Appendix 33 Quantum Correspondence Tests 33.1 Mapping to Hilbert Space Formalism 33.1.1 Objective This section establishes the requirement that BSFM admits a well-defined mapping into a Hilbert space structure in regimes where quantum mechanical behavior is observed. The purpose is to ensure compatibility with the standard mathematical framework of quantum theory. 33.1.2 Hilbert Space Embedding Requirement BSFM must allow a mapping such that: • each physical state S ∈ S corresponds to a vector |Ψ⟩ ∈ H • the set of admissible states maps into a complete inner-product space • superposition structure is preserved under state combination This implies the existence of a well-defined embedding: • Φ : S → H 33.1.3 Linearity Condition Within the mapped representation: • linear combinations of states must correspond to valid physical superpositions • evolution must be representable as a linear (or effectively linear) operator in the quantum regime • interference effects must be preserved under the mapping This ensures consistency with quantum mechanical structure. 33.1.4 Inner Product Consistency The mapping must define an inner product structure such that: • physical distinguishability corresponds to orthogonality relations • probability amplitudes are derived from inner product magnitudes • normalization conditions are preserved under evolution Formally, physical probabilities must be recoverable from: • ⟨Ψ|Ψ⟩ 33.1.5 State Completeness Requirement The mapped Hilbert space representation must be: • complete under limits of convergent sequences of states • closed under allowed linear operations in the quantum regime • sufficient to represent all physically distinguishable configurations This ensures no loss of physical information under mapping. 33.1.6 Branch-to-State Correspondence Within BSFM, branching structure must map consistently into quantum structure such that: • branches correspond to components of a superposed state • branch weights correspond to amplitude coefficients • coherence relations between branches map to phase structure This provides the structural link between BSFM evolution and quantum formalism. 33.1.7 Failure Condition The mapping to Hilbert space fails if: • no consistent vector space representation exists for BSFM states • inner product structure cannot be defined or preserved • branching cannot be represented as superposition without contradiction • linearity or normalization is violated in the quantum regime 33.1.8 Summary BSFM is compatible with quantum formalism only if its state structure can be consistently embedded into a Hilbert space where superposition, inner products, and linear evolution reproduce standard quantum mechanical behavior in the appropriate limit. Appendix 33 — Quantum Correspondence Tests 33.2 Unitarity and Norm Preservation Conditions 33.2.1 Objective This section establishes the requirement that BSFM must preserve total probability (norm) under evolution in the quantum correspondence regime. This ensures compatibility with unitary evolution in standard quantum mechanics and prevents loss or creation of probability mass. 33.2.2 Norm Definition Requirement In the Hilbert space representation: • each physical state |Ψ⟩ ∈ Hmust satisfy a well-defined norm • probability interpretation requires normalization of state vectors Formally: • ⟨Ψ|Ψ⟩ =1 for all physically valid states after normalization. 33.2.3 Unitary Evolution Condition For an evolution operator U acting in the quantum regime: • |Ψₙ₊₁⟩ = U|Ψₙ⟩ BSFM must ensure that: • U†U = 1 This guarantees that evolution preserves inner products between states and maintains probability consistency over time. 33.2.4 Norm Preservation Requirement Under BSFM evolution in the quantum correspondence regime: • total probability must remain invariant under iteration • branching weights (when mapped to amplitudes) must conserve norm globally • no net gain or loss of probability mass is permitted Formally: • ⟨Ψₙ₊₁|Ψₙ₊₁⟩ = ⟨Ψₙ|Ψₙ⟩ 33.2.5 Branch-Amplitude Consistency Condition If BSFM branching maps to quantum amplitudes: • the sum of squared amplitude magnitudes must remain constant • interference structure must not alter total probability • redistribution among branches is permitted only under unitary constraints Formally: • Σᵢ |cᵢ⁽ⁿ⁺¹⁾|² = Σᵢ |cᵢⁿ|² 33.2.6 Conservation of Probability Flow In dynamic evolution: • probability may redistribute across states or branches • but total probability within the Hilbert space must remain fixed • no leakage outside the admissible state space is allowed This ensures closed-system behavior in the quantum regime. 