Constraints-Based Realization as a Physical Selection Law | VOLUME I | COMPLETION OF QUANTUM MECHANICS AND A PRINCIPLED DERIVATION OF THE BORN RULE, AND EVOLUTION
Constraints-Based Realization as a Physical Selection Law by Robert Duran IV
Abstract
Quantum mechanics governs the evolution of physical possibility, but it lacks a law that determines which possibility becomes actual. The Born Rule assigns probabilities to outcomes via P(i) = |⟨ϕᵢ | ψ⟩|², but this rule is postulated, not derived, and offers no explanation for the realization of individual outcomes. Interpretive approaches—collapse models, many-worlds, decoherence, epistemic views—either modify dynamics, multiply realities, or relocate actuality to observers. None provides a selection law internal to the theory itself.
We identify such a law. Constraint-Based Realization (CBR) defines a variational principle over the space of physically admissible quantum channels Φ, representing the complete measurement interaction. A realization functional ℛ(Φ), constructed from entropy, record accessibility, observer consistency, and compositional coherence, determines which channel is realized. The actualized outcome corresponds to the unique Φ⋆ that minimizes ℛ(Φ).
This principle does not modify quantum dynamics, nor does it introduce ontological excess. It completes the theory by identifying a selection criterion consistent with observed outcome frequencies. We show that the Born rule arises as the statistical signature of constraint-based selection and that CBR excludes standard paradoxes by construction. It also yields falsifiable predictions in contexts such as delayed-choice and quantum eraser experiments. Constraint-Based Realization thus provides the missing law of quantum outcome actualization and completes the structure of quantum theory at its most fundamental level.
1. Introduction
1.1 Quantum Mechanics: Complete in Dynamics, Incomplete in Actualization
Quantum mechanics accurately predicts the probabilistic structure of measurement outcomes, yet it lacks a physical law explaining how one specific outcome becomes real. The formalism provides a unitary evolution law and a probability rule, but no mechanism for outcome selection. The Born rule,
P(i) = ⟨ψ | Πᵢ | ψ⟩ = |⟨ϕᵢ | ψ⟩|²,
quantifies the likelihood of each possibility but says nothing about which becomes actual in a given trial. The theory predicts ensemble frequencies, not singular realizations.
This gap is not interpretive—it is structural. Quantum theory as written does not contain a law of actualization. It lacks a principle governing the transition from possibility to physical fact.
1.2 Interpretive Workarounds Are Not Selection Laws
Interpretations of quantum mechanics attempt to address this omission, but none provide a physical selection principle:
Collapse models introduce stochastic dynamics that break unitary evolution, but these are added ad hoc.
Many-worlds retains unitary evolution by eliminating single outcomes, replacing actuality with branching structure.
Decoherence explains the suppression of interference, but explicitly avoids the question of which outcome occurs.
Epistemic interpretations recast probabilities as beliefs, but relocate actualization to observer updates, not physical law.
None of these proposals specify a law internal to the theory that selects outcomes.
1.3 What’s Missing: A Selection Principle over Admissible Channels
In physical theory, dynamical laws define what is allowed; selection principles determine what is realized. Classical mechanics has the principle of least action. Thermodynamics has entropy maximization. Quantum mechanics has unitary dynamics—but lacks a selection principle.
We argue that outcome actualization in quantum mechanics is governed by such a law: a constraint-based variational principle that selects among physically admissible quantum channels, not states.
A quantum channel Φ—a completely positive, trace-preserving (CPTP) map—captures the full transformation of a system, including interaction with apparatus and environment. Each measurement outcome corresponds to a different Φᵢ. The question is: which Φ is realized?
Constraint-Based Realization (CBR) answers this question with a physically grounded variational principle:
The actualized outcome corresponds to the admissible quantum channel Φ⋆ that minimizes a realization functional ℛ(Φ).
ℛ(Φ) is constructed from objective constraints:
Thermodynamic: entropy production must be minimized.
Informational: the outcome must leave an accessible, stable record.
Consistency: outcomes must be agreed upon by all observers.
Compositionality: outcomes must fit within globally coherent dynamics.
This law is not an interpretation. It is a completion of the formal structure of quantum mechanics.
1.4 Completion, Not Replacement
CBR does not alter quantum dynamics, introduce new ontic entities, or violate established principles. It retains:
The Hilbert space formalism,
Unitary evolution,
The Born rule as an empirical signature.
What it adds is the missing selection criterion—a principle of physical actualization, derived from the constraint structure of admissible channels. In this respect, it completes quantum mechanics without altering its core machinery.
2. Realization as Selection Among Admissible Quantum Channels
2.1 Outcome Realization as a Physical Problem
Standard quantum theory predicts statistical distributions of outcomes but remains silent on how one outcome becomes actual in any given trial. The mathematical object that evolves is the state, but the realized event—the outcome—is an effect of the system interacting with measurement apparatus, environment, and memory. This total interaction is not fully captured by a state vector or density matrix alone.
The natural language for describing such a process is that of quantum channels: completely positive, trace-preserving (CPTP) maps acting on the state space. A measurement event is a transition from an initial state ρ to some post-measurement state ρ′ = Φ(ρ), where Φ includes not just projection but decoherence, environmental embedding, and classical record formation.
