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\hypersetup{
  pdftitle={Exact Causal Cones for Partitioned Enter Dynamics},
  pdfauthor={Ngo Cheung},
  pdfsubject={Schedule locality, readout semantics, relay attainability,
              velocity polytopes, and conditional continuum limits}
}
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\title{Exact Causal Cones for Partitioned Enter Dynamics:\\
Snapshot--Channel Comparison, Sharp Relay Speed,\\
and Conditional Continuum Limits}
\author{Ngo Cheung}
\date{October 2026}

\begin{document}
\maketitle

\begin{abstract}
We study complete quantum--classical instruments executed on a
locally finite metric substrate under partitioned updates of diameter
at most \(r\). We give a self-contained comparison between snapshot
factorization, Heisenberg support containment, and reference-preserving
channel no-signalling. The schedule past is an upper bound on
operational influence, not an assertion that every scheduled ancestor
affects a readout. An explicit relay attains the metric bound, giving
\(c_{\mathrm{lat}}=r\) on the stated lattice substrates. Arbitrary
block instruments and calculus-generated instruments are distinguished;
finite-program soundness requires a compositional interface, and
simulation requires routing from every starting tick.

Destructive Enter readout, coherent recording, operational discard,
and complete residual response are kept separate. Transmissive
exceptional blocks violate the original-metric cone, whereas inert
exceptional blocks admit a local refinement. Long-range completely
positive channels remain affine and cannot universally clone unknown
states. Entanglement-assisted state transfer is stated with its
communication and timing resources.

The comparison with periodic dependency graphs supplies a sharp
isotropy obstruction: a fixed finite periodic transport model has a
velocity polytope, and the cardinal model has an \(\ell^1\) velocity
ball rather than a Euclidean ball. We also identify a limitation of
the earlier boost-closure formulation: together with unrestricted
composition and well-founded executability, it excludes ordinary
bidirectional one-tick relays as well as super-cone devices.

Finally, continuum domain of dependence, operational no-signalling,
algebraic microcausality, and frame-invariant maximal speed are proved
only under explicit geometric, operational, observable-algebra, and
frame-covariance hypotheses. The operational hypotheses include matched
realizations of local modifications; the algebraic hypotheses concern
observables, not merely commuting superoperators. Existence of a family
satisfying the full package remains open. No physical metric, Einstein
dynamics, or invariant physical speed is derived.
\end{abstract}

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

Partitioned finite-range dynamics have an exact finite domain of
dependence. Their geometry supplies the propagation bound; the
semantics of a programming calculus supplies a separate interface
between programs and complete instruments. These two statements should
not be conflated.

This manuscript develops the comparison proposed for En05-A and
revises the supplied manuscript \oplabel{085VvAv3}.
The discrete cone, its relay attainment, and the distinction between
semantic and syntactic admissibility are retained. The additional
development is a precise alignment with channel-locality interfaces and
a velocity-polytope obstruction to the simplest isotropic scaling
proposal.

Cheung's En03-A gives one-layer adjoint support, schedule locality,
local-record laws, and invariance under legal reorderings of fixed
labelled channels \cite{CheungEn03}.
VII-J separates scheduled reachability, operational influence, and an
independently declared physical causal relation, and gives a
phase-explicit cycle-velocity description \cite{CheungVIIJ}.
Psy04-B emphasizes that scheduled ancestry need not equal influence
\cite{CheungPsy04B}. We prove the comparison statements needed here
rather than importing those manuscripts as theorem premises.

The measurement accounting follows the distinctions in Landmark I,
En04-A, and Hawk04-B: destructive readout returns a classical record;
discarding that record gives a partial trace; and recovery from a
complete output is not recovery from a selected residual
\cite{CheungI,CheungEn04,CheungHawk04B}.
The syntax-specific results of Landmark I are optional semantic input
once the carrier is a qubit interface. They are not needed to prove
the geometric cone.

\paragraph{Contributions.}
\begin{enumerate}[label=(\roman*)]
\item We formulate snapshots as unnormalized hybrid states and prove
their equivalence with reduced-channel factorization and adjoint
support containment.
\item We prove exact schedule locality and its metric envelope, with
complete classical controls and untouched references included.
\item We retain the sharp relay witness \(c_{\mathrm{lat}}=r\), while
making finite-program soundness and tail-routing simulation precise.
\item We distinguish readout laws, residual response, classical-record
reconstruction, and entanglement-assisted transport.
\item We state and prove the timed velocity-polytope comparison and
the fixed-periodic-model isotropy obstruction.
\item We correct the boost-closure and continuum interfaces. In
particular, we do not retain a physical retiming-stability conclusion
from an axiom that is incompatible with ordinary bidirectional relays.
\end{enumerate}

\paragraph{Interpretation boundary.}
The metric, schedule, register interfaces, and update rules are
declared inputs. The results do not derive a metric from entanglement,
Einstein dynamics, a Page curve, an island, a horizon, or a holographic
identification. Results about reversible quantum cellular automata are
related but have additional hypotheses; no classification theorem for
arbitrary instruments is asserted
\cite{SchumacherWerner,Arrighi}.
The present exact support bound is structurally different from the
exponentially suppressed propagation outside an effective cone in
Hamiltonian Lieb--Robinson estimates \cite{LiebRobinson}.

\section{Complete hybrid instruments and operational snapshots}
\label{sec:hybrid}

\subsection{Finite local interfaces}

All statements concern finite time windows and finite experiments.
At each site and tick, reserve a finite interface large enough for that
experiment. An interface may contain quantum registers, classical
records, occupation and placement labels, and controller memory.

A local observable algebra has the form
\begin{equation}
 \Alg_{x,s}
 =
 \bigoplus_{\lambda\in L_{x,s}}
 \mathcal{B}(\Hil_{x,s,\lambda}),
 \label{eq:localalgebra}
\end{equation}
where \(L_{x,s}\) is a finite classical label set and each
\(\Hil_{x,s,\lambda}\) is finite dimensional. Different occupation
patterns may have different quantum dimensions. Inactive reserved
slots are not accessible surviving quantum systems.

For a finite region \(I\), write
\[
 \Alg_s(I)=\bigotimes_{x\in I}\Alg_{x,s}.
\]
A state is represented by positive blocks
\begin{equation}
 \omega_I=\bigoplus_{\lambda}\omega_{I,\lambda},
 \qquad
 \sum_\lambda\Tr\omega_{I,\lambda}=1.
 \label{eq:hybridstate}
\end{equation}
The trace is the ordinary sum of matrix traces.

On a countable substrate we require a compatible family of states on
finite local algebras. No density operator on an unspecified infinite
tensor product is required. All calculations below reduce to finite
regions.

\begin{definition}[Complete instrument]
A finite-outcome instrument is a family
\(\{\Phi_k\}_k\) of completely positive maps such that
\(\sum_k\Phi_k\) is trace preserving. Its recorded channel is
\begin{equation}
 \widehat\Phi(\omega)
 =
 \bigoplus_k\Phi_k(\omega).
 \label{eq:recordedchannel}
\end{equation}
A \emph{complete channel} includes every reported outcome and every
failure outcome. Classical discard sums blocks without conditional
normalization.
\end{definition}

The channel language is the standard finite-dimensional
completely positive map language \cite{Choi,Watrous}.
A local adaptive instruction is one complete channel on its quantum
registers and its classical controller blocks.

\begin{assumption}[Complete local control]\label{ass:completecontrol}
Every state-dependent input to an update is present in its declared
local interface. This includes records, program counters, random
seeds, controller memory, and resource registers.

An update may depend on its own classical input block, but not on an
unrepresented global history, an unknown quantum-state description,
or a remote controller. Its complete map is fixed before the unknown
input state is supplied.
\end{assumption}

Shared randomness is allowed by preparing correlated classical
registers at the relevant sites. It is not an unaccounted global
scheduler input. This support convention is essential for adaptive
locality; compare Cheung VII-G \cite{CheungVIIG}.

\subsection{Snapshots and normalization}

Let \(R_{I,s}\) denote restriction to the interface in \(I\): quantum
factors outside \(I\) are traced out and external classical labels are
summed over.

\begin{definition}[Operational snapshot]\label{def:snapshot}
The snapshot at tick \(s\) in \(I\) is
\[
 \Sigma_s(I)=R_{I,s}(\omega_s).
\]
Its classical labels include the identities and placements of live
names, record values, and all retained controller data.

If \(p_\lambda=\Tr\omega_{I,\lambda}>0\), its normalized conditional
quantum state is
\(\rho_{I\mid\lambda}=\omega_{I,\lambda}/p_\lambda\).
At \(p_\lambda=0\), no conditional state is assigned.
\end{definition}

Thus the snapshot determines both the local classical law and the
conditional reduced quantum states after non-selectively averaging
external records. Conversely, those data determine the snapshot by
\(\omega_{I,\lambda}=p_\lambda\rho_{I\mid\lambda}\).

\begin{remark}[The affine object]
The affine object is the array of unnormalized blocks
\((\omega_{I,\lambda})_\lambda\), not the normalized conditional states
separately. Conditional normalization generally introduces nonlinear
ratios. All factorization identities in this paper are identities of
complete hybrid states.
\end{remark}

\subsection{Destructive Enter and coherent recording}

For a measured register \(S\) and a surviving register \(T\), set
\(M_x=\bra{x}_S\otimes I_T\). Destructive computational readout is
\begin{equation}
 \Ent_S(\rho_{ST})
 =
 \sum_x\kb{x}{x}_K\otimes M_x\rho_{ST}M_x^\dagger.
 \label{eq:enter}
\end{equation}
The measured names in \(S\) are consumed; \(K\) is classical.
The surviving quantum output is \(T\), not a renamed copy of \(S\).
This is the readout accounting used in
\cite{CheungI,CheungEn03,CheungEn04,CheungHawk04B}.

\begin{proposition}[Readout, recording, and discard]\label{prop:drop}
Define the coherent recording isometry
\[
 W:\ket{x}_S\longmapsto\ket{x}_S\ket{x}_K.
\]
Then
\begin{equation}
 \Tr_S\!\left[
 (W\otimes I_T)\rho_{ST}(W^\dagger\otimes I_T)
 \right]
 =
 \Ent_S(\rho_{ST}),
 \label{eq:recordequivalence}
\end{equation}
and
\begin{equation}
 \Tr_K\Ent_S(\rho_{ST})=\Tr_S\rho_{ST}.
 \label{eq:drop}
\end{equation}
Both identities hold with an arbitrary untouched reference.
\end{proposition}

\begin{proof}
Expand
\(\rho_{ST}=\sum_{x,y}\kb{x}{y}_S\otimes\rho_{xy}\).
Tracing the recorded \(S\) factor kills the terms with \(x\ne y\)
and leaves
\(\sum_x\kb{x}{x}_K\otimes\rho_{xx}\), which is
\eqref{eq:enter}. Tracing \(K\) gives \(\sum_x\rho_{xx}\), the partial
trace of the original state. Tensoring the calculation with a
reference identity changes neither equality.
\end{proof}

\begin{remark}[The context restriction is necessary]
Equation \eqref{eq:recordequivalence} permits deferred canonical
readout of the record after the measured factor has been made
inaccessible and the record is used only through basis-diagonal
classical controls and readout. It is not an equivalence with an
unrestricted coherent context that may access \(S\), erase \(K\),
or interfere different record branches.
\end{remark}

\section{Partitioned dynamics and the exact comparison theorem}
\label{sec:cone}

\subsection{Substrates and admissibility}

\begin{definition}[Substrate]\label{def:substrate}
A substrate is \((\Lat,d,r)\), where \(\Lat\) is countable,
\(d\) is an integer-valued metric with finite balls, and
\(r\ge1\) is an integer.
\end{definition}

\begin{definition}[Block schedule]\label{def:blocks}
A block schedule is a sequence
\((\mathcal{P}_s)_s\) of partitions of \(\Lat\) into finite blocks
\(b\) satisfying \(\diam_d b\le r\).
Tick \(s\) maps the tick-\(s\) interface to the tick-\((s+1)\)
interface.
\end{definition}

\begin{definition}[Semantic admissibility]\label{def:semantic}
A semantically admissible update is
\[
 \Phi_s=\bigotimes_{b\in\mathcal{P}_s}\Phi_{s,b},
\]
where each \(\Phi_{s,b}\) is a complete channel between the declared
input and output interfaces in \(b\), satisfying
Assumption~\ref{ass:completecontrol}.