33.2.7 Non-Unitary Failure Detection Any deviation from unitary behavior indicates breakdown of quantum correspondence. This includes: • loss of normalization under evolution • growth or decay of total probability • non-invertible evolution operators in the quantum regime • inconsistent amplitude mapping from branching structure 33.2.8 Boundary Condition for Emergent Unitarity If unitarity is not fundamental in BSFM but emergent: • there must exist a regime where effective dynamics become unitary • non-unitary effects must be suppressed or canceled in that limit • deviations must vanish under appropriate coarse-graining or symmetry conditions 33.2.9 Summary BSFM is consistent with quantum mechanics only if its mapped evolution preserves norm and satisfies unitary dynamics in the Hilbert space representation, ensuring conservation of probability and reversible evolution in the quantum correspondence regime. Appendix 33 — Quantum Correspondence Tests 33.3 Superposition and Interference Consistency 33.3.1 Objective This section establishes that BSFM must reproduce the defining quantum features of superposition and interference when mapped into its Hilbert space correspondence. The requirement is that linear combination of states and phase-dependent effects must emerge consistently from the underlying branching structure. 33.3.2 Superposition Principle Requirement In the quantum correspondence regime, BSFM must allow state combinations such that: • if |Ψ₁⟩ and |Ψ₂⟩ are valid states, then a|Ψ₁⟩ + b|Ψ₂⟩ • is also a valid physical state where a, b ∈ ℂ This ensures closure under linear combination within the mapped Hilbert space structure. 33.3.3 Branch-State Correspondence In BSFM terms: • individual branches correspond to components of a composite state • branching structure maps to decomposition of a superposed state • branch weights correspond to amplitude magnitudes and phases The mapping must preserve distinguishability and recombination behavior of branches. 33.3.4 Interference Requirement BSFM must reproduce interference effects such that: • combined amplitudes produce non-classical probability distributions • phase relationships between components affect observable outcomes • constructive and destructive interference emerge naturally from the structure Formally, observable probabilities must depend on: • cross terms of amplitude combinations, not only individual contributions 33.3.5 Coherence Preservation Condition For interference to remain valid: • phase relationships between components must be preserved under evolution • coherence between branches must not collapse prematurely in the quantum regime • decoherence must only emerge under classical-limit conditions (Appendix 32 regime boundary) This ensures correct separation between quantum and classical behavior. 33.3.6 Non-Additive Probability Structure BSFM must reproduce the quantum rule that: • probabilities are not simply additive over branches • interference terms modify total outcomes • classical probability emerges only when coherence is lost This distinguishes quantum behavior from classical stochastic aggregation. 33.3.7 Consistency Requirement The mapping must ensure: • superposition in Hilbert space corresponds consistently to branching combinations in BSFM • interference patterns remain invariant under valid transformations • no hidden classical averaging removes quantum phase dependence 33.3.8 Failure Condition Superposition and interference consistency fails if: • linear combinations of states are not preserved in the mapping • interference terms cannot be recovered from branch structure • probabilities reduce to purely additive classical form in the quantum regime • phase information is lost or undefined under evolution 33.3.9 Summary BSFM is consistent with quantum superposition only if its branching structure maps into a linear state space where phase-dependent interference effects are preserved, producing non-classical probability behavior consistent with standard quantum mechanics. Appendix 33 — Quantum Correspondence Tests 33.4 Path Integral Equivalence Validation 33.4.1 Objective This section establishes the requirement that BSFM must reproduce, in an appropriate limit, the structure and predictions of the path integral formulation of quantum mechanics. The goal is to ensure equivalence between BSFM branching evolution and summation over histories. 33.4.2 Path Integral Correspondence Requirement In the quantum regime, BSFM must admit an equivalence such that: • the total amplitude for transition between states is given by a sum over all allowed evolution histories • each history contributes with a well-defined weight and phase structure • the full evolution is recoverable as an aggregate over all admissible paths Formally, BSFM must support a mapping consistent with: • “sum over histories” structure in the quantum correspondence limit 33.4.3 Branch-History Mapping Condition BSFM branching must correspond to path decomposition such that: • each branch corresponds to a distinct dynamical history • the full set of branches represents the complete set of allowed paths • recombination of branches reproduces full transition amplitudes This ensures structural eq
