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\hypersetup{
  pdftitle={Landmark VIII-C: Certified Flow, Guarded Events,
            and Finite Reversible Composition},
  pdfsubject={Conditional scheduling theorem for spinless conical geometry},
  pdfkeywords={conical geometry, affine matching, guarded scheduling,
               source token, reversible computation, forward domain}
}

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\title{\textbf{Landmark VIII-C}\\[3pt]
Certified Fixed-Packet Flow, Guarded Events,\\
and Finite Reversible Composition}
% Insert verified author and affiliation information before submission.
\author{}
\date{}

\begin{document}
\maketitle

\begin{abstract}
We give a conditional composition theorem for polygonal descriptions
of a declared spinless conical spacetime. At a fixed particle packet,
explicit realizations produce correlated affine edge-length laws.
Incoming slice advance preserves the finite lengths, while outgoing
advance changes genuine crease lengths. A separately installed collar
module changes subdivision lengths without changing the embedded slice.

Exact parameter clocks, profile-specific guards, separate incidence
and source permissions, and complete output obligations define legal
schedules. Relative to explicitly stated realization and transport
interfaces, every legal schedule returns an accepted five-component
affine matching certificate after every command. Geometric commands
preserve the installed packet and reference-frame signature; regular
source-table replacement installs a complete new realization.

An explicit five-command history performs source replacement at
\(s=-1/2\), reaches the incoming particle germ, changes the particle
link at a frozen deficit, and continues along the outgoing family.
Finitely encoded words have reversible implementations on their
composite canonical forward domains, with retained input histories,
identity action on internal matter, and explicit resource bounds for
this history. Independently, prescribed block channels obey an exact
backwards-support inclusion, including untouched references.

The source token is supplied under a semantic promise, not acquired
or tested. Local obstruction and separation arguments prevent
incomplete packet updates, expectation extraction, record--residual
identification, schedule--causality identification, and confusion of
code decoding with quantum recovery. No mixed incidence--source germ,
physical embedding of the schedule, polygon Hamiltonian, spacetime
emergence, or holographic identification is constructed.
\end{abstract}

\section{Problem, contribution, and logical scope}
\label{sec:introduction}

Landmark VIII-A defines a finite affine matching interface for a
declared spinless conical geometry
\cite{CheungVIIIA}. Landmark VIII-B constructs a variable-deficit
realization family and a finite promised source table
\cite{CheungVIIIB}. The present paper supplies a scheduling layer:
fixed-packet movement, guarded incidence, regular source replacement,
and finite reversible composition.

The polygon literature studies evolving descriptions of
\(2+1\)-dimensional gravity
\cite{Hooft1992,Hooft1993,Kadar2005}. The construction here is narrower.
We prescribe realizations and derive their length laws; we do not
identify those laws with a polygon Hamiltonian or import a general
transition catalogue. The spinless conical interpretation is a
declared geometric dictionary, motivated by the locally flat
\(2+1\)-dimensional setting \cite{DJH1984}, not a consequence of the
scheduling theorem.

\paragraph{Contributions.}
\begin{enumerate}
\item Explicit fixed-packet realizations yield correlated affine length
laws, including nonzero outgoing crease rates and a coplanar
length-changing collar refinement.
\item A guarded grammar composes regular source replacement and
fixed-packet incidence while returning complete matching outputs.
\item One sourced five-command word has exact packet data, a permission
trace, canonical report checks, and finite reversible resource bounds.
\item Local proofs state the observation, support, and decoding
boundaries needed to interpret this composition correctly.
\end{enumerate}

The contribution is a certified composition framework, not a new
particle geometry, source law, measurement primitive, or gravitational
dynamics.

\subsection{Conditional theorem architecture}

The required geometric realization and location-transport interfaces
are stated in \cref{sec:families,sec:commands}. They are hypotheses of
the conditional scheduling theorem. A citation does not replace an
interface statement or independently validate its implementation.

By contrast, the completeness, token, record, support, recovery, and
non-identification statements added here are proved locally. They are
not imported as additional premises from companion manuscripts.
Their citations identify corresponding results and comparisons in
Cheung's series.

The two substrate profiles are alternatives. Their moving cores are
not glued together, and no profile-switching command is supplied.
A collar-refined representation must already be installed before
collar flow acts. Source and incidence remain separate command kinds.

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.34\linewidth}X@{}}
\toprule
Object & Status\\
\midrule
Realization and transport interfaces
 & Explicit hypotheses of the conditional composition theorem\\
T0 source token
 & Supplied with a semantic promise; not acquired or verified\\
Particle incidence
 & At a fixed installed packet in profile \(\Pprof\)\\
Four-face exchange
 & In the separate particle-free moving patch of profile \(\Aprof\)\\
Collar flow
 & A separately installed module; absent from the hybrid dictionary\\
Mixed incidence--source germ
 & Not constructed or admitted\\
Backwards schedule support
 & A computational upper bound; not a physical causal past\\
Quantum recovery and holographic reconstruction
 & Not conclusions of reversible code decoding\\
\bottomrule
\end{tabularx}
\end{center}

The dependency and source-locator tables in
\cref{app:dependencies} distinguish inherited hypotheses from locally
proved statements.

\section{Complete affine matching at every command boundary}
\label{sec:matching}

\subsection{Packets and affine frames}

Use developed coordinates
\[
 z=(T,X,Y)^{\mathsf T},\qquad
 \eta=\diag(-1,1,1),\qquad e_0=(1,0,0)^{\mathsf T}.
\]
Affine isometries act and compose by
\[
 (L,b)z=Lz+b,\qquad
 (L,b)(L',b')=(LL',Lb'+b),
\]
and
\[
 (L,b)^{-1}=(L^{-1},-L^{-1}b).
\]
Define
\begin{equation}
 R(\mu)=
 \begin{pmatrix}
 1&0&0\\
 0&\cos\mu&-\sin\mu\\
 0&\sin\mu&\cos\mu
 \end{pmatrix},
 \qquad J(\mu)=(R(\mu),0).
 \label{eq:rotation}
\end{equation}
The reference packet is
\begin{equation}
 \Pi_\mu=(p,\mu,\cW_0,e_0,\id,J(\mu)),
 \qquad \cW_0=\RR e_0.
 \label{eq:packet}
\end{equation}
Here \(p\) is a geometric particle identifier.

For a declared affine frame \(A=(L_A,b_A)\), the coherently transported
packet has
\[
 \cW_\Pi=A\cW_0,\qquad
 u_\Pi=L_Ae_0,\qquad
 H_\Pi=AJ(\mu)A^{-1}.
\]
In particular,
\[
 H_\Pi(A(te_0))=A(te_0)
 \quad\text{for every }t\in\RR.
\]
The frame, worldline, rest vector, deficit, and holonomy are transported
together.

\subsection{The five-component output}

\begin{definition}[Complete matching output]
\label{def:matching}
For a realized description \(g\), write
\[
 \Ein(g)=
 \bigl(\cV(g),\cP(g),\cH(g),\cK(g),\cW(g)\bigr).
\]
An accepted output supplies the finite tables, exact equalities, and
topological evidence for all of the following obligations:
\begin{enumerate}[label=(E\arabic*)]
\item \textbf{Vertex and overlap matching \(\cV\).}
Every proposed transition equals its prescribed affine map on the
declared overlap component. Reverse maps are inverses; paired endpoints
and shared segments match; face boundaries close; regular small
vertex words are affine identities.
\item \textbf{Particle matching \(\cP\).}
The declared deficit, oriented rest frame, affine worldline, and
peripheral map agree with the installed packet.
\item \textbf{Ordered holonomy evidence \(\cH\).}
The ordered words, overlap-component indices, and winding classes
are supplied. A positive primitive particle meridian has winding one,
not merely a nonidentity matrix.
\item \textbf{Core--collar matching \(\cK\).}
The core and collar share their prescribed boundary realization and
transported framing. Their primitive words give the same installed
peripheral map.
\item \textbf{Worldline data \(\cW\).}
All charts describe the same singular worldline. Its holonomy fixes
that worldline pointwise, rest vectors are transported consistently,
and the declared slice intersection lies on it.
\end{enumerate}
Embedding, spacelikeness, and nondegeneracy of active cells are separate
realization requirements. A germ is accepted as a specified one-sided
limiting object, not as a regular cellulation with extra active
zero-size cells.
\end{definition}

This is the finite matching interface of VIII-A
\cite[Definition \texttt{A:def:Ein}]{CheungVIIIA}, restated here.
The notation \(\Ein\) does not mean that a Hamiltonian constraint
algebra or a general Einstein initial-data theorem has been proved.

For fixed face charts \(\chart_f\), seam map \(J(\mu)\), and
overlap-component exponent \(n_{gf}\), the prescribed transition is
\begin{equation}
 t^{\mathrm{can}}_{g\leftarrow f}
 =\chart_g J(\mu)^{n_{gf}}\chart_f^{-1}.
 \label{eq:canonical}
\end{equation}
Elementary overlap components have \(n_{gf}\in\{-1,0,1\}\); longer
paths retain their individual component data.

\begin{proposition}[Output completeness and two independent failures]
\label{prop:complete-output}
The following statements hold for every command output, at every
installed \(\mu\), in a fixed declared presentation.
\begin{enumerate}
\item With realization, charts, overlap components, reference framing,
and prescribed seam maps fixed, two different proposed values of a
required transition cannot both satisfy (E1).
\item If \(\mu_{\mathrm{old}}\ne\mu_{\mathrm{new}}\) belong to the
interval used below, retaining \(J(\mu_{\mathrm{old}})\) at the new
deficit violates particle/seam matching, although it still fixes
\(\cW_0\) pointwise.
\item For \(\delta\ne0\), the affine map
\[
 \bigl(R(\mu),\delta e_0\bigr)
\]
preserves \(\cW_0\) as a set and preserves the rest vector in its linear
part, but does not fix \(\cW_0\) pointwise. It therefore fails (E5).
\end{enumerate}
All three statements remain true after coherent conjugation by any
declared affine frame \(A\).
\end{proposition}

\begin{proof}
For the first assertion, (E1) requires equality with the single map
in \cref{eq:canonical}. Two accepted proposed maps must therefore
equal one another.

The interval below is contained in \((0,2\pi)\), so distinct deficits
in that interval give distinct oriented spatial rotations. Thus
\(J(\mu_{\mathrm{old}})\ne J(\mu_{\mathrm{new}})\).
Every \(J(\mu_{\mathrm{old}})\) nevertheless sends \(te_0\) to \(te_0\).
The failure is deficit/seam matching, not pointwise worldline fixation.

For the third assertion,
\[
 \bigl(R(\mu),\delta e_0\bigr)(te_0)=(t+\delta)e_0\ne te_0.
\]
Conjugation is injective. Moreover, the conjugated time-shifted map sends
\(A(te_0)\) to \(A((t+\delta)e_0)\). Hence the same distinctions hold
in every coherently transported frame.
\end{proof}

These are local versions of VIII-A's fixed-presentation completeness
and rejected updates
\cite[Theorems \texttt{A:thm:A1},
\texttt{A:thm:A4-stale}, and \texttt{A:thm:A4-time}]{CheungVIIIA}.
They do not assert that holonomy alone determines all chart transitions.
In particular, matching a few endpoints or a peripheral product is
not a substitute for \cref{eq:canonical}.

\subsection{Reference-frame signature and worldline distinctions}

A chart-based primitive product is generally conjugate to \(J(\mu)\).
After transport back to the declared reference frame it is exactly
\(J(\mu)\). We use the latter convention in the persistent signature
\begin{equation}
 \Sig(g)=
 \bigl(p,\mu,\cW_0,e_0,\text{reference framing},
 [\gamma]_{\mathrm{primitive}},J(\mu),\text{collar matching}\bigr).
 \label{eq:signature}
\end{equation}
Its preservation does not require literal equality of every
chart-based matrix.

Three different statements are kept separate:
\[
 \cW_0\text{ is a fixed ambient set},\qquad
 J(\mu)|_{\cW_0}=\id,\qquad
 P_s\in\cW_0\text{ may vary with }s.
\]
Fixed-packet slice advance preserves the first two, not generally the
third.

\section{Named access and the supplied-token boundary}
\label{sec:token}

\subsection{Matter, source algebras, and policy T0}

A finite named matter interface \(\Gamma\) carries one joint state
\[
 \rho\in\Dens(\cH_\Gamma).
\]
Names specify access to declared factors. Classical registers are
jointly block diagonal in their distinguished bases, including their
correlations with other factors.

