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\title{A sourced spinless conical packet with a persistent\\
three-to-six polyhedral slicing event\\[4pt]
\large Landmark VIII-B}
\author{N.~Cheung}
\date{}

\begin{document}
\maketitle

\begin{abstract}
We construct an explicit spinless conical family on a compact
deficit interval about $\pi/2$ and prove persistence of one declared
three-to-six polyhedral slicing event. Three sectors of angle
$(2\pi-\mu)/3$ are used, with coefficients retuned in the physical
oblique metric. The two nonconstant gradient norms are exactly
$2\sqrt2/3$ and $3/5$. The slices are embedded, uniformly spacelike
on regular faces, globally acausal, and strictly ordered. At every
admitted fixed deficit, the declared cellulation has exactly one
incidence-event parameter. Vertices, lengths, one-sided laws, areas,
adapted affine charts, winding-marked words, and primitive core and
collar holonomies are explicit.

A fixed Hermitian observable on a named surviving matter interface
selects a member of this preconstructed family through its
expectation. The resulting deterministic semantic constructor leaves
matter unchanged and is invariant under spectator extensions that
preserve the named source reduction. Fixed-presentation matching
uniqueness, stale-rotation rejection, and time-shift rejection are
proved for the whole deficit interval.

A separate finite restriction uses an algebraic packet table and a
supplied classical source token. On canonical promised inputs, it
has an exact NOT--CNOT--Toffoli realization with retained history,
clean work returned to zero, and identity action on internal matter.
The token's agreement with the expectation is a representation
promise, not an implemented measurement. Affine packet selection,
expectation acquisition, classical records, and quantum recovery
are distinguished by local proofs.

No mixed incidence--deficit germ, hybrid evolution theorem,
field-theoretic source admission, mass law, or derivation of Einstein
dynamics is asserted.
\end{abstract}

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

Changing a conical deficit requires more than changing a label.
The sector opening, seam identification, realized slices, adapted
charts, and matching witnesses must change coherently. This paper
supplies such a constructor on an explicit compact interval and
proves that a specified particle-link event persists throughout it.

The argument has three independent stages:
\begin{enumerate}
\item construct and certify the geometry for every scalar deficit
      in the interval;
\item substitute the expectation of a fixed named Hermitian matrix
      into that scalar construction;
\item implement a finite algebraic restriction on classical codes
      carrying a compatible, externally supplied source token.
\end{enumerate}
The geometric theorem does not depend on the source law. The finite
implementation does not acquire the expectation and does not turn
the continuum semantic rule into a quantum channel.

The affine matching interface of Landmark VIII-A
\cite{CheungVIIIA} is restated here. Its fixed-presentation uniqueness
and two rejected updates are proved locally for variable deficit.
The accessible-current and residual distinctions reported in
VII-E \cite{CheungVIIE}, and the affinity distinction reported in
VII-G \cite{CheungVIIG}, are also proved in the present notation.
None of these companion results is a formal premise.

The physical setting is motivated by classical $2+1$-dimensional
gravity with zero cosmological constant: source-free regions are
locally flat \cite{DJH1984}, and polygon descriptions can be
reconstructed from first-order gravity \cite{Kadar2005}.
These references provide context. We import neither a mass
normalization nor a Hamiltonian or constraint-algebra theorem.

Faces in the present construction may be nonconvex, and regular
vertices need not be trivalent. The results concern the explicit
matching contract below, not every more restrictive polygon gauge.

\begin{center}
\begin{tabularx}{\textwidth}{@{}lX@{}}
\toprule
Object & Result established here\\
\midrule
Variable-deficit cone & Explicit spinless dictionary on a compact interval\\
Retuned slicing family & Embedded, acausal, ordered, and uniformly spacelike\\
Three-to-six event & Unique event of the declared cellulation\\
Matching certificate & Realized witnesses for all five checks\\
Rejected incomplete updates & Stale rotation and time shift rejected locally\\
Continuous source law & Expectation-dependent semantic constructor\\
Spectator extension & Invariance under unchanged named source reduction\\
Finite table & Exact promised-domain reversible realization\\
Expectation acquisition & Not implemented; an obstruction is proved\\
Quantum recovery & Separate comparison criteria, not a table-decoding claim\\
\bottomrule
\end{tabularx}
\end{center}

\paragraph{Notation and conventions.}
Matter states are $\varrho$; the sector radial coordinate is $\rho$.
Developed spatial rays are $e_i\in\R^2$, whereas
$\tvec=(1,0,0)^{\mathsf T}$ is timelike.
The second oblique coordinate is $\zeta$, distinct from the
Minkowski metric $\eta$.
All matrix operator norms are Euclidean spectral norms, including
norms of homogeneous affine matrices; $\|\cdot\|_1$ is the trace norm.
Report precision, clock precision, and permission width are
$p_{\rep}$, $p_{\clk}$, and $w_{\src}$.
Lengths and the slicing parameter are expressed in fixed normalized
units.

\section{Named sources and observable agreement}
\label{sec:source-interface}

\subsection{The surviving source algebra}

Let $\Gamma$ be a finite named matter interface and
$\varrho\in\calD(\calH_\Gamma)$.
Registers declared classical are jointly block diagonal in their
distinguished bases, including their correlations with other
factors.

For destructive basis readout of a factor $S$, with complement $B$,
the output channel is
\begin{equation}
\label{eq:readout}
\mathcal M_S(\varrho)=
\sum_z |z\rangle\langle z|_R\otimes
(\langle z|_S\otimes I_B)\varrho(|z\rangle_S\otimes I_B).
\end{equation}
The name $S$ is absent from the output interface. Forgetting $R$
gives $\Tr_S\varrho$, by summing the diagonal blocks.

Choose a unital $*$-subalgebra
\[
\calA_{\src}\subseteq L(\calH_{\Gamma_{\src}})
\]
on declared surviving names, with a fixed embedding into the complete
current interface. Fix
\begin{equation}
\label{eq:current}
j[f]=j[f]^\dagger\in\calA_{\src},
\qquad
I_f(\varrho)=\Tr(\varrho j[f]).
\end{equation}
The embedding is suppressed in formulas. Equivalently, the source
input is the restricted state functional
\[
\omega_\varrho|_{\calA_{\src}},
\qquad
\omega_\varrho(a)=\Tr(\varrho a).
\]

The notation $j[f]$ permits a declared real linear combination of
smearings, but only the resulting fixed finite matrix is used.
Calling it a current is a declared interpretation, not a hypothesis
needed to evaluate the trace.

A consumed factor, an inaccessible coherent carrier, spectator
cardinality, and implementation history are not source variables.
A produced record is eligible as a source only when its surviving
register is explicitly named.

\subsection{Agreement on one current and on an algebra}

For a channel $\Phi$, its adjoint is defined by
\[
\Tr(\Phi(X)a)=\Tr(X\Phi^*(a)).
\]
The following finite-dimensional observation criterion makes the
meaning of source agreement explicit.

\begin{proposition}[Accessible-current criterion]
\label{prop:accessible}
Let $\Phi,\Psi$ be channels with the same input and output
interfaces, and let $j=j^\dagger$ be a fixed output observable.
On the full input state space,
\begin{equation}
\label{eq:accessible-current}
\Tr(\Phi(\varrho)j)=\Tr(\Psi(\varrho)j)
\quad\hbox{for every }\varrho
\quad\Longleftrightarrow\quad
\Phi^*(j)=\Psi^*(j).
\end{equation}
Moreover,
\begin{equation}
\label{eq:current-error}
\sup_\varrho
\left|\Tr\bigl(j[\Phi(\varrho)-\Psi(\varrho)]\bigr)\right|
=
\normop{\Phi^*(j)-\Psi^*(j)}.
\end{equation}
Agreement on every observable in a unital $*$-algebra
$\calA_{\src}$ is equivalent to equality of the adjoints on that
algebra.

If inputs are restricted to the complete classical block-diagonal
state space, replace adjoint equality by
\[
\mathcal P_{\mathrm{cl}}\bigl(\Phi^*(j)-\Psi^*(j)\bigr)=0,
\]
and replace the norm in \eqref{eq:current-error} by the norm of this
pinched operator. Here $\mathcal P_{\mathrm{cl}}$ is the input block
pinching.
\end{proposition}

\begin{proof}
Put $\Delta=\Phi^*(j)-\Psi^*(j)$, which is Hermitian.
The expectation difference is $\Tr(\varrho\Delta)$.
Pure states in the eigenspaces of $\Delta$ show that all these
expectations vanish precisely when $\Delta=0$, and that the largest
absolute expectation is $\|\Delta\|_{\op}$.
Every algebra element is a complex linear combination of Hermitian
elements, proving the algebra statement.

For block-diagonal $\varrho$,
\[
\Tr(\varrho\Delta)
=\Tr(\varrho\mathcal P_{\mathrm{cl}}(\Delta)).
\]
All density operators within every classical block are admitted.
Spectral testing within those blocks proves both restricted
statements.
\end{proof}

This is the local current version of the observation-boundary
criterion documented in VII-E \cite{CheungVIIE}. Equality on one
current does not imply equality on the whole source algebra, equality
of channels, or equality of residual instruments.

\begin{proposition}[Record agreement does not determine residuals]
\label{prop:residual}
Two channels can have identical classical records for every input
and different surviving residual states.
\end{proposition}

\begin{proof}
For a qubit, let
\[
\Delta_Z(X)=P_0XP_0+P_1XP_1,
\qquad P_z=|z\rangle\langle z|,
\]
and define channels to a one-label record and a surviving qubit by
\[
\mathcal G(X)=|*\rangle\langle *|_R\otimes X,
\qquad
\mathcal F(X)=|*\rangle\langle *|_R\otimes\Delta_Z(X).
\]
Both records are always $*$.
For $|+\rangle=(|0\rangle+|1\rangle)/\sqrt2$,
the residual states are $|+\rangle\langle+|$ and $I/2$.
Their difference is the Pauli matrix $X_{\mathrm P}/2$, with
eigenvalues $\pm1/2$. Thus their trace distance is $1/2$.
The residual observable $X_{\mathrm P}$ distinguishes them.
\end{proof}

The example restates the residual distinction in VII-E
\cite{CheungVIIE}. In particular, agreement of classical token values
does not certify agreement of an unobserved residual algebra or
verify a token's agreement with an expectation.

\section{Spinless packets and the matching contract}
\label{sec:interface}

\subsection{Affine conventions and packets}

Use coordinates $z=(T,X,Y)$ and
\[
\eta=\diag(-1,1,1),\qquad \tvec=(1,0,0)^{\mathsf T}.
\]
Affine maps and their composition are
\[
(L,b)z=Lz+b,\qquad
(L,b)(L',b')=(LL',Lb'+b).
\]
Define
\begin{equation}
\label{eq:rotation}
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).
\end{equation}
The reference worldline and rest vector are
\[
\calW_0=\R\tvec,\qquad u_0=\tvec.
\]

The declared frame is restricted to
\[
A=(C,a_0)\in\ISO^+(2,1),\qquad C\in\SO^+(2,1).
\]
It preserves the metric, orientation, and time orientation.
For a fixed particle identifier $\mathsf p$, put
\begin{equation}
\label{eq:packet}
\Pi_A(\mu)=
\bigl(\mathsf p,\mu,A\calW_0,C\tvec,A,H_A(\mu)\bigr),
\qquad
H_A(\mu)=AJ(\mu)A^{-1}.
\end{equation}
Explicitly,
\begin{equation}
\label{eq:conjugated-packet}
H_A(\mu)=
\bigl(CR(\mu)C^{-1},
a_0-CR(\mu)C^{-1}a_0\bigr).
\end{equation}

\begin{lemma}[Spinless matching]
\label{lem:spinless}
For $0<\mu<2\pi$, the fixed-point set of $J(\mu)$ is
$\calW_0$, its Lorentz part fixes $\tvec$, and its oriented
rest-frame spatial rotation is $\mu$.
Consequently $H_A(\mu)$ fixes $A\calW_0$ pointwise and its
Lorentz part fixes $C\tvec$.

For $\delta\ne0$, the map $(R(\mu),\delta\tvec)$ has no fixed
point and is not an admitted spinless peripheral map.
\end{lemma}

\begin{proof}
The spatial rotation has no nonzero fixed spatial vector for
$0<\mu<2\pi$, while its time component is unchanged.
Affine conjugation transports the fixed set.
A fixed point of $(R(\mu),\delta\tvec)$ would require
$T=T+\delta$, which is impossible.
\end{proof}

\subsection{Fixed presentations and complete acceptance}

\begin{definition}[Fixed developed presentation]
\label{def:presentation}
A developed presentation fixes:
\begin{enumerate}
\item the developed realization and all face, edge, and vertex labels;
\item every oriented overlap component;
\item the affine extension $\chart_f$ of each face-coordinate map;
\item its prescribed developed ambient identification $S_{gf}$,
      including the seam identification;
\item a frame and topological winding convention.
\end{enumerate}
An accepted transition on that component must satisfy the
\emph{ambient affine-map equality}
\begin{equation}
\label{eq:ambient-matching}
t_{g\leftarrow f}\chart_f=\chart_g S_{gf}.
\end{equation}
Its restriction must also match the realized shared segment.
\end{definition}

The ambient equality is important. Agreement on the endpoints of
one segment does not generally determine a Lorentz-affine map:
a nontrivial boost can fix a spacelike line pointwise.

