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  pdftitle={En08-B: Resource-controlled operational approximation
    of null boundary transport}
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\title{\textbf{En08-B}\\[3pt]
Resource-controlled operational approximation\\
of null boundary transport}
\author{Ngo Cheung}
\date{}

\begin{document}
\maketitle

\begin{abstract}
We specify a Euclidean-calibrated family of finite quantum circuits
and prove one operational boundary-approximation theorem.
At refinement \(j\), the spatial mesh is \(\ell/j^2\), the permitted
interaction radius is \(\ell/j\), and one circuit layer takes
\(\ell/(cj)\), where \(c>0\) is supplied.
Every prescribed future-null displacement is approximated by a
payload-independent sequence of local SWAP operations and waits.
The spatial error is \(O(j^{-2})\), the nonnegative arrival delay
is \(O(j^{-1})\), and the ideal logical channel is exactly the
identity on a fixed finite-dimensional payload, including every
untouched reference.
All bounds are uniform on bounded-distance target families.
The construction uses \(O(j)\) ticks and \(O(j)\) initialized
registers; a matching leading-order register lower bound holds
in the declared static-site model.
Complete implementation errors are charged by channel telescoping.
In particular, per-gate half-diamond error \(o(j^{-1})\) is sufficient
when preparations, spectators, waits, and output identifications
are exact.
The result concerns operational approximation of a prescribed
causal boundary. It does not derive the metric, the speed,
Lorentz covariance, a continuum field theory, or gravitational
dynamics.
\end{abstract}

\noindent
\textbf{Keywords:}
quantum transport; circuit locality; causal boundary;
diamond norm; lattice refinement; resource accounting.

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

A geometrically specified causal cone and an operational transport
certificate are different objects. A boundary vector in a cone,
or in the closure of a set of displacements, does not by itself
supply a finite protocol that transmits an unknown quantum state.
The supplied description of Cheung's En08-A emphasizes this
distinction, including the possibility that a generated null
boundary contains no attained null event pair
\cite{CheungEn08A}.

The present paper supplies a connecting theorem in one explicitly
declared model:
\[
\begin{gathered}
 \text{prescribed future-null displacement}\\
 \Longrightarrow\\
 \text{finite local transport protocols with complete-channel}\\
 \text{correctness, uniform geometric control, and a resource ledger}.
\end{gathered}
The implication is constructed directly. No result concerning
cone selection is used as a premise.

\paragraph{Relationship to earlier transport constructions.}
The supplied En05-B description already includes
direction-uniform relays, reference-preserving quantum transport,
and matched finite experiments. The supplied En06-B description
already includes prescribed-speed finite-network transport
\cite{CheungEn05B,CheungEn06B}.
Accordingly, En08-B is an independently proved quantitative
specialization and synthesis, not a claim that either
reference-preserving transport or increasing-stencil isotropy
is new in general.
Its particular content is a single-payload, two-site-gate model
with explicit endpoint bounds, a future-directed arrival convention,
a static-site memory lower bound, and complete noise accounting.

The distinction between scheduled ancestry and actual influence
is also retained
\cite{CheungEn05A,CheungVIIJ,CheungVIIIC}.
The support argument below gives an upper bound for every admissible
circuit. A separately constructed SWAP circuit gives the positive
transport witness.

\paragraph{External context.}
Finite periodic transport structures generally have polyhedral
large-scale velocity sets; Fritz gives a systematic treatment
\cite{Fritz2013}.
Our directional permission set therefore grows with refinement.
This does not contradict a fixed-stencil obstruction.

There are also constructive representation results relating
unitary quantum causality to local circuit implementation
\cite{ArrighiNesmeWerner}.
We do not invoke such a representation theorem: the local circuits
used here are specified explicitly.

\paragraph{Inputs, rather than conclusions.}
Throughout, the following are externally supplied:
the Euclidean spatial metric, spatial and temporal calibration,
the length unit \(\ell\), the speed \(c\), finite-dimensional quantum
channel semantics, and the permission rule.
The target cone is
\begin{equation}
 K_c=\{(T,x)\in\RR_{\ge0}\times\RR^d:
                  \norm{x}_2\le cT\}.
 \label{eq:target-cone}
\end{equation}
We approximate its prescribed null boundary. We do not infer this
cone from scheduler consistency or from quantum computation.

\section{The frozen model}
\label{sec:model}

\subsection{Geometry and refinement}

Fix
\[
 d\ge2,\qquad D\ge2,\qquad \ell>0,\qquad c>0,
 \qquad \kappa=\sqrt d.
\]
The restriction \(D\ge2\) makes unknown-state transport and signalling
nontrivial. The geometric estimates also hold for \(D=1\), but a
one-dimensional payload does not witness nontrivial influence.

For each integer \(j\ge1\), declare
\begin{equation}
 h_j=\frac{\ell}{j^2},\qquad
 R_j=j,\qquad
 a_j=h_jR_j=\frac{\ell}{j},\qquad
 \delta_j=\frac{a_j}{c}=\frac{\ell}{cj}.
 \label{eq:scales}
\end{equation}
The spatial permission lattice is
\[
 \Lambda_j=h_j\ZZ^d,
\]
and the displacement stencil is
\begin{equation}
 S_j=\{s\in\ZZ^d:\norm{s}_2\le j\}.
 \label{eq:stencil}
\end{equation}
In particular, \(0\in S_j\).

The stencil is finite at each \(j\). Its physical radius \(a_j\)
tends to zero, whereas its radius in lattice units grows.
No fixed-stencil or fixed-degree substrate is asserted.

\subsection{Finite realizations and one-layer ticks}

\begin{definition}[Admissible finite realization]
\label{def:realization}
At refinement \(j\), a finite realization consists of:
\begin{enumerate}[label=\textnormal{(\roman*)},leftmargin=*]
\item a finite set \(X\subset\Lambda_j\) containing a source \(0\)
      and the declared destination;
\item one register \(A_v\cong\mathbb C^D\) at each \(v\in X\);
\item a prescribed finite sequence of partitions
      \(\mathcal B_1,\ldots,\mathcal B_n\) of \(X\);
\item for each tick \(k\), a channel
      \[
       C_k=\bigotimes_{B\in\mathcal B_k}C_{k,B}
      \]
      on \(\Lin(\HH_X)\), where
      \[
       \HH_X=\bigotimes_{v\in X}A_v;
      \]
\item blocks \(B\) of cardinality at most two, with
      \(\diam(B)\le a_j\), and completely positive
      trace-preserving maps \(C_{k,B}\) on the fixed block registers.
\end{enumerate}
Each tick is exactly one such partition layer and takes time
\(\delta_j\).
The partitions and channels are fixed before the payload is supplied
and do not depend on its state.
\end{definition}

A two-site block \(\{u,v\}\) is permitted precisely when
\[
 (v-u)/h_j\in S_j.
\]
Singleton channels include the identity and local state replacement.
A circuit may use arbitrary finite \(X\), but no physical register
outside \(X\) is available during that realization.

The phrase ``one tick'' is important.
A sequence of overlapping two-site gates is not silently included
inside one tick. If such gates require successive layers, each layer
is charged its own \(\delta_j\).

\begin{remark}[Controls and observations]
\label{rem:controls}
A target-dependent route may be selected before injection.
The target and route are classical task data, not payload data.
Preinstalled command lists do not receive the payload or measurement
results during execution.

Every channel is unconditional and trace preserving.
No successful branch, normalized postselection, instantaneous
external feedback, or uncounted measurement-record wire is admitted.
A more general adaptive model would have to place every quantum
and classical control wire inside a separately specified local
interface.