Allocation, destructive readout, classical record creation and
discard, and relocation of names and records are permitted only
within \(b\). No output of the block is placed outside \(b\).
\end{definition}

Relocation changes the spatial address of a wire and is part of the
local channel interface. It need not be a reduction step of the
calculus. The propagation clock counts block updates, not the number
of gates evaluated inside a block. There is no additional gate-depth
or per-site capacity bound unless explicitly imposed.

For an observable with finite output support, only finitely many
block adjoints are relevant. Their product defines its pullback,
including on a countable substrate.

\subsection{Factorization and adjoint support}

An observable is supported in \(I\) when it is the identity on the
complementary interface. We identify local observables with their
extensions by identity.

\begin{lemma}[Snapshot--channel--adjoint equivalence]
\label{lem:comparison}
Let \(\Phi\) be a complete channel between finite tensor-product
hybrid interfaces. Fix an input region \(I\) and output region \(Y\).
The following are equivalent:
\begin{enumerate}[label=(\roman*)]
\item There is a complete channel \(K\) such that
\[
 R_Y\Phi=K R_I.
\]
\item Equal input snapshots in \(I\) give equal output snapshots in
\(Y\), for all input states.
\item
\[
 \Phi^*(\Alg_{\mathrm{out}}(Y))
 \subseteq\Alg_{\mathrm{in}}(I).
\]
\end{enumerate}
When they hold, factorization is stable under every untouched
reference: equal input reductions on \(I\) together with that
reference give equal output reductions on \(Y\) together with it.
\end{lemma}

\begin{proof}
(i) implies (ii). For (ii) implies (i), choose a fixed complementary
state \(\sigma\) and define
\[
 E_\sigma(\omega_I)=\omega_I\otimes\sigma,
 \qquad K=R_Y\Phi E_\sigma.
\]
The map \(K\) is complete and trace preserving. By (ii),
\(R_Y\Phi(\omega)=K R_I(\omega)\) on every state. States linearly
span the hybrid input space, so the equality holds as a linear-map
identity.

Taking adjoints in (i) gives
\[
 \Phi^*R_Y^*=R_I^*K^*,
\]
which is (iii). Conversely, (iii) implies that expectations of every
output observable depend only on the input snapshot in \(I\), giving
(ii).

Finally, tensor the linear-map identity in (i) with a reference
identity. This proves the reference statement. Equality on \(I\)
alone is not enough to conclude equality jointly with a reference
unless the corresponding joint input reductions also agree.
\end{proof}

This is the factorization form of the no-signalling interface in
Cheung VII-J \cite[\oplabel{thm:nosignal}]{CheungVIIJ}.
Its adjoint form matches En03-A
\cite[\oplabel{lem:one-layer}, \oplabel{thm:locality}]{CheungEn03}.
The proof above is independent of those source records.

\begin{lemma}[One-layer support]\label{lem:onelayer}
For finite \(Y\), put
\[
 U_s(Y)=
 \bigcup\{b\in\mathcal{P}_s:b\cap Y\ne\varnothing\}.
\]
Then
\[
 \Phi_s^*(\Alg_{s+1}(Y))\subseteq\Alg_s(U_s(Y)).
\]
The reduced channel on \(Y\) is determined by the channels of the
blocks meeting \(Y\).
\end{lemma}

\begin{proof}
The complement of \(U_s(Y)\) is a union of blocks. An observable on
\(Y\) is the identity on their output interfaces. Since the adjoint
of a trace-preserving channel is unital, each complementary block
adjoint contributes an identity on its input interface. The remaining
adjoints act only in \(U_s(Y)\). Their composition determines the
reduced output channel.
\end{proof}

\subsection{The schedule past and the metric envelope}

\begin{definition}[Schedule past]\label{def:schedulepast}
For a finite terminal region \(Y\) and terminal tick \(T\), set
\begin{equation}
 \begin{aligned}
 N_T(Y)&=Y,\\
 N_s(Y)&=
 \bigcup\{b\in\mathcal{P}_s:
              b\cap N_{s+1}(Y)\ne\varnothing\},
 \qquad s<T.
 \end{aligned}
 \label{eq:schedulepast}
\end{equation}
The scheduled past between \(t_0\) and \(T\) is
\[
 J^-_{\Sch}(Y,T)
 =
 \{(x,s):t_0\le s\le T,\ x\in N_s(Y)\}.
\]
\end{definition}

Each \(N_s(Y)\) is finite. Moreover,
\begin{equation}
 N_s(Y)\subseteq
 \{x:d(x,Y)\le r(T-s)\}.
 \label{eq:metricenvelope}
\end{equation}
Indeed, each backward layer adds at most one block diameter.

For a terminal site \(A\), let
\[
 I_s=\Ball(A,r(T-s)),\qquad
 J^-_d(A,T)=\{(x,s):t_0\le s\le T,\ x\in I_s\}.
\]
A block is \emph{inner} at tick \(s\) if it is contained in \(I_s\).

\begin{lemma}[Block geometry]\label{lem:geometry}
If \(b\in\mathcal{P}_s\) meets \(I_{s+1}\), then \(b\subseteq I_s\).
Consequently, no block that is not inner can place an output into
\(I_{s+1}\).
\end{lemma}

\begin{proof}
For \(z\in b\cap I_{s+1}\) and \(w\in b\),
\[
 d(w,A)\le d(w,z)+d(z,A)
 \le r+r(T-s-1)=r(T-s).
\]
Every output remains in its block.
\end{proof}

\begin{lemma}[One-step domain of dependence]\label{lem:dod}
There is a complete affine channel \(K_s\), determined only by the
schedule and the inner block channels at tick \(s\), such that
\begin{equation}
 \Sigma_{s+1}(I_{s+1})=K_s(\Sigma_s(I_s)).
 \label{eq:onesnapshot}
\end{equation}
\end{lemma}

\begin{proof}
The blocks meeting \(I_{s+1}\) are contained in \(I_s\) by
Lemma~\ref{lem:geometry}. Restrict the input snapshot to their union,
apply their product channel, and restrict its output to
\(I_{s+1}\). This is a complete channel \(K_s\).

All other block adjoints act on identity factors when testing an
observable supported in \(I_{s+1}\); they are therefore invisible by
Lemma~\ref{lem:onelayer}. Lemma~\ref{lem:comparison} gives
\eqref{eq:onesnapshot}. External outcomes are summed, not
postselected.
\end{proof}

\begin{remark}[What the comparison identifies]
Equation \eqref{eq:onesnapshot} is the snapshot form of one-layer
adjoint locality. The normalized conditional states and their
probabilities are recovered from its unnormalized blocks. This
formulation does not add an assumption that the initial state is
unentangled across the cone boundary.
\end{remark}

\begin{definition}[Readout]\label{def:readout}
A readout at \((Y,T)\) is a prescribed complete instrument on the
tick-\(T\) interface in \(Y\). Its outcome law is obtained from
\(\Sigma_T(Y)\). A readout may return both a classical record and a
quantum residual; the \emph{readout law} means its classical marginal.
\end{definition}

\begin{theorem}[Exact schedule locality and metric cone]
\label{thm:cone}
Let two semantically admissible runs have the same schedule over
\([t_0,T]\).

If their initial snapshots on \(N_{t_0}(Y)\) agree and their block
channels agree whenever
\(b\cap N_{s+1}(Y)\ne\varnothing\), then their terminal snapshots on
\(Y\) agree. Equivalently,
\begin{equation}
 \Phi_{t_0}^*\Phi_{t_0+1}^*\cdots\Phi_{T-1}^*
       (\Alg_T(Y))
 \subseteq\Alg_{t_0}(N_{t_0}(Y)).
 \label{eq:adjointcone}
\end{equation}

In particular, for \(Y=\{A\}\), agreement of the initial snapshots on
\(\Ball(A,r(T-t_0))\) and agreement of every inner block channel imply
\[
 \Sigma_T(\{A\})=\Sigma'_T(\{A\}).
\]
Every common readout at \((A,T)\) then has the same outcome law.
These conclusions are reference preserving in the sense of
Lemma~\ref{lem:comparison}.
\end{theorem}

\begin{proof}
Apply Lemma~\ref{lem:onelayer} backward through the ticks.
Definition~\ref{def:schedulepast} gives \eqref{eq:adjointcone}.
At each step the reduced channel depends only on the blocks meeting
the next scheduled slice, so the two runs have the same reduced
evolution through the scheduled past.

For the metric statement, use \eqref{eq:metricenvelope} and induct
forward using Lemma~\ref{lem:dod}. The initial snapshots agree on
\(I_{t_0}\), and the common inner channels give the same \(K_s\) at
every step. Thus the snapshots agree on \(I_T=\{A\}\).
Readout laws are functions of that terminal snapshot.
\end{proof}

\begin{figure}[t]
\centering
\begin{tikzpicture}[scale=0.8]
 \draw[->] (-4.5,0)--(4.5,0) node[right]{space};
 \draw[->] (-4.2,-0.25)--(-4.2,4.1) node[above]{tick};
 \draw[thick] (-3.7,0)--(0,3.7)--(3.7,0);
 \foreach \j in {0,1,2,3}{
   \pgfmathsetmacro{\w}{3.7-\j}
   \draw[blue!60] (-\w,\j)--(\w,\j);
 }
 \draw[very thick,orange] (-1.9,0)--(-1.9,1)--(-0.9,1)
                         --(-0.9,2)--(0,2)--(0,3.7);
 \fill (0,3.7) circle (2pt);
 \node[above] at (0,3.7){\((A,T)\)};
 \node[below] at (0,0){\(I_{t_0}=\Ball(A,r(T-t_0))\)};
 \node[orange!80!black] at (2.8,2.5){schedule past};
 \node[blue!70!black] at (-2.1,3.3){metric envelope};
\end{tikzpicture}
\caption{The scheduled past lies inside the metric cone.
The drawing is schematic: the schedule past depends on the actual
partitions and need not fill the metric envelope. Neither region is
asserted to be the minimal influence region of a particular channel.}
\label{fig:cone}
\end{figure}

\subsection{Ancestry is not influence}

\begin{definition}[Operational influence]
For a fixed complete network and terminal output \(Y\), an external
input port is influential if two trace-preserving local interventions
there, for some admitted initial state, produce different
unconditioned output laws on \(Y\). Denote the set of such ports by
\(\Inf^-(Y)\).
\end{definition}

\begin{corollary}[Influence containment]\label{cor:influence}
For the complete dependency network,
\[
 \Inf^-(Y)\subseteq\Sch^-(Y).
\]
The inclusion can be strict.
\end{corollary}

\begin{proof}
A trace-preserving intervention outside the scheduled ancestors
preserves their reduced state, including when the initial state is
entangled. Theorem~\ref{thm:cone} excludes an output change.
For strict inclusion, give a channel a declared memory input and let
it return the constant classical answer \(0\). The memory port is a
scheduled ancestor but is not influential.
\end{proof}

This is the distinction made explicitly in Cheung Psy04-B
\cite[\oplabel{thm:schedule}]{CheungPsy04B}. Selectively normalized
branches and correlations caused by common ancestors are not
counterexamples to this unconditioned intervention statement.

\section{Finite programs, syntactic admissibility, and routing}
\label{sec:programs}

\subsection{The compositional requirement}

\begin{assumption}[Finite compositional program interface]
\label{ass:programinterface}
A selected calculus has well-typed finite first-order quantum and
classical interfaces. It supports sequential composition, tensor
composition on disjoint owned interfaces, finite classical control,
allocation, its declared gate library, and destructive readout.

