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% No successful compilation, numerical execution, hardware compilation,
% proof-assistant verification, or independent certification is asserted.
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% Principal corrections incorporated:
% 1. The 3+1-dimensional conical defect is a worldsheet.
% 2. The three-to-six statement concerns a transverse link.
% 3. The source VII-J apex-register benchmark is not a regular-part
%    experiment. The present benchmark replaces it by regular ports.
% 4. A distance infimum on the punctured cone is not automatically a
%    realized causal route at equality. Actual regular routes are supplied.
% 5. Every controller, record wire, encoder, and decoder is included.
% 6. Joint-output support, not only single-output support, is certified.
% 7. Full- and half-diamond conventions are distinguished explicitly.
% 8. Cycle velocities do not certify short-time causal safety.
% 9. Hamiltonian tails do not establish an exact relativistic cutoff.
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% 11. The geometric incidence timestamp is not the completion time of
%     every metadata computation.
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  pdftitle={VII-J-ST: Scheduled dependencies and certified causality on a prescribed conical spacetime},
  pdfsubject={Finite Enter clocks, complete channels, directional reachability, and buffered causal transport}
}

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\title{\textbf{VII-J-ST: Scheduled dependencies and certified causality\\
on a prescribed conical spacetime}\\[5pt]
\large A local Grade S Enter adaptation with complete clock instruments,\\
directional certificates, and buffered regular-part transport}
\author{Ngo Cheung}
\date{}

\begin{document}
\maketitle

\begin{abstract}
We give a self-contained local Grade S adaptation of VII-J's
finite-network causality programme to I-B's prescribed
\(3+1\)-dimensional conical product. The metric, deficit
\(\pi/2\), causal relation, slicing family, preparations, and
communication permissions are inputs. Complete timed dependency
graphs, including classical controllers, measurements, records,
encoders, and decoders, bound the Heisenberg support of every
jointly observed output. A causal embedding then bounds
reference-assisted intervention influence by the prescribed
spacetime causal relation.

The operational addition is an explicit regular-part realization
through the parameter of VII-F's transverse three-to-six incidence.
Unlike the source apex-register benchmark, no quantum register,
gate, or communication path meets the singular worldsheet.
A destructive two-qubit clock controls finite event metadata;
a buffered gather--operate--distribute protocol implements a
nontrivial two-qubit exchange, preserves arbitrary retained
reference correlations, and has a unit signalling witness on an
allowed route. The complete recorded channel, positive durations,
blank registers, controller storage, copied records, and finite
instruction reserves are specified.

Directional certificates retain both cycle-velocity polytopes
and finite-time offsets. A finite potential inequality exposes
transient directional costs that an asymptotic velocity inclusion
can miss. Approximate simulation is treated separately:
signalling changes by at most twice the half-diamond error,
while a finite-volume Hamiltonian tail is reference-stable but
not an exact Lorentzian cutoff. These assembled certificates may
provide a useful reproducible benchmark for distinguishing
logical simulation accuracy, operational influence, and elementary
causal realizability. The novelty claimed is the supported
assembly and regular-part correction, not the general principles
of channel locality, periodic-graph velocities, or destructive
measurement. No continuum-field simulation, spacetime emergence,
Einstein dynamics, or inherited quantum response coefficient is
constructed.
\end{abstract}

\statementbox{Release boundary}{%
This manuscript proves a finite operational certificate on an
installed classical geometry. The geometric event is a prescribed
incidence change. It is not caused, detected at the singularity,
or reconstructed by the finite clock.

The source VII-J benchmark permits a register on the apex.
That permission is incompatible with the regular-part laboratory
used here. We therefore construct a different, explicitly regular
benchmark rather than silently transferring the apex construction.}

\tableofcontents

%=====================================================================
\section{Scope, Enter philosophy, and the two targets}
\label{sec:scope}
%=====================================================================

VII-J separates scheduled dependencies, intervention influence,
and an independently supplied causal structure
\cite{CheungVIIJ}. I-B equips a prescribed conical product with
finite names, accessible algebras, clock calibration, and
communication permissions \cite{CheungIB}. We retain those
distinctions and prove the interfaces needed for their local
combination.

\subsection{The Enter discipline}

Cheung's Enter framework treats destructive readout as an
input--output and access discipline: measured quantum names are
consumed, classical outcomes are returned, and specified
unmeasured factors survive \cite{CheungEnter,CheungI}.
This does not identify a classical record with the return of an
unknown quantum input. Nor does a channel equality identify a
thermal implementation, a geometric interpretation, or a
gravitational dynamical law \cite{CheungPhilosophy}.

We adopt an explicit finite Born instrument. Its channel identity
is proved below. We do not rederive the structural measurement
characterization of Manuscript I from an abbreviated hypothesis
list, and do not extend that characterization to fields.

The operational motivation can be summarized as four questions:
what is supplied, what may be accessed, what is returned, and
what additional construction supports the proposed interpretation?
The present answer supplies a classical background and a finite
network; it returns classical records and surviving finite
quantum payloads.

\begin{definition}[Local Grade S]
\label{def:grade}
A local Grade S certificate consists of:
\begin{enumerate}[label=\textup{(S\arabic*)}]
\item prescribed classical geometry and a supported regular
operational presentation;
\item explicit finite channels and complete control interfaces;
\item a finite destructive-clock instrument with its calibration;
\item proved support, influence, and causal-embedding statements;
\item a nonempty finite transport realization with a resource ledger.
\end{enumerate}
Derivation of the geometry from operational records, called
Grade E here, is not included.
\end{definition}

\begin{definition}[Targets A and B]
\label{def:targets}
Let \(\Psi\) be a declared finite target channel and
\[
 \Phi=\mathcal D\circ\mathcal C\circ\mathcal E
\]
an implementation on the same comparison interfaces.

\emph{Target A} requires a support or signalling statement for
\(\Phi\), together with a separate accuracy comparison to \(\Psi\).

\emph{Target B} additionally realizes every preparation, gate,
communication, clock operation, controller read, record, reset,
and decoder by the admitted local and causal primitives on the
supplied spacetime.
\end{definition}

Endpoint channel correctness does not by itself construct
Target B. A physical embedding of dependency endpoints is a
sufficient support certificate; an actual elementary realization
also needs the declared implementation primitives.

\paragraph{Provenance.}
VII-J is the source causality manuscript; VII-F supplies the
prescribed slicing family; I-B supplies the spacetime-interface
pattern \cite{CheungVIIJ,CheungVIIF,CheungIB}.
Other sheets are documentary comparisons. Their citation does
not import unprinted proof hypotheses. Every positive mathematical
claim used in this release has a proof below.

%=====================================================================
\section{Prescribed spacetime, support, and the incidence input}
\label{sec:geometry}
%=====================================================================

Fix a length unit \(\ell>0\), a speed \(c>0\), and a time origin
\(T_*\). Write
\[
 \tau=\frac{c(T-T_*)}{\ell},\qquad
 \mu=\frac{\pi}{2},\qquad \alpha=\frac34.
\]
The regular background is
\begin{equation}
 M=\RR_T\times C_\mu^\circ\times\RR_z,\qquad
 g=-c^2\dd T^2+\dd r^2+\alpha^2r^2\dd\phi^2+\dd z^2,
 \label{eq:metric}
\end{equation}
where \(r>0\), \(\phi\) has period \(2\pi\), and the stipulated
completion has singular locus
\begin{equation}
 \WW=\RR_T\times\{o\}\times\RR_z.
 \label{eq:worldsheet}
\end{equation}
The defect is a worldsheet. Only its fixed-\(z\) sections are
worldlines. No smooth chart, pointwise Einstein tensor, or field
boundary condition is asserted on \(\WW\).

On a lifted regular angular interval,
\[
 (T,r,\phi,z)\longmapsto
 \bigl(c(T-T_*),r\cos(\alpha\phi),r\sin(\alpha\phi),z\bigr)
\]
is a local Minkowski development. Static observers have
\(\dd\tau_{\rm proper}=\dd T\). The metric lapse in \(T\) is one.
The spinless seam gluing fixes \(T,z\) and rotates the transverse
developed plane by \(\pi/2\) in the chosen gluing convention.
These are the prescribed I-B data \cite{CheungIB}.

\begin{proposition}[Non-global hyperbolicity]
\label{prop:nonGH}
The regular spacetime \(M\) is not globally hyperbolic.
\end{proposition}

\begin{proof}
Fix \(r_0>0\), an angle, and \(z=0\). Take two events at that
spatial point with times \(T_*\pm2r_0/c\). Events at time \(T_*\)
and radii \(r_n\downarrow0\), at the same angle and axial
coordinate, lie in their causal diamond by radial causal paths.
They have no convergent subsequence in \(M\).
The diamond is therefore noncompact.
\end{proof}

No global Green or Cauchy theorem is inferred. All operational
paths below lie in a relatively compact regular developing tube.

\subsection{The prescribed transverse event}

Use a cut wedge of angle \(3\pi/2\), divided into three
right-angle sectors. In each sector let the dimensionless
coordinates be \(\xi,\eta\ge0\), and put
\begin{equation}
 \rho=\xi+\eta,\qquad
 h=\frac23(\xi+\eta),\qquad
 k=\frac35\min\{\xi,\eta\},\qquad
 F_s=s+\min\{2,\max\{h,s+k\}\},
 \quad -1\le s\le1.
 \label{eq:family}
\end{equation}
The functions agree on identified sector rays.
The prescribed product markings are
\begin{equation}
 \widehat\Sigma_s:
 \quad \tau=F_s(x),\qquad z=\ell\zeta,
 \label{eq:slices}
\end{equation}
with no axial dependence in \(F_s\).
The formula is VII-F's transverse family, explicitly lifted
rather than identified with a three-dimensional cell count
\cite{CheungVIIF}.

\begin{lemma}[Estimates used by the causal certificate]
\label{lem:slices}
Set
\[
 \lambda=\frac{2\sqrt2}{3}<1.
\]
On every regular developed region, \(F_s\) is spatially
\(\lambda\)-Lipschitz, including the axial variable. For \(s'>s\),
\begin{equation}
 s'-s\le F_{s'}(x)-F_s(x)\le2(s'-s).
 \label{eq:time-order}
\end{equation}
The graphs are embedded, and their regular affine pieces are
spacelike with induced dimensionless metric bounded below by
\(I/9\).

