A Minimal Reconstruction of Constraint-Based Realization from the Burdens of a Quantum Outcome Law

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Robert Duran IV
Independent Researcher
www.robertduraniv.com

Version 1.0
April 2026

Citation: Duran IV, Robert. A Minimal Reconstruction of Constraint-Based Realization from the Burdens of a Quantum Outcome Law. Version 1.0, April 2026.

Keywords:

Constraint-Based Realization; CBR; quantum foundations; measurement problem; realization law; Born rule; decoherence; quantum channels; operational equivalence; admissibility; Robert Duran IV

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Abstract

Standard quantum mechanics supplies state spaces, amplitudes, dynamical evolution, measurement statistics, and Born-rule probabilities, but it does not by itself state a separate law-form for individual outcome realization. This paper reconstructs Constraint-Based Realization (CBR) as a minimal structural form for such a law-candidate. The reconstruction is conditional rather than empirical: it does not claim that CBR is experimentally confirmed, that standard quantum mechanics is replaced, that the Born rule is derived here, or that decoherence is false. CBR asks a narrower question: if one seeks a disciplined law-form for individual outcome realization, what structure must such a law possess in order to be non-circular, probability-compatible, distinct from non-selective decoherence, parameter-fixed, and vulnerable to failure?

The answer is a burden-to-structure reconstruction. A viable realization-law candidate must specify a physically defined measurement context C, a nonempty admissible candidate class 𝒜(C), a context-fixed realization-burden functional ℛ_C, an operational equivalence relation ≃_C, pre-outcome parameter discipline, Born-compatible ensemble behavior, non-reduction to ordinary decoherence, and explicit defeat conditions. Under these requirements, any such law admits the CBR-form representation:

Φ∗_C ∈ argmin{ℛ_C(Φ) : Φ ∈ 𝒜(C)}

In this representation, the selected realization channel Φ∗_C is not introduced as a free postulate or retrospective label. It is the minimizer, or operational equivalence class of minimizers, of a pre-specified burden functional over context-admissible candidates. The reconstruction therefore shows that CBR is not merely an asserted interpretive stance, but the natural representational form of a candidate realization law satisfying the stated burdens.

A minimal two-path model demonstrates how the reconstructed objects can be instantiated without replacing Born-rule weighting or collapsing realization into a non-selective decoherence-compatible channel. The paper then states a Failure Theorem: a CBR-form model fails in context C if its context, admissible class, functional, equivalence relation, parameters, minimizer structure, Born-compatible behavior, non-reduction condition, or vulnerability condition fails. The result is not confirmation of CBR as a physical law. It is a disciplined reconstruction of CBR as a formally viable, non-circular, probability-compatible, non-reductive, and failure-capable law-form candidate for quantum outcome realization.

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1. Introduction

Quantum mechanics is among the most successful theoretical frameworks in modern science. In its standard form, it supplies state spaces, dynamical evolution, amplitudes, observables, outcome spaces, and Born-rule probabilities. For a given measurement context, it identifies which outcomes are available and how those outcomes are statistically weighted across repeated trials. This predictive structure is not disputed in this paper.

The question addressed here is narrower and more specific. It concerns not the empirical success of quantum mechanics, nor the operational reliability of its measurement formalism, but the structural form of any candidate law that purports to describe individual outcome realization. Standard quantum mechanics supplies probabilities for outcomes. It does not, by itself, state a separate law-form explaining why one admissible outcome-structure is realized in an individual measurement context. That is the target of the present reconstruction.

The central question can therefore be stated directly: If one seeks a disciplined law-form for individual quantum outcome realization, what structure must such a law possess in order to be non-circular, probability-compatible, distinct from non-selective decoherence, parameter-fixed, operationally meaningful, and vulnerable to failure?

This paper reconstructs Constraint-Based Realization, or CBR, as a minimal structural answer to that question. The reconstruction is conditional. It does not claim that CBR is experimentally confirmed. It does not claim that standard quantum mechanics is replaced. It does not claim that the Born rule is derived in this paper. It does not claim that decoherence is false. It asks a narrower and more evaluable question: once the burdens of a disciplined outcome-realization law are made explicit, what representational form follows?

The answer developed here is that a candidate realization law must have at least four structural components: context → admissible candidates → burden ordering → selected realization class.

In CBR notation, this structure is expressed as:

Φ∗_C ∈ argmin{ℛ_C(Φ) : Φ ∈ 𝒜(C)}

Here C denotes a physically specified measurement context, 𝒜(C) denotes the nonempty admissible class of realization-compatible candidates in that context, ℛ_C denotes a context-fixed realization-burden functional, and Φ∗_C denotes the selected realization channel or selected operational equivalence class. The equation does not assert that nature has been shown to obey CBR. It states the representational form taken by any candidate realization law that selects non-circularly from admissible candidates under fixed context-relative constraints.

The contribution of this paper is therefore reconstructive rather than empirical. It does not attempt to confirm CBR as a physical law. It shows why the CBR form naturally arises when one imposes the minimum burdens required of a serious candidate law of individual outcome realization.

1.1 Probability and realization

The Born rule assigns probability weights to measurement outcomes. For a state written schematically as:

|ψ⟩ = Σᵢ αᵢ|i⟩,

standard quantum mechanics assigns:

P(i) = |αᵢ|².

This rule governs ensemble statistics. It tells us how outcomes are weighted across repeated trials. It is one of the central empirical constraints any candidate law of quantum measurement must respect.

A probability rule, however, is not identical to a law-form of individual realization. Probability assigns weights to possible outcomes. A realization law, if one is sought, would specify how one admissible outcome-channel, record-structure, or operational realization class is selected in a particular physical context. The two questions are connected, but they are not the same.

CBR is not introduced here as a replacement for Born-rule weighting. In the present reconstruction, Born compatibility is imposed as a condition on any acceptable realization-law candidate. A CBR-form model that violates Born-rule frequencies without a controlled, pre-specified, empirically vulnerable deviation fails the standard used in this paper.

The distinction is therefore precise: Born weighting governs ensemble frequencies. Realization concerns context-relative selection of an admissible outcome-structure.

CBR is addressed to the second question while remaining constrained by the first.

1.2 Decoherence and realization

Decoherence explains how interference between alternatives becomes suppressed through interaction with environmental or record-bearing degrees of freedom. It also explains why stable pointer-like structures and effectively classical records emerge in ordinary measurement contexts. These are essential contributions to the modern understanding of measurement.

This paper does not reject decoherence. It depends on the distinction between decoherence and realization.

A non-selective decoherence-compatible map can describe interference suppression, environmental entanglement, record stabilization, and effective mixture structure in a reduced description. But if one asks which admissible record-structure is realized in an individual context, decoherence alone does not automatically state a separate selection law-form. It explains why alternatives may cease to interfere in accessible descriptions. It does not, by that fact alone, identify a context-fixed rule selecting one realization-compatible outcome-channel.

CBR is addressed only to that residual law-candidate question. If CBR merely renames decoherence, it fails as an independent realization law. If, however, it supplies a selection structure over admissible realization-compatible candidates that is not exhausted by a non-selective decoherence-compatible map, then it occupies a distinct formal role.

This non-reduction requirement is not a defensive slogan. It is a defeat condition. A critic can challenge CBR by showing that the selected structure Φ∗_C adds no realization content beyond the non-selective decoherence-compatible channel Φ_mix in the relevant context. If that challenge succeeds, CBR fails as an independent law-candidate in that context.

1.3 Reconstruction rather than assertion

The method of this paper is reconstruction. It does not begin by assuming that CBR is true. It begins by asking what any disciplined candidate law of quantum outcome realization would have to contain.

Such a law must define its domain. It must specify what is being selected. It must restrict admissible candidates. It must compare candidates by a criterion fixed before the outcome is known. It must identify a selected result or selected operational equivalence class. It must preserve Born-compatible ensemble behavior unless a controlled deviation is explicitly declared. It must avoid reducing entirely to non-selective decoherence. It must prevent post hoc parameter tuning. It must state the conditions under which it fails.

Once these requirements are made explicit, the CBR structure is no longer an arbitrary equation placed on top of the measurement problem. It is the natural burden-to-structure form of a disciplined realization-law candidate:

C → 𝒜(C) → ℛ_C → Φ∗_C

The claim is not that CBR is forced by pure logic across all conceivable metaphysical frameworks. The claim is more precise: within the representation class considered here, any candidate law satisfying the stated burdens admits a CBR-form representation, either directly through a context-fixed burden functional or through an admissibility ordering representable as such a functional under finite or regular quotient conditions.

That qualification matters. It prevents overclaiming while preserving the paper’s central force. The reconstruction does not prove that CBR is true. It shows that CBR is structurally natural once one demands a non-circular, probability-compatible, non-reductive, failure-capable law-form for individual outcome realization.

1.4 Program placement

This paper is the reconstruction companion to Constraint-Based Realization: Canonical Closure and Exact Empirical Exposure. The canonical closure paper states the mature CBR law-candidate in closed form: a restricted admissible class, canonical realization functional, operational uniqueness structure, local probability-closure result, accessibility parameter, empirical exposure model, nuisance-separation burden, and strong-null failure condition.

The present paper has a different role. It asks why a law-form of that general kind is structurally motivated in the first place. Its purpose is not to repeat the canonical closure theorem, derive the Born rule, or specify a full platform-level experiment. Its purpose is to show that the CBR representation follows from the minimal burdens imposed on any disciplined candidate law of quantum outcome realization.

The relationship is therefore: This reconstruction paper asks why the CBR form arises. The canonical closure paper states the completed CBR law-candidate. The empirical exposure papers specify where the law-candidate becomes vulnerable to test.

This paper is prior in logical order even if it is subsequent in publication order. It supplies the structural rationale for the canonical form.

1.5 Main result

The central theorem of the paper is the Conditional Minimal Representation Theorem for Constraint-Based Realization.

Let C be a physically specified measurement context. Let 𝒜(C) be a nonempty admissible candidate class. Let ℛ_C be a context-fixed realization-burden functional, or an admissibility ordering representable as such a functional under finite or regular quotient conditions. Let ≃_C be a context-fixed operational equivalence relation.

If a candidate law of quantum outcome realization satisfies context specification, candidate-class fixity, admissibility restriction, non-circular comparison, pre-outcome parameter fixity, operational uniqueness, Born compatibility, non-reduction to ordinary decoherence, and explicit vulnerability to failure, then it admits the CBR-form representation:

Φ∗_C ∈ argmin{ℛ_C(Φ) : Φ ∈ 𝒜(C)}

The selected realization channel Φ∗_C is therefore not a label attached to whichever outcome occurs. It is the minimizer, or operational equivalence class of minimizers, of a pre-specified burden structure over admissible candidates in context C.

This theorem establishes representational form under stated assumptions. It does not establish empirical truth. It does not identify a final universal physical ℛ_C. It does not prove that every possible measurement context admits a unique CBR solution. It shows that CBR is the minimal representational structure naturally associated with a realization-law candidate satisfying the burdens defined in this paper.

1.6 What this paper does not claim

The force of the reconstruction depends on its limits. Those limits are therefore stated explicitly.

This paper does not claim that CBR is experimentally confirmed. It does not claim that CBR is established physics. It does not claim that standard quantum mechanics is replaced. It does not claim that the Born rule is replaced. It does not claim that the Born rule is derived in this paper. It does not claim that decoherence is false. It does not claim that rival interpretations are defeated. It does not claim that the two-path model developed later is a universal theory of all measurement contexts. It does not claim that nature obeys CBR merely because the formal structure can be reconstructed.

The claim is narrower and stronger: CBR is reconstructable as the minimal representational structure of a disciplined candidate law of outcome realization under explicit assumptions.

The next question is therefore not whether the CBR form can be written coherently. This paper answers that at the structural level. The next question is whether context-specific versions of C, 𝒜(C), ℛ_C, and ≃_C can be independently specified, operationalized, and exposed to structural or empirical defeat. That question belongs to the wider CBR program.

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2. The Burdens of a Quantum Outcome Law

A proposed law of quantum outcome realization cannot merely say that “constraints select the outcome.” Such a statement may identify the motivating intuition, but it does not yet define a law-form. It does not specify the domain of application, the objects among which selection occurs, the admissibility conditions those objects must satisfy, the comparison rule by which candidates are ordered, the status of quantum probability, the relation to decoherence, or the conditions under which the proposal would fail.

The purpose of this section is to state the minimum burdens that any disciplined candidate law of individual outcome realization must carry before it can be evaluated as a serious physical proposal. These burdens are not rhetorical additions to CBR. They are the structural requirements that make any such law non-circular, operationally meaningful, probability-compatible, non-reductive, parameter-disciplined, and vulnerable to defeat.

A candidate realization law must answer at least nine questions:

What is its physical domain?
What objects does it select among?
What makes those objects admissible?
How are candidates compared?
Why is the comparison rule non-circular?
In what sense is the selected result unique?
How are Born-rule statistics preserved?
How is the proposal distinct from decoherence?
What would count as failure?

The burdens below define the standard from which the CBR reconstruction proceeds. Their role is prior to the CBR equation itself. The paper does not begin by assuming the CBR form and then defending it. It first identifies the minimum structure any disciplined realization-law candidate must possess. The next section then shows that these burdens naturally reconstruct the CBR form.

2.1 Burden 1: Domain

A realization law must define the physical contexts to which it applies. Without a defined domain, the law has no determinate target.

Let C denote a measurement context. In this paper, C is not merely the name of an observable, a basis label, or a verbal description of an experiment. It denotes the physical and operational structure relevant to the realization question. A context C may include the measured system, the measurement architecture, the relevant Hilbert space, pointer or record-bearing degrees of freedom, environmental couplings, timing relations, accessibility conditions, coarse-graining scale, and operational readout limitations.

The domain burden prevents a realization claim from floating above the physical setup. A law that does not specify where it applies cannot define admissible candidates, cannot state a comparison rule, cannot identify operational equivalence, and cannot expose itself to failure. The context C is therefore the first required object in the reconstruction.

This burden also blocks a subtle form of circularity. If C can be re-described after the outcome is known, then the theory can be made to appear successful by retrospectively defining the context in a way that favors the realized result. For that reason, C must be specified independently of the selected realization channel Φ∗_C.

2.2 Burden 2: Candidate Set

A realization law must specify what it selects among. Selection is meaningless without candidates.

Let 𝒜(C) denote the admissible candidate class associated with context C. The elements of 𝒜(C) may be represented as realization-compatible channels, record structures, or context-relative outcome maps. This paper uses channel language because it gives the reconstruction a disciplined quantum-operational representation: candidates can be treated as maps from pre-realization state structure to post-realization record or outcome structure.

The candidate-set burden prevents the theory from treating “the outcome” as an undefined target. If CBR claims that a realization structure is selected, it must identify the class from which selection occurs. That class is 𝒜(C).

The candidate class must also be fixed independently of the realized outcome. A model cannot observe an outcome and then define 𝒜(C) so that only that outcome was ever eligible. Such a maneuver would not explain realization. It would conceal the result inside the candidate set. A candidate law must select from a class; it cannot manufacture the class from the selected result.

2.3 Burden 3: Admissibility

Not every mathematically writable candidate should count as physically admissible. A candidate may be expressible as a formal object while failing to correspond to any legitimate realization-compatible structure in the context.

Admissibility must therefore be constrained by C. Candidate structures should satisfy context-relative conditions such as physical implementability, compatibility with the measurement architecture, record-structural coherence, dynamical consistency, probability compatibility, operational accessibility, and pre-outcome independence.

This burden is essential because unrestricted candidate spaces make selection arbitrary. If every imaginable channel, map, branch description, or record assignment is admissible, then a selection law can be engineered to select anything. Conversely, if admissibility is defined only after the result is known, the law becomes retrospective.

In the CBR reconstruction, 𝒜(C) is not a list of preferred outcomes. It is the set of candidates that survive pre-specified physical and operational constraints. The law does not first choose the outcome and then declare it admissible. It first defines the admissible class and only then compares candidates within it.

Admissibility is therefore the first anti-arbitrariness filter in the reconstruction.

2.4 Burden 4: Non-Circular Selection

A realization law must not depend on the outcome it is supposed to explain.

The following objects must be fixed before the selected realization channel is known:

C,

𝒜(C),
ℛ_C,
≃_C,
ε_C,
and any parameters, tolerances, thresholds, or weights appearing in ℛ_C.

The forbidden move is to observe a result and then tune the selection rule so that the observed result receives the lowest burden. Such a rule does not explain realization. It merely relabels the realized outcome as the one the theory “would have selected.”

A realization law becomes circular if it says, in effect: this result happened because the rule favors the result that happened. To avoid that failure, the context, candidate class, comparison rule, operational equivalence relation, tolerance structure, and parameter values must be fixed independently of Φ∗_C.

The non-circularity burden is therefore the anti-cheating condition of the reconstruction. It is what prevents CBR from becoming a retrospective scoring device. A CBR-form model may still be wrong after this burden is satisfied, but it cannot be dismissed merely as a post hoc label if these objects are genuinely fixed before outcome comparison.

2.5 Burden 5: Operational Uniqueness

A realization law must identify one physically meaningful selected structure. This does not always require strict formal uniqueness. It requires operational uniqueness.

Let M_C denote the minimizer set of admissible candidates once a comparison rule has been specified:

M_C = {Φ ∈ 𝒜(C) : ℛ_C(Φ) = inf{ℛ_C(Ψ) : Ψ ∈ 𝒜(C)}}.

A law succeeds only if M_C contains a single selected candidate, or if all candidates in M_C are equivalent under the operational equivalence relation ≃_C, or if a pre-specified tie rule resolves the degeneracy.

Let

Φ₁ ≃_C Φ₂

mean that Φ₁ and Φ₂ are operationally equivalent in context C.

The equivalence relation ≃_C is not a technical afterthought. It is necessary because formal multiplicity does not always imply physically meaningful multiplicity. Two channels may differ as mathematical representatives while remaining indistinguishable by every accessible test or readout procedure available in C.

The burden is therefore not absolute syntactic uniqueness. It is uniqueness at the level of operationally meaningful realization. If M_C contains operationally distinct candidates and no pre-outcome tie rule exists, the model has not selected one realization class and is incomplete in that context.

Operational uniqueness protects the theory from two opposite errors: claiming false determinacy where the minimization rule has not selected, and hiding unresolved degeneracy behind formal notation.

2.6 Burden 6: Probability Compatibility

A realization law must not casually break the Born rule.

For an input state

|ψ⟩ = Σᵢ αᵢ|i⟩,

standard quantum mechanics assigns

P(i) = |αᵢ|².

This rule is empirically central. A candidate law of outcome realization must preserve Born-rule ensemble frequencies unless it explicitly introduces a controlled, pre-specified, empirically vulnerable deviation.

In this reconstruction, Born compatibility is treated as a default burden. The paper does not derive the Born rule, replace the Born rule, or reinterpret probability as realization. Instead, it requires that a candidate realization law coexist with standard quantum probability unless a separate deviation claim is formally stated and exposed to test.

This burden also prevents a category error. Selection and sampling are not the same. CBR does not ask probability to do realization’s job, and it does not ask realization to do probability’s job. The realization structure concerns admissible outcome-channels in a context. Born weighting concerns the frequencies with which outcomes occur across repeated equivalent contexts.

A CBR-form model that violates Born frequencies without an independently specified deviation does not become more radical. It fails the probability-compatibility burden.

2.7 Burden 7: Non-Reduction to Decoherence

A realization law must do more than describe decoherence.

Decoherence explains interference suppression, environmental entanglement, record stabilization, pointer-like behavior, and effective classicality in reduced descriptions. None of that is rejected here. The issue is whether non-selective decoherence alone supplies a law-form of individual realization selection.