The denotation of a composite program is the corresponding composition
of complete instruments. Every selected local generating program
terminates and has a strategy-independent denotation. Programs and
controllers do not inspect a description of the unknown quantum state.
\end{assumption}

For a concrete strongly normalizing calculus, the relevant
metatheoretic clauses are recorded in
Appendix~\ref{app:interface}. Assumption~\ref{ass:programinterface}
is explicit because separate evaluability of block terms does not by
itself imply that arbitrarily concatenated block evaluations are
reductions of one fixed original term.

\begin{definition}[Syntactic admissibility]\label{def:syntactic}
A finite block-channel network is syntactically admissible when every
block channel is the denotation of a selected generating program and
the network is well typed under
Assumption~\ref{ass:programinterface}. Interfaces, record dependencies,
ownership, and branch continuations must match.
\end{definition}

The geometric theorem applies to arbitrary semantically admissible
channels. Syntactic admissibility is a stronger implementation
condition; it does not assert that every arbitrary instrument is
expressible over the available gate library.

\begin{proposition}[Finite path accounting]\label{prop:path}
Every finite acyclic program built from the specified primitives has
linear path maps \(K_h\) satisfying
\[
 \sum_h K_h^\dagger K_h=I.
\]
For reported outcome \(y\), its selective map is
\[
 \Phi_y(\rho)=
 \sum_{h:\operatorname{out}(h)=y}K_h\rho K_h^\dagger.
\]
The sum over reported outcomes is a complete channel, including every
untouched reference.
\end{proposition}

\begin{proof}
A complete history fixes the classical choices. Compose allocation
isometries, gates, wiring maps, and destructive measurement bras
along that history to obtain \(K_h\). Successive branch
normalizations cancel against their probabilities.

At a readout node,
\(\sum_xM_x^\dagger M_x=I\). Allocation is isometric and gates are
unitary. Backward induction over the finite instruction tree gives
the completeness identity. Grouping histories gives the reported
maps. Tensoring every path map with a reference identity proves the
referenced statement.
\end{proof}

This is the finite-program accounting in Landmark I,
En04-A, Hawk04-B, and En02-C
\cite{CheungI,CheungEn04,CheungHawk04B,CheungEn02C};
the proof is included to make the present fragment self-contained.

\begin{proposition}[Soundness of finite syntactic networks]
\label{prop:sound}
Every finite syntactically admissible network denotes the same complete
instrument as a well-typed composite program. Legal within-tick
orderings of its disjoint block evaluations have the same denotation.
\end{proposition}

\begin{proof}
At a tick, compose the generating programs in parallel on their
disjoint owned interfaces. Compose these tick programs sequentially,
using finite classical control for the retained histories.
Assumption~\ref{ass:programinterface} supplies the composite program
and identifies its denotation with the block network.

Channels on disjoint interfaces commute, so legal within-tick
orderings give the same channel. Evaluation of the composite program
then supplies a valid finite execution in the selected calculus.
\end{proof}

\begin{remark}[Finite network versus one strongly normalizing term]
The proposition does not turn an indefinitely driven substrate into a
reduction sequence of one fixed term. For example, an external driver
may request a new gate at every tick forever, although every requested
local program terminates. Strong normalization of individual terms
does not prohibit that external drive.
\end{remark}

\subsection{Routing from every starting tick}

\begin{definition}[Tail-routing property]\label{def:routing}
A schedule has the tail-routing property if, from every starting tick,
every finite collection of names and records at declared initial
locations can be brought to any declared target site in finitely many
ticks by within-block relocations alone. The finite reserved interface
must have sufficient capacity for the routed collection.
\end{definition}

\begin{example}[Fair coordinate schedules]\label{ex:relay}
On \(\ZZ^D\) with the \(\ell^1\) metric, define, along one coordinate
axis, the two partitions with intervals
\[
 [(2k+p)r,(2k+p+1)r]\cap\ZZ,\qquad k\in\ZZ,\quad p\in\{0,1\},
\]
and singletons at all remaining sites. Lift these partitions along
each coordinate line.

A schedule that repeats both phases on every axis is tail routing.
Every nearest-neighbor edge lies in an interval of one of the two
phases. A name can therefore follow any finite coordinate path by
waiting for the required phase. Previously routed names may remain at
the target while the remaining names are transported.
\end{example}

\begin{proposition}[Simulation under tail routing]\label{prop:sim}
Suppose the selected program has a finite input interface and a finite
complete evaluation, and its gates belong to the specified library.
On a tail-routing schedule with sufficient finite capacity, its
instrument can be realized by a syntactically admissible run.
\end{proposition}

\begin{proof}
Choose a target site. Route the program's input names and classical
input records to that site, using a state-independent transport plan.
Transport is a wire-address change and leaves their joint logical
state unchanged.

At a later tick, evaluate the generating program inside the block
containing the target site. All required fresh slots are reserved
there. The block's denotation is the instrument of the original
program. If a prescribed final placement is required, route the
returned names and records after evaluation.
\end{proof}

This is a simulation of the complete finite instrument. It is not an
exact finite-gate compilation theorem for arbitrary specified
unitaries, an efficiency theorem, or a claim that each administrative
redex is itself the terminal evaluation of an isolated term.

\begin{corollary}[Completed program records]\label{cor:terminal}
Under the hypotheses of Theorem~\ref{thm:cone}, if a common completed
program's reported outcome records are in the terminal region, their
law is the same in both runs.
\end{corollary}

\begin{proof}
The reported records are part of the terminal snapshot.
\end{proof}

\begin{remark}[Termination is not a tick bound]
Strong normalization bounds reductions, not externally inserted idle
ticks or unsuccessful routing attempts. A finite completion horizon
requires a chosen finite scheduled execution, a finite progress
condition, or the explicit routing construction above.
\end{remark}

\section{Sharp signalling speed}\label{sec:speed}

\begin{definition}[Perturbation pairs]\label{def:speed}
For a finite horizon, two runs form an \(x\)-perturbation pair when
they share the schedule, their initial snapshots agree on every finite
region not containing \(x\), and their channels agree on every block
not containing \(x\).

The site \(x\) signals to \((A,\tau)\), where \(\tau\ge1\), if such a
pair produces different laws for a common readout at
\((A,t_0+\tau)\). Define
\[
 c_{\mathrm{lat}}
 =
 \sup\left\{
 \frac{d(x,A)}{\tau}:
 x\text{ signals to }(A,\tau)
 \text{ for some admissible schedule}
 \right\},
\]
with supremum \(0\) if there is no positive-distance signalling.
\end{definition}

\begin{proposition}[Universal upper bound]\label{prop:upper}
Every substrate satisfies \(c_{\mathrm{lat}}\le r\).
\end{proposition}

\begin{proof}
If \(d(x,A)>r\tau\), the initial cone ball excludes \(x\), as does
every inner block. An \(x\)-perturbation pair therefore satisfies the
metric hypotheses of Theorem~\ref{thm:cone}, so its readout laws agree.
\end{proof}

\begin{proposition}[Exact relay attainment]\label{prop:attain}
On \(\ZZ^D\) with the \(\ell^1\) metric, for every \(\tau\ge1\)
there is an \(x\)-perturbation pair with \(d(x,A)=r\tau\) and readout
laws \(\delta_0,\delta_1\).
Consequently \(c_{\mathrm{lat}}=r\).
\end{proposition}

\begin{proof}
Take \(A=0\), \(x=r\tau e_1\), and place a classical bit
\(\beta\in\{0,1\}\) at \(x\). At step \(j=0,\ldots,\tau-1\),
choose the phase containing the interval
\[
 [(\tau-j-1)r,(\tau-j)r]e_1.
\]
Relocate the record from its right endpoint to its left endpoint.
All other updates may be identities.

After \(\tau\) ticks, the record is at \(A\). Reading it gives
\(\delta_\beta\). The two runs differ only in the initial bit at \(x\)
and use identical rules. The lower bound \(r\) combines with
Proposition~\ref{prop:upper}.
\end{proof}

\begin{remark}[Sharpness has an existential quantifier]
The speed is a supremum over schedules and admitted channels.
Attainment does not say that every fixed schedule, every block
instrument, or every site in the metric cone transmits information.
\end{remark}

\begin{proposition}[Geometric separation of propagated supports]
\label{prop:cert}
Let two initial regions \(X_0,Y_0\) satisfy
\(d(X_0,Y_0)>2r\tau\). Propagate each region forward by adjoining
all outputs of every block meeting its current support. Up to tick
\(t_0+\tau\), the two propagated families are disjoint, and no update
before that terminal tick meets both families.
\end{proposition}

\begin{proof}
After \(j\) ticks, each family lies within distance \(rj\) of its
initial region. If a block at step \(j<\tau\) met both families,
the triangle inequality would give
\[
 d(X_0,Y_0)\le rj+r+rj\le(2\tau-1)r,
\]
a contradiction. At the terminal tick, a common site would give
\(d(X_0,Y_0)\le2r\tau\), also a contradiction.
\end{proof}

This proposition certifies separation of complete propagated
interfaces, not merely separation of the original name labels.
For syntactic subcomputations with the corresponding ownership
interfaces, it supplies the geometric disjointness needed to apply
the selected calculus's disjoint-support results.

\section{Readout, residual response, and quantum reconstruction}
\label{sec:readout}

\subsection{A classical record is not a quantum decoder}

\begin{proposition}[Classical-record reconstruction obstruction]
\label{prop:classical}
Let \(Q\) be a \(d\)-dimensional quantum input, \(d\ge2\).
Every channel from \(Q\) to a finite classical record followed by a
channel from that record to a quantum output is entanglement breaking.
It cannot equal \(\id_Q\), including on reference-entangled inputs.
Its half-diamond distance from \(\id_Q\) is at least \(1-1/d\).
\end{proposition}

\begin{proof}
The composite has measure-and-prepare form
\[
 \mathcal{D}\mathcal{M}(\rho)
 =
 \sum_x\Tr(E_x\rho)\sigma_x.
\]
For an arbitrary reference, its output is a sum of positive product
operators and is therefore separable.

Apply the channel to one half of
\(\ket{\Phi_d}=d^{-1/2}\sum_{j=1}^d\ket{j}\ket{j}\).
Every separable state has overlap at most \(1/d\) with
\(\kb{\Phi_d}{\Phi_d}\): this follows for pure product states from
Cauchy--Schwarz and then for separable mixtures by convexity.
Testing the Bell projector gives trace distance at least \(1-1/d\)
from the ideal output. The half-diamond distance dominates this
particular referenced-input distance.
\end{proof}

The general measure-and-prepare characterization is standard
\cite{HorodeckiEB}. The present application agrees with Cheung
Landmark I \oplabel{prop:classical-recovery}, En04-A
\oplabel{thm:classical}, and Hawk04-B \oplabel{thm:classical}
\cite{CheungI,CheungEn04,CheungHawk04B}.

\begin{remark}[Entanglement assistance is a different interface]
The proposition concerns a decoder whose entire operational input is
the classical record and independent fixed ancillas. It does not apply
to a receiver that also holds a quantum system entangled with a system
at the sender. That additional system is exactly the resource used by
teleportation.
\end{remark}

\subsection{Complete residual response}

\begin{definition}[Residual response]\label{def:response}
For an isometry \(V:C\to R\otimes B\), let
\[
 \mathcal{N}(\rho)=\Tr_B(V\rho V^\dagger),
 \qquad
 \mathcal{C}(\rho)=\Tr_R(V\rho V^\dagger).
\]
The complete residual \(B\) responds to the code when
\(\mathcal{C}\) is not a constant channel on the code.
The residual must include all inaccessible systems and preparation
purifications belonging to the stated dilation.
\end{definition}

\begin{proposition}[Complete restoration does not identify a residual]
\label{prop:restore}
Every finite-dimensional isometry \(V:C\to R\otimes B\) has a CPTP
left inverse on its complete output. This does not imply recovery
from \(R\), recovery from \(B\), or constancy of either complement.
\end{proposition}

\begin{proof}
Choose a state \(\tau_C\) and define
\[
 \mathcal{L}(X)
 =
 V^\dagger X V+
 \Tr((I-VV^\dagger)X)\tau_C.
\]
Both summands are completely positive and their traces sum to
\(\Tr X\). Thus \(\mathcal{L}\) is CPTP and
\(\mathcal{L}(V\rho V^\dagger)=\rho\), including references.