The prescribed transverse cellulation has one incidence-event
parameter, \(s=0\), where its singular transverse link changes
from three incident faces to six.
\end{lemma}

\begin{proof}
The active gradients are
\[
 (2/3,2/3,0),\quad(3/5,0,0),\quad(0,3/5,0),\quad0,
\]
up to sector rotations. Minima and maxima preserve their common
Lipschitz bound. The graph metric is \(I-aa^{\mathsf T}\);
its least eigenvalue is at least \(1-\lambda^2=1/9\).
Graph projection gives embeddedness. For fixed \(x\), the term
after the initial \(s\) in \eqref{eq:family} is nondecreasing
and 1-Lipschitz in \(s\), proving \eqref{eq:time-order}.

For the incidence, use
\[
 h-k\ge\frac{11}{30}\rho.
\]
When \(s\le0\), the \(h\)-branch dominates and one face in each
sector meets the apex. When \(s>0\), in the half-sector
\(\xi\ge\eta\), the new boundary is
\(10\xi+\eta=15s\); its symmetric counterpart is
\(\xi+10\eta=15s\). Their vertices are
\[
 B_i=\frac{3s}{2}e_i,\qquad
 C_i=\frac{15s}{11}(e_i+e_{i+1}).
\]
They define two distinct new apex triangles in each sector.
Their outermost radial coordinate is \(30s/11<3\), so they do
not meet the persistent crest \(\rho=3\).
Within each open epoch these boundaries have fixed combinatorics,
and all new coordinates vanish only at zero. Thus the normal
transverse link changes exactly once from three to six.
\end{proof}

The source's full lengths, areas, affine chart words, primitive
winding, and peripheral matching remain documentary geometric
data \cite{CheungVIIF}. They are not needed to infer the
finite-channel support theorem. In the product, the event locus
is the line \(\{\tau=0,x=o\}\subset\WW\); it is not an isolated
point event of the \(3+1\)-dimensional background.

\begin{definition}[Supported operational presentation]
\label{def:support}
A geometric atlas may describe a finite transverse patch and a
finite axial slab in the conical completion, including its
stipulated singular stratum.

The operational support is different: it is the closure of a
finite time interval times a bounded convex developed spatial
tube \(U\), with \(\overline U\cap\WW=\varnothing\).
All initial carriers, blanks, programme words, controller
registers, and ancillary inputs are declared there.
There is no undeclared incoming boundary data.
\end{definition}

This is not an initial-boundary-value problem for a field.
A meridian surrounding the defect needs a developing atlas;
a single regular laboratory chart need not and does not contain
the apex.

%=====================================================================
\section{Complete channels and intervention-based influence}
\label{sec:influence}
%=====================================================================

For finite input factors indexed by \(X\), write
\[
 \HH_X=\bigotimes_{x\in X}\HH_x,\qquad
 \AA_X(S)=\BB(\HH_S)\otimes I_{X\setminus S}.
\]
Classical registers have a distinguished basis and diagonal
observable algebra. Their canonical matrix extension first
pinches that basis. Every controlled branch map is fixed
independently of the unknown input density operator.

A channel is completely positive and trace preserving (CPTP).
We use
\[
 \dtr(\rho,\sigma)=\frac12\norm{\rho-\sigma}_1,\qquad
 \ddia(\Phi,\Psi)=\frac12\norm{\Phi-\Psi}_\diamond.
\]
Standard finite-channel conventions may be found in
\cite{Watrous}. Instruments retain their complete records
\cite{DaviesLewis}.

\begin{definition}[Reference-assisted signalling]
\label{def:signal}
For input region \(A\), output region \(B\), and
\(\Phi_B=\Tr_{B^c}\Phi\), define
\begin{equation}
 \Sig_\Phi(A\to B)=
 \sup_{E,\rho,\mathcal I_A,\mathcal J_A}
 \dtr\!\left(
 (\Phi_B\otimes\id_E)(\rho_{\mathcal I}),
 (\Phi_B\otimes\id_E)(\rho_{\mathcal J})
 \right),
 \label{eq:signal}
\end{equation}
where \(E\) is any finite untouched reference,
\(\rho\) is any input-reference state, and
\[
 \rho_{\mathcal I}
 =(\mathcal I_A\otimes\id_{A^cE})(\rho).
\]
The interventions \(\mathcal I_A,\mathcal J_A\) are deterministic
CPTP maps on the same interface. They do not act on \(E\).
\end{definition}

Thus \(0\le\Sig_\Phi\le1\). Postselected normalization is not an
intervention in this definition.

\begin{theorem}[Exact no-signalling criterion]
\label{thm:nosignal}
For a finite channel \(\Phi\), the following are equivalent:
\begin{enumerate}[label=\textup{(\roman*)}]
\item \(\Sig_\Phi(A\to B)=0\);
\item a channel \(\Xi:\BB(\HH_{A^c})\to\BB(\HH_B)\) satisfies
\(\Phi_B=\Xi\Tr_A\);
\item
\[
 \Phi_B^*(\BB(\HH_B))
 \subseteq I_A\otimes\BB(\HH_{A^c}).
\]
\end{enumerate}
The factorization holds with every untouched reference.
\end{theorem}

\begin{proof}
For (ii), local trace preservation leaves the \(A^cE\)
reduction unchanged, proving (i).
Conversely, compare the identity intervention with the local
replacement
\[
 \mathcal R_A(Z)=\Tr(Z)\frac{I_A}{\dim\HH_A}.
\]
Condition (i), even without a reference, implies
\(\Phi_B=\Phi_B(\mathcal R_A\otimes\id)\) on all states and
therefore on all operators. Set
\[
 \Xi(Z)=\Phi_B\!\left(\frac{I_A}{\dim\HH_A}\otimes Z\right).
\]
It is CPTP and proves (ii).
Taking adjoints gives (iii). Conversely, (iii) makes every output
expectation depend only on the \(A^c\) reduction; the same
replacement argument gives (ii). Tensor the factorization with
\(\id_E\).
\end{proof}

This is the precise finite criterion used in VII-J
\cite[\loc{thm:nosignal}]{CheungVIIJ}. It does not identify
correlation with signalling.

\begin{example}[A joint output can reveal a signal hidden in marginals]
\label{ex:parity}
Encode a classical input bit \(b\) as
\((R,R\mathbin{\oplus}b)\), with \(R\) uniform.
Each individual share is independent of \(b\), but their joint
parity equals \(b\). Separate marginal tests cannot replace a
joint-output certificate.
\end{example}

\begin{example}[Postselection does not refute deterministic locality]
For an entangled pair, selecting a local measurement outcome can
change a conditional remote state. Summing the outcomes leaves
the remote marginal unchanged. The latter is the deterministic
comparison in Definition~\ref{def:signal}; an outcome used
remotely requires a declared classical wire.
\end{example}

%=====================================================================
\section{The complete timed network and exact scheduled support}
\label{sec:network}
%=====================================================================

\begin{definition}[Complete finite network]
\label{def:network}
A complete network is a finite directed acyclic circuit of CPTP
maps, including:
\begin{itemize}
\item quantum and classical carrier ports and identity wires;
\item fixed independent ancillary preparations;
\item gates, storage, allocations, resets, and discards;
\item destructive measurements and all retained outcomes;
\item every controller variable read to choose a gate, guard,
route, reset, or availability time;
\item the encoder and decoder, with their actual support.
\end{itemize}
A finite adaptive instruction tree is padded to a finite common
envelope containing every branch and its selecting controls.
There is no hidden state-functional scheduler.
\end{definition}

For a conservative dependency graph, connect every input of a
gate to every output. A reduced dependency graph is admissible
only if it has a proved support statement for \emph{every joint
subset} of that gate's outputs, including references.
Single-output no-signalling alone is insufficient.

For joint terminal output \(B\), let \(\Sch^-(B)\) be the model
input factors with a directed dependency path to \(B\).

\begin{theorem}[Complete scheduled-support bound]
\label{thm:schedule}
For the channel \(\Phi\) of a complete network,
\begin{equation}
 \Phi^*(\AA_Y(B))\subseteq\AA_X(\Sch^-(B)).
 \label{eq:schedule}
\end{equation}
Inputs outside \(\Sch^-(B)\) cannot signal to \(B\), including
when the observer also retains an arbitrary untouched reference.
Replacing a gate outside the backward region by any CPTP map
on the same ports does not change that output reduction.
\end{theorem}

\begin{proof}
Pull an output observable backward in reverse topological order.
A gate whose outputs all carry identity is invisible, because
its adjoint is unital. Otherwise the conservative graph adds
only that gate's input ports. Certified reduced graphs use
their joint-output support hypothesis instead.
A fixed preparation creates no unknown model-input predecessor.
Measurements, classical feed-forward, and resets are ordinary
complete channel nodes in this argument.
Induction gives \eqref{eq:schedule}.
Apply Theorem~\ref{thm:nosignal}. The same reasoning with reference
observables proves complete equality. On an excluded gate the
pulled-back observable is identity, also preserved by any
replacement channel.
\end{proof}

This reproduces the support mechanism of VII-J and the
partitioned-channel comparisons of En03-A and En05-A
\cite{CheungVIIJ,CheungLocality}.

\begin{corollary}[Partitioned metric envelope]
\label{cor:metric-envelope}
For layers
\(\Phi_j=\bigotimes_{B\in\mathcal B_j}\Phi_{j,B}\)
on a finite metric site set \((X,d_{\rm comp})\), assume
\(\diam B\le\Delta\).
Define backward sets by
\[
 N_{t,t}(Y)=Y,\qquad
 N_{s,t}(Y)=
 \bigcup_{\substack{B\in\mathcal B_{s+1}\\
                   B\cap N_{s+1,t}(Y)\ne\varnothing}}B.
\]
Then
\[
 \supp(\Phi_{s+1}^*\cdots\Phi_t^*(O_Y))
       \subseteq N_{s,t}(Y),\qquad
 N_{s,t}(Y)\subseteq
 \{x:d_{\rm comp}(x,Y)\le(t-s)\Delta\}.
\]
\end{corollary}

\begin{proof}
The support assertion is Theorem~\ref{thm:schedule}.
Each backward layer adds at most one block diameter.
Apply the triangle inequality inductively.
\end{proof}

The radius is \((t-s)\Delta\), not \(t-s\) without the
normalization \(\Delta=1\). The computational metric is an input;
it is not inferred from the conical metric.

\begin{example}[Strictness of the first distinction]
Two successive SWAPs have a diagram with cross-register
dependencies but total channel \(\id\). The first input has no
influence on the second output. Thus scheduled ancestry can
strictly overestimate actual influence.
\end{example}

\begin{proposition}[What retiming preserves]
\label{prop:retiming}
Swapping adjacent independent channel nodes acting on disjoint
factors preserves the complete channel, including records and
references. It need not preserve an assigned physical timing
certificate.
\end{proposition}

\begin{proof}
The two tensor-factor maps commute, also after amplification.
A finite sequence of such swaps preserves their composition.
Their new assigned port times need not satisfy the original
causal inequalities, which are separate hypotheses.
\end{proof}

Logical depth, hardware execution time, slicing parameter \(s\),
geometric time \(T\), and clock-code labels are not identified.