Only tick-boundary input and output access is defined.
No continuous-time Hamiltonian implementation or within-tick
access theorem is claimed.
\end{remark}

\subsection{Preparation and the comparison interface}

Let \(M\cong\mathbb C^D\) be the logical payload space.
At every site use the same declared computational basis and the
same blank state
\[
 \sigma=\kb{0}{0}.
\]
For a realization \(X\), the encoding channel is
\begin{equation}
 E_X(Z)=Z_{A_0}\otimes
          \bigotimes_{v\in X\setminus\{0\}}\sigma_{A_v}.
 \label{eq:encoder}
\end{equation}
All auxiliary registers are prepared before time zero, independently
of the payload and of any reference. Transport time is measured
from availability of this primed finite network and the source
payload. Preparation time and physical apparatus construction
are not included in the transport-time estimate.

For a destination \(b\in X\), define
\begin{equation}
 D_{X,b}(Y)=
       \Tr_{X\setminus\{b\}}Y,
 \label{eq:decoder}
\end{equation}
followed by the fixed basis identification \(A_b\cong M\).
This is an output-reduction convention, not a physical command
that transports discarded systems to the destination.

The logical channel of a realization is
\begin{equation}
 \Phi_{X,b}
   =D_{X,b}\,C_n\cdots C_1\,E_X.
 \label{eq:logical-channel}
\end{equation}
Thus the physical interface may grow with \(j\), while the comparison
interface \(M\) remains fixed.

\subsection{Reference-stable distances}

For finite-dimensional compatible channels \(\Phi,\Psi\), use
\begin{equation}
 \ddia(\Phi,\Psi)=\frac12\norm{\Phi-\Psi}_{\diamond},
 \qquad
 \dtr(\rho,\omega)=\frac12\norm{\rho-\omega}_1.
 \label{eq:distances}
\end{equation}
One definition of the diamond norm is
\[
 \norm{\Theta}_{\diamond}
 =
 \sup_{r\ge1}\ \sup_{\norm{Z}_1=1}
       \norm{(\Theta\otimes\id_r)(Z)}_1.
\]
In finite input dimension the reference size can be bounded by
the input dimension. Channels have diamond norm one, and compatible
pre- and post-composition by channels are contractions
\cite[Chapter~3]{Watrous2018}.

The use of a complete channel, rather than selected input tests,
is compatible with the finite-channel conventions in the supplied
Cheung manuscripts
\cite{CheungEn05C,CheungEn02C}.
The corresponding finite telescoping estimate is proved below.

\section{Support, signalling, and necessary resources}
\label{sec:locality}

For \(Y\subseteq X\), write
\[
 \cA(Y)=\Lin(\HH_Y)\otimes I_{X\setminus Y}.
\]
For a circuit of \(n\) ticks and an output set \(Y\), define its
scheduled ancestor sets backwards:
\begin{align}
 N_n(Y)&=Y, \nonumber\\
 N_{k-1}(Y)&=
 \bigcup\{B\in\mathcal B_k:B\cap N_k(Y)\ne\varnothing\},
 \qquad 1\le k\le n.
 \label{eq:ancestors}
\end{align}

\begin{lemma}[One-layer support bound]
\label{lem:support}
For every admissible realization,
\begin{equation}
 (C_1^*\cdots C_n^*)(\cA(Y))
       \subseteq\cA(N_0(Y)),
 \label{eq:adjoint-support}
\end{equation}
and
\begin{equation}
 N_0(Y)\subseteq
       \{v\in X:\dist(v,Y)\le na_j\}.
 \label{eq:metric-support}
\end{equation}
Equal input reductions on \(N_0(Y)\) give equal output reductions
on \(Y\).
With an untouched reference \(R\), equality of the joint
\(N_0(Y)\)--\(R\) input reductions gives equality of the joint
\(Y\)--\(R\) output reductions.
\end{lemma}

\begin{proof}
At one backwards tick, a block disjoint from the current support
acts on an identity factor. Its adjoint is unital and therefore
preserves that factor. All other blocks can enlarge the support
only to their union. Induction proves
\eqref{eq:adjoint-support}.

Every point added at a backwards step is within distance \(a_j\)
of a point in the preceding backwards set. Following these steps
and applying the triangle inequality proves
\eqref{eq:metric-support}.

The adjoint inclusion makes every output expectation depend only
on the indicated input reduction. Output expectations determine
the reduced output state. Tensoring the inclusion with the identity
on \(R\) proves the referenced assertion.
\end{proof}

This proof is included in full. The similar prescribed-block
statement in VIII-C is a comparison, not an imported hypothesis
\cite[Theorem \texttt{thm:C5}]{CheungVIIIC}.

\begin{definition}[Payload influence]
\label{def:influence}
For a fixed circuit and fixed auxiliary preparation,
the payload influences the destination at tick \(n\) if its
logical channel is not a replacement channel.
Equivalently, some two payload states produce different
unconditioned destination states.
The circuit, auxiliary preparation, and output interface are held
fixed under this intervention.
\end{definition}

\begin{corollary}[Exact locality lower bound]
\label{cor:speed}
If \(\norm{b}_2>na_j\), the source-to-\(b\) logical channel is a
replacement channel. In particular, no admissible protocol can
transmit payload influence, or implement \(\id_D\), in time
strictly less than
\[
 \frac{\norm{b}_2}{c}.
\]
\end{corollary}

\begin{proof}
For \(Y=\{b\}\), the source does not belong to \(N_0(Y)\).
Every encoded payload state has the same reduction on that
ancestor set, namely the fixed auxiliary reduction.
Lemma~\ref{lem:support} therefore makes the destination state
independent of the payload.

Since \(a_j=c\delta_j\), the condition
\(n\delta_j<\norm{b}_2/c\) is exactly
\(na_j<\norm{b}_2\).
A replacement channel cannot equal \(\id_D\) when \(D\ge2\).
\end{proof}

\begin{remark}[References do not evade the lower bound]
For an encoded state \(\rho_{MR}\), the auxiliary systems are
independent of \(MR\). If the source lies outside the ancestor set,
the joint destination--reference output is
\(\omega_b\otimes\rho_R\).
For local trace-preserving payload interventions, \(\rho_R\) is
unchanged. The reference is not an extra communication wire.
\end{remark}

\begin{proposition}[Static-site register lower bound]
\label{prop:memory-lower}
Suppose an admissible realization uses \(q=\abs{X}\) sites and
transmits payload influence from \(0\) to \(b\).
Then
\begin{equation}
 q\ge1+\ceil{\frac{\norm{b}_2}{a_j}}
      =1+\ceil{\frac{j\norm{b}_2}{\ell}}.
 \label{eq:memory-lower}
\end{equation}
Also, its number \(n\) of ticks satisfies
\[
 n\ge\ceil{\frac{\norm{b}_2}{a_j}}.
\]
\end{proposition}

\begin{proof}
Influence requires \(0\in N_0(\{b\})\), since otherwise the argument
of Corollary~\ref{cor:speed} gives a replacement channel.
Unwinding the ancestor recursion supplies a scheduled walk
connecting \(0\) to \(b\), with one displacement of length at most
\(a_j\) per tick.

Form the undirected graph on \(X\) containing every two-site block
used by the circuit. The walk places \(0\) and \(b\) in the same
connected component. A shortest graph path between them is simple,
so it uses at most \(q-1\) edges. Each edge has Euclidean length
at most \(a_j\). Consequently
\[
 \norm{b}_2\le(q-1)a_j.
\]
This gives \eqref{eq:memory-lower}. The scheduled walk similarly
gives \(\norm{b}_2\le na_j\).
\end{proof}

The bound is specific to the static-site realization.
Moving carriers, undeclared storage, or allocation at new locations
would change the model and its resource question.