For a counterexample to residual constancy, take
\(V\ket{\psi}=\ket{0}_R\otimes\ket{\psi}_B\).
The accessible output is constant and the residual is the identity
channel. Complete restoration uses \(R B\) and returns a new code
output; it does not assert that the code was relocated into either
chosen output by the restore map.
\end{proof}

The distinction is the one emphasized in Cheung En04-A and Hawk04-A
\cite{CheungEn04,CheungHawk04A}.

\begin{remark}[Residual response is not a signalling path]
In the counterexample, place the input and its unchanged residual at
a site outside the scheduled past of a readout of \(R\). The residual
responds to the input while the readout law on \(R\) remains constant.
The cone theorem does not require all systems outside its readout
cone to be input independent. Nor does nonconstant residual response
establish geometric backreaction.
\end{remark}

\section{Exceptional blocks and original-metric necessity}
\label{sec:necessity}

\subsection{A genuine repartitioning}

\begin{definition}[Exceptional block]\label{def:exception}
Fix sites \(P,Q\) with \(\ell=d(P,Q)>r\).
Their ordinary blocks \(b_P,b_Q\) are distinct.
Replace them by
\[
 b_P\setminus\{P\},\qquad b_Q\setminus\{Q\},\qquad\{P,Q\},
\]
discarding empty sets. This is a partition.

Channels on the shrunken blocks must be specified anew; a channel on
\(b_P\) does not automatically restrict to one on
\(b_P\setminus\{P\}\). Channels of unaffected blocks are retained.

The exceptional block may have a complete instrument supported on its
two-site interface. It is record transmissive or name transmissive
when it admits the corresponding lossless one-tick transport.
It is \emph{inert} when its complete channel is
\(\Phi_P\otimes\Phi_Q\), with no cross-site relocation or classical
control.
\end{definition}

\begin{proposition}[Inert range is harmless]\label{prop:inert}
An inert exceptional block can be refined into the two singletons
\(\{P\},\{Q\}\) with the same complete update. The refined dynamics is
admissible in the original metric \(d\).
\end{proposition}

\begin{proof}
The factorization identifies the exceptional update with the product
of the two singleton channels. All remaining blocks have diameter
at most \(r\).
\end{proof}

\begin{remark}[Non-product is not synonymous with transmissive]
The constant channel
\(\rho\mapsto\kb{\Phi^+}{\Phi^+}_{PQ}\Tr\rho\) has a non-product
output but transmits no input dependence in either direction.
Transmission must be established by an operational witness or a
failure of reduced-channel factorization, not by block diameter or
non-product structure alone.
\end{remark}

\begin{proposition}[The shortened metric]\label{prop:wormmetric}
Define
\[
 d_W(x,y)=
 \min\{d(x,y),\ d(x,P)+r+d(Q,y),\
                  d(x,Q)+r+d(P,y)\}.
\]
This is the shortest-path metric obtained by adjoining an undirected
edge of length \(r\) between \(P,Q\). It is integer valued, has finite
balls, and all repartitioned blocks have \(d_W\)-diameter at most \(r\).
\end{proposition}

\begin{proof}
Represent the original metric by the complete weighted graph with
edge lengths \(d(x,y)\). The triangle inequality allows every path
not using the added edge to be shortened to one original edge.
A shortest path uses the new edge at most once, since a repeated use
contains a positive-length loop that can be removed. This gives the
displayed formula and proves that it is a metric.

A \(d_W\)-ball is contained in a union of three finite \(d\)-balls:
one about its center and ones about \(P,Q\), with radius no larger
than the specified ball radius. Finally \(d_W\le d\) and
\(d_W(P,Q)\le r\).
\end{proof}

All necessity statements below concern the \emph{original} metric.
The cone theorem still holds for the modified metric \(d_W\).

\subsection{Transmission and complete operational input}

\begin{theorem}[Original-metric cone violation]\label{thm:longrange}
A lossless one-tick record transport from \(P\) to \(Q\) gives a
\(P\)-perturbation pair with readout laws \(\delta_0,\delta_1\) at
\((Q,t_0+1)\). Hence the original-metric signalling speed of the
extension is at least \(\ell>r\).

A lossless one-tick name transport has the same conclusion by
transporting a computational-basis bit and reading it at \(Q\).
\end{theorem}

\begin{proof}
Prepare the two possible bit values at \(P\), use the same transport
channel in both runs, and apply the common terminal readout at \(Q\).
Initial snapshots agree away from \(P\). The signal arrives in one
tick over original-metric distance \(\ell\).
\end{proof}

\begin{theorem}[Equal quantum marginals do not give equal local input]
\label{thm:autonomy}
In the exceptional-block model there are runs with identical initial
states and identical channels on all original-metric inner blocks for
\((Q,t_0+1)\), but different readout laws at that event. In both runs
the non-selective quantum marginal at \(Q\) is \(I/2\).
\end{theorem}

\begin{proof}
Prepare a Bell pair with halves at \(P,Q\), using a declared
prehistory before the exceptional update is activated.
In the exceptional block, measure the \(P\)-half destructively and
deliver its record \(b\) to \(Q\). Use the \(Z\) basis in the first
run and the \(X\) basis in the second. These are admissible choices
of complete exceptional-block instruments.

The resulting record--qubit states at \(Q\) are
\begin{align*}
 \omega_Z&=
 \tfrac12\left(
 \kb0 0_b\otimes\kb0 0_Q+
 \kb1 1_b\otimes\kb1 1_Q
 \right),\\
 \omega_X&=
 \tfrac12\left(
 \kb0 0_b\otimes\kb+ +_Q+
 \kb1 1_b\otimes\kb- -_Q
 \right).
\end{align*}
Their quantum marginals are both \(I/2\), but
\(\omega_Z\ne\omega_X\).

Read \(b\), measure the qubit in the \(Z\) basis with outcome \(a\),
and output \(a\oplus b\). The law is \(\delta_0\) on \(\omega_Z\)
and uniform on \(\{0,1\}\) on \(\omega_X\).
The exceptional block is not original-metric inner because
\(d(P,Q)>r\); all inner channels and all initial data agree.
\end{proof}

The theorem violates original-metric autonomy, not locality relative
to the complete operational input. The arriving record is part of
that input.

\begin{proposition}[Affine long-range channels do not universally clone]
\label{prop:linear}
For any finite extension made of complete instruments and wire
relocations:
\begin{enumerate}[label=(\alph*)]
\item the input-to-readout-law map is affine;
\item equal complete operational input states give equal laws for a
fixed readout;
\item no deterministic program universally clones unknown pure
states.
\end{enumerate}
\end{proposition}

\begin{proof}
(a) follows from linearity of complete channels and classical
marginalization. For (b), a fixed readout is a channel on its accessed
quantum and classical factors.

For (c), let \(\mathcal{C}\) be a hypothetical CPTP cloner.
Use root fidelity
\(\Fid(\rho,\sigma)=\norm{\sqrt{\rho}\sqrt{\sigma}}_1\).
CPTP maps do not decrease fidelity \cite{Watrous}.
For two pure states with \(0<f=\abs{\langle\psi,\phi\rangle}<1\),
cloning would give output fidelity \(f^2<f\), contradicting
monotonicity.
\end{proof}

For universal qubit cloning there is also a direct affine proof:
the two decompositions of \(I/2\) into the \(Z\) and \(X\) basis
ensembles would give different two-copy outputs. Landmark I uses
that universal-input argument \cite[\oplabel{thm:nocloning}]{CheungI}.
The fidelity argument additionally explains why a physical
deterministic map cannot exactly clone even a specified pair of
nonorthogonal pure states \cite{WoottersZurek,Dieks}.

\subsection{State transfer with explicit resources and timing}

\begin{theorem}[Entanglement-assisted sector transfer]\label{thm:teleport}
Suppose the receiver at \(Q\) holds one half of a Bell pair whose
preparation and distribution were completed before the unknown input
is supplied. Suppose also that:
\begin{enumerate}[label=(\roman*)]
\item a Bell-basis measurement at \(P\) is available;
\item its two outcome bits can be delivered from \(P\) to \(Q\) in
\(m\) ticks;
\item the receiver can apply the corresponding Pauli correction.
\end{enumerate}
If measurement and correction are each charged one separate tick,
the unknown qubit is transported in \(m+2\) ticks, including when it
is entangled with an untouched reference.
The transport is outside the original-metric cone whenever
\(\ell>r(m+2)\).
\end{theorem}

\begin{proof}
The Bell measurement has four equally weighted corrected branches.
In each branch, the \(Q\)-half is \(X^bZ^a\ket{\chi}\), up to the
fixed Bell-basis phase convention. Delivery of \((a,b)\), followed
by \(Z^aX^b\), leaves \(\ket{\chi}\) at \(Q\).
The corrected branch maps from input to output are
\(I/2\), up to phase; summing their four CP maps gives the identity
channel. Therefore the calculation preserves every reference.

The charged stages are one measurement tick, \(m\) communication
ticks, and one correction tick. Comparison with the cone requires
the stated distance inequality.
\end{proof}

This is the standard teleportation resource identity
\cite{BennettTeleport}. If two bits can be batched, \(m=1\);
if only one bit per tick is available, \(m=2\). If local wrappers
and transport are admitted as one compound exceptional-block
instrument, a different timing convention may charge fewer ticks,
but that closure must be stated.

Bell-pair distribution requires a certified ordinary route. On a
suitable \(\ZZ^D\) schedule it can take
\(\lceil\ell/r\rceil\) ticks; an arbitrary metric substrate need not
admit such a route. A constant-time transfer protocol is not
automatically super-cone at every distance \(\ell>r\) when its local
processing stages are charged separately.

\section{Velocity sets and the discrete isotropy obstruction}
\label{sec:velocity}

\subsection{Physical containment is an additional comparison}

Let sites be embedded by \(z:\Lat\to\RR^D\), and let \(K\subset\RR^D\)
be a declared compact convex velocity set containing \(0\).
A dependency edge of displacement \(\Delta\) and duration \(u>0\)
is physically contained when \(\Delta\in uK\).
A nonzero displacement at duration \(0\) is not contained in this
future cone.

\begin{proposition}[Edgewise velocity certificate]\label{prop:velocity}
If every edge of a complete dependency network is physically
contained, every dependency path is contained in the same convex
velocity cone. Operational influence obeys that containment.
\end{proposition}

\begin{proof}
For a path with edges \((\Delta_j,u_j)\),
\[
 \sum_j\Delta_j
 =
 \left(\sum_j u_j\right)
 \sum_j\frac{u_j}{\sum_i u_i}\frac{\Delta_j}{u_j}
 \in\left(\sum_j u_j\right)K.
\]
Operational influence is contained in scheduled ancestry by
Corollary~\ref{cor:influence}.
\end{proof}

This is the local-velocity and physical-containment language of
Cheung VII-J \cite{CheungVIIJ}.
A transmissive exceptional block supplies an actual edge outside the
declared physical cone when its displacement exceeds the admitted
one-tick velocity set. An inert exceptional block instead admits the
singleton refinement of Proposition~\ref{prop:inert}. It is not
correct to assign an inert operation a physical transport velocity
merely from the diameter of its conservative two-site envelope.

\subsection{Timed periodic dependency graphs}

\begin{definition}[Finite periodic transport model]\label{def:periodic}
Let \(Q\) be a finite strongly connected directed quotient graph.
Vertices include register type and schedule phase. Every edge \(e\)
has a displacement \(\Delta_e\in\RR^D\) and duration
\(u_e\ge u_{\min}>0\). Its periodic lift permits translated copies
of the transitions. Waiting without displacement is allowed.