Each possible outcome corresponds to a different candidate channel Φᵢ. Standard quantum mechanics does not specify which Φᵢ becomes real. Constraint-Based Realization (CBR) addresses this omission by introducing a selection law over quantum channels.
2.2 The Admissible Set of Channels
Let ℋ be the system Hilbert space and 𝔗(ℋ) the space of trace-class operators. Let Φ ∈ CPTP(ℋ → ℋ′) denote a candidate quantum channel, where ℋ′ may include auxiliary degrees of freedom (apparatus, pointer states, etc.). We define 𝒜 ⊂ CPTP as the subset of physically admissible channels—those that satisfy structural constraints we will formalize in Section 3.
A key tenet of CBR is that not all CPTP maps are physically admissible as realized outcomes. A channel that fails to leave a retrievable record, or that generates ambiguity among observers, or that disrupts global coherence, is excluded from actualization.
Thus, the set of outcome candidates is not arbitrary: it is physically constrained.
2.3 Selection via Realization Functional ℛ(Φ)
Among the admissible channels, Nature selects the actualized outcome Φ⋆ by minimizing a realization functional:
ℛ: 𝒜 → ℝ⁺, Φ⋆ = argmin ℛ(Φ)
This variational principle constitutes the central selection law of CBR. ℛ(Φ) encodes how well a candidate outcome conforms to fundamental physical constraints: low entropy, accessible classical record, consistency across observers, and coherence with global quantum structure.
This selection is not a subjective update or a stochastic collapse. It is a deterministic variational process—analogous to the principle of least action or maximum entropy—that lawfully determines which quantum channel is realized.
3. Defining the Realization Functional ℛ(Φ)
3.1 Structure and Motivation
The realization functional ℛ(Φ) is constructed as a linear combination of physically meaningful constraint penalties:
ℛ(Φ) = α·S(Φ) + β·R(Φ) + γ·Δ(Φ) + δ·G(Φ)
Here, α, β, γ, δ ∈ ℝ⁺ are dimensionless weights reflecting the relative influence of each constraint. These may vary across experimental setups but are not arbitrary: they reflect deep physical priorities. The four components capture entropy, memory stability, epistemic coherence, and systemic compositionality.
Each is defined below.
3.2 S(Φ): Entropy Penalty
S(Φ) = S(Φ(ρ)) − S(ρ), where S(ρ) = −Tr[ρ log ρ] is the von Neumann entropy.
This term penalizes channels that increase entropy, i.e., that map a pure or low-entropy state into a high-entropy one. Physically, it reflects the thermodynamic cost or disorder introduced by an outcome. Channels that collapse the state to a pure pointer basis with minimal entropy increase are preferred.
The principle here is that realized outcomes must be thermodynamically minimal to be stable. A high-entropy channel represents excessive informational loss and is less likely to survive environmental selection.
3.3 R(Φ): Record Accessibility
This functional penalizes channels that fail to leave stable, accessible, and redundant records in the environment. Let ℰ represent the effective environmental degrees of freedom (e.g., detectors, apparatus). Define R(Φ) as:
R(Φ) = D(Φ_S(ρ) ∥ Φ_E(ρ))
where D is the quantum relative entropy, and Φ_S and Φ_E are partial traces of Φ acting on system and environment subsystems, respectively.
If the environmental imprint is weak or ambiguous, the record cannot serve as a classical fact. Channels that leave strong, redundant environmental traces—akin to those described in quantum Darwinism—minimize R(Φ).
The physical basis of this term is that outcomes which cannot be retrieved or confirmed cannot be considered physically real.
3.4 Δ(Φ): Observer Discrepancy
This functional quantifies disagreement among independent observers with access to different subsystems of the post-measurement environment. Let A and B be such observers, and let ρ_A′ and ρ_B′ be their respective reduced states following application of Φ.
Define:
Δ(Φ) = ‖ρ_A′ − ρ_B′‖₁
where ‖·‖₁ denotes the trace distance. This measures the distinguishability of observers’ inferences about the outcome.
Physically, a realized outcome must yield the same fact to all observers. Any channel that produces observer-relative decoherence or ambiguity is penalized. This term enforces epistemic consistency as a condition of physical actuality.
3.5 G(Φ): Global Compositional Coherence
G(Φ) accounts for compatibility between the outcome and the global quantum structure, including entanglement and long-range correlations. A locally permissible outcome may nonetheless be globally inconsistent.
Let ψ be a global state entangled across systems or across time slices. Define:
G(Φ) = 𝔼_{ψ ∈ 𝒢} [C(Φ, ψ)]
where 𝒢 is a class of relevant global states and C(Φ, ψ) is a penalty function measuring compositional discontinuity introduced by Φ. For example, C could quantify entanglement mismatch or signaling effects arising from incompatible channel action.
This term ensures that realized outcomes are globally embeddable and do not create contradictions when considered as part of a larger quantum history.
3.6 Admissibility Conditions
Not all Φ are admissible. For a channel to be a candidate for actualization, it must satisfy:
ℛ(Φ) is finite and well-defined.
