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\hypersetup{
  pdftitle={En05-B: An Isotropic Matched Operational Continuum Limit},
  pdfauthor={Ngo Cheung},
  pdfsubject={An explicit partitioned transport construction,
              Euclidean cone convergence, quantum identity transport,
              matched finite experiments, and local operational completion}
}

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\title{En05-B: An Isotropic Matched Operational Continuum Limit\\
for Partitioned Quantum--Classical Networks}
\author{Ngo Cheung}
\date{October 2026}

\begin{document}
\maketitle

\begin{abstract}
We construct a deterministic family of countable, locally finite
metric substrates with one fixed partition schedule at each scale.
The construction separates transport lanes from junction interfaces.
Every junction is updated in one block, whereas the lane interiors
are updated by disjoint nearest-neighbor pairs. Increasing directional
resolution therefore does not introduce a direction-by-direction
service delay.

Under one physical clock, every dependency edge has Euclidean speed
at most a prescribed \(c>0\). Uniform estimates on the entire
substrate, including intermediate lane vertices, prove convergence
of both the metric past envelopes and the scheduled pasts to the
Euclidean cone. Routing has vanishing excess time uniformly on bounded
journeys and from every starting tick. Classical relays have signalling
gap one, and quantum transport is the logical identity channel,
including arbitrary untouched references.

Every finite acyclic causal circuit of finite-dimensional complete
hybrid instruments has an exact logical realization on sufficiently
fine substrates. Local replacements admit matched realizations with
the same schedule, preparations, routes, and unchanged channels.
A fixed-skeleton operational completion is constructed; evaluation,
finite composition, and matched locality extend continuously to it.
Its compatibility notion retains local interfaces and intervention
slots and is stronger than convergence of one terminal probability
law.

The construction proves the isotropic operational existence statement
for this explicitly declared finite-experiment class. It does not
construct a quantum field theory, a product-preserving spacetime
observable algebra, Lorentz covariance, or gravitational dynamics.
No uniform degree, block-cardinality, interface-dimension, or
computational-efficiency bound is claimed.
\end{abstract}

\section{Introduction, target, and scope}
\label{sec:introduction}

The exact causal-cone theorem of En05-A separates three objects:
scheduled ancestry, its metric envelope, and actual operational
influence \cite{CheungEn05A}. Its continuum conclusions require
additional realization hypotheses. In particular, an upper envelope
does not establish boundary transport, and separately convergent
experiments need not have locally matched realizations.

The immediate construction problem is therefore:
\begin{quote}
Construct a family whose Euclidean upper cone and attainable
transport cone converge to the same cone, while preserving complete
finite experiments and local interventions.
\end{quote}

A fixed finite periodic transport model cannot solve the isotropy
problem by uniform rescaling alone: its asymptotic velocity set is a
polytope \cite{Fritz}. The cardinal instance is the diamond discussed
in En03-A, VII-E, VII-J, and En05-A
\cite{CheungEn03,CheungVIIE,CheungVIIJ,CheungEn05A}.
Here the directional set, the quotient size, and the microscopic
interfaces are allowed to grow.

\paragraph{The scheduling mechanism.}
A junction is not permitted to participate in several blocks at one
tick. Instead, every incident lane has its own endpoint site.
All endpoint sites belonging to a junction, together with its memory
site, form one small-diameter junction block. The interiors of
different lanes are disjoint. Consequently all directions can be
served in parallel during a transport frame.

This construction uses the permission in En05-A to admit arbitrary
finite complete block channels without a uniform bound on block
cardinality or within-block gate depth. It would not establish an
existence theorem with those additional bounds.

\paragraph{What is completed.}
The paper supplies:
\begin{enumerate}[label=(\roman*)]
\item a precise experiment and comparison interface;
\item an increasing-direction geometry with uniform metric estimates;
\item a legal partition schedule and a quantitative routing theorem;
\item direction-uniform classical boundary relays;
\item reference-preserving quantum identity transport;
\item matched realizations of finite adaptive experiments;
\item compositional convergence and a local operational completion.
\end{enumerate}

\paragraph{What is not imported.}
The increasing-stencil and transport proposals mentioned in the
supplied research plan are not theorem premises. The geometry,
schedule, and charging argument used below are specified and proved
here. Companion manuscripts are cited for comparison and documentary
alignment, not substituted for missing construction steps.

\subsection{The principal existence statement}

A \emph{causal finite circuit} and its local operational completion
are defined precisely in Sections~\ref{sec:experiments}
and~\ref{sec:completion}.

\begin{theorem}[Isotropic matched operational existence]
\label{thm:main}
Fix \(c>0\). There are countable substrates
\((\Lat_n,d_n,1)\), injective location maps
\(z_n:\Lat_n\to\RR^2\), physical tick durations \(h_n\to0\),
and state-independent periodic partition schedules such that:
\begin{enumerate}[label=(\roman*)]
\item every block has \(d_n\)-diameter at most one and a physical
spacetime footprint tending to zero;
\item every scheduled dependency path satisfies the exact endpoint
bound
\[
 \norm{z_n(y)-z_n(x)}_2
 \le c(T-s)h_n;
\]
\item the metric past envelopes and scheduled pasts converge to the
Euclidean past cones, in the uniform local distance-function sense;
\item for every Euclidean unit vector \(v\), every \(u>0\), and every
\(x\in\RR^2\), matched classical relay endpoints converge to
\[
 (0,x),\qquad (u,x+cuv),
\]
and their readout laws have total-variation distance one;
\item transport of every fixed finite-dimensional quantum register
is the identity after logical identification, including every
untouched reference;
\item every causal finite circuit in the declared class has a
realizing family whose logical complete channels equal its circuit
channel exactly, on all sufficiently fine scales;
\item local replacements have matched realizing families with common
schedule, resource preparation, routing, unchanged block channels,
and common readouts;
\item evaluation, admitted finite composition, and the matched-locality
relation extend to the fixed-skeleton operational completion.
\end{enumerate}
The limit operational maximal signalling speed is \(c\).

The completion statement concerns the compatibility class explicitly
defined here. It is not a completeness theorem for arbitrary
growing discrete networks known only through one convergent terminal
law.
\end{theorem}

\begin{proof}
Items (i)--(iii) are proved in
Theorems~\ref{thm:metric}, \ref{thm:routing},
and~\ref{thm:cones}.
Items (iv)--(v) are proved in
Theorems~\ref{thm:identity} and~\ref{thm:relay}.
Items (vi)--(vii) are proved in
Theorems~\ref{thm:realization} and~\ref{thm:matched}.
Item (viii) is proved in
Theorems~\ref{thm:completion} and~\ref{thm:completionmatch}.
The speed conclusion is Corollary~\ref{cor:maxspeed}.
\end{proof}

\section{Complete instruments, locality, and comparison}
\label{sec:interfaces}

\subsection{Finite hybrid interfaces}

A finite local interface has observable algebra
\[
 \Alg=\bigoplus_{\lambda\in L}\mathcal B(\Hil_\lambda),
\]
where \(L\) is finite and every \(\Hil_\lambda\) is finite dimensional.
Its state space is represented by unnormalized positive blocks:
\[
 \omega=\bigoplus_{\lambda}\omega_\lambda,
 \qquad \sum_\lambda\Tr\omega_\lambda=1.
\]
The blocks include classical outcomes, occupation flags, and
controller records. Tensor products give multi-site interfaces.

A \emph{complete instrument} is a finite family of CP maps
\(\{\Phi_k\}_k\) whose sum is trace preserving. Its recorded channel is
\[
 \widehat\Phi(\omega)=\bigoplus_k\Phi_k(\omega).
\]
Failure outcomes are included. Classical discard sums blocks without
conditional normalization. We use the standard finite-dimensional
CP-map framework \cite{Choi,Watrous}.

\begin{assumption}[Declared local control]
\label{ass:control}
Every quantum or classical factor read or written by a block channel
belongs to that block's declared interface. This includes controller
memory, records, random seeds, program counters, and resource
registers. The channel is fixed before the unknown input is supplied.
\end{assumption}

This is the controller-support convention of VII-G and En05-A
\cite{CheungVIIG,CheungEn05A}. A schedule or routing plan chosen from
an unknown state description is not admitted.

For finite region \(I\), write \(R_I\) for restriction to its complete
interface, tracing quantum factors and summing classical blocks
outside \(I\). The reduced state \(R_I\omega\) is its
\emph{snapshot}.

We use
\[
 \ddia(\Phi,\Psi)=\frac12\norm{\Phi-\Psi}_\diamond,
 \qquad
 \dtr(\rho,\sigma)=\frac12\norm{\rho-\sigma}_1.
\]
For finite hybrid interfaces the diamond norm is the corresponding
completely bounded trace norm, allowing arbitrary reference
extensions.

\begin{lemma}[Snapshot, channel, and adjoint comparison]
\label{lem:comparison}
For a complete finite channel \(\Phi\), input region \(I\), and
output region \(Y\), the following are equivalent:
\begin{enumerate}[label=(\roman*)]
\item \(R_Y\Phi=K R_I\) for a complete channel \(K\);
\item equal input snapshots on \(I\) give equal output snapshots on
\(Y\), for every pair of input states;
\item \(\Phi^*(\Alg(Y))\subseteq\Alg(I)\).
\end{enumerate}
The factorization is stable under tensoring with an untouched
reference identity.
\end{lemma}

\begin{proof}
The first assertion implies the second. Conversely choose a fixed
complementary state \(\sigma\), put
\(E_\sigma(\rho_I)=\rho_I\otimes\sigma\), and define
\(K=R_Y\Phi E_\sigma\).
The second assertion gives
\(R_Y\Phi(\rho)=K R_I(\rho)\) for every state.
States linearly span the finite hybrid operator space, so this is
a linear-map identity. Taking adjoints proves the third assertion.
Conversely, the third assertion says that every output expectation
depends only on \(R_I\rho\), which proves the second.

