Constraint-Based Realization: Canonical Law Form, Operational Uniqueness, and an Accessibility-Based Failure Criterion
Abstract
This paper formulates the quantum measurement problem as a realization problem: unitary evolution and decoherence describe dynamical branching, environmental stabilization, and effective record formation, but do not by themselves specify which admissible registration channel becomes the realized outcome. Constraint-Based Realization, or CBR, treats realization as a constrained selection law over physically available registration channels Γ, evaluated relative to a preparation 𝒮, constraint structure 𝒞, experimental context ℰ, and calibrated record-accessibility parameter η.
The central result is conditional. Given admissibility assumptions A1–A7 and a fixed admissible preorder ⪯ over Γ, any admissible realization law ℛ ∈ 𝒜 is operationally equivalent to the canonical law
ℛ∗ = arg min over γ ∈ Γ of 𝓕(γ; 𝒮, 𝒞, ℰ, η),
where 𝓕 represents the admissible constraint ordering. The theorem establishes uniqueness only up to operational equivalence and does not claim that nature uses CBR.
The framework is empirically exposed through calibrated η-variation. A valid test must pre-register Γ, ⪯, 𝓕 or W, η-grid, baseline comparator ℬ, total uncertainty ε_total, and the exclusion criterion. If the observed residual Data − ℬ tracks the predicted Δ_CBR(η) in sign, shape, and scale above ε_total, CBR receives platform-specific support. If a strong null holds across the calibrated η-grid under adequate sensitivity, canonical CBR is ruled out in the tested admissible class and regime.
1. Introduction
1.1 The realization problem
Quantum theory gives a precise account of state evolution and measurement statistics. Under unitary evolution, a prepared state develops continuously according to the dynamical laws of the theory. Under decoherence, interactions among system, apparatus, and environment suppress observable interference between effectively distinguishable branches. These mechanisms explain why measurement records become stable, why classical descriptions become practically available, and why interference between macroscopically distinct alternatives is ordinarily inaccessible.
They do not, by themselves, specify which registered outcome is realized as the single observed result.
This paper isolates that remaining issue as the realization problem. The problem is not whether quantum theory predicts correct statistics; it does. The problem is whether there exists a law-governed account of the transition from multiple physically available registration channels to one realized registration.
Four notions must be kept distinct.
Unitary evolution describes the dynamical propagation of the quantum state.
Decoherence describes the suppression of interference and stabilization of effective records.
Registration describes the formation of physically available detector or environmental records.
Realization describes the occurrence of one registered channel as the observed outcome.
Constraint-Based Realization, or CBR, treats realization as a constrained selection problem over admissible registration channels. The available channels are not arbitrary. They belong to a contextually specified set Γ. The relevant physical restrictions are encoded in 𝒞. The experimental context is ℰ. The preparation is 𝒮. The accessibility of record constraints is η. The candidate realization law is expressed through a functional 𝓕.
The central question is therefore:
Given 𝒮, 𝒞, ℰ, η, and Γ, is there a unique admissible realization law form, up to operational equivalence?
This paper answers conditionally: yes, if the realization law belongs to the admissible class 𝒜 defined by A1–A7.
1.2 Contribution of the paper
The paper’s main contribution is not the claim that CBR is true. The main contribution is a conditional classification result:
If realization is required to be operationally invariant, contextually admissible, constraint-sensitive, probabilistically normalized, stable under coarse-graining, compositionally consistent, and baseline-recovering, then its canonical law form is fixed up to operational equivalence.
The canonical law is:
ℛ∗ = arg min over γ ∈ Γ of 𝓕(γ; 𝒮, 𝒞, ℰ, η).
The paper contributes five linked results.
First, it formulates realization as a selection problem over admissible registration channels.
Second, it defines the admissible class 𝒜 through assumptions A1–A7.
Third, it states the uniqueness target operationally:
For every ℛ ∈ 𝒜,
ℛ ≃ ℛ∗.
Fourth, it introduces η as an experimentally calibrated record-accessibility control. η is not a free fit parameter. It must be fixed by independent calibration before the outcome data are analyzed.
Fifth, it states a strong-null failure criterion. If η is calibrated, varied, and not absorbed by the baseline comparator ℬ, yet no deviation appears above ε_total, then canonical CBR fails in that tested regime.
Here ℬ is not idealized textbook quantum theory alone. ℬ denotes standard quantum mechanics plus the platform-specific model of decoherence, detector inefficiency, apparatus noise, environmental drift, and known nuisance effects.
1.3 Theorem preview
The central theorem can be previewed as follows.
Theorem Preview.
Let 𝒜 be the class of realization laws satisfying A1–A7. Let Γ be the admissible set of registration channels for 𝒮, 𝒞, ℰ, and η. If ℛ ∈ 𝒜, then ℛ admits an extremal representation over Γ and is operationally equivalent to the canonical law
ℛ∗ = arg min over γ ∈ Γ of 𝓕(γ; 𝒮, 𝒞, ℰ, η).
Thus:
ℛ ≃ ℛ∗.
This is a conditional theorem. Its force depends on the admissibility assumptions. Its empirical risk depends on η being independently calibrated and on ℬ being specified before testing.
1.4 Scope and non-claims
This paper does not prove that nature uses CBR. It proves only that, within the stated admissible class 𝒜, the canonical realization law is forced up to operational equivalence.
It does not replace standard quantum mechanics in ordinary regimes. In screened or inaccessible regimes, CBR must recover ℬ.
It does not claim that a null result in one experiment falsifies every possible realization theory. A strong null rules out canonical CBR only within the tested admissible class, platform, η-range, and sensitivity regime.
It does not derive the full Born rule. The present paper requires baseline compatibility with Born statistics in the appropriate limit. The deeper classification of admissible probability weighting belongs to the companion probability analysis.
The paper is therefore deliberately narrow. It asks whether realization can be made precise, constrained, and experimentally vulnerable.
2. Operational Setup
2.1 Experimental context
An experimental context ℰ is the full operational specification of the measurement situation.
ℰ includes the preparation procedure, transformations applied to the system, measurement settings, record degrees of freedom, detector configuration, environmental couplings, available registration channels, and platform-specific noise sources.
The role of ℰ is to prevent realization from being detached from the apparatus. CBR does not select from all imaginable outcomes. It selects only from channels physically available in the context.
A candidate channel γ is admissible only if γ ∈ Γ, where Γ is determined by ℰ together with 𝒮, 𝒞, and η.
2.2 Preparation state
Let 𝒮 denote the preparation state or operational preparation class.
In simple pure-state examples, one may write:
|ψ⟩ = α|0⟩ + β|1⟩.
This expression is illustrative. The formalism does not require 𝒮 to be pure. 𝒮 may represent a density operator, an equivalence class of preparation procedures, or any operationally specified input state.
The preparation 𝒮 matters because the available registration channels, baseline predictions, and constraint functional are evaluated relative to it.
2.3 Registration channels
Let Γ denote the set of admissible registration channels.
Each γ ∈ Γ represents a physically available channel through which a measurement record can become registered. A registration channel is not merely an outcome label. It includes the physical route by which an outcome becomes stabilized, recorded, and made available in ℰ.
For example, in a two-detector setup, Γ may contain γ₀ and γ₁. In a more complex platform, Γ may contain coarse-grained detector states, environmental record classes, or macroscopic registration modes.
The channel set may depend on the full context:
Γ = Γ(𝒮, 𝒞, ℰ, η).
However, in any proposed test, Γ must be fixed before outcome analysis. Otherwise the theory would risk becoming adjustable after the data are known.
2.4 Constraint structure
Let 𝒞 denote the constraint structure relevant to realization.
𝒞 may include boundary conditions, conservation constraints, detector constraints, environmental constraints, record-stability constraints, contextual compatibility constraints, and constraints associated with accessible which-record information.
The role of 𝒞 is to make realization law-governed. CBR does not assert that one outcome is selected arbitrarily. It asserts that candidate channels are evaluated relative to the constraints imposed by the physical context.
2.5 Accessibility parameter
Let η ∈ [0, 1] denote operational record accessibility.
η = 0 represents no accessible record constraint.
η = 1 represents a fully accessible stable record constraint.
Intermediate values represent partial record accessibility.
