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\title{\textbf{En10-A}\\[0.4em]
A Finite Enter Instrument on a Qubit--Antiqubit Interface:\\
Classical Records, Source-Weight Estimation, and Non-Identification}
\author{}
\date{}

\begin{document}
\maketitle

\begin{abstract}
We construct a complete finite instrument for a two-zone
qubit--antiqubit toy model. The antiqubit designation means a
declared inverse response to one specified unitary generator;
it does not introduce antimatter, an antiunitary operation, or
backward physical time. Both data qubits are read destructively
in the computational basis. The resulting channel returns a
classical word, consumes the corresponding quantum names, and
agrees exactly with its projective recorded-channel specification
on all operators, including arbitrary untouched references.

For an independently supplied one-parameter source family, a
separate classical inference map estimates the normalized source
weight. Its finite-shot confidence radius, sensitivity denominator,
Fisher-information ledger, and implementation-error allowance are
explicit. A concrete instance has entirely rational outcome
probabilities and estimator values. We also prove a record-only
quantum-recovery obstruction and non-identifiability of parameters
omitted from the frozen model.

The philosophical contribution is an application of Enter's
interface discipline: a record is not a quantum return; an inference
map is not the record; an inaccessible complement is not destroyed
information; a channel is not a thermal machine; and a finite analog
is not an astrophysical identification. No Galactic quantum state,
511\,keV morphology, absolute Galactic rate, continuum transport
law, or spacetime geometry is reconstructed.
\end{abstract}

\paragraph{Identifier and provenance.}
The identifier \textbf{En10-A} is used here as requested for this
finite-model specialization. The separately supplied
\texttt{En10-A.tex}, version~1, concerns relational clock readout
\cite{CheungRelational}. References to that source are written as
\emph{source En10-A@1}; references to the present main theorem use
Theorem~\ref{thm:main}. This document neither replaces that source
nor asserts that it is a registered revision of it.

\paragraph{Mathematical status.}
All mathematical claims needed for the instrument and estimation
contract are proved below from explicitly stated finite-dimensional
and statistical assumptions. The companion manuscripts are
comparisons, not substitutes for these proofs. Born measurement
semantics are assumed here; their characterization within
Cheung's stated operational class is not independently re-proved.

\section{Motivation and the philosophical background of Enter}
\label{sec:philosophy}

\subsection{From a physical analogy to an operational question}

The Galactic positron problem motivates questions about sources,
transport, and annihilation observables. Actual analyses of the
511\,keV signal involve instrumental backgrounds, spatial models,
spectra, and component-dependent inference
\cite{Siegert2016}. None of those structures is supplied by the
four-dimensional data Hilbert space used here.
% Astrophysical motivation and source attribution:
% ([arxiv.org](https://arxiv.org/abs/1512.00325))

The present question is consequently narrower:
\begin{quote}
Given a declared finite preparation family and a declared finite
channel, what does destructive readout return, and what may a
separately declared statistical procedure infer from its records?
\end{quote}
The words ``bulge'' and ``disk'' label two basis values. They do not
identify those values with regions of the Milky Way.

The term \emph{antiqubit} is motivated by the platform-specific
unitary-inversion construction of Song and collaborators
\cite{Song2025}. Their work distinguishes an inverse unitary
response from the ordinary response of a qubit. Here that
distinction is installed as a mathematical input. We do not
construct their transmon controls or import their experimental
metrological performance.
% Antiqubit terminology and platform-specific inversion:
% ([arxiv.org](https://arxiv.org/abs/2506.04315?utm_source=openai))

\subsection{Enter as an interface discipline}

Cheung's Enter calculus treats runtime quantum names as access
rights and destructive computational-basis measurement as a
consuming operation. The published article separates concrete Born
semantics from an operational characterization within a specified
strategy class \cite{CheungEnter,CheungI}.
% Published Enter article and the scope of its characterization:
% ([frontiersin.org](https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2026.1954943/abstract?utm_source=openai))

The introductory essay \emph{The Philosophy of Enter} emphasizes
the distinctions between destructive readout, quantum recovery,
implementation ledgers, and physical identification
\cite{CheungPhilosophy}. We use that essay as philosophical
orientation, not as a theorem premise.
% Philosophical essay, author, version, and non-premise status:
% ([enter.grok.me](https://enter.grok.me/enter))

For the present construction, this orientation has five precise
consequences.

\begin{enumerate}[label=\textup{(\roman*)},leftmargin=*]
\item
\textbf{A quantum name is an operational entitlement, not an
ontology of the apparatus.}
Consumption means that a continuation cannot use the measured
name as an available quantum input. It does not mean that every
microscopic carrier, environmental correlation, or physical
degree of freedom has ceased to exist.

\item
\textbf{A classical record is not a quantum return.}
The output word belongs to a commutative record algebra.
It may be copied or used for classical control, but that does not
restore arbitrary coherence or reference entanglement of the
measured register.

\item
\textbf{Inference requires an additional contract.}
A recorded word does not estimate a parameter by itself.
The preparation family, likelihood, calibration constants,
sample count, and inference map must be declared. Calling that map
a statistical decoder does not make it a quantum-return decoder.

\item
\textbf{The observation boundary is an access boundary.}
It specifies which record, spectator, and residual observables
may be inspected and which continuations are permitted.
It is not a light cone, a distance, or a Galactic radius.

\item
\textbf{An analog requires a separate identification theorem.}
A finite channel may be useful without being the interstellar
medium or a continuum transport law. Physical interpretation
cannot be strengthened merely by changing the labels on its
basis states.
\end{enumerate}

These are methodological distinctions with mathematical
consequences below. They are not a new measurement postulate,
a claim about consciousness, or a resolution of every
interpretational question in quantum mechanics.

\subsection{Corrections to the proposed theorem}

Three qualifications are necessary before a valid theorem can be
stated.

\paragraph{Residuals can sometimes continue.}
Destructive readout of one factor does not imply that every
unmeasured factor becomes incapable of further evolution.
Source En10-A@1 explicitly retains a conditional system, and under
additional covariance assumptions that system has a relational
unitary evolution \cite{CheungRelational}.
The correct statement here is that \emph{neither measured data
name is returned}. Any surviving spectator or inaccessible
environment is a different interface.

\paragraph{Two qubits do not encode every proposed observable.}
Two position qubits cannot simultaneously provide independent
position, two-particle spin, energy-bin, and annihilation-flag
registers. We therefore freeze a position-only interface.
The relaxation channel below is a declared surrogate, not a
physical annihilation event. The primary records are occupation
words, not annihilation or escape counts.

\paragraph{Identifiability is a hypothesis, not a consequence of
readout.}
An arbitrary finite measurement need not distinguish a proposed
source parameter. We select one family for which the accessible
probability is affine with a strictly positive slope. The slope
appears explicitly in every estimation bound.

\section{The frozen finite interface}
\label{sec:interface}

\subsection{Registers, basis, and externally supplied source family}

Let
\[
 \mathcal H_N=\mathbb C^2,\qquad
 \mathcal H_A=\mathbb C^2,\qquad
 \mathcal H_B=\mathcal H_N\otimes\mathcal H_A.
\]
The register \(N\) is the normal qubit; \(A\) is the antiqubit.
For both registers, the basis values \(0\) and \(1\) are called
bulge and disk, respectively. They are finite labels only.

The sole unknown parameter is the normalized source weight
\(\theta\in[0,1]\), defined by
\begin{equation}
 \rho_\theta
 =
 \theta\,\kb{00}{00}
 +(1-\theta)\,\kb{11}{11}.
 \label{eq:source}
\end{equation}
Thus \(\theta\) is the relative weight of the first source component,
not an absolute injection rate.

\begin{assumption}[Source access]
\label{ass:source}
The observer receives independent preparations of
\(\rho_\theta\) and does not receive the source-component label or
a classical token containing the unknown value of \(\theta\).
The preparation mechanism and any purification of
\(\rho_\theta\) belong to the supplier's interface.
\end{assumption}

This assumption prevents an expectation-value oracle from being
hidden in the notation. If a simulator prepares
\eqref{eq:source} from a known numerical value of \(\theta\), the
same theorem is a sampling or validation theorem for that
simulator; it is not acquisition of previously unknown Galactic
information. This is the token-versus-acquisition distinction
recorded in XIX-C \cite{CheungXIXC}.