Appendix 33 Quantum Correspondence Tests 33.1 Mapping to Hilbert Space Formalism 33.1.1 Objective This section establishes the requirement that BSFM admits a well-defined mapping into a Hilbert space structure in regimes where quantum mechanical behavior is observed. The purpose is to ensure compatibility with the standard mathematical framework of quantum theory. 33.1.2 Hilbert Space Embedding Requirement BSFM must allow a mapping such that: • each physical state S ∈ S corresponds to a vector |Ψ⟩ ∈ H • the set of admissible states maps into a complete inner-product space • superposition structure is preserved under state combination This implies the existence of a well-defined embedding: • Φ : S → H 33.1.3 Linearity Condition Within the mapped representation: • linear combinations of states must correspond to valid physical superpositions • evolution must be representable as a linear (or effectively linear) operator in the quantum regime • interference effects must be preserved under the mapping This ensures consistency with quantum mechanical structure. 33.1.4 Inner Product Consistency The mapping must define an inner product structure such that: • physical distinguishability corresponds to orthogonality relations • probability amplitudes are derived from inner product magnitudes • normalization conditions are preserved under evolution Formally, physical probabilities must be recoverable from: • ⟨Ψ|Ψ⟩ 33.1.5 State Completeness Requirement The mapped Hilbert space representation must be: • complete under limits of convergent sequences of states • closed under allowed linear operations in the quantum regime • sufficient to represent all physically distinguishable configurations This ensures no loss of physical information under mapping. 33.1.6 Branch-to-State Correspondence Within BSFM, branching structure must map consistently into quantum structure such that: • branches correspond to components of a superposed state • branch weights correspond to amplitude coefficients • coherence relations between branches map to phase structure This provides the structural link between BSFM evolution and quantum formalism. 33.1.7 Failure Condition The mapping to Hilbert space fails if: • no consistent vector space representation exists for BSFM states • inner product structure cannot be defined or preserved • branching cannot be represented as superposition without contradiction • linearity or normalization is violated in the quantum regime 33.1.8 Summary BSFM is compatible with quantum formalism only if its state structure can be consistently embedded into a Hilbert space where superposition, inner products, and linear evolution reproduce standard quantum mechanical behavior in the appropriate limit. Appendix 33 — Quantum Correspondence Tests 33.2 Unitarity and Norm Preservation Conditions 33.2.1 Objective This section establishes the requirement that BSFM must preserve total probability (norm) under evolution in the quantum correspondence regime. This ensures compatibility with unitary evolution in standard quantum mechanics and prevents loss or creation of probability mass. 33.2.2 Norm Definition Requirement In the Hilbert space representation: • each physical state |Ψ⟩ ∈ Hmust satisfy a well-defined norm • probability interpretation requires normalization of state vectors Formally: • ⟨Ψ|Ψ⟩ =1 for all physically valid states after normalization. 33.2.3 Unitary Evolution Condition For an evolution operator U acting in the quantum regime: • |Ψₙ₊₁⟩ = U|Ψₙ⟩ BSFM must ensure that: • U†U = 1 This guarantees that evolution preserves inner products between states and maintains probability consistency over time. 33.2.4 Norm Preservation Requirement Under BSFM evolution in the quantum correspondence regime: • total probability must remain invariant under iteration • branching weights (when mapped to amplitudes) must conserve norm globally • no net gain or loss of probability mass is permitted Formally: • ⟨Ψₙ₊₁|Ψₙ₊₁⟩ = ⟨Ψₙ|Ψₙ⟩ 33.2.5 Branch-Amplitude Consistency Condition If BSFM branching maps to quantum amplitudes: • the sum of squared amplitude magnitudes must remain constant • interference structure must not alter total probability • redistribution among branches is permitted only under unitary constraints Formally: • Σᵢ |cᵢ⁽ⁿ⁺¹⁾|² = Σᵢ |cᵢⁿ|² 33.2.6 Conservation of Probability Flow In dynamic evolution: • probability may redistribute across states or branches • but total probability within the Hilbert space must remain fixed • no leakage outside the admissible state space is allowed This ensures closed-system behavior in the quantum regime. 