, and channel lists in this section are
prescribed. No state-dependent controller changes them.

\subsection{Exact backwards support}

Let \((X,d)\) be a finite metric space, with site Hilbert spaces
\(\cH_x\). For \(Y\subseteq X\), set
\[
 \cH_X=\bigotimes_{x\in X}\cH_x,
 \qquad
 \cA(Y)=L\!\left(\bigotimes_{x\in Y}\cH_x\right)\otimes I_{X\setminus Y}.
\]
At tick \(t\), let \(\mathcal B_t\) be a partition of \(X\), and suppose
\[
 \diam B\le\Delta,
 \qquad
 \Phi_t=\bigotimes_{B\in\mathcal B_t}\Phi_{t,B},
\]
where every block map is a channel. If ticks have duration \(\tau>0\),
the resulting scalar schedule bound is \(\Delta/\tau\).

Define the backwards sets recursively by
\begin{equation}
 \begin{split}
 N_{t,t}(Y)&=Y,\\
 N_{s,t}(Y)&=
 \bigcup\{B\in\mathcal B_{s+1}:B\cap N_{s+1,t}(Y)\neq\varnothing\},
 \qquad s<t.
 \end{split}
 \label{eq:schedule-past}
\end{equation}

\begin{lemma}[One-layer support]
\label{lem:one-layer}
For a block channel \(\Phi=\bigotimes_{B\in\mathcal B}\Phi_B\),
\[
 \Phi^*(\cA(Y))
 \subseteq
 \cA\!\left(\bigcup_{B\cap Y\neq\varnothing}B\right).
\]
\end{lemma}

\begin{proof}
The complement of the displayed union is a union of blocks on which
the input observable is the identity. Each corresponding adjoint
maps the identity to itself. The other blocks can enlarge support
only within their own union.
\end{proof}

\begin{theorem}[Prescribed schedule locality]
\label{thm:locality}
For \(A\in\cA(Y)\) and \(0\le s\le t\),
\begin{equation}
 \supp\bigl((\Phi_{s+1}^*\circ\cdots\circ\Phi_t^*)(A)\bigr)
 \subseteq N_{s,t}(Y),
 \label{eq:support-bound}
\end{equation}
and
\begin{equation}
 N_{s,t}(Y)\subseteq
 \{x:d(x,Y)\le(t-s)\Delta\}.
 \label{eq:metric-bound}
\end{equation}
The output reduction on \(Y\) depends on the state at tick \(s\)
only through its reduction to \(N_{s,t}(Y)\).
\end{theorem}

\begin{proof}
Repeated application of \cref{lem:one-layer} in reverse tick order
proves \cref{eq:support-bound}.

If \(x\in N_{s,t}(Y)\), its block meets \(N_{s+1,t}(Y)\) at some \(y\).
Hence
\[
 d(x,Y)\le d(x,y)+d(y,Y)
 \le\Delta+(t-s-1)\Delta.
\]
Induction proves \cref{eq:metric-bound}.

Finally, every output observable on \(Y\) pulls back to an observable
on \(N_{s,t}(Y)\). Equal input reductions there give equal expectations
for all output observables, and therefore equal output reductions.
\end{proof}

\begin{corollary}[Readout dependence]
\label{cor:local-readout}
A prescribed readout on \(Y\) at tick \(t\) depends only on the initial
reduction to \(N_{0,t}(Y)\). Replacing a prescribed block channel outside
its backwards dependency region does not change its law, provided
the supports and schedule remain fixed.
\end{corollary}

\begin{proof}
Apply \cref{thm:locality} to the readout effects. A replacement outside
the backwards region acts on an identity factor in the pulled-back
effect, and every channel adjoint preserves that identity.
\end{proof}

Readout can be represented on fixed site spaces by reserving output
record registers. An inactive placeholder after destructive measurement
is not a surviving accessible measured factor.

The backwards region is an exact finite-step upper bound, not
necessarily the smallest support for a particular list of channels.
Joint record laws require the backwards region of the complete
record-retaining network.

\subsection{A relay attaining the scalar bound}

\begin{proposition}[One-dimensional relay]
\label{prop:relay}
Let \(R\ge1\) be an integer. On \(\Z\), with
\(d(i,j)=a\abs{i-j}\), use the blocks
\begin{equation}
 B_{t,k}=\{k(R+1)+(t-1)R+j:0\le j\le R\}.
 \label{eq:relay}
\end{equation}
Swap the endpoint factors of every block at each tick. A factor
initially at \(0\) is at \(tR\) after \(t\) ticks, attaining speed
\(Ra/\tau\). In units \(R=a=\tau=1\), the attained speed is \(1\).
\end{proposition}

\begin{proof}
At each tick the blocks are disjoint consecutive intervals of diameter
\(Ra\). At tick \(t\), the transported factor is at the left endpoint
\((t-1)R\) of \(B_{t,0}\). The endpoint swap moves it to \(tR\), which
is the left endpoint of \(B_{t+1,0}\). Induction proves the claim.

For any finite horizon, retain a finite interval containing all blocks
used by this relay. Keep complete blocks inside that interval and
complete cut boundary fragments with singleton blocks. No infinite
tensor-product state is needed.
\end{proof}

The transported factor may be entangled with an untouched reference;
the argument is an identity of wire permutations.

\subsection{Retiming fixed labelled channels}

\begin{definition}[Admissible labelled gate order]
A finite collection of fixed labelled channels has declared supports
\(S_g\subseteq X\). A strict partial order is admissible if channels
with intersecting supports are comparable. A legal execution is a
linear extension of this order.
\end{definition}

\begin{theorem}[Retiming invariance]
\label{thm:retiming}
All legal executions of an admissibly ordered labelled channel
collection have the same total channel.
\end{theorem}

\begin{proof}
Incomparable channels have disjoint supports, so their extensions by
identities commute. Any two linear extensions are connected by adjacent
swaps of incomparable elements.

For completeness, compare two extensions at their first disagreement.
Move the desired element leftwards in the first extension. Each
intervening element is incomparable with it: either order relation
would contradict one of the two extensions. Repeating this procedure
makes the extensions equal. Every swap preserves the composed channel.
\end{proof}

Records are included in gate supports. The theorem does not move a gate
past another gate that reads its record. Nor does it change a geometric
parameter while treating the resulting channel as the same labelled
gate.

\subsection{Directional dependence}

\begin{proposition}[A scalar speed does not determine a velocity set]
\label{prop:direction}
In a square-lattice dependency model permitting at most one cardinal
step of length \(a\) per tick, or a wait, the asymptotic velocity set is
\[
 \mathcal V=\{(v_x,v_y):\abs{v_x}+\abs{v_y}\le a/\tau\}.
\]
An isotropic physical speed \(a/\tau\) is not dominated in every
direction by this set.
\end{proposition}

\begin{proof}
After \(n\) ticks,
\(\abs{\Delta x}+\abs{\Delta y}\le na\).
Conversely, suitable proportions of cardinal steps and waits approximate
every velocity in the displayed diamond. But
\[
 \left(\frac{a}{\sqrt2\,\tau},\frac{a}{\sqrt2\,\tau}\right)
\]
has Euclidean norm \(a/\tau\) and lies outside that diamond.
\end{proof}

This proposition concerns the stated cardinal-hop model, not every
block schedule. It illustrates why a directional inclusion theorem
cannot be replaced by a comparison of two scalar speeds. More general
periodic-graph velocity polytopes are studied in~\cite{Fritz2013}.

\section{Spectator-extension invariance}
\label{sec:spectators}

A spectator extension of \(\rho_\Gamma\) to a disjoint interface \(N\)
is any admissible state satisfying
\begin{equation}
 \Tr_N\widetilde\rho_{\Gamma N}=\rho_\Gamma.
 \label{eq:extension}
\end{equation}
The extension may be entangled. Existing classical registers must remain
classical in the full joint state.

\begin{lemma}[Restriction commutes with an untouched spectator]
\label{lem:partial-trace}
For a channel
\(\Phi:L(\cH_\Gamma)\to L(\cH_{\Gamma'})\),
\begin{equation}
 \Tr_N\bigl((\Phi\otimes\id_N)(\widetilde\rho)\bigr)
 =\Phi(\Tr_N\widetilde\rho).
 \label{eq:trace-channel}
\end{equation}
\end{lemma}

\begin{proof}
For every \(A\in L(\cH_{\Gamma'})\),
\begin{align*}
 \Tr\!\left(A\Tr_N[(\Phi\otimes\id_N)(\widetilde\rho)]\right)
 &=\Tr((\Phi^*(A)\otimes I_N)\widetilde\rho)\\
 &=\Tr(\Phi^*(A)\Tr_N\widetilde\rho)\\
 &=\Tr(A\Phi(\Tr_N\widetilde\rho)).
\end{align*}
Equality against every output observable proves the result.
\end{proof}

\begin{theorem}[Spectator-extension invariance]
\label{thm:spectators}
Consider a finite certified execution
\[
 C_k=(g_k;\Gamma_k;\rho_k;\loc_k;h_k),
 \qquad 0\le k\le T,
\]
and an admissible initial extension satisfying \cref{eq:extension},
with the same geometry, finite control state, and original-name
locations. Assume:
\begin{enumerate}
\item each original matter map in the extended execution is tensored
with \(\id_N\);
\item each coupling algebra is embedded as \(\cA_v\otimes I_N\), or its
input is a specified function only of the original reduced state;
\item spectator-blind control and resources,
\cref{ass:blind}, hold;
\item spectator locations have the required certified transport, and
no spectator outcome is used during the comparison.
\end{enumerate}
Then a paired extended execution is legal and, at every step,
\begin{equation}
 \widetilde g_k=g_k,\qquad
 \widetilde h_k=h_k,\qquad
 \Tr_N\widetilde\rho_k=\rho_k.
 \label{eq:spectator-conclusion}
\end{equation}
Original-name locations also agree.
\end{theorem}

\begin{proof}
The assertions hold initially. Assume they hold at step \(k\).
By \cref{ass:blind}, both executions select the same rule, guard result,
support, and resource certificate.

For a geometric rule, the geometric and resource inputs agree.
Its output geometry and original-name transport therefore agree.
The matter states are unchanged.

For a matter rule, the geometric parameter is common. By
\cref{lem:partial-trace},
\[
 \Tr_N\widetilde\rho_{k+1}
 =\Phi_{g_k}(\Tr_N\widetilde\rho_k)
 =\Phi_{g_k}(\rho_k)
 =\rho_{k+1}.
\]

For a coupling rule, every \(A\in\cA_v\) has equal expectation:
\[
 \Tr((A\otimes I_N)\widetilde\rho_k)=\Tr(A\rho_k).
\]
The rule receives identical state functionals and returns identical
certified geometry and original-name transport. The same conclusion
holds when its input is a specified function of the original reduction.

