A Durable-Record Certificate Ladder from Detector Semantics to Shared-Fact Conditions

Detector tomography answers a specific operational question: what measurement semantics does a device implement? In modern form, this is the reconstruction or certification of a POVM or equivalent outcome map linking quantum inputs to reported outcome probabilities [citation]. Recent measurement-characterization work extends this semantic layer by benchmarking mid-circuit measurement error and by deriving precision bounds for detector estimation, but these advances still characterize measurement semantics rather than durable record formation or shared-fact status [citation].

A different line of work asks when information about a system becomes redundantly accessible to many observers through disjoint fragments of an environment. This literature includes environment-as-witness and quantum-Darwinism formulations, operational refinements of spectrum broadcast structure and strong quantum Darwinism, and recent work on redundancy, consensus, branching structure, Hamiltonian criteria, measurement compatibility, and laboratory demonstrations in photonic, spin, and superconducting platforms [citation]. These results address classical objectivity, not detector-side event formation.

A detector trigger need not become a durable record, and a durable local record need not become a shared fact. The separation is organized below as a certificate ladder. Once measurement semantics have been declared, the analysis asks which triggers become durable records and which durable records become shared facts. The resulting ladder contains a semantic certificate, a readout certificate, and an objectivity certificate. Cross-platform composition is admitted only for one declared pointer variable and one declared outcome class, under explicit pointer maps and reporting-compatible conventions fixed before certificate evaluation.

Three distinct layers

definition: Semantic, readout, and objectivity layers. A measurement chain is described on three distinct layers:

- the semantic layer, specified by a declared measurement semantics such as a POVM $\{\Pi_m\}$TeX\{\Pi_m\}; - the readout layer, specified by whether triggers enter a durable record-bearing sector on declared reporting and persistence windows; - the objectivity layer, specified by whether many disjoint fragments redundantly encode one common pointer variable with low disagreement, low discord leakage, and pointer stability.

This declared three-layer stratification is the fixed reference-system family for the present paper, and all downstream certificate claims are read only relative to it.

The three layers answer different operational questions. The semantic layer fixes what the outcome labels mean. The readout layer asks whether a reported event is durably recorded rather than merely triggered. The objectivity layer asks whether the durable record is redundantly and consensually accessible to many observers.

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Claim boundary

No detector micro-Hamiltonian is derived here, and no derivation of the Born rule or appeal to intrinsic collapse is attempted. The claim boundary is limited to when the chain

\text{measurement semantics} \longrightarrow \text{durable record} \longrightarrow \text{shared fact}

is certifiable for one declared outcome class.

The three layers need not be obtained on one device or in one experiment. Nor is it assumed that every platform supports the full objectivity diagnostic set. The point is operational rather than ontological: the three layers answer different questions, require different diagnostics, and cannot be interchanged without loss of observable structure.

proposition: Measurement semantics is not enough. Within the declared three-layer reference-system family, a realized measurement semantics, even when certified by detector tomography or an equivalent calibration protocol, does not by itself certify either durable record formation or shared-fact status. Additional readout-layer and objectivity-layer diagnostics are required.

proof. A semantic certificate determines how input states are mapped to outcome probabilities [citation]. That mapping does not determine whether a trigger is stabilized as a durable record, nor whether many disjoint fragments redundantly encode one common pointer variable with low disagreement and low discord leakage. Those are separate operational questions and require separate diagnostics.

remark: Layer-wise rather than single-apparatus certification. A semantic certificate may be imported from detector tomography or an equivalent calibration protocol, a readout certificate from time-resolved detector characterization, and an objectivity certificate from fragment-accessibility diagnostics on another platform class. The logical claim concerns the admissibility of a shared-fact report for one declared outcome class. It is not a universal single-device tomography claim. No de Broglie or wavefunction-semantics claim is made here, and no dynamical source law for classical objectivity is derived.

