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\hypersetup{
  pdftitle={En09-A: Realization of Relational Time by Destructive Enter on a Clock Register},
  pdfsubject={Relational quantum time, destructive instruments, and linear quantum resources},
  pdfkeywords={Enter calculus, Page-Wootters, quantum clocks, history states, quantum instruments}
}

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\title{\textbf{En09-A}\\[0.5em]
Realization of Relational Time by Destructive Enter\\
on a Clock Register}
\author{Ngo Cheung}
\date{}

\begin{document}
\maketitle

\begin{abstract}
We construct a finite operational realization of Page--Wootters
clock conditioning using destructive Enter on a named clock register.
The complete output consists of a classical clock code and the
surviving system; the measured clock names are consumed.
An explicit channel identity proves equality of the Born record law
and the unnormalized conditional system states with their
Page--Wootters counterparts, including arbitrary untouched references.
Under a separately stated covariance condition for the clock basis,
stationarity of the joint preparation implies unitary relational
evolution at the calibrated readings. An exact finite spectral
construction establishes that these hypotheses are jointly
inhabited.

The probability and update formulas have two distinct justifications:
they follow algebraically from the specified instrument, and,
when the full Enter characterization hypotheses are satisfied,
they are the measurement semantics selected by the previously
established operational characterization. We do not rederive that
characterization from abbreviated axiom names.

We also give a typed hybrid propagator, complete-channel stability
bounds, and refinement independence within an explicitly matched
class. The philosophy of Enter is expressed as an interface
discipline: a destructive record is not a surviving clock, a decoder,
a thermodynamic ledger, or a spacetime geometry. The construction
realizes relational extraction, not clock-free quantum gravity,
a dynamical causal cone, or spacetime emergence.
\end{abstract}

\paragraph{Keywords.}
Enter calculus; Page--Wootters conditioning; quantum clocks;
history states; destructive instruments; linear resources;
reference-stable channels.

%=====================================================================
\section{Introduction and exact scope}
\label{sec:introduction}
%=====================================================================

The Page--Wootters proposal describes system dynamics through
correlations with an internal clock in a stationary joint
description \cite{PageWootters}.
Finite clock history states and their circuit realizations provide
a concrete setting in which this conditioning can be studied
\cite{BoetteRossignoli,McClean,Diaz}.
The present manuscript supplies a particular operational extraction
interface for such correlations: destructive Enter on the clock.

Enter is not an alternative name for an arbitrary measurement.
Its defining access discipline is that the measured quantum names
do not survive as quantum outputs. The readout of those names is
classical. Unmeasured system names can nevertheless survive in the
same complete process. This distinction is central to Cheung's
Enter calculus and to the accompanying philosophy of destructive
readout and refusal of identification
\cite{CheungEnter,CheungI,CheungPhilosophy}.

Our claim is therefore an interface realization, not an identification
of all relational-time formalisms with a programming primitive.
Mathematical conditioning on a clock projector need not itself
consume a clock. The construction below chooses a destructive
implementation, proves its complete channel, and accounts for the
resource that this implementation consumes.

\paragraph{Three claims that must remain separate.}
\begin{enumerate}
\item \emph{Extraction:} a clock-conditioned state is returned as the
      residual system accompanying a classical record.
\item \emph{Dynamics:} those conditional states satisfy a specified
      evolution law. This requires clock covariance in addition to
      stationarity.
\item \emph{Foundational semantics:} the measurement rule is selected
      by an operational characterization inside the full strategy
      class of the Enter theorem.
\end{enumerate}
The first is an explicit channel identity. The second is proved
under stated mathematical hypotheses. The third invokes an
identified predecessor result and is not inferred merely by writing
down a Born-form instrument.

\paragraph{Non-claims installed at the outset.}
No theorem below constructs a residual matter clock, returns the
consumed clock as an unknown quantum state, produces an exact
all-real reading register, derives a metric or Einstein dynamics,
or supplies a Page curve. External calibration, preparation,
scheduling, and any subsequently supplied physical embedding remain
declared inputs. In particular, finite operational consistency does
not determine a Lorentz cone \cite[Theorem \texttt{thm:main}]
{CheungEn08A}.

The manuscript is independent of the unresolved characteristic
quantum-product programme described in the supplied XIV--XVI
records. A finite readout cannot supply an absent field-theoretic
coefficient, its representation, or its admission.

%=====================================================================
\section{Frozen native data and measurement semantics}
\label{sec:data}
%=====================================================================

\subsection{Registers, names, and interfaces}

Fix disjoint finite sets of quantum names \(C\) and \(S\), with
\[
 m=\abs{C}\geq 1,\qquad n=\abs{S},\qquad d=2^m.
\]
Their Hilbert spaces are
\[
 \HH_C=(\CC^2)^{\otimes m},
 \qquad
 \HH_S=(\CC^2)^{\otimes n}.
\]
A register \(Q\) denotes an arbitrary finite untouched reference.
It may be entangled with \(C\otimes S\). It is not available to the
clock-reading program.

Write
\[
 \ClockCodes=\{0,\ldots,d-1\},
 \qquad
 \HH_R=\CC^d.
\]
The output record algebra is the diagonal algebra in the
distinguished basis \(\{\ket{k}_R\}_{k\in\ClockCodes}\).
Thus a recorded output has the form
\[
 \Omega_{RSQ}
 =
 \sum_{k\in\ClockCodes}\kb{k}{k}_R\otimes \sigma_{SQ,k},
 \qquad
 \sigma_{SQ,k}\geq 0,
 \qquad
 \sum_k\Tr\sigma_{SQ,k}=1.
\]
This is a classical--quantum state, not a coherent quantum return of
the clock.

The frozen operational interfaces are:
\begin{enumerate}
\item a preparation of \(\rho_{CS}\), or a declared preparation
      channel with this output;
\item the accessible clock input \(C\);
\item the surviving system output \(S\);
\item the retained classical record \(R\);
\item all untouched references, acted on by \(\id_Q\).
\end{enumerate}
If a purification, auxiliary token, encoding apparatus, or fresh
clock supply is used, its ownership and access are declared
separately. Nothing in the readout theorem acquires a state-dependent
expectation token.

\subsection{Clock basis and calibration}

Choose an orthonormal clock basis
\[
 \{\ket{\tau_k}_C:k\in\ClockCodes\},
 \qquad
 P_k=\kb{\tau_k}{\tau_k}_C.
\]
Choose an injective calibration
\[
 c:\ClockCodes\longrightarrow\RR,\qquad c(k)=t_k,
 \qquad t_0=0.
\]
The stored object is the finite code \(k\). The real number \(t_k\)
is its declared interpretation; no finite register is asserted to
contain every real number exactly.

Let \(B_C\) be the calibrated basis-change unitary satisfying
\[
 B_C\ket{\tau_k}_C=\ket{k}_C.
\]
Reading the clock means applying \(B_C\), then destructive
computational-basis Enter to every name in \(C\).
If the clock basis is already the computational basis, \(B_C=I_C\).

The logical clock Hamiltonian need not be diagonal in the
computational presentation. Indeed, confusing an energy basis with
a time-reading basis can destroy the desired dynamical
interpretation; see Example~\ref{ex:energy-basis}.

\subsection{Stationarity and the stronger constraint}

Fix self-adjoint finite matrices \(H_C\) and \(H_S\), and put
\begin{equation}
 K=H_C\otimes I_S+I_C\otimes H_S.
 \label{eq:constraint}
\end{equation}
A density operator is \emph{stationary} when
\begin{equation}
 [K,\rho_{CS}]=0.
 \label{eq:stationarity}
\end{equation}
A stronger zero-constraint preparation satisfies
\begin{equation}
 K\rho_{CS}=0.
 \label{eq:strong-constraint}
\end{equation}
For a positive density operator, \eqref{eq:strong-constraint} means
that its support lies in \(\ker K\), and implies
\eqref{eq:stationarity}. For a pure preparation the stronger
condition is \(K\ket{\Psi}=0\).

These two conditions are not silently identified. The extraction
identity requires neither. Stationarity is used only in the
relational-evolution clause.

\begin{definition}[Sampled clock covariance]
\label{def:covariance}
The calibrated clock basis is covariant for the readings
\(\{t_k\}\) when
\begin{equation}
 P_k=U_C(t_k)P_0U_C(t_k)^\dagger,
 \qquad
 U_C(t)=e^{-iH_Ct/\hbar}.
 \label{eq:clock-covariance}
\end{equation}
This is an explicit compatibility condition between the clock
generator, reading basis, and external calibration.
\end{definition}

Covariant clock observables are an established ingredient of
relational quantum dynamics \cite{Trinity}. Here we require only
the finite sampled condition \eqref{eq:clock-covariance}; we do not
import an infinite-dimensional physical-inner-product or
Dirac-observable construction.

