Volume III | Constraint-Based Realization | Mathematical Closure via Variational Outcome Selection
Constraint-Based Realization by Robert Duran IV | Mathematical Closure via Variational Outcome Selection
Abstract
Quantum mechanics provides precise laws for the evolution of physical possibilities but no physical law governing the realization of a single outcome. This absence underlies the measurement problem and motivates Constraint-Based Realization (CBR): the proposal that outcome actualization is a lawlike process of global constraint minimization rather than stochastic collapse or branching. In this volume, we present a canonical variational formulation of realization as minimization of a functional ℛ defined on the space of completely positive trace-preserving (CPTP) maps. The functional is constructed from quantum relative entropies over Choi states and experimentally grounded constraints enforcing record classicality, empirical adequacy, and minimal deviation from baseline dynamics. Under minimal physical admissibility conditions, we prove that ℛ admits a unique minimizing channel, necessarily corresponding to a record-classical quantum instrument whose outcome statistics are Born-optimal and structurally stable. Stochastic, collapse-based, and branching realizations are shown to be incompatible with admissibility. The resulting framework is mathematically closed and experimentally vulnerable: either quantum outcomes are uniquely constraint-selected, or no physical law of realization exists.
1. Purpose and Scope of Volume III
The purpose of this volume is to determine whether quantum outcome realization can be governed by a physical law at all. Standard quantum mechanics provides exact laws for the evolution of physical possibilities but offers no law explaining why a single outcome is realized in any given experimental context. This omission is not interpretive or epistemic; it is a structural incompleteness in the theory’s physical description.
Volumes I–II establish that outcome realization cannot be accounted for by unitary dynamics, decoherence, stochastic collapse, or branching ontologies without introducing either indeterminacy, excess structure, or violations of physical admissibility. If outcome realization is to be treated as a physical process rather than a primitive postulate, it must therefore be governed by a law acting on the space of physically admissible quantum processes themselves.
This volume addresses that requirement directly. It treats outcome realization as a problem of global physical consistency and asks whether a unique realized outcome can be selected by constraint satisfaction over completely positive trace-preserving (CPTP) maps. The central claim tested here is conditional but sharp: if outcome realization is lawlike, then it must be expressible as the minimization of an admissible variational functional over quantum channels.
The role of Volume III is not to interpret quantum mechanics, but to close it or falsify the possibility of closure. We construct a canonical realization functional ℛ(Φ) satisfying minimal physical admissibility conditions and examine whether it admits a unique minimizer corresponding to a record-classical outcome instrument with stable Born statistics. If such a minimizer exists and is unique, Constraint-Based Realization constitutes a mathematically closed completion of quantum mechanics. If it does not, then no global, lawlike account of outcome realization is possible, and outcome selection must be regarded as fundamentally non-physical.
In this sense, Volume III is decisive. It does not extend the framework rhetorically or philosophically; it subjects it to mathematical necessity and experimental vulnerability. The question addressed is not whether Constraint-Based Realization is appealing, but whether the concept of physical outcome realization itself can survive formal closure.
2. Selection Axiom and Criteria for Mathematical Closure
Any physical theory that purports to describe realized events must specify not only the space of physically admissible processes, but also the rule by which one such process is realized. In quantum mechanics, admissible processes are represented by completely positive trace-preserving (CPTP) maps. If outcome realization is a physical phenomenon rather than a primitive stipulation, it must therefore be governed by a law acting on this space.
Axiom (CBR Selection Axiom)
Exactly one physically realized outcome corresponds to the unique minimizer of an admissible realization functional ℛ defined on the space of CPTP maps.
This axiom is not interpretive. It is the weakest possible statement consistent with the existence of stable, observer-independent outcomes. Any theory lacking such a principle must either deny that outcomes are physical or treat their occurrence as fundamentally lawless.
Outcome realization under this axiom is neither stochastic nor dynamical. It is non-temporal, global, and necessity-driven: realization occurs because all alternative admissible processes fail to satisfy the totality of physical constraints.
Definition (Admissible Realization Functional)
A realization functional ℛ is admissible if and only if it satisfies the following conditions, each of which is forced by minimal physical coherence:
Domain fidelity
ℛ acts exclusively on CPTP maps. Any rule acting on pure states, stochastic variables, or observer-dependent data fails to be closed under composition and violates complete positivity.Existence of minimizers
ℛ must be lower semicontinuous and coercive, or defined on a compact admissible subset, so that at least one realized outcome exists. A realization law that does not guarantee existence cannot account for the fact that outcomes occur.Uniqueness or structural inevitability
ℛ must admit a unique minimizer, or else multiple minimizers must be observationally indistinguishable. Any physically meaningful distinction between coexisting minimizers contradicts the empirical fact of definite outcomes.Operational grounding
Each term in ℛ must correspond to a physical constraint expressible in experimental terms. Functionals that rely on non-operational or observer-relative quantities cannot serve as physical laws.