Minimal ready--activated--latched model

The readout layer distinguishes generic activation from durable record formation. We use a minimal three-sector description with a metastable ready state $R$TeXR, an activated transient sector $A$TeXA, and a latched record-bearing sector $L$TeXL. Let $p_R,p_A,p_L$TeXp_R,p_A,p_L denote the corresponding populations. We write

\dot p_R = - (\Gsig+\Gdark)p_R + \Grel p_A + \Greset p_L,

\dot p_A = (\Gsig+\Gdark)p_R - (\Glat+\Grel)p_A,

\dot p_L = \Glat p_A - \Greset p_L.

Here $\Gsig$TeX\Gsig is the signal-assisted activation rate, $\Gdark$TeX\Gdark is the dark-trigger rate, $\Glat$TeX\Glat is the latching rate into the record-bearing sector, $\Grel$TeX\Grel is the non-recording relaxation rate back to readiness, and $\Greset$TeX\Greset controls the reset path. The model is a certificate-level template for readout status; it is not a derivation of detector microphysics. This ready--activated--latched model is the declared signal-transfer class for the present paper, and all downstream readout claims are read only relative to it.

remark: Activation is not yet a record. The transition $R\to A$TeXR\to A is an activation event. The transition $A\to R$TeXA\to R is a failed transient. Only the transition $A\to L$TeXA\to L forms a durable record-bearing output class.

definition: Durable record on declared windows. For one declared outcome class, one declared reporting window $[0,\tauread]$TeX[0,\tauread], and one predeclared persistence window $0<\taupers\le \tauread$TeX0<\taupers\le \tauread, a trigger is counted as a durable record only if the corresponding latched output remains discriminable on a time interval of length $\taupers$TeX\taupers under the same threshold convention used for reporting and is compatible with the same platform's reset/dead-time architecture. All downstream readout-layer claims are read only relative to this declared reporting window, persistence window, threshold convention, and reset/dead-time architecture.

Local-commit current, record-formation efficiency, and event-accounting mismatch

definition: Local-commit current. The detector-facing local-commit current is the coarse-grained current into the latched record-bearing sector:

\Rc(t) \equiv \Glat p_A(t).

All downstream readout-rate and event-accounting claims are read only relative to this declared current on the same ready--activated--latched model and reporting convention.

The generic trigger or switching flux is instead

\Rsw(t) \equiv (\Gsig+\Gdark)p_R(t).

The two quantities need not coincide. A threshold event can activate the detector without producing a durable record.

proposition: Within the minimal ready--activated--latched model, durable-record current is $\Rc$TeX\Rc, not generic switching flux. For one declared outcome class and the same reporting convention as in the readout-layer definitions above, the detector-facing recorded-event current in the minimal model reference--reference is $\Rc$TeX\Rc, not $\Rsw$TeX\Rsw. In general, switching statistics need not coincide with durable recorded-event statistics.

proof. Only the transition $A\to L$TeXA\to L contributes to the durable record-bearing sector, so the recorded-event current is $\Rc=\Glat p_A$TeX\Rc=\Glat p_A. By contrast, $\Rsw$TeX\Rsw counts all activations, including activations that relax through $A\to R$TeXA\to R without producing a durable record. Therefore $\Rsw$TeX\Rsw and $\Rc$TeX\Rc are generically distinct.

In the quasi-steady activated sector, $\dot p_A\approx0$TeX\dot p_A\approx0, so

p_A \approx \frac{\Gsig+\Gdark}{\Glat+\Grel}\,p_R.

This motivates the record-formation efficiency

\etarec \equiv \frac{\Glat}{\Glat+\Grel},

which yields

\Rc \approx \etarec(\Gsig+\Gdark)p_R = \etarec\,\Rsw.

corollary: Quasi-steady window leading-order overcount relation. On declared quasi-steady windows with $\Rsw>0$TeX\Rsw>0 and $\Grel>0$TeX\Grel>0, the leading-order relation $\Rc \approx \etarec\,\Rsw$TeX\Rc \approx \etarec\,\Rsw holds with $0<\etarec<1$TeX0<\etarec<1. Accordingly, on that same declared quasi-steady approximation/window, raw switching statistics are a leading-order overcount proxy for durable recorded events rather than an exact durable-record count.

proof. If $\Grel>0$TeX\Grel>0, then $\etarec=\Glat/(\Glat+\Grel)$TeX\etarec=\Glat/(\Glat+\Grel) obeys $0<\etarec<1$TeX0<\etarec<1. The quasi-steady relation reference is therefore a leading-order relation with coefficient strictly below one, so under the declared quasi-steady approximation/window one reads $\Rc$TeX\Rc as smaller than the corresponding raw switching count. No exact pointwise inequality is claimed beyond that approximation/window.

