Constraint-Based Realization | VOLUME II | Necessity, Dynamical Inevitability, and the No-Alternative Structure of Quantum Outcome Selection
Constraint-Based Realization | VOLUME II | By Robert Duran IV
1. Introduction: From Proposal to Necessity
1.1 What Volume I Established
Volume I introduced Constraint-Based Realization (CBR) as a physical completion of quantum mechanics. Its central claim was that the standard formalism lacks a law selecting which admissible quantum outcome becomes physically realized, and that this gap can be closed by a variational principle acting on the space of quantum outcome channels. In that work, outcomes were treated not as abstract eigenvalues but as complete physical processes—quantum channels encompassing system, apparatus, environment, and record formation. A realization functional ℛ(Φ) was proposed to encode physical constraints such as thermodynamic viability, stable record formation, intersubjective consistency, and global compositional coherence. Minimization of this functional was shown to reproduce Born statistics under realistic conditions.
Volume I thus demonstrated that a constraint-based selection law is coherent, compatible with unitary dynamics, and capable of reproducing standard quantum statistics without invoking collapse postulates, many-world branching, hidden variables, or observer-relative probabilities. However, it left open a deeper question: whether such a law is merely one possible completion of quantum mechanics, or whether it is forced by the basic structural commitments of the theory.
The purpose of the present volume is to answer that question.
1.2 What Remains Logically Undecided
Despite the internal consistency of Volume I, three logically significant possibilities remained open:
That alternative outcome-selection laws—stochastic, contextual, or observer-indexed—might satisfy the same physical requirements.
That Born statistics might arise contingently from the particular form of ℛ rather than being structurally unavoidable.
That the constraint-based character of outcome selection might be an optional modeling choice rather than a necessity.
These possibilities cannot be excluded by motivation or analogy alone. They require eliminative arguments: demonstrations that any alternative violates at least one minimal physical requirement.
Volume II is devoted entirely to closing these logical gaps.
1.3 Criteria for Theoretical Closure
We adopt the following criteria for a completed theory of quantum outcome selection:
Single-Outcome Realism: Each experimental run yields exactly one realized outcome.
Intersubjective Agreement: Distinct observers can consistently agree on that outcome through physical interaction.
Unitary Dynamics: Quantum evolution between outcome events is unitary.
Compositional Closure: Independent systems compose consistently via tensor products.
No Auxiliary Ontology: Outcome selection does not rely on hidden variables or observer-dependent primitives.
A theory satisfying these criteria must not merely reproduce observed statistics; it must do so stably, objectively, and without contextual dependence on unrealized alternatives.
The central claim of this volume is that these criteria uniquely force a constraint-based realization structure.
1.4 The Necessity Chain for Single Quantum Outcomes
We now state explicitly the logical structure that governs the remainder of this work.
Necessity Chain
Single outcomes + intersubjective agreement
⟹ physical realizability constraints
⟹ constraint-based outcome selection
⟹ unique realized outcome channels
⟹ Born statistical regularities
Equivalently, written as a structured implication chain:
Single outcomes + intersubjective agreement
⟹ physical realizability constraints
⟹ constraint-based outcome selection
⟹ unique realized outcome channels
⟹ Born statistical regularities
Each implication in this chain is forced, not postulated. The task of this volume is to make this necessity explicit and unavoidable.
1.5 Reductio: Absence of Realizability Constraints Is Impossible
Assume, for the sake of contradiction, that no realizability constraints exist on quantum outcome processes. Then every mathematically admissible quantum channel compatible with unitary dynamics would be eligible for physical realization as an outcome.
Under this assumption, there exist admissible outcome processes that fail to generate stable, accessible records in the environment. Such processes either do not imprint redundantly, degrade under environmental interaction, or fail to remain distinguishable across time. Observers interacting with different environmental fragments would therefore be unable, even in principle, to converge on a common account of what outcome occurred.
This directly contradicts the empirical fact of intersubjective agreement: that distinct observers can and do consistently agree on measurement outcomes. Since intersubjective agreement is given, the assumption that no realizability constraints exist must be false.
Therefore, realizability constraints are necessary conditions for the existence of single, objective quantum outcomes.
1.6 Reductio: Non-Constraint-Based Outcome Selection Is Impossible
Assume, again for the sake of contradiction, that realizability constraints exist but that outcome selection does not proceed by a constraint-based principle. Then the selection of a realized outcome must be either stochastic without physical discrimination, contextual with respect to unrealized alternatives, or observer-indexed.
If outcome selection is stochastic without reference to realizability constraints, then outcomes that violate record stability or intersubjective accessibility are selected with nonzero probability. Such outcomes cannot persist as objective facts, contradicting intersubjective agreement.
If outcome selection is contextual with respect to unrealized measurements, then the realized outcome depends on counterfactual experimental configurations. This renders outcome realization non-compositional and observer-relative, contradicting the assumption of compositional closure.
If outcome selection is observer-indexed, then distinct observers may realize incompatible outcomes under identical physical conditions, contradicting both intersubjective agreement and single-outcome realism.
In all cases, rejecting constraint-based selection leads to contradiction. Hence, any physically viable outcome-selection law must discriminate among admissible processes according to realizability constraints. Such discrimination necessarily induces an ordering over candidate outcome channels, which is precisely what is meant by constraint-based realization.
1.7 Reductio: Non-Born Outcome Statistics Are Physically Unstable
Assume that outcome selection proceeds by a realizability-preserving, constraint-based principle, but that the resulting outcome statistics do not converge to Born weights under repeated realization of identically prepared systems.
Under repeated outcome realization, environmental interaction amplifies outcome records and enforces compositional consistency across trials. If the induced outcome statistics differ from Born weights, then at least one of the following must occur: (i) some outcomes fail to generate redundantly stable records under amplification, (ii) outcome frequencies differ across environmental partitions, or (iii) outcome statistics depend on the order or grouping of independent subsystems.
Each of these possibilities contradicts realizability, intersubjective agreement, or compositional closure. Therefore, the assumption that non-Born statistics can be stable under realizability-preserving dynamics is false.
It follows that Born statistics are not contingently produced but are enforced as the only dynamically stable outcome distribution compatible with realizability constraints.
1.8 Strategy and Structure of This Volume
The remainder of this volume proceeds as follows. Sections 2–5 formalize realizability constraints and structural axioms, demonstrating that any violation leads to contradiction or observer-relative reality. Sections 6–7 establish the effective uniqueness of realized outcome channels. Sections 8–9 introduce the realizability flow on the probability simplex and prove the dynamical instability of non-Born statistics. Section 10 presents a no-go theorem excluding all non-constraint-based outcome laws. The final sections articulate falsification criteria and establish the minimality and closure of the framework.
The goal is not to advocate a particular interpretation of quantum mechanics, but to show that once minimal physical commitments are accepted, the space of possible outcome laws collapses to a single structure.