\section{Explicit routing and complete quantum transport}
\label{sec:routing}

\subsection{A deterministic rounding convention}

Define coordinatewise nearest-lattice rounding by
\begin{equation}
 (\Qj z)_i
     =h_j\floor{\frac{z_i}{h_j}+\frac12},
 \qquad z\in\RR^d.
 \label{eq:rounding}
\end{equation}
Ties are rounded towards the larger coordinate.
This map is coordinatewise monotone, fixes \(\Lambda_j\), and satisfies
\begin{equation}
 \norm{\Qj z-z}_2\le\epsj,
 \qquad
 \epsj=\frac{\kappa h_j}{2}
      =\frac{\kappa\ell}{2j^2}.
 \label{eq:rounding-error}
\end{equation}

\begin{lemma}[Uniform lattice routing]
\label{lem:routing}
Let \(j>\kappa\) and \(b\in\Lambda_j\).
Set \(r=\norm{b}_2\).
For \(b=0\), set \(N_j(b)=0\).
For \(b\ne0\), set
\begin{equation}
 N_j(b)=
 \ceil{\frac{r}{h_j(j-\kappa)}}.
 \label{eq:route-count}
\end{equation}
There is an explicit route
\[
 p_0=0,p_1,\ldots,p_N=b,\qquad N=N_j(b),
\]
whose increments belong to \(h_jS_j\).
Its time
\[
 \tau_j(b)=N_j(b)\delta_j
\]
satisfies
\begin{equation}
 0\le\tau_j(b)-\frac{r}{c}
   \le
   \frac{\kappa r}{c(j-\kappa)}+\delta_j.
 \label{eq:lattice-delay}
\end{equation}
If \(r\le L\) and \(j\ge2\kappa\), then
\begin{equation}
 N_j(b)\le\frac{2Lj}{\ell}+1.
 \label{eq:uniform-ticks}
\end{equation}
\end{lemma}

\begin{proof}
The zero case uses the single point \(0\), no gates, and zero time.

For \(b\ne0\), let \(N\) be given by \eqref{eq:route-count} and define
\begin{equation}
 p_k=\Qj\left(\frac{k}{N}b\right),
 \qquad 0\le k\le N.
 \label{eq:route-points}
\end{equation}
Because rounding fixes lattice points, \(p_0=0\) and \(p_N=b\).
Write
\[
 e_k=p_k-\frac{k}{N}b,\qquad \norm{e_k}_2\le\epsj.
\]
Then
\[
 \norm{p_k-p_{k-1}}_2
 \le \frac{r}{N}+2\epsj
 \le h_j(j-\kappa)+\kappa h_j
 =h_jj=a_j.
\]
Both points are lattice points, so every increment is in \(h_jS_j\).
An increment may be zero; it is a permitted wait.

The triangle inequality along the route gives
\(r\le Na_j=cN\delta_j\), proving the lower bound.
Also,
\[
 N\le\frac{r}{h_j(j-\kappa)}+1,
\]
so
\[
 N\delta_j
 \le\frac{rj}{c(j-\kappa)}+\delta_j.
\]
Subtracting \(r/c\) proves \eqref{eq:lattice-delay}.
Finally, \(j-\kappa\ge j/2\) gives
\[
 N\le\frac{rj^2}{\ell(j-\kappa)}+1
   \le\frac{2Lj}{\ell}+1.
\]
\end{proof}

\begin{lemma}[Repeated vertices and the register count]
\label{lem:repetitions}
For the route \eqref{eq:route-points}, every coordinate sequence
\(k\mapsto(p_k)_i\) is monotone.
If \(p_k=p_l\) for \(k<l\), then every intermediate vertex equals
that same point.
Consequently, after deleting consecutive repetitions, the route
has distinct vertices.
If \(m\) is its number of nonzero increments and \(q\) its number
of distinct sites, then
\begin{equation}
 q=m+1\le N+1.
 \label{eq:distinct-sites}
\end{equation}
\end{lemma}

\begin{proof}
The affine coordinate \(kb_i/N\) is monotone, and the scalar
rounding rule is monotone. A monotone scalar sequence with equal
values at two indices has the same value at every intermediate
index. Applying this to every coordinate proves the assertion
about repeated vertices. Each nonzero increment then introduces
one new site.
\end{proof}

\subsection{An exact SWAP protocol}

Let \(X\) be the set of distinct route sites.
Prepare \eqref{eq:encoder}.
At tick \(k\):
\begin{itemize}[leftmargin=*]
\item if \(p_k=p_{k-1}\), apply the identity to all sites;
\item otherwise, use the block \(\{p_{k-1},p_k\}\), apply its SWAP
      channel, and apply identity singleton channels elsewhere.
\end{itemize}
This is one partition layer. The block diameter is at most \(a_j\),
and the schedule is determined by the target, not by the payload.

\begin{lemma}[Exact reference-preserving transport]
\label{lem:identity}
The route protocol implements
\[
 \Phi_{j,b}=\id_D.
\]
More strongly, for every untouched reference \(R\) and every state
\(\rho_{MR}\), its physical state after tick \(k\) is
\begin{equation}
 \rho_{A_{p_k}R}\otimes
       \bigotimes_{v\in X\setminus\{p_k\}}\sigma_{A_v},
 \label{eq:state-invariant}
\end{equation}
under the fixed register identifications.
Every non-output route register is therefore in a
payload-independent state, jointly independent of the payload
and reference, at the terminal tick.
\end{lemma}

\begin{proof}
At tick zero, \eqref{eq:state-invariant} is precisely the declared
product preparation.

A wait preserves the expression.
For a nonzero increment, the target register is blank and independent
of every other factor. SWAP exchanges the entire current payload
factor, including its correlations with \(R\), with that target
factor. The former payload site becomes blank.
This proves the invariant by induction.

At \(k=N\), the payload-reference state is \(\rho_{A_bR}\), and
all other factors are blank. Taking the destination reduction and
identifying \(A_b\) with \(M\) gives
\[
 (\Phi_{j,b}\otimes\id_R)(\rho_{MR})=\rho_{MR}.
\]
The argument is an identity of tensor-factor permutations, so it
holds for arbitrary reference Hilbert spaces carrying density
operators, not only for references of a particular finite size.
\end{proof}

\begin{remark}[Residuals and records]
The payload is not measured, and the protocol produces no destructive
classical payload record. Its non-output quantum registers remain
part of the resource ledger until terminal reduction.
The distinction between a record and a residual system, emphasized
in the supplied VIII-A material, is respected directly
\cite{CheungVIIIA}. Classical influence--reachability equality is
not used to infer Lemma~\ref{lem:identity}.
\end{remark}

\section{The connecting theorem}
\label{sec:main}

\subsection{Null targets and future-directed rounding}

Let \((T,x)\in K_c\) be null:
\[
 \norm{x}_2=cT.
\]
Define
\begin{equation}
 x_j=\Qj x,\qquad
 N=N_j(x_j),\qquad
 n_j(T,x)=\max\left\{N,\ceil{\frac{T}{\delta_j}}\right\},
 \qquad
 T_j=n_j(T,x)\delta_j.
 \label{eq:null-construction}
\end{equation}
First execute the route to \(x_j\).
If necessary, wait there until tick \(n_j(T,x)\).
Thus \(T_j\ge T\).

This final waiting convention repairs a possible sign issue:
rounding \(x\) inward can make the unpadded route arrive slightly
before the original continuum target time. Spatial rounding alone
does not prove a nonnegative delay relative to \(T\).

