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

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Abstract

Standard quantum mechanics provides outcome spaces, amplitudes, 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, or that decoherence is false. Instead, it asks what any disciplined realization law must contain if it is to avoid circularity, preserve probability, distinguish itself from decoherence, and remain vulnerable to failure. We show that these burdens require a physically specified context C, an admissible candidate class 𝒜(C), a pre-outcome realization-burden functional ℛ_C, an operational equivalence relation ≃_C, Born-compatible ensemble behavior, parameter fixity, and explicit defeat conditions. Under these assumptions, any such law admits the CBR-form representation Φ∗C ∈ argmin{Φ ∈ 𝒜(C)} ℛ_C(Φ), where the selected realization channel belongs to the admissible class minimizing the context-relative realization burden. A minimal two-path model demonstrates how the reconstructed structure can be instantiated without replacing Born-rule weighting or reducing realization to non-selective decoherence. The paper also states a Failure Theorem: CBR fails in a context if its domain, candidate set, functional, equivalence relation, parameters, minimizers, probability behavior, non-reduction condition, or vulnerability criteria fail. The result is not confirmation of CBR as physical law, but a disciplined reconstruction of CBR as a formally viable, non-circular, probability-compatible, and failure-capable candidate law of quantum outcome realization.

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1. Introduction

Quantum mechanics gives an extraordinarily successful formal account of physical systems. In its standard mathematical form, it specifies state spaces, state evolution, amplitudes, observables, outcome spaces, and probability weights. For a measurement context, the theory can identify which outcomes are possible and how frequently those outcomes should occur across repeated trials. This predictive structure is not in question here.

The question addressed in this paper is narrower. It concerns not the statistical adequacy of quantum mechanics, but the formal structure of any candidate law that purports to describe individual outcome realization. Stated directly: what physical law-form, if any, governs the realization of one actual outcome-structure in an individual measurement context?

This paper uses the term “realization” operationally. It refers to the selection of a context-compatible record or outcome-structure represented by a realization-compatible channel. The argument does not require adding a metaphysical thesis about what it means for an outcome to “become real.” It asks whether a disciplined formal structure can be given for the context-relative selection of an admissible record or outcome-channel.

The central claim is conditional. If one seeks a non-circular law-form for individual outcome realization, and if such a law must define a physical context, select from admissible candidates, apply a pre-outcome comparison criterion, produce operational uniqueness, preserve Born-rule statistics, distinguish realization from decoherence, control parameter tuning, and expose itself to failure, then a CBR-form representation naturally follows.

The reconstructed form is:

Φ∗C ∈ argmin{Φ ∈ 𝒜(C)} ℛ_C(Φ).

This expression means that the selected realization channel Φ∗_C belongs to the set of admissible channels Φ in context C that minimize the realization-burden functional ℛ_C. It is not presented as an empirically confirmed law of nature. It is presented as the conditional form a disciplined realization-law candidate takes under explicit assumptions.

1.1 Probability and realization

The Born rule gives probability weights. For a state written schematically as

|ψ⟩ = Σᵢ αᵢ|i⟩,

standard quantum mechanics assigns

P(i) = |αᵢ|².

This equation says that the probability of observing outcome i is given by the squared magnitude of the corresponding amplitude. It governs ensemble frequencies across repeated trials.

A probability distribution, however, is not itself a law of individual realization. It tells us how outcomes are weighted; it does not, by itself, specify the physical law-form by which one admissible outcome-structure is selected in a particular context. This distinction does not weaken the Born rule. It clarifies the role the Born rule plays.

CBR is not introduced here as a replacement for Born-rule weighting. It is reconstructed as a candidate structure for realization that must remain Born-compatible unless it explicitly introduces a controlled and testable deviation. In the present paper, Born compatibility is imposed as a requirement, not claimed as a derived result.

1.2 Decoherence and realization

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

Decoherence is not rejected here. The claim is narrower: a non-selective decoherence map does not, by itself, supply the additional law-form of individual realization selection. A decohered mixture may describe the suppression of interference and the stability of records, but it does not alone identify which admissible record-structure is selected as the realized outcome in an individual context.

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

1.3 Reconstruction rather than assertion

This paper does not simply assert CBR. It reconstructs CBR from the burdens that any disciplined quantum outcome-realization law must satisfy.

A candidate realization law must define its domain. It must specify what is being selected. It must impose admissibility conditions. It must compare candidates by a rule fixed before the outcome is known. It must produce a physically meaningful selected structure. It must remain compatible with Born-rule statistics. It must not reduce entirely to decoherence. It must prevent post hoc parameter tuning. It must be capable of failure.

Once these requirements are made explicit, the following structure emerges:

context → admissible candidates → burden/constraint ordering → selected realization class.

This structure is the CBR law-form.

The point is not that CBR is uniquely forced by logic in all possible frameworks. The point is that any law satisfying the stated burdens admits a CBR-form representation. The strength of the result is therefore its disciplined conditionality.

1.4 Main result

The central theorem of the paper will be stated as the Conditional Minimal Representation Theorem for Constraint-Based Realization.

Given:

C = a physically specified measurement context,
𝒜(C) = a nonempty admissible candidate class,
ℛ_C = a context-fixed realization-burden functional,
≃_C = a context-fixed operational equivalence relation,

any candidate law satisfying non-circularity, operational uniqueness, Born compatibility, non-reduction to decoherence, parameter fixity, and failure vulnerability admits the form:

Φ∗C ∈ argmin{Φ ∈ 𝒜(C)} ℛ_C(Φ).

This theorem establishes representational form under stated assumptions. It does not establish empirical truth, universal applicability, or the uniqueness of any particular physical ℛ_C. It shows that CBR is structurally natural under the burdens imposed on a disciplined realization-law candidate.

1.5 What this paper does not claim

The paper does not claim that CBR is experimentally confirmed. It does not claim that CBR is already established physics. It does not claim that CBR replaces standard quantum mechanics, replaces the Born rule, or proves decoherence 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.

Instead, it claims that CBR is reconstructable as a minimal representational structure of a disciplined, non-circular, probability-compatible candidate law of outcome realization under explicit assumptions.

The argument is therefore conditional and structural. If one seeks a law-form for individual outcome realization that satisfies the stated burdens, then a CBR-form representation naturally follows. The next question is not whether the structure can be written coherently; it is whether context-specific versions of 𝒜(C), ℛ_C, and ≃_C can be independently specified and made vulnerable to empirical or structural test.

2. The Burdens of a Quantum Outcome Law

A proposed realization law cannot merely say that “constraints select the outcome.” Such a statement may be suggestive, but it is not yet a law-form. It does not specify the domain of application, the objects being selected, the admissibility criteria, the comparison rule, the relation to quantum probability, or the conditions under which the proposal would fail.

A serious law-candidate must carry clear burdens. These burdens are not rhetorical additions. They are structural requirements that determine whether the proposal is precise enough to be evaluated.

A disciplined outcome-realization law must answer at least the following questions: What problem does it address? What objects does it select among? What makes those objects admissible? How does selection occur? Why is the selection rule non-circular? In what sense is the selected result unique? How are Born-rule statistics preserved? How is the proposal distinct from decoherence? How is parameter tuning controlled? What would count as failure?