33.2.7 Non-Unitary Failure Detection Any deviation from unitary behavior indicates breakdown of quantum correspondence. This includes: • loss of normalization under evolution • growth or decay of total probability • non-invertible evolution operators in the quantum regime • inconsistent amplitude mapping from branching structure 33.2.8 Boundary Condition for Emergent Unitarity If unitarity is not fundamental in BSFM but emergent: • there must exist a regime where effective dynamics become unitary • non-unitary effects must be suppressed or canceled in that limit • deviations must vanish under appropriate coarse-graining or symmetry conditions 33.2.9 Summary BSFM is consistent with quantum mechanics only if its mapped evolution preserves norm and satisfies unitary dynamics in the Hilbert space representation, ensuring conservation of probability and reversible evolution in the quantum correspondence regime. Appendix 33 — Quantum Correspondence Tests 33.3 Superposition and Interference Consistency 33.3.1 Objective This section establishes that BSFM must reproduce the defining quantum features of superposition and interference when mapped into its Hilbert space correspondence. The requirement is that linear combination of states and phase-dependent effects must emerge consistently from the underlying branching structure. 33.3.2 Superposition Principle Requirement In the quantum correspondence regime, BSFM must allow state combinations such that: • if |Ψ₁⟩ and |Ψ₂⟩ are valid states, then a|Ψ₁⟩ + b|Ψ₂⟩ • is also a valid physical state where a, b ∈ ℂ This ensures closure under linear combination within the mapped Hilbert space structure. 33.3.3 Branch-State Correspondence In BSFM terms: • individual branches correspond to components of a composite state • branching structure maps to decomposition of a superposed state • branch weights correspond to amplitude magnitudes and phases The mapping must preserve distinguishability and recombination behavior of branches. 33.3.4 Interference Requirement BSFM must reproduce interference effects such that: • combined amplitudes produce non-classical probability distributions • phase relationships between components affect observable outcomes • constructive and destructive interference emerge naturally from the structure Formally, observable probabilities must depend on: • cross terms of amplitude combinations, not only individual contributions 33.3.5 Coherence Preservation Condition For interference to remain valid: • phase relationships between components must be preserved under evolution • coherence between branches must not collapse prematurely in the quantum regime • decoherence must only emerge under classical-limit conditions (Appendix 32 regime boundary) This ensures correct separation between quantum and classical behavior. 33.3.6 Non-Additive Probability Structure BSFM must reproduce the quantum rule that: • probabilities are not simply additive over branches • interference terms modify total outcomes • classical probability emerges only when coherence is lost This distinguishes quantum behavior from classical stochastic aggregation. 33.3.7 Consistency Requirement The mapping must ensure: • superposition in Hilbert space corresponds consistently to branching combinations in BSFM • interference patterns remain invariant under valid transformations • no hidden classical averaging removes quantum phase dependence 33.3.8 Failure Condition Superposition and interference consistency fails if: • linear combinations of states are not preserved in the mapping • interference terms cannot be recovered from branch structure • probabilities reduce to purely additive classical form in the quantum regime • phase information is lost or undefined under evolution 33.3.9 Summary BSFM is consistent with quantum superposition only if its branching structure maps into a linear state space where phase-dependent interference effects are preserved, producing non-classical probability behavior consistent with standard quantum mechanics. Appendix 33 — Quantum Correspondence Tests 33.4 Path Integral Equivalence Validation 33.4.1 Objective This section establishes the requirement that BSFM must reproduce, in an appropriate limit, the structure and predictions of the path integral formulation of quantum mechanics. The goal is to ensure equivalence between BSFM branching evolution and summation over histories. 33.4.2 Path Integral Correspondence Requirement In the quantum regime, BSFM must admit an equivalence such that: • the total amplitude for transition between states is given by a sum over all allowed evolution histories • each history contributes with a well-defined weight and phase structure • the full evolution is recoverable as an aggregate over all admissible paths Formally, BSFM must support a mapping consistent with: • “sum over histories” structure in the quantum correspondence limit 33.4.3 Branch-History Mapping Condition BSFM branching must correspond to path decomposition such that: • each branch corresponds to a distinct dynamical history • the full set of branches represents the complete set of allowed paths • recombination of branches reproduces full transition amplitudes This ensures structural eq
