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% Scope: constructed finite-model results and a proved conditional bridge.
% The unconditional physical positron acquisition-and-identification target
% remains unconstructed. A declared Hamiltonian is not evidence that an
% actual positron source implements it.

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\title{\textbf{En10-B}\\[0.4em]
Physical Acquisition and Identification for a Declared
Positron-Source Observable\\[0.4em]
\large A Derived Finite-Model Instrument, Calibrated Inference,
and a Conditional Bridge from En10-A}
\author{Ngo Cheung}
\date{}

\begin{document}
\maketitle

\begin{abstract}
We separate three obligations that cannot be discharged by
renaming a finite quantum circuit: deriving a channel from a
specified generator, acquiring an unknown source through a
complete instrument, and identifying the acquired parameter with
a physical source observable. On a two-vertex graph we construct
a finite synthetic source model with an opposite-sign kick,
finite random-kick nuisance, and a collision-induced relaxation
surrogate. Its channel and destructive Enter instrument are
derived explicitly, including arbitrary untouched references.
The likelihood depends on two kick moments. We prove calibrated
finite-shot estimation, complete-record model compatibility, and
discrimination certificates for predeclared competing families.
An explicit equal-record example shows that different source
weights can become indistinguishable when the nuisance is
allowed to vary.

A separate conditional theorem transfers these results to a
physical source observable when a source encoding, a
microscopically derived acquisition instrument, and certified
comparison allowances are supplied. Those physical matching data
are not present in the supplied materials. Consequently the
finite-model and conditional results are proved, but the
unconditional physical positron acquisition-and-identification
theorem remains unconstructed. No continuum transport law,
Galactic 511\,keV morphology or rate, thermodynamic reset cost,
or spacetime embedding is inferred.
\end{abstract}

\paragraph{Release boundary.}
This manuscript supplies proofs, not an asserted physical
identification. Its positive results are
Theorems~\ref{thm:derived}, \ref{thm:calibrated},
\ref{thm:competition}, and \ref{thm:main}.
The last is explicitly conditional on a physical acceptance
package. Section~\ref{sec:obligations} records the remaining
physical obligations O1--O6.

\paragraph{Predecessor and identifier correction.}
The supplied \texttt{En10-A.tex} concerns the finite
qubit--antiqubit instrument \cite{CheungEn10A}.
The relational clock manuscript in the supplied excerpts is
\texttt{En09-A.tex}, version~1 \cite{CheungEn09A}.
The predecessor's references to a relational
``source En10-A@1'' are therefore read as references to
\emph{En09-A@1}, not as a second manuscript with the same
En10-A identifier. No unprovided registered revision is inferred.

\section{The question, its philosophical background, and necessary corrections}
\label{sec:scope}

\subsection{Acquisition, derivation, and identification are different}

En10-A proves an estimation contract on a declared finite
interface. Its unknown source weight is a parameter of a
specified preparation family. That result does not by itself
construct a positron injection apparatus or identify its
parameter with a physical source.

The present question has three layers:
\[
\begin{gathered}
 \text{specified finite dynamics}
 \longrightarrow
 \text{derived finite record channel},\\
 \text{physical source and apparatus}
 \longrightarrow
 \text{validated acquisition channel},\\
 \text{acquired records}
 \longrightarrow
 \text{inference about a named physical observable}.
\end{gathered}
The first arrow can be proved from a finite Hamiltonian.
The second requires additional physical data and comparison
certificates. The third requires identifiability, calibration,
and a statistical contract.

The observational context is positron-source inference, not a
new Galactic analysis. Actual 511\,keV analyses use spatial and
spectral component models and instrumental-background modelling
\cite{Siegert2016}. These structures are absent from the graph
constructed below.
% Primary astrophysical source:
% ([arxiv.org](https://arxiv.org/abs/1512.00325))

\subsection{Enter and the interpretation of the theorem}

Cheung's operational formulation treats runtime quantum names
as access rights and destructive computational-basis Enter as a
consuming operation. The operational characterization of its
Born law is stated within a specified strategy class
\cite{CheungEnter,CheungI}. Here Born semantics are assumed;
that characterization is not re-proved.
% Primary article:
% ([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} separates
readout from quantum recovery, records from quantum-return
decoders, channels from thermal machines, and analogs from
physical identifications \cite{CheungPhilosophy}.
These distinctions provide the methodological background, not
additional theorem hypotheses.
% Author's philosophical essay:
% ([enter.grok.me](https://enter.grok.me/enter))

In this manuscript those distinctions acquire precise forms:
\begin{enumerate}[label=\textup{(\roman*)}]
\item
An instrument is a linear, complete collection of
subnormalized CP branches. Normalizing a selected successful
branch does not construct a deterministic acquisition channel.
\item
Consumption concerns the measured quantum names and their
output interface. It does not assert that every inaccessible
physical degree of freedom ceases to exist.
\item
A source parameter is acquired from unknown inputs, not from a
classical token already containing its numerical value.
\item
Calibration and competing explanations constrain what a record
identifies. More shots do not distinguish identical record laws.
\item
A physical correspondence must identify the source observable
and acquisition channel; philosophical vocabulary cannot supply
that correspondence.
\end{enumerate}

\subsection{Four debugging corrections}

\paragraph{A declared generator is not an established positron model.}
Exponentiating a specified finite Hamiltonian genuinely derives
its channel. It does not establish that an actual positron
source, magnetic environment, or detector implements that
Hamiltonian. We construct a finite synthetic pilot and state the
physical matching requirements separately.

\paragraph{A magnetic spin observable is not a position observable.}
The qubits below encode graph positions. A direction vector in
their Hamiltonian describes coefficients of an effective
two-level generator. Its identification with a physical
magnetic-field orientation is an additional obligation.
No positronium spin or singlet measurement is silently added.

\paragraph{Unknown preparation cannot be implemented from an unknown number.}
The apparatus receives a source system. It does not prepare
\(\rho_\theta\) by consulting an unavailable exact value of
\(\theta\). A parameterized source family describes the supplier,
not an expectation oracle.

\paragraph{Uncertain nuisance can destroy point identification.}
The finite likelihood is injective in the source weight when
the nuisance is fixed and known. It need not remain injective
when the nuisance varies. The correct general output is a
calibrated confidence set or a discrimination certificate under
a positive separation margin.

\section{The frozen finite source and generator model}
\label{sec:model}

\subsection{Graph and supplier input}

Freeze the graph
\[
 G_2=(\{b,d\},\{\{b,d\}\}),
\]
with basis \(0=b\), \(1=d\). The labels may be read as bulge and
disk in the analogy, but carry no Galactic radius or metric.

Let
\[
 \mathcal H_J=\mathbb C^2,\qquad
 \mathcal H_N=\mathcal H_A=\mathbb C^2,\qquad
 \mathcal H_B=\mathcal H_N\otimes\mathcal H_A.
\]
The supplier delivers a source qubit \(J\) in
\begin{equation}
 \tau_\theta=\theta\kb{0}{0}+(1-\theta)\kb{1}{1},
 \qquad 0\le\theta\le1.
 \label{eq:source-input}
\end{equation}
Its source observable is \(F_J=\kb{0}{0}\), so that
\(\Tr(F_J\tau_\theta)=\theta\).

Define the fixed preparation isometry
\begin{equation}
 S:\mathcal H_J\longrightarrow\mathcal H_B,
 \qquad S\ket{j}=\ket{jj},
 \qquad
 \mathcal P(X)=SXS^\dagger.
 \label{eq:preparation}
\end{equation}
Operationally, the supplied name becomes \(N\), a fresh qubit
\(A\) is allocated in \(0\), and
\(\CNOT_{N\to A}\) is applied. Consequently
\begin{equation}
 \mathcal P(\tau_\theta)
 =\rho_\theta
 =\theta\kb{00}{00}+(1-\theta)\kb{11}{11}.
 \label{eq:source-B}
\end{equation}
The map \(\mathcal P\) is independent of \(\theta\).
It does not clone arbitrary source states: a coherent input
generally produces an entangled pair.

\begin{assumption}[Supplier access]
\label{ass:supplier}
The source systems are delivered without an exported
source-component label and without a token containing the exact
unknown value of \(\theta\). Source purifications remain
supplier-side. Fresh trial inputs have the stated independent
preparation law when a fixed-shot theorem is invoked.
\end{assumption}

This implements the distinction between a supplied token and
an acquisition instrument in XIX-C \cite{CheungXIXC}.
If \(\theta\) is supplied numerically to a simulator, the
experiment is instead a simulation-validation task.

\subsection{Finite nuisance and calibration set}

Fix a positive integer \(m\). A nuisance value is
\[
 \eta=(\phi,\gamma,\widehat n,\xi_1,\ldots,\xi_m,w_1,\ldots,w_m),
\]
where
\[
 \widehat n=(n_x,n_y,n_z)\in S^2,\qquad
 w_k\ge0,\qquad \sum_{k=1}^m w_k=1.
\]
A fresh hidden label \(k\) is drawn with probabilities \(w_k\)
on each trial, independently of the source input.
It selects one kick, not an unrecorded adaptive stopping rule.

