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\hypersetup{
  pdftitle={I-B-v1: A Finite Enter Spacetime},
  pdfsubject={Prescribed Lorentzian geometry and calibrated destructive clock readout},
  pdfkeywords={Enter calculus, spacetime, destructive measurement, finite clock,
    causal transport, relational lapse}
}

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\title{\textbf{I-B-v1: A Finite Enter Spacetime}\\[4pt]
\large Calibrated Destructive Time Readout\\
on a Prescribed Conical Product}
\author{Ngo Cheung}
\date{Version 1}

\begin{document}
\maketitle

\begin{abstract}
We construct a finite operational model on the regular part of an
explicit \(3+1\)-dimensional conical Lorentzian background. Its additional
data are finite quantum names, locations, accessible algebras, a complete
control and resource interface, and a calibration from a finite clock
basis to geometric time labels. Destructive Enter consumes the clock
names and returns a classical code together with the surviving matter.
We prove the complete instrument identity, the distinction between
event-time calibration and history-label calibration, the absence of a
universal quantum return, and a finite-dimensional canonical-commutator
obstruction. Scheduled influence is contained in the prescribed metric
causal relation. A target-adapted finite chain supplies exact
reference-preserving SWAP transport for a prescribed local null
displacement, with an explicit resource ledger.

Several necessary corrections are incorporated. The conical defect in
the product is a worldsheet, not a worldline. An arbitrary clock-basis
change is not a spacetime coordinate transformation. A stochastic reading
at one fixed event cannot equal several distinct coordinates of that
event. A relational lapse cannot replace an entire cosmological metric,
and spatial homogeneity does not imply flatness. The construction is a
spacetime model with an Enter interface, not emergence of spacetime from
records. No inherited quantum response coefficient is admitted and no
standing analytic gate is discharged.
\end{abstract}

\tableofcontents

%=====================================================================
\section{Scope, provenance, and necessary corrections}
\label{sec:scope}
%=====================================================================

\subsection{Enter as an interface discipline}

The motivating discipline is Cheung's separation of destructive
readout, quantum recovery, physical implementation, and geometric
identification. Enter removes measured quantum names from the returned
quantum interface and supplies a classical record. This is an access
statement, not annihilation of every implementing degree of freedom.
Likewise, equality of channels is not a thermodynamic account, and an
analogy is not a mathematical dictionary
\cite{CheungPhil,CheungI,CheungEn03}.
% Online documentary source: ([enter.grok.me](https://enter.grok.me/enter))

We adopt specified finite Born semantics. We do not claim to rederive
Landmark I's characterization from an abbreviated list of requirements.
That characterization has its own structural class and hypotheses
\cite{CheungI}. All mathematical assertions needed for the model below
are stated with their hypotheses and proved locally.

The construction follows En03-A's four-part geometric interface:
external classical geometry, locations, accessible coupling algebras,
and complete control and resource data \cite{CheungEn03}.
Geometry is supplied, not extracted from a count of names or records.
% Online documentary source: ([enter.grok.me](https://enter.grok.me/browse/En03-A))

\subsection{What ``finite'' means}

The background is a continuum. An experiment is finite: it has finitely
many active names, finite-dimensional quantum factors, finitely many
classical outcomes, and finitely many instructions. Classical geometric
parameters may be prescribed real numbers. No assertion is made that
arbitrary such numbers have been determined by a finite-bit physical
calibration procedure.

The main release chooses the conical branch with unit lapse. A
cosmological attachment interface is given in
Appendix~\ref{app:lapse}; its input geometry is not silently identified
with the conical product.

\subsection{Corrections to the proposed architecture}

\begin{center}
\small
\renewcommand{\arraystretch}{1.16}
\begin{tabularx}{\linewidth}{@{}p{.28\linewidth}Y@{}}
\toprule
Potential identification & Correction used here\\
\midrule
The product defect is a worldline &
After adjoining \(z\), the full defect has coordinates \((T,z)\)
and is a worldsheet. Fixing \(z\) gives one worldline.\\
Clock outcome equals the time of one fixed event &
For distinct calibrated labels, exact event-time agreement forces a
deterministic clock outcome. Nontrivial history conditioning instead
refers to different labelled events.\\
Changing the clock basis changes time coordinates &
A general basis change changes the observable. Coordinate relabelling
is a much more restrictive operation.\\
Scheduled past lies in metric future &
An ancestor of an output lies in its metric past; equivalently, the
output lies in the ancestor's metric future.\\
A lapse jet supplies a cosmological metric &
A spatial metric and shift are also required. The exact background
and the formal perturbation parameter remain distinct.\\
Spatially constant metric components imply flatness &
Spacetime-constant components imply flatness. Time-dependent,
spatially homogeneous metrics need not be flat.\\
Enter consumes the geometric time axis &
Enter consumes a finite clock encoding. The classical manifold,
coordinate, and future cone remain present.\\
Exact null transport follows on a fixed lattice &
Our exact result uses a declared target-adapted chain. It is not an
exact-attainment claim for a previously fixed lattice.\\
\bottomrule
\end{tabularx}
\end{center}

These corrections restrict the theorem to a coherent, nonempty model;
they are not additional emergence assumptions.

\paragraph{Standing boundaries.}
Throughout this paper, ``XV-B is not opened'' means that the inherited
downstream admission exit is not activated by this construction.
It does not deny the existence of a technical manuscript bearing that
identifier. XVI-N is not discharged; O.1 remains
\status{NOT\_CLOSED} for the purposes of this release; no inherited
quantum coefficient is admitted; and no compilation, numerical
execution, physical implementation, or machine verification is asserted.
The narrower internal-amplitude advance in XIV-I is not withdrawn, but
it is not the full joint characteristic map absent at I3/J2
\cite{CheungXIVIJ,CheungStops}.

%=====================================================================
\section{The geometric background}
\label{sec:geometry}
%=====================================================================

\subsection{The cone, its regular part, and its completion}

Fix
\[
 0<\mu<2\pi,\qquad
 \alpha=1-\frac{\mu}{2\pi},\qquad c>0.
\]
Let \(C_\mu\) be the completion of
\[
 C_\mu^\circ=(0,\infty)\times(\R/2\pi\mathbb Z)
\]
with spatial line element
\[
 h_{C_\mu}=\dd r^2+\alpha^2r^2\dd\phi^2.
\]
Write \(o\) for its apex. The circumference of the circle \(r=R\) is
\((2\pi-\mu)R\), so the opening angle is \(2\pi-\mu\).

Set
\begin{align}
 \overline M&=\R_T\times C_\mu\times\R_z,\\
 M&=\R_T\times C_\mu^\circ\times\R_z,\\
 h&=\dd r^2+\alpha^2r^2\dd\phi^2+\dd z^2.
\end{align}
For any smooth positive lapse \(N\) on \(M\), prescribe
\begin{equation}
 g_N=-c^2N^2\dd T^2+h.
 \label{eq:metric-general}
\end{equation}
The main model uses
\begin{equation}
 \boxed{
 g=-c^2\dd T^2+\dd r^2+\alpha^2r^2\dd\phi^2+\dd z^2.
 }
 \label{eq:metric}
\end{equation}
In geometrized units \(c=1\), this is the metric in the proposed
conical-product branch. Equivalently, with
\(\theta=\alpha\phi\), the angular coordinate has period \(2\pi-\mu\)
and the angular term is \(r^2\dd\theta^2\).

The singular completion locus is
\begin{equation}
 \W=\R_T\times\{o\}\times\R_z.
 \label{eq:worldsheet}
\end{equation}
It is two-dimensional. For fixed \(z_0\), the subset
\(\W_{z_0}=\R_T\times\{o\}\times\{z_0\}\) is a worldline.

The smooth Lorentzian manifold in our theorem is \(M\), not
\(\overline M\) equipped with a purported smooth metric at its apex.
The completion records the conical defect. No smooth chart or ordinary
pointwise Einstein tensor is asserted on \(\W\).

\begin{proposition}[Lorentzian character and normalized static time]
\label{prop:geometry}
For \(N>0\), \(g_N\) has signature \((-,+,+,+)\), and
\[
 g_N^{-1}(\dd T,\dd T)=-\frac{1}{c^2N^2}.
\]
Choose the time orientation for which \(\partial_T\) is future directed.
For
\[
 v=a\partial_T+w,\qquad w\in T(C_\mu^\circ\times\R),
\]
a nonzero future-causal vector satisfies
\begin{equation}
 a>0,\qquad |w|_h\le cNa.
 \label{eq:tangent-cone}
\end{equation}
A static observer has proper-time increment
\[
 \dd\tau=N\dd T.
\]
In particular, \(T\) is proper time up to the chosen origin for the
static observers of \eqref{eq:metric}.
\end{proposition}

\begin{proof}
In the displayed coordinates the metric matrix is diagonal, with one
negative and three positive entries. Its inverse gives the stated
norm of \(\dd T\). The causal inequality is
\[
 g_N(v,v)=-c^2N^2a^2+|w|_h^2\le0.
\]
A nonzero causal vector cannot have \(a=0\); the chosen orientation
therefore gives \(a>0\). Along a static curve,
\(-g_N/c^2=N^2\dd T^2\), which proves the proper-time formula.
\end{proof}

\subsection{Local rods and holonomy}

On a sufficiently small angular interval choose a real lift of \(\phi\).
The developing map
\begin{equation}
 \operatorname{dev}(T,r,\phi,z)
 =
 \bigl(T,r\cos(\alpha\phi),r\sin(\alpha\phi),z\bigr)
 \label{eq:development}
\end{equation}
is a local isometry from \eqref{eq:metric} to Minkowski space with metric
\(-c^2\dd T^2+\dd x^2+\dd y^2+\dd z^2\).
Its spatial coordinates are three local rods.

\begin{proposition}[Regular flatness and nontrivial gluing]
\label{prop:holonomy}
The metric \eqref{eq:metric} is locally flat on \(M\).
In the convention
\[
 \operatorname{dev}\circ\mathsf T^{-1}
     =J_\mu\circ\operatorname{dev},
 \qquad \mathsf T(\phi)=\phi+2\pi,
\]
the gluing map \(J_\mu\) fixes \(T,z\) and rotates the developed
\((x,y)\)-plane by \(\mu\). Thus local flatness does not remove the
global conical gluing.
\end{proposition}

\begin{proof}
Differentiating \eqref{eq:development} gives
\[
 \dd x^2+\dd y^2=\dd r^2+\alpha^2r^2\dd\phi^2.
\]
For the deck transformation, the developed spatial angle changes by
\(-2\pi\alpha=\mu-2\pi\), which is the same rotation as \(\mu\).
\end{proof}