A surviving source boundary is an explicitly embedded unital
\(^{*}\)-subalgebra
\[
 \cA_\src\subseteq\Lin(\cH_{\Gamma_\src}).
\]
Its accessible state functional is
\[
 \omega_\rho|_{\cA_\src}:B\longmapsto\Tr(\rho B).
\]
For a fixed Hermitian \(B_\src\in\cA_\src\) and fixed \(\kappa\), a
table token \(m_\src\) can carry the semantic promise
\begin{equation}
 \mu_0+\kappa\Tr(\rho B_\src)=\mu_{m_\src}.
 \label{eq:general-promise}
\end{equation}
The example below uses \(B_\src=Z\) and \(\kappa=1\).

\paragraph{Policy T0.}
Every sourced history begins with an already supplied classical token.
At each sourced command the token must satisfy its declared promise
for the current surviving source state. No token-acquisition operation
or operational promise test occurs in the history.

``Current token'' means correct for the current source functional.
It does not mean that the selected packet is already installed before
source replacement.

For a fixed token, the promised state class may be convex. The union
of classes for different deterministic expectation labels need not be
convex. Nothing here implements a state-only deterministic source
selection on the full quantum state space.

\begin{proposition}[Expectation and affine-label obstructions]
\label{prop:token-obstruction}
\begin{enumerate}
\item Let \(\mathcal S\) be a convex state set. If an affine map assigns
to every \(\rho\in\mathcal S\) a classical distribution concentrated
at one label,
\[
 T(\rho)=\delta_{\gamma(\rho)},
\]
then \(\gamma\) is constant on \(\mathcal S\).
The statement also holds for a classical outcome space whose distinct
labels can be separated by a measurable event.
\item A circuit acting only on code registers and as the identity on
matter cannot, for a fixed independent code input, make its code output
depend on an unencoded matter expectation.
\end{enumerate}
Requiring that a putative label-producing channel also preserve the
matter input does not remove the first obstruction.
\end{proposition}

\begin{proof}
If two states have distinct labels, affinity gives, for \(0<t<1\),
\[
 T(t\rho+(1-t)\sigma)
 =t\delta_{\gamma(\rho)}+(1-t)\delta_{\gamma(\sigma)}.
\]
This is not concentrated at one label. In a general outcome space,
an event separating the labels has probability strictly between zero
and one. Either conclusion contradicts the assumed deterministic
output on the mixture.

For the second assertion, a code-only circuit \(U_{\mathrm c}\)
acts as \(U_{\mathrm c}\otimes I_\Gamma\). On an independent input
\(c\otimes\rho\), its code marginal depends on \(c\), not on \(\rho\).
Different unencoded expectations therefore cannot select different
code outputs. An additional matter-preservation requirement cannot
invalidate either argument.
\end{proof}

The two arguments correspond to VIII-B's expectation-label obstruction
and VII-G's affine obstruction
\cite[Proposition \texttt{prop:obstruction}]{CheungVIIIB},
\cite[Proposition \texttt{prop:affine}]{CheungVIIG}.
They are proved here rather than assumed. They do not prohibit
deterministic processing of an already supplied classical token,
stochastic instruments, or separately controlled approximations.

\subsection{Records do not determine residual access}

For destructive computational readout of \(S\), with surviving
complement \(B\), a specified output is
\begin{equation}
 \mathcal M_S(\rho)
 =\sum_z \kb{z}{z}_R\otimes
   (\bra{z}_S\otimes I_B)\rho(\ket{z}_S\otimes I_B).
 \label{eq:destructive}
\end{equation}
Its codomain contains \(R\) and \(B\), not the consumed factor \(S\).
A pullback of an output observable is an input observable; it does
not restore \(S\) as a surviving output name
\cite{CheungI,CheungVIIIA}.

\begin{proposition}[Accessible expectations and residual separation]
\label{prop:accessible}
\begin{enumerate}
\item For channels \(\Phi,\Psi\) with common input and output spaces
and a fixed output Hermitian \(B\),
\[
 \Tr(B\Phi(\rho))=\Tr(B\Psi(\rho))
 \quad\text{for every input state }\rho
\]
if and only if
\[
 \Phi^*(B)=\Psi^*(B).
\]
On a restricted state class, only the corresponding annihilator
condition follows; operator equality need not follow.
\item Identical classical record channels do not imply identical
channels on the record--residual output algebra.
\end{enumerate}
\end{proposition}

\begin{proof}
Trace duality turns the first equality into
\[
 \Tr\bigl(\rho(\Phi^*(B)-\Psi^*(B))\bigr)=0.
\]
A Hermitian operator with zero expectation in every state is zero.
Restricting the tested states restricts this conclusion to their
real linear span.

For the second assertion, let
\(\Delta_Z(\rho)=P_0\rho P_0+P_1\rho P_1\) on a qubit and define
\[
 \mathcal G(\rho)=\kb{*}{*}_R\otimes\rho,\qquad
 \mathcal F(\rho)=\kb{*}{*}_R\otimes\Delta_Z(\rho).
\]
Their record marginals are the same constant record.
For \(\rho=\kb{+}{+}\), their residual difference is \(X/2\), so
\[
 \frac12\norm{\mathcal G(\rho)-\mathcal F(\rho)}_1=\frac12.
\]
The full residual algebra distinguishes them.
\end{proof}

These statements sharpen the source boundary in the notation used
here, in agreement with VII-E's accessible-observable and
residual-obstruction results
\cite[Proposition \texttt{prop:accessible} and
Theorem \texttt{thm:residual-obstruction}]{CheungVIIE}.
A produced record can become a source only when its surviving register
and source algebra are explicitly named. Record agreement alone does
not justify using an unnamed residual observable as a source.

No readout or residual-changing instrument occurs in the T0 word
constructed below.

\section{Configurations, clocks, and command domains}
\label{sec:commands}

\begin{assumption}[Deterministic spectator-blind control]
\label{ass:control}
Equal geometry, controller data, permissions, declared geometric
resources, source embedding, and explicitly accessible inputs select
the same command, applicability result, geometric support, complete
output certificate, original-name transport, and successor controller
state.

Geometric commands do not read internal matter, occupancy, source
tokens, or spectator number. A source-table command reads its declared
classical token. Spectator locations and implementation histories have
separately reserved storage and do not affect geometric capacity guards.
\end{assumption}

This is a declared controller interface, not a general no-signalling
theorem for nonlinear feedback. The continuum source law of VIII-B
remains its nonlinear Option N. Its promised table restriction is not
an expectation extractor or an implementation of that continuum law
on arbitrary unknown states.

\begin{definition}[Scheduled configuration]
\label{def:configuration}
A scheduled configuration is
\[
 C=
 \bigl(g,\Pi,m,\mathsf{profile},s,\sigma,\tau,h,
       \nu_G,\nu_\src,m_\src;
       \Gamma,\rho,\ell;\cR,\mathsf{history}\bigr),
\]
where:
\begin{enumerate}
\item \(g\) is a certified regular geometry or certified one-sided germ;
\item \(\Pi\) is its installed packet, and \(m\) is a table index,
or \(*\) in a semantic continuum description;
\item \(\mathsf{profile}\in\{\Pprof,\Aprof\}\) fixes the substrate;
\item \(s,\sigma\) are the particle-family and four-face parameters;
an unused parameter is inactive metadata;
\item \(\tau\) is an execution/flow clock;
\item \(h\) includes incidence mode, source stage, schedule stage,
module, and finite controller data;
\item \(\nu_G,\nu_\src\) are separate nonnegative integer permissions;
\item \(m_\src\) is an optional classical source token;
\item \(\ell:\Gamma\to\Cell(g)\) specifies abstract locations;
\item \(\cR\) reserves geometric resources separately from spectator,
location, and implementation-history storage.
\end{enumerate}
For profile \(\Pprof\),
\[
 \Module\in\{\mathsf{slice},\mathsf{collar},\mathsf{none}\}.
\]
Collar mode has an active parameter \(t_\col\), holds \(s\) fixed and
regular, and uses its already installed refined dictionary.
Profile \(\Aprof\) uses its own slice module and dictionary.

No command changes profile, silently refines the dictionary, or uses
implementation history as a source observable.
\end{definition}

\subsection{Exact clocks and germ modes}

Finite codes use exact dyadic parameters
\begin{equation}
 s=j2^{-d},\qquad
 \sigma=j_A2^{-d_A},\qquad
 t_\col=j_\col2^{-d_\col}.
 \label{eq:clocks}
\end{equation}
Recorded execution slots satisfy
\[
 \tau=k\Delta\tau,\qquad \Delta\tau>0.
\]
A geometric segment can be parametrized by
\begin{equation}
 s=s_a+v_s(\tau-\tau_a),
 \label{eq:s-flow}
\end{equation}
or the analogous law for \(\sigma\) or \(t_\col\).
Only designated endpoint codes need lie on the finite grid.

Incoming and outgoing germs at zero have the same exact parameter
but different incidence modes. We write \(0^-\) and \(0^+\) for those
modes, not for rounded real numbers.

A discrete incidence or source command may consume an execution slot
while leaving its slicing parameter fixed. No continuous interpolation
of that discrete command is asserted. The execution clock is not
identified with proper time or a relational lapse.

\begin{contract}[Four admitted command kinds]
\label{contract:commands}
Each command returns the complete output of \cref{def:matching},
canonical finite reports when applicable, and certified original-name
transport.
\begin{enumerate}
\item \(\Gflow\): fixed-packet flow within one declared module and
incidence epoch. Slice advance has nonnegative parameter speed.
A particle slice segment may end at the incoming germ or begin at the
outgoing germ, but may not cross zero internally.
Collar flow requires profile \(\Pprof\), a regular fixed \(s\), and
the prescribed collar window.
\item \(\Eparticle\): profile \(\Pprof\), unrefined slice module,
incoming mode, exact \(s=0\), and \(\nu_G\ge1\).
It selects the outgoing germ at the same packet, decreases \(\nu_G\)
by one, and recomputes the incidence and matching tables.
\item \(\EA\): profile \(\Aprof\), incoming mode, exact \(\sigma=0\),
and \(\nu_G\ge1\). It performs the certified four-face exchange away
from the particle, decreases \(\nu_G\), and preserves the ambient packet.
\item \(\Stab\): profile \(\Pprof\), unrefined slice module,
source stage zero, exact regular parameter \(s=-1/2\), a supplied
table token satisfying its semantic promise, and \(\nu_\src\ge1\).
It installs the complete realization at the token's packet and the
same \(s\), decreases \(\nu_\src\), and sets source stage to one.
\end{enumerate}
No command combines source replacement and incidence.
Every admitted command acts as the identity on internal matter in a
T0 history. Acceptance is conjunctive: every realization requirement
and every obligation (E1)--(E5) is required.
\end{contract}

The contract specifies admitted domains and outputs. Its geometric
existence and transport content is supplied by the explicit interfaces
in the next section. Source replacement is evaluation of the complete
particle constructor at the selected table entry, not a deficit-label
write with stale charts.

\section{Realization interfaces and explicit data}
\label{sec:families}

\subsection{Particle-family geometry}

Set
\begin{align}
 \mu_0&=\frac{\pi}{2},&
 I&=\left[\frac{\pi}{2}-\frac{\pi}{20},
          \frac{\pi}{2}+\frac{\pi}{20}\right],\nonumber\\
 \alpha&=\frac{2\pi-\mu}{3},&
 a&=\frac{2\sqrt2}{3}\cos\frac{\alpha}{2},&
 b_\mu&=\frac35\sin\alpha.
 \label{eq:coefficients}
\end{align}
The subscript distinguishes \(b_\mu\) from report precision \(b\).

In the cut development put
\[
 e_i=\begin{pmatrix}\cos(i\alpha)\\ \sin(i\alpha)\end{pmatrix},
 \quad i=0,1,2,3.
\]
Sector \(i=0,1,2\) has coordinates
\[
 x=\xi e_i+\zeta e_{i+1},\qquad \xi,\zeta\ge0.
\]
The last ray is identified with the first by \(J(\mu)\), since
\(3\alpha=2\pi-\mu\). Cell indices are cyclic modulo three after seam
identification; the developed rays \(e_0,\ldots,e_3\) are kept distinct
before that identification.

Define
\begin{align}
 \rho_c&=\xi+\zeta,&
 h_{\mathrm{lin}}&=a(\xi+\zeta),&
 k_\mu&=b_\mu\min\{\xi,\zeta\},\nonumber\\
 F_s^{(\mu)}
 &=s+\min\{2,\max\{h_{\mathrm{lin}},s+k_\mu\}\}.
 \label{eq:graph}
\end{align}
The ambient regular metric is the ultrastatic cone metric
\(-dT^2+ds_{C_\mu}^2\); the graph is \(T=F_s^{(\mu)}\).