For \(L>0\), define
\begin{align}
 B_j(L)&=
 \frac{\epsj}{c}
 +\frac{\kappa(L+\epsj)}{c(j-\kappa)}
 +\delta_j,
 \label{eq:B}\\
 H_j(L)&=
 \ceil{\frac{L+\epsj}{h_j(j-\kappa)}}+1.
 \label{eq:H}
\end{align}

\begin{lemma}[Endpoint, waiting, and uniform resource bounds]
\label{lem:endpoint}
For \(j>\kappa\) and every null target with \(\norm{x}_2\le L\),
the construction \eqref{eq:null-construction} satisfies
\begin{align}
 \norm{x_j-x}_2&\le\epsj,
 \label{eq:space-bound}\\
 0\le T_j-T&\le B_j(L).
 \label{eq:time-bound}
\end{align}
At most one extra terminal waiting tick is needed.
The total tick count and the number \(q_j\) of initialized sites obey
\begin{equation}
 n_j(T,x)\le H_j(L),\qquad q_j\le H_j(L).
 \label{eq:resource-H}
\end{equation}
For \(j\ge2\kappa\),
\begin{align}
 B_j(L)&\le
 \frac{2\kappa L+\ell}{cj}
 +\frac{\kappa\ell}{2cj^2}
 +\frac{\kappa^2\ell}{cj^3},
 \label{eq:B-expanded}\\
 H_j(L)&\le\frac{2Lj}{\ell}+3.
 \label{eq:H-expanded}
\end{align}
\end{lemma}

\begin{proof}
The spatial estimate is \eqref{eq:rounding-error}.
Put \(r=\norm{x}_2=cT\) and \(r_j=\norm{x_j}_2\).
Then
\[
 \abs{r_j-r}\le\epsj.
\]
By Lemma~\ref{lem:routing},
\[
 \tau_j(x_j)\ge\frac{r_j}{c}
      \ge T-\frac{\epsj}{c},
\]
and
\[
 \tau_j(x_j)-T
 \le \frac{\epsj}{c}
     +\frac{\kappa(L+\epsj)}{c(j-\kappa)}
     +\delta_j
 =B_j(L).
\]
The upward-rounded target time has delay less than \(\delta_j\).
Taking the maximum of these two tick-boundary times proves
\eqref{eq:time-bound}.

If \(\tau_j(x_j)<T\), then
\[
 0<T-\tau_j(x_j)
 \le\frac{\epsj}{c}
 =\frac{\kappa}{2j}\delta_j
 <\delta_j.
\]
Since \(\tau_j(x_j)\) is already a tick-boundary time, one additional
tick suffices. Otherwise no additional tick is needed.

The route uses at most
\[
 \ceil{\frac{L+\epsj}{h_j(j-\kappa)}}
\]
ticks and one more site than its number of nonzero increments.
Including the possible terminal wait gives \eqref{eq:resource-H}.
The zero target uses one source register and zero ticks and also
satisfies these estimates.

For \(j\ge2\kappa\), substitute \(j-\kappa\ge j/2\) and
\(\epsj=\kappa\ell/(2j^2)\) into \eqref{eq:B}.
This gives \eqref{eq:B-expanded}.
Similarly,
\[
 H_j(L)
 \le\frac{2(L+\epsj)j}{\ell}+2
 =\frac{2Lj}{\ell}+\frac{\kappa}{j}+2
 \le\frac{2Lj}{\ell}+3.
\]
\end{proof}

\subsection{A complete-channel error lemma}

\begin{lemma}[Finite complete-channel telescoping]
\label{lem:telescoping}
Let \(E,D,G_1,\ldots,G_n\) be compatible channels, and let their
implemented counterparts be compatible channels on the same
comparison interfaces. Then
\begin{align}
 &\ddia\left(
 \widetilde D\widetilde G_n\cdots\widetilde G_1\widetilde E,
 DG_n\cdots G_1E\right)
 \nonumber\\
 &\qquad\le
 \ddia(\widetilde E,E)
 +\sum_{k=1}^n\ddia(\widetilde G_k,G_k)
 +\ddia(\widetilde D,D).
 \label{eq:telescoping}
\end{align}
The left-hand side is also at most one.
\end{lemma}

\begin{proof}
Subtract the two compositions and replace one factor at a time.
Each term contains exactly one difference of corresponding factors,
with channels before and after it. Diamond-norm contraction under
those compositions bounds that term by the diamond norm of the
factor difference. The triangle inequality and division by two
give \eqref{eq:telescoping}.
The distance between two channels in half-diamond normalization
is at most one.
\end{proof}

The corresponding Heisenberg complete-bounded-norm ledger is
recorded in En05-C
\cite[Proposition \texttt{prop:telescoping}]{CheungEn05C}.
Equation~\eqref{eq:telescoping} is used on the actual compatible
interfaces, including any spectator registers.

\begin{theorem}[Resource-controlled operational null-boundary approximation]
\label{thm:main}
In the model of Section~\ref{sec:model}, the following hold.

\begin{enumerate}[label=\textnormal{(\roman*)},leftmargin=*]
\item \textbf{Exact locality ceiling.}
For every admissible \(n\)-tick realization, payload influence
from \(0\) to \(b\) requires
\[
 \norm{b}_2\le cn\delta_j.
\]

\item \textbf{Uniform lattice transport.}
For \(j>\kappa\) and every \(b\in\Lambda_j\), the explicit route
implements \(\id_D\), including every untouched reference, at time
\(\tau_j(b)\), with
\[
 0\le\tau_j(b)-\frac{\norm{b}_2}{c}
 \le\frac{\kappa\norm{b}_2}{c(j-\kappa)}+\delta_j.
\]
It uses \(N_j(b)\) ticks, \(m_j(b)\le N_j(b)\) SWAPs, and
\(q_j(b)=m_j(b)+1\le N_j(b)+1\) initialized sites.

\item \textbf{Uniform null-target approximation.}
For every \(L>0\), every null target
\(\norm{x}_2=cT\le L\), and every \(j>\kappa\), there is a finite
payload-independent admissible protocol with destination \(x_j\),
arrival time \(T_j\), and logical channel \(\Phi_j(T,x)\), such that
\[
 \norm{x_j-x}_2\le\epsj,\qquad
 0\le T_j-T\le B_j(L),\qquad
 \Phi_j(T,x)=\id_D.
\]
It uses at most \(H_j(L)\) ticks and \(H_j(L)\) initialized sites.

\item \textbf{Conditional implementation convergence.}
Suppose the encoding and decoding errors of these protocols are
at most \(\alpha_j\) and \(\beta_j\), uniformly over the target family,
and every complete tick map has error at most \(e_j\).
Then
\begin{equation}
 \sup_{\norm{x}_2=cT\le L}
 \ddia\bigl(\widetilde\Phi_j(T,x),\id_D\bigr)
 \le
 \min\{1,\alpha_j+H_j(L)e_j+\beta_j\}.
 \label{eq:main-noise}
\end{equation}
In particular, \(\alpha_j\to0\), \(\beta_j\to0\), and \(je_j\to0\)
are sufficient for uniform complete-channel convergence.
\end{enumerate}

For a fixed nonzero null target \(r=\norm{x}_2>0\), the constructed
tick and initialized-site counts additionally satisfy
\begin{equation}
 \frac{n_j(T,x)}{j}\longrightarrow\frac r\ell,
 \qquad
 \frac{q_j(T,x)}{j}\longrightarrow\frac r\ell.
 \label{eq:leading-resources}
\end{equation}
The site count has the optimal leading coefficient in the
declared static-site model.
\end{theorem}

\begin{proof}
Part (i) is Corollary~\ref{cor:speed}.
Part (ii) combines Lemmas~\ref{lem:routing},
\ref{lem:repetitions}, and \ref{lem:identity}.
Terminal identity waits preserve the complete transport identity,
so Lemma~\ref{lem:endpoint} gives part (iii).
Lemma~\ref{lem:telescoping} and the bound on the total number of
ticks give part (iv).