Write the certificate as
\[
\Ein(g)=
\bigl(\mathcal V(g),\mathcal P(g),\mathcal H(g),
\mathcal K(g),\mathcal W(g)\bigr).
\]
Its components contain the following five checks:
\begin{enumerate}[label=\textbf{(E\arabic*)}]
\item \emph{Realized local matching:}
      endpoints, shared segments, polygon closure, overlap
      components, canonical ambient transitions
      \eqref{eq:ambient-matching}, and regular local vertex words.
\item \emph{Packet matching:}
      deficit, oriented rotation, and spinless peripheral affine map.
\item \emph{Marked ordered words:}
      affine products with their overlap data and integer winding
      marks.
\item \emph{Peripheral agreement:}
      core and collar primitive maps in the declared transported frame.
\item \emph{Particle data:}
      worldline, rest vectors, slice intersection, and pointwise
      fixation of the worldline.
\end{enumerate}
Embedding, spacelikeness, and active-cell nondegeneracy are separate
geometric preconditions. Matrix matching alone proves none of them.

\begin{proposition}[Fixed-presentation transition uniqueness]
\label{prop:matching-unique}
In a fixed presentation, two distinct proposed values for the same
required transition cannot both be accepted.
This is not uniqueness of transitions from peripheral holonomy alone.
\end{proposition}

\begin{proof}
Since $\chart_f$ is invertible, \eqref{eq:ambient-matching} requires
\[
t_{g\leftarrow f}=\chart_gS_{gf}\chart_f^{-1}.
\]
Both accepted proposals must equal this map.

For the qualification, changing face coordinates by invertible
maps $B_f$ changes transitions to
$B_gt_{g\leftarrow f}B_f^{-1}$ and changes a based peripheral
word by conjugation at its base. If that base map is the identity,
the peripheral word is unchanged despite changes in individual
transitions. Such a change is excluded only after the presentation
has been fixed.
\end{proof}

This is the precise local completeness statement used here, in the
sense of VIII-A \cite{CheungVIIIA}; it is not a classification of
Einstein initial data.

At an incidence change, the raw cell tables change.
For a fixed deficit, the preserved data are the particle and
peripheral signature, not an unchanged list of vertex equations.

\subsection{Control and the sourced-step signature}

\begin{assumption}[Spectator-blind deterministic control]
\label{ass:control}
For a fixed named source embedding, equal incoming geometry,
controller state, source permission, geometric resource reservation,
declared frame, interpolation flag, and accessible source functional
select equal applicability results, sourced rules, realizations,
certificates, abstract original-name transports, and successor
controller states.

Spectator locations and history storage are reserved separately.
Their number and state do not change the geometric or sourced
update budget.
\end{assumption}

The sourced step is performed at
\begin{equation}
\label{eq:source-clock}
s_c=-\frac12.
\end{equation}
Its formal input is
\[
\bigl(
g_{\mathrm{in}},\text{ source embedding},
\omega_\varrho|_{\calA_{\src}},
\text{ controller},\nu_{\src}\ge1,
A,\text{ geometric reservation},\theta
\bigr).
\]
On the admitted domain, it returns:
\begin{enumerate}
\item a new packet and ambient cone, with seam, rest data, complete
      slicing family, charts, collar, and certificate;
\item the new slice at $s=s_c$;
\item permission $\nu_{\src}-1$ and the selected successor controller;
\item unchanged internal matter and identity transport of valid
      incoming abstract cell labels.
\end{enumerate}
Identity label transport is not a pointwise identification of old
and new cones.

This step consumes no incidence permission and fires no incidence
rule. Future incidence germs are returned as family data; their
installed deficit stays fixed when traversed.

\section{The variable-deficit cone}
\label{sec:cone}

Set
\begin{equation}
\label{eq:interval}
\mu_0=\frac{\pi}{2},\qquad
I=\left[\frac{\pi}{2}-\frac{\pi}{20},
         \frac{\pi}{2}+\frac{\pi}{20}\right],
\qquad
\alpha(\mu)=\frac{2\pi-\mu}{3}.
\end{equation}
Then
\[
\frac{\pi}{2}-\frac{\pi}{60}
\le\alpha\le
\frac{\pi}{2}+\frac{\pi}{60}.
\]
Write
\[
z_\alpha=\cos\alpha,\qquad S_\alpha=\sin\alpha,\qquad
\lambda_*=\frac{2\sqrt2}{3},
\]
and retune the coefficients by
\begin{equation}
\label{eq:coefficients}
\boxed{
a(\mu)=\lambda_*\cos\frac{\alpha(\mu)}2,\qquad
b(\mu)=\frac35\sin\alpha(\mu).
}
\end{equation}

The developed rays are
\begin{equation}
\label{eq:rays}
e_i(\mu)=
\begin{pmatrix}
\cos(i\alpha(\mu))\\
\sin(i\alpha(\mu))
\end{pmatrix},
\qquad i=0,1,2,3.
\end{equation}
Sector $i$, for $i=0,1,2$, has coordinates
\[
x=\xi e_i+\zeta e_{i+1},\qquad \xi,\zeta\ge0.
\]
Physical cone labels are cyclic modulo three.
The fourth developed ray is an upper seam copy, not a fourth sector.

\begin{theorem}[Cone dictionary]
\label{thm:dictionary}
For every $\mu\in I$, the three sectors have total opening
$2\pi-\mu$. Identifying the upper boundary with the lower boundary
using $J(\mu)$ produces a spinless spatial cone $C_\mu$.
The ray data, metric coefficients, seam, and packet vary continuously
with $\mu$.
The particle axis is fixed pointwise and the oriented rest-frame
rotation is $\mu$.
\end{theorem}

\begin{proof}
Since $0<\alpha<\pi$, each sector is a convex wedge with independent
boundary rays. Its opening is $\alpha$, so the total is
$3\alpha=2\pi-\mu$.
The upper ray has angle $3\alpha$; rotation by $\mu$ sends it to
angle $2\pi$, the lower ray.
All formulas are continuous, and $I\subset(0,2\pi)$, so
Lemma~\ref{lem:spinless} applies.
\end{proof}

Continuity across different cones can be understood on one common
topological template: glue three copies of the nonnegative
$(\xi,\zeta)$ quadrant by their corresponding boundary parameters.
Only their physical Gram matrices and developments vary.

The ambient spacetime is
\[
\R\times C_\mu,\qquad -dT^2+ds_{C_\mu}^2
\]
on its regular part. The singular time axis is the particle.

\begin{proposition}[Two rejected updates on the whole interval]
\label{prop:rejections}
Let $\mu,\nu\in I$ with $\mu\ne\nu$.
Using the stale rotation $J(\nu)$ for the $\mu$-cone fails seam
and packet matching, although it still fixes $\calW_0$ pointwise.

For every $\mu\in I$ and $\delta\ne0$, replacing $J(\mu)$ by
$(R(\mu),\delta\tvec)$ fails pointwise worldline fixation.
Both statements persist under a common declared frame $A$.
\end{proposition}

\begin{proof}
The stale rotation sends the upper ray $e_3(\mu)$ to angle
\[
3\alpha(\mu)+\nu=2\pi+\nu-\mu.
\]
For this to be the lower ray, $\nu-\mu$ must be a multiple of
$2\pi$. But $|\nu-\mu|\le\pi/10$, so it must be zero, contrary
to the hypothesis. Also $R(\nu)\ne R(\mu)$.
Hence (E1) and (E2) fail.
The stale rotation nevertheless fixes the time axis.

The time-shifted map sends $T\tvec$ to $(T+\delta)\tvec$.
It fixes the rest vector in its linear part but not a single point
of the worldline. Thus (E5) fails; unchanged seam heights also
cannot match the shifted map.
Conjugation by $A$ preserves these failures.
\end{proof}

The two omissions are independent, as emphasized in VIII-A
\cite{CheungVIIIA}. Worldline fixation alone does not detect a stale
spatial angle, and a correct spatial angle alone does not exclude
a time shift.

\section{The retuned polyhedral slicing family}
\label{sec:geometry}

Fix $\mu\in I$ throughout this section and let $s\in[-1,1]$.
Abbreviate $a=a(\mu)$, $b=b(\mu)$, and $\alpha=\alpha(\mu)$.
Define sectorwise
\begin{equation}
\label{eq:family}
\rho=\xi+\zeta,\qquad
h_\mu=a\rho,\qquad
k_\mu=b\min\{\xi,\zeta\},\qquad
F_s^{(\mu)}
=s+\min\{2,\max\{h_\mu,s+k_\mu\}\}.
\end{equation}
The slice is the graph
\begin{equation}
\label{eq:graph}
\Sigma_s^{(\mu)}
=\{(F_s^{(\mu)}(x),x):x\in C_\mu\}.
\end{equation}
The function $\rho$ is a continuous piecewise-linear radial
coordinate; it is not the Euclidean radius $|x|$.

\subsection{Descent and the physical gradient identities}

\begin{lemma}[Descent and constant physical norms]
\label{lem:gradients}
The functions in \eqref{eq:family} agree across sector rays and the
seam. The oblique spatial metric is
\begin{equation}
\label{eq:oblique}
G_\alpha=
\begin{pmatrix}1&z_\alpha\\z_\alpha&1\end{pmatrix},
\qquad
G_\alpha^{-1}
=\frac1{S_\alpha^2}
\begin{pmatrix}1&-z_\alpha\\-z_\alpha&1\end{pmatrix}.
\end{equation}
For a covector $p$, its physical squared norm is
$p^{\mathsf T}G_\alpha^{-1}p$. In particular,
\begin{equation}
\label{eq:gradient-identities}
\begin{aligned}
\|\nabla h_\mu\|^2
&=\frac{2a^2}{1+z_\alpha}
 =\frac{a^2}{\cos^2(\alpha/2)}=\frac89,\\
\|\nabla(b\xi)\|^2
&=\|\nabla(b\zeta)\|^2
 =\frac{b^2}{S_\alpha^2}=\frac9{25}.
\end{aligned}
\end{equation}
\end{lemma}

\begin{proof}
On a sector ray, $\rho$ is its ray parameter,
$h_\mu=a\rho$, and $k_\mu=0$. Adjacent sectors and the
seam-identified boundaries therefore agree.

The Gram matrix of the unit boundary rays is $G_\alpha$.
Substitute $(a,a)^{\mathsf T}$, $(b,0)^{\mathsf T}$, and
$(0,b)^{\mathsf T}$ into its inverse. Equation
\eqref{eq:coefficients} gives the constant values.
\end{proof}

Keeping the coefficients fixed while varying $\alpha$ would not
preserve these physical norms.

\subsection{Exact localization bounds}

Put
\begin{equation}
\label{eq:radii}
c=a-\frac b2,\qquad
R_*=\frac1c,\qquad
r=\frac2a,\qquad
\beta=\frac1a,\qquad
\chi=\frac1{2a-b}=\frac1{2c}.
\end{equation}

\begin{lemma}[Bounds and resting collar]
\label{lem:bounds}
Uniformly for $\mu\in I$,
\begin{equation}
\label{eq:bounds}
\frac{129}{200}<a<\frac{137}{200},\qquad
0<b\le\frac35<a,\qquad
c>\frac{69}{200}.
\end{equation}
Consequently,
\begin{equation}
\label{eq:loop-clearance}
R_*<
\frac{200}{69}<
\frac{29}{10}<
\frac{400}{137}<
r<
\frac{400}{129}<
\frac72<
\frac{15}{4}<4.
\end{equation}
For $s>0$, the region $s+k_\mu\ge h_\mu$ has largest
$\rho$-coordinate $s/c<r$ and is not cut by the cap.
For $\rho\ge r$, the graph is resting:
\[
T=s+2.
\]
In particular, $7/2\le\rho\le4$ is a fixed resting collar.
\end{lemma}

\begin{proof}
Write $\alpha/2=\pi/4+t$, with
$|t|\le\pi/120<27/1000$. Then
\[
a=\frac23(\cos t-\sin t).
\]
The inequalities
$\cos t\ge1-t^2/2$, $|\sin t|\le|t|$, and $\cos t\le1$
give the stated strict bounds on $a$.
Since $0<\alpha<\pi$,
$b=(3/5)\sin\alpha\in(0,3/5]$, whence
$c>a-3/10>69/200$.

The comparisons in \eqref{eq:loop-clearance} follow from
$r=2/a$ and $R_*=1/c$, together with direct rational
comparisons. In particular,
$200/69<29/10$ is equivalent to $2000<2001$.