For a simple directed cycle \(c\), set
\[
 \Delta(c)=\sum_{e\in c}\Delta_e,\quad
 u(c)=\sum_{e\in c}u_e,\quad
 v(c)=\frac{\Delta(c)}{u(c)}.
\]
Define
\begin{equation}
 K_{\Sch}
 =
 \conv\bigl(\{0\}\cup\{v(c):c\text{ a simple directed cycle}\}\bigr).
 \label{eq:velocitypolytope}
\end{equation}
\end{definition}

\begin{theorem}[Cycle velocities and finite-time offset]
\label{thm:polytope}
Let \(q=\abs{V(Q)}\) and
\(D_{\max}=\max_e\norm{\Delta_e}_2\).
Every dependency path completed within time \(u\) satisfies
\begin{equation}
 \Delta_{\mathrm{path}}
 \in uK_{\Sch}+
 \overline{\Ball}_2(0,(q-1)D_{\max}).
 \label{eq:finiteoffset}
\end{equation}
The set of limiting path velocities as elapsed time tends to infinity
is exactly \(K_{\Sch}\).
This is a statement about potential scheduled reachability, not
transmission by every path.
\end{theorem}

\begin{proof}
Remove directed cycles from a finite walk until the remainder has
no repeated vertex. Decompose the removed cycles into simple cycles.
The remainder has at most \(q-1\) edges, so its displacement has norm
at most \((q-1)D_{\max}\).

The cycle contribution is
\[
 \sum_c n_c\Delta(c)=\sum_c n_cu(c)v(c).
\]
Its total duration is no greater than \(u\). Since \(0\in K_{\Sch}\),
the contribution belongs to \(uK_{\Sch}\), proving
\eqref{eq:finiteoffset}. Dividing by elapsed time proves the upper
inclusion for limiting velocities.

Conversely, approximate a convex combination of cycle velocities
and \(0\) by spending the corresponding time proportions on repeated
cycles and waiting. Strong connectivity supplies fixed connector
walks between their base vertices. Choose repetition numbers tending
to infinity; connector durations and displacements become negligible.
Integer repetition numbers approximate the desired time proportions,
giving every velocity in \(K_{\Sch}\).
\end{proof}

The untimed periodic-graph mechanism is studied by Fritz
\cite{Fritz}. The timed finite-offset version agrees with Cheung
VII-J \oplabel{thm:polytope} \cite{CheungVIIJ}.
Without strong connectivity, the ordering of reachable strongly
connected components must be retained; taking all cycle velocities
in the graph can overstate attainability from a specified input.

\begin{example}[Asymptotic speed does not control the first jump]
Take two quotient states with displacements \(L,-L\), each of duration
one, and allow waiting. All cycles have zero displacement, so
\(K_{\Sch}=\{0\}\). A first transition still moves a dependency by
\(L\). The finite-time offset cannot be omitted.
\end{example}

\subsection{The cardinal velocity ball}

\begin{proposition}[Cardinal anisotropy]\label{prop:diamond}
In the cardinal transport model with displacements
\(\pm a e_j\), each of duration \(a/c\), and waiting,
\[
 K_{\Sch}=\{v:\norm{v}_1\le c\}.
\]
For a Euclidean unit direction \(u\), its maximal scheduled radial
speed is
\begin{equation}
 \rho_{\Sch}(u)=\frac{c}{\norm{u}_1}.
 \label{eq:radialspeed}
\end{equation}
In two dimensions it is \(c\) on an axis and \(c/\sqrt2\) on a
diagonal.
\end{proposition}

\begin{proof}
The cardinal edge velocities are \(\pm ce_j\).
Their convex hull with \(0\) is the \(\ell^1\) ball of radius \(c\).
Time-sharing cardinal movements and waiting attains every point in
that hull asymptotically. A vector \(\rho u\) lies in the hull exactly
when \(\rho\norm{u}_1\le c\).
\end{proof}

For a prescribed partition schedule whose phases restrict the
available movements further, its velocity set may be smaller.
Equality above concerns the specified cardinal transport model,
not every coordinate schedule.

\begin{figure}[t]
\centering
\begin{tikzpicture}[scale=1.55]
 \draw[->] (-1.35,0)--(1.45,0) node[right]{\(v_x/c\)};
 \draw[->] (0,-1.3)--(0,1.4) node[above]{\(v_y/c\)};
 \draw[blue,thick] (0,0) circle (1);
 \draw[orange,very thick] (1,0)--(0,1)--(-1,0)--(0,-1)--cycle;
 \draw[dashed] (0,0)--(0.7071,0.7071);
 \fill[orange] (0.5,0.5) circle (1.4pt);
 \fill[blue] (0.7071,0.7071) circle (1.4pt);
 \node[orange!80!black,anchor=west] at (0.58,0.35){cardinal boundary};
 \node[blue,anchor=west] at (0.78,0.83){Euclidean boundary};
\end{tikzpicture}
\caption{The cardinal diamond is contained in, but does not fill,
the Euclidean disk of radius \(c\). The metric upper bound and its
attainment along axes do not establish Euclidean attainment in every
direction.}
\label{fig:diamond}
\end{figure}

\begin{theorem}[Fixed-periodic-model isotropy obstruction]
\label{thm:isotropyobstruction}
Let \(D\ge2\). A fixed finite periodic transport model, rescaled with
a fixed length-to-time ratio, cannot have a nonzero Euclidean ball as
its limiting scheduled velocity set.

More generally, a family of compact velocity polytopes with uniformly
bounded numbers of vertices cannot converge in Hausdorff distance to a
nonzero Euclidean ball.
\end{theorem}

\begin{proof}
For the fixed model, uniform length and time rescaling leaves
\eqref{eq:velocitypolytope} unchanged. It is a finite polytope.
A nonzero Euclidean ball in \(D\ge2\) has infinitely many extreme
points and is not a finite polytope.

For the second statement, let the vertex count be bounded by \(M\).
Pad each vertex list to length \(M\) by repetition. Hausdorff
convergence to a compact ball bounds all vertices. A subsequence has
coordinatewise convergent vertex lists. Its limiting convex hull is
a polytope with at most \(M\) vertices and also equals the asserted
Hausdorff limit, a contradiction.
\end{proof}

This is an obstruction within a specified periodic class, not a
nonexistence theorem for arbitrary refinements, nonperiodic
substrates, or increasing directional complexity.

\begin{corollary}[A finite-scale certificate for a candidate limit]
\label{cor:velocitymargin}
For a scaling family of periodic models, suppose
\[
 K_{\Sch,\varepsilon}\longrightarrow K,\qquad
 b_\varepsilon=
 (q_\varepsilon-1)D_{\max,\varepsilon}\longrightarrow0.
\]
Then paths with elapsed physical times tending to \(u>0\) have
limiting displacements in \(uK\).
\end{corollary}

\begin{proof}
Apply \eqref{eq:finiteoffset} in physical units and pass to the
Hausdorff limit. The offset vanishes.
\end{proof}

The corollary is an upper certificate. It does not supply a channel
on each path, a lower relay bound, operational convergence, or frame
covariance. If the quotient complexity grows, the physical offset
must be controlled as well as the polytope shape.

\section{Two retiming statements and a boost-closure limitation}
\label{sec:retiming}

\subsection{Legal reorderings of fixed channels}

\begin{proposition}[Fixed-channel ordering invariance]
\label{prop:gateorder}
Let a finite collection of fixed labelled channels have declared
complete supports. Suppose a strict partial order makes all channels
with intersecting supports comparable. Every legal linear extension
has the same total channel.
\end{proposition}

\begin{proof}
Incomparable channels have disjoint supports and commute after
extension by identities. Any two linear extensions of a finite
partial order are connected by adjacent swaps of incomparable
elements. Every swap preserves the channel composition.
\end{proof}

Records and resource wires belong to the supports. The proposition
does not move a gate before a gate that creates a record it reads,
and it does not identify channels obtained by changing a geometric
parameter. This is the fixed-channel retiming statement in
Cheung VII-E and En03-A \cite{CheungVIIE,CheungEn03}.

\subsection{Coordinate retiming is not device realizability}

\begin{definition}[Integer coordinate retiming]\label{def:retiming}
For \(\varphi:\Lat\to\ZZ\), let
\[
 \beta_\varphi(x,s)=(x,s+\varphi(x)).
\]
An edge from \((x,s)\) to \((y,s+1)\) has retimed duration
\(1+\varphi(y)-\varphi(x)\).
The retiming is \(r\)-compatible if
\[
 \abs{\varphi(x)-\varphi(y)}\le1
 \quad\text{whenever }d(x,y)\le r.
\]
\end{definition}

\begin{lemma}[A coordinate change preserves the dependency graph]
\label{lem:noloops}
Applying one bijection \(\beta_\varphi\) to a complete network does
not create a directed graph cycle. For a spatial round trip, the sum
of retimed durations equals the original total duration.
\end{lemma}

\begin{proof}
A graph isomorphism preserves directed cycles. On a spatial round
trip, the correction terms
\(\varphi(y)-\varphi(x)\) telescope.
\end{proof}

An \(r\)-compatible retiming can assign duration \(0\) to an ordinary
edge. Therefore it does not preserve a numerical bound of slope \(r\)
when one retains the old spatial metric and uses the new tick
coordinate. The causal relation must be transformed with the
coordinates. Negative or zero coordinate duration is not, by itself,
a realizable feedback device.

\subsection{The earlier strong closure hypothesis}

We record the following hypothesis for analysis, not as a standing
assumption of the discrete or continuum theorems.

\begin{hypothesis}[Unrestricted device retiming closure]
\label{hyp:boost}
Every \(r\)-compatible retiming image of every realizable transport
step type is realizable as a device in one common substrate.
Devices may be translated in tick time and finitely composed without
additional compatibility restrictions. Ordinary waiting steps are
available.
\end{hypothesis}

\begin{assumption}[E7: Finite causal executability]\label{ass:E7}
\begin{enumerate}[label=(\alph*)]
\item Every admitted finite wired device network has a well-founded
linearization consistent with its input--output dependencies.
Composition along any legal linearization gives the same complete
outcome law.
\item If the network is calculus generated and well typed under
Assumption~\ref{ass:programinterface}, its linearization is realized
by a finite composite program with that denotation.
\end{enumerate}
\end{assumption}

\begin{proposition}[Executability of admissible finite networks]
\label{prop:wf}
Every finite semantically admissible block network satisfies
E7(a). Every finite syntactically admissible network satisfies E7(b).
\end{proposition}

\begin{proof}
Dependencies increase the original tick, so the finite graph is
acyclic. Topological orderings exist. Within a tick, disjoint complete
channels commute, giving ordering-independent composition.
The syntactic statement is Proposition~\ref{prop:sound}.
\end{proof}

\begin{theorem}[Formal super-cone cycle construction]\label{thm:cycles}
Assume Hypothesis~\ref{hyp:boost}. Suppose lossless classical-bit
transport is available in both directions between \(P,Q\), in one
original tick, with \(\rho=d(P,Q)>r\).
Then an admitted composition has a directed closed chain through one
event.
\end{theorem}

\begin{proof}
Put \(G=\lfloor\rho/r\rfloor\ge1\) and define
\[
 \varphi(x)=
 \max\left\{0,\ G-\left\lceil d(x,P)/r\right\rceil\right\}.
\]
If \(d(x,y)\le r\), the two ceiling terms differ by at most one.
Taking the positive part preserves that bound, so \(\varphi\) is
compatible. Also \(\varphi(P)=G\), \(\varphi(Q)=0\).

The \(\beta_\varphi\)-image of the forward transport and the
\(\beta_{-\varphi}\)-image of the reverse transport both have duration
\(1-G\). By the closure hypothesis, they are independently realizable
in the common substrate.