On declared windows with $\Rsw>0$TeX\Rsw>0, the event-accounting mismatch is defined by

\DeltaEA \,\equiv\, 1-\frac{\Rc}{\Rsw}
\approx 1-\etarec.

If $\Rsw=0$TeX\Rsw=0, $\DeltaEA$TeX\DeltaEA is left undefined and no event-accounting mismatch is assigned on that window under the present event-accounting convention. Such a window is not a positive readout-certificate window unless a separate no-trigger reporting convention is explicitly declared. Thus $\DeltaEA$TeX\DeltaEA quantifies, under one declared event-accounting convention, the leading-order mismatch by which raw switching statistics can overcount durable recorded events on the admitted quasi-steady window. The quantity is especially relevant when dead time, afterpulsing, non-recording relaxation, or finite-window reporting distort the relation between trigger counts and durable recorded events [citation].

remark: Analytical role of $\DeltaEA$TeX\DeltaEA. The quantity $\DeltaEA$TeX\DeltaEA keeps event accounting from conflating threshold crossing, dark-triggered activation, failed transients, and stable readout. Any detector report that identifies recorded-event statistics with raw switching statistics must therefore state the convention under which $\DeltaEA$TeX\DeltaEA is neglected, bounded, or measured.

Threshold-only models and the joint readout-envelope test

A threshold crossing is at most one ingredient of a readout theory. By itself it does not determine persistence, afterpulse behavior, or reset compatibility. This motivates the following fail-closed test.

definition: Joint readout-envelope test. A threshold-only model is accepted as a full readout description only if it reproduces, for one and the same platform,

- trigger and inter-arrival statistics, - the dark-count / afterpulse envelope, - latched-output persistence on the declared window $\taupers$TeX\taupers, - and the reset/dead-time architecture on the same reporting convention.

If any of these fail, threshold crossing is an incomplete readout description even when trigger statistics are reproduced. All downstream full-readout claims are read only relative to this declared joint readout-envelope test on one platform and one reporting convention.

definition: Accepted readout certificate. For one declared outcome class, a readout certificate $\CR$TeX\CR is accepted on the declared reporting window $[0,\tauread]$TeX[0,\tauread] only if

- $\Rc>0$TeX\Rc>0 is reported or bounded away from zero on that window, - $0<\etarec\le 1$TeX0<\etarec\le 1 is reported or bounded on the same detector-envelope convention, - on any declared window with $\Rsw>0$TeX\Rsw>0, $\DeltaEA$TeX\DeltaEA is reported or conservatively upper-bounded on the same event-accounting convention; if $\Rsw=0$TeX\Rsw=0, no mismatch is assigned on that window unless a separate no-trigger reporting convention is explicitly declared, - and the joint readout-envelope test is passed for the same platform, including trigger/inter-arrival statistics, dark-count / afterpulse behavior, persistence on the declared $\taupers$TeX\taupers, and reset/dead-time compatibility.

All downstream readout and cross-platform claims are read only relative to this declared reporting window, persistence window, event-accounting convention, and accepted readout-envelope rule.

The rationale is experimentally standard. APD/SPAD detectors require quenching and recharge to suppress self-sustaining avalanches and dead-time distortions [citation], while SNSPD-class detectors require an electrothermal cycle whose latching and reset behavior constrains usable readout [citation]. A threshold event is therefore not yet a readout certificate.

Fragment inference and admissible diagnostics

Once a durable record exists, the next question is whether many observers can independently access the same pointer information from disjoint fragments. Let

\Ptr = \sum_z z\,\Pi_z

be the declared pointer observable, and let

\rho_S = \Tr_{\Env}\rho_{S\Env}

be the reduced state of the declared system. The corresponding pointer distribution is

p_z = \Tr(\Pi_z\rho_S),
\qquad
H(\Ptr) \equiv -\sum_{z:\,p_z>0} p_z \log p_z.