Since $\min\{\xi,\zeta\}\le\rho/2$,
\[
h_\mu-k_\mu\ge c\rho.
\]
Thus $s+k_\mu\ge h_\mu$ implies $\rho\le s/c$.
Equality occurs on the bisector.
The inequality $b<a$ gives $a<2a-b=2c$, hence
$1/c<2/a=r$.
Finally, $h_\mu\ge2$ when $\rho\ge r$, so the outer minimum
in \eqref{eq:family} equals $2$.
\end{proof}

\subsection{Embedding, acausality, and ordering}

Causality at the singular axis is understood in the ultrastatic
length-space sense: a future causal curve has nondecreasing time
and spatial speed at most its time speed. This agrees with ordinary
Lorentzian causality on the regular part.

\begin{lemma}[Global graph properties]
\label{lem:embedding}
For every $\mu\in I$ and $s\in[-1,1]$:
\begin{enumerate}
\item $F_s^{(\mu)}$ is globally $\lambda_*$-Lipschitz in the
      intrinsic cone distance;
\item graph projection $\Sigma_s^{(\mu)}\to C_\mu$ is a
      homeomorphism;
\item on each regular face, the induced metric is at least
      $1/9$ times the spatial metric;
\item the graph is acausal.
\end{enumerate}
For $s_1<s_2$,
\begin{equation}
\label{eq:ordering}
s_2-s_1
\le F_{s_2}^{(\mu)}(x)-F_{s_1}^{(\mu)}(x)
\le2(s_2-s_1).
\end{equation}
The particle intersection is
\begin{equation}
\label{eq:particle-time}
P_s=(T_P(s),0,0),\qquad
T_P(s)=
\begin{cases}
s,&s\le0,\\
2s,&s\ge0.
\end{cases}
\end{equation}
No regular Minkowski tangent plane is asserted at the particle.
\end{lemma}

\begin{proof}
The affine spatial branches have gradient norm at most
$\lambda_*<1$, by Lemma~\ref{lem:gradients}; the constant
branch has norm zero.
Minimum and maximum preserve a common Lipschitz bound.
Integrating along paths, using continuity at sector rays, the
seam, and the apex, proves the intrinsic global bound.

The inverse projection is
$x\mapsto(F_s^{(\mu)}(x),x)$, so projection is a homeomorphism.
For a graph plane with Cartesian gradient $g$, its induced metric is
\[
I_2-gg^{\mathsf T}
\ge(1-\lambda_*^2)I_2=\frac19I_2.
\]
This is a coordinate-independent lower bound relative to the
spatial metric.

A causal curve between distinct spatial points has time difference
at least their intrinsic spatial distance. The graph's time
difference is strictly smaller, since $\lambda_*<1$.
Distinct graph points therefore cannot be causally related.

For fixed $x$, the function
$\min\{2,\max\{h_\mu(x),s+k_\mu(x)\}\}$ is nondecreasing
and $1$-Lipschitz in $s$. Adding $s$ proves
\eqref{eq:ordering}. At the apex $h_\mu=k_\mu=0$, giving
\eqref{eq:particle-time}.
\end{proof}

\begin{remark}[Parameter and kinematical lapse]
\label{rem:lapse}
The source clock $s_c=-1/2$ is a parameter value, not a lapse.
On a regular branch, the future unit normal is
\[
n=\frac{(1,g)}{\sqrt{1-|g|^2}},
\]
so the normal component of the deformation $(\partial_sF,0)$ is
\[
N_s=\frac{\partial_sF}{\sqrt{1-|g|^2}}.
\]
It equals $3$ on a core branch, $5/2$ on an inner branch, and $1$
on a resting branch.
These are local kinematical quantities of this prescribed family.
No matter-defined relational clock or relational lapse is constructed.
The relational expansion reported in Em02-E
\cite{CheungEm02E} is a different object.
\end{remark}

\subsection{The declared cellulation}

The compact patch is $\rho\le4$. Define
\begin{equation}
\label{eq:fixed-vertices}
A_i=re_i,\qquad D_i=4e_i,\qquad
T(A_i)=T(D_i)=2+s.
\end{equation}
For $s>0$, define
\begin{equation}
\label{eq:born-vertices}
\begin{aligned}
B_i&=\beta s\,e_i,& T(B_i)&=2s,\\
C_i&=\chi s(e_i+e_{i+1}),&
T(C_i)&=2s+b\chi s.
\end{aligned}
\end{equation}
Vertex symbols mean their lifted graph points when used in cells.

For $s\le0$, the active core face in sector $i$ is
\[
V_i=[P,A_i,A_{i+1}],\qquad T=s+a(\xi+\zeta).
\]
For $s>0$, the sector faces are
\begin{equation}
\label{eq:faces}
\begin{aligned}
U_i^-&=[P,B_i,C_i],
& T&=2s+b\zeta,\\
U_i^+&=[P,C_i,B_{i+1}],
& T&=2s+b\xi,\\
V_i&=[B_i,A_i,A_{i+1},B_{i+1},C_i],
& T&=s+a(\xi+\zeta).
\end{aligned}
\end{equation}
For all $s$, retain
\begin{equation}
\label{eq:collar-faces}
Q_i=[A_i,D_i,D_{i+1},A_{i+1}],\qquad T=2+s,
\end{equation}
and the three resting unbounded sector faces beyond $\rho=4$,
with the exterior ray subdivisions.

\begin{lemma}[Exact decomposition]
\label{lem:decomposition}
The displayed active cells cover the graph, with disjoint face
interiors. The incoming core faces are triangles; the outgoing
$V_i$ are simple pentagons and may be nonconvex.
\end{lemma}

\begin{proof}
For $s\le0$,
$s+k_\mu\le h_\mu$, because $h_\mu-k_\mu\ge c\rho\ge0$.
Thus the uncapped core is $T=s+h_\mu$.

For $s>0$, in the half-sector $\xi\ge\zeta$, equality of the
nonconstant branches is
\[
a\xi+(a-b)\zeta=s.
\]
Its endpoints are $(\beta s,0)$ and $(\chi s,\chi s)$.
The other half-sector has boundary
\[
(a-b)\xi+a\zeta=s.
\]
The dominance region is the union of the two stated inner triangles.
Its complement within $\rho\le r$ is the stated pentagon.
The inequalities $a-b>0$ and
$\rho(C_i)=s/c<r$ establish the placement and prevent a cap
intersection. Outside $\rho=r$, the graph is resting.
\end{proof}

\subsection{Lengths, areas, and nondegeneracy}

Set
\begin{equation}
\label{eq:length-constants}
\begin{aligned}
L_\alpha&=\sqrt{2(1-z_\alpha)},\\
W_\alpha&=
\sqrt{(\chi-\beta)^2+\chi^2
+2z_\alpha\chi(\chi-\beta)-b^2\chi^2}.
\end{aligned}
\end{equation}
The persistent radial slot has epoch-dependent interpretation
\[
r_i^{\rad}=
\begin{cases}
[P,A_i],&s\le0,\\
[B_i,A_i],&s>0.
\end{cases}
\]

\begin{table}[htbp]
\centering
\small
\begin{tabular}{@{}lll@{}}
\toprule
Finite edge slot & $s<0$ & $s>0$\\
\midrule
$r_i^{\rad}$ &
$r\sqrt{1-a^2}$ &
$(r-\beta s)\sqrt{1-a^2}$\\
$r_i^{\crest}=[A_i,A_{i+1}]$ &
$rL_\alpha$ & $rL_\alpha$\\
$r_i^{\conn}=[A_i,D_i]$ &
$4-r$ & $4-r$\\
$r_i^{\outer}=[D_i,D_{i+1}]$ &
$4L_\alpha$ & $4L_\alpha$\\
$u_i=[P,B_i]$ &
inactive & $\beta s$\\
$v_i=[P,C_i]$ &
inactive & $\chi s\sqrt{2(1+z_\alpha)-b^2}$\\
$w_i^-=[B_i,C_i]$ &
inactive & $W_\alpha s$\\
$w_i^+=[C_i,B_{i+1}]$ &
inactive & $W_\alpha s$\\
\bottomrule
\end{tabular}
\caption{The 24 finite edge slots: three cyclic copies of each row.
The first four types are active at $s=0$; the final four types are
outgoing-only slots.}
\label{tab:lengths}
\end{table}

\begin{lemma}[Metric data and positivity]
\label{lem:lengths}
The finite lengths are those of Table~\ref{tab:lengths}.
Every active finite edge has positive length.
The compact face areas are
\begin{equation}
\label{eq:areas}
\begin{aligned}
\Area(V_i)&=
\begin{cases}
S_\alpha r^2/6,&s\le0,\\[2pt]
\displaystyle\frac{S_\alpha}{3}
\left(\frac{r^2}{2}-\beta\chi s^2\right),&s>0,
\end{cases}\\
\Area(U_i^\pm)&=\frac25S_\alpha\beta\chi s^2,
\qquad s>0,\\
\Area(Q_i)&=\frac{S_\alpha}{2}(16-r^2).
\end{aligned}
\end{equation}
Every active compact face has positive area.
\end{lemma}

\begin{proof}
For oblique spatial displacement $(\Delta\xi,\Delta\zeta)$ and
time displacement $\Delta T$,
\begin{equation}
\label{eq:edge-metric}
\ell^2=
(\Delta\xi)^2+(\Delta\zeta)^2
+2z_\alpha\Delta\xi\Delta\zeta-(\Delta T)^2.
\end{equation}
Substitution gives the first four rows and $PB_i$.
For $PC_i$, the spatial displacement is
$(\chi s,\chi s)$ and the time displacement is $b\chi s$.
For $B_iC_i$, these are $((\chi-\beta)s,\chi s)$ and
$b\chi s$; the reflected edge has the same length.

Every active spatial edge is nonzero and tangent to a spacelike
face, so its induced length is positive.
In particular the displayed radicals are positive.
Also
\[
r-\beta s=\beta(2-s)>0.
\]

The spatial area element is $S_\alpha\,d\xi\,d\zeta$.
Its graph multiplier is $\sqrt{1-|g|^2}$:
$1/3$ on $V_i$, $4/5$ on $U_i^\pm$, and $1$ on $Q_i$.
The incoming projected triangle has area $S_\alpha r^2/2$.
Each projected inner triangle has area
$S_\alpha\beta\chi s^2/2$.
Subtracting both inner triangles gives the outgoing core area.
The collar is the difference of projected triangles of radial sizes
$4$ and $r$.

Since $b<a$,
\[
0<\chi=\frac1{2a-b}<\frac1a=\beta,\qquad r=2\beta.
\]
Therefore, for $0<s\le1$,
\[
\frac{r^2}{2}-\beta\chi s^2
\ge2\beta^2-\beta\chi>\beta^2>0.
\]
Finally $S_\alpha>0$ and $r<4$, proving all area assertions.
\end{proof}

The persistent radial slot has one-sided laws
\begin{equation}
\label{eq:radial-derivatives}
\partial_s\ell(r_i^{\rad})(0^-)=0,\qquad
\partial_s\ell(r_i^{\rad})(0^+)
=-\beta\sqrt{1-a^2}.
\end{equation}
The crest, connector, and outer lengths have zero derivative.
The born slots have right derivatives
\begin{equation}
\label{eq:born-derivatives}
\begin{aligned}
\partial_s\ell(u_i)(0^+)&=\beta,\\
\partial_s\ell(v_i)(0^+)&=
\chi\sqrt{2(1+z_\alpha)-b^2},\\
\partial_s\ell(w_i^\pm)(0^+)&=W_\alpha.
\end{aligned}
\end{equation}
Born slots have no incoming geometric edge derivative, since they
are inactive rather than zero-length incoming edges.

The physical normal speed of the moving half-sector boundary is
\[
\left((a,a-b)G_\alpha^{-1}(a,a-b)^{\mathsf T}\right)^{-1/2};
\]
the other half-sector has the reflected expression.
These are slicing-parameter laws, not matter-velocity laws.

\subsection{The unique event and its germs}

\begin{definition}[Incidence epochs and germs]
An incidence epoch is an open parameter interval on which the
declared active-cell pattern is constant.
The germs at $s=0$ consist of the one-sided labelled cell data,
their limits, and their one-sided laws.
Outgoing cells whose size tends to zero are germ metadata, not
additional active zero-size cells of the limiting slice.
\end{definition}

\begin{lemma}[Unique incidence-event parameter]
\label{lem:event}
For the declared cellulation, the only incidence-event parameter
in $[-1,1]$ is $s=0$. At that parameter, six outgoing vertex slots,
six particle-star radial edge slots, six perimeter edge slots, and
six inner face slots collapse at $P$.
The particle link changes from three to six faces.
\end{lemma}

\begin{proof}
Lemma~\ref{lem:decomposition} gives a constant pattern for each sign
of $s$. For $s>0$, the three $B_i$ lie on distinct cone rays and the
three $C_i$ lie strictly in their sectors. Their coordinates are
positive multiples of $s$, so they coalesce only at $s=0$.

All particle-star edges are genuine creases.
Across $PC_i$, the gradients are those of $b\xi$ and $b\zeta$,
which are distinct. Across a cone ray, unfolding the adjacent
sectors gives inner transverse gradients of opposite nonzero signs.
Incoming core gradients also have opposite nonzero transverse
components after unfolding: their magnitude is
$a\tan(\alpha/2)>0$.