η is not subjective knowledge. It is not what an observer happens to know. It is a calibrated physical control describing how accessible, stable, and recoverable the relevant record degrees of freedom are in the experimental context.
A valid CBR test requires η to be estimated independently of the final outcome statistics. The analysis may not choose η after seeing the data.
2.6 Baseline comparator
Let ℬ denote the baseline comparator.
ℬ is standard quantum mechanics supplemented by the platform-specific account of decoherence, detector inefficiency, apparatus noise, environmental drift, calibration uncertainty, and known systematic effects.
CBR is not tested against an unrealistically clean quantum model. It is tested against ℬ.
The empirical burden is therefore:
CBR must predict an η-dependent deviation from ℬ that exceeds ε_total.
If no such deviation appears under calibrated η-variation, canonical CBR fails in that tested regime.
2.7 Realization law
A deterministic realization law may be written:
ℛ: (𝒮, 𝒞, ℰ, η) ↦ γ.
A probabilistic realization law may be written:
ℛ: (𝒮, 𝒞, ℰ, η) ↦ P(γ | 𝒮, 𝒞, ℰ, η).
The probabilistic form is more general. It assigns normalized realization statistics over Γ.
The canonical CBR law is the law generated by extremizing 𝓕 over Γ:
ℛ∗ = arg min over γ ∈ Γ of 𝓕(γ; 𝒮, 𝒞, ℰ, η).
3. Operational Equivalence
3.1 Definition
Two realization laws ℛ₁ and ℛ₂ are operationally equivalent when they generate identical observable registration statistics in every admissible experiment.
Write:
ℛ₁ ≃ ℛ₂.
More explicitly, ℛ₁ ≃ ℛ₂ iff, for every admissible ℰ, every applicable 𝒮, every relevant 𝒞, every calibrated η, and every γ ∈ Γ,
P₁(γ | 𝒮, 𝒞, ℰ, η) = P₂(γ | 𝒮, 𝒞, ℰ, η).
Operational equivalence is reflexive, symmetric, and transitive. It therefore defines an equivalence relation over admissible realization laws.
3.2 Why uniqueness must be operational
The theorem does not require ontological uniqueness. That would be too strong and unnecessary.
Two theories may differ in interpretation, hidden structure, or metaphysical description while producing identical observable registration statistics. If no admissible experiment can distinguish them, then they are equivalent for the purposes of this theorem.
The relevant uniqueness claim is therefore:
For every ℛ ∈ 𝒜,
ℛ ≃ ℛ∗.
This means that canonical CBR is unique within 𝒜 only up to observable consequences. It does not exclude empirically indistinguishable reformulations. It does exclude any operationally distinct law that satisfies A1–A7 while producing different registration statistics.
3.3 Why operational equivalence strengthens the paper
Operational equivalence makes the claim sharper and safer.
It avoids metaphysical overreach.
It gives the theorem a precise target.
It clarifies what would refute the theorem: an admissible ℛ ∈ 𝒜 that is not operationally equivalent to ℛ∗.
It also clarifies what would refute the physical model: calibrated η-variation producing no deviation from ℬ above ε_total when canonical CBR predicts one.
4. Admissible Realization Laws
4.1 Definition of admissible class 𝒜
The admissible class 𝒜 is the class of realization laws satisfying A1–A7.
These assumptions are not presented as metaphysical necessities. They are minimal operational constraints in the following sense: removing any one permits a recognizable pathology.
A law violating A1 becomes representation-dependent.
A law violating A2 can select unavailable outcomes.
A law violating A3 makes η empirically empty.
A law violating A4 fails as a probability law.
A law violating A5 depends on arbitrary coarse-graining.
A law violating A6 risks artificial correlations or signaling.
A law violating A7 conflicts with ordinary quantum experiments.
Thus A1–A7 define the standard of admissibility for the theorem.
4.2 A1. Operational invariance
Operationally indistinguishable descriptions must yield the same realization statistics.
If ℰ₁ and ℰ₂ differ only by notation, representation, labeling convention, or inaccessible descriptive detail, then a physical realization law cannot assign different observable statistics to them.
If ℰ₁ ≃ ℰ₂, then:
ℛ(𝒮, 𝒞, ℰ₁, η) ≃ ℛ(𝒮, 𝒞, ℰ₂, η).
A law violating A1 is not merely inelegant. It is representation-dependent.
4.3 A2. Contextual admissibility
The realized channel must belong to the physically available channel set Γ.
Thus:
γ∗ ∈ Γ.
A law violating A2 can assign realization weight to an outcome that the apparatus cannot register. Such a law is detached from ℰ and therefore physically inadmissible.
4.4 A3. Constraint monotonicity
If η is part of the physical context, then changes in η cannot be declared dynamically irrelevant by definition.
The realization functional must allow η to affect channel evaluation:
𝓕 = 𝓕(γ; 𝒮, 𝒞, ℰ, η).
A3 does not say every η-change produces a measurable deviation. It says η must be a genuine physical control, not a decorative symbol.
A law violating A3 makes accessibility empirically empty.
4.5 A4. Normalization
Realization weights must define a normalized probability law:
Σ over γ ∈ Γ of P(γ | 𝒮, 𝒞, ℰ, η) = 1.
A law violating A4 does not produce coherent outcome statistics. It cannot be compared consistently with ℬ.
4.6 A5. Stability under coarse-graining
Coarse-graining physically equivalent records cannot change observable realization statistics.
If γ₁ and γ₂ are merged into a coarse-grained channel γ₁₂, then:
P(γ₁₂) = P(γ₁) + P(γ₂).
A law violating A5 allows arbitrary relabeling or grouping to alter predictions. Such a law is operationally unstable.
4.7 A6. Composition consistency
Independent systems must compose without artificial correlations, signaling, or probability inconsistency.
For independent contexts A and B, with no coupling and no shared constraint:
P(γ_A, γ_B) = P(γ_A)P(γ_B).
A6 does not deny entanglement correlations. It applies only when the contexts are independent. A law violating A6 risks generating correlations from the realization rule itself rather than from physical interaction or shared preparation.
4.8 A7. Baseline recovery
When η is inaccessible, screened, or dynamically irrelevant, CBR must recover ℬ:
ℛ → ℬ.
A law violating A7 conflicts with ordinary quantum behavior. Since ℬ already accounts for standard quantum predictions plus platform-specific nuisance effects, CBR can only be viable if it agrees with ℬ in the regimes where ℬ is known to work.
A7 is also what makes the proposed empirical test sharp. CBR is not allowed to reinterpret every ordinary quantum experiment as evidence for itself. Its distinctive content appears only where calibrated η-dependence is predicted to exceed ε_total.
5. Assumption Audit
5.1 Minimality of the assumptions
A1–A7 are not claimed to be the only conceivable assumptions one could impose on a realization theory. They are claimed to be minimal in a weaker operational sense: each assumption blocks a specific failure mode.
The admissible class 𝒜 is therefore not arbitrary. It is the class of realization laws that avoid representation dependence, unavailable outcomes, empty accessibility parameters, malformed probabilities, coarse-graining ambiguity, pathological composition, and conflict with established baseline quantum behavior.
This audit matters because the theorem is conditional. If one rejects an assumption, the theorem’s scope narrows. But rejecting an assumption also requires accepting the pathology that the assumption prevents, or replacing it with a better condition.
5.2 Why A1 is needed
A1 is required because physical laws must not depend on arbitrary representation.
If two contexts are operationally indistinguishable, a realization law that assigns them different statistics is sensitive to description rather than experiment. Such a law could change its predictions under relabeling, reformulation, or inaccessible descriptive variation.
A1 therefore enforces representation independence.
5.3 Why A2 is needed
A2 is required because realization must occur through available physical channels.
Without A2, a law could select a channel outside Γ. That would detach realization from the apparatus and make the theory empirically undefined.
A2 therefore anchors the law to ℰ.
5.4 Why A3 is needed
A3 is required because η is introduced as a physical accessibility control.
If η cannot influence 𝓕 even in principle, then η has no empirical role. The theory would mention accessibility while preventing accessibility from mattering.
A3 therefore makes CBR testable through calibrated η-variation.
5.5 Why A4 is needed
A4 is required because realization statistics must be probabilities.