For each fragment $\Frag_a$TeX\Frag_a and each pointer label $z$TeXz with $p_z>0$TeXp_z>0, define the conditional fragment state

\rho_{\Frag_a}^{(z)} =
\frac{\Tr_S[(\Pi_z\otimes\mathbb{I}_{\Frag_a})\rho_{S\Frag_a}]}{p_z},
\qquad
\rho_{\Frag_a}=\sum_{z:\,p_z>0} p_z\rho_{\Frag_a}^{(z)}.

Zero-weight pointer labels are omitted from the conditional-fragment diagnostics. The probabilities $p_z$TeXp_z are those of the same declared pointer variable; fragment dependence enters through the conditional states and the induced fragment posteriors, not through a fragment-specific redefinition of the pointer distribution. The fragment classical information is the Holevo quantity

\Holevo(\Frag_a) =
\Svn(\rho_{\Frag_a}) - \sum_{z:\,p_z>0} p_z\,\Svn(\rho_{\Frag_a}^{(z)}),

and the discord leakage is

\Discord(\Frag_a) = I(S:\Frag_a)-\Holevo(\Frag_a).

To compare fragment inferences, let $p_a(z)$TeXp_a(z) and $p_b(z)$TeXp_b(z) denote fragment-conditioned posteriors for the pointer value and define the pairwise disagreement

\epsab = \frac12 \sum_z \abs{p_a(z)-p_b(z)}.

Pointer stability on the observation window is encoded by

\Sigptr = \sup_{t\in[0,\tauobs]} \sum_{z\neq z'} \left|\matrixel{z}{\rho_S(t)}{z'}\right|.

Because real environments can be inhomogeneous, the relevant fragment quantities are averaged over a declared family $\FragFam$TeX\FragFam of disjoint fragments of fixed size fraction $f$TeXf:

\AvgChi(f) = \mathbb E_{\Frag\in\FragFam}[\Holevo(\Frag)],

\AvgDisc(f) = \mathbb E_{\Frag\in\FragFam}[\Discord(\Frag)].

Only the diagnostics needed to type $\CO^{(1)}$TeX\CO^{(1)}--$\CO^{(3)}$TeX\CO^{(3)} are fixed below. The pointer variable, fragment family, observation window, and any surrogate map used for reporting are fixed before certificate evaluation; post hoc repartitioning is inadmissible [citation].

definition: Admissible witness or proxy. A reported witness or proxy $W_X$TeXW_X for a target quantity $X\in\{\AvgChi,\AvgDisc,\epsab,\Sigptr\}$TeXX\in\{\AvgChi,\AvgDisc,\epsab,\Sigptr\} is admissible only if the report states either

- an explicit bound of the form $X\le f(W_X)$TeXX\le f(W_X) or $X\ge g(W_X)$TeXX\ge g(W_X), or - a monotone calibration $X=h(W_X)$TeXX=h(W_X),

together with the calibration domain, uncertainty or tolerance, the same pointer variable, fragment family, and observation window, and any declared fragment-access rule required by the witness construction. Absent such a declaration, the witness or proxy does not certify $X$TeXX. All downstream witness-based objectivity claims are read only relative to this declared witness admissibility rule on the same pointer variable, fragment family, observation window, and fragment-access rule.

Redundancy, consensus, and the objectivity ladder

definition: Redundancy and consensus. For tolerances $0<\delta\ll1$TeX0<\delta\ll1 and $0<\epsilon\ll1$TeX0<\epsilon\ll1, a family of disjoint fragments $\{\Frag_a\}_{a=1}^m$TeX\{\Frag_a\}_{a=1}^m is called $\delta$TeX\delta-redundant for the pointer variable $\Ptr$TeX\Ptr if, for every $a$TeXa,

\Holevo(\Frag_a)\ge (1-\delta)H(\Ptr),
\qquad
\Discord(\Frag_a)\le \delta H(\Ptr).