This geometry is motivated by the conical constructions in VII-E,
VII-F, and VIII-B \cite{CheungVIIEF,CheungVIIIB}. The additional \(z\)
direction and the lapse interface are declared here; they are not
conclusions transferred from a \(2+1\)-dimensional theorem.

\begin{remark}[The slicing-event boundary]
VII-F's particle-supported cellulation event is a geometric
construction in its own declared family
\cite[Theorem \sourceid{thm:event}]{CheungVIIEF}.
We do not import its cell counts or uniqueness conclusion into a
three-dimensional spatial cellulation. Our operational events below
lie in the regular part. Attaching that separate slicing event would
require its full geometric data and a clock-and-record dictionary.
A slicing event is not already an Enter event.
\end{remark}

\subsection{The local causal chart used operationally}

Choose an open spatial set \(U\subset C_\mu^\circ\times\R\) whose
development is an open convex Euclidean set, and write
\[
 \chi:U\longrightarrow V\subset\R^3
\]
for its spatial developing chart. All finite protocols in the main
construction remain inside \(U\).

\begin{lemma}[A local causal certificate]
\label{lem:local-causal}
Let \(x,y\in U\) and \(T_1\ge T_0\). If
\[
 \|\chi(y)-\chi(x)\|_2\le c(T_1-T_0),
\]
then \((T_1,y)\in J_g^+((T_0,x))\). If equality holds and \(x\ne y\),
the straight developed segment, traversed at constant speed, is null.
\end{lemma}

\begin{proof}
Convexity keeps the straight spatial segment in \(V\). Its lift has
constant coordinate velocity of norm at most \(c\), so
\eqref{eq:tangent-cone} applies with \(N=1\). Equality makes its tangent
null.
\end{proof}

No global formula for \(J^\pm\) across the singular completion is needed.

%=====================================================================
\section{Finite names, access, control, and destructive Enter}
\label{sec:interface}
%=====================================================================

\subsection{Four declared structures}

\begin{definition}[A finite Enter experiment on the background]
\label{def:experiment}
An experiment consists of the following data.
\begin{enumerate}
\item \textbf{External geometry.}
The fixed configuration
\[
 (\overline M,M,g,N=1,\W,U,\chi,c).
\]

\item \textbf{Names and locations.}
At each semantic cut there is a finite set of live quantum names.
Their Hilbert spaces are finite tensor products of qubits.
A location map assigns each accessible port to a point of \(M\).
Several names may be co-located.

\item \textbf{Accessible algebras.}
For a declared local collection \(A\) of live names, the accessible
algebra is the full matrix algebra on those names, tensored with the
identity on other live names. A smaller algebra may be declared, but
no subsequent operation may silently use an observable outside it.

\item \textbf{Complete control and resources.}
A finite instruction tree specifies allocations, permitted unitaries,
destructive readouts, classical controls, input and output ports,
schedules, location data, retained storage, and all auxiliary factors.
Every classical control dependency is either preinstalled task data
or a declared causal record wire.
\end{enumerate}
The background metric has no state-dependent update rule in this model.
\end{definition}

A fixed classical history has one normalized pure state on its complete
live quantum names. Unitary gates and pure allocations preserve this
property. At a rank-one destructive readout, each supported outcome has
a normalized pure residual. Density operators and recorded channels
describe ensembles, inaccessible restrictions, and complete
input--output comparisons; they do not assert that a classical mixture
is one pure branch.

An untouched reference \(E\) may be included in a complete input. It
has no available control or communication wire. Its storage is listed
separately from the laboratory storage.

\subsection{The clock instrument}

Fix disjoint clock and matter names, with
\[
 H_C\cong\C^D,\qquad D=2^q,\qquad q\ge1,
\]
and a finite-dimensional surviving space \(H_X\).
We consistently order the input factors as \(C\otimes X\).

Choose an orthonormal clock basis
\[
 \{\ket{\tau_k}:0\le k<D\},\qquad P_k=\kb{\tau_k}{\tau_k},
\]
and the unitary
\[
 B_C\ket{\tau_k}=\ket{k}.
\]
The program applies \(B_C\), then computational-basis Enter to all
clock names. It returns the \(q\)-bit code \(k\), not the names \(C\).

Define
\[
 D_k=\bra{\tau_k}_C\otimes I_X:
 H_C\otimes H_X\longrightarrow H_X.
\]
The record has diagonal observable algebra
\[
 \Alg_R=\left\{\sum_{k=0}^{D-1}a_k\kb{k}{k}_R:a_k\in\C\right\}.
\]

\begin{theorem}[Complete destructive clock instrument]
\label{thm:instrument}
The specified readout has channel
\begin{align}
 \Mclock(Z)
 &=\sum_{k=0}^{D-1}\kb{k}{k}_R\otimes D_kZD_k^\dagger
 \label{eq:instrument}\\
 &=\sum_{k=0}^{D-1}\kb{k}{k}_R\otimes
   \Tr_C\bigl[(P_k\otimes I_X)Z(P_k\otimes I_X)\bigr].
 \nonumber
\end{align}
It is completely positive and trace preserving. Its \(k\)-th
subnormalized matter block is
\[
 \sigma_k=D_k\rho D_k^\dagger,\qquad
 p_k=\Tr\sigma_k.
\]
For \(p_k>0\), the conditional matter state is \(\sigma_k/p_k\).
No normalized state is assigned to a zero-probability outcome.
Every identity holds after tensoring with \(\id_E\).
\end{theorem}

\begin{proof}
After applying \(B_C\), the destructive measurement bra for outcome
\(k\) is
\[
 (\bra{k}B_C)\otimes I_X=D_k.
\]
The Kraus operator including the classical output is
\[
 L_k=\ket{k}_R\otimes D_k.
\]
Consequently the channel is \eqref{eq:instrument}, and
\[
 \sum_kL_k^\dagger L_k
 =\sum_kP_k\otimes I_X=I_{CX}.
\]
This proves complete positivity and trace preservation.
The partial-trace expression follows by expanding in the clock basis.
The block probabilities and conditional states are the specified Born
semantics. Tensoring each operator identity with \(I_E\) proves the
reference versions.
\end{proof}

Theorem~\ref{thm:instrument} is the finite interface used in En09-A;
its conditioning block is not, merely by its form, a metric or a
stress tensor \cite{CheungEn09}.
% Online documentary source: ([enter.grok.me](https://enter.grok.me/browse/En09-A))

\subsection{Three interfaces, not one}

For comparison, define the coherent recording isometry
\[
 W_{\mathrm{rec}}
 =
 \sum_k P_k\otimes\ket{k}_A,
\]
where the output retains \(C\), adds a quantum carrier \(A\), and
retains \(X\). Define also the represented nondestructive instrument
\[
 \mathcal I^{\mathrm{nd}}(\rho)
 =
 \sum_k\kb{k}{k}_R\otimes
 (P_k\otimes I_X)\rho(P_k\otimes I_X).
\]

\begin{proposition}[Interface separation]
\label{prop:interfaces}
Among these three interfaces, only \(\Mclock\) removes \(C\) from the
returned quantum context. Moreover,
\[
 (\id_R\otimes\Tr_C)\mathcal I^{\mathrm{nd}}=\Mclock.
\]
Coherent recording returns a quantum carrier and is not itself a
classical recorded channel.
\end{proposition}

\begin{proof}
The assertions about returned names follow from the displayed domains
and codomains. The partial-trace identity is
Theorem~\ref{thm:instrument}. For a superposition of two clock basis
states, coherent recording contains off-diagonal terms between the
corresponding \(C\otimes A\) states. Hence it is not generally
block diagonal in \(A\).
\end{proof}

The equality after discarding \(C\) does not identify the larger
nondestructive interface with destructive Enter
\cite{CheungI,CheungEn03}.

%=====================================================================
\section{The dictionary and its calibration limits}
\label{sec:dictionary}
%=====================================================================

\subsection{Time labels and reading events}

Fix distinct ordered real numbers
\[
 t_0<t_1<\cdots<t_{D-1}.
\]
The stored object is \(k\). Its interpretation as \(t_k\) is classical
metadata. The finite operator
\[
 \clockobs=\sum_kt_kP_k
\]
is a pointer observable, not the continuum coordinate function \(T\).

There are two different admissible calibrations.

\begin{definition}[Event and history calibration]
\label{def:calibration}
\begin{enumerate}
\item \textbf{Event-time calibration.}
For a reading at the fixed event \(e\), an input is exactly calibrated
when
\begin{equation}
 p_k>0\quad\Longrightarrow\quad t_k=T(e).
 \label{eq:event-calibration}
\end{equation}
Thus the returned label is the coordinate of the reading event.

\item \textbf{History-label calibration.}
Choose a finite event map
\[
 \kappa:\{0,\ldots,D-1\}\longrightarrow M,
 \qquad T(\kappa(k))=t_k.
\]
A readout at an independently specified event \(e_{\mathrm r}\)
returns the label of \(\kappa(k)\). It need not return
\(T(e_{\mathrm r})\). In the examples below all \(\kappa(k)\) are
earlier points on the sender's static worldline.
\end{enumerate}
\end{definition}

\begin{proposition}[Exact event matching at a fixed event is deterministic]
\label{prop:fixed-event}
Under distinct-label calibration, \eqref{eq:event-calibration} holds
if and only if there is a unique \(j\) with \(t_j=T(e)\) and
\[
 \rho=(P_j\otimes I_X)\rho(P_j\otimes I_X).
\]
The readout then returns \(j\) with probability one.
If \(T(e)\notin\{t_k\}\), no normalized exactly calibrated input exists.
\end{proposition}

\begin{proof}
Distinctness permits at most one index \(j\) satisfying
\(t_j=T(e)\). Calibration and normalization give \(p_j=1\).
Put \(Q=I-P_j\otimes I_X\). Positivity and
\(\Tr(Q\rho)=0\) imply
\[
 0=\Tr(\rho^{1/2}Q\rho^{1/2})
   =\|Q\rho^{1/2}\|_{\mathrm{HS}}^2.
\]
Therefore \(Q\rho=\rho Q=0\), proving the support statement.
The converse follows immediately from \eqref{eq:instrument}.
\end{proof}

\begin{remark}[Why the distinction is necessary]
A coherent history with several supported readings cannot make the
single classical number \(T(e_{\mathrm r})\) equal several distinct
\(t_k\). Calling \(t_k\) a geometric label is legitimate after
\(\kappa\) is supplied. Calling every such outcome the actual time of
one fixed event would be false. Random classical selection among
different reading events is another valid construction, but the event
selection must then be included as classical input data.
\end{remark}

\subsection{The three dictionary maps}

\begin{definition}[Dictionary on admitted ports]
\label{def:dictionary}
The dictionary is the following explicit collection of maps.
\begin{enumerate}
\item \textbf{Clock map.}
The basis calibration \(B_C\), the finite map \(k\mapsto t_k\),
and either the event support condition
\eqref{eq:event-calibration} or the history-event map \(\kappa\).