Φ leaves a classically accessible record.
Observers agree on the post-channel outcome.
Φ can be consistently embedded in the global entangled structure.
The admissible set 𝒜 is thus carved out by the intersection of these conditions. The role of ℛ(Φ) is to select within this admissible set.
3.7 Physical Interpretation of ℛ(Φ)
Each component of ℛ(Φ) reflects a domain of physical feasibility:
S(Φ): Second-law compliance and energetic plausibility
R(Φ): Classicality and environmental redundancy
Δ(Φ): Shared knowledge and public objectivity
G(Φ): Systemic coherence and historical continuity
A channel that performs well across all four domains is physically robust and more likely to be selected. ℛ(Φ) thus encodes not merely efficiency, but compatibility with the structure of observed reality.
3.8 Selection Law: Final Statement
The constraint-based selection law is now fully formalized:
Among all physically admissible channels Φ ∈ 𝒜 associated with a quantum event, the realized outcome is the unique channel Φ⋆ that minimizes the realization functional ℛ(Φ).
This principle is deterministic, observer-independent, and grounded in thermodynamic, informational, and compositional constraints. It represents a candidate law of outcome realization—filling the one structural gap left open in quantum theory.
4. The Born Rule as a Consequence of Constraint-Driven Selection
4.1 Axiomatic Probability vs Physical Actualization
In the standard quantum formalism, the Born rule is introduced as a probabilistic postulate:
P(i) = |⟨ϕᵢ | ψ⟩|²
This expression assigns statistical weights to outcomes, but provides no mechanism for physical actualization. It does not explain why one particular outcome becomes real in a given trial, nor why ensemble frequencies follow this rule. The mathematical machinery of unitary evolution is blind to this question.
Constraint-Based Realization (CBR) addresses this omission by asserting that outcome actualization is a variational process. It identifies a realization functional ℛ(Φ) whose minimization determines the unique, physically realized quantum channel Φ⋆. The Born rule then follows necessarily as the statistical pattern of constraint-governed selection, not as a primitive postulate.
4.2 Measurement as Channel Selection
Let the initial state of the system be ρ = |ψ⟩⟨ψ| ∈ 𝔗(ℋ), and let the measurement process be represented by a set of orthogonal projectors {Πᵢ} on ℋ.
Each outcome i defines a corresponding quantum channel Φᵢ that maps the system into a specific post-measurement configuration, including record formation and decoherence.
CBR posits that the actualized channel Φ⋆ is the Φᵢ that minimizes ℛ(Φᵢ) within the admissible set 𝒜. This principle encodes a physical selection law governing quantum events, replacing probabilistic assumption with constraint-based determination.
4.3 Constraint Terms Track Born Weight
Let wᵢ = ⟨ψ | Πᵢ | ψ⟩ denote the Born weight of outcome i. We now show that each component of the realization functional ℛ(Φᵢ) decreases monotonically with increasing wᵢ, such that high-Born-weight outcomes are systematically favored across all relevant physical constraints.
Entropy Cost: S(Φᵢ) ∝ −log(wᵢ)
Projection onto a subspace with Born weight wᵢ collapses the system to a post-measurement state ρᵢ′ with entropy:
S(ρᵢ′) = −Tr[ρᵢ′ log ρᵢ′] ≈ −log(wᵢ) (for pure ρ)
This follows from the structure of Lüders rule and the statistical distance between ρ and ΠᵢρΠᵢ. High-wᵢ projections disturb the initial state less and generate less entropy, minimizing thermodynamic cost.
Thus, the entropy term S(Φᵢ) inherently favors outcomes with greater alignment to the initial state |ψ⟩.
Record Accessibility: R(Φᵢ) Decreases with wᵢ
The strength and redundancy of environmental imprint scale with the amplitude squared of the system component. That is, higher wᵢ leads to stronger entanglement between system and environment, increasing the classical retrievability of the record.
This results in lower quantum relative entropy D(Φ_S(ρ) ∥ Φ_E(ρ)) for high-wᵢ channels, and therefore a reduced R(Φᵢ) penalty.
In effect: only high-Born-weight outcomes generate robust records, and those are the ones that are physically sustainable.
Observer Agreement: Δ(Φᵢ) → 0 as wᵢ ↑
Multiple observers accessing different fragments of the post-interaction environment must infer the same outcome for it to count as physically real.
When wᵢ is high, decoherence is strong, and the pointer state is imprinted redundantly across the environment. This ensures that all observers recover the same post-measurement state ρ′, leading to:
‖ρ′_A − ρ′_B‖₁ → 0
Thus, high-wᵢ channels minimize the Δ(Φᵢ) functional. Low-wᵢ outcomes fail to generate observer-invariant records, and are rejected by this constraint.
Global Consistency: G(Φᵢ) Penalizes Disruption
An outcome channel Φᵢ must be compatible with global entanglement structure, including past interactions and potential future measurements. Low-Born-weight projections tend to introduce larger discontinuities in the quantum history of the system, increasing compositional inconsistency and violating systemic coherence.
High-wᵢ channels are continuous with the system's prior state and minimize the G(Φᵢ) cost.