Use
\begin{equation}
 r=\frac2a,\qquad B=\frac1a,\qquad
 C=\frac1{2a-b_\mu},\qquad
 z_\alpha=\cos\alpha,\qquad S_\alpha=\sin\alpha.
 \label{eq:scalars}
\end{equation}
The particle intersection and outer vertices are
\[
 P_s=
 \begin{cases}
 (s,0,0),&s\le0,\\
 (2s,0,0),&s\ge0,
 \end{cases}
 \qquad A_i=re_i,\qquad D_i=4e_i,
\]
with \(T(A_i)=T(D_i)=2+s\). Spatial vertex notation is supplemented
by its displayed time coordinate.

For \(s>0\),
\[
 B_i=Bs\,e_i,\quad T(B_i)=2s,\qquad
 C_i=Cs(e_i+e_{i+1}),\quad T(C_i)=2s+b_\mu Cs.
\]
Incoming core faces are
\[
 V_i=[P,A_i,A_{i+1}],\qquad T=s+a(\xi+\zeta).
\]
Outgoing core faces are
\begin{align}
 U_i^-&=[P,B_i,C_i],& T&=2s+b_\mu\zeta,\nonumber\\
 U_i^+&=[P,C_i,B_{i+1}],& T&=2s+b_\mu\xi,\nonumber\\
 V_i&=[B_i,A_i,A_{i+1},B_{i+1},C_i],&
 T&=s+a(\xi+\zeta).
 \label{eq:faces}
\end{align}
The collar-side faces
\[
 Q_i=[A_i,D_i,D_{i+1},A_{i+1}]
\]
and exterior are horizontal at \(T=2+s\).
The annular collar \(7/2\le\rho_c\le4\) has resting plane orientation.
Its height varies with \(s\).

\begin{lemma}[Uniform coefficients and cap separation]
\label{lem:bounds}
For every \(\mu\in I\),
\begin{equation}
 0<b_\mu\le\frac35<a,\qquad
 \frac{129}{200}<a<\frac{137}{200},\qquad
 \frac1{a-b_\mu/2}<\frac{29}{10}<r<\frac72.
 \label{eq:bounds}
\end{equation}
For \(0<s\le1\), the outgoing star is strictly inside the cap:
\[
 \rho_c(C_i)=2Cs=\frac{s}{a-b_\mu/2}<\frac{29}{10}<r.
\]
\end{lemma}

\begin{proof}
Write \(\alpha/2=\pi/4+\varepsilon\), where
\(|\varepsilon|\le\pi/120<11/420\). Then
\[
 a=\frac23(\cos\varepsilon-\sin\varepsilon).
\]
The inequalities
\(\cos\varepsilon\ge1-\varepsilon^2/2\) and
\(|\sin\varepsilon|\le|\varepsilon|\) give
\[
 \frac23\left(1-\frac12(11/420)^2-\frac{11}{420}\right)
 < a <
 \frac23\left(1+\frac{11}{420}\right).
\]
The left bound exceeds \(129/200\), and the right bound is below
\(137/200\). Since \(0<\alpha<\pi\), \(b_\mu>0\), and
\(b_\mu\le3/5<129/200<a\). Consequently,
\[
 \frac1{a-b_\mu/2}<\frac{200}{69}<\frac{29}{10},
 \qquad
 \frac{400}{137}<r<\frac{400}{129}<\frac72.
\]
The remaining claims follow from \cref{eq:scalars}.
\end{proof}

\begin{contract}[Particle realization and germ transport]
\label{contract:particle}
For every \(\mu\in I\) and \(s\in[-1,1]\), the particle constructor
supplies the realization above, with:
\begin{enumerate}
\item embedded spacelike faces and exterior, with physical branch
gradient norms \(2\sqrt2/3\), \(3/5\), and \(0\);
\item positive active cells on each regular epoch;
\item exactly one incidence value, \(s=0\), and the specified incoming
and outgoing germs;
\item the affine charts and prescribed overlap components below;
\item complete accepted (E1)--(E5), including identity regular words
and positive primitive core and collar words;
\item identity abstract transport through \(\Eparticle\) of the
persistent labels specified in \cref{sec:locations}.
\end{enumerate}
This is the realization interface corresponding to VIII-B's persistence
theorem, specializing at \(\mu_0\) to VII-F
\cite[Theorem \texttt{thm:persistence}]{CheungVIIIB},
\cite{CheungVIIF}. The interface is an explicit hypothesis; no
independent validation of all predecessor certificate tables is
asserted here.
\end{contract}

The displayed graph also explains the absence of an additional
branch event. For \(s\le0\), \(h_{\mathrm{lin}}\ge s+k_\mu\).
For \(s>0\), the new triangles are bounded by the vertices \(B_i,C_i\);
\cref{lem:bounds} separates them from the cap.

For reference, the induced areas of compact faces are
\begin{align}
 \Area(V_i)&=
 \begin{cases}
 S_\alpha r^2/6,&s\le0,\\
 \dfrac{S_\alpha}{3}(r^2/2-BCs^2),&s>0,
 \end{cases}\nonumber\\
 \Area(U_i^\pm)&=\frac25S_\alpha BCs^2\quad(s>0),&
 \Area(Q_i)&=\frac12S_\alpha(16-r^2).
 \label{eq:areas}
\end{align}
These follow by multiplying projected areas by the constant
spacelike area factors \(1/3\), \(4/5\), and \(1\), respectively.
The outgoing star lies strictly inside the incoming core triangle,
so the displayed active areas are positive.

\subsection{Finite particle lengths}

Put
\begin{align}
 L_\alpha&=\sqrt{2(1-z_\alpha)},\nonumber\\
 W_\alpha&=
 \sqrt{(C-B)^2+C^2+
       2z_\alpha C(C-B)-b_\mu^2C^2}.
 \label{eq:length-coefficients}
\end{align}
The unrefined finite dictionary has three slots of each of eight types:
\begin{equation}
\begin{array}{c|c|c}
 \text{type}&s<0&s>0\\ \hline
 e_i&r\sqrt{1-a^2}&(r-Bs)\sqrt{1-a^2}\\
 c_i=[A_i,A_{i+1}]&rL_\alpha&rL_\alpha\\
 t_i=[A_i,D_i]&4-r&4-r\\
 o_i=[D_i,D_{i+1}]&4L_\alpha&4L_\alpha\\
 u_i=[P,B_i]&\text{inactive}&Bs\\
 v_i=[P,C_i]&\text{inactive}
 &Cs\sqrt{2(1+z_\alpha)-b_\mu^2}\\
 w_i^-=[B_i,C_i]&\text{inactive}&W_\alpha s\\
 w_i^+=[C_i,B_{i+1}]&\text{inactive}&W_\alpha s
\end{array}
\label{eq:lengths}
\end{equation}
The persistent slot \(e_i\) realizes \([P,A_i]\) before the event and
\([B_i,A_i]\) after it.

Inactive slots are not incoming edges. At a germ, born-slot lengths
have zero outgoing limit but the slots are metadata, not additional
active zero-size cells. Their zero reports are coding conventions,
not intrinsic incoming derivatives.

\subsection{Charts and ordered words}

For a spatial gradient \(g\), \(|g|<1\), let
\begin{equation}
 \mathcal B(g)=
 \begin{pmatrix}
 \gamma&-\gamma g^{\mathsf T}\\
 -\gamma g&I_2+(\gamma-1)gg^{\mathsf T}/|g|^2
 \end{pmatrix},
 \qquad \gamma=(1-|g|^2)^{-1/2},
 \label{eq:boost}
\end{equation}
with \(\mathcal B(0)=I_3\).
In the first sector,
\[
 g_h=\begin{pmatrix}a\\a\tan(\alpha/2)\end{pmatrix},\quad
 g_-=\begin{pmatrix}0\\b_\mu/S_\alpha\end{pmatrix},\quad
 g_+=\begin{pmatrix}b_\mu\\-b_\mu z_\alpha/S_\alpha\end{pmatrix}.
\]
Their norms are \(2\sqrt2/3,3/5,3/5\), and their Lorentz factors are
\(3,5/4,5/4\).

With \(O_i=\diag(1,\Rot(i\alpha))\), the adapted charts are
\begin{equation}
\begin{array}{c|l}
 \text{face}&\chart_f(z)\\ \hline
 V_i&O_i\mathcal B(g_h)O_i^{-1}z-3s e_0\\
 U_i^-&O_i\mathcal B(g_-)O_i^{-1}z-\frac52s e_0\\
 U_i^+&O_i\mathcal B(g_+)O_i^{-1}z-\frac52s e_0\\
 Q_i,\ \text{exterior}&z
\end{array}
\label{eq:charts}
\end{equation}
The first three rows flatten their plane intercepts to zero.
The identity charts on horizontal faces retain the constant face
height \(2+s\); adaptation does not require the same intercept in
every chart.

Canonical overlaps are \cref{eq:canonical}. For a closed ordered word
based in \(f_0\), telescoping gives
\begin{equation}
 \Hol(\gamma)=
 \chart_{f_0}J(\mu)^{N(\gamma)}\chart_{f_0}^{-1}.
 \label{eq:word}
\end{equation}
The integer \(N(\gamma)\) is supplied by the winding evidence, not
inferred from the matrix alone. For example, at \(\mu_0=\pi/2\),
different windings can have the same rotational matrix.

Within an epoch, linear chart parts remain fixed, while translations
and plane intercepts change. Thus orientations remain fixed, not
literal spacetime locations.

\subsection{The separate four-face profile}

Profile \(\Aprof\) uses the four-face patch of VII-E at the ambient
packet \(\Pi_{\mu_0}\) \cite{CheungVIIE}. In local coordinates \((x,y)\),
set
\[
 p_A=\frac35,\quad q_A=\frac5{13},\quad
 a_A=\frac45,\quad b_A=\frac{12}{13},\quad
 \delta_A=b_A-a_A=\frac8{65},
\]
and
\begin{align}
 f_E&=p_Ax+a_A\sigma,& f_W&=-p_Ax+a_A\sigma,\nonumber\\
 f_N&=q_Ay+b_A\sigma,& f_S&=-q_Ay+b_A\sigma,\nonumber\\
 H_\sigma&=\min\{2+\sigma,\max(f_E,f_W,f_N,f_S)\}.
 \label{eq:A-graph}
\end{align}
The rectangle is
\[
 [-14/3,14/3]\times[-6,6],
\]
placed by \((X,Y)=(20+x,20+y)\), away from the particle and seam.
The central lengths are
\begin{equation}
 \ell_{EW}=-\frac{16}{25}\sigma\quad(\sigma<0),
 \qquad
 \ell_{NS}=\frac{16}{39}\sigma\quad(\sigma>0).
 \label{eq:A-central}
\end{equation}
Each of four mixed arms has length
\begin{equation}
 \ell_{\mathrm{arm}}=
 \begin{cases}
 (10+\sigma)\sqrt{1921}/75,&\sigma<0,\\
 (26+\sigma)\sqrt{1921}/195,&\sigma>0.
 \end{cases}
 \label{eq:A-arms}
\end{equation}
The inner cap rectangle has half-widths
\begin{equation}
 X_\sigma=\frac{10+\sigma}{3},\qquad
 Y_\sigma=\frac{26+\sigma}{5}.
 \label{eq:A-cap}
\end{equation}

For a fixed corner-spoke subdivision of its horizontal annulus, each
spoke to the corresponding outer corner has length
\[
 \sqrt{(14/3-X_\sigma)^2+(6-Y_\sigma)^2}
 =\frac{\sqrt{34}}{15}(4-\sigma).
\]
The cap sides have lengths \(2X_\sigma,2Y_\sigma\), and outer rectangle
sides have lengths \(28/3,12\). These supply the finite patch length
dictionary. Any additional ambient dictionary is part of the following
interface and is not silently identified with the particle dictionary.

\begin{contract}[Four-face realization and exchange]
\label{contract:A}
For \(\sigma\in[-1,1]\), the installed profile-\(\Aprof\) constructor
supplies:
\begin{enumerate}
\item an embedded spacelike capped graph, with slope bound \(3/5\),
the displayed patch dictionary, and a complete declared ambient
finite dictionary with affine within-epoch length laws;
\item exactly one four-face adjacency exchange, at \(\sigma=0\),
with positive noncentral cells;
\item adapted affine charts, identity contractible patch words,
and complete accepted ambient (E1)--(E5);
\item a certified boundary interface with its resting exterior,
with particle and seam outside the moving incidence patch;
\item fixed ambient packet \(\Pi_{\mu_0}\), fixed exterior particle
incidence, and certified original-name transport through the exchange.
\end{enumerate}
The transport may identify junction labels. A finite implementation
must retain enough input information to reverse that identification.
No source command is admitted in this profile.
\end{contract}

The interface includes the required ambient matching, not merely
identity words inside the contractible patch. A resting exterior
orientation does not imply that every exterior slice point is
stationary.