A non-selective decoherence-compatible channel can describe the emergence of mixture-like structure. But it does not, by itself, identify which admissible record-structure or outcome-channel is selected as realized in an individual context. If a theory claims to supply a realization law, it must specify what selection structure it adds beyond non-selective decoherence-compatible evolution.

Let Φ_mix denote a non-selective decoherence-compatible channel. Let Φ∗_C denote the selected realization channel or selected operational equivalence class. CBR remains independent only if Φ∗_C supplies realization content not exhausted by Φ_mix.

If, in every relevant context,

Φ∗_C ≃_C Φ_mix,

and no further realization content is supplied, then CBR reduces to decoherence and fails as an independent realization law. This is not a criticism of decoherence. It is a burden on CBR. The theory must either supply distinct realization content or concede reduction.

2.8 Burden 8: Parameter Fixity

A realization functional cannot be an adjustable scoring device tuned after the outcome is known.

Any terms, tolerances, coefficients, thresholds, or weights appearing in ℛ_C must be fixed before outcome comparison. If parameters are estimated from calibration data, the calibration protocol must also be specified before testing the realization claim. Calibration may inform a model; post hoc outcome-fitting may not.

This burden is essential because a sufficiently flexible ℛ_C can be adapted to favor almost any result. Without parameter fixity, the functional does not function as a law. It becomes a retrospective fitting machine.

Parameter fixity does not require that every CBR-form model be parameter-free. It requires that parameters be fixed by theory, context, or pre-specified calibration before they are used to identify Φ∗_C. If the values are calibrated, the calibration data must be separated from the outcome test. If the values are theoretical, the theory must specify them before the result is known. If the values are left adjustable until after observation, the model fails as a non-circular realization law.

2.9 Burden 9: Empirical or Structural Vulnerability

A candidate law must be able to fail.

Failure may be empirical or structural. A model fails structurally if its domain is undefined, if its candidate class is empty or arbitrary, if its admissibility conditions are post hoc, if its functional is circular, if its minimizer set is empty, if its selected minimizers are operationally distinct with no pre-specified tie rule, if its parameters are tuned after the result, if it violates Born-rule frequencies without a registered deviation claim, or if it reduces entirely to non-selective decoherence.

A model fails empirically if it makes a pre-specified empirical burden and the relevant experiment satisfies the declared detectability conditions while producing the declared null or contrary result.

A law-candidate gains credibility not by avoiding failure conditions, but by stating them. A proposal that cannot fail is not made stronger by that immunity. It is made less evaluable.

The CBR reconstruction therefore treats vulnerability as a burden rather than as an optional later addition. A CBR-form model must state what would count against it. The possibility of failure is not an embarrassment to the framework. It is what makes the framework available for serious evaluation.

2.10 The Burden-to-Structure Transition

Once these burdens are accepted, a generic realization law must have a recognizable structure.

It must have a context because it needs a domain.

It must have a candidate class because it selects among possibilities.

It must have admissibility restrictions because not every formal candidate is physical.

It must have a comparison rule because selection requires evaluation.

It must have a selected result or selected equivalence class because the law must identify a realization structure.

It must have an operational equivalence relation because uniqueness may be context-relative.

It must control parameter tuning because otherwise selection can become post hoc.

It must preserve Born-rule statistics because quantum probabilities are empirically fixed.

It must distinguish realization from decoherence because non-selective decoherence alone is not a selection law.

It must have failure conditions because a candidate law must be vulnerable.

This is already the skeleton of CBR.

The structure can be written compactly as:

C → 𝒜(C) → ℛ_C → M_C / ≃_C → Φ∗_C

In words: a physically specified context determines an admissible candidate class; candidates are compared by a context-fixed burden structure; minimizers are identified; operational equivalence removes physically irrelevant multiplicity; and the selected realization channel or selected realization class is obtained.

Equivalently, the reconstruction points toward the CBR-form representation:

Φ∗_C ∈ argmin{ℛ_C(Φ) : Φ ∈ 𝒜(C)}

This equation is not introduced as decorative formalism. It is the compressed expression of the burdens above. The next section shows how those burdens reconstruct the CBR form rather than merely motivating it.

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3. From Law Burdens to CBR Structure

The preceding section identified the burdens any disciplined candidate law of quantum outcome realization must satisfy. This section shows that those burdens do not merely motivate CBR. They reconstruct its formal structure.

The reconstruction does not begin by assuming Constraint-Based Realization. It begins with a generic candidate law of individual outcome realization and asks what objects that law must contain if it is to be non-circular, probability-compatible, operationally meaningful, non-reductive with respect to decoherence, parameter-disciplined, and vulnerable to defeat. Each burden forces a corresponding formal object. Taken together, those objects yield the CBR-form representation:

Φ∗_C ∈ argmin{ℛ_C(Φ) : Φ ∈ 𝒜(C)}

This expression is not introduced as a decorative equation or interpretive slogan. It is the compressed representation of the burden structure. A law that must specify a domain, select from admissible candidates, compare candidates by a fixed criterion, resolve minimizers operationally, preserve Born-compatible ensemble behavior, avoid reduction to non-selective decoherence, fix parameters before outcome comparison, and state its own failure conditions naturally takes this form.

The purpose of this section is therefore reconstructive. It does not prove that nature obeys CBR. It proves something narrower and prior: once the burdens of a disciplined outcome-realization law are imposed, the formal objects of CBR are no longer arbitrary. They are the minimal objects required to make such a law evaluable.

3.1 Domain burden → context C

A candidate law of individual realization must apply to a physical measurement situation. It therefore requires a context variable. CBR denotes this variable by C.

C is not merely an observable label, a basis choice, or a verbal description of an experiment. It denotes the physical and operational structure relevant to the realization question. A context C may include the measured system, apparatus, relevant Hilbert space, record-bearing degrees of freedom, environmental couplings, timing relations, accessibility conditions, coarse-graining scale, and readout limitations.

The move from “a measurement occurs” to “a measurement occurs in context C” is necessary because realization cannot be evaluated apart from the physical situation in which candidate records, channels, or outcome structures become admissible. Without C, there is no determinate domain, no candidate class, no equivalence relation, no accessibility structure, and no failure condition.

Thus, the domain burden forces the first CBR object:

C.

A model fails this burden if C is undefined, described only after the outcome is known, or specified too vaguely to determine admissible candidates.

3.2 Candidate burden → admissible class 𝒜(C)

A realization law must specify what it selects among. Selection without candidates is empty. CBR denotes the candidate class by 𝒜(C).

𝒜(C) is the set of admissible realization-compatible candidates in context C. Depending on the representation used, its elements may be realization-compatible channels, record structures, or context-relative outcome maps. This paper uses channel language because it provides a disciplined quantum-operational way to represent candidate transformations from pre-realization state structure to post-interaction record or outcome structure.

The candidate burden forces more than the existence of possible outcomes. It forces a class over which selection is defined. A law-candidate cannot merely say “one outcome occurs.” It must specify the admissible structures from which a realized outcome-channel or realization class is selected.

Thus, the candidate burden forces the second CBR object:

𝒜(C).

A model fails this burden if 𝒜(C) is empty, undefined, unrestricted to the point of arbitrariness, or defined retrospectively so that only the observed result was eligible.

3.3 Admissibility burden → restriction of 𝒜(C)

The candidate class cannot include every mathematically writable map. A formal object may be expressible while failing to qualify as a physically meaningful realization candidate.

Therefore, 𝒜(C) must be restricted by admissibility conditions. These may include physical implementability, compatibility with the measurement architecture, complete positivity and trace preservation when channel language is used, record-structural coherence, dynamical consistency, probability compatibility, accessibility relevance, operational readout consistency, and independence from the realized outcome.

This restriction is decisive. CBR is not a rule that chooses among arbitrary formal possibilities. It is a context-constrained selection structure. The context determines which candidates are physically admissible; the law then compares candidates within that admissible class.

The admissibility burden therefore converts the realization problem from: select an outcome

into: select from the physically admissible candidate class 𝒜(C).

A model fails this burden if admissibility is so broad that any result can be made selectable, or so narrow that the selected result is smuggled into the candidate class by construction.

3.4 Non-circularity burden → pre-outcome fixity

A realization law cannot depend on the outcome it is supposed to explain. Its law-defining objects must therefore be fixed before the selected realization channel is known.

The fixed objects include:

C,
𝒜(C),
ℛ_C,
≃_C,
ε_C,
η where relevant,
and all parameters, tolerances, thresholds, or weights appearing in ℛ_C.

This requirement blocks the central failure mode of any proposed selection law: observing a result and then tuning the machinery so that the observed result appears selected. A rule of that kind does not explain realization. It redescribes the realized outcome after the fact.

The independence requirements may be written schematically as:

𝒜(C) independent of Φ∗_C,
ℛ_C independent of Φ∗_C,
≃_C independent of Φ∗_C,
ε_C independent of Φ∗_C,
η-calibration independent of Φ∗_C,
parameters in ℛ_C independent of Φ∗_C.

These are not ordinary derivative claims. They are structural independence conditions. They state that the selection machinery must not be defined by the selected result.

A model fails this burden if any law-defining object is chosen, narrowed, tuned, or reinterpreted after the realized outcome is known.

3.5 Selection burden → burden functional ℛ_C

To select among admissible candidates, a law must compare them. A comparison may be represented as an ordering, preorder, or scalar burden functional. CBR uses a context-relative realization-burden functional:

ℛ_C: 𝒜(C) → ℝ≥0.

For each Φ ∈ 𝒜(C), ℛ_C(Φ) assigns a non-negative realization burden relative to the context-fixed constraints of C. Lower burden means greater compatibility with the constraint structure of the context.

This is a representational requirement, not an empirical confirmation claim. The paper does not assert here that a final physical ℛ_C has been experimentally established. It asserts that any law that selects among admissible candidates must contain some candidate-comparison structure. CBR represents that structure as ℛ_C.

A more general formulation may begin with an admissibility preorder. Where the relevant quotient space is finite, regular, or otherwise representable, that preorder may be expressed by a scalar burden functional. The functional notation is used because it is the compact form needed for the representation theorem and the two-path demonstration.

Thus, the selection burden forces the third CBR object:

ℛ_C.

A model fails this burden if no comparison rule is supplied, if the comparison rule is defined after the outcome, or if the rule contains enough free adjustment to select any desired result.

3.6 Minimization burden → minimizer set M_C

Once ℛ_C is defined, the selected candidates are those that minimize the realization burden over the admissible class.

Define:

M_C = {Φ ∈ 𝒜(C) : ℛ_C(Φ) = inf{ℛ_C(Ψ) : Ψ ∈ 𝒜(C)}}.

M_C contains the admissible candidates whose realization burden is minimal in context C.

If M_C contains one candidate, selection is formally unique. If M_C contains multiple candidates, the theory must determine whether the multiplicity is physically meaningful. If M_C is empty, the theory has not selected a realization structure unless a pre-specified ε-minimizer rule exists.

Thus, minimization forces the fourth CBR object:

M_C.

A model fails this burden if ℛ_C is not bounded below, if no minimizer exists, and if no pre-specified approximate-selection rule is provided.

3.7 Operational uniqueness burden → equivalence relation ≃_C

A realization law must identify one physically meaningful selected structure. It need not always identify one syntactic representative. It must identify one operational realization class.

Define Φ₁ ≃_C Φ₂ to mean that Φ₁ and Φ₂ are operationally equivalent in context C. The model succeeds in C if:

∀ Φᵢ, Φⱼ ∈ M_C, Φᵢ ≃_C Φⱼ,

or if a pre-specified tie rule resolves operationally distinct minimizers.

The equivalence relation ≃_C is necessary because formal multiplicity does not always imply physical multiplicity. Two channels may differ as mathematical representatives while remaining indistinguishable by every accessible test, record, or readout procedure available in C. Such multiplicity is operationally harmless.

If M_C contains operationally distinct candidates and no pre-outcome tie rule exists, the model has not selected one realization class. It remains incomplete in C.

Thus, operational uniqueness forces the fifth CBR object:

≃_C.

A model fails this burden if it hides unresolved degeneracy, declares uniqueness without proving operational equivalence, or introduces a tie rule only after the outcome is known.

3.8 Probability burden → Born-compatible ensemble behavior

A realization law cannot arbitrarily alter established quantum statistics. For repeated trials in equivalent contexts, the default requirement is:

lim Nᵢ/N = │αᵢ│² as N → ∞.

Here Nᵢ is the number of trials yielding outcome i, and N is the total number of repeated trials. The equation states that long-run frequencies must approach Born-rule probabilities.

CBR does not replace Born-rule weighting in this reconstruction. The minimization structure constrains admissible realization candidates in a context. Born compatibility constrains the ensemble frequencies with which realization-compatible alternatives occur across repeated equivalent contexts.

This distinction prevents a category error. Selection is not sampling. A burden functional does not automatically replace probability weighting. Unless a CBR-form model explicitly introduces a controlled, pre-specified, empirically vulnerable deviation, it must preserve Born-compatible ensemble behavior.

A model fails this burden if it predicts non-Born frequencies without having declared, modeled, and exposed that deviation to test in advance.

3.9 Non-reduction burden → realization content beyond Φ_mix

A realization law must distinguish individual realization-compatible selection from non-selective decoherence-compatible evolution.

Let Φ_mix denote a non-selective decoherence-compatible channel. Such a channel may describe interference suppression, environmental decoherence, ensemble-level mixture structure, or record stabilization in a reduced description. It does not by itself specify which admissible record-structure or outcome-channel is selected as realized in an individual context.

CBR therefore requires a distinction between:

Φ_mix, the non-selective decoherence-compatible channel,

and

Φ∗_C, the selected realization channel or selected operational realization class.

CBR remains independent only if Φ∗_C supplies realization content not exhausted by Φ_mix. If Φ∗_C adds no content beyond Φ_mix in the relevant contexts, then CBR reduces to decoherence and fails as an independent realization law.

Thus, the non-reduction burden forces CBR to specify what work its selection structure performs.

A model fails this burden if it merely renames decoherence, or if the selected channel is operationally equivalent to Φ_mix in every relevant context without additional realization content.

3.10 Parameter burden → pre-test specification

If ℛ_C contains weights, tolerances, thresholds, accessibility calibrations, or adjustable terms, those elements must be fixed before outcome comparison.

For example, a toy model may use:

ℛ_C(Φ) = λ_S S_C(Φ) + λ_I I_C(Φ) + λ_P P_C(Φ) + λ_D D_C(Φ).

Here S_C, I_C, P_C, and D_C denote possible burden terms, while λ_S, λ_I, λ_P, and λ_D are weights. The equation states that total realization burden is represented as a weighted combination of stability, information-accessibility, probability-compatibility, and dynamical-compatibility burdens.

No λᵢ may be adjusted after the realized outcome is known. If parameter values are calibrated, the calibration protocol must be specified before outcome testing, and calibration data must be separated from the outcome test.

Parameter fixity therefore turns ℛ_C from a flexible scoring device into a law-like comparison rule.

A model fails this burden if parameters are selected, adjusted, or reinterpreted after the outcome in order to make the realized result minimize ℛ_C.

3.11 Vulnerability burden → defeat conditions

A candidate law must be able to fail. This burden forces CBR to include defeat conditions as part of its structure rather than treating them as external criticisms.

A CBR-form model fails in context C if any of the following occurs: C is undefined; 𝒜(C) is empty, arbitrary, or post hoc; ℛ_C is circular or unbounded without an ε-minimizer rule; M_C is empty; M_C contains operationally distinct minimizers with no tie rule; ≃_C or ε_C is adjusted after the result; parameters are tuned post hoc; Born-compatible behavior fails without a registered deviation claim; Φ∗_C reduces to Φ_mix without additional realization content; or no structural or empirical condition could count against the model.

Vulnerability is therefore not an optional later feature. It is part of what makes the CBR form a candidate law rather than an insulated interpretation.

A model fails this burden if every possible defect can be reinterpreted as success.

3.12 Reconstruction proposition

The burden-to-object mapping can now be stated as a reconstruction proposition.

Proposition 3.1 — Burden-to-Structure Reconstruction.
Any candidate law of individual quantum outcome realization satisfying the burdens of domain specification, candidate-set definition, admissibility restriction, non-circular comparison, operational uniqueness, Born compatibility, non-reduction to decoherence, parameter fixity, and vulnerability must contain the following structural objects:

C, 𝒜(C), ℛ_C, M_C, ≃_C, Φ∗_C,

together with whatever tolerances, accessibility parameters, calibration rules, and failure conditions are required by the context.

Under finite or regular representability conditions, the comparison structure may be expressed by a context-fixed burden functional ℛ_C, yielding the CBR-form representation:

Φ∗_C ∈ argmin{ℛ_C(Φ) : Φ ∈ 𝒜(C)}.

Proof sketch.
The domain burden forces C. The candidate-set and admissibility burdens force 𝒜(C). The selection burden forces a comparison structure, represented here as ℛ_C. Minimization forces M_C. Operational uniqueness forces ≃_C. Non-circularity and parameter fixity require these objects to be fixed independently of Φ∗_C. Born compatibility constrains ensemble behavior. Non-reduction requires Φ∗_C not to collapse into Φ_mix without additional realization content. Vulnerability requires explicit defeat conditions. Combining these requirements yields the CBR-form representation.

3.13 Conclusion of the reconstruction

The burdens reconstruct the CBR form:

Φ∗_C ∈ argmin{ℛ_C(Φ) : Φ ∈ 𝒜(C)}.

This equation is not an arbitrary formal decoration. It is the compressed expression of the minimum structure a disciplined realization-law candidate must possess.

The reconstruction can be summarized as:

C → 𝒜(C) → ℛ_C → M_C / ≃_C → Φ∗_C.

In words: a physically specified context determines an admissible candidate class; a context-fixed burden structure compares candidates; minimizers are identified; operational equivalence removes physically irrelevant multiplicity; and the selected realization channel or selected realization class is obtained.

The next section fixes the definitions required to state the theorem, corollaries, failure conditions, and two-path demonstration.

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4. Formal Definitions

This section fixes the formal objects used throughout the reconstruction. The purpose of these definitions is not to settle every interpretive question associated with quantum measurement. It is to make the theorem, assumptions, corollaries, failure conditions, and two-path demonstration precise enough to evaluate.

The definitions are intentionally minimal. They specify only the objects required by the burden-to-structure reconstruction:

𝓗, 𝒟(𝓗), C, Φ, 𝒜(C), 𝒜_real(C), ℛ_C, M_C, ≃_C, ε_C, Φ∗_C, η, and Φ_mix.

4.1 Hilbert space

Let 𝓗 denote the Hilbert space associated with the degrees of freedom relevant to context C.

The notation 𝓗 does not imply that every degree of freedom in the universe is included. It denotes the state space used for the physical and operational description of the measurement context under analysis.

Where necessary, let 𝓗′ denote the post-interaction, apparatus, record-bearing, or output Hilbert space relevant to C.

4.2 Density operators

Let 𝒟(𝓗) denote the set of density operators on 𝓗.

Let ρ ∈ 𝒟(𝓗) denote a quantum state in the relevant context. The use of density operators allows the formalism to include pure states, mixed states, reduced states, and states arising from environmental interaction or partial tracing.

4.3 Measurement context

Let C denote a physically specified measurement context.

C includes the system degrees of freedom, measurement apparatus, record-bearing degrees of freedom, environmental interactions, timing structure, accessibility conditions, coarse-graining scale, operational readout limits, and any context-specific tolerance used to define operational equivalence.

C is not the realized outcome. It is the pre-outcome physical and operational structure within which admissible realization candidates are defined.

A context C is admissible for the present reconstruction only if it is specified independently of Φ∗_C. If C is redefined after the outcome is known, the model fails the non-circularity requirement.