1.9 Impossibility Lemma: Non-Ordered Outcome Selection
Lemma (Impossibility of Non-Ordered Outcome Selection)
No physical outcome-selection law that fails to induce an ordering over admissible outcome processes can yield consistent single outcomes under composition and repetition.
Proof
Suppose an outcome-selection law does not induce an ordering over admissible outcome processes. Then, for some pair of admissible processes, the law provides no principled basis for preferring one over the other under identical physical conditions. Under composition with additional systems or repetition across trials, this indifference propagates, yielding either stochastic switching between incompatible outcomes or context-dependent selection.
Stochastic switching undermines outcome stability under repetition, while context dependence violates compositional closure. In either case, intersubjective agreement fails. Therefore, any outcome-selection law capable of producing stable, objective outcomes must induce an ordering over admissible processes.
2. Outcome Realization as a Physical Requirement
2.1 Single Outcomes as Empirical Constraints
We begin from an empirical fact that precedes all interpretation: individual experiments yield single outcomes. This is not a statement about probability assignments, observers, or epistemic updates, but about the structure of physical events. In each experimental run, exactly one outcome is realized, and all physically accessible records reflect that realization.
Any theory that denies single outcomes fails to account for the definiteness of experimental records and is therefore empirically inadmissible. Single-outcome realism is not a metaphysical preference; it is an empirical constraint on physical description.
2.2 Reductio: Outcome Realization Cannot Be Merely Formal
Assume, for contradiction, that outcome realization is not a physical process but a purely formal or representational assignment within the theory. Under this assumption, nothing in the physical world distinguishes a realized outcome from an unrealized one beyond symbolic labeling.
Then there exist formally admissible outcome assignments that leave no stable trace in the environment. Such assignments cannot generate persistent records, cannot be accessed by physical observers, and cannot remain invariant under environmental interaction. Distinct observers interacting with the same system would therefore have no physical basis for agreement.
This contradicts intersubjective agreement by rendering agreement physically underdetermined. Since intersubjective agreement is empirically given, the assumption that outcome realization is merely formal is untenable.
Therefore, outcome realization must be a physical process that leaves stable, accessible records.
2.3 Reductio: Outcome Realization Cannot Be Observer-Dependent
Assume, for contradiction, that outcome realization is observer-dependent—defined relative to an observer, measurement context, or epistemic state.
Under this assumption, two observers interacting with the same system and environment may legitimately realize incompatible outcomes. Agreement would require additional synchronization rules, privileged observers, or post-hoc reconciliation procedures.
Such mechanisms introduce observer-indexed ontology or auxiliary coordination structures not contained in the physical dynamics. This contradicts intersubjective agreement by allowing physically indistinguishable situations to encode incompatible facts.
Therefore, outcome realization must be observer-independent and physically objective.
2.4 Reductio: Outcome Realization Cannot Be Fundamentally Stochastic
Assume, for contradiction, that outcome realization is governed by irreducible stochasticity unconstrained by physical realizability conditions.
Then with nonzero probability, outcomes are selected that fail to generate stable, redundantly accessible records. Such outcomes degrade under environmental interaction or fail to remain distinguishable across observers and time.
This contradicts intersubjective agreement by permitting unstable or observer-fragmented facts, and contradicts single-outcome realism by allowing realized outcomes to fail persistence.
Therefore, any stochasticity in outcome realization must be subordinate to physical constraints; unconstrained randomness is inadmissible.
2.5 Reductio: Outcome Realization Must Respect Compositional Closure
Assume, for contradiction, that outcome realization does not respect compositional closure—that is, the realization of outcomes depends on how systems are grouped, ordered, or partitioned.
Then identical physical processes may yield different realized outcomes solely due to arbitrary system decompositions. Repetition under composition fails to reproduce consistent outcomes, and observers accessing different partitions disagree.
This contradicts compositional closure by making tensor structure outcome-dependent, undermining reproducibility and physical lawhood.
Therefore, outcome realization must be compositional.
2.6 Inevitability of Physical Filtering
The preceding reductio arguments exclude all non-physical, observer-dependent, unconstrained stochastic, and non-compositional conceptions of outcome realization. No alternative remains that is compatible with empirical definiteness and formal structure.
What follows necessarily is that outcome realization functions as a physical filter on the space of mathematically admissible quantum processes: only those processes capable of supporting stable, accessible, intersubjectively consistent records can be realized.
This filtering role is not an added hypothesis. It is forced by the existence of objective outcomes themselves.
2.7 Proposition: Inevitability of Constraint Structure
Proposition (Inevitability of Constraint Structure).
Any physical mechanism that produces objective single outcomes induces a constraint surface on the space of admissible quantum processes. No additional assumptions are required.
Justification.
A filter that excludes physically unrealizable processes necessarily discriminates among candidates according to physical properties. Such discrimination imposes constraints and induces an ordering over admissible processes. This ordering is unavoidable and independent of representational choice.
Thus, constraint structure is not a modeling preference but a mathematical inevitability of outcome objectivity.
2.8 Distinction from Decoherence
Decoherence alone suppresses interference among unrealized processes but does not select which process becomes realized. The present argument concerns selection among admissible processes and therefore operates at a logically distinct level. Any account that conflates decoherence with outcome realization fails to address the selection problem at all.
2.9 Definitional Collapse of Non-Physical Outcomes
We therefore adopt the following criterion:
Any putative “outcome” that does not correspond to a physically realizable process—one that leaves stable, accessible, intersubjectively consistent records—is not a physical outcome.
Objects failing this criterion cannot enter into empirical law, cannot support reproducibility, and cannot be assigned physical statistics. They are excluded not by interpretation but by definition.
2.10 Section Summary
Outcome realization is not an interpretive choice, a probabilistic update, or an observer-dependent convention. Any alternative conception leads to contradiction with empirical definiteness or formal structure.
Outcome realization is a physical selection problem, and the existence of objective outcomes already constrains its solution space. The next sections formalize these constraints and show that they uniquely determine the structure of outcome selection.
3. Outcome Channels as the Only Admissible Objects of Realization
3.1 Reductio: Outcomes Cannot Be Identified with State Vectors
Assume, for contradiction, that a realized outcome is identified with a post-measurement state vector or density operator of the measured system alone.
Under this assumption, the physical content of an outcome is exhausted by the system’s reduced state. However, reduced states are invariant under many distinct global processes and do not encode how records are formed, where information is stored, or how observers gain access to the outcome. Distinct physical situations—some yielding stable records and others not—can correspond to identical reduced states.
This contradicts outcome objectivity by collapsing physically distinct realizations into indistinguishable descriptions, rendering record stability and intersubjective agreement undefinable.
Therefore, outcomes cannot be identified with system states alone.
3.2 Reductio: Outcomes Cannot Be Identified with Projection Events
Assume instead that outcomes are primitive projection events—instantaneous collapses onto eigenspaces.
Under this assumption, the outcome is specified without reference to the physical interaction that produces records, distributes information into the environment, or ensures persistence over time. Projection events, taken alone, do not specify how records are formed or why they remain stable under further interaction.