Freeze a nonempty compact calibration set \(\Kcal\) satisfying
\begin{equation}
 \begin{gathered}
 0\le\gamma_-\le\gamma\le\gamma_+<1,\qquad
 |\phi-\phi_0|\le\epsilon_\phi,\qquad
 |\xi_k|\le\kappa,\qquad |n_z|\le z_*<1,\\
 0<\phi_0-\epsilon_\phi-\kappa,\qquad
 \phi_0+\epsilon_\phi+\kappa<\frac{\pi}{4}.
 \end{gathered}
 \label{eq:calibration-set}
\end{equation}
The nominal loss is \(\gamma_0\in[\gamma_-,\gamma_+]\), and
\[
 \epsilon_\gamma
 =\sup_{\eta\in\Kcal}|\gamma-\gamma_0|.
\]
The set may impose additional correlations among its variables.
A rectangular calibration set is not required.

The finite list \((\xi_k,w_k)\) is a random-kick nuisance model.
It is not an exact continuum turbulence spectrum.

\subsection{Named generators}

Let \(X,Y,Z\) be the Pauli matrices in the graph basis and set
\[
 K_{\widehat n}=n_xX+n_yY+n_zZ.
\]
The \(X,Y\) terms are edge couplings; the \(Z\) term is an
on-site contrast. The vector \(\widehat n\) is an effective
generator axis until a physical correspondence is supplied.

Freeze positive externally scheduled durations
\(T_{\rm k},T_{\rm c},T_{\rm e}\), and \(\hbar>0\).
For label \(k\), prescribe
\begin{equation}
 H_{{\rm k},k}
 =
 \frac{\hbar(\phi+\xi_k)}{T_{\rm k}}
 \bigl(K_{\widehat n}\otimes I-I\otimes K_{\widehat n}\bigr).
 \label{eq:kick-H}
\end{equation}
Its normal and antiqubit factors have opposite signs.

The interaction stage has generator
\begin{equation}
 H_{\rm c}=\frac{\pi\hbar}{T_{\rm c}}\kb{11}{11}.
 \label{eq:CZ-H}
\end{equation}

For relaxation, allocate \(E\) in \(\ket0\).
On \(A\otimes E\), define the self-adjoint exchange generator
\[
 K_{\rm e}
 =
 i\bigl(\kb{01}{10}-\kb{10}{01}\bigr),
 \qquad
 \chi_\gamma=\arcsin\sqrt{\gamma},
\]
and prescribe
\begin{equation}
 H_{\rm e}=\frac{\hbar\chi_\gamma}{T_{\rm e}}K_{\rm e}.
 \label{eq:exchange-H}
\end{equation}
These are finite, piecewise scheduled generator data.
They are not derived from an unprovided Galactic action.

\begin{remark}[Meaning of antiqubit]
The inverse response in \eqref{eq:kick-H} is synthetic and
platform-specific in interpretation. Song and collaborators
provide a separate superconducting antiqubit construction
\cite{Song2025}. Their apparatus and sensing advantage are not
implemented or imported here.
% Primary antiqubit paper:
% ([arxiv.org](https://arxiv.org/abs/2506.04315?utm_source=openai))

For the single constant kick, the two local propagators are
inverse unitaries. This does not invert the subsequent
interaction, relaxation, or complete acquisition instrument.
It also does not assert inverse time ordering for an arbitrary
sequence of noncommuting, time-dependent fields.
\end{remark}

\section{Derived channel and complete acquisition instrument}
\label{sec:derived}

\subsection{Propagators and channel construction}

Define
\[
 V_k=e^{-i(\phi+\xi_k)K_{\widehat n}},
 \qquad U_k=\CZ(V_k\otimes V_k^\dagger).
\]
For \(h=1-\gamma\), put
\[
 L_0=\kb{0}{0}+\sqrt h\,\kb{1}{1},
 \qquad L_1=\sqrt\gamma\,\kb{0}{1}.
\]
The channel to be derived is
\begin{equation}
 \Phi_\eta(X)
 =
 \sum_{k=1}^m\sum_{\ell=0}^1
 H_{k\ell}XH_{k\ell}^\dagger,
 \qquad
 H_{k\ell}=\sqrt{w_k}(I\otimes L_\ell)U_k.
 \label{eq:derived-channel}
\end{equation}

Let \(\Rcal=\{00,01,10,11\}\).
To permit comparison with complete attempted-run records,
use the common alphabet
\[
 \Rcal_+=\Rcal\cup\{\perp\},
 \qquad \mathcal H_C=\mathbb C^5.
\]
The ideal failure branch is zero. Real failure or rejection
branches, if present, must remain in \(\Rcal_+\) or in a
predeclared larger alphabet.

For any finite retained factor \(T\), define
\begin{align}
 D_r&=\bra r_B\otimes I_T,\\
 \Mcal_B^T(X)
 &=
 \sum_{r\in\Rcal}
 \kb r r_C\otimes D_rXD_r^\dagger,
 \label{eq:Enter}\\
 \mathcal J_B^T(X)
 &=
 \sum_{r\in\Rcal}
 \kb r r_C\otimes
 \Tr_B[(\kb r r_B\otimes I_T)X(\kb r r_B\otimes I_T)].
 \label{eq:projective-target}
\end{align}
The complete acquisition map from the supplier interface is
\begin{equation}
 \Acal_\eta^T
 =
 \Mcal_B^T\circ(\Phi_\eta\otimes\id_T)
 \circ(\mathcal P\otimes\id_T).
 \label{eq:acquisition}
\end{equation}

\begin{theorem}[Derived finite-model acquisition]
\label{thm:derived}
For the frozen generators and source interface:
\begin{enumerate}[label=\textup{(\roman*)}]
\item
The scheduled propagators of
\eqref{eq:kick-H}--\eqref{eq:exchange-H} yield
\eqref{eq:derived-channel}. The maps
\(\mathcal P,\Phi_\eta,\Acal_\eta\) are CPTP.
\item
For every finite retained factor \(T\),
\(\Mcal_B^T=\mathcal J_B^T\) on all operators, and
\begin{equation}
 \Tr_C\Acal_\eta^T(X)=\Tr_JX.
 \label{eq:reference-marginal}
\end{equation}
This includes arbitrary untouched references.
\item
Under destructive name semantics, \(N,A,E\) are consumed.
The newly exported output is the classical record; no measured
data name is a quantum output.
\end{enumerate}
\end{theorem}

\begin{proof}
The two summands of \(H_{{\rm k},k}\) commute, so its propagator
is \(V_k\otimes V_k^\dagger\).
Since \(e^{-i\pi\kb{11}{11}}=\CZ\), the next propagator is
\(\CZ\).

On the exchange subspace, \(K_{\rm e}^2=I\), whereas it
annihilates \(00,11\). Thus its propagator satisfies
\[
 \ket{0}_A\ket{0}_E\longmapsto\ket{0}_A\ket{0}_E,
\]
\[
 \ket{1}_A\ket{0}_E\longmapsto
 \sqrt h\,\ket{1}_A\ket{0}_E
 +\sqrt\gamma\,\ket{0}_A\ket{1}_E.
\]
Taking the environment matrix elements gives \(L_0,L_1\).
Drawing the independent label \(k\), and summing all its
unexported branches, yields \eqref{eq:derived-channel}.
Moreover,
\[
 \sum_\ell L_\ell^\dagger L_\ell=I,\qquad
 \sum_{k,\ell}H_{k\ell}^\dagger H_{k\ell}
 =\sum_kw_kI=I.
\]
The source map is isometric, so these maps are CPTP.

For readout, the Kraus operators are
\(\ket r_C\otimes D_r\), and their squared moduli sum to
\(I_B\otimes I_T\).
If
\(X=\sum_{r,s}\kb r s_B\otimes X_{rs}\), both readout
specifications have \(r\)-block \(X_{rr}\).
Therefore the maps agree on all operators, and
\[
 \Tr_C\Mcal_B^T(X)=\sum_rX_{rr}=\Tr_BX.
\]
Local trace-preserving maps on \(J\), followed by local
preparation and evolution, preserve the retained marginal.
This proves \eqref{eq:reference-marginal}.

Finally, the name rule removes a measured name from the live
quantum context. Enter of \(E\) is unexported; Enter of \(N,A\)
exports two bits. A subsequent unitary preserves live names,
classical processing introduces none, and allocation introduces
only fresh names. Induction over a continuation proves that no
consumed name is reused.
\end{proof}

\begin{remark}[Residuals and the observation boundary]
The accessible algebra is
\(\operatorname{Diag}(C)\otimes\Bop(\mathcal H_S)\)
if a spectator \(S\) is supplied. Source purifications, random-kick
labels, and detector complements are not observer assistance.
A conditional spectator or reference state can change when a
record is selected, although \eqref{eq:reference-marginal}
preserves its unconditional marginal.