Admissibility Physics — Quantum Structure from Finite Enforceability (Paper 5) This Zenodo community collects research outputs associated with Admissibility Physics, with primary focus on Paper 5 — Quantum Structure from Finite Enforceability and the derivation of quantum mechanics as an admissible representational regime. Paper 5 shows that the mathematical structure of quantum mechanics—Hilbert space, linearity, inner products, tensor products, unitary evolution, CPTP maps, and the Born rule—is not postulated, but forced by admissibility once coherent, jointly enforceable distinctions exist under finite correlation capacity. Quantum structure emerges as the unique bookkeeping required to represent interference, refinement, and recombination while preserving admissibility. The derivation does not assume probability, measurement postulates, dynamics, or spacetime geometry. Probability enters only as a secondary representation, with the Born rule arising uniquely from admissibility invariance and refinement consistency. Measurement is treated as a regime transition associated with saturation and record formation, not as a fundamental collapse mechanism. Quantum mechanics is shown to be conditionally valid: exact in regimes where coherence and non-saturation permit admissible composition, and structurally inapplicable where capacity is exhausted. Monogamy, entanglement limits, and decoherence follow from non-closure under composition and finite interface capacity, rather than from quantum axioms. All subsequent works in this community rely on the quantum-admissible representation established in Paper 5. Later papers show how variational dynamics and spacetime geometry arise as further conditional representations layered on top of quantum structure, and how quantum description itself fails at saturation boundaries, horizons, and measurement-like events. This community serves as a curated, citable archive centered on the derivation and limits of quantum structure within Admissibility Physics. Paper 5 functions as the representational keystone of the program: the point at which finite enforceability yields the quantum formalism without postulation. Core papers are frozen where noted; companion papers, mathematical appendices, figures, and computational artifacts document derivations, regime conditions, and testable consequences.
This work presents a foundational derivation of the Born probability rule of quantum mechanics. Instead of positing the quadratic probability measure (P=∣Ψ∣2P=∣Ψ∣2) as an axiom, this framework demonstrates its emergence from a set of physically motivated principles governing the composition of amplitudes on spacetime regions. The core axioms are: Spacetime Gluing (Functoriality): Amplitudes for adjacent spacetime regions compose via integration over their shared boundary, in a manner consistent with the Feynman path integral. Boundary States: Quantum states are fundamentally defined as complex functionals on the boundaries (or slices) of spacetime regions. Spacelike Factorization: The Hilbert space of a composite system of spacelike-separated regions is the tensor product of the individual Hilbert spaces, ensuring locality. Unitary Invariance: Probabilities are invariant under unitary transformations of the state space. It is shown that these axioms, particularly the requirement of consistency under different choices of spacelike slicing (foliation-invariance), uniquely select the standard inner product norm ⟨ψ∣ψ⟩⟨ψ∣ψ⟩ as the only valid, continuous, and multiplicative probability functional. This approach reframes the probabilistic nature of quantum mechanics as an intrinsic consequence of spacetime's geometric and causal structure, bridging concepts from quantum field theory and the General Boundary Formulation. This document is submitted to Zenodo to establish a permanent, citable, and timestamped record of this specific derivation, thereby planting a flag for the principle of deriving quantum probabilities from first principles of spacetime composition.