\item \textbf{Rod map.}
At each cut, a surviving carrier name \(a\) has a prescribed spatial
location \(x(a)\in U\), with rod label \(\chi(x(a))\). In polar notation
the corresponding local labels are \((r,\alpha\phi,z)\).

\item \textbf{Causal map.}
The complete scheduled port graph is mapped to \(M\) so that every
directed dependency edge maps to a future-causal pair. Ancestors of
an output therefore map into its metric past.
\end{enumerate}
These maps are model data. They are not inferred from output
probabilities alone.
\end{definition}

A pre-read and post-read germ \(s_*^-\prec s_*^+\) describes semantic
order at one reading event. It need not denote two distinct geometric
times. In particular, strict semantic sequencing does not imply a
positive proper-time duration for an idealized local instruction.

\subsection{The five time-related objects}

The distinctions in XIX-A motivate the following ledger
\cite{CheungXIX}.

\begin{center}
\small
\renewcommand{\arraystretch}{1.13}
\begin{tabularx}{\linewidth}{@{}p{.35\linewidth}Y@{}}
\toprule
Object & Status in this construction\\
\midrule
Finite-time harmonic relational scalar &
A separate classical prescription; not used to determine the
conical lapse.\\
Inflaton clock and original slice-harmonic rods &
An optional cosmological attachment, conditional on full input
geometry; Appendix~\ref{app:lapse}.\\
Conical incidence parameter &
Metadata of a prescribed slicing family, not the finite clock
factor or an automatically normalized time coordinate.\\
Density ramp &
A prescribed regulator, not an independently evolving clock
degree of freedom.\\
Finite Enter clock \(C\) &
A live quantum factor, with a finite reading basis and an
explicit calibration to geometric labels.\\
\bottomrule
\end{tabularx}
\end{center}

\begin{lemma}[Elementary non-identifications]
\label{lem:nonidentifications}
In the present model class:
\begin{enumerate}
\item even a harmonic scalar profile does not determine the lapse;
\item a finite pointer is not the all-real coordinate \(T\);
\item an externally specified density ramp does not, by its
specification alone, supply an independent dynamical clock;
\item a slicing-family parameter is not identified with \(T\)
without an additional calibration.
\end{enumerate}
\end{lemma}

\begin{proof}
For (i), take any two positive constant lapses \(N=a\) and \(N=b\).
The scalar \(\xi=T\) is harmonic for both metrics:
the coefficients and volume density are independent of \(T\), so
\(\Box_{g_N}T=0\). The scalar profile is identical, but
\[
 -g_N^{-1}(\dd T,\dd T)=\frac{1}{c^2N^2}
\]
depends on the lapse.

For (ii), the pointer has at most \(D\) distinct eigenvalues, whereas
\(T\) takes every real value on \(M\).

For (iii), a fixed regulator
\(r_\varepsilon(T)=r(T/\varepsilon)\) adds a prescribed function of
external labels, not a new state variable or evolution equation.
It can be used in a calibration if further data are supplied; that
does not make an independent dynamical degree of freedom appear.

For (iv), a family of slices may be reindexed by any bijection of its
parameter interval without changing the collection of slices.
Neither the background metric nor the finite clock instrument selects
one such indexing.
\end{proof}

The original-versus-modified rod distinction is stronger than an
indexing issue: Appendix~\ref{app:rod-witness} gives a compact
perturbative witness separating their quadratic lapses.

\subsection{Nonuniqueness of the dictionary}

\begin{proposition}[Records do not uniquely determine the geometric dictionary]
\label{prop:nonunique}
Finite record laws and complete matter channels in this construction
do not determine the rod frame, the lapse, or the global deficit.
\end{proposition}

\begin{proof}
First, apply a spatial Euclidean isometry to the chosen developing
chart and all prescribed carrier locations. The abstract name
permutations and channels are unchanged, while their numerical rod
coordinates change.

Second, retain the same event labels, sites, and scheduled channels,
but compare the constant lapses \(N=1\) and \(N=2\).
Every edge satisfying
\(\|\Delta x\|\le c\Delta T\) is causal for both.
The same matrices produce exactly the same complete channels and
record probabilities. Nevertheless, a static interval of coordinate
duration \(\Delta T\) has proper duration \(\Delta T\) in the first
calibration and \(2\Delta T\) in the second.

Third, any two deficits in \((0,2\pi)\) admit regular developing patches
containing isometric sufficiently small Euclidean balls. Install the
same finite experiment in such balls. The complete channels coincide,
although the global circumference-to-radius ratios and conical gluing
angles differ.
\end{proof}

This is nonuniqueness relative to the supplied finite operational data.
It does not assert that all the displayed geometric configurations
are inequivalent under every possible change of coordinates.

\subsection{Signature is an additional choice}

\begin{proposition}[No equivariant signature change from an inner product alone]
\label{prop:signature}
For \(n\ge2\), no rule assigns to each positive-definite inner product
on an \(n\)-dimensional real vector space a Lorentzian bilinear form
on that same space, using only that inner product and equivariantly
under all linear changes of coordinates.
\end{proposition}

\begin{proof}
At the Euclidean inner product \(I\), equivariance makes the assigned
form invariant under every orthogonal transformation. Coordinate
reflections force its off-diagonal entries to vanish. Coordinate
permutations force its diagonal entries to coincide. It is therefore
\(\lambda I\), which is definite or degenerate, not Lorentzian.
\end{proof}

This reproduces the elementary signature refusal recorded in
Oai365-A \cite[Proposition \sourceid{prop:signature}]{CheungOai365}.
Here we add a time factor, a negative sign, and a time orientation
explicitly. A velocity diamond or polytope is not a substitute for
that choice.

%=====================================================================
\section{What destructive clock reading does and does not do}
\label{sec:clock}
%=====================================================================

\subsection{Consumption, nonreturn, and the surviving record}

\begin{proposition}[No universal quantum return]
\label{prop:no-return}
For \(D\ge2\), \(\Mclock\) has no channel left inverse on its full
\(C\otimes X\) input space.
\end{proposition}

\begin{proof}
Let
\[
 \ket{\chi_\pm}
 =\frac{\ket{\tau_0}\pm\ket{\tau_1}}{\sqrt2},
\]
and fix any matter state \(\eta_X\). The distinct inputs
\[
 \kb{\chi_+}{\chi_+}\otimes\eta_X,\qquad
 \kb{\chi_-}{\chi_-}\otimes\eta_X
\]
both have output
\[
 \frac12\bigl(\kb{0}{0}_R+\kb{1}{1}_R\bigr)\otimes\eta_X.
\]
A left inverse would send one output to two different inputs.
\end{proof}

\begin{remark}[The quantifier matters]
The proposition concerns the full clock--matter input space. It does
not forbid preparing a fresh clock in a known basis state from a
classical code. Indeed, on the block-diagonal class
\[
 \rho=\sum_kP_k\otimes\sigma_k,
\]
a channel can reconstruct the displayed density operator by preparing
\(\ket{\tau_k}\) conditional on \(k\). This is recovery of a classical
sector, not recovery of arbitrary clock coherence and not return of
the consumed names.
\end{remark}

\begin{proposition}[A record is not a residual-state certificate]
\label{prop:record-residual}
Identical clock record laws do not determine the surviving matter
state, even on exactly event-calibrated inputs.
\end{proposition}

\begin{proof}
Choose orthogonal matter vectors \(\ket{a},\ket{b}\). Both inputs
\[
 P_0\otimes\kb{a}{a},\qquad
 P_0\otimes\kb{b}{b}
\]
return the deterministic record \(0\). Their matter residuals are
orthogonal. More generally, postcomposing every selective block with
either of two different trace-preserving matter channels preserves
the clock probabilities while changing the residual instrument.
\end{proof}

Record compatibility and residual compatibility are therefore
different assertions, as emphasized in I, VIII-A, and En09-A
\cite{CheungI,CheungVIIIA,CheungEn09}.

\begin{proposition}[Forgetting the record]
\label{prop:forget}
For every input operator,
\[
 \Tr_R\Mclock(Z)=\Tr_C Z.
\]
This identity is reference preserving.
\end{proposition}

\begin{proof}
Writing
\(Z=\sum_{j,k}\kb{\tau_j}{\tau_k}\otimes Z_{jk}\),
both sides equal \(\sum_kZ_{kk}\). The same computation retains any
reference factor.
\end{proof}

The geometric coordinate has not disappeared: for example, an
unchanged classical record may remain stored on a static worldline
while \(T\) increases along that worldline. A fixed record of an
earlier reading is not a replacement for all later external time.

\subsection{No finite canonical commutator}

\begin{proposition}[Finite-dimensional CCR obstruction]
\label{prop:no-ccr}
For nonzero finite-dimensional \(H_C\) and \(\hbar>0\), no matrices
\(A,H\) satisfy
\[
 [A,H]=i\hbar I_C.
\]
\end{proposition}

\begin{proof}
The trace of a matrix commutator is zero. The trace of the right side
is \(i\hbar D\ne0\).
\end{proof}

Thus \(\clockobs\) is not an exact canonical time operator conjugate
to a finite clock Hamiltonian. The continuum coordinate \(T\) remains
a classical function on the supplied manifold.

\subsection{Basis change is not generally coordinate change}

Suppose a unitary \(U_C\) is applied before the calibrated readout.
Its effective pointer observable on the previous input is
\[
 U_C^\dagger\clockobs U_C.
\]

\begin{proposition}[When a basis change can be a label relabelling]
\label{prop:basis}
Assume the \(t_k\) are distinct. If
\[
 U_C^\dagger\clockobs U_C=f(\clockobs)
\]
for a function \(f\) on the finite spectrum, then \(U_C\) permutes the
one-dimensional eigenspaces of \(\clockobs\), up to phases.
Consequently, a general unitary basis change is not even a finite
relabeling of the original pointer, much less a spacetime coordinate
transformation.
\end{proposition}

\begin{proof}
The conjugated operator has simple spectrum. Hence \(f(\clockobs)\)
must also have distinct eigenvalues. Its eigenvectors are the
original \(\ket{\tau_k}\), whereas those of the conjugated operator
are \(U_C^\dagger\ket{\tau_k}\). Simplicity identifies these two
collections of one-dimensional eigenspaces by a permutation.