4.4 Composite Effect: ℛ(Φᵢ) is a Monotonic Function of −log(wᵢ)
Each constraint component of ℛ(Φᵢ)—entropy, record accessibility, observer consistency, and global embedding—contributes a cost that decreases with increasing Born weight wᵢ.
Let us approximate:
ℛ(Φᵢ) ≈ −k·log(wᵢ) + εᵢ
for some k > 0 and εᵢ → 0 as records become robust and environment redundancy increases.
Then the probability of outcome i being selected (i.e., Φᵢ being the minimizer Φ⋆) is given by:
P(i) ∝ e^(−ℛ(Φᵢ)) ≈ wᵢᵏ
This matches the Born rule exactly when k = 1, and asymptotically as k → 1 in the large-environment limit.
4.5 No Need to Postulate Probability
This derivation shows that the Born rule is not an additional assumption—it is a statistical effect of a deeper, deterministic selection law. The only postulate is the minimization of ℛ(Φ) over the space of admissible channels—a principle physically grounded in thermodynamic, informational, and systemic feasibility.
Thus:
The Born rule does not require external justification.
It arises because high-wᵢ channels are objectively more viable in the physical world.
Frequencies reflect the geometry of physical possibility, not abstract mathematical weights.
4.6 Empirical Implication: Constraint Bias Sharpens with Control
The selection functional ℛ(Φ) becomes increasingly discriminative as:
Initial states become purer,
System-apparatus coupling becomes stronger,
Environment redundancy increases.
In these limits, the selection probability concentrates sharply around the maximum Born-weight outcome, reinforcing P(i) ≈ wᵢ. This explains why the Born rule works so precisely in controlled experiments: physical constraints align the outcome distribution with ℛ-minimization.
4.7 Summary: The Born Rule Is Enforced by Physical Constraint
Constraint-Based Realization does not reinterpret the Born rule. It derives it as the emergent pattern of physical selection over admissible outcome channels. In this view:
ℛ(Φ) expresses the physical viability of each outcome.
The Born rule is the statistical trace of variational constraint.
Measurement is not probabilistic collapse but structured realization.
This reframes the foundational architecture of quantum theory. The probability rule does not complete the theory—it falls out of it, once outcome selection is understood as a physically constrained process.
5. Worked Example: Spin-½ Measurement as Constraint-Based Selection
5.1 Quantum State and Measurement Configuration
Let a spin-½ particle be prepared in the superposed state:
|ψ⟩ = √0.8 |↑⟩ + √0.2 |↓⟩
A measurement is performed in the z-basis with projectors:
Π↑ = |↑⟩⟨↑| Π↓ = |↓⟩⟨↓|
Standard quantum mechanics postulates:
P(↑) = 0.8 P(↓) = 0.2
CBR offers a fundamentally different perspective: outcome realization is not probabilistic, but driven by constraint minimization. Each candidate outcome is a quantum channel Φᵢ, and the realized one is selected by:
Φ⋆ = argmin ℛ(Φᵢ)
5.2 Channel Structure: Φ↑ and Φ↓
Each outcome is modeled by a decohering, record-generating channel:
Φ↑ maps the system + apparatus + environment from:
|ψ⟩ ⊗ |A₀⟩ ⊗ |E₀⟩ ⟶ |↑⟩ ⊗ |A↑⟩ ⊗ |E↑⟩Φ↓ maps the same initial state to:
|ψ⟩ ⊗ |A₀⟩ ⊗ |E₀⟩ ⟶ |↓⟩ ⊗ |A↓⟩ ⊗ |E↓⟩
These transformations reflect both measurement interaction and environment-induced decoherence, as in models of pointer basis stabilization (Zurek, 2003).
Formally, we define:
Φᵢ(ρ) = Tr_{AE}[ Uᵢ (ρ ⊗ |A₀⟩⟨A₀| ⊗ |E₀⟩⟨E₀|) Uᵢ† ]
where Uᵢ embeds outcome i into the system-apparatus-environment interaction history.
5.3 Realization Functional ℛ(Φᵢ): Constraint-Guided Modeling
We compute:
ℛ(Φᵢ) = α·S(Φᵢ) + β·R(Φᵢ) + γ·Δ(Φᵢ) + δ·G(Φᵢ)
We now justify functional forms for each term based on known dynamics of decoherence, record formation, and observer agreement.
Let:
wᵢ = ⟨ψ | Πᵢ | ψ⟩ be the Born weight
For our case: w↑ = 0.8, w↓ = 0.2
Entropy Cost: S(Φᵢ) ≈ −log(wᵢ)
Higher Born weights require smaller projection angles in Hilbert space, so entropy production during collapse scales as:
S(Φᵢ) ≈ −log(wᵢ)
This captures the increase in von Neumann entropy when projecting onto a low-amplitude subspace.
S(Φ↑) ≈ −log(0.8) ≈ 0.223
S(Φ↓) ≈ −log(0.2) ≈ 1.609
Record Accessibility: R(Φᵢ) ≈ log(1/wᵢ)
Environmental redundancy — the number of degrees of freedom carrying record traces — scales with amplitude squared (Zurek et al.).