\subsection{Declared opposite response and relaxation}

Fix, before collecting data,
\begin{equation}
 0<\phi<\frac{\pi}{4},\qquad 0\le\gamma<1.
 \label{eq:calibration}
\end{equation}
These are known calibration inputs, not parameters estimated by the
main inference map.

Let
\[
 Y=
 \begin{pmatrix}
 0&-i\\ i&0
 \end{pmatrix},
 \qquad
 V_\phi=e^{-i\phi Y}
 =
 \begin{pmatrix}
 \cos\phi&-\sin\phi\\
 \sin\phi&\cos\phi
 \end{pmatrix}.
\]
The declared local response is
\begin{equation}
 W_\phi=V_\phi\otimes V_\phi^\dagger.
 \label{eq:inverse}
\end{equation}
The antiqubit designation means precisely the inverse factor
\(V_\phi^\dagger\) in this equation. It is a linear unitary on an
ordinary two-dimensional Hilbert space. No antiunitary
transformation, unknown-unitary inversion algorithm, or reversal
of physical time is asserted.

We include the fixed entangling gate
\[
 \CZ=\operatorname{diag}(1,1,1,-1),
 \qquad
 U_\phi=\CZ W_\phi.
\]
The inverse-response claim concerns the two local factors of
\(W_\phi\), not inversion of the complete interacting channel.

Put \(h=1-\gamma\), and define
\[
 L_0=\kb0 0+\sqrt h\,\kb1 1,
 \qquad
 L_1=\sqrt\gamma\,\kb0 1.
\]
The one-step channel is
\begin{equation}
 \Phi(X)
 =
 \sum_{\ell=0}^1 H_\ell XH_\ell^\dagger,
 \qquad
 H_\ell=(I_N\otimes L_\ell)U_\phi.
 \label{eq:channel}
\end{equation}

The coherent rotation is a two-zone mixing analog. The map
\(L_1\) transfers the antiqubit basis value \(1\) to \(0\).
Because these basis values encode zones, this is a
\emph{declared relaxation surrogate}. It is not simultaneously
an independently encoded energy-loss process. No vacuum,
positronium spin, photon, or physical annihilation sector has been
installed.

\begin{lemma}[Validity of the frozen step]
\label{lem:step}
The map \(\Phi:\Bop(\mathcal H_B)\to\Bop(\mathcal H_B)\)
is completely positive and trace preserving.
\end{lemma}

\begin{proof}
The displayed Kraus form proves complete positivity.
Moreover,
\[
 L_0^\dagger L_0+L_1^\dagger L_1
 =
 \kb0 0+h\kb1 1+\gamma\kb1 1
 =I_A.
\]
Since \(U_\phi\) is unitary,
\[
 \sum_{\ell=0}^1H_\ell^\dagger H_\ell
 =
 U_\phi^\dagger
 \bigl(I_N\otimes I_A\bigr)U_\phi
 =I_B.
\]
This proves trace preservation.
\end{proof}

\subsection{Access and observation boundary}

The accessible data factor is \(B=NA\), read only through its
computational-basis occupation projectors
\[
 P_r=\kb r r,\qquad r\in\mathcal R:=\{00,01,10,11\}.
\]
The record register \(C\) has basis
\(\{\ket r_C:r\in\mathcal R\}\), and its observable algebra is
\[
 \operatorname{Diag}(C)
 =
 \left\{\sum_{r\in\mathcal R}c_r\kb r r_C:c_r\in\mathbb C\right\}.
\]

An optional laboratory spectator \(S\) is untouched. An optional
reference \(Q\) is also untouched and is included for complete
channel comparisons, not as assistance to the observer.
The accessible recorded-spectator algebra is
\[
 \operatorname{Diag}(C)\otimes\Bop(\mathcal H_S).
\]
The supplier's source labels, channel environments, and
measurement-device complements are outside this access boundary.

The newly returned object is \(C\) alone. An existing spectator
may remain on a comparison interface, but neither \(N\) nor \(A\)
is a quantum output. Allowed continuations may read \(C\), act on
\(S\), and allocate fresh names. They may not reuse the consumed
names of \(N\) or \(A\).

\section{The destructive instrument and its complete identity}
\label{sec:instrument}

\subsection{Record and residual are specified together}

For any retained factor \(T\), including \(T=S\otimes Q\), define
\begin{align}
 D_r&=\bra r_B\otimes I_T,
 \\
 \mathcal M_B^T(X)
 &=
 \sum_{r\in\mathcal R}
 \kb r r_C\otimes D_rXD_r^\dagger,
 \label{eq:enter}
 \\
 \mathcal J_B^T(X)
 &=
 \sum_{r\in\mathcal R}
 \kb r r_C\otimes
 \Tr_B\!\left[
 (P_r\otimes I_T)X(P_r\otimes I_T)
 \right].
 \label{eq:target}
\end{align}
When \(T\) is one-dimensional, the \(T\)-blocks are scalars and
the output is a classical record state.

The complete one-shot instrument is
\begin{equation}
 \mathcal I^T
 =
 \mathcal M_B^T\circ(\Phi\otimes\id_T).
 \label{eq:instrument}
\end{equation}
Its selective \(r\)-branch is the corresponding subnormalized
block. All four outcomes are retained. There is no postselection.

\begin{lemma}[Complete destructive Enter identity]
\label{lem:identity}
For every finite retained factor \(T\),
\[
 \mathcal M_B^T=\mathcal J_B^T
\]
as linear maps on all operators. The map
\(\mathcal M_B^T\) is CPTP, and
\begin{equation}
 \Tr_C\mathcal M_B^T(X)=\Tr_BX.
 \label{eq:drop}
\end{equation}
Consequently \(\mathcal I^T\) is CPTP and
\begin{equation}
 \Tr_C\mathcal I^T(X)=\Tr_BX.
 \label{eq:drop-step}
\end{equation}
\end{lemma}

\begin{proof}
Let
\[
 A_r=\ket r_C\otimes D_r.
\]
Then
\[
 \mathcal M_B^T(X)=\sum_r A_rXA_r^\dagger,
 \qquad
 \sum_rA_r^\dagger A_r
 =
 \sum_rP_r\otimes I_T
 =I_B\otimes I_T.
\]
Thus the map is completely positive and trace preserving.

Expand an arbitrary operator as
\[
 X=\sum_{r,s\in\mathcal R}\kb r s_B\otimes X_{rs}.
\]
Both the \(r\)-block of \(\mathcal M_B^T(X)\) and the \(r\)-block
of \(\mathcal J_B^T(X)\) are \(X_{rr}\). Hence the two maps agree
on every operator, not merely on the source family.

Discarding the record gives \(\sum_rX_{rr}=\Tr_BX\), proving
\eqref{eq:drop}. Lemma~\ref{lem:step} proves that
\(\mathcal I^T\) is CPTP. Finally, any local trace-preserving map
on \(B\) leaves the partial trace over \(B\) unchanged. In the
present Kraus representation,
\[
 \Tr_B[(\Phi\otimes\id_T)(X)]
 =
 \Tr_B\!\left[
 \left(\sum_\ell H_\ell^\dagger H_\ell\otimes I_T\right)X
 \right]
 =\Tr_BX,
\]
which proves \eqref{eq:drop-step}.
\end{proof}

The form of this proof matches the complete recorded-channel
identity in source En10-A@1 \cite{CheungRelational}; the proof
above is local and does not import its clock assumptions.

\begin{remark}[References are untouched unconditionally]
For a state \(\rho_{BT}\), the conditional \(T\)-state on outcome
\(r\) is the normalized \(r\)-block when that block has positive
trace. It may depend on \(r\). Equation~\eqref{eq:drop-step}
states that the \emph{unconditional} \(T\)-marginal is unchanged.
Conditional steering is not a failure of the complete identity.
No normalized state is assigned to a zero-probability branch.
\end{remark}

\subsection{Linear consumption is an additional operational declaration}

\begin{definition}[Name discipline]
\label{def:names}
A quantum context is a finite set of pairwise distinct live
quantum names. A unitary instruction acts on names in the
context and preserves them. Allocation introduces a globally
fresh name. The instruction \(\Enter(q)\) removes \(q\) from
the quantum context and returns one classical basis value.
Subsequent instructions may use only live quantum names.
Classical computation does not introduce a quantum name.
\end{definition}

\begin{lemma}[Consumption]
\label{lem:consumption}
Under Definition~\ref{def:names}, destructive readout of both
names in \(B=NA\) has the interface
\[
 \mathsf Q_N\otimes\mathsf Q_A\otimes\mathsf Q_S
 \multimap
 \mathsf{Bit}^{\,2}\otimes\mathsf Q_S.
\]
No legal continuation uses the consumed \(N\)- or \(A\)-name
as a quantum input.
\end{lemma}

\begin{proof}
The first Enter instruction removes its measured name.
The second removes the other measured name. The resulting
quantum context contains only the previously retained spectator
names. Induction over any subsequent instruction sequence
shows that unitary operations preserve this context, classical
operations introduce no quantum names, and allocations introduce
only globally fresh names. Neither consumed name can reappear.
\end{proof}

Consumption is therefore not inferred from complete positivity
alone. It follows from the declared name discipline and agrees
with the instrument codomain. A physical implementation that
retains a measured carrier but removes access to it may realize
the same interface; it must not present that carrier as an
unpriced quantum return.