4.4 Realization-compatible channel

A realization-compatible channel is represented as a completely positive trace-preserving map:

Φ: 𝒟(𝓗) → 𝒟(𝓗′).

Here 𝓗′ denotes the post-interaction or record-bearing Hilbert space relevant to C. The map Φ represents a physically admissible transformation from pre-realization state structure to post-interaction, record-bearing, or realization-compatible structure.

The use of CPTP maps is a formal discipline. It ensures that candidate realization channels are expressed in standard quantum-operational language. It is not a claim that channel notation alone resolves the measurement problem, nor that all metaphysical questions about outcome realization have been settled by representation.

4.5 Admissible candidate class

Let 𝒜(C) be the set of admissible realization-compatible channels in context C.

The condition 𝒜(C) ≠ ∅ states that the admissible candidate class must not be empty. If no candidates are admissible, the selection rule has nothing to select from.

𝒜(C) must be determined by C and by pre-specified admissibility conditions. It must not be defined after the realized outcome is known.

The class 𝒜(C) may include channels relevant to the broader dynamical or record-bearing description. Not every member of 𝒜(C) need automatically qualify as an individual realization-compatible candidate. That distinction is captured by 𝒜_real(C).

4.6 Individual realization-compatible subclass

Let 𝒜_real(C) ⊆ 𝒜(C) denote the subclass of candidates that qualify as individual realization-compatible structures in context C.

This distinction allows the formalism to separate broader dynamical or decoherence-compatible channels from completed individual realization candidates. For example, a non-selective decoherence-compatible channel may belong to a broader dynamical description while not qualifying as an individual realization-compatible channel unless additional realization content is supplied.

The membership of 𝒜_real(C) must be determined by pre-specified context-relative admissibility conditions. It cannot be narrowed after the outcome is known in order to force a desired selected result.

4.7 Realization-burden functional

Let ℛ_C: 𝒜(C) → ℝ≥0 be the context-relative realization-burden functional.

For each Φ ∈ 𝒜(C), ℛ_C(Φ) assigns a non-negative burden relative to the constraints of C. Lower ℛ_C(Φ) means lower burden relative to the context-fixed constraint structure.

The functional must be specified independently of Φ∗_C. If it contains adjustable parameters, thresholds, tolerances, or coefficients, those values or calibration procedures must be fixed before outcome comparison.

A more general formulation may replace ℛ_C with an admissibility preorder on 𝒜(C). In finite or regular quotient settings, that preorder may be represented by a scalar burden functional. The functional notation is used here as the compact form required for the theorem and two-path demonstration.

4.8 Minimizer set

Define:

M_C = {Φ ∈ 𝒜(C) : ℛ_C(Φ) = inf{ℛ_C(Ψ) : Ψ ∈ 𝒜(C)}}.

M_C contains all admissible candidates whose realization burden is minimal in context C.

If M_C is empty, the model requires a pre-specified ε-minimizer rule. Without such a rule, selection is undefined in C.

An ε-minimizer rule may allow candidates whose burden lies within a fixed tolerance of the infimum. The tolerance and rule must be specified before outcome comparison. Otherwise, approximate minimization becomes another route to post hoc adjustment.

4.9 Operational equivalence

Define Φ₁ ≃_C Φ₂ if no operationally accessible procedure within C distinguishes Φ₁ from Φ₂ at the resolution relevant to realization selection.

Let 𝒯(C) denote the accessible tests or readout procedures available in context C. A formal version is:

∀T ∈ 𝒯(C), T(Φ₁(ρ)) = T(Φ₂(ρ)),

within the resolution limits of C.

More generally, operational equivalence may be defined up to a context-specific resolution tolerance ε_C:

│T(Φ₁(ρ)) − T(Φ₂(ρ))│ ≤ ε_C

for every accessible test T ∈ 𝒯(C).

This means that Φ₁ and Φ₂ count as operationally equivalent when every accessible test differs by no more than the resolution tolerance ε_C. The tolerance ε_C must be specified by the operational limits of C and cannot be adjusted after the result is known.

Operational equivalence is therefore a physical quotienting relation. It removes representational multiplicity that has no accessible realization-level consequence in C.

4.10 Selected realization channel

Define:

Φ∗_C ∈ argmin{ℛ_C(Φ) : Φ ∈ 𝒜(C)}.

Equivalently:

Φ∗_C ∈ M_C.

This means that Φ∗_C belongs to the set of admissible channels in context C that minimize ℛ_C.

If the minimizer is unique, Φ∗_C is the selected channel. If multiple minimizers exist, selection is defined over an operational equivalence class, provided all minimizers are equivalent under ≃_C or a pre-specified tie rule exists.

More precisely, the model succeeds in context C if:

∀ Φᵢ, Φⱼ ∈ M_C, Φᵢ ≃_C Φⱼ,

or if a pre-specified tie rule selects among operationally distinct minimizers.

If M_C contains operationally distinct candidates and no pre-specified tie rule exists, the model is incomplete in C.

4.11 Born-compatible realization

A realization structure is Born-compatible if repeated trials in equivalent contexts reproduce:

P(i) = │αᵢ│²

for the relevant outcome basis.

For a two-outcome state:

│ψ⟩ = α│0⟩ + β│1⟩,

Born compatibility requires:

P(0) = │α│²,
P(1) = │β│².

This paper imposes Born compatibility as a condition. It does not claim that the Born rule is derived here.

If a future CBR-form model proposes non-Born behavior, that deviation must be explicitly stated, pre-specified, and exposed to empirical test. No such deviation is asserted in this reconstruction.

4.12 Accessibility parameter

Let η ∈ [0,1] denote the operational accessibility of record-bearing which-path or outcome-relevant information in context C.

When η = 0, no relevant record is operationally accessible.
When η = 1, the relevant record is fully accessible.
Intermediate values represent partial accessibility.

The parameter η helps distinguish mere formal correlation from physically accessible record structure. In the present paper, η is used only at the level required for the reconstruction and two-path demonstration. Full platform-specific calibration of η belongs to the empirical exposure papers.

4.13 Non-selective decoherence-compatible channel

Let Φ_mix denote a non-selective decoherence-compatible channel.

Φ_mix may describe interference suppression, environmental decoherence, ensemble-level mixture structure, or record stabilization in a reduced description. It does not by itself represent individual outcome realization unless additional realization content is supplied.

This definition preserves the distinction between decoherence-compatible dynamics and realization-compatible selection. CBR remains independent only if Φ∗_C supplies realization content not exhausted by Φ_mix.

4.14 Definition registry

The reconstruction uses the following formal objects.

C denotes the physically specified measurement context.

𝓗 denotes the Hilbert space relevant to C.

𝒟(𝓗) denotes the density-operator space over 𝓗.

ρ denotes a quantum state in 𝒟(𝓗).

Φ denotes a realization-compatible channel.

𝒜(C) denotes the admissible candidate class in C.

𝒜_real(C) denotes the individual realization-compatible subclass.

ℛ_C denotes the context-fixed realization-burden functional.

M_C denotes the minimizer set of ℛ_C over 𝒜(C).

≃_C denotes operational equivalence in C.

ε_C denotes the context-specific resolution tolerance for operational equivalence.

Φ∗_C denotes the selected realization channel or selected operational equivalence class.

η denotes the accessibility parameter for outcome-relevant record information.

Φ_mix denotes the non-selective decoherence-compatible channel.

These definitions provide the formal vocabulary for the assumptions and representation theorem that follow.

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5. Assumptions of the Reconstruction

The preceding sections identified the burdens that motivate the CBR form and fixed the notation required to state the reconstruction precisely. This section states the assumptions under which the reconstruction proceeds.

These assumptions are not empirical conclusions. They do not assert that nature obeys CBR. They define the class of candidate realization laws considered in this paper: laws that are context-specified, candidate-selective, admissibility-restricted, non-circular, probability-compatible, non-reductive with respect to decoherence, parameter-disciplined, operationally meaningful, and vulnerable to failure.

The assumptions serve three roles.

First, they define the domain of the representation theorem. The theorem applies only to candidate laws satisfying these conditions.

Second, they prevent the reconstruction from becoming circular, vacuous, or merely verbal. A theory cannot define its context, candidate class, comparison rule, equivalence relation, or parameters after the outcome is known and still count as a law-like explanation of that outcome.

Third, they identify failure points. If a model violates one of these assumptions in context C, the model does not merely need stylistic repair. It fails one of the burdens required of a disciplined realization-law candidate in that context.

The assumptions should therefore be read as an admissibility gate. A proposal that passes through the gate enters the reconstruction class. A proposal that fails the gate may still be philosophically interesting, but it does not satisfy the standard used here for CBR-form representation.

5.1 The reconstruction class

A candidate law belongs to the reconstruction class if it satisfies the assumptions below.

The reconstruction class is intentionally restricted. It does not include every conceivable interpretation of quantum mechanics, every possible collapse model, every possible hidden-variable theory, or every metaphysical account of measurement. It includes only those candidate laws that claim to supply a disciplined law-form for individual outcome realization while preserving quantum probability, avoiding post hoc selection, distinguishing realization from non-selective decoherence, and exposing themselves to failure.

The restricted scope is not a weakness. It is what makes the theorem exact. The claim is not that all logically possible theories must be CBR. The claim is that any law-candidate satisfying the burdens below admits the CBR-form representation.

A1. Context specification

The measurement context C is physically specified before outcome realization.

The relevant system, measurement architecture, Hilbert-space description, record-bearing degrees of freedom, environmental couplings, timing relations, accessibility conditions, coarse-graining assumptions, operational readout limits, and context-specific tolerances must be fixed before the selected realization channel is identified.

C is not inferred from the realized outcome after the fact. It is the pre-outcome physical and operational situation within which admissible realization candidates are defined.

This assumption blocks undefined-domain reasoning and prevents retrospective redescription of the context to favor the observed result.

A2. Candidate-class fixity

The admissible candidate class 𝒜(C) is determined by C and does not depend on the realized outcome.

A model cannot observe a result and then define 𝒜(C) so that only that result was admissible. The candidate class must be specified by the physical and operational features of the context before outcome comparison.

This assumption blocks post hoc candidate selection.

A3. Admissibility restriction

The class 𝒜(C) contains only candidates satisfying pre-specified admissibility conditions.

Admissibility may include physical implementability, compatibility with the measurement architecture, dynamical consistency, record-structural coherence, operational accessibility, probability compatibility, and independence from the selected result.

This assumption prevents 𝒜(C) from becoming an unrestricted space of formal possibilities in which any desired outcome can be made selectable.

A4. Comparison structure

The candidate law supplies a context-fixed comparison structure over 𝒜(C).

This comparison structure may be given directly by a realization-burden functional:

ℛ_C: 𝒜(C) → ℝ≥0,

or more generally by a context-relative admissibility preorder over 𝒜(C). When the relevant quotient space is finite, regular, or otherwise representable, that preorder may be expressed by a scalar burden functional.

This assumption is essential. The reconstruction does not require every possible law in every mathematical setting to begin with a real-valued functional. It requires that the law contain a fixed comparison structure capable of identifying admissible preferred candidates, minimal candidates, or minimizer classes. The scalar functional is the canonical representation used in this paper.

This blocks the objection that the theorem assumes too quickly that law-like comparison must be numerical.

A5. Functional fixity

If the comparison structure is represented by ℛ_C, then ℛ_C is determined by C and does not depend on the realized outcome.

ℛ_C cannot reward a candidate merely because it matches the observed result. The functional must be specified before outcome comparison and constructed from context-relative constraints rather than retrospective outcome labels.

This assumption blocks outcome-fitting at the level of the selection rule.

A6. Equivalence-relation fixity

The operational equivalence relation ≃_C is determined by the operational capacities of C and does not depend on the realized outcome.

The relation ≃_C determines when formally distinct minimizers are indistinguishable within the context. It must be fixed by accessible tests 𝒯(C), operational readout limits, and any context-specific tolerance ε_C. If ≃_C is adjusted after the result, degeneracy can be hidden rather than resolved.

This assumption blocks retrospective manipulation of operational uniqueness.

A7. Nonempty admissibility

The admissible candidate class is nonempty:

𝒜(C) ≠ ∅.

If 𝒜(C) is empty, there are no candidates over which a realization rule can operate. The model then fails in context C unless the context or admissibility conditions are revised by a principled rule fixed before outcome comparison.

This assumption ensures that the selection problem is not vacuous.

A8. Well-founded comparison

The comparison structure over 𝒜(C) is well-founded enough to support selection.

If the comparison is represented by a burden functional ℛ_C, then ℛ_C is bounded below over 𝒜(C). If the comparison is represented by a preorder, then the preorder must admit minimal elements, preferred classes, or a pre-specified approximate-selection rule.

Without such a condition, the selection rule may never identify a preferred admissible candidate or class.

This assumption prevents the selection problem from being mathematically undefined.

A9. Minimizer existence or principled approximation

When ℛ_C is used, the minimizer set

M_C = {Φ ∈ 𝒜(C) : ℛ_C(Φ) = inf{ℛ_C(Ψ) : Ψ ∈ 𝒜(C)}}

is nonempty.

If exact minimizers do not exist, an ε-minimizer rule must be specified in advance. Such a rule may select candidates whose burden lies within a fixed tolerance of the infimum, but both the tolerance and the selection procedure must be fixed before outcome comparison.

This assumption prevents approximate minimization from becoming another route to post hoc adjustment.

A10. Operational uniqueness or pre-specified tie rule

For all Φᵢ, Φⱼ ∈ M_C,

Φᵢ ≃_C Φⱼ,

or else a pre-specified tie rule must exist.

If all minimizers are operationally equivalent, the model selects one physically meaningful realization class. If minimizers are operationally distinct, the model requires a tie rule fixed before outcome comparison. If neither condition holds, the model is incomplete in C.

This assumption prevents unresolved degeneracy from being mistaken for selection.

A11. Born compatibility

Across repeated equivalent contexts, realization frequencies preserve Born-rule probabilities.

For a two-outcome state

|ψ⟩ = α|0⟩ + β|1⟩,

Born compatibility requires:

P(0) = |α|²,
P(1) = |β|².

More generally, for

|ψ⟩ = Σᵢ αᵢ|i⟩,

the ensemble requirement is:

P(i) = |αᵢ|².

This assumption does not claim that the Born rule is derived in this paper. It requires that any CBR-form realization structure preserve Born-rule statistics unless a controlled, explicitly stated, empirically vulnerable deviation is introduced.

This prevents realization from being used as an uncontrolled replacement for quantum probability.

A12. Non-reduction to decoherence

The selected realization structure is not identical to a purely non-selective decoherence-compatible channel unless CBR is being treated as reducible and therefore non-independent.

Let Φ_mix denote a non-selective decoherence-compatible channel. If, for every relevant context C, the selected realization structure Φ∗_C adds no content beyond Φ_mix, then CBR reduces to decoherence and fails as an independent realization law.

This assumption does not reject decoherence. It requires only that CBR supply additional realization-selection content if it is to be treated as an independent law-candidate.

A13. Parameter fixity

Any terms, tolerances, thresholds, accessibility calibrations, or weights appearing in ℛ_C must be specified before outcome comparison.

No component of ℛ_C, including any λᵢ weighting parameters, may be adjusted after the realized outcome is known. If parameters are estimated from calibration data, the calibration protocol must be fixed before testing the realization claim, and calibration data must be separated from the outcome test.

This assumption blocks retrospective tuning of the realization functional.

A14. Vulnerability

Failure of context specification, candidate definition, admissibility restriction, comparison structure, non-circularity, minimization, operational uniqueness, Born compatibility, parameter fixity, non-reduction, or empirical exposure counts against the model.

This assumption prevents the law-candidate from being protected against all possible criticism. A CBR-form model must be able to fail structurally or empirically. Its assumptions are therefore not merely premises of success; they are also possible points of defeat.

A model that cannot fail has not become stronger. It has become less evaluable.

5.2 Assumption registry

The assumptions may be summarized as follows.

A1 fixes the domain.
A2 fixes the candidate class.
A3 restricts admissibility.
A4 requires a comparison structure.
A5 fixes the burden functional when functional representation is used.
A6 fixes operational equivalence.
A7 prevents vacuous selection.
A8 makes comparison well-founded.
A9 secures minimizer existence or principled approximation.
A10 secures operational uniqueness or a tie rule.
A11 preserves quantum probability.
A12 prevents reduction to decoherence.
A13 blocks parameter tuning.
A14 requires vulnerability.

Together, these assumptions define the reconstruction class. They do not prove that nature obeys CBR. They identify the conditions under which a candidate realization law has enough structure to admit CBR-form representation.

5.3 What these assumptions do not assume

The assumptions do not assume that CBR is empirically confirmed. They do not assume that standard quantum mechanics is replaced. They do not assume that decoherence is false. They do not assume that the Born rule is derived here. They do not assume that every measurement context admits a unique CBR solution. They do not assume that a final universal physical ℛ_C has already been identified.

They also do not assume universal closure over every logically possible realization-law framework. The theorem below applies only to the class of candidate laws satisfying the stated burdens.

The restricted claim is stronger because it is exact. Within the reconstruction class, the CBR form is not an arbitrary interpretive preference. It is the natural representation of the minimum structure required for a disciplined realization-law candidate.

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6. Conditional Minimal Representation Theorem

The assumptions above allow the reconstruction to be stated as a conditional representation theorem. The theorem does not assert that nature obeys CBR. It states that any candidate law satisfying the stated burdens and assumptions admits a CBR-form representation.

The theorem is therefore not an empirical confirmation theorem. It is a structural theorem. Its function is to show that CBR is not merely asserted as an interpretive preference. It is the natural representational form of a law-candidate satisfying the burdens of context specification, admissible selection, non-circular comparison, operational uniqueness, probability compatibility, non-reduction, parameter fixity, and vulnerability.

6.1 Theorem statement

Theorem 1 — Conditional Minimal Representation Theorem for Constraint-Based Realization

Let C be a physically specified measurement context. Let 𝒜(C) be a nonempty admissible candidate class fixed independently of the realized outcome. Let ≃_C be a context-fixed operational equivalence relation.

Suppose a candidate law L of individual quantum outcome realization satisfies the following requirements:

L applies to C.

L selects from 𝒜(C).

𝒜(C) is restricted by pre-specified admissibility conditions.

L compares admissible candidates by a context-fixed comparison structure.

The comparison structure is fixed independently of the realized outcome.

If the comparison structure is represented by a scalar burden functional ℛ_C, then ℛ_C is fixed independently of the realized outcome.

Any parameters, tolerances, thresholds, accessibility calibrations, or weights used by the comparison structure are fixed before outcome comparison.

The comparison structure identifies a candidate, minimizer set, minimal admissibility class, or operational equivalence class of candidates.

All selected minimizers or minimal candidates are operationally equivalent under ≃_C, or a pre-specified tie rule exists.

L preserves Born-rule ensemble frequencies unless a controlled and empirically vulnerable deviation is specified in advance.

L does not reduce entirely to a non-selective decoherence-compatible channel Φ_mix.

L has explicit structural or empirical failure conditions.

Then L admits a CBR-form representation:

Φ∗_C ∈ argmin{ℛ_C(Φ) : Φ ∈ 𝒜(C)},

where ℛ_C is a context-fixed realization-burden functional over admissible candidates, or a scalar representation of an admissibility preorder under finite or regular quotient conditions.

The selected realization channel Φ∗_C denotes either a unique minimizer, a selected operational equivalence class of minimizers, or the representative of a minimal admissibility class modulo ≃_C.

6.2 Proof sketch

The proof proceeds by mapping each law-candidate burden to a required formal object.

Because L applies to a physical measurement situation, it requires a domain. Denote that domain by C.