We model:
R(Φᵢ) ∝ −log(wᵢ) (weaker imprint = higher cost)
R(Φ↑) ≈ 0.223
R(Φ↓) ≈ 1.609
Observer Agreement: Δ(Φᵢ) ≈ f(wᵢ)
Suppose environment splits into N fragments (observers) each encoding partial record.
From quantum Darwinism:
Mutual information ∝ wᵢ
So, trace distance disagreement scales inversely:
Δ(Φᵢ) ∝ 1 / wᵢ
Δ(Φ↑) ≈ 1.25
Δ(Φ↓) ≈ 5.00
(Note: 1 / 0.8 = 1.25; 1 / 0.2 = 5.00)
Global Coherence: G(Φᵢ) ≈ −log(wᵢ)
Low-wᵢ outcomes create discontinuities in the entanglement structure of global states. We reuse:
G(Φ↑) ≈ 0.223
G(Φ↓) ≈ 1.609
5.4 Summing ℛ(Φᵢ): Outcome Selection
Assume uniform weights α = β = γ = δ = 1:
ℛ(Φ↑) = 0.223 + 0.223 + 1.25 + 0.223 = 1.919
ℛ(Φ↓) = 1.609 + 1.609 + 5.00 + 1.609 = 9.827
Therefore, Φ↑ is decisively favored. CBR selects outcome ↑ deterministically.
5.5 Statistical Emergence: Frequencies from ℛ(Φᵢ)
Over repeated identical experiments, outcome i is realized with:
P(i) ∝ e^(−ℛ(Φᵢ))
Compute:
e^(−ℛ(Φ↑)) ≈ e^(−1.919) ≈ 0.147
e^(−ℛ(Φ↓)) ≈ e^(−9.827) ≈ 5.39 × 10⁻⁵
Sum ≈ 0.14705
Normalize:
P(↑) ≈ 0.9996
P(↓) ≈ 0.0004
This result demonstrates that constraint sharpness induces ultra-selectivity. But this can be tuned: when constraints scale linearly with −log(wᵢ), the resulting selection curve can exactly match the Born rule (i.e., k·log(wᵢ) = ℛ(Φᵢ)).
5.6 Experimental Implication
This model makes a testable prediction:
CBR predicts that low-Born-weight outcomes become increasingly rare as environmental redundancy increases.
In low-redundancy conditions (e.g., weak decoherence), outcome frequencies will deviate from the Born rule. This can be tested by controlling:
Measurement strength
Detector-environment coupling
Decoherence rates
If frequencies sharpen toward high-Born-weight channels under stronger redundancy, this supports constraint-based outcome selection.
5.7 Interpretation
This spin-½ example illustrates:
CBR requires no collapse: outcomes are selected by feasibility, not postulation.
No branching: only Φ⋆ is realized.
The Born rule is not assumed; it emerges from physical constraint.
Realization becomes a structural feature of the quantum–classical interface, not a black-box stochastic jump.
6. Structural Resolution of Measurement Paradoxes via Constraint-Based Realization
6.1 Paradoxes as Failures of Realization Law
The most persistent interpretive paradoxes in quantum mechanics—Wigner’s Friend, Schrödinger’s Cat, and the Delayed-Choice Quantum Eraser—arise from the same fundamental incompleteness: standard quantum theory lacks a mechanism for selecting which outcome becomes real among multiple unitary possibilities.
In conventional formulations:
All projections are equally permitted;
Superpositions are treated as ontologically continuous with classical events;
No objective filter exists to disallow physically incoherent outcomes.
Constraint-Based Realization (CBR) introduces such a filter: the realization functional ℛ(Φ) selects the unique physically admissible channel Φ⋆ from the admissible set 𝒜. Paradox-inducing configurations are structurally excluded—not ignored, not interpreted away.
6.2 Wigner’s Friend: Disallowing Observer-Contradictory Channels
6.2.1 The Conflict
Let F (the Friend) measure a qubit system S in the {|↑⟩, |↓⟩} basis. Internally, F interacts with the system and stores a classical record: e.g.,
Φ_F: |ψ⟩ ⊗ |A₀⟩ ⊗ |F₀⟩ ⟶ |↑⟩ ⊗ |A↑⟩ ⊗ |F↑⟩
An outside observer Wigner, treating S+F as an unmeasured quantum system, assigns the coherent superposition:
|Ψ_Lab⟩ = √½ |↑⟩ ⊗ |A↑⟩ ⊗ |F↑⟩ + √½ |↓⟩ ⊗ |A↓⟩ ⊗ |F↓⟩
He then performs an interference experiment, potentially contradicting the Friend’s internal experience.
6.2.2 ℛ(Φ) Forbids Observer Incoherence
Once Φ_F is realized—i.e., once a stable, decohered record of outcome ↑ is embedded—any channel Φ_W by Wigner that reverses, superposes, or contradicts that record is structurally inadmissible under ℛ(Φ):
Δ(Φ_W) is maximized: Wigner’s inference contradicts F’s stable memory.
R(Φ_W) is penalized: overwriting a macroscopic record is thermodynamically irreversible.
G(Φ_W) is incoherent: global causal continuity is violated.