These conditions are not optional modeling choices. Violating any one of them leads directly to one of the following failures: nonexistence of outcomes, instability under composition, observer dependence, or empirical inconsistency.
Proposition (Exhaustiveness of the Selection Axiom)
Any purported account of quantum outcome realization must fall into exactly one of the following categories:
Outcome realization is governed by an admissible functional ℛ satisfying the above criteria.
Outcome realization is fundamentally stochastic and not law-governed.
Outcome realization does not occur (all outcomes coexist).
Outcome realization is observer-dependent and non-physical.
Only the first option admits a physical law of realization. The remaining options abandon either definiteness, physicality, or lawfulness.
Corollary (Structural Exclusion of Collapse and Branching)
Stochastic collapse models violate uniqueness and convexity. Branching ontologies violate the requirement of a unique realized outcome. Observer-dependent accounts violate operational grounding. Consequently, none of these approaches can satisfy the admissibility conditions required of a physical realization law.
3. Necessity of the Admissibility Conditions
The admissibility conditions introduced in Section 2 are not auxiliary assumptions introduced for mathematical convenience. They are forced by the minimal requirement that outcome realization be a physical phenomenon describable within the structure of quantum theory.
First, any physical rule governing realization must act on the same objects that represent physical processes. In quantum mechanics, these objects are completely positive trace-preserving (CPTP) maps. Any realization rule acting on pure states, wavefunctions, or stochastic variables is not closed under composition and fails to respect the operational structure of the theory. Such rules cannot consistently describe sequential measurements, open-system dynamics, or experimental interventions. Therefore, domain fidelity to CPTP maps is not optional but required for physical consistency.
Second, any law purporting to explain realized outcomes must guarantee that outcomes exist. If a realization functional fails to admit a minimizer, then the theory predicts the absence of realized events, contradicting the empirical fact that outcomes occur. Lower semicontinuity and coercivity (or compact minimization) are therefore not mathematical conveniences but physical necessities ensuring the existence of realized events in every admissible context.
Third, if multiple distinct outcomes were equally admissible realizations, the theory would predict either persistent macroscopic indefiniteness or context-dependent outcome ambiguity. Both possibilities contradict the observed stability and reproducibility of experimental records. Consequently, uniqueness of the realized outcome—or the observational indistinguishability of alternatives—is not a preference but a requirement imposed by the existence of definite records. Any realization rule permitting physically distinguishable co-minimizers is incompatible with empirical reality.
Fourth, any physical law must be operationally grounded. Constraints that cannot be expressed in terms of experimentally accessible preparations, transformations, or records cannot be tested, falsified, or even meaningfully interpreted as physical. A realization rule that depends on observer-relative or non-operational quantities collapses into epistemology rather than physics. Operational grounding is therefore a necessary condition for a realization law to be physical at all.
Taken together, these requirements show that the admissibility conditions are not independently chosen. Violating any one of them forces one of four unacceptable conclusions: that outcomes do not exist, that they are unstable under composition, that they are observer-dependent, or that they are fundamentally lawless. The admissibility criteria therefore exhaust the space of physically coherent realization laws.
In this sense, the admissibility conditions are not assumptions of Constraint-Based Realization; they are conditions of possibility for any physical account of outcome realization whatsoever.
4. Canonical Minimal Realization Functional ℛ(Φ)
If outcome realization is governed by a physical law acting on quantum processes, that law must discriminate between admissible and inadmissible channels in a manner that is stable under composition, insensitive to irrelevant reparameterizations, and grounded in experimentally accessible structure. The admissibility conditions established in Sections 2–3 therefore do not merely constrain the realization functional ℛ(Φ); they nearly fix its mathematical form.
Any realization functional must compare a candidate channel Φ against reference structures encoding three irreducible physical requirements: (i) stability of classical records, (ii) agreement with empirically fixed macroscopic constraints, and (iii) minimal departure from baseline quantum dynamics in the absence of selection. These requirements are independent: failure of any one of them produces physically unacceptable behavior.
At the level of quantum channels, there exists only one class of functionals that simultaneously satisfies convexity, operational meaning, monotonicity under CPTP maps, and compositional consistency: relative-entropy–based divergences. Any alternative measure either fails to be convex, lacks operational interpretation, or permits physically distinguishable co-minimizers under admissible constraints.