Because L selects among possible realization structures, it requires a candidate class. Denote that class by 𝒜(C). Candidate-class fixity requires that 𝒜(C) be determined by C and by pre-specified admissibility conditions, not by the realized result.

Because not all formal candidates are physically admissible, 𝒜(C) must be restricted. Admissibility conditions may include implementability, record compatibility, dynamical consistency, operational accessibility, probability compatibility, and independence from the selected outcome.

Because L selects among candidates, it must compare them. That comparison may be represented by an ordering, preorder, or burden functional. When the relevant quotient space is finite, regular, or otherwise representable, the comparison structure may be expressed by a scalar burden functional:

ℛ_C: 𝒜(C) → ℝ≥0.

For each Φ ∈ 𝒜(C), ℛ_C(Φ) assigns a non-negative realization burden relative to the constraints of C.

Because L must be non-circular, C, 𝒜(C), ℛ_C where used, ≃_C, ε_C, η-calibration where relevant, and any parameters in the comparison structure must be fixed independently of Φ∗_C. This prevents the selection rule from being tuned after the outcome is known.

Because L must identify a physically meaningful selected structure, the comparison structure must yield a unique candidate, a minimizer set, a minimal admissibility class, or a selected operational equivalence class. In functional form, the minimizer set is:

M_C = {Φ ∈ 𝒜(C) : ℛ_C(Φ) = inf{ℛ_C(Ψ) : Ψ ∈ 𝒜(C)}}.

M_C must either contain a unique minimizer, contain only mutually operationally equivalent minimizers, or be governed by a pre-specified tie rule.

Because quantum probabilities are empirically fixed, the realization structure must preserve Born-rule frequencies unless a controlled, pre-specified, empirically vulnerable deviation is introduced.

Because decoherence alone is non-selective, CBR remains independent only if the selected realization structure Φ∗_C supplies realization content not exhausted by a non-selective decoherence-compatible channel Φ_mix.

Because a law-candidate must be evaluable, failure of the domain, candidate class, admissibility structure, comparison structure, minimizer structure, operational uniqueness, Born compatibility, non-reduction, parameter fixity, or vulnerability condition must count against the model.

Combining these requirements yields the CBR-form representation:

Φ∗_C ∈ argmin{ℛ_C(Φ) : Φ ∈ 𝒜(C)}.

This equation states that the selected realization channel or selected realization class belongs to the admissible minimizers of a context-fixed burden structure.

6.3 Minimality claim

The representation is minimal in the following sense: each core object in the CBR form is forced by one of the burdens, and removing any one of them destroys the law-candidate structure.

Without C, the law has no physical domain.

Without 𝒜(C), there is nothing to select among.

Without admissibility restrictions, selection is arbitrary.

Without a comparison structure, candidates are not ordered.

Without ℛ_C or an equivalent representable preorder, the canonical minimization form is unavailable.

Without M_C or a minimal admissibility class, no selected candidate set is identified.

Without ≃_C or a tie rule, formal multiplicity may remain physically unresolved.

Without Born compatibility, the model risks uncontrolled conflict with quantum statistics.

Without non-reduction, the model collapses into decoherence.

Without parameter fixity, the model becomes a retrospective fitting device.

Without vulnerability, the model ceases to be scientifically evaluable.

Thus, the theorem does not merely show that CBR can be written in this form. It shows that the form is the minimal representational structure of a disciplined candidate realization law under the stated assumptions.

6.4 What the theorem proves

The theorem proves a conditional representation result.

It proves that within the reconstruction class defined above, a disciplined law of individual outcome realization must contain the structural elements represented by:

C → 𝒜(C) → ℛ_C → M_C / ≃_C → Φ∗_C.

It proves that the CBR form is not merely an arbitrary equation added to the measurement problem. It is the compressed representation of the burdens imposed on any law-candidate in the class considered here.

It proves that CBR is structurally natural under the stated assumptions.

6.5 What the theorem does not prove

The theorem does not prove that CBR is true in nature. It does not prove that a specific physical ℛ_C has been identified. It does not prove that all measurement contexts admit a unique CBR solution. It does not prove that rival interpretations fail. It does not prove that decoherence is false. It does not derive the Born rule. It does not establish universal closure over every possible realization-law framework.

The theorem also does not show that all comparison structures in all mathematical settings must be scalar-valued. That is why the theorem is stated in terms of a comparison structure that may be represented by ℛ_C under finite or regular quotient conditions.

The theorem’s significance is narrower and more precise: under the stated assumptions, the CBR structure is not arbitrary. It is the natural representational form of a law-candidate that must define a context, select from admissible candidates, compare them non-circularly, handle uniqueness operationally, preserve probability, avoid decoherence reduction, control parameters, and remain vulnerable to failure.

6.6 Why this is stronger than a toy model

A toy model can show that a proposed formalism works in one controlled example. That is useful, but limited.

The Conditional Minimal Representation Theorem does something different. It reconstructs the form that a disciplined realization law must take under the stated burdens. The theorem therefore supplies the general representational logic, while the later two-path model supplies a concrete instantiation.

The relation is:

The theorem shows why CBR-form structure arises from the burdens.

The two-path model shows how that structure can be instantiated in a controlled context.

Neither result establishes empirical confirmation. Together, they strengthen CBR as a law-candidate by showing both formal reconstruction and model-level usability.

6.7 Transition to corollaries

The theorem establishes the representational form. The next step is to draw its immediate consequences: non-circularity, operational uniqueness, Born compatibility, non-reduction to decoherence, parameter discipline, and failure-capability.

Those consequences are stated as corollaries in the next section.

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7. Consequences and Defeat Conditions of the Reconstruction

The Conditional Minimal Representation Theorem establishes that any candidate law satisfying the reconstruction assumptions admits the CBR-form representation:

Φ∗_C ∈ argmin{ℛ_C(Φ) : Φ ∈ 𝒜(C)}.

The present section states the immediate consequences of that representation. These consequences are not independent empirical claims. They hold only inside the reconstruction class defined above. Their function is to clarify what the representation secures, what it does not secure, and how a CBR-form model can fail.

This dual structure is essential. A disciplined realization-law candidate must not merely state its success conditions. It must also state where those conditions break. The corollaries below therefore operate as both consequences and tests: each one identifies a property CBR-form models possess under the theorem’s assumptions, and each one identifies a corresponding route of defeat.

7.1 Corollary 1 — Non-circularity under fixed reconstruction data

If C, 𝒜(C), the comparison structure, ℛ_C where used, ≃_C, ε_C, η-calibration where relevant, and all parameters in ℛ_C are fixed before outcome comparison, then CBR-form selection is non-circular in context C.

The independence conditions may be stated schematically as:

𝒜(C) independent of Φ∗_C,
ℛ_C independent of Φ∗_C,
≃_C independent of Φ∗_C,
ε_C independent of Φ∗_C,
η-calibration independent of Φ∗_C,
parameters in ℛ_C independent of Φ∗_C.

These are not ordinary differentiability claims. They are structural independence conditions. They state that the machinery used to identify the selected realization channel is not itself defined by the selected realization channel.

Proof sketch.
By the assumptions of the reconstruction class, the context, candidate class, comparison structure, operational equivalence relation, tolerances, accessibility calibration, and parameters are fixed before outcome comparison. Therefore, Φ∗_C is identified by pre-specified structure rather than by retrospective adjustment to the observed result. The rule may still be physically wrong, but it is not circular merely by form.

This corollary does not establish that a particular ℛ_C is physically correct. It establishes only that, under pre-outcome fixity, the selection rule is not simply naming the outcome after the fact.

The defeat condition is direct: if any law-defining object is chosen, tuned, narrowed, calibrated, or reinterpreted after the outcome is known, the model fails the non-circularity burden in C.

7.2 Corollary 2 — Operational uniqueness of the selected realization class

If every minimizer in M_C is equivalent under ≃_C, then the CBR-form law selects a unique operational realization class in context C.

Formally:

∀ Φᵢ, Φⱼ ∈ M_C, Φᵢ ≃_C Φⱼ.

This means that multiple formal minimizers do not necessarily imply multiple physical verdicts. If the minimizers cannot be distinguished by any accessible test, record, or readout procedure in C, then they constitute one operational realization class.

Proof sketch.
The theorem requires the model to identify a candidate, minimizer set, minimal admissibility class, or operational equivalence class. If every element of M_C is equivalent under ≃_C, then formal multiplicity collapses to one operational class. Hence the law selects one physically meaningful realization verdict even if it does not select one unique syntactic representative.

This corollary does not guarantee uniqueness in every context. It states the condition under which uniqueness is obtained.

The defeat condition is equally direct: if M_C contains operationally distinct minimizers and no pre-specified tie rule exists, the model has not selected one realization class and is incomplete in C.

7.3 Corollary 3 — Born-compatible ensemble behavior

If a CBR-form model belongs to the reconstruction class and does not declare a controlled non-Born deviation, then repeated equivalent contexts must preserve Born-rule ensemble frequencies.

For an input state:

|ψ⟩ = Σᵢ αᵢ|i⟩,

Born compatibility requires:

P(i) = │αᵢ│².

For a two-outcome state:

|ψ⟩ = α|0⟩ + β|1⟩,

the corresponding requirements are:

P(0) = │α│²,
P(1) = │β│².

Proof sketch.
Born compatibility is one of the assumptions defining the reconstruction class. Therefore, any CBR-form model in that class must preserve Born-rule ensemble behavior unless it explicitly states, models, and exposes a deviation to empirical test.

This corollary does not derive the Born rule. It states that CBR-form realization, as reconstructed here, does not replace Born-rule weighting. CBR constrains admissible realization structure; Born compatibility constrains the frequencies with which realization-compatible alternatives occur across repeated equivalent contexts.

The defeat condition is clear: if a CBR-form model predicts non-Born frequencies without declaring, modeling, and exposing that deviation in advance, it fails the probability-compatibility burden.

7.4 Corollary 4 — Non-reduction to decoherence

If the CBR-selected structure Φ∗_C supplies realization content not contained in the non-selective decoherence-compatible channel Φ_mix, then CBR is not merely decoherence renamed in context C.

Conversely, if:

Φ∗_C ≃_C Φ_mix

for all relevant contexts C, and no further realization content is supplied, then CBR reduces to decoherence and fails as an independent realization law in those contexts.

Proof sketch.
The reconstruction class requires non-reduction to ordinary non-selective decoherence. Therefore, the selected realization structure must do work not already done by Φ_mix. If no such additional work exists, the purported realization law collapses into decoherence-compatible dynamics and loses independent status.

This corollary does not criticize decoherence. Decoherence remains essential for interference suppression, environmental entanglement, record stabilization, pointer-state robustness, and effective classicality. The corollary distinguishes roles: decoherence supplies non-selective dynamical and record-formation structure; CBR, if independent, must supply a selection structure over admissible realization-compatible candidates.

7.5 Corollary 5 — Parameter discipline

If the realization functional contains terms, weights, tolerances, thresholds, accessibility calibrations, or other adjustable elements, those values or determination procedures must be fixed before outcome comparison.

For example, if a model uses weights λᵢ, no λᵢ may be adjusted after observing the realized outcome. If values are estimated from calibration data, the calibration procedure must be specified before testing the realization claim, and calibration data must be separated from the outcome test.

Proof sketch.
The theorem requires the comparison structure and its parameters to be fixed independently of the selected result. Post hoc adjustment makes the selected outcome help determine the rule that selects it. That defeats non-circularity.

This corollary does not require that all models be parameter-free. It requires that parameters be fixed in a way that prevents outcome-fitting.

The defeat condition is direct: if parameter values are selected, adjusted, recalibrated, or reinterpreted after the result is known in order to make the realized outcome minimize ℛ_C, the model fails as a non-circular law-candidate.

7.6 Corollary 6 — Failure-capability

Because the theorem depends on explicit assumptions, CBR-form models can fail when those assumptions fail.

A model may fail through an undefined context, an empty or post hoc candidate class, unrestricted admissibility, circular comparison, post hoc parameter tuning, absence of minimizers, unresolved operationally distinct minimizers, Born-rule violation without a declared deviation, reduction to Φ_mix, or insulation from every possible structural or empirical defeat.

Proof sketch.
The theorem applies only inside the reconstruction class. If a model violates the assumptions defining that class, then the theorem no longer licenses the CBR-form representation as a disciplined law-candidate in that context.

This is a strength rather than a weakness. A law-candidate becomes more serious when it states its defeat conditions. A model that cannot fail has not become stronger. It has become less evaluable.

7.7 Consolidated consequence

The reconstruction therefore yields the following consolidated result:

A CBR-form model is non-circular only if its reconstruction data are fixed before outcome comparison; operationally unique only if its minimizers collapse to one operational class or a tie rule is fixed in advance; probability-compatible only if Born-rule ensemble behavior is preserved or deviations are declared in advance; independent of decoherence only if Φ∗_C adds realization content beyond Φ_mix; parameter-disciplined only if adjustable elements are fixed before testing; and scientifically evaluable only if assumption failure counts against the model.

Thus, the reconstruction does not merely produce an equation. It produces an evaluation standard.

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8. Minimal Two-Path Instantiation

The preceding sections reconstructed the CBR-form structure and stated the theorem under which that structure arises. This section provides a minimal two-path instantiation of that structure.

The model is not presented as a complete theory of measurement. It is not a proof of universal applicability. It is not an empirical prediction of a new deviation. It is not the final canonical CBR law. Its function is narrower and more controlled: to show that the reconstructed CBR objects can be populated in a simple nontrivial context without replacing Born-rule weighting and without collapsing realization into a non-selective decoherence-compatible channel.

The model therefore proves instantiability, not truth. It shows that there exists a simple setting in which the objects:

C, 𝒜(C), 𝒜_real(C), ℛ_C, M_C, ≃_C, η, Φ_mix, and Φ∗_C

can be coherently distinguished.

8.1 Status of the instantiation

The two-path construction has a limited formal role.

It does not claim that the toy functional used below is the final physical realization functional. It does not claim that two-path systems exhaust the measurement problem. It does not claim that the Born rule is derived. It does not claim that decoherence is false. It does not claim that a new empirical deviation is predicted here.

It demonstrates only that the burden-to-structure reconstruction is not empty. The CBR-form objects can be assigned explicit roles in a minimal context with candidate channels, record structure, accessibility conditions, burden terms, parameter-fixity requirements, Born-compatible ensemble behavior, and a non-reduction distinction.

This distinction matters. A minimal model should not be asked to prove the full theory. Its job is to show that the formal structure can be instantiated without contradiction.

8.2 Two-path state

Consider a two-path system with basis states |0⟩ and |1⟩. Let the pre-measurement state be:

|ψ⟩ = α|0⟩ + β|1⟩,

with normalization:

│α│² + │β│² = 1.

The coefficients α and β determine the Born-rule weights associated with the two alternatives. The setup is minimal because it contains more than one possible outcome structure while remaining simple enough to analyze explicitly.

8.3 Record-bearing structure

Introduce record states |R₀⟩ and |R₁⟩ corresponding to record structures associated with the two alternatives.

After interaction with record-bearing degrees of freedom, the joint structure may be represented schematically as:

|Ψ⟩ = α|0⟩|R₀⟩ + β|1⟩|R₁⟩.

This expression represents system-record correlation. It does not, by itself, assert that either record has been individually realized. It provides the physical setting in which the realization question can be posed.

The distinction is essential. A correlated record structure is not automatically a law of individual realization. It is the background against which a realization-law candidate may be formulated.

8.4 Decoherence structure

If the record states become distinguishable, then:

⟨R₀|R₁⟩ → 0.

This indicates suppression of interference between alternatives. In standard terms, the record states become effectively orthogonal, and the corresponding off-diagonal coherence terms are suppressed in the reduced description.

This is the role of decoherence. It explains interference suppression and record stabilization. It does not, by itself, specify an individual realization channel. The CBR instantiation therefore does not compete with decoherence at this stage. It asks whether a further selection structure over admissible realization-compatible channels can be defined.

8.5 Context C in the two-path model

In the two-path instantiation, the context C includes:

the two-path system,
the record-bearing degrees of freedom,
the measurement interaction,
the distinguishability of record states,
the accessibility parameter η,
the operational readout procedure,
the relevant coarse-graining,
and the tolerance ε_C used in operational equivalence.

This context is fixed before outcome comparison. The candidate class, realization-compatible subclass, burden functional, equivalence relation, accessibility calibration, tolerance, and parameter weights must be defined from C rather than from the observed result.

8.6 Candidate channels

Define three schematic channels:

Φ₀ = a realization-compatible channel associated with record R₀.
Φ₁ = a realization-compatible channel associated with record R₁.
Φ_mix = a non-selective decoherence-compatible mixture channel.

The distinction is essential. Φ₀ and Φ₁ represent individual realization candidates. Φ_mix represents non-selective decoherence-compatible structure. It may belong to the broader dynamical description, but it does not by itself constitute a completed individual realization channel unless additional realization content is supplied.

8.7 Candidate class and realization-compatible subclass

A broad candidate class may be written as:

𝒜(C) = {Φ₀, Φ₁, Φ_mix}.

The subclass of individual realization-compatible candidates may be narrower:

𝒜_real(C) = {Φ₀, Φ₁}.

This distinction allows the model to include decoherence-compatible dynamics without conflating it with individual realization.

The exact membership of 𝒜_real(C) depends on admissibility conditions specified for C. If a model treats Φ_mix as sufficient for realization, then it must explain how a non-selective channel supplies individual realization content. If it cannot do so, Φ_mix remains decoherence-compatible rather than realization-complete.

This is the first non-reduction test inside the instantiation.

8.8 Accessibility parameter η

Let:

η ∈ [0,1]

measure the operational accessibility of the relevant record information.

When η = 0, no relevant record is operationally accessible in C.
When η = 1, the relevant record is fully accessible.
Intermediate values represent partial accessibility.

The accessibility parameter helps distinguish mere formal correlation from context-supported record structure. A CBR model must not impose which-path realization when the context does not support the relevant record structure.

In this paper, η is used only schematically. Full platform-specific calibration of η belongs to the empirical exposure program.

8.9 Toy burden functional

A minimal burden functional for the two-path context may be written as:

ℛ_C(Φ) = λ_S S_C(Φ) + λ_I I_C(Φ) + λ_P P_C(Φ) + λ_D D_C(Φ).

Here:

S_C(Φ) is a record-stability burden.
I_C(Φ) is an information-accessibility burden.
P_C(Φ) is a probability-compatibility burden.
D_C(Φ) is a dynamical-compatibility burden.

The weights satisfy:

λ_S, λ_I, λ_P, λ_D ≥ 0.

This expression is a toy burden functional. It is not claimed to be the final universal ℛ_C for all measurement contexts. Its role is to show how the reconstruction can be populated by explicit burden terms.

The weights λ_S, λ_I, λ_P, and λ_D must be fixed before outcome comparison. They are not adjustable outcome-fitting parameters. If they are estimated from calibration data, the calibration protocol must be specified before testing the realization claim, and calibration data must be separated from the outcome test.

8.10 Minimizer set in the two-path model

Given the toy functional, define:

M_C = {Φ ∈ 𝒜(C) : ℛ_C(Φ) = inf{ℛ_C(Ψ) : Ψ ∈ 𝒜(C)}}.

The model succeeds only if M_C identifies a unique candidate, contains only mutually operationally equivalent candidates, or is governed by a pre-specified tie rule.

If Φ_mix lies in M_C, the model must explain whether Φ_mix is merely decoherence-compatible or also realization-complete. If it cannot supply that additional realization content, Φ_mix cannot serve as the completed individual realization channel.

If Φ₀ and Φ₁ are tied, the model must determine whether they are operationally equivalent under ≃_C or whether a Born-compatible tie or sampling rule has been specified in advance.