The burdens below define the standard that motivates the CBR reconstruction.

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 target.

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

This burden forces the theory to state where it applies. A realization claim that is not attached to a specified physical context cannot define admissible candidates or a selection rule. The context C is therefore the first required object in the reconstruction.

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 candidates may be represented as realization-compatible channels, record structures, or context-relative outcome maps. This paper uses channel language because it allows the proposal to be expressed in standard quantum-operational terms.

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 state the class of structures from which selection occurs. That class is 𝒜(C).

2.3 Burden 3: Admissibility

Not every mathematically imaginable candidate should be allowed. A candidate may be formally expressible but physically inadmissible.

Admissibility must be constrained by the context. Candidate structures should satisfy conditions such as physical implementability, compatibility with the measurement architecture, record-structure consistency, dynamical consistency, probability compatibility, and pre-outcome independence.

This burden prevents arbitrary selection. If every imagined candidate is admissible, the law can be made to select anything. If admissibility is defined after the outcome is known, the law becomes post hoc. Therefore, admissibility must be fixed by C and by pre-specified criteria.

In the CBR reconstruction, 𝒜(C) is not a list of preferred results. It is the set of candidates that survive context-relative admissibility constraints.

2.4 Burden 4: Non-circular selection

The rule must not depend on the outcome it is supposed to explain.

The following objects must be fixed before the realized outcome is known:

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

The forbidden move is to observe an outcome and then define or tune the selection rule so that the observed outcome is favored. Such a rule would not explain realization. It would merely relabel the result.

A realization law becomes circular if it says, in effect, “this result happened because the rule favors the result that happened.” To avoid this, the context, candidate set, comparison rule, equivalence relation, and parameter values must be fixed independently of the selected result.

2.5 Burden 5: Operational uniqueness

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

If several candidates minimize the relevant burden functional, the law must show that they are indistinguishable within the operational capacities of C, or else provide a pre-specified tie rule. Let

Φ₁ ≃_C Φ₂

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

The equivalence relation ≃_C is not an afterthought. It is necessary because formal multiplicity does not always imply physically meaningful multiplicity. Two channels may be distinct as mathematical objects but indistinguishable by any accessible procedure in C.

A law succeeds in this respect when all selected minimizers belong to the same operational equivalence class, or when a pre-outcome tie rule handles the degeneracy.

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 probability rule is empirically central. A candidate realization law must preserve Born-rule frequencies at the ensemble level unless it explicitly introduces a controlled, testable deviation. In the present reconstruction, Born compatibility is treated as a default requirement.

This burden also prevents confusion between selection and sampling. 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 trials.

2.7 Burden 7: Non-reduction to decoherence

A realization law must do more than describe decoherence.

Decoherence suppresses interference and supports stable record formation. It is not being rejected. However, a non-selective decoherence map does not, by itself, specify an individual realization selection rule.

CBR must therefore address the additional realization question: which admissible record or outcome-structure becomes actual in context C?

If CBR reduces entirely to an ordinary non-selective decoherence map with no additional realization content, it fails as an independent realization law. The burden is not to deny decoherence, but to distinguish realization-selection structure from decoherence-compatible evolution.

2.8 Burden 8: Parameter fixity

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

Any terms 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.

This burden is essential because a flexible functional can otherwise be adapted to favor any result. A post hoc ℛ_C is not a law-candidate. It is an outcome-fitting device. Parameter fixity blocks this failure mode.

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 if its domain is undefined, if its candidate set is arbitrary or empty, if its selection rule is circular, if parameters are tuned post hoc, if no minimizer exists, if non-equivalent minimizers remain unresolved, if Born-rule frequencies are violated without a testable deviation claim, if the theory reduces entirely to decoherence, or if no observation or structural critique could count against it.

A law-candidate gains credibility not by avoiding failure conditions, but by stating them. The CBR reconstruction therefore includes failure vulnerability as a burden rather than as a later optional feature.

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 all candidates are physical.
It must have a comparison rule, because selection requires evaluation.
It must have a selected result or selected equivalence class.
It must have an operational equivalence relation, because uniqueness may be context-relative.
It must control parameter tuning, because otherwise selection may be post hoc.
It must preserve Born-rule statistics, because quantum probabilities are empirically fixed.
It must distinguish realization from decoherence, because decoherence alone is non-selective.
It must have failure conditions, because a law-candidate must be vulnerable.

This is already the skeleton of CBR. The next section shows how the burdens reconstruct the CBR form rather than merely motivating it.

3. From Law Burdens to CBR Structure

The preceding section identified the burdens any disciplined quantum outcome-realization law must carry. This section shows how those burdens reconstruct the CBR structure.

The reconstruction does not begin by assuming CBR. It begins from requirements on any candidate realization law. Each requirement forces a corresponding formal object. Taken together, these objects yield the CBR form:

Φ∗C ∈ argmin{Φ ∈ 𝒜(C)} ℛ_C(Φ).

This equation will later be formalized more precisely. At this stage, the goal is to show why the structure arises.

3.1 Domain forces context C

Because a realization law must apply to a physical measurement situation, it requires a context variable. CBR denotes this variable by C.

C is not merely an observable label. It is the full physical and operational measurement structure relevant to the realization question. It includes the system, apparatus, record-bearing degrees of freedom, environmental structure, timing, accessibility, and readout limits.

The move from “a measurement occurs” to “a measurement occurs in context C” is necessary. It prevents the law from floating above the physical setup. Realization is treated as context-relative, and admissibility is defined relative to that context.

Thus, the domain burden forces the introduction of C.

3.2 Candidate burden forces 𝒜(C)

Because a realization law must select among possible realization structures, it requires a candidate set. CBR denotes this candidate set by 𝒜(C).

𝒜(C) is the set of physically admissible realization-compatible candidates in context C. If candidates are represented as channels, then the elements of 𝒜(C) are candidate realization-compatible maps. If candidates are represented as record structures, then 𝒜(C) contains the admissible record-structures associated with C. The present paper uses channel language because it provides a precise operational representation.

Without 𝒜(C), selection is empty. With a post hoc 𝒜(C), selection is circular. Therefore, the candidate burden forces a context-fixed admissible candidate class.

3.3 Admissibility forces physical restrictions

Because not every formal candidate is physically allowed, the candidate class must be constrained. Thus, 𝒜(C) must be defined by admissibility conditions.

These conditions may include complete positivity and trace preservation when candidates are represented as quantum channels, compatibility with record-bearing degrees of freedom, stability of record structure, dynamical consistency, Born compatibility, and operational readout consistency.

This point is central. CBR is not merely a rule that chooses among arbitrary candidates. It is a context-constrained selection structure. The context determines which candidates are physically admissible; the selection rule then compares candidates within that admissible class.

3.4 Non-circularity forces pre-outcome fixity

Because a realization law cannot depend on the result it is meant to explain, its key objects must be fixed before realization.

The fixed objects include:

C,
𝒜(C),
the selection criterion,
ℛ_C,
≃_C,
and any parameters in ℛ_C.