\subsection{The imported Enter characterization}

The distinction between a specified instrument and a derived
measurement semantics is important enough to make explicit.

\begin{hypothesis}[Applicability of the Enter characterization]
\label{hyp:enter}
Whenever a realization is described as using \emph{derived Enter
semantics}, its measurement strategy belongs to the full structural
class of Cheung's characterization theorem and satisfies its
composability, no-signaling, nullity, continuity, and typed record
requirements in their stated forms.

The imported conclusion is that, for a pure finite configuration
\(\ket{\Xi}\), a measured subset, and outcome \(k\), the destructive
component \(D_k\ket{\Xi}\) has probability
\[
 p_k=\norm{D_k\ket{\Xi}}^2,
\]
and, if \(p_k>0\), residual ray
\[
 \left[D_k\ket{\Xi}/\sqrt{p_k}\right].
\]
No residual is prescribed on a zero-probability branch.
\end{hypothesis}

This is the conclusion of Landmark I, version 2,
\texttt{thm:main}, with its stated strategy class
\cite{CheungI}; Cheung's journal article distinguishes specified
Born semantics from their subsequent operational characterization
\cite{CheungEnter}.
Hypothesis~\ref{hyp:enter} retains the full predecessor hypotheses:
state sufficiency, coherent dynamics, evaluation conventions,
finite configurations, and the precise record condition are not
replaced by informal slogans.

We use the characterization as a cited semantic result, not as a
new theorem proved from the five abbreviated requirement names.
An arbitrary laboratory apparatus is not certified to belong to
that class by this manuscript.

\begin{lemma}[Recorded density-operator semantics]
\label{lem:density-semantics}
Under Hypothesis~\ref{hyp:enter}, destructive readout in the clock
basis has unnormalized branch map
\begin{equation}
 \PhiBranch_k(X)=D_kXD_k^\dagger,
 \qquad
 D_k=\bra{\tau_k}_C\otimes I_S.
 \label{eq:branch-map}
\end{equation}
The same formula holds with every untouched reference.
\end{lemma}

\begin{proof}
For a pure input, multiplication of the normalized positive-branch
residual projector by its probability gives
\[
 \norm{D_k\ket{\Xi}}^2
 \frac{D_k\kb{\Xi}{\Xi}D_k^\dagger}
      {\norm{D_k\ket{\Xi}}^2}
 =
 D_k\kb{\Xi}{\Xi}D_k^\dagger.
\]
A zero component gives the zero operator, without assigning a
normalized residual.

For a mixed input, purify it with a reference \(E\), apply the pure
formula with \(D_k\otimes I_E\), and take the required reduced view.
The result is \(D_kXD_k^\dagger\). This expression depends only on
\(X\), so it is independent of the purification or ensemble
presentation. Keeping rather than tracing a reference gives the
same formula with its identity factor.
\end{proof}

Thus the density-operator channel used below is a consequence of the
imported semantics when Hypothesis~\ref{hyp:enter} applies. Independently,
it is an explicitly specified finite quantum instrument.

%=====================================================================
\section{The complete relational Enter theorem}
\label{sec:main}
%=====================================================================

\subsection{The instrument and the conditioning target}

Define the complete destructive clock-readout channel by
\begin{equation}
 \Mclock(X)
 =
 \sum_{k\in\ClockCodes}
 \kb{k}{k}_R\otimes D_kXD_k^\dagger.
 \label{eq:readout}
\end{equation}
Equivalently, in the computational presentation,
\begin{equation}
 \Mclock(X)
 =
 \sum_k\kb{k}{k}_R\otimes
 (\bra{k}_C\otimes I_S)
 (B_C\otimes I_S)X(B_C^\dagger\otimes I_S)
 (\ket{k}_C\otimes I_S).
 \label{eq:computational-readout}
\end{equation}

For a preparation \(\rho_{CS}\), define
\begin{align}
 \sigma_{S,k}
 &=
 \Tr_C\!\left[
 (P_k\otimes I_S)\rho_{CS}(P_k\otimes I_S)
 \right],
 \label{eq:pw-subnormalized}\\
 p_k&=\Tr\sigma_{S,k},
 \label{eq:pw-probability}\\
 \rho_S(t_k)&=\frac{\sigma_{S,k}}{p_k}
 \quad\text{when }p_k>0.
 \label{eq:pw-normalized}
\end{align}
Equation~\eqref{eq:pw-normalized} is the clock-conditioned
Page--Wootters state for this finite observable.

The appropriate \emph{channel} target is not the normalized quotient
in \eqref{eq:pw-normalized}. It is the linear recorded map
\begin{equation}
 \Jclock(X)
 =
 \sum_k\kb{k}{k}_R\otimes
 \Tr_C\!\left[
 (P_k\otimes I_S)X(P_k\otimes I_S)
 \right].
 \label{eq:ideal-relational-instrument}
\end{equation}
Keeping subnormalized blocks prevents postselection from being
mistaken for a deterministic channel.

\begin{theorem}[Relational Enter]
\label{thm:main}
For the frozen data of Section~\ref{sec:data}, the following
statements hold.
\begin{enumerate}
\item \textbf{Complete instrument identity.}
      The map \(\Mclock\) is completely positive and trace
      preserving, and
      \begin{equation}
       \Mclock=\Jclock
       \label{eq:exact-channel-identity}
      \end{equation}
      as linear maps on all clock--system operators. This equality
      remains valid after tensoring with \(\id_Q\) for every
      untouched reference.

\item \textbf{Record law and residual.}
      On input \(\rho_{CS}\), the probability of code \(k\), hence of
      calibrated reading \(t_k\), is
      \begin{equation}
       p_k=\Tr\!\left[(P_k\otimes I_S)\rho_{CS}\right].
       \label{eq:born-clock}
      \end{equation}
      On a positive branch, the surviving system is exactly
      \(\rho_S(t_k)\) of \eqref{eq:pw-normalized}.
      Under Hypothesis~\ref{hyp:enter}, these are the probabilities
      and updates selected by the Enter characterization, not an
      additional measurement postulate introduced here.

\item \textbf{Linear consumption.}
      The readout consumes every measured quantum name in \(C\).
      Its quantum output context contains \(S\), not \(C\).
      A continuation in the same process cannot use a consumed
      clock name as a quantum input.

\item \textbf{Relational evolution under covariance.}
      If \(\rho_{CS}\) satisfies \eqref{eq:stationarity} and the
      reading basis satisfies \eqref{eq:clock-covariance}, then
      \begin{equation}
       \sigma_{S,k}
       =
       U_S(t_k)\sigma_{S,0}U_S(t_k)^\dagger,
       \qquad
       U_S(t)=e^{-iH_St/\hbar}.
       \label{eq:subnormalized-evolution}
      \end{equation}
      In this complete covariant basis, \(p_k=1/d\), and therefore
      \begin{equation}
       \rho_S(t_k)
       =
       U_S(t_k)\rho_S(0)U_S(t_k)^\dagger.
       \label{eq:relational-evolution}
      \end{equation}

\item \textbf{Complete operational realization.}
      The record, surviving system, and reference extension
      realize precisely the declared relational extraction.
      Discarding the record yields
      \begin{equation}
       \Tr_R[(\Mclock\otimes\id_Q)(X)]
       =
       \Tr_C X.
       \label{eq:drop-record}
      \end{equation}
      No quantum clock output or spacetime identification is part
      of this channel.
\end{enumerate}
\end{theorem}

\begin{proof}
Define Kraus operators from \(\HH_C\otimes\HH_S\) to
\(\HH_R\otimes\HH_S\) by
\[
 A_k=\ket{k}_R\otimes D_k.
\]
Then \(\Mclock(X)=\sum_k A_kXA_k^\dagger\), which is completely
positive. Orthonormality of the clock basis gives
\[
 \sum_k A_k^\dagger A_k
 =
 \sum_k P_k\otimes I_S
 =
 I_C\otimes I_S.
\]
Hence \(\Mclock\) is trace preserving.

For an arbitrary operator \(X\), expand in the clock basis:
\[
 X=\sum_{j,\ell}
 \kb{\tau_j}{\tau_\ell}_C\otimes X_{j\ell}.
\]
Both
\[
 D_kXD_k^\dagger
 \quad\text{and}\quad
 \Tr_C[(P_k\otimes I_S)X(P_k\otimes I_S)]
\]
equal \(X_{kk}\). This proves
\eqref{eq:exact-channel-identity} on all operators, not only on the
frozen preparation. Replacing \(I_S\) by \(I_S\otimes I_Q\) proves
the reference statement.

The \(R\)-marginal is
\[
 \Tr_S\Mclock(\rho_{CS})
 =
 \sum_k p_k\kb{k}{k}_R.
\]
Cyclicity of the full trace and \(P_k^2=P_k\) give
\[
 p_k
 =
 \Tr[(P_k\otimes I_S)\rho_{CS}(P_k\otimes I_S)]
 =
 \Tr[(P_k\otimes I_S)\rho_{CS}],
\]
proving \eqref{eq:born-clock}. Dividing the positive \(k\)-block by
its trace gives exactly \eqref{eq:pw-normalized}.
Lemma~\ref{lem:density-semantics} supplies the semantic attribution
under Hypothesis~\ref{hyp:enter}.

