En10-B Physical Acquisition and Identification for a Declared Positron-Source Observable A Derived Finite-Model Instrument, Calibrated Inference, and a Conditional Bridge from En10-A

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.

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

and identifier correction. The supplied En10-A.tex concerns the finite qubit–antiqubit instrument CheungEn10A. The relational clock manuscript in the supplied excerpts is En09-A.tex, version 1 CheungEn09A. The predecessor's references to a relational “source En10-A@1” are therefore read as references to En09-A@1, not as a second manuscript with the same En10-A identifier. No unprovided registered revision is inferred.

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: \[

specified finite dynamics derived finite record channel,
physical source and apparatus validated acquisition channel,
acquired records inference about a named physical observable.

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 Siegert2016. These structures are absent from the graph constructed below.

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 CheungEnter,CheungI. Here Born semantics are assumed; that characterization is not re-proved.

The introductory essay The Philosophy of Enter separates readout from quantum recovery, records from quantum-return decoders, channels from thermal machines, and analogs from physical identifications CheungPhilosophy. These distinctions provide the methodological background, not additional theorem hypotheses.

In this manuscript those distinctions acquire precise forms:

1. An instrument is a linear, complete collection of subnormalized CP branches. Normalizing a selected successful branch does not construct a deterministic acquisition channel.

2. Consumption concerns the measured quantum names and their output interface. It does not assert that every inaccessible physical degree of freedom ceases to exist.

3. A source parameter is acquired from unknown inputs, not from a classical token already containing its numerical value.

4. Calibration and competing explanations constrain what a record identifies. More shots do not distinguish identical record laws.

5. A physical correspondence must identify the source observable and acquisition channel; philosophical vocabulary cannot supply that correspondence.

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.

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.

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.

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.

Freeze the graph

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

The supplier delivers a source qubit $J$ in

Its source observable is $F_J=|0⟩⟨0|$, so that $\mathrm{Tr}(F_J{\tau }_{\theta })=\theta$.

Define the fixed preparation isometry

Operationally, the supplied name becomes $N$, a fresh qubit $A$ is allocated in $0$, and ${\mathrm{CNOT}}_{N\to A}$ is applied. Consequently

The map $\mathcal{P}$ is independent of $\theta$. It does not clone arbitrary source states: a coherent input generally produces an entangled pair.

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.

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

Fix a positive integer $m$. A nuisance value is

where

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 $\mathcal{K}$ satisfying

The nominal loss is ${\gamma }_0\in [{\gamma }_{-},{\gamma }_{+}]$, and

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.

Let $X,Y,Z$ be the Pauli matrices in the graph basis and set

The $X,Y$ terms are edge couplings; the $Z$ term is an on-site contrast. The vector $\hat{n}$ is an effective generator axis until a physical correspondence is supplied.

Freeze positive externally scheduled durations $T_{\mathrm{k}},T_{\mathrm{c}},T_{\mathrm{e}}$, and $\hslash >0$. For label $k$, prescribe

Its normal and antiqubit factors have opposite signs.

The interaction stage has generator

For relaxation, allocate $E$ in $|0⟩$. On $A\otimes E$, define the self-adjoint exchange generator

and prescribe

These are finite, piecewise scheduled generator data. They are not derived from an unprovided Galactic action.

The inverse response in (eq:kick-H) is synthetic and platform-specific in interpretation. Song and collaborators provide a separate superconducting antiqubit construction Song2025. Their apparatus and sensing advantage are not implemented or imported here.

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.

Define

For $h=1-\gamma$, put

The channel to be derived is

Let $\mathcal{R}=\{00,01,10,11\}$. To permit comparison with complete attempted-run records, use the common alphabet

The ideal failure branch is zero. Real failure or rejection branches, if present, must remain in ${\mathcal{R}}_{+}$ or in a predeclared larger alphabet.

For any finite retained factor $T$, define

The complete acquisition map from the supplier interface is

For the frozen generators and source interface:

1. The scheduled propagators of (eq:kick-H)–(eq:exchange-H) yield (eq:derived-channel). The maps $\mathcal{P},{\Phi }_{\eta },{\mathcal{A}}_{\eta }$ are CPTP.