This requirement forces the law away from post hoc explanation. A CBR model is not valid if it observes the result and then adjusts the admissible class, functional, equivalence relation, or parameter values to favor that result.

Pre-outcome fixity is therefore not a technical preference. It is the condition that makes the selection rule non-circular.

3.5 Selection forces an ordering or burden structure

To select among candidates, the law must compare them. A comparison can be represented as a functional:

ℛ_C: 𝒜(C) → ℝ_{\ge 0}.

This functional assigns each candidate Φ a non-negative realization burden relative to context C. Lower burden means greater compatibility with the full constraint structure of C.

This is not yet a claim that any particular physical ℛ_C has been empirically confirmed. It is a representational requirement. If a law selects among admissible candidates, it must have some way of comparing them. CBR represents that comparison as a context-relative burden functional.

A more general theory might use a preorder rather than a real-valued functional. For the purposes of this reconstruction, the real-valued functional provides a compact and mathematically tractable representation.

3.6 Operational uniqueness forces equivalence classes

Selection may not produce a single formal minimizer. Several candidates may minimize ℛ_C. The law must then determine whether this multiplicity is physically meaningful.

Define the minimizer set:

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

This set contains the admissible candidates whose realization burden is minimal.

CBR succeeds when:

∀ Φᵢ, Φⱼ ∈ M_C, Φᵢ ≃_C Φⱼ.

This means that all minimizers are operationally equivalent in context C. If M_C contains operationally distinct minimizers and no pre-specified tie rule exists, the model is incomplete in C.

Operational uniqueness therefore forces the introduction of both M_C and ≃_C.

3.7 Probability compatibility forces Born preservation

The selection structure cannot arbitrarily alter observed quantum frequencies. For repeated trials, the default requirement is:

lim_{N→∞} Nᵢ/N = |αᵢ|².

Here Nᵢ is the number of trials in which outcome i occurs, and N is the total number of repeated trials. The equation states that the long-run frequency of outcome i must approach the Born-rule probability.

CBR does not replace Born-rule weighting. The minimization structure constrains admissible realization channels in a context; Born compatibility constrains the ensemble frequencies with which realization-compatible alternatives occur.

Any version of CBR that predicts non-Born frequencies must state that prediction explicitly and expose it to empirical test. In the present reconstruction, no such deviation is asserted.

3.8 Non-reduction forces realization content

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

This distinction will later be represented by contrasting Φ_mix, a non-selective decoherence-compatible channel, with Φ∗_C, a selected realization-compatible channel or class.

If Φ∗_C adds no content beyond Φ_mix for every relevant context C, then CBR reduces to decoherence and fails as an independent realization law. If Φ∗_C supplies individual realization structure not contained in the non-selective decoherence map, then CBR remains formally distinct.

Non-reduction therefore forces CBR to specify what work its selection structure performs.

3.9 Parameter fixity forces pre-test specification

If ℛ_C contains weights or adjustable terms, those terms must be fixed before outcome comparison.

For example, a toy model may later use a burden functional of 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).

Here S_C, I_C, P_C, and D_C denote possible burden terms, and λ_S, λ_I, λ_P, and λ_D are weights. The equation says that the total realization burden is 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. Without this requirement, ℛ_C could be tuned to fit the result and would fail as a non-circular law-candidate.

3.10 Conclusion of reconstruction

The burdens reconstruct the CBR form:

Φ∗C ∈ argmin{Φ ∈ 𝒜(C)} ℛ_C(Φ).

This is not introduced as an arbitrary equation. It is the natural representational form of any candidate law that must define a context, select from admissible candidates, use a non-circular comparison criterion, produce operational uniqueness, preserve probability, avoid reduction to decoherence, control parameter tuning, and remain failure-capable.

The next section fixes the notation required for the theorem and the later two-path demonstration.

4. Formal Definitions

This section introduces the notation used throughout the paper. The goal is not to settle all physical interpretation questions at the level of definition. The goal is to make the reconstruction precise enough to state assumptions, theorem conditions, and failure modes.

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.

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, and reduced states arising from environmental interaction or partial tracing.

4.3 Measurement context

Let C denote a physically specified measurement context.

C includes system degrees of freedom, measurement apparatus, record-bearing degrees of freedom, environmental interactions, timing structure, accessibility parameter η, coarse-graining, and operational readout limits.

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

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-realization record 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 this paper has solved all metaphysical questions associated with measurement.

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.

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 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.

4.7 Realization functional

Let

ℛ_C: 𝒜(C) → ℝ_{\ge 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 the realized outcome. If it contains adjustable parameters, those parameters must be fixed before outcome comparison.

4.8 Selected realization channel

Define:

Φ∗C ∈ argmin{Φ ∈ 𝒜(C)} ℛ_C(Φ).

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.

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

4.9 Minimizer set

Define:

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

M_C contains all admissible candidates whose realization burden is minimal.

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

4.10 Operational equivalence

Define Φ₁ ≃_C Φ₂ if no operationally accessible procedure within C distinguishes Φ₁ from Φ₂.

A formal version is:

∀T ∈ 𝒯(C), T(Φ₁(ρ)) = T(Φ₂(ρ))

within the resolution limits of C.

Here 𝒯(C) denotes the accessible tests or readout procedures available in context 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.

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.

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

4.12 Accessibility parameter

Let

η ∈ [0,1]

denote the accessibility of record-bearing which-path or outcome information.

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 will later help characterize the two-path demonstration and the distinction between record accessibility, decoherence, and realization.

4.13 Non-selective decoherence-compatible channel

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

Φ_mix may describe interference suppression or ensemble-level mixture structure. 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.

5. Assumptions of the Reconstruction

The preceding sections introduced the burdens that motivate the CBR form and defined the notation required to state it precisely. This section collects the assumptions under which the reconstruction proceeds. These assumptions are not empirical conclusions. They are the conditions a candidate realization law must satisfy in order to admit the CBR-form representation developed in the next section.

The assumptions are deliberately stated as constraints. Their purpose is to prevent the reconstruction from becoming circular, unconstrained, or immune from criticism. A candidate law that violates these assumptions may still be an object of philosophical interest, but it does not satisfy the standard of a disciplined, non-circular, probability-compatible law-candidate as used here.

A1. Context specification

The measurement context C is physically specified before outcome realization.

This means that the relevant system, measurement architecture, record-bearing degrees of freedom, accessibility conditions, timing relations, operational readout limits, and coarse-graining assumptions are fixed before the selected realization channel is identified. The context is not inferred from the realized outcome after the fact.

A2. Candidate-class fixity

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

This assumption blocks the most direct form of post hoc selection. A model cannot observe a result and then define 𝒜(C) so that only that result is admissible. The candidate class must be specified by the physical and operational features of the context.

A3. Functional fixity

The realization-burden functional ℛ_C is determined by C and does not depend on the realized outcome.

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

A4. Equivalence-relation fixity

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

The equivalence relation is used to determine when formally distinct minimizers are indistinguishable within the operational limits of the context. If ≃_C were adjusted after the result, degeneracy could be hidden rather than resolved. Therefore, ≃_C must be fixed in advance by the accessible tests 𝒯(C) and any context-specific tolerance ε_C.