\section{Derived fixed-packet flow}
\label{sec:flow}

\subsection{Correlated length rates}

For local flow time \(t=\tau-\tau_a\), write
\begin{equation}
 \ell_e(t)=\ell_e(0)+tv_e,\qquad
 \beta_e^{\mathrm{rate}}=\arsinh(v_e).
 \label{eq:flow-law}
\end{equation}
Chosen units make the rate parameter dimensionless. It is not
identified with a Lorentz rapidity between face frames.

For particle slice advance \(s=s_a+v_st\), \(v_s\ge0\), incoming active
finite lengths have zero rates. In the outgoing epoch,
\begin{equation}
\begin{array}{c|c}
 \text{type}&v_e\\ \hline
 e_i&-v_sB\sqrt{1-a^2}\\
 u_i&v_sB\\
 v_i&v_sC\sqrt{2(1+z_\alpha)-b_\mu^2}\\
 w_i^-,w_i^+&v_sW_\alpha
\end{array}
\label{eq:particle-rates}
\end{equation}
and \(c_i,t_i,o_i\) are frozen.

For example, on \(e_i=[B_i,A_i]\),
\[
 \Delta x=(r-Bs)e_i,\qquad
 \Delta T=(2+s)-2s=2-s=a(r-Bs).
\]
Hence
\[
 \ell(e_i)
 =\sqrt{|\Delta x|^2-(\Delta T)^2}
 =(r-Bs)\sqrt{1-a^2},
\]
and its rate is the first row of \cref{eq:particle-rates}.
On \(u_i=[P,B_i]\), both endpoints have time \(2s\), so
\(\ell(u_i)=Bs\). The shrinking and growing rates are correlated by
the same vertices and parameter; they are not independently assigned.

For \(\sigma=\sigma_a+v_At\), \(v_A\ge0\), the four-face central rates are
\[
 -\frac{16}{25}v_A,\qquad \frac{16}{39}v_A
\]
on their respective epochs. The arm rates are
\[
 \frac{v_A\sqrt{1921}}{75},\qquad
 \frac{v_A\sqrt{1921}}{195}.
\]
Cap-side rates are \(2v_A/3,2v_A/5\), corner-spoke rates are
\(-v_A\sqrt{34}/15\), and outer rectangle sides are frozen.
Additional ambient rates are the declared laws of
\cref{contract:A}.

\subsection{A separately installed collar module}
\label{sec:collar}

Fix \(\mu\in I\) and a regular \(s=s_a\).
In an already refined collar representation, let
\begin{equation}
 E_i(t_\col)=R_\col(t_\col)e_i,\qquad
 T(E_i)=2+s_a,\qquad
 R_\col(t_\col)=\frac{15}{4}+\frac{t_\col}{8},
 \quad -1\le t_\col\le1.
 \label{eq:collar-flow}
\end{equation}
Replace each \(Q_i\), in this installed representation, by
\[
 Q_i^{\mathrm{in}}=[A_i,E_i,E_{i+1},A_{i+1}],\qquad
 Q_i^{\mathrm{out}}=[E_i,D_i,D_{i+1},E_{i+1}].
\]
For \(t_\col=t_{\col,a}+v_\col(\tau-\tau_a)\),
\begin{equation}
\begin{array}{c|c|c}
 \text{edge}&\ell_e&d\ell_e/d\tau\\ \hline
 [A_i,E_i]&R_\col-r&v_\col/8\\
 [E_i,D_i]&4-R_\col&-v_\col/8\\
 [E_i,E_{i+1}]&R_\col L_\alpha&v_\col L_\alpha/8
\end{array}
\label{eq:collar-lengths}
\end{equation}
with multiplicity three per row. All other edges are frozen.

\begin{lemma}[Collar realization]
\label{lem:collar}
The collar module is embedded and nondegenerate on its full window,
has no incidence event, and preserves the installed packet and
reference-frame signature. Its embedded slice and particle intersection
are unchanged.
\end{lemma}

\begin{proof}
By \cref{lem:bounds},
\[
 r<\frac72<\frac{29}{8}\le R_\col\le\frac{31}{8}<4.
\]
Thus \(R_\col-r>1/8\) and \(4-R_\col\ge1/8\).
Ring lengths are positive because \(0<\alpha<\pi\).
The horizontal areas are
\[
 \frac12S_\alpha(R_\col^2-r^2),\qquad
 \frac12S_\alpha(16-R_\col^2),
\]
both positive. The two refined faces partition exactly the original
horizontal face, so their union has not moved.

All new faces have the identity reference chart. Ordinary transitions
are identities and seam transitions remain \(J(\mu)\). Shared segments,
regular words, and primitive core--collar matching are therefore
unchanged. Since \(s_a\) is fixed, so is \(P_{s_a}\). These checks give
the complete matching output.
\end{proof}

Adjacent collar faces have zero relative boost, although their
subdivision lengths may have nonzero rates. This separates
\(\beta_e^{\mathrm{rate}}\) from a canonical boost variable.

\begin{ctheorem}[Certified fixed-packet flow]
\label{thm:C1}
Relative to \cref{contract:particle,contract:A}, the following are
spectator-blind geometric commands:
\begin{enumerate}
\item particle-family slice advance at fixed \(\mu\in I\), within one
incidence epoch;
\item four-face advance within one epoch of its installed window;
\item collar advance at fixed regular \(s\), within its installed
\(t_\col\)-window.
\end{enumerate}
Their represented lengths have the derived affine laws
\cref{eq:flow-law,eq:particle-rates,eq:collar-lengths} and the declared
four-face laws.

On every compact regular subinterval, active finite lengths and compact
face areas have positive minima, and no additional collision occurs.
Particle slice advance may end at the incoming germ or begin at the
outgoing germ. Collapsed born slots there remain unoccupiable metadata.

Every output returns complete accepted \(\Ein\), with the same installed
packet and reference-frame signature. No flow reads internal matter,
occupancy, a source token, or spectator number.
\end{ctheorem}

\begin{proof}
Substitution of the affine parameter law into \cref{eq:lengths}
gives the particle rates. The preceding endpoint calculation derives
representative shrinking and growing laws directly.
The four-face and collar laws follow by the same substitution.

Positive continuous lengths and areas on a regular epoch have positive
minima on compact subintervals. The cap-separation estimate of
\cref{lem:bounds} excludes contact of the particle star with the cap.
The four-face interface excludes noncentral collisions, and
\cref{lem:collar} proves the collar assertion.

For particle flow, chart translations update by \cref{eq:charts};
canonical overlaps telescope to \cref{eq:word}. Regular words have
winding zero, and positive primitive core and collar words have winding
one. Transport to the reference frame therefore yields \(J(\mu)\).
The inherited interface supplies the remaining realization and
worldline checks. The ambient worldline stays fixed even though its
slice intersection changes.

The four-face interface supplies complete ambient matching, not only
patch identities. The collar matching was checked directly.
In every case, the rule depends only on declared geometric data,
parameters, resources, and transport. The inaccessible matter and
spectator data are not read.
\end{proof}

Theorem C1 is kinematics within certified realizations. It does not
derive a Hamiltonian, prescribe a relational lapse, or impose equal
proper-time increments in different face frames.

\section{Guarded incidence and location transport}
\label{sec:events}

\begin{ctheorem}[Restricted catalogue and isolation]
\label{thm:C2}
Within the admitted windows:
\begin{enumerate}
\item the unrefined particle slice module has only
\(\Eparticle\), at \(s=0\), with the installed packet frozen;
\item the four-face profile has only \(\EA\), at \(\sigma=0\),
with the particle outside the moving incidence patch and its
packet frozen;
\item regular within-epoch flow introduces no additional incidence;
\item the collar module has no incidence event on its full window.
\end{enumerate}
This is isolation in the prescribed families, not genericity among
arbitrary slicings.
\end{ctheorem}

\begin{proof}
The particle constructor supplies the two decompositions and their
germs. Born vertices are proportional to \(s\), their nonzero length
coefficients are positive by spacelikeness, and the star remains
strictly inside the cap. The supplied incidence value is therefore
the sole coalescence in this window.

In profile \(\Aprof\), each central length in
\cref{eq:A-central} has its sole zero at \(\sigma=0\).
Noncentral cells remain positive by the four-face interface.
Within-epoch flow preserves these incidence diagrams.
The collar conclusion is \cref{lem:collar}.
\end{proof}

A particle hop, two-particle collision, or another classical transition
does not enter this catalogue merely by being named. It requires its
own realization and full output interface.

\subsection{Persistent labels and germ metadata}
\label{sec:locations}

The particle persistent labels are
\[
 P,\ A_i,\ D_i,\ e_i,\ c_i,\ t_i,\ o_i,\ V_i,\ Q_i
\]
and the declared exterior labels. Their abstract transports through
\(\Eparticle\) are identities.

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.32\linewidth}p{.22\linewidth}X@{}}
\toprule
Label class & At the particle germ & Location convention\\
\midrule
Persistent labels above
 & Valid persistent limiting cells
 & Valid on both sides; identity abstract transport\\
\(B_i,C_i,U_i^\pm,u_i,v_i,w_i^\pm\)
 & Outgoing germ metadata
 & Not incoming locations; active only for \(s>0\)\\
Collapsed new slots as zero-size cells
 & Not active cells
 & Not occupiable at the germ\\
Four-face junction labels
 & Separate profile
 & Use its certified transport; retain input information if needed\\
\bottomrule
\end{tabularx}
\end{center}

The identity label \(e_i\) changes realization from \([P,A_i]\) to
\([B_i,A_i]\). Identity abstract transport is not a stationary developed
trajectory or unchanged incidence relation.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[scale=.75,every node/.style={font=\small}]
 \begin{scope}[xshift=-3cm]
  \fill (0,0) circle(2pt);
  \node[below] at (0,0) {$P$};
  \foreach \a in {30,150,270}{
   \draw[thick] (0,0)--(\a:1.3);
   \fill (\a:1.3) circle(1.5pt);
  }
  \node at (0,1.8) {incoming: three links};
 \end{scope}
 \begin{scope}[xshift=3cm]
  \fill (0,0) circle(2pt);
  \node[below] at (0,0) {$P$};
  \foreach \a in {0,60,120,180,240,300}{
   \draw[thick] (0,0)--(\a:1.3);
   \fill (\a:1.3) circle(1.5pt);
  }
  \node at (0,1.8) {outgoing: six links};
 \end{scope}
 \draw[-{Latex},thick] (-.9,0)--(.9,0)
   node[midway,above] {$\Eparticle$};
\end{tikzpicture}
\caption{Schematic link diagrams, not developed coordinates.
The incidence command holds the installed deficit fixed.}
\end{figure}

\section{Legal words and complete-output induction}
\label{sec:schedules}

\begin{definition}[Legal schedule]
\label{def:legal}
A legal schedule is a finite chronological word in
\[
 \{\Gflow,\Eparticle,\EA,\Stab\}
\]
with declared parameters, modules, and supports, satisfying:
\begin{enumerate}
\item every input lies in its command's certified domain;
\item flow preserves the packet and module and does not cross an
incidence value internally;
\item source replacement satisfies its regular boundary, stage,
token-promise, module, and source-permission conditions;
\item incidence satisfies its exact profile, germ-mode, clock, and
incidence-permission conditions;
\item no command combines source replacement and incidence;
\item all commands after source replacement use the newly installed
packet, charts, collar, realization, and report dictionary;
\item original names occupy valid input locations and receive their
certified transport;
\item no unsupported profile or module conversion occurs;
\item every output satisfies all realization requirements and
(E1)--(E5), including \cref{prop:complete-output}.
\end{enumerate}
Canonical code predicates concern finite indices, exact clocks,
reports, permissions, locations, and stages. The token promise is
instead a hypothesis about the matter state, not a circuit-tested
predicate.
\end{definition}

\begin{ctheorem}[Scheduling and complete-output induction]
\label{thm:C3}
Every legal schedule returns complete accepted \(\Ein\) and valid
original-name locations after every command. Geometric commands
preserve the currently installed packet and reference-frame signature.
A source command returns the complete newly installed packet and
signature.

Under \cref{ass:control}, two aligned legal executions with equal
initial geometric/controller inputs, resources, permissions, declared
accessible inputs, and original matter reductions have equal
geometric outputs, controller outputs, and original-name transports.
Their original matter reductions remain equal. Inaccessible spectator
extensions do not change geometric decisions.