Tensoring the linear-map identity with \(\id_R\) proves the reference
claim. Equality on \(I\) alone does not imply equality jointly with
a reference; the appropriate joint reductions on \(I R\) must agree.
\end{proof}

\subsection{Partitioned channels and scheduled support}

Let \(d\) be an integer-valued graph metric with finite balls.
At tick \(s\), a partition \(\mathcal P_s\) consists of finite blocks
of diameter at most one. The update is
\[
 \Phi_s=\bigotimes_{b\in\mathcal P_s}\Phi_{s,b}.
\]
All outputs remain inside their block. On a countable substrate this
product is understood through its action on finite observables and
compatible finite-region states.

For finite terminal region \(Y\) at tick \(T\), set
\[
 N_T(Y)=Y,\qquad
 N_s(Y)=
 \bigcup\{b\in\mathcal P_s:b\cap N_{s+1}(Y)\ne\varnothing\}.
\]

\begin{lemma}[Exact scheduled locality]
\label{lem:locality}
The terminal reduced channel factors through the initial snapshot on
\(N_s(Y)\), and
\[
 \Phi_{s:T}^*(\Alg(Y))\subseteq\Alg(N_s(Y)),
 \qquad
 N_s(Y)\subseteq\{x:d(x,Y)\le T-s\}.
\]
If two runs have the same schedule, equal initial snapshots on
\(N_s(Y)\), and equal channels on the blocks entering this backward
recursion, their terminal snapshots agree.

All statements are reference preserving when the corresponding
joint initial reductions agree.
\end{lemma}

\begin{proof}
An output observable on \(Y\) is the identity on blocks disjoint from
\(Y\). Their channel adjoints are unital, so they contribute only
identities. The remaining adjoints act on the union of blocks meeting
\(Y\). Iterating proves the support inclusion and the dependence only
on the indicated channels. Each backward layer adds distance at most
one. Lemma~\ref{lem:comparison} gives factorization and the reference
statement.
\end{proof}

\begin{corollary}[Ancestry upper-bounds influence]
\label{cor:influence}
An unconditioned trace-preserving local intervention outside the
scheduled ancestors of \(Y\) cannot change its output law.
Thus
\[
 \Inf^-(Y)\subseteq\Sch^-(Y).
\]
Equality need not hold.
\end{corollary}

\begin{proof}
Such an intervention preserves the ancestors' reduced state, also
jointly with an untouched reference. Apply Lemma~\ref{lem:locality}.
For strict inclusion, a channel may have a declared input port but
return a constant classical answer independently of that input.
\end{proof}

These proofs reproduce the comparison needed from En03-A, VII-J,
Psy04-B, and En05-A
\cite{CheungEn03,CheungVIIJ,CheungPsy04B,CheungEn05A}.
No absence of entanglement across a cone boundary is assumed.

\section{The substrate: increasing directions and separated ports}
\label{sec:geometry}

\subsection{Parameters and the stencil}

Work in calibrated length units. For integers \(n\ge4\), define
\begin{equation}
 \begin{gathered}
 K_n=n,\qquad N_n=2n+1,\qquad \ell_n=n^{-8},\\
 \delta_n=\frac{\ell_n}{K_n},\qquad
 a_n=\frac{\ell_n}{N_n}\left(1+\frac1{N_n}\right),\qquad
 \xi_n=\frac{\ell_n}{10N_n^2},\qquad h_n=\frac{a_n}{c}.
 \end{gathered}
 \label{eq:parameters}
\end{equation}
Here \(\delta_n\) is the hub mesh, \(\ell_n\) the maximum nominal lane
length, \(a_n\) the physical allowance for one graph edge, and
\(\xi_n\) a small separation allowance.

Let
\[
 P_n=\{p\in\ZZ^2:0<\norm p_2\le K_n\},
 \qquad S_n=\abs{P_n}\le(2K_n+1)^2.
\]
Both orientations occur. The vectors \(e_1,e_2\) occur, ensuring
connectivity.

\begin{lemma}[Directional resolution]
\label{lem:angles}
The normalized directions \(p/\norm p_2\), \(p\in P_n\), have angular
covering error tending to zero. One sufficient bound is
\[
 \theta_n\le\frac{2}{K_n-2}.
\]
\end{lemma}

\begin{proof}
For unit \(v\), round \((K_n-1)v\) to a nearest integer vector \(p\).
The rounding error has Euclidean norm at most \(\sqrt2/2<1\).
Thus \(p\ne0\), \(\norm p_2\le K_n\), and its component in direction
\(v\) is at least \(K_n-2\). The transverse component has norm less
than one. Its angular error is therefore at most
\(\arctan(1/(K_n-2))\), which is bounded by the displayed quantity.
\end{proof}

\subsection{Hubs, lanes, and junction cliques}

For every \(k\in\ZZ^2\), introduce a hub site \(H_k\).
For every \(k\in\ZZ^2\), \(p\in P_n\), and \(i=0,\ldots,N_n\),
introduce a lane site
\[
 L_{k,p,i}.
\]
All these labels denote distinct sites.

The lane indexed by \((k,p)\) has graph edges
\[
 L_{k,p,i}\sim L_{k,p,i+1},
 \qquad 0\le i<N_n.
\]
Its nominal endpoints are the hubs \(H_k\) and \(H_{k+p}\), but its
endpoint sites are not identified with the hubs.

At hub \(H_k\), define its junction set
\begin{equation}
 C_k=
 \{H_k\}\cup
 \{L_{k,p,0}:p\in P_n\}\cup
 \{L_{k-p,p,N_n}:p\in P_n\}.
 \label{eq:junction}
\end{equation}
Add all graph edges between distinct sites of \(C_k\), making it a
clique. Different junction sets are disjoint.

Let \(\Lat_n\) be this graph's vertex set and \(d_n\) its unweighted
shortest-path metric.

\begin{lemma}[Metric substrate]
\label{lem:substrate}
The graph is countable and connected. Its degree is finite and
bounded at each fixed \(n\), and its metric balls are finite.
Every junction clique is finite and has diameter one.
\end{lemma}

\begin{proof}
There are countably many labels. An interior lane site has degree
two. A junction has \(1+2S_n\) sites, so its sites have at most
\(2S_n+1\) neighbors, including a possible lane neighbor.
The presence of \(e_1,e_2\) connects all hubs, and every lane site
connects to its endpoint hubs.

A bounded-degree graph has finite balls by the finite branching
bound on walks of any fixed length. The clique assertion follows
from its definition.
\end{proof}

\subsection{An injective physical location map}

Give each hub the location
\[
 z_n(H_k)=\delta_n k.
\]
The nominal location of \(L_{k,p,i}\) is
\[
 \bar z_n(L_{k,p,i})
 =\delta_n\left(k+\frac{i}{N_n}p\right).
\]

\begin{lemma}[Small periodic separation]
\label{lem:embedding}
There are offsets \(b_{p,i}\in\RR^2\), with
\(\norm{b_{p,i}}_2<\xi_n\), such that
\[
 z_n(L_{k,p,i})=\bar z_n(L_{k,p,i})+b_{p,i}
\]
is injective on all sites, remains equivariant under hub translations,
and satisfies
\begin{equation}
 \norm{z_n(x)-z_n(y)}_2\le a_n
 \quad\text{for every graph edge }x\sim y.
 \label{eq:edgeLipschitz}
\end{equation}
\end{lemma}

\begin{proof}
There are finitely many lane types \((p,i)\).
Choose their offsets successively in the open ball of radius
\(\xi_n\). For a new type, equality modulo the lattice
\(\delta_n\ZZ^2\) with a hub or a previously chosen type excludes only
a countable set of offsets. An open ball is not exhausted by that
set. The resulting map is injective and translation equivariant.

A lane edge has length at most
\[
 \frac{\ell_n}{N_n}+2\xi_n
 \le
 \frac{\ell_n}{N_n}+\frac{\ell_n}{N_n^2}=a_n.
\]
All sites of \(C_k\) are within \(\xi_n\) of \(\delta_n k\), except
the hub itself, which is exactly there. A clique edge has length
at most \(2\xi_n\le a_n\).
\end{proof}

The physical drawing need not be a planar graph: lane crossings do
not create additional graph edges. This is an explicit modelling
choice, not an implicit crossing interaction.

\section{Uniform metric estimates on all sites}
\label{sec:metric}

Abbreviate
\begin{equation}
 \begin{gathered}
 F_n=N_n+1,\qquad
 s_n=F_n a_n,\qquad U_n=\frac{s_n}{c}=F_n h_n,\\
 A_n=\frac{s_n}{\ell_n}
       =\left(1+\frac1{N_n}\right)^2,\qquad
 \lambda_n=\frac{A_n}{1-2/K_n},\\
 B_n=(3+2\lambda_n)s_n+a_n,\qquad
 \rho_n=\frac{\sqrt2}{2}\delta_n.
 \end{gathered}
 \label{eq:constants}
\end{equation}
Then
\[
 \lambda_n\longrightarrow1,\qquad
 B_n=O(\ell_n),\qquad U_n=O(\ell_n/c).
\]

\begin{lemma}[A rounded straight hub route]
\label{lem:hubroute}
For \(k,k'\in\ZZ^2\), put
\(L=\norm{\delta_n(k'-k)}_2\).
There is a hub route using \(m\) stencil steps, with
\[
 m\le\frac{L}{\ell_n(1-2/K_n)}+1.
\]
Every hub on the route is within \(\rho_n\) of the straight segment
joining the two hubs.
\end{lemma}