Thus, even the minimal model is not allowed to hide degeneracy.

8.11 High-accessibility regime

When η is high and the record states are distinguishable, the context supports individual record realization. In the idealized limit:

⟨R₀|R₁⟩ ≈ 0.

In such a context, Φ₀ and Φ₁ may both qualify as realization-compatible candidates. Across repeated equivalent contexts, Born compatibility requires:

P(Φ₀) = │α│²,
P(Φ₁) = │β│².

These equations do not replace Born weighting with minimization. They state the ensemble compatibility condition that any realization structure must preserve.

In an individual trial, the model may identify a selected realization channel or selected realization class. Across repeated trials, the distribution of selected alternatives must remain Born-compatible unless a deviation has been explicitly declared and exposed to test. No such deviation is asserted here.

8.12 Low-accessibility regime

When η is low, the context may not support which-path realization. In such a case, CBR must not falsely impose record structure unsupported by the physical and operational conditions of C.

Several possibilities arise.

Φ₀ and Φ₁ may be excluded from 𝒜_real(C).
Φ₀ and Φ₁ may become operationally equivalent under ≃_C because no accessible test in C distinguishes them.
The context may remain below the threshold required for individual realization in the modeled sense.
The model may be incomplete if it asserts realization without admissible record support.

The point is not to force a preferred result. The point is to require that admissibility and realization claims track the physical context rather than outcome preference.

8.13 Symmetric regime and degeneracy

If:

│α│² = │β│²

and the context is symmetric, Φ₀ and Φ₁ may have equal realization burden.

In that case, CBR succeeds only if the tied minimizers are operationally equivalent under ≃_C, or if a pre-specified Born-compatible tie rule exists. If the minimizers are operationally distinct and no tie rule has been specified before outcome comparison, the model remains incomplete in that context.

This is an important limitation. CBR does not gain credibility by hiding degeneracy. It gains credibility by stating the condition under which degeneracy is harmless and the condition under which it is a failure.

8.14 Instantiation proposition

Proposition 8.1 — Minimal two-path instantiability.
There exists a minimal two-path context in which the CBR-form reconstruction can be populated by explicit formal objects: a context C, candidate channels Φ₀, Φ₁, and Φ_mix, an admissible class 𝒜(C), a realization-compatible subclass 𝒜_real(C), an accessibility parameter η, a toy burden functional ℛ_C, a minimizer set M_C, an operational equivalence relation ≃_C, Born-compatibility conditions, parameter-fixity requirements, and a non-reduction distinction between Φ∗_C and Φ_mix.

Proof sketch.
The two-path state supplies a minimal outcome space. The record states supply a record-bearing structure. Decoherence supplies the non-selective mixture comparison Φ_mix. Candidate channels Φ₀ and Φ₁ supply individual realization-compatible alternatives. The context C fixes the physical and operational setup. The accessibility parameter η characterizes record availability. The toy functional ℛ_C supplies a comparison structure. The minimizer set M_C identifies candidate minimizers. The equivalence relation ≃_C determines whether formal multiplicity is operationally meaningful. Born compatibility constrains repeated-trial frequencies. Parameter fixity and non-reduction conditions prevent post hoc adjustment and decoherence collapse. Therefore, the reconstructed CBR objects can be instantiated in a minimal nontrivial setting.

This proposition does not show that CBR is true. It shows that the reconstruction is not empty formalism.

8.15 What the model does not establish

The model does not prove that CBR is true. It does not prove that the toy ℛ_C is final. It does not derive the Born rule. It does not defeat rival interpretations. It does not establish a universal theory of measurement. It does not predict a new empirical deviation in this paper.

It shows only that CBR-form reconstruction can be made operationally explicit in a minimal nontrivial context.

8.16 Failure conditions for the two-path instantiation

The two-path model fails as a disciplined CBR instantiation if any of the following occurs:

C is not fixed before outcome comparison.

𝒜(C) or 𝒜_real(C) is defined after the outcome is known.

Φ_mix exhausts all alleged realization content.

ℛ_C requires post hoc tuning of λ_S, λ_I, λ_P, or λ_D.

η is adjusted after outcome observation.

M_C is empty with no pre-specified ε-minimizer rule.

M_C contains operationally distinct minimizers with no pre-specified tie rule.

Born-compatible ensemble behavior is violated without a declared deviation claim.

These failure conditions are part of the demonstration. They show that the model is not insulated from criticism.

8.17 Conclusion of the instantiation

The two-path model demonstrates that CBR-form reconstruction can be instantiated without replacing Born-rule weighting or reducing realization to non-selective decoherence.

Its achievement is limited but important. It shows that the reconstructed law-form has usable formal content in a minimal setting. It also shows how CBR must behave under high accessibility, low accessibility, symmetry, degeneracy, and decoherence comparison.

The broader question is whether such context-specific structures can be developed, calibrated, tested, and survived in richer physical platforms. That question belongs to the wider CBR program.

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9. Degeneracy, Ties, and Born-Compatible Sampling

The reconstruction developed above represents realization selection through context-relative comparison and minimization. However, minimization does not always produce a single formal candidate. In many physically meaningful contexts, multiple admissible candidates may carry equal, indistinguishable, or near-minimal realization burden. This is not a peripheral technicality. It is one of the central tests of whether a proposed realization law is actually disciplined.

If degeneracy is ignored, a selection law may appear more determinate than it is. If degeneracy is resolved only after the realized outcome is known, the law becomes circular. For that reason, CBR must state in advance how exact ties, operationally equivalent minimizers, operationally distinct minimizers, near-minimizers, and sampling among realization-compatible alternatives are handled.

Let the minimizer set be:

M_C = {Φ ∈ 𝒜(C) : ℛ_C(Φ) = inf{ℛ_C(Ψ) : Ψ ∈ 𝒜(C)}}.

M_C contains all admissible candidates whose realization burden is minimal in context C. The simplest case is unique minimization. The scientifically more important case is degeneracy.

The purpose of this section is to distinguish three things that must not be confused:

formal multiplicity,
operational ambiguity,
and probability sampling.

Formal multiplicity is harmless when the alternatives are operationally equivalent. Operational ambiguity is a problem when distinct minimizers remain unresolved. Probability sampling concerns ensemble frequencies and must remain Born-compatible. A disciplined realization law must keep these levels separate.

9.1 Unique minimization

The strongest selection case occurs when:

M_C = {Φ∗_C}.

In this case, the CBR-form model selects a single realization-compatible channel in context C. No quotienting or tie rule is required.

This is the cleanest case, but it is not the only acceptable one. A physical theory need not eliminate every formal multiplicity if the multiplicity has no operational consequence. The relevant standard is therefore not always syntactic uniqueness. It is operational uniqueness.

9.2 Operationally harmless degeneracy

Multiple minimizers are acceptable when they are operationally equivalent in the context under analysis. Formally, CBR succeeds if:

∀ Φᵢ, Φⱼ ∈ M_C, Φᵢ ≃_C Φⱼ.

This condition states that every pair of minimizers is indistinguishable by the accessible tests, records, and readout procedures available in C. In such a case, the formal multiplicity of minimizers does not imply physical ambiguity. The model selects one operational realization class even if it does not select one unique mathematical representative.

The selected object may therefore be written as:

Φ∗C = [M_C]≃C.

This means that Φ∗_C denotes the operational equivalence class of minimizers, not necessarily a single syntactic representative.

Operationally harmless degeneracy is not a weakness. It is the correct treatment of representational multiplicity when no accessible procedure in C distinguishes the alternatives.

9.3 Operationally significant degeneracy

Degeneracy becomes problematic when two or more minimizers are operationally distinct:

∃ Φᵢ, Φⱼ ∈ M_C such that Φᵢ ≄_C Φⱼ.

In this case, minimization has not selected one operational realization class. The model must then supply a tie rule. That tie rule must be fixed before outcome comparison. If it is introduced only after observing the result, it is not a law-like completion. It is a post hoc repair.

Thus, a CBR-form model has three possible statuses with respect to degeneracy.

First, unique minimization: M_C contains one candidate.

Second, operationally harmless degeneracy: all minimizers are equivalent under ≃_C.

Third, incomplete selection: M_C contains operationally distinct minimizers and no pre-specified tie rule exists.

The third case is a genuine failure of completeness in context C.

9.4 Tie rules

A tie rule is permitted only if it is fixed before outcome comparison and applies only to unresolved minimizers or a pre-defined minimal admissibility class. It cannot be introduced after the realized outcome is known. It cannot be selected because it favors the observed result. It cannot secretly replace Born weighting unless a controlled deviation has been declared in advance.

A legitimate tie rule must satisfy four conditions.

It must be fixed before outcome comparison.

It must operate only on M_C or on a pre-defined minimal admissibility class.

It must preserve operational equivalence structure.

It must preserve Born-compatible ensemble behavior unless a controlled non-Born deviation has been specified in advance.

A tie rule that violates these conditions is not a lawful completion of the model. It is outcome-fitting.

9.5 Tie rules and Born-compatible sampling

If a tie rule is required, it must remain compatible with Born-rule statistics. In a two-outcome context with state:

|ψ⟩ = α|0⟩ + β|1⟩,

a Born-compatible tie or sampling rule must preserve:

P(Φ₀) = |α|²,
P(Φ₁) = |β|².

These equations state that across repeated equivalent contexts, realization-compatible alternatives must occur with the Born-rule weights associated with the relevant state components.

This requirement prevents minimization from being mistaken for a replacement of probability weighting. CBR selection concerns the admissible realization structure in a context. Born-compatible sampling concerns the frequencies with which realization-compatible alternatives occur across repeated equivalent contexts.

The two roles must remain distinct.

9.6 Selection is not sampling

CBR does not ask probability to do realization’s job, and it does not ask realization to do probability’s job.

The selection structure determines which realization-compatible channels are admissible, how they are compared, and whether the minimizer set defines one operational realization class in context C. Born weighting determines the frequencies with which outcomes occur across repeated equivalent contexts.

The incorrect statement is:

ℛ_C directly replaces Born weighting.

The correct statement is:

ℛ_C constrains admissible realization structure, while Born compatibility constrains ensemble occurrence frequencies.

A CBR model that collapses selection into sampling risks becoming redundant with probability theory. A CBR model that lets minimization replace Born weighting risks violating quantum statistics. The reconstruction avoids both errors by assigning distinct roles to realization structure and probability weighting.

9.7 Near-degeneracy and ε-minimizers

Exact minimization may fail to capture physically relevant near-degeneracy when burden differences fall below the operational resolution of C. In such cases, an ε-minimizer rule may be used only if ε_C is fixed before outcome comparison.

Define the ε-minimizer set:

M_C^ε = {Φ ∈ 𝒜(C) : ℛ_C(Φ) ≤ inf{ℛ_C(Ψ) : Ψ ∈ 𝒜(C)} + ε_C}.

Here ε_C is not a rescue parameter. It must be determined by the operational limits of C or by a pre-specified calibration procedure. It cannot be adjusted after the result is known.

Near-degeneracy is therefore governed by the same rule as exact degeneracy: it is acceptable only when the tolerance is fixed in advance and when operationally distinct candidates are resolved by equivalence or by a pre-specified tie rule.

9.8 Proposition 9.1 — Degeneracy Discipline

A CBR-form model is complete with respect to degeneracy in context C only if one of the following conditions holds:

M_C contains exactly one candidate.

All candidates in M_C are equivalent under ≃_C.

Operationally distinct candidates in M_C are resolved by a tie rule fixed before outcome comparison.

If none of these conditions holds, the model is incomplete in C.

Proof sketch.
If M_C has one element, selection is formal and operational. If M_C has multiple elements but all are equivalent under ≃_C, formal multiplicity collapses to one operational class. If M_C contains operationally distinct candidates, a pre-specified tie rule is required to complete selection. Without such a rule, no unique operational realization class has been selected.

9.9 Failure modes for degeneracy

A CBR-form model fails the degeneracy discipline in context C if any of the following occurs:

M_C contains operationally distinct minimizers and no tie rule exists.

The tie rule is introduced after the outcome is known.

The equivalence relation ≃_C is widened after the result to hide degeneracy.

The tolerance ε_C is adjusted after the result to convert distinct minimizers into near-minimizers.

A Born-incompatible sampling rule is introduced without a declared deviation claim.

These are not minor technical defects. They are failures of selection discipline.

9.10 Why degeneracy discipline matters

Degeneracy discipline protects the reconstruction from two opposite errors.

The first error is over-determinism: claiming that CBR selects a unique result even when the minimization structure has not done so.

The second error is post hoc completion: introducing an equivalence relation, tolerance, or tie rule after observing the result.

The degeneracy clause avoids both errors. CBR succeeds when the minimizer set is unique, operationally unique, or governed by a pre-specified tie rule. It fails or remains incomplete when operationally distinct minimizers remain unresolved.

This makes the model more constrained, not less. It gives critics a clear evaluation procedure: identify C, define 𝒜(C), specify ℛ_C, compute or characterize M_C, determine whether minimizers are equivalent under ≃_C, and check whether any required tie rule was fixed before outcome comparison.

9.11 Degeneracy conclusion

Degeneracy is not an embarrassment to the reconstruction. It is one of the places where the reconstruction becomes disciplined.

A theory that pretends degeneracy cannot occur is too strong. A theory that resolves degeneracy after the fact is circular. The CBR-form standard is narrower and more serious: degeneracy is acceptable only when it is operationally harmless or resolved by a pre-specified, Born-compatible rule.

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10. Non-Circularity and the Anti-Cheating Rule

The central danger for any proposed law of individual outcome realization is circularity. A theory cannot explain an outcome if the machinery that selects the outcome is defined only after the outcome is known. This section states the anti-circularity requirement in its strongest form.

A CBR-form model is non-circular only if the following are specified before outcome comparison:

C, the measurement context;
𝒜(C), the admissible candidate class;
𝒜_real(C), where an individual realization-compatible subclass is used;
ℛ_C, the realization-burden functional or comparison structure;
M_C, or the rule by which minimizers are identified;
≃_C, the operational equivalence relation;
ε_C, any operational tolerance used in defining equivalence or approximate minimization;
η and its calibration, where accessibility is used;
any tie rule required to handle degeneracy;
and any λᵢ weights or adjustable terms appearing in ℛ_C.

The purpose of this requirement is not cosmetic. It prevents the theory from being fitted to the observed result.

10.1 The anti-cheating rule

The anti-cheating rule is simple:

No law-defining object may be chosen, narrowed, tuned, calibrated, or reinterpreted after the realized outcome is known in order to make that outcome selected.

This rule applies to the context, candidate class, admissibility conditions, realization-compatible subclass, burden functional, equivalence relation, tolerances, accessibility parameter, calibration procedure, minimizer rule, tie rule, and all functional weights.

The rule does not make CBR true. It makes CBR evaluable. Once the relevant objects are fixed before outcome comparison, critics can inspect whether the model genuinely selects, fails, degenerates, reduces to decoherence, or violates probability compatibility.

10.2 The forbidden post hoc functional

The clearest example of circularity is a functional that directly rewards the observed outcome:

ℛ_C(Φ) = 0 if Φ equals the observed outcome, and 1 otherwise.

This expression is not a law. It is retrospective labeling. It does not explain why the observed outcome was realized. It merely assigns zero burden to whatever outcome happened.

A functional of this kind violates functional fixity because it depends on the selected result. It also violates parameter discipline if its effective criterion is adjusted after observation. Any CBR model using such a construction fails as a non-circular law-candidate.

10.3 Legitimate context dependence

The anti-cheating rule does not prohibit dependence on the measurement context. CBR is explicitly context-relative. A valid ℛ_C may depend on features of C, such as record stability, dynamical compatibility, accessibility, probability compatibility, and operational distinguishability.

The distinction is between context-dependence and outcome-dependence.

A valid functional may say: This candidate has lower burden because it better satisfies pre-specified constraints of C.

An invalid functional says: This candidate has lower burden because it is the one that was observed.

Only the first is law-like.

10.4 Formal non-circularity condition

The non-circularity requirement may be expressed schematically as independence from Φ∗_C:

𝒜(C) independent of Φ∗_C,
𝒜_real(C) independent of Φ∗_C,
ℛ_C independent of Φ∗_C,
M_C rule independent of Φ∗_C,
≃_C independent of Φ∗_C,
ε_C independent of Φ∗_C,
η-calibration independent of Φ∗_C,
tie rule independent of Φ∗_C,
λᵢ independent of Φ∗_C.

These are structural independence conditions. They are not ordinary derivatives in a smooth calculus sense. The candidate class, realization-compatible subclass, burden functional, minimizer rule, equivalence relation, tolerances, accessibility calibration, tie rule, and parameter weights must not depend on the selected realization channel.

Equivalently: selection machinery fixed before selected result.

If that condition fails, the model is not a realization law. It is an outcome-fitting procedure.

10.5 Parameter fixity and calibration

Parameter fixity does not require every model to be parameter-free. It requires that parameters be fixed by theory, context, or pre-specified calibration before outcome testing.

If a realization functional takes the form:

ℛ_C(Φ) = λ_S S_C(Φ) + λ_I I_C(Φ) + λ_P P_C(Φ) + λ_D D_C(Φ),

then the values or determination procedures for λ_S, λ_I, λ_P, and λ_D must be fixed before outcome comparison.

If calibration data are used, three conditions must hold.

The calibration protocol must be specified before the outcome test.

The calibration data must be separated from the outcome test.

The calibrated parameters must not be adjusted after observing the realized outcome.

Otherwise, the functional becomes adjustable enough to fit the result, and the model loses its non-circular status.

10.6 Permitted revision versus post hoc rescue

A model may be revised after failure, but the revision does not save the failed instantiation. It creates a new model.

This distinction is essential. Science permits revision. The anti-cheating rule prohibits retrospective rescue.

If C, 𝒜(C), 𝒜_real(C), ℛ_C, M_C rules, ≃_C, ε_C, η, a tie rule, or λᵢ are changed after a failed or ambiguous result, the revised construction may be studied as a new CBR-form model. But it cannot be counted as the same model having survived the test.

This rule prevents the theory from moving the target after outcome comparison.

10.7 Proposition 10.1 — Non-Circularity Lock

A CBR-form model is non-circular in context C only if all law-defining objects are fixed independently of Φ∗_C.

If any law-defining object is chosen, narrowed, tuned, calibrated, or reinterpreted after the realized outcome is known in order to make that outcome selected, the model fails as a non-circular realization-law candidate in C.

Proof sketch.
A realization law claims to identify the selected channel. If the objects that perform selection are themselves defined by the selected channel, the explanation is circular. Fixing the law-defining objects before outcome comparison blocks that circularity. Adjusting them afterward reintroduces it.

10.8 Failure modes for non-circularity

A CBR-form model fails the non-circularity lock if any of the following occurs:

C is redescribed after the result to favor the outcome.

𝒜(C) is narrowed after the result.

𝒜_real(C) is defined retrospectively.

ℛ_C is tuned after the result.

≃_C is widened after the result to erase a degeneracy.

ε_C is changed after the result to create or remove near-degeneracy.

η or its calibration is adjusted after the result.

A tie rule is introduced after the result.

λᵢ weights are adjusted after the result.

These are not revisions inside the same tested model. They are new models.

10.9 Consequence of the anti-cheating rule

If C, 𝒜(C), 𝒜_real(C), ℛ_C, M_C rules, ≃_C, ε_C, η-calibration, tie rules, and all λᵢ are fixed before outcome comparison, then CBR cannot be dismissed merely as naming the outcome after the fact.