For the last assertion, set \(r_j=\norm{x_j}_2\).
The route has \(m_j\) nonzero increments, so
\[
 \frac{r_j}{a_j}\le m_j\le N_j(x_j).
\]
For nonzero \(x_j\),
\[
 \frac{N_j(x_j)}{j}
 \le\frac{r_jj}{\ell(j-\kappa)}+\frac1j.
\]
Since \(r_j\to r>0\), these inequalities imply
\[
 \frac{m_j}{j}\longrightarrow\frac r\ell,
 \qquad
 \frac{N_j(x_j)}{j}\longrightarrow\frac r\ell.
\]
There is at most one additional wait, and \(q_j=m_j+1\).
This proves \eqref{eq:leading-resources}.

For any other admissible static-site protocol that influences the
same rounded destination, Proposition~\ref{prop:memory-lower} gives
\[
 q\ge1+\ceil{\frac{jr_j}{\ell}}.
\]
Therefore its asymptotic site count cannot have a smaller leading
coefficient.
\end{proof}

\subsection{The precise comparison topology}

\begin{definition}[Endpoint operational comparison]
\label{def:comparison}
For a target \((T,x,\id_D)\), define the dimensionless comparison
error of an implemented witness by
\begin{equation}
 \mathsf{Err}_j(T,x)=
 \frac{\norm{x_j-x}_2+c\abs{T_j-T}}{\ell}
 +\ddia\bigl(\widetilde\Phi_j(T,x),\id_D\bigr).
 \label{eq:operational-error}
\end{equation}
\end{definition}

\begin{corollary}[Uniform endpoint operational limit]
\label{cor:operational-limit}
Under the assumptions of Theorem~\ref{thm:main}(iv),
\begin{align}
 \sup_{\norm{x}_2=cT\le L}\mathsf{Err}_j(T,x)
 \le&
 \frac{\epsj+cB_j(L)}{\ell}
 \nonumber\\
 &+\min\{1,\alpha_j+H_j(L)e_j+\beta_j\}.
 \label{eq:operational-limit}
\end{align}
The right-hand side tends to zero whenever
\(\alpha_j,\beta_j\to0\) and \(je_j\to0\).
In the ideal construction, the channel-error term is zero at every
finite refinement.
\end{corollary}

\begin{proof}
Substitute the three estimates from Theorem~\ref{thm:main} into
\eqref{eq:operational-error}.
Equations~\eqref{eq:B-expanded} and \eqref{eq:H-expanded}
give the stated convergence.
\end{proof}

This is an endpoint transport limit on a common payload interface.
It is not a construction of a spatial continuum observable algebra,
and it does not supply a common family of global evolution channels
at all intermediate times. In particular, it is not silently
identified with the operational evolution-limit definition in
XIX-C
\cite[Definition \texttt{def:limit}]{CheungXIXC}.

\section{Resource ledger and effective specification}
\label{sec:ledger}

\subsection{Quantum registers and gate resources}

For the constructed route, let \(N\) be its unpadded tick count,
\(m\) the number of nonzero increments, \(q=m+1\) the number of
distinct route sites, and \(w\in\{0,1\}\) its terminal padding count.
The total number of ticks is \(n=N+w\).

All \(q\) registers exist at time zero and are retained until the
terminal output reduction. Thus the global accounted peak is
\(q\), even though only two registers participate in a nontrivial
gate at a given tick.

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{0.36\linewidth}X@{}}
\toprule
Quantity & Exact statement or uniform bound\\
\midrule
Logical payload dimension & \(D\), fixed independently of \(j\).\\
Supplied payload registers & One, at the source.\\
Fresh auxiliary registers & \(q-1=m\), each prepared once in \(\sigma\).\\
Total initialized sites & \(q=m+1\le N+1\).\\
Global retained-register peak & \(q\).\\
Terminal quantum output & One \(D\)-dimensional destination register.\\
Non-output registers ignored at the end & \(q-1\), all blank in the ideal protocol.\\
Two-site SWAP count & \(m\le N\).\\
Tick count & \(n=N+w\), with \(w\le1\) for null targets.\\
Largest active block & Two registers; block Hilbert dimension \(D^2\).\\
Physical interaction radius & \(a_j=\ell/j\).\\
Ideal logical channel error & Zero.\\
Null-target spatial error & At most \(\kappa\ell/(2j^2)\).\\
Null-target arrival delay & At most \(B_j(L)=O(j^{-1})\).\\
Uniform ticks and sites & At most \(H_j(L)\le2Lj/\ell+3\).\\
\bottomrule
\end{tabularx}
\end{center}

The total quantum Hilbert dimension is \(D^q\).
A fixed logical dimension is therefore not a constant global-memory
bound. If a finite reference of dimension \(r_R\) is included in
a global Hilbert-space count, that dimension becomes \(r_RD^q\).
The reference is not initialized or operated on by the transport
protocol.

The ledger concerns quantum register ownership and circuit resources.
It assigns no heat, work, hardware fabrication cost, or Hamiltonian
energy bound to a SWAP or to blank-state preparation.

\subsection{Stencil size and permission data}

The permission stencil obeys
\begin{equation}
 \left(2\floor{\frac{j}{\kappa}}+1\right)^d
 \le\abs{S_j}\le(2j+1)^d.
 \label{eq:stencil-size}
\end{equation}
Indeed, the integer cube with coordinate bound
\(\floor{j/\kappa}\) lies inside the Euclidean radius-\(j\) ball,
and that ball lies inside the coordinate radius-\(j\) cube.
Consequently
\[
 \abs{S_j}=\Theta_d(j^d).
\]

Permission is tested by the integer inequality
\[
 \sum_{i=1}^d s_i^2\le j^2.
\]
The model may use this fixed membership rule instead of storing
a fully enumerated stencil.
If the stencil is explicitly materialized, its entry count is
\(\abs{S_j}\), and its displacement-bit storage is
\(O_d(j^d\log(j+1))\).
These documentary permission choices are not additional quantum
registers.

Only the \(q\) route sites are initialized in a given realization.
The theorem does not require a quantum register at every site in a
bounded continuum region, nor does it count a full infinite lattice
as an initialized finite apparatus.

\subsection{Route description and compilation boundary}

For a lattice destination \(b\) with \(\norm{b}_2\le L\), every
route coordinate in lattice units has magnitude at most
\(\ceil{L/h_j}+1\).
A signed coordinate can therefore be stored with
\[
 b_j(L)=1+\ceil{\log_2\bigl(\ceil{L/h_j}+2\bigr)}
\]
bits. An explicit route list uses at most
\[
 d(N+1)b_j(L)
\]
coordinate bits, together with \(O(\log(N+2))\) bits for its length
and an execution index. For fixed \(d,L,\ell\), this is
\(O(j\log(j+1))\) classical storage.

Given an exact lattice destination, the route is finitely
computable:
\(N\) is determined by comparisons of algebraic numbers, and
\eqref{eq:route-points} uses rational-coordinate rounding.
No optimization oracle is needed.

For an arbitrary real continuum target, \(\Qj x\) specifies a
mathematical family. Effective compilation additionally requires
a target representation sufficient to certify the relevant
rounding choices. No algorithmic access to arbitrary noncomputable
real coordinates is asserted.

\section{Implementation errors and their boundaries}
\label{sec:noise}

\subsection{The gate-only allowance}

\begin{corollary}[Per-SWAP half-diamond allowance]
\label{cor:gate-noise}
Assume exact encoding, exact output reduction, exact waits, and
exact spectator storage. Suppose each implemented SWAP is a
channel on the same two-site interface and differs from the ideal
SWAP channel by at most \(\eta_j\) in half-diamond distance.
Then
\begin{equation}
 \ddia(\widetilde\Phi_j(T,x),\id_D)
 \le\min\{1,m_j(T,x)\eta_j\}
 \le\min\{1,H_j(L)\eta_j\}.
 \label{eq:gate-noise}
\end{equation}
Thus \(j\eta_j\to0\) is sufficient for uniform convergence on
bounded-distance null targets.
\end{corollary}

\begin{proof}
Tensoring a local gate difference with identity channels on the
other registers does not enlarge its diamond norm.
All wait ticks have zero error under the stated assumptions.
Apply Lemma~\ref{lem:telescoping} to the \(m_j\) nontrivial gates.
\end{proof}

A state-specific gate fidelity is not the hypothesis in this
corollary. The allowance must hold for the complete local channel,
including arbitrary extensions by the other route registers and
the untouched reference.