The common control selection gives the same successor control state.
Admissibility and the transport certificates establish legality of the
next paired configurations. Induction completes the proof.
\end{proof}

\begin{corollary}[Cardinality blindness within the declared rule class]
\label{cor:cardinality}
Under the hypotheses of \cref{thm:spectators}, inserting any finite
product spectator state \(\rho_0\otimes\phi_N\) leaves the geometric
trace and original accessible matter reductions unchanged.
\end{corollary}

\begin{proof}
A product insertion is a special case of \cref{eq:extension}.
\end{proof}

\subsection{The capacity leak}

\begin{example}[A guard can reveal an otherwise inaccessible spectator]
\label{ex:capacity}
Let \(g\in\{0,1\}\), and let a geometric rule request \(g\mapsto1\).
Suppose its capacity guard is
\[
 \abs{\Gamma}\le K.
\]
At \(\abs{\Gamma}=K\), the update is permitted. Adding a single product
spectator blocks it, even when every coupling algebra is \(\C I\)
and no matter operation touches the spectator.
\end{example}

The example violates \cref{ass:blind}; it does not contradict
\cref{thm:spectators}. Similar leaks can occur through queue priority,
repartitioning, support selection, or a spectator-dependent choice of
successor location.

Within the theorem's rule class, direct inspection of spectator state
or cardinality is excluded as well. It would be inaccurate to infer
that every possible leak is a capacity leak. The point is that coupling
blindness alone is insufficient: all control and resource observations
must also be declared.

\subsection{Terminal readout and selective states}

A spectator may be returned or consumed by terminal \(\Enter\).
The partial trace in \cref{eq:spectator-conclusion} describes the
accessible observation; it is not a term-level weakening rule.

Nonselective terminal readout of an entangled spectator preserves the
original reduced state. Its conditional states need not be preserved.
For
\[
 \ket{\Omega}
 =\frac{\ket{00}+\ket{11}}{\sqrt2},
\]
readout of the second factor leaves the first marginal \(I/2\) when
the outcome is ignored, but conditioning on the outcome gives
\(\ket0\bra0\) or \(\ket1\bra1\).

Thus the theorem neither identifies the enlarged joint output with the
original output nor permits a later geometric rule to read the spectator
outcome without changing the comparison.

\section{Restricted destructive deferred readout}
\label{sec:deferred}

Immediate \(\Enter_S\) removes \(S\), whereas coherent recording retains
it. Their outputs can be compared only after specifying a common
accessible interface.

\begin{lemma}[Coherent recording followed by restriction]
\label{lem:recording-restriction}
For every state on \(S\otimes B\),
\begin{equation}
 \Tr_S(V\rho V^\dagger)
 =\sum_z\ket z\bra z_A\otimes M_z\rho M_z^\dagger.
 \label{eq:copy-trace}
\end{equation}
\end{lemma}

\begin{proof}
Write
\[
 \rho=\sum_{z,z'}\ket z\bra{z'}_S\otimes\rho_{zz'}.
\]
Then
\[
 V\rho V^\dagger
 =\sum_{z,z'}
 \ket z\bra{z'}_S\otimes
 \ket z\bra{z'}_A\otimes\rho_{zz'}.
\]
Tracing out \(S\) removes precisely the terms with \(z\neq z'\).
\end{proof}

Let \(\cN_A\) denote destructive computational-basis readout of the
carrier \(A\), returning a classical record \(R\).

\begin{lemma}[Continuation identity]
\label{lem:continuation}
For every channel
\(\Lambda:L(\cH_B)\to L(\cH_{B'})\),
\begin{equation}
 \Tr_S\!\left[
 \cN_A\bigl((\id_{SA}\otimes\Lambda)(V\rho V^\dagger)\bigr)
 \right]
 =(\id_R\otimes\Lambda)\cM_S(\rho).
 \label{eq:continuation}
\end{equation}
\end{lemma}

\begin{proof}
A channel acting only on \(B\) commutes with restriction over \(S\)
and with carrier readout. Apply \cref{lem:recording-restriction}
and identify the carrier's diagonal basis with the classical record
basis.
\end{proof}

\begin{theorem}[Restricted destructive deferred readout]
\label{thm:deferred}
Start from geometry \(g_0\) and a state \(\rho\) on \(S\otimes B\).
Compare:
\begin{enumerate}
\item immediate \(\Enter_S\), followed by a finite continuation;
\item coherent recording into a fresh \(A\), the corresponding
continuation, and terminal \(\Enter_A\).
\end{enumerate}
During the continuation assume:
\begin{enumerate}[label=(D\arabic*)]
\item matter commands act only on the current complement \(B_k\),
never on the retained \(S\), carrier \(A\), or immediate record \(R\);
\item coupling inputs and state-dependent guards depend only on
the reduced state on \(B_k\);
\item geometric rules inspect only their declared geometric labels;
\item control, support, and resources are blind to the excluded
registers, as in \cref{ass:blind}.
\end{enumerate}
Align executions by their common continuation events. Their geometric
traces and complement reductions coincide at every aligned event.
After terminal carrier readout, restricting the deferred output away
from the retained factor \(S\) gives the immediate output on
\(R\otimes B_T\).
\end{theorem}

\begin{proof}
Immediately after the two initial operations,
\[
 \Tr_R\cM_S(\rho)=\Tr_S\rho,
 \qquad
 \Tr_{SA}(V\rho V^\dagger)=\Tr_S\rho.
\]
Thus the complement reductions agree.

Suppose they agree at a continuation event, together with geometry
and control. Conditions (D2)--(D4) and \cref{ass:blind} give the same
guard result, rule, resource certificate, and geometric update.
Condition (D1) gives the same matter channel on the complement.
By \cref{lem:partial-trace}, its application preserves equality of
complement reductions. Induction establishes the common trace.

For this fixed pair of inputs, compose the matter channels selected
along the common trace into a channel \(\Lambda\) on the continuation
interface. Equation \cref{eq:continuation} gives equality of the final
outputs on \(R\otimes B_T\).
\end{proof}

\begin{remark}[The factor removed in the comparison]
The terminal operation has already consumed the carrier \(A\).
The extra quantum output is the retained original factor \(S\).
It is \(S\), not \(A\), that is omitted from the common observation
interface.
\end{remark}

The selected \(\Lambda\) can depend on the input through the coupled
geometric trace. The theorem is pairwise and does not assert that the
complete coupled map is linear.

The deferred circuit must still return \(S\) or consume it in a
well-typed continuation. Observational inaccessibility does not create
a discard rule. Allowing intermediate access to \(R\), \(S\), or \(A\)
requires a different theorem. Standard deferred-measurement constructions
can treat more general record-dependent circuits
\cite{GurevichBlass2022}, but their hypotheses are not silently imported
into the present geometric comparison.

\section{Record compatibility, residuals, and composition}
\label{sec:records}

This section concerns genuine finite-dimensional channels.
It does not regard the nonlinear coupling in \cref{eq:coupling}
as a channel.

\subsection{Effect pullback}

Let
\[
 T:L(\cH_{\mathrm f})\longrightarrow L(\cH_{\mathrm c})
\]
be a channel. It may include prescribed evolution followed by a
coarse-graining channel. Let \(\{F_z:z\in Z\}\) be a fine POVM,
\(\{E_w:w\in W\}\) a coarse POVM, and \(q:Z\to W\) a label map.
Define
\begin{equation}
 \Pi_w=\sum_{z:q(z)=w}F_z.
 \label{eq:aggregated}
\end{equation}
Empty sums are zero. The two record laws are
\begin{equation}
 p_T(w\mid\rho)=\Tr(E_wT(\rho)),
 \qquad
 p_{\mathrm f}(w\mid\rho)=\Tr(\Pi_w\rho).
 \label{eq:record-laws}
\end{equation}

\begin{theorem}[Exact record compatibility]
\label{thm:pullback}
The two record laws agree for every state if and only if
\begin{equation}
 T^*(E_w)=\Pi_w\qquad(w\in W).
 \label{eq:pullback}
\end{equation}
\end{theorem}

\begin{proof}
Their difference at outcome \(w\) is
\[
 \Tr\bigl(\rho[T^*(E_w)-\Pi_w]\bigr).
\]
The bracket is Hermitian. If its expectation vanishes in every state,
testing its eigenvectors shows that every eigenvalue is zero.
The reverse implication is the adjoint identity.
\end{proof}

For example, if \(T=\Tr_{\mathrm r}\) and
\(\Pi_w=E_w\otimes I_{\mathrm r}\), compatibility is exact because
\(T^*(E)=E\otimes I_{\mathrm r}\).
Readout at another time must first be pulled back through the
appropriate evolution; a label aggregation is not an evolution map.

\subsection{The exact unassisted record error}

Set
\[
 D_w=T^*(E_w)-\Pi_w,
 \qquad
 D_A=\sum_{w\in A}D_w.
\]
These operators are Hermitian and satisfy \(\sum_wD_w=0\).
Define
\begin{equation}
 \delta_{\mathrm{rec}}
 =\sup_{\rho\in D(\cH_{\mathrm f})}
 d_{\mathrm{TV}}\bigl(p_T(\cdot\mid\rho),
                         p_{\mathrm f}(\cdot\mid\rho)\bigr).
 \label{eq:record-error}
\end{equation}

\begin{theorem}[Event operator-norm formula]
\label{thm:event-norm}
With the conventions in \cref{eq:distance-conventions},
\begin{equation}
 \boxed{
 \delta_{\mathrm{rec}}
 =\max_{A\subseteq W}\norm{D_A}_\infty
 \le\frac12\sum_w\norm{D_w}_\infty.}
 \label{eq:event-norm}
\end{equation}
The maximum is attained by a pure input state.
\end{theorem}

\begin{proof}
For real numbers \(d_w\) with zero sum,
\[
 \frac12\sum_w\abs{d_w}
 =\max_{A\subseteq W}\abs{\sum_{w\in A}d_w}.
\]
Use \(d_w=\Tr(\rho D_w)\). Since the event set is finite,
\[
 \delta_{\mathrm{rec}}
 =\max_{A\subseteq W}\sup_\rho\abs{\Tr(\rho D_A)}.
\]
For a Hermitian operator \(H\), the latter supremum is
\(\norm{H}_\infty\), attained at an eigenvector whose eigenvalue has
maximal absolute value. This proves the equality and attainment.
The bound follows from
\(\abs{\Tr(\rho D_w)}\le\norm{D_w}_\infty\).
\end{proof}

\begin{example}[Why individual outcomes are insufficient]
\label{ex:event-not-singleton}
On a one-dimensional input space, take the two four-outcome POVMs
\[
 p=(1/2,1/2,0,0),\qquad
 q=(0,0,1/2,1/2).
\]
Their total-variation distance is \(1\), whereas the largest
individual-outcome discrepancy is \(1/2\).
The event \(\{1,2\}\) attains the correct value.
Thus \(\max_w\norm{D_w}_\infty\) is not the general formula.
For two outcomes, it does suffice because \(D_1=-D_0\).
\end{example}

\begin{example}[A qubit incompatibility]
Let \(T(\rho)=H\rho H\), where \(H\) is the Hadamard unitary.
Use computational-basis measurements before and after \(T\), with
identity label aggregation. Then
\[
 D_0=\frac{X-Z}{2},\qquad D_1=-D_0,
 \qquad
 \delta_{\mathrm{rec}}=\frac1{\sqrt2}.
\]
The state \(\ket0\bra0\) gives error \(1/2\), so it is not a worst-case
input.
\end{example}

The optimization in \cref{eq:record-error} is unassisted.
Equation \cref{eq:event-norm} is not a diamond-norm formula.