No inner region meets the cap, by Lemma~\ref{lem:bounds}.
Persistent radial lengths remain positive, all born edges and
inner areas are positive for $s>0$, the outgoing core area is bounded
below by a positive coefficient, and the collar never degenerates,
by Lemma~\ref{lem:lengths}.
The exterior is resting with fixed incidence.
Thus no other collision, edge collapse, face collapse, or crease
loss occurs in the window.

For $s<0$ the three $V_i$ meet $P$.
For $s>0$ the six $U_i^\pm$ meet $P$, proving the link change.
\end{proof}

\begin{center}
\begin{tabularx}{\textwidth}{@{}lX@{}}
\toprule
Possible extra event & Exclusion\\
\midrule
Inner cells reach the cap & $\rho\le s/c<r$\\
Persistent radial edge collapses & $r-\beta s=\beta(2-s)>0$\\
Inner triangle degenerates off $s=0$ & Positive coefficient times $s^2$\\
Outgoing pentagon degenerates & $\Area(V_i)>S_\alpha\beta^2/3$\\
Collar degenerates & $r<7/2<4$\\
New vertices coincide off $P$ & Distinct ray/interior placement for $s>0$\\
Particle-star edge loses its crease & Distinct unfolded gradients\\
\bottomrule
\end{tabularx}
\end{center}

The common limiting slice consists of the regular core triangles,
collar, and resting exterior. The labels $P,A_i,D_i$, the persistent
slots, and the core/collar/exterior face labels persist.
The outgoing-only $B_i,C_i$, $PB_i,PC_i$, $B_iC_i,C_iB_{i+1}$,
and $U_i^\pm$ belong to outgoing germ metadata.
A zero report in an inactive slot is a coding convention, not an
additional geometric cell.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[scale=.60,
 every node/.style={font=\small},
 core/.style={fill=blue!8},
 collar/.style={fill=gray!13},
 inner/.style={fill=orange!20}]
\begin{scope}[xshift=-4.5cm]
 \coordinate (P) at (0,0);
 \foreach \i/\ang in {0/0,1/90,2/180,3/270}{
  \coordinate (A\i) at (\ang:3);
  \coordinate (D\i) at (\ang:4);
 }
 \foreach \i/\j in {0/1,1/2,2/3}{
  \fill[core] (P)--(A\i)--(A\j)--cycle;
  \fill[collar] (A\i)--(D\i)--(D\j)--(A\j)--cycle;
  \draw (P)--(A\i)--(A\j)--cycle;
  \draw (A\i)--(D\i)--(D\j)--(A\j);
 }
 \draw (P)--(A3)--(D3);
 \node[below right] at (P) {$P$};
 \node at (.8,.8) {$V_0$};
 \node at (-.8,.8) {$V_1$};
 \node at (-.8,-.8) {$V_2$};
 \node at (0,-4.7) {Incoming: three particle faces};
\end{scope}
\begin{scope}[xshift=4.5cm]
 \coordinate (P) at (0,0);
 \foreach \i/\ang in {0/0,1/90,2/180,3/270}{
  \coordinate (A\i) at (\ang:3);
  \coordinate (D\i) at (\ang:4);
  \coordinate (B\i) at (\ang:1.2);
 }
 \coordinate (C0) at (1.09,1.09);
 \coordinate (C1) at (-1.09,1.09);
 \coordinate (C2) at (-1.09,-1.09);
 \foreach \i/\j in {0/1,1/2,2/3}{
  \fill[core] (B\i)--(A\i)--(A\j)--(B\j)--(C\i)--cycle;
  \fill[collar] (A\i)--(D\i)--(D\j)--(A\j)--cycle;
  \fill[inner] (P)--(B\i)--(C\i)--cycle;
  \fill[inner] (P)--(C\i)--(B\j)--cycle;
  \draw (P)--(B\i)--(C\i)--(B\j)--cycle;
  \draw (P)--(C\i);
  \draw (B\i)--(A\i)--(A\j)--(B\j);
  \draw (A\i)--(D\i)--(D\j)--(A\j);
 }
 \draw (B3)--(A3)--(D3);
 \node[below right] at (P) {$P$};
 \node[below] at (B0) {$B_0$};
 \node[right] at (B1) {$B_1$};
 \node[above right] at (C0) {$C_0$};
 \node at (1.9,.65) {$V_0$};
 \node at (0,-4.7) {Outgoing: six particle faces};
\end{scope}
\end{tikzpicture}
\caption{Spatial developments at $\mu=\mu_0$; the outgoing sketch
uses $s=4/5$. Upper and lower developed boundaries are seam-identified.
The outgoing core faces have inward notches. Heights and exact
coordinates are supplied by the formulas, not by the sketch.}
\label{fig:cells}
\end{figure}

\section{Adapted charts and winding-marked affine matching}
\label{sec:charts}

\subsection{Physical gradients and boosts}

In the first developed sector,
\[
e_0=(1,0)^{\mathsf T},\qquad
e_1=(z_\alpha,S_\alpha)^{\mathsf T}.
\]
They are orthonormal only at $\alpha=\pi/2$.
The Cartesian gradients are
\begin{equation}
\label{eq:physical-gradients}
g_h=
\begin{pmatrix}a\\a\tan(\alpha/2)\end{pmatrix},
\qquad
g_-=
\begin{pmatrix}0\\b/S_\alpha\end{pmatrix}
=\begin{pmatrix}0\\3/5\end{pmatrix},
\qquad
g_+=
\begin{pmatrix}b\\-bz_\alpha/S_\alpha\end{pmatrix}.
\end{equation}
For $0<|g|<1$, define
\begin{equation}
\label{eq:boost}
\calB(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}.
\end{equation}
In an orthonormal spatial basis along and perpendicular to $g$,
this is a standard two-dimensional boost plus an unchanged
perpendicular coordinate. Consequently,
\[
\calB(g)^{\mathsf T}\eta\calB(g)=\eta,\qquad
\det\calB(g)=1,\qquad \calB(g)^0{}_0=\gamma>0.
\]
Thus $\calB(g)\in\SO^+(2,1)$.
Here $\gamma_h=3$ and $\gamma_\pm=5/4$.

Put $O_i=\diag(1,\Rot(i\alpha))$. The affine extensions of the
adapted face-coordinate maps are
\begin{equation}
\label{eq:chart-table}
\begin{array}{c|l}
\text{Face}&\chart_f(z)\\
\hline
V_i&O_i\calB(g_h)O_i^{-1}z-3s\tvec\\
U_i^-&O_i\calB(g_-)O_i^{-1}z-\frac52s\tvec\\
U_i^+&O_i\calB(g_+)O_i^{-1}z-\frac52s\tvec\\
Q_i\text{ and resting exterior}&z.
\end{array}
\end{equation}
If a face is $T=q+g\cdot x$, its first boost coordinate is
$\gamma(T-g\cdot x)=\gamma q$.
Therefore nonresting faces have adapted time $T'=0$,
whereas resting faces have $T'=2+s$.

These are regular ambient affine coordinates away from the singular
axis and algebraic extensions to face closures. They do not supply
a regular Minkowski neighbourhood of the apex.

\subsection{Transitions and winding convention}

Positive traversal means increasing developed angle.
For each oriented overlap component, let $n_{gf}=1$ for an
upper-to-lower marked seam crossing, $n_{gf}=-1$ for its reverse,
and $n_{gf}=0$ for an ordinary crossing.
The fixed presentation prescribes
$S_{gf}=J(\mu)^{n_{gf}}$. Hence
\begin{equation}
\label{eq:transitions}
t_{g\leftarrow f}
=\chart_gJ(\mu)^{n_{gf}}\chart_f^{-1}.
\end{equation}
The last chronological transition acts on the left.

For a closed ordered chart word based in $f_0$,
\begin{equation}
\label{eq:word}
\Hol(\gamma)
=\chart_{f_0}J(\mu)^{N(\gamma)}\chart_{f_0}^{-1},
\qquad
N(\gamma)=\sum n_{gf}.
\end{equation}
The integer $N$ is retained as a topological mark, not reconstructed
from the matrix. In the angular covering of the punctured cone,
a loop of winding $N$ has angular lift increment
$N(2\pi-\mu)$.

Transport from the base-face coordinates to the developed rest
reference frame is conjugation by $\chart_{f_0}^{-1}$.
Transport onward to the declared frame is conjugation by $A$.

\begin{lemma}[Complete local matching and primitive words]
\label{lem:holonomy}
The transitions \eqref{eq:transitions} match every shared developed
segment, satisfy \eqref{eq:ambient-matching}, and have inverse
reverse transitions. Face polygons close and regular local vertex
words are affine identities.

The loops $\rho=29/10$ and $\rho=15/4$ are primitive positive
meridians, lying respectively in the core and resting collar.
Their products, transported to the reference frame, are both
$J(\mu)$. For $s>0$, the six-face particle star has the same
primitive peripheral product.
\end{lemma}

\begin{proof}
An ordinary overlap shares one developed segment.
On a seam overlap, $J(\mu)$ identifies the upper segment with the
lower segment, including its height by descent.
Applying the two face maps gives
\eqref{eq:transitions} and \eqref{eq:ambient-matching}.
Reversing the overlap gives the inverse map.

Every polygon is realized by the displayed endpoint coordinates.
Its oriented endpoint displacements telescope to zero.
Shared endpoints coincide, including seam copies.
In an ordered word, consecutive chart factors cancel, leaving
\eqref{eq:word}. A sufficiently small loop about a regular vertex
has winding zero, so its word is the identity.

The bounds
\[
R_*<\frac{29}{10}<r<\frac{15}{4}<4
\]
place the core loop outside all inner regions and inside the cap,
and place the collar loop in the resting region.
Write $\chart_i=\chart_{V_i}$.
The core's chronological crossings give
\begin{equation}
\label{eq:core-word}
(\chart_0J\chart_2^{-1})
(\chart_2\chart_1^{-1})
(\chart_1\chart_0^{-1})
=\chart_0J\chart_0^{-1}.
\end{equation}
Transport gives $J$.
The collar charts are identities, so its chronological crossings
are $I,I,J$, with product $J$.
Both loops traverse all three sectors once and have angular lift
increment $3\alpha=2\pi-\mu$, proving primitivity independently
of matrix values.

For the outgoing star use $\rho=s$, where $s>0$.
Since $a<1$,
\[
h_\mu-k_\mu\le h_\mu=as<s,
\]
so this loop lies strictly in the inner region.
In angular order set
\[
F_0=U_0^-,\ F_1=U_0^+,\ F_2=U_1^-,
\ F_3=U_1^+,\ F_4=U_2^-,\ F_5=U_2^+,
\qquad \chart_j^\star=\chart_{F_j}.
\]
Its six-factor word is
\begin{equation}
\label{eq:star-word}
\begin{aligned}
&(\chart_0^\star J(\chart_5^\star)^{-1})
(\chart_5^\star(\chart_4^\star)^{-1})
(\chart_4^\star(\chart_3^\star)^{-1})\\
&\quad{}\cdot
(\chart_3^\star(\chart_2^\star)^{-1})
(\chart_2^\star(\chart_1^\star)^{-1})
(\chart_1^\star(\chart_0^\star)^{-1})
=\chart_0^\star J(\chart_0^\star)^{-1}.
\end{aligned}
\end{equation}
It has one positive seam crossing, winding one, and transported
product $J$. At $s=0$ the nonvanishing core meridian certifies
the common peripheral signature of both germs.
\end{proof}

A power $J(\mu)^k$ can be the identity at a rational rotation angle.
That equality does not erase winding: $k$ remains part of the
marked certificate.

\begin{corollary}[Peripheral reduction in this presentation]
\label{cor:reduction}
After the realized geometry, overlap components, charts, and seam
have been fixed, every closed word is determined by its winding
through \eqref{eq:word}. Regular words have winding zero and one
primitive meridian determines all peripheral powers.
Core/collar agreement must still be checked in the same transported
frame.
\end{corollary}

\begin{proof}
Equation \eqref{eq:word} gives the first two assertions.
The punctured cone is topologically a punctured plane; angular
lifting identifies its loop classes with integer winding.
The core and collar checks are the two explicit winding-one words
of Lemma~\ref{lem:holonomy}.
\end{proof}

This reduction is a statement \emph{after realization}. It does not
recover embedding or individual transitions from one holonomy.

\section{Persistence and complete realization}
\label{sec:persistence}

\begin{theorem}[Principal persistence theorem]
\label{thm:persistence}
For every $\mu\in I$, formulas \eqref{eq:coefficients},
\eqref{eq:rays}, and \eqref{eq:family} construct a spinless conical
slicing family with the following properties:
\begin{enumerate}
\item all affine spatial branch gradients have norm at most
      $\lambda_*<1$, uniformly in $\mu$;
\item graphs descend through rays and the seam, are embedded,
      and are globally acausal;
\item regular-face induced metrics are at least $1/9$ times the
      corresponding spatial metrics;
\item the family is strictly ordered by \eqref{eq:ordering};
\item exactly one event of the declared cellulation occurs,
      at $s=0$, changing the particle link from three to six;
\item every regular slice, the common limiting slice, and both
      fixed-deficit incidence germs have complete certificates
      (E1)--(E5).
\end{enumerate}
The particle intersection is \eqref{eq:particle-time}.
Primitive core, collar, and outgoing-star maps are $J(\mu)$
in the reference framing and $AJ(\mu)A^{-1}$ in the declared
framing.