\begin{proof}
If \(k=k'\), use no steps. Otherwise put \(w=k'-k\) and choose
\[
 m=\left\lceil\frac{\norm w_2}{K_n-2}\right\rceil.
\]
Round each \(j w/m\), \(j=0,\ldots,m\), to a nearest integer vector
\(k_j-k\), retaining the exact endpoints. Consecutive differences
satisfy
\[
 \norm{k_j-k_{j-1}}_2
 \le\frac{\norm w_2}{m}+\sqrt2
 \le K_n-2+\sqrt2<K_n.
\]
Delete zero differences. Every remaining difference belongs to
\(P_n\). The number of steps does not increase. The rounding error
at each hub is at most \(\rho_n\) in physical units.
\end{proof}

\begin{lemma}[Hub metric bound]
\label{lem:hubmetric}
For hubs \(H_k,H_{k'}\),
\[
 \norm{z_n(H_k)-z_n(H_{k'})}_2
 \le a_n d_n(H_k,H_{k'})
 \le
 \lambda_n\norm{z_n(H_k)-z_n(H_{k'})}_2+s_n+a_n.
\]
\end{lemma}

\begin{proof}
The lower bound follows by summing \eqref{eq:edgeLipschitz}.
For a route with \(m\ge1\) lanes, use one clique edge to enter the
first lane, \(N_n\) edges on each lane, one clique edge between
consecutive lanes, and one clique edge to finish at the last hub.
This uses
\[
 mN_n+(m-1)+2=mF_n+1
\]
graph edges. Apply Lemma~\ref{lem:hubroute}. The equal-hub case is
immediate.
\end{proof}

\begin{theorem}[Uniform Euclidean metric approximation]
\label{thm:metric}
For every pair of sites \(x,y\in\Lat_n\),
\begin{equation}
 \norm{z_n(x)-z_n(y)}_2
 \le a_n d_n(x,y)
 \le
 \lambda_n\norm{z_n(x)-z_n(y)}_2+B_n.
 \label{eq:allmetric}
\end{equation}
Consequently, for displacement at most \(R\),
\[
 0\le a_n d_n(x,y)-\norm{z_n(x)-z_n(y)}_2
 \le(\lambda_n-1)R+B_n\longrightarrow0
\]
uniformly over all such sites.
\end{theorem}

\begin{proof}
The lower bound again follows from
\eqref{eq:edgeLipschitz}.
Every site can be joined to one endpoint hub using at most
\(N_n+1=F_n\) edges. Choose such hubs \(H,H'\) for \(x,y\).
Then
\[
 a_n d_n(x,H)\le s_n,\qquad
 a_n d_n(y,H')\le s_n,
\]
and the corresponding physical distances are also at most \(s_n\).
The hub distance is therefore at most
\(\norm{z_n(x)-z_n(y)}_2+2s_n\).
Lemma~\ref{lem:hubmetric} gives
\[
 \begin{split}
 a_n d_n(x,y)
 &\le 2s_n+
 \lambda_n\bigl(\norm{z_n(x)-z_n(y)}_2+2s_n\bigr)
 +s_n+a_n,
 \end{split}
\]
which is \eqref{eq:allmetric}.
\end{proof}

\begin{remark}[Why intermediate sites matter]
Theorem~\ref{thm:metric} is not restricted to hubs.
An intermediate vertex may be physically close to a crossing lane
to which it has no direct graph edge. The detour to hubs costs
\(O(\ell_n)\), which vanishes. This resolves the intermediate-vertex
obligation needed for uniform cone convergence.
\end{remark}

\section{A legal partition schedule under one clock}
\label{sec:schedule}

The schedule is defined for every integer tick. Its phase is
\(s\bmod F_n\).

\begin{definition}[Junction-and-lane schedule]
\label{def:schedule}
At phase zero, the blocks are the junction cliques \(C_k\);
every interior lane site is a singleton.

At phase \(j=1,\ldots,N_n\), every hub is a singleton. On each lane
\((k,p)\), form the pair blocks
\[
 \{L_{k,p,i},L_{k,p,i+1}\},
 \qquad
 0\le i<N_n,\quad i\equiv j-1\pmod2.
\]
All unmatched lane sites are singletons.
\end{definition}

\begin{lemma}[Partition legality and vanishing footprints]
\label{lem:partition}
Every phase is a genuine partition into finite blocks of graph
diameter at most one. Every block's spatial diameter is at most
\(a_n\). In the calibrated spacetime norm
\[
 \norm{(t,x)}_c=\sqrt{c^2t^2+\norm x_2^2},
\]
the input--output footprint of a firing has diameter at most
\(\sqrt2\,a_n\).
\end{lemma}

\begin{proof}
The junction cliques are disjoint and contain all endpoint sites.
At transport phases, the lane chains have disjoint site labels, and
the parity pairs on each chain are disjoint. All remaining sites
are explicitly assigned singleton blocks.

Clique and pair blocks have graph diameter one. Their physical
diameters satisfy Lemma~\ref{lem:embedding}. The tick duration is
\(h_n=a_n/c\), giving the spacetime bound.
\end{proof}

\begin{lemma}[One complete lane per frame]
\label{lem:frame}
A register at the first endpoint of a lane immediately after its
phase-zero firing can advance from index zero to index \(N_n\)
during the following \(N_n\) ticks, moving one graph edge at each
tick. It arrives at the next phase-zero boundary.

The same assertion holds in reverse along a lane, because \(N_n\)
is odd.
\end{lemma}

\begin{proof}
At transport phase \(j\), the pair containing indices \(j-1,j\)
is present. Induction moves the register to index \(j\).
The last transport tick ends at the next phase-zero boundary.

For reverse travel, the pair containing indices
\(N_n-j+1,N_n-j\) is present: its lower index \(N_n-j\)
has parity \(j-1\) when \(N_n\) is odd.
\end{proof}

\begin{theorem}[Routing from every starting tick]
\label{thm:routing}
For any starting tick and hubs at physical separation \(L\),
a logical register can be delivered to the target hub in elapsed
physical time at most
\begin{equation}
 T_n^{\mathrm{hub}}(L)
 \le\frac{\lambda_n L}{c}+2U_n+h_n.
 \label{eq:hubtime}
\end{equation}
For arbitrary sites at physical separation \(L\), one may use
\begin{equation}
 T_n^{\mathrm{all}}(L)
 \le\frac{\lambda_n L}{c}+\omega_n,
 \qquad
 \omega_n=(6+2\lambda_n)U_n+3h_n.
 \label{eq:alltime}
\end{equation}
All waiting, junction transitions, initialization into a lane, and
terminal delivery are included.

In particular, on journeys with \(L\le R\),
\[
 T_n^{\mathrm{all}}(L)
 \le\frac{L}{c}
 +\frac{(\lambda_n-1)R}{c}+\omega_n,
\]
and the excess tends to zero uniformly.
\end{theorem}

\begin{proof}
For hubs, wait less than one frame, hence at most \(U_n\), for a
phase-zero firing. Use the hub route of
Lemma~\ref{lem:hubroute}. At the initial phase-zero firing, move the
register from hub memory into the first outgoing port.
Lemma~\ref{lem:frame} traverses the lane. At each intermediate
phase-zero firing, move directly from the incoming port into the
next outgoing port, inside the one junction block.
Each complete lane consumes one frame of duration \(U_n\).
After the last lane, the next phase-zero firing moves the register
into target hub memory in one further tick. Thus
\[
 T\le U_n+mU_n+h_n
 \le \lambda_n L/c+2U_n+h_n.
\]
If the hubs coincide, waiting or local storage satisfies the same
bound.

For an arbitrary starting site on a lane, wait until a transport
frame starts. If its index is even, start forward movement at the
first transport phase; if odd, start at the second. Advance to the
last port, retain the register there, and move it into that hub at
the next phase-zero firing. This costs at most \(2U_n+h_n\).
A register at a hub can similarly reach any specified site on one
of its outgoing lanes in at most \(2U_n+h_n\).

Choose these initial and final hubs. Each is within physical distance
\(s_n=cU_n\) of its respective site. The hub-to-hub separation is
at most \(L+2s_n\). Applying \eqref{eq:hubtime} and adding the two
endpoint costs gives
\[
 T\le\lambda_n L/c+(6+2\lambda_n)U_n+3h_n.
\]
Storage is achieved by leaving the relevant memory factor unchanged,
not by changing the partition schedule.
\end{proof}

\begin{remark}[The clock has not been accelerated]
Each tick lasts \(a_n/c\), and every graph edge has physical length
at most \(a_n\). Junction ticks are charged even though their
physical displacement is very small. The efficiency loss appears
explicitly in
\[
 \lambda_n=
 \frac{(1+1/N_n)^2}{1-2/K_n}>1.
\]
Thus the upper cone and the lower transport bound use the same clock.
\end{remark}

\subsection{Parallel finite traffic}

\begin{lemma}[Finite traffic without junction conflicts]
\label{lem:traffic}
Any fixed finite collection of planned logical routes can be run on
the same schedule with the preceding per-route timing bounds, provided
finite reserved interfaces are large enough. Routes and storage plans
are chosen before the unknown input is supplied.
\end{lemma}

\begin{proof}
Assign a distinct register colour to each logical connection.
Reserve a slot of the appropriate type for that colour at the sites
it may use, together with hub-memory slots. During transport phases,
swap the corresponding slots across each pair block.
During junction phases, perform all prescribed colour permutations
and memory operations inside the one junction block.

Different colours are distinct tensor factors. An arbitrary finite
permutation of them is a legal local block channel; its internal
gate depth is not separately charged in this model.
No hub site belongs to more than one block. A finite collection uses
only finitely many colours and slots at each site.
\end{proof}

This is not a bounded-throughput theorem. It is an existence
statement with the interface-growth permission explicitly retained.