Without normalization, the law cannot consistently assign outcome weights over Γ. It also cannot be compared meaningfully with ℬ, which generates normalized observable statistics.
A4 therefore enforces probability discipline.
5.6 Why A5 is needed
A5 is required because observable predictions must not depend on arbitrary resolution of labels.
If two physically indistinguishable subchannels are merged, their total probability must be preserved. Otherwise the theory would predict different outcomes depending on how finely an experimenter describes the detector.
A5 therefore enforces coarse-graining stability.
5.7 Why A6 is needed
A6 is required because independent contexts must remain independent unless connected by physical coupling, shared preparation, or shared constraint.
Without A6, CBR could introduce artificial correlations between unrelated systems. In extreme cases, this could mimic signaling or violate probability consistency.
A6 therefore enforces compositional discipline.
5.8 Why A7 is needed
A7 is required because ordinary quantum experiments already strongly constrain any proposed realization law.
If CBR did not recover ℬ in screened regimes, it would already be empirically excluded. The theory must agree with ℬ where η is inaccessible, screened, or dynamically irrelevant.
A7 therefore enforces empirical compatibility.
5.9 The strong-null consequence
The same assumptions that make CBR admissible also make it vulnerable.
If Γ, 𝓕, η, ℬ, and ε_total are fixed before testing, and if η is independently calibrated, then CBR cannot freely absorb a null result.
The decisive strong-null condition is:
For every calibrated η,
|Data − ℬ| ≤ ε_total.
If canonical CBR predicts an accessibility-sensitive deviation and the strong-null condition holds, then canonical CBR in the tested admissible class is ruled out for that platform and constraint regime.
This is the paper’s central empirical discipline. CBR is not protected by vagueness. It succeeds only if calibrated η-dependence appears beyond the baseline comparator. It fails if that dependence is absent under adequate sensitivity.
6. Canonical Law Form
6.1 Candidate channel set
For fixed 𝒮, 𝒞, ℰ, and η, define the admissible registration set as
Γ = Γ(𝒮, 𝒞, ℰ, η).
Γ contains exactly those channels γ that are physically available in ℰ and compatible with 𝒞. A channel is not admissible merely because it can be named. It must be a possible registration route in the specified apparatus, detector, environment, or record-bearing system.
In any empirical test, Γ must be fixed before outcome analysis. If Γ is adjusted after the data are known, the framework loses falsifiability.
6.2 Admissible constraint ordering
To avoid making 𝓕 an arbitrary scoring device, introduce an admissible preorder ⪯ over Γ.
Write
γ₁ ⪯ γ₂
to mean that γ₁ is no more realization-costly than γ₂ relative to 𝒮, 𝒞, ℰ, and η.
The preorder ⪯ must be complete, transitive, operationally invariant, stable under coarse-graining, composition-consistent, and baseline-compatible.
Thus, the primitive structure is not an arbitrary function. It is the admissible constraint ordering induced by the physical context.
6.3 Realization functional
The realization functional
𝓕(γ; 𝒮, 𝒞, ℰ, η)
represents the admissible preorder ⪯.
That is,
γ₁ ⪯ γ₂ iff 𝓕(γ₁; 𝒮, 𝒞, ℰ, η) ≤ 𝓕(γ₂; 𝒮, 𝒞, ℰ, η).
This makes 𝓕 a representation of constraint ordering, not a post hoc fitting rule. Different functionals may represent the same preorder. Such functionals are operationally equivalent if they induce the same minimizers and registration statistics.
6.4 Canonical selection law
The canonical realization law is
ℛ∗ = arg min over γ ∈ Γ of 𝓕(γ; 𝒮, 𝒞, ℰ, η).
Equivalently, ℛ∗ selects the minimal channel or minimal equivalence class under ⪯.
If the minimizer is unique, ℛ∗ selects γ∗. If multiple minimizers are operationally indistinguishable, ℛ∗ selects their operational equivalence class. If distinguishable minimizers remain, the probabilistic version must be used.
6.5 Probabilistic version
A probabilistic realization law assigns weights over Γ:
P(γ | 𝒮, 𝒞, ℰ, η) = W(γ; 𝒮, 𝒞, ℰ, η) ÷ Σ over γ′ ∈ Γ of W(γ′; 𝒮, 𝒞, ℰ, η).
The weight function W is admissible only if it is nonnegative, operationally invariant, stable under coarse-graining, composition-consistent, and baseline-recovering.
Arbitrary W is not allowed. W must represent the same admissible constraint ordering ⪯ or a permitted probabilistic refinement of it.
In screened regimes,
P(γ) → P_Born(γ),
so that ℛ∗ → ℬ.
6.6 Operational uniqueness target
The strengthened target is:
For a fixed admissible constraint ordering ⪯, every ℛ ∈ 𝒜 representing ⪯ satisfies
ℛ ≃ ℛ∗.
This avoids overclaiming. The theorem does not say every imaginable realization rule collapses to CBR. It says that once A1–A7 and the admissible ordering ⪯ are fixed, the canonical representative is unique up to operational equivalence.
7. Main Theorems
7.1 Theorem 1: Existence
Theorem 1.
Let Γ be nonempty and compact. Let 𝓕 be bounded below and lower-semicontinuous on Γ. Then there exists at least one minimizing channel γ∗ ∈ Γ such that
𝓕(γ∗) ≤ 𝓕(γ)
for every γ ∈ Γ.
This ensures that ℛ∗ is well-defined under ordinary regularity conditions.
7.2 Theorem 2: Representation of the admissible preorder
Theorem 2.
Let ⪯ be a complete, transitive, operationally invariant, coarse-graining-stable, composition-consistent, baseline-compatible preorder over Γ. Then there exists an admissible functional 𝓕 representing ⪯ such that
γ₁ ⪯ γ₂ iff 𝓕(γ₁; 𝒮, 𝒞, ℰ, η) ≤ 𝓕(γ₂; 𝒮, 𝒞, ℰ, η).
This is the key strengthening. The theorem does not claim that any arbitrary rule can be disguised as minimization. It claims that an admissible physical ordering over Γ can be represented by an admissible functional.
7.3 Theorem 3: Minimizer equivalence
Theorem 3.
Let 𝓕₁ and 𝓕₂ be admissible functionals representing the same preorder ⪯ over Γ. Then their minimizing channels are operationally equivalent.
Thus, if
ℛ₁∗ = arg min over γ ∈ Γ of 𝓕₁(γ; 𝒮, 𝒞, ℰ, η)
and
ℛ₂∗ = arg min over γ ∈ Γ of 𝓕₂(γ; 𝒮, 𝒞, ℰ, η),
then
ℛ₁∗ ≃ ℛ₂∗.
Theorem 3 prevents the canonical law from depending on arbitrary numerical representation of the same physical ordering.
7.4 Theorem 4: Canonical reduction
Theorem 4.
For a fixed admissible preorder ⪯, every realization law ℛ ∈ 𝒜 representing ⪯ is operationally equivalent to the canonical law ℛ∗.
Therefore,
ℛ ≃ ℛ∗.
This is the central result in its strongest defensible form. The uniqueness is not absolute across all possible orderings. It is uniqueness within the admissible class once the physically motivated constraint ordering is fixed.
7.5 Corollary 1: Baseline recovery
In the screened-accessibility limit,
η → η_baseline,
the canonical law satisfies
ℛ∗ → ℬ.
This follows because baseline compatibility is part of admissibility. If η is inaccessible, screened, or dynamically irrelevant, CBR must recover the baseline comparator.
7.6 Corollary 2: Strong-null falsification
Let Γ, ⪯, 𝓕 or W, η-grid, ℬ, ε_total, and the exclusion criterion be fixed before outcome analysis.
If calibrated η-variation produces
|Data − ℬ| ≤ ε_total
for every calibrated η, then canonical CBR in the tested admissible class, platform, and constraint regime is ruled out.
This is the strong-null criterion.
8. Proof Sketch
8.1 Equivalence-class reduction
A1 implies that realization laws cannot depend on representational differences. If two contexts are operationally indistinguishable, they must produce the same registration statistics.
Thus ℛ is defined over operational equivalence classes, not arbitrary descriptions.