The maximal number of disjoint fragments satisfying reference is the redundancy count $\Red$TeX\Red. The maximal number of disjoint fragments that are $\delta$TeX\delta-redundant and satisfy $\max_{a\neq b}\epsab\le\epsilon$TeX\max_{a\neq b}\epsab\le\epsilon is the consensus count $\Con$TeX\Con. All downstream objectivity and shared-fact claims are read only relative to these declared tolerances on the same pointer variable and fragment family.

Redundancy and consensus are distinct operational notions [citation]. Redundancy counts how many fragments encode the information nearly classically. Consensus counts how many such fragments also support agreement about the pointer value. The distinction matters whenever fragment quality is inhomogeneous or when mutual-information saturation coexists with significant discord leakage or disagreement [citation].

definition: Accepted objectivity certificates. Fix one declared pointer variable $\Ptr$TeX\Ptr, one declared fragment family $\FragFam$TeX\FragFam, one observation window $[0,\tauobs]$TeX[0,\tauobs], and one predeclared pointer-stability tolerance $\sigma_*\ge0$TeX\sigma_*\ge0. Then:

- $\CO^{(1)}$TeX\CO^{(1)} is accepted if $\AvgChi\ge(1-\delta)H(\Ptr)$TeX\AvgChi\ge(1-\delta)H(\Ptr), $\AvgDisc\le\delta H(\Ptr)$TeX\AvgDisc\le\delta H(\Ptr), $\Red\ge2$TeX\Red\ge2, and $\Sigptr\le \sigma_*$TeX\Sigptr\le \sigma_*. - $\CO^{(2)}$TeX\CO^{(2)} is accepted if $\CO^{(1)}$TeX\CO^{(1)} is accepted and, in addition, $\max_{a\neq b}\epsab\le\epsilon$TeX\max_{a\neq b}\epsab\le\epsilon and $\Con\ge2$TeX\Con\ge2 on the same fragment family. - $\CO^{(3)}$TeX\CO^{(3)} is accepted if a strong imported certificate---such as approximate spectrum broadcast structure or strong quantum Darwinism with strong independence---is reported on the same reduced state and fragment family with an explicit metric, tolerance, and fragment-access rule, and if that import is compatible with the same pointer declaration, fragment family, observation window, stability condition, and underlying $\CO^{(2)}$TeX\CO^{(2)}-level diagnostics.

All downstream shared-fact admissibility claims are read only relative to these declared certificate levels on the same pointer variable, fragment family, observation window, and stability tolerance.

remark: Certificate-only status of objectivity diagnostics. The diagnostics entering $\CO^{(1)}$TeX\CO^{(1)}--$\CO^{(3)}$TeX\CO^{(3)} are used only as certificate predicates on a declared pointer variable, fragment family, and observation window. They do not by themselves identify the full dynamical mechanism by which redundant encoding is formed.

proposition: Durable records are not yet shared facts. Within the declared objectivity ladder for one declared pointer variable, a durable local record is not sufficient for classical objectivity. Shared-fact status requires, in addition, redundant fragment encoding of one common pointer variable, low discord leakage, low pairwise disagreement, and pointer stability on the declared observation window.

proof. A durable record is a local readout property. Objectivity requires independent accessibility by many observers consulting disjoint fragments. Local persistence does not by itself guarantee high fragment classical information, low discord leakage, low disagreement, or stable pointer structure. Therefore durability is necessary but not sufficient for shared-fact status.

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The ladder in reference is asymmetric. The semantic and readout layers answer detector-facing questions; the objectivity layer answers shared-fact questions. The level $\CO^{(1)}$TeX\CO^{(1)} establishes redundancy only. Shared-fact admissibility starts at $\CO^{(2)}$TeX\CO^{(2)}. The level $\CO^{(3)}$TeX\CO^{(3)} is stronger, but only as an imported certificate with a declared approximation metric, tolerance, and fragment-access rule [citation].

remark: Meaning of $\CO$TeX\CO in the final acceptability rule. In the final acceptability rule, an accepted objectivity certificate $\CO$TeX\CO means $\CO\in\{\CO^{(2)},\CO^{(3)}\}$TeX\CO\in\{\CO^{(2)},\CO^{(3)}\}. The level $\CO^{(1)}$TeX\CO^{(1)} is not sufficient for shared-fact status.