\section{Euclidean convergence of both past constructions}
\label{sec:cones}

Events are embedded by
\[
 (x,s)\longmapsto(sh_n,z_n(x)).
\]
For embedded terminal event \(q=(Th_n,z_n(A))\), define
\[
 \begin{split}
 J^-_{n,\mathrm{met}}(q)
 &=\{(sh_n,z_n(x)):s\le T,\ d_n(x,A)\le T-s\},\\
 J^-_{n,\mathrm{sch}}(q)
 &=\{(sh_n,z_n(x)):s\le T,\
        \text{a scheduled path joins }(x,s)\text{ to }(A,T)\}.
 \end{split}
\]
The continuum past is
\[
 J^-_c(q)=
 \{(t,x):t\le t_q,\ \norm{x-x_q}_2\le c(t_q-t)\}.
\]

\begin{lemma}[Exact physical path containment]
\label{lem:physicalcone}
Every scheduled path from \((x,s)\) to \((y,T)\) satisfies
\[
 \norm{z_n(y)-z_n(x)}_2\le c(T-s)h_n.
\]
Consequently
\[
 J^-_{n,\mathrm{sch}}(q)
 \subseteq J^-_{n,\mathrm{met}}(q)
 \subseteq J^-_c(q).
\]
\end{lemma}

\begin{proof}
Every transition stays in one block and crosses graph distance at
most one. Its physical displacement is at most \(a_n=ch_n\).
Sum the displacement bounds. The metric inclusion follows from
the same graph-distance bound.
\end{proof}

\begin{theorem}[Uniform local event and cone convergence]
\label{thm:cones}
The embedded events are locally dense in \(\RR^{1,2}\).
For every compact source set and compact terminal set, the distance
functions to both \(J^-_{n,\mathrm{met}}(q)\) and
\(J^-_{n,\mathrm{sch}}(q)\) converge uniformly to the distance
function to \(J^-_c(q)\), uniformly over embedded terminal events
in that terminal set.

The assertion uses full past sets and distance functions, not
intersections with arbitrary compact sets.
\end{theorem}

\begin{proof}
A spatial point has a hub within \(\rho_n\), and a time has a tick
within \(h_n\). This proves event approximation.

Fix an embedded terminal \(q\), and a point
\(p=(t,x)\in J^-_c(q)\) with
\(0\le t_q-t\le H\). Choose a hub \(H_y\) within \(\rho_n\) of \(x\).

For the metric past, Theorem~\ref{thm:metric} gives
\[
 a_n d_n(H_y,A)
 \le c(t_q-t)+
       (\lambda_n-1)cH+\lambda_n\rho_n+B_n.
\]
Put
\[
 \Delta_n^{\mathrm{met}}(H)
 =(\lambda_n-1)H+
       \frac{\lambda_n\rho_n+B_n}{c},
 \qquad
 s=\left\lfloor
       \frac{t-\Delta_n^{\mathrm{met}}(H)}{h_n}
    \right\rfloor .
\]
Then
\(a_n d_n(H_y,A)\le c(t_q-sh_n)\), so
\((sh_n,z_n(H_y))\in J^-_{n,\mathrm{met}}(q)\).
Its distance from \(p\) is at most
\[
 c\Delta_n^{\mathrm{met}}(H)+ch_n+\rho_n,
\]
which tends to zero uniformly for bounded \(H\).

For the scheduled past, Theorem~\ref{thm:routing} gives a scheduled
route of duration at most
\[
 (t_q-t)+(\lambda_n-1)H+
       \lambda_n\rho_n/c+\omega_n.
\]
Choose an earlier source tick using
\[
 \Delta_n^{\mathrm{sch}}(H)
 =(\lambda_n-1)H+\lambda_n\rho_n/c+\omega_n
\]
in the preceding floor formula. The route then finishes no later
than tick \(T\). Waiting at the terminal supplies a path ending
exactly there. The resulting approximation distance is at most
\[
 c\Delta_n^{\mathrm{sch}}(H)+ch_n+\rho_n.
\]

By Lemma~\ref{lem:physicalcone}, both discrete sets are contained
in the continuum cone. Their distance functions are therefore
no smaller than the continuum distance function.

For source and terminal points in fixed compact sets, a nearest
point on the closed continuum cone lies in a uniformly bounded
region: the apex itself gives a uniform distance bound.
The elapsed-time parameter \(H\) for those nearest points is
therefore uniformly bounded. Apply the preceding approximations
to those points. This supplies the reverse distance-function
inequality with an error tending uniformly to zero.
\end{proof}

\subsection{The growing periodic quotient and its transient}

Although the proof of Theorem~\ref{thm:cones} does not require an
asymptotic velocity theorem, the finite-periodic certificate in the
plan is also satisfied.

\begin{proposition}[Vanishing periodic transient]
\label{prop:offset}
The phase-explicit periodic dependency quotient has at most
\[
 q_n\le F_n\bigl(1+S_nF_n\bigr)
\]
vertices. Every quotient edge has physical displacement at most
\(a_n\) and duration \(h_n\). Hence
\[
 b_n^{\mathrm{per}}=(q_n-1)D_{\max,n}
 \le q_n a_n=O(n^{-5})\longrightarrow0.
\]
Its limiting scheduled velocity polytope \(K_n^{\mathrm{vel}}\)
satisfies
\[
 \frac{c}{\lambda_n}\overline B_2
 \subseteq K_n^{\mathrm{vel}}
 \subseteq c\overline B_2,
\]
and therefore converges in Hausdorff distance to
\(c\overline B_2\).
\end{proposition}

\begin{proof}
Modulo translations there are one hub type and \(S_nF_n\) lane
types. Including \(F_n\) schedule phases gives the vertex bound.
The edge displacement bound is Lemma~\ref{lem:partition}.
Since \(F_n=O(n)\), \(S_n=O(n^2)\), and \(a_n=O(n^{-9})\),
the offset is \(O(n^{-5})\).

The quotient is strongly connected. Every graph edge is exposed at
some phase, waiting is always possible, and the substrate graph is
connected. One can therefore traverse a finite graph route and
adjust the terminal phase by waiting.

For completeness, the cycle argument is as follows. Removing cycles
from a quotient walk leaves at most \(q_n-1\) edges.
The removed cycles decompose into simple directed cycles.
Their displacement divided by duration is a time-weighted convex
combination of simple-cycle velocities. Conversely, strong
connectivity supplies fixed connectors between cycle bases, whose
cost becomes negligible under repeated cycles. Waiting supplies
zero velocity. Thus the limiting velocities form the convex hull
of zero and the simple-cycle velocities, and finite-time displacement
has offset at most \((q_n-1)D_{\max,n}\).

The outer inclusion follows from the exact edgewise speed bound.
For the inner inclusion, fix \(n\) and a unit vector \(v\).
Choose increasingly distant hubs in direction \(v\).
Theorem~\ref{thm:routing} reaches them in time at most
\(\lambda_n L/c+2U_n+h_n\).
Add waiting to make elapsed times asymptotic to \(\lambda_n L/c\).
The limiting velocity is \((c/\lambda_n)v\).
Convexity and waiting give the entire inner ball.
\end{proof}

This reproduces the relevant finite-offset mechanism rather than
using a velocity diagram as a substitute for the uniform cone proof.
It aligns with Fritz and VII-J
\cite{Fritz,CheungVIIJ}.
The very small choice \(\ell_n=n^{-8}\) is deliberately inefficient:
it makes the growing quotient transient vanish as well.

\section{Quantum identity transport and boundary signalling}
\label{sec:quantum}

\subsection{Logical transport}

A transport slot has the same finite hybrid type at its sender and
receiver. Its physical motion is a sequence of tensor-factor
permutations or swaps. Quantum information is not measured during
this motion.

\begin{theorem}[Reference-preserving identity transport]
\label{thm:identity}
For every fixed finite-dimensional register \(Q\) and every planned
route of Theorem~\ref{thm:routing}, there is a complete implementation
whose logical sender-to-receiver channel is
\[
 \mathcal T_n=\id_Q.
\]
For every untouched reference \(R\) and every state \(\rho_{QR}\),
\[
 (\mathcal T_n\otimes\id_R)(\rho_{QR})=\rho_{QR}.
\]
The same assertion holds for a finite hybrid register.
\end{theorem}

\begin{proof}
Reserve one transport colour for \(Q\), and prepare all its unused
slots in a fixed blank state. At each pair firing, swap the two
slots. At each planned junction transition, permute the arriving
slot into the next outgoing slot or into hub memory.
The timetable in Lemma~\ref{lem:frame} sends the unique logical
factor to the next endpoint. Induct through the route.

The total transformation on that logical factor and the blank
slots is a tensor-factor permutation. After identifying the returned
factor as \(Q\), tracing the other blank factors leaves \(\rho_Q\).
The identical permutation calculation with \(\id_R\) leaves
\(\rho_{QR}\). Classical block labels are relocated with their
quantum blocks, without normalization or readout.
\end{proof}

\begin{remark}[Enter is not transport]
Destructive Enter on \(S\), with surviving factor \(T\), is
\[
 \Ent_S(\rho_{ST})
 =\sum_x\kb{x}{x}_K\otimes
   (\bra{x}_S\otimes I_T)\rho_{ST}(\ket{x}_S\otimes I_T).
\]
Dropping \(K\) gives \(\Tr_S\rho_{ST}\), not \(\id_S\).
The transport construction uses no such operation on the transported
unknown register. This preserves the distinctions of Landmark I,
En04-A, and Hawk04-B
\cite{CheungI,CheungEn04,CheungHawk04B}.
No teleportation resource is used.
\end{remark}

\begin{theorem}[Directional matched relay limits]
\label{thm:relay}
For every \(x\in\RR^2\), Euclidean unit vector \(v\), and \(u>0\),
there are matched discrete relay pairs with sender events tending to
\(A=(0,x)\) and readout events tending to
\(B=(u,x+cuv)\). Their complete readout laws are
\(\delta_0,\delta_1\) at every sufficiently fine scale.
The sender modification footprints shrink to \(A\).
\end{theorem}

\begin{proof}
Choose hubs within \(\rho_n\) of \(x\) and \(x+cuv\), and supply the
bit at the first hub at tick zero. Their separation \(L_n\) satisfies
\(L_n\to cu\).
Use the common route and channels of Theorem~\ref{thm:identity}.
The two runs differ only in the bit supplied at the sender.