Consumption does not imply destruction of all microscopic
information, nor does it forbid an unmeasured spectator from
undergoing later dynamics. This is the residual distinction in
Landmark I and En09-A@1 \cite{CheungI,CheungEn09A}.
\end{remark}

\subsection{Why this does not yet derive a physical positron channel}

\begin{proposition}[Opposite charge does not generally invert graph evolution]
\label{prop:charge}
There are finite charge-phase graph generators for which changing
the charge sign does not give the inverse propagator.
Consequently the antiqubit inverse response cannot be justified
solely by assigning opposite-charge labels.
\end{proposition}

\begin{proof}
Consider the permissible two-site family
\[
 H_q
 =
 J_*\bigl(e^{iq\alpha}\kb{0}{1}
          +e^{-iq\alpha}\kb{1}{0}\bigr),
 \qquad J_*>0.
\]
It is self-adjoint. Its sum with \(H_{-q}\) is
\(2J_*\cos(q\alpha)X\), which need not vanish.
At \(\alpha=0\), both charge signs give \(J_*X\).
For \(0<J_*T/\hbar<\pi/2\),
\[
 e^{-iTH_{-q}/\hbar}
 =e^{-iTJ_*X/\hbar}
 \ne e^{+iTJ_*X/\hbar}.
\]
Hence charge-sign reversal is not, in this family, full
Hamiltonian inversion.
\end{proof}

This is a counterexample to an automatic implication, not a
claim that every physical opposite-charge response is identical.
A physical derivation must identify which generator is inverted,
which terms remain unchanged, and the validity of its finite
encoding.

\section{Likelihood, source sensitivity, and nuisance information}
\label{sec:likelihood}

\subsection{The derived four-outcome law}

Define
\begin{equation}
 t_k=(1-n_z^2)\sin^2(\phi+\xi_k),\qquad
 u=\sum_kw_kt_k,\qquad v=\sum_kw_kt_k^2,\qquad
 d=1-2u.
 \label{eq:moments}
\end{equation}
The calibration set implies
\begin{equation}
 \begin{split}
 t_-&=(1-z_*^2)\sin^2(\phi_0-\epsilon_\phi-\kappa)>0,\\
 t_+&=\sin^2(\phi_0+\epsilon_\phi+\kappa)<\frac12,\\
 t_-&\le u\le t_+,\qquad d>0,\qquad h=1-\gamma>0.
 \end{split}
 \label{eq:moment-bounds}
\end{equation}

\begin{lemma}[Exact record law]
\label{lem:law}
On input \(\tau_\theta\), the derived acquisition probabilities are
\begin{align}
 p_{00}(\theta,\eta)&=\gamma u+hv+\theta d,\nonumber\\
 p_{01}(\theta,\eta)&=h(u-v),\nonumber\\
 p_{10}(\theta,\eta)
 &=\gamma+(1-2\gamma)u-hv-\gamma\theta d,\nonumber\\
 p_{11}(\theta,\eta)
 &=h(1-2u+v)-h\theta d,\qquad p_\perp=0.
 \label{eq:likelihood}
\end{align}
All four nonfailure probabilities are strictly positive.
For the complete-record score
\begin{equation}
 f(r)=
 \begin{cases}
 1,&r=00\text{ or }10,\\
 0,&r=01,11,\perp,
 \end{cases}
 \label{eq:score}
\end{equation}
its ideal mean is
\begin{equation}
 q_\eta(\theta)=a_\eta+b_\eta\theta,\qquad
 a_\eta=\gamma+hu,\qquad b_\eta=h(1-2u)>0.
 \label{eq:affine}
\end{equation}
\end{lemma}

\begin{proof}
Since \(K_{\widehat n}^2=I\),
\[
 V_k=\cos(\phi+\xi_k)I
 -i\sin(\phi+\xi_k)K_{\widehat n}.
\]
Both \(V_k\) and \(V_k^\dagger\) have squared matrix elements
\[
 \begin{pmatrix}1-t_k&t_k\\t_k&1-t_k\end{pmatrix}.
\]
The diagonal interaction changes no occupation population.

Conditional on source component \(00\) and label \(k\), the
four probabilities after relaxation are
\[
 ((1-t_k)(1-ht_k),\ h(t_k-t_k^2),\
 t_k-ht_k^2,\ ht_k^2).
\]
Conditional on source component \(11\), they are
\[
 (\gamma t_k+ht_k^2,\ h(t_k-t_k^2),\
 \gamma+(1-2\gamma)t_k-ht_k^2,\
 h(1-t_k)^2).
\]
The population update follows directly from the two Kraus
operators \(L_\ell\); coherences do not contribute to these
computational probabilities.
Mixing with source weights \(\theta,1-\theta\) and then weights
\(w_k\) proves \eqref{eq:likelihood}.
Every conditional entry is positive because \(0<t_k<1/2\) and
\(h>0\). The averaged entries are therefore positive.
Summing \(p_{00}+p_{10}\) gives \eqref{eq:affine}.
\end{proof}

For fixed known \(\eta\), the probability \(q_\eta\) determines
the model parameter:
\[
 \theta=\frac{q_\eta-a_\eta}{b_\eta}.
\]
This is population-level identification, not exact acquisition
of a probability from one record.

The deterministic specialization
\[
 m=1,\quad \xi_1=0,\quad \widehat n=(0,1,0)
\]
has \(u=\sin^2\phi,\ v=u^2\) and reproduces the finite likelihood
of En10-A. The construction above derives that specialization
from the named pilot generators.

\subsection{Fisher-information ledger}

\begin{proposition}[Information at a fixed known nuisance value]
\label{prop:FI}
For \(0<\theta<1\), fixed \(\eta\), and independent repetitions,
the one-shot score information and complete-record information are
\begin{align}
 F_f(\theta,\eta)
 &=\frac{b_\eta^2}{q_\eta(1-q_\eta)},\label{eq:FI-score}\\
 F_C(\theta,\eta)
 &=d^2\left(
 \frac1{p_{00}}+\frac{\gamma^2}{p_{10}}
 +\frac{h^2}{p_{11}}\right).
 \label{eq:FI-record}
\end{align}
They satisfy
\begin{equation}
 F_f(\theta,\eta)\le F_C(\theta,\eta)
 \le \frac1{\theta(1-\theta)}.
 \label{eq:FI-bound}
\end{equation}
Independent repetition multiplies each information by \(n\).
The upper bound applies to every measurement of the derived
post-preparation system at fixed nuisance.
\end{proposition}

\begin{proof}
The score has a Bernoulli law with derivative \(b_\eta\).
Differentiating \eqref{eq:likelihood} gives
\[
 (\partial_\theta p_{00},\partial_\theta p_{01},
 \partial_\theta p_{10},\partial_\theta p_{11})
 =(d,0,-\gamma d,-hd),
\]
which proves both formulas.

The hidden source component has score \(1/\theta\) on \(0\)
and \(-1/(1-\theta)\) on \(1\), with squared-score expectation
\(1/[\theta(1-\theta)]\).
Any measurement after a \(\theta\)-independent channel has
an outcome score equal to the conditional expectation of this
source score given the outcome. Conditional Jensen bounds its
squared expectation by the source information.
The statistic \(f\) is a deterministic coarsening of the full
record; a second conditional Jensen inequality gives
\(F_f\le F_C\).
Independent scores have mean zero, so their information adds.
\end{proof}

\begin{remark}[Known-nuisance information is not a nuisance-robust bound]
The quantities in Proposition~\ref{prop:FI} hold the nuisance
fixed. They must not be interpreted as information available
after unrestricted nuisance fitting.
The equal-record construction below shows why a positive
fixed-nuisance source sensitivity is insufficient for that claim.
\end{remark}

\section{Calibration and complete-record finite-shot inference}
\label{sec:statistics}

\subsection{A complete statistical contract}

For states and channels use
\[
 \dtr(\rho,\sigma)=\frac12\norm{\rho-\sigma}_1,\qquad
 \ddia(\Lambda,\Gamma)=\frac12\norm{\Lambda-\Gamma}_\diamond.
\]
For classical laws, \(\TV(P,Q)=\frac12\sum_r|P(r)-Q(r)|\).
An expectation of a function in \([0,1]\) changes by at most
total variation.

\begin{assumption}[Conditional fixed-shot acquisition]
\label{ass:statistics}
A calibration record \(\mathcal Z\) is collected before validation.
Conditional on \(\mathcal Z\):
\begin{enumerate}[label=\textup{(\roman*)}]
\item
\(n\ge1\), the calibration set \(\Kcal\), all scores, and all
comparison allowances are fixed before the \(n\) trial records
are observed.
\item
The trial records are independent. Their actual laws
\(\widetilde p_i\) may differ, but on a calibration-measurable
event \(G\) they obey
\[
 \eta\in\Kcal,\qquad
 \TV(\widetilde p_i,p_{\theta,\eta})\le e
 \quad(i=1,\ldots,n).
\]
Here \(\theta,\eta\) describe one fixed target family member.
\item
The event has \(\Pr(G^c)\le\beta\).
Every attempted trial produces a complete record, including
failure or rejection. There is no unaccounted postselection,
validation-driven tuning, or adaptive stopping.
\end{enumerate}
\end{assumption}

A sufficient way to supply the law allowance is a
half-diamond comparison on the same complete record alphabet.
That comparison does not establish independence; independence
is a separate assumption.