The mathematical structure of quantum theory is usually postulated rather than explained—Hilbert spaces, operators, and probabilistic structure—without explaining why this arena should exist in the first place. In this work we address a more primitive question: starting from the absence of any prior structure, what conditions must be satisfied for coherent persistence to be possible at all? We begin from a state of zero determination (the Void), understood not as physical emptiness but as the absence of admissibility constraints. From this starting point, we show that the possibility of persistence forces a minimal triadic structure consisting of reference, transformation, and return. This structure enables self-verification without external supervision and distinguishes genuine continuity from accidental resemblance. The requirement of self-verifying persistence forces deviation to be representable quantitatively and to compose across histories. This necessity leads uniquely to a linear vector space structure. Stability further imposes a knife-edge viability condition: identity-preserving and identity-violating contributions must equalize at closure. From this condition, an inner product and norm are forced, yielding Hilbert space geometry and operator representations as consequences of viability rather than axioms. Within this framework, familiar features of quantum kinematics—such as uncertainty, unitary evolution, and probabilistic weighting—arise as structural necessities rather than postulated rules. The present work establishes the kinematic foundations of coherence, deriving the mathematical arena of quantum theory from the requirement that something persist at all. Applications to mass relations, symmetry, temporal ordering, and composite systems are developed separately.Version 1.0. This preprint establishes the structural foundations of Coherence Algebra and derives Hilbert space from self-verification and persistence.
Quantum mechanics represents physical states through complex amplitudes, state vectors, and density operators, yet the ontological status of these objects remains interpretation-dependent. This paper develops a closure-based formulation in which a quantum state is interpreted as a representation of contextually partial relational closure: the state possesses definite mathematical and relational structure while remaining unresolved relative to one or more observational contexts. Within this language, decoherence differentiates closure channels and redistributes locally accessible coherence into relational structure; environmental redundancy stabilizes records; and classical facts emerge as approximately fixed, effectively commuting, redundantly recorded structures. Measurement is separated into three problems—context, weight, and outcome—so that decoherence, Born weighting, and unique outcome selection are not conflated. The paper then distinguishes two stronger research programs from the exact reformulation. First, it asks whether Hilbert-space quantum mechanics can be reconstructed from primitive closure, incompatibility, composition, and consistency principles. Second, it formulates a falsifiable closure-history extension in which physically relevant information might exist beyond the present density operator. Quantum Partial Closure Theory is therefore presented as a relational interpretation and information calculus, a reconstruction program, and a prospective experimental framework rather than as an established replacement for quantum mechanics. Keywords: quantum foundations; closure; wavefunction ontology; coherence; entanglement; decoherence; measurement; classical emergence; quantum information; Hilbert-space reconstruction The wavefunction question Quantum theory is operationally clear about what a wavefunction does. For a normalized pure-state vector ψ∈H and a projective measurement with alternatives {Π_i}, the Born weights are p_i=Tr(ρ_ψ Π_i ), ρ_ψ=ψψ^†. For a rank-one outcome Π_i=e_i e_i^†, this reduces to p_i=|e_i^† ψ|^2. Yet the mathematical clarity of the formalism does not by itself settle what the state vector is ontologically. Interpretations of quantum theory differ over whether the state should be regarded as physical, epistemic, relational, branch-relative, dispositional, or otherwise. The present paper proposes a specific interpretive and reconstructive language: partial relational closure. The central claim is not that the wavefunction is literally a new material substance called closure. Rather, ▭(ψ "is interpreted as a Hilbert-space representation of coherent partial relational closure" .) The word partial is crucial. It does not mean that a state is a scalar fraction of being real or complete. It means that information and constraint are distributed across local, coherent, and relational sectors, and that closure is always specified relative to a context. A Bell state, for example, can be locally indefinite while being relationally exact. A decohered state can be closed with respect to one effective macroscopic context while retaining a larger quantum