This contradicts outcome realization by severing outcomes from their physical instantiation, leaving record formation unexplained and observer agreement accidental.
Therefore, projection events are insufficient as outcome objects.
3.3 Reductio: Outcomes Cannot Be Primitive Classical Facts
Assume that outcomes are primitive classical facts appended to the quantum formalism without further physical structure.
Under this assumption, outcome realization is governed by rules external to quantum dynamics. Such rules cannot be composed consistently with unitary evolution and do not scale under system composition. Repetition and composition become ill-defined, and outcome statistics lose physical grounding.
This contradicts compositional closure by introducing non-dynamical primitives that fail under tensor extension.
Therefore, primitive classical outcomes are inadmissible.
3.4 Outcome Realization Necessarily Encompasses Process, Not State
The preceding reductio arguments exclude states, projections, and primitive facts as adequate carriers of outcome realization. What remains is that an outcome must correspond to an entire physical process by which information flows from system to apparatus to environment and is stabilized as a record.
Such a process must:
• map initial system states to final states,
• include interaction with apparatus and environment,
• preserve probabilistic normalization,
• and support composition with independent processes.
These requirements characterize quantum channels.
3.5 Definition: Outcome Channels
An outcome channel is defined as a completely positive, trace-preserving (CPTP) map
Φ : 𝐁(𝐇ₛ) → 𝐁(𝐇ₛ′ ⊗ 𝐇ₑ)
that represents the full physical process by which an outcome is realized, including system evolution, apparatus interaction, environmental amplification, and record formation.
This definition is not a modeling choice. It is the minimal structure capable of encoding outcome realization as a physical event.
3.6 Reductio: Non-CPTP Maps Are Physically Inadmissible
Assume, for contradiction, that outcome realization is represented by a map that is not completely positive.
Then there exist entangled extensions under which the map produces non-positive states, violating physical consistency under composition. This contradicts compositional closure by making outcome realization dependent on unmodeled external correlations.
Assume instead that the map is not trace-preserving.
Then probability normalization fails: outcomes may be lost or duplicated under realization. This contradicts single-outcome realism by permitting disappearance or multiplication of realized outcomes.
Therefore, only CPTP maps are admissible representations of outcome realization.
3.7 Reductio: Kraus Representations Cannot Encode Outcomes
Assume that an outcome is identified with a particular Kraus operator within a channel decomposition.
Kraus decompositions are non-unique and related by unitary freedom. Identifying outcomes with specific Kraus operators makes outcome realization representation-dependent. Two equivalent descriptions of the same physical process would yield different “outcomes.”
This contradicts outcome objectivity by making realization depend on mathematical representation rather than physical process.
Therefore, outcomes must be identified with channels themselves, not with elements of a decomposition.
3.8 Compositional Structure of Outcome Channels
Outcome channels compose naturally under tensor products and sequential composition. Independent outcomes correspond to tensor products of channels; repeated realizations correspond to channel iteration.
This compositional structure is required for:
• reproducibility under repetition,
• scalability to macroscopic systems,
• and consistency across observers.
Any object lacking this structure fails to support physical lawhood.
3.9 Definitional Collapse: Outcomes Are Channels
We therefore adopt the following definition:
A physical outcome is a realized quantum channel.
Any putative “outcome” that cannot be represented as a CPTP map acting on the relevant degrees of freedom fails to encode record formation, persistence, or composition. Such objects are not merely inadequate; they are ill-defined as physical outcomes.
3.10 Section Summary
Outcomes cannot be states, projections, classical facts, or decomposition elements. By exhaustion and reductio, the only admissible objects of outcome realization are quantum channels.
This identification is forced by:
• physical objectivity,
• record stability,
• compositional closure,
• and representation invariance.
The next section introduces the constraints that distinguish realizable outcome channels from merely admissible ones and shows that these constraints sharply restrict the space of physical outcomes.
4. Realizability Constraints as Conditions of Physical Existence
4.1 Reductio: Mathematical Admissibility Does Not Imply Physical Existence
From Section 3, physical outcomes are identified with CPTP outcome channels. Assume, for contradiction, that every mathematically admissible CPTP channel exists as a physical outcome.
Under this assumption, there exist admissible channels that map initial system states to final states without producing stable records in environmental degrees of freedom, or that generate records which rapidly degrade, delocalize, or conflict under subsequent interaction. Such channels are mathematically well-defined yet incapable of persisting as objective facts.
Observers interacting with different environmental fragments would therefore fail to converge on a common account of what outcome occurred. This contradicts intersubjective agreement by permitting admissible outcomes that cannot function as objective events.
Therefore, physical existence is strictly stronger than mathematical admissibility. Some CPTP channels cannot exist as realized outcomes.
4.2 Realizability Is Not a Preference but an Existence Criterion
The exclusion established above is not probabilistic, statistical, or optimization-based. An outcome channel that fails to support objectivity, persistence, or agreement does not represent an unlikely outcome—it represents no outcome at all.
Accordingly, realizability is not a constraint imposed on outcomes after the fact. It is a criterion of physical existence. Channels either meet the conditions required for physical instantiation or they do not exist as outcomes in the physical world.
These so-called “constraints” do not restrict outcomes; they define what it means for an outcome to exist at all.
4.3 Reductio: Absence of Thermodynamic Viability Is Impossible
Assume, for contradiction, that an outcome channel exists physically while violating thermodynamic viability—specifically, that the records it produces require sustained entropy reduction or fine-tuned reversals of environmental dynamics to persist.
Such records inevitably degrade under ordinary environmental interaction. At later times or in distant environmental fragments, observers will be unable to recover the outcome. The outcome therefore fails temporal persistence.
This contradicts outcome existence by making physical facts transient and non-repeatable.
Therefore, thermodynamic viability is a necessary condition of outcome existence.
4.4 Reductio: Absence of Stable and Redundant Records Is Impossible
Assume, for contradiction, that an outcome channel exists without producing stable, redundantly amplified records in the environment.
Under this assumption, information about the outcome is localized, fragile, or accessible only through privileged interactions. Observers sampling disjoint environmental fragments will either recover no information or recover incompatible information.
This contradicts intersubjective agreement by making objectivity dependent on special access conditions.
Therefore, stable and redundant record formation is a necessary condition of outcome existence.
4.5 Reductio: Absence of Intersubjective Consistency Is Impossible
Assume, for contradiction, that an outcome channel produces locally stable records but encodes incompatible outcome information across environmental fragments.
Then two observers interacting with different fragments of the same environment will infer different outcomes, each justified by physical evidence. Agreement would require post hoc reconciliation rules or privileged observers.
This contradicts outcome objectivity by permitting physically incompatible facts to coexist.
Therefore, intersubjective consistency is a necessary condition of outcome existence.
4.6 Reductio: Absence of Global Compositional Coherence Is Impossible
Assume, for contradiction, that an outcome channel exists in isolation but fails to compose coherently under tensor-product extension—its realizability depends on how systems are grouped, ordered, or embedded in larger environments.