8.2 Channel restriction
A2 restricts realization to Γ. The law may select only physically available registration channels.
This makes realization a constrained problem rather than an unconstrained outcome assignment.
8.3 Ordering structure
The admissible preorder ⪯ ranks channels by realization cost relative to 𝒮, 𝒞, ℰ, and η.
Completeness ensures that admissible channels can be compared.
Transitivity prevents cyclic realization preferences.
Operational invariance prevents representation dependence.
Coarse-graining stability prevents label-resolution artifacts.
Composition consistency prevents artificial correlations.
Baseline compatibility anchors the ordering to ℬ in screened regimes.
8.4 Functional representation
Once ⪯ is fixed, 𝓕 represents the ordering. It does not create the ordering after the fact.
Any two admissible 𝓕 functions representing the same ⪯ induce the same minimal operational class. Therefore the canonical law is invariant under harmless reparameterizations of 𝓕.
8.5 Probability discipline
For probabilistic CBR, W must be nonnegative, normalized after summation, operationally invariant, coarse-graining stable, composition-consistent, and baseline-recovering.
This prevents arbitrary weighting. W must either represent ⪯ or refine it in an admissible way.
8.6 Baseline anchoring
A7 requires
ℛ → ℬ
when η is inaccessible, screened, or dynamically irrelevant.
This prevents CBR from conflicting with ordinary quantum experiments and prevents post hoc reinterpretation of all data as CBR-supporting.
8.7 Uniqueness
For fixed ⪯, the canonical law is unique up to operational equivalence:
ℛ ≃ ℛ∗.
This is a precise, conditional uniqueness claim. It is strong enough to be meaningful and narrow enough to avoid overreach.
9. Formal Proofs
9.1 Definition 1: Operational context
An operational context ℰ is an equivalence class of experimentally indistinguishable descriptions specifying preparation, transformations, measurement setting, detector configuration, record degrees of freedom, environmental couplings, nuisance structure, and available registration channels.
9.2 Definition 2: Registration channel
A registration channel γ is a physically available route through which a record can become stabilized and observed in ℰ.
Γ is the set of all admissible γ.
9.3 Definition 3: Constraint structure
𝒞 is the collection of physical and operational restrictions relevant to realization, including conservation constraints, boundary constraints, detector constraints, environmental constraints, record-stability constraints, and accessibility-sensitive record constraints.
9.4 Definition 4: Accessibility
η ∈ [0, 1] denotes calibrated record accessibility.
η = 0 means inaccessible record constraint.
η = 1 means fully accessible stable record constraint.
In practice,
η = η̂ ± δη,
where δη is calibration uncertainty.
The uncertainty δη must be propagated into ε_total.
9.5 Definition 5: Realization law
A realization law ℛ assigns registration statistics over Γ:
ℛ: (𝒮, 𝒞, ℰ, η) ↦ P(γ | 𝒮, 𝒞, ℰ, η).
A deterministic law is the limiting case where all probability mass is assigned to one γ∗.
9.6 Definition 6: Operational equivalence
ℛ₁ ≃ ℛ₂ iff, for every admissible 𝒮, 𝒞, ℰ, η, and γ ∈ Γ,
P₁(γ | 𝒮, 𝒞, ℰ, η) = P₂(γ | 𝒮, 𝒞, ℰ, η).
9.7 Definition 7: Admissible preorder
An admissible preorder ⪯ over Γ is a complete and transitive relation satisfying operational invariance, coarse-graining stability, composition consistency, and baseline compatibility.
γ₁ ⪯ γ₂ means γ₁ is no more realization-costly than γ₂.
9.8 Lemma 1: Invariance reduction
Lemma 1.
A1 implies that ℛ is defined over operational equivalence classes.
Proof.
If ℰ₁ and ℰ₂ are operationally indistinguishable, A1 requires
ℛ(𝒮, 𝒞, ℰ₁, η) ≃ ℛ(𝒮, 𝒞, ℰ₂, η).
Therefore ℛ cannot depend on representational differences between ℰ₁ and ℰ₂.
9.9 Lemma 2: Channel restriction
Lemma 2.
A2 implies γ∗ ∈ Γ.
Proof.
A2 states that realized channels must be physically available. Γ is the set of physically available registration channels. Therefore any admissible γ∗ lies in Γ.
9.10 Lemma 3: Coarse-graining preservation
Lemma 3.
A4 and A5 imply probability preservation under record merging.
Proof.
If γ₁ and γ₂ are merged into γ₁₂, coarse-graining stability requires
P(γ₁₂) = P(γ₁) + P(γ₂).
Normalization ensures the total probability over Γ remains one.
9.11 Lemma 4: Product consistency
Lemma 4.
A6 implies independent contexts compose without artificial signaling.
Proof.
For independent contexts A and B, with no coupling, no shared preparation, and no shared constraint,
P(γ_A, γ_B) = P(γ_A)P(γ_B).
Thus ℛ introduces no correlation by itself.
9.12 Lemma 5: Baseline limit
Lemma 5.
A7 implies ℛ → ℬ in screened regimes.
Proof.
A7 requires baseline recovery when η is inaccessible, screened, or dynamically irrelevant. Therefore ℛ approaches ℬ in that limit.
9.13 Lemma 6: Functional representation
Lemma 6.
An admissible preorder ⪯ over Γ admits representation by an admissible functional 𝓕.
Proof sketch.
Because ⪯ is complete and transitive, it defines a consistent ranking of admissible channels. Because it is operationally invariant, the ranking depends only on physical context, not representation. Because it is coarse-graining stable and composition-consistent, it respects observable probability structure. Because it is baseline-compatible, it agrees with ℬ in screened regimes.
Therefore ⪯ may be represented by a functional 𝓕 such that
γ₁ ⪯ γ₂ iff 𝓕(γ₁; 𝒮, 𝒞, ℰ, η) ≤ 𝓕(γ₂; 𝒮, 𝒞, ℰ, η).
9.14 Lemma 7: Representation invariance
Lemma 7.
If 𝓕₁ and 𝓕₂ represent the same admissible preorder ⪯, then they induce operationally equivalent minimizers.
Proof.
If both functionals represent ⪯, then they agree on the ordering of channels. Therefore the set of minimal channels under 𝓕₁ is the same operational minimal class as the set of minimal channels under 𝓕₂. Hence their induced canonical laws are operationally equivalent.
9.15 Proof of Theorem 1
Since Γ is nonempty and compact, and 𝓕 is bounded below and lower-semicontinuous, the minimum of 𝓕 over Γ is attained.
Therefore there exists γ∗ ∈ Γ such that
𝓕(γ∗) ≤ 𝓕(γ)
for every γ ∈ Γ.
9.16 Proof of Theorem 2
Let ⪯ be a complete, transitive, operationally invariant, coarse-graining-stable, composition-consistent, baseline-compatible preorder over Γ.
By Lemma 6, ⪯ admits representation by an admissible functional 𝓕.
Therefore
γ₁ ⪯ γ₂ iff 𝓕(γ₁; 𝒮, 𝒞, ℰ, η) ≤ 𝓕(γ₂; 𝒮, 𝒞, ℰ, η).
9.17 Proof of Theorem 3
Let 𝓕₁ and 𝓕₂ represent the same admissible preorder ⪯.
By Lemma 7, their minimizers belong to the same operational minimal class. Therefore the canonical laws induced by 𝓕₁ and 𝓕₂ generate identical observable registration statistics.
Thus
ℛ₁∗ ≃ ℛ₂∗.
9.18 Proof of Theorem 4
Let ℛ ∈ 𝒜 represent the fixed admissible preorder ⪯.
By Theorem 2, ⪯ admits an admissible functional representation 𝓕.
By definition,
ℛ∗ = arg min over γ ∈ Γ of 𝓕(γ; 𝒮, 𝒞, ℰ, η).
By Theorem 3, all admissible representations of ⪯ induce operationally equivalent minimizers. Therefore ℛ and ℛ∗ produce the same observable registration statistics.
Hence
ℛ ≃ ℛ∗.
9.19 Proof of Corollary 1
By A7, every admissible law recovers ℬ when η is inaccessible, screened, or dynamically irrelevant.
Since ℛ∗ ∈ 𝒜,
ℛ∗ → ℬ
as η → η_baseline.