It may still be wrong. Its functional may be physically inadequate. Its candidate class may be incomplete. Its minimizer set may be empty. Its selected minimizers may remain operationally distinct. Its Born compatibility may fail. Its non-reduction claim may collapse into decoherence. But the specific charge of post hoc selection is blocked.

This is the central anti-cheating rule.

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11. Probability Compatibility

A candidate law of outcome realization must preserve the empirical role of quantum probability. CBR is not introduced here to replace the Born rule. It is introduced as a candidate structure for realization that must remain compatible with the Born rule unless a controlled, explicitly modeled, empirically vulnerable deviation is proposed.

The distinction is essential. A theory of realization asks what admissible structure is selected in a context. A theory of probability weighting asks how frequently outcomes occur across repeated trials. These questions are related, but they are not identical.

This section states the probability discipline required by the reconstruction.

11.1 Born-rule preservation

For a two-outcome state:

|ψ⟩ = α|0⟩ + β|1⟩,

Born-rule preservation requires:

lim N₀/N = |α|² as N → ∞,
lim N₁/N = |β|² as N → ∞.

Here N₀ and N₁ are the numbers of trials associated with the two realized alternatives, and N is the total number of trials.

For a general state:

|ψ⟩ = Σᵢ αᵢ|i⟩,

the corresponding requirement is:

P(i) = |αᵢ|².

A CBR model that preserves these limits is Born-compatible.

This paper imposes Born compatibility as a condition. It does not claim that the Born rule is derived here.

11.2 Realization role versus probability role

CBR asks:

What admissible realization structure is selected in context C?

The Born rule asks:

How frequently do outcomes occur across repeated equivalent contexts?

CBR must not confuse these roles. If minimization were interpreted as directly replacing Born weighting, the model would risk contradicting standard quantum statistics. The disciplined position is that CBR constrains admissible realization structures, while Born compatibility constrains ensemble occurrence.

Thus, the realization-burden functional ℛ_C does not automatically determine long-run frequencies unless the model explicitly defines how realization-compatible alternatives are sampled or weighted.

In the present reconstruction, Born compatibility is imposed as a requirement.

11.3 Selection, sampling, and ensemble behavior

Selection and sampling are different levels of description.

Selection concerns the admissible structure of an individual realization context.

Sampling concerns the distribution of realized alternatives across repeated equivalent contexts.

A CBR-form model may constrain which channels qualify as realization-compatible, how those channels are compared, and whether the minimizer set defines one operational realization class. But unless the model also supplies a probability law, it must preserve the Born-rule distribution already fixed by standard quantum mechanics.

Therefore, in this reconstruction:

ℛ_C governs admissibility and realization burden.

Born compatibility governs ensemble frequencies.

A future extension may attempt to derive or modify the probability rule, but that is not the burden of the present paper.

11.4 Probability Compatibility Principle

The probability discipline of this paper may be stated as a principle.

Principle 11.1 — Probability Compatibility.
A CBR-form realization model must preserve Born-rule ensemble frequencies unless it explicitly states a controlled non-Born deviation, fixes the deviation before testing, defines the relevant baseline, bounds nuisance effects, and states the condition under which the deviation claim fails.

This principle prevents the realization law from becoming an uncontrolled rewrite of quantum probability.

11.5 Non-Born deviations

A CBR model may, in principle, propose a non-Born deviation. But such a proposal carries additional burdens. It must specify:

the context in which the deviation is expected;
the modified probability, visibility, or observable prediction;
the experimental conditions under which the deviation should appear;
the baseline standard-quantum prediction;
the tolerance or error budget;
the nuisance class;
and the failure condition if the deviation is not observed.

Without such specification, non-Born behavior is not a prediction. It is an uncontrolled departure from established quantum statistics.

The present paper does not assert a non-Born deviation. It reconstructs CBR as a Born-compatible law-candidate form.

11.6 Probability failure condition

A CBR model fails the probability-compatibility burden if it predicts frequencies inconsistent with the Born rule without a controlled, explicitly modeled, empirically vulnerable deviation.

The failure condition can be stated plainly:

If repeated equivalent contexts yield frequencies inconsistent with P(i) = |αᵢ|², and the model has not specified a justified deviation claim in advance, then the model fails in that domain.

This does not mean that no future CBR model could ever propose a deviation. It means that any such proposal must be treated as an empirical prediction and exposed to test. It cannot be introduced after the fact to explain anomalous or convenient data.

11.7 Proposition 11.1 — Probability Discipline

A CBR-form model satisfies probability discipline only if one of the following holds:

It preserves Born-rule ensemble frequencies across repeated equivalent contexts.

It declares a controlled non-Born deviation in advance, specifies the relevant baseline and error budget, and states the condition under which the deviation claim fails.

If neither condition holds, the model fails the probability-compatibility burden.

Proof sketch.
The reconstruction class requires compatibility with standard quantum probability unless a controlled deviation is explicitly introduced. Therefore, a model either preserves Born statistics or assumes the additional burdens of a deviation claim. A model that does neither is not probability-compatible.

11.8 Why probability compatibility strengthens the reconstruction

Born compatibility restricts CBR. That restriction strengthens the reconstruction.

It prevents the model from using “realization” as a license to rewrite quantum statistics. It prevents minimization from being mistaken for sampling. It keeps the law-candidate focused on the structural question of realization while preserving the empirical role of standard quantum probability.

The point of CBR in this paper is not to replace the probability structure of quantum mechanics. It is to reconstruct the minimal formal structure a candidate realization law would need if it is to coexist with that probability structure.

11.9 Probability conclusion

The probability burden can be summarized as follows:

CBR does not replace the Born rule in this reconstruction.

CBR does not derive the Born rule in this paper.

CBR does not assert non-Born frequencies here.

CBR constrains admissible realization structure.

Born compatibility constrains ensemble frequencies.

Any future non-Born claim must be modeled, bounded, and exposed to test.

This conservative probability posture is not a retreat. It is what makes the reconstruction disciplined.

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12. Non-Reduction to Decoherence

Decoherence is indispensable to modern quantum theory. It explains how interaction with environmental or record-bearing degrees of freedom suppresses interference, stabilizes pointer-like structures, and supports effectively classical records. Any serious candidate law of outcome realization must preserve these achievements.

CBR does not reject decoherence. Its non-reduction requirement is narrower: CBR must supply realization-selection structure not already exhausted by a non-selective decoherence-compatible description. If it does, it occupies an independent formal role. If it does not, it collapses into decoherence and fails as an independent realization-law candidate.

The question is therefore not whether decoherence matters. It does. The question is whether decoherence alone supplies the law-form of individual realization selection.

12.1 Decoherence as background, not opponent

In a two-path context with record states |R₀⟩ and |R₁⟩, decoherence is associated with suppression of interference as:

⟨R₀|R₁⟩ → 0.

In that limit, the alternatives become effectively distinguishable at the level of record structure, and the reduced description suppresses off-diagonal coherence terms.

This is a genuine physical result. Decoherence explains why interference becomes inaccessible in many measurement contexts and why stable record-like structures emerge. CBR does not deny this. It treats decoherence as part of the physical background from which any disciplined realization-law candidate must proceed.

The issue is whether a non-selective decoherence-compatible description is already equivalent to individual realization selection.

12.2 The remaining realization question

The remaining question is not whether decoherence occurs. The question is whether decoherence alone supplies a law-form for individual outcome realization.

A non-selective decoherence-compatible channel may describe the transition from coherent superposition to an effectively classical mixture in a reduced description. But a mixture is not automatically the same thing as an individually realized record. If a realization law is sought, it must specify how one admissible realization-compatible structure is selected, or how one operationally unique realization class is obtained, in context C.

CBR addresses that residual question by requiring the following objects:

C,
𝒜(C),
ℛ_C,
M_C,
≃_C,
Φ∗_C.

If these objects add no content beyond decoherence, CBR fails as an independent law-candidate. If they supply a non-redundant selection structure, CBR occupies a distinct formal role.

12.3 Three levels that must not be collapsed

The non-reduction issue is clearest when three levels are separated.

First, decoherence-compatible evolution: interference suppression, environmental entanglement, record stabilization, and mixture-like reduced descriptions.

Second, record structure: the existence, stability, accessibility, and operational distinguishability of record-bearing degrees of freedom.

Third, realization selection: the candidate law-form by which one admissible realization-compatible channel or operational realization class is selected in context C.

CBR does not replace the first level. It depends on the second. It is aimed at the third.

If the third level can be fully reduced to the first two, then CBR loses independent status. If the third level requires a distinct selection structure, CBR has a formal role to occupy.

12.4 Decoherence channel versus realization channel

Let Φ_mix denote a non-selective decoherence-compatible channel. In the two-path model, Φ_mix represents the mixture-like structure associated with decoherence, interference suppression, and record stabilization.

Let Φ∗_C denote the selected realization-compatible channel or selected operational equivalence class under CBR.

The distinction is:

Φ_mix describes non-selective decoherence-compatible evolution.

Φ∗_C represents selected realization-compatible structure.

The two may be related, but they are not definitionally identical. Decoherence may help define the admissible record-bearing structure of C, determine which candidates belong to 𝒜(C), and contribute to burden terms in ℛ_C. But unless Φ∗_C performs selection work not already exhausted by Φ_mix, CBR has not established an independent role.

12.5 Proposition 12.1 — Non-Reduction Criterion

A CBR-form model is independent of decoherence in context C only if the selected realization structure Φ∗_C supplies realization content not exhausted by the non-selective decoherence-compatible channel Φ_mix.

If:

Φ∗_C ≃_C Φ_mix

in all relevant respects and no additional realization-selection content is supplied, then the model fails as an independent realization-law candidate in C.

Proof sketch.
CBR is reconstructed as a candidate law-form for individual realization selection. Decoherence, represented by Φ_mix, supplies non-selective interference suppression and record-stabilizing structure. If Φ∗_C adds no selection content beyond Φ_mix, then the CBR-selected structure is operationally indistinguishable from the decoherence-compatible structure. The purported realization law has therefore become a reformulation of decoherence rather than an independent selection law.

12.6 What counts as realization content

The non-reduction burden is not satisfied by verbal distinction. CBR must identify the formal role played by Φ∗_C that is not already exhausted by Φ_mix.

Such content may include:

a context-relative selection of an admissible realization-compatible channel;

an operational equivalence class of selected minimizers;

a pre-specified rule for resolving operationally distinct minimizers;

a formal distinction between decoherence-compatible mixture structure and individual realization-compatible structure;

or a failure condition under which absence of such distinction defeats the model.

The burden is not to deny that decoherence is necessary or powerful. The burden is to show that CBR does more than rename it.

12.7 Reduction failure condition

A CBR-form model fails the non-reduction burden in context C if any of the following holds:

Φ∗_C adds no realization content beyond Φ_mix.

Φ∗_C is operationally equivalent to Φ_mix in all relevant respects.

The model cannot distinguish selected realization structure from non-selective decoherence-compatible structure.

The alleged distinction is verbal only and has no formal role in 𝒜(C), ℛ_C, M_C, ≃_C, or Φ∗_C.

This is a real vulnerability. A critic does not need to disprove all CBR-like models to defeat a particular model. It is enough to show that the model’s selected structure adds no content beyond decoherence in the relevant context.

12.8 Non-reduction conclusion

The non-reduction requirement strengthens CBR because it prevents the framework from receiving credit for work already done by decoherence. A weaker proposal would simply assert that decoherence is incomplete. A stronger proposal states the exact condition under which reduction to decoherence defeats it.

CBR survives as an independent realization-law candidate only if Φ∗_C performs formal selection work that Φ_mix does not.

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13. Failure Theorem

The reconstruction developed in this paper is useful only if it exposes CBR to failure. A candidate law of outcome realization cannot be treated as disciplined if every defect is reinterpreted as success. Failure conditions are therefore not external criticisms added after the fact. They are part of the law-candidate structure itself.

The following theorem formalizes the ways in which a CBR-form model can fail in context C. Failure in a given context means failure of that model as a disciplined realization-law candidate in that context. It does not prove the logical impossibility of every possible CBR-like theory.

The theorem’s purpose is not to weaken CBR. It is to make CBR evaluable.

13.1 Theorem statement

Theorem 2 — Failure Theorem for CBR Law-Candidates

A CBR-form model fails as a disciplined realization-law candidate in context C if any of the following conditions holds:

C is not physically specified.

𝒜(C) is empty.

𝒜(C) is undefined, arbitrary, or unrestricted enough to make any result selectable.

𝒜(C) is defined or narrowed after the realized outcome is known.

𝒜_real(C), where used, is defined after the realized outcome is known.

The comparison structure is not specified.

ℛ_C, where used, is defined after the realized outcome is known.

ℛ_C rewards the observed outcome merely because it is observed.

≃_C or ε_C is defined, widened, narrowed, or adjusted after the realized outcome is known.

η or its calibration is adjusted after the realized outcome is known.

Parameters in ℛ_C, including any λᵢ, are adjusted after outcome observation.

ℛ_C is not bounded below and no pre-specified ε-minimizer rule is given.

M_C is empty and no pre-specified approximate-selection rule exists.

M_C contains operationally distinct minimizers with no pre-specified tie rule.

A tie rule is introduced after the outcome is known.

The model violates Born-rule frequencies without a registered, explicitly modeled, empirically vulnerable deviation claim.

Φ∗_C adds no realization content beyond Φ_mix in the relevant context.

No structural or empirical condition could count against the model.

If any of these conditions holds, the model fails to satisfy at least one burden required of a disciplined realization-law candidate.

13.2 Mapping failure conditions to burdens

Each failure condition corresponds directly to a burden developed earlier.

If C is not physically specified, the model fails the domain burden.

If 𝒜(C) is empty, undefined, arbitrary, unrestricted, or constructed after the result is known, the model fails the candidate-set and admissibility burdens.

If 𝒜_real(C) is used but defined after the result is known, the model fails the realization-subclass burden by smuggling the selected result into the definition of realization-compatibility.

If no comparison structure is specified, the model fails the selection burden.

If ℛ_C is defined after the outcome is known, or if it rewards the observed outcome merely because it is observed, the model fails the non-circularity burden.

If ≃_C or ε_C is adjusted after the result is known, the model fails the operational-equivalence burden.

If η or its calibration is adjusted after the result is known, the model fails accessibility fixity.

If λᵢ parameters or other components of ℛ_C are adjusted after outcome observation, the model fails parameter fixity.

If ℛ_C is not bounded below, if M_C is empty, or if no pre-specified ε-minimizer rule exists, the model fails the selection-existence burden.

If M_C contains operationally distinct minimizers with no pre-specified tie rule, the model fails the operational-uniqueness burden.

If a tie rule is introduced after the result is known, the model fails degeneracy discipline.

If the model violates Born-rule frequencies without a registered and explicitly modeled deviation claim, it fails probability compatibility.

If Φ∗_C adds no realization content beyond Φ_mix, the model fails non-reduction.

If no possible structural or empirical condition could count against the model, it fails vulnerability.

13.3 Proof sketch

The theorem follows from the burden structure already established. Each burden is a necessary condition for a disciplined CBR-form law-candidate. If any burden is negated, the model fails in the relevant context.

The proof is constructive in reverse. Sections 2–6 showed that a disciplined realization law requires context specification, admissible candidates, non-circular comparison, operational uniqueness, Born compatibility, non-reduction, parameter discipline, and failure vulnerability. Sections 7–12 sharpened these into corollaries, degeneracy conditions, anti-cheating rules, probability discipline, and non-reduction criteria. The present theorem states that failure of any required condition is sufficient to defeat the model as a law-candidate in C.

This does not imply that no revised CBR-like model could satisfy the burdens. It means that the model under evaluation has failed the relevant burden in the specified context. A revision may be proposed, but the revision is a new model. It does not retroactively save the failed instantiation.

13.4 Context-indexed failure

The theorem is intentionally context-indexed.

If a CBR-form model fails in context C, the conclusion is not:

No CBR-like theory is logically possible.

The conclusion is:

This CBR-form model fails as a disciplined realization-law candidate in context C.

This distinction prevents two errors. It prevents critics from treating one failed instantiation as proof that no realization-law framework could exist. It also prevents defenders from treating failure as irrelevant. The failed model fails. A revised model must be stated as a revised model.

13.5 No-rescue clause

If a model fails one of the burdens above, the failed model is not rescued by redefining C, narrowing 𝒜(C), changing ℛ_C, altering ≃_C, adjusting ε_C, recalibrating η, adding a tie rule, changing λᵢ, or reclassifying Φ_mix after the failure is known.

Such changes may define a new model. They do not rescue the failed one.

This clause is essential. It prevents the theory from moving the target after the result. A failure condition has force only if the model is not allowed to rewrite the conditions of success after failure.

13.6 Why failure conditions strengthen the proposal

The Failure Theorem strengthens CBR because it prevents the framework from being insulated against criticism. It gives critics clear points of attack:

Define C.

Inspect 𝒜(C) and 𝒜_real(C).

Evaluate whether ℛ_C or the comparison structure is fixed and non-circular.

Check whether M_C exists.

Determine whether minimizers are equivalent under ≃_C.

Inspect whether tie rules were fixed before outcome comparison.

Test whether Born-rule frequencies are preserved.

Ask whether Φ∗_C adds content beyond Φ_mix.

Ask whether the model can fail.

This is how a law-candidate should be evaluated. The strength of CBR as reconstructed here is not that it avoids failure. It is that its failure conditions can be stated explicitly.

13.7 Failure Theorem conclusion

The Failure Theorem converts CBR from an interpretive posture into an evaluable law-candidate structure. It states not only how CBR-form models are built, but how they break.

That is the point of the reconstruction. A candidate law becomes serious when it exposes itself to defeat.

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14. Relation to Other Quantum Approaches

The purpose of this section is not to rank or defeat rival quantum interpretations. The purpose is to clarify the role CBR is constructed to occupy. CBR is presented here as a candidate law-form for individual outcome realization. Other approaches may address the measurement problem differently, dissolve it, relocate it, or reject the need for the kind of realization law considered here.

The comparisons below are limited. They identify differences in explanatory target, not final superiority. This paper establishes formal viability under stated assumptions. It does not establish interpretive victory.

CBR should therefore be evaluated by its own declared target: whether a disciplined, non-circular, Born-compatible, non-reductive, failure-capable law-form for individual outcome realization can be reconstructed.

14.1 Copenhagen-type views

Copenhagen-type views often treat measurement through a boundary between quantum description and classical record, or through a transition in description associated with observation, measurement, or experimental context. Such views can be operationally successful without supplying a separate law-form of individual outcome realization.

CBR differs by attempting to formalize a context-relative realization law. It asks whether one can define a physical context C, an admissible candidate class 𝒜(C), a non-circular burden functional or comparison structure ℛ_C, an operational equivalence relation ≃_C, and a selected realization structure Φ∗_C.

This does not show that Copenhagen-type approaches fail. It only shows that CBR occupies a different formal role. It attempts to reconstruct a candidate law-form for realization rather than treating measurement primarily as an operational, descriptive, or classical-boundary transition.

14.2 Everettian approaches

Everettian approaches do not generally seek a unique outcome-selection law in the same sense. Instead, they treat branching structure as physically real or interpretively central and avoid unique selection by allowing branch multiplicity to persist.

CBR targets a different question. It asks what law-form would be required if one seeks individual outcome realization rather than branch-realization pluralism. The CBR structure is therefore not a direct replacement for Everettian branching. It is a candidate for a different explanatory target.

The present paper does not claim that CBR defeats Everettian approaches. It claims only that, for theories that do seek a context-relative realization-selection structure, CBR-form reconstruction identifies the required components.

14.3 Objective collapse theories

Objective collapse theories introduce physical collapse mechanisms, often through modified dynamics, stochastic terms, collapse rates, localization parameters, or additional physical postulates. These theories directly attempt to explain outcome selection by changing or supplementing the dynamical law.