Broadcast versus encoding phases

Recent work sharpens a distinction that matters for certificate logic: decohering environments need not be information-broadcasting environments. Branching-state structure identifies global-state organization compatible with low discord and redundant classical accessibility [citation]. By contrast, broadcast-versus-encoding transitions in many-body and circuit models show that environments can scramble or delocalize information rather than supply small, redundant, independently readable fragments [citation]. Accordingly, decoherence-only signatures are not accepted as objectivity certificates unless fragment accessibility, low disagreement, low discord leakage, and pointer stability are also established.

Measurement-chain characterization tuple

The three layers are certified separately before a shared-fact claim is admitted. This motivates the following reporting object.

definition: Measurement-chain characterization tuple. For a declared pointer variable $\Ptr$TeX\Ptr, together with declared outcome labels, reporting windows, persistence windows, and threshold conventions, the measurement-chain characterization tuple is

\mathfrak{C}_{\mathrm{meas}}
=
\bigl(\{\Pi_m\},\,\Rc,\,\etarec,\,\DeltaEA,\,\AvgChi,\,\AvgDisc,\,\Red,\,\Con,\,\Sigptr\bigr),

possibly together with platform-specific envelope observables such as inter-arrival distributions, dark-count or afterpulse envelopes, persistence traces, and reset/dead-time observables. All downstream cross-layer and cross-platform composition claims are read only relative to this declared tuple for the same pointer variable, outcome class, and reporting convention.

The tuple is a declared cross-layer reporting object for one candidate outcome class. It is not a raw single-experiment observable vector.

definition: Modular certificate composability. Certificates established on different declared platform classes are compositionally admissible only if each imported platform declares how its native outcome labels or surrogate observable map onto the same declared pointer variable $\Ptr$TeX\Ptr, or onto the same declared coarse-grained surrogate of that pointer variable, and if the outcome labels, reporting windows, persistence windows, threshold conventions, fragment-access rules, and any imported approximation metrics or tolerances are reporting-compatible across the imported layers. For any composed claim, those declarations are fixed before certificate composition and are not assigned post hoc. All downstream cross-platform composition claims are read only relative to this declared composability rule and the fixed pointer, outcome-label, reporting-window, persistence-window, threshold, fragment-access, and approximation conventions.

remark: Type-level rather than token-level composition. Compositional admissibility is a type-level claim. It certifies one declared outcome class under one declared pointer map. It does not assert numerical identity matching of individual detection tokens across distinct devices.

remark: Admitted surrogate variables. A declared coarse-grained surrogate is admissible only when the coarse-graining is stated explicitly, preserves the reported outcome class across the imported layers, and does not reverse the acceptance ordering of an imported certificate on the declared reporting convention.

proposition: Type-level cross-platform certificate composition for one declared pointer variable and one declared outcome class. For the declared measurement-chain characterization tuple and the declared composability rule above, the entries of $\mathfrak{C}_{\mathrm{meas}}$TeX\mathfrak{C}_{\mathrm{meas}} may be assembled from distinct declared platform classes only as a logical composition of certificates for that same declared pointer variable and that same declared outcome class. Such composition is admissible only under the composability conditions above and does not imply that a single device realizes the entire certificate ladder.

proof. The semantic, readout, and objectivity layers are anchored to different diagnostics and may be realized on different platform classes. Without an explicit map from each imported platform's native labels or surrogate observable to the same pointer variable, together with reporting-compatible conventions, the imported entries do not refer to one outcome class and cannot support one acceptability claim. When those declarations are fixed, the result is a logical certification claim rather than a single-device realizability statement.