A5. Nonempty admissibility

The admissible candidate class is nonempty:

𝒜(C) ≠ ∅.

If 𝒜(C) is empty, there are no admissible candidates over which a realization rule can operate. In that case, the model fails in context C unless the context or admissibility conditions are revised in a principled, pre-specified way.

A6. Lower-bounded burden

The realization-burden functional ℛ_C is bounded below over 𝒜(C).

This assumption ensures that minimization is mathematically meaningful. Without a lower bound, the expression

argmin_{Φ ∈ 𝒜(C)} ℛ_C(Φ)

may be undefined. Lower-boundedness does not by itself guarantee an exact minimizer, but it is a necessary condition for a disciplined minimization structure.

A7. Minimizer existence or pre-specified approximate selection

The minimizer set

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

is nonempty.

If exact minimizers do not exist, an ε-minimizer rule must be specified in advance. An ε-minimizer rule may select candidates whose burden lies within a fixed tolerance of the infimum, but the tolerance and selection procedure must be fixed before outcome comparison. Otherwise, approximate minimization can become another route to post hoc adjustment.

A8. 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.

This assumption handles degeneracy. If all minimizers are operationally equivalent, then 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 is met, the model is incomplete in C.

A9. Born compatibility

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

For a two-outcome case with state

|ψ⟩ = α|0⟩ + β|1⟩,

Born compatibility requires long-run frequencies consistent with

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 here. It requires that any CBR-form realization structure preserve Born-rule statistics unless a controlled, explicitly stated, empirically vulnerable deviation is introduced.

A10. Non-reduction to decoherence

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

In later sections, the non-selective decoherence-compatible channel is denoted Φ_mix. 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. The reconstruction does not reject decoherence; it requires only that CBR supply additional realization-selection content if it is to be independent.

A11. Vulnerability

Failure of candidate definition, non-circularity, minimization, uniqueness, Born compatibility, parameter fixity, or non-reduction counts against the model.

This assumption is methodological. It prevents the law-candidate from being protected against all possible criticism. A CBR model must be able to fail structurally or empirically. Its assumptions are therefore not decorative; they are also potential failure points.

A12. Parameter fixity

Any terms 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.

This assumption is essential. Without parameter fixity, ℛ_C could be tuned to favor any observed result. That would convert a proposed law-form into a retrospective fitting device.

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 a candidate law satisfying the stated burdens admits a CBR-form representation.

6.1 Theorem statement

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 the following requirements:

1. It applies to a context C.

2. It selects from a nonempty admissible candidate class 𝒜(C).

3. 𝒜(C) is fixed independently of the realized outcome.

4. The law compares admissible candidates by a context-fixed criterion.

5. The comparison criterion is fixed independently of the realized outcome.

6. Any parameters in the comparison criterion are fixed prior to outcome comparison.

7. The law selects one candidate or one operational equivalence class of candidates.

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

9. The law preserves Born-rule ensemble frequencies.

10. The law does not reduce entirely to ordinary decoherence.

11. The law has explicit failure conditions.

Then the candidate law admits a CBR-form representation:

Φ∗C ∈ argmin{Φ ∈ 𝒜(C)} ℛ_C(Φ),

where ℛ_C is a context-fixed realization-burden functional over admissible candidates.

This theorem establishes representational form under stated assumptions. It does not establish empirical truth, universal applicability, or the uniqueness of any particular physical ℛ_C.

6.2 Proof sketch

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

First, because the law applies to a physical measurement situation, it requires a domain. Denote that domain by the context C. This gives the reconstruction its context-relative structure.

Second, because the law 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, not by the realized result.

Third, because not all formal candidates are physically admissible, 𝒜(C) must be restricted by context-relative admissibility conditions. These may include implementability, record compatibility, dynamical consistency, operational accessibility, and probability compatibility.

Fourth, because the law selects among candidates, it must compare them. Any comparison criterion that ranks candidates can be represented, for the purposes of this reconstruction, by a context-relative burden functional:

ℛ_C: 𝒜(C) → ℝ_{\ge 0}.

This functional assigns to each candidate a non-negative realization burden. Lower values indicate lower burden relative to the constraints of C.

Fifth, because the law must be non-circular, C, 𝒜(C), ℛ_C, ≃_C, and any parameters in ℛ_C must be fixed independently of Φ∗_C. This prevents the selection rule from being tuned after the outcome is known.

Sixth, because the law must select a physically meaningful result, the minimizer set

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

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

Seventh, because quantum probabilities are empirically fixed, the realization structure must preserve Born-rule frequencies unless a controlled and testable deviation is explicitly introduced.

Eighth, 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.

Combining these requirements yields the CBR-form representation:

Φ∗C ∈ argmin{Φ ∈ 𝒜(C)} ℛ_C(Φ).

This equation states that the selected realization channel or selected realization class belongs to the set of admissible candidates that minimize the context-relative realization burden.

6.3 Scope of the theorem

The theorem establishes a conditional representation result. It shows that any disciplined non-circular realization law satisfying the stated burdens can be represented in CBR form.

It 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 or that the Born rule is replaced.

The theorem’s significance is narrower: 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.

6.4 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:

Theorem: shows why CBR-form structure arises from the burdens.
Toy 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.

7. Corollaries of the Reconstruction

The following corollaries draw modest consequences from the Conditional Minimal Representation Theorem. They should not be read as independent empirical claims. Each corollary follows only under the theorem’s assumptions.

7.1 Corollary 1: Non-circularity

If C, 𝒜(C), ℛ_C, ≃_C, and all parameters in ℛ_C are fixed before realization, then the CBR-form selection rule is non-circular in context C.

A symbolic way to express this requirement is:

∂𝒜(C)/∂Φ∗_C = 0,
∂ℛ_C/∂Φ∗_C = 0,
∂≃_C/∂Φ∗_C = 0.

These expressions mean that the candidate class, burden functional, and operational equivalence relation do not depend on the selected result.

This corollary establishes only non-circularity under pre-outcome fixity. It does not establish that a particular proposed ℛ_C is physically correct. It states that if the machinery is fixed before outcome comparison, the selection rule is not merely naming the result after the fact.

7.2 Corollary 2: Operational uniqueness

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 even if there are multiple formal minimizers, they are indistinguishable by accessible tests within C. The selected result is therefore unique at the operational level.

If this condition fails and no pre-specified tie rule exists, the model is incomplete in C. This corollary does not guarantee uniqueness in every possible context. It states the condition under which operational uniqueness is obtained.

7.3 Corollary 3: Born compatibility

If selected realization structures occur across repeated equivalent contexts with frequencies given by the Born rule, then CBR-form selection is compatible with standard quantum probability.

For an input state

|ψ⟩ = Σᵢ αᵢ|i⟩,

Born compatibility requires:

P(i) = |αᵢ|².

This corollary does not claim that the Born rule is derived. It states that the realization structure does not replace Born-rule weighting. CBR constrains admissible realization structure; Born compatibility constrains the ensemble frequencies with which realization-compatible alternatives occur.