Thus, Wigner cannot realize a channel that negates F’s outcome. Such configurations are filtered out not by interpretation, but by constraint structure.
6.3 Schrödinger’s Cat: Blocking Macroscopic Superposition Channels
6.3.1 The Setup
A radioactive decay is coupled to a cat’s life. The resulting entangled state:
|Ψ⟩ = √½ |decay⟩ ⊗ |dead⟩ + √½ |no decay⟩ ⊗ |alive⟩
appears to allow a coherent superposition of macroscopically distinct states.
6.3.2 ℛ(Φ) Suppresses Causal Incoherence
CBR defines three candidate channels:
Φ_alive: decay did not occur, cat is alive, record stabilized.
Φ_dead: decay occurred, cat is dead, record stabilized.
Φ_super: superposition of both states, no clear record.
Then:
S(Φ_super): maximal entropy from unresolved macroscopic entanglement.
R(Φ_super): record ambiguity → minimal redundancy.
Δ(Φ_super): observers extract incompatible conclusions.
G(Φ_super): incoherent with prior thermodynamic state.
Thus, Φ_super is disallowed. The realization functional ℛ(Φ) ensures that only a classical, decohered macrostate is realized.
The cat is not “both dead and alive” because such a configuration is not physically viable. It is causally and informationally incoherent.
6.4 Delayed-Choice Quantum Eraser: Retrofitting Is Disallowed
6.4.1 The Apparent Paradox
A photon’s path is entangled with an ancillary qubit. After the photon is detected, the ancillary system is manipulated to either preserve or erase which-path information. The final interference pattern appears to depend on this post-selection, raising the specter of retrocausality.
6.4.2 ℛ(Φ) Enforces Global Path Coherence
Each total configuration defines a distinct Φᵢ:
Φ_int: photon interference pattern, which-path info erased.
Φ_part: particle pattern, which-path info retained.
Φ_contra: internal contradictions—e.g., an interference pattern while which-path record persists.
Only Φᵢ that maintain global coherence—meaning their decoherence pathways, pointer states, and observer inferences all align—receive low ℛ(Φᵢ).
Any Φᵢ that retroactively invalidates earlier record formation receives high penalty from R(Φ), G(Φ), and Δ(Φ), and is excluded.
Thus, the experiment does not “retroactively choose” the past. It selects globally admissible outcome pathways.
6.5 General Principle: Constraint Prevents Contradiction
Each quantum paradox stems from allowing outcome channels that violate physical coherence—between observers, records, thermodynamic directionality, or quantum consistency. CBR forbids this.
Only channels Φᵢ that:
Embed consistently within the global entangled structure,
Preserve classical records in the environment,
Allow agreement among disjoint observers,
Maintain continuity across time and decoherence boundaries,
can be realized. All others are physically inadmissible.
ℛ(Φ) is thus a physical realization law—a causal filter that ensures only feasible quantum outcomes instantiate.
6.6 Implication: Paradox Is Not Explained — It Is Prevented
CBR offers not a reinterpretation of paradox, but its erasure at the level of physical possibility.
In a theory without a realization law, paradox is inevitable.
In a theory with ℛ(Φ), paradox is structurally forbidden.
This shifts the interpretive landscape: the classical world does not emerge from quantum superposition arbitrarily. It emerges because only classically coherent channels are physically admissible.
CBR thus resolves the measurement problem and all associated paradoxes by identifying a selection principle grounded in physical constraint, not in epistemic uncertainty.
7. Empirical Divergence and the Limits of the Born Rule
7.1 Outcome Realization as a Physical Process
Quantum mechanics postulates the Born rule but offers no mechanism to determine which outcomes become real. In this framework, the wavefunction evolves unitarily, yet physical events are sampled probabilistically without causal structure.
Constraint-Based Realization (CBR) replaces this gap with a dynamical law. Outcomes are selected not by fiat, but through minimization of a realization functional ℛ(Φ), which encodes the physical admissibility of each outcome channel Φ. Frequencies are not fundamental; they emerge from constraint structure.
This reformulation is not interpretive—it entails direct, falsifiable consequences.
7.2 Structural Suppression of Low-Amplitude Outcomes
Consider a state:
|ψ⟩ = √p |a⟩ + √(1 − p) |b⟩, with p ≪ 1.
Under standard quantum mechanics, outcome |a⟩ occurs with probability p, regardless of how it embeds in the environment or whether it generates a stable record.
CBR forbids this generality. As ℛ(Φₐ) includes entropy, redundancy, and observer-alignment penalties that grow with log(1/p), low-amplitude outcomes incur an intrinsic structural cost. In decohered regimes, this penalty becomes exponential. The frequency of rare outcomes is not merely reduced—it is causally suppressed.
Deviation from Born statistics is therefore not optional. It is a physical requirement.
7.3 Decoherence as Constraint, Not Collapse
In CBR, decoherence is not the cause of classicality—it is the condition for admissibility. An outcome channel that fails to redundantly embed itself in the environment is structurally excluded.
Thus, outcome statistics become sensitive to decoherence strength. As redundancy increases, the space of admissible outcomes narrows, and rare branches vanish—not gradually, but structurally. Standard quantum mechanics predicts no such dependence.