For consumption, the register-level readout has the first-order
interface
\begin{equation}
 \Qubit^{\otimes m}\otimes\Qubit^{\otimes n}
 \lto
 \Bit^{\otimes m}\otimes\Qubit^{\otimes n}.
 \label{eq:linear-interface}
\end{equation}
It is implemented by the calibrated basis change and destructive
Enter on the \(m\) clock names. Each elementary readout removes its
measured name from the quantum context. After all \(m\) readouts,
the surviving quantum context is \(S\). Tensor permutation and
classical packaging do not recreate a quantum name. This proves
the linear-resource assertion. The system output in
\eqref{eq:linear-interface} is unmeasured residual output; it is not
a quantum return of any name in \(C\).

To prove the dynamical clause, fix \(t=t_k\), write
\(U_C=U_C(t)\), \(U_S=U_S(t)\), and use stationarity:
\[
 (U_C\otimes U_S)\rho_{CS}(U_C^\dagger\otimes U_S^\dagger)
 =
 \rho_{CS}.
\]
Consequently,
\begin{equation}
 (U_C^\dagger\otimes I_S)\rho_{CS}(U_C\otimes I_S)
 =
 (I_C\otimes U_S)\rho_{CS}(I_C\otimes U_S^\dagger).
 \label{eq:stationarity-rearranged}
\end{equation}
For partial trace over \(C\), the two-projector expression in
\eqref{eq:pw-subnormalized} equals the corresponding single-projector
expression. Using covariance and unitary invariance of the partial
trace,
\begin{align*}
 \sigma_{S,k}
 &=
 \Tr_C[(U_CP_0U_C^\dagger\otimes I_S)\rho_{CS}]\\
 &=
 \Tr_C[(P_0\otimes I_S)
       (U_C^\dagger\otimes I_S)\rho_{CS}(U_C\otimes I_S)]\\
 &=
 U_S\Tr_C[(P_0\otimes I_S)\rho_{CS}]U_S^\dagger\\
 &=
 U_S\sigma_{S,0}U_S^\dagger.
\end{align*}
Thus all \(p_k\) coincide. Their sum is one and there are \(d\)
outcomes, so \(p_k=1/d\). Normalization proves
\eqref{eq:relational-evolution}.

Finally, the clock-basis expansion gives
\[
 \Tr_R\Mclock(X)=\sum_kX_{kk}=\Tr_CX.
\]
The same calculation includes \(Q\). The complete channel therefore
returns exactly the record and residual specified above. Its
codomain has no quantum \(C\) factor, and no geometric object occurs
in its definition.
\end{proof}

\begin{remark}[What is untouched under reference extension]
The operation acts as \(\id_Q\), but the \emph{conditional} reference
state may change when a clock outcome is selected. Such steering is
compatible with \eqref{eq:drop-record}: the unconditional reference
marginal is unchanged.

The relational-evolution assertion on \(S\otimes Q\) additionally
requires stationarity of the full preparation under
\(K\otimes I_Q\). Stationarity of the \(CS\)-marginal alone does not
imply stationarity of every extension.
\end{remark}

\begin{remark}[Extraction is weaker than dynamics]
The channel identity holds for every preparation and every calibrated
orthonormal basis. Stationarity alone does not make an arbitrary
basis a time observable. Covariance is the extra hypothesis that
justifies the Schr\"odinger-type interpretation in
\eqref{eq:relational-evolution}.
\end{remark}

\begin{example}[An energy readout is not automatically a time readout]
\label{ex:energy-basis}
Take one clock qubit and one system qubit, with
\[
 H_C=-\hbar\omega\kb{1}{1},
 \qquad
 H_S=\hbar\omega\kb{1}{1},
 \qquad
 \ket{\Psi}=\frac{\ket{00}+\ket{11}}{\sqrt2},
 \qquad \omega>0.
\]
Then \(K\ket{\Psi}=0\). Enter in the clock energy basis produces
system states \(\ket0\) and \(\ket1\), each with probability \(1/2\).
However, \(U_S(t)\kb0 0U_S(t)^\dagger=\kb0 0\) for every \(t\);
it cannot turn the first conditional state into the second.

The problem is not failure of the extraction identity. It is failure
of the proposed basis to satisfy clock covariance for two distinct
readings. Changing to the clock basis
\(\ket{\tau_0}=(\ket0+\ket1)/\sqrt2\),
\(\ket{\tau_1}=(\ket0-\ket1)/\sqrt2\), with
\(t_1=\pi/\omega\), produces the conditional system states
\(\ket+\) and \(\ket-\), which do satisfy the asserted evolution.
\end{example}

%=====================================================================
\section{An exact finite stationary realization}
\label{sec:inhabitation}
%=====================================================================

The theorem's covariance and constraint assumptions are compatible,
but not automatic. We now construct a nonempty exact class.

\begin{proposition}[Commensurate-spectrum realization]
\label{prop:spectral-realization}
Let \(d=2^m\), \(\omega>0\), and let
\(\{\Pi_a\}_{a=0}^{d-1}\) be mutually orthogonal projections on
\(\HH_S\) with \(\sum_a\Pi_a=I_S\); zero projections are allowed.
Set
\[
 H_S=\hbar\omega\sum_{a=0}^{d-1}a\Pi_a.
\]
Choose a clock energy basis \(\{\ket{e_a}\}_{a=0}^{d-1}\) and define
\begin{align}
 H_C&=-\hbar\omega\sum_{a=0}^{d-1}a\kb{e_a}{e_a},
 \label{eq:clock-spectrum}\\
 \Delta&=\frac{2\pi}{d\omega},
 \qquad t_k=k\Delta,
 \label{eq:spectral-calibration}\\
 \ket{\tau_k}
 &=
 \frac1{\sqrt d}
 \sum_{a=0}^{d-1}e^{2\pi iak/d}\ket{e_a},
 \label{eq:fourier-clock}\\
 W\ket{\phi}
 &=
 \sum_{a=0}^{d-1}\ket{e_a}\otimes\Pi_a\ket{\phi}.
 \label{eq:stationary-encoding}
\end{align}
Then \(W:\HH_S\to\HH_C\otimes\HH_S\) is an isometry,
\(KW=0\), and the clock basis is covariant at the readings \(t_k\).
Moreover,
\begin{equation}
 W
 =
 \frac1{\sqrt d}
 \sum_{k=0}^{d-1}\ket{\tau_k}\otimes U_S(t_k),
 \label{eq:history-encoding}
\end{equation}
and, for every operator \(X\),
\begin{equation}
 \Mclock(WXW^\dagger)
 =
 \frac1d\sum_{k=0}^{d-1}
 \kb{k}{k}_R\otimes U_S(t_k)XU_S(t_k)^\dagger.
 \label{eq:spectral-recorded-channel}
\end{equation}
All identities hold with every reference.
\end{proposition}

\begin{proof}
Orthogonality and completeness of the projections imply
\[
 W^\dagger W=\sum_a\Pi_a=I_S.
\]
On the term \(\ket{e_a}\otimes\Pi_a\ket{\phi}\), the clock and
system energies sum to zero, so \(KW=0\).

The finite Fourier identity
\[
 \frac1d\sum_{a=0}^{d-1}e^{2\pi ia(k-\ell)/d}
 =
 \delta_{k\ell}
\]
proves orthonormality of \(\{\ket{\tau_k}\}\). The sign in
\eqref{eq:clock-spectrum} gives
\[
 U_C(t_k)\ket{\tau_0}=\ket{\tau_k},
\]
hence covariance.

Expanding the right side of \eqref{eq:history-encoding}, using
\[
 U_S(t_k)=\sum_a e^{-2\pi iak/d}\Pi_a,
\]
and summing over \(k\), leaves precisely \eqref{eq:stationary-encoding}.
Applying \(D_k\) to \eqref{eq:history-encoding} gives
\(D_kW=d^{-1/2}U_S(t_k)\). Substitution in
\eqref{eq:readout} proves \eqref{eq:spectral-recorded-channel}.
Tensoring the operator identities with an identity proves the
reference versions.
\end{proof}

\begin{remark}[Spectral matching and periodicity are explicit]
This construction is exact for the stated commensurate spectrum.
It does not encode an arbitrary prescribed Hamiltonian into the
same finite additive constraint without an additional approximation
or spectral-matching theorem. Its readings are periodic with period
\(2\pi/\omega\); an absolute winding number would require an
additional classical interface.