Compatible per-step CPTP channels also constitute a model
assumption. Arbitrary time-correlated environmental memory is not
automatically covered by local channel benchmarks. Such memory
must be included in a declared complete interface on which the
comparison maps and error estimates actually hold.

\subsection{Preparations, storage, waits, and decoding}

The full estimate \eqref{eq:main-noise} charges complete tick maps,
not merely active SWAP gates. This distinction is necessary when
idle storage or the terminal wait is imperfect.

For example, suppose the \(q-1\) auxiliary preparations are
independent states \(\widetilde\sigma_v\) with
\[
 \dtr(\widetilde\sigma_v,\sigma)\le\xi_j.
\]
Product-state telescoping gives
\[
 \dtr\left(
 \bigotimes_{v\ne0}\widetilde\sigma_v,
 \bigotimes_{v\ne0}\sigma\right)
 \le(q-1)\xi_j.
\]
The corresponding encoding-channel half-diamond error has the
same bound: its difference tensors the input with this fixed
state difference.

Consequently, if each complete tick map has error at most \(e_j\)
and the final output map has error at most \(\beta_j\), then
\begin{equation}
 \ddia(\widetilde\Phi_j(T,x),\id_D)
 \le
 \min\{1,(q-1)\xi_j+n_je_j+\beta_j\}.
 \label{eq:preparation-ledger}
\end{equation}
Uniform conditions
\[
 j\xi_j\to0,\qquad je_j\to0,\qquad \beta_j\to0
\]
are sufficient.

No independence between payload and reference is imposed.
Independence is imposed on auxiliary preparation.
Other preparation models may instead use a separately certified
complete encoding allowance.

\subsection{Small error is not exact causal containment}

\begin{proposition}[A small nonlocal perturbation can signal]
\label{prop:near-local}
For every \(0<\epsilon\le1\), there is a channel within
half-diamond distance at most \(\epsilon\) of the identity that
signals across an arbitrarily large spatial separation.
\end{proposition}

\begin{proof}
Place two \(D\)-dimensional registers \(A,B\) at the chosen
separation and define
\[
 \widetilde C
 =(1-\epsilon)\id_{AB}
       +\epsilon\,\Ad_{\SWAP_{AB}}.
\]
It is a channel, and
\[
 \ddia(\widetilde C,\id_{AB})
 \le\epsilon,
\]
because the half-diamond distance between any two channels is
at most one.

Prepare \(B\) in \(\ket0\). Inputs \(\ket0\) and \(\ket1\) at \(A\)
produce respective \(B\)-outputs
\[
 \kb00,\qquad
 (1-\epsilon)\kb00+\epsilon\kb11.
\]
Their trace distance is \(\epsilon>0\).
The separation can exceed \(a_j\), making this channel inadmissible
as one tick of the frozen model.
\end{proof}

Thus implementation closeness alone is not a support theorem.
Exact locality for noisy gates follows when their actual complete
maps remain on the permitted blocks. Without that structural
assumption, only an approximate observational statement follows.
This is consistent with the approximate certificate distinctions
in En05-A and En05-C
\cite[Proposition \texttt{prop:approx}]{CheungEn05A}
\cite[Proposition \texttt{prop:approx}]{CheungEn05C}.

\subsection{Vanishing local errors can accumulate}

\begin{proposition}[A vanishing per-gate error need not vanish globally]
\label{prop:accumulation}
Fix a nonzero null target of distance \(r>0\) and a number
\(\lambda>0\).
There are local gate implementations with per-gate half-diamond
error at most \(\lambda/j\), for all sufficiently large \(j\),
whose logical channels do not converge to \(\id_D\).
\end{proposition}

\begin{proof}
Let
\[
 \mathcal R_D(Z)=\Tr(Z)\frac{I_D}{D},
 \qquad
 \mathcal P_p=(1-p)\id_D+p\mathcal R_D.
\]
At each nonzero route step, apply the ideal SWAP and then
\(\mathcal P_{p_j}\) to the arriving payload register, where
\(p_j=\lambda/j\) for sufficiently large \(j\).
This is a channel on the same permitted two-site block.
Its distance from the ideal SWAP channel is at most \(p_j\).

All other registers remain blank. If the route has \(m_j\) SWAPs,
the logical channel is
\[
 \mathcal P_{p_j}^{\,m_j}
 =(1-p_j)^{m_j}\id_D+
        \bigl(1-(1-p_j)^{m_j}\bigr)\mathcal R_D.
\]
Theorem~\ref{thm:main} gives \(m_j/j\to r/\ell\), so
\[
 (1-\lambda/j)^{m_j}\longrightarrow e^{-\lambda r/\ell}<1.
\]

To see a positive complete-channel discrepancy, apply the channel
to one half of
\[
 \ket{\Omega_D}
 =\frac1{\sqrt D}\sum_{a=0}^{D-1}\ket a\ket a.
\]
The replacement component has overlap \(D^{-2}\) with
\(\ket{\Omega_D}\).
Testing the projector \(\kb{\Omega_D}{\Omega_D}\) therefore gives
\[
 \ddia(\mathcal P_{p_j}^{\,m_j},\id_D)
 \ge
 \bigl(1-(1-p_j)^{m_j}\bigr)(1-D^{-2}),
\]
whose limit is positive.
\end{proof}

The condition \(j\eta_j\to0\) is sufficient, not necessary for every
possible implementation: errors can cancel or have additional
structure. Proposition~\ref{prop:accumulation} shows why merely
assuming \(\eta_j\to0\) is insufficient.

\section{Failure tests and finite-scale attainment}
\label{sec:failures}

\subsection{Fixed normalized stencils}

\begin{proposition}[Fixed-stencil directional obstruction]
\label{prop:fixed-stencil}
Suppose the available one-tick velocities at every refinement lie
in one fixed finite set \(V\subset c\overline B_2(0,1)\).
If all positions are obtained by concatenating those steps,
then every positive-time endpoint velocity lies in
\[
 P=\conv(V).
\]
For \(d\ge2\), these permissions cannot approximate every point
of the Euclidean speed-\(c\) sphere at its null travel time.
\end{proposition}

\begin{proof}
The endpoint velocity of an \(n\)-step route is the arithmetic mean
of its \(n\) step velocities, hence belongs to \(P\).

The finite convex hull \(P\) is a polytope contained in the Euclidean
ball. It cannot contain the entire sphere when \(d\ge2\), since that
would force it to contain the ball and hence equal the ball.
Choose a unit vector \(u\) with \(cu\notin P\).
If endpoints and positive arrival times approached
\((ru,r/c)\), their endpoint velocities would approach \(cu\),
contradicting closedness of \(P\).
\end{proof}

This is the elementary version of the velocity-polytope issue
treated more generally by Fritz
\cite{Fritz2013}.
The growing \(S_j\) in \eqref{eq:stencil} is an explicit resource
choice, not a hidden exception to a fixed-stencil hypothesis.

\subsection{Fixed resource caps}

A fixed tick cap \(n\le n_{\max}\) gives
\[
 \norm{b}_2\le n_{\max}a_j\longrightarrow0.
\]
A fixed initialized-site cap \(q\le q_{\max}\), even with unrestricted
tick count, gives
\[
 \norm{b}_2\le(q_{\max}-1)a_j\longrightarrow0
\]
for every destination influenced by the source.
The second statement follows from
Proposition~\ref{prop:memory-lower}.
Thus the declared construction cannot be reinterpreted as a
fixed-resource approximation of nonzero null distances.