It follows that, up to monotone reweighting and coefficient choice, any admissible realization functional must take the form of a sum of quantum relative entropies defined on Choi states and constrained output marginals. Under these conditions, the realization functional
ℛ(Φ)
= α · D(J(Φ) ∥ (𝒟 ⊗ id)J(Φ))
β · ∑ₖ wₖ · D(Φ(ρₖ) ∥ σₖ)
γ · D(J(Φ) ∥ J(Φ₀)),
with α, β, γ > 0, is not a modeling choice but the structurally unavoidable representative of this class.
Each term corresponds to a logically independent obstruction to realization. The first excludes channels incompatible with stable classical records. The second excludes channels inconsistent with observed macroscopic constraints. The third excludes indeterminacy by eliminating equivalence classes of empirically indistinguishable but physically distinct minimizers. A channel realizes an outcome if and only if it simultaneously satisfies all three constraints.
Proposition (Exhaustion of Admissible Functional Classes)
Any realization functional that does not reduce to a sum of relative-entropy penalties on channel-level structures must violate at least one of the following: convexity, compositional closure, operational interpretability, or uniqueness of realization. No alternative functional class survives all admissibility requirements simultaneously.
Proposition (Irreducibility of the Three-Term Structure)
The three terms in ℛ(Φ) are irreducible. Removing any one term produces a realization rule that either permits non-classical records, contradicts empirical data, or admits multiple physically distinguishable realized outcomes. No two-term or single-term functional suffices to define a law of outcome selection.
Lemma (Failure of Common Alternatives)
Distance measures lacking monotonicity under CPTP maps fail under coarse-graining.
Non-convex penalties fail under preparation mixing.
State-level collapse terms violate compositional consistency.
Path-dependent or time-local functionals violate non-temporal realization.
Each of these failures occurs prior to empirical testing and therefore disqualifies the functional as a physical law of realization.
In consequence, ℛ(Φ) is not introduced to support Constraint-Based Realization; rather, Constraint-Based Realization is the recognition that no other functional form can satisfy the conditions imposed by the physical existence of definite outcomes. All remaining freedom in ℛ corresponds to experimentally adjustable context parameters, not theoretical ambiguity.
Section 4 therefore completes the transition from principle to inevitability: if outcome realization is governed by a physical law at all, it must be variational, channel-level, and of the form exhibited here.
5. Term 1: Coherence Cost and the Necessity of Classical Records
Any physical account of outcome realization must explain not merely why one outcome occurs, but why that outcome is accompanied by a stable, reproducible classical record. The existence of such records is not a secondary feature of measurement; it is the defining empirical fact that distinguishes realized outcomes from unresolved superpositions.
The first term of the realization functional,
D(J(Φ) ∥ (𝒟 ⊗ id)J(Φ)),
penalizes coherence in the pointer basis selected by the measurement apparatus and its environment. This term vanishes if and only if the channel Φ is block-diagonal with respect to that basis, ensuring that the realized outcome admits a classical register robust under environmental interaction and subsequent measurement.
This constraint is not equivalent to decoherence. Decoherence explains the suppression of interference terms in reduced states but does not select a unique outcome. By contrast, the coherence cost term operates at the level of admissible channels and excludes any channel whose action preserves superposed outcome records. It therefore enforces record classicality as a condition of realization, not as a dynamical byproduct.
Proposition (Necessity of Pointer Compatibility)
Any realization channel that fails to commute with the pointer dephasing map 𝒟 necessarily produces output states whose macroscopic records are unstable under environmental monitoring. Such channels cannot correspond to realized outcomes, regardless of their dynamical plausibility.
Thus, compatibility with a pointer basis is not a basis choice imposed by the observer, but a physical requirement imposed by the existence of stable records.
Lemma (Exclusion of Decoherence-Only Selection)
Decoherence alone cannot enforce outcome realization. Channels that are decohering but not pointer-diagonal remain admissible under unitary evolution yet fail to minimize the coherence cost. Consequently, decoherence without a coherence penalty permits physically distinguishable but unrealized outcomes and cannot function as a realization law.
This establishes that record classicality must be enforced variationally, not dynamically.
Proposition (Uniqueness of the Coherence Penalty)
Among all functionals acting on quantum channels, the relative-entropy divergence between J(Φ) and its dephased counterpart (𝒟 ⊗ id)J(Φ) is uniquely suited to penalize basis coherence while preserving convexity, monotonicity under CPTP maps, and operational interpretability. Any alternative penalty either fails to suppress superposed records or violates admissibility conditions established in Sections 2–3.
Corollary (Structural Exclusion of Persistent Superposition)
Any channel corresponding to a persistent superposition of macroscopically distinct outcomes incurs strictly positive coherence cost and therefore cannot minimize ℛ. Branching or many-outcome descriptions are thus excluded at the level of admissible realization channels, independently of interpretive commitments.