At $\mu=\mu_0$, all displayed data specialize to the reference
formulas in Appendix~\ref{app:reference}.
\end{theorem}

\begin{proof}
Lemmas~\ref{lem:gradients}, \ref{lem:bounds}, and
\ref{lem:embedding} establish descent, uniform gradients,
localization, embedding, the metric lower bound, acausality,
ordering, and particle times.
Lemmas~\ref{lem:decomposition}, \ref{lem:lengths}, and
\ref{lem:event} establish the realized cellulation, nondegeneracy,
and unique event.

Lemma~\ref{lem:holonomy} verifies endpoints, segments, closure,
canonical transitions, regular local words, marked ordered words,
and primitive core/collar agreement.
Lemma~\ref{lem:spinless} verifies spinless packet matching and
pointwise worldline fixation.
The graph supplies the slice intersection, and the rest vector is
the declared transported $\tvec$. These are all five checks.

A common $A\in\ISO^+(2,1)$ transports the spacetime, faces,
rest splitting, and affine words coherently.
Spacelikeness, embedding, and acausality are preserved.
The metric comparison is with the transported spatial metric;
it is not an assertion about an unrelated fixed Cartesian
time splitting after a boost.

At the event, outgoing collapsed slots are germ metadata, while
the limiting slice uses the active incoming triangles.
The core meridian supplies its noncollapsing peripheral witness.
Finally, substituting $\alpha=\pi/2$ proves the appendix identities.
\end{proof}

\begin{center}
\begin{tabularx}{\textwidth}{@{}p{.16\textwidth}Xp{.26\textwidth}@{}}
\toprule
Requirement & Witness & Proof\\
\midrule
Geometric preconditions &
Graph, physical gradients, exact lengths, and areas &
Lemmas~\ref{lem:gradients}--\ref{lem:event}\\
(E1) &
Developed polygons, canonical transitions, regular words &
Lemma~\ref{lem:holonomy}\\
(E2) &
Deficit, oriented seam rotation, fixed axis &
Theorem~\ref{thm:dictionary};
Lemma~\ref{lem:spinless}\\
(E3) &
Ordered words and retained integer winding &
Equations~\eqref{eq:word}--\eqref{eq:star-word}\\
(E4) &
Primitive core and collar words in one frame &
Lemma~\ref{lem:holonomy}\\
(E5) &
$P_s$, worldline, rest vector, pointwise fixation &
Lemmas~\ref{lem:embedding}, \ref{lem:spinless}\\
\bottomrule
\end{tabularx}
\end{center}

\begin{corollary}[Parameter regularity of labelled data]
\label{cor:continuity}
On each incidence epoch and its one-sided germ, the finite
coefficient, ray, vertex, gradient, chart, length, and area tables
are Lipschitz in $\mu$, uniformly over the allowed $s$.
The graphs are jointly continuous on the common topological
template.
\end{corollary}

\begin{proof}
The finite table formulas are smooth in $\mu$ on a neighbourhood
of $I$. Denominators stay away from zero by
Lemma~\ref{lem:bounds}.
Each active edge radical has a positive continuous coefficient
on the compact interval, so its square root is smooth there.
Born lengths are smooth coefficients times $s$ and born areas
are smooth coefficients times $s^2$.
Bounded derivatives prove the uniform Lipschitz assertions.
Joint graph continuity follows directly from continuous
minimum and maximum on the glued coordinate template.
\end{proof}

This compares labelled data, not identical incoming and outgoing
raw cell lists.

\section{Expectation selection and spectator invariance}
\label{sec:source}

Fix $\kappa\in\R$ and define the admitted state domain
\begin{equation}
\label{eq:source-domain}
\calD_{\src}
=\{\varrho\in\calD(\calH_\Gamma):
\mu_0+\kappa I_f(\varrho)\in I\},
\end{equation}
with the declared classical restrictions when present.
For a fixed interpolation flag $\theta\in[0,1]$, put
\begin{equation}
\label{eq:sourced-deficit}
\mu_\theta(\varrho)=\mu_0+\theta\kappa I_f(\varrho).
\end{equation}
The principal rule uses $\theta=1$.
The domain is convex, although it need not be nonempty for every
choice of fixed source and coupling.

\begin{theorem}[Sourced semantic constructor]
\label{thm:source-map}
On $\calD_{\src}$, the rule
\[
\varrho\longmapsto\Pi_A(\mu_\theta(\varrho))
\]
is well defined. For a fixed declared frame, its deficit and
homogeneous affine matrix data are Lipschitz in the input state.
In particular,
\begin{equation}
\label{eq:source-lipschitz}
\begin{aligned}
|\mu_\theta(\varrho)-\mu_\theta(\sigma)|
&=\theta|\kappa|
\left|\Tr((\varrho-\sigma)j[f])\right|\\
&\le\theta|\kappa|\normop{j[f]}\normtr{\varrho-\sigma}.
\end{aligned}
\end{equation}
Substitution into Theorem~\ref{thm:persistence} returns the complete
new geometry and certificate, evaluated at $s_c=-1/2$,
with internal matter unchanged.

If $I_f(\varrho)=0$, $\kappa=0$, or $\theta=0$, the geometric
output is the reference family at $\mu_0$.
\end{theorem}

\begin{proof}
The interval $I$ is convex and contains $\mu_0$.
Thus $\mu_\theta(\varrho)\in I$ whenever
$\varrho\in\calD_{\src}$.
Trace duality proves \eqref{eq:source-lipschitz}.

For the spectral norm,
\[
\normop{R(\mu)-R(\nu)}
=2\left|\sin\frac{\mu-\nu}{2}\right|
\le|\mu-\nu|.
\]
With tildes denoting homogeneous affine matrices,
\[
\normop{\widetilde A
(\widetilde J(\mu)-\widetilde J(\nu))
\widetilde A^{-1}}
\le
\normop{\widetilde A}\normop{\widetilde A^{-1}}|\mu-\nu|.
\]
The remaining fixed packet components do not vary.

The geometric construction was proved independently for every
scalar $\mu\in I$, so substitution is not circular.
The stated controls set the selected deficit to $\mu_0$.
\end{proof}

Source-off does not mean identity on an arbitrary old deficit:
it selects the reference family. Permission and history may still
change.

\begin{theorem}[Spectator invariance]
\label{thm:spectators}
Suppose $\widetilde\varrho$ extends $\varrho$ by spectator names $N$
and $\Tr_N\widetilde\varrho=\varrho$. Then:
\begin{enumerate}
\item the named source functional, including $I_f$, is unchanged;
\item for fixed declared geometry and frame, the constructed
      geometry and certificate are unchanged;
\item under Assumption~\ref{ass:control}, equal complete declared
      inputs and accessible source functionals give equal
      applicability, output certificates, original-name transport,
      and successor controller data.
\end{enumerate}
Entangled spectator extensions are allowed.
The sourced step preserves the original matter reduction.
\end{theorem}

\begin{proof}
For every $a\in\calA_{\src}$,
\[
\Tr((a\otimes I_N)\widetilde\varrho)=\Tr(a\varrho).
\]
The geometry depends on matter only through the selected scalar
deficit, proving the first two assertions.
The additional controller conclusion is exactly conditional on
Assumption~\ref{ass:control}.
Identity action on matter preserves its original reduction.
\end{proof}

Equality of the one scalar $I_f$ suffices for this geometric
constructor. Equality of an arbitrary controller's output requires
its declared input agreement; it cannot be inferred merely from
one current unless that controller is declared to factor through
that current.

A consumed name is absent from the codomain of
\eqref{eq:readout}. Pulling an observable back through a readout
channel does not make the consumed input name a surviving output
name. If a produced record is explicitly included in the source
algebra, invariance refers to that record-containing interface;
it does not assert a record-blind deferred-readout comparison.

\section{Affinity and expectation-acquisition obstructions}
\label{sec:obstructions}

The semantic deficit in \eqref{eq:sourced-deficit} is affine as a
scalar expectation. Assigning a \emph{definite classical packet}
with that deficit is a different operation.

\begin{proposition}[Affine obstruction to definite packet selection]
\label{prop:affine}
Let $\mathcal S$ be a convex state set and suppose an affine map
assigns each state a probability measure concentrated at one
classical packet:
\[
T(\varrho)=\delta_{\gamma(\varrho)}.
\]
If distinct packets are separated by measurable events, then
$\gamma$ is constant on $\mathcal S$.

Consequently, if $\theta\kappa\ne0$ and $I_f$ takes two distinct
values on $\calD_{\src}$, the sourced constructor cannot be
the definite classical output of an affine channel on that domain.
\end{proposition}

\begin{proof}
If $\gamma(\varrho)\ne\gamma(\sigma)$, choose a measurable event
containing one packet and not the other.
For $0<t<1$, affinity gives
\[
T(t\varrho+(1-t)\sigma)
=t\delta_{\gamma(\varrho)}
 +(1-t)\delta_{\gamma(\sigma)}.
\]
The chosen event has probability strictly between zero and one,
so this measure is not concentrated at one packet.
The mixture belongs to $\mathcal S$, a contradiction.
For the application, distinct selected deficits distinguish the
packets.
\end{proof}

This locally proves the obstruction reported in VII-G
\cite{CheungVIIG}. It does not prohibit deterministic processing
of an already supplied classical label.

\begin{proposition}[Deterministic expectation-label obstruction]
\label{prop:obstruction}
A channel with fixed auxiliary input cannot output a deterministic
classical label $\ell(\varrho)$ for every state on the full convex
state space if at least two labels occur.
The impossibility remains if the matter output is additionally
required to equal the input.

A circuit acting only on classical code registers, and as identity
on internal matter, cannot make a fixed code input's output depend
on an unencoded matter expectation.
\end{proposition}

\begin{proof}
Choose states assigned distinct labels $\ell_1,\ell_2$.
Linearity forces the label marginal on their equal mixture to be
\[
\frac12|\ell_1\rangle\langle\ell_1|
+\frac12|\ell_2\rangle\langle\ell_2|,
\]
which is not a deterministic label.
The additional identity requirement on matter cannot remove this
contradiction.

For a fixed code input, a code-only circuit has the same code
output for every separate matter state. Therefore it cannot
distinguish expectations not represented in that input.
\end{proof}

Proposition~\ref{prop:affine} concerns the channel status of a
definite semantic packet assignment.
Proposition~\ref{prop:obstruction} concerns acquisition of a
deterministic label from unencoded matter.
They do not exclude stochastic instruments, repeated-preparation
estimation, or approximation with separately specified errors.

\section{A finite algebraic table and reversible realization}
\label{sec:finite}

The finite construction has four layers:
\[
\text{semantic constructor}
\ \longleftarrow\
\text{algebraic packet table}
\ \longleftarrow\
\text{canonical classical map}
\ \longleftarrow\
\text{reversible circuit}.
\]
Decoder equality is asserted only on canonical promised forward
inputs, not on arbitrary intermediate circuit words.

\subsection{Algebraic packet entries}

Fix an integer $q\ge6$ and define
\begin{equation}
\label{eq:packet-table}
\begin{aligned}
K_q&=\left\lfloor\frac{2^q}{40}\right\rfloor,&
\calT_q&=\{-K_q,\ldots,K_q\},&
T_q&=2K_q+1,\\
t_m&=m2^{-q},&
\mu_m&=\mu_0+6\arctan t_m.
\end{aligned}
\end{equation}
Since $|t_m|\le1/40$,
\[
|\mu_m-\mu_0|
\le6\arctan(1/40)<\frac3{20}<\frac{\pi}{20},
\]
so every table entry lies in $I$.

Its geometric coefficients are exactly algebraic:
\begin{equation}
\label{eq:algebraic-table}
\begin{aligned}
\alpha_m&=\frac{\pi}{2}-2\arctan t_m,\\
z_{\alpha_m}&=\frac{2t_m}{1+t_m^2},
&
S_{\alpha_m}&=\frac{1-t_m^2}{1+t_m^2},\\
a_m&=\frac23\frac{1+t_m}{\sqrt{1+t_m^2}},
&
b_m&=\frac35\frac{1-t_m^2}{1+t_m^2}.
\end{aligned}
\end{equation}
Indeed,
\[
e^{2i\arctan t}=\frac{(1+it)^2}{1+t^2}.
\]
Thus $e^{i\alpha_m}$ has rational real and imaginary parts,
as do the powers giving the rays.
Also
\[
e^{i\mu_m}
=i\left(\frac{(1+it_m)^2}{1+t_m^2}\right)^3
\]
has rational components, so the seam matrix is rational.