\subsection{Untouched entangled references}

Let \(\mathcal M_E\) and \(\mathcal M_\Pi\) denote the record-only
measurement channels associated with \(E_w\) and \(\Pi_w\).

\begin{corollary}[Exact compatibility with an untouched reference]
\label{cor:reference}
Under \cref{eq:pullback}, for every finite reference system \(N\),
\begin{equation}
 (\mathcal M_E\circ T)\otimes\id_N
 =\mathcal M_\Pi\otimes\id_N.
 \label{eq:reference-equality}
\end{equation}
In particular, the joint record--reference outputs agree for entangled
inputs.
\end{corollary}

\begin{proof}
For every operator \(X\),
\[
 (\mathcal M_E\circ T)(X)
 =\sum_w\ket w\bra w\,\Tr(T^*(E_w)X)
 =\sum_w\ket w\bra w\,\Tr(\Pi_wX)
 =\mathcal M_\Pi(X).
\]
Equal linear maps remain equal after tensoring with an identity.
\end{proof}

An untouched reference is not the residual output produced by an
instrument. Equal record channels determine the former comparison,
not the latter. Also, an approximate unassisted bound does not become
an ancillary-assisted bound merely because exact equality tensors.

\subsection{Accessible output algebras}

For a unital residual algebra
\(\cB\subseteq L(\cH_{\mathrm c})\), define
\begin{equation}
 \cO_{\cB}
 =\left\{
 \sum_w\ket w\bra w_R\otimes B_w:B_w\in\cB
 \right\}.
 \label{eq:record-algebra}
\end{equation}
The record register is classical. Thus \(\cO_{\cB}\) is the named
record--residual observation algebra.

For channels \(\Theta,\Xi\) with a common output space and a unital
output algebra \(\cO\), define
\begin{equation}
 d_{\cO}(\Theta,\Xi)
 =\frac12\sup_\rho
 \sup_{\substack{A=A^\dagger\in\cO\\\norm A_\infty\le1}}
 \abs{\Tr(A[\Theta(\rho)-\Xi(\rho)])}.
 \label{eq:observable-distance}
\end{equation}

\begin{proposition}[Accessible-output criterion]
\label{prop:accessible}
The channels agree on \(\cO\) for every input if and only if
\[
 \Theta^*(A)=\Xi^*(A)\qquad(A\in\cO).
\]
Moreover,
\begin{equation}
 d_{\cO}(\Theta,\Xi)
 =\frac12
 \sup_{\substack{A=A^\dagger\in\cO\\\norm A_\infty\le1}}
 \norm{\Theta^*(A)-\Xi^*(A)}_\infty.
 \label{eq:observable-norm}
\end{equation}
\end{proposition}

\begin{proof}
For Hermitian \(A\), equality of all input expectations is equality
of the Hermitian adjoint images. General algebra elements are complex
linear combinations of Hermitian elements. For the distance, interchange
the two suprema and optimize the Hermitian expectation over states.
\end{proof}

For \(\cO_{\C I}\), this is record-law comparison.
For \(\cO_{L(\cH_{\mathrm c})}\), it compares complete classical-record
instrument outputs. Intermediate algebras specify intermediate
observation boundaries.

\subsection{Record agreement does not determine residuals}

Now let the fine measurement be a PVM \(\{P_z\}\).
Define
\begin{equation}
 \cF(X)
 =\sum_w\ket w\bra w_R\otimes
 T\!\left(\sum_{z:q(z)=w}P_zXP_z\right).
 \label{eq:fine-instrument}
\end{equation}
For a coarse instrument with outcome maps \(\cI_w\), put
\begin{equation}
 \cG(X)
 =\sum_w\ket w\bra w_R\otimes\cI_w(T(X)),
 \qquad E_w=\cI_w^*(I).
 \label{eq:coarse-instrument}
\end{equation}
These channels have the same record--residual output interface.

\begin{theorem}[Residual-state obstruction]
\label{thm:residual-obstruction}
Record compatibility does not imply agreement of \(\cF\) and \(\cG\)
on \(\cO_{L(\cH_{\mathrm c})}\), even when \(T=\id\).
\end{theorem}

\begin{proof}
Take one qubit, computational projectors \(P_0,P_1\), and a single
coarse outcome \(*\). Let the coarse instrument be the identity.
Then
\[
 E_*=I=\Pi_*,
\]
so the record error is zero. For \(\rho=\ket+\bra+\),
\[
 \cG(\rho)=\ket*\bra*_R\otimes\ket+\bra+,
 \qquad
 \cF(\rho)=\ket*\bra*_R\otimes I/2.
\]
Their difference has residual block \(X/2\), with eigenvalues
\(\pm1/2\). Therefore
\[
 \norm{\cG(\rho)-\cF(\rho)}_1=1,
 \qquad
 d_{\mathrm{tr}}(\cG(\rho),\cF(\rho))=\frac12.
\]
The full residual algebra distinguishes the outputs.
\end{proof}

If two implementations approximate these targets on this input with
trace-distance errors at most \(\varepsilon_{\mathrm f}\) and
\(\varepsilon_{\mathrm c}\), their mutual trace distance is at least
\[
 \frac12-\varepsilon_{\mathrm f}-\varepsilon_{\mathrm c}.
\]
This follows from the triangle inequality. More accurate implementation
cannot eliminate an incompatibility of the ideal targets.

\subsection{Sharp-sector intertwining}

Assume now that \(\{E_w\}\) is also a PVM and the coarse instrument is
the L\"uders instrument
\[
 \cI_w(X)=E_wXE_w.
\]
The aggregated \(\Pi_w=\sum_{q(z)=w}P_z\) are mutually orthogonal
projections.

\begin{lemma}[Intertwining forced by sharp pullback]
\label{lem:intertwining}
If \(T^*(E_w)=\Pi_w\), then every Kraus operator \(K_\alpha\) in
any Kraus representation of \(T\) satisfies
\begin{equation}
 E_wK_\alpha=K_\alpha\Pi_w.
 \label{eq:kraus-intertwining}
\end{equation}
Consequently,
\begin{equation}
 E_wT(X)E_w=T(\Pi_wX\Pi_w)
 \qquad(X\in L(\cH_{\mathrm f})).
 \label{eq:corner-identity}
\end{equation}
\end{lemma}

\begin{proof}
For every vector \(v\),
\begin{align*}
 \sum_\alpha
 \norm{(I-E_w)K_\alpha\Pi_wv}^2
 &=\pair{v}{\Pi_wT^*(I-E_w)\Pi_wv}\\
 &=\pair{v}{\Pi_w(I-\Pi_w)\Pi_wv}=0.
\end{align*}
Every summand is nonnegative. Hence
\((I-E_w)K_\alpha\Pi_w=0\) for every \(\alpha\).
Similarly,
\begin{align*}
 \sum_\alpha
 \norm{E_wK_\alpha(I-\Pi_w)v}^2
 &=\pair{v}{(I-\Pi_w)T^*(E_w)(I-\Pi_w)v}\\
 &=0,
\end{align*}
so \(E_wK_\alpha(I-\Pi_w)=0\).
These two identities give
\[
 E_wK_\alpha=E_wK_\alpha\Pi_w=K_\alpha\Pi_w.
\]
Substitution into the Kraus representation proves
\cref{eq:corner-identity}.
\end{proof}

Define the two dephasing channels
\begin{equation}
 \cD_P(X)=\sum_zP_zXP_z,
 \qquad
 \cD_\Pi(X)=\sum_w\Pi_wX\Pi_w.
 \label{eq:dephasings}
\end{equation}

\begin{theorem}[Exact L\"uders residual criterion]
\label{thm:luders}
Assume that the fine and coarse measurements are PVMs, the coarse
instrument is L\"uders, and \(T^*(E_w)=\Pi_w\) for every \(w\).
Then the following statements are equivalent:
\begin{enumerate}[label=(\roman*)]
\item \(\cF=\cG\), equivalently they agree on
\(\cO_{L(\cH_{\mathrm c})}\) for every input;
\item
\begin{equation}
 T\circ\cD_\Pi=T\circ\cD_P;
 \label{eq:luders-criterion}
\end{equation}
\item for every \(X\) and every distinct \(z,z'\) with
\(q(z)=q(z')\),
\begin{equation}
 T(P_zXP_{z'})=0.
 \label{eq:intra-bin}
\end{equation}
\end{enumerate}
\end{theorem}

\begin{proof}
By \cref{lem:intertwining}, the difference of outcome maps is
\begin{align}
 \cG_w(X)-\cF_w(X)
 &=T\!\left(\Pi_wX\Pi_w
       -\sum_{z:q(z)=w}P_zXP_z\right)\notag\\
 &=T\!\left(
   \sum_{\substack{z\neq z'\\q(z)=q(z')=w}}P_zXP_{z'}
   \right).
 \label{eq:block-difference}
\end{align}
Each such difference is supported in the \(E_w\)-corner. Its sum over
\(w\) is
\[
 T(\cD_\Pi(X))-T(\cD_P(X)).
\]
Since the \(E_w\) are orthogonal, this sum vanishes if and only if
every corner vanishes: multiply the sum on both sides by \(E_w\).
This proves the equivalence of (i) and (ii).

Condition (iii) makes every summand in \cref{eq:block-difference}
zero, proving (i). Conversely, assume (ii) and take
\(X=P_zYP_{z'}\), with \(z\neq z'\) in the same coarse outcome.
Then \(\cD_\Pi(X)=X\) and \(\cD_P(X)=0\), so \(T(X)=0\).
This proves (iii).
\end{proof}

\begin{corollary}[Two limiting cases]
\label{cor:luders-cases}
Under the hypotheses of \cref{thm:luders}:
\begin{enumerate}
\item if \(q\) is injective, exact record compatibility already implies
full L\"uders compatibility;
\item if there is one coarse outcome, full compatibility is equivalent
to \(T=T\circ\cD_P\).
\end{enumerate}
\end{corollary}

\begin{proof}
For injective \(q\), the two dephasings in \cref{eq:dephasings} coincide.
For one coarse outcome, \(\Pi_*=I\), so \(\cD_\Pi=\id\).
Apply \cref{thm:luders}.
\end{proof}

Thus \(T=T\circ\cD_P\) is sufficient in general but stronger than
necessary. Coherence between distinct coarse outcomes need not be
destroyed by \(T\), because the coarse L\"uders instrument removes it.
The obstruction is precisely fine coherence within a single coarse
outcome.