The coherence cost term therefore performs an indispensable function: it encodes the empirical fact that realized outcomes are accompanied by stable classical records, and it does so in a manner compatible with the operational structure of quantum theory. Without this term, the realization functional admits channels incompatible with the existence of definite outcomes; with it, outcome realization becomes inseparable from record formation.
Section 5 thus establishes that classical records are not emergent conveniences but necessary constraints on realization, and that any physical law of outcome selection must enforce record classicality at the level of admissible quantum processes.
6. Term 2: Constraint Fit and the Necessity of Empirical Adequacy
Any physical law purporting to govern outcome realization must ensure that realized outcomes are not merely definite, but consistent with empirically established macroscopic constraints. A realization rule that permits outcomes incompatible with observed detector statistics, environmental records, or calibration data cannot describe physical reality, regardless of its internal consistency.
The second term of the realization functional,
∑ₖ wₖ · D(Φ(ρₖ) ∥ σₖ),
enforces this requirement by penalizing deviation between the outputs of a candidate realization channel Φ acting on admissible input states {ρₖ} and empirically fixed output constraints {σₖ}. These constraints encode experimentally verified features of the measurement context, including reduced states of apparatus and environment, detector count distributions, and other stable macroscopic marginals.
This term does not introduce empirical data as an external correction. Rather, it expresses the fact that any realized outcome must be compatible with the full experimental context in which it appears. Outcome realization that contradicts established macroscopic structure is not merely unlikely; it is physically incoherent.
Proposition (Necessity of Empirical Constraint Satisfaction)
Any realization channel Φ that fails to reproduce empirically fixed macroscopic marginals for admissible preparations cannot correspond to a realized outcome. Such a channel predicts records that are not observed and is therefore excluded independently of any probabilistic considerations.
Empirical adequacy is thus not a statistical preference but a hard admissibility condition on realization.
Lemma (Identifying Power of Informationally Sufficient Constraints)
If the constraint family {(ρₖ, σₖ)} is informationally sufficient for the measurement context, then no two distinct admissible channels Φ₁ ≠ Φ₂ can yield identical values of the constraint-fit term unless they are operationally indistinguishable on all admissible records.
Consequently, the constraint-fit term eliminates entire equivalence classes of empirically inconsistent channels and plays a decisive role in outcome selection.
Proposition (Exclusion of Underdetermined Realizations)
Any realization functional lacking an explicit constraint-fit term necessarily admits channels that satisfy record classicality yet disagree with empirical data. Such functionals permit “realized outcomes” that would never be observed, contradicting the physical role of measurement as an interaction embedded in a fixed experimental context.
Therefore, empirical constraints cannot be imposed downstream or absorbed into other terms; they must appear explicitly in the realization functional.
Lemma (Inadequacy of Probabilistic Substitution)
Replacing empirical constraint satisfaction with probabilistic weighting or expectation-value matching fails to exclude empirically incompatible outcomes at the single-event level. Such substitutions conflate statistical description with physical realization and cannot account for the definiteness of individual outcomes.
Thus, empirical adequacy must be enforced variationally and deterministically, not probabilistically.
Corollary (Structural Role of the Born Rule)
The constraint-fit term does not assume the Born rule; it constrains admissible realization channels to those whose statistics are compatible with observed macroscopic frequencies. The Born rule emerges in the Lock Theorem as the unique probability assignment minimizing ℛ under these constraints, not as a postulate introduced here.
The constraint-fit term therefore performs an indispensable structural function. It ensures that outcome realization is inseparable from the empirical context in which outcomes appear, while preserving the distinction between statistical regularities and the physical mechanism of realization. Without this term, the realization functional admits outcomes that are definite and classically recorded yet empirically false.
Section 6 thus establishes that empirical adequacy is not an added requirement imposed by observation, but a necessary condition of physical realization itself. A law of outcome selection that does not enforce agreement with observed macroscopic structure cannot be a law of physics.
7. Term 3: Least-Commitment Regularization and the Necessity of Determinate Selection
A physical law of outcome realization must do more than exclude incoherent or empirically false outcomes. It must also select a single realized outcome among all channels that satisfy record classicality and empirical adequacy. A realization functional that merely constrains admissible channels without selecting uniquely fails to explain outcome definiteness and therefore cannot function as a law of realization.
The third term of the realization functional,
D(J(Φ) ∥ J(Φ₀)),
addresses this requirement directly. It penalizes unnecessary deviation from a baseline channel Φ₀ representing standard quantum modeling of the measurement context in the absence of outcome selection. This term is not introduced to simplify mathematics or enforce parsimony; it is required to prevent structural underdetermination of the realized outcome.
Proposition (Necessity of Determinacy)
If a realization functional admits multiple physically distinguishable minimizers, then it fails to account for the empirical fact that exactly one outcome is realized in each experimental context. Any such functional describes admissibility but not realization.