For a concrete failure, take \(D=2\), use the computational reading
basis, and apply Hadamard to the input \(\ket{0}\). The originally
deterministic reading becomes a two-outcome distribution.
A deterministic relabelling of the original reading cannot do this.
\end{proof}

Even an allowed finite relabelling requires more to define a smooth,
time-oriented coordinate transformation. If an actual coordinate
change \(T'=f(T)\), with \(f'>0\), is supplied, tensor transformation
gives in the zero-shift case
\[
 -c^2N(T,x)^2\dd T^2
 =
 -c^2
 \left(\frac{N(f^{-1}(T'),x)}
 {f'(f^{-1}(T'))}\right)^2\dd T'^2.
\]
No such transformation law follows from applying \(U_C\) alone.
A simultaneous passive transformation of states and observable
matrices is instead a change of representation.

\subsection{A nontrivial finite history readout}

The Page--Wootters literature motivates distinguishing extraction of
conditional blocks from additional dynamical hypotheses
\cite{PageWootters,CheungEn09}. The following finite example proves
exact conditioning without claiming a stationary-universe
construction.

Choose any finite-dimensional \(H_X\), a self-adjoint \(H_X^{\rm gen}\),
and
\[
 U_k=\exp(-it_kH_X^{\rm gen}/\hbar).
\]
Define the isometry
\begin{equation}
 J:H_X\longrightarrow H_C\otimes H_X,\qquad
 J\ket{\psi}
 =
 \frac{1}{\sqrt D}\sum_{k=0}^{D-1}
 \ket{\tau_k}\otimes U_k\ket{\psi}.
 \label{eq:history-isometry}
\end{equation}

\begin{proposition}[Complete history-conditioning identity]
\label{prop:history}
For every operator \(Z\) on \(H_X\),
\begin{equation}
 \Mclock(JZJ^\dagger)
 =
 \frac1D\sum_{k=0}^{D-1}
 \kb{k}{k}_R\otimes U_kZU_k^\dagger.
 \label{eq:history-channel}
\end{equation}
This equality holds with every untouched reference. A record-controlled
application of \(U_k^\dagger\) returns the original matter input,
including its reference correlations. It does not invert
\(\Mclock\) on the full clock--matter input space.
\end{proposition}

\begin{proof}
Orthogonality gives
\[
 J^\dagger J=\frac1D\sum_kU_k^\dagger U_k=I_X,
 \qquad D_kJ=D^{-1/2}U_k.
\]
Substitution into \eqref{eq:instrument} proves
\eqref{eq:history-channel}. Controlled inversion gives
\(D^{-1}\sum_k\kb{k}{k}\otimes Z\).
These are operator identities and remain true with a reference.
The inverse acts only on the encoded matter task; the phase-erasure
counterexample of Proposition~\ref{prop:no-return} still applies to
the full input space.
\end{proof}

For a concrete preparation, allocate the clock in \(\ket{\tau_0}\),
rotate it to \(D^{-1/2}\sum_k\ket{\tau_k}\), and apply the controlled
unitary \(\sum_kP_k\otimes U_k\). All these operations can be declared
at the sender's location before the reading.

Choose a static sender \(x_0\in U\), set
\[
 \kappa(k)=(t_k,x_0),
 \qquad T(e_{\mathrm r})>\max_kt_k,
\]
and use history-label calibration. The labels refer to the declared
points \(\kappa(k)\), not to different actual locations of
\(e_{\mathrm r}\). The displayed preparation does not, by itself,
prove that a physical matter trajectory visited all the conditional
states at those earlier events. It constructs a calibrated finite
history encoding.

\subsection{Fresh allocation and thermodynamic boundaries}

After consumption, a later reading may allocate a new factor \(C'\).
For example,
\[
 \eta_{RX}\longmapsto
 \eta_{RX}\otimes\kb{0}{0}_{C'}
\]
preserves the existing record and supplies a new blank. It neither
returns the old name nor recovers an unknown phase lost by Enter.

A channel specification does not determine a heat or work bill
\cite{CheungEn07}. For instance, resetting a memory by swapping with a
blank ancilla has the same reduced reset channel for every common
energy gap \(\varepsilon>0\). With
\[
 H_M=H_A=\varepsilon\kb{1}{1},
\]
an input excited-state probability \(p\) produces energy transfers
\[
 \Delta E_M=-\varepsilon p,\qquad
 \Delta E_A=\varepsilon p.
\]
The blank is consumed and the original information moves to the
ancilla. Restoring that ancilla, its preparation resources, and all
controls is a separate cyclic-restoration task.

%=====================================================================
\section{Schedule locality and the metric causal dictionary}
\label{sec:locality}
%=====================================================================

\subsection{A finite local execution model}

Let \(\Lambda\) be a finite set of static sites in \(U\), with
finite-dimensional site spaces \(H_a\).
Fix times
\[
 T_m=T_{\rm in}+m\delta,\qquad \delta>0.
\]
At each tick \(m\), a partition \(\mathcal P_m\) of \(\Lambda\) is
supplied. Each block obeys
\begin{equation}
 \|\chi(x_a)-\chi(x_b)\|_2\le c\delta
 \quad(a,b\in B,\ B\in\mathcal P_m).
 \label{eq:block-permission}
\end{equation}
A tick is one complete product channel
\[
 \Phi_m=\bigotimes_{B\in\mathcal P_m}\Phi_{m,B}.
\]
All quantum ports, records, controller memories, and resource wires
that are available during execution must be included in the finite
network. An external reference has no execution wire.

The access contract is stroboscopic: input and output access is
specified at tick boundaries. No continuous-time Hamiltonian or
within-tick apparatus dynamics is asserted. A two-site gate is not
treated as an instantaneous equal-time interaction. Its dependencies
run from input ports at \(T_m\) to output ports at \(T_{m+1}\).

For \(Y\subseteq\Lambda\), write
\[
 \Alg(Y)=
 \B\left(\bigotimes_{a\in Y}H_a\right)\otimes I_{\Lambda\setminus Y}.
\]
For a terminal cut \(n\), define backward sets by
\begin{equation}
 N_n(Y)=Y,\qquad
 N_m(Y)=\bigcup\{B\in\mathcal P_m:B\cap N_{m+1}(Y)\ne\varnothing\}.
 \label{eq:ancestors}
\end{equation}

\begin{theorem}[Complete schedule locality]
\label{thm:locality}
For the complete scheduled channels,
\begin{equation}
 \Phi_0^*\cdots\Phi_{n-1}^*(\Alg(Y))
 \subseteq\Alg(N_0(Y)).
 \label{eq:adjoint-locality}
\end{equation}
The terminal reduction to \(Y\) factors through the initial reduction
to \(N_0(Y)\). The factorization is reference preserving.
Changing a complete gate outside the ancestor subnetwork does not
change the terminal reduction, when wiring and included gates are
held fixed.
\end{theorem}

\begin{proof}
At one backward step, an output observable is the identity on every
block disjoint from its current support. The adjoint of a
trace-preserving channel is unital, so such a block remains an
identity. Other blocks can enlarge support only to their union.
Induction gives \eqref{eq:adjoint-locality}.

To obtain a reduced channel, append any fixed state on the initial
complement of \(N_0(Y)\), apply the complete evolution, and take the
output reduction to \(Y\). The adjoint inclusion shows that this
result is independent of the appended state and agrees with the
actual output for every input, including correlated inputs.
The same adjoint argument tensored with \(\id_E\) proves the
reference version.

On an excluded gate all pulled-back observables are identities.
Any replacement complete channel preserves those identities.
\end{proof}

This is the local proof underlying the prescribed-network comparisons
in En03-A, En05-A, and En06-A
\cite{CheungEn03,CheungEn05A,CheungEn06A}. It concerns complete
unconditioned channels with records retained, not normalized
postselection.

\begin{theorem}[Metric containment]
\label{thm:metric-containment}
Every directed scheduled dependency path in this execution model
maps to a future-causal curve in \(M\). In particular,
\[
 a\in N_m(\{b\})
 \quad\Longrightarrow\quad
 (T_m,x_a)\in J_g^-((T_n,x_b)).
\]
Operational influence under trace-preserving local interventions
is contained in the scheduled dependency relation and hence in
the prescribed metric causal relation.
\end{theorem}

\begin{proof}
Every elementary dependency connects two sites in one permitted
block across duration \(\delta\). Condition
\eqref{eq:block-permission} and Lemma~\ref{lem:local-causal}
supply a future-causal segment for that pair. Concatenation supplies
a causal curve along a dependency path.

If an intervention acts outside the ancestors, its adjoint acts on
an identity in the backward calculation, and therefore cannot
change an unconditioned terminal expectation. This proves the
influence inclusion.
\end{proof}

\begin{remark}[An upper bound, not equality]
Declare a two-site scheduled block but choose its channel to be the
identity. Its formal block ancestry includes both input sites for
each output, although no cross-site influence occurs. Thus the
scheduled region can strictly exceed the actual influence region.
An entangled input does not invalidate the theorem, but comparing
only individual marginals is insufficient for a claim involving
joint outputs and references.
\end{remark}

\subsection{Scheduler consistency does not select the Lorentz cone}

\begin{proposition}[A non-Lorentzian scheduler countermodel]
\label{prop:scheduler}
Acyclic dependency, finite local updates, and deterministic
schedule-independent records do not imply a Lorentzian generated cone.
\end{proposition}

\begin{proof}
Take events indexed by \(\N^4\), with directed edges
\(n\to n+e_i\), \(i=1,\ldots,4\).
Use a deterministic bit relay on any finite predecessor-closed
subnetwork: the recorded value is the Boolean OR of incoming values
and a local input bit. Every update occurs after its predecessors.
The graph is acyclic, has bounded degree, and every finite
experiment terminates. Associativity and determinism make records
independent of the legal interleaving. A source bit can be detected
at every reachable descendant by setting other inputs to zero.

The closed cone generated by edge displacements is
\[
 K=\R_{\ge0}^4.
\]
It is not linearly isomorphic to the four-dimensional Lorentz cone.
Indeed, \(K\) has four extreme rays, whereas the Lorentz cone has
infinitely many. An invertible linear map preserves extreme rays.
\end{proof}

This is a local counterexample to the proposed inference, not a
reproduction of every additional clause of En08-A's stronger
countermodel \cite{CheungEn08A}.