2. For every finite retained factor $T$, ${{\mathcal{M}}_{\mathcal{B}}}^T={{\mathcal{J}}_{\mathcal{B}}}^T$ on all operators, and

   $${\mathrm{Tr}}_C{{\mathcal{A}}_{\eta }}^T(X)={\mathrm{Tr}}_{J}X.$$

   This includes arbitrary untouched references.

3. 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.

Proof. The two summands of $H_{\mathrm{k},k}$ commute, so its propagator is $V_k\otimes {V_k}^{†}$. Since $e^{-i\pi |11⟩⟨11|}=\mathrm{CZ}$, the next propagator is $\mathrm{CZ}$.

On the exchange subspace, ${K_{\mathrm{e}}}^2=I$, whereas it annihilates $00,11$. Thus its propagator satisfies

Taking the environment matrix elements gives $L_0,L_1$. Drawing the independent label $k$, and summing all its unexported branches, yields (eq:derived-channel). Moreover,

The source map is isometric, so these maps are CPTP.

For readout, the Kraus operators are $|r{⟩}_C\otimes D_r$, and their squared moduli sum to $I_B\otimes I_T$. If $X={\sum }_{r,s}|r⟩⟨s{|}_B\otimes X_{rs}$, both readout specifications have $r$-block $X_{rr}$. Therefore the maps agree on all operators, and

Local trace-preserving maps on $J$, followed by local preparation and evolution, preserve the retained marginal. This proves (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.

The accessible algebra is $\mathrm{Diag}(C)\otimes \mathcal{L}({\mathcal{H}}_S)$ 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 (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 CheungI,CheungEn09A.

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.

Proof. Consider the permissible two-site family

It is self-adjoint. Its sum with $H_{-q}$ is $2J_{\ast }\cos (q\alpha )X$, which need not vanish. At $\alpha =0$, both charge signs give $J_{\ast }X$. For $0<J_{\ast }T/\hslash <\pi /2$,

Hence charge-sign reversal is not, in this family, full Hamiltonian inversion.

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.

Define

The calibration set implies

On input ${\tau }_{\theta }$, the derived acquisition probabilities are

All four nonfailure probabilities are strictly positive. For the complete-record score

its ideal mean is

Proof. Since ${K_{\hat{n}}}^2=I$,

Both $V_k$ and ${V_k}^{†}$ have squared matrix elements

The diagonal interaction changes no occupation population.

Conditional on source component $00$ and label $k$, the four probabilities after relaxation are

Conditional on source component $11$, they are

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 (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 (eq:affine).

For fixed known $\eta$, the probability $q_{\eta }$ determines the model parameter:

This is population-level identification, not exact acquisition of a probability from one record.

The deterministic specialization

has $u={\sin }^2\varphi ,v=u^2$ and reproduces the finite likelihood of En10-A. The construction above derives that specialization from the named pilot generators.

For $0<\theta <1$, fixed $\eta$, and independent repetitions, the one-shot score information and complete-record information are

They satisfy

Independent repetition multiplies each information by $n$. The upper bound applies to every measurement of the derived post-preparation system at fixed nuisance.

Proof. The score has a Bernoulli law with derivative $b_{\eta }$. Differentiating (eq:likelihood) gives

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.

The quantities in Proposition 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.

For states and channels use

For classical laws, $d_{\mathrm{TV}}(P,Q)=\frac{1}{2}{\sum }_r|P(r)-Q(r)|$. An expectation of a function in $[0,1]$ changes by at most total variation.

A calibration record $\mathcal{Z}$ is collected before validation. Conditional on $\mathcal{Z}$:

1. $n\ge 1$, the calibration set $\mathcal{K}$, all scores, and all comparison allowances are fixed before the $n$ trial records are observed.

2. The trial records are independent. Their actual laws ${\tilde{p}}_i$ may differ, but on a calibration-measurable event $G$ they obey

   $$\eta \in \mathcal{K},d_{\mathrm{TV}}({\tilde{p}}_i,p_{\theta ,\eta })\le e(i=1,\dots ,n).$$

   Here $\theta ,\eta$ describe one fixed target family member.