If a CBR model predicts non-Born frequencies, that prediction must be explicitly stated, modeled, and exposed to empirical test.

7.4 Corollary 4: Non-reduction to decoherence

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

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.

This corollary does not criticize decoherence. It distinguishes roles. Decoherence explains interference suppression and record stabilization. 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 or weights, those values 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.

Post hoc parameter adjustment 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.

7.6 Corollary 6: Failure-capability

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

This is a strength rather than a weakness. A law-candidate becomes more serious when it states its defeat conditions. Failure may occur through an undefined context, empty candidate class, circular functional, post hoc parameter tuning, absent minimizer, unresolved non-equivalent minimizers, Born-rule violation without a testable deviation, reduction to decoherence, or insulation from possible criticism.

The later Failure Theorem will formalize these conditions.

8. Minimal Two-Path Demonstration

The preceding sections reconstructed the CBR-form structure and stated the theorem under which that structure arises. This section begins a controlled demonstration of how the reconstructed structure can be instantiated.

The two-path model is not intended as a complete theory of measurement. It is not a proof of universal applicability. Its role is narrower: to show that the reconstructed CBR structure can be applied in a simple context with explicitly defined candidates, accessibility conditions, burden terms, parameter-fixity requirements, Born compatibility, and decoherence distinction.

Generalization to broader measurement contexts requires additional context-specific definitions of 𝒜(C), ℛ_C, ≃_C, and the relevant admissibility conditions.

8.1 The 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. This setup is chosen because it is minimal: it contains more than one possible outcome structure while remaining simple enough to analyze explicitly.

8.2 Measurement records

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

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

|Ψ⟩ = α|0⟩|R₀⟩ + β|1⟩|R₁⟩.

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

8.3 Decoherence structure

If the record states become distinguishable, then

⟨R₀|R₁⟩ → 0.

This indicates suppression of interference between the 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 the individual realization channel selected in the context. The CBR demonstration therefore does not compete with decoherence at this stage; it asks whether a further selection structure over admissible realization-compatible channels can be defined.

8.4 Context C in the two-path model

In the two-path demonstration, 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,
and the relevant coarse-graining.

This context is fixed before outcome comparison. The candidate class, burden functional, equivalence relation, and any parameter weights must be defined from this context rather than from the observed result.

8.5 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 important. Φ₀ and Φ₁ represent individual realization candidates. Φ_mix represents the 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.

8.6 Candidate class and realization-compatible subclass

A broad candidate class may be written as

𝒜(C) = {Φ₀, Φ₁, Φ_mix}.

However, 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 them with individual realization. The exact membership of 𝒜_real(C) depends on the admissibility conditions specified for the context.

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 a decoherence-compatible channel rather than a completed realization candidate.

8.7 Accessibility parameter η

Let

η ∈ [0,1]

measure the accessibility of the relevant record information.

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

The accessibility parameter helps distinguish mere formal alternatives from context-supported record structures. A CBR model must not impose a which-path realization claim when the context does not support such a record structure.

8.8 Toy realization functional

A minimal burden functional for the two-path model 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 equation states that the realization burden of a candidate channel is modeled as a weighted combination of stability, accessibility, probability, and dynamical burdens. It is a toy-model functional, not a claimed final universal functional for all measurement contexts.

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.

8.9 High-accessibility case

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. Born compatibility then requires that, across repeated equivalent contexts,

P(Φ₀) = |α|²,
P(Φ₁) = |β|².

These equations are not a replacement of Born weighting by minimization. They state the ensemble compatibility condition that any realization structure must preserve.

8.10 Low-accessibility case

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

Several possibilities arise. The channels Φ₀ and Φ₁ may be excluded from 𝒜_real(C). Alternatively, they may become operationally equivalent under ≃_C because no accessible test in C distinguishes them. Or the context may remain below the threshold required for individual realization in the modeled sense.

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

8.11 Symmetric case 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.

8.12 What the two-path model demonstrates

The two-path model demonstrates that the reconstructed structure can be instantiated in a controlled setting. It shows how to define a context C, candidate channels, a realization-compatible subclass, an accessibility parameter, a toy burden functional, parameter-fixity requirements, Born-compatibility conditions, and a decoherence distinction.

It does not demonstrate that CBR is universally true. It does not demonstrate that the toy ℛ_C is final. It does not defeat rival interpretations. It shows only that CBR-form reconstruction can be made operationally explicit in a minimal nontrivial context.

That is sufficient for the demonstration’s purpose. The broader question is whether such context-specific structures can be developed, tested, and survived in richer physical platforms.

9. Degeneracy, Ties, and Born-Compatible Sampling

The reconstruction developed above treats minimization as a formal representation of context-relative candidate comparison. However, minimization does not always produce a single formal candidate. In many physically meaningful contexts, multiple admissible candidates may have equal or indistinguishably low realization burden. This possibility is not a minor technicality. It is a central requirement for any disciplined realization law.

If degeneracy is ignored, a proposed selection law may appear more determinate than it actually is. If degeneracy is handled after the outcome is known, the law becomes vulnerable to circularity. For this reason, CBR must state in advance how tied or co-minimizing candidates are treated.

Let the minimizer set be:

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

This set contains all admissible candidates whose realization burden is minimal in context C. CBR is straightforward when M_C contains exactly one candidate. The more important case is when M_C contains more than one.

9.1 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 and readout procedures available in C. In such a case, the formal multiplicity of minimizers does not imply a physically meaningful ambiguity. The model selects a unique operational realization class even if it does not select a unique mathematical representative.

The distinction matters. A physical theory need not eliminate distinctions that cannot be operationally accessed in the relevant context. But it must state when such distinctions are operationally irrelevant. That is the role of ≃_C.

9.2 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, the minimization rule has not selected one operational realization class. The model must then provide a tie rule. That tie rule must be specified before outcome comparison. If it is introduced only after observing the result, it is not a law-like completion of the model; it is a post hoc repair.

Thus, in any context C, CBR has three possible statuses with respect to degeneracy:

1. Unique minimizer:M_C contains one candidate.

2. Operationally harmless degeneracy: all minimizers are equivalent under ≃_C.

3. Incomplete selection:M_C contains operationally distinct minimizers and no pre-specified tie rule exists.

The third case is a failure of completeness in that context.

9.3 Tie rules and Born compatibility

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

|ψ⟩ = α|0⟩ + β|1⟩,

a Born-compatible tie 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. Born-compatible sampling concerns the frequencies with which realization-compatible alternatives occur across repeated trials.

The two roles must remain distinct.

9.4 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 and how they are compared within context C. Born weighting determines the frequencies with which outcomes occur across repeated equivalent contexts. A CBR model that collapses these roles into one another risks either becoming redundant with probability theory or violating quantum statistics.

Therefore, the proper statement is not:

ℛ_C directly replaces Born weighting.

The proper statement is:

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

If a future CBR model proposes a deviation from Born weighting, that deviation must be explicitly modeled, specified in advance, and exposed to empirical test. No such deviation is asserted in the present reconstruction.

9.5 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 a tie rule after observing the result.