\subsection{Record-only transmission}

For completeness, the relevant quantitative distinction can be
proved on the same payload interface.

\begin{proposition}[Record-only quantum recovery obstruction]
\label{prop:record}
If a payload is converted into a finite classical record and a
decoder prepares a \(D\)-dimensional quantum output from that
record, then the resulting channel \(\Lambda\) satisfies
\begin{equation}
 \ddia(\Lambda,\id_D)\ge1-\frac1D.
 \label{eq:record-bound}
\end{equation}
\end{proposition}

\begin{proof}
The composite channel has measure-and-prepare form
\[
 \Lambda(Z)=\sum_a\Tr(F_aZ)\omega_a,
 \qquad F_a\ge0,\quad\sum_aF_a=I.
\]
Its output on half of \(\kb{\Omega_D}{\Omega_D}\) is separable
between reference and output.

For unit product vectors,
\[
 \abs{\langle\Omega_D\mid u\otimes v\rangle}^2
 \le\frac1D
\]
by Cauchy--Schwarz. Convexity gives the same overlap bound for every
separable state. Testing the Bell projector yields trace distance
at least \(1-1/D\) from \(\kb{\Omega_D}{\Omega_D}\).
This is an admissible referenced channel test and proves
\eqref{eq:record-bound}.
\end{proof}

Measure-and-prepare channels are the standard entanglement-breaking
class \cite{HorodeckiShorRuskai2003}.
The quantitative distinction is also explicitly present in the
supplied VIII-A and VIII-C material
\cite[Theorem \texttt{A:thm:classical}]{CheungVIIIA}
\cite[Proposition \texttt{prop:recovery}]{CheungVIIIC}.
The SWAP construction avoids the premise of
Proposition~\ref{prop:record}: the payload remains a quantum register.

\subsection{Exact discrete null transport is exceptional, not forbidden}

The main theorem is an approximation theorem.
However, this particular permission family does have some exactly
attained finite-scale null pairs. For example, the displacement
\(a_je_1\) is permitted in one tick.

\begin{proposition}[Characterization of finite-scale null attainment]
\label{prop:exact-null}
Fix \(j\ge1\), \(n\ge1\), and \(b\in\Lambda_j\) with
\[
 \norm{b}_2=na_j.
\]
There exists an admissible \(n\)-tick protocol transmitting payload
influence from \(0\) to \(b\) if and only if
\begin{equation}
 b=nh_js
 \quad\text{for some }s\in\ZZ^d
 \text{ satisfying }\norm{s}_2=j.
 \label{eq:exact-null}
\end{equation}
Whenever \eqref{eq:exact-null} holds, exact referenced identity
transport is attainable.
\end{proposition}

\begin{proof}
Influence requires an ancestor walk from \(0\) to \(b\).
Write its \(n\) displacements, including possible zero steps, as
\(v_1,\ldots,v_n\), where
\[
 \norm{v_k}_2\le a_j,\qquad \sum_kv_k=b.
\]
The null equality gives
\[
 na_j=\norm{b}_2
 \le\sum_k\norm{v_k}_2\le na_j.
\]
Hence every step has length \(a_j\), and equality holds in the
Euclidean triangle inequality. All steps have the same direction
and, because their lengths agree, the same vector.
Each is a lattice displacement \(h_js\) with \(\norm{s}_2=j\).
This proves necessity.

Conversely, initialize the \(n+1\) sites
\(0,h_js,\ldots,nh_js\), and apply a SWAP along each successive
edge in one tick. Lemma~\ref{lem:identity} proves complete identity
transport.
\end{proof}

At each fixed \(j\), the attained null directions form a finite set.
Across all \(j\), they form a countable set with rational coordinate
directions \(s/j\). Targets in other directions are nevertheless
approximated by Theorem~\ref{thm:main}.
Accordingly, the correct conclusion is neither
``all null pairs are attained'' nor
``no discrete null pair is attained.''

\section{Limit selection and comparison boundaries}
\label{sec:comparison}

\subsection{What is uniquely selected here}

Let \(t_j^*(b)\) be the smallest arrival time of any admissible
finite protocol implementing \(\id_D\) at \(b\).
This minimum exists: coordinate steps provide a finite route,
and the possible times are nonnegative integer multiples of
\(\delta_j\).

\begin{corollary}[Uniform limit of optimal lattice arrival times]
\label{cor:optimal-arrival}
For \(j>\kappa\) and \(b\in\Lambda_j\),
\[
 \frac{\norm{b}_2}{c}
 \le t_j^*(b)\le\tau_j(b).
\]
Consequently, for every \(L>0\),
\[
 \sup_{\substack{b\in\Lambda_j\\\norm{b}_2\le L}}
 \abs{t_j^*(b)-\norm{b}_2/c}
 \le\frac{\kappa L}{c(j-\kappa)}+\delta_j
 \longrightarrow0.
\]
For rounded continuum endpoints \(x_j=\Qj x\), the same conclusion
holds uniformly for \(\norm{x}_2\le L\), after adding the endpoint
allowance \(\epsj/c\).
\end{corollary}

\begin{proof}
The lower bound is Corollary~\ref{cor:speed}; the explicit route
gives the upper bound. Lemma~\ref{lem:routing} supplies the uniform
estimate. For rounded targets,
\(\abs{\norm{x_j}_2-\norm{x}_2}\le\epsj\).
\end{proof}

The uniquely selected object in this statement is an arrival-time
function inside the declared model.
It is not an unrestricted uniqueness theorem for continuum
evolutions.

The supplied XIX-C example exhibits different refinement choices
with a common coarse channel but different limiting evolutions;
its interpolation statement likewise prevents finitely many
readings from selecting an unprescribed interpolation
\cite[Propositions \texttt{prop:two-limits} and
\texttt{prop:interpolation}]{CheungXIXC}.
Here the refinement, calibration, rounding map, comparison interface,
and route rule are fixed before the limit.

\subsection{Non-transfer statements}

\begin{enumerate}[label=\textnormal{(\roman*)},leftmargin=*]
\item \textbf{No predecessor proof transfer.}
The geometric, support, transport, and error arguments needed
for Theorem~\ref{thm:main} are proved locally.
En05-B and En06-B are documentary comparisons
\cite{CheungEn05B,CheungEn06B}.
Their networks are not identified with the present route family.

\item \textbf{No universal network realization.}
The positive construction transports one fixed-dimensional
payload along one target-dependent route.
It does not establish simultaneous routing, universal adaptive
experiment matching, or a target-independent autonomous schedule.

\item \textbf{No native incidence realization.}
No conical packet, three-to-six incidence structure,
affine-conical certificate, source-dependent lapse, or
residual-matter instrument is used.
The finite support and recovery comparisons in VIII-A and VIII-C
do not identify their geometric models with this Euclidean lattice
\cite{CheungVIIIA,CheungVIIIC}.

\item \textbf{No derived relativity or gravity.}
No spatial metric, numerical speed, observer-invariant speed,
Lorentz action on experiments, field regulator limit, Einstein
equation, or quantum-gravitational coefficient is derived.
The time parameter is external.

\item \textbf{No catalogue premise.}
The supplied OpenAI catalogue descriptions, including Family~271,
are not mathematical inputs to any argument here.
No catalogue claim is verified or transferred.