The negative clock generator in \eqref{eq:clock-spectrum} is a
mathematical constraint convention, not a thermodynamic claim.
One may shift its spectrum upward and replace \(K=0\) by a fixed
total-energy constraint without changing the covariance or readout
calculation.
\end{remark}

\begin{proposition}[No finite canonical time commutator]
\label{prop:no-ccr}
For nonzero finite-dimensional \(\HH_C\), no matrices \(T_C,H_C\)
satisfy
\[
 [T_C,H_C]=i\hbar I_C.
\]
\end{proposition}

\begin{proof}
The trace of a matrix commutator is zero, whereas the trace of
\(i\hbar I_C\) is \(i\hbar\dim\HH_C\neq0\).
\end{proof}

This obstruction does not contradict
Proposition~\ref{prop:spectral-realization}. Sampled covariance of
a finite periodic reading basis is not an exact canonical
commutation relation.

%=====================================================================
\section{Quantum-computing analogs and selective extensions}
\label{sec:analogs}
%=====================================================================

\subsection{Circuit history states}

History-state algorithms are established quantum-computing
constructions \cite{McClean,BoetteRossignoli,Diaz}. Their connection
to Enter can be expressed without presuming that every history
state satisfies the additive constraint \eqref{eq:constraint}.

\begin{corollary}[Circuit-history extraction]
\label{cor:circuit-history}
Let \(V_0,\ldots,V_{d-1}\) be unitaries on \(\HH_S\), and define
\[
 W_V=\frac1{\sqrt d}\sum_{k=0}^{d-1}\ket{k}_C\otimes V_k.
\]
Then \(W_V\) is an isometry, and computational-basis Enter on the
clock realizes the channel
\begin{equation}
 X\longmapsto
 \frac1d\sum_{k=0}^{d-1}\kb{k}{k}_R\otimes V_kXV_k^\dagger.
 \label{eq:circuit-extraction}
\end{equation}
The clock names are consumed and the system names survive.
\end{corollary}

\begin{proof}
Orthogonality of the clock basis gives
\[
 W_V^\dagger W_V=\frac1d\sum_kV_k^\dagger V_k=I_S.
\]
The \(k\)-branch destructive component is
\[
 (\bra{k}\otimes I_S)W_V=d^{-1/2}V_k.
\]
Equation~\eqref{eq:circuit-extraction} follows from
Theorem~\ref{thm:main}. Its resource assertion is the same typing
calculation.
\end{proof}

A preparation interface for \(W_V\) consists of
\[
 \ket{+}^{\otimes m}\otimes\ket{\phi}
 \quad\longmapsto\quad
 \left(\sum_k\kb{k}{k}_C\otimes V_k\right)
 \bigl(\ket{+}^{\otimes m}\otimes\ket{\phi}\bigr).
\]
The controlled unitaries are declared operations. Exact synthesis
from a particular finite gate library, its approximation error,
and its storage cost are separate implementation obligations.

If \(V_{k+1}=G_kV_k\), define maps
\[
 L_k=
 \frac1{\sqrt2}
 \left(\bra{k+1}\otimes I_S-\bra{k}\otimes G_k\right),
 \qquad 0\leq k<d-1.
\]
Then the positive propagation Hamiltonian
\begin{equation}
 H_{\mathrm{prop}}=\sum_{k=0}^{d-2}L_k^\dagger L_k
 \label{eq:propagation-H}
\end{equation}
annihilates \(W_V\), because \(L_kW_V=0\).
This is a legitimate history-state constraint, but generally not
the noninteracting sum \(H_C+H_S\). A propagation constraint is not
silently substituted for the frozen additive constraint.

\subsection{The grounded comparison}

\begin{table}[tbp]
\centering
\small
\begin{tabularx}{\linewidth}{@{}p{0.17\linewidth}YY@{}}
\toprule
Aspect & Relational quantum description & Enter realization\\
\midrule
Time label &
A calibrated clock outcome \(t_k\) indexing a conditional state. &
A finite classical code \(k\), interpreted by the declared map
\(c(k)=t_k\).\\[4pt]

Extraction &
Conditioning on the clock projector \(P_k\). &
A calibrated basis change followed by destructive Enter; every
measured clock name is consumed.\\[4pt]

Joint description &
A clock--system history state; an additive stationary constraint
when its compatibility hypotheses hold. &
A declared history-state preparation and a finite recorded
readout channel.\\[4pt]

System evolution &
The family \(\rho_S(t_k)\); unitary evolution under sampled
covariance and stationarity. &
The surviving system block conditioned on the classical record.\\[4pt]

Measurement &
A specified instrument or represented POVM. &
Enter, together with declared coherent preprocessing and complete
classical coarse-graining.\\[4pt]

Resource &
The clock's quantum degrees of freedom. &
Linear quantum access names, with explicit consumption and residual
ownership.\\[4pt]

Born rule &
Probabilities associated with the clock effects. &
The same probabilities; their derived status requires the full
Enter characterization hypotheses.\\[4pt]

Non-claims &
Clock conditioning is not itself a derivation of spacetime geometry. &
The record is not a clock-return decoder, and the finite channel
does not construct spacetime emergence.\\
\bottomrule
\end{tabularx}
\caption{Analogs grounded in Theorem~\ref{thm:main} and
Corollary~\ref{cor:circuit-history}. A correspondence between
interfaces is not an identity between physical theories.}
\label{tab:analogs}
\end{table}

\subsection{Coarse records, failure flags, and local POVMs}

\begin{proposition}[Complete classical coarse-graining]
\label{prop:coarse}
Let \(q:\ClockCodes\to\mathsf Y\) be a declared finite
coarse-graining. Reporting only \(y=q(k)\), while retaining \(S\),
gives branch maps
\[
 \PhiBranch_y^q(X)=\sum_{k:q(k)=y}D_kXD_k^\dagger.
\]
Their sum is trace preserving. On a positive branch,
\[
 \rho_S(y)
 =
 \frac{\sum_{k:q(k)=y}p_k\rho_S(t_k)}
      {\sum_{k:q(k)=y}p_k}.
\]
This is generally a mixture of fine conditional states, not the
state at a single exact time.
\end{proposition}

\begin{proof}
Classical relabeling adds the corresponding diagonal output blocks.
Completeness follows by summing over all \(y\). The normalized formula
is division of the resulting subnormalized block by its trace.
\end{proof}

Failure or rejection is treated in the same way: its flag is part
of the complete output alphabet. If the rejected residual is
discarded or replaced, that action is an additional declared
channel. A successful normalized branch alone is not the complete
experiment.

For a local clock instrument with Kraus operators
\(\{F_{y,a}:\HH_C\to\HH_{C'}\}_a\), suppose the entire output
\(C'\) is also discarded or consumed, while \(S\) is untouched.
Its residual system branch is
\begin{equation}
 \sum_a\Tr_{C'}\!\left[
 (F_{y,a}\otimes I_S)X(F_{y,a}^\dagger\otimes I_S)
 \right]
 =
 \Tr_C[(E_y\otimes I_S)X],
 \quad
 E_y=\sum_aF_{y,a}^\dagger F_{y,a}.
 \label{eq:destructive-povm}
\end{equation}
The equality follows by pairing both sides with every system
observable and using trace cyclicity. Thus a represented local
POVM can provide a further destructive conditioning interface.

Equation~\eqref{eq:destructive-povm} does not say that effects determine
arbitrary residual instruments. It uses the special assumptions
that only the clock is acted on and that all quantum clock output
is removed. Measuring an ancilla pointer while returning a quantum
clock is a different interface.

The supplied X-Enter selective-readout theorem is an appropriate
comparison for retaining failure branches and proving selective-map,
rather than effect-only, agreement
\cite[Theorem \texttt{thm:relational}]{CheungXEnter}.
Its encoded phase model is not a premise or a specialization of the
clock Hamiltonian construction here.

%=====================================================================
\section{The philosophy of Enter as an interface discipline}
\label{sec:philosophy}
%=====================================================================

The philosophical contribution is not an extra dynamical axiom.
It is the requirement that an interpretation remain attached to
the input, access rights, outputs, and continuation that actually
support it \cite{CheungPhilosophy}.
For the present construction this discipline has five precise
consequences.

\subsection{A reading is a record-producing event}

Before the readout, the preparation contains quantum correlations
between a clock and a system. After the readout, the accessible
interface contains a finite classical label and a conditional
system. The correlations supply the relational description;
Enter supplies the chosen operation that makes a classical label
available.

The operation does not create the family \(U_S(t_k)\) from nothing.
Its dynamical content was already fixed by the preparation,
constraint, and clock calibration. Enter realizes extraction of
that content.

\subsection{Consumption is an access statement}

A consumed quantum name is absent from the continuation's quantum
context. This does not assert annihilation of the laboratory device,
of its environment, or of every physical degree of freedom that
implemented the channel.