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

This makes the model more constrained, not less. It gives critics a clear point of evaluation: identify a context C, define 𝒜(C) and ℛ_C, compute or characterize M_C, and determine whether minimizers are equivalent under ≃_C or governed by a legitimate pre-outcome tie rule.

10. Non-Circularity and the Anti-Cheating Rule

The central danger for any proposed law of 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 model is non-circular only if the following are specified before outcome comparison:

C, the measurement context;
𝒜(C), the admissible candidate class;
ℛ_C, the realization-burden functional;
≃_C, the operational equivalence relation;
ε_C, any operational tolerance used in defining equivalence;
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 forbidden post hoc functional

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

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 that uses such a construction fails as a non-circular law-candidate.

10.2 Legitimate dependence on context

The anti-cheating rule does not prohibit dependence on the measurement context. On the contrary, 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.

10.3 Formal non-circularity condition

The non-circularity requirement may be expressed symbolically as:

∂𝒜(C)/∂Φ∗_C = 0,
∂ℛ_C/∂Φ∗_C = 0,
∂≃_C/∂Φ∗_C = 0.

These expressions are not meant as ordinary derivatives in a smooth calculus sense. They are compact notation for independence requirements. The candidate class, burden functional, and equivalence relation must not depend on the selected realization channel.

If an operational tolerance ε_C or parameters λᵢ are used, the same independence requirement applies:

∂ε_C/∂Φ∗_C = 0,
∂λᵢ/∂Φ∗_C = 0.

The tolerance and weights cannot be adjusted after the result is known.

10.4 Parameter fixity and calibration

Parameter fixity does not require that every model be parameter-free. It requires that parameters be fixed by context, theory, 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 prior to outcome comparison. If calibration data are used, the calibration data and protocol must be separated from the outcome test.

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

10.5 Consequence of the anti-cheating rule

If C, 𝒜(C), ℛ_C, ≃_C, ε_C, and all λᵢ are fixed before outcome comparison, then CBR cannot be dismissed as merely naming the outcome after the fact. It may still be wrong. Its functional may still be physically inadequate. Its candidate class may still be incomplete. Its predictions may still fail. But the specific charge of post hoc selection is blocked.

If any of these objects are defined or adjusted after the realized outcome is known, CBR fails as a non-circular law-candidate in that context.

This is the central anti-cheating rule.

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.

11.1 Born-rule preservation

For a two-outcome state

|ψ⟩ = α|0⟩ + β|1⟩,

Born-rule preservation requires:

lim_{N→∞} N₀/N = |α|²,
lim_{N→∞} N₁/N = |β|².

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

These equations state that long-run frequencies must approach the Born-rule weights. A CBR model that preserves these limits is Born-compatible in the two-outcome case.

For a general state

|ψ⟩ = Σᵢ αᵢ|i⟩,

the corresponding requirement is:

P(i) = |αᵢ|².

11.2 Realization role versus probability role

CBR asks:

What is the physically admissible structure of realization in context C?

The Born rule asks:

How frequently do outcomes occur across repeated trials?

CBR must not confuse these roles. If minimization were interpreted as directly replacing Born weighting, the model would risk contradicting standard quantum statistics. The safer and more 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 Non-Born deviations

A CBR model may, in principle, propose a non-Born deviation. But such a proposal would carry additional burdens. It would need to specify:

the context in which the deviation is expected,
the modified probability or visibility prediction,
the experimental conditions under which the deviation should appear,
the baseline standard quantum prediction,
the tolerance or error budget,
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.

11.4 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.

11.5 Why probability compatibility strengthens the reconstruction

Born compatibility restricts CBR. It prevents the model from using “realization” as a license to rewrite quantum statistics. This makes the reconstruction more conservative and more referee-facing.

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.

12. Non-Reduction to Decoherence

Decoherence is central to modern quantum theory. It explains how interactions with environmental or record-bearing degrees of freedom suppress interference between alternatives. It also explains how stable pointer-like records and effectively classical structures can emerge.

CBR does not reject decoherence. The non-reduction requirement is narrower. It asks whether a candidate realization law supplies any additional selection structure beyond a non-selective decoherence-compatible map.

12.1 Decoherence role

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

⟨R₀|R₁⟩

approaches zero. 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 significant physical process. It helps explain why stable records emerge and why interference becomes inaccessible in ordinary measurement contexts.

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 realization selection.

A non-selective decoherence-compatible channel may describe the transition to an effectively classical mixture. 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 an operationally unique realization class is obtained.

This is the role CBR attempts to formalize.

12.3 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.

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.

CBR remains independent only if Φ∗_C supplies realization content not already contained in Φ_mix.

12.4 Reduction failure condition

The reduction failure condition is straightforward.

If, for every relevant context C,

Φ∗_C ≃_C Φ_mix,

and no further realization content is supplied, then CBR reduces to decoherence. In that case, CBR fails as an independent realization law.

This is not a failure of decoherence. It is a failure of CBR to add independent law-candidate content.

12.5 Non-reduction condition

CBR remains distinct from decoherence if the selected realization structure does work that the non-selective decoherence map does not do. This may involve selecting an admissible realization-compatible channel, defining a context-relative realization class, or specifying conditions under which multiple minimizers are operationally equivalent.

The burden is not merely verbal. CBR must be able to identify the formal role played by Φ∗_C that is not already exhausted by Φ_mix.

If it cannot, the theory should be treated as a reformulation of decoherence rather than an independent realization law.

12.6 Why non-reduction strengthens the proposal

The non-reduction condition makes CBR more vulnerable. It gives critics a clear test: show that the CBR-selected structure adds no content beyond decoherence. If that demonstration succeeds across the relevant contexts, CBR loses its independent status.

This is exactly how a disciplined law-candidate should behave. It should not be protected from reduction. It should state the condition under which reduction defeats it.

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. For this reason, the failure conditions are 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 model can fail in a context C. The theorem should be read carefully: failure in a given context means failure of that CBR model as a law-candidate in that context. It does not prove the logical impossibility of every possible CBR-like theory.

13.1 Theorem statement

Theorem 2: Failure Theorem for CBR Law-Candidates

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

1. C is not physically specified.

2. 𝒜(C) is empty.

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

4. ℛ_C is defined after the realized outcome is known.

5. ≃_C or ε_C is defined or adjusted after the realized outcome is known.

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

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

8. M_C is empty.

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

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

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

12. 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. A law without a defined context has no determinate target.

If 𝒜(C) is empty, undefined, arbitrary, or constructed after the result is known, the model fails the candidate-set and admissibility burdens. A selection law cannot operate without a candidate class fixed independently of the selected result.

If ℛ_C is defined after the outcome is known, the model fails the non-circularity burden. In that case, the functional does not select the outcome; it is fitted to the outcome.

If ≃_C or ε_C is adjusted after the result is known, the model fails the operational-equivalence burden. Operational equivalence must be determined by the accessible tests and resolution limits of C, not by the desired outcome of the analysis.

If λᵢ parameters or other components of ℛ_C are adjusted after outcome observation, the model fails the parameter-fixity burden. This converts the burden functional into an outcome-fitting device.