9.20 Proof of Corollary 2
Assume Γ, ⪯, 𝓕 or W, η-grid, ℬ, ε_total, and the exclusion criterion are fixed before outcome analysis.
Assume η is independently calibrated as
η = η̂ ± δη,
with δη included in ε_total.
If for every calibrated η,
|Data − ℬ| ≤ ε_total,
then no accessibility-sensitive deviation is observed above the declared bound.
Therefore canonical CBR in the tested admissible class, platform, and constraint regime is ruled out.
10. Operational Definition of η
10.1 η as record accessibility
η measures operational record accessibility.
It is not observer belief. It is not subjective information. It is not a post hoc fitting parameter. It is a calibrated physical control describing how recoverable, stable, and dynamically coupled the relevant record-bearing degrees of freedom are.
CBR becomes empirically meaningful only if η can be independently calibrated and varied.
10.2 Normalization
η is normalized as
η ∈ [0, 1].
η = 0 denotes inaccessible record constraint.
η = 1 denotes fully accessible stable record constraint.
Intermediate values denote partial accessibility.
The normalization may be platform-specific, but it must be fixed before the final CBR signature test.
10.3 Estimator with uncertainty
A platform-level estimator is
η̂ = 𝒜_record ÷ 𝒜_max.
The calibrated accessibility is
η = η̂ ± δη.
Here 𝒜_record is measured record accessibility, 𝒜_max is maximum calibrated accessibility under the same procedure, and δη is calibration uncertainty.
The uncertainty δη must be propagated into ε_total. Thus η-calibration error cannot be ignored or hidden.
10.4 Component model
A more detailed model may write
η = f(I_record, τ_stability, κ_coupling, ν_noise),
where I_record is retrievable record information, τ_stability is record persistence time, κ_coupling is coupling strength to the registration environment, and ν_noise is effective record-degrading noise.
The function f must be specified before outcome analysis. It must be monotone in the intended accessibility variables and must include uncertainty propagation.
10.5 Pre-registration requirement
Before testing, the following must be fixed:
Γ.
⪯.
𝓕 or W.
η-grid.
η-calibration method.
δη propagation.
ℬ.
ε_total.
Strong-null criterion.
Success criterion.
No element of this list may be adjusted after outcome data are inspected.
10.6 Strong-null role of η
The empirical test proceeds as follows.
Calibrate η.
Fix Γ, ⪯, 𝓕 or W, ℬ, and ε_total.
Predict the η-dependent CBR deviation.
Measure Data.
Compare Data to ℬ.
If, for every calibrated η,
|Data − ℬ| ≤ ε_total,
then the strong-null condition holds.
Thus η is the bridge between the formal theorem and empirical risk. Without calibrated η, CBR remains a formal law-candidate. With calibrated η, it becomes testable and falsifiable.
11. η Calibration Protocol
11.1 Calibration goal
The goal of calibration is to determine η̂ from independently measurable record accessibility before testing the CBR signature.
The calibrated accessibility is written
η = η̂ ± δη,
where η̂ is the estimated accessibility and δη is calibration uncertainty. The uncertainty δη must be propagated into ε_total.
η calibration must satisfy five acceptance conditions.
First, it must be independent of the final CBR residuals.
Second, it must be repeatable across calibration runs.
Third, it must be monotone in the intended record-accessibility control.
Fourth, it must remain stable over the test window.
Fifth, its uncertainty must be included in the final sensitivity bound.
If these conditions are not met, η is not a valid experimental control.
11.2 Calibration steps
Begin with a controlled two-channel system.
Introduce a tunable record degree of freedom coupled to the two alternatives.
Vary record distinguishability across a pre-registered η-grid.
Measure record recoverability using a procedure independent of the final CBR outcome test.
Estimate
η̂ = 𝒜_record ÷ 𝒜_max.
Assign δη from calibration spread, apparatus drift, finite sampling, and model uncertainty.
Lock η̂, δη, Γ, ⪯, 𝓕 or W, ℬ, ε_total, and the exclusion criterion before testing the CBR signature.
11.3 Blinding and anti-free-parameter rule
η cannot be retrofitted after the outcome data are known.
Whenever possible, the analysis should be blinded. η calibration and ℬ fitting should be completed without access to the final residuals Data − ℬ.
The permitted sequence is:
Calibrate η.
Freeze η̂ and δη.
Freeze ℬ.
Freeze ε_total.
Predict Δ_CBR(η).
Analyze Data.
Compare Data − ℬ against Δ_CBR(η).
Any adjustment of η after inspecting the residuals invalidates the test as evidence for CBR.
12. Worked Model 1: Two-Path Interferometer
12.1 State
Consider a two-path state:
|ψ⟩ = α|0⟩ + β|1⟩.
The states |0⟩ and |1⟩ represent two alternatives before final registration.
12.2 Record states
Introduce record states |R₀⟩ and |R₁⟩ correlated with the two paths:
|0⟩ → |0⟩|R₀⟩,
|1⟩ → |1⟩|R₁⟩.
The joint state is
|Ψ⟩ = α|0⟩|R₀⟩ + β|1⟩|R₁⟩.
12.3 Accessibility proxy
A simple illustrative proxy is
η ≈ 1 − |⟨R₀|R₁⟩|.
If |R₀⟩ and |R₁⟩ are nearly identical, η ≈ 0.
If |R₀⟩ and |R₁⟩ are nearly orthogonal, η ≈ 1.
In an actual test, this proxy is insufficient by itself. The experiment must use calibrated
η = η̂ ± δη.
12.4 Baseline behavior
Under ℬ, when η ≈ 0, path information is inaccessible and interference is recovered.
When η ≈ 1, path information is accessible and interference is suppressed.
This transition is ordinary quantum behavior. Loss of visibility as |⟨R₀|R₁⟩| decreases is not a CBR signature.
12.5 CBR signature
A CBR signature must be a residual η-dependent deviation after subtracting ℬ.
Define
Δ_obs(η) = Data(η) − ℬ(η).
A claimed CBR detection requires
Δ_obs(η) ≈ Δ_CBR(η)
in sign, shape, and scale across the pre-registered η-grid, with deviations exceeding ε_total.
It is not enough that interference changes with η. ℬ already predicts that. Only a residual pattern beyond decoherence, detector inefficiency, noise, drift, and calibration uncertainty can count.
13. Worked Model 2: Noisy Detector
13.1 Detector channels
Consider two detector channels:
Γ = {γ₀, γ₁}.
These may represent detector clicks, macroscopic records, or stable environmental registration classes.
13.2 Functional
Use the illustrative functional
𝓕(γ) = λ₁ mismatch(γ, ℬ) + λ₂ constraint violation(γ, η) + λ₃ instability(γ).
The coefficients λ₁, λ₂, and λ₃ are not free post hoc parameters. They must be pre-registered, physically motivated, bounded, and estimated from calibration or independent apparatus characterization.
If the coefficients cannot be independently constrained, the model is not a valid CBR test.
13.3 Selection
The selected channel is
γ∗ = arg min over γ ∈ Γ of 𝓕(γ).
In probabilistic form,
P(γ | 𝒮, 𝒞, ℰ, η) = W(γ) ÷ Σ over γ′ ∈ Γ of W(γ′).
W must be nonnegative, pre-specified, operationally invariant, stable under coarse-graining, composition-consistent, and baseline-recovering.
13.4 Purpose
This model demonstrates calculability while exposing the burden on CBR.
The model is not allowed to explain any detector anomaly after the fact. It must predict, before outcome analysis, how η changes the realization statistics relative to ℬ.
The relevant detection condition is
Δ_obs(η) matches Δ_CBR(η)
in sign, shape, and scale, with residuals exceeding ε_total.
14. Worked Model 3: Delayed-Choice Record Erasure
14.1 Setup
Create path-record entanglement:
|Ψ⟩ = α|0⟩|R₀⟩ + β|1⟩|R₁⟩.
Then delay the choice of whether the record remains accessible or is made inaccessible.
Finally, perform registration.
This model tests whether η-sensitive realization can be defined without retrocausal signaling, postselection artifacts, or composition failure.
14.2 No-signaling constraint
CBR must not permit controllable signaling backward in time.