%=====================================================================
\section{Causal embedding, realized routes, and robustness}
\label{sec:embedding}
%=====================================================================

Give each port \(v\) a prescribed event \(p_v\in M\).
Every dependency edge \(u\to v\) must have a certified
future-causal route in the operational tube. For uncertain
regions \(L_u,L_v\), a conservative sufficient condition is
\[
 p\preceq_g q\quad
 \text{for all }p\in L_u,\ q\in L_v.
\]
A weaker allowance must specify the correlated endpoint
choices and their actual routes.

\begin{theorem}[Encoded physical containment]
\label{thm:physical}
Suppose the full encoded--computed--decoded network is complete,
and every dependency edge has the stated causal certificate.
For a joint output \(B\), let \(\Phys^-(B)\) contain input
factors whose assigned events can reach at least one output
event of \(B\). Then
\begin{equation}
 \Inf_\Phi^-(B)\subseteq
 \Sch_{\rm full}^-(B)\subseteq\Phys^-(B).
 \label{eq:physical}
\end{equation}
Every intervention supported outside \(\Phys^-(B)\) has zero
signalling to \(B\), with arbitrary untouched references.
\end{theorem}

\begin{proof}
Theorem~\ref{thm:schedule} gives the first inclusion.
Concatenate the supplied causal routes along a scheduled path.
Transitivity gives the second inclusion. Apply
Theorem~\ref{thm:nosignal} to the complement of the physical past.
\end{proof}

The distinction in \eqref{eq:physical} is the central VII-J
certificate \cite[\loc{thm:physical}]{CheungVIIJ}.
A physically possible path with no installed wire shows that
the second inclusion need not be equality.

\subsection{Admitted implementation primitives}

\begin{assumption}[Finite routing architecture]
\label{ass:routing}
The model permits exact carriage of a finite register along a
specified future-causal curve; positive-duration local gates and
complete instruments on co-located registers; local independent
preparation; local storage, resets, and discards; and classical
records carried on explicitly scheduled causal wires.

These are ideal finite-model permissions, not a field-theoretic
or laboratory construction of the communication mechanism.
\end{assumption}

\begin{proposition}[Gather--operate--distribute]
\label{prop:gather}
Suppose all gate inputs have supplied causal routes to one
station event \(r_-\), all outputs have supplied routes from a
later station event \(r_+\), and a finite local implementation
fits between \(r_-\) and \(r_+\).
Then Assumption~\ref{ass:routing} realizes that gate, with its
records, ancillary factors, and references, as Target B.
\end{proposition}

\begin{proof}
Carry the input factors to the station, execute the stipulated
local channel, and distribute its outputs.
Identity carriage and the local channel give the intended
complete composition on all operators. Every elementary
dependency has an explicit causal route.
\end{proof}

Pairwise causal input--output possibilities alone do not
establish this common-station hypothesis.

\begin{proposition}[Composition through a reindexing or event]
\label{prop:composition}
If \(\Phi=\Phi_+\mathcal R\Phi_-\) and the three channels have
predecessor maps \(P_+,P_{\mathcal R},P_-\), then
\[
 \Phi^*(\AA(B))
 \subseteq
 \AA\bigl(P_-(P_{\mathcal R}(P_+(B)))\bigr).
\]
Causal certificates compose in the same order.
Independent allocations have no model-input predecessors;
discards and all remapping controls must be included.
\end{proposition}

\begin{proof}
Compose the adjoint inclusions and then the corresponding
causal paths.
\end{proof}

\subsection{The conservative slicing-parameter speed}

All spatial coordinates in the next statements are dimensionless.
Let a regular spatial route in the supported presentation have
length \(D\).

\begin{theorem}[Realized-route cone certificate]
\label{thm:conegates}
A spatial route parametrized over \(s\in[a,b]\), with metric
speed at most
\begin{equation}
 v_*=\frac{1}{1+\lambda}
       =\frac{1}{1+2\sqrt2/3},
 \label{eq:vstar}
\end{equation}
lifts under \(s\mapsto(F_s(x(s)),x(s))\) to a future-causal
curve. In particular, if an actual regular route of length
\(D\) joins \(x\) to \(y\), then
\[
 b-a\ge(1+\lambda)D
\]
is sufficient for a causal route between its corresponding
slice ports. The statement holds across \(s=0\).
\end{theorem}

\begin{proof}
Lemma~\ref{lem:slices} gives, for an increment \(\delta s>0\),
\[
 F_{s+\delta s}(x(s+\delta s))-F_s(x(s))
 \ge\delta s-\lambda\,d(x(s+\delta s),x(s)).
\]
The lift is Lipschitz. At almost every parameter its time
derivative is at least \(1-\lambda|\dot x|\).
When \(|\dot x|\le v_*\), this is at least \(|\dot x|\), so
the lifted tangent is future causal.
For the final assertion, traverse the supplied spatial route
at constant arc-length speed \(D/(b-a)\), allowing waits.
\end{proof}

The source's conservative speed is retained
\cite[\loc{cor:conespeed}]{CheungVIIJ}.
It is an inner sufficient bound in the slicing parameter,
not a derived invariant speed or the full physical velocity set.

\begin{remark}[Distance equality on the punctured cone]
A spatial distance defined as an infimum need not be realized
by a path in the regular part: the limiting minimizer could
meet the removed apex. Consequently a statement at exact
distance equality needs a realized regular path.
Our benchmark supplies such paths in a convex regular chart.
\end{remark}

\begin{proposition}[Robust port margin]
\label{prop:robust}
In a convex regular developed chart, let a nominal slice-time
gap be \(h\), nominal spatial endpoint distance at most \(D\),
spatial localization radii \(r_-,r_+\), slicing-time errors at
most \(\varepsilon_s\) at each endpoint, and additional
dimensionless geometric-time errors at most
\(\varepsilon_\tau\) at each endpoint.
All endpoint choices remain causally admissible if
\begin{equation}
 h-(1+\lambda)D
 \ge
 (1+\lambda)(r_-+r_+)+2\varepsilon_s+2\varepsilon_\tau.
 \label{eq:robust}
\end{equation}
Strict inequality gives positive slack.
\end{proposition}

\begin{proof}
The actual spatial distance is at most \(D+r_-+r_+\), and
the actual slicing gap is at least \(h-2\varepsilon_s\).
The two-family estimate gives actual geometric-time difference
at least
\[
 h-2\varepsilon_s-\lambda(D+r_-+r_+)
                         -2\varepsilon_\tau.
\]
Under \eqref{eq:robust} it exceeds the actual spatial distance.
The straight developed segment is available within the chart.
\end{proof}

This is an availability certificate, not permission to replace
an exact event guard by a rounded geometric report.

%=====================================================================
\section{Directional reachability and finite-time certificates}
\label{sec:directions}
%=====================================================================

These are kinematic graph statements. An arbitrarily long
periodic lift is not asserted to be embedded in the one bounded
conical laboratory. Only finite prefixes supplied with actual
routes enter Theorem~\ref{thm:physical}.

\subsection{Cycle velocities with a transient remainder}

Let \(Q\) be a finite strongly connected directed multigraph.
Its vertices include schedule phase and register type.
Edge \(e\) has displacement \(\Delta_e\in\RR^d\) and duration
\(t_e\ge t_{\min}>0\). The translation lift permits the declared
copies of these transitions. Waiting is permitted in a manner
compatible with the phase model.

For each simple directed cycle \(C\), define
\[
 v(C)=\frac{\sum_{e\in C}\Delta_e}{\sum_{e\in C}t_e},
 \qquad
 K_Q=\conv\bigl(\{0\}\cup\{v(C):C\text{ simple}\}\bigr).
\]
This is the timed version of the periodic-graph velocity
construction \cite{Fritz,CheungVIIJ}.

\begin{theorem}[Finite-time offset and limiting velocities]
\label{thm:polytope}
Let \(q=|V(Q)|\), \(D_{\max}=\max_e|\Delta_e|\), and
\(R_Q=(q-1)D_{\max}\).
Every path completed within elapsed time \(t\) has displacement
\begin{equation}
 \Delta_{\rm path}\in tK_Q+\overline B(0,R_Q).
 \label{eq:offset}
\end{equation}
From any specified initial quotient vertex, the set of
subsequential limiting displacement/time ratios, for unbounded
elapsed times, is exactly \(K_Q\).
\end{theorem}

\begin{proof}
Remove closed subwalks until the remaining walk has no repeated
vertex. Decompose removed closed walks into simple cycles.
Their total displacement is a time-weighted sum of cycle
velocities, with total cycle time at most \(t\).
The zero vector accounts for unused time. The remaining simple
walk has at most \(q-1\) edges and displacement norm at most
\(R_Q\), proving \eqref{eq:offset}.

Divide by \(t\) to obtain one inclusion for limiting ratios.
Conversely, take any convex combination of finitely many cycle
velocities and zero. Repeat the cycles in integer counts whose
durations approximate the chosen proportions on a scale tending
to infinity. Strong connectivity supplies fixed connecting
walks from the chosen start and between cycles.
Their total cost is bounded for this fixed combination.
Waiting realizes its zero component.
The normalized displacement tends to the desired combination.
\end{proof}

For a graph with several strongly connected components, only
cycles that can be visited in the allowed component order may
be combined. The convex hull of every cycle can overestimate
attainability from a specified start.