At every command boundary, the incomplete outputs of
\cref{prop:complete-output} are rejected. This conclusion is not
restricted to the initial reference packet.
\end{ctheorem}

\begin{proof}
Proceed by induction from the certified initial configuration.
A flow output is complete by Theorem C1. An incidence output is
complete by the appropriate realization/germ interface and Theorem C2,
and uses the certified transport in \cref{sec:locations}.

At \(\Stab\), the regular parameter remains \(s=-1/2\).
Evaluation of the particle constructor at the promised table entry
returns the full new realization, including new rays, seam, charts,
collar, and matching data. Persistent incoming labels remain valid.
Only the source permission and source stage change in addition to
the installed geometric data. Legality places this complete output
in the next command's domain. This proves certificate and location
induction.

For paired executions, deterministic control selects equal commands,
guards, geometric outputs, transports, and successor controller states
from equal declared inputs. Every T0 command is the identity on internal
matter. For a spectator extension \(\widetilde\rho\) with original
reduction \(\rho\),
\[
 \Tr((B\otimes I_N)\widetilde\rho)=\Tr(B\rho)
\]
for each accessible original observable \(B\).
Equal original reductions thus give equal accessible functionals.
The classical tokens are separately required to agree and satisfy
their promises; no extraction step is used. Separately reserved
spectator storage prevents a spectator-number capacity guard.

Finally, acceptance includes the complete obligations after each
command. The local calculations in \cref{prop:complete-output} apply
at the packet installed at that boundary, including after a source
replacement.
\end{proof}

A hybrid word need not preserve its initial signature from start to
finish. Its geometric subwords preserve whichever packet is currently
installed; a source command can change that packet.

\section{An explicit promised sourced history}
\label{sec:hybrid}

\subsection{Finite packet table}

For \(q\ge6\), define
\begin{equation}
 K_q=\left\lfloor\frac{2^q}{40}\right\rfloor,\quad
 \cT_q=\{-K_q,\ldots,K_q\},\quad
 t_m=m2^{-q},\quad
 \mu_m=\mu_0+6\arctan t_m.
 \label{eq:table}
\end{equation}
This is the promised table of VIII-B
\cite[Theorem \texttt{thm:finite}]{CheungVIIIB}.
Every entry lies in \(I\), since
\[
 |\mu_m-\mu_0|
 \le6\arctan(1/40)<3/20<\pi/20.
\]
At \(q=6\), \(\cT_6=\{-1,0,1\}\).

For \(\alpha_m=(2\pi-\mu_m)/3\),
\begin{align}
 z_{\alpha_m}&=\frac{2t_m}{1+t_m^2},&
 S_{\alpha_m}&=\frac{1-t_m^2}{1+t_m^2},\nonumber\\
 a_m&=\frac23\frac{1+t_m}{\sqrt{1+t_m^2}},&
 b_m&=\frac35\frac{1-t_m^2}{1+t_m^2}.
 \label{eq:algebraic}
\end{align}
The rays and seam matrices have rational entries; the finite coordinates,
charts, and lengths are algebraic. The real number \(\mu_m\) itself is
specified symbolically, not approximated by a floating-point angle.
Exact derivations appear in \cref{app:checks}.

\subsection{Initial state, token, and permissions}

Use a surviving source qubit with
\[
 \cA_\src=\Lin(\CC^2),\qquad B_\src=Z,\qquad \kappa=1.
\]
For the \(q=6\) history, supply token \(m_\src=1\), and take
\begin{equation}
 z_+=6\arctan(1/64),\qquad
 \rho_+=\frac12(I+z_+Z).
 \label{eq:source-state}
\end{equation}
Because \(0<z_+<6/64<1\), \(\rho_+\) is a state and
\[
 \Tr(\rho_+Z)=z_+,\qquad \mu_0+\Tr(\rho_+Z)=\mu_1.
\]
The same word is defined on every state satisfying this fixed-token
promise, including compatible spectator extensions.

Initially install \(\Pi_{\mu_0}\), independently of the source state,
and set
\[
 s=-1,\quad \nu_G=\nu_\src=1,\quad
 \Module=\mathsf{slice},
\]
with incoming mode and source stage zero.
All original names occupy valid persistent incoming labels.
Choose \(d\ge1\), \(\Delta\tau=1\), and slots \(k=0,\ldots,5\).
Each flow segment has \(v_s=1/2\).

The chronological word is
\begin{equation}
 W=
 \mathsf G_{-1\to-1/2}\ ;
 \Stab\ ;
 \mathsf G_{-1/2\to0^-}\ ;
 \Eparticle\ ;
 \mathsf G_{0^+\to1/2}.
 \label{eq:hybrid}
\end{equation}

\subsection{Installed packet and final lengths}

After source replacement,
\begin{align}
 \mu_1&=\frac{\pi}{2}+6\arctan(1/64),\nonumber\\
 a_1&=\frac{130}{3\sqrt{4097}},&
 b_1&=\frac35\frac{4095}{4097},\nonumber\\
 r_1&=\frac{3\sqrt{4097}}{65},&
 B_1&=\frac{3\sqrt{4097}}{130},&
 C_1&=\frac1{2a_1-b_1}.
 \label{eq:installed}
\end{align}
The ray generator is
\begin{equation}
 e^{i\alpha_1}=\frac{128+4095i}{4097}.
 \label{eq:ray}
\end{equation}
Every subsequent face, chart, overlap, and report uses these installed
coefficients rather than the reference table.

At \(s=1/2\),
\[
 T(P)=1,\quad B_i=\frac{B_1}{2}e_i,\quad
 C_i=\frac{C_1}{2}(e_i+e_{i+1}),\quad
 T(A_i)=T(D_i)=\frac52,
\]
and
\begin{align}
 \ell(e_i)&=\frac34r_1\sqrt{1-a_1^2},&
 \ell(u_i)&=\frac{B_1}{2},\nonumber\\
 \ell(v_i)&=\frac{C_1}{2}
   \sqrt{2(1+z_{\alpha_1})-b_1^2},&
 \ell(w_i^\pm)&=\frac{W_{\alpha_1}}2.
 \label{eq:final-lengths}
\end{align}

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}cllllX@{}}
\toprule
Step & Command & \(s\) & Packet & \((\nu_G,\nu_\src)\)
 & Certificate and intersection\\
\midrule
0 & Initial & \(-1\) & \(\mu_0\) & \((1,1)\)
 & Reference realization; \(P=(-1,0,0)\)\\
1 & \(G\) & \(-1/2\) & \(\mu_0\) & \((1,1)\)
 & Regular source input; \(P=(-1/2,0,0)\)\\
2 & \(\Stab\) & \(-1/2\) & \(\mu_1\) & \((1,0)\)
 & Complete new family; \(P=(-1/2,0,0)\)\\
3 & \(G\) & \(0^-\) & \(\mu_1\) & \((1,0)\)
 & Incoming germ; \(P=(0,0,0)\)\\
4 & \(\Eparticle\) & \(0^+\) & \(\mu_1\) & \((0,0)\)
 & Outgoing germ, frozen packet; \(P=(0,0,0)\)\\
5 & \(G\) & \(1/2\) & \(\mu_1\) & \((0,0)\)
 & Outgoing realization; \(P=(1,0,0)\)\\
\bottomrule
\end{tabularx}
\end{center}

\begin{ctheorem}[The five-command T0 history]
\label{thm:C4}
The data above make \cref{eq:hybrid} a legal five-command history.
Every command returns complete accepted \(\Ein\). Its only deficit
change occurs at the regular source command \(s=-1/2\).
The particle-link event occurs at the frozen installed deficit \(\mu_1\).

Internal matter is unchanged, persistent original locations remain
valid under identity abstract transport, and permissions have exactly
the displayed trace. The uniform particle branch-gradient bound is
\(2\sqrt2/3<1\).
\end{ctheorem}

\begin{proof}
The first flow remains in the incoming reference epoch and ends at
the regular source boundary. The token satisfies the promise,
source stage is zero, and source permission is one. Thus \(\Stab\)
installs the complete \(\mu_1\) family at the same parameter, decreases
only source permission, and changes source stage to one.

The next flow reaches the incoming germ of the installed family.
The incidence guard is exact, incidence permission is one, and
\(\Eparticle\) holds \(\mu_1\) fixed while selecting the outgoing germ.
The final flow lies in the outgoing epoch.
Theorem C3 supplies all matching and location conclusions.
\end{proof}

For source-off token \(m_\src=0\) with \(\Tr(\rho Z)=0\), the analogous
source command returns the reference family. Source-off does not mean
identity on an arbitrary nonreference initial packet.

\paragraph{Finer-table encoding of the same history.}
The symbol \(\mu_1\) above denotes the \(q=6\) entry.
At \(q>6\), the identical physical packet has index
\begin{equation}
 m_\star=2^{q-6},\qquad t_{m_\star}=1/64.
 \label{eq:finer}
\end{equation}
Since \(m_\star\le2^q/40\) and is an integer, \(m_\star\le K_q\).
Finer encodings of this history use token and installed index
\(m_\star\), not index one.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[x=1.45cm,y=1.1cm,
 every node/.style={font=\small}]
 \draw[-{Latex}] (-.2,-1.55)--(5.35,-1.55) node[right] {$k$};
 \foreach \k in {0,...,5}{
  \draw (\k,-1.6)--(\k,-1.5);
  \node[below] at (\k,-1.6) {\k};
 }
 \draw[-{Latex}] (-.5,-1.1)--(-.5,.9) node[above] {$s$};
 \foreach \y/\lab in {-1/{-1},-.5/{-1/2},0/{0},.5/{1/2}}{
  \draw (-.55,\y)--(-.45,\y);
  \node[left] at (-.55,\y) {$\lab$};
 }
 \draw[very thick] (0,-1)--(1,-.5)--(2,-.5)--(3,0)--(4,0)--(5,.5);
 \foreach \x/\y in {0/-1,1/-.5,2/-.5,3/0,4/0,5/.5}{
  \fill (\x,\y) circle(2pt);
 }
 \node[above] at (.5,-.7) {$G$};
 \node[above] at (1.5,-.5) {$\Stab$};
 \node[above] at (2.5,-.25) {$G$};
 \node[above] at (3.5,0) {$\Eparticle$};
 \node[above] at (4.5,.25) {$G$};
 \node[below] at (3,-.04) {$0^-$};
 \node[below] at (4,-.04) {$0^+$};
\end{tikzpicture}
\caption{Execution and slicing clocks are distinct.
Horizontal source and incidence steps are discrete commands;
the drawing asserts no continuous mixed interpolation.}
\end{figure}

\section{Prescribed schedule support}
\label{sec:support}

A geometric word does not itself specify a local quantum substrate.
For matter computations between geometric commands, independently
supply a finite metric site space \((X,d_X)\), site spaces \(\cH_x\),
and
\[
 \cH_X=\bigotimes_{x\in X}\cH_x,\qquad
 \cA(Y)=\Lin(\cH_Y)\otimes I_{X\setminus Y}.
\]
Every classical controller or record wire read by a local operation
must be included in its declared site interface. All channels in
this section are unconditional; selective normalized updates are
not substituted for trace-preserving channels.

\begin{lemma}[Regional determination is adjoint inclusion]
\label{lem:regional}
Let \(\Psi\) be a channel on the finite site interface, and let
\(N,Y\subseteq X\). Its output reduction on \(Y\) is determined by
the input reduction on \(N\), for all input states, if and only if
\[
 \Psi^*(\cA(Y))\subseteq\cA(N).
\]
The implication from inclusion remains valid with an untouched
reference, provided equality is imposed on the joint
\(N\)--reference input reductions.
\end{lemma}

\begin{proof}
The inclusion makes every output expectation an input expectation
of an operator in \(\cA(N)\), so equal reductions give equal outputs.

Conversely, suppose a Hermitian output observable has pullback
\(B\notin\cA(N)\). Let \(K\ne0\) be the Hermitian Hilbert--Schmidt
orthogonal component of \(B\) outside \(\cA(N)\). Since the identity
belongs to \(\cA(N)\), \(\Tr K=0\). For sufficiently small
\(\varepsilon>0\),
\[
 \rho_\pm=I/\dim\cH_X\pm\varepsilon K
\]
are states with equal reductions on \(N\), but their tested output
expectations differ by
\(2\varepsilon\Tr(KB)=2\varepsilon\Tr(K^2)>0\).
This contradicts determination. Hermitian observables span the algebra.