The angle label $\mu_m$ need not be an algebraic number.
The geometric data reconstructed from its symbolic identity are
algebraic. At dyadic $s$, gradients, boosts, vertices, lengths, and
areas are real algebraic.

Exact report floors are therefore decidable by algebraic-number
comparison with dyadic thresholds, including equality.
The finite length bound below gives a finite threshold range.
These comparisons are a mathematical construction of the table;
no executed table generation is claimed.

\subsection{The source promise}

For a concrete nontrivial example, take one named surviving qubit,
with
\[
\calA_{\src}=L(\C^2),\qquad j[f]=Z,\qquad
\kappa=\theta=1,
\]
where $Z=\diag(1,-1)$.
A supplied token $m_{\src}\in\calT_q$ has the promise
\begin{equation}
\label{eq:promise}
\Tr(\varrho Z)=6\arctan t_{m_{\src}}.
\end{equation}
Every token has a nonempty promised matter class. For example,
\[
\varrho_m
=\frac12\bigl(I+6\arctan(t_m)Z\bigr)
\]
is a density operator because the coefficient has absolute value
less than $3/20$.
Spectator extensions preserving the source reduction are admitted.

For another fixed source and nonzero $\kappa$, retain only entries
whose required expectation $(\mu_m-\mu_0)/\kappa$ is attainable.

\begin{definition}[Canonical promised forward configuration]
\label{def:forward-domain}
A forward configuration consists of a fixed canonical classical
word $c$, with definite token and packet indices, together with
an arbitrary matter state $\varrho$ satisfying
\eqref{eq:promise} for that token. Spectators and external
references may be included in $\varrho$.
The required history and work words are zero.
\end{definition}

A block-diagonal mixture of such configurations can be processed
branchwise by the circuit. Decoder equality then means equality
of the \emph{branchwise decoded ensemble}.
It does not identify that ensemble with applying the nonlinear
semantic constructor once to the unconditional averaged matter
state. In a correlated code--matter mixture, the promise is
conditional on each occupied classical block.

\subsection{Canonical blocks and reports}

The cell dictionary in Appendix~\ref{app:coding} has 55 entries,
so six location bits suffice. At the source boundary, only valid
incoming labels are admitted.

Fix integers
\[
p_{\rep}\ge0,\qquad p_{\clk}\ge1,\qquad w_{\src}\ge1.
\]
The clock stores $s=j2^{-p_{\clk}}$.
A source input has the exact integer
\[
j=-2^{p_{\clk}-1},
\]
representing $s=-1/2$.
No event decision is inferred from a rounded length report.

For each of the 24 finite slots, store
\[
Q_e=\lfloor2^{p_{\rep}}\ell_e\rfloor;
\]
inactive slots have report zero.
All finite lengths are less than 16.
Indeed, compact endpoint spatial norms are at most 4:
\[
|A_i|=r<4,\qquad
|B_i|\le\beta<4,\qquad
|C_i|\le2\chi<200/69<4,\qquad |D_i|=4.
\]
The developed spatial distance between two endpoints is at most 8,
and a spacelike edge's induced length is no larger than its
spatial length. Hence $p_{\rep}+4$ report bits suffice.

\begin{table}[htbp]
\centering
\begin{tabular}{@{}lr@{}}
\toprule
Field in one geometric block & Bits\\
\midrule
Source-step stage & $1$\\
Incidence mode & $1$\\
Source permission & $w_{\src}$\\
Exact signed clock & $p_{\clk}+2$\\
Current packet index & $q+2$\\
Source token & $q+2$\\
Fixed metadata & $11$\\
24 edge reports & $24(p_{\rep}+4)$\\
\midrule
Total $B$ & $24p_{\rep}+p_{\clk}+2q+w_{\src}+115$\\
\bottomrule
\end{tabular}
\caption{The metadata consists of five fixed one-bit identifiers
and one six-bit particle-cell label.}
\label{tab:block-bits}
\end{table}

Fix injective encodings of $\calT_q$ into the index fields and of
$j\in[-2^{p_{\clk}},2^{p_{\clk}}]$ into the clock field.
The stated widths suffice. All admitted metadata values are fixed.

A canonical geometric input $x$ has stage 0, incoming incidence
mode, the exact source clock, indices
$m_{\mathrm{old}},m_{\src}\in\calT_q$, permission
$1\le\nu_{\src}\le2^{w_{\src}}-1$, fixed metadata, and the
canonical reports for $\mu_{m_{\mathrm{old}}}$.

A second $B$-bit block stores complete geometric input history.
Each original matter name has six location bits and six name-history
bits, reserved separately.

The decoder reconstructs the entire family from its packet index,
clock, and fixed symbolic formulas. Reports are consistency data;
they are not the underlying exact geometry. The decoder ignores
implementation history and returns the matter state unchanged.

\subsection{The classical update and permutation extension}

Define $f(x)$ by setting stage to 1, replacing the packet index
by $m_{\src}$, decreasing permission by one, and recomputing
reports. Token, clock, incidence mode, and metadata are retained.
Prescribe
\begin{equation}
\label{eq:forward-map}
(x,0_B)\longmapsto(f(x),x).
\end{equation}
The number of canonical \emph{geometric} inputs is
\begin{equation}
\label{eq:Q}
Q=T_q^2(2^{w_{\src}}-1).
\end{equation}
Matter states and name locations are not counted by $Q$.

Outputs are distinct because they retain $x$.
Input and output sets are disjoint because their first-block stage
bits differ. Exchange each input word with its prescribed output
and fix every other $2B$-bit word.
This defines a product $P_{\geo}$ of $Q$ disjoint transpositions.

Retaining input history is the elementary reversible-computation
principle associated with Bennett \cite{Bennett1973};
the gate basis is the classical reversible basis associated with
Toffoli \cite{Toffoli1980}. The bound needed here is proved directly.

\begin{lemma}[Clean-work transposition synthesis]
\label{lem:synthesis}
Let $N\ge3$. A permutation of $N$ logical bits that is a product
of $Q$ specified transpositions has a NOT--CNOT--Toffoli
realization using $N-2$ clean work bits and at most
\begin{equation}
\label{eq:synthesis-bound}
4N(2N-1)Q
\end{equation}
gates. On work-zero inputs, its logical action is the permutation
and work returns to zero.
\end{lemma}

\begin{proof}
Join the endpoints of one transposition by a shortest Hamming path
of length $h\le N$. Exchange successive path words going forward,
then going backward along all but the final edge.
This uses $2h-1\le2N-1$ adjacent-word transpositions and restores
every intermediate path word.

An adjacent-word transposition flips its differing bit controlled
on one specified pattern of the other $N-1$ bits.
Conjugating negative controls by NOT gates costs at most $2(N-1)$
gates. Compute the conjunction of the positive controls into
$N-2$ clean work bits by a Toffoli chain, use one CNOT to flip
the target, and reverse the chain.
This costs another $2(N-2)+1$ gates and restores work.
The total is $4N-5<4N$ per adjacent exchange.
Summing proves \eqref{eq:synthesis-bound}.
\end{proof}

\subsection{Implementation theorem}

\begin{theorem}[Finite promised-table realization]
\label{thm:finite}
Fix integers $p_{\rep}\ge0$, $p_{\clk}\ge1$, $q\ge6$,
$w_{\src}\ge1$, and $n\ge1$ original matter names, including
the named source qubit.

On the canonical promised forward domain of
Definition~\ref{def:forward-domain}, with valid incoming locations
and clean name history, the table step has an exact
NOT--CNOT--Toffoli realization $U$ using
\begin{equation}
\label{eq:total-bits}
\begin{aligned}
M&=4B-2+12n\\
 &=96p_{\rep}+4p_{\clk}+8q+4w_{\src}+458+12n
\end{aligned}
\end{equation}
classical bits, excluding internal matter and external references,
and at most
\begin{equation}
\label{eq:gate-bound}
G\le4(2B)(4B-1)T_q^2(2^{w_{\src}}-1)+6n
\end{equation}
gates.

Clean synthesis work returns to zero.
Internal matter, its external references, and valid location words
are unchanged. Complete geometric input history is retained.
Writing $U$ also for its identity extension to matter, the decoder
satisfies
\begin{equation}
\label{eq:decoder}
\Dec\bigl(U(c,\varrho)\bigr)
=G_{\src}\bigl(\Dec(c,\varrho)\bigr)
\end{equation}
on this domain, with $\kappa=\theta=1$ and the fixed table metadata.
\end{theorem}

\begin{proof}
Apply Lemma~\ref{lem:synthesis} to $P_{\geo}$ with $N=2B$.
It realizes \eqref{eq:forward-map} on the forward domain and
restores $2B-2$ work bits.

The promise gives
\[
\mu_0+\Tr(\varrho Z)=\mu_{m_{\src}}.
\]
The decoder therefore reconstructs exactly the geometry and
certificate of Theorems~\ref{thm:persistence} and
\ref{thm:source-map}.
Exact algebraic floors provide the canonical output reports.
The parameter stays at $s=-1/2$, so valid incoming abstract
cell labels remain valid after packet replacement.

For each name, copy each of its six location bits into its
clean name-history bit, controlled by the output stage bit.
Each copy is one Toffoli with stage and location as controls,
using $6n$ gates. Locations are unchanged and no name bit
influences a geometric output.

There are $2B$ logical geometric/history bits, $2B-2$ work
bits, and $12n$ location/history bits, giving
\eqref{eq:total-bits}.
The synthesis bound and name copies give \eqref{eq:gate-bound}.
No gate acts on internal matter or references.
The decoder ignores implementation history, proving
\eqref{eq:decoder}.
\end{proof}

The theorem is an existence construction with explicit resource
bounds, not an efficient compilation claim or an execution record.
It neither acquires nor verifies the token.
Intermediate gate words and arbitrary words in the permutation
completion need not decode as geometric configurations.

\begin{center}
\begin{tabular}{@{}lr@{}}
\toprule
Bit inventory & Amount\\
\midrule
Current geometry & $B$\\
Retained geometric history & $B$\\
Clean synthesis work & $2B-2$\\
Locations and name histories & $12n$\\
Internal matter & Untouched; excluded from $M$\\
\bottomrule
\end{tabular}
\qquad
\begin{tabular}{@{}lr@{}}
\toprule
Gate inventory & Bound\\
\midrule
Geometric permutation & $4(2B)(4B-1)Q$\\
Name-history copies & $6n$\\
Internal matter gates & $0$\\
\bottomrule
\end{tabular}
\end{center}

\paragraph{Smallest table example.}
For $q=6$, $K_6=1$ and $\calT_6=\{-1,0,1\}$.
Take old index $0$, token $+1$, stage $0$, and permission $1$.
The promise is $\Tr(\varrho Z)=6\arctan(1/64)$.
The update installs $\pi/2+6\arctan(1/64)$, sets stage to $1$
and permission to $0$, recomputes reports at $s=-1/2$, and retains
the original word in history. Matter and locations are unchanged.

\section{Classical table decoding is not quantum recovery}
\label{sec:recovery}

The decoder $\Dec$ in \eqref{eq:decoder} is a mathematical
interpretation of exact classical codes. It is not a channel
recovering an unknown quantum state from a classical record.
For precision, the two relevant finite-dimensional comparisons
are proved here.

For channels with the same input and output quantum spaces, use
the half-diamond convention
\[
\ddiamond(\Phi,\Psi)=\frac12\|\Phi-\Psi\|_\diamond.
\]
The diamond norm permits an untouched reference; in the statements
below it suffices to test a maximally entangled input for a lower
bound.

\subsection{Full-code recovery and the complete complement}

\begin{proposition}[Exact full-code criterion]
\label{prop:complement}
Let $\mathcal N:L(\calH_Q)\to L(\calH_B)$ be a channel
with Kraus operators $E_i:\calH_Q\to\calH_B$.
The following are equivalent:
\begin{enumerate}[label=(\roman*)]
\item a channel $\mathcal D$ satisfies $\mathcal D\mathcal N=\id_Q$;
\item $E_i^\dagger E_j=c_{ij}I_Q$ for scalar coefficients;
\item a complete complementary channel is a replacement channel,
      $\widehat{\mathcal N}(X)=\Tr(X)\omega$ for a fixed state
      $\omega$.
\end{enumerate}
The criterion concerns recovery of the whole input matrix algebra,
not merely a subsystem.
\end{proposition}

\begin{proof}
The standard complementary channel has matrix entries
\[
[\widehat{\mathcal N}(X)]_{ij}
=\Tr(E_iXE_j^\dagger)
=\Tr(XE_j^\dagger E_i).
\]
This proves equivalence of (ii) and (iii).