\subsection{A finite potential certificate}

\begin{proposition}[Directional transient potentials]
\label{prop:potential}
Fix \(n\in\RR^d\) and \(\beta\ge0\).
The following are equivalent:
\begin{enumerate}[label=\textup{(\roman*)}]
\item \(n\cdot v(C)\le\beta\) for every simple cycle;
\item finite potentials \(\pi_v\) exist such that
\begin{equation}
 n\cdot\Delta_e-\beta t_e
       \le\pi_{\mathrm{head}(e)}-\pi_{\mathrm{tail}(e)}
 \quad(e\in E(Q)).
 \label{eq:potential}
\end{equation}
\end{enumerate}
Every path from \(u\) to \(v\), of duration at most \(t\), then
satisfies
\begin{equation}
 n\cdot\Delta_{\rm path}
       \le\beta t+\pi_v-\pi_u.
 \label{eq:potentialpath}
\end{equation}
\end{proposition}

\begin{proof}
Summing \eqref{eq:potential} around a cycle proves (i).
Conversely, set \(w_e=n\cdot\Delta_e-\beta t_e\).
Every closed walk has nonpositive weight by cycle decomposition.
Add a root with zero-weight edges to all vertices.
Let \(\pi_v\) be the maximum weight of a root-to-\(v\) walk.
Removing nonpositive closed subwalks does not decrease its weight,
so a maximum exists among finitely many simple walks.
Appending an edge gives \(\pi_{\rm head}\ge\pi_{\rm tail}+w_e\),
which proves (ii).
Summation gives \eqref{eq:potentialpath}; use \(\beta\ge0\)
for a path completed before \(t\).
\end{proof}

This is a finite difference-inequality witness, not a new
general graph-theoretic principle. It supplements the cycle
polytope by explicit endpoint-dependent transient costs.
For a physical directional bound \(b(n)t\), the sufficient
endpoint check is
\[
 (b(n)-\beta)t\ge\pi_v-\pi_u.
\]
A constant potential would instead certify every edge in that
direction directly. Neither potential feasibility nor a cycle
average supplies the missing physical routes.

\begin{example}[Zero asymptotic velocity does not prevent a fast first hop]
Two quotient states have transitions \(+L\) and \(-L\), each
of duration one, and waits. Every cycle velocity is zero.
The first hop still travels \(L\) in one unit.
For a prescribed speed \(c_0<L\), it is unsafe.
The finite offset or potential difference detects the transient.
\end{example}

\subsection{Safety, coverage, and support functions}

For compact convex \(K\subset\RR^d\), define
\(h_K(n)=\sup_{v\in K}n\cdot v\).

\begin{proposition}[Directional inclusion]
\label{prop:supportfunctions}
For nonempty compact convex sets \(K,L\),
\[
 K\subseteq L
 \quad\Longleftrightarrow\quad
 h_K(n)\le h_L(n)\ \text{for all }n.
\]
If \(L\) is a polytope, its defining facet normals suffice.
If true support functions have certified errors
\(\delta_K(n),\delta_L(n)\), the nominal gap
\(h_L(n)-h_K(n)\ge\delta_K(n)+\delta_L(n)\)
certifies true inclusion.
\end{proposition}

\begin{proof}
Inclusion implies the inequalities. If \(v\in K\setminus L\),
choose its nearest point \(z\in L\).
Then \((v-z)\cdot(w-z)\le0\) for all \(w\in L\), which strictly
separates \(v\) from \(L\).
The facet and error statements follow by substitution.
\end{proof}

For cardinal hops of length \(a\), one per interval \(\delta\),
or a wait, in two dimensions,
\[
 K_{\rm Sch}=\{v:|v_1|+|v_2|\le a/\delta\}
 \subseteq
 K_{\rm phys}=\{v:|v|_2\le a/\delta\}.
\]
Their support functions are
\((a/\delta)\norm n_\infty\) and
\((a/\delta)\norm n_2\).
The disk is not contained in the diamond:
\((a/(\sqrt2\delta),a/(\sqrt2\delta))\) is a counterexample.
Thus the diamond is safe but does not cover all physical
directions.

If a constant physical velocity set \(K_{\rm phys}\) contains
\(K_Q+\overline B(0,R_Q/t)\), then \eqref{eq:offset}
certifies endpoint reachability within time \(t\).
It does not certify the intermediate implementation paths.
For Target B, check the actual route velocities locally.

\subsection{Earliest arrivals and coordinate-varying examples}

On a finite time-expanded DAG, earliest arrivals are obtained
by the recurrence
\[
 A(v)=\min_{u\to v}\{\text{allowed arrival at }v
                         \text{ after arrival }A(u)\},
\]
including all waits and availability restrictions.
Topological induction proves the recurrence exactly.
The value is \(+\infty\) when no path exists.

Two closed-form comparisons can be made without changing the
prescribed local Minkowski geometry.
In a one-dimensional regular rod coordinate \(x\), the time
reparametrization
\(\tau=a\log(1+u)\), \(a>0\), gives
\[
 \left|\frac{\dd x}{\dd u}\right|\le\frac{a}{1+u}.
\]
Therefore
\begin{equation}
 |y-x|\le a\log\frac{1+u_1}{1+u_0},\qquad
 u_{\rm arr}=(1+u_0)e^{|y-x|/a}-1.
 \label{eq:timevarying}
\end{equation}
This is time-varying, not spatially varying.

For a genuinely spatially varying coordinate speed, set
\(x=a\log(1+q)\), \(q>-1\). Then
\[
 |\dot q|\le\frac{1+q}{a},\qquad
 \tau_{\rm arr}=\tau_0+
 a\left|\log\frac{1+q_1}{1+q_0}\right|.
\]
Both formulas follow by integrating the rod-distance bound;
monotone null motion attains equality.
They are coordinate descriptions of the installed regular
geometry, not derived gravitational dynamics.

%=====================================================================
\section{Approximate simulation and the Hamiltonian comparison}
\label{sec:approximation}
%=====================================================================

\begin{theorem}[Signalling stability with consistent constants]
\label{thm:signal-stability}
For channels on the same interfaces,
\begin{equation}
 \left|\Sig_\Phi(A\to B)-\Sig_\Psi(A\to B)\right|
 \le\norm{\Phi_B-\Psi_B}_\diamond
 =2\ddia(\Phi_B,\Psi_B).
 \label{eq:signal-stability}
\end{equation}
In particular, if \(\Psi\) is nonsignalling and
\(\ddia(\Phi_B,\Psi_B)\le\varepsilon\), then
\(\Sig_\Phi\le\min\{1,2\varepsilon\}\).
\end{theorem}

\begin{proof}
For fixed reference, state, and interventions, the two comparison
errors each have trace norm at most
\(\norm{\Phi_B-\Psi_B}_\diamond\).
The triangle inequality and the \(1/2\) trace-distance factor
give one such full-diamond norm for the difference of the two
distinguishabilities. Take suprema and exchange the channels.
\end{proof}

This retains VII-J's norm convention
\cite[\loc{thm:signal-stability}]{CheungVIIJ}.
An expectation bound for one state or one observable does not
supply its diamond-norm hypothesis.

\begin{proposition}[Complete-channel error ledger]
\label{prop:telescope}
For compatible sequences of channels,
\[
 \ddia(\widetilde\Phi_N\cdots\widetilde\Phi_1,
             \Phi_N\cdots\Phi_1)
 \le\min\left\{1,\sum_{j=1}^N
                    \ddia(\widetilde\Phi_j,\Phi_j)\right\}.
\]
Every encoding, clock, controller, transition, communication,
decoding, and failure output must be compared on matching
interfaces.
\end{proposition}

\begin{proof}
Replace one factor at a time. The resulting telescoping sum has
one channel difference in each term.
Composition with channels contracts the diamond norm.
Use the triangle inequality and divide by two.
\end{proof}

\begin{corollary}[Accuracy and causal safety are independent obligations]
A causally embedded implementation has exact zero signalling
outside its certified physical past, irrespective of its
accuracy. If its target has signalling \(s_0\) on a forbidden
pair, any half-diamond approximation to that target has error
at least \(s_0/2\).
Conversely, an arbitrarily accurate approximation to a
nonsignalling target need not be exactly nonsignalling.
\end{corollary}

\begin{proof}
Use Theorems~\ref{thm:physical} and
\ref{thm:signal-stability}.
For the last assertion,
\[
 \Phi_\epsilon=(1-\epsilon)\mathcal R_0+\epsilon\id
\]
on a qubit differs from the constant channel \(\mathcal R_0\)
by half-diamond distance at most \(\epsilon\), but has signalling
\(\epsilon\) between its input and output.
\end{proof}

\subsection{A finite-volume influence tail}

Let \(G=(X,E_G)\) be finite and
\[
 H(t)=\sum_x h_x(t)+\sum_{e\in E_G}h_e(t),
 \qquad
 \Lambda(s,t)=\frac1\hbar\int_s^t
                        \sum_e\norm{h_e(r)}\,\dd r,
\]
with bounded piecewise-continuous self-adjoint onsite and
two-site edge terms. Let \(\Psi_{t,s}\) be its unitary channel.

\begin{theorem}[Reference-stable finite-volume tail]
\label{thm:tail}
For disjoint nonempty \(A,B\), let \(d=d_G(A,B)\ge1\).
Then
\begin{equation}
 \Sig_{\Psi_{t,s}}(A\to B)
 \le
 \min\left\{1,\sum_{m=d}^{\infty}
                         \frac{(2\Lambda(s,t))^m}{m!}\right\}.
 \label{eq:tail}
\end{equation}
For every \(\gamma>0\), the series is at most
\[
 \exp\bigl(-\gamma d+2e^\gamma\Lambda(s,t)\bigr).
\]
The estimate is independent of the untouched reference dimension,
but is not uniform in the number of edges.
\end{theorem}

\begin{proof}
Pass to the interaction picture for the onsite propagator,
a product of onsite unitaries. It preserves support and all
edge-term norms.

For a Hermitian output observable \(O_{BE}\) of norm at most one,
expand the Heisenberg evolution in its norm-convergent ordered
nested-commutator series. At order \(m\), summing edge choices and
ordered integrals bounds the norm by
\((2\Lambda)^m/m!\).
Each nonzero commutator must meet the current site support,
and can extend its graph distance from \(B\) by at most one.
Every term of order below \(d\) is therefore identity on \(A\),
also when its coefficients are arbitrary operators on \(E\).