\subsection{Compositional observation boundaries}

The preceding comparisons concern specified output channels.
For a continuation, a further question is whether input
indistinguishability remains indistinguishability.

\begin{theorem}[Compositional observation principle]
\label{thm:composition}
Let \(\cA\subseteq L(\cH_{\mathrm{in}})\) and
\(\cB\subseteq L(\cH_{\mathrm{out}})\) be unital
\(^{*}\)-subalgebras, and let \(\Theta\) be a channel.
On the unrestricted matrix-state spaces, the following are equivalent:
\begin{enumerate}[label=(\roman*)]
\item for every pair of states,
\[
 \omega_\rho|_{\cA}=\omega_\sigma|_{\cA}
 \quad\Longrightarrow\quad
 \omega_{\Theta(\rho)}|_{\cB}
 =\omega_{\Theta(\sigma)}|_{\cB};
\]
\item
\begin{equation}
 \Theta^*(\cB)\subseteq\cA.
 \label{eq:composition}
\end{equation}
\end{enumerate}
\end{theorem}

\begin{proof}
Condition (ii) implies (i) by pulling each output observable back.

Conversely, assume (i). Let \(H=H^\dagger\) satisfy
\[
 \Tr(HA)=0\qquad(A\in\cA).
\]
Since \(I\in\cA\), \(\Tr H=0\). With
\(d=\dim\cH_{\mathrm{in}}\), choose \(\varepsilon>0\) small enough that
\[
 \rho_\pm=I/d\pm\varepsilon H
\]
are positive. They are states agreeing on \(\cA\).
For every Hermitian \(B\in\cB\), assumption (i) gives
\[
 0=\Tr(B[\Theta(\rho_+)-\Theta(\rho_-)])
  =2\varepsilon\Tr(H\Theta^*(B)).
\]
Thus \(\Theta^*(B)\) is orthogonal, in the real Hilbert space of Hermitian
matrices, to the orthogonal complement of the Hermitian part of
\(\cA\). Finite-dimensional double orthogonal complementation puts it
in that Hermitian part. Complex linearity proves
\(\Theta^*(\cB)\subseteq\cA\).
\end{proof}

If successive continuations obey
\(\Theta_k^*(\cA_{k+1})\subseteq\cA_k\), their composition preserves
the initial observation boundary. This follows by composing the
adjoint inclusions.

\subsection{Classical block-diagonal state spaces}

Let
\[
 \mathcal C=\bigoplus_j L(\cH_j)
 \subseteq L\!\left(\bigoplus_j\cH_j\right)
\]
be a prescribed block-diagonal state algebra, and let
\[
 \mathcal E_{\mathcal C}(X)=\sum_jQ_jXQ_j
\]
be its trace-orthogonal pinching projection.
Admissible input states are the density operators in \(\mathcal C\).
The accessible input algebra \(\cA\) need not lie inside \(\mathcal C\).

\begin{proposition}[Composition modulo unobservable directions]
\label{prop:classical-composition}
For all admissible input pairs, agreement on \(\cA\) is preserved as
agreement on \(\cB\) if and only if
\begin{equation}
 \mathcal E_{\mathcal C}\Theta^*(\cB)
 \subseteq \mathcal E_{\mathcal C}(\cA).
 \label{eq:classical-composition}
\end{equation}
Equivalently,
\[
 \Theta^*(\cB)\subseteq\cA+\mathcal C^\perp,
\]
where \(\mathcal C^\perp\) is the Hilbert--Schmidt orthogonal complement.
\end{proposition}

\begin{proof}
An admissible state sees \(A\) only through
\(\mathcal E_{\mathcal C}(A)\). Therefore the displayed inclusion is
sufficient.

For necessity, repeat the perturbation proof of
\cref{thm:composition} with Hermitian \(H\in\mathcal C\) annihilating
\(\mathcal E_{\mathcal C}(\cA)\). The states \(I/d\pm\varepsilon H\)
are admissible. The resulting orthogonality puts
\(\mathcal E_{\mathcal C}\Theta^*(B)\) in
\(\mathcal E_{\mathcal C}(\cA)\) for every Hermitian \(B\), and then
for every \(B\). Finally,
\(\ker\mathcal E_{\mathcal C}=\mathcal C^\perp\), proving the
equivalent formulation.
\end{proof}

The image \(\mathcal E_{\mathcal C}(\cA)\) need not itself be an algebra.
The statement uses it as the actual observable linear space on the
restricted states.

\section{A dictionary for distributional observations}
\label{sec:dictionary}

This section gives three linear inference rules for a separate
distributional problem. It assumes no particular graph kernel and
proves no finite-part existence theorem for a graph.

Let \(X,Y\) be smooth manifolds, with a fixed test-function convention.
Write \(\cD(X)=C_c^\infty(X)\) and \(\cD'(X)\) for complex distributions,
using the complex-bilinear distribution--test pairing.
Convergence below means distributional convergence.

The names VII-C and VII-D refer only to the separate Wick/graph track.
Nothing from that track is needed to state or prove the results here.

\subsection{Descent through declared invisible directions}

In finite dimensions, the inclusion
\(\Theta^*(\cB)\subseteq\cA\) is dual to
\[
 \Theta(\cA^\perp)\subseteq\cB^\perp.
\]
Indeed, both say that
\(\Tr(B\Theta(H))=\Tr(\Theta^*(B)H)=0\) whenever \(H\) annihilates
\(\cA\). The following quotient criterion has the same formal role.

\begin{proposition}[Descent]
\label{prop:descent}
Let \(P:\cD'(X)\to\cD'(Y)\) be continuous and linear, and let
\(\mathfrak C\subseteq\cD'(X)\) and
\(\mathfrak C_P\subseteq\cD'(Y)\) be declared real linear subspaces.
The assignment
\[
 [u]\longmapsto[Pu]
\]
is well defined on the real vector-space quotients if and only if
\begin{equation}
 P\mathfrak C\subseteq\mathfrak C_P.
 \label{eq:descent}
\end{equation}
With quotient topologies, the induced map is continuous.
\end{proposition}

\begin{proof}
If \(u-v\in\mathfrak C\), then \(Pu-Pv=P(u-v)\).
Condition \cref{eq:descent} therefore makes the output class independent
of the representative. Conversely, well-definedness applied to
\([c]=[0]\), \(c\in\mathfrak C\), gives \(Pc\in\mathfrak C_P\).
Continuity follows from the defining property of the quotient topology.
\end{proof}

For \(\mathfrak C=\R r\), this is a real quotient, even though the
ambient distributions are complex. If source and target use the same
nonzero residue \(r\), descent requires \(Pr\in\R r\).
Failure of this inclusion is failure of descent for that specified
quotient; it is not a proof that the original finite part fails to
exist.

No multiplication is defined on these quotients.

\subsection{Counterterm-blind tests}

\begin{proposition}[Exact one-dimensional counterterm test]
\label{prop:blind-tests}
Let \(r\neq0\) and \(L\) belong to \(\cD'(X)\).
Choose \(\Psi\in\cD(X)\) with \(\pair r\Psi=1\), and set
\begin{equation}
 \Phi^\circ=\Phi-\pair r\Phi\,\Psi.
 \label{eq:blind-test}
\end{equation}
Then
\begin{equation}
 L\in\R r
 \quad\Longleftrightarrow\quad
 \begin{cases}
 \pair L{\Phi^\circ}=0&\text{for every }\Phi\in\cD(X),\\
 \pair L\Psi\in\R.
 \end{cases}
 \label{eq:blind-test-criterion}
\end{equation}
\end{proposition}

\begin{proof}
Nonzero \(r\) permits a test with nonzero pairing, which can be rescaled
to obtain \(\Psi\). By construction,
\(\pair r{\Phi^\circ}=0\).

If \(L=cr\), \(c\in\R\), both conditions follow. Conversely, if they
hold, then
\[
 \pair L\Phi
 =\pair L{\Phi^\circ}+\pair r\Phi\,\pair L\Psi
 =c\pair r\Phi,
 \qquad c=\pair L\Psi\in\R.
\]
This holds for every compact test, so \(L=cr\).
\end{proof}

For real distributions and real tests, the reality condition is
automatic. Over complex tests it must not be omitted: tests annihilating
\(r\) alone cannot distinguish real from nonreal multiples of \(r\).

Suppose a complete regulator difference has already been shown to
converge,
\[
 u_\varepsilon^a-u_\varepsilon^b\longrightarrow L_{ab}
 \quad\text{in }\cD'(X).
\]
Then a single compact test \(\Phi^\circ\) with
\[
 \pair r{\Phi^\circ}=0,
 \qquad
 \pair{L_{ab}}{\Phi^\circ}\neq0
\]
obstructs removal of \(L_{ab}\) by an allowed finite counterterm in
\(\R r\). These tests are diagnostics. Existence of a finite part
still means convergence against all compact tests, not only
against \(\ker r\).

For an incidence representation, suppose a pushforward is defined and
\[
 r=\psi_*\nu,\qquad L_{ab}=\psi_*(\gamma_{ab}\nu).
\]
For example, \(\gamma_{ab}\) may be smooth and \(\psi\) proper on the
relevant supports. The correct condition is
\begin{equation}
 L_{ab}=cr
 \quad\Longleftrightarrow\quad
 \psi_*((\gamma_{ab}-c)\nu)=0.
 \label{eq:incidence-pushforward}
\end{equation}
Nonconstancy of \(\gamma_{ab}\) by itself is not an obstruction unless
the needed injectivity or separation property of the pushforward has
also been proved.