There is a relevant positive classification: a proper,
full-dimensional, homogeneous, strictly convex cone in dimension at
least three is linearly isomorphic to a Lorentz cone
\cite{ItoLourenco}. Here ``proper'' means closed, convex, pointed,
and with nonempty interior; strict convexity is understood in the
cone sense. This theorem classifies a cone satisfying its additional
hypotheses; it does not construct those hypotheses from a scheduler.
% Primary classification source: ([arxiv.org](https://arxiv.org/abs/1701.05801))

XXII-F records the additional extraction, invariance, approximation,
resource, and layer-separation obligations for an operational
instantiation \cite{CheungXXIIF}. We do not claim to discharge that
existence target. Declaring \eqref{eq:metric} is not a
metric-independent operational extraction rule.

%=====================================================================
\section{One finite complete-channel null transport protocol}
\label{sec:transport}
%=====================================================================

\subsection{The target-adapted contract}

Fix \(x_0,x_1\in U\), \(x_0\ne x_1\), with
\[
 L=\|\chi(x_1)-\chi(x_0)\|_2.
\]
Fix a departure time \(T_{\rm d}\). The prescribed target pair is
\[
 p=(T_{\rm d},x_0),\qquad
 q=(T_{\rm d}+L/c,x_1).
\]
The developed straight segment is future null.

Choose any integer \(m\ge1\). Install the \(m+1\) sites
\begin{equation}
 \chi(x_j)=
 \chi(x_0)+\frac{j}{m}\bigl(\chi(x_1)-\chi(x_0)\bigr),
 \qquad 0\le j\le m,
 \label{eq:chain-sites}
\end{equation}
where the notation \(x_m=x_1\) is understood.
Each site has a register \(A_j\cong\C^d\), with \(d=2^p\ge2\).
Set
\[
 a=\frac{L}{m},\qquad
 \delta=\frac{L}{mc}.
\]
At tick \(j=1,\ldots,m\), apply SWAP to
\(A_{j-1}\otimes A_j\) and identities to all other sites.

The station locations and tick duration are task data. They are not
selected by the unknown payload or inferred from its entanglement.

\begin{theorem}[Exact target-adapted null transport]
\label{thm:transport}
Prepare \(A_1,\ldots,A_m\) in the fixed blank
\(\sigma=\kb{0}{0}\), independently of the payload and any reference.
For every state \(\rho_{A_0E}\), after tick \(j\) the state is
\begin{equation}
 \rho_{A_jE}\otimes
 \bigotimes_{\substack{0\le i\le m\\i\ne j}}\sigma_{A_i},
 \label{eq:swap-invariant}
\end{equation}
under the fixed register identifications. Consequently:
\begin{enumerate}
\item the logical source-to-destination channel is exactly
\(\id_d\), including every untouched reference;
\item all non-output route registers end in a jointly
payload-independent blank state;
\item the arrival event is exactly \(q\);
\item every scheduled dependency is metric causal;
\item the protocol uses \(m\) two-register SWAP gates,
\(m+1\) payload-sized registers, and \(m\) ticks.
\end{enumerate}
\end{theorem}

\begin{proof}
At \(j=0\), \eqref{eq:swap-invariant} is the preparation.
At tick \(j\), register \(A_j\) is blank and independent.
SWAP exchanges it with the entire current payload factor
\(A_{j-1}\), including all correlations with \(E\).
The previous site becomes blank, proving the invariant by induction.

At \(j=m\), the destination-reference state is the original payload
state under the fixed identification. The remaining registers are
blank, proving the complete-channel assertions.

Each active pair has separation \(a=c\delta\). Hence its input-output
dependencies satisfy the local causal certificate. The payload
positions at successive ticks trace the prescribed null segment.
Finally \(m\delta=L/c\), and the gate and register counts follow
from the explicit schedule.
\end{proof}

\begin{remark}[Exactness and its boundary]
This theorem uses a target-adapted substrate. It is not a claim that
an arbitrary null displacement is exactly attained on one fixed
lattice. En08-B instead supplies resource-controlled approximation
for a prescribed null boundary on its declared lattice family
\cite{CheungEn08B}. Our construction has the same essential
reference-preserving permutation identity, but a different spatial
permission contract. No resource formula is transferred between
these models.
% Online transport comparison: ([enter.grok.me](https://enter.grok.me/browse/En08-B))
\end{remark}

For increasing \(m\), the gate radius and tick duration tend to zero,
while total travel time remains \(L/c\) and register count grows.
This family is not a continuum quantum field theory.

\begin{remark}[Geometric embedding versus microscopic implementation]
The theorem certifies the complete finite channel and the causal
embedding of its tick-boundary dependencies. It supplies no
continuous-time Hamiltonian that implements a two-site SWAP inside
one tick, no detector model for its intermediate dynamics, and no
field-theoretic derivation of the channel. Those are additional
physical-realization obligations, not consequences of the endpoint
permutation.
\end{remark}

\subsection{Composition with the clock instrument}

Let the surviving clock-conditioned matter be the register \(A_0\).
Write \(\mathcal T_m\) for the complete route channel with its blank
preparation, retaining all route outputs. Let \(\iota\) identify
\(A_0\) with \(A_m\).

\begin{corollary}[Complete clock-readout and transport identity]
\label{cor:clock-transport}
For every input state on \(C\otimes A_0\otimes E\), clock readout at
the sender followed by the route has complete output
\begin{equation}
 \sum_k\kb{k}{k}_{R_{\rm sender}}
 \otimes
 (\iota\otimes\id_E)(\sigma_{A_0E,k})
 \otimes
 \bigotimes_{i=0}^{m-1}\sigma_{A_i},
 \label{eq:composed-channel}
\end{equation}
where
\[
 \sigma_{A_0E,k}
 =
 (D_k\otimes I_E)\rho(D_k^\dagger\otimes I_E).
\]
The clock factor \(C\) is absent.
\end{corollary}

\begin{proof}
Apply Theorem~\ref{thm:transport} to each subnormalized positive block
of Theorem~\ref{thm:instrument}, and sum the blocks. Linearity gives
\eqref{eq:composed-channel}.
\end{proof}

The record in \eqref{eq:composed-channel} remains at the sender.
Writing a joint output state is not instantaneous delivery of that
record to the receiver. If a receiver operation is to depend on
\(k\), a causal classical record wire must be added, or the
record-controlled operation must be performed locally before departure.

\subsection{Resource and robustness ledgers}

With a \(q\)-qubit clock and \(p\)-qubit payload registers, the explicit
laboratory quantum peak is at most
\begin{equation}
 Q_{\rm lab}=q+p(m+1).
 \label{eq:peak}
\end{equation}
After clock consumption it is \(p(m+1)\) until the route residuals
are released. The clock record has \(q\) classical bits. An untouched
\(r\)-qubit reference contributes \(r\) to a separate external-storage
ledger, or to a total ledger if that convention is chosen. The
preinstalled schedule, site metadata, and blank preparations are
declared resources, not outputs supplied by the record.

The peak in \eqref{eq:peak} does not include undeclared synthesis
ancillas: the named local unitaries are permitted primitives of this
model. No compilation into a smaller universal gate library is
asserted.

\begin{proposition}[Finite composition error]
\label{prop:noise}
For compatible channels define
\[
 d_\diamond(\Phi,\Psi)=\frac12\|\Phi-\Psi\|_\diamond.
\]
If the implemented preparation, readout, \(m\) route gates, and output
interface differ from their ideal versions by at most
\(\varepsilon_{\rm prep}\), \(\varepsilon_{\rm read}\),
\(\varepsilon_1,\ldots,\varepsilon_m\), and
\(\varepsilon_{\rm out}\), respectively, then the complete output
channel differs by at most
\[
 \varepsilon_{\rm prep}+\varepsilon_{\rm read}
 +\sum_{j=1}^m\varepsilon_j+\varepsilon_{\rm out}.
\]
\end{proposition}

\begin{proof}
Insert and subtract hybrid compositions in which one component at a
time is replaced by its ideal version. The triangle inequality and
contraction of the diamond norm under pre- and postcomposition by
channels give the bound.
\end{proof}

This estimate is not exact causal containment of an erroneous device.
Nor does a per-gate error tending to zero imply a vanishing total
error for an arbitrarily growing number of gates.

%=====================================================================
\section{Dynamics without an invented source}
\label{sec:dynamics}
%=====================================================================

\subsection{The Einstein comparator}

For a declared smooth geometric metric \(q\), a declared covariant
source tensor \(\mathsf T_{\mu\nu}\), constants \(\Lambda\) and
\(\kappa_{\rm grav}\ne0\), define the comparison equation
\begin{equation}
 G_{\mu\nu}(q)+\Lambda q_{\mu\nu}
 =\kappa_{\rm grav}\mathsf T_{\mu\nu}.
 \label{eq:einstein}
\end{equation}
This equation is not a rule already contained in
\eqref{eq:instrument}. A finite conditional density matrix is not a
covariant spacetime tensor until a field, observable, state,
renormalization prescription, and map to tensor-valued distributions
are supplied.

\begin{proposition}[Conservation and trace do not imply sourcing]
\label{prop:not-source}
A symmetric, conserved, traceless tensor on a background need not
source that background through \eqref{eq:einstein}.
\end{proposition}

\begin{proof}
In a flat four-dimensional chart with metric
\(\eta=\diag(-1,1,1,1)\), let
\[
 \mathsf t_{\mu\nu}
 =\diag(3e,e,e,e),\qquad e>0.
\]
It is constant and symmetric, its covariant divergence vanishes, and
\(\eta^{\mu\nu}\mathsf t_{\mu\nu}=0\).
For \(\Lambda=0\), the left side of \eqref{eq:einstein} is zero while
the right side with \(\mathsf T=\mathsf t\) is nonzero.
\end{proof}

This is the four-dimensional version of the elementary refusal in
VIII-C; local Ward identities likewise do not, by themselves, supply
Einstein dynamics \cite{CheungVIIIC,CheungWard}.

\begin{proposition}[The precise constant-metric flatness statement]
\label{prop:constant-flat}
If the components of a nondegenerate metric are constant in all
spacetime coordinates, its connection coefficients and curvature
tensors vanish. Positive constant rescaling preserves this conclusion.
A family of such constant metrics has zero linearized Einstein tensor
for its parameter variation.
\end{proposition}

\begin{proof}
Every spacetime derivative of every component vanishes, so
\[
 \Gamma^\rho_{\mu\nu}
 =\frac12q^{\rho\sigma}
 \left(\partial_\mu q_{\sigma\nu}
      +\partial_\nu q_{\sigma\mu}
      -\partial_\sigma q_{\mu\nu}\right)=0.
\]
The curvature formulas then vanish. Constant rescaling and
differentiation with respect to an external family parameter retain
spacetime-constant components.
\end{proof}

The needed statement is the spacetime-constant one
\cite{CheungXXIIE}. The phrase ``spatially constant'' alone would be
insufficient: in units \(c=1\),
\[
 q=-\dd T^2+e^{2HT}
       (\dd x^2+\dd y^2+\dd z^2),\qquad H\ne0,
\]
is spatially homogeneous but has
\[
 R_{00}=-3H^2,\qquad R_{ij}=3H^2q_{ij},\qquad R=12H^2.
\]
These follow directly from
\(\Gamma^0_{ij}=Hq_{ij}\) and
\(\Gamma^i_{0j}=H\delta^i_j\).