The two intervention adjoints agree on that partial sum.
They are operator-norm contractions, so their difference on the
remainder is bounded by twice the series in \eqref{eq:tail}.
Trace-norm duality and the \(1/2\) normalization in
Definition~\ref{def:signal} give \eqref{eq:tail}.
Insert \(1\le e^{\gamma(m-d)}\) for \(m\ge d\) and sum the
exponential series for the final bound.
\end{proof}

This is the conservative finite-volume route of VII-J
\cite[\loc{thm:tail}]{CheungVIIJ}. Sharper quasi-local estimates
require additional interaction hypotheses
\cite{BHV,NachtergaeleSims}. We do not claim a new
Lieb--Robinson theorem.

\begin{example}[A Hamiltonian tail is not an exact light cone]
For two assigned separated sites, take \(H=JX_AX_B\), \(J>0\).
Prepare \(B\) in \(\ket0\), and replace \(A\) by \(\ket+\) or
\(\ket-\). Their output states on \(B\) are
\(e^{\mp iJtX/\hbar}\ket0\), with trace distance
\[
 |\sin(2Jt/\hbar)|.
\]
It is nonzero for arbitrarily small positive \(t\).
Such a Hamiltonian cannot be identified with an exact
geometrically local communication primitive at positive separation.
\end{example}

\subsection{Constructive finite simulation estimates}

\begin{proposition}[Duhamel and product-formula bounds]
\label{prop:hamiltonian-errors}
For bounded self-adjoint generators on a fixed finite space,
\[
 \ddia(\Ad_U,\Ad_{\widetilde U})
 \le\norm{U-\widetilde U}
 \le\frac1\hbar\int_s^t\norm{H(r)-\widetilde H(r)}\,\dd r.
\]
For time-independent \(H=\sum_{j=1}^m H_j\), put
\[
 V_n(t)=
 \left(e^{-itH_m/(n\hbar)}\cdots
       e^{-itH_1/(n\hbar)}\right)^n,\qquad
 C_H=\sum_{j<k}\norm{[H_j,H_k]}.
\]
Then
\begin{equation}
 \norm{e^{-itH/\hbar}-V_n(t)}
 \le\frac{t^2C_H}{2n\hbar^2},\qquad
 \ddia(\Ad_{e^{-itH/\hbar}},\Ad_{V_n(t)})
 \le\min\left\{1,\frac{t^2C_H}{2n\hbar^2}\right\}.
 \label{eq:trotter}
\end{equation}
\end{proposition}

\begin{proof}
Differentiate the mixed propagator
\(U(t,r)\widetilde U(r,s)\), integrate, and use unitarity.
The channel comparison follows by expanding
\(UXU^\dagger-VXV^\dagger\) into two terms, also after
amplification.

For two summands, the double integral comparing the joint
exponential and ordered product has norm at most
\(\delta^2\norm{[H_1,H_2]}/(2\hbar^2)\).
Iterating over the finite sum gives one-step error
\(\delta^2C_H/(2\hbar^2)\).
Telescoping \(n\) unitary steps with \(\delta=t/n\) proves
\eqref{eq:trotter}.
\end{proof}

The product-formula channel is a logical approximation. Its
elementary implementation has its own positive travel and gate
times. Approximating a Hamiltonian to accuracy
\eqref{eq:trotter} does not embed its numerical clock into the
physical clock.

%=====================================================================
\section{Destructive clocks, programme records, and calibration}
\label{sec:clock}
%=====================================================================

Let \(C\cong\CC^{2^m}\) have an installed orthonormal basis
\(\{\ket{\tau_a}\}\). The surviving space \(S\) includes matter
and every retained control or location/history register.
Set
\[
 P_a=\proj{\tau_a},\qquad D_a=\bra{\tau_a}_C\otimes I_S.
\]

\begin{theorem}[Complete destructive-clock instrument]
\label{thm:clock}
The basis change \(\ket{\tau_a}\mapsto\ket a\), followed by
destructive Enter on all clock names, has channel
\begin{equation}
 \MM_C(X)=\sum_a\proj a_R\otimes D_aXD_a^\dagger
 =
 \sum_a\proj a_R\otimes
 \Tr_C[(P_a\otimes I)X(P_a\otimes I)].
 \label{eq:clock}
\end{equation}
It is CPTP on all operators, including every reference extension.
Its output has no quantum \(C\) factor, and
\[
 \Tr_R\MM_C(X)=\Tr_CX.
\]
For branch probability \(p_a>0\), the conditional surviving
state is \(D_a\rho D_a^\dagger/p_a\).
Zero-probability branches are not normalized.
\end{theorem}

\begin{proof}
Kraus operators are \(\ket a_R\otimes D_a\), with squared
moduli summing to
\(\sum_aP_a\otimes I=I\).
Expansion in the clock basis shows that both expressions in
\eqref{eq:clock} return its \(aa\) block.
Summation gives the partial trace. The calculation is unchanged
after adjoining an arbitrary untouched factor.
The destructive output type removes the measured quantum names.
\end{proof}

This is the I-B/En09-A finite interface
\cite{CheungIB,CheungEn09}; its conditional block is not a metric,
stress tensor, or singular detector.

For fixed continuation channels \(\mathcal F_a\), the full
programme channel is
\begin{equation}
 \mathcal J(X)=
 \sum_a\proj a_R\otimes\mathcal F_a(D_aXD_a^\dagger).
 \label{eq:programme}
\end{equation}
It is CPTP because each continuation preserves the trace of its
unnormalized block. Routing choices and record delivery must be
implemented before the receiver uses \(a\).
An arbitrary state-functional selection of \(\mathcal F_a\) is
not this channel.

\begin{proposition}[Calibration boundaries]
\label{prop:calibration}
Let distinct numbers \(\theta_a\) label clock codes.
At one actual reading event \(e\), exact event-time calibration
\[
 p_a>0\Longrightarrow\theta_a=T(e)
\]
forces a single deterministic outcome. If no label equals
\(T(e)\), it is impossible for a normalized input.

A history or programme dictionary may instead associate \(a\)
with a different labeled event, arrival, route, or stored
incidence parameter. It need not equal the actual reading time.
\end{proposition}

\begin{proof}
Distinctness allows at most one matching index and the
probabilities sum to one. The dictionary interpretation makes
no claim that several different labels are coordinates of the
same reading event.
\end{proof}

A programme label for a future arrival is a planned target, not
evidence that the arrival already occurred. A history claim
requires its independently specified preparation and provenance.

\begin{proposition}[No universal quantum return or finite CCR]
\label{prop:clock-obstructions}
The channel \(\MM_C\) is not invertible on all clock inputs.
Finite matrices cannot satisfy
\([T_C,H_C]=i\hbar I\) when \(\hbar\ne0\).
Consuming \(C\) does not remove \(T\), \(M\), or \(g\).
\end{proposition}

\begin{proof}
The clock states
\((\ket{\tau_0}\pm\ket{\tau_1})/\sqrt2\), with the same
payload, have identical outputs. A left inverse is impossible.
Taking the trace proves the commutator assertion.
The final statement follows because these geometric objects
are prescribed parameters, not output factors of \(\MM_C\).
\end{proof}

\subsection{Optional relational interpretation}

For completeness, the additional hypothesis needed to turn
conditioning into relational evolution can be stated precisely
\cite{PageWootters,CheungEn09}.

\begin{proposition}[Sampled covariance and stationarity]
\label{prop:relational}
Suppose finite Hermitian \(H_C,H_S\) and \(\rho_{CS}\) satisfy
\[
 [H_C\otimes I+I\otimes H_S,\rho]=0,\qquad
 P_a=U_C(t_a)P_0U_C(t_a)^\dagger.
\]
Then
\[
 D_a\rho D_a^\dagger
 =U_S(t_a)(D_0\rho D_0^\dagger)U_S(t_a)^\dagger.
\]
For a complete orthonormal reading basis all probabilities
equal \(2^{-m}\).
For an extended \(SE\) assertion, stationarity of the full
extension is required.
\end{proposition}

\begin{proof}
Stationarity gives
\[
 (U_C^\dagger\otimes I)\rho(U_C\otimes I)
 =(I\otimes U_S)\rho(I\otimes U_S^\dagger).
\]
Insert the covariant projector into the clock partial trace
and use unitary invariance of that trace.
The resulting blocks are unitarily related, so their traces
are equal and sum to one. The same calculation applies to an
extension exactly under the extended stationarity hypothesis.
\end{proof}

No such dynamical covariance is assumed for the programme table
below. The external schedule remains an input.

\subsection{Records and residuals remain different certificates}

Two record-retaining instruments may have the same outcome law
and different residuals. For example, with one coarse outcome,
the identity qubit channel and computational dephasing have
identical records but, on \(\proj+\), output \(\proj+\) and
\(I/2\). Their trace distance is \(1/2\).
Implementation errors \(\varepsilon_1,\varepsilon_2\) leave a
gap at least \(1/2-\varepsilon_1-\varepsilon_2\), by the triangle
inequality. This is the observation-boundary discipline compared
with VII-E, VII-G, and VII-E-ST \cite{CheungObservation}.

More generally, a complete recorded instrument that recovers
every input in the full quantum algebra can have only
input-independent record effects. Indeed, for Kraus operators
\(A_i\) of the instrument and \(B_j\) of a proposed recovery,
all \(B_jA_i\) are scalar multiples of \(I\), because the identity
channel has rank-one Choi matrix. Completeness of the recovery
then gives \(A_i^\dagger A_k=\lambda_{ik}I\).
Summing the diagonal terms for one record makes its effect
scalar. This full-input criterion is consistent with standard
quantum error correction \cite{KnillLaflamme,Watrous}.
It does not prohibit transport of the surviving payload.

%=====================================================================
\section{Exact regular-part benchmark through the incidence parameter}
\label{sec:benchmark}
%=====================================================================

We now give a nonempty Target B instance. It deliberately
replaces VII-J's singular-axis rendezvous
\cite[\loc{thm:eventbenchmark}]{CheungVIIJ}.

\subsection{Persistent spatial marks and exact times}

In sector zero choose the dimensionless transverse mark
\[
 x_\perp=(1/64,1/128).
\]
At that mark,
\[
 h(x_\perp)=1/64,\qquad k(x_\perp)=3/640.
\]
Choose two regular persistent ports
\[
 A=(x_\perp,\zeta=0),\qquad
 B=(x_\perp,\zeta=L),\qquad L=\frac1{64},
\]
and slice parameters
\[
 s_-=-\frac14,\qquad s_+=\frac14.
\]
Both ports have identical time coordinates on each marking.
Their input and output times are
\begin{equation}
 \tau_{\rm in}=-\frac{15}{64},\qquad
 \tau_{\rm out}=\frac{323}{640}.
 \label{eq:benchmark-times}
\end{equation}

Take a convex developed spatial tube containing
\[
 [3/256,5/256]\times[1/256,3/256]
                \times[-1/256,5/256].
\]
Its closure is entirely regular and lies inside the first
sector. Choose a finite time interval containing
\([\tau_{\rm in},\tau_{\rm out}]\).
All operational paths and gates remain in this tube.

The marks \(A,B\) are spatial locations, not evolving abstract
cell labels. The local branch at \(x_\perp\) changes at
\(s=h-k=7/640\), but the mark and its quantum register do not
cease to exist.

\subsection{A fixed finite event-code restriction}

The complete VII-F metadata circuit is not silently inferred
from these few bits. We use its following explicit
fixed-instance restriction.