\section{The finite Enter theorem and estimation contract}
\label{sec:main}

Put
\begin{equation}
 t=\sin^2\phi,\qquad
 d=1-2t=\cos(2\phi),\qquad
 h=1-\gamma.
 \label{eq:constants}
\end{equation}
Under \eqref{eq:calibration},
\[
 0<t<\frac12,\qquad d>0,\qquad h>0.
\]
Define the calibrated affine coefficients
\begin{equation}
 a=\gamma+ht,\qquad b=hd>0.
 \label{eq:affine}
\end{equation}

\begin{assumption}[Fixed-shot statistical model]
\label{ass:statistics}
A positive integer \(n\) is chosen before data collection.
The \(n\) source preparations satisfy Assumption~\ref{ass:source};
the one-shot device is repeated with independent fresh
environments and without access to the hidden source labels.
The reduced input on the data registers is
\(\rho_\theta^{\otimes n}\), and the ideal repeated record map is
\(\mathcal I^{\otimes n}\).
There is no unaccounted adaptive stopping, record deletion,
or selection of successful trials.
\end{assumption}

For the complete word \(r_i=(x_i,y_i)\), define
\begin{equation}
 Z_i=\one_{\{y_i=0\}},\qquad
 K=\sum_{i=1}^nZ_i,\qquad
 \overline Z=\frac Kn.
 \label{eq:statistic}
\end{equation}
The separate classical inference map returns
\begin{equation}
 \widehat\theta
 =
 \clamp\!\left(\frac{\overline Z-a}{b}\right),
 \qquad
 \clamp(u)=\max\{0,\min\{1,u\}\}.
 \label{eq:estimator}
\end{equation}

For definiteness, this is a classical channel from word lists to
integer labels \(k\in\{0,\ldots,n\}\), with the declared
interpretation
\[
 k\longmapsto
 \widehat\theta_k
 =
 \clamp\!\left(\frac{k/n-a}{b}\right).
\]
A CPTP matrix extension is
\begin{equation}
 \mathcal D_n(X)
 =
 \sum_{\mathbf r\in\mathcal R^n}
 \bra{\mathbf r}X\ket{\mathbf r}\,
 \kb{K(\mathbf r)}{K(\mathbf r)}.
 \label{eq:decoder-channel}
\end{equation}
Its Kraus operators are
\(\ket{K(\mathbf r)}\bra{\mathbf r}\), whose squared moduli
sum to the identity. This map does not return a data qubit.

\begin{theorem}[Finite Enter instrument]
\label{thm:main}
Freeze the interface of Section~\ref{sec:interface}, adopt Born
semantics and Definition~\ref{def:names}, and impose
Assumption~\ref{ass:statistics} for the estimation statements.
Then the following hold.

\begin{enumerate}[label=\textup{(\roman*)},leftmargin=*]
\item
\textbf{Complete instrument identity.}
The instrument \(\mathcal I\) is a complete CP instrument with
classical output. For every finite untouched retained factor
\(T\), including an arbitrary reference,
\[
 \mathcal I^T
 =
 \mathcal J_B^T\circ(\Phi\otimes\id_T)
\]
on all input operators. Discarding its record leaves the
unconditional \(T\)-marginal unchanged.

\item
\textbf{Consumption and residual scope.}
Both measured data names are consumed. The newly returned object
is the classical word \(r\); no quantum \(N\)- or \(A\)-output
belongs to the instrument. A conditional spectator or inaccessible
implementation complement is not a return of those names.

\item
\textbf{Declared likelihood.}
The one-shot probabilities are
\begin{align}
 p_{00}(\theta)
 &=
 \theta(1-t)(1-ht)
 +(1-\theta)t(\gamma+ht),
 \nonumber\\
 p_{01}(\theta)
 &=ht(1-t),
 \nonumber\\
 p_{10}(\theta)
 &=
 \theta t(1-ht)
 +(1-\theta)(1-t)(\gamma+ht),
 \nonumber\\
 p_{11}(\theta)
 &=
 h\bigl[\theta t^2+(1-\theta)(1-t)^2\bigr].
 \label{eq:probabilities}
\end{align}
In particular, \(Z_i\) are independent Bernoulli variables with
\begin{equation}
 q_\theta=\Pr_\theta\{Z_i=1\}=a+b\theta.
 \label{eq:bernoulli}
\end{equation}

\item
\textbf{Separate finite-shot inference.}
The estimator \eqref{eq:estimator} is the maximum-likelihood
estimator for the \emph{coarsened Bernoulli data}
\(Z_1,\ldots,Z_n\). It is not asserted to be the
maximum-likelihood estimator for the complete four-outcome
record.

For every \(\varepsilon>0\),
\begin{equation}
 \Pr_\theta\{|\widehat\theta-\theta|>\varepsilon\}
 \le
 2e^{-2nb^2\varepsilon^2}.
 \label{eq:tail}
\end{equation}
For \(0<\alpha<1\), define
\begin{equation}
 R_n(\alpha)
 =
 \min\left\{1,\frac1b
 \sqrt{\frac{\log(2/\alpha)}{2n}}\right\}.
 \label{eq:radius}
\end{equation}
Uniformly over \(\theta\in[0,1]\),
\begin{equation}
 \Pr_\theta\{|\widehat\theta-\theta|\le R_n(\alpha)\}
 \ge1-\alpha.
 \label{eq:coverage}
\end{equation}
Thus a sufficient shot budget for a radius at most
\(0<\varepsilon<1\) is
\begin{equation}
 n\ge
 \left\lceil
 \frac{\log(2/\alpha)}{2b^2\varepsilon^2}
 \right\rceil.
 \label{eq:shots}
\end{equation}

\item
\textbf{Fisher-information ledger.}
For \(0<\theta<1\), the achieved one-shot information in the
coarsened data is
\begin{equation}
 F_A(\theta)=
 \frac{b^2}{q_\theta(1-q_\theta)}.
 \label{eq:FA}
\end{equation}
The complete word has information
\begin{equation}
 F_C(\theta)
 =
 d^2\left[
 \frac1{p_{00}(\theta)}
 +\frac{\gamma^2}{p_{10}(\theta)}
 +\frac{h^2}{p_{11}(\theta)}
 \right].
 \label{eq:FC}
\end{equation}
They satisfy
\begin{equation}
 F_A(\theta)\le F_C(\theta)
 \le B_\theta:=\frac1{\theta(1-\theta)}.
 \label{eq:FI-ledger}
\end{equation}
The last bound also applies to any POVM on the post-step data
state \(\Phi(\rho_\theta)\). Independent repetition multiplies
the respective Fisher informations by \(n\).
\end{enumerate}

The output interpretation is
\emph{finite-model estimate under declared access}.
The theorem supplies neither a quantum-return decoder nor a
physical identification with Galactic transport.
\end{theorem}

\begin{proof}
We prove each claim on the declared interface.

\paragraph{Instrument and consumption.}
Lemma~\ref{lem:identity} proves the complete identity and its
reference extensions. Equivalently, the \(r\)-branch on a
record-only interface has Kraus rows
\[
 B_{r\ell}=\bra rH_\ell.
\]
Their completeness is
\[
 \sum_{r,\ell}B_{r\ell}^\dagger B_{r\ell}
 =
 \sum_\ell H_\ell^\dagger
 \left(\sum_rP_r\right)H_\ell
 =I_B.
\]
Lemma~\ref{lem:consumption} proves the name statement.