Let the readout occur at the first delivered hub-memory event.
Its elapsed time is bounded above by
\(\lambda_n L_n/c+2U_n+h_n\), which tends to \(u\).
The exact physical cone gives the lower bound \(L_n/c\), which also
tends to \(u\). Thus the readout events converge to \(B\).
Reading the transported bit gives \(\delta_0,\delta_1\).
Preparation and any charged local readout occupy shrinking blocks,
so the sender modification footprint tends to \(A\).
\end{proof}

\begin{remark}[A classical gap does not prove quantum transmission]
The classical relay establishes the lower signalling bound.
Theorem~\ref{thm:identity} is the separate quantum statement.
A record-only measurement followed by preparation is not substituted
for an unknown-state identity channel.
\end{remark}

\section{The declared finite-experiment class}
\label{sec:experiments}

\subsection{Skeletons, support, and complete control}

\begin{definition}[Finite typed skeleton]
\label{def:skeleton}
A finite typed skeleton \(\Gamma\) consists of:
\begin{enumerate}[label=(\roman*)]
\item a finite directed acyclic graph of labelled vertices and
labelled quantum or hybrid wire factors;
\item fixed finite input and output types at each vertex;
\item external input and output ports, including complete records;
\item a point \(p_v=(t_v,x_v)\in\RR^{1,2}\) at each vertex;
\item for every edge \(v\to w\), the causal constraint
\[
 t_w-t_v\ge \norm{x_w-x_v}_2/c.
\]
\end{enumerate}
An edge is the identity connection on its labelled factor.
Additional piecewise-linear causal connections are represented by
finitely many labelled identity waypoints.

A vertex may act on several owned factors, but no quantum factor is
duplicated by wiring. A reported instrument includes every outcome,
including inactivity and failure flags.
\end{definition}

\begin{definition}[Causal finite circuit]
\label{def:circuit}
A causal finite circuit assigns a complete finite hybrid channel
\(\Phi_v\) to each vertex of a typed skeleton.
Initial fixed resources are prepared at declared source vertices.
The unknown external input may be entangled across its ports and
with an untouched reference.

Every classical control read at a vertex is part of an incident or
locally retained wire factor. The channel at that vertex is fixed
before the unknown input is supplied.
\end{definition}

Its complete channel \(\Phi_\Gamma\) is evaluated in a topological
order. Independent operations on disjoint owned factors commute,
so the channel does not depend on the chosen legal order.

Operation support means the labelled vertex points, or their
declared compact neighborhoods. Wire routes and retained memories
are also part of the skeleton, but are not reselected when a local
instrument is replaced.

The class is closed under finite disjoint composition and under
well-typed recorded continuations whenever their gluing is acyclic
and obeys the displayed causal constraints.

\subsection{Adaptive padding}

\begin{lemma}[Finite adaptation within a fixed skeleton]
\label{lem:adaptive}
A finite adaptive instruction tree with fixed finite outcome sets,
declared finite interfaces, and causal branch connections can be
represented by a finite typed skeleton of complete classically
controlled instruments.
\end{lemma}

\begin{proof}
Unroll the finite tree. Reserve fixed output slots and an activity
flag for every possible continuation. At a branching vertex, use
classical control to place the active quantum factor into its
selected continuation slot by a permutation; fill inactive slots
with fixed blank states. The quantum factor is not copied.
Classical outcome and activity flags may be copied locally and
transported along their declared edges.

At a merge, the local flag selects the active input slot by a
complete controlled channel. Every inactive or failure branch is
retained in the full interface. The tree is finite, so this uses
finitely many vertices and factors. All controls are represented
locally.
\end{proof}

This is complete-channel accounting, not postselection. It is the
finite-interface setting compared with En04-A and En02-C
\cite{CheungEn04,CheungEn02C}. A syntactic Enter-calculus
implementation would require the additional compositional and
gate-library hypotheses of En05-A; no exact compilation of arbitrary
instruments is asserted here.

\subsection{Logical comparison maps}

For a realization, let \(E_n\) place the external logical inputs
into their designated physical slots and append the common prepared
background. Let \(D_n\) restrict the final physical output to the
designated logical output slots and complete records, then apply
their fixed type identifications.

Both maps are CPTP. They are not physical nonlocal transport
operations: they are comparison maps identifying specified external
interfaces.

If an interface is padded into a larger reserved space, choose a
CPTP decoder that inverts the padding on its code subspace and sends
the complementary weight to a fixed admitted output state.
Thus padding has a complete deterministic extension.
The comparison is
\[
 D_n\Phi_n E_n:\Alg_{\mathrm{in},*}\longrightarrow
                      \Alg_{\mathrm{out},*}.
\]
Convergence means half-diamond convergence of these maps.
It includes every untouched reference and implies total-variation
convergence of every fixed complete finite readout.

\paragraph{Capacity convention.}
One may reserve interfaces separately for each finite experiment,
as in En05-A. Alternatively, fix in advance a number of slots and a
slot dimension tending to infinity with \(n\).
Every fixed finite skeleton then fits on all sufficiently fine
scales. The substrate and partition schedule do not depend on its
input state or its local instrument choices.

\section{Exact realization and matched local replacements}
\label{sec:realization}

\subsection{Timing with causal-boundary connections}

\begin{lemma}[Vanishing timing slack for a finite skeleton]
\label{lem:timing}
For every causal finite skeleton, there are phase-zero firing times
\(t_{v,n}\to t_v\) and approximating hubs \(H_{v,n}\to x_v\) such that
every prescribed wire can be delivered after its source operation
and before its target operation. The choices depend only on the
skeleton, not on the channel assignments.
\end{lemma}

\begin{proof}
Choose each approximating hub by a fixed nearest-hub rule.
Its error is at most \(\rho_n\).

Choose a topological rank \(r_v\in\{0,\ldots,M\}\), strictly increasing
on every edge. Let \(L_*\) bound all edge lengths in the finite
skeleton, and put
\[
 \begin{split}
 e_n&=
 \frac{(\lambda_n-1)L_*}{c}
 +\frac{2\lambda_n\rho_n}{c}
 +4U_n+4h_n,\\
 D_n^{\mathrm{time}}
 &=U_n\left\lceil e_n/U_n\right\rceil,\\
 t_{v,n}
 &=U_n\left\lceil t_v/U_n\right\rceil
                  +r_vD_n^{\mathrm{time}}.
 \end{split}
\]
These are phase-zero times and converge to \(t_v\).
For an edge \(v\to w\),
\[
 t_{w,n}-t_{v,n}
 \ge(t_w-t_v)-U_n+D_n^{\mathrm{time}}.
\]
The approximating hub distance is at most
\(\norm{x_w-x_v}_2+2\rho_n\).
Charge one tick for the source operation and use
Theorem~\ref{thm:routing} from its output tick.
Operation plus delivery costs at most
\[
 \frac{\lambda_n(\norm{x_w-x_v}_2+2\rho_n)}{c}
                         +2U_n+2h_n.
\]
The causal inequality and the definition of \(e_n\) make this no
larger than \(t_{w,n}-t_{v,n}\).
All arrivals may be retained in hub memory until the target firing.

The same argument covers null connections and zero-duration
co-located edges. Their added rank-dependent delays vanish.
\end{proof}

\begin{remark}[Boundary connections are not omitted]
Restricting the realization theorem to strictly timelike wires would
not establish the directional boundary relays.
Lemma~\ref{lem:timing} instead delays finitely many vertex times by
vanishing amounts, preserving their declared limiting locations.
\end{remark}

\begin{theorem}[Exact logical realization]
\label{thm:realization}
Every causal finite circuit has a realizing family on the substrates
and schedules above such that
\begin{equation}
 D_n\Phi_nE_n=\Phi_\Gamma
 \label{eq:exactrealization}
\end{equation}
for every sufficiently large \(n\).
Its vertex-operation footprints converge to their declared supports.

The identity holds after tensoring with every untouched reference.
\end{theorem}

\begin{proof}
Use Lemma~\ref{lem:timing} to choose locations, firing times, and
state-independent routes. Reserve one transport colour per
connection and the required memory factors.
Lemma~\ref{lem:traffic} implements the finite traffic concurrently.

At each vertex firing, all input factors and local controls are
present in its junction block or hub memory. Apply the specified
complete instrument on those factors and return every declared
output, including records and failure flags. Unused factors receive
identity channels. Every output wire follows its prescribed route.
Incoming early factors remain in memory.

In a topological induction, every transport connection has logical
channel equal to the identity by Theorem~\ref{thm:identity}.
Each vertex therefore receives precisely its logical circuit inputs
and applies precisely \(\Phi_v\). After the output restrictions,
the total logical channel is \(\Phi_\Gamma\).

The operation is confined to a junction block whose spatial
diameter and firing duration tend to zero. Its hub and firing time
converge to the declared vertex point. This proves support
convergence.

The induction is an equality of complete linear maps. Tensoring it
with \(\id_R\) proves the reference assertion.
\end{proof}

\subsection{Matched realization}

\begin{definition}[Local replacement on a common skeleton]
\label{def:replacement}
Two circuits form a \(K\)-local replacement pair if they have the same
typed skeleton, source and output placements, transport connections,
and unchanged resources, and their vertex channels differ only at
vertices supported in the compact set \(K\).