3. The event has $\mathrm{Pr}(G^c)\le \beta$. Every attempted trial produces a complete record, including failure or rejection. There is no unaccounted postselection, validation-driven tuning, or adaptive stopping.

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 CheungOai265B,CheungEn01C. Their device models and data are not transferred here.

Set

Define the exact mean-level calibration allowance

This supremum exists because the set is compact and its likelihood is continuous.

Uniformly for $\eta \in \mathcal{K},\theta \in [0,1]$,

Moreover,

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 (eq:calibration-allowance).

The function ${\sin }^{2}x$ is globally $1$-Lipschitz. Hence

Averaging gives the same bound on $|u-u_0|$. Writing

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 (eq:moment-bounds).

Let

This is a separate classical post-processing of the record list. It does not prepare a returned quantum source.

Under Assumption ass:statistics, for $0<\alpha <1$ with $\alpha +\beta <1$, define

Then, uniformly over the promised source and calibration family,

Thus

is a confidence interval with that coverage.

For $0<\epsilon <1$, a sufficient finite shot budget is

This is a certificate for the named model parameter, not a Galactic-source identification.

Proof. Conditional on the calibration record, the bounded independent scores obey

For completeness, a bounded variable $W\in [0,1]$ has

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 Hoeffding1963.

On $G$, the law allowance gives

Lemma lem:calibration then gives, on the concentration event,

Projection onto $[0,1]$ cannot increase distance from $\theta \in [0,1]$, so division by $b_0$ proves the radius. The bound $|{\hat{\theta }}_0-\theta |\le 1$ permits its cap at one. Integrating the conditional statistical failure probability and adding $\mathrm{Pr}(G^c)\le \beta$ proves (eq:source-coverage). Rearranging the radius requirement proves (eq:shot-budget).

The displayed nominal certificate need not shrink below $({\Delta }_{\mathrm{cal}}+e)/b_0$ 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.

Let $\mathcal{H}=\{0,\dots ,L\}$ be a finite predeclared list of competing families. Family $j$ has a compact parameter domain ${\Theta }_j$ and a continuous complete record law

Family $0$ may be the derived family (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

provided the corresponding channel ${{\Phi }^j}_{\eta }$ has been constructed. Merely writing this expression does not derive the alternative channel.

Let

Define the complete-record compatibility set

All counts, including $\perp$, enter this set.

Suppose the conditional fixed-shot contract holds for a true member $(j_{\ast },{\lambda }_{\ast })$, with

on $G$. Then

For two promised families, define the complete-record separation

If

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

Proof. For each coordinate indicator, conditional Hoeffding gives failure probability at most $2e^{-2n{c}_n^2}=\alpha /D$. A union bound yields simultaneous deviation at most $c_n$ from the average actual law. Total variation at most $e_{j_{\ast }}$ bounds every coordinate bias by $e_{j_{\ast }}$. This proves (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

contradicting (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.

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 CheungEn01B,CheungOai265A.

Put

Equation (eq:likelihood) becomes

Thus at fixed $\gamma$ the complete record depends only on $r,c$, not separately on $\theta ,u,v$.

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 ${\alpha }_0+{\alpha }_1=1$.

Proof. Take $\hat{n}=(0,1,0)$, $\gamma =1/4$, and three possible transition values

They are realized by $\varphi =\arcsin \sqrt{1/5}$ and ${\xi }_k=\arcsin \sqrt{t_k}-\varphi$. For an explicit admissible angle bound, choose

The other offset has smaller absolute value. For instance, the relevant elementary angle inequalities give

Hence $\varphi -\kappa >0$ and $\varphi +\kappa <0.72<\pi /4$. A compact set containing these fixed angles and the weight simplex satisfies (eq:calibration-set).

For the first member choose

Then $u^{(0)}=1/5,v^{(0)}=1/25$. For the second choose

Direct rational arithmetic gives $u^{(1)}=1/4,v^{(1)}=9/100$. Both members have

Their common complete law is

Let a randomized test declare the second member with probability $\psi (\mathbf{r})\in [0,1]$. The record-list laws are identical, so

For the pair in Proposition prop:overlap, no record-only estimator has both probabilities

greater than $1/2$ when $\epsilon <|{\theta }^{(0)}-{\theta }^{(1)}|/2=1/100$.