The conditional form follows the statistical discipline of
Oai265-B and En01-C \cite{CheungOai265B,CheungEn01C}.
Their device models and data are not transferred here.

\subsection{An explicit calibration allowance}

Set
\[
 u_0=\sin^2\phi_0,\qquad h_0=1-\gamma_0,\qquad
 a_0=\gamma_0+h_0u_0,\qquad b_0=h_0(1-2u_0)>0.
\]
Define the exact mean-level calibration allowance
\begin{equation}
 \Delta_{\rm cal}
 =
 \sup_{\eta\in\Kcal}\max\left\{
 |\gamma+hu-a_0|,\ |hu-h_0u_0|
 \right\}.
 \label{eq:calibration-allowance}
\end{equation}
This supremum exists because the set is compact and its
likelihood is continuous.

\begin{lemma}[Calibration conversion]
\label{lem:calibration}
Uniformly for \(\eta\in\Kcal,\ \theta\in[0,1]\),
\begin{equation}
 |q_\eta(\theta)-(a_0+b_0\theta)|
 \le\Delta_{\rm cal}
 \le \epsilon_\gamma+\epsilon_\phi+\kappa+z_*^2.
 \label{eq:calibration-conversion}
\end{equation}
Moreover,
\[
 b_\eta\ge b_*:=(1-\gamma_+)(1-2t_+)>0.
\]
\end{lemma}

\begin{proof}
The difference of the two affine mean functions is affine in
\(\theta\). Its largest absolute value on \([0,1]\) occurs
at an endpoint. At \(\theta=0\) and \(1\) those differences
are precisely the two expressions in
\eqref{eq:calibration-allowance}.

The function \(\sin^2x\) is globally \(1\)-Lipschitz. Hence
\[
 |t_k-u_0|
 \le|\phi-\phi_0+\xi_k|+n_z^2
 \le\epsilon_\phi+\kappa+z_*^2.
\]
Averaging gives the same bound on \(|u-u_0|\).
Writing
\[
 q_\eta(\theta)
 =\gamma+(1-\gamma)[u+(1-2u)\theta],
\]
its derivative with respect to \(u\) has absolute value at
most one, and its derivative with respect to \(\gamma\)
lies in \([0,1]\). Comparing first \(u\), then \(\gamma\),
proves the displayed upper bound.
The slope bound follows from \eqref{eq:moment-bounds}.
\end{proof}

\subsection{The calibrated estimation theorem}

Let
\[
 \overline f=\frac1n\sum_{i=1}^nf(r_i),\qquad
 \widehat\theta_0
 =\clip\left(\frac{\overline f-a_0}{b_0}\right),
 \qquad
 \clip(x)=\max\{0,\min\{1,x\}\}.
\]
This is a separate classical post-processing of the record list.
It does not prepare a returned quantum source.

\begin{theorem}[Calibrated finite-model source-weight inference]
\label{thm:calibrated}
Under Assumption~\ref{ass:statistics}, for
\(0<\alpha<1\) with \(\alpha+\beta<1\), define
\begin{equation}
 s_n(\alpha)=\sqrt{\frac{\log(2/\alpha)}{2n}},
 \qquad
 R_n=\min\left\{1,
 \frac{\Delta_{\rm cal}+e+s_n(\alpha)}{b_0}\right\}.
 \label{eq:radius}
\end{equation}
Then, uniformly over the promised source and calibration family,
\begin{equation}
 \Pr\{|\widehat\theta_0-\theta|\le R_n\}
 \ge1-\alpha-\beta.
 \label{eq:source-coverage}
\end{equation}
Thus
\[
 [\,\max\{0,\widehat\theta_0-R_n\},
    \min\{1,\widehat\theta_0+R_n\}\,]
\]
is a confidence interval with that coverage.

For \(0<\varepsilon<1\), a sufficient finite shot budget is
\begin{equation}
 b_0\varepsilon>\Delta_{\rm cal}+e,\qquad
 n\ge
 \left\lceil
 \frac{\log(2/\alpha)}
 {2(b_0\varepsilon-\Delta_{\rm cal}-e)^2}
 \right\rceil.
 \label{eq:shot-budget}
\end{equation}
This is a certificate for the named model parameter, not a
Galactic-source identification.
\end{theorem}

\begin{proof}
Conditional on the calibration record, the bounded independent
scores obey
\[
 \Pr\left\{
 \left|\overline f-\frac1n\sum_i\mathbb E f(r_i)\right|
 >s_n(\alpha)\ \middle|\ \mathcal Z
 \right\}\le\alpha.
\]
For completeness, a bounded variable \(W\in[0,1]\) has
\[
 \log\mathbb E e^{\lambda(W-\mathbb EW)}
 \le\lambda^2/8.
\]
Indeed the second derivative of its log moment-generating
function is the variance of a tilted law on \([0,1]\), at
most \(1/4\). Integration from zero proves the bound.
Independence, exponential Markov inequalities, and optimization
in \(\lambda\) give the stated two-sided concentration bound.
This is the bounded-variable form of Hoeffding's inequality
\cite{Hoeffding1963}.
% Original concentration paper:
% ([cs.rpi.edu](https://www.cs.rpi.edu/academics/courses/spring06/random/hoefding.pdf))

On \(G\), the law allowance gives
\[
 \left|
 \frac1n\sum_i\mathbb E f(r_i)-q_\eta(\theta)
 \right|\le e.
\]
Lemma~\ref{lem:calibration} then gives, on the concentration event,
\[
 |\overline f-(a_0+b_0\theta)|
 \le s_n(\alpha)+e+\Delta_{\rm cal}.
\]
Projection onto \([0,1]\) cannot increase distance from
\(\theta\in[0,1]\), so division by \(b_0\) proves the radius.
The bound \(|\widehat\theta_0-\theta|\le1\) permits its cap at one.
Integrating the conditional statistical failure probability
and adding \(\Pr(G^c)\le\beta\) proves
\eqref{eq:source-coverage}. Rearranging the radius requirement
proves \eqref{eq:shot-budget}.
\end{proof}

\begin{remark}[Scope of the calibration floor]
The displayed nominal certificate need not shrink below
\((\Delta_{\rm cal}+e)/b_0\) as \(n\) increases. This is a
limitation of this certificate, not a universal minimax lower
bound. Complete-record fitting or a new sensing protocol may
improve it. Equal-law alternatives, however, cannot be removed
by additional repetitions of the same acquisition.
\end{remark}

\section{Competing explanations and complete-record discrimination}
\label{sec:competition}

\subsection{Predeclared model families}

Let \(\mathscr H=\{0,\ldots,L\}\) be a finite predeclared list
of competing families. Family \(j\) has a compact parameter
domain \(\Theta_j\) and a continuous complete record law
\[
 p^j_\lambda\in\Delta(\Rcal_+),\qquad\lambda\in\Theta_j.
\]
Family \(0\) may be the derived family
\eqref{eq:likelihood}; other families may have different
source templates or finite generators. Their likelihoods and
allowances \(e_j\) must be supplied, not fitted without restriction
after the validation records are inspected.

For example, replacing the two source templates by specified
density matrices \(\sigma_{j,0},\sigma_{j,1}\) gives the
finite-model likelihood
\[
 p^j_r(\theta,\eta)
 =
 \Tr\left[
 P_r\Phi^j_\eta
 \bigl(\theta\sigma_{j,0}+(1-\theta)\sigma_{j,1}\bigr)
 \right],
\]
provided the corresponding channel \(\Phi^j_\eta\) has been
constructed. Merely writing this expression does not derive
the alternative channel.