\subsection{Presentation defects before regulator removal}

Fix \(\mu>0\), \(\varepsilon>0\), and
\[
 \ell_\mu(\varepsilon)=\log\!\left(\frac1{\mu\varepsilon}\right).
\]
Let \(P:\cD'(X)\to\cD'(Y)\) be continuous for distributional convergence.

\begin{proposition}[Presentation-defect identity]
\label{prop:presentation}
Suppose the two complete finite parts exist:
\begin{align}
 u_{\mathrm{fin}}
 &=\lim_{\varepsilon\downarrow0}
   [u_\varepsilon-\ell_\mu(\varepsilon)r],
 \label{eq:finite-source}\\
 u_{\mathrm{fin}}^{(P)}
 &=\lim_{\varepsilon\downarrow0}
   [u_\varepsilon^{(P)}
     -\ell_\mu(\varepsilon)r^{(P)}].
 \label{eq:finite-target}
\end{align}
Then the presentation defect is
\begin{align}
 \mathcal A_P
 &:=
 u_{\mathrm{fin}}^{(P)}-Pu_{\mathrm{fin}}\notag\\
 &=
 \lim_{\varepsilon\downarrow0}
 \left[
 u_\varepsilon^{(P)}-Pu_\varepsilon
 -\ell_\mu(\varepsilon)(r^{(P)}-Pr)
 \right].
 \label{eq:presentation-defect}
\end{align}
\end{proposition}

\begin{proof}
Continuity of \(P\) gives
\[
 P[u_\varepsilon-\ell_\mu(\varepsilon)r]
 \longrightarrow Pu_{\mathrm{fin}}.
\]
Subtract this convergence from \cref{eq:finite-target} and collect
terms.
\end{proof}

The identity separates:
\begin{enumerate}
\item the finite-cutoff comparison
\(u_\varepsilon^{(P)}-Pu_\varepsilon\);
\item the residue comparison \(r^{(P)}-Pr\);
\item the finite normalization left after both subtractions.
\end{enumerate}
If both finite-cutoff and residue identities hold literally, then
\(\mathcal A_P=0\). Otherwise their failures must be computed in the
specified presentation. A formal unregulated identity is not a substitute.

Transported differentiation is an immediate application: a fixed
differential operator is continuous on distributions, so it transports
an established finite part together with its residue.
Cubic parenthesization is different. Intermediate multiplication of
singular distributions is not automatically a defined continuous
linear map \(P\).

Tests on a product manifold must not be restricted to a selected
family \(f\otimes g\otimes f'\) without a proved extension argument.
Convergence on simple product tests alone is not the asserted
distributional convergence on all compact joint tests. The correct
organization of a future graph estimate may separate regular regions,
transition regions, matching, and uniform control near an incidence
intersection such as \(\mathcal R\cap\Delta_{13}\). That organization
does not supply those estimates.

\section{One certified four-face slicing exchange}
\label{sec:geometry}

We now give one complete geometric-rule certificate. The ambient
spacetime is fixed; only its prescribed slicing and the associated
cell incidence vary. No matter channel, readout, or coupling update
fires at the event.

\subsection{Fixed cone and oriented holonomy}

Use Minkowski coordinates \(z=(T,x,y)\) and
\[
 \eta=\operatorname{diag}(-1,1,1).
\]
For a prescribed deficit \(0<\mu_{\mathrm c}<2\pi\), set
\(\alpha=1-\mu_{\mathrm c}/(2\pi)\). The punctured cone metric is
\begin{equation}
 ds^2=-dT^2+dr^2+\alpha^2r^2d\theta^2,
 \qquad r>0,\quad\theta\ \mathrm{mod}\ 2\pi.
 \label{eq:cone}
\end{equation}
Its completion has a timelike conical axis. Its spinless point-particle
interpretation is standard~\cite{DeserJackiwHooft1984}; here the metric
is prescribed rather than derived.

On the universal cover, let
\[
 D(T,r,\widetilde\theta)
 =(T,r\cos(\alpha\widetilde\theta),
      r\sin(\alpha\widetilde\theta)),
 \qquad
 \mathsf T(\widetilde\theta)=\widetilde\theta+2\pi.
\]
With
\[
 R(\mu_{\mathrm c})=
 \begin{pmatrix}
 1&0&0\\
 0&\cos\mu_{\mathrm c}&-\sin\mu_{\mathrm c}\\
 0&\sin\mu_{\mathrm c}&\cos\mu_{\mathrm c}
 \end{pmatrix},
\]
we have
\begin{equation}
 D\circ\mathsf T^{-1}=R(\mu_{\mathrm c})D.
 \label{eq:development}
\end{equation}

Choose angular chart lifts on
\[
 U_1=(-\epsilon,\pi+\epsilon),\qquad
 U_2=(\pi-\epsilon,2\pi+\epsilon),
 \qquad 0<\epsilon<\pi/4.
\]
Refine if needed to injective developing charts. The two overlap
components have different transitions:
\[
 \Lambda_{2\leftarrow1}^{\pi}=I,\qquad
 \Lambda_{1\leftarrow2}^{0}=R(\mu_{\mathrm c}).
\]
The positively oriented particle loop has ordered affine word
\begin{equation}
 J=(R(\mu_{\mathrm c}),0)(I,0)=(R(\mu_{\mathrm c}),0).
 \label{eq:particle-word}
\end{equation}
The overlap-component index is essential. No flat chart is asserted
to contain the axis.

These matrices act on spacetime tangent vectors. More generally, if
\[
 D\circ\mathsf T_\gamma^{-1}
 =h_\gamma D,\qquad h_\gamma(z)=L_\gamma z+b_\gamma,
\]
a lifted parallel vector has constant developed components.
At the endpoint, differentiation of this identity gives
\[
 dD_{\widetilde x}\,d\mathsf T_\gamma^{-1}(v')
 =L_\gamma dD_{\mathsf T_\gamma\widetilde x}(v')
 =L_\gamma dD_{\widetilde x}(v).
\]
Thus the tangent holonomy in the starting developed frame is
\(L_\gamma\). This is gluing data for the spacetime, not an unrelated
internal fibre; compare the polygon-model formulations in
\cite{tHooft1992,Kadar2005}.

For the certificate fix
\[
 \mu_{\mathrm c}=\pi/2.
\]
Its value is independent of the slicing slopes introduced next.
The future unit holonomy axis is \(e_0=(1,0,0)\).
For a marked spacelike face with future unit normal \(n\), define
\[
 \chi(p,n)=\arccosh(-\eta(p,n)).
\]
Proper future-preserving chart changes transform both vectors and
conjugate the holonomy, preserving this quantity. It is a
particle--slice observable, not a particle invariant alone.

\subsection{Four moving affine faces}

Choose
\begin{equation}
 p=\frac35,\quad q=\frac5{13},\quad
 a=\sqrt{1-p^2}=\frac45,\quad
 b=\sqrt{1-q^2}=\frac{12}{13},\quad
 \delta=b-a=\frac8{65}.
 \label{eq:constants}
\end{equation}
For \(-1\le s\le1\), let
\begin{align}
 f_E&=px+as,& f_W&=-px+as,\notag\\
 f_N&=qy+bs,& f_S&=-qy+bs,
 \label{eq:planes}\\
 F_s(x,y)&=\max\{f_E,f_W,f_N,f_S\}.&&
 \label{eq:maximum}
\end{align}
A core face is the region where the corresponding affine function
attains the maximum, with its interior given by strict dominance.

\begin{proposition}[Spacelike ordering and central exchange]
\label{prop:core}
The graph \(T=F_s(x,y)\) is continuous, piecewise affine, and
\(p\)-Lipschitz. Every face is spacelike, and for \(s_2>s_1\),
\[
 F_{s_2}(x,y)-F_{s_1}(x,y)\ge a(s_2-s_1)>0.
\]
For \(s<0\), the central adjacency is \(EW\), with edge
\begin{equation}
 x=0,\quad
 \abs y\le\frac{\delta(-s)}q,\quad
 T=as,\qquad
 \ell_-(s)=\frac{2\delta}{q}(-s)=\frac{16}{25}(-s).
 \label{eq:incoming}
\end{equation}
For \(s>0\), the central adjacency is \(NS\), with edge
\begin{equation}
 y=0,\quad
 \abs x\le\frac{\delta s}p,\quad
 T=bs,\qquad
 \ell_+(s)=\frac{2\delta}{p}s=\frac{16}{39}s.
 \label{eq:outgoing}
\end{equation}
The central incidence changes as
\[
 \{EWN,EWS\}\longrightarrow\{EWNS\}
 \longrightarrow\{NSE,NSW\}.
\]
\end{proposition}

\begin{proof}
Each affine function has spatial gradient of norm at most \(p<1\).
Their maximum has the same Lipschitz bound. On a face of slope \(v\),
a nonzero tangent vector has squared norm
\[
 \eta((v\cdot u,u),(v\cdot u,u))
 =\abs u^2-(v\cdot u)^2>0.
\]
Every affine function increases in \(s\) at rate at least \(a\), proving
the ordering bound.

For \(EW\), equality of the two affine functions forces \(x=0\).
Their common value \(as\) dominates the other two precisely when
\(q\abs y\le-\delta s\). This gives \cref{eq:incoming} and its triple
endpoints. The endpoint difference has constant \(T,x\), so its
spacelike length is the displayed \(y\)-distance.

For \(NS\), equality forces \(y=0\), and dominance is
\(p\abs x\le\delta s\), giving \cref{eq:outgoing}.

At \(s=0\),
\(F_0=\max\{p\abs x,q\abs y\}\). All four faces meet only at the origin,
and each contains an open strict-dominance cone. Thus the central
edge collapses, but the two-dimensional patch does not.
\end{proof}

The exchange is transverse:
\begin{equation}
 \dot\ell_-(0^-)=-\frac{16}{25}<0,\qquad
 \dot\ell_+(0^+)=\frac{16}{39}>0.
 \label{eq:transverse}
\end{equation}
The mixed adjacencies \(EN,ES,WN,WS\) persist. In their respective
quadrants they satisfy
\[
 p\abs x-q\abs y=\delta s.
\]
The only remaining core pairs are \(EW\) and \(NS\), already determined.
The exchange is forced by this prescribed maximum rule, not by a
uniqueness theorem for all slicings of the spacetime.

\subsection{Adapted charts and exact products}

For each face, use the following Lorentz transformation:
\begin{equation}
\begin{aligned}
 C_E&=\begin{pmatrix}
 5/4&-3/4&0\\-3/4&5/4&0\\0&0&1
 \end{pmatrix},
&
 C_W&=\begin{pmatrix}
 5/4&3/4&0\\3/4&5/4&0\\0&0&1
 \end{pmatrix},\\[3pt]
 C_N&=\begin{pmatrix}
 13/12&0&-5/12\\0&1&0\\-5/12&0&13/12
 \end{pmatrix},
&
 C_S&=\begin{pmatrix}
 13/12&0&5/12\\0&1&0\\5/12&0&13/12
 \end{pmatrix}.
\end{aligned}
\label{eq:charts}
\end{equation}
Direct multiplication gives \(C_i^T\eta C_i=\eta\), determinant \(1\),
and future time orientation. For the slope \(v_i\) of face \(i\),
\[
 (C_i z)^0
 =\frac{T-v_i\cdot(x,y)}{\sqrt{1-\abs{v_i}^2}}
 =s
\]
on that face.