At the controller \(A\), the exact stored incidence parameter
is \(j=0\); the fixed geometric packet specifies the prescribed
cone and zero germ. Reserve classical mode, permission, and
history bits
\[
 (M,N,H)=(0,1,0),
\]
and one scratch bit \(E=0\).
A two-qubit computational-basis clock returns
\(a=(a_1,a_0)\in\{00,01,10,11\}\).
Only \(a=01\) requests the event update. Set
\[
 e(a)=(1-a_1)a_0.
\]
On the promised initial bits the update is
\begin{equation}
 (M,N,H)=(0,1,0)
   \longmapsto(e(a),1-e(a),e(a)).
 \label{eq:event-code}
\end{equation}
Thus the canonical programme \(01\) performs the exact
incoming-to-outgoing zero-germ mode update, spends its one
permission, and retains a history bit.
Other programmes request no update. No length report is used
as a clock guard.

The permutation is implemented by:
\begin{enumerate}[label=\textup{\arabic*.}]
\item NOT on \(a_1\), Toffoli \((a_1,a_0;E)\), NOT on \(a_1\);
\item CNOT from \(E\) to each of \(M,N,H\);
\item the inverse of step 1.
\end{enumerate}
There are nine elementary NOT/CNOT/Toffoli gates.
The record is restored and \(E\) returns to zero.
Off the promised domain this remains a reversible bit
permutation, but is not assigned a geometric interpretation.

\begin{remark}[A full supplied event circuit]
Any separately specified finite source circuit \(U_{\rm code}\)
with \(k\) local code/history/work carriers and \(G\) gates can
replace this restriction. Co-locate its carriers at \(A\) and
give each gate a positive duration \(1/(16(G+4))\) in the buffer
below. This localizes the supplied finite map; it does not
prove an unsupplied decoder theorem or its forward promise.
The explicitly proved numerical ledger below concerns
\eqref{eq:event-code}.
\end{remark}

\subsection{Target-adapted chains and the buffer}

Fix an integer \(n\ge1\). Install a chain of qubit registers
\(Q_0,\ldots,Q_n\) at
\[
 (x_\perp,\zeta_j),\qquad \zeta_j=jL/n.
\]
Here \(Q_0\) is a temporary register at \(A\);
\(Q_n\) is the input register at \(B\).
A separate qubit \(P\) is the input at \(A\).
Initialize \(Q_0,\ldots,Q_{n-1}\) in \(\ket0\).

Let
\begin{equation}
 D=\frac{4L}{v_*}=\frac{1+\lambda}{16}<\frac18.
 \label{eq:D}
\end{equation}
During the inbound interval of length \(D\), use \(n\) successive
content exchanges
\[
 (Q_n,Q_{n-1}),\ldots,(Q_1,Q_0),
\]
each of duration \(D/n\), realized by opposite causal carriage
of the two contents along the regular axial segment.
Their physical dimensionless speed is \(v_*/4<1\).

The unknown qubit reaches \(A\) by
\[
 \tau_{\rm in}+D<-\frac7{64}<0.
\]
Both matter qubits then remain stored at \(A\).
Destroy the two clock qubits in two local slots of duration
\(1/64\), from \(\tau=-1/32\) to \(0\).
The complete record is available at \(\tau=0\).

Wait until \(\tau=1/16\).
Run the nine event-code gates, three local CNOTs implementing
\(\operatorname{SWAP}_{P,Q_0}\), and the classical-record
controlled gate \(Z_P^{a_1}\).
The thirteen local slots each have duration \(1/208\), so
they finish at \(\tau=1/8\).

Finally move \(Q_0\)'s content outward by the \(n\) successive
exchanges
\[
 (Q_0,Q_1),\ldots,(Q_{n-1},Q_n).
\]
Allow duration \(D/n\) per outbound stage.
Its first half copies the five-bit classical word
\[
 r_a=(a_1,a_0,e(a),1-e(a),e(a))
\]
into the next site's blank record buffer by five sequential
classical CNOTs.
Each copy slot has positive duration \(D/(10n)\).
The second half carries the qubit exchange and the copied
record over the segment of length \(L/n\).
The communication speed is \(v_*/2<1\).

All records and payloads arrive by
\[
 \frac18+D<\frac14<\tau_{\rm out}.
\]
Wait at the destination ports until \(\tau_{\rm out}\).
Records at earlier sites are retained, not reset.

\begin{lemma}[Every benchmark operation is regular and causal]
\label{lem:benchmark-causal}
The specified execution is a finite Target B schedule.
The matter is buffered across both the worldsheet incidence
time \(\tau=0\) and the local slice parameter \(s=0\).
No operation meets \(\WW\).

The moving routes also satisfy the sufficient slicing speed
\eqref{eq:vstar}.
\end{lemma}

\begin{proof}
The entire axial chain is in the displayed regular tube.
Every local operation is on one station worldline with strictly
positive duration. The inbound exchange routes have speed
\(v_*/4\), and the outbound communication portions have speed
\(v_*/2\); both are future timelike.

The incoming branch has \(\partial_sF_s=1\).
On the outgoing communication interval,
\(\partial_sF_s=2\).
Thus the route speeds in \(s\) are respectively \(v_*/4\)
and \(v_*\). Stationary waits and gates have zero spatial speed.
Theorem~\ref{thm:conegates} also certifies them.

The qubits are at \(A\) before \(\tau=0\), and remain there until
\(\tau=1/16\). At \(A\), the \(s=0\) slice has time \(1/64\),
which lies inside that storage interval.
All control copies occur before their communication departures,
and every remote use waits for the record to arrive.
\end{proof}

The classical metadata describes the zero germ.
Its positive-duration computation finishes later.
It is neither an instantaneous global update of geometry nor
acquisition of data from the singularity.

\subsection{Complete channel and signalling witness}

Write the input matter order as \(P,Q_n\), and define
\[
 V_a=(Z^{a_1}\otimes I)\operatorname{SWAP}.
\]
The programme \(a=01\) has \(V_{01}=\operatorname{SWAP}\).
The other programmes provide declared phase-controlled variants,
not alternative spacetime metrics.

\begin{theorem}[Exact complete regular benchmark]
\label{thm:benchmark}
For every input operator \(X\) on \(C\otimes P\otimes Q_n\),
the complete output of the benchmark is
\begin{equation}
 \mathcal J_n(X)=
 \sum_{a\in\{0,1\}^2}
 \bigl(\proj{r_a}\bigr)^{\otimes(n+1)}
 \otimes
 V_a D_aXD_a^\dagger V_a^\dagger
 \otimes\omega_{\rm blank},
 \label{eq:benchmark-channel}
\end{equation}
where \(D_a=\bra a_C\otimes I_{PQ_n}\),
and \(\omega_{\rm blank}\) contains the final blank
\(Q_0,\ldots,Q_{n-1}\) and scratch \(E=0\).
The two matter output factors are at \(A,B\), respectively.
Equation~\eqref{eq:benchmark-channel} holds with every untouched
reference.

On the canonical clock preparation \(\proj{01}\), the matter
channel is exactly \(\Ad_{\operatorname{SWAP}}\).
It has
\[
 \Sig(A_{\rm in}\to B_{\rm out})=1.
\]
No input outside the physically certified backward region signals
to the declared joint output.
\end{theorem}

\begin{proof}
Successive exchanges with blank chain registers move the entire
\(Q_n\) factor, with all its correlations, into \(Q_0\).
The other chain registers become blank.
The clock identity gives \(D_aXD_a^\dagger\) on branch \(a\).
The nine-gate calculation gives \eqref{eq:event-code}.
The three CNOTs exchange the local matter factors and the
record-controlled phase gives \(V_a\).
Outbound exchanges move the second factor into \(Q_n\), again
returning all intermediate qubits to their initial blanks.
The record is already classical; copying its specified basis
word creates precisely the repeated projectors in
\eqref{eq:benchmark-channel}, including their correlations
with matter. No unknown quantum state is copied.
This proves the formula on all operators and references.

For the witness, fix the clock to \(01\), hold the initial
\(B\) qubit fixed, and replace the \(A\) qubit by \(\ket0\) or
\(\ket1\). The \(B\)-outputs are orthogonal, hence their trace
distance is one. Theorem~\ref{thm:physical} gives the final
no-signalling assertion.
\end{proof}

The pre-existing reference can be entangled with the clock and
both inputs. A conditional reference state may change at clock
readout, but the channel acts as identity on the reference.
No gravity-generated entanglement is claimed.

\begin{proposition}[No instantaneous exchange on one slice]
\label{prop:instant}
A zero-duration remote SWAP between distinct ports of a single
regular marking cannot satisfy this causal architecture.
\end{proposition}

\begin{proof}
Lemma~\ref{lem:slices} makes the marking strictly spacelike:
its time difference is bounded by \(\lambda\) times spatial
path length. The ports are spacelike.
A SWAP would have the unit intervention witness just used,
contradicting physical containment.
\end{proof}

\subsection{An independent exact null-transport contract}

The preceding buffer is deliberately timelike. Exact local null
transport is also available under the I-B routing permission,
but only for a supplied target-adapted chain.

\begin{proposition}[Target-adapted reference-preserving transport]
\label{prop:nulltransport}
Let a developed segment of length \(L_0>0\) remain in a convex
regular chart. Install \(m+1\) equal finite payload registers
along it and initialize all but the source in a fixed blank.
Successive content exchanges over intervals \(L_0/(mc)\)
implement the logical identity at the null-related target,
including every reference, using \(m\) exchanges.

An already available classical record and finite location/history
payload may be included in the transported packet.
Nontrivial preparation, copying, or decoding time is additional.
\end{proposition}

\begin{proof}
At each hop the complete source factor replaces the next blank,
and the previous site becomes blank. Induction proves the
complete identity. Each communication path is the regular
straight segment traversed at speed \(c\).
The same argument applies to a joint quantum--classical packet.
\end{proof}

This is not an exact-attainment theorem for a previously fixed
lattice or for records produced too late to enter the packet.

%=====================================================================
\section{Finite tables, resources, and checkable obligations}
\label{sec:ledger}
%=====================================================================

\subsection{Finite evaluation table}

For the benchmark's two-qubit clock, the exact table is
\[
\begin{array}{c|c|c|c|c}
a&e(a)&(M,N,H)_{\rm out}&V_a&
       \Sig(A_{\rm in}\to B_{\rm out})\\ \hline
00&0&(0,1,0)&\operatorname{SWAP}&1\\
01&1&(1,0,1)&\operatorname{SWAP}&1\\
10&0&(0,1,0)&(Z\otimes I)\operatorname{SWAP}&1\\
11&0&(0,1,0)&(Z\otimes I)\operatorname{SWAP}&1
\end{array}
\]
Each signalling entry is evaluated for that fixed programme.
The full programme channel retains the record and is
\eqref{eq:benchmark-channel}.