\paragraph{Likelihood.}
The squared moduli of both \(V_\phi\) and
\(V_\phi^\dagger\) are the transition matrix
\[
 \begin{pmatrix}
 1-t&t\\ t&1-t
 \end{pmatrix}.
\]
The diagonal gate \(\CZ\) changes phases but not computational
populations.

Conditional on source component \(00\), the normal output has
probabilities \(1-t,t\). The antiqubit's probability of value
\(1\), after relaxation, is \(ht\). Conditional on source component
\(11\), the normal output has probabilities \(t,1-t\), and the
antiqubit's probability of value \(1\) is \(h(1-t)\).

The diagonal output populations conditional on either source
component factor accordingly. This remains true despite
coherences in the quantum state: amplitude damping transfers
diagonal populations as stated, and its off-diagonal matrix
elements do not contribute to these computational probabilities.
Mixing the two source components gives
\eqref{eq:probabilities}. Summing the outcomes with \(y=0\)
gives
\[
 q_\theta
 =
 \gamma+h\bigl[t+(1-2t)\theta\bigr]
 =a+b\theta.
\]
The product preparation and product device assumptions give
independence.

\paragraph{Likelihood maximization.}
For a count \(K=k\), the Bernoulli likelihood is proportional to
\[
 q^k(1-q)^{n-k},
 \qquad q\in[a,a+b].
\]
Its unconstrained maximizing value is \(k/n\); restricting it to
the interval gives the projection of \(k/n\) onto \([a,a+b]\).
Applying the affine inverse gives
\eqref{eq:estimator}, including endpoint counts.

\paragraph{Concentration.}
For completeness, let \(Z\) be Bernoulli with mean \(q\), and set
\[
 g(u)=\log\mathbb E e^{u(Z-q)}.
\]
Then \(g(0)=g'(0)=0\), and \(g''(u)\) is the variance of a
Bernoulli variable under an exponentially tilted law. Hence
\(g''(u)\le1/4\), and integration yields
\[
 g(u)\le\frac{u^2}{8}
 \qquad(u\in\mathbb R).
\]
Independence and exponential Markov bounds therefore give
\[
 \Pr\{|\overline Z-q|>v\}\le2e^{-2nv^2}.
\]
This is the Bernoulli instance of Hoeffding's inequality
\cite{Hoeffding1963}.
% Concentration inequality and its statistical assumptions:
% ([cs.rpi.edu](https://www.cs.rpi.edu/academics/courses/spring06/random/hoefding.pdf?utm_source=openai))

Projection onto \([0,1]\) cannot increase distance from
\(\theta\in[0,1]\). Consequently,
\[
 |\widehat\theta-\theta|
 \le
 \left|
 \frac{\overline Z-a}{b}-\theta
 \right|
 =
 \frac{|\overline Z-q_\theta|}{b}.
\]
Taking \(v=b\varepsilon\) proves \eqref{eq:tail}.
Taking
\(v=\sqrt{\log(2/\alpha)/(2n)}\), and using
\(|\widehat\theta-\theta|\le1\), proves
\eqref{eq:radius}--\eqref{eq:coverage}.
Equation~\eqref{eq:shots} follows directly.

\paragraph{Achieved Fisher information.}
For a Bernoulli law with derivative \(q'_\theta=b\),
\[
 F_A
 =
 \frac{b^2}{q_\theta}
 +\frac{b^2}{1-q_\theta}
 =
 \frac{b^2}{q_\theta(1-q_\theta)}.
\]
Differentiating \eqref{eq:probabilities} gives
\[
 (p'_{00},p'_{01},p'_{10},p'_{11})
 =
 (d,0,-\gamma d,-hd),
\]
which proves \eqref{eq:FC}. All four probabilities are strictly
positive under the calibration assumptions.

\paragraph{Information comparisons.}
Let \(J\) denote the hidden source component, with
\(\Pr_\theta\{J=0\}=\theta\). Its score is
\[
 s_\theta(J)=
 \begin{cases}
 1/\theta,&J=0,\\
 -1/(1-\theta),&J=1.
 \end{cases}
\]
Thus
\[
 \mathbb E_\theta s_\theta(J)=0,\qquad
 \mathbb E_\theta s_\theta(J)^2=B_\theta.
\]

For any outcome \(R\) obtained through a
\(\theta\)-independent conditional law given \(J\), the outcome
score equals
\[
 \partial_\theta\log p_\theta(R)
 =
 \mathbb E_\theta[s_\theta(J)\mid R].
\]
Conditional Jensen gives
\[
 F_R
 =
 \mathbb E_\theta
 \left(\mathbb E_\theta[s_\theta(J)\mid R]\right)^2
 \le B_\theta.
\]
This applies to the complete word \(C\). It also applies to
any POVM on \(\Phi(\rho_\theta)\): its probabilities are
\[
 \theta\,\Tr(E_j\Phi(\kb{00}{00}))
 +(1-\theta)\,\Tr(E_j\Phi(\kb{11}{11})),
\]
which have exactly the required mixture form.

Finally, \(Z\) is a deterministic coarsening of \(C\).
Its score is the conditional expectation of the complete-word
score given \(Z\). A second application of conditional Jensen
gives \(F_A\le F_C\).

For independent repetitions, scores add and have mean zero,
so the squared-score expectations add. This proves the
\(n\)-shot information statements and completes the proof.
\end{proof}

\begin{remark}[Quantum-information comparison]
The input family has symmetric logarithmic derivative, on its
support,
\[
 L_\theta^{\rm src}
 =
 \frac1\theta\kb{00}{00}
 -\frac1{1-\theta}\kb{11}{11},
\]
and therefore input quantum Fisher information
\[
 \Tr(\rho_\theta(L_\theta^{\rm src})^2)=B_\theta.
\]
The preceding proof establishes the post-step measurement bound
directly. Its relation to the standard optimization of classical
Fisher information over measurements is consistent with the
Braunstein--Caves framework \cite{BraunsteinCaves1994}.
No equality of \(F_A\), \(F_C\), and the post-step quantum Fisher
information is asserted.
% Quantum Fisher information and measurement optimization:
% ([doi.org](https://doi.org/10.1103/PhysRevLett.72.3439?utm_source=openai))
\end{remark}

\begin{remark}[Confidence is not exact recovery]
A confidence statement is a repeated-sampling coverage guarantee.
It is not a posterior probability without additional Bayesian
assumptions. The Fisher-information comparison is not by itself
a finite-shot confidence theorem.

The unclipped estimator
\[
 \widetilde\theta=\frac{\overline Z-a}{b}
\]
is unbiased and has variance
\[
 \operatorname{Var}_\theta(\widetilde\theta)
 =
 \frac{q_\theta(1-q_\theta)}{nb^2}
 =
 \frac1{nF_A(\theta)}.
\]
Clipping may introduce bias, but decreases squared error relative
to every \(\theta\in[0,1]\). Thus
\[
 \mathbb E_\theta(\widehat\theta-\theta)^2
 \le
 \frac{q_\theta(1-q_\theta)}{nb^2}.
\]
No unbiased-estimator assertion is made for the clipped estimator.
\end{remark}

\subsection{An explicit rational instance}

Freeze
\begin{equation}
 \phi=\frac{\pi}{6},\qquad
 \gamma=\frac14.
 \label{eq:benchmark}
\end{equation}
Then
\[
 t=\frac14,\qquad h=\frac34,\qquad d=\frac12,
 \qquad a=\frac7{16},\qquad b=\frac38.
\]
The complete probabilities become
\begin{equation}
 \begin{array}{c|cccc}
 r&00&01&10&11\\ \hline
 p_r(\theta)
 &\dfrac{7+32\theta}{64}
 &\dfrac9{64}
 &\dfrac{21-8\theta}{64}
 &\dfrac{27-24\theta}{64}.
 \end{array}
 \label{eq:rational-law}
\end{equation}
They are nonnegative and sum to one for every
\(\theta\in[0,1]\).