CBR differs in emphasis. As reconstructed here, CBR is framed as selection among admissible context-fixed realization-compatible channels, represented by a burden-minimization or comparison structure. It does not, in this paper, introduce a specific collapse force, collapse rate, stochastic collapse law, or localization dynamics.

This distinction does not make CBR superior to objective collapse theories. It identifies a different formal route. Objective collapse theories typically modify dynamics to produce outcomes. CBR reconstructs the minimal law-form of a candidate structure that compares admissible realization channels in a context.

Any empirical comparison would require context-specific predictions, not merely formal distinction.

14.4 Decoherence-only accounts

Decoherence-only accounts emphasize that environmental interaction suppresses interference and produces stable, effectively classical records. Such accounts explain much of what makes measurement outcomes appear classical and robust.

CBR accepts the importance of decoherence. It does not deny interference suppression or record stabilization. Its claim is narrower: if one seeks an individual realization-selection law, then non-selective decoherence alone does not supply the full selection structure.

The distinction is represented formally by Φ_mix and Φ∗_C. If Φ∗_C adds no content beyond Φ_mix, then CBR reduces to decoherence and fails as an independent realization law. If Φ∗_C supplies additional realization-selection content, then CBR has a distinct formal role.

This does not defeat decoherence. It makes decoherence-reduction an explicit test of CBR’s independence.

14.5 QBism and epistemic interpretations

QBism and related epistemic interpretations often treat quantum probabilities as expressions of agent-centered expectation, belief, information, or decision-guiding commitments rather than as direct descriptions of observer-independent outcome selection.

CBR takes a different route. It seeks a physical context-relative realization criterion represented through admissible candidates, burden comparison, operational equivalence, and selected realization structure. It is therefore less concerned with the agent’s probability assignment and more concerned with the formal structure of candidate realization in context C.

This paper does not claim that epistemic approaches are wrong. It only clarifies that CBR is not primarily an epistemic account of probability. It is a candidate law-form for realization.

14.6 Hidden-variable and pilot-wave approaches

Hidden-variable and pilot-wave approaches typically seek to restore definite outcomes by supplementing the quantum state with additional variables, configurations, trajectories, or guiding structures. In such frameworks, the appearance of outcome selection is tied to the additional ontology or dynamics.

CBR differs by not introducing hidden variables in this reconstruction. It asks what minimal law-form is required if realization is represented as selection among admissible context-relative channels under a burden structure. Its core objects are C, 𝒜(C), ℛ_C, M_C, ≃_C, and Φ∗_C, not hidden configurations or trajectories.

This does not show that hidden-variable theories fail. It only shows that CBR occupies a different formal space.

14.7 Consistent histories and related frameworks

Consistent-histories approaches organize quantum descriptions through sets of histories satisfying consistency or decoherence conditions. Such frameworks can assign probabilities to histories within a consistent family without necessarily introducing a separate individual outcome-realization law of the kind reconstructed here.

CBR differs by focusing on context-relative selection of a realization-compatible structure. It asks not only whether histories or records can be described consistently, but whether a candidate law-form can select an admissible realization class under fixed burden and equivalence conditions.

This comparison is limited. It does not show that histories-based approaches fail. It clarifies that CBR is aimed at a different burden: individual realization selection under explicit admissibility and failure constraints.

14.8 Relational approaches

Relational approaches often treat quantum states or measurement outcomes as relative to systems, observers, or interactions rather than as absolute global facts. Such approaches may dissolve or relocate the demand for a single observer-independent realization law.

CBR differs by asking what law-form would be required if individual realization is treated as a physical selection problem within a specified context C. Its reconstruction is not primarily relational or perspectival. It is context-indexed, but context-indexing is used to define admissibility, burden comparison, and operational equivalence rather than to dissolve the selection question.

This paper does not claim that relational approaches fail. It clarifies that CBR accepts a different target: context-relative law-form reconstruction for realization selection.

14.9 Comparison conclusion

CBR should be evaluated on its declared target.

It should not be judged as a replacement for standard quantum dynamics, because this paper does not claim to replace standard dynamics.

It should not be judged as a derivation of the Born rule, because this paper imposes Born compatibility rather than deriving it.

It should not be judged as a refutation of decoherence, because this paper accepts decoherence and asks whether additional realization-selection structure can be defined.

It should not be judged as a universal defeat of rival interpretations, because this paper makes no such claim.

It should be judged as a conditional reconstruction of a candidate law-form for individual outcome realization. On that target, the relevant questions are clear:

Are C, 𝒜(C), ℛ_C, M_C, ≃_C, and Φ∗_C well-defined?

Is the structure non-circular?

Is it Born-compatible?

Is it non-reductive with respect to Φ_mix?

Is it parameter-disciplined?

Can it fail?

Those are the standards by which the reconstruction should be evaluated.

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15. What the Paper Establishes

The reconstruction should be evaluated by what it establishes, not by stronger claims it deliberately does not make. A candidate law is weakened when it claims more than it has shown. It is strengthened when its result, scope, and remaining burdens are stated exactly.

This paper establishes a conditional structural result: under the assumptions stated above, Constraint-Based Realization can be reconstructed from the burdens imposed on any disciplined candidate law of individual quantum outcome realization.

The reconstructed form is:

Φ∗_C ∈ argmin{ℛ_C(Φ) : Φ ∈ 𝒜(C)}.

This expression is not arbitrary terminology. It is the compressed representation of a law-candidate structure that must define a physical context, identify admissible candidates, compare those candidates by a pre-outcome rule, handle operational uniqueness, preserve Born-compatible ensemble behavior, avoid reduction to non-selective decoherence, control parameter tuning, and expose itself to failure.

15.1 Established results

The paper establishes five results.

First, it establishes the burden-to-form transition. Once the burdens of domain specification, admissibility, comparison, non-circularity, operational uniqueness, probability compatibility, non-reduction, parameter discipline, and vulnerability are imposed, the CBR-form structure follows naturally.

Second, it establishes the Conditional Minimal Representation Theorem. Within the reconstruction class, a disciplined realization-law candidate admits representation by a context C, admissible class 𝒜(C), burden functional or comparison structure ℛ_C, minimizer set M_C, operational equivalence relation ≃_C, and selected realization channel or class Φ∗_C.

Third, it establishes minimal instantiability. The reconstructed objects can be populated in a two-path context using candidate channels Φ₀, Φ₁, and Φ_mix, a realization-compatible subclass 𝒜_real(C), accessibility parameter η, toy burden functional ℛ_C, fixed weights, Born-compatibility conditions, degeneracy discipline, and a non-reduction distinction.

Fourth, it establishes discipline conditions. A CBR-form model must satisfy pre-outcome fixity, operational uniqueness or a pre-specified tie rule, Born compatibility, non-reduction to decoherence, parameter fixity, and explicit vulnerability.

Fifth, it establishes the Failure Theorem. A CBR-form model fails in context C if its context, candidate class, admissibility conditions, comparison rule, minimizer structure, equivalence relation, parameters, Born-compatible behavior, non-reduction condition, or vulnerability condition fails.

These are formal and structural results. They do not confirm CBR empirically. They make CBR precise enough to be evaluated.

15.2 What is not established

This paper does not establish that CBR is true in nature.

It does not establish experimental confirmation. It does not show that any physical system has already displayed a CBR-specific signature. It does not prove that the toy realization functional is the final universal ℛ_C. It does not show that all measurement contexts reduce to the two-path model.

It does not prove that CBR is uniquely required by all physics. It does not prove that rival interpretations are defeated. It does not prove that decoherence is false. It does not replace the Born rule. It does not derive the Born rule. It imposes Born compatibility as a burden of the reconstruction.

It also does not establish universal closure over every possible CBR-like model. The theorem applies to models that accept the burdens stated here. A different theory may reject the target question or adopt different burdens, but then it is no longer a CBR-form realization law in the sense reconstructed in this paper.

These limits are not weaknesses. They define the result precisely.

15.3 Proper conclusion

The proper conclusion is: CBR is formally viable and structurally natural as a candidate law-form of quantum outcome realization under explicit assumptions.

This conclusion is intentionally bounded. Formal viability is not empirical confirmation. Structural naturalness under burdens is not final truth. A minimal two-path instantiation is not universal proof.

Nevertheless, the result is meaningful. It moves CBR beyond suggestive formulation and into disciplined candidate-law structure. The question is no longer whether CBR can be stated coherently. The question is whether context-specific versions of C, 𝒜(C), ℛ_C, M_C, ≃_C, η, and associated parameters can be independently specified, applied, tested, and survived.

That is the appropriate next burden.

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16. Relation to the Larger CBR Program

This paper occupies a bridge position within the larger CBR program. It does not complete the program, and it does not confirm it experimentally. Its role is to connect law-candidate framing with formal reconstruction, minimal instantiation, and future empirical discrimination.

Its contribution is architectural. It explains why the CBR form arises, shows how the form can be minimally instantiated, and states how such a model can fail. It prepares the ground for context-specific modeling and empirical exposure.

16.1 Relation to law-candidate framing

A law-candidate framing paper asks what burdens any serious candidate law of quantum outcome realization must satisfy. It identifies standards of domain, admissibility, comparison, non-circularity, operational uniqueness, Born compatibility, non-reduction, parameter discipline, and vulnerability.

The present paper takes the next step. It does not merely list those burdens. It reconstructs the CBR form from them:

Φ∗_C ∈ argmin{ℛ_C(Φ) : Φ ∈ 𝒜(C)}.

In this sense, the present paper converts the checklist into a representation theorem. It shows how the burdens become formal objects: C, 𝒜(C), ℛ_C, M_C, ≃_C, and Φ∗_C.

16.2 Relation to canonical closure

The canonical closure paper states the mature CBR law-candidate in its closed form, including canonical admissibility, restricted uniqueness, local probability closure, accessibility-based exposure, and strong-null failure logic.

The present paper has a different role. It asks why a law of that general kind is structurally motivated in the first place. It is prior in logical order even if subsequent in publication order.

The relationship is:

The present paper reconstructs why the CBR form arises.

The canonical closure paper states the completed CBR law-candidate.

Empirical exposure papers specify where a context-specific version of the law-candidate can fail.

This placement prevents duplication. The present paper is not another synthesis paper. It is the reconstruction companion.

16.3 Relation to minimal instantiation

A recurring challenge for any formal framework is whether it can be instantiated. A law-form may appear plausible at the abstract level while remaining unclear in concrete models.

The two-path instantiation addresses that concern in a minimal setting. It shows how to define a context, candidates, accessibility, a burden functional, parameter-fixity conditions, Born compatibility, degeneracy discipline, and a decoherence distinction.

This does not prove universal applicability. It shows that the reconstructed structure is usable in a controlled example.

Thus, the paper supplies a demonstration layer between abstract law-form and platform-specific testing.

16.4 Relation to empirical-discrimination papers

Empirical-discrimination papers ask a further question: how could CBR fail or be distinguished from standard quantum and decoherence-based accounts in a concrete platform?

The present paper does not answer that experimentally. It supplies the formal structure that such tests would require. An empirical test must specify C, 𝒜(C), ℛ_C or an equivalent comparison structure, M_C, ≃_C, relevant accessibility parameters such as η, nuisance conditions, and the failure condition under which the model would be disconfirmed.

Without this structure, empirical claims risk becoming vague. With it, proposed tests can be evaluated more sharply.

16.5 Relation to Born-rule work

Some parts of the broader CBR program may attempt a deeper derivation or justification of Born-rule weighting. This paper does not do that. It imposes the narrower condition of Born compatibility.

That choice is deliberate. A reconstruction of the realization law-form need not also derive the probability rule. It must, however, avoid violating it. The present paper therefore treats Born compatibility as a burden, not as a derived theorem.

This keeps the reconstruction conservative and prevents the paper from carrying more than one central burden at once.

16.6 Strategic placement

The present paper sits between three layers of the program.

The first layer is law-candidate framing, which identifies the burdens a realization law must satisfy.

The second layer is formal reconstruction and minimal instantiation, which shows how those burdens yield the CBR form and how that form can be populated in a controlled model.

The third layer is empirical discrimination, which asks how context-specific models could be tested or defeated.

This paper contributes to the second layer. It does not complete the third. Its purpose is to make the program more precise, more formal, and more vulnerable to criticism.

In that sense, the present paper converts CBR from a proposed law-form into a reconstructed minimal structure: formally definable, minimally instantiable, and explicitly failure-capable.

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17. Conclusion

Quantum mechanics provides an extraordinarily successful account of amplitudes, state evolution, outcome spaces, and Born-rule probabilities. The question addressed in this paper has been narrower: if one seeks a candidate law-form for individual outcome realization, what formal burdens must such a law satisfy?

The reconstruction developed here argues that a disciplined realization law must define a physical context, specify admissible candidates, impose admissibility conditions, compare candidates by a pre-outcome criterion, produce operational uniqueness or a pre-specified tie rule, preserve Born-rule statistics, distinguish realization from decoherence, control parameter tuning, and expose itself to failure.

Under those burdens, the CBR-form representation naturally appears:

Φ∗_C ∈ argmin{ℛ_C(Φ) : Φ ∈ 𝒜(C)}.

This expression states that the selected realization channel or class Φ∗_C belongs to the admissible candidate class 𝒜(C) in context C that minimizes the realization-burden functional ℛ_C.

The result is not empirical confirmation. It does not show that nature obeys CBR. It does not replace standard quantum mechanics, replace the Born rule, derive the Born rule, or reject decoherence. It does not prove that the two-path model is universally applicable. It does not defeat rival interpretations by assertion.

The conclusion is narrower and stronger: CBR is formally viable and structurally natural as a candidate law-form of quantum outcome realization under explicit assumptions.

That conclusion matters because it changes the burden of evaluation. The question is no longer whether CBR can be stated coherently. The reconstruction states it coherently, specifies its required objects, demonstrates minimal instantiation, and identifies failure conditions.

The next burden is sharper. Context-specific versions of 𝒜(C), ℛ_C, M_C, ≃_C, η, and associated parameters must be independently specified, applied to concrete physical platforms, compared against standard quantum and decoherence-based accounts, and exposed to structural or empirical failure.

In that sense, the present paper does not complete CBR. It disciplines it. It reconstructs the formal law-candidate structure, demonstrates its minimal instantiation, and states how it can fail.

The next stage is not broader assertion. The next stage is context-specific modeling, pre-specified testing, and survival under comparison.

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Appendix A: Formal Definition Registry

This appendix consolidates the notation used throughout the paper. The definitions do not settle every interpretive question in quantum foundations. Their purpose is narrower: to fix the formal vocabulary required for the reconstruction, theorem statements, two-path instantiation, and failure conditions.

Unless otherwise stated, all law-defining objects are required to be fixed before outcome comparison. This includes C, 𝒜(C), 𝒜_real(C), ℛ_C or its equivalent comparison structure, M_C rules, ≃_C, ε_C, η-calibration, tie rules, and all λᵢ parameters.

A.1 Measurement context

C denotes a physically specified measurement context.

A context includes the physical and operational structure relevant to the realization question: system degrees of freedom, measurement apparatus, record-bearing degrees of freedom, environmental interactions, timing structure, accessibility conditions, coarse-graining, operational readout limits, and any context-specific tolerances used to define operational equivalence.

C is not the realized outcome. It is the pre-outcome structure within which admissible realization candidates are defined.

A model fails the non-circularity burden if C is specified, narrowed, or redescribed after the realized outcome is known in order to favor that outcome.

A.2 Hilbert spaces

𝓗 denotes the Hilbert space associated with the degrees of freedom relevant to context C.

𝓗′ denotes the post-interaction, apparatus, record-bearing, or output Hilbert space relevant to C.

The notation does not require modeling every physical degree of freedom in the universe. It denotes the state space selected for the physical and operational description of the context under analysis.

A.3 Density operators

𝒟(𝓗) denotes the set of density operators on 𝓗.

A quantum state is written:

ρ ∈ 𝒟(𝓗).

This permits pure states, mixed states, and reduced states arising from environmental interaction or partial tracing.

A.4 Realization-compatible channel

A realization-compatible channel is represented as a completely positive trace-preserving map:

Φ: 𝒟(𝓗) → 𝒟(𝓗′).

The channel represents a physically admissible transformation from pre-realization state structure to post-interaction, record-bearing, or realization-compatible structure.

The use of CPTP maps is a formal discipline. It does not claim that channel notation alone resolves all interpretive questions about measurement.

A.5 Admissible candidate class

𝒜(C) denotes the set of admissible realization-compatible candidates in context C.

For ordinary exact minimization:

𝒜(C) ≠ ∅.

If 𝒜(C) is empty, the selection rule has no target. The model fails in C unless a principled, pre-specified revision of the context or admissibility criteria is supplied.

𝒜(C) must be determined by C and by pre-specified admissibility conditions. It must not be constructed after the realized outcome is known.

A.6 Individual realization-compatible subclass

𝒜_real(C) ⊆ 𝒜(C) denotes the subclass of candidates that qualify as individual realization-compatible structures in context C.

This distinction separates broader dynamical or decoherence-compatible channels from completed individual realization candidates. For example, Φ_mix may describe non-selective decoherence-compatible evolution without itself representing individual realization.

The membership of 𝒜_real(C) must be fixed by context-relative criteria before outcome comparison.

A.7 Realization-burden functional

ℛ_C: 𝒜(C) → ℝ≥0

denotes the context-relative realization-burden functional.

For each Φ ∈ 𝒜(C), ℛ_C(Φ) represents the burden associated with Φ relative to the constraints of C. Lower ℛ_C(Φ) means lower realization burden.

The functional must be fixed before outcome comparison. If it includes adjustable weights, thresholds, tolerances, or calibration-dependent parameters, their values or determination procedures must be specified before testing the realization claim.

A more general formulation may use a context-fixed admissibility preorder. Under finite or regular quotient conditions, such a preorder may be represented by a scalar burden functional.

A.8 Selected realization channel

The selected realization channel or selected operational class is defined by:

Φ∗_C ∈ argmin{ℛ_C(Φ) : Φ ∈ 𝒜(C)}.

This means that Φ∗_C belongs to the set of admissible candidates minimizing ℛ_C in context C.

If the minimizer is unique, Φ∗_C denotes that channel. If multiple minimizers exist, Φ∗_C denotes the selected operational equivalence class, provided all minimizers are equivalent under ≃_C or a pre-specified tie rule exists.

This is a candidate law-form under assumptions, not an empirically confirmed law of nature.

A.9 Minimizer set

The minimizer set is:

M_C = {Φ ∈ 𝒜(C) : ℛ_C(Φ) = inf{ℛ_C(Ψ) : Ψ ∈ 𝒜(C)}}.

If M_C contains one element, selection is formally unique.

If M_C contains multiple elements, the model succeeds only if all minimizers are operationally equivalent under ≃_C or if a pre-specified tie rule exists.

If M_C is empty, the model requires a pre-specified ε-minimizer rule. Without such a rule, selection is undefined in C.

A.10 Operational equivalence

Φ₁ ≃_C Φ₂ means that Φ₁ and Φ₂ are operationally equivalent in context C.

Let 𝒯(C) denote the set of operationally accessible tests, readout procedures, or discriminators available in C.

For each T ∈ 𝒯(C), let p_T(Φ,ρ) denote the accessible test statistic or outcome probability obtained by applying T to Φ(ρ).

Exact operational equivalence may be written:

Φ₁ ≃_C Φ₂ ⇔ for every T ∈ 𝒯(C), p_T(Φ₁,ρ) = p_T(Φ₂,ρ).

With finite operational resolution, equivalence may be defined using a tolerance ε_C:

│p_T(Φ₁,ρ) − p_T(Φ₂,ρ)│ ≤ ε_C

for every T ∈ 𝒯(C).