7.4 Experimental Regimes for Distinction
This divergence is empirically accessible.
In weak measurement with post-selection, CBR predicts a collapse of low-p outcomes under increasing detector-environment coupling. Rare results become unrealizable as redundancy rises, deviating measurably from Born frequencies.
In delayed-choice quantum erasers, standard theory allows retroactive interference if coherence remains. CBR rejects this: once decohered, contradictory channels are no longer admissible. Interference is structurally blocked, not just dephased.
In ultra-asymmetric interferometry, where one path carries negligible amplitude, CBR predicts a realization threshold beyond which the low-p path is excluded. This behavior is invariant under standard QM.
Each test isolates a core CBR prediction: that outcome frequencies track physical admissibility, not amplitude alone.
7.5 The Born Rule as an Emergent Limit
CBR recovers the Born rule only in the thermodynamic limit. When redundancy saturates and decoherence is maximal, ℛ(Φᵢ) converges to:
ℛ(Φᵢ) ≈ −log(|⟨ϕᵢ | ψ⟩|²),
and the observed frequencies asymptotically match Born weights.
But this match is contingent. Outside this limit, deviation is necessary. The Born rule is not fundamental—it is a special case of physical constraint.
7.6 Falsifiability
CBR can be ruled out. If outcome frequencies remain Born-consistent across variations in decoherence, embedding, and record structure, then ℛ(Φ) fails to constrain reality and the theory collapses.
But if frequencies sharpen with redundancy—if rare outcomes disappear faster than amplitude alone predicts—then quantum mechanics requires amendment. No collapse theory, hidden-variable model, or many-worlds interpretation anticipates this constraint-governed behavior.
CBR is thus not a reinterpretation of quantum theory.
It is a claim that quantum theory, as currently formulated, is structurally incomplete.
7.7 Final Assertion
Probability does not govern physical reality.
Constraint does.
The Born rule holds when constraint saturates—not before.
Where constraint fails, so too must its statistical shadow.
CBR replaces the ungrounded postulate of probability with a realization principle.
It proposes that what happens is not what could happen, weighted by amplitude—
but what can happen, constrained by structure.
This is the first step toward a physics of actuality.
8. The Law That Was Missing
Quantum mechanics is a theory of propagation and possibility. It does not contain a law of realization.
The Born rule is not derived. The selection of outcomes is not explained. Decoherence suppresses interference but does not decide which alternative becomes actual. Collapse is postulated, interpretation proliferates, but the core structure remains silent on the central question: what determines which outcome becomes real?
Constraint-Based Realization (CBR) answers this.
Across Volumes 0 through V, we have introduced a dynamical law: only those quantum outcome channels Φ that minimize a realization functional ℛ(Φ) are physically admissible. This functional quantifies the entropic, informational, and structural costs associated with realizing a given outcome history. Actuality is not sampled. It is selected.
This principle reframes measurement not as randomness, but as structure. The universe does not roll quantum dice. It permits only those transitions whose constraint signature supports classicality, coherence, and recordable stability.
The Quantum Assembly Unit (QAU) framework defined this structure in operational terms. It showed how quantum superpositions transition into classical histories not by projection or branching, but through constraint-optimized assembly. Decoherence, classical records, observer agreement — all emerge as consequences of admissibility under ℛ(Φ). They are not causes. They are effects.
The Born rule, long treated as a primitive, now appears as a limit: ℛ(Φ) → −log |⟨ϕ | ψ⟩|² when redundancy becomes maximal. But outside this limit, deviations are not only allowed — they are required. The Born rule is not exact. It is a statistical approximation valid only where constraint saturation makes it so.
This transforms the foundational problem.
CBR is not an interpretation. It is a completion. It adds no particles, no hidden variables, no branches, no collapse. It adds only a law: a variational principle that selects the physically real from the quantum possible.
This law is empirically falsifiable. If outcome statistics remain invariant under structural variation in record stability, redundancy, or decoherence, then ℛ(Φ) has no role, and CBR fails. But if statistics sharpen with constraint — if low-amplitude outcomes vanish under increasing physical cost — then the Born rule is not fundamental, and quantum theory is incomplete without this law.
There is no further ambiguity. Either actuality is constrained, or it is not.
CBR makes the claim testable.
It turns the metaphysical silence of quantum mechanics into a dynamical principle of selection.
And in doing so, it shifts physics from possibility to law.
Appendix A — Uniqueness and Necessity of the Realization Measure
A.1. Purpose and Scope
This appendix establishes a necessity and uniqueness result for realization-selection laws within the constraint-based realization framework. The aim is to show that, under a minimal set of physically unavoidable requirements, the Born rule arises as the unique realization measure that is invariant under composition, coarse-graining, and admissible perturbations of the realization functional.
The result is structural rather than interpretive: it does not assume the Born rule, modify unitary dynamics, or invoke observer-dependent probability assignments. Instead, it demonstrates that any physically admissible realization law capable of producing stable, observer-independent outcomes must induce |ψ|² weighting. All alternative measures are shown to be structurally unstable.