Starting at \((P,s_0)\), use the forward image, wait at \(Q\) for
\(2(G-1)\) ticks, and use the reverse image. The final event is
\((P,s_0)\), because the total duration is
\[
 (1-G)+2(G-1)+(1-G)=0.
\]
The devices are composed from two different retiming images; this is
not the application of one coordinate change to an original network.
\end{proof}

The new edges have nonpositive duration and nonzero displacement.
They lie outside a future velocity cone; they are not inputs to the
positive-duration velocity-polytope theorem.

\begin{proposition}[Closed dependencies conflict with E7(a)]
\label{prop:cyclecontradiction}
A directed dependency cycle has no well-founded linearization.
For lossless classical-bit legs, a NOT rule on one leg gives no
pathwise consistent bit assignment; identity rules allow multiple
assignments with the same external initial data.
\end{proposition}

\begin{proof}
A topological ordering of a directed cycle would require an event
to precede itself. For the classical examples, the consistency
equations are respectively \(x=1-x\) and \(x=x\).
The first has no bit-valued solution; the second has both constant
bit solutions. These are statements about pathwise assignments,
not about selecting a stationary marginal as a substitute for an
acyclic instrument execution.
\end{proof}

\begin{theorem}[The strong closure also excludes ordinary relays]
\label{thm:boostobstruction}
Hypothesis~\ref{hyp:boost} and E7(a) are incompatible with any
bidirectional lossless one-tick transport between distinct sites,
even when their distance is at most \(r\).
\end{theorem}

\begin{proof}
Let \(P\ne Q\) be such sites. Define
\(\varphi(P)=1\) and \(\varphi(x)=0\) for \(x\ne P\).
This function is \(r\)-compatible for every metric because every
difference of values is at most one.

The \(\beta_\varphi\)-image of the \(P\)-to-\(Q\) step has duration
zero. The \(\beta_{-\varphi}\)-image of the \(Q\)-to-\(P\) step also
has duration zero. Unrestricted composition makes
\((P,s)\to(Q,s)\to(P,s)\) an admitted dependency cycle.
Proposition~\ref{prop:cyclecontradiction} contradicts E7(a).
\end{proof}

\begin{remark}[Correct conclusion]
The formal implication ``super-cone bidirectional transport plus
strong closure contradicts E7(a)'' is valid, but it is not a selective
physical exclusion: the same assumptions exclude the ordinary relay
used for sharp attainment. Accordingly, unrestricted device retiming
closure is not used as an additional axiom of an admissible relay
substrate and is not included in the continuum existence conjectures.

A useful physical boost interface would need frame-labelled devices,
admissible inter-frame compositions, and a causal-order-preserving
notion of frame change. The fixed-channel ordering proposition supplies
none of those data. The construction above also does not prove a
general evasion theorem for one-way channels under unrestricted closure
of an entire device class.
\end{remark}

\section{A conditional continuum package with matched realizations}
\label{sec:continuum}

\subsection{Geometric hypotheses}

A scaling family consists of admissible substrates
\((\Lat_\varepsilon,d_\varepsilon,r)\), site embeddings
\(z_\varepsilon:\Lat_\varepsilon\to\RR^D\), and physical tick
durations \(h_\varepsilon\to0\). Events embed as
\[
 (x,s)\longmapsto(s h_\varepsilon,z_\varepsilon(x)).
\]
Assume block footprints in physical spacetime have diameter tending
to zero on bounded windows. This includes both input and output
interfaces of each block firing.

For the cubic example,
\[
 \Lat_\varepsilon=\varepsilon\ZZ^D,\qquad
 d_\varepsilon(x,y)=\frac{\norm{x-y}_1}{\varepsilon},
 \qquad h_\varepsilon=\frac{r\varepsilon}{c}.
\]
Its metric envelope has physical slope \(c\) in the \(\ell^1\) norm.

\begin{hypothesis}[C1: Local event approximation]\label{hyp:C1}
Let \(E_\varepsilon\) be the embedded event set. For every compact
\(K\subset\RR^{1,D}\),
\[
 \sup_{p\in K}\dist(p,E_\varepsilon)\longrightarrow0.
\]
\end{hypothesis}

This distance-function formulation avoids requiring
\(E_\varepsilon\cap K\) to approximate every compact set, which would
fail, for example, for a singleton not lying on a particular mesh.

For an embedded terminal event \(q=(T h_\varepsilon,z_\varepsilon(A))\),
define the full metric past envelope
\[
 J^-_\varepsilon(q)=
 \left\{
 (s h_\varepsilon,z_\varepsilon(x)):
 s\le T,\quad d_\varepsilon(x,A)\le r(T-s)
 \right\}.
\]
A finite experiment uses its restriction to the experiment's starting
time.

\begin{hypothesis}[C2: Uniform local cone convergence]\label{hyp:C2}
There is a norm \(\nu\) on \(\RR^D\) and \(c>0\) such that, for
\[
 J^-_\nu(q)=
 \{(t,x):t\le t_q,\ \nu(x-x_q)\le c(t_q-t)\},
\]
the distance functions to \(J^-_\varepsilon(q)\) converge to those
of \(J^-_\nu(q)\), uniformly for sources in a fixed compact set and
embedded terminal events \(q\) in a fixed compact set.
\end{hypothesis}

The uniformity is needed when terminal events or operation supports
move within compact sets. It gives the positive-margin exclusion used
below. The cubic example has \(\nu=\norm{\cdot}_1\), not the Euclidean
norm.

\begin{remark}[An upper envelope is not an attainable cone]
C2 concerns a metric envelope. It does not assert that a given
schedule realizes every direction on its boundary.
Theorem~\ref{thm:isotropyobstruction} constrains actual scheduled
reachability. The lower operational hypothesis C5 is therefore
separate from C2.
\end{remark}

\subsection{Operational hypotheses}

Every reported finite experiment has a fixed finite outcome alphabet
and includes its complete attempted-run law. Probabilities are compared
in total variation on that common alphabet.

A modification footprint includes changed complete local rules and
changed initial input sites. An initial modification confined to a
region means that the two initial snapshots agree on every finite
region disjoint from it. It does not mean that arbitrary correlated
initial states with the same site labels are interchangeable.

\begin{hypothesis}[C3: Locality-compatible operational realization]
\label{hyp:C3}
There is a class \(\mathcal{X}\) of limit experiments with declared
compact operation and readout supports, satisfying:
\begin{enumerate}[label=(\roman*)]
\item Every experiment has a realizing family of finite discrete
experiments whose complete outcome laws converge in total variation
and whose operation footprints approach the declared supports.
\item Realization is compatible with finite disjoint composition and
with the declared recorded continuations.
\item If two limit experiments differ only by a modification in a
compact set \(K'\), they admit \emph{matched} realizing families:
the schedule and unchanged background are common, all discrete
modification footprints approach \(K'\), and their initial snapshots
agree outside the declared discrete initial modifications.
Common readouts have common discrete realizations.
\end{enumerate}
The same clauses may be required with an untouched reference as an
additional operational interface.
\end{hypothesis}

Clause (iii) is essential. Separate convergence of two experiment
laws, without preservation of their local agreement, is not enough
to apply a discrete domain-of-dependence theorem.

\begin{hypothesis}[C4: Local completeness]\label{hyp:C4}
A compatible family of finite discrete experiments with convergent
declared supports in a fixed compact set and total-variation-convergent
complete outcome laws determines an experiment in \(\mathcal{X}\).
Membership and identifications use the stated operational realization,
not support alone.
\end{hypothesis}

C4 supplies membership of newly constructed convergent families.
It is not a compactness theorem ensuring that arbitrary experiment
families converge.

\begin{theorem}[Continuum domain of dependence]\label{thm:contDOD}
Assume C1--C3. Suppose two limit experiments have a common readout at
\(a\) and differ only by a compact modification footprint \(K'\) with
\[
 \dist(K',J^-_\nu(a))\ge\delta>0.
\]
Then their readout laws agree.
\end{theorem}

\begin{proof}
Use C3(iii) to choose matched discrete realizations with terminal
events \(a_\varepsilon\to a\). Uniform cone convergence and convergence
of the modification footprints imply that, for sufficiently small
\(\varepsilon\), every modification lies outside
\(J^-_\varepsilon(a_\varepsilon)\), with a positive margin.

Because complete block footprints shrink, no changed block meeting
the next cone slice is hidden outside the declared modification
footprint. The discrete initial cone snapshots agree and the required
inner block channels agree. Theorem~\ref{thm:cone} gives equal discrete
readout laws. Their total-variation limits are equal by C3(i).
\end{proof}

\begin{corollary}[Continuum operational no-signalling]\label{cor:contNS}
Assume C1--C3. Let two compact operation/readout regions be spacelike
separated at positive margin:
\[
 \nu(x_p-x_q)\ge c\abs{t_p-t_q}+\delta
 \quad\text{for every }p\in K_1,\ q\in K_2.
\]
Replacing one experiment by another in its region does not change
the unconditioned joint record law of the other.
\end{corollary}

\begin{proof}
By compactness and continuity, every point of either region is at
positive distance from the causal past of every point in the other.
For sufficiently small meshes, uniform C2 excludes all dependency
paths in either direction between the two complete discrete regions.
The discrete schedule-locality argument applies to the full set of
the other experiment's reported records, not only to each marginal
record separately. C3(ii)--(iii) supplies matched composites.
Pass to the total-variation limit.
\end{proof}

Conditioning on a record in the remote region may change a conditional
law; that does not contradict this unconditioned statement.

\subsection{A non-degenerate operational lower bound}

\begin{hypothesis}[C5: Directional relay limits]\label{hyp:C5}
For every \(\nu\)-unit vector \(v\) and duration \(u>0\), there are
matched discrete relay pairs whose sender modifications approach
\(A=(0,x)\), whose readouts approach
\(B=(u,x+cu v)\), and whose limits belong to \(\mathcal{X}\).
Their two complete readout laws converge in total variation and satisfy
\[
 \dtv(p_\varepsilon^{(0)},p_\varepsilon^{(1)})
 \ge\gamma>0
\]
for all sufficiently small meshes.
The sender modifications can be localized in arbitrarily small
neighborhoods of \(A\).
\end{hypothesis}

\begin{theorem}[Sharp continuum speed]\label{thm:contspeed}
Assume C1--C5. Define \(c_{\mathrm{cont}}\) as the supremum of
\[
 \frac{\nu(x_B-x_A)}{t_B-t_A}
\]
over positive-duration signalling pairs of limit events, with sender
modifications supported arbitrarily close to \(A\).
Then \(c_{\mathrm{cont}}=c\).
\end{theorem}

\begin{proof}
If the ratio exceeds \(c\), a sufficiently small neighborhood of
\(A\) is at positive distance from \(J^-_\nu(B)\).
Theorem~\ref{thm:contDOD} excludes signalling, so
\(c_{\mathrm{cont}}\le c\).

C5 gives pairs on the boundary. Total-variation convergence and the
triangle inequality preserve a gap at least \(\gamma\) in their
limit laws. Thus \(c_{\mathrm{cont}}\ge c\).
\end{proof}

\begin{proposition}[A sufficient relay-gap certificate]
\label{prop:relaygap}
Suppose an ideal discrete relay pair has laws \(\delta_0,\delta_1\).
If its two implemented laws are each within total variation
\(\eta_\varepsilon\) of the corresponding ideal law, then
\[
 \dtv(\widetilde p_\varepsilon^{(0)},
       \widetilde p_\varepsilon^{(1)})
 \ge1-2\eta_\varepsilon.
\]
For a finite composition of complete channels with local half-diamond
errors \(\delta_{\varepsilon,j}\), one may take
\(\eta_\varepsilon\le\sum_j\delta_{\varepsilon,j}\).
\end{proposition}

\begin{proof}
The first claim is the triangle inequality.
For the second, telescope the channel compositions one factor at a
time. CPTP pre- and postcomposition contract diamond distance.
The resulting total channel error is at most the sum of local errors;
a terminal readout cannot increase it.
\end{proof}

This is the complete-channel error accounting in Cheung En02-B
\oplabel{thm:B6} \cite{CheungEn02B}. For a fixed physical journey
requiring \(O(\varepsilon^{-1})\) ticks, per-tick error
\(o(\varepsilon)\) is one sufficient condition for a vanishing total
allowance. It is not necessary in every error model.

Ideal classical relays already have gap \(1\) at every discrete
scale. The open task is to realize the required directional endpoint
families and compatible operational limits, or to control errors
uniformly in an implementation. Discrete relay attainment alone
does not establish those combined requirements.