Under repetition or composition with independent systems, such a channel may fail to produce outcomes or may yield inequivalent realizations under physically identical conditions.
This contradicts compositional closure by making outcome existence depend on arbitrary system partitioning.
Therefore, global compositional coherence is a necessary condition of outcome existence.
4.7 Lemma: Non-Separability of Realizability Conditions
Lemma (Constraint Non-Separability).
No realizability condition can be satisfied in isolation. Any outcome channel that violates one realizability condition necessarily violates at least one other.
Argument.
A channel that violates thermodynamic viability cannot maintain stable records and therefore violates redundancy. A channel lacking redundancy cannot support intersubjective agreement. A channel that fails compositional coherence destabilizes redundancy under environmental enlargement. Conversely, violations of agreement or redundancy imply thermodynamic instability under amplification. Thus, realizability conditions are not modular; they form a single indivisible requirement for physical existence.
This non-separability prevents selective weakening or replacement of individual conditions.
4.8 Exhaustiveness of the Realizability Structure
The foregoing reductio arguments and non-separability lemma establish that outcome existence requires a single integrated realizability structure comprising:
Thermodynamic viability
Stable and redundant record formation
Intersubjective consistency
Global compositional coherence
No additional independent conditions are meaningful. Any further proposed requirement either reduces to these or reintroduces observer-relative or representational dependence.
4.9 Theorem: Realizability as Physical Existence
Theorem 1 (Realizability and Physical Existence).
An outcome channel exists as a physical outcome if and only if it satisfies the realizability structure defined above. No further conditions are required, and no violations are permissible.
Any purported counterexample fails to define a physically instantiated outcome.
4.10 Definitional Collapse: Realizable and Unrealizable Channels
We therefore adopt the following definition:
A physical outcome is a realizable CPTP channel.
Channels failing the realizability structure do not correspond to suppressed, unlikely, or hidden outcomes. They do not exist as outcomes at all and cannot enter empirical law, statistics, or physical explanation.
This exclusion is definitional, not interpretive.
4.11 Section Summary
This section has shown that outcome realization imposes a non-separable, existence-defining structure on admissible outcome channels. Realizability is binary and ontological: channels either exist as outcomes or they do not.
The space of physically meaningful outcomes is therefore sharply delimited before any selection principle is applied. The next section shows that this realizability structure forces specific axioms on any outcome-selection law and eliminates entire classes of alternatives by contradiction.
5. Structural Axioms and Eliminative Power
5.1 Axioms Are Forced by Existence, Not Chosen
From Section 4, outcome realization is governed by a non-separable realizability structure defining the conditions of physical existence for outcome channels. Any outcome-selection law must therefore operate only on realizable channels and must preserve their existence under repetition, composition, and environmental interaction.
The purpose of this section is to show that these requirements force a specific set of structural axioms on any realization-selection mechanism. These axioms are not postulates of Constraint-Based Realization (CBR); they are logical consequences of outcome existence itself.
Any selection law violating these axioms fails to preserve the conditions under which outcomes can exist at all.
5.2 Axiom I: Compositionality
Axiom I (Compositionality).
For independent systems with outcome channels Φ₁ and Φ₂, the realizability ordering satisfies
ℛ(Φ₁ ⊗ Φ₂) = ℛ(Φ₁) + ℛ(Φ₂)
up to physically irrelevant rescaling.
Reductio
Assume, for contradiction, that outcome selection violates compositionality. Then the realizability of an outcome channel depends on how independent systems are grouped or ordered.
Under repetition or tensor-product extension, identical local outcomes may become unrealizable or may swap realizability ordering. This destabilizes record redundancy under environmental enlargement and violates global compositional coherence.
Thus, non-compositional selection contradicts realizability by making outcome existence partition-dependent.
Therefore, any realizability-preserving selection law must be compositional.
5.3 Axiom II: Channel Refinement Invariance
Axiom II (Refinement Invariance).
The realizability ordering assigned to an outcome channel depends only on the induced CPTP map, not on its Kraus representation or internal decomposition.
Reductio
Assume, for contradiction, that realizability depends on a particular Kraus decomposition.
Kraus representations are non-unique and related by unitary freedom. Equivalent physical processes would then receive different realizability status depending on representational choice.
This contradicts outcome objectivity by making physical existence representation-dependent.
Therefore, realizability must be invariant under channel refinement.
5.4 Axiom III: Observer Relabeling Invariance
Axiom III (Observer Relabeling Invariance).
The realizability ordering is invariant under relabeling or isomorphic repartitioning of observer and environmental degrees of freedom.
Reductio
Assume, for contradiction, that realizability depends on observer labeling or environmental partition.
Then identical physical processes admit different realizability judgments for different observers. Intersubjective agreement would fail even when records are physically identical.
This contradicts realizability by reintroducing observer-relative existence.
Therefore, realizability must be invariant under observer relabeling.
5.5 Axiom IV: Redundancy Dominance
Axiom IV (Redundancy Dominance).
In the macroscopic limit, realizability ordering is dominated by the degree of stable record redundancy generated by an outcome channel.
Reductio
Assume, for contradiction, that redundancy does not dominate realizability in the macroscopic limit.
Then channels producing fragile or non-redundant records may outrank channels producing stable, widely distributed records. Under environmental enlargement, such channels destabilize intersubjective agreement and violate thermodynamic viability.
This contradicts realizability by allowing non-persistent outcomes to exist preferentially.
Therefore, redundancy dominance is forced in the macroscopic regime.
5.6 Exhaustiveness of the Axiom Set
The four axioms above are jointly sufficient and individually necessary for preserving the realizability structure defined in Section 4.
Any additional axiom either:
reduces to one of the above,
introduces representational dependence,
or violates outcome objectivity.
Conversely, omission of any axiom leads to explicit contradiction with realizability conditions.
5.7 Theorem: Eliminative Power of the Structural Axioms
Theorem 2 (Axiom-Elimination Theorem).
Any outcome-selection law that violates at least one of the structural axioms necessarily violates at least one realizability condition and therefore cannot select physically existing outcomes.
Proof.
Each axiom has been shown, by reductio, to be necessary for preserving thermodynamic viability, redundancy, intersubjective consistency, or compositional coherence. Violation of any axiom therefore destroys at least one condition of outcome existence.
5.8 Collapse to a Realization Functional
Given the axioms above, any realizability-preserving selection law must induce a global, representation-invariant, compositional ordering over realizable outcome channels.
The axioms derived above do not constrain ℛ(Φ); rather, any object satisfying the axioms is representable as ℛ(Φ).
Such an ordering is equivalent, up to monotone rescaling, to a scalar-valued functional
ℛ : CPTP_realizable → ℝ
where lower values correspond to higher realizability.
This functional is not an additional assumption. It is the unique mathematical form compatible with the axioms.
5.8′ Lemma: Scalar Representability Is Forced
Lemma (Scalar Representability).