Let
\[
 \widehat p_r=\frac1n\sum_i\one_{\{r_i=r\}},
 \qquad D=|\Rcal_+|=5,\qquad
 c_n(\alpha)=\sqrt{\frac{\log(2D/\alpha)}{2n}}.
\]
Define the complete-record compatibility set
\begin{equation}
 \Omega_n
 =
 \left\{(j,\lambda):
 \norm{\widehat p-p^j_\lambda}_\infty
 \le c_n(\alpha)+e_j
 \right\}.
 \label{eq:compatibility}
\end{equation}
All counts, including \(\perp\), enter this set.

\begin{theorem}[Complete-record compatibility and discrimination]
\label{thm:competition}
Suppose the conditional fixed-shot contract holds for a true
member \((j_*,\lambda_*)\), with
\[
 \TV(\widetilde p_i,p^{j_*}_{\lambda_*})\le e_{j_*}
\]
on \(G\). Then
\begin{equation}
 \Pr\{(j_*,\lambda_*)\in\Omega_n\}\ge1-\alpha-\beta.
 \label{eq:compatibility-coverage}
\end{equation}

For two promised families, define the complete-record separation
\begin{equation}
 g=\inf_{\lambda_0\in\Theta_0,\lambda_1\in\Theta_1}
 \norm{p^0_{\lambda_0}-p^1_{\lambda_1}}_\infty.
 \label{eq:gap}
\end{equation}
If
\begin{equation}
 g>e_0+e_1,\qquad
 2c_n(\alpha)<g-e_0-e_1,
 \label{eq:separation-condition}
\end{equation}
the rule that reports the uniquely compatible family, and
otherwise reports ``inconclusive'', reports the true family
with probability at least \(1-\alpha-\beta\), uniformly on
either promise class.

One conservative sufficient sample size is
\begin{equation}
 n\ge
 \left\lceil
 \frac{8\log(2D/\alpha)}{(g-e_0-e_1)^2}
 \right\rceil.
 \label{eq:discrimination-shots}
\end{equation}
\end{theorem}

\begin{proof}
For each coordinate indicator, conditional Hoeffding gives
failure probability at most
\(2e^{-2nc_n^2}=\alpha/D\).
A union bound yields simultaneous deviation at most \(c_n\)
from the average actual law. Total variation at most \(e_{j_*}\)
bounds every coordinate bias by \(e_{j_*}\).
This proves \eqref{eq:compatibility-coverage} after calibration
failure is added.

On that simultaneous event, the true parameter is compatible.
If a parameter of the other family were also compatible, the
triangle inequality would give
\[
 \norm{p^0_{\lambda_0}-p^1_{\lambda_1}}_\infty
 \le2c_n+e_0+e_1,
\]
contradicting \eqref{eq:separation-condition}.
Thus exactly the true family is compatible.
The conservative sample bound gives
\(c_n\le(g-e_0-e_1)/4\), and therefore a strict separation.
\end{proof}

For family \(0\), projection of \(\Omega_n\) onto its
\(\theta\)-coordinate gives a nuisance-robust confidence set.
It need not be an interval or a singleton. If another physical
template interpretation remains compatible, the output must
retain that ambiguity.

The guarantee is relative to the predeclared candidate families.
An unlisted explanation has not been excluded. Failure of a
scalar threshold margin is not automatically equality of
complete laws; conversely, an absolute mean difference with
changing orientation does not define one common threshold.
These cautions parallel En01-B and Oai265-A
\cite{CheungEn01B,CheungOai265A}.

\subsection{An explicit complete-law ambiguity}

Put
\begin{equation}
 r=u+\theta(1-2u),\qquad c=u-v.
 \label{eq:reduced-parameters}
\end{equation}
Equation~\eqref{eq:likelihood} becomes
\begin{equation}
 \begin{array}{ll}
 p_{00}=r-hc,&p_{01}=hc,\\
 p_{10}=\gamma(1-r)+hc,&p_{11}=h(1-r-c).
 \end{array}
 \label{eq:reduced-law}
\end{equation}
Thus at fixed \(\gamma\) the complete record depends only on
\(r,c\), not separately on \(\theta,u,v\).

\begin{proposition}[Different source weights with identical complete records]
\label{prop:overlap}
There is an admissible finite-kick calibration set containing
two members with different source weights and identical
complete laws. For that pair, every randomized binary test
based on \(n\) ideal independent records has errors satisfying
\(\alpha_0+\alpha_1=1\).
\end{proposition}

\begin{proof}
Take \(\widehat n=(0,1,0)\), \(\gamma=1/4\), and three possible
transition values
\[
 t_1=\frac1{20},\qquad t_2=\frac15,\qquad t_3=\frac{31}{80}.
\]
They are realized by
\(\phi=\arcsin\sqrt{1/5}\) and
\(\xi_k=\arcsin\sqrt{t_k}-\phi\).
For an explicit admissible angle bound, choose
\[
 \kappa=\phi-\arcsin\sqrt{1/20}.
\]
The other offset has smaller absolute value. For instance,
the relevant elementary angle inequalities give
\[
 0.46<\phi<0.47,\quad
 0.22<\arcsin\sqrt{1/20}<0.23,\quad
 0.67<\arcsin\sqrt{31/80}<0.68.
\]
Hence \(\phi-\kappa>0\) and
\(\phi+\kappa<0.72<\pi/4\).
A compact set containing these fixed angles and the weight
simplex satisfies \eqref{eq:calibration-set}.

For the first member choose
\[
 \theta^{(0)}=\frac25,\qquad w^{(0)}=(0,1,0).
\]
Then \(u^{(0)}=1/5,\ v^{(0)}=1/25\).
For the second choose
\[
 \theta^{(1)}=\frac{19}{50},\qquad
 w^{(1)}=\left(\frac{11}{27},0,\frac{16}{27}\right).
\]
Direct rational arithmetic gives
\(u^{(1)}=1/4,\ v^{(1)}=9/100\).
Both members have
\[
 r=\frac{11}{25},\qquad c=\frac4{25}.
\]
Their common complete law is
\[
 (p_{00},p_{01},p_{10},p_{11},p_\perp)
 =
 \left(\frac8{25},\frac3{25},\frac{13}{50},\frac3{10},0\right).
\]

Let a randomized test declare the second member with probability
\(\psi(\mathbf r)\in[0,1]\). The record-list laws are identical,
so
\[
 \alpha_0+\alpha_1
 =
 \mathbb E_0\psi+\mathbb E_1(1-\psi)=1.
\]
\end{proof}

\begin{corollary}[An estimation obstruction for this pair]
\label{cor:overlap-estimation}
For the pair in Proposition~\ref{prop:overlap}, no record-only
estimator has both probabilities
\[
 \Pr_j\{|\widehat\theta-\theta^{(j)}|\le\varepsilon\}
\]
greater than \(1/2\) when
\(\varepsilon<|\theta^{(0)}-\theta^{(1)}|/2=1/100\).
\end{corollary}

\begin{proof}
The two success intervals are disjoint. Under the common data
law, and any common randomized estimator, their probabilities
sum to at most one.
\end{proof}

This is an obstruction for an explicit allowed pair, not a
universal obstruction to narrower calibrated families.
It prevents the claim that nuisance-robust point identification
follows automatically from the positive slope \(b_\eta\).

\section{The physical source-observable bridge}
\label{sec:bridge}

\subsection{The acceptance package that must actually be supplied}

The intended physical source observable is a relative
injection-template weight on the two-zone graph. Its physical
realization requires the following independent data.

\begin{definition}[Physical acquisition and identification package]
\label{def:physical-package}
A package for this target consists of:
\begin{enumerate}[label=\textup{(P\arabic*)}]
\item
A named physical source model on a specified finite input space
\(\mathcal H_{\rm P}\), with fixed injection templates
\(\sigma_b,\sigma_d\) and an effect \(F_b^{\rm P}\) satisfying
\[
 \Tr(F_b^{\rm P}\sigma_b)=1,\qquad
 \Tr(F_b^{\rm P}\sigma_d)=0.
\]
Its promised source family is
\[
 \sigma_\theta=\theta\sigma_b+(1-\theta)\sigma_d.
\]
Therefore the named source observable obeys
\(\Tr(F_b^{\rm P}\sigma_\theta)=\theta\).
This excludes source-dependent selection from being silently
identified with injection weight.
\item
A \(\theta\)-independent CPTP source comparison map
\(\mathcal E:\Bop(\mathcal H_{\rm P})\to\Bop(\mathcal H_J)\)
and a proved allowance
\[
 \dtr(\mathcal E(\sigma_\theta),\tau_\theta)
 \le e_{\rm src}
 \quad\text{uniformly in }\theta.
\]
It is sufficient to prove this allowance on the two source
templates, with the corresponding targets \(0,1\).
\item
A named microscopic physical evolution, its auxiliary
preparations, transport stages, nuisance domain, and complete
detector instrument, yielding a CPTP record channel
\[
 \mathcal B_\eta:
 \Bop(\mathcal H_{\rm P})\longrightarrow\Bop(\mathcal H_C).
\]
All attempted outcomes and all physical loss, failure, or
rejection branches are retained. Its measured-name access rule
is specified separately from its matrix representation.
\item
A derived or independently certified complete comparison
\begin{equation}
 \ddia(\mathcal B_\eta,\Acal_\eta\mathcal E)
 \le e_{\rm acq},
 \label{eq:physical-comparison}
\end{equation}
uniformly over the admitted nuisance and input interface.
The map \(\mathcal E\) in this comparison need not be a
physically executed preprocessing, but its domain and meaning
must be explicit.
\item
A calibration event and independent attempted-run contract of
the form in Assumption~\ref{ass:statistics}.
The correspondence between physical field/turbulence parameters
and \(\eta\) is part of the package. It is not inferred from the
notation \(\widehat n\).
\item
If competing physical source interpretations are to be excluded,
their complete promised record families and allowances are
supplied, together with the scope of the candidate list.
\end{enumerate}
\end{definition}

The stated source templates, physical generator, actual
detector, and comparison certificate are not contained in the
supplied En10-A model. Definition~\ref{def:physical-package}
fixes their required interface; it does not assert their
existence for a positron apparatus.