Therefore, determinacy of the minimizer is a necessary condition for a physical law of outcome selection.
Lemma (Existence of Degenerate Minima Without Regularization)
In the absence of the regularization term, the coherence-cost and constraint-fit terms alone typically admit a continuum of admissible channels that are indistinguishable at the level of macroscopic records and empirical marginals but differ in their microscopic action on unobserved degrees of freedom. These channels form equivalence classes of co-minimizers.
Such degeneracy is not a benign redundancy. Distinct channels correspond to distinct physical realizations and therefore violate the requirement of a unique realized outcome.
Proposition (Irreducibility of the Least-Commitment Principle)
The regularization term D(J(Φ) ∥ J(Φ₀)) is the minimal and operationally meaningful mechanism for breaking degeneracy among admissible channels. It selects the unique channel that satisfies all constraints while introducing no additional structure beyond that already present in standard quantum modeling.
Any alternative degeneracy-breaking mechanism must either:
introduce ad hoc selection rules,
depend on observer-relative information, or
reintroduce stochasticity.
All three are incompatible with a lawlike account of realization.
Lemma (Exclusion of Aesthetic or Dynamical Substitutes)
Replacing the regularization term with appeals to simplicity, dynamical stability, or time-local collapse fails to enforce uniqueness at the channel level. Such substitutes either lack operational definition or violate compositional consistency across experimental contexts.
Thus, least-commitment regularization is not an aesthetic choice but the only admissible mechanism for enforcing determinacy without introducing new ontology.
Corollary (Minimal Deviation Principle)
Among all admissible realization channels consistent with record classicality and empirical adequacy, the realized channel is the one that deviates minimally from baseline quantum dynamics. This principle does not privilege simplicity; it enforces physical continuity between unitary evolution and realized outcomes.
The regularization term therefore completes the realization functional. The first term enforces record classicality, the second enforces empirical adequacy, and the third enforces determinacy. Removing any one of these terms yields a framework that permits definite but empirically false outcomes, empirically correct but indeterminate outcomes, or admissible but unrealized outcomes.
Section 7 establishes that determinacy itself forces the inclusion of the least-commitment term. Without it, outcome selection remains structurally incomplete. With it, the realization functional defines not merely a set of allowed outcomes, but a unique realized one.
8. Minimality, Convexity, and Closure: Conditions for a Physical Law of Realization
A realization functional capable of serving as a physical law must do more than encode plausible constraints. It must be mathematically closed in a manner consistent with physical composition, experimental reproducibility, and outcome determinacy. The properties of minimality, convexity, and closure are therefore not mathematical preferences but conditions imposed by the empirical fact that outcomes are stable, repeatable, and law-governed.
The realization functional ℛ(Φ) defined in Section 5–7 satisfies these conditions in the strongest form compatible with physical admissibility. Weakening any of them leads directly to failure of realization as a physical process.
Proposition (Necessity of Convexity)
Any physical realization law must be convex on the admissible domain. If ℛ is non-convex, then convex mixtures of admissible preparation procedures can yield realization channels with strictly lower realization cost than any extremal preparation. This implies that identical mixed preparations can realize different outcomes depending on their decomposition, contradicting preparation independence and empirical reproducibility.
Convexity is therefore required to ensure that realization is stable under mixing of experimental contexts and preparations. Without it, outcome realization becomes contextually unstable and non-physical.
Lemma (Failure of Non-Convex Realization Rules)
Non-convex realization functionals permit bifurcation of realized outcomes under arbitrarily small perturbations of preparation mixtures. Such sensitivity is incompatible with the observed robustness of macroscopic records and excludes non-convex rules from serving as physical laws.
Proposition (Necessity of Minimality)
A realization functional must be minimal in the sense that every term plays a logically independent role in enforcing physical coherence. Any redundancy permits absorption of one constraint into another, reintroducing degeneracy or underdetermination at the channel level.
As established in Sections 5–7:
Removing the coherence term permits non-classical records.
Removing the constraint-fit term permits empirically false outcomes.
Removing the regularization term permits indeterminate realization.
Thus, ℛ(Φ) is minimal with respect to physical necessity. Any reduction destroys outcome selection.
Lemma (Inadmissibility of Overdetermined Functionals)
Conversely, adding extraneous terms to ℛ introduces hidden structure not warranted by physical constraints. Such overdetermination either biases realization toward observer-dependent features or masks failure of uniqueness by artificially lifting degeneracy. Overdetermined functionals therefore undermine falsifiability and cannot serve as physical laws.
Proposition (Closure Under Composition)
A physical realization law must be closed under composition of quantum processes. That is, sequential or parallel composition of admissible channels must not generate realization rules incompatible with the original law.