The inference map is
\begin{equation}
 \widehat\theta
 =
 \clamp\!\left(\frac{16K/n-7}{6}\right),
 \qquad
 R_n(\alpha)
 =
 \min\left\{1,\frac83
 \sqrt{\frac{\log(2/\alpha)}{2n}}\right\}.
 \label{eq:rational-estimator}
\end{equation}
Its estimate values are rational at every finite \(n\).
A sufficient shot count is
\begin{equation}
 n\ge
 \left\lceil
 \frac{32}{9\varepsilon^2}\log\frac2\alpha
 \right\rceil.
 \label{eq:rational-shots}
\end{equation}
The information ledger is
\[
 F_A(\theta)
 =
 \frac{36}{(7+6\theta)(9-6\theta)},
\]
and
\[
 F_C(\theta)
 =
 \frac{16}{7+32\theta}
 +\frac1{21-8\theta}
 +\frac9{27-24\theta}.
\]
These are analytic formulas, not reported numerical execution.

\section{Record-only recovery and the correct residual distinction}
\label{sec:recovery}

\subsection{No unrestricted quantum return from the record alone}

For compatible channels \(\Lambda,\Gamma\), use half-diamond
distance
\[
 \ddia(\Lambda,\Gamma)
 =
 \frac12\norm{\Lambda-\Gamma}_{\diamond}.
\]

\begin{proposition}[Record-only quantum-recovery obstruction]
\label{prop:no-return}
Let
\(\mathcal R:\Bop(\mathcal H_C)\to\Bop(\mathcal H_B)\)
be any CPTP map whose only input is the classical record.
Then neither \(\mathcal R\mathcal M_B\) nor
\(\mathcal R\mathcal I\) is the identity channel on an unrestricted
four-dimensional input. In fact,
\begin{equation}
 \ddia(\mathcal R\mathcal M_B,\id_B)\ge\frac34,
 \qquad
 \ddia(\mathcal R\mathcal I,\id_B)\ge\frac34.
 \label{eq:no-return-bound}
\end{equation}
\end{proposition}

\begin{proof}
On diagonal record inputs, any such map has the form
\[
 \mathcal R\left(\sum_rp_r\kb r r_C\right)
 =
 \sum_rp_r\sigma_r,
 \qquad
 \sigma_r=\mathcal R(\kb r r_C).
\]
Consequently,
\[
 (\mathcal R\mathcal I)(X)
 =
 \sum_r\Tr(E_rX)\sigma_r,
 \qquad
 E_r=\sum_\ell H_\ell^\dagger P_rH_\ell.
\]
The effects \(E_r\) are positive and sum to \(I_B\).
The same expression with \(E_r=P_r\) describes
\(\mathcal R\mathcal M_B\).

Such a measure-and-prepare channel has separable output across
an untouched-reference/output cut. This is the
entanglement-breaking structure studied in
\cite{HorodeckiShorRuskai2003}; here it follows immediately
from the displayed form.
% Measure-and-prepare channels and entanglement breaking:
% ([arxiv.org](https://arxiv.org/abs/quant-ph/0302031?utm_source=openai))

Take a four-dimensional reference and
\[
 \ket{\Omega_4}
 =
 \frac12\sum_{j=0}^3\ket j_Q\ket j_B.
\]
For any product state \(\tau_Q\otimes\sigma_B\),
\[
 \bra{\Omega_4}(\tau_Q\otimes\sigma_B)\ket{\Omega_4}
 =
 \frac14\Tr(\tau_Q^{\mathsf T}\sigma_B)
 \le\frac14.
\]
The same bound holds for every separable state by convexity.
The identity channel preserves \(\kb{\Omega_4}{\Omega_4}\),
whose overlap with itself is one. Testing its projector
therefore gives trace distance at least \(1-1/4=3/4\).
This particular referenced input lower-bounds the half-diamond
distance, proving both inequalities.
\end{proof}

The proposition concerns unrestricted quantum-state recovery
from the record \emph{alone}. It does not rule out parameter
estimation, recovery of a restricted classical algebra, or a
different protocol with an additional quantum receiver,
entangled assistance, or access to a complete environment.

\subsection{Residual information is not physically annihilated}

The instrument is a map on an observer's accessible interface.
A dilation may retain information in its inaccessible complement.
For example, the isometry
\[
 \ket r_B\longmapsto\ket r_C\ket r_F
\]
followed by tracing out \(F\) realizes computational record
readout. Before the partial trace, joint coherences can be
present in \(CF\). They are not available as a quantum output
of the record-only channel.

Thus the statement
\begin{quote}
``The quantum names are consumed''
\end{quote}
must not be replaced by
\begin{quote}
``All quantum information has been destroyed everywhere.''
\end{quote}

Likewise, an unmeasured spectator may continue to evolve.
For example, readout of \(B\) on
\(\kb{00}{00}_B\otimes\rho_S\) leaves \(\rho_S\) unchanged,
and a subsequently declared channel on \(S\) can act on it.
The positive statement is about the absence of measured data
names from the output context, not a universal prohibition on
residual dynamics.

This correction preserves the record-versus-residual distinction
in Landmark I and source En10-A@1
\cite{CheungI,CheungRelational}. The static-record versus
recovery comparison in Oai265-A is a documentary parallel,
not a premise of Proposition~\ref{prop:no-return}
\cite{CheungOai265}.

\subsection{No exact finite-shot source recovery}

\begin{proposition}[Finite samples do not recover every source weight exactly]
\label{prop:no-exact-theta}
At any fixed finite \(n\), no classical inference procedure on
the complete word list returns the exact value of \(\theta\)
with probability one for every \(\theta\in[0,1]\).
\end{proposition}

\begin{proof}
Under the calibration assumptions every one-shot outcome has
strictly positive probability for every \(\theta\in[0,1]\).
Hence every finite word list has positive probability under
every such \(\theta\).

Suppose a possibly randomized inference procedure returned
\(\theta_0\) almost surely when the parameter was \(\theta_0\).
For each word list its conditional output law would then have
to be concentrated at \(\theta_0\), since each list has positive
probability. The same conditional output laws are used at any
different parameter \(\theta_1\), so the procedure would still
return \(\theta_0\) almost surely. It cannot also return
\(\theta_1\) almost surely.
\end{proof}

The finite-confidence contract is therefore the substantive
positive result; it is not a disguised exact-recovery claim.

\section{Implementation tolerance on the same interface}
\label{sec:tolerance}

\subsection{A finite-shot allowance}

\begin{proposition}[Robust estimation under complete-channel error]
\label{prop:robust}
Keep the nominal coefficients \(a,b\), source family, and fixed
sample size. Suppose shot \(i\) uses a classical-output CPTP map
\(\widetilde{\mathcal I}_i\) on the same comparison interface and
\[
 \ddia(\widetilde{\mathcal I}_i,\mathcal I)\le\delta
 \qquad(i=1,\ldots,n).
\]
Assume the resulting shot records are independent; they need not
be identically distributed. Apply the nominal inference map
\eqref{eq:estimator}. Then
\begin{equation}
 \Pr_\theta\left\{
 |\widehat\theta-\theta|
 \le
 \min\left\{1,\frac{\delta+
 \sqrt{\log(2/\alpha)/(2n)}}{b}\right\}
 \right\}
 \ge1-\alpha.
 \label{eq:robust}
\end{equation}
\end{proposition}

\begin{proof}
Let \(q_i=\mathbb E_\theta Z_i\) for the implemented shot.
Trace-distance control of the output states bounds every event
probability difference, so
\[
 |q_i-q_\theta|\le\delta.
\]
Consequently
\[
 \left|\frac1n\sum_iq_i-q_\theta\right|\le\delta.
\]
The bounded-independent-variable argument in the main proof
applies without identical means and gives
\[
 \Pr\left\{
 \left|\overline Z-\frac1n\sum_iq_i\right|
 >
 \sqrt{\frac{\log(2/\alpha)}{2n}}
 \right\}
 \le\alpha.
\]
Combine these inequalities and apply the same projection
argument as before.
\end{proof}

For a target radius \(\varepsilon\), the displayed certificate
has a finite sufficient shot budget only when
\(b\varepsilon>\delta\), namely
\begin{equation}
 n\ge
 \left\lceil
 \frac{\log(2/\alpha)}
 {2(b\varepsilon-\delta)^2}
 \right\rceil.
 \label{eq:robust-shots}
\end{equation}
This is a limitation of this nominal certificate, not a
universal impossibility theorem for alternative calibrated
procedures.

If these assumptions hold conditional on calibration information
and on an event \(G\) with \(\Pr(G^c)\le\beta\), the same conditional
proof followed by integration gives unconditional coverage at
least \(1-\alpha-\beta\). The conditional independence and bias
bounds must be justified; they do not follow from naming a device
``calibrated''. This is the same statistical caution illustrated
by the finite-confidence contracts in Oai265-B
\cite{CheungOai265}.