A reversible dilation may retain information in inaccessible
degrees of freedom. Such degrees of freedom are not thereby
returned through the declared clock-reading interface.

\begin{proposition}[No universal quantum return from the readout]
\label{prop:no-return}
For \(d\geq2\), \(\Mclock\) has no channel left inverse on its full
clock--system input space. In particular, with a fixed system
preparation, its output cannot universally return an arbitrary
unknown quantum clock state.
\end{proposition}

\begin{proof}
Let
\[
 \ket{\chi_\pm}
 =
 \frac{\ket{\tau_0}\pm\ket{\tau_1}}{\sqrt2},
\]
and let \(\eta_S\) be any fixed system state. The distinct inputs
\(\kb{\chi_+}{\chi_+}\otimes\eta_S\) and
\(\kb{\chi_-}{\chi_-}\otimes\eta_S\) have the same output:
\[
 \frac12\bigl(\kb0 0_R+\kb1 1_R\bigr)\otimes\eta_S.
\]
A left inverse would have to map this single output to both inputs,
which is impossible.
\end{proof}

This does not exclude recovery of information from a separately
declared code in the surviving system or another accessible factor.
For example, a record-controlled inverse of
\eqref{eq:spectral-recorded-channel} recovers its encoded
\emph{system input}. That is not a return of the consumed clock
names, and it does not invert \(\Mclock\) on the full input space.

\subsection{The record is not the residual}

The classical marginal contains \(\{p_k\}\). The complete output
also contains the blocks \(\{\sigma_{S,k}\}\). Agreement of the
first does not imply agreement of the second.

\begin{proposition}[Record laws do not certify conditional evolution]
\label{prop:record-not-residual}
There are recorded channels with the same clock outcome law on
every input and different conditional system states.
\end{proposition}

\begin{proof}
Let \(S\) contain a qubit, let \(Z_S\) be its Pauli \(Z\), and compare
\[
 \mathcal F(X)=\Mclock(X),
 \qquad
 \mathcal G(X)=
 \sum_k\kb{k}{k}_R\otimes
 Z_SD_kXD_k^\dagger Z_S.
\]
Unitary conjugation preserves each block trace, so their record
laws agree on every input. For input
\(\kb{\tau_0}{\tau_0}\otimes\kb+ +\), the conditional qubit output
is \(\ket+\) for \(\mathcal F\) and \(\ket-\) for \(\mathcal G\).
The residual states are orthogonal.
\end{proof}

This elementary example expresses the same interface separation
emphasized by VIII-A's residual-obstruction theorem and XIX-C's
record--residual analysis
\cite{CheungVIII,CheungXIXC}. It explains why complete recorded
channels, rather than record frequencies alone, appear in the
stability statements below.

\subsection{A stochastic clock reading is not an expectation oracle}

\begin{proposition}[Affine obstruction to nonconstant definite labels]
\label{prop:affine}
Let \(\mathcal S\) be a convex state set. If an affine channel assigns
each \(\rho\in\mathcal S\) a classical distribution concentrated at
one label \(\ell(\rho)\), and distinct labels are distinguishable,
then \(\ell\) is constant.
\end{proposition}

\begin{proof}
If \(\ell(\rho)\neq\ell(\sigma)\), affinity forces the classical
output on \(a\rho+(1-a)\sigma\), \(0<a<1\), to be
\[
 a\delta_{\ell(\rho)}+(1-a)\delta_{\ell(\sigma)}.
\]
This distribution is not concentrated at one label, a contradiction.
\end{proof}

Accordingly, the record \(k\) is a sample from a law determined by
the clock preparation. It is not an exact single-run label of an
arbitrary unknown expectation value. Finite supplied tokens,
repeated-preparation estimation, and approximate acquisition are
different tasks, as stressed in XIX-C's
\texttt{prop:token} and \texttt{prop:affine}
\cite{CheungXIXC}.

\subsection{Neither a heat ledger nor a spacetime metric follows}

Equation~\eqref{eq:drop-record} is a channel equality. It does not
assign measurement work, heat, reset cost, an energy budget, or
environmental leakage. A thermodynamic implementation requires its
own apparatus and resource specification.

Likewise, the code \(k\), calibrated scalar \(t_k\), and program
schedule are not a metric tensor or a matter-defined relational
lapse. Em02-E's classical lapse construction uses its own background
and relational-map hypotheses \cite{CheungEm02E}. No comparison map
identifying that lapse with the present clock code has been
constructed.

Thus the analog in Table~\ref{tab:analogs} is useful precisely because
it does not promote a finite interface correspondence into an
identity with gravitational spacetime.

%=====================================================================
\section{Typed hybrid evolution and the two notions of time}
\label{sec:hybrid}
%=====================================================================

The relational label \(t_k\) and the external operational schedule
parameter \(s\) have different roles.
The former labels a conditional state extracted from one
preparation. The latter locates preparation, readout, and later
operations in a declared experiment.

\subsection{Live and recorded interfaces}

Before readout, use the live operator space
\[
 \mathcal A_-=\BB(\HH_C\otimes\HH_S).
\]
After readout, use
\[
 \mathcal A_+
 =
 \left\{
 \sum_k\kb{k}{k}_R\otimes X_k:
 X_k\in\BB(\HH_S)
 \right\}.
\]
The space \(\mathcal A_+\) is a finite classical--quantum algebra.
There is no surviving clock factor in it.

Choose a readout event with ordered germs
\[
 s_*^-<s_*^+.
\]
They denote the interfaces immediately before and after the event.
They may share a numerical scheduling coordinate with distinct
event-order tags; no positive physical duration is inferred from
this notation.

Let
\[
 \mathcal U_-(s,u)
 =
 \Ad_{e^{-iK(s-u)/\hbar}}
\]
be the declared pre-event propagator.
On a stationary preparation it leaves the preparation unchanged.

For the post-event interface, let \(\mathcal V_k(s,u)\) be any
declared CPTP propagator satisfying
\[
 \mathcal V_k(s,v)\mathcal V_k(v,u)=\mathcal V_k(s,u),
 \qquad
 \mathcal V_k(u,u)=\id.
\]
Define
\begin{equation}
 \mathcal U_+(s,u)
 \left(\sum_k\kb{k}{k}_R\otimes X_k\right)
 =
 \sum_k\kb{k}{k}_R\otimes\mathcal V_k(s,u)(X_k).
 \label{eq:recorded-propagator}
\end{equation}
The minimal construction chooses every \(\mathcal V_k=\id\).
Time-dependent Hamiltonians or other admitted post-event dynamics
are also allowed. No time-homogeneous semigroup is assumed.

\begin{proposition}[Hybrid propagator and composition]
\label{prop:hybrid}
Define the typed operational propagator by
\begin{equation}
 T(s,u)=
 \begin{cases}
 \mathcal U_-(s,u),&
        u\leq s<s_*^-,\\[2pt]
 \mathcal U_+(s,s_*^+)\Mclock\mathcal U_-(s_*^-,u),&
        u\leq s_*^-<s_*^+\leq s,\\[2pt]
 \mathcal U_+(s,u),&
        s_*^+\leq u\leq s.
 \end{cases}
 \label{eq:hybrid-T}
\end{equation}
With the event germs included as endpoints, every expression is a
CPTP map between its declared interfaces and
\begin{equation}
 T(s,v)T(v,u)=T(s,u)
 \label{eq:hybrid-composition}
\end{equation}
for every compatible ordered triple.
\end{proposition}

\begin{proof}
The pre-event and post-event maps have their stipulated composition
laws. A crossing interval contains exactly one readout. Splitting
such an interval before the readout composes two pre-event factors;
splitting it after the readout composes two post-event factors.
In either case the factors reduce to the middle line of
\eqref{eq:hybrid-T}. No composition introduces a second \(\Mclock\).
Complete positivity and trace preservation are preserved under
composition.
\end{proof}

The type distinction is necessary. Writing \(\Mclock\Mclock\) would
be meaningless on these interfaces: the first readout has already
consumed its clock input.

\subsection{Relational transport is not a second clock measurement}

Under the covariance hypotheses, the mathematical family
\[
 \mathcal T_{\mathrm{rel}}(t,u)
 =
 \id_R\otimes\Ad_{U_S(t-u)}
\]
acts on the recorded interface and satisfies a composition law.
It transports a conditional system state from the represented
reading \(u\) to \(t\) while leaving the stored record unchanged.
It does not create a new observation of the consumed clock or
rewrite the original event as if it had occurred at a new reading.

Multiple physical clock readings require additional clock resources
or a different nondestructive protocol, with their own complete
instruments. One destructive readout of a history state is not,
by itself, a multi-time intervention experiment.

%=====================================================================
\section{Complete-channel stability and refinement independence}
\label{sec:stability}
%=====================================================================

\subsection{Reference-stable comparison}

For compatible finite linear maps define
\[
 \norm{\Lambda}_\diamond
 =
 \sup_{r\geq1}\sup_{X\neq0}
 \frac{\norm{(\Lambda\otimes\id_r)(X)}_1}{\norm{X}_1},
 \qquad
 \ddiam(\Phi,\Psi)=\frac12\norm{\Phi-\Psi}_\diamond.
\]
For states, \(\dtr(\rho,\sigma)=\frac12\norm{\rho-\sigma}_1\).
The half-diamond normalization gives
\(0\leq\ddiam(\Phi,\Psi)\leq1\) for channels and includes arbitrary
reference-entangled inputs \cite[Chapter 3]{Watrous}.