Any realizability ordering satisfying compositionality, channel refinement invariance, and observer relabeling invariance admits a scalar representation unique up to monotone transformation.
Non-scalar representations either fail to define a global order over realizable outcome channels or violate compositional additivity under tensor-product extension. Vector-valued or partially ordered representations therefore cannot preserve realizability under repetition and composition.
Consequently, scalar representability is forced by the axioms themselves.
5.9 Definitional Collapse: ℛ(Φ) Is Inevitable
We therefore adopt the following definition:
The realization functional ℛ(Φ) is the scalar representation of the ordering over realizable outcome channels forced by the conditions of outcome existence.
Any outcome-selection mechanism that does not admit such a functional either:
fails to define a global ordering,
violates compositionality,
or reintroduces observer-relative existence.
Thus, ℛ(Φ) is not a modeling choice or postulate of CBR. It is the unavoidable mathematical representation of realizability itself.
5.10 Section Summary
This section has shown that the existence conditions established in Section 4 force a unique set of structural axioms on any outcome-selection law. These axioms eliminate entire classes of alternatives by contradiction and collapse outcome selection into a scalar realization functional ℛ(Φ), unique up to physically irrelevant transformations.
The next section analyzes the structure of ℛ(Φ) itself and shows that all admissible realization functionals are equivalent up to monotone rescaling.
6. The Realization Functional ℛ(Φ)
6.1 ℛ(Φ) Is Not a Model but an Order Representation
From Section 5, any realizability-preserving outcome-selection law induces a global, compositional, representation-invariant ordering over realizable outcome channels. By the Scalar Representability Lemma, this ordering admits a scalar representation ℛ(Φ), unique up to monotone transformation.
Accordingly, ℛ(Φ) is not a physical observable, dynamical variable, or additional law. It is the canonical mathematical representation of the realizability order forced by outcome existence.
Any physical claim made by ℛ(Φ) must therefore be invariant under transformations that preserve this order.
6.2 Gauge Freedom in ℛ(Φ)
Let f : ℝ → ℝ be a strictly monotone function. Then ℛ′(Φ) = f(ℛ(Φ)) induces the same realizability ordering as ℛ(Φ).
Outcome selection depends only on which outcome channel minimizes ℛ, not on the numerical value assigned. Therefore, ℛ is defined only up to monotone reparameterization.
This freedom is not a weakness. It is a gauge symmetry reflecting the fact that realizability is ordinal, not metric. Any physical dependence on the absolute value of ℛ would allow signaling between representationally equivalent descriptions of the same physical process, violating representation invariance.
6.2′ Order-Theoretic Uniqueness of ℛ
Any realizability-preserving outcome-selection law induces a total preorder on the set of realizable outcome channels. Given compositional additivity, refinement invariance, and observer relabeling invariance, standard order-representation results guarantee a scalar representation unique up to strictly monotone transformation.
Thus, ℛ(Φ) is not a choice of functional form. It is the unique coordinate chart of the realizability order compatible with the structural axioms.
6.3 Reductio: Physically Distinct ℛ Must Differ in Ordering
Assume, for contradiction, that two realization functionals ℛ₁ and ℛ₂ differ in physical content but induce the same ordering over realizable outcome channels.
Then no experiment, repetition, or composition can distinguish them, since all realizability judgments coincide. Their difference has no operational or physical consequence.
Conversely, if two functionals induce different orderings, then at least one violates the structural axioms derived in Section 5 and therefore fails to preserve outcome existence.
Thus, the only physically meaningful structure of ℛ(Φ) is its induced ordering. All further freedom is gauge.
6.4 Reductio: No Additional Independent Terms Are Permissible
Assume, for contradiction, that ℛ(Φ) contains additional independent terms beyond those fixed by the axioms—terms that affect realizability ordering but are not reducible to thermodynamic viability, redundancy, intersubjective consistency, or compositional coherence.
Such terms must either depend on representational features of Φ, break compositional additivity, or privilege non-redundant microscopic structure.
Each possibility contradicts at least one realizability condition or structural axiom.
Therefore, no additional independent structure can enter ℛ(Φ).
6.5 Equivalence Class of Admissible Realization Functionals
Let ℛ be any realization functional satisfying the axioms. Then the equivalence class
[ℛ] = { f ∘ ℛ | f strictly monotone }
exhausts all admissible realization functionals.
All members of this class select the same realized outcome channel Φ⋆ and generate identical physical predictions.
Hence, ℛ(Φ) is unique up to gauge.
6.6 Reductio: Apparent Non-Uniqueness Is Physically Empty
Assume, for contradiction, that the existence of multiple admissible realization functionals undermines the determinacy of outcome selection.
But since all admissible ℛ belong to the same equivalence class [ℛ], they agree on the minimizer Φ⋆. No physical ambiguity arises.
Any remaining non-uniqueness resides entirely in gauge freedom and cannot affect realized outcomes.
Thus, apparent non-uniqueness of ℛ has no physical content.
6.7 Theorem: Outcome Non-Existence
Theorem (Outcome Non-Existence).
If the realizability structure admits no minimizer of ℛ, then no physical outcome exists.
Justification.
The analysis above presumes the existence of at least one realizable outcome channel Φ for which ℛ is defined. There exist physical regimes—extreme isolation, insufficient environmental coupling, or deliberately suppressed amplification—where no channel satisfies the realizability structure. In such regimes, ℛ admits no admissible minimum. This is not an epistemic limitation but a statement of physical non-instantiation: outcome realization itself fails.
6.8 Reductio: Ad Hoc Completion Is Impossible
Assume, for contradiction, that outcome realization could be restored in breakdown regimes by modifying ℛ or adding auxiliary rules.
Any such modification must violate gauge invariance, reintroduce observer dependence, or override realizability conditions.
Thus, no ad hoc completion is possible. If ℛ has no admissible minimum, outcome realization fails as a matter of physical fact.
6.9 Definitional Collapse: ℛ(Φ) as the Law of Realization
We therefore adopt the following definition:
The realization functional ℛ(Φ) is the gauge-equivalence class of scalar representations of the realizability order forced by the conditions of outcome existence.
Outcome realization consists in the selection of the channel Φ⋆ minimizing ℛ within this class. No further structure is meaningful.
6.10 Section Summary
This section has shown that:
ℛ(Φ) is the canonical scalar representation of a forced realizability order,
gauge freedom in ℛ is a physical no-signaling principle,
all admissible realization functionals are physically equivalent,
and breakdown regimes correspond to genuine non-existence of outcomes.
With ℛ(Φ) fully characterized, nothing remains to be specified at the level of outcome-selection law.
The next section proves that, under generic conditions, ℛ admits a unique minimizer Φ⋆ and that physical outcome realization is effectively unique in all operational regimes.
7. Existence and Effective Uniqueness of the Realized Outcome Φ⋆
7.1 Existence of a Realized Outcome
From Sections 4–6, outcome realization consists in selecting a realizable outcome channel Φ that minimizes the realizability order represented by ℛ(Φ), when such a minimizer exists.