Define \(\Lambda_{i\leftarrow j}=C_iC_j^{-1}\).
The opposite pairs are
\begin{equation}
 \Lambda_{E\leftarrow W}
 =\begin{pmatrix}
 17/8&-15/8&0\\-15/8&17/8&0\\0&0&1
 \end{pmatrix},
 \qquad
 \Lambda_{N\leftarrow S}
 =\begin{pmatrix}
 97/72&0&-65/72\\0&1&0\\-65/72&0&97/72
 \end{pmatrix}.
 \label{eq:opposite}
\end{equation}
Writing \(X_+=E,X_-=W,Y_+=N,Y_-=S\), the mixed pairs are
\begin{equation}
 \Lambda_{X_\epsilon\leftarrow Y_\nu}
 =
 \begin{pmatrix}
 65/48&-\epsilon\,3/4&\nu\,25/48\\
 -\epsilon\,13/16&5/4&-\epsilon\nu\,5/16\\
 \nu\,5/12&0&13/12
 \end{pmatrix}.
 \label{eq:mixed}
\end{equation}
Reverse transitions are inverses. Core translations are zero.

\begin{proposition}[Local ordered products]
\label{prop:products}
Every closed core transition word is the identity. In particular,
the incoming words
\[
 \Lambda_{E\leftarrow W}
 \Lambda_{W\leftarrow N}
 \Lambda_{N\leftarrow E},
 \qquad
 \Lambda_{E\leftarrow W}
 \Lambda_{W\leftarrow S}
 \Lambda_{S\leftarrow E}
\]
and the outgoing words
\[
 \Lambda_{N\leftarrow S}
 \Lambda_{S\leftarrow E}
 \Lambda_{E\leftarrow N},
 \qquad
 \Lambda_{N\leftarrow S}
 \Lambda_{S\leftarrow W}
 \Lambda_{W\leftarrow N}
\]
equal \(I\). The four-face event word also equals \(I\).
\end{proposition}

\begin{proof}
Every word is a product
\[
 (C_{i_0}C_{i_1}^{-1})
 (C_{i_1}C_{i_2}^{-1})\cdots
 (C_{i_{m-1}}C_{i_0}^{-1}),
\]
which telescopes.
\end{proof}

The nonidentity adapted-frame transitions are pure changes of frame
in one flat development. They do not create spacetime curvature.

\subsection{A compact collar}

Set \(H=2\), and cap the graph by
\begin{equation}
 g_s(x,y)=\min\{F_s(x,y),H+s\}.
 \label{eq:cap}
\end{equation}
Its non-resting region is
\[
 \mathscr R_s=[-X_s,X_s]\times[-Y_s,Y_s],
 \qquad
 X_s=\frac{10+s}{3},\quad Y_s=\frac{26+s}{5}.
\]
Indeed, \(F_s\le H+s\) is equivalent to the two displayed coordinate
bounds. Fix the outer rectangle
\begin{equation}
 K=[-R_x,R_x]\times[-R_y,R_y],
 \qquad R_x=\frac{14}{3},\quad R_y=6.
 \label{eq:outer-rectangle}
\end{equation}
Its collar margins are
\begin{equation}
 R_x-X_s=\frac{4-s}{3}>0,\qquad
 R_y-Y_s=\frac{4-s}{5}>0.
 \label{eq:margins}
\end{equation}

\begin{proposition}[Spacelike cap and fixed boundary]
\label{prop:cap}
The capped graph is continuous, piecewise flat, \(p\)-Lipschitz, and
strictly time-ordered with lower increment \(a(s_2-s_1)\).
It equals \(T=H+s\) on a neighbourhood of \(\partial K\).
The central edge laws in \cref{eq:incoming,eq:outgoing} are unchanged.
\end{proposition}

\begin{proof}
Minimum and maximum preserve a common Lipschitz bound.
Both arguments of the minimum increase by at least
\(a(s_2-s_1)\), so the minimum does also.
The strict margins in \cref{eq:margins} give the resting boundary
neighbourhood.

On the incoming central edge, the cap-height margin is
\(2+s/5\ge9/5\); on the outgoing edge it is
\(2+s/13\ge25/13\). Hence neither central edge is clipped.
\end{proof}

There are four core faces inside \(\mathscr R_s\).
Partition \(K\setminus\mathscr R_s\) into four resting collar polygons.
The north-east polygon has counterclockwise spatial vertices
\begin{equation}
 (X_s,0),\ (R_x,0),\ (R_x,R_y),\
 (0,R_y),\ (0,Y_s),\ (X_s,Y_s).
 \label{eq:hexagon}
\end{equation}
The other three are reflections, ordered counterclockwise.
Every collar vertex has \(T=H+s\).
The inner and outer side midpoints divide each rectangular boundary
into eight segments. These coplanar subdivisions are part of the
declared cell interface.

For affine pairs, use
\[
 (L,b)z=Lz+b,\qquad
 (L,b)(L',b')=(LL',Lb'+b).
\]
The collar chart is
\[
 A_0(z)=z-He_0=(T-H,x,y).
\]
The core chart is \(A_i=(C_i,0)\), so a core-to-collar transition is
\begin{equation}
 A_0A_i^{-1}=(C_i^{-1},-He_0).
 \label{eq:collar-transition}
\end{equation}
Collar-to-collar transitions are identities.

\begin{proposition}[Closure and collar holonomy]
\label{prop:collar}
Every collar or mixed core--collar vertex word is the affine identity.
All cells close and all matching edges have identical induced lengths.
\end{proposition}

\begin{proof}
Every transition is \(A_iA_j^{-1}\) for globally defined affine
isometries of the patch, so closed words telescope in the affine group.

Core cells are intersections of their affine dominance half-spaces
with \(\mathscr R_s\). The collar polygons are the explicitly listed
simple polygons and their reflections. Adjacent cells share the same
developed endpoints and segment. Affine isometries preserve its induced
length.
\end{proof}

\subsection{Complete vertices, lengths, and isolation}

The central endpoints are those of
\cref{eq:incoming,eq:outgoing}. All other vertices have \(T=H+s\)
and spatial coordinates in the following list:
\[
 (\epsilon X_s,\nu Y_s),\quad
 (\epsilon X_s,0),\quad(0,\nu Y_s),\quad
 (\epsilon R_x,\nu R_y),\quad
 (\epsilon R_x,0),\quad(0,\nu R_y),
 \qquad \epsilon,\nu\in\{-1,1\}.
\]
The four mixed core edges join their appropriate central triple
junctions to the corresponding inner corners. Their common length is
\begin{equation}
 L_{\mathrm{arm}}(s)=
 \begin{cases}
 \displaystyle\frac{10+s}{75}\sqrt{1921},&-1\le s\le0,\\[4pt]
 \displaystyle\frac{26+s}{195}\sqrt{1921},&0\le s\le1.
 \end{cases}
 \label{eq:arm-length}
\end{equation}
The two expressions agree at \(s=0\).

\begin{table}[htbp]
\centering
\small
\begin{tabularx}{\textwidth}{@{}lclX@{}}
\toprule
Edge class & Number & Exact length & Incidence\\
\midrule
Central & \(1\) & \(\ell_-\) or \(\ell_+\) & \(EW\) or \(NS\)\\
Mixed core arms & \(4\) & \(L_{\mathrm{arm}}(s)\) & \(EN,ES,WN,WS\)\\
Horizontal cap segments & \(4\) & \(X_s\) & Inner horizontal half-sides\\
Vertical cap segments & \(4\) & \(Y_s\) & Inner vertical half-sides\\
Horizontal collar joins & \(2\) & \((4-s)/3\) & Inner to outer side midpoint\\
Vertical collar joins & \(2\) & \((4-s)/5\) & Inner to outer side midpoint\\
Outer horizontal segments & \(4\) & \(14/3\) & Fixed boundary\\
Outer vertical segments & \(4\) & \(6\) & Fixed boundary\\
\bottomrule
\end{tabularx}
\caption{The 25 actual patch edges on either regular epoch.
A representation retaining both central slots has 26 length reports.
At the event the central edge is replaced geometrically by one junction.}
\label{tab:edges}
\end{table}

\begin{proposition}[All noncentral edges remain positive]
\label{prop:edge-certificate}
The list in \cref{tab:edges} is complete. Every actual noncentral
edge has length greater than \(1/2\) throughout \([-1,1]\).
All eight faces remain two-dimensional, and the only incidence event
in the patch is the central exchange.
\end{proposition}

\begin{proof}
For the incoming north-east arm, the displacement from the central
endpoint to the inner corner is
\[
 X_s(p,1,p/q).
\]
Its squared Minkowski norm is
\[
 X_s^2\bigl(1+(p/q)^2-p^2\bigr)
 =X_s^2\,\frac{1921}{625}.
\]
For the outgoing arm the displacement is
\(Y_s(q,q/p,1)\), with norm \(Y_s\sqrt{1921}/39\).
These give \cref{eq:arm-length}; symmetry gives the other arms.

The remaining edges lie in the resting plane and have the Euclidean
lengths listed in the table. The smallest such length over the window
is \(3/5>1/2\). The arms have length at least
\(3\sqrt{1921}/25>1/2\).

The central endpoints remain strictly inside \(\mathscr R_s\).
Each core face contains a strict-dominance neighbourhood just inside
its inner side midpoint. Each collar polygon contains a nonempty open
region because the margins are positive. The mixed arms and all
peripheral incidences therefore persist. The central laws determine
the only exchange.
\end{proof}

No general assertion that affine endpoint motion always gives affine
length is used. Here the arm directions are fixed on each epoch, and
all other lengths follow directly from their displayed segments.

\subsection{Attachment to one frozen conical exterior}

Use the cone wedge of angle \(3\pi/2\), and place the patch in its first
quadrant by
\[
 (X,Y)=(20+x,20+y).
\]
The translated \(K\) lies strictly inside that quadrant, away from the
axis and seam. On the rest of the cone use \(T=H+s\).
By \cref{prop:cap}, the pieces agree on an open boundary neighbourhood.

Here is a finite exterior cellulation with exactly the required patch
boundary subdivision. Put
\[
 A=\{20-R_x,20,20+R_x\},\qquad
 B=\{20-R_y,20,20+R_y\}.
\]
In the first quadrant cut by \(X\in A\) and \(Y\in B\).
In the second quadrant, using coordinates \((-X,Y)\), use cuts
\((-X)\in A\), \(Y\in B\).
In the third quadrant, using \((-X,-Y)\), use cuts
\((-X)\in A\), \((-Y)\in A\).
The positive \(Y\)-axis subdivisions match through \(B\), the negative
\(X\)-axis subdivisions match through \(A\), and the two wedge-seam
subdivisions both use \(A\). Keep the cells outside the translated
\(K\) and replace its four grid rectangles by the eight patch cells.
This gives finitely many cell labels; unbounded exterior cells are
allowed.