Changing record accessibility in a delayed-choice setting may affect admissible registration statistics only in ways compatible with A6 and ℬ. It cannot transmit information to the past, create inconsistent records, or generate controllable superluminal signaling.
Any version of CBR that permits such signaling is inadmissible.
14.3 Postselection constraint
No conditioning on final outcomes may be introduced unless it is pre-registered and included in ℬ.
Postselection cannot be used after the fact to isolate an apparent η-dependent effect. Any allowed selection rule must be fixed before data analysis and applied equally to ℬ and CBR.
This blocks a common loophole in delayed-choice protocols.
14.4 Allowed signature
The only allowed signature is a pre-registered accessibility-sensitive deviation within the calibrated protocol.
It must track η.
It must exceed ε_total.
It must match Δ_CBR(η) in sign, shape, and scale.
It must survive nuisance modeling.
It must respect no-signaling.
It must be reproducible.
A generic delayed-choice anomaly is not enough.
14.5 Failure mode
If apparent η-dependence enables signaling, violates composition consistency, depends on unregistered postselection, or disappears under improved nuisance modeling, then that version of CBR is inadmissible or empirically unsupported.
If
|Data − ℬ| ≤ ε_total
for every calibrated η, then the strong-null condition holds.
15. Baseline Comparator ℬ
15.1 Definition
ℬ is the baseline comparator against which CBR is tested.
ℬ is not idealized textbook quantum mechanics. It is standard quantum mechanics plus decoherence, apparatus noise, detector inefficiency, environmental drift, calibration uncertainty, and known systematic effects.
More precisely, ℬ has two levels.
First, there is a permitted baseline model family: all non-CBR models allowed by standard quantum mechanics and independently measured apparatus behavior.
Second, there is a frozen baseline instance: the specific ℬ used for final comparison, fixed before CBR residuals are inspected.
15.2 Role
CBR is not compared against an unrealistically clean model. It is compared against the best controlled frozen baseline instance ℬ.
This prevents false positives. Many apparent deviations can arise from ordinary effects: imperfect erasure, phase drift, detector bias, visibility loss, environmental noise, thermal instability, or calibration error.
A result supports CBR only if it survives comparison with ℬ and exceeds ε_total in the pre-registered η-dependent pattern.
15.3 Baseline fitting discipline
The baseline model family may be fit using calibration data, control data, or blinded portions of the experiment.
But the final ℬ must be frozen before inspecting the CBR-sensitive residuals.
After that point, ℬ cannot be adjusted to erase or create an apparent CBR signal.
This protects against both baseline overfitting and CBR overfitting.
15.4 Prediction requirement
CBR must produce a pre-registered η-dependent prediction:
Δ_CBR(η).
The observed residual is
Δ_obs(η) = Data(η) − ℬ(η).
A claimed detection requires
Δ_obs(η) ≈ Δ_CBR(η)
in sign, shape, and scale across the pre-specified η-grid, with residuals exceeding ε_total.
A strong null occurs when
|Data − ℬ| ≤ ε_total
for every calibrated η.
In that case, canonical CBR fails in the tested admissible class, platform, and constraint regime.
16. Sensitivity and Decisive Inequality
16.1 Total uncertainty bound
Define the total uncertainty bound as
ε_total = ε_stat + ε_sys + ε_model + ε_drift + ε_η.
Here ε_stat is statistical uncertainty, ε_sys is systematic uncertainty, ε_model is baseline-model uncertainty, ε_drift is apparatus or environmental drift, and ε_η is the propagated uncertainty from η calibration.
If η is calibrated as
η = η̂ ± δη,
then δη must enter ε_total through a pre-registered uncertainty-propagation rule.
ε_total must be fixed before inspecting CBR-sensitive residuals. It may be conservative, but it cannot be adjusted after the data are known.
16.2 Detectability versus confirmation
Detectability and confirmation are distinct.
A predicted CBR effect is detectable only if
|Δ_CBR(η)| > ε_total
over the pre-specified η-region.
This means the platform has enough sensitivity to test the prediction.
Confirmation requires more. The observed residual
Δ_obs(η) = Data(η) − ℬ(η)
must match Δ_CBR(η) in sign, shape, scale, and robustness across the pre-registered η-grid.
Thus, a random deviation from ℬ is not confirmation. A correct η-tracking residual is required.
16.3 Three possible outcomes
A properly designed test has three possible outcomes.
Detection occurs when
Δ_obs(η) ≈ Δ_CBR(η)
in sign, shape, and scale, with residuals exceeding ε_total and surviving nuisance controls.
A strong null occurs when, for every calibrated η,
|Data − ℬ| ≤ ε_total,
despite |Δ_CBR(η)| being above sensitivity.
An inconclusive result occurs when the predicted effect is below sensitivity, η is not adequately calibrated, ℬ is not stable, or uncertainties are too large.
This distinction prevents underpowered tests from being misread as falsifications.
16.4 Strong-null consequence
A strong null rules out canonical CBR in the tested admissible class, platform, η-range, and constraint regime.
The conclusion is limited but decisive. It does not refute all possible realization theories. It refutes the pre-registered canonical CBR instance defined by Γ, ⪯, 𝓕 or W, η, ℬ, ε_total, and the exclusion criterion.
17. Success Criteria
A successful CBR signature must satisfy all of the following.
It must be reproducible across repeated runs.
It must track calibrated η in the pre-registered direction or pattern.
It must exceed ε_total.
It must match Δ_CBR(η) in sign, shape, and scale.
It must survive nuisance modeling.
It must not be absorbed by decoherence.
It must not be detector drift.
It must not be postselection bias.
It must appear under blinded or locked analysis.
It must preserve no-signaling.
It should survive out-of-sample replication. A discovery run may motivate attention, but a locked replication run is required for strong evidential status.
The standard is intentionally strict. CBR is not supported by any anomaly. It is supported only by the specific accessibility-sensitive residual it predicts.
18. Failure Criteria
Failure criteria must distinguish test failure from model failure.
A test fails if η cannot be calibrated, if δη is too large, if ℬ cannot be stabilized, if ε_total is too large, or if the predicted effect is below sensitivity. In these cases, the experiment is inconclusive rather than a decisive refutation.
Canonical CBR fails in the tested regime if the test is adequately powered and a strong null holds:
|Data − ℬ| ≤ ε_total
for every calibrated η.
Canonical CBR also fails if the apparent signal disappears under improved controls, is absorbed by ℬ, depends on unregistered postselection, violates no-signaling, or violates A1–A7.
Thus, failure has two levels. Poor calibration invalidates the test. A strong null under adequate calibration and sensitivity invalidates the tested canonical model.
19. Relation to the Born Rule
19.1 Baseline compatibility
In screened regimes, CBR must recover Born statistics.
When η is inaccessible, screened, or dynamically irrelevant, the realization law must approach ℬ. Since ℬ includes standard quantum predictions, this requires
P_CBR(γ) → P_Born(γ).
This is assumed here as part of A7. It is not derived in this core theorem paper.
19.2 Limited claim
The present paper proves no standalone derivation of the Born rule.
Its claim is narrower: under A1–A7 and a fixed admissible preorder ⪯, the canonical law form is unique up to operational equivalence.
Born recovery is a boundary condition on admissibility, not the conclusion of the core theorem.
19.3 Bridge to probability analysis
The companion probability paper addresses the separate question of whether admissible realization weighting forces quadratic Born weighting.
The present paper supplies the law-form structure:
ℛ∗ = arg min over γ ∈ Γ of 𝓕(γ; 𝒮, 𝒞, ℰ, η).
The probability analysis supplies the weighting constraint for W.
Together, the intended division is:
core theorem paper: law form and falsifiability.
probability paper: admissible weighting and Born recovery.
19.4 Required limit
The required limit is
P_CBR(γ) → P_Born(γ)
as η → η_baseline or as accessibility-sensitive constraints are dynamically screened.
If this limit fails, canonical CBR fails before any novel η-dependent signature is considered.
20. Relation to Existing Interpretations
20.1 Relation to many-worlds
CBR shares with many-worlds the acceptance that unitary dynamics may describe branching structure prior to registration.
It departs from many-worlds by introducing a realization law over admissible registration channels. CBR does not treat every branch as equally realized in the same sense. It asks which channel becomes the realized registration.