\item \textbf{No external obligation is discharged.}
The result makes no admission into an inherited characteristic
map and does not discharge the separate obligations designated
O.1, XV-B, XVI-N, or the series Route~B.
Those designations refer to external manuscript projects, not to
additional conclusions of this theorem.
\end{enumerate}

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

In one frozen Euclidean-calibrated model, every prescribed future-null
target admits finite local complete-channel witnesses with
\[
 \text{spatial error }O(j^{-2}),\qquad
 \text{arrival delay }O(j^{-1}),\qquad
 \text{ideal logical error }0.
\]
Ticks and initialized registers grow linearly with refinement;
their leading coefficient is necessary in the declared static-site
model. Compatible complete implementation errors are accumulated
explicitly, rather than hidden in ideal identity transport.

The theorem supplies operational approximability of a prescribed
causal boundary. It does not derive that boundary, identify it
with another manuscript's generated cone, or establish spacetime
emergence.

\appendix

\section{Technical audit of the formulation}
\label{app:audit}

The following points are part of the mathematical specification,
rather than informal implementation assumptions.

\begin{enumerate}[label=\textnormal{(\arabic*)},leftmargin=*]
\item \textbf{One layer per tick.}
The support bound requires a partition layer.
It would be false if arbitrarily many overlapping gates were
allowed inside an uncharged tick.

\item \textbf{One preparation per distinct site.}
Repeated route vertices are waits under the chosen monotone
rounding convention. A fresh auxiliary is prepared at each
distinct non-source site, not at every repeated occurrence.

\item \textbf{Future-directed endpoint timing.}
The padded time \eqref{eq:null-construction}, rather than the raw
route time alone, ensures \(T_j\ge T\).
At most one padding tick is required for null targets.

\item \textbf{Nontrivial payload dimension.}
The operational lower-bound and memory statements use \(D\ge2\).
A trivial one-dimensional register cannot signal.

\item \textbf{Complete versus reduced residual accounting.}
The ideal whole-output factorization
\eqref{eq:state-invariant} is proved explicitly.
Ignoring non-output factors does not mean that they were never
initialized or retained.

\item \textbf{Gate error versus full implementation error.}
Equation~\eqref{eq:gate-noise} assumes exact preparations,
spectators, waits, and output maps.
Equation~\eqref{eq:main-noise} is the complete-tick alternative.
Small channel error alone does not establish causal containment.

\item \textbf{Exact null attainment.}
Some finite-scale null pairs occur in this family.
Proposition~\ref{prop:exact-null} characterizes them.
The approximation theorem does not require their absence.

\item \textbf{Restricted limit claim.}
The established limit compares endpoints and logical payload
channels. It is not an unrestricted interpolation theorem,
a global evolution semigroup, or a continuum observable-net limit.

\item \textbf{Resource scope.}
The ledger includes initialized and retained quantum registers,
the finite route description, and stencil permissions.
It does not claim a physical energy, fabrication, or preparation-time
budget.

\item \textbf{Verification scope.}
The paper gives finite analytic proofs.
No proof-assistant certificate, numerical execution,
successful pdfLaTeX build, or laboratory realization is asserted.
\end{enumerate}

\section*{Bibliographic and documentary status}

The external references below identify standard published or
author-posted sources.
Cheung's manuscripts are cited by the identifiers and versions
given in the supplied material.
Their dates are recorded as undated where no bibliographic date
was supplied. The Enter Calculus Companion is a documentary locator,
not an independent verification of those manuscripts.
No comparison citation substitutes for a missing proof in this paper.

\begin{thebibliography}{99}

\bibitem{Watrous2018}
J.~Watrous,
\emph{The Theory of Quantum Information},
Cambridge University Press, 2018.
Author's manuscript and chapter files:
\url{https://cs.uwaterloo.ca/~watrous/TQI/}.

\bibitem{Fritz2013}
T.~Fritz,
Velocity polytopes of periodic graphs and a no-go theorem for
digital physics,
\emph{Discrete Mathematics} \textbf{313} (2013), 1289--1301.
doi: \href{https://doi.org/10.1016/j.disc.2013.02.010}
{10.1016/j.disc.2013.02.010}.
Author's revised preprint:
\url{https://arxiv.org/abs/1109.1963v3}.

\bibitem{ArrighiNesmeWerner}
P.~Arrighi, V.~Nesme, and R.~Werner,
\emph{Unitarity plus causality implies localizability},
arXiv:0711.3975, 2007; revised version~3, 2009.
\url{https://arxiv.org/abs/0711.3975v3}.

\bibitem{HorodeckiShorRuskai2003}
M.~Horodecki, P.~W.~Shor, and M.~B.~Ruskai,
Entanglement breaking channels,
\emph{Reviews in Mathematical Physics} \textbf{15} (2003), 629--641.
doi: \href{https://doi.org/10.1142/S0129055X03001709}
{10.1142/S0129055X03001709}.
Preprint record, titled ``General Entanglement Breaking Channels'':
\url{https://arxiv.org/abs/quant-ph/0302031}.

\bibitem{CheungEn08A}
N.~Cheung,
\emph{En08-A}, version~1,
undated manuscript.
Documentary comparison: scheduler consistency, conditional
Lorentz-cone representation, and the distinction between a generated
null boundary and attained null pairs.
Enter Calculus Companion: \companionurl.

\bibitem{CheungEn05A}
N.~Cheung,
\emph{En05-A}, version~1,
undated manuscript.
Relevant supplied labels:
\texttt{thm:cone} and \texttt{prop:approx}.
Enter Calculus Companion: \companionurl.

\bibitem{CheungEn05B}
N.~Cheung,
\emph{En05-B}, version~1,
undated manuscript.
Relevant supplied section:
``Introduction, target, and scope'';
documentary comparison with increasing-direction relays,
reference-preserving transport, and matched finite experiments.
Enter Calculus Companion: \companionurl.

\bibitem{CheungEn06B}
N.~Cheung,
\emph{En06-B}, version~1,
undated manuscript.
Documentary comparison with prescribed-speed finite-network transport;
relevant supplied label: \texttt{thm:T}.
Enter Calculus Companion: \companionurl.

\bibitem{CheungVIIJ}
N.~Cheung,
\emph{VII-J},
undated manuscript; version not specified in the supplied material.
Relevant supplied label: \texttt{thm:release}.
Documentary distinction between scheduled ancestry, operational
influence, and a separately supplied causal relation.
Enter Calculus Companion: \companionurl.

\bibitem{CheungXIXC}
N.~Cheung,
\emph{XIX-C}, version~1,
undated manuscript.
Relevant supplied section:
``Why finite certification does not select a continuum evolution'';
labels \texttt{def:limit}, \texttt{prop:two-limits},
and \texttt{prop:interpolation}.
Enter Calculus Companion: \companionurl.

\bibitem{CheungEn05C}
N.~Cheung,
\emph{En05-C}, version~1,
undated manuscript.
Relevant supplied section:
``Operational no-signalling and approximate certificates'';
labels \texttt{prop:approx} and \texttt{prop:telescoping}.
Enter Calculus Companion: \companionurl.

\bibitem{CheungVIIIA}
N.~Cheung,
\emph{VIII-A}, version~3,
undated manuscript.
Relevant supplied labels:
\texttt{A:thm:classical} and \texttt{A:thm:complement};
documentary comparison concerning records, residual systems,
and complete quantum recovery.
Enter Calculus Companion: \companionurl.

\bibitem{CheungVIIIC}
N.~Cheung,
\emph{VIII-C}, version~3,
undated manuscript.
Relevant supplied labels:
\texttt{thm:C5} and \texttt{prop:recovery};
sections ``Prescribed schedule support'' and
``Code decoding is not a quantum-return decoder''.
Enter Calculus Companion: \companionurl.

\bibitem{CheungEn02C}
N.~Cheung,
\emph{En02-C}, version~1,
undated manuscript.
Relevant supplied label:
\texttt{thm:C1}, reference-preserving sequential soundness and
actual-schedule storage ledgers.
Enter Calculus Companion: \companionurl.

\end{thebibliography}

\end{document}