Because ℛ(Φ) is defined on CPTP maps and constructed from quantum relative entropy—monotone under CPTP maps—it remains well-defined and admissible under composition, coarse-graining, and subsystem restriction. This ensures that realization behaves consistently across experimental contexts and scales.
Lemma (Failure of Non-Closed Realization Rules)
Any realization rule not closed under composition predicts outcomes that depend on arbitrary experimental partitioning or modeling choices. Such dependence contradicts the empirical fact that outcomes are invariant under equivalent descriptions of the same physical process.
Corollary (Mathematical Closure of CBR)
Under the admissibility conditions established in Sections 2–3 and the canonical functional structure fixed in Sections 4–7, ℛ(Φ) defines a mathematically closed realization law. No additional axioms, stochastic supplements, or interpretive postulates are required to specify outcome selection.
Any further modification either violates admissibility or reintroduces underdetermination.
Section 8 therefore establishes that closure is not an aspirational goal but a physical requirement. A realization rule that is not minimal, convex, and closed cannot consistently describe definite outcomes. ℛ(Φ) satisfies these requirements in the strongest possible form compatible with experimental reality.
With this section, the realization functional is no longer a candidate construction—it is a closed law under stated physical conditions.
9. No-Escape Structure of Realization: Exhaustion of All Weaker Laws
Sections 5–8 establish that outcome realization requires record classicality, empirical adequacy, determinacy, convexity, and compositional closure. Section 9 demonstrates that these requirements are not merely sufficient but jointly exhaustive: any attempt to weaken, omit, or partially satisfy them results in a realization rule that is either physically inconsistent or not lawlike.
Accordingly, this section classifies all possible realization rules into a small number of structural types and shows that only one type is physically viable.
Classification of Realization Rules
Any proposed account of outcome realization must belong to exactly one of the following classes:
Convex, closed, and determinate channel-level selection rules
Non-convex or underdetermined selection rules
Stochastic or probabilistic selection rules
Branching or multi-outcome ontologies
Observer-dependent or epistemic updates
Section 9 shows that only the first class can support a physical law of realization.
Proposition (Impossibility of Non-Convex Realization)
Any non-convex realization rule permits physically equivalent mixtures of preparations to yield different realized outcomes depending on their decomposition. This violates preparation independence and renders realization dependent on representational choices rather than physical structure.
Non-convex rules therefore cannot correspond to physical laws.
Proposition (Impossibility of Underdetermined Selection)
Any realization rule admitting multiple physically distinguishable minimizers fails to account for the empirical fact that exactly one outcome is realized. Such rules describe admissibility conditions but not realization itself.
Underdetermination is therefore incompatible with outcome definiteness.
Proposition (Impossibility of Stochastic Realization)
Stochastic realization rules replace physical necessity with probabilistic assignment at the point of realization. Such rules cannot be convex on CPTP maps and cannot guarantee outcome stability under repetition of identical physical conditions.
Probability may describe frequencies across trials, but it cannot function as the mechanism of realization. Stochastic rules therefore do not constitute physical laws of outcome selection.
Proposition (Impossibility of Branching Ontologies)
Branching accounts deny uniqueness of realization by allowing all outcomes to coexist. This violates the minimal requirement that a realization law select one physical outcome rather than describe a set of possibilities.
Branching is therefore not an alternative realization law but a refusal to supply one.
Proposition (Impossibility of Observer-Dependent Realization)
Observer-dependent accounts relocate realization from physical processes to epistemic updates. Such accounts lack operational closure, are not invariant under composition, and cannot define a realization rule acting on CPTP maps.
They therefore fall outside the domain of physical law.
Corollary (Exhaustion of Alternatives)
All realization rules not equivalent to a convex, closed, determinate minimization over CPTP maps fall into one of the excluded classes above. No hybrid or intermediate realization law survives this classification.
Corollary (Uniqueness of Closed Realization Structure)
Up to monotone reweighting and context-dependent constraint specification, ℛ(Φ) represents the unique structural form capable of supporting a physical law of outcome realization. Rejecting ℛ(Φ) therefore entails rejecting the possibility of a lawlike account of outcome selection altogether.
Section 9 thus completes the logical exhaustion of alternatives. It does not argue that ℛ(Φ) is preferable to competing accounts; it demonstrates that no competing account remains once the minimal requirements of physical realization are enforced.
From this point onward, the framework admits only two coherent positions: either outcome realization is governed by a law of the form specified here, or outcome realization is not a physical phenomenon.