\subsection{Application to the constructed background}

By Proposition~\ref{prop:holonomy}, the main conical metric has
\(G_{\mu\nu}=0\) on its regular part. This says nothing about a
distributional defect source on \(\W\), which is not constructed here.
Its conical gluing is prescribed rather than sourced by the Enter
residual.

More generally, if a cosmological Einstein--scalar background is
adjoined, its equations must be imposed on the geometric side.
A quotient or relational scalar construction does not insert a
missing metric source equation. This is the equation-space discipline
illustrated by III and VI \cite{CheungIII,CheungVI}.

\begin{remark}[No measurement backreaction law]
The experiment has no map
\[
 \sigma_k\longmapsto g_k
 \quad\hbox{or}\quad
 \sigma_k\longmapsto\mathsf T_{\mu\nu,k}
\]
in its specification. Consequently conditioning, changing a reading
basis, forgetting a record, or allocating a fresh clock does not
change the prescribed metric. Any such update would be an extension
of the model with its own consistency and dynamics obligations.
\end{remark}

%=====================================================================
\section{Existence and exact mathematical release}
\label{sec:release}
%=====================================================================

\begin{theorem}[Finite Enter spacetime: I-B-v1]
\label{thm:main}
Fix \(0<\mu<2\pi\), \(c>0\), a regular convex developing patch \(U\),
a clock dimension \(D=2^q\ge2\), and a payload dimension \(d=2^p\ge2\).
Then the following finite operational package exists.
\begin{enumerate}
\item Its smooth background is the time-oriented
\(3+1\)-dimensional Lorentzian manifold
\[
 M=\R_T\times C_\mu^\circ\times\R_z,\qquad
 g=-c^2\dd T^2+\dd r^2+\alpha^2r^2\dd\phi^2+\dd z^2.
\]
Its completion has the declared conical worldsheet \(\W\).

\item It has one designated live clock factor \(C\), finite matter
names, a location map, accessible algebras, and a complete finite
control and resource interface.

\item Destructive clock Enter is exactly
\eqref{eq:instrument}, including every untouched reference.
It consumes \(C\), writes a classical code, and retains the
subnormalized conditional matter block.

\item An explicit dictionary matches the code to geometric time
labels and the surviving carrier names to three spatial rods.
Exact equality with the actual reading-event coordinate holds on the
event-calibrated support class of
Proposition~\ref{prop:fixed-event}. Nontrivial multi-outcome history
readout instead uses the separate event map \(\kappa\).

\item Scheduled dependencies are contained in the prescribed metric
causal relation. Actual intervention influence is contained in
scheduled dependency.

\item Every prescribed straight future-null displacement inside the
patch has the finite target-adapted SWAP protocol of
Theorem~\ref{thm:transport}, with exact complete-channel correctness
and the resource ledger \eqref{eq:peak}. Its composition with clock
readout is \eqref{eq:composed-channel}.

\item The clock instrument has no universal left inverse on its full
input space, and no finite canonical time commutator is available.
The supplied metric and its time coordinate remain after clock
consumption.
\end{enumerate}
Signature, spatial chart, speed, lapse, calibration, and transport
permissions are inputs. Einstein dynamics are not derived.
\end{theorem}

\begin{proof}
The geometric assertions are
Propositions~\ref{prop:geometry} and~\ref{prop:holonomy}.
Choose any finite set of sites in \(U\), assign qubit registers,
and prescribe the finite primitives of
Definition~\ref{def:experiment}; this supplies item 2.

Theorem~\ref{thm:instrument} proves item 3.
For item 4, choose distinct \(t_k\) and a static sender \(x_0\).
The family
\[
 e_k=(t_k,x_0),\qquad
 \rho_k=P_k\otimes\rho_X
\]
is a nonempty event-calibrated family. A classical task input selects
which \(e_k\) and preparation are used. For a pure matter input these
are pure complete laboratory preparations. Thus actual event-time
matching is exhibited, not merely postulated.

For a separate nontrivial history experiment, choose
\(\kappa(k)=(t_k,x_0)\), read at a later fixed event, and use the
finite preparation \eqref{eq:history-isometry}.
Proposition~\ref{prop:history} gives all \(D\) outcomes and their
complete conditional outputs. The rod map is the explicit developing
chart \(\chi\).

Theorems~\ref{thm:locality} and~\ref{thm:metric-containment}
give item 5.
The sites \eqref{eq:chain-sites}, their partition schedule,
Theorem~\ref{thm:transport}, and
Corollary~\ref{cor:clock-transport} give item 6.
All preparations, clocks, site factors, records, and references have
the declared ledger.

Propositions~\ref{prop:no-return} and~\ref{prop:no-ccr} prove item 7.
The geometry is fixed metadata throughout the specification, so no
instrument operation removes or changes its coordinate function.
No source-extraction or metric-update map has been introduced.
\end{proof}

\begin{corollary}[The precise sense of time-axis manipulation]
\label{cor:manipulation}
In this model, Enter acts on a finite quantum encoding calibrated to
the metric's time axis. It can change the selected clock observable
through preceding local gates, extract a time label, condition
surviving matter, and enable declared causal classical control.
It does not consume a spacetime dimension, move the readout event
backward in \(T\), or derive a new metric.
\end{corollary}

\begin{proof}
The positive statements are the specified unitary and instrument
operations. Their outputs and locations are those already proved.
The negative statements follow because \(M,g,T\) are unchanged
classical components of the specification.
\end{proof}

\subsection{What remains outside the release}

The following six stronger implications have not been supplied.
\begin{enumerate}
\item A chart reconstructed from records without a predeclared
location map.
\item Lapse coefficients reconstructed from records without a
geometric clock and background prescription.
\item A Lorentz cone selected by scheduler consistency alone.
The conditional classification and XXII-F's operational existence
target remain distinct from our prescribed cone.
\item A covariant stress tensor extracted from the residual that
sources the same metric through a proved dynamics law.
\item The native quantum-lapse construction and its compatible joint
regulator removal.
\item A calibration to proper time derived from records rather than
supplied and checked against a declared metric and observer.
\end{enumerate}

The last item does not retract
Proposition~\ref{prop:geometry}: for the chosen unit-lapse static
observers, equality of coordinate and proper-time increments is
proved. What is not derived is the calibration of a physical device
to that ideal geometric clock.

\paragraph{Release.}
A prescribed spacetime and a finite Enter experiment have been paired
by written maps. This is a spacetime model under Enter in the stated
operational sense. It is not spacetime emergence, holographic
reconstruction, a quantum field theory, or a derivation of gravity.
No operational wormhole certificate is identified with a geometric
bridge. No catalogue family is a premise.

%=====================================================================
\appendix
\section{Optional original-rod lapse attachment}
\label{app:lapse}
%=====================================================================

This appendix gives a self-contained classical composition interface.
It does not replace the conical metric by a polynomial lapse or claim
to reproduce every native input of XX-A or Em02-E. Those manuscripts
supply their own frozen background, clock, rods, and quantum admission
contracts \cite{CheungXXA,CheungEm02E}.

Throughout the appendix set \(c=1\), and use \(\varepsilon\) for a
perturbation parameter. It is neither a schedule tick nor a clock
reading.

\subsection{The full geometric input}

Suppose a smooth Lorentzian metric \(q_\varepsilon\) and a scalar
clock \(\mathcal T_\varepsilon\) are supplied on a region where
\(\dd\mathcal T_\varepsilon\) is timelike. Define the positive raw lapse
\[
 N_\varepsilon
 =
 \left(
 -q_\varepsilon^{-1}
    (\dd\mathcal T_\varepsilon,\dd\mathcal T_\varepsilon)
 \right)^{-1/2}.
\]
Supply original slice-harmonic rods and suppose the resulting
coordinate map
\[
 \mathcal C_\varepsilon:
 x\longmapsto
 \bigl(\mathcal T_\varepsilon(x),Y_\varepsilon^1(x),
       Y_\varepsilon^2(x),Y_\varepsilon^3(x)\bigr)
\]
is locally invertible, with inverse \(\alpha_\varepsilon\).
Then the relational lapse is
\[
 n_\varepsilon=N_\varepsilon\circ\alpha_\varepsilon.
\]

In these relational coordinates the full metric has ADM form
\begin{equation}
 q_\varepsilon
 =
 -n_\varepsilon^2\dd\tau^2
 +\gamma_{\varepsilon,ij}
  (\dd y^i+\beta_\varepsilon^i\dd\tau)
  (\dd y^j+\beta_\varepsilon^j\dd\tau).
 \label{eq:adm}
\end{equation}
Indeed, writing the metric as
\(A\dd\tau^2+2B_i\dd\tau\dd y^i+\gamma_{ij}\dd y^i\dd y^j\),
set \(\beta^i=\gamma^{ij}B_j\) and
\(n^2=-A+\gamma^{ij}B_iB_j\); completing the square gives
\eqref{eq:adm}, and inversion gives \(q^{\tau\tau}=-n^{-2}\).

Thus a lapse alone does not specify the metric. Along a curve of
constant rods, proper-time rate is
\[
 \frac{\dd\tau_{\rm proper}}{\dd\tau}
 =
 \sqrt{n_\varepsilon^2
       -\gamma_{\varepsilon,ij}\beta_\varepsilon^i
                                      \beta_\varepsilon^j},
\]
when the curve is timelike. Along a normal curve
\(\dd y^i=-\beta_\varepsilon^i\dd\tau\), it is
\(n_\varepsilon\). Constant rods and normal observers must not be
silently identified.

If \(q_\varepsilon,\varphi_\varepsilon\) are required to satisfy
Einstein--scalar equations, those equations are additional geometric
input. Merely supplying \(n_\varepsilon\) does not impose them.