CBR is also more directly exposed to η-dependent null tests. A strong null under calibrated accessibility variation rules out the tested canonical CBR model.
20.2 Relation to objective collapse
CBR shares with collapse approaches the aim of accounting for single realized outcomes.
It departs from standard objective-collapse theories because the core theorem does not require modifying global unitary dynamics. The canonical law is a realization-selection law, not necessarily a dynamical collapse equation.
A collapse mechanism could be added only as an extra hypothesis, not as part of the core theorem.
20.3 Relation to hidden variables
CBR shares with contextual hidden-variable approaches the idea that outcome structure depends on the measurement context.
It departs from ordinary hidden-variable models because it does not require pre-existing definite values for all observables. The realization law is evaluated relative to 𝒮, 𝒞, ℰ, η, and Γ.
CBR is therefore contextual, but not automatically a hidden-variable theory.
20.4 Relation to decoherence
CBR fully uses the lesson of decoherence: record stability and interference suppression are physical, not merely epistemic.
It departs from decoherence-only accounts by addressing the remaining realization question. Decoherence explains why branches become effectively noninterfering. It does not, by itself, specify which registered channel is realized as the observed outcome.
CBR’s empirical burden is not to reproduce ordinary decoherence. It must predict a residual η-dependent deviation from ℬ.
20.5 Relation to superdeterminism
CBR does not require conspiratorial correlations between measurement settings and initial conditions.
Its intended empirical handle is calibrated record accessibility, not denial of experimental freedom.
Any CBR variant relying on setting-initial-condition conspiracy would fall outside the clean admissible class unless it explicitly preserved A1–A7 and independent testability.
20.6 Relation to retrocausal signaling
CBR may be tested in delayed-choice or record-erasure settings, but it must preserve no-signaling.
Delayed accessibility variation must not permit controllable backward-in-time communication, superluminal signaling, or inconsistent records.
The allowed signature is narrow:
Δ_obs(η) must track Δ_CBR(η), exceed ε_total, survive ℬ, and preserve no-signaling.
CBR is therefore not a license for causal paradoxes. It is a constrained realization framework with a specific accessibility-based empirical exposure.
Appendix A: Formal Definitions
A.1 Preparation state 𝒮
𝒮 denotes the preparation state or operational preparation class. It specifies the physical input to the experiment before registration.
𝒮 may be represented by a pure state, mixed state, density operator, or equivalence class of preparation procedures.
Two preparations belong to the same 𝒮 when no admissible measurement in the relevant context distinguishes them.
A.2 Experimental context ℰ
ℰ denotes the full operational context of the experiment.
It includes the preparation procedure, transformations, measurement settings, detector configuration, environmental couplings, record degrees of freedom, nuisance structure, and available registration channels.
Two descriptions belong to the same ℰ when they generate identical observable registration statistics under the same admissible conditions.
A.3 Constraint structure 𝒞
𝒞 denotes the physical and operational constraints relevant to realization.
𝒞 may include conservation laws, boundary conditions, detector constraints, environmental constraints, record-stability constraints, contextual compatibility constraints, and accessibility-sensitive record constraints.
A.4 Registration channel γ
A registration channel γ is a physically available route through which a record can become stabilized and observed in ℰ.
A channel is not merely an outcome label. It represents an admissible physical route to registration.
A.5 Channel set Γ
Γ denotes the set of admissible registration channels:
Γ = Γ(𝒮, 𝒞, ℰ, η).
A channel γ is admissible only if
γ ∈ Γ.
Γ must be fixed before outcome analysis in any empirical test.
A.6 Accessibility η
η ∈ [0, 1] denotes calibrated record accessibility.
η = 0 means inaccessible record constraint.
η = 1 means fully accessible stable record constraint.
Experimentally,
η = η̂ ± δη,
where η̂ is the calibrated estimate and δη is calibration uncertainty.
δη must be propagated into ε_total.
A.7 Realization law ℛ
A realization law ℛ assigns registration statistics over Γ:
ℛ: (𝒮, 𝒞, ℰ, η) ↦ P(γ | 𝒮, 𝒞, ℰ, η).
A deterministic realization law is the limiting case where all probability mass is assigned to one γ∗.
A.8 Admissible preorder ⪯
⪯ is the admissible constraint ordering over Γ.
Write
γ₁ ⪯ γ₂
to mean that γ₁ is no more realization-costly than γ₂ relative to 𝒮, 𝒞, ℰ, and η.
The preorder must be complete, transitive, operationally invariant, coarse-graining stable, composition-consistent, and baseline-compatible.
A.9 Realization functional 𝓕
𝓕(γ; 𝒮, 𝒞, ℰ, η) is an admissible functional representation of ⪯.
It satisfies
γ₁ ⪯ γ₂ iff 𝓕(γ₁; 𝒮, 𝒞, ℰ, η) ≤ 𝓕(γ₂; 𝒮, 𝒞, ℰ, η).
The canonical realization law is
ℛ∗ = arg min over γ ∈ Γ of 𝓕(γ; 𝒮, 𝒞, ℰ, η).
A.10 Baseline comparator ℬ
ℬ is the frozen baseline comparator.
It consists of standard quantum mechanics plus decoherence, apparatus noise, detector inefficiency, environmental drift, calibration uncertainty, and known systematic effects.
ℬ must be specified and frozen before CBR-sensitive residuals are inspected.
A.11 Total uncertainty ε_total
ε_total is the total declared uncertainty bound:
ε_total = ε_stat + ε_sys + ε_model + ε_drift + ε_η.
The combination rule must be pre-registered.
The conservative default is additive. A quadrature rule may be used only if independence assumptions are pre-registered and justified.
A.12 Operational equivalence ≃
Two realization laws ℛ₁ and ℛ₂ are operationally equivalent, written
ℛ₁ ≃ ℛ₂,
iff they generate identical observable registration statistics for every admissible 𝒮, 𝒞, ℰ, η, and γ ∈ Γ.
A.13 Admissible class 𝒜
𝒜 is the class of realization laws satisfying A1–A7.
A1: operational invariance.
A2: contextual admissibility.
A3: constraint monotonicity.
A4: normalization.
A5: coarse-graining stability.
A6: composition consistency.
A7: baseline recovery.
A realization law outside 𝒜 is not covered by the core theorem.
Appendix B: Full Proofs
B.1 Standing assumptions
The proofs are conditional on the admissibility assumptions A1–A7 and on a representability condition for ⪯.
The representability condition states that the admissible preorder ⪯ over Γ admits a real-valued functional representation 𝓕 preserving the ordering.
This condition is required because not every abstract preorder automatically has a physically meaningful real-valued representation without regularity assumptions.
B.2 Lemma 1: Invariance reduction
If ℰ₁ and ℰ₂ are operationally indistinguishable, A1 requires
ℛ(𝒮, 𝒞, ℰ₁, η) ≃ ℛ(𝒮, 𝒞, ℰ₂, η).
Therefore ℛ cannot depend on arbitrary representation. It is defined over operational equivalence classes.
B.3 Lemma 2: Channel restriction
By A2, a realized channel must be physically available. Since Γ is the set of physically available registration channels, any selected γ∗ satisfies
γ∗ ∈ Γ.
B.4 Lemma 3: Coarse-graining preservation
By A4, probabilities over Γ are normalized.
By A5, admissible coarse-graining preserves observable statistics.
If γ₁ and γ₂ merge into γ₁₂, then
P(γ₁₂) = P(γ₁) + P(γ₂).
B.5 Lemma 4: Product consistency
For independent contexts A and B, A6 requires
P(γ_A, γ_B) = P(γ_A)P(γ_B).
Thus ℛ cannot introduce artificial correlations between independent systems.
B.6 Lemma 5: Baseline limit
By A7, when η is inaccessible, screened, or dynamically irrelevant,
ℛ → ℬ.
B.7 Lemma 6: Functional representation
Let ⪯ be an admissible preorder over Γ satisfying the stated representability condition.
Then there exists an admissible 𝓕 such that
γ₁ ⪯ γ₂ iff 𝓕(γ₁; 𝒮, 𝒞, ℰ, η) ≤ 𝓕(γ₂; 𝒮, 𝒞, ℰ, η).