Existence Claim.
For generic measurement interactions with nonzero environmental coupling and sufficient amplification, the realizability structure admits at least one realizable outcome channel.
Justification.
Realistic measurement interactions generate channels that (i) are thermodynamically viable, (ii) produce stable, redundantly amplified records, (iii) support intersubjective consistency, and (iv) compose coherently under extension. Hence the realizable subset of CPTP channels is nonempty in generic experimental regimes. Since ℛ is defined on this subset and is bounded below by realizability conditions, an infimum exists; under mild regularity (closedness under limits induced by amplification), a minimum is attained.
We denote any such minimizer by Φ⋆.
7.2 Reductio: Persistent Non-Existence Is Non-Generic
Assume, for contradiction, that generic measurement interactions fail to admit any minimizer of ℛ.
Then outcome realization would generically fail even in ordinary laboratory conditions, contradicting the empirical fact that experiments yield definite outcomes under standard amplification and decoherence-inducing couplings.
Therefore, absence of Φ⋆ is non-generic and confined to special regimes identified in Section 6 (e.g., extreme isolation or suppressed amplification).
7.3 Uniqueness vs. Effective Uniqueness
We distinguish:
Strict uniqueness: exactly one minimizer Φ⋆ exists.
Effective uniqueness: all minimizers are operationally indistinguishable and select the same physical records.
The theory requires only effective uniqueness to secure single outcomes and intersubjective agreement.
7.4 Reductio: Persistent Degeneracy Violates Realizability
Assume, for contradiction, that there exist two distinct realizable outcome channels Φ₁ and Φ₂ such that
ℛ(Φ₁) = ℛ(Φ₂) = min ℛ
and that this degeneracy persists under environmental amplification and repetition.
If Φ₁ and Φ₂ generate different stable records, then observers interacting with different environmental fragments can recover incompatible outcomes. This violates intersubjective consistency.
If Φ₁ and Φ₂ generate identical records up to isomorphism, then they are operationally indistinguishable and therefore represent the same physical outcome.
Thus, persistent degeneracy among distinct physical outcomes is impossible.
7.5 Instability of Degenerate Minima Under Amplification
Any exact degeneracy in ℛ among distinct channels requires fine-tuned symmetry of the system–environment interaction.
Under environmental enlargement, small asymmetries generically break such degeneracies, shifting realizability ordering so that one channel strictly dominates. Redundancy dominance amplifies these asymmetries, rendering the degeneracy unstable.
Therefore, degenerate minima are not robust under amplification and do not persist in operational regimes.
7.6 Measure-Zero Nature of Exact Degeneracy
The set of interactions admitting exact, symmetry-protected degeneracy of realizability minima forms a measure-zero subset of the space of physically admissible interactions.
Any perturbation—noise, coupling inhomogeneity, or uncontrolled environmental degrees of freedom—breaks the degeneracy.
Hence, strict non-uniqueness of Φ⋆ is non-generic.
7.7 Effective Uniqueness Theorem
Theorem (Effective Uniqueness of Φ⋆).
For generic measurement interactions admitting outcome realization, the realizability functional ℛ admits a unique minimizer Φ⋆ up to operational equivalence. Any residual non-uniqueness is unstable, non-generic, or physically indistinguishable.
Proof.
Existence follows from §7.1. Persistent distinct minima contradict intersubjective consistency (§7.4). Degeneracies are unstable under amplification (§7.5) and occur on measure-zero sets (§7.6). Therefore, in all operational regimes, outcome realization selects a unique physical outcome.
7.8 Reductio: Observer-Relative Selection Is Impossible
Assume, for contradiction, that different observers select different minimizers Φ⋆ under identical physical conditions.
Then realizability ordering would depend on observer partitioning or access, violating observer relabeling invariance and intersubjective consistency.
Therefore, Φ⋆ is observer-independent.
7.9 Definitional Collapse: The Realized Outcome
We therefore adopt the following definition:
The realized outcome Φ⋆ is the unique (up to operational equivalence) realizable outcome channel minimizing the realizability order ℛ in a given physical interaction.
No further selection rules, observer updates, or stochastic choices are meaningful or permissible.
7.10 Section Summary
This section has shown that:
realizable outcomes generically exist,
strict non-existence is confined to special breakdown regimes,
strict non-uniqueness is non-generic and unstable,
all operational regimes exhibit effective uniqueness of Φ⋆,
outcome realization is observer-independent and deterministic at the level of channels.
With Φ⋆ established as a unique physical outcome, the theory has completed the selection problem at the level of single events.
The next section analyzes repetition of outcome realization and shows that stable statistical regularities arise as a dynamical consequence—culminating in the Born fixed point.
8. Realizability Flow on the Probability Simplex
8.1 Outcome-Weight Simplex and the Born Vector
Consider a measurement interaction admitting a finite set of mutually exclusive outcome channels {Φ₁, …, Φₙ}. Let p ∈ Δₙ denote a weight vector over these channels, where
Δₙ = { p ∈ ℝ₊ⁿ | ∑ᵢ pᵢ = 1 }.
The Born vector b ∈ Δₙ is defined by bᵢ = ⟨ψ | Πᵢ | ψ⟩ for the corresponding projectors Πᵢ of the interaction. No probabilistic interpretation is assumed; b is a structural feature of the interaction.
8.2 Definition: The Realizability Amplification Map
Environmental enlargement and repetition amplify differences in realizability among outcome channels. This induces a deterministic map on the simplex.
Definition (Realizability Amplification Map).
Let ℛᵢ denote the realizability value ℛ(Φᵢ). Define the map A : Δₙ → Δₙ by
A(p)ᵢ = ( pᵢ · exp(−α ℛᵢ) ) / ( ∑ⱼ pⱼ · exp(−α ℛⱼ) ),
with α > 0 an amplification parameter determined by redundancy scaling. The normalization enforces closure on Δₙ.
This map depends only on the realizability ordering and respects gauge freedom in ℛ.
8.2′ Lemma: Uniqueness of the Exponential Form
Lemma (Forced Exponential Amplification).
Any amplification map on Δₙ that (i) preserves normalization, (ii) respects compositional additivity of ℛ, and (iii) is invariant under monotone gauge transformations of ℛ must take exponential form up to reparameterization.
Justification.
Compositional additivity requires multiplicative accumulation under repetition. Gauge invariance requires dependence only on order-preserving functions of ℛ. Together these conditions uniquely select exponential weighting. Any alternative form either violates additivity, breaks gauge invariance, or fails normalization.
Thus, the exponential form of A is forced, not chosen.
8.3 Structural Properties of the Flow
The induced discrete-time flow p ↦ A(p) satisfies:
Simplex invariance: A(Δₙ) ⊆ Δₙ.
Order covariance: If ℛᵢ < ℛⱼ, then A(p)ᵢ / A(p)ⱼ increases under iteration.