Tensoring the inclusion with the reference matrix algebra proves
the final assertion.
\end{proof}

This is the local comparison underlying the schedule statements in
VII-E, En05-A, En03-A, and the algebraic formulation of En05-C
\cite{CheungVIIE,CheungEn05A,CheungEn03A,CheungEn05C}.
No result from those manuscripts is needed as a premise of the proof.

\begin{ctheorem}[Backwards support for prescribed block channels]
\label{thm:C5}
For ticks \(r=1,\ldots,T\), let \(\cB_r\) be prescribed partitions
of \(X\), with \(\diam(B)\le\Delta\), and let
\[
 \Phi_r=\bigotimes_{B\in\cB_r}\Phi_{r,B},
\]
where every \(\Phi_{r,B}\) is CPTP on its fixed block interface.

For \(0\le u\le t\le T\), define
\begin{align}
 N_{t,t}(Y)&=Y,\nonumber\\
 N_{u,t}(Y)&=
 \bigcup\{B\in\cB_{u+1}:B\cap N_{u+1,t}(Y)\ne\varnothing\}
 \quad(u<t).
 \label{eq:backwards}
\end{align}
Then
\begin{equation}
 (\Phi_{u+1}^*\circ\cdots\circ\Phi_t^*)(\cA(Y))
 \subseteq\cA(N_{u,t}(Y)),
 \label{eq:support}
\end{equation}
and
\begin{equation}
 N_{u,t}(Y)\subseteq
 \{x:d_X(x,Y)\le(t-u)\Delta\}.
 \label{eq:metric-support}
\end{equation}
Equal input reductions on \(N_{u,t}(Y)\) give equal output reductions
on \(Y\). With an untouched finite-dimensional reference \(R\),
equal joint input reductions on \(N_{u,t}(Y)\cup R\) give equal
joint output reductions on \(Y\cup R\).

The scheduled ancestor set is an upper bound on dependence. It need
not be the minimal influence region and is not identified with a
spacetime causal past.
\end{ctheorem}

\begin{proof}
At one backwards tick, a block disjoint from the current support
acts on an identity factor. Its adjoint is unital and preserves that
identity. All remaining support can enlarge only within the union
of blocks meeting the current support. Backwards induction gives
\cref{eq:support}.

A point added at one step is within \(\Delta\) of a point in the
previous backwards set. The triangle inequality proves
\cref{eq:metric-support}. For \(Y=\varnothing\), all backwards sets
are empty and the conclusion is trivial.

The reduced-state conclusions follow from
\cref{lem:regional}. In the referenced version, block adjoints act
as identities on \(R\), so the pullback lies in
\(\cA(N_{u,t}(Y))\otimes\Lin(\cH_R)\).
Equality of the joint input reduction is the required hypothesis;
equality on the ancestor sites alone is insufficient for a joint
referenced conclusion.
\end{proof}

\subsection{Ancestry, influence, and physical cones}

If a prescribed two-site block channel is the identity, the schedule
can include both sites in \(N_{u,t}(Y)\), although an observable on
one site pulls back to that site alone. Scheduled reachability
therefore does not imply actual influence. This is the upper-bound
distinction emphasized in En05-A and VII-J
\cite[Theorem \texttt{thm:cone}]{CheungEn05A},
\cite[Theorem \texttt{thm:release}]{CheungVIIJ}.

A physical directional-containment theorem would additionally require
an independently certified map from the timed site network into the
conical spacetime, compatible with every communication operation.
It would need, for example, a proved inclusion of each embedded
ancestor region in the appropriate geometric causal past.
No such embedding is part of Theorem C5.
VII-J's physical containment is conditional on that additional
certification; it does not discharge the present interface
\cite{CheungVIIJ}.

\paragraph{A scalar speed comparison is not directional containment.}
On a square lattice of spacing \(a_0\), a cardinal hop or wait per tick
of duration \(\tau_0\) has asymptotic reachable velocities
\[
 |v_x|+|v_y|\le a_0/\tau_0.
\]
The vector
\[
 \left(\frac{a_0}{\sqrt2\,\tau_0},
       \frac{a_0}{\sqrt2\,\tau_0}\right)
\]
has Euclidean norm \(a_0/\tau_0\), but is outside that diamond.
A scalar norm bound does not provide a missing directional inclusion.

\paragraph{Channel commutation is not algebraic microcausality.}
The equality \(\Phi\Psi=\Psi\Phi\) is a statement about operations.
The relation
\[
 [\cA(O_1),\cA(O_2)]=0
\]
for spacelike regions is a statement about a separately defined
observable net and its spacetime assignment.
The former alone does not define or prove the latter.
Indeed, identity channels commute on a qubit, although its observables
\(X,Z\) do not commute. En06-A accordingly requires a matching
observable-limit hypothesis for its algebraic microcausality statement
\cite[Theorem \texttt{thm:algmicro}]{CheungEn06A}.

Theorem C5 cannot be applied across an arbitrary state-dependent source
law by calling that law a local channel. Nor does spectator-blind
geometric control certify a local controller envelope.
Feedback must include every read control in a separately specified
local hybrid channel, as distinguished in VII-G
\cite[Theorem \texttt{thm:locality}]{CheungVIIG}.

\section{Finite reversible implementation}
\label{sec:implementation}

\subsection{Finite dictionaries and canonical reports}

Fix finite packet tables, parameter grids, controller widths,
location dictionaries, and a prescribed finite word.
A finite word in a continuum family, without such an encoding,
does not by itself define a finite implementation problem.

For particle codes, exact lengths determine canonical reports
\begin{equation}
 Q_e=\lfloor2^b\ell_e\rfloor,\qquad b\ge0.
 \label{eq:reports}
\end{equation}
Inactive incoming born slots have canonical report zero.
At a germ, their limiting reports are zero and germ mode distinguishes
the two codes.

Packet index, exact clock, module, and mode reconstruct the exact
realization and complete matching data. Reports are redundant
consistency data, not independently variable lengths.
Exact algebraic comparisons determine their floors, including equality
cases. The particle bounds in \cref{app:checks} show that \(b+4\)
bits suffice for each of the 24 reports.

Four-face and collar codes use their own installed dictionaries.
They are not silently stored in the unrefined hybrid block.
Unbounded exterior cells add no finite-edge reports to that block.

Retaining classical input information to make a deterministic update
reversible is standard in reversible computation \cite{Bennett1973}.
The following finite synthesis lemma supplies the explicit bound used
here.

\begin{lemma}[Forward-domain permutation completion]
\label{lem:completion}
Let \(H\ge1\), \(D\subseteq\{0,1\}^H\), and
\(f:D\to\{0,1\}^H\) be deterministic. Suppose an allocated stage field
separates designated input and output sets, so that
\[
 I=\{(x,0_H):x\in D\},\qquad
 O=\{(f(x),x):x\in D\}
\]
are disjoint. Put \(N=2H\).

There is a permutation realizing
\begin{equation}
 (x,0_H)\longmapsto(f(x),x)\quad(x\in D).
 \label{eq:completion}
\end{equation}
For \(N\ge3\), NOT, CNOT, and Toffoli gates implement it on clean-work
inputs with at most \(N-2\) additional work bits and
\begin{equation}
 4N(2N-1)|D|
 \label{eq:synthesis}
\end{equation}
elementary gates. Work returns to zero.
For \(N=2\), no work bits are needed and the same bound holds.
An adjacent exchange on a one-bit register, when needed in another
encoding, is a single NOT.
\end{lemma}

\begin{proof}
Retaining \(x\) makes the designated output map injective.
Transpose each disjoint pair
\[
 (x,0_H)\longleftrightarrow(f(x),x)
\]
and fix all other words. This defines a permutation.

For one transposition, choose a shortest Hamming path
\(v_0,\ldots,v_\ell\), with \(\ell\le N\). Its endpoint exchange is
\[
 (v_0v_1)(v_1v_2)\cdots(v_{\ell-1}v_\ell)
 (v_{\ell-2}v_{\ell-1})\cdots(v_0v_1),
\]
using \(2\ell-1\le2N-1\) adjacent-word exchanges.

An adjacent exchange flips one target conditioned on a specified
pattern of all \(N-1\) other logical bits. Surrounding NOTs convert
negative controls, using at most \(2(N-1)\) NOTs.
For \(N\ge3\), a Toffoli chain computes the conjunction into
\(N-2\) clean bits, a CNOT flips the target, and the chain is reversed.
This uses \(2(N-2)\) Toffoli gates and one CNOT.
Thus the total is at most \(4N-5<4N\), with clean work restored.

For \(N=2\), one CNOT, possibly surrounded by two NOTs, suffices.
Multiplication by the path bound and number of transpositions proves
\cref{eq:synthesis}.
\end{proof}

Hamming-path intermediate words need not encode admissible geometries.
The lemma is an existence and synthesis bound, not an efficiency or
physical-time bound.

\subsection{Composite code domains and separate semantic classes}

Let \(F_L\cdots F_1\) denote a prescribed finitely encoded word, with
deterministic canonical code maps. For each command let
\(D_i^{\mathrm{code}}\) be its finite domain.
Define
\begin{equation}
 D_W^{\mathrm{code}}=
 \left\{c:
 c\in D_1^{\mathrm{code}},\
 F_1c\in D_2^{\mathrm{code}},\ldots,
 F_{L-1}\cdots F_1c\in D_L^{\mathrm{code}}
 \right\}.
 \label{eq:composite-domain}
\end{equation}
The domain includes canonical reports, exact clocks, finite indices,
permissions, valid locations, and stage data.

Separately, \(\mathfrak S_W(c)\) is the class of matter states for which
every supplied token along the decoded word satisfies its current
semantic promise. Membership is not a finite circuit test.

Let \(\iota(c)\) append all required clean history and work registers.
Each command receives a fresh history block. Previous histories remain
retained and inaccessible to geometric control. An allocated monotone
schedule-stage field separates each command's designated input and
output sets.

\begin{ctheorem}[Finite reversible composition]
\label{thm:C6}
Fix a finite encoding and a prescribed legal word with deterministic
canonical code maps and certified finite location transports.
There are reversible command circuits \(U_1,\ldots,U_L\), fresh retained
histories, and reusable clean work such that, at every command boundary,
\begin{equation}
 \Dec\bigl(U_i\cdots U_1\iota(c)\bigr)
 =F_i\cdots F_1(\Dec(c)),
 \quad c\in D_W^{\mathrm{code}},\quad 1\le i\le L.
 \label{eq:decoder}
\end{equation}
The decoder ignores implementation histories.

The complete circuits act as the identity on internal matter.
For \(\rho\in\mathfrak S_W(c)\), the decoded outputs are the accepted
outputs of Theorem C3, with valid original locations.
No circuit verifies this semantic membership.

No geometric decoder or admissibility assertion is assigned to
arbitrary off-domain words or intermediate synthesis states.
Theorem C4's word has the bounds in \cref{sec:budget}.
\end{ctheorem}

\begin{proof}
At each boundary, a canonical command has a finite deterministic
output map. Flow changes exact parameters and recomputes canonical
reports. Incidence changes germ mode, permission, and incidence data.
Source replacement evaluates the complete new packet-indexed
representation. None of these maps reads internal matter.

Apply \cref{lem:completion} with stage separation and a fresh geometric
history to every command. Earlier histories are acted on by identities.
Clean work can be reused after uncomputation.

Location transport is a finite deterministic map conditioned on the
geometric input. If it is noninjective, use a separately reserved
history of the input location word and the same completion argument.
The implementation may consult retained geometric input to execute
the already selected location map; this does not make histories
accessible source inputs or permit history-dependent geometric
decisions.

On \cref{eq:composite-domain}, each canonical output is in the next
command's domain, so decoder correctness follows by induction.
Tensoring all code and location gates with the identity on
\(\cH_\Gamma\) preserves internal matter, including its correlations
with untouched references. The semantic promise and Theorem C3 then
give the accepted decoded execution.
\end{proof}

The last hybrid command sets schedule stage and slot to five,
\(s=1/2\), outgoing mode, and permissions \((0,0)\).
It assigns all 24 reports from \cref{eq:lengths}.
Its fresh history retains the preceding geometric word.
The persistent \(e_i\) reports change at every report precision
\(b\ge0\), as proved in \cref{app:checks}; the final command is not
a clock-only update.

\subsection{Explicit hybrid budget}
\label{sec:budget}

Let \(b\ge0\), \(d\ge1\), \(q\ge6\), and
\(r_G,r_\src\ge1\). At \(q>6\), use \cref{eq:finer} to encode the same
physical history.