This lemma does not claim that any arbitrary decision rule can be written as a meaningful physical minimization. It applies only to admissible, representable preorders.
B.8 Lemma 7: Representation invariance
If 𝓕₁ and 𝓕₂ represent the same admissible preorder ⪯, then they induce the same minimal operational class.
Therefore their canonical laws are operationally equivalent:
ℛ₁∗ ≃ ℛ₂∗.
B.9 Theorem 1: Existence
If Γ is nonempty and compact, and 𝓕 is bounded below and lower-semicontinuous, then 𝓕 attains a minimum on Γ.
Therefore there exists γ∗ ∈ Γ such that
𝓕(γ∗) ≤ 𝓕(γ)
for all γ ∈ Γ.
B.10 Theorem 2: Representation of admissible preorder
By Lemma 6, every admissible representable preorder ⪯ over Γ admits an admissible functional representation 𝓕.
Therefore the canonical law may be written as minimization over Γ.
B.11 Theorem 3: Minimizer equivalence
By Lemma 7, any two admissible functionals representing the same preorder ⪯ have operationally equivalent minimizers.
Thus their induced realization laws are equivalent under ≃.
B.12 Theorem 4: Canonical reduction
Let ℛ ∈ 𝒜 represent the fixed admissible preorder ⪯.
By Theorem 2, ⪯ admits representation by 𝓕.
By Theorem 3, all admissible representations of ⪯ induce the same operational minimal class.
Therefore
ℛ ≃ ℛ∗.
B.13 Corollary: Baseline recovery
Since ℛ∗ ∈ 𝒜 and A7 requires baseline recovery,
ℛ∗ → ℬ
as η → η_baseline.
B.14 Corollary: Strong-null falsification
If Γ, ⪯, 𝓕 or W, η, ℬ, ε_total, and the exclusion criterion are fixed in advance, and if
|Data − ℬ| ≤ ε_total
for every calibrated η under adequate sensitivity, then canonical CBR is ruled out in the tested regime.
If sensitivity is inadequate, the result is inconclusive rather than falsifying.
Appendix C: η Calibration Details
C.1 Platform-specific estimator
A generic estimator is
η̂ = 𝒜_record ÷ 𝒜_max.
Here 𝒜_record is measured record accessibility, and 𝒜_max is maximum accessibility under the same calibration procedure.
C.2 Component estimator
A more detailed model may use
η = f(I_record, τ_stability, κ_coupling, ν_noise).
I_record measures recoverable record information.
τ_stability measures persistence time.
κ_coupling measures coupling to the registration environment.
ν_noise measures record-degrading noise.
C.3 Calibration acceptance tests
η calibration is acceptable only if it satisfies repeatability, monotonicity, drift stability, independence from outcome residuals, and bounded δη.
If these conditions fail, the test is inconclusive.
C.4 Calibration timing
Calibration must occur before outcome testing.
The η-grid, η̂ values, δη values, and uncertainty-propagation rule must be frozen before CBR-sensitive residuals are inspected.
C.5 Uncertainty propagation
If η = η̂ ± δη, then η uncertainty contributes
ε_η = |∂Δ_CBR ÷ ∂η| δη
for local propagation.
For nonlocal or nonlinear propagation, ε_η should be defined conservatively as the maximum deviation over the calibrated η uncertainty interval.
Then
ε_total = ε_stat + ε_sys + ε_model + ε_drift + ε_η.
Appendix D: Toy Model Calculations
D.1 Two-path interferometer
Begin with
|ψ⟩ = α|0⟩ + β|1⟩.
After record coupling:
|Ψ⟩ = α|0⟩|R₀⟩ + β|1⟩|R₁⟩.
A simple accessibility proxy is
η ≈ 1 − |⟨R₀|R₁⟩|.
Ordinary visibility loss as |⟨R₀|R₁⟩| decreases is not a CBR signature. That behavior belongs to ℬ.
CBR is tested only by the residual
Δ_obs(η) = Data(η) − ℬ(η).
A valid CBR signature requires
Δ_obs(η) ≈ Δ_CBR(η)
in sign, shape, and scale, with residuals exceeding ε_total.
D.2 Noisy detector
Let
Γ = {γ₀, γ₁}.
Use
𝓕(γ) = λ₁ mismatch(γ, ℬ) + λ₂ constraint violation(γ, η) + λ₃ instability(γ).
The selected channel is
γ∗ = arg min over γ ∈ Γ of 𝓕(γ).
The coefficients λ₁, λ₂, and λ₃ must be pre-registered, bounded, and estimated from independent apparatus characterization.
If they are tuned after outcome analysis, the model is invalid as evidence for CBR.
D.3 Delayed-choice record erasure
Use
|Ψ⟩ = α|0⟩|R₀⟩ + β|1⟩|R₁⟩.
Then vary whether record accessibility is preserved or erased before final registration.
Allowed CBR signatures must satisfy
Δ_obs(η) ≈ Δ_CBR(η).
They must exceed ε_total, survive ℬ, preserve no-signaling, and avoid unregistered postselection.
If η-dependence enables signaling or violates A6, the model is inadmissible.
Appendix E: Nuisance Bound Construction
E.1 Statistical uncertainty ε_stat
ε_stat is determined from sample size, count variance, estimator variance, and the pre-registered confidence rule.
E.2 Systematic uncertainty ε_sys
ε_sys includes detector bias, calibration offset, alignment error, timing uncertainty, and known apparatus imperfections.
E.3 Baseline-model uncertainty ε_model
ε_model captures uncertainty in ℬ, including decoherence modeling, detector-response modeling, and fitted baseline parameters.
E.4 Drift uncertainty ε_drift
ε_drift captures time-dependent apparatus or environmental changes across the test window.
E.5 η-calibration uncertainty ε_η
ε_η captures uncertainty propagated from
η = η̂ ± δη.
It may be estimated locally by
ε_η = |∂Δ_CBR ÷ ∂η| δη,
or conservatively by maximizing deviation over the η uncertainty interval.
E.6 Total bound
The conservative default is additive:
ε_total = ε_stat + ε_sys + ε_model + ε_drift + ε_η.
A quadrature rule may be used only if independence assumptions are pre-registered and justified:
ε_total = √(ε_stat² + ε_sys² + ε_model² + ε_drift² + ε_η²).
The chosen rule must be frozen before residuals are inspected.
Appendix F: Pre-Registration Template
F.1 Versioning and freeze record
Before testing, assign frozen version identifiers and timestamps to:
η-grid.
η-calibration rule.
Γ.
⪯.
𝓕 or W.
ℬ.
ε_total.
Δ_CBR(η).
Exclusion criterion.
Success criterion.
Blinding and unblinding protocol.
F.2 η-grid
Specify
η₁, η₂, η₃, …, ηₙ.
For each value, state
ηᵢ = η̂ᵢ ± δηᵢ.
F.3 Baseline model ℬ
Specify the permitted baseline model family.
Specify the frozen baseline instance ℬ.
State what data were used to fit ℬ.
State when ℬ was frozen.
F.4 CBR prediction
State the predicted residual
Δ_CBR(η).
Specify its sign, shape, scale, and η-dependence.
F.5 Nuisance bounds
Specify ε_stat, ε_sys, ε_model, ε_drift, ε_η, and ε_total.
State whether the total bound is additive or quadrature.
Justify the combination rule.
F.6 Exclusion criterion
State the strong-null criterion:
For every calibrated η,
|Data − ℬ| ≤ ε_total.
If this holds under adequate sensitivity, canonical CBR is ruled out in the tested regime.
F.7 Success criterion
A successful result requires
Δ_obs(η) ≈ Δ_CBR(η)
in sign, shape, and scale, with residuals exceeding ε_total and surviving nuisance controls.
F.8 Inconclusive outcome
A result is inconclusive if η calibration fails, δη is too large, ℬ is unstable, drift is excessive, the predicted effect is below ε_total, or blinding is broken.
Inconclusive results do not confirm CBR and do not decisively falsify it.
F.9 Data-blinding method
State which data are blinded.
State who has access to η calibration, ℬ fitting, and final residuals.
State when unblinding occurs.
State that η, ℬ, ε_total, and Δ_CBR(η) cannot be changed after unblinding.