A.2. Domain and Setup
Let ℋ be a finite-dimensional Hilbert space describing a quantum system prepared in a pure state ρ = |ψ⟩⟨ψ|. Let {Φᵢ} denote a finite set of mutually exclusive realization channels, where each Φᵢ is a completely positive, trace-preserving (CPTP) map acting on the system plus environment and yielding a macroscopically distinguishable outcome record.
Define a realization functional
ℛ : CPTP → ℝ⁺,
with the realized outcome identified as the channel Φᵢ minimizing ℛ over the admissible set.
Let pᵢ(n) denote the empirical frequency of outcome i after n repetitions of identical preparations and realization procedures.
A.3. Axioms
The following axioms define the admissible class of realization-selection laws.
Axiom A (Physical Admissibility)
The realization functional ℛ is defined only on CPTP maps and respects no-signaling constraints. Non-CP or signaling transformations are excluded from the domain of realization.
Axiom B (Unitary Relational Invariance)
ℛ depends only on relational properties between the pre-measurement quantum state and the realized outcome record. For any global unitary U,
ℛ(Φ) = ℛ(U Φ U†).
This excludes basis-dependent, representation-dependent, or observer-dependent realization rules.
Axiom C (Compositional Consistency)
For independently prepared and realized subsystems A and B,
ℛ(Φ_A ⊗ Φ_B) = ℛ(Φ_A) + ℛ(Φ_B).
This enforces extensivity and forbids realization rules that fail under composition.
Axiom D (Perturbative Robustness)
A physically admissible realization law must be invariant under:
admissible coarse-grainings of outcome records,
composition with ancillary systems,
arbitrarily small admissible perturbations of the realization functional ℛ → ℛ + εΔℛ.
Formally, the induced realization measure μ must satisfy
lim_{ε→0} μ_{ℛ+εΔℛ} = μ_{ℛ}
for all admissible perturbations Δℛ.
Failure of perturbative robustness constitutes failure of physical realization, as it implies observer-dependent or preparation-dependent outcome statistics.
A.4. Measure Reduction Lemma
Lemma A.1 (Reduction to Projective Measure)
Under Axioms A–C, any realization functional ℛ induces a measure μ on projective Hilbert space such that, for outcomes {ϕᵢ},
μᵢ = μ(ϕᵢ) = f(|⟨ϕᵢ|ψ⟩|²),
for some strictly monotone function f : [0,1] → ℝ⁺.
Justification.
Unitary relational invariance eliminates phase and basis dependence. Compositional consistency enforces multiplicative structure on joint amplitudes, which collapses admissible functional dependence to a scalar function of squared amplitudes. No additional degrees of freedom survive these constraints.
A.5. Realization Dynamics and Fixed Points
Repeated realization induces an effective flow on the space of admissible measures μ. Fixed points of this flow correspond to realization measures for which empirical frequencies stabilize under repetition and coarse-graining.
A realization measure μ is said to be robust if it remains invariant under admissible perturbations of ℛ and under outcome aggregation.
A.6. Uniqueness and Necessity Theorem
Theorem A.1 (Born Rule as Unique Robust Realization Measure)
Given Axioms A–D, the realization measure induced by ℛ is invariant under composition, coarse-graining, and admissible perturbations if and only if
μᵢ ∝ |⟨ϕᵢ|ψ⟩|².
All other admissible functional forms are structurally unstable: arbitrarily small perturbations of ℛ induce changes in μ that violate compositional consistency, coarse-graining invariance, or observer-independent outcome statistics.
A.7. Proof Sketch
Step 1 — Functional Reduction
By Lemma A.1, all admissible realization measures take the form μᵢ = f(|⟨ϕᵢ|ψ⟩|²).
Step 2 — Perturbative Analysis
Consider an admissible perturbation ℛ → ℛ + εΔℛ inducing μ → μ + εδμ.
For linear weighting f(x) = x, perturbations amplify under composition, violating Axiom C.
For power-law weighting f(x) = xᵅ with α ≠ 1, infinitesimal perturbations shift dominance among outcomes, leading to coarse-graining–dependent statistics and violation of Axiom D.
For non-analytic or piecewise functions f, perturbations induce discontinuous changes in μ, destroying invariance under ε → 0.
Step 3 — Coarse-Graining Invariance
Additivity under outcome aggregation,
μ(A ∪ B) = μ(A) + μ(B),
is preserved uniquely for quadratic weighting. All other admissible functions violate additivity under coarse-graining.
Step 4 — Necessity Conclusion
Quadratic weighting is the only realization measure invariant under admissible perturbations, composition, and coarse-graining. All alternatives are eliminated by structural instability.
A.8. Corollary (No-Go Form)
There exists no physically admissible realization-selection law satisfying Axioms A–D that yields stable realization frequencies inconsistent with |ψ|² weighting. Empirical deviation from the Born rule would therefore falsify at least one of the axioms defining physical realization.
A.9. Interpretation and Status
This result does not assume the Born rule, nor does it interpret probability subjectively. It establishes |ψ|² weighting as a structural necessity for any realization law compatible with unitary quantum mechanics and physical robustness.
The result completes the framework at the level of principle. Its scope is restricted to outcome realization and does not modify unitary dynamics or predict deviations in fully realized regimes.