By \eqref{eq:exact-channel-identity},
\begin{equation}
 \ddiam(\Mclock,\Jclock)=0.
 \label{eq:zero-distance}
\end{equation}
This is equality on the full fixed interface, not merely agreement
at one calibrated preparation.

\begin{proposition}[Recorded and conditional stability]
\label{prop:conditional-stability}
Suppose \(\widetilde{\Mclock}\) is a channel with the same classical
record and system interface, with classical output blocks, and
\[
 \ddiam(\widetilde{\Mclock},\Mclock)\leq\varepsilon.
\]
For every reference-entangled input, write its two outputs as
\[
 \sum_k\kb{k}{k}_R\otimes\widetilde{\sigma}_{SQ,k},
 \qquad
 \sum_k\kb{k}{k}_R\otimes\sigma_{SQ,k}.
\]
Then
\begin{equation}
 \frac12\sum_k
 \norm{\widetilde{\sigma}_{SQ,k}-\sigma_{SQ,k}}_1
 \leq\varepsilon,
 \qquad
 \frac12\sum_k\abs{\widetilde p_k-p_k}\leq\varepsilon.
 \label{eq:record-block-bound}
\end{equation}
Every subsequent complete channel preserves the same error upper
bound. If \(p_k\geq p_{\min}>0\) and
\(\varepsilon<p_{\min}\), then \(\widetilde p_k>0\) and
\begin{equation}
 \dtr\!\left(
 \frac{\sigma_{SQ,k}}{p_k},
 \frac{\widetilde{\sigma}_{SQ,k}}{\widetilde p_k}
 \right)
 \leq
 \min\left\{1,\frac{\varepsilon}{p_{\min}}\right\}.
 \label{eq:conditional-error}
\end{equation}
\end{proposition}

\begin{proof}
The diamond bound gives trace distance at most \(\varepsilon\)
for the two complete outputs. Trace norm is additive on their
orthogonal classical blocks, proving the first inequality in
\eqref{eq:record-block-bound}. Taking the classical marginal is a
channel and proves the second. Channel postprocessing contracts
trace distance, including with a reference.

Fix \(k\), abbreviate
\[
 \sigma=\sigma_{SQ,k},\quad
 \widetilde\sigma=\widetilde{\sigma}_{SQ,k},\quad
 p=p_k,\quad q=\widetilde p_k,\quad
 a=\norm{\sigma-\widetilde\sigma}_1.
\]
The classical marginal bound implies \(\abs{p-q}\leq\varepsilon\);
thus \(q>0\) under the stated lower bound. The sum of the norms of
all other output blocks is at least \(\abs{p-q}\), because their
total trace difference is \(q-p\). Therefore
\[
 a+\abs{p-q}\leq2\varepsilon.
\]
Using positivity and \(\Tr\widetilde\sigma=q\),
\[
 \frac12\norm{\frac{\sigma}{p}-\frac{\widetilde\sigma}{q}}_1
 \leq
 \frac{a+\abs{p-q}}{2p}
 \leq\frac{\varepsilon}{p_{\min}}.
\]
State trace distance is also at most one.
\end{proof}

A positive lower probability bound is essential for a uniform
normalized-branch statement. Complete-channel convergence alone
does not control conditioning on arbitrarily rare outcomes.
The exact covariant finite model has \(p_{\min}=1/d\), but this bound
need not remain uniform if the clock dimension grows.

\subsection{Telescoping on compatible complete interfaces}

\begin{lemma}[Complete-channel telescoping]
\label{lem:telescoping}
Let \(G_1,\ldots,G_N\) and
\(\widetilde G_1,\ldots,\widetilde G_N\) be compatible channels, with
corresponding factors on the same comparison interfaces. Then
\begin{equation}
 \ddiam(\widetilde G_N\cdots\widetilde G_1,G_N\cdots G_1)
 \leq
 \min\left\{1,\sum_{j=1}^N
                  \ddiam(\widetilde G_j,G_j)\right\}.
 \label{eq:telescoping}
\end{equation}
\end{lemma}

\begin{proof}
Subtract the two compositions and replace one factor at a time:
\[
 \widetilde G_N\cdots\widetilde G_1-G_N\cdots G_1
 =
 \sum_{j=1}^N
 \widetilde G_N\cdots\widetilde G_{j+1}
 (\widetilde G_j-G_j)
 G_{j-1}\cdots G_1.
\]
Channels have diamond norm one. Submultiplicativity therefore bounds
each term by \(\norm{\widetilde G_j-G_j}_\diamond\).
The triangle inequality and division by two prove the sum bound.
The distance between channels is at most one.
\end{proof}

This lemma applies to finite adaptive instruction trees after their
classical controls and all reported outcomes are represented on the
complete interfaces. A branch deleted by postselection is not such
a trace-preserving factor.

\subsection{The declared refinement class}

\begin{definition}[Matched refinement of frozen operational data]
\label{def:refinement}
Fix a complete finite target channel
\(\Phi:\BB(\HH_{\mathrm{in}})\to\mathcal A_{\mathrm{out}}\), including
its preparation, calibration, clock Enter instrument, all retained
records, residual ownership, and any acceptance or rejection rule.

An admissible refinement supplies, for each scale \(j\):
\begin{enumerate}
\item an implemented channel \(\widetilde\Phi_j\) on these same
      external comparison interfaces;
\item an ideal factor chain on compatible intermediate interfaces
      whose complete composition is exactly \(\Phi\);
\item corresponding implemented factors, compared with those ideal
      factors on the same intermediate interfaces;
\item an explicit assignment of factor errors to encoding, flow,
      event, classical record processing, and decoding;
\item any needed padding, identification, or finite classical
      coarse-graining maps, included as factors rather than
      suppressed.
\end{enumerate}
Intermediate physical registers may depend on \(j\). The external
comparison algebra does not.
\end{definition}

The existence of the ideal matched factor chains is an assumption
of the refinement class, not a consequence of a shared finite
diagram. In particular, this definition does not infer a continuum
target from the native finite clock.

\begin{theorem}[Refinement stability and class-relative independence]
\label{thm:refinement}
In Definition~\ref{def:refinement}, suppose the complete non-flow
error totals are
\[
 \varepsilon_{\mathrm{enc},j},\quad
 \varepsilon_{\mathrm{event},j},\quad
 \varepsilon_{\mathrm{record},j},\quad
 \varepsilon_{\mathrm{dec},j}.
\]
Suppose the flow factors correspond to intervals of lengths
\(h_{j,\ell}\), with
\[
 \sum_\ell h_{j,\ell}\leq L,
 \qquad
 \max_\ell h_{j,\ell}\leq\delta_j,
\]
and have half-diamond errors bounded by
\(A h_{j,\ell}^{1+\alpha}\), where \(A<\infty\) and \(\alpha>0\)
are independent of \(j,\ell\).
Then
\begin{equation}
 \ddiam(\widetilde\Phi_j,\Phi)
 \leq
 \min\left\{1,\,
 \varepsilon_{\mathrm{enc},j}
 +AL\delta_j^\alpha
 +\varepsilon_{\mathrm{event},j}
 +\varepsilon_{\mathrm{record},j}
 +\varepsilon_{\mathrm{dec},j}
 \right\}.
 \label{eq:refinement-budget}
\end{equation}
If \(\delta_j\to0\) and the four non-flow totals tend to zero,
the complete channels converge to \(\Phi\).

Any two admissible refinement families satisfying these conditions
for the same frozen target converge to the same limit.
\end{theorem}

\begin{proof}
The total flow error satisfies
\[
 \sum_\ell A h_{j,\ell}^{1+\alpha}
 \leq
 A\delta_j^\alpha\sum_\ell h_{j,\ell}
 \leq AL\delta_j^\alpha.
\]
Apply Lemma~\ref{lem:telescoping} to the complete matched factor
chain and group the remaining factor errors by their declared
roles. This proves \eqref{eq:refinement-budget}.

For two families with bounds \(b_j\to0\) and \(b'_j\to0\),
\[
 \ddiam(\widetilde\Phi_j,\widetilde\Phi'_j)
 \leq
 \ddiam(\widetilde\Phi_j,\Phi)
 +\ddiam(\Phi,\widetilde\Phi'_j)
 \leq b_j+b'_j\longrightarrow0.
\]
Both families consequently have the same channel limit.
\end{proof}

This is the precise error decomposition
\[
 \varepsilon_{\mathrm{enc}}
 +\varepsilon_{\mathrm{flow}}
 +\varepsilon_{\mathrm{event}}
 +\varepsilon_{\mathrm{record}}
 +\varepsilon_{\mathrm{dec}}.
\]
A one-step error of order \(O(h)\) alone does not force the accumulated
error to vanish. Nor does a proposed product formula automatically
supply the complete-instrument estimates required here.
The supplied En05-A comparison distinguishes geometry, schedule,
and channel semantics \cite{CheungEn05A}.