\subsection{The second-order composition formula}

Work in background-adapted coordinates for which the background
label map is the identity. Suppose the raw lapse has factorial jets
\[
 N_\varepsilon
 =N_0+\varepsilon N_1+\frac{\varepsilon^2}{2}N_2
 \pmod{\varepsilon^3},
 \qquad N_0=N_0(t).
\]
At fixed clock label, let the forward spatial rod map be
\[
 R_\varepsilon(\tau,x)
 =x+\varepsilon Y_1(\tau,x)
       +\frac{\varepsilon^2}{2}Y_2(\tau,x)
 \pmod{\varepsilon^3}.
\]
Let the inverse time map at fixed spatial argument be
\[
 \theta_\varepsilon(\tau,x)
 =\tau+\varepsilon u_1(\tau,x)
       +\frac{\varepsilon^2}{2}u_2(\tau,x)
 \pmod{\varepsilon^3}.
\]
The full inverse first inverts \(R_\varepsilon\), then substitutes
the resulting argument into \(\theta_\varepsilon\).

\begin{lemma}[Inverse-map coefficients]
\label{lem:inverse-jets}
Writing
\[
 \alpha_\varepsilon
 =\id+\varepsilon\alpha_1
       +\frac{\varepsilon^2}{2}\alpha_2
 \pmod{\varepsilon^3},
\]
one obtains
\[
 \alpha_1^0=u_1,\qquad \alpha_1^i=-Y_1^i,
\]
and
\[
 \alpha_2^0=u_2-2Y_1^i\partial_i u_1,\qquad
 \alpha_2^i=2Y_1^j\partial_jY_1^i-Y_2^i.
\]
\end{lemma}

\begin{proof}
Substitution into \(R_\varepsilon(\tau,S_\varepsilon(\tau,y))=y\)
gives
\[
 S_\varepsilon
 =y-\varepsilon Y_1
  +\frac{\varepsilon^2}{2}
       (2Y_1^j\partial_jY_1-Y_2).
\]
Expanding
\(\theta_\varepsilon(\tau,S_\varepsilon(\tau,y))\)
gives the time component, including
\(-2Y_1^i\partial_i u_1\).
\end{proof}

\begin{theorem}[Complete classical composition through second order]
\label{thm:lapse-composition}
Under these hypotheses,
\[
 n_\varepsilon
 =N_0+\varepsilon n_1+\frac{\varepsilon^2}{2}n_2
 \pmod{\varepsilon^3},
\]
where
\begin{align}
 n_1&=N_1+\dot N_0u_1,
 \label{eq:n1}\\
 n_2&=N_2+2u_1\dot N_1+\dot N_0u_2
          +\ddot N_0u_1^2
          -2Y_1^i\partial_i n_1.
 \label{eq:n2}
\end{align}
The dot is differentiation in the background coordinate used in this
expansion. The coefficient \(Y_2\) does not enter \(n_2\).
\end{theorem}

\begin{proof}
Taylor composition gives
\begin{align*}
 n_1&=N_1+\alpha_1^\mu\partial_\mu N_0,\\
 n_2&=N_2+2\alpha_1^\mu\partial_\mu N_1
       +\alpha_2^\mu\partial_\mu N_0
       +\alpha_1^\mu\alpha_1^\nu
                            \partial_\mu\partial_\nu N_0.
\end{align*}
Since \(N_0\) depends only on time, insertion of
Lemma~\ref{lem:inverse-jets} yields
\[
 n_2
 =N_2+2u_1\dot N_1+\dot N_0u_2+\ddot N_0u_1^2
 -2Y_1^i\partial_iN_1
 -2\dot N_0Y_1^i\partial_i u_1.
\]
The final two terms are
\(-2Y_1^i\partial_i(N_1+\dot N_0u_1)\), giving
\eqref{eq:n2}. The coefficient containing \(Y_2\) multiplies
\(\partial_iN_0=0\).
\end{proof}

The shape agrees with the supplied classical formulas in XX-A and
Em02-E. Equality of this algebraic shape is not identification of
every clock normalization, background coordinate, raw coefficient,
or boundary condition in those manuscripts.

A formal two-jet is also not, by itself, an exact positive lapse or
an exact Einstein solution. When an exact smooth input family is
supplied, positivity and local coordinate invertibility persist on
a fixed compact comparison region for sufficiently small
\(\varepsilon\). When only formal data are supplied, the conclusion
of Theorem~\ref{thm:lapse-composition} remains a formal one.

\subsection{The gauge composition identity}

\begin{proposition}[Gauge identity under the stated covariance contract]
\label{prop:gauge}
Suppose a coordinate change \(F\) acts on the supplied objects so that
\[
 N_\varepsilon'=N_\varepsilon\circ F,\qquad
 \mathcal C_\varepsilon'=\mathcal C_\varepsilon\circ F.
\]
Then
\[
 \alpha_\varepsilon'=F^{-1}\circ\alpha_\varepsilon,\qquad
 n_\varepsilon'=n_\varepsilon.
\]
The same statement holds coefficientwise modulo
\(\varepsilon^3\).
\end{proposition}

\begin{proof}
Invert the second covariance identity, then compose:
\[
 N_\varepsilon'\circ\alpha_\varepsilon'
 =
 N_\varepsilon\circ F\circ F^{-1}\circ\alpha_\varepsilon
 =
 N_\varepsilon\circ\alpha_\varepsilon.
\]
Formal composition gives the jet statement.
\end{proof}

This proposition assumes the stated covariance of the actual clock
and rod prescription, including its boundary data. It is not a
quantum Ward identity and does not establish those covariance
hypotheses for an unspecified prescription.

\subsection{A compact rod-detection witness}
\label{app:rod-witness}

The following local calculation reproduces the relevant distinction
in XIX-A without using it as an unexamined premise
\cite[Theorem \sourceid{thm:rod-detection}]{CheungXIX}.

Let \(a(t)>0\), and assume \(H=\dot a/a\ne0\) on a comparison interval.
Choose a nonconstant
\(\psi\in C_c^\infty(\R^3;\R)\) and a nonzero smooth time bump \(b\)
supported strictly inside that interval. Set
\[
 A=b(t)\psi,\qquad D=-b(t)\Delta\psi,
\]
and consider the off-shell metric jets
\begin{equation}
 q_\varepsilon
 =-(1+2\varepsilon A)\dd t^2
   +a(t)^2(1+2\varepsilon D)\delta_{ij}\dd x^i\dd x^j
 \pmod{\varepsilon^3},
 \qquad \mathcal T_\varepsilon=t.
 \label{eq:rod-witness-metric}
\end{equation}
Original rods are harmonic on the \(t\)-slices with asymptotic
Cartesian boundary condition. Modified rods are harmonic on the
slices of
\[
 \Theta_\varepsilon
 =t+\varepsilon D/H+O(\varepsilon^2),
\]
with the same spatial boundary condition. The time label used in
both relational lapse observables remains \(t\).

\begin{proposition}[First rod variations and a detected quadratic difference]
\label{prop:rod-witness}
Let \(Y_1\) and \(Z_1\) be the first original and modified rod
variations. Then
\[
 Y_1^i=b\,\partial_i\psi,\qquad Z_1^i=0,
\]
and the quadratic relational lapses satisfy
\begin{equation}
 n_{2,Y}-n_{2,Z}=-2b(t)^2|\nabla\psi|^2.
 \label{eq:rod-difference}
\end{equation}
A nonnegative compact spacetime test detects a strictly negative
pairing.
\end{proposition}

\begin{proof}
For
\(\gamma_{ij}=a^2(1+2\varepsilon D)\delta_{ij}\),
expansion of
\[
 \Delta_\gamma f
 =(\det\gamma)^{-1/2}
   \partial_j\bigl((\det\gamma)^{1/2}\gamma^{jk}
                                      \partial_kf\bigr)
\]
gives
\(\Delta_\gamma x^i=\varepsilon a^{-2}\partial_iD
+O(\varepsilon^2)\).
Hence
\[
 \Delta Y_1^i=-\partial_iD
             =b\,\partial_i\Delta\psi,
\]
whose decaying solution is \(b\,\partial_i\psi\).

A modified slice through background time \(\tau\) has
\(t=\tau-\varepsilon D/H+O(\varepsilon^2)\).
The change in \(a(t)^2\) cancels the first-order factor
\(2\varepsilon D\). Terms involving spatial derivatives of the
embedding are second order. Thus its induced spatial metric is
\(a(\tau)^2\delta_{ij}+O(\varepsilon^2)\), and the decaying first
harmonic correction is \(Z_1^i=0\).

The common raw lapse is
\[
 N_\varepsilon=\sqrt{1+2\varepsilon A}
 =1+\varepsilon A-\frac{\varepsilon^2}{2}A^2
 \pmod{\varepsilon^3}.
\]
The unchanged time coordinate gives \(u_1=u_2=0\),
\(n_1=A=b\psi\), and the same raw \(N_2\) for both prescriptions.
Equation~\eqref{eq:n2} therefore gives
\[
 n_{2,Y}-n_{2,Z}
 =-2(Y_1^i-Z_1^i)\partial_i n_1
 =-2b^2|\nabla\psi|^2.
\]
Choose a nonnegative compact test supported where
\(b\ne0\) and \(\nabla\psi\ne0\), positive on a nonempty open set.
Its pairing is strictly negative.
\end{proof}

The witness is off shell. It proves that the functional prescriptions
differ; it is not a claim that every on-shell cosmological
configuration distinguishes them.

\subsection{Quantum admission is not supplied by the classical formula}

Before a quantum version could be attached, one would need, at
minimum:
\begin{enumerate}
\item a normalized coupled free-field and gauge-fixed input with
stated domains and propagators;
\item an admissible state or covariance with positivity and the
required singularity conditions;
\item defined composite lapse and rod insertions, with the complete
quadratic terms and a specified renormalization freedom;
\item compatible gauge or Ward operations on those same objects,
including the transformation of all required jets;
\item uniform estimates controlling the nonlocal rod tails,
ultraviolet operations, and the proposed regulator removals in one
common output topology.
\end{enumerate}
This is an attachment checklist, not a claim to replace the full
native admission contracts of XX-A or Em02-E. Their classical
second-order release is not a quantum coefficient
\cite{CheungXXA,CheungEm02E}.

The clock channel \eqref{eq:instrument} neither defines an absent
field product nor proves any of these estimates. The
characteristic-product programme at I3/J2 is a different analytic
problem. No result of this appendix opens its downstream admission
exit.

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

\paragraph{Mathematical contribution.}
The release is the explicit prescribed-background finite model,
its calibration correction, its complete destructive instrument,
its locality certificate, and its target-adapted null transport
protocol. The appendix proves a classical lapse-composition formula
and a compact rod-detection witness under stated hypotheses.

\paragraph{Physical scope.}
The construction supplies a continuum classical metric and finite
operational semantics. It does not supply microscopic within-tick
dynamics, a quantum field-theoretic scaling limit, gravity-generated
entanglement, a Page curve, a horizon, an island, or holographic
emergence.