\subsection{Observable algebras require their own interface}

\begin{hypothesis}[C6: Product-preserving observable realization]
\label{hyp:C6}
For each scale there is a represented observable algebra
\(\mathfrak{B}_\varepsilon\) containing local subalgebras associated
with bounded spacetime operation regions. It has the following
properties:
\begin{enumerate}[label=(\roman*)]
\item Generators assigned to two regions with no scheduled dependency
path in either direction commute in the chosen complete observable
realization.
\item There are linear comparison maps
\(j_\varepsilon:\mathfrak{B}_\varepsilon\to\mathfrak{B}\)
into a common \(C^*\)-algebra. For the admitted uniformly bounded
generator sequences and their finite products, these maps are
asymptotically product preserving:
\[
 \norm{
 j_\varepsilon(a_\varepsilon b_\varepsilon)
 -j_\varepsilon(a_\varepsilon)j_\varepsilon(b_\varepsilon)
 }\longrightarrow0.
\]
They also preserve the adjoint asymptotically. Admitted generator
images converge in norm.
\item The norm limits generate isotone \(C^*\)-subalgebras
\(\mathfrak{A}(O)\subseteq\mathfrak{B}\).
The operational laws extend to states on their generated algebra.
\end{enumerate}
The realization must specify how allocation, consumed wires,
discarded systems, and dilation ancillas are represented.
\end{hypothesis}

\begin{remark}[Channels and observables are not interchangeable]
Commutation of two CP superoperators does not imply commutation of
elements of an observable algebra, and a channel adjoint need not be
multiplicative. C6 is an additional observable-level construction
hypothesis. A common local unitary dilation can motivate such an
interface, but that dilation and its spacetime localization must be
provided. They are not constructed by the operational cone theorem.
\end{remark}

\begin{theorem}[Conditional algebraic microcausality]
\label{thm:microcausality}
Assume C1--C3 and C6. For bounded open regions \(O_1,O_2\) whose
closures are spacelike separated at positive margin,
\[
 [\mathfrak{A}(O_1),\mathfrak{A}(O_2)]=\{0\}.
\]
Their admitted non-selective operations also obey operational
no-signalling.
\end{theorem}

\begin{proof}
Uniform C2, the positive margin, and shrinking block footprints exclude
scheduled dependency paths between the two regions at sufficiently
small scales. By C6(i), corresponding discrete generators
\(a_\varepsilon,b_\varepsilon\) commute.

Let their comparison images converge to \(a,b\).
Asymptotic multiplicativity gives
\[
 [a,b]
 =
 \lim_{\varepsilon\to0}
 j_\varepsilon(a_\varepsilon b_\varepsilon
                -b_\varepsilon a_\varepsilon)=0.
\]
Finite polynomials in each set of generators commute with finite
polynomials in the other. Norm continuity of multiplication extends
commutation to the generated \(C^*\)-algebras.
Operational no-signalling is Corollary~\ref{cor:contNS}.
\end{proof}

The theorem is a transfer theorem under C6, not a construction of a
quasi-local algebra or a quantum field theory from arbitrary block
instruments.

\subsection{Frame closure and Alexandrov--Zeeman}

\begin{hypothesis}[C7: Calibrated frame covariance]\label{hyp:C7}
The limit theory has a group \(\mathcal{G}\) of bijections of
\(\RR^{1,D}\) such that:
\begin{enumerate}[label=(\roman*)]
\item transformations carry admitted experiments and their supports
to admitted experiments and preserve their complete outcome laws;
\item they preserve the order defined by \(J^-_\nu\) in both directions;
\item they preserve calibrated spacetime volume in the fixed time and
length units, and hence allow no change of the common scale;
\item they act transitively on future timelike inertial directions,
represented by parallel classes of future timelike lines.
\end{enumerate}
\end{hypothesis}

Clause (iii) is stronger than saying that no \emph{pure} dilation is
a group element. The latter alone would not exclude transformations
that combine a dilation with a Lorentz transformation.

\begin{theorem}[Conditional frame-invariant maximal speed]
\label{thm:invariantspeed}
Let \(D\ge2\). Assume C1--C5 and C7, with Euclidean \(\nu\).
Every \(g\in\mathcal{G}\) is an orthochronous Poincare transformation.
The operational maximal speed \(c_{\mathrm{cont}}=c\) has the same
value in every admitted inertial frame.
\end{theorem}

\begin{proof}
Rescale the time coordinate to \(ct\). The cone order is the Minkowski
causal order. The Alexandrov--Zeeman theorem identifies its bijective
order automorphisms, in spacetime dimension at least three, as
\[
 g(p)=a+\lambda Lp,
\]
where \(\lambda>0\) and \(L\) is an orthochronous Lorentz
transformation, possibly reversing spatial orientation
\cite{Alexandrov,Zeeman}.

Such a map multiplies spacetime volume by \(\lambda^{D+1}\).
C7(iii) therefore forces \(\lambda=1\).
C7(iv) connects the distinguished inertial direction with every
admitted one. By C7(i), signalling pairs and their law gaps transform
with the frame. Poincare transformations preserve the boundary cone,
and Theorem~\ref{thm:contspeed} gives its attained maximal speed \(c\).
\end{proof}

\begin{remark}[One spatial dimension]
For \(D=1\), arbitrary increasing bijections of the two null
coordinates preserve causal order, so the cited causal-automorphism
reconstruction does not give the same rigidity \cite{Zeeman}.
Additional affine and calibration assumptions are required.
\end{remark}

C7 is a covariance hypothesis of a limit theory, not unrestricted
retiming closure of discrete devices. It is not affected by retaining
the original tick grading in admissible discrete networks.

\subsection{Separated existence obligations}

\begin{conjecture}[Isotropic geometric and relay family]
\label{conj:family}
There exists a scaling family of admissible substrates and schedules
satisfying C1--C2 with Euclidean limiting norm and admitting the
directional matched relay families required by C5.
\end{conjecture}

\begin{conjecture}[Operational, algebraic, and frame-covariant limit]
\label{conj:opcov}
A family as in Conjecture~\ref{conj:family} can be equipped with
realizations satisfying C3--C4 and C6, and its limit theory satisfies
C7. Finite experiments in each admitted frame have complete acyclic
instrument semantics.
\end{conjecture}

Unrestricted device retiming closure
(Hypothesis~\ref{hyp:boost}) is deliberately not included:
Theorem~\ref{thm:boostobstruction} makes it incompatible with the
ordinary bidirectional relay class.

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.14\linewidth}p{.35\linewidth}X@{}}
\toprule
Requirement & Discrete comparison or certificate & Missing construction\\
\midrule
C1--C2 &
Metric envelopes; timed velocity polytopes; cardinal diamond &
A Euclidean envelope together with sufficiently isotropic actual
reachability. Fixed finite periodic rescaling is excluded.\\
\addlinespace
C3--C4 &
Complete local-record factorization &
Matched operational realizations, convergence, and local
completeness.\\
\addlinespace
C5 &
Exact relay gap \(1\); implementation gap certificate &
Boundary relays in every limiting direction with admitted
total-variation limits.\\
\addlinespace
C6 &
Discrete complete supports &
A localized observable realization and product-preserving algebraic
limit.\\
\addlinespace
C7 &
Conditional causal-automorphism theorem &
An outcome-preserving calibrated frame group for the admitted
limit experiments.\\
\bottomrule
\end{tabularx}
\end{center}

\begin{remark}[Status of suggested routes]\label{rem:routes}
Record-boost constructions, interleaving arguments, and
Gromov--Hausdorff or reachable-set comparisons remain possible
research proposals, not proofs of the conjectures.
Their logical targets must be specified:
\begin{enumerate}[label=(\roman*)]
\item Anisotropy alone does not force a positive probability for a
Born-null event. A record-boost contradiction needs an independently
justified operational covariance interface.
\item Interleaving that remains within the declared block diameter
still obeys Theorem~\ref{thm:cone}. A claimed violation must identify
which locality or clock hypothesis changes.
\item Persistence of anisotropy can obstruct Euclidean frame
covariance. It does not contradict no-signalling relative to the
anisotropic cone itself.
\end{enumerate}
Increasing directional or quotient complexity, or abandoning the
fixed periodic class, may evade Theorem~\ref{thm:isotropyobstruction}.
Neither evasion establishes the full limit package.
\end{remark}

\section{Finite verification and comparison boundaries}
\label{sec:discussion}

\subsection{A finite verification programme}

The proposal to check finite graphs with \(N\le20\) and one
exceptional block remains a programme, not an execution claim.
An exhaustive computation must specify:
\begin{enumerate}[label=(\roman*)]
\item a finite catalogue of graphs, schedules, horizons, and
register interfaces;
\item a finite exact gate/instrument library, or a symbolic
parameterization with proved algebraic coverage;
\item complete outcomes, failure flags, and controller supports;
\item exact or validated numerical arithmetic and its error bounds.
\end{enumerate}
The bound \(N\le20\) alone does not make the class of all quantum
instruments finite or justify an exhaustive enumeration.

For a fixed finite network, the factorization identity can be checked
on a basis of matrix units in each classical block. For a chosen
complementary state \(\sigma\), form
\(K=R_Y\Phi E_\sigma\) and test
\[
 R_Y\Phi-KR_I=0.
\]
That is a channel identity, not a sample check on selected input states.

\begin{proposition}[Approximate cone certificate]\label{prop:approx}
Suppose a complete implemented channel \(\widetilde\Phi\) satisfies
\(\ddia(\widetilde\Phi,\Phi)\le e\), where \(\Phi\) factors through
the initial cone snapshot. Two inputs with equal cone snapshots have
implemented output reductions at trace distance at most
\(\min\{1,2e\}\). The estimate is reference stable when the corresponding
joint cone--reference reductions agree.
\end{proposition}

\begin{proof}
The two ideal outputs agree. Insert that common output between the
implemented outputs and use the triangle inequality. Each
implemented--ideal discrepancy is at most \(e\). Trace distance is
at most one. The diamond allowance includes references.
\end{proof}

For finite input dimension \(d\), a normalized Choi comparison gives
\[
 \ddia(\Phi,\Psi)
 \le\frac d2\norm{J_{\mathrm n}(\Phi)-J_{\mathrm n}(\Psi)}_1.
\]
A separate already-certified diamond allowance is added without
another factor \(d\). This is the certificate accounting emphasized
in Cheung Hawk02-E and Hawk04-A
\cite{CheungHawk02E,CheungHawk04A}; see also \cite{Watrous}.

Flagged truncations must remain deterministic. For a projection \(P\),
the map
\[
 X\longmapsto PXP+\Tr((I-P)X)\kb{\perp}{\perp}
\]
is CPTP. For states commuting with \(P\), the trace-distance error in
a common enlarged output space is exactly the discarded probability.
That equality is not asserted for arbitrary states with low--high
coherences. Cheung En02-E gives this accounting for its specified
Gaussian packet states \cite[\oplabel{thm:cutoff}]{CheungEn02E}.

A finite-graph certificate is neither a continuum limit nor evidence
that the operational, algebraic, or covariance construction gaps
have been discharged.

\subsection{Relations to other geometric and analytic manuscripts}

Cheung VIII-B constructs a variable-deficit spatial-cone dictionary
on its declared background
\cite[\oplabel{thm:dictionary}]{CheungVIIIB}.
That cone is not the scheduled past defined in
\eqref{eq:schedulepast}, and its construction is not an
Alexandrov--Zeeman reconstruction.

Cheung Em02-E gives a classical relational lapse on a declared
background \cite[\oplabel{thm:lapse}]{CheungEm02E}.
It does not supply the frame group in C7 or a derivation of the metric
from the hybrid input.

The characteristic-product programme in XIV-H--XIV-J is also
separate. The earlier internal-amplitude stop in XIV-H is followed
by the internal construction reported at I2 in XIV-I; the full joint
characteristic map remains unresolved at I3 and J2 in the supplied
releases \cite{CheungXIVH,CheungXIVI,CheungXIVJ}.
A finite channel implementation or a causal-cone proof cannot define
that missing distributional operation. No inherited quantum
coefficient is admitted by this manuscript.

\subsection{Conclusion}

The discrete results establish the following chain:
\[
 \text{partitioned bounded-range channels}
 \ \Longrightarrow\
 \text{snapshot and adjoint locality}
 \ \Longrightarrow\
 \text{readout domain of dependence}.
\]
The explicit relay supplies the separate lower bound and makes the
metric lattice speed exactly \(r\) on the stated substrates.
Neither scheduled ancestry nor the metric envelope is identified
with the minimal influence region of every channel.