This separation between assumed transfer interfaces and
microscopically derived ones is also explicit in Hawk02-A;
the named-effect comparison in Em03-D concerns a different
physical model \cite{CheungHawk02A,CheungEm03D}.
Neither source supplies the missing positron interface.

\subsection{A proved conditional physical identification theorem}

\begin{theorem}[Conditional acquisition of the named physical observable]
\label{thm:main}
Suppose a physical package satisfying
Definition~\ref{def:physical-package} is supplied.
Then the records of its complete acquisition instrument
estimate
\[
 \theta=\Tr(F_b^{\rm P}\sigma_\theta)
\]
with Theorem~\ref{thm:calibrated}, using
\begin{equation}
 e=\min\{1,e_{\rm src}+e_{\rm acq}\}.
 \label{eq:physical-error}
\end{equation}
The resulting radius is
\begin{equation}
 R_n^{\rm P}
 =
 \min\left\{1,
 \frac{\Delta_{\rm cal}+e+s_n(\alpha)}{b_0}\right\},
 \qquad
 \Pr\{|\widehat\theta_0-\Tr(F_b^{\rm P}\sigma_\theta)|
       \le R_n^{\rm P}\}
 \ge1-\alpha-\beta.
 \label{eq:physical-radius}
\end{equation}
If the additional competing-family package has a positive
complete-record margin, Theorem~\ref{thm:competition} supplies
the corresponding physical discrimination certificate.

The theorem identifies one promised source observable.
It does not reconstruct the source quantum state, establish a
Galactic source law, or validate its own physical-package
hypotheses.
\end{theorem}

\begin{proof}
For each source state,
\[
 \dtr\bigl(
 \mathcal B_\eta(\sigma_\theta),
 \Acal_\eta\mathcal E(\sigma_\theta)\bigr)
 \le e_{\rm acq}
\]
by the complete channel comparison.
Trace-distance contraction under the CPTP map \(\Acal_\eta\)
gives
\[
 \dtr\bigl(
 \Acal_\eta\mathcal E(\sigma_\theta),
 \Acal_\eta(\tau_\theta)\bigr)
 \le e_{\rm src}.
\]
Their triangle inequality bounds the distance from the ideal
record law by \(e_{\rm acq}+e_{\rm src}\).
For classical outputs that distance is total variation.
The source-observable identity follows from the two template
normalizations in (P1). The assumptions now match those of
Theorem~\ref{thm:calibrated}, proving
\eqref{eq:physical-radius}.

The complete-record comparison and statistical proof apply to
each supplied competing family in exactly the same way.
Theorem~\ref{thm:competition} then supplies the conditional
discrimination conclusion.
\end{proof}

The trace-distance and complete-channel framework is standard
finite-dimensional quantum information
\cite{Watrous2018}; the composition and inference arguments
needed here have been written explicitly.
% Primary author/publisher record:
% ([cs.uwaterloo.ca](https://cs.uwaterloo.ca/~watrous/TQI/?utm_source=openai))

\begin{remark}[What a successful fit does not certify]
Finite-frequency agreement does not prove (P1)--(P5), the
source-access promise, the microscopic generator, or an
unrestricted channel bound. An empirical fit can be consistent
with several physically different templates.
Theorem~\ref{thm:main} is an implication from separately supplied
certificates, not a procedure that manufactures them from the
same source records.
\end{remark}

\section{Complete-channel tolerance and layer separation}
\label{sec:layers}

\subsection{Compatible interfaces are required for telescoping}

\begin{lemma}[Finite comparison ledger]
\label{lem:telescoping}
For compatible CPTP factors \(G_1,\ldots,G_s\) and implemented
factors \(\widetilde G_1,\ldots,\widetilde G_s\) on identified
matching interfaces,
\[
 \ddia(\widetilde G_s\cdots\widetilde G_1,
       G_s\cdots G_1)
 \le
 \min\left\{1,\sum_{j=1}^s
 \ddia(\widetilde G_j,G_j)\right\}.
\]
\end{lemma}

\begin{proof}
Subtract the two compositions and replace one factor at a time.
Every term contains one factor difference, before and after
which are channels. Diamond norm is contractive under those
compositions. The triangle inequality gives the sum.
Half-diamond distance between channels is at most one.
\end{proof}

This provides an allowance only after the comparison maps and
their interfaces exist. It does not derive a physical encoding
or identify a detector with ideal Enter.
The same finite-protocol comparison architecture is used in
En08-B \cite{CheungEn08B}.

A small tolerance is not exact independence. If
\(\mathcal T\) is a constant record channel and
\(\mathcal Z\) is computational readout, then
\[
 \Lambda_\varepsilon
 =(1-\varepsilon)\mathcal T+\varepsilon\mathcal Z
\]
has \(\ddia(\Lambda_\varepsilon,\mathcal T)\le\varepsilon\),
but for every \(\varepsilon>0\) it retains some input
dependence. Thus a tolerance guard must not be renamed an
exact non-signalling or containment guard.

The plan's XXII-B labels identify possible comparisons.
No unprovided statement from XXII-B is used as a premise;
the allowances used here are stated and proved locally.

\subsection{Ancestry, influence, and geometry}

\begin{proposition}[Scheduled support is an upper bound]
\label{prop:schedule}
In a finite circuit of prescribed local CPTP operations, include
every quantum wire, classical control, random-kick selector,
and record wire in the operation's input interface.
The Heisenberg pullback of a terminal record observable is
supported within its backwards scheduled ancestor set.
A replacement of an operation outside that set leaves the
terminal law unchanged.

Neither a minimal influence region nor a spacetime causal
embedding follows from this support statement.
\end{proposition}

\begin{proof}
At a backwards step, a channel on wires disjoint from the
current observable support acts on an identity factor.
Its adjoint is unital, so it preserves that factor.
A channel meeting the support can enlarge it only into its
declared input wires. Backwards induction gives the scheduled
ancestor bound. A disjoint replacement still acts on an
identity and changes no expectation.

The construction contains no location map into spacetime,
metric, or causal-curve certificate. Those additional data are
therefore not conclusions of this support proof.
\end{proof}

For the pilot, \(G_2\) is a graph of basis positions, not an
asserted spatial arrangement of the hardware qubits \(N,A,E\).
Conflating those two graphs would introduce an unproved
embedding.

This maintains the three-relation distinction of VII-J and
the scheduler-versus-geometry cautions in En08-A and XIX-C
\cite{CheungVIIJ,CheungEn08A,CheungXIXC}.
Any geometric comparison is a subsequent obligation, not
part of source estimation.

\section{Resources and residual non-identification}
\label{sec:ledgers}

\subsection{Finite-model resource ledger}

The following schedule uses the admitted primitive library:
CNOT, one-qubit propagators, CZ, the exchange propagator of
\eqref{eq:exchange-H}, fresh allocation, classical finite
random selection, and destructive Enter.

\begin{enumerate}[label=\textup{(\arabic*)}]
\item Receive \(J\), name it \(N\), allocate \(A\) in \(0\),
and apply the preparation CNOT.
\item Draw the independent hidden kick label and apply the
two local kick propagators in parallel.
\item Apply CZ.
\item Allocate \(E\) in \(0\) and apply its exchange propagator
with \(A\).
\item Enter \(E\), retain no exported environment outcome, and
Enter \(N,A\) to export the occupation word.
\end{enumerate}

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{0.30\linewidth}X@{}}
\toprule
Resource & Declared finite-model ledger\\
\midrule
Supplier input &
One unknown source qubit \(J\) per trial; no exact source-weight
token.\\
Fresh laboratory qubits &
One preparation qubit \(A\), one collision qubit \(E\).\\
Peak live laboratory qubits &
\(3+s\), with \(s\) retained laboratory spectator qubits.\\
Coherent operations &
One preparation CNOT, two local kicks, one CZ, and one
two-qubit exchange propagator.\\
Coherent depth &
At most four layers in this primitive library.\\
Destructive readouts &
Three one-qubit Enter operations, of which two export data.\\
Random-kick controller &
One classical \(m\)-valued selector per trial; its law and
independence are part of the supplier/device model.\\
Selector purification &
If the mixed selector is realized coherently, its purification
and storage must be counted separately; it is not free
laboratory quantum memory.\\
Source purification &
Supplier-side, inaccessible, separately accounted.\\
Validation shots &
Predeclared \(n\), as priced by
\eqref{eq:shot-budget} or
\eqref{eq:discrimination-shots}.\\
Classical records &
All exported attempted-run records and calibration metadata.
Failures are not deleted.\\
Classical processing &
Counts, the nominal estimator, or the predeclared compatibility
set. No quantum-return output.\\
Physical apparatus &
Not constructed by this primitive count. A positron source,
encoding, detector, and physical matching certificate remain
separate.\\
\bottomrule
\end{tabularx}
\end{center}

A coherent preparation of the random selector from prescribed
amplitudes is itself a resource. An unknown nuisance law cannot
be acquired exactly merely by naming those amplitudes.