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 burden. The minimization structure must actually identify a minimizer, an equivalence class of minimizers, or a principled approximate selection rule.

If M_C contains operationally distinct minimizers with no pre-specified tie rule, the model fails the operational-uniqueness burden. The model has not selected one physically meaningful realization class.

If the model violates Born-rule frequencies without a registered and explicitly modeled deviation claim, it fails the probability-compatibility burden. A realization law cannot casually override established quantum statistics.

If Φ∗_C adds no realization content beyond Φ_mix, the model fails the non-reduction burden. In that case, CBR has become a restatement of non-selective decoherence rather than an independent realization law.

If no possible structural or empirical condition could count against the model, it fails the vulnerability burden. A law-candidate must expose itself to defeat.

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 therefore 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. The present theorem states that failure of any of these conditions 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.

13.4 Why failure conditions strengthen the proposal

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

Define C.
Inspect 𝒜(C).
Evaluate whether ℛ_C is fixed and non-circular.
Check whether M_C exists.
Determine whether minimizers are equivalent under ≃_C.
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.

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 therefore limited. They identify differences in explanatory target, not final superiority.

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 the description associated with observation, measurement, or experimental context. Such views can be operationally successful without supplying a separate dynamical law 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 ℛ_C, and an operational equivalence relation ≃_C such that a selected realization structure is obtained.

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

14.2 Everett / Many-Worlds

Everettian approaches do not generally seek a unique outcome-selection law in the same sense. Instead, they treat branching structure as physically real and avoid unique selection by allowing multiple branches 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 Everett. 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 dynamics, often with stochastic terms or modified evolution laws. These theories directly attempt to explain outcome selection by changing or supplementing the dynamics.

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

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

Again, 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, or information rather than as direct descriptions of observer-independent physical propensities.

CBR takes a different route. It seeks a physical context-relative realization criterion, represented through admissible candidates, burden comparison, and operational equivalence. 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 Purpose of comparison

These comparisons show that CBR occupies a specific law-candidate space:

physical outcome realization through context-relative admissibility, non-circular comparison, operational uniqueness, Born compatibility, non-reduction, and failure vulnerability.

The claim is not that all other approaches fail. The claim is that CBR addresses a specific target that many approaches either treat differently, avoid, dissolve, or relocate.

This restraint is important. The present paper establishes formal viability under assumptions. It does not establish interpretive victory.

15. What the Paper Establishes

The reconstruction should be evaluated by what it actually establishes, not by stronger claims it does not make. This section separates the established result from claims that remain outside the scope of the paper.

15.1 Established result

Under the assumptions stated in Section 5, this paper establishes that CBR can be reconstructed from general law-candidate burdens.

It establishes that the CBR form

Φ∗C ∈ argmin{Φ ∈ 𝒜(C)} ℛ_C(Φ)

is not arbitrary terminology. It is a compact representation of the structure required by a candidate realization law that must define a context, select from admissible candidates, use a pre-outcome comparison rule, handle operational uniqueness, preserve Born-rule statistics, avoid reduction to decoherence, control parameter tuning, and expose itself to failure.

The paper also establishes that the reconstructed structure can be instantiated in a minimal two-path context. The two-path model defines a context C, candidate channels Φ₀, Φ₁, and Φ_mix, an admissible realization-compatible subclass 𝒜_real(C), an accessibility parameter η, and a toy burden functional ℛ_C with fixed weights.

The paper further establishes that CBR can state explicit non-circularity conditions, operational-uniqueness conditions, Born-compatibility requirements, non-reduction conditions, and failure conditions.

These are formal and structural achievements. They make CBR more precise as a law-candidate.

15.2 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 particular 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.

The paper 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 requirement for this reconstruction.

The paper also does not show that every possible CBR-like model satisfies the failure theorem. The theorem applies to models that accept the burdens stated here. A different theory could reject the target question or adopt different burdens, but then it would no longer be a CBR-form realization law in the sense reconstructed in this paper.

15.3 The proper conclusion

The proper conclusion is this:

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

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

Nevertheless, the result is meaningful. It shifts the question from whether CBR can be stated coherently to whether the reconstructed structure can be specified, tested, and survived in concrete physical platforms.

That is the appropriate next burden.

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 demonstration and future empirical discrimination.

16.1 Relation to the law-candidate/checklist paper

A law-candidate or checklist paper defines the burdens a realization law must satisfy. Such a paper asks: what must any serious candidate law of outcome realization define, preserve, distinguish, and risk?

The present paper takes the next step. It reconstructs the CBR form from those burdens. Instead of merely listing criteria, it shows how the criteria lead to the formal structure:

Φ∗C ∈ argmin{Φ ∈ 𝒜(C)} ℛ_C(Φ).

In this sense, the present paper converts the checklist into a representational theorem.

16.2 Relation to the toy-model/demonstration need

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

The two-path demonstration addresses this concern in a minimal setting. It shows how to define a context, candidates, accessibility, a burden functional, parameter-fixity conditions, Born compatibility, and decoherence distinction. This does not prove universal applicability, but 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.

16.3 Relation to empirical-discrimination papers

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

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

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

16.4 Relation to Born-rule derivation work

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

16.5 Strategic placement

The present paper should be understood as a bridge between three layers:

1. Law-candidate framing: what burdens a realization law must satisfy.

2. Formal reconstruction and demonstration: how those burdens yield the CBR form and how that form can be instantiated.

3. Empirical discrimination: how context-specific models could be tested or defeated.

The 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.

The present paper therefore converts CBR from a proposed law-form into a reconstructed minimal structure: one that is formally definable, demonstrable in a controlled model, and vulnerable to failure.

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{Φ ∈ 𝒜(C)} ℛ_C(Φ).

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, or reject decoherence. It does not prove that the two-path model is universally applicable. It does not defeat rival interpretations by assertion.

The paper’s conclusion is more limited:

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

That conclusion is meaningful because it shifts the burden of evaluation. The question is no longer merely whether CBR can be stated coherently. It can. The next burden is whether context-specific versions of 𝒜(C), ℛ_C, ≃_C, and any associated parameters can be independently specified, applied to concrete physical platforms, 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 with standard quantum and decoherence-based accounts.

Appendix A: Complete Formal Definitions

This appendix consolidates the notation used throughout the paper.

A.1 Measurement context

C denotes a physically specified measurement context.

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

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

A.2 Hilbert space

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

𝓗′ denotes the post-interaction or record-bearing Hilbert space relevant to the context.

The notation does not require modeling every physical degree of freedom in the universe. It denotes the state space selected for 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.

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-realization record structure.

The use of CPTP maps is a formal discipline. It does not claim that all interpretive questions about measurement are thereby resolved.

A.5 Admissible candidate class

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

The class must satisfy:

𝒜(C) ≠ ∅

for ordinary exact minimization. If 𝒜(C) is empty, the selection rule has no target and the model fails in C unless the context or admissibility conditions are revised in a principled, pre-specified way.

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 allows the formalism to distinguish broader dynamical channels from completed individual realization candidates. For example, Φ_mix may describe non-selective decoherence-compatible evolution without itself representing individual realization.