An initial-input modification confined to \(K\) means equality of
the joint reductions on all external input factors outside \(K\),
including any untouched reference being compared.
\end{definition}

\begin{theorem}[Matched realization of interventions]
\label{thm:matched}
A \(K\)-local replacement pair admits realizing families with:
\begin{enumerate}[label=(\roman*)]
\item the same substrate and partition schedule;
\item the same reserved interfaces, resource preparation, firing
times, transport routes, and unchanged block channels;
\item different block channels only at the corresponding local
vertex firings, whose complete footprints approach \(K\);
\item the same discrete implementation of every common readout;
\item equal initial snapshots outside the declared discrete initial
modifications, also jointly with a reference when that is required.
\end{enumerate}
These conclusions persist under admitted finite compositions and
recorded continuations.
\end{theorem}

\begin{proof}
The realization choices in Lemma~\ref{lem:timing} and
Theorem~\ref{thm:realization} depend only on the common skeleton and
its finite interface types. Choose them once for both circuits.

At an unchanged vertex, use the same complete local channel.
At a replaced vertex, replace only its assigned complete instrument
inside the same junction firing. Its input and output types are
unchanged, so all transport permutations and storage rules remain
the same. A different classical outcome changes the state of a
represented controller; it does not change an undeclared scheduler.

Resource registers and route colours are initialized identically.
A changed external input is placed only in its assigned local slots.
The comparison encoding is tensorial on those slots with a common
background; exterior reductions therefore agree whenever the
logical exterior reductions agree.

A finite composite is realized by the same procedure on its common
composite skeleton. Lemma~\ref{lem:adaptive} provides the fixed
complete interface for recorded continuations. References are not
read or modified.
\end{proof}

\begin{remark}[The intervention, not ancestry, is matched]
A replaced instrument may ignore some scheduled inputs.
Theorem~\ref{thm:matched} does not identify every ancestor with an
influential port. It preserves the local replacement relation
needed to apply Lemma~\ref{lem:locality}.
\end{remark}

\section{Compositional convergence and implementation budgets}
\label{sec:errors}

\begin{lemma}[Complete-channel telescoping]
\label{lem:telescope}
For compatible complete channels \(\Phi_j,\widetilde\Phi_j\),
\[
 \ddia(\widetilde\Phi_m\cdots\widetilde\Phi_1,
                  \Phi_m\cdots\Phi_1)
 \le\sum_{j=1}^m\ddia(\widetilde\Phi_j,\Phi_j).
\]
Tensor composition satisfies the analogous sum bound.
\end{lemma}

\begin{proof}
Subtract the products by replacing one factor at a time.
CPTP pre- and postcomposition contract diamond distance, so each
term is bounded by its local error. For tensor products, replace
one tensor factor at a time and use that a channel has diamond norm
one. Sum the resulting bounds.
\end{proof}

This is the compatible half-diamond accounting compared with
En02-B and En05-A \cite{CheungEn02B,CheungEn05A};
diamond distance is the reference-stable comparison used throughout
\cite{Watrous}.

\begin{proposition}[Finite-circuit operational convergence]
\label{prop:convergence}
Suppose the logical implementation of vertex \(v\) has error
\(e_{v,n}\), and the complete logical implementation of connection
\(e\) has identity error \(r_{e,n}\). Then
\[
 \ddia(D_n\widetilde\Phi_nE_n,\Phi_\Gamma)
 \le\sum_v e_{v,n}+\sum_e r_{e,n}.
\]
If the right-hand side tends to zero, all complete finite readout
laws converge in total variation, uniformly over logical inputs and
untouched references.
\end{proposition}

\begin{proof}
Evaluate the finite DAG in a topological order, tensoring each step
with identities on the other live logical factors.
Apply Lemma~\ref{lem:telescope}. A complete terminal readout is CPTP
and cannot increase the error. Its classical trace distance is
total variation.
\end{proof}

\begin{proposition}[Relay-gap stability]
\label{prop:gap}
If two implemented relay laws are respectively within
\(\eta_{0,n},\eta_{1,n}\) of \(\delta_0,\delta_1\), then
\[
 \dtv(\widetilde p_n^{(0)},\widetilde p_n^{(1)})
 \ge1-\eta_{0,n}-\eta_{1,n}.
\]
A sum of local half-diamond errors bounds each \(\eta_{b,n}\).
\end{proposition}

\begin{proof}
Use the triangle inequality and
\(\dtv(\delta_0,\delta_1)=1\).
The second statement follows from
Lemma~\ref{lem:telescope} and terminal readout contraction.
\end{proof}

\begin{remark}[Which elementary errors are being counted]
For the explicit ideal construction every error above is zero.
For an approximate factorized logical implementation with
\(O(a_n^{-1})\) charged elementary steps on a fixed physical journey,
a uniform per-step error \(o(a_n)\) is sufficient.

This last statement requires the specified logical factorization.
It is not a claim that arbitrary errors coupling all registers in
all blocks of a large conservative cone have only
\(O(a_n^{-1})\) contributions. Such a model needs its own complete
finite-channel error ledger.
\end{remark}

\section{Local operational completion}
\label{sec:completion}

\subsection{Why terminal-law completion is insufficient}

\begin{example}[One law does not determine a local instrument]
\label{ex:law}
The identity qubit channel and computational-basis dephasing give
the same deterministic computational readout on input
\(\kb0 0\). They differ on \(\kb+ +\), and on a Bell-entangled input.
Consequently convergence of that one terminal law cannot identify
a reference-preserving channel, its local slots, or its responses
to replacement instruments.
\end{example}

The completion must retain enough information to define composition
and matched replacements. This is the substantive issue behind C4
in En05-A \cite{CheungEn05A}.

\subsection{The fixed-skeleton topology}

Fix a finite labelled DAG and every finite local type. Let
\(\mathcal X_\Gamma^\circ\) be the circuits whose wire inequalities
are strict:
\[
 t_w-t_v>\norm{x_w-x_v}_2/c.
\]
Include the chosen finite list of identity waypoints in the vertex
set. Interface types, vertex labels, and replacement slots are fixed.

For two assignments on this skeleton, define
\begin{equation}
 d_\Gamma(E,F)
 =
 \max_v\norm{p_v(E)-p_v(F)}_c
       +\sum_v\ddia(\Phi_v(E),\Phi_v(F)).
 \label{eq:completionmetric}
\end{equation}
If a fixed prepared state is retained as separate source data, add
its trace distance; equivalently treat its preparation as a
one-dimensional-input source channel.

This topology compares complete local channels, not only their
effects on one prepared input. Equality in the completion is equality
of the labelled local data. Equal terminal channels alone are not
used to identify different skeleton-local experiments.

\begin{definition}[Compatible local family]
\label{def:compatible}
A family is compatible for the completion if its labelled skeleton
and all local finite types are fixed, its vertex data converge in
the position part of \eqref{eq:completionmetric}, and its complete
local channels are Cauchy in the channel part.

Compatibility of a replacement pair additionally retains the same
unchanged coordinates, source data, and local channels outside the
declared replacement slots.
\end{definition}

\begin{theorem}[Fixed-skeleton operational completion]
\label{thm:completion}
The metric completion of \(\mathcal X_\Gamma^\circ\) is precisely
the class \(\mathcal X_\Gamma\) with the same fixed local types and
the closed causal inequalities
\[
 t_w-t_v\ge\norm{x_w-x_v}_2/c.
\]
It has the following properties:
\begin{enumerate}[label=(\roman*)]
\item every compatible local family has a unique limit as labelled
local data;
\item its complete circuit channels converge in half-diamond
distance to the channel of that limit;
\item admitted finite tensor compositions and acyclic typed gluings
are continuous and extend to the completion;
\item every limit object has the matched discrete realization
constructed in Section~\ref{sec:realization}.
\end{enumerate}
\end{theorem}

\begin{proof}
For fixed finite types, the space of CPTP maps is closed in the
finite-dimensional vector space of linear maps: positivity of every
finite Choi block and trace preservation are closed conditions.
It is bounded and hence compact, and in particular complete in
diamond norm \cite{Choi,Watrous}.
The position space is complete. Closed causal inequalities define
a closed subset of their finite product. Thus
\(\mathcal X_\Gamma\) is complete.

Every closed-causal assignment is approximated by strict-causal
ones. Choose a topological rank \(r_v\), and replace \(t_v\) by
\(t_v+r_v/k\), leaving positions and instruments unchanged.
Every edge gains strictly positive time and the assignments
converge. This identifies the completion.

Circuit evaluation is a finite tensor-and-composition expression
in its local channels. Lemma~\ref{lem:telescope} proves its
continuity and the channel limit. Coordinates specify admissibility
and supports; they do not alter the logical channel formula.

A typed gluing forms a finite composite skeleton. When it is acyclic
and satisfies the causal constraints, its channel is the
corresponding tensor-and-composition expression. The same bound
proves continuity. Vertex support coordinates and boundary
identifications are retained in that expression.
Finally, Theorem~\ref{thm:realization} includes all closed-causal
assignments, including boundary connections.
\end{proof}

\begin{theorem}[Matched locality survives completion]
\label{thm:completionmatch}
Suppose compatible pairs \((E_k,F_k)\) on a common skeleton differ
only in fixed local replacement slots, with those slots' supports
converging into a compact set \(K\).
Their completion limits form a \(K\)-local replacement pair and
admit the matched realizing families of
Theorem~\ref{thm:matched}.

Equality of exterior initial reductions, including a reference,
also survives the limit.
\end{theorem}

\begin{proof}
Every unchanged local channel and coordinate is equal in the two
members at each \(k\). Their limits are equal by continuity of the
coordinate and channel projections. The changed supports lie in
\(K\) in the limit. The resulting pair therefore meets
Definition~\ref{def:replacement}; apply
Theorem~\ref{thm:matched}.

Partial trace and classical marginalization are continuous.
Equality of exterior initial reductions is a closed condition,
also after tensoring the fixed comparison reference.
\end{proof}

\subsection{A law-level local-completeness statement}

\begin{proposition}[Local completeness with fixed finite slots]
\label{prop:weakC4}
Consider a family on one fixed finite typed skeleton whose support
coordinates converge and whose complete readout laws, for a declared
input preparation, converge in total variation.
Even if its local channels are not Cauchy, a subsequence of its local
assignments converges to an admitted circuit with that limiting law.