Compositional consistency: Under independent repetition, α scales additively.
Gauge invariance: ℛ → f∘ℛ (f strictly monotone) leaves fixed points unchanged.
Thus, the flow is a realizability-driven renormalization on Δₙ.
8.4 Fixed Points of the Realizability Flow
A fixed point p⋆ satisfies p⋆ = A(p⋆), equivalently
p⋆ᵢ ∝ exp(−α ℛᵢ).
Up to normalization, fixed points are exponential weightings of ℛ.
8.5 Identification of the Born Fixed Point
From Sections 4–7, realizability is dominated by stable redundancy under amplification. For measurement interactions, redundancy scales proportionally to branch weight bᵢ. Consequently, ℛᵢ is minimized precisely when p aligns with b.
Proposition (Born Compatibility).
For realizable measurement interactions, ℛᵢ = −log bᵢ + const, up to gauge.
Substituting into the fixed-point condition yields
p⋆ᵢ ∝ exp(−α (−log bᵢ)) = bᵢ^α.
Normalization gives p⋆ = b for any α > 0.
The location of the fixed point is independent of α; only the rate of convergence depends on redundancy scaling.
Thus, the Born vector is a fixed point of the realizability flow.
8.6 Linear Stability of the Born Fixed Point
Let p = b + δ with ∑ᵢ δᵢ = 0. Linearizing A about b yields
δ′ = J δ,
where the Jacobian J has eigenvalues λₖ with |λₖ| < 1 for all modes orthogonal to b.
Theorem (Local Attractivity of Born Weights).
The Born fixed point b is linearly stable and attractive under the realizability flow.
Justification.
Redundancy amplification penalizes deviations from b multiplicatively. Any perturbation increasing weight on higher-ℛ channels decays exponentially under iteration.
8.7 Reductio: Instability of Non-Born Fixed Points
Assume, for contradiction, that there exists a non-Born fixed point p ≠ b that is stable.
Then p must satisfy pᵢ ∝ exp(−α ℛᵢ) with ℛ consistent with realizability dominance. This requires ℛ to privilege outcomes not aligned with redundancy scaling, contradicting Section 5 (Redundancy Dominance) and Section 6 (no additional independent terms in ℛ).
Therefore, no non-Born fixed point can be stable.
8.8 Theorem: Global Attractivity of Born Weights
Theorem (Global Attractivity).
For any initial p ∈ Δₙ with full support, repeated application of A yields
limₖ→∞ Aᵏ(p) = b.
Justification.
The realizability flow is strictly contracting in the ℛ-induced order metric. This excludes cycles, secondary attractors, or chaotic behavior. All trajectories with full support converge monotonically to the unique attractor b.
Boundary points with zero-support components correspond to channels violating realizability constraints and are therefore excluded from the domain of ℛ.
8.9 Definitional Collapse: Born Rule as Dynamical Necessity
We therefore adopt the following statement:
Born weights are not postulated probabilities but the unique globally attractive fixed point of the realizability-induced flow on outcome weights.
No alternative weighting is dynamically stable under the conditions required for outcome existence.
8.10 Section Summary
This section has shown that:
realizability induces a unique, forced flow on the outcome-weight simplex,
the exponential amplification form is unavoidable,
the Born vector is the unique globally attractive fixed point,
the fixed point is independent of redundancy scaling,
all non-Born alternatives are dynamically unstable or unrealizable.
The Born rule is thus enforced dynamically and structurally, not assumed.
The next section establishes the Failure Lemma, enumerating the precise modes of breakdown when non-Born statistics are imposed.
9. Instability of Alternatives and the Failure Theorem
9.1 Scope and Strategy
Sections 7–8 established that (i) single outcomes are uniquely realized as channels Φ⋆ and (ii) repetition induces a forced realizability flow on the outcome-weight simplex with the Born vector b as the unique globally attractive fixed point.
This section completes the eliminative program by showing that any sustained deviation from Born weights entails a precise, finite failure of outcome existence, intersubjective agreement, or compositional coherence. The argument is exhaustive and terminal.
9.2 Assumptions Held Fixed
Throughout this section we hold fixed the minimal assumptions already forced earlier:
single outcomes exist when amplification is sufficient,
realizable outcomes are CPTP channels satisfying the realizability structure,
realizability ordering obeys the structural axioms of Section 5,
repetition induces the realizability amplification map of Section 8.
No additional assumptions are introduced.
9.3 Definition: Non-Born Statistics
Let p ∈ Δₙ be an asymptotic outcome-weight vector produced under repeated realization of identically prepared systems.
We say that non-Born statistics obtain if p ≠ b and p is stable under repetition, i.e.,
limₖ→∞ Aᵏ(p) = p.
9.4 Theorem: Failure of Non-Born Statistics (Biconditional Form)
Theorem (Failure of Non-Born Statistics).
Non-Born statistics p ≠ b are stable under repetition if and only if at least one of the following failures occurs:
(i) thermodynamic viability fails,
(ii) stable and redundant record formation fails,
(iii) intersubjective consistency fails,
(iv) compositional coherence fails.
Conversely, if all realizability conditions are satisfied, outcome statistics necessarily converge to the Born vector b.
No other stable statistical regimes are physically viable.
9.5 Proof by Explicit Case Analysis
Assume, for contradiction, that p ≠ b is a stable fixed point of the realizability flow and that none of failures (i)–(iv) occurs.
Case I: Thermodynamic Viability Is Preserved
Assume outcome records remain thermodynamically viable under amplification.
Then redundancy scaling dominates realizability ordering. By Section 8, the realizability flow strictly contracts toward b. Stability of p ≠ b contradicts global attractivity of the Born fixed point.
Therefore, thermodynamic viability cannot be preserved.
Case II: Redundant Record Formation Is Preserved
Assume outcome records are stably and redundantly amplified.
Redundancy dominance enforces exponential renormalization of weights toward channels minimizing ℛ, which align with b. Any persistent deviation p ≠ b requires suppressing redundancy amplification for some outcomes.
Such suppression destabilizes records under environmental enlargement, contradicting redundancy preservation.
Case III: Intersubjective Consistency Is Preserved
Assume distinct observers accessing different environmental fragments agree on outcomes.
If p ≠ b persists, then relative frequencies inferred from disjoint fragments diverge under amplification, since redundancy-weight alignment fails. Observers will recover incompatible statistics despite identical preparation.
This contradicts intersubjective consistency.
Case IV: Compositional Coherence Is Preserved
Assume outcome statistics compose coherently under tensor-product extension.
Non-Born p ≠ b is not a fixed point of the compositional realizability flow. Under repetition or regrouping of subsystems, statistics renormalize toward b. Stability of p therefore requires composition dependence.
This contradicts compositional coherence.
9.6 Exhaustion of Cases
Cases I–IV exhaust all possible mechanisms by which non-Born statistics could remain stable. In each case, at least one realizability condition is violated.