The event indicator is the finite matrix
\[
 A_{\rm event}=\diag(0,1,0,0).
\]
A nominal label, route length, duration, cycle reward, or
potential value can similarly be stored as a diagonal matrix
over a finite programme set. Such matrices encode declared
evaluations; they are not measurements of an unknown field.

A finite time-expanded dependency graph with \(N\) vertices can
be represented by its Boolean adjacency matrix \(A\).
Its reachability table is the Boolean matrix
\[
 I\vee A\vee\cdots\vee A^{N-1}.
\]
Every reachable pair has a path of at most \(N-1\) edges in the
DAG, which proves this finite test without numerical execution.

\subsection{Quantum and classical resource ledger}

For the explicit \(n\)-segment benchmark:
\[
\begin{array}{l|l}
\text{resource}&\text{sufficient reserve or count}\\ \hline
\text{matter and routing qubits}&n+2\\
\text{clock qubits before destruction}&2\\
\text{quantum peak}&n+4\\
\text{initial blank routing qubits}&n\\
\text{classical record/code buffers}&5(n+1)\text{ bits}\\
\text{event scratch}&1\text{ bit, returned to zero}\\
\text{event-code elementary gates}&9\\
\text{local quantum CNOT gates}&3\\
\text{local record-controlled phase}&1\\
\text{outbound classical copying}&5n\text{ CNOTs}\\
\text{destructive clock operations}&2\\
\text{qubit content exchanges}&2n\\
\text{classical record transmissions}&n\\
\text{resets during the run}&0\\
\text{retained record words}&n+1
\end{array}
\]
The bits \(M,N,H\) are part of the five-bit sender word,
not extra uncharged buffers.
Their initial permission bit is one.
History and record copies are retained outputs, not clean work.

For an explicit conservative finite controller reserve, put
\[
 b_n=\left\lceil\log_2(8n+32)\right\rceil.
\]
Allow one \(b_n\)-bit local instruction counter per station and
at most \(8n+32\) programme rows per station.
An instruction row can be encoded with a four-bit opcode and
three \(b_n\)-bit indices, so the programme-ROM reserve is
\[
 B_{\rm ROM}\le(n+1)(8n+32)(4+3b_n)
\]
bits, in addition to \((n+1)b_n\) counter bits.
All station times are evaluated from the fixed algebraic
schedule formulas, not acquired as unknown real parameters.
A single remote global counter is not used as a hidden
instantaneous control wire.

The ledger is sufficient rather than optimal.
It assumes parallel carriage of the stated finite packet
capacity on each admitted link.
Different bandwidth, gate, or reset permissions require a new
ledger. External reference storage is not charged as laboratory
storage. No energy, thermal-machine, or Landauer-cost claim
follows.

\subsection{Exact geometric checks}

The following inequalities are symbolic certificates, not
reported numerical runs:
\[
\begin{array}{l|l}
\text{obligation}&\text{witness}\\ \hline
\text{spacelike margin}&1-\lambda^2=1/9\\
\text{conservative slice speed}&v_*=(1+\lambda)^{-1}\\
\text{regular transverse port}&(1/64,1/128)\ne o\\
\text{axial route length}&L=1/64\\
\text{inbound duration}&D=(1+\lambda)/16<1/8\\
\text{gather before clock readout}&
 -15/64+D<-7/64<-1/32\\
\text{local gate buffer}&
 13(1/208)=1/16\\
\text{distribution before output}&
 1/8+D<1/4<323/640\\
\text{event scratch cleanup}&E_{\rm out}=0\\
\text{event flags}&(M,N,H)=(e,1-e,e)\\
\text{canonical matter target}&V_{01}=\operatorname{SWAP}
\end{array}
\]
All nontrivial communication and local-operation intervals have
positive causal slack. For any fixed finite \(n\), sufficiently
small port and timing errors preserve these checks by
Proposition~\ref{prop:robust}, or directly by the strict
developed time-minus-distance margins.
Shared input/output ports are the same physical event, not
independently perturbed copies of an event.

\subsection{Finite approximation of a fixed instrument}

Suppose the benchmark's clock, local gates, record copies, and
communication channels have half-diamond errors with sum
\(\varepsilon_{\rm tot}\) on their complete stated interfaces.
Then Proposition~\ref{prop:telescope} gives
\[
 \ddia(\widetilde{\mathcal J}_n,\mathcal J_n)
 \le\min\{1,\varepsilon_{\rm tot}\}.
\]
The unit signalling witness on an allowed route remains at least
\(1-2\varepsilon_{\rm tot}\).
Physical containment is exact only when the implemented
dependencies retain their own causal certificates; channel
accuracy alone supplies at most the leakage bound of
Theorem~\ref{thm:signal-stability}.

No normalization of a rare selected branch is used in this
error statement. The comparison is the complete recorded
instrument.

%=====================================================================
\section{Assembly, significance, and standing non-claims}
\label{sec:release}
%=====================================================================

\begin{theorem}[VII-J-ST local Grade S release]
\label{thm:release}
For the declared conical geometry and finite permissions:
\begin{enumerate}[label=\textup{(\roman*)}]
\item complete finite dependency graphs bound joint-output
Heisenberg support and deterministic reference-assisted influence;
\item certified causal routes yield
\(\Inf^-\subseteq\Sch^-\subseteq\Phys^-\) for the full
encoded--computed--decoded channel;
\item directional cycle polytopes have the finite-time remainder
\eqref{eq:offset}, and the finite potential certificate
\eqref{eq:potential} records directional transients;
\item signalling stability has the constant
\(2\ddia\), and the finite Hamiltonian comparison has the
reference-stable tail \eqref{eq:tail};
\item destructive clocks realize the complete instrument
\eqref{eq:clock}, with event-time and programme/history
calibrations distinguished;
\item the benchmark \eqref{eq:benchmark-channel} is an exact,
nontrivial, reference-preserving Target B realization entirely
in the regular part, buffered through the prescribed incidence
parameter and supported by the displayed resource ledger.
\end{enumerate}
These statements constitute a nonempty local Grade S package.
\end{theorem}

\begin{proof}
Theorems~\ref{thm:nosignal} and \ref{thm:schedule} prove (i).
Theorem~\ref{thm:physical} proves (ii).
Theorem~\ref{thm:polytope} and
Proposition~\ref{prop:potential} prove (iii).
Theorems~\ref{thm:signal-stability} and \ref{thm:tail}, with
Proposition~\ref{prop:hamiltonian-errors}, prove (iv).
Theorem~\ref{thm:clock} and
Proposition~\ref{prop:calibration} prove (v).
Lemma~\ref{lem:benchmark-causal},
Theorem~\ref{thm:benchmark}, and
Section~\ref{sec:ledger} prove (vi).
These provide all clauses of Definition~\ref{def:grade}.
\end{proof}

\subsection{What the adaptation contributes}

The general support theorem, cycle-velocity mechanism,
instrument calculus, and Hamiltonian norm methods have
predecessors \cite{CheungVIIJ,Fritz,DaviesLewis,Watrous,BHV}.
The adaptation does not claim priority for them.

Its contribution is their explicit supported assembly with:
\begin{itemize}
\item a regular-part replacement for the singular-axis benchmark;
\item actual regular paths rather than an unrealized metric
distance equality;
\item complete records, controllers, and joint-output support;
\item a finite transient-potential check in the directional ledger;
\item an exact clock-controlled recorded channel and positive
execution buffer;
\item charged blank registers, history, record copies, local
controllers, and programme storage.
\end{itemize}
This combination is potentially useful as a test case for a later
proposal to add field acquisition, continuum compatibility, or
dynamical geometry. Those later proposals would need their own
same-model interfaces and proofs.

\subsection{Documentary source relationships}

VII-E's block locality, legal retiming, record pullback, and
residual obstruction are distinct source results; VII-G's
finite hybrid-channel interfaces do not themselves supply a
physical embedding \cite{CheungObservation}.
En03-A and En05-A compare prescribed schedule support with
separate metric envelopes \cite{CheungLocality}.
En09-A distinguishes conditioning from its additional
dynamical covariance hypotheses \cite{CheungEn09}.

VI-B and VII-A-ST supply supported-presentation comparisons,
not imported Einstein--scalar equations or descent theorems
\cite{CheungSupported}.
VIII-A/B/C distinguish preconstructed cone packets and
source interfaces from gravitational dynamics
\cite{CheungVIII}. In particular,
\[
 \text{matter-controlled slicing}
 \ne
 \text{matter-driven gravitational dynamics}.
\]
A location word is not the located quantum register.
The fixed incidence changes a marking, not the background metric.

The OpenAI catalogue is a separate documentary collection
attributed to OpenAI \cite{OpenAIRecord}.
It supplies no geometry, channel, clock, formal premise,
or verification of this manuscript.
A documentary separation is not a proof transfer.

\subsection{Version-specific analytical boundaries}

The historical H2 ending in XIV-H is not discharged by this
adaptation. It must be distinguished from XIV-I's constructed
internal I2 amplitude, which is not the full characteristic
map still absent at I3/J2 \cite{CheungAnalytic}.
No internal amplitude or supported-sector reference is treated
as the complete characteristic quantum composition.

The downstream quantum-admission exit called XV-B is not
opened. This is not a claim that no technical manuscript with
that identifier exists. XVI-N's universal obligations and
Route B are not discharged. Within the stated XVI-O
prescription,
\[
 \mathrm{O.1}=\loc{NOT_CLOSED}.
\]
Separately scoped enlarged-prescription subsector statements
do not update that particular record
\cite{CheungStops}. The quantum certificate remains
\loc{RG-open}; no quantum renormalization separator is proved.

\statementbox{Explicit non-claims}{%
No global conical Green/Cauchy theorem, intrinsic conical field
theory, sourced Einstein--scalar solution, continuum-field
acquisition, or quantum-field discretization is constructed.

No metric or Lorentz cone is derived from records or scheduler
consistency. No Einstein dynamics, spacetime emergence,
cosmological bounce, vacuum-energy stability, ultraviolet
completion, Page curve, island, interior, or gravity-generated
entanglement test is supplied.

No quantum Ward/BRST/BV identity, radiative coefficient,
characteristic quantum response coefficient, or quantum
evolution without external scheduling is admitted.