\paragraph{Source scope.}
Cheung's manuscripts are cited with identifiers and supplied versions.
Source references motivate comparisons; they are not automatic
transfers of models or proof hypotheses. Bibliographic descriptions
for internal manuscripts are not invented publication titles or
dates. The introductory philosophy and the directly accessed
companion records are documentary sources, not a proof-checking
service.

\paragraph{Analytic scope.}
No full inherited characteristic composition, joint section,
transverse estimate package, contact gluing, or quantum output
wavefront theorem is constructed. No inherited quantum coefficient
is admitted. XV-B's downstream exit is not opened, XVI-N is not
discharged, and O.1 remains \status{NOT\_CLOSED} within this release.

\paragraph{Verification scope.}
The source is self-contained for pdfLaTeX and requires no external
figures or bibliography database. No successful compilation,
proof-assistant check, independently executed symbolic calculation,
laboratory realization, referee approval, or journal acceptance
is asserted.

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

\bibitem{CheungPhil}
N.~Cheung,
\emph{The Philosophy of Enter: Destructive Readout, Quantum Recovery,
and the Refusal of Identification},
\sourceid{EntPhil-v2.tex}, version 2.
Introductory essay, Enter Calculus Companion.
\url{https://enter.grok.me/enter}.
Documentary orientation, not a theorem premise.

\bibitem{CheungI}
N.~Cheung,
\emph{Manuscript I}, \sourceid{I.tex}, version 2.
Supplied manuscript excerpts, Enter Calculus Companion.
Relevant items: destructive Enter characterization, pure finite
configurations, name ownership, and the distinction between
record and residual compatibility.
\url{https://enter.grok.me/}.

\bibitem{CheungEn03}
N.~Cheung,
\emph{Observation Boundaries for Destructive Quantum Readout and
External Geometry},
\sourceid{En03-A.tex}, version 1.
Enter Calculus Companion.
Relevant items: four geometric interface structures, three readout
interfaces, and prescribed-network locality.
\url{https://enter.grok.me/browse/En03-A}.

\bibitem{CheungVIIEF}
N.~Cheung,
\emph{Manuscripts VII-E and VII-F},
\sourceid{VII-E.tex} and \sourceid{VII-F.tex}, version 1.
Supplied manuscript excerpts, Enter Calculus Companion.
Relevant items: fixed conical geometry, oriented holonomy,
observation boundaries, and VII-F
Theorem \sourceid{thm:event}.
\url{https://enter.grok.me/}.

\bibitem{CheungVIIIB}
N.~Cheung,
\emph{Manuscript VIII-B},
\sourceid{VIII-B-v3.tex}, version 3.
Supplied manuscript excerpts, Enter Calculus Companion.
Relevant item: Theorem \sourceid{thm:dictionary}, the variable-deficit
spinless cone dictionary.
\url{https://enter.grok.me/}.

\bibitem{CheungXIX}
N.~Cheung,
\emph{Manuscript XIX-A},
\sourceid{XIX-A.tex}, version 1.
Supplied manuscript excerpts, Enter Calculus Companion.
Relevant items: separation of time prescriptions and
Theorem \sourceid{thm:rod-detection}.
\url{https://enter.grok.me/}.

\bibitem{CheungEn09}
N.~Cheung,
\emph{Realization of Relational Time by Destructive Enter on a
Clock Register},
\sourceid{En09-A.tex}, version 1.
Enter Calculus Companion.
Relevant items: complete clock instrument,
Propositions \sourceid{prop:no-return},
\sourceid{prop:record-not-residual}, and
\sourceid{prop:no-ccr}.
\url{https://enter.grok.me/browse/En09-A}.

\bibitem{CheungVIIIA}
N.~Cheung,
\emph{Manuscript VIII-A},
\sourceid{VIII-A-v3.tex}, version 3.
Supplied manuscript excerpts, Enter Calculus Companion.
Relevant item: complete instrument agreement versus classical
record agreement.
\url{https://enter.grok.me/}.

\bibitem{PageWootters}
D.~N.~Page and W.~K.~Wootters,
``Evolution without evolution: Dynamics described by stationary
observables,''
\emph{Physical Review D} \textbf{27} (1983), 2885--2892.
\url{https://doi.org/10.1103/PhysRevD.27.2885}.
Background comparison; the finite extraction identities in this
paper are proved explicitly.

\bibitem{CheungEn07}
N.~Cheung,
\emph{Manuscript En07-A},
\sourceid{En07-A.tex}, version 1.
Supplied manuscript excerpts, Enter Calculus Companion.
Relevant item: Proposition
\sourceid{prop:no-landauer-from-channel}.
\url{https://enter.grok.me/}.

\bibitem{CheungEn05A}
N.~Cheung,
\emph{Manuscript En05-A},
\sourceid{En05-A.tex}, version 1.
Supplied manuscript excerpts, Enter Calculus Companion.
Relevant item: Theorem \sourceid{thm:cone}, complete schedule
locality and metric envelopes.
\url{https://enter.grok.me/}.

\bibitem{CheungEn06A}
N.~Cheung,
\emph{Manuscript En06-A},
\sourceid{En06-A.tex}, version 1.
Supplied manuscript excerpts, Enter Calculus Companion.
Relevant item: Theorem \sourceid{thm:order}, complete wired-network
ancestry and its influence bound.
\url{https://enter.grok.me/}.

\bibitem{CheungEn08A}
N.~Cheung,
\emph{Manuscript En08-A},
\sourceid{En08-A.tex}, version 1.
Supplied manuscript excerpts, Enter Calculus Companion.
Relevant items: Theorem \sourceid{thm:main}, the scheduler
obstruction, and the conditional cone-representation statements.
\url{https://enter.grok.me/}.

\bibitem{ItoLourenco}
M.~Ito and B.~F.~Louren\c{c}o,
\emph{The \(p\)-cones in dimension \(n\ge3\) are not homogeneous
when \(p\ne2\)},
arXiv:1701.05801.
\url{https://arxiv.org/abs/1701.05801}.
Source for the homogeneous strictly convex cone classification;
not a construction of an operational selector.

\bibitem{CheungXXIIF}
N.~Cheung,
\emph{Manuscript XXII-F},
\sourceid{XXII-F.tex}, version 1.
Supplied manuscript excerpts, Enter Calculus Companion.
Relevant items: conditional assembly and completion
obligations F1--F6.
\url{https://enter.grok.me/}.

\bibitem{CheungEn08B}
N.~Cheung,
\emph{Resource-controlled operational approximation of null
boundary transport},
\sourceid{En08-B.tex}, version 1.
Enter Calculus Companion.
Relevant items: prescribed metric and speed, lattice approximation,
reference-preserving SWAP identity, resources, and error accounting.
\url{https://enter.grok.me/browse/En08-B}.

\bibitem{CheungVIIIC}
N.~Cheung,
\emph{Manuscript VIII-C},
\sourceid{VIII-C-v3.tex}, version 3.
Supplied manuscript excerpts, Enter Calculus Companion.
Relevant item: Proposition \sourceid{prop:not-einstein}.
\url{https://enter.grok.me/}.

\bibitem{CheungWard}
N.~Cheung,
\emph{Manuscripts Oai282-A and Oai282-B},
\sourceid{Oai282-A.tex} and \sourceid{Oai282-B.tex}, version 1.
Supplied manuscript excerpts, Enter Calculus Companion.
Local Ward comparisons, not Einstein dynamics or verification of
an OpenAI catalogue family.
\url{https://enter.grok.me/}.

\bibitem{CheungXXIIE}
N.~Cheung,
\emph{Manuscript XXII-E},
\sourceid{XXII-E.tex}, version 1.
Supplied manuscript excerpts, Enter Calculus Companion.
Relevant item: Theorem \sourceid{thm:flat};
used here only for spacetime-constant affine representatives.
\url{https://enter.grok.me/}.

\bibitem{CheungOai365}
N.~Cheung,
\emph{Manuscript Oai365-A},
\sourceid{Oai365-A.tex}, version 1.
Supplied manuscript excerpts, Enter Calculus Companion.
Relevant items: Propositions \sourceid{prop:signature}
and \sourceid{prop:lapse}.
The comparison manuscript is distinct from the separately attributed
OpenAI catalogue.
\url{https://enter.grok.me/}.

\bibitem{CheungIII}
N.~Cheung,
\emph{Manuscript III}, \sourceid{III.tex}, version 2.
Supplied manuscript excerpts, Enter Calculus Companion.
Classical equation-space and descent distinctions.
\url{https://enter.grok.me/}.

\bibitem{CheungVI}
N.~Cheung,
\emph{Manuscript VI}, \sourceid{VI.tex}, version 1.
Supplied manuscript excerpts, Enter Calculus Companion.
Relevant item: Theorem \sourceid{thm:descent}, classical
finite-time harmonic relational descent.
\url{https://enter.grok.me/}.

\bibitem{CheungXXA}
N.~Cheung,
\emph{Original-Rod Lapse: Classical Freeze and Quantum Admission
Contract},
\sourceid{XX-A.tex}, version 1.
Supplied manuscript excerpts, Enter Calculus Companion.
Relevant items: Theorem \sourceid{thm:lapse}, original-rod
classical second-order coefficients, and the separate quantum
admission contract.
\url{https://enter.grok.me/}.

\bibitem{CheungEm02E}
N.~Cheung,
\emph{Manuscript Em02-E},
\sourceid{Em02-E-v1.tex}, version 1.
Supplied manuscript excerpts, Enter Calculus Companion.
Relevant items: Theorem \sourceid{thm:lapse}, classical relational
lapse through second order, and the incomplete quantum
regulator-removal audit.
\url{https://enter.grok.me/}.

\bibitem{CheungXIVIJ}
N.~Cheung,
\emph{Manuscripts XIV-I and XIV-J},
\sourceid{XIV-I.tex} and \sourceid{XIV-J.tex}, version 1.
Supplied manuscript excerpts, Enter Calculus Companion.
Relevant items: XIV-I's constructed internal amplitude at I2;
the missing full characteristic map at I3/J2; XIV-J's
reconstruction-descent criterion, supported-sector
off-characteristic reference, and no-admission conclusion.
\url{https://enter.grok.me/}.

\bibitem{CheungStops}
N.~Cheung,
\emph{Supplied technical and standing-boundary records:
XV-B, XVI-N, and the O.1 ledger},
including \sourceid{XV-B.tex}, version 2, and
\sourceid{XVI-N.tex}, revision r12.
Enter Calculus Companion.
Documentary scope only: existence of technical interfaces is not
downstream quantum admission, and no gate is discharged by the
finite model of this paper.
\url{https://enter.grok.me/}.

\end{thebibliography}

\end{document}