Final acceptability rule

proposition: Within the present certification scheme, shared-fact acceptability for one declared outcome class requires all three certificates. For one declared outcome class and the fixed certificate predicates introduced above, a shared-fact acceptability claim for that outcome class is admitted only if the chain carries an accepted semantic certificate $\CM$TeX\CM, an accepted readout certificate $\CR$TeX\CR in the sense of reference, and an accepted objectivity certificate $\CO\in\{\CO^{(2)},\CO^{(3)}\}$TeX\CO\in\{\CO^{(2)},\CO^{(3)}\} in the sense of reference. Neither $\CM$TeX\CM alone, nor $\CM\wedge\CR$TeX\CM\wedge\CR, nor $\CM\wedge\CR\wedge\CO^{(1)}$TeX\CM\wedge\CR\wedge\CO^{(1)} is sufficient.

proof. By reference, $\CM$TeX\CM does not certify durable records or shared facts. By reference, $\CM\wedge\CR$TeX\CM\wedge\CR does not certify shared facts because a durable record need not be redundantly and consensually accessible from disjoint fragments. The level $\CO^{(1)}$TeX\CO^{(1)} establishes redundancy only. Shared-fact status begins at $\CO^{(2)}$TeX\CO^{(2)}, or at a stronger imported level $\CO^{(3)}$TeX\CO^{(3)} satisfying the same compatibility requirements. Therefore a shared-fact claim requires $\CM\wedge\CR\wedge\CO$TeX\CM\wedge\CR\wedge\CO with $\CO\in\{\CO^{(2)},\CO^{(3)}\}$TeX\CO\in\{\CO^{(2)},\CO^{(3)}\}.

A measurement chain can therefore fail in three distinct ways:

- semantic failure, because the declared measurement semantics is not realized; - readout failure, because triggers do not reliably become durable records on the declared reporting convention; - objectivity failure, because durable records do not become redundant shared facts.

Conflating these failures hides diagnostics that are experimentally distinguishable.

Semantic anchors

The semantic layer is anchored by detector tomography and by modern measurement-characterization protocols. Quantum detector tomography remains the standard route to POVM-level characterization [citation]. Recent work on mid-circuit measurement benchmarking and precision bounds for detector characterization sharpens the same layer by quantifying readout semantics and measurement-induced error without thereby certifying durability or objectivity [citation]. These are the natural anchors for $\CM$TeX\CM.

Readout anchors

The readout layer is anchored by detector studies in which persistence, afterpulsing, dead time, latching, reset, and finite-window effects are explicit observables. For APD/SPAD families, quenching, afterpulse statistics, and finite-window calibration are central [citation]. For SNSPD-class and related superconducting detectors, electrothermal latching, reset, and trace-resolved output structure are central [citation]. These observables anchor $\CR$TeX\CR.

Objectivity anchors

The objectivity layer is anchored by experiments and theory on redundant environmental encoding. Photonic cluster-state and photonic-simulator experiments supply fragment-wise information-saturation and redundancy anchors [citation]; nitrogen-vacancy and superconducting-circuit experiments provide direct demonstrations of redundant information becoming independently accessible in controllable environments [citation]. Theoretical work sharpens the same layer by distinguishing redundancy from consensus, by identifying branching-state structure, and by deriving Hamiltonian and measurement-compatibility criteria for the emergence of classical objectivity [citation]. These are the natural anchors for $\CO$TeX\CO.

Reporting protocols

Readout-layer protocol..

Collect time-tagged trigger and output traces on a declared reporting convention; infer or bound $\Gsig,\Gdark,\Glat,\Grel,\Greset$TeX\Gsig,\Gdark,\Glat,\Grel,\Greset; evaluate $\Rc$TeX\Rc, $\etarec$TeX\etarec, and $\DeltaEA$TeX\DeltaEA; and test whether a threshold-only model reproduces trigger/inter-arrival statistics, the dark-count / afterpulse envelope, persistence on $\taupers$TeX\taupers, and reset/dead-time behavior.

Objectivity-layer protocol..

Choose a candidate pointer variable; fix a fragment family, fragment-access rule, and observation window; estimate $\AvgChi$TeX\AvgChi and $\AvgDisc$TeX\AvgDisc or calibrated bounds thereto; evaluate pairwise disagreement and pointer stability or calibrated witnesses thereof; and place the platform on the ladder in reference.

Composition protocol..

When the layers are imported from different platform classes, declare the pointer map, outcome-class map, reporting window, persistence window, threshold convention, fragment-access rule, and any surrogate variable before composing the certificates. Any imported detector-performance figure or benchmark is reported together with its characterization protocol and operating window [citation]. The result is a type-level admissibility claim for one declared outcome class.