The XXI-A--C identifiers in the development plan motivate this
hybrid and complete-experiment interface
\cite{CheungXXIOutline}. Their full texts were not supplied.
The estimates used here are proved in this manuscript rather than
attributed to unavailable proposition numbers.

\subsection{Three convergence questions}

Refinement must distinguish:
\begin{enumerate}
\item convergence of externally calibrated classical readings;
\item convergence of the quantum residual channel after forgetting
      some or all records;
\item convergence of the complete recorded experiment on a common
      record--residual interface.
\end{enumerate}
The third implies the relevant marginal channel convergences.
Neither the first nor the second, by itself, implies the third.
Proposition~\ref{prop:record-not-residual} illustrates the residual
gap even when the record laws agree exactly.

Finite classical codes also do not converge to an exact
all-real record in total variation merely because their calibrated
values become dense. For example, let \(\mu_j\) be any probability
measure with finite support in \([0,1]\), and let \(\lambda\) be
uniform Lebesgue probability. If \(F_j=\supp\mu_j\), then
\[
 \mu_j(F_j)=1,\qquad \lambda(F_j)=0.
\]
Hence their total-variation distance is one at every scale.
A weak limit can still exist, but it is a different convergence
claim. Fixed finite bins or a specified bounded-statistic interface
can be used instead; the binning or testing map must be declared.

Finally, common finite samples do not select a unique interpolation,
generator, or continuum limit. XIX-C's
\texttt{prop:two-limits} and \texttt{prop:interpolation} record
precisely this non-determination \cite{CheungXIXC}.
Theorem~\ref{thm:refinement} avoids it by freezing the target and
restricting the comparison class explicitly.

%=====================================================================
\section{Optional causal embedding}
\label{sec:causal}
%=====================================================================

No physical causal geometry is needed for the channel identity.
If a realization is assigned locations, a causal embedding can be
added as a separate contract.

Let the complete finite dependency graph include every actual
quantum input, classical control, preparation, routing operation,
and retained record. Assign each event a location \(p\) in a set
with a supplied transitive causal relation \(\preceq\).
Require every dependency edge \(p\to q\) to satisfy \(p\preceq q\).

\begin{proposition}[Containment relative to a supplied causal relation]
\label{prop:causal}
Under this edge condition, every dependency path lies in
\(\preceq\). For local channel realizations of the graph,
Heisenberg support of a terminal observable is contained in its
scheduled ancestor region. Thus operational influence is bounded
by that region and, through the certified embedding, by the
supplied causal relation.
\end{proposition}

\begin{proof}
Path containment is transitivity. For support containment, pull the
terminal observable backward through the circuit. The adjoint of a
trace-preserving local channel sends an identity on an untouched
factor to the identity, and can enlarge support only to the input
factors of that event. Induction through the finite graph places
the support in the ancestor region. An intervention on a factor
outside that region cannot change the terminal reduced output,
because its pulled-back observables act as the identity there.
\end{proof}

Scheduled ancestry, actual operational influence, and
\(\preceq\) remain three different relations. Channels may fail to
use a scheduled edge, while entanglement may produce correlations
without intervention-induced signalling. The embedding contract
does not reconstruct the supplied relation from the program.

These distinctions match VII-J's two targets: complete decoded
simulation and physical embedding of the actual elementary
operations \cite{CheungVIIJ}. They do not derive a Lorentz-invariant
speed or select a Lorentz cone; En08-A's scheduler-consistency
separation remains applicable \cite{CheungEn08A}.

If a fixed geometric packet or a source-replacement controller is
attached, its matching certificates, guards, and residual ownership
must also be supplied. The VIII constructions are comparisons for
such contracts, not a geometric conclusion of the clock-readout
theorem \cite{CheungVIII}.

%=====================================================================
\section{Exact release and standing boundaries}
\label{sec:release}
%=====================================================================

\begin{corollary}[En09-A release]
\label{cor:release}
For one declared finite clock--system preparation and one complete
destructive clock-readout event:
\begin{enumerate}
\item the retained classical code supplies the calibrated relational
      label;
\item the surviving system block is exactly the Page--Wootters
      conditional state;
\item the measured clock names are consumed;
\item the complete recorded instrument agrees with its relational
      extraction target on every input and with every reference;
\item sampled covariance and stationarity imply the stated
      conditional unitary evolution;
\item under the full Enter characterization hypotheses, the Born
      record law and projective residual update have the inherited
      derived status;
\item matched refinements obey the complete-channel budget and
      class-relative uniqueness theorem.
\end{enumerate}
\end{corollary}

\begin{proof}
Items (i)--(vi) are Theorem~\ref{thm:main}, with
Lemma~\ref{lem:density-semantics} for the semantic attribution.
Proposition~\ref{prop:spectral-realization} gives an exact inhabited
finite constraint class. Item (vii) is
Theorem~\ref{thm:refinement}.
\end{proof}

\paragraph{What the release does not do.}
It does not remove external preparation or scheduling time.
It does not construct an internal matter clock that remains
available after the declared destructive event. It does not derive
a dynamical cone, a continuum metric, Einstein dynamics, a
field-theoretic measurement apparatus, or fully quantized
gravitational relational observables. A finite history state and
its record are not automatically spacetime emergence.

The classical relational-observable and lapse results associated
with VI and Em02-E retain their own equation spaces, backgrounds,
and quantum obligations \cite{CheungBoundaries,CheungEm02E}.
No Enter readout closes those additional obligations.

\paragraph{Analytic non-transfer.}
No characteristic joint product, quantum coefficient, interacting
Ward identity, or downstream admission is constructed.
The XIV-I internal amplitude and XIV-J supported-sector reference
are not replaced by a finite clock instrument.
XV-B version 2 may exist as a technical continuation while the
distinct downstream quantum-admission exit remains unopened.
This manuscript does not change that status and does not discharge
XVI-N \cite{CheungBoundaries}.

\paragraph{Documentary negative controls.}
The supplied OpenAI catalogue and the identifier ``Family 271''
remain documentary negative controls only. No claim or proof from
that family is used, reproduced, or verified here. No authorship,
formalization certificate, or physical interpretation is transferred
from an OpenAI document to this construction.
The standing documentary entry
\[
 \text{O.1}=\NOTCLOSED
\]
is unchanged. These statements record scope; they are not an
independent audit or a universal nonexistence theorem.

\paragraph{Conclusion.}
The philosophical and mathematical content can be summarized as
\[
 \boxed{
 \begin{gathered}
 \text{declared history correlations}
 \;+\;
 \text{calibrated destructive Enter}\\
 \Longrightarrow
 \text{classical relational label}
 \;+\;
 \text{correct conditional system residual},\\[2pt]
 \text{clock names consumed},\qquad
 \Mclock=\Jclock,\qquad
 \ddiam(\Mclock,\Jclock)=0.
 \end{gathered}}
\]
The label is a record, not a surviving clock.
The record is not a decoder.
The channel is not a thermal machine.
The analog is not an identification with spacetime.

%=====================================================================
\appendix
\section{Corrections to the proposed theorem architecture}
\label{app:debug}
%=====================================================================

The following corrections are necessary for the completed
statement to be mathematically and operationally sound.

\begin{enumerate}
\item \textbf{A clock basis must be declared.}
      Computational-basis Enter realizes the intended observable
      only when the clock basis is already computational or a
      calibrated basis change is included.

\item \textbf{Stationarity does not imply the desired evolution for
      an arbitrary observable.}
      Sampled covariance is explicit in
      Definition~\ref{def:covariance};
      Example~\ref{ex:energy-basis} supplies a counterexample without
      it.

\item \textbf{Stationarity and a zero-energy constraint differ.}
      Equations~\eqref{eq:stationarity} and
      \eqref{eq:strong-constraint} are not interchangeable. The
      exact finite construction satisfies the stronger condition.

\item \textbf{Uniform readings are a consequence in the exact
      covariant complete basis.}
      One cannot simultaneously assume arbitrary clock weights
      and exact covariance of every basis projector for a
      stationary preparation.

\item \textbf{Only the measured register's quantum output is absent.}
      The complete process still returns the unmeasured system.
      Saying that the entire process has ``only a classical
      output'' would incorrectly discard its residual.

\item \textbf{The Born attribution is conditional on the full
      predecessor class.}
      Writing the instrument in Born form proves its algebraic
      law, not the predecessor's operational characterization.
      Hypothesis~\ref{hyp:enter} identifies the imported result and
      its applicability condition.

\item \textbf{Normalized conditioning is not a linear channel.}
      The instrument uses subnormalized blocks. Zero-probability
      branches have no prescribed normalized state.

\item \textbf{Reference stability is an operator statement.}
      Equality on one frozen state would not establish diamond
      equality. Equation~\eqref{eq:exact-channel-identity} holds on
      every operator. Relational evolution of a reference-inclusive
      preparation additionally requires its own full stationarity.

\item \textbf{A consumed clock cannot be measured again in the same
      continuation.}
      The hybrid composition crosses the readout exactly once.
      Further physical readings need additional resources or a
      different instrument.