A.7 Realization-burden functional

ℛ_C: 𝒜(C) → ℝ_{\ge 0}

denotes the context-relative realization-burden functional.

For a candidate Φ ∈ 𝒜(C), the value ℛ_C(Φ) represents the burden associated with that candidate 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 or parameters, those values must be specified before testing the realization claim.

A.8 Selected realization channel

The selected realization channel or class is defined by:

Φ∗C ∈ argmin{Φ ∈ 𝒜(C)} ℛ_C(Φ).

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

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_{Ψ ∈ 𝒜(C)} ℛ_C(Ψ)}.

If M_C contains one element, selection is 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.

A.10 Operational equivalence relation

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

A strict definition is:

Φ₁ ≃_C Φ₂ ⇔ ∀T ∈ 𝒯(C), T(Φ₁(ρ)) = T(Φ₂(ρ)).

Here 𝒯(C) is the set of operationally accessible tests or readout procedures in context C.

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

|T(Φ₁(ρ)) − T(Φ₂(ρ))| ≤ ε_C

for all T ∈ 𝒯(C).

The tolerance ε_C must be fixed by the operational limits of C before outcome comparison.

A.11 Accessibility parameter

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

η = 0 means no relevant record is operationally accessible.
η = 1 means the relevant record is fully accessible.
Intermediate values indicate partial accessibility.

A.12 Non-selective decoherence-compatible channel

Φ_mix denotes a non-selective decoherence-compatible channel.

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

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

A.13 Fixed burden weights

λᵢ ≥ 0 denotes a fixed burden weight.

In the two-path toy model, the weights may be written:

λ_S, λ_I, λ_P, λ_D ≥ 0.

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

No λᵢ may be adjusted after the realized outcome is known.

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_{Ψ ∈ 𝒜(C)} ℛ_C(Ψ) + ε.

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

Appendix B: Full Proof Structure of the Conditional Minimal Representation Theorem

This appendix expands the proof logic for Theorem 1.

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:

1. context specification,

2. nonempty admissible candidate selection,

3. pre-outcome candidate-class fixity,

4. context-fixed candidate comparison,

5. pre-outcome comparison-rule fixity,

6. pre-outcome parameter fixity,

7. selection of one candidate or one operational equivalence class,

8. operational equivalence of minimizers or a pre-specified tie rule,

9. Born-rule ensemble compatibility,

10. non-reduction to decoherence,

11. explicit failure conditions.

Then the law admits a CBR-form representation:

Φ∗C ∈ argmin{Φ ∈ 𝒜(C)} ℛ_C(Φ),

where ℛ_C is a context-fixed realization-burden functional over admissible candidates.

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 any 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.

Since admissibility depends on the 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 imposes restrictions on 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

A law that selects among candidates must compare them.

For the reconstruction, any context-fixed ordering or ranking of admissible candidates may be represented by a non-negative burden functional:

ℛ_C: 𝒜(C) → ℝ_{\ge 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.

B.6 Non-circularity requires pre-outcome fixity

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

Therefore:

𝒜(C) must be fixed before outcome comparison.
ℛ_C must be fixed before outcome comparison.
≃_C must be fixed before outcome comparison.
Any λᵢ or ε_C must be fixed before outcome comparison.

Symbolically:

∂𝒜(C)/∂Φ∗_C = 0,
∂ℛ_C/∂Φ∗_C = 0,
∂≃_C/∂Φ∗_C = 0.

This expresses independence from the selected result.

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{Φ ∈ 𝒜(C)} ℛ_C(Φ).

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_{Ψ ∈ 𝒜(C)} ℛ_C(Ψ)}.

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 sampling

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 proof 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,
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:

CBR is true in nature,
a particular ℛ_C is correct,
the Born rule is derived,
decoherence is false,
or rival interpretations are defeated.

Appendix C: Two-Path Model Calculations

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

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 similarly for |1R₁⟩.

C.3 Decoherence condition

When the record states become distinguishable,

⟨R₀|R₁⟩ → 0.

If the record degrees of freedom are traced out, the reduced state of the system 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 the 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:

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

Where:

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, not the replacement of probability by 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.

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.

D.2 Exact operational equivalence

Define:

Φ₁ ≃_C Φ₂ ⇔ ∀T ∈ 𝒯(C), T(Φ₁(ρ)) = 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 ⇔ ∀T ∈ 𝒯(C), |T(Φ₁(ρ)) − 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.

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_{N→∞} N₀/N = |α|²,
lim_{N→∞} N₁/N = |β|².

E.2 General Born compatibility

For:

|ψ⟩ = Σᵢ αᵢ|i⟩,

Born compatibility requires:

P(i) = |αᵢ|².

Across repeated equivalent contexts:

lim_{N→∞} Nᵢ/N = |αᵢ|².

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 model may propose a non-Born deviation only if the deviation is:

explicitly stated,
mathematically modeled,
context-specific,
pre-specified before testing,
and empirically vulnerable.

Without these conditions, non-Born behavior is a failure of probability compatibility.

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.

First, 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.

Second, CBR fails at the level of the candidate set if 𝒜(C) is empty, arbitrary, undefined, 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.

Third, CBR fails at the level of admissibility if the subclass of individual realization-compatible candidates, 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.

Fourth, CBR fails at the level of non-circularity if the realization-burden functional ℛ_C 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. The comparison rule must be fixed before outcome comparison.

Fifth, CBR fails at the level of operational equivalence if the equivalence relation ≃_C or the operational tolerance ε_C is adjusted after the outcome is observed. Operational equivalence must be determined by the accessible tests and resolution limits of context C, not by the desire to make a particular minimizer appear unique.

Sixth, 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.

Seventh, CBR fails at the level of selection if no minimizer exists and no pre-specified ε-minimizer rule has been provided. The model must be able to identify a minimizing candidate, a minimizing equivalence class, or a principled approximate minimizer. Without that, the selection rule is undefined in context C.

Eighth, CBR fails at the level of operational uniqueness if the minimizer set 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.

Ninth, 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.

Tenth, 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.

Finally, 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.

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.

Final Continuity/Completion Notes

The completed manuscript establishes a formal, conditional reconstruction of CBR as a candidate law-form for quantum outcome realization. It shows that if a realization law must define a context, select from admissible candidates, compare them by a pre-outcome burden functional, produce operational uniqueness, preserve Born-rule statistics, avoid reduction to decoherence, control parameter tuning, and expose itself to failure, then it admits the CBR-form representation:

Φ∗C ∈ argmin{Φ ∈ 𝒜(C)} ℛ_C(Φ).

The manuscript does not establish that CBR is true in nature. It does not experimentally confirm CBR. It does not replace quantum mechanics, replace the Born rule, reject decoherence, or defeat rival interpretations. It does not prove that the toy ℛ_C is universal.

The next paper or empirical program must do the following:

define context-specific 𝒜(C),
specify a non-circular physical ℛ_C,
fix all λᵢ parameters before outcome testing,
define ≃_C and ε_C,
state platform-specific Born-compatible or deviation predictions,
compare against standard quantum/decoherence baselines,
and identify precise failure conditions.

That is the proper next burden.