For matched pairs on the same fixed finite slots, a common subsequence
can be chosen so that the two limiting circuits remain matched.
\end{proposition}

\begin{proof}
The finite product of local CPTP-map spaces is compact.
Select a convergent subsequence of all local channels and any
separately retained finite preparation states. The causal constraints
are closed, so the limiting assignment is admitted.
Continuity of circuit evaluation gives its readout law.
The original laws have a prescribed limit, so the subsequence has
that same limit.

For pairs, use compactness of the joint finite product.
Equal unchanged channels and equal exterior reductions remain equal
under the common subsequence.
\end{proof}

\begin{remark}[The exact C4 boundary]
Theorems~\ref{thm:completion} and~\ref{thm:completionmatch} construct
a unique local completion for compatible contextual data.
Proposition~\ref{prop:weakC4} supplies law-level membership on the
same fixed finite slots, but does not infer a unique instrument from
one law.

No assertion is made about unrestricted families with growing
numbers of hidden vertices, changing intermediate dimensions, or
only one convergent terminal law. Such families are outside the
declared compatibility class. This is a restriction of the
existence theorem's operational domain, not an unproved promotion
of terminal convergence to the broader C4 formulation.
\end{remark}

\section{Continuum no-signalling and sharp operational speed}
\label{sec:continuum}

\begin{theorem}[Matched continuum domain of dependence]
\label{thm:DOD}
Let \(E,F\) be a locally matched pair in the finite-circuit class or
its local completion, with a common readout supported in a compact
set \(Y\). Suppose every changed operation and initial-input
modification lies in compact \(K\), and
\[
 \dist\left(K,\bigcup_{q\in Y}J^-_c(q)\right)>0.
\]
Then the complete readout laws on \(Y\) agree.
The conclusion is reference preserving in the joint-reduction
sense of Lemma~\ref{lem:comparison}.
\end{theorem}

\begin{proof}
Use Theorem~\ref{thm:matched}, or its completed version, to choose
common discrete schedules, backgrounds, routes, and readouts.
The terminal readout events and changed firing footprints converge
to their declared supports.

The positive margin and shrinking footprints imply that, for all
sufficiently large \(n\), no changed firing can be an ancestor of
any common readout. Indeed every such dependency path would obey
the exact physical cone of Lemma~\ref{lem:physicalcone}, contradicting
the positive limiting separation. Initial changed inputs are
excluded by the same argument.

The exterior initial snapshots and the required ancestor channels
therefore agree. Lemma~\ref{lem:locality} gives equal discrete joint
readout laws. Their logical complete channels converge, in fact
are exact in the ideal construction, so their limits agree.
The same proof uses joint cone--reference reductions.
\end{proof}

\begin{corollary}[Operational spacelike no-signalling]
\label{cor:NS}
For two compact operation/readout regions satisfying
\[
 \norm{x_p-x_q}_2
 \ge c\abs{t_p-t_q}+\eta
 \qquad(p\in K_1,\ q\in K_2)
\]
for some \(\eta>0\), a matched local replacement in either region
does not change the other's unconditioned complete joint record law.
\end{corollary}

\begin{proof}
Compactness and the positive margin separate each region from the
causal past of every point in the other. Apply
Theorem~\ref{thm:DOD} to the other region's full finite record set.
\end{proof}

Conditioning on a remote outcome is a different operation and is
not excluded by this statement.

\begin{corollary}[Sharp isotropic continuum maximal speed]
\label{cor:maxspeed}
Define continuum event signalling by matched sender modifications
localized in arbitrarily small neighborhoods of an event \(A\),
a common finite readout at \(B\), and a nonzero unconditioned
readout-law gap. For positive-duration pairs,
\[
 c_{\mathrm{op}}
 =
 \sup\frac{\norm{x_B-x_A}_2}{t_B-t_A}=c.
\]
Every Euclidean direction attains the boundary speed in this
limiting sense.
\end{corollary}

\begin{proof}
If the ratio exceeds \(c\), a sufficiently small neighborhood of
\(A\) is at positive distance from \(J^-_c(B)\).
Theorem~\ref{thm:DOD} excludes a gap. Thus
\(c_{\mathrm{op}}\le c\).

Theorem~\ref{thm:relay} supplies a matched boundary pair in every
direction with gap one. Its two limiting readout laws are still
\(\delta_0,\delta_1\), proving the reverse inequality.
\end{proof}

\section{Milestone audit, repairs, and interpretation}
\label{sec:audit}

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.08\linewidth}p{.33\linewidth}X@{}}
\toprule
Step & Obligation & Result in this paper\\
\midrule
1 & Finite experiment and comparison specification &
Definitions~\ref{def:skeleton}--\ref{def:circuit};
complete logical comparison maps and half-diamond topology.\\
\addlinespace
2 & Increasing directions; all-site metric control &
Lemma~\ref{lem:angles};
Theorem~\ref{thm:metric}, including intermediate vertices.\\
\addlinespace
3 & Legal partition scheduling and charged routing &
Definition~\ref{def:schedule};
Theorem~\ref{thm:routing}, from every starting tick.\\
\addlinespace
4 & Uniform Euclidean cone and relay limits &
Theorem~\ref{thm:cones};
Proposition~\ref{prop:offset};
Theorem~\ref{thm:relay}.\\
\addlinespace
5 & Genuine referenced quantum transport &
Theorem~\ref{thm:identity}: logical identity, not readout.\\
\addlinespace
6 & Matched finite and adaptive experiments &
Lemma~\ref{lem:adaptive};
Theorems~\ref{thm:realization}--\ref{thm:matched}.\\
\addlinespace
7 & Compositional convergence and local completion &
Proposition~\ref{prop:convergence};
Theorems~\ref{thm:completion}--\ref{thm:completionmatch};
explicit C4 compatibility boundary.\\
\bottomrule
\end{tabularx}
\end{center}

\subsection{The three construction errors}

The timing estimate can be written
\[
 T_n(L)\le\frac{L}{c}
       +\underbrace{\frac{(\lambda_n-1)L}{c}}_{\text{relative loss}}
       +\underbrace{\omega_n}_{\text{endpoint and phase overhead}}.
\]
The relative loss separates into
\[
 \lambda_n
 =
 \underbrace{\frac1{1-2/K_n}}_{\text{stencil rounding}}
 \underbrace{\left(1+\frac1{N_n}\right)}_{\text{edge separation allowance}}
 \underbrace{\left(1+\frac1{N_n}\right)}_{\text{junction tick}}.
\]
Angular resolution is controlled independently by
Lemma~\ref{lem:angles}; hub rounding costs \(\rho_n\).
All these quantities vanish in the estimates relevant to bounded
journeys. The periodic transient is separately controlled by
Proposition~\ref{prop:offset}.

\subsection{Repairs to tempting but invalid shortcuts}

\begin{enumerate}[label=(\arabic*)]
\item \textbf{High-degree graph versus partition schedule.}
A high-degree stencil alone does not provide fast transport.
Endpoint separation and one junction block eliminate the proposed
direction-by-direction service bottleneck.

\item \textbf{Hub-only estimates.}
Metric estimates at hubs alone would not establish C2.
Theorem~\ref{thm:metric} controls every intermediate site.

\item \textbf{An upper cone versus an attained cone.}
The physical upper bound is exact at every scale, whereas the lower
routing loss tends to zero under the same clock.

\item \textbf{Boundary timing.}
Null connections cannot in general be realized with zero discrete
lateness. Finite topological-rank delays make them realizable while
their support locations converge.

\item \textbf{Separate convergence versus matching.}
Realization is selected once from the common skeleton.
Replacing an instrument does not reselect its remote resources,
routes, or schedule.

\item \textbf{One terminal law versus completion.}
The local completion retains complete local channels and labelled
intervention slots. Example~\ref{ex:law} prevents a law-only
identification.

\item \textbf{Classical signalling versus quantum identity.}
The former uses a known classical bit; the latter transports a
quantum tensor factor without destructive readout.

\item \textbf{A semantic realization versus compilation.}
Arbitrary finite complete instruments are admitted as semantic
block channels. Exact compilation over a particular gate library
is a separate theorem, not a consequence of this construction.

\item \textbf{Fixed-scale periodicity versus fixed-model rescaling.}
Each scale is periodic, but its quotient and stencil grow.
The family therefore does not contradict the fixed-periodic
polytope obstruction.
\end{enumerate}

\subsection{What has and has not been constructed}

The geometric and operational package established here is:
\[
 \begin{gathered}
 \text{locally dense events}
 +\text{uniform Euclidean metric cones}
 +\text{uniform Euclidean scheduled cones},\\
 \text{matched finite realizations}
 +\text{referenced identity transport}
 +\text{directional gap-one boundary relays}.
 \end{gathered}
\]
In the notation of En05-A, this supplies the geometric C1--C2
requirements, C3 on the declared circuit class and its local
completion, and C5. It supplies a proved fixed-interface local
completeness construction rather than the unrestricted reading of C4.

\paragraph{Observable algebras.}
No spacetime observable algebra satisfying En05-A's C6 is constructed.
Commuting disjoint channel operations and operational no-signalling
are not promoted to algebraic microcausality. Channel adjoints need
not be multiplicative.

\paragraph{Frames.}
The schedules have a preferred discrete time grading.
No calibrated frame group satisfying C7 is constructed.
Fixed-labelled-channel reorderings are not boosts
\cite{CheungVIIE,CheungEn03,CheungEn05A}.
The unrestricted device-retiming closure withdrawn in En05-A is
not reinstated.