\subsection{Gate errors, repeated records, and exact causality}

\begin{lemma}[Complete-channel telescoping]
\label{lem:telescoping}
For compatible CPTP maps \(G_1,\ldots,G_m\) and implemented maps
\(\widetilde G_1,\ldots,\widetilde G_m\) on matching interfaces,
\[
 \ddia(\widetilde G_m\cdots\widetilde G_1,
       G_m\cdots G_1)
 \le
 \min\left\{1,\sum_{j=1}^m
 \ddia(\widetilde G_j,G_j)\right\}.
\]
\end{lemma}

\begin{proof}
Subtract the two compositions and replace one factor at a time.
Each term contains one difference of corresponding maps,
composed before and after with channels. Diamond norm is
contractive under those channel compositions. The triangle
inequality gives the sum, and half-diamond distance between
channels is at most one.
\end{proof}

This proves a tolerance statement only after the comparison
interfaces are fixed. It is the local counterpart of the
resource-controlled complete-channel allowance used in En08-B
\cite{CheungEn08B}.

For \(n\) independent repetitions, the complete word-list channel
has the separate bound
\[
 \ddia\left(
 \bigotimes_{i=1}^n\widetilde{\mathcal I}_i,
 \mathcal I^{\otimes n}\right)
 \le\min\{1,n\delta\}.
\]
This is not the same ledger as the per-shot mean-bias allowance
\(\delta/b\) in \eqref{eq:robust}.

Nor does a small allowance prove exact non-signalling or exact
geometric containment. For example, let \(\mathcal T\) be the
constant qubit-output channel and \(\mathcal Z\) the channel
that measures the input in the computational basis and prepares
the corresponding basis state. Then
\[
 \Lambda_\eta=(1-\eta)\mathcal T+\eta\mathcal Z,
 \qquad
 \ddia(\Lambda_\eta,\mathcal T)\le\eta,
\]
but for every \(\eta>0\) the outputs on inputs \(0\) and \(1\)
can differ. Approximate channel agreement is not exact
independence from an input.

A hardware failure alphabet, if needed, must be added to a common
comparison interface with all attempted outcomes retained.
Discarding failed trials without an explicit statistical
adjustment does not satisfy Assumption~\ref{ass:statistics}.

\section{Resource, estimation, and residual ledgers}
\label{sec:ledgers}

\subsection{An explicit finite dilation and schedule}

A fresh environment qubit \(E\), initially in \(\ket0\), realizes
the antiqubit relaxation by the isometry
\begin{align}
 \ket0_A\ket0_E&\longmapsto\ket0_A\ket0_E,
 \nonumber\\
 \ket1_A\ket0_E&\longmapsto
 \sqrt h\,\ket1_A\ket0_E
 +\sqrt\gamma\,\ket0_A\ket1_E.
 \label{eq:dilation}
\end{align}
An explicit unitary extension is obtained by applying:
\begin{enumerate}[label=\textup{(\arabic*)},leftmargin=*]
\item a rotation
\(R_y(2\arcsin\sqrt\gamma)\) on \(E\), controlled by \(A=1\);
\item \(\CNOT_{E\to A}\).
\end{enumerate}
Here \(R_y(\omega)=e^{-i\omega Y/2}\).
The two formulas in \eqref{eq:dilation} follow directly.

Destructive Enter of \(E\), with its outcome not exported,
gives the two Kraus branches \(L_0,L_1\) and consumes the
environment's laboratory name. Destructive Enter of \(N,A\)
then produces the exported word.

A complete one-shot schedule, after the supplier delivers the
data registers, is:
\begin{enumerate}[label=\textup{(\arabic*)},leftmargin=*]
\item allocate \(E\) in \(\ket0\);
\item apply \(V_\phi\) and \(V_\phi^\dagger\) in parallel;
\item apply \(\CZ\);
\item apply the controlled environment rotation;
\item apply \(\CNOT_{E\to A}\);
\item Enter \(E\) and forget its unexported outcome;
\item Enter \(N,A\) and export both outcome bits.
\end{enumerate}
Forgetting the unexported classical value is an interface
operation. No physical cyclic erasure cost is assigned to it.

\subsection{Computational ledger}

The following counts refer to the declared primitive library:
single-qubit rotations, \(\CZ\), controlled rotations, CNOT,
fresh allocation, and destructive Enter. They are not counts in
a synthesized hardware gate set.

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{0.28\linewidth}X@{}}
\toprule
Resource & Declared quantity or boundary\\
\midrule
Data input per shot &
Two supplied qubits, \(N,A\).\\
Fresh laboratory ancilla &
One environment qubit \(E\) per shot.\\
Peak live laboratory qubits &
\(3+s\), if \(S\) consists of \(s\) retained laboratory qubits.\\
Coherent operations per shot &
Two one-qubit rotations and three two-qubit operations.\\
Coherent gate depth &
At most four layers: parallel local responses, \(\CZ\),
controlled rotation, CNOT.\\
Destructive readouts &
Three one-qubit Enter operations: one unexported environment
outcome and two exported data outcomes.\\
Readout rounds &
Two rounds if the two final data readouts are parallel.\\
Quantum data output &
None. Previously retained spectators remain; they are not
measured-data returns.\\
Shots &
The predeclared \(n\), satisfying the selected confidence budget.\\
Sequential laboratory allocations &
\(n\) fresh environment names and \(2n\) supplied data names.\\
Classical record &
Two exported bits per shot. A complete histogram contains four
counts summing to \(n\).\\
Classical inference storage &
The declared estimator needs the integer count \(K\);
the complete histogram may also be retained.\\
Supplier preparation &
External and not priced by the laboratory gate count.
Any source purification is inaccessible and separately held.\\
External reference &
Included in correctness comparisons. If physically laboratory-held,
its storage must be added to the laboratory ledger.\\
Primitive measurement apparatus &
Not microscopically constructed or thermodynamically priced.
Enter is an admitted primitive of
\end{tabularx}
\end{center}

Sequential reuse of laboratory capacity does not imply reuse of
a consumed name or cost-free restoration of a physical device.
No storage of all \(n\) quantum inputs at once is assumed.

\subsection{Estimation ledger}

The named parameter and known calibration inputs are
\[
 \text{estimated: }\theta,\qquad
 \text{known: }\phi,\gamma,a,b.
\]
The achieved and comparison quantities are
\[
 nF_A(\theta)
 \le nF_C(\theta)
 \le nB_\theta,
\]
together with the finite-confidence radius
\(R_n(\alpha)\), or its implementation-adjusted counterpart.

The denominator \(b\) is indispensable. If the calibration were
changed so that \(b=0\), the selected Bernoulli statistic would
be independent of \(\theta\), and its inference contract would
fail. This is a statement about that statistic; some other
record factor could still contain information in a different
interface.

In particular, \(\gamma=1\) would make the antiqubit readout
constant, while the normal-qubit marginal could remain
source-dependent. The main theorem avoids this degeneracy and
does not conflate loss of one statistic with loss of every
accessible parameter.

\subsection{Residual and omitted-parameter ledger}

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{0.31\linewidth}X@{}}
\toprule
Quantity & Status in the frozen interface\\
\midrule
Measured \(N,A\) quantum names &
Consumed; no quantum output port.\\
Conditional spectator state &
Retained only if explicitly supplied; not determined by the
record alone on arbitrary correlated inputs.\\
Source-component label &
Not supplied to the observer.\\
Source purification &
Supplier-side complement; inaccessible.\\
Relaxation and readout complements &
Outside record-only access; not declared annihilated.\\
Exact turbulence spectrum &
Absent from the preparation family and channel.\\
Galactic magnetic-field direction \(\hat n\) &
Absent. The matrix \(Y\) is a prescribed control generator,
not an acquired Galactic field direction.\\
Absolute Galactic normalization &
Absent. Fixed trial count is not a clock-calibrated source rate.\\
Dimensionful diffusion length or coefficient &
Not estimated. No length scale, time step, or continuum
transport equation is installed.\\
Positronium singlet probability &
Not measured: these qubits encode position labels, not two
independent physical spin factors.\\
Annihilation, escape, or photon counts &
No corresponding output flags or sectors are installed.\\
Physical energy, heat, work, and reset cost &
No complete thermodynamic implementation is supplied.\\
\bottomrule
\end{tabularx}
\end{center}

\section{What the record does not identify}
\label{sec:nonidentification}

\subsection{Omitted nuisance parameters have zero information}

\begin{proposition}[Non-identifiability of omitted parameters]
\label{prop:nuisance}
Let \(\zeta\) denote any additional parameter that is absent from
\(\rho_\theta\), \(\Phi\), the readout, and the declared repeated-shot
model. Then the complete record law is independent of \(\zeta\).
Where derivatives are defined, every classical Fisher-information
entry involving \(\zeta\) is zero.