Exact finite channel equality is not a thermodynamic
implementation. Clock consumption does not consume the
geometric time axis. A buffered code update does not resolve
the conical singularity.}

%=====================================================================
\appendix
\section{Suggested registry-facing record}

The following is documentary text, not an assertion that an
external registry or interface was changed.
\begin{verbatim}
identifier: VII-J-ST
grade: local Grade S
problemConnectionsFor: ["time"]
emergenceStatus: not constructed
quantumCertificate: RG-open

relationshipRole: separation
relationshipDistinction:
  A documentary separation is not a proof transfer.

scope:
  Complete finite dependency and causal-embedding certificates,
  finite destructive clock control, and exact buffered regular
  transport through one prescribed transverse incidence parameter.

corrections:
  No apex quantum register is used.
  The 3+1 defect is a worldsheet.
  The 3-to-6 assertion is transverse.
  All controllers and joint-output dependencies are included.
  Half-diamond simulation error contributes twice that amount
  to the signalling bound.

standingBoundaries:
  O.1 remains NOT_CLOSED in the stated XVI-O prescription.
  The downstream XV-B quantum-admission exit is not opened.
  XVI-N and Route B are not discharged.
  Historical H2 and full I3/J2 records are not discharged.
  No inherited quantum coefficient is admitted.
\end{verbatim}

A suitable emergence-walk sentence is:
\begin{quote}
VII-J-ST certifies finite scheduled influence and one buffered
regular-part transport on a prescribed conical spacetime.
It does not extract the causal cone, reconstruct the metric,
or turn slicing incidence into gravitational dynamics.
\end{quote}

%=====================================================================
\section*{Declarations}
\addcontentsline{toc}{section}{Declarations}

\paragraph{Anonymous review.}
The author field is intentionally empty. Authorized author,
affiliation, and contribution metadata are required for a
signed submission. Attribution of documentary records to
N.~Cheung or OpenAI does not assign authorship of this adaptation.

\paragraph{Sources and bibliographic status.}
Registered manuscripts are identified by their supplied version
rather than an invented publication date. Their statements are
not independent verification of other programmes.
The journal citation to Cheung's Enter article follows the
bibliographic record supplied with the question; no unverified
update of its publication status is asserted here.

\paragraph{Verification and implementation.}
The document is one conventional pdfLaTeX source with an internal
bibliography. No successful pdfLaTeX execution, numerical run,
symbolic-computation execution, hardware compilation,
experiment, proof-assistant check, independent referee
certification, or journal acceptance of this adaptation is
asserted. The mathematical release consists of the stated
definitions and proofs.

%=====================================================================
\begin{thebibliography}{99}
\addcontentsline{toc}{section}{References}

\bibitem{CheungVIIJ}
N.~Cheung,
\emph{Landmark VII-J: Scheduled dependencies and certified physical
causality in localized quantum simulation.
Exact support, directional reachability, approximation, and
continuation through a fixed-cone slicing event}.
\loc{VII-J.tex}, version 1, Enter Calculus Companion,
unpublished manuscript.
Relevant locators include \loc{thm:nosignal},
\loc{thm:schedule}, \loc{thm:physical},
\loc{thm:polytope}, \loc{thm:signal-stability},
\loc{thm:tail}, \loc{thm:conegates},
\loc{cor:conespeed}, and \loc{thm:eventbenchmark}.
\url{https://enter.grok.me/browse/VII-J}.

\bibitem{CheungIB}
N.~Cheung,
\emph{I-B-v1: A Finite Enter Spacetime:
Calibrated Destructive Time Readout on a Prescribed Conical Product}.
\loc{I-B-v1.tex}, version 1, Enter Calculus Companion,
unpublished manuscript.
\url{https://enter.grok.me/browse/I-B}.

\bibitem{CheungEnter}
N.~Cheung,
``The Enter Calculus: A Linear Quantum Lambda Calculus with an
Operational Characterization of Measurement,''
\emph{Frontiers in Physics} \textbf{14} (2026),
article 1954943.
Bibliographic information as supplied; no publication-status
update is inferred.
DOI: \doi{10.3389/fphy.2026.1954943}.

\bibitem{CheungI}
N.~Cheung,
\emph{Manuscript I}.
\loc{I.tex}, version 2, Enter Calculus Companion,
supplied unpublished manuscript.
Finite measurement characterization within its declared
structural class.
\url{https://enter.grok.me/browse/I}.

\bibitem{CheungPhilosophy}
N.~Cheung,
\emph{The Philosophy of Enter: Destructive Readout, Quantum
Recovery, and the Refusal of Identification}.
\loc{EntPhil-v2.tex}, version 2.
Introductory essay, Enter Calculus Companion.
\url{https://enter.grok.me/enter}.

\bibitem{CheungVIIF}
N.~Cheung,
\emph{Manuscript VII-F: A particle-supported slicing event on a
fixed spinless cone}.
\loc{VII-F.tex}, version 1, Enter Calculus Companion,
unpublished manuscript.
The geometric certificate and finite event-code interface have
their own declared domains.
\url{https://enter.grok.me/browse/VII-F}.

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

\bibitem{DaviesLewis}
E.~B.~Davies and J.~T.~Lewis,
``An operational approach to quantum probability,''
\emph{Communications in Mathematical Physics}
\textbf{17} (1970), 239--260.
DOI: \doi{10.1007/BF01647093}.

\bibitem{Fritz}
T.~Fritz,
``Velocity polytopes of periodic graphs and a no-go theorem for
digital physics,''
\emph{Discrete Mathematics} \textbf{313} (2013), 1289--1301.
DOI: \doi{10.1016/j.disc.2013.02.010}.
\url{https://arxiv.org/abs/1109.1963}.
The revised preprint records earlier cycle-figure work;
no priority claim for the general polytope mechanism is made here.

\bibitem{BHV}
S.~Bravyi, M.~B.~Hastings, and F.~Verstraete,
``Lieb--Robinson bounds and the generation of correlations and
topological quantum order,''
\emph{Physical Review Letters} \textbf{97} (2006), 050401.
DOI: \doi{10.1103/PhysRevLett.97.050401}.
\url{https://arxiv.org/abs/quant-ph/0603121}.

\bibitem{NachtergaeleSims}
B.~Nachtergaele and R.~Sims,
``Lieb--Robinson bounds in quantum many-body physics,''
in \emph{Entropy and the Quantum},
Contemporary Mathematics \textbf{529},
American Mathematical Society, 2010, pp.~141--176.
\url{https://arxiv.org/abs/1004.2086}.

\bibitem{CheungLocality}
N.~Cheung,
\emph{En03-A}, \loc{En03-A.tex}, version 1; and
\emph{En05-A}, \loc{En05-A.tex}, version 1.
Enter Calculus Companion, supplied unpublished source records.
Complete channel locality, schedule envelopes, and their
separate physical interpretations.
\url{https://enter.grok.me/browse/En03-A};
\url{https://enter.grok.me/browse/En05-A}.

\bibitem{CheungEn09}
N.~Cheung,
\emph{En09-A: Realization of Relational Time by Destructive Enter
on a Clock Register}.
\loc{En09-A.tex}, version 1, Enter Calculus Companion,
supplied unpublished manuscript.
Complete conditioning and separately stated dynamical
covariance hypotheses.
\url{https://enter.grok.me/browse/En09-A}.

\bibitem{PageWootters}
D.~N.~Page and W.~K.~Wootters,
``Evolution without evolution: Dynamics described by stationary
observables,''
\emph{Physical Review D} \textbf{27} (1983), 2885--2892.
DOI: \doi{10.1103/PhysRevD.27.2885}.

\bibitem{KnillLaflamme}
E.~Knill and R.~Laflamme,
``Theory of quantum error-correcting codes,''
\emph{Physical Review A} \textbf{55} (1997), 900--911.
DOI: \doi{10.1103/PhysRevA.55.900}.

\bibitem{CheungObservation}
N.~Cheung,
\emph{VII-E}, \loc{VII-E.tex}, version 1;
\emph{VII-G}, \loc{VII-G.tex}, version 1; and
\emph{VII-E-ST}, \loc{VII-E-ST.tex}, version 1.
Enter Calculus Companion, supplied unpublished source records.
Observation boundaries, complete hybrid channels, residual
distinctions, and recovery scope.
\url{https://enter.grok.me/browse/VII-E};
\url{https://enter.grok.me/browse/VII-G};
\url{https://enter.grok.me/browse/VII-E-ST}.

\bibitem{CheungSupported}
N.~Cheung,
\emph{VI-B}, \loc{VI-B-v1.tex}, version 1; and
\emph{VII-A-ST}, \loc{VII-A-ST.tex}, version 1.
Enter Calculus Companion, supplied unpublished source records.
Supported Grade S comparisons only.
\url{https://enter.grok.me/browse/VI-B};
\url{https://enter.grok.me/browse/VII-A-ST}.

\bibitem{CheungVIII}
N.~Cheung,
\emph{VIII-A}, \emph{VIII-B}, and \emph{VIII-C},
version 3.
Enter Calculus Companion, supplied unpublished source records.
Fixed-presentation conical packets and separately scoped
matter--geometry interfaces.
\url{https://enter.grok.me/browse/VIII-A};
\url{https://enter.grok.me/browse/VIII-B};
\url{https://enter.grok.me/browse/VIII-C}.

\bibitem{CheungAnalytic}
N.~Cheung,
\emph{XIV-H}, version 1;
\emph{XIV-I}, version 1;
\emph{XIV-J}, version 1; and technical
\emph{XV-B}, version 2.
Enter Calculus Companion, supplied unpublished source records.
Historical H2, internal I2, full I3/J2, and reconstruction or
extension interfaces are distinct records.
\url{https://enter.grok.me/browse/XIV-H};
\url{https://enter.grok.me/browse/XIV-I};
\url{https://enter.grok.me/browse/XIV-J};
\url{https://enter.grok.me/browse/XV-B}.

\bibitem{CheungStops}
N.~Cheung,
\emph{XVI-N}, revision r12;
\emph{XVI-O}, revision o70-sync; and
\emph{XVI-Q}, release Q11.
Enter Calculus Companion, supplied unpublished source records.
The inherited insertion and stated XVI-O prescription are not
updated by separately scoped enlarged-prescription sectors.
\url{https://enter.grok.me/browse/XVI-N};
\url{https://enter.grok.me/browse/XVI-O};
\url{https://enter.grok.me/browse/XVI-Q}.

\bibitem{OpenAIRecord}
OpenAI,
\emph{OpenAI catalogue}.
Separate documentary collection supplied with the question and
recorded by the Enter Calculus Companion, 2026.
Not a premise; no family proof, formalization scope, or
verification status is imported.
\url{https://enter.grok.me}.

\end{thebibliography}

\end{document}