Use shifted cone developments
\[
 D_H=(T-H,r\cos(\alpha\widetilde\theta),
             r\sin(\alpha\widetilde\theta)).
\]
Time translation commutes with the seam rotation, so the particle word
is unchanged. From a collar chart to the first-quadrant cone development,
the transition is the fixed spatial translation
\[
 (I,(0,20,20)).
\]

\begin{proposition}[Frozen exterior and particle data]
\label{prop:exterior}
Both event germs use the same boundary labels, eight fixed spatial
boundary segments, constant boundary lengths, resting boundary time
law \(T-H=s\), and fixed outgoing transitions.
The exterior cellulation, cone metric, and resting particle
neighbourhood are not rewritten.

On both sides of the event,
\begin{equation}
 \mu_{\mathrm c}=\pi/2,\qquad
 p_{\mathrm{particle}}=e_0,\qquad
 \chi(e_0,e_0)=0,\qquad
 J=(R(\pi/2),0).
 \label{eq:frozen-observables}
\end{equation}
\end{proposition}

\begin{proof}
The construction matches the same resting plane near the same fixed
boundary. The exterior grid is independent of \(s\) in spatial
coordinates, and its affine time law is prescribed once.

A particle loop can be chosen in the unchanged neighbourhood of the
axis and has exactly the word \cref{eq:particle-word}. A path that
enters the patch acquires a telescoping affine word determined by its
endpoint charts. Changing internal incidence cannot change that word.
The particle's slice normal remains \(e_0\), proving the remaining
claims.
\end{proof}

``Frozen'' here means that the declared exterior labels and affine-time
laws are not changed by the event. It does not mean that the time
coordinate fails to advance with \(s\).

\subsection{Transition stratum, exact guard, and transport}

Let \(\cQ_-\) and \(\cQ_+\) be the regular families for
\(-1\le s<0\) and \(0<s\le1\). Let \(\cQ_0^-\) and \(\cQ_0^+\)
be their certified event germs.

Both germs realize the same capped surface at \(s=0\), with one
four-face junction. The incoming germ retains an \(EW\) edge slot
\(c_-\) and its coincident endpoint labels \(j_N,j_S\).
The outgoing germ retains an \(NS\) slot \(c_+\) and its coincident
endpoint labels \(j_E,j_W\). These are event metadata, not positive
length edges.

Include a mode bit and a nonnegative permission label.
The semantic event rule
\[
 G_{\mathrm{ex}}:\cQ_0^-\longrightarrow\cQ_0^+
\]
has guard
\begin{equation}
 \mode=0,\qquad \perm\ge1,\qquad s=0.
 \label{eq:guard}
\end{equation}
It changes mode to \(1\), decreases permission by \(1\), and selects
outgoing incidence. It does not advance \(s\). The no-other-collapse
certificate is \cref{prop:edge-certificate}, not a numerical test.

No active physical matter is placed inside \(K\) in this example.
Inaccessible spectator names may nevertheless have abstract cell labels
there. Their transport is the fixed map
\begin{equation}
 \tau(c_-)=\tau(j_N)=\tau(j_S)=c_+,
 \label{eq:transport}
\end{equation}
with persistent face, edge, vertex, collar, boundary, and exterior
labels unchanged. This map factors through the event junction.
It does not inspect states or infer developed positions.

\begin{proposition}[Certified geometric rule]
\label{prop:geometric-rule}
The event is a certified \(G\)-rule. Its geometric guard, support,
permission update, and original-name transport are spectator-blind
under separately reserved spectator location and history storage.
Thus \cref{ass:blind} holds at the semantic event boundaries.
\end{proposition}

\begin{proof}
The one-sided families, common event realization, all local products,
positive peripheral lengths, and fixed boundary data have been proved
in \cref{prop:core,prop:products,prop:cap,prop:collar,prop:edge-certificate,prop:exterior}.
They certify the outgoing germ and its continuation.

The guard uses only the exact event parameter, mode, permission, and
the fixed family certificate. The transport is a fixed function of
cell labels. None of these operations tests matter state or name
cardinality. Separate reservation makes additional spectator storage
irrelevant to the geometric capacity check and leaves original-name
transport unchanged.
\end{proof}

\subsection{Finite code and reversible implementation}
\label{sec:event-code}

\paragraph{Exact timing versus numerical reports.}
If an incoming length is \(v(-s)\), floor quantization at step \(h\)
reports zero throughout \(0\le -s<h/v\). A zero-report guard can
therefore fire prematurely.

Fix an integer \(d\ge0\) and store the exact parameter
\begin{equation}
 s=j2^{-d},\qquad -2^d\le j\le2^d,
 \label{eq:clock}
\end{equation}
in a signed \((d+2)\)-bit word. The event tests \(j=0\).

For report precision \(b_{\mathrm r}\ge1\), store each length in
an unsigned \((b_{\mathrm r}+4)\)-bit word:
\begin{equation}
 \widehat L
 =2^{-b_{\mathrm r}}\lfloor2^{b_{\mathrm r}}L\rfloor,
 \qquad
 0\le L-\widehat L<2^{-b_{\mathrm r}}.
 \label{eq:reports}
\end{equation}
Every length is less than \(16\), so four integer bits suffice.
Both central slots are retained; the inactive one is reported as zero.
At the event both are zero.

The exact decoder reconstructs geometry from the fixed family,
the mode, and exact \(s\), not from independent numerical edge labels.
Reports must agree with \cref{eq:reports} on its domain.
The irrational arm reports can be precomputed exactly on the finite
grid by comparisons of squared rational numbers.

\paragraph{Fixed chart data.}
The code includes identifiers for the charts in \cref{eq:charts},
the collar chart, the prescribed transitions, the fixed exterior
translation, and \(R(\pi/2)\). All entries are rational.
Only these listed data and products are needed; no finite closure
under arbitrary Lorentz multiplication is asserted.

\paragraph{Location and history words.}
Fix a finite named matter interface of size \(n\).
Let \(\ell\) bits accommodate the union of pre- and post-event cell
labels. Encode each location in an \(\ell\)-bit word and provide a
second \(\ell\)-bit history word, initially zero.
Invalid binary labels lie outside the decoder domain.

Let \(x\) be the tuple of valid incoming location words.
On the forward event domain, prescribe
\begin{equation}
 \begin{split}
 &(0,k,j=0,\mathrm{reports},x,0_{\mathrm{hist}})\\
 &\hspace{8mm}\longmapsto
 (1,k-1,j=0,\mathrm{reports},\tau(x),x_{\mathrm{hist}}),
 \qquad 1\le k\le2^r-1,
 \end{split}
 \label{eq:event-code-map}
\end{equation}
where the permission word has \(r\ge1\) bits.
All fixed family identifiers are retained.

\begin{lemma}[Injectivity with retained history]
\label{lem:event-injection}
The map \cref{eq:event-code-map} is injective, and its input and output
sets are disjoint.
\end{lemma}

\begin{proof}
The output history determines \(x\), and the output permission determines
\(k\). All other words are retained. Input and output modes differ.
\end{proof}

\begin{proposition}[Decoder-correct finite permutation]
\label{prop:decoder}
Exchange each valid input in \cref{eq:event-code-map} with its specified
output, and fix every other basis string. This defines a permutation
\(U_{\mathrm{ex}}\) of the complete finite code space. On the declared
forward domain,
\begin{equation}
 \Dec(U_{\mathrm{ex}}c)
 =G_{\mathrm{ex}}(\Dec(c)).
 \label{eq:decoder}
\end{equation}
The decoded equality includes geometry, permission, event germ, and
original-name locations. History remains an output of the implementation.
\end{proposition}

\begin{proof}
By \cref{lem:event-injection}, all exchanged pairs are disjoint.
Their exchanges and the identity on the remaining strings form a
permutation.

At \(j=0\), both central lengths are zero; the two arm formulas agree;
and every peripheral formula is identical on the two germs. Hence
the unchanged report words remain valid.
The mode selects outgoing incidence, permission decreases, all fixed
chart data remain unchanged, and the location tuple becomes \(\tau(x)\).
These are precisely the semantic event outputs.

Ignoring history in the geometric decoder does not discard it from
the circuit interface.
\end{proof}

\begin{proposition}[Exact finite reversible circuit]
\label{prop:circuit}
The permutation \(U_{\mathrm{ex}}\) has an exact finite
NOT--CNOT--Toffoli realization with clean work bits returned to zero.
Code, location, and history wires are returned. The matter density
operator is acted on by the identity.
\end{proposition}

\begin{proof}
Every finite permutation is a product of transpositions.
For two \(M\)-bit strings, choose a Gray path
\(v_0,\ldots,v_D\) between them, changing one differing bit at a time.
If \(s_j=(v_j\,v_{j+1})\), the product
\[
 s_0s_1\cdots s_{D-2}s_{D-1}s_{D-2}\cdots s_1s_0
\]
exchanges the endpoints and fixes the intermediate strings.

An adjacent transposition is a target NOT controlled on a specified
pattern of the other bits. Negative controls use surrounding NOT gates.
A multiple-control NOT is implemented by computing the conjunction
of its controls into clean work bits with Toffoli gates, flipping the
target, and reversing the conjunction computation. Zero, one, or two
controls use NOT, CNOT, or Toffoli directly. At most \(M\) clean work
bits suffice.

Apply this construction to a transposition decomposition of
\(U_{\mathrm{ex}}\). Each gate consumes and returns its interface.
The work is uncomputed, but the history required by
\cref{eq:event-code-map} is deliberately retained.
Tensor the circuit with the identity on matter.
\end{proof}

Excluding fixed table storage, a sufficient number of data bits is
\begin{equation}
 M=26(b_{\mathrm r}+4)+1+r+(d+2)+2n\ell.
 \label{eq:storage}
\end{equation}
The generic synthesis uses at most \(2M\) bits including clean work.
This is an existence bound, not an efficiency claim.

Valid location words and initially zero history are representation
preconditions, not geometric guards that inspect spectator state.
Additional spectators receive separately reserved location/history
storage. The semantic theorem compares decoded event endpoints;
it does not assert identical low-level gate lists for different
interface sizes. Intermediate compiler states need not lie in the
geometric decoder domain and are not exposed to the geometric
controller. No claim about retiming an event through such intermediate
states is made.

The reverse action of \(U_{\mathrm{ex}}\) is a reversible completion,
not an additional forward geometric rule. The forward rule still
requires \cref{eq:guard}. The circuit implements the discrete event,
not continuous affine evolution or arbitrary real geometric data.