No inference procedure on these records distinguishes two
values of \(\zeta\) with different output distributions.
\end{proposition}

\begin{proof}
The complete record probability is
\[
 \Pr_\theta\{r_1,\ldots,r_n\}
 =
 \prod_{i=1}^np_{r_i}(\theta),
\]
which contains no \(\zeta\). Its \(\zeta\)-derivative is zero.
A classical post-processing of identical input laws has
identical output laws.
\end{proof}

Thus absence of access is not asserted merely by verbal decree.
For the omitted quantities listed in the ledger it is witnessed
by their absence from the likelihood.

Conversely, if an unknown field direction, loss law, turbulence
parameter, or calibration uncertainty is inserted into the
channel, it may affect the records indirectly. Its
identifiability and nuisance-parameter information must then be
analyzed in the enlarged model. The present theorem cannot
declare it inaccessible while silently using its influence.

The nominal scale
\(5\times10^{43}\,\mathrm{s}^{-1}\) mentioned in the motivating
plan is not a parameter of this construction. No value of an
absolute Galactic rate, including that benchmark, is derived.

\subsection{These occupation records do not certify inversion}

\begin{proposition}[Inversion is an input, not a conclusion of this record]
\label{prop:inversion}
Replace \(V_\phi^\dagger\) on \(A\) in \eqref{eq:inverse} by
\(V_\phi\), keeping the same source family, diagonal \(\CZ\),
relaxation channel, and computational readout. The complete
record law on every \(\rho_\theta\) is unchanged.

Accordingly, the records used by the main theorem do not
certify the sign of the antiqubit response.
\end{proposition}

\begin{proof}
The two rotations have identical squared matrix elements.
The source family is diagonal. The computational populations
after the local rotations therefore agree. The diagonal
\(\CZ\) leaves those populations unchanged, and amplitude
damping transforms them by the same population rule.
Computational readout consequently gives the same
\eqref{eq:probabilities}.
\end{proof}

This equality is asserted on the declared source family.
It is not an equality of the two channels on all coherent,
reference-entangled input operators. A protocol that validates
unitary inversion must add suitable coherent probes and its own
comparison contract.

The construction is not empty of quantum coherence: before
relaxation, for example, the amplitude matrix of
\(U_\phi\ket{00}\) is
\[
 \begin{pmatrix}
 \cos^2\phi&-\cos\phi\sin\phi\\
 \cos\phi\sin\phi&\sin^2\phi
 \end{pmatrix},
\]
whose determinant is
\(2\cos^2\phi\sin^2\phi\ne0\).
That branch is entangled. Nevertheless the selected occupation
likelihood has an exact classical sampling description:
sample the hidden source component and then sample the two
conditional population transitions derived in the proof.
No computational or metrological quantum advantage for this
source-weight task is claimed.

\subsection{A channel is not a thermal machine}

\begin{proposition}[Thermodynamic underdetermination]
\label{prop:thermal}
The visible channel \(\Phi\), the destructive record channel,
and their complete reference identities do not determine a
unique physical energy transfer, bath heat, work cost, or cyclic
memory-reset cost.
\end{proposition}

\begin{proof}
In the dilation \eqref{eq:dilation}, assign the environment
Hamiltonian
\[
 H_E^{(\epsilon)}=\epsilon\kb1 1_E,\qquad \epsilon>0.
\]
The reduced channel on the data is unchanged as \(\epsilon\)
varies. If the pre-relaxation data state has antiqubit population
\(p_1\) in value \(1\), the environment's energy increase is
\[
 \Delta E_E^{(\epsilon)}=\epsilon\gamma p_1.
\]
Hence the visible channel alone fixes neither that energy
transfer nor the Hamiltonian needed to interpret it.
A pure blank environment has not thereby been identified as a
thermal reservoir.

Even after chosen Hamiltonians are fixed, an independent
unobserved process can alter a designated reservoir ledger
without altering the data or record channel. For example, take
a fuel qubit initially excited and a Gibbs reservoir qubit,
both with gap \(\epsilon\). At inverse temperature \(\beta\),
the reservoir excitation probability is
\[
 r=\frac1{1+e^{\beta\epsilon}}.
\]
Swapping the two qubits increases the designated reservoir
energy by \(\epsilon(1-r)\), while acting trivially on the
visible interface. Repeating this independent process changes
that ledger without changing the visible channel.

These examples consume blanks or fuel. They are not cyclic
erasure protocols unless those resources and controls are also
restored. They establish underdetermination, not a violation of
a thermal inequality.
\end{proof}

The proposition is a local version of the channel-versus-ledger
separation in En07-B \cite{CheungEn07B}. Landauer-type statements
require a further physical setup, including the reservoir,
initial correlations, Hamiltonians, process, and restoration
requirements; a rigorous finite-reservoir treatment is given
by Reeb and Wolf \cite{ReebWolf2014}.
% Thermal inequalities require a specified physical setup:
% ([arxiv.org](https://arxiv.org/abs/1306.4352))

\subsection{Finite approximation is not a continuum identification}

Any later larger-graph or continuum comparison must specify,
at minimum:
\begin{enumerate}[label=\textup{(\roman*)},leftmargin=*]
\item the target state and channel domains;
\item encoding and decoding maps;
\item the output and residual comparison interface;
\item the error topology and a proved error bound;
\item an actual resource schedule;
\item independently supplied spatial and temporal calibration;
\item a bridge connecting the named finite observables to the
physical target.
\end{enumerate}
Lemma~\ref{lem:telescoping} can price a finite approximation
once these interfaces exist. It does not create them.

This maintains the separation between scheduled ancestry,
operational influence, and independently supplied causal
geometry in VII-J, and the finite-data cautions of En08-A and
XIX-C \cite{CheungVIIJ,CheungEn08A,CheungXIXC}.
An observation boundary defined by available operations is not
a derived causal cone.

\section{Relationship to the supplied manuscript programme}
\label{sec:comparisons}

The following comparisons identify methodological relations;
they do not transfer hypotheses between models.

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{0.27\linewidth}X@{}}
\toprule
Source & Relationship to the present result\\
\midrule
Landmark I, version 2 &
Consumption and Born/residual characterization within its
specified class. The present name rule and instrument are
defined and proved locally.\\
Source En10-A@1 &
Template for complete destructive readout with subnormalized
residual blocks and arbitrary references. Its clock-conditioned
residual demonstrates why ``no residual can evolve'' would be
too strong.\\
XIX-C, version 1 &
Distinction between a supplied token and an acquisition
instrument, and between finite determinacy and continuum
evolution.\\
En08-B, version 1 &
Complete-channel approximation and resource-ledger pattern
after compatible comparison interfaces are supplied.\\
En08-A and VII-J &
Finite schedule, operational influence, and geometric comparison
remain distinct.\\
Oai265-A and Oai265-B &
Static records do not imply quantum recovery; finite-confidence
certificates require explicit complete-record statistical
assumptions.\\
En07-B, version 1 &
Channel equality alone does not determine a thermal cost.
Proposition~\ref{prop:thermal} supplies the local witness here.\\
XXII-B Candidate D, as named in the plan &
A proposed comparison label, not an imported theorem.
Exact identity and tolerance statements used here are proved
locally.\\
Inherited characteristic-graph programme &
No characteristic distribution, quantum gravitational
coefficient, interacting differential, or Ward identity is
constructed by this finite instrument.\\
\bottomrule
\end{tabularx}
\end{center}

The supplied XIV-J and XV-B excerpts concern separate analytic
domains and unresolved graph-specific constructions
\cite{CheungAnalytic}. A four-dimensional instrument does not
construct those missing products or supply their output-space
admission. This document does not open the downstream XV-B
exit, discharge XVI-N or Route B, or alter any version-specific
O.1 verdict. In particular, a scoped later subsector result is
not silently replaced by an older global label.