\item \textbf{General circuit histories need not satisfy the
      additive constraint.}
      Their propagation Hamiltonian
      \eqref{eq:propagation-H} is kept distinct from
      \(H_C+H_S\).

\item \textbf{Refinement uniqueness is class-relative.}
      The ideal target, calibration, complete output interface,
      and local consistency estimates are frozen. Shared finite
      samples alone do not prove uniqueness.

\item \textbf{No unsupplied theorem is silently imported.}
      The XXI-A--C material available here is an outline-level
      comparison. The hybrid and telescoping results needed for
      En09-A are stated and proved above.
\end{enumerate}

%=====================================================================
\section{Source provenance and verification status}
\label{app:provenance}
%=====================================================================

The bibliography distinguishes journal articles, internal manuscript
records, and outline-only material. Internal documents are cited
by the supplied identifier and version rather than by an invented
publication date, DOI, or formal title.

The only imported nonstandard semantic theorem used in the main
argument is the Enter characterization identified in
Hypothesis~\ref{hyp:enter}. Its full proof is not reconstructed here.
The finite channel identity, covariance argument, spectral
realization, hybrid composition, stability estimates, and
class-relative refinement result have explicit proofs in this
manuscript.

The public publisher record for \cite{CheungEnter}, consulted for
bibliographic metadata, gives acceptance on 28 September 2026 and
states that the final formatted article is forthcoming.
This status is not converted into a publication date for any
internal manuscript.

The supplied internal records support the stated comparisons and
standing non-claims. They are not treated as independent
machine verification, and similarities in notation do not transfer
their mathematical hypotheses to En09-A.

No successful \texttt{pdflatex} execution, proof-assistant check,
numerical execution, hardware experiment, independent referee
review, or journal acceptance is asserted. The source is arranged
as a single conventional \texttt{pdflatex} manuscript with its
bibliography included.

%=====================================================================
\begin{thebibliography}{99}

\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}.

\bibitem{BoetteRossignoli}
A.~Boette and R.~Rossignoli,
``History states of systems and operators,''
\emph{Physical Review A} \textbf{98} (2018), 032108.
\url{https://doi.org/10.1103/PhysRevA.98.032108}.

\bibitem{McClean}
J.~R. McClean, J.~A. Parkhill, and A.~Aspuru-Guzik,
``Feynman's clock, a new variational principle, and parallel-in-time
quantum dynamics,''
\emph{Proceedings of the National Academy of Sciences}
\textbf{110} (2013), E3901--E3909.
\url{https://doi.org/10.1073/pnas.1308069110}.

\bibitem{Diaz}
N.~L. Diaz, P.~Braccia, M.~Larocca, J.~M. Matera,
R.~Rossignoli, and M.~Cerezo,
``Parallel-in-time quantum simulation via Page and Wootters quantum
time,''
\emph{Physical Review Research} \textbf{7} (2025), 033294.
\url{https://doi.org/10.1103/wpnf-4nnn}.

\bibitem{CheungEnter}
N.~Cheung,
``The Enter Calculus: A Linear Quantum Lambda Calculus with an
Operational Characterization of Measurement,''
\emph{Frontiers in Physics} \textbf{14} (2026),
article 1954943.
Accepted 28 September 2026; final formatted version forthcoming
according to the publisher record consulted on 13 October 2026.
\url{https://doi.org/10.3389/fphy.2026.1954943}.

\bibitem{CheungI}
N.~Cheung,
\emph{Manuscript I},
\texttt{I.tex}, version 2, n.d.
Supplied manuscript record and theorem excerpts.
In particular, \texttt{thm:main} characterizes Born probabilities
and positive-branch destructive residual rays within its declared
strategy class.
Enter Calculus Companion,
\url{https://enter.grok.me/}.

\bibitem{CheungPhilosophy}
N.~Cheung,
\emph{The Philosophy of Enter: Destructive Readout, Quantum Recovery,
and the Refusal of Identification},
\texttt{EntPhil-v2.tex}, version 2, n.d.
Introductory philosophical map, not a registered theorem premise.
\url{https://enter.grok.me/enter}.

\bibitem{CheungEn08A}
N.~Cheung,
\emph{Manuscript En08-A},
\texttt{En08-A.tex}, version 1, n.d.
Supplied manuscript record and excerpts, especially
\texttt{thm:main}, ``Failure of reconstruction from scheduler
consistency.''
Enter Calculus Companion,
\url{https://enter.grok.me/}.

\bibitem{Trinity}
P.~A. H\"ohn, A.~R.~H. Smith, and M.~P.~E. Lock,
``Trinity of relational quantum dynamics,''
\emph{Physical Review D} \textbf{104} (2021), 066001.
\url{https://doi.org/10.1103/PhysRevD.104.066001}.

\bibitem{CheungXEnter}
N.~Cheung,
\emph{Manuscript X-Enter},
\texttt{X-Enter.tex}, version 1, n.d.
Supplied manuscript record and excerpts, especially
\texttt{thm:relational}, ``Conditional selective relational readout.''
Its encoded phase model is cited as a comparison only.
Enter Calculus Companion,
\url{https://enter.grok.me/}.

\bibitem{CheungVIII}
N.~Cheung,
\emph{Manuscripts VIII-A, VIII-B, and VIII-C},
\texttt{VIII-A-v3.tex}, \texttt{VIII-B-v3.tex}, and
\texttt{VIII-C-v3.tex}, versions 3, n.d.
Supplied records and excerpts concerning fixed-presentation
matching, record--residual separation, affine acquisition
obstructions, guarded packet flow, and residual ownership.
In VIII-A, see \texttt{A:thm:residual}.
These are comparison records, not geometric premises of En09-A.
Enter Calculus Companion,
\url{https://enter.grok.me/}.

\bibitem{CheungXIXC}
N.~Cheung,
\emph{Manuscript XIX-C},
\texttt{XIX-C.tex}, version 1, n.d.
Supplied record and excerpts, especially
\texttt{thm:main}, ``Finite incidence does not determine continuum
evolution,'' and the identified
\texttt{prop:token}, \texttt{prop:affine},
\texttt{prop:two-limits}, and \texttt{prop:interpolation}.
Enter Calculus Companion,
\url{https://enter.grok.me/}.

\bibitem{CheungEm02E}
N.~Cheung,
\emph{Manuscript Em02-E},
\texttt{Em02-E-v1.tex}, version 1, n.d.
Supplied manuscript record and excerpts, especially
\texttt{thm:lapse}, ``Complete relational lapse through second order.''
Its classical background-dependent lapse is not identified with
the present clock record.
Enter Calculus Companion,
\url{https://enter.grok.me/}.

\bibitem{Watrous}
J.~Watrous,
\emph{The Theory of Quantum Information},
Cambridge University Press, 2018.
Chapter 3: similarity and distance among states and channels.
Author's manuscript and errata interface:
\url{https://cs.uwaterloo.ca/~watrous/TQI/}.

\bibitem{CheungEn05A}
N.~Cheung,
\emph{Manuscript En05-A},
\texttt{En05-A.tex}, version 1, n.d.
Supplied manuscript record and introductory excerpts.
Finite-range schedule geometry, complete channel locality,
and continuum-comparison cautions are cited as distinctions,
not as automatic transfer theorems.
Enter Calculus Companion,
\url{https://enter.grok.me/}.

\bibitem{CheungXXIOutline}
N.~Cheung,
\emph{XXI-A--C development-plan references}, n.d.
Outline-level material supplied with the En09-A proposal,
concerning finite clock codes, hybrid event germs, and complete
recorded-channel convergence.
Full manuscript texts, versions, and publication metadata were not
supplied. No unsupplied XXI theorem is used as a premise; the
required budget is proved in Section~\ref{sec:stability}.

\bibitem{CheungVIIJ}
N.~Cheung,
\emph{Manuscript VII-J},
\texttt{VII-J.tex}, version 1, n.d.
Supplied manuscript record and scope excerpts.
Scheduled reachability, operational influence, complete decoded
simulation, and separately certified physical embedding remain
distinct.
Enter Calculus Companion,
\url{https://enter.grok.me/}.

\bibitem{CheungBoundaries}
N.~Cheung,
\emph{Supplied analytic and classical boundary records}, n.d.
Manuscript VI, \texttt{VI.tex}, version 1;
XIV-I, \texttt{XIV-I.tex}, version 1;
XIV-J, \texttt{XIV-J.tex}, version 1;
XV-B, \texttt{XV-B.tex}, version 2; and
XVI-N, \texttt{XVI-N.tex}, version r12.
The cited excerpts distinguish classical descent, internal
amplitude construction, the unresolved full characteristic map,
technical continuation, downstream admission, and the conditional
massless obstruction.
They are cited solely to preserve non-transfer and standing scope.
Enter Calculus Companion,
\url{https://enter.grok.me/}.

\end{thebibliography}

\end{document}