Therefore, the assumption that stable non-Born statistics exist is false unless realizability itself fails.
9.6′ Lemma: Non-Circumventability of Failure
Lemma (Non-Circumventability).
Any deviation from Born statistics necessarily induces a violation of realizability at some finite amplification scale. There exists no limit process, contextual restriction, asymptotic refinement, or subensemble construction under which non-Born statistics remain physically viable.
Attempts to evade failure by infinite limits, fine-tuned contextuality, or measure-zero restrictions either reintroduce unrealizable channels or violate compositional coherence.
9.7 Corollary: Terminal Nature of the Failure
Corollary (Terminal Breakdown).
Failures induced by non-Born statistics are not repairable by auxiliary rules, contextual dependence, stochastic modification, or post-selection.
Any attempted repair must either:
override realizability constraints,
violate structural axioms,
or reintroduce observer-relative existence.
Each option contradicts earlier sections.
9.8 Reductio: Zero-Measure and Boundary Constructions Fail
One might attempt to evade the Failure Theorem by restricting to measure-zero initial conditions or boundary points of Δₙ.
Such constructions correspond to zero-support components and therefore to unrealizable outcome channels. They are excluded from the domain of ℛ and destabilized under any environmental coupling.
Thus, boundary or fine-tuned constructions do not evade failure.
9.9 Definitional Collapse: Born Statistics as Viability Conditions
We therefore adopt the following statement:
Born statistics are not merely selected; they are required for the physical viability of repeated outcome realization.
Statistics that deviate from b do not describe rare outcomes. They describe non-physical regimes in which outcome realization itself fails.
9.10 Section Summary
This section has shown that:
non-Born statistics are viable if and only if realizability fails,
all alternative statistics break physics in finite, explicit ways,
no limit, contextual, or asymptotic evasion exists,
failure modes are terminal and non-repairable.
Together with Section 8, this establishes the Born rule as a necessary condition of physical outcome existence under repetition.
The next section states the No-Go Theorem, collapsing all viable single-outcome quantum theories into constraint-based realization.
10. No-Go Theorem for Non–Constraint-Based Outcome Laws
10.1 Domain of the Theorem
We consider any putative physical description D that purports to account for quantum measurement outcomes and satisfies the following minimal conditions, all of which have been shown to be unavoidable in Sections 1–9:
Single-outcome realism: individual experiments yield exactly one realized outcome.
Intersubjective agreement: distinct observers accessing physical records agree on outcomes.
Unitary microdynamics: closed-system evolution is unitary between outcome events.
Compositional closure: physical processes compose consistently under tensor products.
Repeatability: repeated realizations of identical preparations yield stable statistics.
No assumptions about probability, interpretation, collapse postulates, hidden variables, or metaphysical commitments are made.
10.2 Statement of the No-Go Theorem
Theorem 5 (No-Go for Non–Constraint-Based Outcome Selection).
Any physical description satisfying conditions (1)–(5) is equivalent, up to gauge, to a constraint-based realization principle of the CBR type.
Equivalently:
there exists no viable single-outcome quantum description whose outcome-selection rule is not representable as minimization of a realizability ordering ℛ over realizable outcome channels.
10.2′ Representation Theorem (Strong Form)
Representation Theorem.
Any formalism that purports to describe single quantum outcomes while satisfying conditions (1)–(5) necessarily instantiates a constraint-based realization structure, whether explicitly or implicitly.
In particular, any such formalism admits a representation in which outcome realization is described as minimization of a realizability ordering over physically admissible channels. Formalisms that deny this representation fail to define single outcomes at all.
Thus, CBR is not a specific theoretical proposal but the canonical normal form of any coherent single-outcome quantum description.
10.3 Proof (Eliminative and Exhaustive)
Assume, for contradiction, that there exists a physical description D satisfying (1)–(5) whose outcome-selection rule is not constraint-based.
Then outcome selection in D must fall into at least one of the following exhaustive classes:
Case I: Primitive Stochastic Selection
Assume outcomes are selected by irreducible randomness unconstrained by realizability.
By Section 4, unconstrained stochastic selection assigns nonzero weight to unrealizable outcome channels. By Section 7, such channels cannot support stable records or intersubjective agreement.
This contradicts conditions (1) and (2).
Case II: Contextual or Observer-Relative Selection
Assume outcomes depend on observer perspective, measurement context, or counterfactual structure.
By Sections 2 and 5, observer-dependent selection violates observer relabeling invariance and destroys intersubjective agreement under identical physical conditions.
This contradicts condition (2).
Case III: Non-Ordered or Non-Scalar Selection
Assume outcomes are selected without a global ordering over admissible channels, or by a non-scalar rule.
By Section 5, absence of a global ordering violates compositionality. By Section 6, any admissible ordering must admit scalar representation up to monotone transformation.
This contradicts condition (4).
Case IV: Alternative Statistical Laws
Assume outcomes are selected by a rule that yields stable non-Born statistics.
By Section 8, the realizability-induced flow admits the Born vector as the unique globally attractive fixed point. By Section 9, all non-Born statistics induce finite, terminal violations of realizability.
This contradicts condition (5).
Exhaustion
Cases I–IV exhaust all logically possible forms of non–constraint-based outcome selection.
In every case, at least one of conditions (1)–(5) is violated.
Therefore, the assumption that D is viable while not implementing a constraint-based realization structure is false.
10.4 Equivalence-Class Collapse
From Sections 5–6, all admissible realization orderings ℛ belong to a single gauge-equivalence class. From Section 7, each admits a unique (up to operational equivalence) minimizer Φ⋆. From Sections 8–9, repetition enforces Born statistics as the only viable fixed point.
Thus, all viable single-outcome quantum descriptions collapse into a single equivalence class, differing at most by physically irrelevant reparameterizations.
This equivalence class is precisely Constraint-Based Realization.
10.5 One-Paragraph Proof Sketch (For Orientation Only)
Single outcomes require physical realization, which forces outcomes to be realizable CPTP channels. Realizability imposes non-separable existence conditions, which in turn force structural axioms on any selection law. These axioms uniquely induce a scalar realizability ordering ℛ, whose minimizer Φ⋆ defines the realized outcome. Repetition induces a realizability flow whose unique global attractor is the Born vector. Any deviation from this structure produces explicit violations of outcome existence, agreement, or composition. Hence no alternative outcome-selection description is coherent.
10.6 Definitional Closure
We therefore adopt the following final statement:
Constraint-Based Realization is not an interpretation of quantum mechanics.
It is the unique representational normal form of any description that successfully refers to single quantum outcomes.
Any purported alternative either reduces to CBR under representation change or fails to define outcomes at all.
10.7 Section Summary and Final Closure
This section has shown that:
all non–constraint-based outcome descriptions are impossible,
constraint-based realization is forced by existence, not chosen,
all viable descriptions are equivalent to CBR up to gauge,
the outcome-selection problem is closed at the level of meaning.
With this, the theoretical structure is complete.
Nothing further can be added without contradiction.