\subsection{Parameters included, omitted, and only partially identified}

\begin{proposition}[Residual identifiability ledger]
\label{prop:residual}
The following statements hold for the frozen acquisition:
\begin{enumerate}[label=\textup{(\roman*)}]
\item
Any parameter absent from the source states, generators,
selector law, detector, and attempted-run model has an
independent record law; every Fisher-information entry
involving it is zero wherever derivatives exist.
\item
The included direction and kick nuisance affect the likelihood
through \(u,v,\gamma\). They are not generally zero-information
parameters.
\item
At fixed angles and direction, distinct finite kick-weight
vectors with the same \(u,v\) have identical records.
For \(m\ge4\) distinct transition values, such distinct
positive weight vectors exist locally around every strictly
positive weight vector.
\item
The occupation records do not certify inversion of the local
kick. Replacing \(V_k^\dagger\) on \(A\) by \(V_k\) leaves
the record law on the promised source family unchanged.
\end{enumerate}
\end{proposition}

\begin{proof}
For (i), the complete likelihood contains no such parameter.
Any post-processing of equal record laws again has equal laws.

Statement (ii) is the explicit dependence in
\eqref{eq:moments}--\eqref{eq:likelihood}.
For (iii), the weight constraints depend only on the three
linear functionals
\[
 \sum_kw_k,\qquad \sum_kw_kt_k,\qquad \sum_kw_kt_k^2.
\]
The matrix with rows \(1,t_k,t_k^2\) has rank three when at
least three transition values are distinct. With \(m\ge4\)
it has a nonzero kernel vector \(z\). For a strictly positive
\(w\), both \(w+\epsilon z\) and \(w-\epsilon z\) remain
positive for sufficiently small \(\epsilon>0\), preserve
normalization and both moments, and are distinct.

For (iv), \(V_k\) and \(V_k^\dagger\) have the same
computational transition probabilities. The source is diagonal,
CZ changes no populations, and the relaxation population rule
is unchanged. The proof of Lemma~\ref{lem:law} therefore gives
the same likelihood.
\end{proof}

The resulting ledger is:
\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{0.31\linewidth}X@{}}
\toprule
Quantity & Status\\
\midrule
Relative source weight \(\theta\) &
Identifiable at fixed nuisance; calibrated confidence set with
uncertainty; explicit ambiguity when nuisance varies.\\
Effective kick direction &
Included nuisance, not independently sensed. Its azimuth and
the sign of \(n_z\) are invisible to this diagonal-source
occupation likelihood.\\
Finite kick distribution &
Only its relevant two moments enter these records.
Distinct distributions can be indistinguishable.\\
Continuum turbulence spectrum &
Not installed. No continuum reconstruction follows from finite
kick moments.\\
Absolute Galactic rate &
Not installed. Neither arrival-time exposure nor absolute-flux
instrument is present.\\
Dimensionful diffusion length &
Not estimated. No physical length calibration or diffusion law
has been supplied.\\
Singlet/positronium spin &
Not represented by these position factors.\\
Annihilation or escape counts &
No corresponding physical sectors or flags are identified.
A detector failure label is not an annihilation label.\\
Source and apparatus complements &
Inaccessible; not declared physically annihilated.\\
Spacetime embedding &
Absent; observation boundary is operational.\\
Thermal cost &
Not fixed by the recorded channel or its reference identity.\\
\bottomrule
\end{tabularx}
\end{center}

The benchmark
\(5\times10^{43}\,\mathrm{s}^{-1}\) mentioned in the motivating
plan is not a parameter of this experiment.
An externally supplied normalization token would remain
external input, not an acquired rate.

\subsection{Thermodynamic underdetermination}

\begin{proposition}[No unique thermal ledger from the visible channel]
\label{prop:thermal}
Equality of the visible acquisition channels does not determine
a unique bath heat, work cost, or cyclic reset cost.
\end{proposition}

\begin{proof}
Append to any implementation an independent fuel qubit,
initially excited, and a designated Gibbs reservoir qubit,
both with energy gap \(\epsilon>0\).
At inverse temperature \(\beta\), the reservoir excitation
probability is \(r=1/(1+e^{\beta\epsilon})\).
Swapping these two qubits changes the reservoir energy by
\(\epsilon(1-r)\), while leaving the entire visible acquisition
channel unchanged. Appending more such independent processes
changes that designated ledger again.

These processes consume resources. They are not cyclic reset
protocols unless the fuel, auxiliaries, and controls are also
restored. The example proves underdetermination, not a violation
of a thermodynamic inequality.
\end{proof}

This locally proves the channel-versus-ledger distinction
recorded in En07-B \cite{CheungEn07B}.
Landauer-type bounds belong to an additional physical setup;
Reeb and Wolf give a precise finite-reservoir formulation
\cite{ReebWolf2014}.
The prescribed pilot control generators are not a complete
apparatus, reservoir, and restoration ledger.
% Primary thermal source:
% ([arxiv.org](https://arxiv.org/abs/1306.4352))

\section{The six physical obligations and the exact remaining gap}
\label{sec:obligations}

\subsection{Obligations O1--O6}

\begin{description}[style=nextline,leftmargin=2.5em]
\item[O1: Physical derived channel.]
Supply one actual finite positron source model, its microscopic
generator, source delivery and encoding, auxiliary preparations,
validity domain, and detector. Derive its acquisition channel
and prove the comparison required in
Definition~\ref{def:physical-package}.
Exponentiating the pilot generators discharges the
\emph{finite-model} part only.

\item[O2: Record-to-source identification.]
Identify \(F_b^{\rm P}\), its source templates, and their encoded
meaning. Prove the parameter relation on the promised family.
If nuisance or competing templates vary, use a confidence set
or establish a complete-law separation. Proposition~\ref{prop:overlap}
shows that unconditional point identification over a broad
nuisance class is false.

\item[O3: Calibration uncertainty.]
Supply a calibration set and its joint coverage event.
Compute a justified allowance, not one fitted away by arbitrary
nuisance adjustment. The nominal calibration conversion and
radius are proved here; their applicability to a positron
apparatus is not established.

\item[O4: Complete-record finite-shot guarantee.]
Supply independent attempted trials, fixed validation counts,
failure scoring, and a predeclared candidate family. Once those
hypotheses hold, the statistical guarantees in this manuscript
apply. No source-dependent postselection or unaccounted
adaptive stopping is licensed.

\item[O5: Residual and non-identification ledger.]
Assign every source purification, selector, collision
environment, detector complement, and retained spectator.
Distinguish omitted parameters from included but unresolved
nuisance. No absolute Galactic rate, continuum turbulence
spectrum, or sky map is identified.

\item[O6: Layer separation.]
Specify the actual controller and schedule. Scheduled support
is an upper bound on dependence. Any physical causal embedding
requires independently supplied geometry and location data;
it is not a conclusion of the estimation certificate.
\end{description}

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{0.08\linewidth}p{0.27\linewidth}X@{}}
\toprule
Gate & Finite-model result & Physical target status\\
\midrule
O1 &
Explicit finite generator, propagator, and acquisition constructed &
Positron dynamics, encoding, detector, and comparison not supplied.\\
O2 &
Likelihood, sensitivity, information, and equal-law ambiguity proved &
Physical source observable and its template correspondence required.\\
O3 &
Worst-case calibration conversion proved &
Physical calibration event and allowances required.\\
O4 &
Fixed-shot estimation and competing-family certificates proved &
Independence and complete physical attempted-run law required.\\
O5 &
Finite residual and omitted-parameter distinctions proved &
Actual source/apparatus complement assignment required.\\
O6 &
Finite scheduled-support statement proved &
No geometric embedding constructed.\\
\bottomrule
\end{tabularx}
\end{center}

\subsection{Mathematical release versus physical realization}

\begin{theorem}[En10-B release]
\label{thm:release}
On the explicitly defined interfaces:
\begin{enumerate}[label=\textup{(\roman*)}]
\item
The finite synthetic generator produces a complete,
reference-stable destructive acquisition instrument.
\item
Its likelihood and fixed-nuisance Fisher-information ledger
are explicit.
\item
Calibration uncertainty and complete-law allowances yield
the finite-shot source-weight certificate.
\item
Predeclared competing families admit complete-record
compatibility and discrimination certificates under their
stated separation margin.
\item
Different source weights can have identical complete records
under allowed finite nuisance variation.
\item
A supplied physical acceptance package implies the physical
source-observable certificate of Theorem~\ref{thm:main}.
\end{enumerate}
The unconditional physical positron
acquisition-and-identification target is not established in
this release.
\end{theorem}

\begin{proof}
Theorem~\ref{thm:derived} proves (i).
Lemma~\ref{lem:law} and Proposition~\ref{prop:FI} prove (ii).
Theorem~\ref{thm:calibrated} proves (iii).
Theorem~\ref{thm:competition} proves (iv).
Proposition~\ref{prop:overlap} proves (v).
Theorem~\ref{thm:main} proves (vi) under its explicitly stated
package.