The boundary conditions divide into two classes. The conditions F0--F0b are inadmissibility conditions: they block certificate import or substitution. The conditions F1--F6 are boundary-collapse tests: if any of them succeeds, the three-layer separation loses discriminative force.

F0. Pointer-identity and reporting incompatibility..

If semantic, readout, and objectivity certificates cannot be mapped to one declared pointer variable and one reporting-compatible outcome class by explicit platform-to-pointer maps or declared surrogate maps, the composed measurement-chain tuple is inadmissible.

F0a. Uncalibrated witness substitution..

If a witness or proxy is reported for $\AvgChi$TeX\AvgChi, $\AvgDisc$TeX\AvgDisc, $\epsab$TeX\epsab, or $\Sigptr$TeX\Sigptr without an explicit bound or monotone calibration on the same pointer variable, fragment family, and observation window, it does not certify that diagnostic.

F0b. Unspecified imported approximation standard..

If an approximate SBS or strong-quantum-Darwinism certificate is invoked without an explicit metric, tolerance, and fragment-access rule on the same declared state and fragment family, $\CO^{(3)}$TeX\CO^{(3)} is inadmissible.

F1. Semantic sufficiency..

If a semantic certificate alone determines readout persistence, event accounting, and objectivity diagnostics without any additional readout-layer or fragment-layer observables, the layer separation collapses and the three-layer scheme ceases to distinguish measurement semantics, durable record formation, and shared-fact status.

F2. Threshold-only readout sufficiency..

If a threshold-only model reproduces not only trigger statistics but also persistence on $\taupers$TeX\taupers, the dark-count / afterpulse envelope, and the reset/dead-time architecture of the same platform, the readout layer loses discriminative force.

F3. No operational distinction between activation and durable record..

If $\Rc$TeX\Rc, $\etarec$TeX\etarec, and $\DeltaEA$TeX\DeltaEA cannot distinguish generic activation from durable record-bearing events in any realistic platform class, the readout layer reduces to relabeling.

F4. Decoherence without fragment accessibility..

If a platform exhibits suppressed pointer coherences but no redundant fragment accessibility, no low-disagreement consensus, and no admitted strong imported certificate, the objectivity layer must not be certified.

F5. Redundancy without consensus..

If a claimed objectivity signature reduces to mutual-information or Holevo saturation while discord leakage or inter-fragment disagreement remain large, the objectivity certificate is not satisfied.

F6. Broadcast-versus-encoding indistinguishability..

If environments that redundantly broadcast pointer information cannot be operationally distinguished from environments that merely scramble or encode it nonlocally, the objectivity layer loses bite.

The boundary conditions above keep semantic, readout, and objectivity claims fail-closed and experimentally noninterchangeable.

Detector tomography answers a semantic question. Readout certification answers a durability question. Objectivity certification answers a shared-fact question. These are distinct operational layers and none is reducible to the others without loss of observable structure.

The readout layer is characterized by the local-commit current $\Rc$TeX\Rc, the record-formation efficiency $\etarec$TeX\etarec, the event-accounting mismatch $\DeltaEA$TeX\DeltaEA, persistence on a declared window $\taupers$TeX\taupers, and the full readout envelope. The objectivity layer is characterized by fragment-wise classical information, discord leakage, disagreement, redundancy, consensus, and pointer stability, together with imported strong certificates only when their approximation metric, tolerance, and fragment-access rule are declared. A trigger is not yet a durable record, and a durable record is not yet a shared fact.

Shared-fact admissibility is therefore fail-closed. One declared outcome class is admitted only when it carries a semantic certificate, a readout certificate, and an objectivity certificate at level $\CO^{(2)}$TeX\CO^{(2)} or stronger. Cross-platform composition remains admissible only under explicit pointer maps, reporting-compatible conventions, fragment-access rules, and declared imported tolerances, and only as a type-level claim. No detector microphysics, no de Broglie or wavefunction-semantics claim, and no dynamical source law for classical objectivity are derived here.

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Funding and competing interests..

No external funding was received for this work. The author declares no competing interests.