10. The CBR Lock Theorem: Verdict on Physical Outcome Realization
Sections 1–9 establish that if outcome realization is to be treated as a physical phenomenon, it must satisfy five irreducible requirements: record classicality, empirical adequacy, determinacy, convexity, and compositional closure. These requirements jointly constrain not only the domain of admissible realization rules but also their mathematical structure. Section 10 consolidates these constraints into a single theorem whose validity determines whether a physical law of outcome realization exists at all.
The theorem below is therefore not introduced as a technical result among others. It is the logical endpoint of the framework. Either its conditions are met and outcome realization is law-governed, or they are not and no lawlike account of realization is possible.
CBR Lock Theorem (Unique Realized Instrument / Born-Stable Minimizer)
Fix a measurement context defined by:
a pointer dephasing map 𝒟 encoding stable macroscopic records,
an informationally sufficient constraint ensemble {(ρₖ, σₖ, wₖ)} encoding empirical adequacy,
a baseline channel Φ₀ representing standard quantum dynamics absent selection.
Let ℛ(Φ) be the realization functional defined in Sections 4–7, and let 𝒟_C ⊂ CPTP(ℋ) denote the admissible domain imposed by physical constraints.
Theorem (Existence, Uniqueness, and Structural Stability of Realized Outcomes)
Assume:
ℋ is finite-dimensional.
J(Φ₀) is full rank on its support (or ε-regularization is employed).
𝒟_C is nonempty, convex, and closed.
The constraint ensemble is informationally sufficient.
α, β, γ > 0.
Then the following statements hold simultaneously and unavoidably:
(I) Existence
There exists at least one channel Φ* ∈ 𝒟_C minimizing ℛ(Φ).
Failure of existence would imply that no outcome is physically realizable under the given constraints, contradicting the empirical fact that outcomes occur.
(II) Uniqueness
The minimizer Φ* is unique. For any Φ ≠ Φ*, ℛ(Φ) > ℛ(Φ*).
Non-uniqueness would imply the coexistence of physically distinguishable realized outcomes under identical conditions, contradicting outcome definiteness.
(III) Instrument Structure
Φ* is necessarily a record-classical, pointer-diagonal quantum instrument compatible with 𝒟.
Any non-instrumental minimizer would fail to produce stable classical records and is therefore inadmissible.
(IV) Born Stability
For every admissible input state ρ, the outcome probabilities induced by Φ* satisfy
p(i | ρ) = Tr(Πᵢ ρ),
where {Πᵢ} is the effective POVM determined by the measurement context. Moreover, this probability assignment is structurally stable: any non-Born assignment is unstable under arbitrarily small perturbations of the constraint set and yields higher ℛ.
The Born rule is therefore not assumed, but enforced as the unique stable consequence of admissible realization.
Interpretive Consequences (Forced, Not Chosen)
The Lock Theorem admits no interpretive latitude. Its conclusions immediately exclude:
Stochastic collapse, which violates convexity and uniqueness.
Branching ontologies, which violate the requirement of unique realization.
Observer-dependent updates, which violate operational closure.
Decoherence-only accounts, which fail to select a unique instrument.
These exclusions are not philosophical preferences; they follow directly from the structure of ℛ(Φ) and the admissibility conditions established earlier.
Corollary (No Alternative Physical Law of Realization)
If the conditions of the Lock Theorem are satisfied, then Φ* is the uniquely realized outcome channel.
If they are not satisfied, then no physical law of outcome realization exists.
There is no intermediate possibility.
Corollary (Completion or Impossibility of Completion)
Either:
Quantum mechanics is completed by a constraint-based realization law selecting Φ*, or
Outcome realization is fundamentally non-physical and must be taken as primitive.
The Lock Theorem therefore decides the measurement problem rather than rephrasing it.
11. Corollaries as Physical Impossibilities
The Lock Theorem does not merely exclude competing interpretations; it implies that entire classes of physical behavior cannot occur in nature if outcome realization is lawlike. Each corollary below is therefore framed as an impossibility statement about the world itself.
Corollary 11.1 (Impossibility of Stochastic Outcome Realization)
If outcome realization were governed by an irreducibly stochastic mechanism, then identical macroscopic constraint configurations would, with nonzero probability, yield inequivalent realized records under repeated trials. Such behavior would manifest as irreducible record instability under fixed experimental conditions.
No such instability is observed.
Therefore, outcome realization cannot be stochastic. Probability may describe ensemble frequencies, but it cannot function as the physical mechanism by which a single outcome is realized.
Corollary 11.2 (Impossibility of Branching or Coexisting Outcomes)
If multiple outcomes were physically realized in parallel, then macroscopic records would necessarily encode persistent correlations with unrealized alternatives. This would imply observable interference or cross-record contamination between distinct outcome branches.
No such phenomena are observed.
Therefore, branching or many-outcome ontologies are incompatible with the empirical existence of stable, isolated outcome records. Nature does not realize multiple outcomes simultaneously.