\subsection{Certificate theorem and intrinsic slice geometry}

\begin{theorem}[One complete four-face certificate]
\label{thm:certificate}
In the fixed conical spacetime of deficit \(\pi/2\), the capped slicing
\cref{eq:cap} has exactly one central adjacency exchange in
\([-1,1]\). Its certificate contains:
\begin{enumerate}
\item spacelike, time-ordered slices and transverse incoming/outgoing
edge laws;
\item explicit adapted affine charts and exact ordered local products;
\item an admitted event stratum and an exact parameter guard;
\item a compact collar matching one fixed exterior cellulation;
\item fixed abstract location transport on declared cell labels;
\item spectator-blind control under separately reserved spectator
location/history storage;
\item a decoder-correct finite event permutation and exact reversible
circuit with clean work uncomputed and matter unchanged.
\end{enumerate}
The particle data are those in \cref{eq:frozen-observables} throughout.
\end{theorem}

\begin{proof}
Items 1--4 follow from
\cref{prop:core,prop:products,prop:cap,prop:collar,prop:edge-certificate,prop:exterior}.
The germ and transport construction and
\cref{prop:geometric-rule} give items 3, 5, and 6.
Item 7 is \cref{prop:decoder,prop:circuit}.
The particle statement is \cref{prop:exterior}.
\end{proof}

The exchange has the four-region incidence of the type-A exchange
in~\cite{tHooft1993}. Its scope is:
\begin{quote}
Type-A incidence only. Not the transition catalogue. Not a particle hop.
Not a field equation. Collar subdivisions are declared interface,
not a minimal polygon gauge.
\end{quote}

Although the patch has trivial ambient holonomy, its marked slice is
not intrinsically flat at every vertex. The calculation in
\cref{app:angles} gives the event defect
\[
 \kappa_0
 =2\pi-2\arccos(-1121/1921)-2\arccos(671/1921)<0.
\]
Each incoming triple junction and each outgoing triple junction has
defect \(\kappa_0/2\). Thus the exchange redistributes the central
intrinsic angle defect while leaving the spacetime solution unchanged.

The event can be used in \cref{thm:spectators} only with
\cref{ass:blind}. To use \cref{thm:locality}, a computational metric
and a block-diameter certificate must additionally be supplied.
Physical patch size alone is not such a certificate.

\subsection{A non-equivalent collapsed quadrilateral}

Consider
\[
 A=(-1,0),\quad B=(1,0),\quad
 C=(0,1-t),\quad D=(0,t-1).
\]
For \(t\neq1\), the written boundary order \(ABCD\) has crossing
segments \(AB\) and \(CD\). The signed triangle areas are
\[
 \operatorname{Area}_{\mathrm{sgn}}(A,C,D)=t-1,\qquad
 \operatorname{Area}_{\mathrm{sgn}}(B,C,D)=1-t.
\]
The convex hull has area \(2\abs{1-t}\), which vanishes at \(t=1\);
the entire two-dimensional patch collapses to a segment.
The proposed replacement \(AB\) has constant length \(2\), rather
than being born from zero. Relabelling orientations does not force
an exchange of geometric diagonals.

With spacetime coordinate \(T=t\), the \(A,B\) trajectories are timelike
and the \(C,D\) trajectories are null. These causal classifications
are not by themselves a prohibition on polygonal coordinates.
The failures relevant here are the crossing boundary, the collapse
of the whole patch, and the absence of a replacement edge emerging
from zero. This quadrilateral is not the certified exchange.

\section{Limitations and successor problems}
\label{sec:limitations}

\subsection{What the interface does not imply}

The schedule results require a prescribed metric, support structure,
and channel list. The directional obstruction in
\cref{prop:direction} prevents replacing causal-region inclusion by a
scalar comparison. Periodic graphs can have anisotropic velocity
polytopes~\cite{Fritz2013}. Random sprinkling is not an automatic cure:
the obstruction to intrinsically associating a nontrivial finite-valency
graph with a Minkowski sprinkling in a Lorentz-equivariant manner
requires separate attention~\cite{BombelliHensonSorkin2009}.
Likewise, preferred-frame effects under interactions are a separate
question; the radiative mechanism studied in~\cite{Collins2004}
does not follow from, or disappear because of, our channel identities.

The geometric example has no particle in its moving support.
Its internal affine transitions are pure gauge in one flat development.
It therefore does not test an event that changes gluing in a region
with nontrivial spacetime monodromy.

A deterministic nonlinear state-functional coupling is not automatically
a linear classical--quantum evolution. One could instead specify
classical--quantum states
\[
 \sigma=\sum_g\ket g\bra g\otimes\rho_g
\]
and CP maps \(\Psi_{g'|g}\) satisfying
\(\sum_{g'}\Psi_{g'|g}\) trace preserving for each \(g\), with
\[
 \mathfrak F(\sigma)
 =\sum_{g,g'}\ket{g'}\bra{g'}\otimes\Psi_{g'|g}(\rho_g).
\]
Such a model would need its own geometric admissibility conditions.
It is not an automatic rewriting of \cref{eq:coupling}.

Approximate spectator or deferred-readout results also require
continuity estimates for the coupling and stability margins for the
controller. An arbitrarily small change of an input functional can
cross a discontinuous guard. The exact theorems contain no hidden
small-error estimate.

\subsection{Capacity and target compatibility}

Let \(\mathfrak I_K\) be the permitted implementations at capacity \(K\),
with \(\mathfrak I_K\subseteq\mathfrak I_{K'}\) for \(K\le K'\).
For a fixed task error, put
\[
 \varepsilon_K
 =\inf_{\mathcal J\in\mathfrak I_K}\Err(\mathcal J),
 \qquad \inf\varnothing=+\infty.
\]
Then \(\varepsilon_{K'}\le\varepsilon_K\), simply because the infimum
is taken over a larger set. There is no implication that
\(\varepsilon_K\to0\).

Feasibility, approximation of a specified target, and compatibility
of two ideal targets are separate questions. The last depends on
the named observation algebra. The residual obstruction in
\cref{thm:residual-obstruction} cannot be cured merely by allocating
more work bits.

\subsection{Successor problems}

\begin{openproblem}[A transition with nontrivial monodromy]
Construct a certified transition whose support contains a particle
or nontrivial spacetime monodromy. Supply complete affected labels,
boundary matching, ordered affine products, location transport,
control assumptions, and a decoder equation. Approximate event
detection must distinguish geometric error from event-time and
event-resolution error.
\end{openproblem}

\begin{openproblem}[Controlled approximation on a named algebra]
For a specified observation algebra and resource model, construct
nested implementation families with an explicit approximation rate.
Separate target incompatibility from implementation error.
\end{openproblem}

\begin{openproblem}[Directional physical dominance]
For a moving geometric model and a computational substrate, prove
directional inclusion of physical dependency regions in scheduled
regions. A scalar speed bound alone is insufficient.
\end{openproblem}

\begin{openproblem}[Observation after a distributional construction]
After a renormalized composite has been independently constructed
on all compact tests, specify its accessible observable structure
and test descent of its mandatory presentations through the declared
counterterm equivalence. Establish any required products separately;
do not assume multiplication on a quotient of distributions.
\end{openproblem}

These problems are successors, not missing premises of the finite
interface theorems. In particular, the present results do not settle
the interacting-product problem of the separate Wick track.

\section{Conclusion}
\label{sec:conclusion}

A destructive measurement primitive fixes an output signature, not
a geometric observation policy. The latter must declare coupling
observables, control, resource checks, locations, and accessible
residual outputs.

With those declarations in place, arbitrary spectator extensions
preserve geometry and accessible reductions under spectator-blind
control. Restricted coherent deferral reproduces destructive readout
on the common record--matter interface. Record compatibility is exactly
an effect pullback, with an exact unassisted event-norm error.
Residual compatibility is stronger: for sharp readouts and a coarse
L\"uders instrument, its remaining obstruction is intra-outcome
coherence. Continuations preserve observation boundaries precisely
when their adjoints preserve the declared observable spaces.

The four-face certificate realizes one discrete geometric event on
a prescribed slicing of a fixed spacetime, including its boundary,
guard, transport, and finite reversible code. It neither derives
geometry from measurement nor changes the ambient gravitational
solution.

The observation-boundary problem is closed within the stated finite
model. The distributional graph problem and gravitational dynamics
remain separate tasks.

\appendix

\section{Intrinsic angle defect at the exchange}
\label{app:angles}

The angle at a vertex of a spacelike planar face is computed using
the positive definite restriction of \(\eta\) to that face.
At \(s=0\), the two edge rays of the \(E\)-face may be represented by
\[
 u_\pm=(p,1,\pm p/q).
\]
Their squared norms are equal to
\[
 N_E^2=1-p^2+(p/q)^2,
\]
and their inner product is \(1-p^2-(p/q)^2\). Thus
\begin{equation}
 \cos\alpha_E
 =\frac{1-p^2-(p/q)^2}{1-p^2+(p/q)^2}
 =-\frac{1121}{1921}.
 \label{eq:alpha-E}
\end{equation}
Reflection gives the same angle for \(W\).

For the \(N\)-face, use
\[
 v_\pm=(q,\pm q/p,1).
\]
Then
\begin{equation}
 \cos\alpha_N
 =\frac{1-q^2-(q/p)^2}{1-q^2+(q/p)^2}
 =\frac{671}{1921},
 \label{eq:alpha-N}
\end{equation}
and \(S\) has the same angle. Both angles lie in \((0,\pi)\).

The event junction therefore has defect
\begin{equation}
 \kappa_0=2\pi-2\alpha_E-2\alpha_N.
 \label{eq:kappa}
\end{equation}
Since
\[
 \cos\alpha_N=\frac{671}{1921}
 <\frac{1121}{1921}
 =\cos(\pi-\alpha_E),
\]
strict decrease of cosine on \([0,\pi]\) gives
\(\alpha_N>\pi-\alpha_E\). Hence \(\kappa_0<0\).

At the incoming northern triple junction, the central edge points in
direction \(d=(0,0,-1)\). Its angle \(\beta_E\) with the north-east
arm satisfies
\[
 \cos\beta_E
 =-\frac{p/q}{\sqrt{1-p^2+(p/q)^2}}
 =-\sin(\alpha_E/2).
\]
Because \(\beta_E\in(0,\pi)\),
\[
 \beta_E=\frac{\pi+\alpha_E}{2}.
\]
The \(W\)-angle is the same, and the \(N\)-angle is \(\alpha_N\).
The triple-junction defect is consequently
\[
 2\pi-2\beta_E-\alpha_N
 =\pi-\alpha_E-\alpha_N
 =\frac{\kappa_0}{2}.
\]
Reflection gives the same value at the southern incoming junction.

At an outgoing triple junction the roles of \(E\) and \(N\) exchange:
the two equal angles are
\(\beta_N=(\pi+\alpha_N)/2\), and the third is \(\alpha_E\).
Its defect is again \(\kappa_0/2\).
All relevant edge directions are constant within their respective
epochs, so these defects are independent of \(s\) there.

Thus the sum of the two central triple defects equals the single
event defect on both sides. This is a statement about intrinsic
geometry of the marked piecewise-flat slice. It is consistent with
the identity ambient transition products proved in
\cref{prop:products,prop:collar}.

\begin{thebibliography}{99}

\bibitem{Cheung2026Enter}
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\end{thebibliography}

\end{document}