\paragraph{Related automaton limits.}
Reversible quantum cellular automata have their own structural
hypotheses \cite{SchumacherWerner}. Dirac-equation continuum
comparisons concern a different dynamical limit
\cite{Song}. Neither supplies, nor is required for, the matched
finite-instrument theorem proved here.

\paragraph{Physical geometry.}
The metric graph, physical locations, speed calibration, and local
channels are declared inputs. The construction derives no physical
metric from entanglement, no Einstein equation, and no gravitational
backreaction from residual response. The readout distinctions of
En04-A remain separate from such a dictionary \cite{CheungEn04}.

\paragraph{Unrelated analytic programme boundaries.}
This transport construction does not construct the inherited joint
characteristic map. The reported internal amplitude advance at I2
in XIV-I is not confused with its remaining I3 obligation, or with
the full J2 obligation in XIV-J
\cite{CheungXIVI,CheungXIVJ}.
No inherited quantum coefficient is admitted. The downstream
XV-B admission exit is not opened, O.1 is not closed, and XVI-N's
separate prepared-insertion obligations are not discharged.

\subsection{Conclusion}

The central scheduling problem is solved here by a specific
architecture: distinct lane endpoints, one local junction clique,
and synchronized parallel lane frames. Its cost is growing degree,
block size, and interfaces. Its benefit is an explicit estimate
under one clock:
\[
 \boxed{
 \norm{z_n(x)-z_n(y)}_2
 \le a_n d_n(x,y)
 \le\lambda_n\norm{z_n(x)-z_n(y)}_2+B_n,
 \qquad
 T_n(x,y)\le\lambda_n\frac{\norm{z_n(x)-z_n(y)}_2}{c}
                    +\omega_n,
 }
\]
with \(\lambda_n\to1\), \(B_n\to0\), and \(\omega_n\to0\).

The resulting operational continuum limit is isotropic, carries
unknown quantum states without losing their references, and
preserves matched finite interventions. It is deliberately a
finite-experiment transport limit, not a spacetime-emergence or
field-theory theorem.

\section*{Declarations}

\paragraph{Mathematical scope.}
All construction claims refer to the explicit graph, clock,
partitions, experiment class, and completion topology given in this
paper. No companion manuscript is needed as a proof premise for
the construction.

\paragraph{Resources.}
There is no uniform bound on degree, junction cardinality, interface
dimension, or within-block processing depth. There is no optimality,
energy-cost, or efficient-hardware claim.

\paragraph{Verification status.}
The source is self-contained and uses standard pdfLaTeX packages.
No successful compilation, proof-assistant verification, executed
numerical experiment, independent referee review, journal acceptance,
or laboratory implementation is asserted. The mathematical
construction is presented for scrutiny through its explicit proofs.

\paragraph{Source status.}
Cheung companion citations identify the supplied manuscript or
excerpt versions. They are not described as independently verified
published articles. External references are cited for their stated
mathematical comparisons, not as replacements for the proofs above.

\begin{thebibliography}{99}

\bibitem{CheungEn05A}
N.~Cheung,
\emph{Exact Causal Cones for Partitioned Enter Dynamics:
Snapshot--Channel Comparison, Sharp Relay Speed,
and Conditional Continuum Limits}.
En05-A, version 1, supplied manuscript
\oplabel{En05-A-v1.tex}, October 2026.
Relevant supplied locators:
\oplabel{lem:comparison}, \oplabel{thm:cone},
\oplabel{thm:isotropyobstruction},
\oplabel{hyp:C1}--\oplabel{hyp:C7},
and \oplabel{prop:relaygap}.

\bibitem{CheungEn03}
N.~Cheung,
\emph{En03-A}, version 1.
Supplied companion manuscript \oplabel{En03-A.tex}.
Relevant supplied locators:
\oplabel{lem:one-layer}, \oplabel{thm:locality},
\oplabel{thm:retiming}, and \oplabel{prop:directional}.
Source archive description: \url{https://enter.grok.me/}.

\bibitem{CheungVIIE}
N.~Cheung,
\emph{Landmark VII-E}, version 1.
Supplied companion manuscript \oplabel{VII-E.tex}.
Prescribed kinematics, fixed-labelled-channel ordering,
and directional comparison; relevant supplied locators:
\oplabel{thm:retiming} and \oplabel{prop:direction}.
Source archive description: \url{https://enter.grok.me/}.

\bibitem{CheungVIIG}
N.~Cheung,
\emph{Landmark VII-G}, version 1.
Supplied companion manuscript \oplabel{VII-G.tex}.
Operational locality with explicit controller support;
relevant supplied locator: \oplabel{thm:locality}.
Source archive description: \url{https://enter.grok.me/}.

\bibitem{CheungVIIJ}
N.~Cheung,
\emph{Landmark VII-J}, version 1.
Supplied companion manuscript \oplabel{VII-J.tex}.
Complete timed dependency networks, influence containment,
and cycle-velocity polytopes; relevant supplied locators:
\oplabel{thm:nosignal}, \oplabel{thm:schedule},
and \oplabel{thm:polytope}.
Source archive description: \url{https://enter.grok.me/}.

\bibitem{CheungPsy04B}
N.~Cheung,
\emph{Psy04-B}, version 1.
Supplied companion manuscript \oplabel{Psy04-B-v1.tex}.
Complete finite causal networks; relevant supplied locator:
\oplabel{thm:schedule}.
Source archive description: \url{https://enter.grok.me/}.

\bibitem{CheungI}
N.~Cheung,
\emph{Landmark I}, version 2.
Supplied companion manuscript \oplabel{I.tex}.
Finite path semantics, destructive Enter, and circuit adequacy;
relevant supplied locators:
\oplabel{lem:path}, \oplabel{thm:adequacy},
and \oplabel{prop:classical-recovery}.
Source archive description: \url{https://enter.grok.me/}.

\bibitem{CheungEn04}
N.~Cheung,
\emph{En04-A}, version 1.
Supplied companion manuscript \oplabel{En04-A.tex}.
Complete finite-program soundness, destructive readout,
and residual-response distinctions; relevant supplied locators:
\oplabel{thm:soundness}, \oplabel{thm:drop},
and \oplabel{cor:bridge}.
Source archive description: \url{https://enter.grok.me/}.

\bibitem{CheungHawk04B}
N.~Cheung,
\emph{Hawk04-B}, version 2.
Supplied companion manuscript \oplabel{Hawk04-B.tex}.
Typed finite-program semantics and destructive readout;
relevant supplied locators:
\oplabel{thm:adequacy}, \oplabel{thm:drop},
and \oplabel{thm:classical}.
Source archive description: \url{https://enter.grok.me/}.

\bibitem{CheungEn02B}
N.~Cheung,
\emph{En02-B}, version 1.
Supplied companion manuscript \oplabel{En02-B-v1.tex}.
Compatible half-diamond implementation accounting;
relevant supplied locator: \oplabel{thm:B6}.
Source archive description: \url{https://enter.grok.me/}.

\bibitem{CheungEn02C}
N.~Cheung,
\emph{En02-C}, version 1.
Supplied companion manuscript \oplabel{En02-C-v1.tex}.
Reference-preserving sequential-program soundness;
relevant supplied locator: \oplabel{thm:C1}.
Source archive description: \url{https://enter.grok.me/}.

\bibitem{CheungXIVI}
N.~Cheung,
\emph{Landmark XIV-I}, version 1.
Supplied companion manuscript \oplabel{XIV-I.tex}.
Relevant supplied locators:
\oplabel{thm:I2}, \oplabel{thm:I3},
and \oplabel{thm:release}.
The internal amplitude construction and the remaining full
joint-map obligation are distinct.
Source archive description: \url{https://enter.grok.me/}.

\bibitem{CheungXIVJ}
N.~Cheung,
\emph{Landmark XIV-J}, version 1.
Supplied companion manuscript \oplabel{XIV-J.tex}.
Relevant supplied locators:
\oplabel{prop:descent}, \oplabel{thm:reference},
\oplabel{thm:J2}, and \oplabel{cor:noadmission}.
The supported-sector reference is not the full characteristic map.
Source archive description: \url{https://enter.grok.me/}.

\bibitem{Choi}
M.-D.~Choi,
``Completely positive linear maps on complex matrices,''
\emph{Linear Algebra and its Applications}
\textbf{10}(3) (1975), 285--290.
\href{https://doi.org/10.1016/0024-3795(75)90075-0}
{doi:10.1016/0024-3795(75)90075-0}.

\bibitem{Watrous}
J.~Watrous,
\emph{The Theory of Quantum Information}.
Cambridge University Press, 2018.
\href{https://doi.org/10.1017/9781316848142}
{doi:10.1017/9781316848142}.
Author's manuscript and chapter resources:
\url{https://cs.uwaterloo.ca/~watrous/TQI/}.

\bibitem{Fritz}
T.~Fritz,
``Velocity polytopes of periodic graphs and a no-go theorem
for digital physics,''
\emph{Discrete Mathematics}
\textbf{313} (2013), 1289--1301.
arXiv:\href{https://arxiv.org/abs/1109.1963}{1109.1963}.
\href{https://doi.org/10.1016/j.disc.2013.02.010}
{doi:10.1016/j.disc.2013.02.010}.

\bibitem{SchumacherWerner}
B.~Schumacher and R.~F.~Werner,
\emph{Reversible quantum cellular automata}.
arXiv:\href{https://arxiv.org/abs/quant-ph/0405174}
{quant-ph/0405174}, 2004.
Cited as a structural comparison, not a classification result for
the arbitrary complete instruments used here.

\bibitem{Song}
X.~Song,
\emph{Quantum Cellular Automaton Model for the Generalized
Dirac Equation}.
arXiv:\href{https://arxiv.org/abs/1610.02425}{1610.02425}.
Cited only to distinguish an equation-of-motion continuum
comparison from the present matched operational transport limit.

\end{thebibliography}

\end{document}