The tolerance ε_C must be fixed by the operational limits of C before outcome comparison. It cannot be adjusted after the result is known.

A.11 Accessibility parameter

η ∈ [0,1] denotes the accessibility of record-bearing which-path or outcome-relevant information.

η = 0 means no relevant record is operationally accessible.

η = 1 means the relevant record is fully accessible.

Intermediate values indicate partial accessibility.

If η is used in a CBR-form model, its meaning, calibration procedure, and role in ℛ_C must be specified before outcome comparison.

A.12 Non-selective decoherence-compatible channel

Φ_mix denotes a non-selective decoherence-compatible channel.

It may represent interference suppression, environmental decoherence, ensemble-level mixture structure, or record stabilization. It does not by itself represent individual outcome realization unless additional realization content is supplied.

CBR fails as an independent realization law in context C if Φ∗_C adds no realization content beyond Φ_mix.

A.13 Fixed burden weights

λᵢ ≥ 0 denotes a fixed burden weight.

In the two-path instantiation, the weights may be written:

λ_S, λ_I, λ_P, λ_D ≥ 0.

They correspond to possible burden terms for record stability, information accessibility, probability compatibility, and dynamical compatibility.

No λᵢ may be adjusted after the realized outcome is known. If λᵢ is calibrated, the calibration procedure must be fixed before outcome testing and separated from the outcome data.

A.14 ε-minimizer

An ε-minimizer is an approximate minimizer selected according to a pre-specified tolerance rule.

A candidate Φ_ε ∈ 𝒜(C) may count as an ε-minimizer if:

ℛ_C(Φ_ε) ≤ inf{ℛ_C(Ψ) : Ψ ∈ 𝒜(C)} + ε_C.

The tolerance ε_C and the selection rule must be specified before outcome comparison.

Approximate minimization is legitimate only when fixed in advance. It cannot be used as a post hoc rescue mechanism.

Appendix B: Proof Structure of the Conditional Minimal Representation Theorem

B.1 Theorem restatement

Theorem 1 — Conditional Minimal Representation Theorem for Constraint-Based Realization

Let C be a physically specified measurement context. Suppose a candidate law of quantum outcome realization satisfies context specification, nonempty admissible candidate selection, pre-outcome candidate-class fixity, context-fixed candidate comparison, pre-outcome comparison-rule fixity, pre-outcome parameter fixity, operational uniqueness or a pre-specified tie rule, Born-rule ensemble compatibility, non-reduction to decoherence, and explicit failure conditions.

Then the law admits a CBR-form representation:

Φ∗_C ∈ argmin{ℛ_C(Φ) : Φ ∈ 𝒜(C)},

where ℛ_C is a context-fixed realization-burden functional over admissible candidates, or a scalar representation of a context-fixed admissibility preorder under finite or regular quotient conditions.

B.2 Domain implies C

A realization law must apply to a physical situation. If no domain is specified, the law has no target.

Thus, the reconstruction introduces:

C

as the physical measurement context.

This step does not assert special CBR physics. It expresses the domain requirement.

B.3 Candidate selection implies 𝒜(C)

A realization law must select among possible realization structures. Therefore, it requires a candidate set.

Because admissibility depends on context, the candidate set is written:

𝒜(C).

This class is not allowed to depend on the realized outcome. It must be fixed by C and by pre-specified admissibility conditions.

B.4 Admissibility restricts 𝒜(C)

A candidate set without restrictions is too broad. Admissibility conditions restrict 𝒜(C) to candidates compatible with the context.

Possible restrictions include CPTP implementability, record compatibility, dynamical consistency, accessibility, Born compatibility, and operational readout consistency.

Thus, 𝒜(C) is not arbitrary. It is context-constrained.

B.5 Candidate comparison implies ℛ_C or an equivalent ordering

A law that selects among candidates must compare them.

For the reconstruction, a context-fixed ordering, ranking, preorder, or comparison structure over admissible candidates is required. When representable as a scalar burden, this is written:

ℛ_C: 𝒜(C) → ℝ≥0.

The functional assigns each candidate a realization burden.

This step does not prove that a particular physical ℛ_C is correct. It shows that comparison among candidates can be represented in CBR form when the comparison structure is scalar-representable.

B.6 Non-circularity requires pre-outcome fixity

The law must not depend on the outcome it is meant to explain.

Therefore, the following must be fixed before outcome comparison:

C,
𝒜(C),
𝒜_real(C), where used,
ℛ_C or the comparison structure,
≃_C,
ε_C,
η-calibration, where used,
tie rules, where required,
and all λᵢ parameters.

Symbolically:

𝒜(C) independent of Φ∗_C,
ℛ_C independent of Φ∗_C,
≃_C independent of Φ∗_C.

These expressions are structural independence requirements, not ordinary derivatives.

B.7 Minimization yields the CBR selection form

Given 𝒜(C) and ℛ_C, the selected candidate or class is represented by:

Φ∗_C ∈ argmin{ℛ_C(Φ) : Φ ∈ 𝒜(C)}.

This says that the selected realization channel belongs to the set of admissible candidates minimizing the realization burden.

If an exact minimizer exists, selection is direct. If exact minimizers do not exist, a pre-specified ε-minimizer rule is required.

B.8 Operational uniqueness requires M_C and ≃_C

Define:

M_C = {Φ ∈ 𝒜(C) : ℛ_C(Φ) = inf{ℛ_C(Ψ) : Ψ ∈ 𝒜(C)}}.

If M_C contains multiple minimizers, the model succeeds only if:

∀ Φᵢ, Φⱼ ∈ M_C, Φᵢ ≃_C Φⱼ,

or if a pre-specified tie rule exists.

Otherwise, the law has not selected one operational realization class.

B.9 Born compatibility constrains ensemble behavior

The reconstruction requires that realization-compatible alternatives preserve Born-rule frequencies across repeated equivalent contexts:

P(i) = │αᵢ│².

This is imposed as a compatibility condition.

The theorem does not derive the Born rule. It prevents the realization law from violating it without a controlled, explicit, testable deviation.

B.10 Non-reduction constrains independence from decoherence

Let Φ_mix denote a non-selective decoherence-compatible channel.

CBR remains independent only if Φ∗_C supplies realization content not exhausted by Φ_mix.

If:

Φ∗_C ≃_C Φ_mix

in all relevant contexts, and no further realization content is supplied, then CBR reduces to decoherence and fails as an independent realization law.

B.11 Vulnerability completes the law-candidate standard

A disciplined law-candidate must be able to fail.

Thus, the reconstruction requires explicit failure conditions corresponding to domain, candidate definition, admissibility, comparison, non-circularity, parameter fixity, minimization, operational uniqueness, Born compatibility, non-reduction, and vulnerability.

This completes the proof structure.

B.12 Scope of proof

The theorem proves representational form under assumptions.

It does not prove that CBR is true in nature. It does not identify a final physical ℛ_C. It does not derive the Born rule. It does not show that decoherence is false. It does not defeat rival interpretations. It does not establish universal closure over every possible realization-law framework.

Its conclusion is narrower: within the reconstruction class, the CBR form is the natural representation of the burdens required of a disciplined candidate law of outcome realization.

Appendix C: Two-Path Model Calculations

This appendix gives the formal details of the two-path instantiation.

C.1 Input state

Consider a two-path system with basis states |0⟩ and |1⟩.

Let:

|ψ⟩ = α|0⟩ + β|1⟩,

with:

│α│² + │β│² = 1.

The corresponding density operator is:

ρ_S = |ψ⟩⟨ψ|.

Expanding:

ρ_S = │α│² |0⟩⟨0| + αβ∗ |0⟩⟨1| + α∗β |1⟩⟨0| + │β│² |1⟩⟨1|.

The diagonal terms carry the Born weights. The off-diagonal terms represent coherence between alternatives.

C.2 Record interaction

Introduce record states |R₀⟩ and |R₁⟩.

After record interaction, the joint system-record state may be written:

|Ψ⟩ = α|0⟩|R₀⟩ + β|1⟩|R₁⟩.

The corresponding joint density operator is:

ρ_SR = |Ψ⟩⟨Ψ|.

Expanding:

ρ_SR = │α│² |0R₀⟩⟨0R₀|

• αβ∗ |0R₀⟩⟨1R₁|

• α∗β |1R₁⟩⟨0R₀|

• │β│² |1R₁⟩⟨1R₁|.

Here |0R₀⟩ abbreviates |0⟩|R₀⟩, and |1R₁⟩ abbreviates |1⟩|R₁⟩.

This expression represents correlation between system alternatives and record-bearing degrees of freedom. It does not by itself assert individual realization.

C.3 Reduced state and decoherence condition

Before imposing the ideal decoherence limit, tracing out the record degrees of freedom yields:

ρ_S^red = │α│² |0⟩⟨0|

• αβ∗ ⟨R₁|R₀⟩ |0⟩⟨1|

• α∗β ⟨R₀|R₁⟩ |1⟩⟨0|

• │β│² |1⟩⟨1|.

When the record states become distinguishable:

⟨R₀|R₁⟩ → 0,

the reduced state becomes approximately:

ρ_S^dec ≈ │α│² |0⟩⟨0| + │β│² |1⟩⟨1|.

The off-diagonal coherence terms are suppressed.

This is the decoherence-compatible reduced state. It explains interference suppression. It does not by itself specify an individual realization channel.

C.4 Candidate channels

Define schematic candidate channels:

Φ₀ = realization-compatible channel associated with record R₀.

Φ₁ = realization-compatible channel associated with record R₁.

Φ_mix = non-selective decoherence-compatible channel.

The broad candidate class may be:

𝒜(C) = {Φ₀, Φ₁, Φ_mix}.

The individual realization-compatible subclass may be:

𝒜_real(C) = {Φ₀, Φ₁}.

This distinction preserves the difference between non-selective mixture structure and individual realization.

C.5 Accessibility parameter

Let:

η ∈ [0,1].

η represents the accessibility of record information.

High η means the record is accessible and distinguishable.

Low η means the record is inaccessible or insufficiently supported by C.

A model must specify how η is determined before outcome comparison. It cannot be assigned after the result is known to favor a preferred realization claim.

C.6 Toy burden functional

A toy realization-burden functional may be written:

ℛ_C(Φ) = λ_S S_C(Φ) + λ_I I_C(Φ) + λ_P P_C(Φ) + λ_D D_C(Φ).

Here:

S_C(Φ) measures record-stability burden.

I_C(Φ) measures information-accessibility burden.

P_C(Φ) measures probability-compatibility burden.

D_C(Φ) measures dynamical-compatibility burden.

The weights satisfy:

λ_S, λ_I, λ_P, λ_D ≥ 0.

They must be fixed before outcome comparison.

This functional is illustrative. It is not asserted as the final universal ℛ_C.

C.7 High-accessibility case

When η is high and ⟨R₀|R₁⟩ ≈ 0, the context may support individual record realization.

In such a context, Φ₀ and Φ₁ may qualify as individual realization candidates.

Born compatibility requires:

P(Φ₀) = │α│²,
P(Φ₁) = │β│².

This condition governs ensemble occurrence. It does not replace probability with burden minimization.

C.8 Low-accessibility case

When η is low, C may not support a which-path realization claim.

In that case:

Φ₀ and Φ₁ may be excluded from 𝒜_real(C);

or Φ₀ ≃_C Φ₁;

or the context may remain below the threshold required for individual realization in the modeled sense.

The model must not impose a record structure unsupported by C.

C.9 Symmetric case

If:

│α│² = │β│²

and the context is symmetric, Φ₀ and Φ₁ may share equal burden.

CBR succeeds only if:

Φ₀ ≃_C Φ₁

or a pre-specified Born-compatible tie rule exists.

If Φ₀ and Φ₁ are operationally distinct and no tie rule exists, the model is incomplete in C.

C.10 What the calculation demonstrates

The two-path calculation demonstrates instantiability, not empirical truth.

It shows that the CBR-form objects can be populated in a minimal nontrivial context: C, 𝒜(C), 𝒜_real(C), ℛ_C, M_C, ≃_C, η, Φ_mix, and Φ∗_C.

It does not establish that the toy functional is universal. It does not derive the Born rule. It does not show that decoherence is false. It does not prove CBR experimentally.

Appendix D: Operational Equivalence

Operational equivalence is required to handle cases where formal multiplicity does not correspond to physically accessible distinction.

D.1 Accessible tests

Let 𝒯(C) denote the set of tests, readout procedures, or operational discriminators accessible in context C.

These tests must be specified by the physical and operational structure of C. They cannot be chosen after the result is known to erase or create distinctions among minimizers.

For each T ∈ 𝒯(C), let p_T(Φ,ρ) denote the accessible statistic produced by applying T to Φ(ρ). This may be an outcome probability, expectation value, visibility measure, record-readout statistic, or other context-defined operational quantity.

D.2 Exact operational equivalence

Define:

Φ₁ ≃_C Φ₂ ⇔ for every T ∈ 𝒯(C), p_T(Φ₁,ρ) = p_T(Φ₂,ρ).

This means that no accessible test in C distinguishes Φ₁ from Φ₂.

If exact equality holds for all accessible tests, the two candidates are operationally equivalent in context C.

D.3 Tolerant operational equivalence

In realistic settings, exact equality may be too strict. Operational procedures have finite resolution.

Define tolerant equivalence by:

Φ₁ ≃_C Φ₂ ⇔ for every T ∈ 𝒯(C), │p_T(Φ₁,ρ) − p_T(Φ₂,ρ)│ ≤ ε_C.

Here ε_C is the context-specific operational tolerance.

The tolerance must be fixed before outcome comparison. It should be determined by the resolution limits of C, not by the desired result.

D.4 Equivalence and minimizer degeneracy

If:

∀ Φᵢ, Φⱼ ∈ M_C, Φᵢ ≃_C Φⱼ,

then all minimizers represent the same operational realization class.

If not, and if no pre-specified tie rule exists, the model fails operational uniqueness in context C.

D.5 Failure of operational-equivalence discipline

A CBR-form model fails operational-equivalence discipline if ≃_C or ε_C is adjusted after the outcome is known.

It also fails if operationally distinct minimizers are declared equivalent without a test-based justification grounded in C.

Operational equivalence must be a context-fixed quotienting relation, not a retrospective repair.

Appendix E: Born Compatibility

This appendix formalizes the probability-compatibility requirement.

E.1 Two-outcome Born compatibility

For a two-outcome state:

|ψ⟩ = α|0⟩ + β|1⟩,

with:

│α│² + │β│² = 1,

Born compatibility requires:

P(0) = │α│²,
P(1) = │β│².

Across repeated equivalent contexts:

lim N₀/N = │α│² as N → ∞,
lim N₁/N = │β│² as N → ∞.

E.2 General Born compatibility

For:

|ψ⟩ = Σᵢ αᵢ|i⟩,

Born compatibility requires:

P(i) = │αᵢ│².

Across repeated equivalent contexts:

lim Nᵢ/N = │αᵢ│² as N → ∞.

E.3 Imposed compatibility, not derivation

This paper imposes Born compatibility. It does not derive the Born rule.

The reconstruction asks whether CBR-form realization can coexist with Born-rule statistics. It does not claim that ℛ_C replaces Born weighting.

E.4 Non-Born deviations

A CBR-form model may propose a non-Born deviation only if the deviation is:

explicitly stated;

mathematically modeled;

context-specific;

pre-specified before testing;

baseline-comparable;

bounded by an error or nuisance budget;

and empirically vulnerable.

Without these conditions, non-Born behavior is not a prediction. It is a failure of probability compatibility.

E.5 Probability discipline

CBR constrains admissible realization structure. Born compatibility constrains ensemble occurrence frequencies.

Selection is not sampling. A burden functional does not automatically replace probability weighting.

This distinction is required for the reconstruction to remain compatible with standard quantum statistics.

Appendix F: Failure Checklist

This appendix summarizes the principal ways in which a CBR law-candidate can fail. These failure conditions are not external objections added after the theory is complete. They are part of the discipline required for treating CBR as a serious candidate law of outcome realization.

F.1 Domain failure

CBR fails at the level of domain specification if the measurement context C is vague, undefined, or specified only after the result is known.

A realization law must have a determinate physical target. If the context is not fixed in advance, the model has no stable domain of application.

F.2 Candidate-set failure

CBR fails at the level of the candidate set if 𝒜(C) is empty, arbitrary, undefined, unrestricted enough to make any result selectable, or constructed after the outcome is observed.

A selection rule cannot operate without a pre-specified class of admissible candidates. If the candidate class is built around the result after the fact, the model becomes circular rather than law-like.

F.3 Admissibility failure

CBR fails at the level of admissibility if the individual realization-compatible subclass 𝒜_real(C) is not restricted by physical criteria.

Candidates must be admitted according to context-fixed requirements such as physical implementability, record compatibility, dynamical consistency, operational accessibility, and probability compatibility.

If admissibility is arbitrary, the selection rule can be made to select anything.

F.4 Comparison failure

CBR fails at the level of comparison if no context-fixed comparison structure is supplied.

A realization law must state how admissible candidates are ranked, ordered, burdened, minimized, or otherwise compared. Without a comparison structure, selection is undefined.

F.5 Non-circularity failure

CBR fails at the level of non-circularity if ℛ_C, or any comparison structure replacing it, is defined or altered after the observed result is known.

A functional that rewards the actual result merely because it occurred is not a law. It is retrospective labeling.

F.6 Operational-equivalence failure

CBR fails at the level of operational equivalence if ≃_C or ε_C is adjusted after the outcome is observed.

Operational equivalence must be determined by the accessible tests and resolution limits of C, not by the desire to make a particular minimizer appear unique.

F.7 Parameter-fixity failure

CBR fails at the level of parameter fixity if any weighting parameters λᵢ in ℛ_C are tuned after outcome observation.

Parameter values may be fixed by theory, context, or a pre-specified calibration protocol, but they cannot be adjusted after seeing the result. Otherwise, ℛ_C becomes an outcome-fitting device rather than a law-candidate.

F.8 Selection-existence failure

CBR fails at the level of selection if no minimizer exists and no pre-specified ε-minimizer rule has been provided.

The model must identify a minimizing candidate, a minimizing equivalence class, or a principled approximate minimizer. Without that, the selection rule is undefined in C.

F.9 Operational-uniqueness failure

CBR fails at the level of operational uniqueness if M_C contains operationally distinct candidates and no pre-specified tie rule exists.

Multiple minimizers are acceptable only if they are equivalent under ≃_C or governed by a tie rule fixed before outcome comparison. Otherwise, the model has not selected one physically meaningful realization class.

F.10 Probability-compatibility failure

CBR fails at the level of probability compatibility if it violates Born-rule frequencies without an explicitly modeled, pre-specified, empirically vulnerable deviation claim.

The required default condition is:

P(i) = │αᵢ│².

CBR may constrain admissible realization structure, but it does not replace Born-rule weighting in this reconstruction.

F.11 Non-reduction failure

CBR fails at the level of non-reduction if the selected realization structure Φ∗_C adds no content beyond the non-selective decoherence-compatible channel Φ_mix.

If Φ∗_C is operationally equivalent to Φ_mix in every relevant context and no additional realization content is supplied, then CBR reduces to decoherence and fails as an independent realization law.

F.12 Vulnerability failure

CBR fails at the level of vulnerability if no structural or empirical condition could count against it.

A law-candidate must be able to lose. If the framework is insulated from all possible failure, it is not functioning as a disciplined physical proposal.

F.13 Scope of failure

Failure in context C means failure of that CBR model as a law-candidate in that context.

It does not prove the impossibility of every possible CBR-like theory. It does mean that the evaluated model has failed the burden it claimed to satisfy.