If $K_a$ are Kraus operators of a left inverse, the operators
$K_aE_i$ form a Kraus family for the identity channel.
Its Choi matrix has rank one, so each $K_aE_i=c_{ai}I_Q$.
Trace preservation of $\mathcal D$ then gives
\[
E_j^\dagger E_i
=\sum_aE_j^\dagger K_a^\dagger K_aE_i
=\sum_a\overline{c_{aj}}c_{ai}I_Q.
\]

Conversely, diagonalize the positive scalar matrix in (ii)
by a Kraus-basis change. Remove zero Kraus operators and write
\[
E_i=\sqrt{\lambda_i}W_i,\qquad
W_i^\dagger W_j=\delta_{ij}I_Q,\qquad
\sum_i\lambda_i=1.
\]
The isometry ranges are orthogonal. Put
$\Pi=\sum_iW_iW_i^\dagger$ and choose a fixed state $\tau$ on $Q$.
Then
\[
\mathcal D(Y)
=\sum_iW_i^\dagger YW_i
+\Tr((I_B-\Pi)Y)\tau
\]
is completely positive and trace preserving, and
$\mathcal D\mathcal N=\id_Q$.
\end{proof}

This is the full-code Knill--Laflamme criterion
\cite{KnillLaflamme1997}, also locally reported in Oai280-C
\cite{CheungOai280C}. ``Complete'' means the environment of a
Stinespring dilation, including purifying factors, not a selected
environmental record.

For example, $\mathcal N=\Tr_S$ on $Q_L\otimes S$ exactly
returns the subsystem $Q_L$, but its complement returns the
input-dependent $S$ marginal. Subsystem recovery is not a left
inverse on the full $Q_L\otimes S$ algebra.

\subsection{Record-only recovery}

\begin{proposition}[Sharp record-only obstruction]
\label{prop:classical-recovery}
Let $\dim\calH_Q=d\ge2$.
If a channel $\mathcal E$ is measure-and-prepare and
$\mathcal D$ is any channel returning $Q$, then
\begin{equation}
\label{eq:record-recovery}
\ddiamond(\mathcal D\mathcal E,\id_Q)\ge1-\frac1d.
\end{equation}
The bound is attained by basis readout followed by basis-state
preparation. It does not apply when an authorized input-dependent
quantum memory is also retained.
\end{proposition}

\begin{proof}
A measure-and-prepare map has the form
\[
\mathcal E(X)=\sum_z\Tr(F_zX)\sigma_z,
\qquad F_z\ge0,\quad \sum_zF_z=I.
\]
Applying it and then $\mathcal D$ to half of
$|\Phi_d\rangle=d^{-1/2}\sum_i|i,i\rangle$ gives a separable
reference--output state. For density operators $\alpha,\beta$,
\[
\langle\Phi_d|\alpha\otimes\beta|\Phi_d\rangle
=\frac1d\Tr(\alpha^{\mathsf T}\beta)\le\frac1d.
\]
Convexity gives the same upper bound for the separable output.
Testing the Bell projector yields trace distance at least
$1-1/d$ from the ideal output, proving \eqref{eq:record-recovery}.

For sharpness, computational dephasing is
\[
\Delta_d=\frac1d\sum_{b=0}^{d-1}\Ad_{Z_d^b},
\qquad
Z_d=\diag(1,e^{2\pi i/d},\ldots,e^{2\pi i(d-1)/d}).
\]
The triangle inequality bounds
$\|\id-\Delta_d\|_\diamond$ by $2(1-1/d)$.
The Bell lower bound gives equality.
\end{proof}

The measure-and-prepare characterization is standard
\cite{HorodeckiShorRuskai2003}; the quantitative comparison is
also reported in Hawk04-B \cite{CheungHawk04B}.
The proof above is independent of that manuscript.

\begin{remark}[Application to the present implementation]
For a fixed classical input, the table circuit acts as identity
on internal matter and every reference. Its matter channel
therefore has a trivial constant complement and trivial exact
recovery. This is not recovery from the source token.

The classical table decoder reconstructs geometry, not matter.
Retained classical history makes the code update injective;
it is not a quantum diary encoder or a record-only quantum
decoder. Neither recovery comparison supplies expectation
acquisition or a physical source instrument.
\end{remark}

\section{Intrinsic slice defects}
\label{sec:intrinsic}

The intrinsic particle defect of a slice need not equal the
spacetime deficit in the packet.

On an incoming core face, the induced oblique metric is
\[
G_h=G_\alpha-
\begin{pmatrix}a\\a\end{pmatrix}
\begin{pmatrix}a&a\end{pmatrix}.
\]
The boundary ray tangents have squared norm $1-a^2$ and
inner product $z_\alpha-a^2$. Thus
\begin{equation}
\label{eq:incoming-defect}
\theta_h(\mu)
=\arccos\left(\frac{z_\alpha-a^2}{1-a^2}\right),
\qquad
\kappa_P^-(\mu)=2\pi-3\theta_h(\mu).
\end{equation}

On $U_i^-$, the ray and bisector tangents have oblique coordinates
$(1,0)$ and $(1,1)$. Their angle satisfies
\[
\cos\theta_k
=\frac{1+z_\alpha}{\sqrt{2(1+z_\alpha)-b^2}},
\qquad
\tan\theta_k
=\frac{\sqrt{S_\alpha^2-b^2}}{1+z_\alpha}
=\frac45\tan\frac\alpha2.
\]
Consequently,
\begin{equation}
\label{eq:outgoing-defect}
\theta_k(\mu)
=\arctan\left(\frac45\tan\frac\alpha2\right),
\qquad
\kappa_P^+(\mu)=2\pi-6\theta_k(\mu).
\end{equation}
Positive-definite face metrics give
$0<\theta_h<\pi$ and $0<\theta_k<\pi/2$.

At $\mu_0$,
\[
\kappa_P^-=2\pi-3\arccos(-4/5),\qquad
\kappa_P^+=2\pi-6\arctan(4/5).
\]
Neither replaces $\mu$ in (E2).

\begin{proposition}[Intrinsic compact-patch audit]
\label{prop:intrinsic-audit}
For every regular slice, the total intrinsic interior angle defect
of $\rho\le4$ is $\mu$.
The same holds for the common limiting slice, counting its active
cells only.
\end{proposition}

\begin{proof}
The patch is a piecewise Euclidean metric disk.
Triangulate the simple faces without changing their metrics.
The triangle angle sum and Euler identity for a disk give
\[
\sum_{v\in\operatorname{int}K_s}\kappa_v
+\sum_{v\in\partial K_s}(\pi-\theta_v)=2\pi.
\]
At each outer vertex $D_i$, the two resting sector corners
have total interior angle $\pi-\alpha$.
Hence boundary turning totals $3\alpha$, and
\[
\sum_{v\in\operatorname{int}K_s}\kappa_v
=2\pi-3\alpha=\mu.
\]
At $s=0$, the active incoming core triangles and collar form the
same limiting disk. Collapsed outgoing slots contribute no extra
vertices or faces.
\end{proof}

Other interior defects compensate for the change in the particle's
slice defect. This audit complements, but does not replace,
the affine peripheral certificate.

\section{Non-identifications and separate successor obligations}
\label{sec:limitations}

\subsection{What the constructor does not derive}

The source rule is a deterministic semantic substitution into a
preconstructed geometric dictionary, called Option N in the series.
When its packet output varies, Proposition~\ref{prop:affine} prevents
interpreting it as a definite classical output of an affine channel.

No relation $\mu=8\pi Gm$ is derived.
No stress-energy normalization, gravitational action, quantum
state prescription for a field source, or renormalized source
instrument is supplied.
The symbol $\Ein$ denotes exactly the five-check matching packet,
not an Einstein tensor or an Einstein equation.

For completeness, even familiar stress-tensor identities do not
supply such a missing equation.

\begin{proposition}[Stress identities do not imply Einstein dynamics]
\label{prop:no-einstein}
Symmetry, conservation, and zero trace of a tensor on a declared
metric do not imply
\[
G_{\mu\nu}(g)=8\pi G_{\mathrm N}t_{\mu\nu}.
\]
\end{proposition}

\begin{proof}
This independent counterexample is in $3+1$ dimensions.
Take $\eta_4=\diag(-1,1,1,1)$ and the massless scalar
$\phi=aT$, with $a\ne0$.
Its improved flat-space stress is
\[
t_{\mu\nu}
=\partial_\mu\phi\,\partial_\nu\phi
-\frac12\eta_{4,\mu\nu}(\partial\phi)^2
+\frac16(\eta_{4,\mu\nu}\Box-\partial_\mu\partial_\nu)\phi^2.
\]
Since $\Box\phi^2=-2a^2$, direct substitution gives
\[
t_{00}=\frac{a^2}{2},\qquad
t_{ii}=\frac{a^2}{6},\qquad
t_{0i}=t_{ij}=0\quad(i\ne j).
\]
This constant tensor is symmetric, conserved, and traceless.
However $G_{\mu\nu}(\eta_4)=0$ and $t\ne0$.
For $G_{\mathrm N}>0$, the proposed equation fails.
\end{proof}

This separates the same implication discussed in
Oai282-A and Oai282-B \cite{CheungOai282A,CheungOai282B}
without importing a field theorem from either.

\subsection{Source, collar, lapse, and catalogue separations}

\begin{enumerate}
\item \emph{Finite source versus field-theoretic insertion.}
      The source $j[f]$ is a fixed finite Hermitian matrix.
      No comparison map identifies it with a sharp insertion,
      a ramp coefficient, a field carrier input, or an inherited
      characteristic coefficient.
      The documentary non-admission record of XIV-J
      \cite{CheungXIVJ} is not repaired by selecting a conical packet.
      The separately named IX-A, X-A, and XII-A objects are not
      premises here; no unprovided theorem about them is asserted.

\item \emph{Collar versus holographic boundary.}
      The resting collar is a geometric annulus with identity
      charts. No boundary state algebra, holographic encoder,
      entanglement wedge, or bulk/boundary identification is
      constructed.
      The comparison in XVII-A \cite{CheungXVIIA} concerns a
      different finite operator interface.
      AdS/CFT error-correction and wedge-reconstruction results
      \cite{ADH2015,DHW2016} are external comparisons, not premises
      of this zero-cosmological-constant cone construction.

\item \emph{Parameter versus relational lapse.}
      The installed incidence parameter remains an input of the
      family. Remark~\ref{rem:lapse} computes only its local
      kinematical normal speed; it is not the relational lapse
      of Em02-E \cite{CheungEm02E}.

\item \emph{Catalogue versus proof transfer.}
      Every OpenAI family mentioned in the revision record remains
      an external catalogue entry, not a theorem used here.
      The separate OpenAI catalogue is not identified with the
      Cheung manuscript series \cite{Catalogue,OpenAICatalogue}.
      No claim in this paper changes the documentary status
      \texttt{O.1 = NOT\_CLOSED}, recorded in XVI-O
      \cite{CheungXVIO}. That label is not a newly proved
      mathematical assertion of this paper.
\end{enumerate}

This construction is not spacetime emergence, a Page curve, an
island construction, or holographic identification. The ambient
geometric dictionary is explicitly supplied before its expectation
selection.

\subsection{Separate successors}

\paragraph{Hybrid schedules.}
A fixed-packet flow, a command grammar, legal permission histories,
and support bounds for prescribed matter channels require their own
theorems. VIII-C \cite{CheungVIIIC} is documented as the successor
that composes regular source replacement with frozen-deficit
incidence. No VIII-C theorem is absorbed as a premise or result
here. A schedule-support region is not thereby a spacetime causal
past.

\paragraph{Mixed incidence--deficit germ.}
Neither Theorem~\ref{thm:persistence} nor the finite table constructs
a simultaneous incidence and deficit change.
A successor would have to provide, at minimum:
\begin{enumerate}
\item one justified common limiting realization and a declared
      variable-deficit seam convention;
\item both one-sided cell germs, with embedding, spacelikeness,
      and active-cell nondegeneracy on their domains;
\item complete overlap maps and winding-marked affine words
      through the mixed stratum;
\item coherent worldline, rest-frame, and abstract-label transports;
\item exact incidence/source guards and permission accounting;
\item a canonical forward-domain representation and decoder
      theorem, including its source promise or acquisition errors.
\end{enumerate}
This is a missing-object checklist, not an existence theorem or a
universal obstruction.

\paragraph{Expectation acquisition.}
Approximate acquisition requires a separate instrument with a
specified preparation class, sample resources, selective residual
states, estimation error, and robust-control analysis.
Proposition~\ref{prop:obstruction} does not provide that instrument.
No such instrument for the current of this paper is supplied by
the finite token table.

\paragraph{Other excluded constructions.}
No tessellation Hamiltonian, global hybrid evolution,
characteristic distributional product, interacting quantum Ward
identity, physical communication schedule, no-signalling theorem
for the nonlinear source rule, or $3+1$-dimensional semiclassical
Einstein equation is constructed.

\subsection{Conclusion}

The closed result is
\[
\boxed{
\begin{gathered}
\text{fixed named Hermitian source}
+\text{explicit scalar deficit rule}\\
+\text{independently realized family on }I\\
\Longrightarrow
\text{a certified sourced spinless cone with one persistent}\\
\text{fixed-deficit three-to-six event of the declared cellulation}.
\end{gathered}
}
\]
The finite theorem adds a representation promise and a reversible
classical realization. It does not silently replace that promise
by expectation measurement.