No physical source templates, positron generator-to-register
correspondence, complete physical detector derivation, or
certificate \eqref{eq:physical-comparison} has been supplied
for the intended positron experiment. The finite-model
construction does not produce those missing physical data.
Consequently (vi) cannot be instantiated as an unconditional
physical identification on the supplied evidence.
This is a boundary of the release, not a universal
nonexistence theorem.
\end{proof}

\paragraph{Precise remaining gap.}
A successor must construct, on one named physical positron
source model, a complete acquisition instrument with a
microscopically justified source encoding and transport/detector
channel, and prove the calibration and comparison package
needed for \eqref{eq:physical-radius}.
It must also specify which competing explanations are included
and either distinguish them or retain their common-law
ambiguities.

Until that package is supplied, En10-A remains a finite analog
with its declared estimation contract. En10-B adds a derived
finite-model pilot and conditional bridge mathematics; it does
not establish physical positron-source identification.

\section{Source relationships and standing non-claims}
\label{sec:relations}

En10-A is the supplied finite-interface predecessor.
Landmark I and En09-A@1 explain the measurement and residual
discipline. XIX-C separates source tokens from acquisition and
finite channels from continuum evolution. En08-B provides a
comparison pattern after compatible interfaces exist.
Oai265-A and Oai265-B provide record/recovery and statistical
comparisons. VII-J, En08-A, and En07-B provide the operational,
geometric, and thermodynamic separations.
Their models are not silently identified with the present one.

The unrelated field, black-hole, Ward, and characteristic-graph
excerpts do not provide a positron apparatus or physical
identification certificate. In particular, no unprovided
statement from XV-B is invented or used.

OpenAI catalogue families remain documentary negative controls.
They supply neither the source model, generator,
encoding, acquisition channel, calibration event, nor the
competing-family identification certificate.

The manuscript does not:
\begin{itemize}
\item derive a continuum transport law or a Galactic diffusion
equation;
\item identify records with the observed 511\,keV morphology,
the interstellar medium, or an absolute Galactic rate;
\item construct a thermal machine or infer a Landauer cost
from channel equality;
\item install a light cone, derive Einstein dynamics, or
establish spacetime emergence;
\item construct a Page curve or quantum evolution without
externally scheduled time;
\item close O.1, open the downstream XV-B admission exit,
discharge XVI-N or Route B, or modify the version-specific
H2, I3, and J2 analytic records;
\item admit an inherited gravitational quantum coefficient.
\end{itemize}
These are non-transfers by this manuscript, not global claims
about every other revision or future construction.

\paragraph{Verification and submission boundary.}
The source is a single standard pdfLaTeX document with an
internal bibliography. No successful pdfLaTeX build, hardware
compilation, proof-assistant check, numerical experiment,
physical acquisition, independent referee verification, or
journal acceptance of this manuscript is asserted.

\appendix
\section{Predecessor audit relevant to this extension}
\label{app:audit}

The extension does not rely on the following documentary or
presentation defects of the supplied En10-A source:
\begin{enumerate}
\item
Its relational-clock references should identify
\texttt{En09-A.tex}, version~1, rather than a second
\texttt{En10-A.tex}.
\item
The final computational-ledger row ends with the incomplete
sentence ``Enter is an admitted primitive of''. A complete row
should end, for example, ``Enter is an admitted primitive of the
declared operational library'', followed by its row terminator
and closing rule.
\item
``Rational probabilities'' for the rational example means
probabilities affine in \(\theta\) with rational coefficients;
their values are rational when \(\theta\) is rational.
The finite-count estimator values are rational in that example.
\item
An \(n\)-shot decoder takes an \(n\)-record list or its declared
histogram, not one isolated word. This manuscript uses that
interface explicitly.
\end{enumerate}
The finite likelihood needed here is re-derived in
Lemma~\ref{lem:law}. These corrections neither supply physical
identification nor change the predecessor's source-access
assumptions.

\begin{thebibliography}{99}

\bibitem{CheungEn10A}
N.~Cheung,
\emph{En10-A: A Finite Enter Instrument on a Qubit--Antiqubit
Interface: Classical Records, Source-Weight Estimation,
and Non-Identification}.
Supplied \texttt{En10-A.tex}.
No separate registered version or physical implementation is
certified by this citation.

\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 stated operational 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 orientation, not a registered theorem premise.

\bibitem{CheungI}
N.~Cheung,
\emph{Landmark I}.
\texttt{I.tex}, version~2.
Supplied excerpts, especially \texttt{thm:main} and
\texttt{thm:update}.
Characterization within the specified measurement strategy class.

\bibitem{CheungEn09A}
N.~Cheung,
\emph{En09-A}.
\texttt{En09-A.tex}, version~1.
Supplied excerpts, especially \texttt{thm:main},
\texttt{eq:exact-channel-identity}, and
\texttt{eq:linear-interface}.
Relational clock readout and conditional residuals.

\bibitem{CheungXIXC}
N.~Cheung,
\emph{XIX-C}.
\texttt{XIX-C.tex}, version~1.
Supplied excerpts, especially \texttt{thm:main}.
Source-token/acquisition separation and finite-to-continuum
non-determination.

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

\bibitem{CheungOai265A}
N.~Cheung,
\emph{Oai265-A}.
\texttt{Oai265-A.tex}, version~1.
Supplied excerpts, especially \texttt{thm:separation}.
Static record, quantum recovery, and physical identification
remain separate.

\bibitem{CheungOai265B}
N.~Cheung,
\emph{Oai265-B}.
\texttt{Oai265-B-v2.tex}, version~2.
Supplied excerpts, especially \texttt{ass:statistics} and
\texttt{thm:main}.
Conditional fixed-sample complete-record certification.

\bibitem{CheungEn01B}
N.~Cheung,
\emph{En01-B}.
\texttt{En01-B-v2.tex}, version~2.
Supplied excerpts, especially \texttt{thm:contrast} and
\texttt{thm:width}.
Oriented nuisance margins and complete-law overlap.

\bibitem{CheungEn01C}
N.~Cheung,
\emph{En01-C}.
\texttt{En01-C.tex}, version~1.
Supplied excerpts, especially \texttt{thm:certificate}.
Complete attempted-run scores, implementation allowances,
and finite-shot margins.

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

\bibitem{CheungEn08A}
N.~Cheung,
\emph{En08-A}.
\texttt{En08-A.tex}, version~1.
Supplied excerpts, especially \texttt{thm:main}.
Scheduler consistency does not by itself select spacetime
geometry.

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

\bibitem{CheungHawk02A}
N.~Cheung,
\emph{Hawk02-A}.
\texttt{Hawk02-A.tex}, version~1.
Supplied excerpts, especially \texttt{ass:physical} and
\texttt{thm:main}.
Comparison of additionally assumed and microscopically derived
transfer interfaces; a different model.

\bibitem{CheungEm03D}
N.~Cheung,
\emph{Em03-D}.
\texttt{Em03-D-v1.tex}, version~1.
Supplied excerpts, especially \texttt{thm:main}.
A named local packet-effect correspondence on its specified
state class; not a positron-source construction.

\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 inversion and a separate 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{Watrous2018}
J.~Watrous,
\emph{The Theory of Quantum Information}.
Cambridge University Press, 2018.
doi:\nolinkurl{10.1017/9781316848142}.
Finite-dimensional channels and complete-channel norms.

\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.
Motivation only; no sky model or rate is imported into the
finite construction.

\end{thebibliography}

\section*{Exact non-identification statement}

\begin{center}
\fbox{\parbox{0.91\linewidth}{
The constructed output is a \textbf{finite-model estimate under
declared access and calibration}. Identification with the
named physical source observable is proved only conditionally
on the independently supplied physical acquisition and
identification package. That package has not been constructed
for the intended positron source in this release.

The instrument does not recover a Galactic quantum state,
identify its records with the interstellar medium or the
observed 511\,keV morphology, derive the absolute Galactic
positron rate, or install a continuum transport law.
Its observation boundary is not a light cone or a Galactic
radius. No thermal machine, Landauer cost, complete physical
energy ledger, Einstein dynamics, spacetime emergence, Page
curve, or quantum evolution without externally scheduled time
is implied. No inherited gravitational quantum coefficient is
admitted, and no compilation is asserted.

The unconditional physical positron
acquisition-and-identification theorem remains unconstructed.
This is a precise remaining obligation, not a universal
nonexistence claim.
}}
\end{center}

\end{document}