One current unrefined particle block contains:
\begin{center}
\begin{tabular}{@{}lr@{}}
\toprule
Field & Bits\\
\midrule
Schedule stage \(0,\ldots,5\) & \(3\)\\
Incidence mode & \(1\)\\
Incidence and source permissions & \(r_G+r_\src\)\\
Signed exact slice clock & \(d+2\)\\
Execution slot \(0,\ldots,5\) & \(3\)\\
Current packet index & \(q+2\)\\
Source token & \(q+2\)\\
Fixed identifier metadata & \(11\)\\
Twenty-four canonical finite reports & \(24(b+4)\)\\
\bottomrule
\end{tabular}
\end{center}
The eleven metadata bits are fixed symbolic identifiers, not variable
geometric parameters. Profile and module are fixed as
\(\Pprof,\mathsf{slice}\). Thus
\begin{equation}
 H=24b+d+2q+r_G+r_\src+120.
 \label{eq:H}
\end{equation}
Source stage is decoded from schedule stage \(a\):
\[
 \chi(a)=
 \begin{cases}
 0,&a=0,1,\\
 1,&a=2,3,4,5.
 \end{cases}
\]
The source command has input schedule stage one and source stage zero.
This is an encoding of the same guard, not an enlarged source domain.

Five fresh \(H\)-bit histories, the current \(H\)-bit block, and
\(2H-2\) reusable clean work bits require \(8H-2\) geometric bits.

For this word, all original locations lie in a fixed persistent
dictionary of at most \(64\) entries. The persistent particle labels
and three exterior sector labels need at most \(28\) entries.
Six bits per location suffice, and every transport is the identity.
Reserve one six-bit initial location history per name; it remains
valid throughout the word. For \(n\) original names, including the
surviving source name,
\begin{align}
 M_{\mathrm{hyb}}
 &=8H-2+12n\nonumber\\
 &=192b+8d+16q+8r_G+8r_\src+958+12n.
 \label{eq:M}
\end{align}
The initial location histories can be copied with \(6n\) CNOT gates.
These are computational-basis copies of classical location words,
not copies of unknown quantum matter.

Put \(T_q=2K_q+1\).
At a fixed prescribed stage, clocks, mode, and metadata are fixed,
and reports are determined by the packet.
A conservative input count per geometric command is
\begin{equation}
 Q_{\max}=T_q^2\,2^{r_G+r_\src}.
 \label{eq:Qmax}
\end{equation}
The table factors allow current packet and token to vary independently.
Validity and permission guards only reduce this count.
With \(N=2H\), a sufficient gate bound is
\begin{equation}
 5\cdot4N(2N-1)Q_{\max}+6n.
 \label{eq:gate-budget}
\end{equation}
At \(q=6,d=1,r_G=r_\src=1\),
\[
 H=24b+135,\qquad Q_{\max}=36.
\]
The actual fixed-input history of Theorem C4 has a smaller domain.

The budget excludes internal matter, offline symbolic table preparation,
token acquisition, and continuous geometric interpolation.
It is not a gate-locality, thermodynamic, or physical-time bound.

\section{Rejected interpretations and independent comparisons}
\label{sec:rejected}

\subsection{Mixed commands and incomplete packet outputs}

The fragment
\[
 \text{incoming }s=0\ ;\ \Stab
\]
is illegal: the source command is certified at the regular boundary
\(s=-1/2\), not at the incidence stratum.

An \(\Eparticle\) command that also writes a new source-selected packet
index is not admitted. Its certificate assumes a frozen packet.
An OR guard involving \(\nu_G\) and \(\nu_\src\) does not construct a
common mixed realization.

Source replacement must update the entire realization.
Retaining the reference seam rotation or old charts fails the complete
output contract at a changed deficit. The stale-rotation and
time-translation failures remain distinct by
\cref{prop:complete-output}, in every declared affine frame.

The four-face exchange cannot be turned into a sourced particle event
by enlarging a written support. Its moving incidence diagram remains
away from the particle; identity contractible patch words do not
replace the primitive ambient matching.

\subsection{Code decoding is not a quantum-return decoder}

The map \(\Dec\) in Theorem C6 interprets canonical classical code words
as geometric descriptions and ignores retained histories.
It is not a CPTP decoder from an independently justified restricted
quantum output. The complete reversible permutation can be inverted
when all its registers are retained; this does not certify recovery
from the current code block after histories are discarded.

For example, the completion
\[
 \ket{x}\ket{0}\longmapsto\ket{f(x)}\ket{x}
\]
retains input distinctions in the second register when \(f\) merges
two inputs. Ignoring that register loses those distinctions.
Even a coherent basis-history copy is not a destructive measurement
performed by the scheduling word.

The following standard finite statements make the distinction precise.
They are independent comparison results, not premises of C1--C6.

\begin{proposition}[Full-code recovery and record-only obstruction]
\label{prop:recovery}
Let \(\mathcal N:\Lin(\mathcal H)\to\Lin(\mathcal Q)\) be CPTP.
\begin{enumerate}
\item A CPTP left inverse on the entire input code exists if and only if
a complete Stinespring complementary channel is a replacement channel
\[
 \widehat{\mathcal N}(X)=\Tr(X)\omega
\]
for a fixed state \(\omega\).
Equivalently, for Kraus operators \(E_i\),
every \(E_j^\dagger E_i\) is a scalar multiple of \(I_{\mathcal H}\).
\item Let \(\dim\mathcal H=D\ge2\). If \(\mathcal E\) is
measure-and-prepare and \(\mathcal D\) is any CPTP decoder to
\(\mathcal H\), then in half-diamond normalization,
\begin{equation}
 \frac12\norm{\mathcal D\mathcal E-\id_{\mathcal H}}_\diamond
 \ge1-\frac1D.
 \label{eq:classical-gap}
\end{equation}
In particular, the bound applies when only a finite classical record
is retained. Computational dephasing attains the bound.
\end{enumerate}
The first statement concerns the full input code, not arbitrary
subsystem or subalgebra recovery.
\end{proposition}

\begin{proof}
A standard complement has entries
\[
 [\widehat{\mathcal N}(X)]_{ij}
 =\Tr(E_iXE_j^\dagger).
\]
It is a replacement channel exactly when
\(E_j^\dagger E_i\) is scalar for every \(i,j\).
If \(\mathcal D\mathcal N=\id\), with decoder Kraus operators \(K_a\),
the rank-one Choi matrix of the identity implies
\(K_aE_i=c_{ai}I\). Hence
\[
 E_j^\dagger E_i
 =\sum_a E_j^\dagger K_a^\dagger K_aE_i
 =\sum_a\overline{c_{aj}}c_{ai}I.
\]
Conversely, diagonalizing the scalar Gram matrix gives
\[
 E_i=\sqrt{\lambda_i}W_i,\qquad
 W_i^\dagger W_j=\delta_{ij}I,\qquad \sum_i\lambda_i=1.
\]
With \(\Pi=\sum_iW_iW_i^\dagger\), the channel
\[
 \mathcal D(Y)=
 \sum_iW_i^\dagger YW_i+\Tr((I-\Pi)Y)\tau
\]
for any fixed state \(\tau\) is a CPTP left inverse.

For the second assertion, a normalized maximally entangled input
\(\ket{\Phi_D}\) gives a separable reference--decoded-output state
under every measure-and-prepare channel followed by a decoder.
Its overlap with \(\Phi_D\) is at most \(1/D\): this follows for
product vectors by Cauchy--Schwarz and for separable states by convexity.
Testing the Bell projector gives \cref{eq:classical-gap}.

A channel to a finite classical register has form
\[
 X\longmapsto\sum_x\Tr(F_xX)\kb{x}{x};
\]
a decoder prepares a state for each \(x\), so the composite is
measure-and-prepare. A finite adaptive process with only a final
classical output has the same channel form.

For attainability, if
\(Z_D=\sum_{j=0}^{D-1}e^{2\pi ij/D}\kb{j}{j}\), computational dephasing is
\[
 \Delta_D=\frac1D\sum_{k=0}^{D-1}\Ad_{Z_D^k}.
\]
The triangle inequality gives half-diamond distance at most
\((D-1)/D\) from the identity, while the Bell test gives the matching
lower bound.
\end{proof}

The full-code criterion is the corresponding specialization of
quantum error correction \cite{KnillLaflamme1997}, restated in
Oai280-C \cite[Theorem \texttt{thm:complement}]{CheungOai280C}.
Measure-and-prepare channels are entanglement breaking
\cite{HorodeckiShorRuskai2003}; the quantitative record-only comparison
appears in Hawk04-B
\cite[Theorem \texttt{thm:classical}]{CheungHawk04B}.

For subsystem recovery the complement need not be constant on the
whole input. For example, tracing out a nuisance subsystem preserves
the logical subsystem while returning the nuisance state to a
nonconstant complement. Neither recovery criterion is silently
identified with geometric decoding.

\subsection{Clock, collar, and Einstein non-identifications}

The flow clock is an execution parameter. No scalar-clock field,
relational observable, or lapse constraint has been defined.
The relational lapse considered in Em02-E has additional background,
clock, and domain data absent here
\cite[Theorem \texttt{thm:lapse}]{CheungEm02E}.

The collar is a geometric annulus used for affine matching and
subdivision. No boundary quantum algebra, entropy extremization,
boundary CFT, or bulk--boundary reconstruction map is supplied.
It is not an entanglement wedge. The finite operator reconstruction
comparison in XVII-A does not identify this collar with its independent
boundary model
\cite[Remark \texttt{rem:no-wedge}]{CheungXVIIA}.

\begin{proposition}[Conservation and trace do not impose Einstein dynamics]
\label{prop:not-einstein}
Symmetry, conservation, and zero trace of a tensor do not imply that
it sources the metric on which it was calculated.
\end{proposition}

\begin{proof}
On three-dimensional Minkowski space, let
\[
 t_{\mu\nu}=\diag(2\varepsilon,\varepsilon,\varepsilon),
 \qquad \varepsilon>0.
\]
It is constant, symmetric, conserved, and traceless for
\(\eta=\diag(-1,1,1)\). Yet \(G_{\mu\nu}(\eta)=0\), while
\(t_{\mu\nu}\ne0\). Thus \(G_{\mu\nu}(\eta)=\kappa t_{\mu\nu}\) fails
for any nonzero \(\kappa\).
\end{proof}

This elementary counterexample is independent of a field-theoretic
source construction. It is the same logical separation emphasized
in Oai282-A and Oai282-B
\cite[Proposition \texttt{prop:einstein}]{CheungOai282A},
\cite[Proposition \texttt{prop:einstein}]{CheungOai282B}.
The symbol \(\Ein\) remains the five-check matching packet of
\cref{def:matching}.

A rounded length is not an incidence clock. A rate parameter is not
a face boost. An abstract location is not a stress tensor, mass
assignment, or developed trajectory. Retained implementation history
is not a source measurement record. A token promise is not an
operational test.

\section{Conclusion and the four missing interfaces}
\label{sec:conclusion}

The conditional composition statement is
\[
 \boxed{
 \begin{gathered}
 \text{certified fixed-packet flow}
 +\text{guarded fixed-packet incidence}\\
 +\text{promised regular source replacement}\\
 \Longrightarrow
 \text{complete accepted }\Ein
 \text{ after every legal command}.
 \end{gathered}}
\]
The explicit word separates its source change from its particle
incidence. Its final flow changes genuine crease lengths and necessarily
changes persistent-edge reports. Its finite implementation retains
histories and leaves internal matter untouched.

Four missing interfaces remain outside this theorem.
\begin{enumerate}
\item \textbf{Mixed germ.}
A successor must construct one common realization, both appropriate
one-sided germs, complete ordered affine words through the mixed
stratum, core--collar and worldline matching, valid location transport,
and a canonical forward-domain decoder. None is supplied by combining
the two existing permission guards.
\item \textbf{Token acquisition.}
A successor must specify an operational instrument, its named output
algebra, residual ownership, current-source interpretation, and
controlled error. T0 supplies a token but constructs no such instrument.
\item \textbf{Directional dominance.}
A successor must provide an independently certified physical embedding
and an exact or controlled directional inclusion for the same timed
network and access interface. Scheduled ancestry and a scalar speed
bound do not provide it.
\item \textbf{Canonical dynamics and a larger catalogue.}
A successor must specify a dynamical law, its proper gauge and time
conventions, and complete certificates for every additional admitted
event. The cited polygon literature is context, not an unstated
Hamiltonian premise.
\end{enumerate}

The package is not a derivation of classical spacetime from quantum
data, a Page curve, an evaporation theorem, or a holographic
identification. No characteristic quantum product or inherited
gravitational quantum coefficient is constructed.