\appendix

\section{Reference specialization at
\texorpdfstring{$\mu_0=\pi/2$}{mu0=pi/2}}
\label{app:reference}

This appendix defines and checks the reference specialization used
in the source-off statement. It is directly verifiable without
access to a separately maintained VIII-A source.
The correspondence with VIII-A is documentary
\cite{CheungVIIIA}; no independent file-to-file comparison is
claimed.

\subsection{Cone, graph, and vertices}

At $\mu=\mu_0$,
\begin{equation}
\label{eq:reference-coefficients}
\alpha=\frac\pi2,\quad
z_\alpha=0,\quad S_\alpha=1,\quad
a=\frac23,\quad b=\frac35,\quad
c=\frac{11}{30},\quad
r=3,\quad\beta=\frac32,\quad\chi=\frac{15}{11}.
\end{equation}
Write $d_i=e_i(\mu_0)$. The graph is
\begin{equation}
\label{eq:reference-family}
F_s^{(0)}
=s+\min\left\{2,
\max\left\{\frac23(\xi+\zeta),
s+\frac35\min\{\xi,\zeta\}\right\}\right\}.
\end{equation}
The rays are successive right-angle boundary rays of three sectors;
$d_3$ is the developed upper seam copy.

The vertices and times become
\[
A_i=3d_i,\qquad D_i=4d_i,\qquad
B_i=\frac32s\,d_i,\qquad
C_i=\frac{15}{11}s(d_i+d_{i+1}),
\]
\[
T(A_i)=T(D_i)=2+s,\qquad
T(B_i)=2s,\qquad
T(C_i)=\frac{31}{11}s.
\]
The particle time remains $s$ for $s\le0$ and $2s$ for $s\ge0$.
The resting collar is $7/2\le\rho\le4$.

With $A=\id$, the packet is
\[
(\mathsf p,\pi/2,\calW_0,\tvec,\id,J_0),
\qquad J_0=J(\pi/2).
\]
The timelike $\tvec$ is not the spatial ray $d_0$.

\subsection{Lengths, one-sided laws, and areas}

At the reference point,
\[
L_\alpha=\sqrt2,\qquad
W_\alpha=\frac{3\sqrt{65}}{22}.
\]
The full finite length table specializes to
\begin{center}
\begin{tabular}{@{}lll@{}}
\toprule
Slot & Incoming & Outgoing\\
\midrule
$r_i^{\rad}$ & $\sqrt5$ & $\sqrt5(1-s/2)$\\
$r_i^{\crest}$ & $3\sqrt2$ & $3\sqrt2$\\
$r_i^{\conn}$ & $1$ & $1$\\
$r_i^{\outer}$ & $4\sqrt2$ & $4\sqrt2$\\
$u_i$ & inactive & $(3/2)s$\\
$v_i$ & inactive & $(3\sqrt{41}/11)s$\\
$w_i^\pm$ & inactive & $(3\sqrt{65}/22)s$\\
\bottomrule
\end{tabular}
\end{center}
For the last two constants,
\[
\chi\sqrt{2-b^2}
=\frac{15}{11}\sqrt{\frac{41}{25}}
=\frac{3\sqrt{41}}{11},
\]
and
\[
W_\alpha^2
=\left(-\frac3{22}\right)^2
+\left(\frac{15}{11}\right)^2
-\frac9{25}\left(\frac{15}{11}\right)^2
=\frac{585}{484}.
\]
The persistent radial derivatives are $0$ and $-\sqrt5/2$.
Born right slopes are
$3/2$, $3\sqrt{41}/11$, and $3\sqrt{65}/22$.

The areas are
\begin{equation}
\label{eq:reference-areas}
\Area(V_i)=
\begin{cases}
3/2,&s\le0,\\
3/2-(15/22)s^2,&s>0,
\end{cases}
\qquad
\Area(U_i^\pm)=\frac9{11}s^2,\qquad
\Area(Q_i)=\frac72.
\end{equation}
These follow from $r=3$, $\beta\chi=45/22$, and $S_\alpha=1$.
The branch-boundary normal speed is
\[
\left((2/3)^2+(1/15)^2\right)^{-1/2}
=\frac{15}{\sqrt{101}}.
\]

\subsection{Charts and words}

The reference gradients are
\[
g_h=(2/3,2/3)^{\mathsf T},\qquad
g_-=(0,3/5)^{\mathsf T},\qquad
g_+=(3/5,0)^{\mathsf T}.
\]
Their boosts are
\[
B_h=
\begin{pmatrix}
3&-2&-2\\
-2&2&1\\
-2&1&2
\end{pmatrix},
\qquad
B_x=
\begin{pmatrix}
5/4&-3/4&0\\
-3/4&5/4&0\\
0&0&1
\end{pmatrix},
\qquad
B_y=
\begin{pmatrix}
5/4&0&-3/4\\
0&1&0\\
-3/4&0&5/4
\end{pmatrix}.
\]
Thus
\[
\begin{aligned}
\chart_{V_i}(z)&=O_iB_hO_i^{-1}z-3s\tvec,\\
\chart_{U_i^-}(z)&=O_iB_yO_i^{-1}z-\frac52s\tvec,\\
\chart_{U_i^+}(z)&=O_iB_xO_i^{-1}z-\frac52s\tvec,
\end{aligned}
\qquad
O_i=\diag(1,\Rot(i\pi/2)).
\]
Resting charts are identities.

The core and collar loops remain $\rho=29/10$ and $\rho=15/4$.
Their winding marks are one; the core and star words are
\eqref{eq:core-word} and \eqref{eq:star-word} with $J=J_0$.
The intrinsic audit specializes to
\[
\theta_h=\arccos(-4/5),\qquad
\theta_k=\arctan(4/5),\qquad
\sum_{v\in\operatorname{int}K_s}\kappa_v=\pi/2.
\]

Hence the source-off specialization includes the complete displayed
graph, cell pattern, vertices, times, lengths, one-sided slopes,
areas, adapted charts, seam, primitive words, collar, worldline,
rest data, and intrinsic audit.

\section{Cell dictionary and encoding}
\label{app:coding}

\begin{center}
\begin{longtable}{@{}ll@{}}
\toprule
Code & Cell slot\\
\midrule
\endfirsthead
\toprule
Code & Cell slot\\
\midrule
\endhead
0 & $P$\\
1--3 & $A_i$\\
4--6 & $D_i$\\
7--9 & $r_i^{\rad}$\\
10--12 & $[A_i,A_{i+1}]$\\
13--15 & $[A_i,D_i]$\\
16--18 & $[D_i,D_{i+1}]$\\
19--21 & $V_i$\\
22--24 & $Q_i$\\
25--27 & $B_i$\\
28--30 & $C_i$\\
31--33 & $u_i$\\
34--36 & $v_i$\\
37--39 & $w_i^-$\\
40--42 & $w_i^+$\\
43--45 & $U_i^-$\\
46--48 & $U_i^+$\\
49--51 & Exterior rays\\
52--54 & Unbounded exterior faces\\
\bottomrule
\end{longtable}
\end{center}

At the incoming source slice, slots 25--48 are invalid locations.
Other listed slots have their incoming meanings.
The radial slot is epoch dependent, but source replacement does
not cross epochs.

This dictionary abstracts location to a cell label. It does not
encode a continuously variable point within a cell.
Source replacement preserves labels, not old developed coordinates.

One possible index encoding stores $m+K_q$ in $q+2$ bits;
one possible clock encoding stores $j+2^{p_{\clk}}$ in
$p_{\clk}+2$ bits. Unused strings are invalid.
Field order, valid metadata, and encodings are fixed before
constructing $P_{\geo}$.

Invalid strings need no geometric interpretation.
They complete the update to a permutation.
Likewise, no geometric decoding is asserted for nonzero-work
inputs or intermediate circuit words.

\section{Dependencies and verification status}
\label{app:verification}

\begin{center}
\begin{tabularx}{\textwidth}{@{}lX@{}}
\toprule
Result & Mathematical inputs\\
\midrule
Accessible-current criterion &
Finite-dimensional adjoints and spectral separation\\
Cone dictionary &
Interval, rays, seam, spinless affine packet\\
Persistence &
Oblique metric, retuning, exact graph decomposition,
lengths, areas, and localization\\
Matching certificate &
Realized polygons, fixed affine extensions, overlap components,
winding marks, and primitive meridians\\
Source constructor &
Fixed named Hermitian matrix and scalar substitution\\
Spectator invariance &
Partial trace; controller conclusion additionally uses
Assumption~\ref{ass:control}\\
Finite realization &
Algebraic table, exact reports, source promise,
canonical codes, retained history, and gate synthesis\\
Recovery comparisons &
Kraus identities, complete complements, and a Bell test\\
\bottomrule
\end{tabularx}
\end{center}

All proofs used for the new constructor are present in this file.
Companion citations identify related interfaces and scope
distinctions, not hidden hypotheses.

The main symbolic audits are:
\begin{enumerate}
\item constant physical gradient norms
      \eqref{eq:gradient-identities};
\item exact cap and loop clearances
      \eqref{eq:bounds}--\eqref{eq:loop-clearance};
\item complete finite lengths and areas
      \eqref{eq:edge-metric}, Table~\ref{tab:lengths},
      and \eqref{eq:areas};
\item active-cell positivity and the unique event;
\item Lorentz membership and adapted face times
      \eqref{eq:boost}--\eqref{eq:chart-table};
\item canonical affine transitions and three- and six-factor words;
\item the reference specialization;
\item rational-tangent table identities;
\item bit and gate totals
      \eqref{eq:total-bits}--\eqref{eq:gate-bound};
\item the distinction between fixed-token correctness,
      branchwise ensembles, and nonlinear unconditional semantics.
\end{enumerate}

Numerical sampling would be diagnostic, not a substitute for the
interval proof. Useful independent checks include polygon
intersection tests, area signs, adapted chart equations, seam
orientation, algebraic report floors, and the reversible word action.

\section*{Declarations}

\paragraph{Data and materials.}
This is a theoretical work. No empirical dataset is used.
The formulas, finite table specification, matching witnesses,
proofs, and reversible resource accounting are included here.

\paragraph{Bibliographic scope.}
Cheung references are identified by manuscript name, version where
supplied, and relevant local labels or sections. They are not
represented as peer-reviewed journal articles.
References to the external catalogue are documentary only.

\paragraph{Verification.}
No successful pdfLaTeX execution, proof-assistant verification,
executed numerical diagnostic, independent source comparison,
independent referee review, or journal acceptance is asserted.
The source is designed as a single-file pdfLaTeX document; a local
two-pass build and independent mathematical review remain necessary.

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version 1, manuscript \texttt{Em02-E-v1.tex}, n.d.
``Complete relational lapse through second order,''
\texttt{thm:lapse}; comparison only.
Enter Calculus Companion, \url{https://enter.grok.me}.

\bibitem{CheungXVIIA}
N.~Cheung,
\emph{Landmark XVII-A},
version 1, manuscript \texttt{XVII-A.tex}, n.d.
``No entanglement-wedge inference,''
\texttt{rem:no-wedge}; comparison only.
Enter Calculus Companion, \url{https://enter.grok.me}.

\bibitem{CheungOai282A}
N.~Cheung,
\emph{Oai282-A},
version 1, manuscript \texttt{Oai282-A.tex}, n.d.
``A conserved traceless tensor need not source its declared
flat metric,'' \texttt{prop:einstein}; supplied excerpt.
Enter Calculus Companion, \url{https://enter.grok.me}.

\bibitem{CheungOai282B}
N.~Cheung,
\emph{Oai282-B},
version 1, manuscript \texttt{Oai282-B.tex}, n.d.
``The gravitational dictionary is not implied,''
\texttt{prop:einstein}; comparison only.
Enter Calculus Companion, \url{https://enter.grok.me}.

\bibitem{CheungXIVJ}
N.~Cheung,
\emph{Landmark XIV-J},
version 1, manuscript \texttt{XIV-J.tex}, n.d.
Reconstruction descent, partial reference construction, and
``No quantum admission in this release,''
\texttt{cor:noadmission}; documentary separation only.
Enter Calculus Companion, \url{https://enter.grok.me}.

\bibitem{CheungXVIO}
N.~Cheung,
\emph{Landmark XVI-O},
revision \texttt{o70-sync}, manuscript \texttt{XVI-O.tex}, n.d.
\texttt{rem:three-verdicts} and \texttt{conv:package};
documentary status and evidence boundaries only.
Enter Calculus Companion, \url{https://enter.grok.me}.

\bibitem{Catalogue}
N.~Cheung,
\emph{Enter Calculus Companion: registered manuscript catalogue},
n.d.
A documentary and retrieval interface, not a proof checker.
\url{https://enter.grok.me}.

\bibitem{OpenAICatalogue}
OpenAI,
\emph{External mathematics catalogue},
catalogue section recorded in the Enter Calculus Companion, n.d.
Separate from the Cheung manuscript series; no listed family
is used as a formal premise here.
\url{https://enter.grok.me}.

\end{thebibliography}

\end{document}