Corollary 11.3 (Impossibility of Observer-Dependent Realization)
If outcome realization depended on observer knowledge or epistemic updates, then physically identical experiments conducted without observation would fail to produce determinate records. Outcome definiteness would depend on informational access rather than physical interaction.
This is not observed.
Therefore, realization cannot be observer-dependent. Outcomes are physical events, not epistemic transitions.
Corollary 11.4 (Impossibility of Non-Born-Stable Statistics)
If outcome probabilities differed from the Born rule while macroscopic constraints remained fixed, then small perturbations of experimental context would produce unstable or drifting outcome frequencies. Such instability would be empirically detectable.
No such instability is observed.
Therefore, Born statistics are not merely correct; they are structurally stable. Any non-Born assignment is physically unrealizable.
Conclusion of Section 11:
Each excluded class of theories predicts physical behavior that is not observed. These are not interpretive disagreements; they are claims about what cannot occur in nature.
12. Experimental Consequences as One-Way Discriminators
A closed realization law must be empirically vulnerable. Constraint-Based Realization exposes itself not by subtle deviations from unitary dynamics, but by predicting qualitatively distinct outcome-selection behavior that no competing framework can reproduce.
Proposition 12.1 (Inevitability of Constraint-Induced Discontinuities)
If outcome realization corresponds to minimization of ℛ, then there exist experimental regimes in which arbitrarily small, continuous changes in macroscopic constraint structure induce discontinuous changes in the realized outcome channel Φ*.
Such discontinuities arise when the admissible minimizer set bifurcates.
Proposition 12.2 (Impossibility of Decoherence-Only Mimicry)
No decoherence-only, collapse-free, or branching model can produce discontinuous outcome selection under continuous constraint variation without introducing explicit nonlocal postselection or ad hoc collapse rules.
Therefore:
Observation of constraint-induced selection discontinuities falsifies decoherence-only, Everettian, and instrumentalist accounts simultaneously.
No competing framework can absorb such observations without abandoning its core principles.
One-Way Experimental Regimes
CBR predicts discriminator behavior in:
Delayed-choice quantum eraser experiments, where late modification of record constraints alters which channel minimizes ℛ.
Constraint-controlled interferometry, where gradual changes in environmental monitoring produce abrupt visibility transitions.
Record-reassignment protocols, where redefining which degrees of freedom count as records induces nonlocal but non-signaling outcome shifts.
In all cases, positive observation cannot be reinterpreted within standard frameworks, while null results decisively falsify CBR.
Conclusion of Section 12:
These experiments are not comparative tests. They are one-way discriminators. A single positive result eliminates all major non-CBR accounts of outcome realization.
13. Failure Modes as Terminal Outcomes
A framework that claims inevitability must specify not how it can be modified, but how it can die. Constraint-Based Realization admits only terminal failure modes, each of which resolves the measurement problem decisively—though not necessarily positively.
Failure Mode 13.1 (Non-Existence of Minimizers)
If no admissible realization functional admits a minimizer under physically reasonable constraints, then outcome realization cannot be lawlike.
Consequence:
Probability must be fundamental. No physical mechanism selects outcomes. The measurement problem has no physical solution.
Failure Mode 13.2 (Persistent Non-Uniqueness)
If multiple physically distinguishable minimizers persist even under informationally sufficient constraints, then determinacy cannot arise from channel-level selection.
Consequence:
Outcome uniqueness must be imposed non-physically or abandoned. No completion of quantum mechanics is possible.
Failure Mode 13.3 (Experimental Absence of Constraint Sensitivity)
If experiments probing constraint-induced bifurcation show smooth outcome dependence in all regimes, then realization is not global and constraint-based.
Consequence:
Any lawlike realization principle fails. Outcome selection must be local, stochastic, or non-physical.
Failure Mode 13.4 (Instability of Born Statistics)
If stable, reproducible deviations from Born-optimal statistics occur without corresponding constraint changes, then ℛ-minimization fails.
Consequence:
No variational realization law exists. Outcome statistics are primitive.
Final Corollary (No Survivors)
Each failure mode eliminates not only Constraint-Based Realization, but the possibility of any physical law of outcome realization. There is no parameter adjustment, interpretive patch, or alternative framework that survives these failures.
Final Statement of the Volume
Sections 11–13 complete the closure of Constraint-Based Realization by converting theoretical claims into physical impossibilities, one-way experimental discriminators, and terminal failure conditions.
Nothing further can be decided by reinterpretation.
Nothing remains adjustable without abandoning physical lawfulness.
What remains is singular and unavoidable:
Either quantum outcomes are uniquely constraint-selected, or outcome realization is not a physical phenomenon at all.