The continuum problem is not a second proof of the discrete cone.
It is a construction problem involving isotropic reachability,
matched operational limits, a product-preserving observable
realization, and calibrated frame covariance.
The velocity-polytope obstruction identifies why fixed finite
periodic rescaling does not solve it.

All continuum conclusions remain conditional.
The coordinate-retiming calculation does not establish physical
boost closure, and the unrestricted closure analyzed here is
incompatible with ordinary bidirectional relays.
No invariant physical speed is claimed.

\appendix

\section{Alignment of the Enter interface}\label{app:interface}

The following clauses describe the optional calculus interface.
They are not premises of Lemma~\ref{lem:onelayer},
Lemma~\ref{lem:dod}, or Theorem~\ref{thm:cone}, which are channel
theorems. A concrete calculus must discharge them on its own stated
typing and evaluation domain.

\begin{description}[style=nextline,leftmargin=2em]
\item[E1: Termination.]
Selected well-typed programs terminate. A strongly normalizing
calculus supplies this at the reduction level; a finite acyclic
instruction tree supplies it for the explicit fragment here.
To obtain a finite evaluation tree from strong normalization one
also uses the finite-branching evaluation interface.
External idle ticks are not reductions.

\item[E2: Confluence and disjoint commutation.]
The complete denotation is independent of legal evaluation strategy.
Disjoint enabled quantum operations commute.
Global confluence must be supplied separately: commutation of only
disjoint redexes does not logically imply confluence for every
possible pair of enabled redexes.
The source comparison is Landmark I
\oplabel{thm:confluence}; fixed-network channel ordering is the
different statement in En03-A \oplabel{thm:retiming}
\cite{CheungI,CheungEn03}.

\item[E3: Internal no-signalling.]
Complete programs on disjoint owned quantum and classical interfaces
cannot change each other's unconditioned output law.
The source comparison is Landmark I
\oplabel{thm:internal-ns} \cite{CheungI}.
On the substrate, complete ownership and controller support remain
explicit modeling conditions.

\item[E4: Records.]
Destructive measurement, coherent recording, and surviving quantum
interfaces are distinguished.
The exact identity required here is
Proposition~\ref{prop:drop}, with its context restriction.
Compare Landmark I \oplabel{def:record} and En03-A
\oplabel{def:recording}, \oplabel{def:nondestructive}
\cite{CheungI,CheungEn03}.
A record-only uniqueness or compatibility theorem is not an
unrestricted residual-instrument theorem.

\item[E5: Local complete-instrument semantics.]
The program's first-order instrument depends only on its complete
input state and is affine on unnormalized hybrid states.
No-cloning and ensemble indistinguishability follow at that semantic
level. Landmark I's \oplabel{lem:path}, \oplabel{thm:adequacy}, and
\oplabel{thm:nocloning} provide the source comparison
\cite{CheungI}.
Its \oplabel{prop:classical-recovery} is an additional classical-record
reconstruction obstruction, not the statement of affinity itself.

\item[E6: Mixed states and admitted adaptive control.]
The instrument extends linearly to mixed inputs and tensorially to
untouched references. Finite adaptive and randomized strategies are
admitted when their complete classical histories and controls are
included in the interface.
Arbitrary state-inspecting or remote-history-inspecting schedulers
are not covered.
Proposition~\ref{prop:path} proves this extension for the explicit
finite fragment; compare En04-A and Hawk04-B
\cite{CheungEn04,CheungHawk04B}.
\end{description}

The implementation results additionally require sequential and
parallel compositionality, well-typed wiring, and sufficient declared
routing capacity. These conditions are not consequences of the
spatial cone. Landmark I's circuit adequacy concerns its available
gate library; it is not an exact compilation theorem for every
arbitrary block instrument.

\section{Corrections to the supplied revision}\label{app:corrections}

The substantive corrections made here are:
\begin{enumerate}[label=(\arabic*)]
\item Snapshots are unnormalized classical--quantum block states.
Affinity is not assigned to normalized conditional states separately.
All retained placement and controller information belongs to the
snapshot.
\item The schedule past is distinguished from its metric envelope and
from actual influence. Reference statements specify agreement of joint
input reductions with the reference.
\item Destructive Enter is not identified with unrestricted coherent
recording. The inaccessible measured factor and the allowed
record contexts are explicit.
\item Soundness is a theorem about finite compositional program
networks, not arbitrary driven runs with finitely many live names.
Routing is required from every starting tick.
\item Strong normalization alone is not claimed to bound elapsed
substrate ticks. The completion and routing hypotheses are retained.
\item Channels on blocks shortened during repartitioning are
specified anew. They are not silently inherited by restriction.
\item Non-product exceptional operations are not assumed transmissive.
Classical-record reconstruction and entanglement-assisted transport
have different receiver interfaces.
\item Teleportation includes its Bell resource, communication capacity,
and charged processing ticks. A constant protocol overhead does not
prove super-cone transport at every distance greater than \(r\).
\item Fixed-gate ordering invariance and coordinate/device retiming
are separated. The floor-based cycle calculation is retained as a
formal implication, while the stronger ordinary-relay incompatibility
is proved and the physical retiming-stability claim is withdrawn.
\item Site homogeneity is not promoted to transitivity on all
Euclidean directions. The explicit cardinal diamond supplies the
directional comparison.
\item Local mesh approximation uses distance functions rather than
intersections with arbitrary compact sets. Cone convergence is uniform
on bounded source and target sets.
\item The continuum operational interface includes matched realizations
that preserve local modifications and unchanged data.
\item Algebraic microcausality requires a localized observable
realization and cross-product-preserving convergence. Commuting
superoperators alone do not provide it.
\item Frame calibration excludes the scale factor of every causal
automorphism, not merely pure dilation elements.
\item Isotropic and frame-covariant existence remain open.
Anisotropic no-signalling is not treated as a contradiction.
\end{enumerate}

\section{Documentary scope and verification status}
\label{app:status}

Companion manuscripts are cited as source records for comparisons;
their model-specific constructions are not transferred as premises.
In particular:
\begin{itemize}
\item This paper does not alter the supplied characteristic-map,
prepared-insertion, or bulk-restoration release boundaries.
It does not open the downstream XV-B admission exit or close O.1.
\item The internal amplitude construction reported in XIV-I is not
the full joint map absent at I3 and J2.
The historical H2 stop in XIV-H is not confused with that later,
narrower advance.
\item The supplied external mathematics catalogue is documentary only.
Its labels concerning factoring, circuit complexity, classical
capacity, or metric Alexandrov curvature are not a block-instrument
compilation theorem, a continuum limit, or the
Alexandrov--Zeeman causal-automorphism theorem.
\item No proof-assistant verification, newly executed finite-graph
enumeration, independently rerun numerical calculation, or successful
\oplabel{pdflatex} execution is asserted here.
The source uses only standard packages and an internal bibliography.
\end{itemize}

\begin{thebibliography}{99}

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N.~Cheung,
\emph{An Exact Causal Cone for Partitioned Enter-Calculus Dynamics:
Domain of Dependence, Necessity, and a Conditional Continuum Limit}.
Supplied manuscript \oplabel{085VvAv3 (1).tex}, August 2026.
Predecessor revised in the present manuscript.

\bibitem{CheungEC}
N.~Cheung,
\emph{Enter Calculus}.
Figshare manuscript, version 2, 2026.
DOI supplied with the predecessor:
\href{https://doi.org/10.6084/m9.figshare.33087515.v2}
{\nolinkurl{10.6084/m9.figshare.33087515.v2}}.

\bibitem{CheungI}
N.~Cheung,
\emph{Landmark I}.
Version 2, supplied companion manuscript \oplabel{I.tex}.
Finite Enter characterization and metatheory; especially
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\oplabel{lem:path}, \oplabel{thm:adequacy},
\oplabel{thm:nocloning}, \oplabel{thm:internal-ns}, and
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Kinematics and causal cuts on a prescribed schedule; especially
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\oplabel{prop:directional}.
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\bibitem{CheungVIIJ}
N.~Cheung,
\emph{Landmark VII-J}.
Version 1, supplied companion manuscript \oplabel{VII-J.tex}.
Scheduled influence, physical containment, and timed
cycle-velocity polytopes; especially
\oplabel{thm:nosignal}, \oplabel{thm:schedule},
\oplabel{thm:physical}, \oplabel{thm:local-velocity}, and
\oplabel{thm:polytope}.
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\bibitem{CheungPsy04B}
N.~Cheung,
\emph{Psy04-B}.
Version 1, supplied companion manuscript \oplabel{Psy04-B-v1.tex}.
Complete finite causal networks and scheduled support; especially
\oplabel{thm:schedule} and the influence-containment corollary.
Source archive: \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;
especially \oplabel{thm:locality}.
Source archive: \url{https://enter.grok.me/}.

\bibitem{CheungVIIE}
N.~Cheung,
\emph{Landmark VII-E}.
Version 1, supplied companion manuscript \oplabel{VII-E.tex}.
Prescribed kinematics and fixed labelled-channel retiming;
especially \oplabel{thm:retiming}.
Source archive: \url{https://enter.grok.me/}.

\bibitem{CheungEn04}
N.~Cheung,
\emph{En04-A}.
Version 1, supplied companion manuscript \oplabel{En04-A.tex}.
Typed preparation, complete finite-program soundness, destructive
readout, and complete residual response; especially
\oplabel{thm:soundness}, \oplabel{thm:drop},
\oplabel{thm:classical}, and \oplabel{def:response}.
Source archive: \url{https://enter.grok.me/}.

\bibitem{CheungHawk04A}
N.~Cheung,
\emph{Hawk04-A}.
Version 1, supplied companion manuscript \oplabel{Hawk04-A.tex}.
Complete-output preservation, complementary channels, and finite
channel certificates.
Source archive: \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, destructive readout, discard, and
classical-record reconstruction; especially
\oplabel{thm:adequacy}, \oplabel{thm:drop}, and
\oplabel{thm:classical}.
Source archive: \url{https://enter.grok.me/}.

\bibitem{CheungEn02B}
N.~Cheung,
\emph{En02-B}.
Version 1, supplied companion manuscript \oplabel{En02-B-v1.tex}.
Finite decoder accounting and compatible half-diamond error budgets;
especially \oplabel{thm:B6}.
Source archive: \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;
especially \oplabel{thm:C1}.
Source archive: \url{https://enter.grok.me/}.

\bibitem{CheungEn02E}
N.~Cheung,
\emph{En02-E}.
Version 2, supplied companion manuscript \oplabel{En02-E-v2.tex}.
Deterministic flagged Fock approximation;
especially \oplabel{thm:cutoff}.
Source archive: \url{https://enter.grok.me/}.

\bibitem{CheungHawk02E}
N.~Cheung,
\emph{Hawk02-E}.
Version 1, supplied companion manuscript \oplabel{Hawk02-E.tex}.
Complete-residual and decoder certificates;
especially \oplabel{thm:certificate} and \oplabel{thm:regulator}.
Source archive: \url{https://enter.grok.me/}.

\bibitem{CheungVIIIB}
N.~Cheung,
\emph{Landmark VIII-B}.
Version 2, supplied companion manuscript \oplabel{VIII-B-v2.tex}.
Variable-deficit cone dictionary;
especially \oplabel{thm:dictionary}.
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N.~Cheung,
\emph{Em02-E}.
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Classical relational lapse through second order;
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N.~Cheung,
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N.~Cheung,
\emph{Landmark XIV-I}.
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Internal retarded-pole amplitudes and the joint-map boundary;
especially \oplabel{thm:I2}, \oplabel{thm:I3}, and
\oplabel{thm:release}.
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N.~Cheung,
\emph{Landmark XIV-J}.
Version 1, supplied manuscript \oplabel{XIV-J.tex}.
Reconstruction descent, supported-sector reference, and the missing
joint characteristic map; especially \oplabel{thm:J2} and
\oplabel{cor:noadmission}.
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\end{document}