Proof. The two success intervals are disjoint. Under the common data law, and any common randomized estimator, their probabilities sum to at most one.

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 $b_{\eta }$.

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

A package for this target consists of:

1. A named physical source model on a specified finite input space ${\mathcal{H}}_{\mathrm{P}}$, with fixed injection templates ${\sigma }_b,{\sigma }_d$ and an effect ${F_b}^{\mathrm{P}}$ satisfying

   $$\mathrm{Tr}({F_b}^{\mathrm{P}}{\sigma }_b)=1,\mathrm{Tr}({F_b}^{\mathrm{P}}{\sigma }_d)=0.$$

   Its promised source family is

   $${\sigma }_{\theta }=\theta {\sigma }_b+(1-\theta ){\sigma }_d.$$

   Therefore the named source observable obeys $\mathrm{Tr}({F_b}^{\mathrm{P}}{\sigma }_{\theta })=\theta$. This excludes source-dependent selection from being silently identified with injection weight.

2. A $\theta$-independent CPTP source comparison map $\mathcal{E}:\mathcal{L}({\mathcal{H}}_{\mathrm{P}})\to \mathcal{L}({\mathcal{H}}_J)$ and a proved allowance

   $$d_{\mathrm{tr}}(\mathcal{E}({\sigma }_{\theta }),{\tau }_{\theta })\le e_{\mathrm{src}}\text{uniformly in }\theta .$$

   It is sufficient to prove this allowance on the two source templates, with the corresponding targets $0,1$.

3. A named microscopic physical evolution, its auxiliary preparations, transport stages, nuisance domain, and complete detector instrument, yielding a CPTP record channel

   $${\mathcal{B}}_{\eta }:\mathcal{L}({\mathcal{H}}_{\mathrm{P}})⟶\mathcal{L}({\mathcal{H}}_C).$$

   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.

4. A derived or independently certified complete comparison

   $$d_{\diamond }({\mathcal{B}}_{\eta },{\mathcal{A}}_{\eta }\mathcal{E})\le e_{\mathrm{acq}},$$

   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.

5. A calibration event and independent attempted-run contract of the form in Assumption ass:statistics. The correspondence between physical field/turbulence parameters and $\eta$ is part of the package. It is not inferred from the notation $\hat{n}$.

6. 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.

The stated source templates, physical generator, actual detector, and comparison certificate are not contained in the supplied En10-A model. Definition 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 CheungHawk02A,CheungEm03D. Neither source supplies the missing positron interface.

Suppose a physical package satisfying Definition def:physical-package is supplied. Then the records of its complete acquisition instrument estimate

with Theorem thm:calibrated, using

The resulting radius is

If the additional competing-family package has a positive complete-record margin, Theorem 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.

Proof. For each source state,

by the complete channel comparison. Trace-distance contraction under the CPTP map ${\mathcal{A}}_{\eta }$ gives

Their triangle inequality bounds the distance from the ideal record law by $e_{\mathrm{acq}}+e_{\mathrm{src}}$. 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 thm:calibrated, proving (eq:physical-radius).

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

The trace-distance and complete-channel framework is standard finite-dimensional quantum information Watrous2018; the composition and inference arguments needed here have been written explicitly.

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 thm:main is an implication from separately supplied certificates, not a procedure that manufactures them from the same source records.

For compatible CPTP factors $G_1,\dots ,G_s$ and implemented factors ${\tilde{G}}_1,\dots ,{\tilde{G}}_s$ on identified matching interfaces,

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.

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 CheungEn08B.

A small tolerance is not exact independence. If $\mathcal{T}$ is a constant record channel and $\mathcal{Z}$ is computational readout, then

has $d_{\diamond }({\Lambda }_{\epsilon },\mathcal{T})\le \epsilon$, but for every $\epsilon >0$ 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.

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.

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.

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 CheungVIIJ,CheungEn08A,CheungXIXC. Any geometric comparison is a subsequent obligation, not part of source estimation.