The separately inherited documentary status
\[
 \text{O.1}=\NOTCLOSED
\]
is unchanged. This is a scope statement, not a new universal
nonexistence theorem or an independent audit of that programme.

\appendix

\section{Exact algebraic and resource checks}
\label{app:checks}

\subsection{Packet-table identities}

Let \(t=t_m\) and \(u=\arctan t\). Then
\[
 \alpha_m=\frac{\pi}{2}-2u.
\]
The double-angle identities give
\[
 \cos\alpha_m=\frac{2t}{1+t^2},\qquad
 \sin\alpha_m=\frac{1-t^2}{1+t^2},
\]
and
\[
 \cos\frac{\alpha_m}{2}
 =\frac{1+t}{\sqrt2\sqrt{1+t^2}}.
\]
These prove \cref{eq:algebraic}.
For rational \(t\), \(e^{i\alpha_m}\) has rational real and imaginary
parts, as do its finite powers.
Furthermore,
\[
 e^{i\mu_m}
 =i(e^{2iu})^3,\qquad
 e^{2iu}=\frac{1-t^2+2it}{1+t^2},
\]
so the seam rotation has rational entries.
The remaining displayed finite data are algebraic.

At \(t=1/64\),
\[
 1+t^2=\frac{4097}{4096},\qquad
 \cos\alpha_1=\frac{128}{4097},\qquad
 \sin\alpha_1=\frac{4095}{4097}.
\]
This proves \cref{eq:installed,eq:ray}.
The finer-table index in \cref{eq:finer} is in \(\cT_q\) and has
exactly the same rational \(t\).

\subsection{Representative born lengths}

For \(v_i=[P,C_i]\),
\[
 \Delta T=b_\mu Cs,\qquad
 \Delta x=Cs(e_i+e_{i+1}),
\]
so
\[
 \ell(v_i)^2=C^2s^2\bigl(2(1+z_\alpha)-b_\mu^2\bigr).
\]
For \(w_i^-=[B_i,C_i]\),
\[
 \Delta T=b_\mu Cs,\qquad
 \Delta x=s\bigl((C-B)e_i+Ce_{i+1}\bigr),
\]
giving \(W_\alpha^2s^2\). The plus edge has the same length by reflection.

The corresponding displacements are nonzero for \(s>0\) and lie in
faces of gradient norm below one. Hence the born coefficients are
strictly positive. The persistent \(e_i\) rate is negative for
positive flow speed; positivity of a length does not require positivity
of its rate.

\subsection{A conservative report-width bound}

By \cref{lem:bounds},
\[
 r<\frac72,\qquad B<\frac{200}{129},\qquad 2C<\frac{29}{10}.
\]
For \(0\le s\le1\),
\[
 \ell(e_i)<\frac72,\quad \ell(c_i)<7,\quad
 \ell(t_i)<4,\quad \ell(o_i)<8,\quad
 \ell(u_i)<\frac{200}{129}.
\]
An induced spacelike length is no larger than its projected spatial
displacement, so
\[
 \ell(v_i)\le|C_i|<\frac{29}{10},
\]
and
\[
 \ell(w_i^\pm)
 \le|B_i|+|C_i|
 <\frac{200}{129}+\frac{29}{10}<8.
\]
The same conservative bound covers incoming active values.
Thus all represented finite particle lengths are below \(8\), and
in particular below \(16\). Unsigned \(b+4\)-bit reports suffice.

\subsection{The final flow changes reports at every precision}

Let
\[
 L_{\mathrm{in}}
 =r_1\sqrt{1-a_1^2}
 =2\sqrt{a_1^{-2}-1}.
\]
The proven coefficient bounds imply
\[
 \frac35<a_1<\frac1{\sqrt2},
 \qquad 2<L_{\mathrm{in}}<\frac83.
\]
Since \(B_1=r_1/2\), the final persistent length is
\[
 L_{\mathrm{final}}
 =\left(r_1-\frac{B_1}{2}\right)\sqrt{1-a_1^2}
 =\frac34L_{\mathrm{in}},
\]
and
\[
 \frac32<L_{\mathrm{final}}<2.
\]
For every integer \(b\ge0\),
\[
 \lfloor2^bL_{\mathrm{in}}\rfloor\ge2^{b+1},
 \qquad
 \lfloor2^bL_{\mathrm{final}}\rfloor\le2^{b+1}-1.
\]
The outgoing germ has the same limiting persistent length
\(L_{\mathrm{in}}\) as the installed incoming epoch.
Therefore the final flow necessarily changes the \(e_i\) reports,
including at \(b=0\).

\subsection{Permission and register arithmetic}

The permission trace is
\[
 (1,1)\to(1,1)\to(1,0)\to(1,0)\to(0,0)\to(0,0).
\]
Only \(\Stab\) consumes source permission, and only
\(\Eparticle\) consumes incidence permission.

The exact slice range \(j\in[-2^d,2^d]\) fits in \(d+2\) signed bits.
The table-index range fits in the allocated \(q+2\) signed bits.
Unused signed words are off-domain.

Summing the hybrid fields,
\begin{align*}
 H
 &=3+1+(r_G+r_\src)+(d+2)+3
   +(q+2)+(q+2)+11+24(b+4)\\
 &=24b+d+2q+r_G+r_\src+120.
\end{align*}
Consequently,
\[
 8H-2+12n
 =192b+8d+16q+8r_G+8r_\src+958+12n.
\]
At \(q=6,d=1,r_G=r_\src=1\),
\[
 H=24b+135,\quad K_6=1,\quad T_6=3,\quad Q_{\max}=36.
\]
These are exact mathematical arithmetic checks, not reported software
execution.

\section{Dependencies and version-qualified source locators}
\label{app:dependencies}

\small
\begin{longtable}{@{}
 >{\raggedright\arraybackslash}p{.10\linewidth}
 >{\raggedright\arraybackslash}p{.39\linewidth}
 >{\raggedright\arraybackslash}p{.44\linewidth}@{}}
\toprule
Result & Required hypothesis & Local work\\
\midrule
\endfirsthead
\toprule
Result & Required hypothesis & Local work\\
\midrule
\endhead
C1 &
Particle and four-face realization interfaces &
Endpoint length derivations, affine rates, collar realization,
reference-frame preservation\\
C2 &
Certified particle and four-face germs and isolation &
Restricted catalogue, cap separation, preservation under flow\\
C3 &
Complete command outputs, certified transports, deterministic control &
Legal-domain composition, complete-output induction, paired executions,
rejection at every installed packet\\
C4 &
Particle interface and supplied token promise &
Explicit word, packet algebra, worldline intersections,
permission trace\\
C5 &
Prescribed finite block channels &
Local adjoint-inclusion and metric proofs, joint-reference statement,
ancestry/influence separation\\
C6 &
Finite deterministic canonical maps and transports &
Permutation completion, composite code domain, separate semantic
class, decoder induction, exact hybrid budget\\
\bottomrule
\end{longtable}
\normalsize

\paragraph{Geometric predecessor locators.}
The main textual locators are:
\begin{itemize}
\item VIII-A, version 2:
\texttt{A:def:Ein}, \texttt{A:thm:A1},
\texttt{A:thm:A4-stale}, \texttt{A:thm:A4-time}.
\item VIII-B, version 2:
\texttt{thm:dictionary}, \texttt{thm:persistence},
\texttt{thm:source-map}, \texttt{thm:finite}.
The persistence and finite-table results were called B3 and B5 in
the supplied earlier revision plan; textual labels avoid dependence
on export numbering.
\item VII-E, version 1: its four-face realization, exchange,
transport, and ambient matching interface.
\item VII-F, version 1: the fixed-deficit particle realization
and persistent-label convention.
\end{itemize}

\paragraph{Comparison locators, not additional hypotheses.}
\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.28\linewidth}X@{}}
\toprule
Manuscript & Local role in VIII-C\\
\midrule
VIII-B; VII-G
 & Expectation-token and affine deterministic-label obstructions\\
VII-E; Landmark I
 & Accessible adjoint images, residual/record distinction,
destructive ownership\\
En05-A; En03-A; En05-C
 & Scheduled support and regional determination\\
VII-J; En06-A
 & Independently certified physical containment; algebraic
microcausality is a separate interface\\
Oai280-C; Hawk04-B
 & Full-code recovery and record-only quantum-return obstruction\\
Em02-E; XVII-A
 & Relational lapse and entanglement-wedge non-identifications\\
Oai282-A; Oai282-B
 & Conservation and trace do not imply Einstein dynamics\\
\bottomrule
\end{tabularx}
\end{center}

The supplied excerpts and realization data support these comparisons.
The Enter Calculus Companion provides manuscript identifiers and
passage locators; it is not a proof checker.
A relationship, citation, registry entry, or restatement is not an
independent verification. Stable full versions of any unpublished
realization certificates relied upon should accompany technical review.

\section{Focused revision and boundary ledger}

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.29\linewidth}X@{}}
\toprule
Issue & Treatment in this revision\\
\midrule
Incomplete output updates
 & Acceptance includes every realization and matching obligation
at every command boundary and every installed deficit\\
Fixed-presentation uniqueness
 & Prescribed transitions are unique only with charts, components,
framing, and seam data fixed; holonomy alone is insufficient\\
Rate interpretation
 & Signed rates are derived and correlated; their \(\arsinh\)
parameters are not canonical boosts\\
Plane motion
 & Chart translations and intercepts update even when orientations
remain fixed\\
Germs and occupancy
 & Collapsed born slots are metadata, not active zero-size cells\\
Token semantics
 & Exact code predicates and matter-state promises are separated;
no expectation extraction or test is constructed\\
Record and residual access
 & Equal records do not determine residual channels; surviving
source algebras must be named\\
Support and references
 & Scheduled ancestry is an upper bound; referenced equality uses
joint ancestor--reference reductions\\
Physical causality
 & Directional containment requires an independent physical embedding;
channel commutation does not define microcausality\\
Reversible implementation
 & Retained input makes code updates injective; ignored history need
not preserve an unknown quantum code\\
Final reports
 & Persistent-edge reports necessarily change for every \(b\ge0\)\\
Finer tables
 & The \(q=6\) index-one packet has index \(2^{q-6}\) at finer precision\\
Unconstructed interfaces
 & Mixed germs, token acquisition, physical directional inclusion,
and canonical polygon dynamics remain outside the package\\
\bottomrule
\end{tabularx}
\end{center}

\section*{Declarations}

\paragraph{Materials.}
This theoretical manuscript uses no empirical dataset.
The local proofs, explicit command word, exact packet formulas,
length derivations, finite completion construction, and resource
arithmetic are contained in this file.
Realization and transport interfaces are stated explicitly;
their predecessor sources are version-qualified below.

\paragraph{Verification and submission boundary.}
All claimed checks here are analytical arguments under the printed
hypotheses. No successful pdfLaTeX build, executed notebook, numerical
certificate, machine proof, independent referee validation, or
laboratory realization is asserted.
An actual submission should supply responsible author metadata,
stable access to the unpublished geometric certificates, a clean build,
and independent technical review. The four missing interfaces cannot
be supplied by stronger wording.

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\url{https://enter.grok.me}.

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N.~Cheung,
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\emph{Landmark I},
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\emph{Landmark VII-J},
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N.~Cheung,
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\url{https://enter.grok.me}.

\bibitem{CheungEm02E}
N.~Cheung,
\emph{Em02-E},
version 1, unpublished manuscript, n.d.
Identifier \texttt{Em02-E-v1.tex}.
Relevant locator: \texttt{thm:lapse}.
Enter Calculus Companion, manuscript Em02-E.
\url{https://enter.grok.me}.

\bibitem{CheungXVIIA}
N.~Cheung,
\emph{Landmark XVII-A},
version 1, unpublished manuscript, n.d.
Identifier \texttt{XVII-A.tex}.
Relevant locators: \texttt{rem:no-wedge}, \texttt{thm:release}.
Enter Calculus Companion, manuscript XVII-A.
\url{https://enter.grok.me}.

\bibitem{CheungOai282A}
N.~Cheung,
\emph{Oai282-A},
version 1, unpublished manuscript, n.d.
Identifier \texttt{Oai282-A.tex}.
Relevant locator: \texttt{prop:einstein}.
Enter Calculus Companion, manuscript Oai282-A.
\url{https://enter.grok.me}.

\bibitem{CheungOai282B}
N.~Cheung,
\emph{Oai282-B},
version 1, unpublished manuscript, n.d.
Identifier \texttt{Oai282-B.tex}.
Relevant locators: \texttt{prop:einstein}, \texttt{thm:release}.
Enter Calculus Companion, manuscript Oai282-B.
\url{https://enter.grok.me}.

\end{thebibliography}

\end{document}