OpenAI catalogue entries remain documentary comparisons.
They supply neither \(\Phi\), the source-access assumption,
the Enter instrument, nor the statistical contract. No catalogue
family is verified here.

\section{Conclusion and verification boundary}
\label{sec:conclusion}

The construction has the explicit chain
\[
 \rho_\theta
 \xrightarrow{\ \Phi\ }
 \Phi(\rho_\theta)
 \xrightarrow{\ \mathcal M_B\ }
 \text{classical occupation word}
 \xrightarrow{\ \mathcal D_n\ }
 \text{finite-confidence source-weight estimate}.
\]
Every arrow has a declared domain and output. The second arrow
consumes the measured names. The third is classical
post-processing and is not part of the readout primitive.

The philosophical discipline is therefore realized rather than
merely repeated:
\begin{itemize}[leftmargin=*]
\item complete instrument equality is proved, not inferred from
agreement on one preparation;
\item the classical record and conditional residual are kept
separate;
\item an estimator is provided with a sensitivity and shot ledger;
\item omitted physical parameters are not recovered by relabeling
finite outputs;
\item thermodynamic and geometric conclusions are withheld
because their interfaces have not been constructed.
\end{itemize}

This is a finite analog with an estimation contract. It is not
a solution of the Galactic positron problem. Increasing the
number of qubits or combining this subroutine with a classical
transport code would require a new declared interface and
would not automatically strengthen the conclusions proved
here.

\paragraph{Verification boundary.}
The source uses standard pdfLaTeX packages and contains its
bibliography internally. The algebraic and statistical
arguments are supplied in the document. No successful
pdfLaTeX execution, proof-assistant verification, experimental
implementation, numerical benchmark, hardware compilation,
independent referee review, or journal acceptance of this
manuscript is asserted.

\begin{thebibliography}{99}

\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.
doi:\nolinkurl{10.3389/fphy.2026.1954943}.
The publisher record identifies an accepted article and a
forthcoming final formatted version. Its stated strategy class
is not enlarged here.

\bibitem{CheungPhilosophy}
N.~Cheung,
\emph{The Philosophy of Enter: Destructive Readout, Quantum
Recovery, and the Refusal of Identification},
\texttt{EntPhil-v2.tex}, version~2.
Enter Calculus Companion, introductory essay.
Philosophical map, not a registered theorem or a proof premise.

\bibitem{CheungI}
N.~Cheung,
\emph{Landmark I},
\texttt{I.tex}, version~2.
Supplied manuscript excerpts, especially
\texttt{thm:main}: characterization of destructive Enter within
the specified strategy class.
Cited with the scope of the supplied text; no independent
verification of the complete manuscript is asserted.

\bibitem{CheungRelational}
N.~Cheung,
\emph{En10-A},
\texttt{En10-A.tex}, version~1.
Supplied manuscript excerpts, especially
\texttt{thm:main}, \texttt{eq:exact-channel-identity},
\texttt{eq:drop-record}, and
\texttt{eq:linear-interface}.
Finite relational clock readout by destructive Enter.
Called ``source En10-A@1'' in the present document.

\bibitem{CheungXIXC}
N.~Cheung,
\emph{XIX-C},
\texttt{XIX-C.tex}, version~1.
Supplied manuscript excerpts, especially
\texttt{thm:main} and the token/acquisition comparison.
Finite incidence, complete-channel semantics, and continuum
non-determination.

\bibitem{CheungEn08B}
N.~Cheung,
\emph{En08-B},
\texttt{En08-B.tex}, version~1.
Supplied manuscript excerpts, especially
\texttt{thm:main} and \texttt{lem:telescoping}.
Resource-controlled operational approximation on a prescribed
comparison interface.

\bibitem{CheungEn08A}
N.~Cheung,
\emph{En08-A},
\texttt{En08-A.tex}, version~1.
Supplied manuscript excerpts, especially \texttt{thm:main}.
Scheduler consistency and conditional geometric representation
are distinct claims.

\bibitem{CheungVIIJ}
N.~Cheung,
\emph{VII-J},
\texttt{VII-J.tex}, version~1.
Supplied manuscript excerpts.
Scheduled ancestry, operational influence, and independently
supplied causal geometry remain separate.

\bibitem{CheungOai265}
N.~Cheung,
\emph{Oai265-A}, \texttt{Oai265-A.tex}, version~1,
especially \texttt{thm:separation};
and \emph{Oai265-B}, \texttt{Oai265-B-v2.tex}, version~2,
especially \texttt{thm:main}.
Supplied manuscript excerpts.
Record/recovery separation and complete-record finite-confidence
certification; neither supplies an astrophysical identification.

\bibitem{CheungEn07B}
N.~Cheung,
\emph{En07-B},
\texttt{En07-B.tex}, version~1.
Supplied manuscript excerpts, especially
\texttt{prop:no-landauer-from-channel}.
Thermodynamic underdetermination from observable-channel
equality alone.

\bibitem{CheungAnalytic}
N.~Cheung,
\emph{XIV-J}, \texttt{XIV-J.tex}, version~1,
especially \texttt{thm:J2} and \texttt{cor:noadmission};
and \emph{XV-B}, \texttt{XV-B.tex}, version~2,
especially \texttt{thm:release}.
Supplied manuscript excerpts.
Separate reconstruction and characteristic-map programme;
documentary comparison only.

\bibitem{Song2025}
X.~Song, S.~S.~Borjigin, F.~Salvati, Y.-X.~Wang,
N.~Yunger Halpern, D.~R.~M.~Arvidsson-Shukur, and K.~Murch,
\emph{Superconducting antiqubits achieve optimal phase estimation
via unitary inversion},
arXiv:2506.04315 (2025).
doi:\nolinkurl{10.48550/arXiv.2506.04315}.
Platform-specific inverse response and a separate
qubit--antiqubit metrology protocol.

\bibitem{Hoeffding1963}
W.~Hoeffding,
``Probability inequalities for sums of bounded random variables,''
\emph{Journal of the American Statistical Association}
\textbf{58}, no.~301 (1963), 13--30.
doi:\nolinkurl{10.1080/01621459.1963.10500830}.

\bibitem{BraunsteinCaves1994}
S.~L.~Braunstein and C.~M.~Caves,
``Statistical distance and the geometry of quantum states,''
\emph{Physical Review Letters} \textbf{72} (1994), 3439--3443.
doi:\nolinkurl{10.1103/PhysRevLett.72.3439}.

\bibitem{HorodeckiShorRuskai2003}
M.~Horodecki, P.~W.~Shor, and M.~B.~Ruskai,
``Entanglement breaking channels,''
\emph{Reviews in Mathematical Physics}
\textbf{15}, no.~6 (2003), 629--641.
doi:\nolinkurl{10.1142/S0129055X03001709}.
arXiv:quant-ph/0302031.

\bibitem{ReebWolf2014}
D.~Reeb and M.~M.~Wolf,
``An improved Landauer principle with finite-size corrections,''
\emph{New Journal of Physics} \textbf{16} (2014), 103011.
doi:\nolinkurl{10.1088/1367-2630/16/10/103011}.
arXiv:1306.4352.

\bibitem{Siegert2016}
T.~Siegert, R.~Diehl, G.~Khachatryan, M.~G.~H.~Krause,
F.~Guglielmetti, J.~Greiner, A.~W.~Strong, and X.~Zhang,
``Gamma-ray spectroscopy of positron annihilation in the
Milky Way,''
\emph{Astronomy \& Astrophysics} \textbf{586} (2016), A84.
doi:\nolinkurl{10.1051/0004-6361/201527510}.
arXiv:1512.00325.
Astrophysical motivation only; no fit, rate, or spatial model
is imported into the finite theorem.

\end{thebibliography}

\section*{Exact non-identification statement}

\begin{center}
\fbox{\parbox{0.91\linewidth}{
The output is a \textbf{finite-model estimate under declared
access}. The instrument does not recover a Galactic quantum
state, does not identify its observation boundary with a light
cone or a Galactic radius, and does not identify its records
with the interstellar medium, the observed 511\,keV morphology,
or the Galactic positron rate. No thermal machine, Landauer
cost, energy ledger, spacetime emergence, Einstein dynamics,
Page curve, or quantum evolution without an external time is
implied. No inherited gravitational quantum coefficient is
admitted, and no compilation is asserted. All stronger
identifications require additional constructions that are not
supplied here.
}}
\end{center}

\end{document}