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

1. Receive $J$, name it $N$, allocate $A$ in $0$, and apply the preparation CNOT.

2. Draw the independent hidden kick label and apply the two local kick propagators in parallel.

3. Apply CZ.

4. Allocate $E$ in $0$ and apply its exchange propagator with $A$.

5. Enter $E$, retain no exported environment outcome, and Enter $N,A$ to export the occupation word.

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.

The following statements hold for the frozen acquisition:

1. 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.

2. The included direction and kick nuisance affect the likelihood through $u,v,\gamma$. They are not generally zero-information parameters.

3. At fixed angles and direction, distinct finite kick-weight vectors with the same $u,v$ have identical records. For $m\ge 4$ distinct transition values, such distinct positive weight vectors exist locally around every strictly positive weight vector.

4. The occupation records do not certify inversion of the local kick. Replacing ${V_k}^{†}$ on $A$ by $V_k$ leaves the record law on the promised source family unchanged.

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 (eq:moments)–(eq:likelihood). For (iii), the weight constraints depend only on the three linear functionals

The matrix with rows $1,t_k,{t_k}^2$ has rank three when at least three transition values are distinct. With $m\ge 4$ it has a nonzero kernel vector $z$. For a strictly positive $w$, both $w+ϵz$ and $w-ϵz$ remain positive for sufficiently small $ϵ>0$, preserve normalization and both moments, and are distinct.

For (iv), $V_k$ and ${V_k}^{†}$ 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 lem:law therefore gives the same likelihood.

The resulting ledger is:

The benchmark $5\times 10^{43}{\mathrm{s}}^{-1}$ 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.

Equality of the visible acquisition channels does not determine a unique bath heat, work cost, or cyclic reset cost.

Proof. Append to any implementation an independent fuel qubit, initially excited, and a designated Gibbs reservoir qubit, both with energy gap $ϵ>0$. At inverse temperature $\beta$, the reservoir excitation probability is $r=1/(1+e^{\beta ϵ})$. Swapping these two qubits changes the reservoir energy by $ϵ(1-r)$, 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.

This locally proves the channel-versus-ledger distinction recorded in En07-B CheungEn07B. Landauer-type bounds belong to an additional physical setup; Reeb and Wolf give a precise finite-reservoir formulation ReebWolf2014. The prescribed pilot control generators are not a complete apparatus, reservoir, and restoration ledger.

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 def:physical-package. Exponentiating the pilot generators discharges the finite-model part only.

O2: Record-to-source identification.

Identify ${F_b}^{\mathrm{P}}$, 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 prop:overlap shows that unconditional point identification over a broad nuisance class is false.

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.

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.

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.

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.

On the explicitly defined interfaces:

1. The finite synthetic generator produces a complete, reference-stable destructive acquisition instrument.

2. Its likelihood and fixed-nuisance Fisher-information ledger are explicit.

3. Calibration uncertainty and complete-law allowances yield the finite-shot source-weight certificate.

4. Predeclared competing families admit complete-record compatibility and discrimination certificates under their stated separation margin.

5. Different source weights can have identical complete records under allowed finite nuisance variation.

6. A supplied physical acceptance package implies the physical source-observable certificate of Theorem thm:main.

The unconditional physical positron acquisition-and-identification target is not established in this release.

Proof. Theorem thm:derived proves (i). Lemma lem:law and Proposition prop:FI prove (ii). Theorem thm:calibrated proves (iii). Theorem thm:competition proves (iv). Proposition prop:overlap proves (v). Theorem thm:main proves (vi) under its explicitly stated package.

No physical source templates, positron generator-to-register correspondence, complete physical detector derivation, or certificate (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.

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 (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.

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:

• derive a continuum transport law or a Galactic diffusion equation;

• identify records with the observed 511\,keV morphology, the interstellar medium, or an absolute Galactic rate;

• construct a thermal machine or infer a Landauer cost from channel equality;

• install a light cone, derive Einstein dynamics, or establish spacetime emergence;

• construct a Page curve or quantum evolution without externally scheduled time;

• 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;

• admit an inherited gravitational quantum coefficient.