embedding. A selective outcome can be locally definite without implying that every larger-scale relation in the universe is globally closed. This formulation is motivated by three observations. First, projective dephasing is naturally closure-like because it is idempotent. Second, modern quantum information already supplies exact measures for local coherence, correlations, and record redundancy. Third, the quantum-to-classical transition is naturally relational: local coherence can disappear from a subsystem while information migrates into system-environment correlations. The paper has three epistemic layers. Layer A uses established quantum mathematics. Layer B derives exact information identities and interprets them in closure language. Layer C states reconstruction conjectures and an explicitly speculative beyond-QM closure-history hypothesis. These layers are kept separate throughout. A quantum system is not structureless before measurement. Its amplitudes, phases, conservation relations, and correlations are tightly organized. What remains unresolved is how that structure will become locally definite relative to a chosen measurement context. The partial-closure view therefore distinguishes three broad regimes: Coherent partial closure. Alternatives remain phase-related and can interfere. Contextual differentiation. Interaction or dephasing suppresses coherence between alternatives relative to a context without necessarily choosing one. Selective local closure. A particular outcome is conditioned or realized relative to the measurement relation. In this language, wave-like behavior corresponds to spatially or otherwise contextually unresolved coherent closure, while particle-like detection corresponds to localized closure. Entanglement is closure stored in relations rather than in isolated subsystems. Decoherence is not simply destruction: in a globally pure description it can be a transfer of local coherence into relational entanglement. Classical objectivity appears when relevant records become stable and redundantly accessible. The framework does not assign consciousness a privileged dynamical role. An observer is a partial closure process for constraining possibilities. Explainer video Watch on YouTube (Roots of Reality): https://www.youtube.com/watch?v=UYx_omk3h3A The video, its thumbnail and description are included in this record.
ABSTRACT Mathematical expressibility is a necessary but not sufficient condition for physical admissibility. A model may be written in well-defined symbols, embedded in formal geometry, or rendered algebraically consistent within a selected framework, and yet fail as physics if it entails unobserved violations of conservation, causal ordering, entropy, memory, measurement stability, or ordinary spacetime structure. This paper proposes the Structural Reality Compatibility Test (SRCT) as a general methodological constraint on proposed physical theories. The test requires that any posited hidden structure — whether additional temporal dimensions, branching ontologies, nonlocal pilot structures, collapse mechanisms, hidden variables, higher-dimensional manifolds, or purely formal histories — must account for the phenomenon it is introduced to explain without producing consequences inconsistent with the established behavior of the observable world. The paper first states the SRCT in plain terms and then develops it formally through scalar-field conservation, Lorentz invariance, Noether symmetry, and the Einstein field equations. Noether's theorem is treated as central: continuous symmetries of the action correspond to conserved quantities, with temporal-translation symmetry grounding energy conservation. Extra-temporal models, many-world branching, and related exotic structures are evaluated under the test. The difficulty is not that such models are mathematically unwritable; it is that they must give a structurally coherent account of all consequences of their additional structures. A second temporal dimension, for example, must explain why conservation, causal order, entropy, memory, and observable time evolution remain stable despite the enlarged temporal symmetry structure. A branching-world ontology must give a physically meaningful account of branching without invoking unobserved thermodynamic or causal consequences. The Information Boundary Principle (IBP) and Minimum Informational Boundary (MIB) are then considered as an example of a theoretical framework attempting to satisfy the SRCT. In that model, nonlocal quantum correlation is treated not as superluminal signaling, extra-temporal communication, or world multiplication, but as a projection-limited manifestation of higher-dimensional geometric convergence constrained by informational compatibility conditions. The point is not that mathematical novelty is forbidden. The point is that new structure must be reality-compatible. The burden of a proposed theory is not merely to explain the targeted anomaly. It must explain the anomaly without breaking the world. Modern theoretical physics has repeatedly advanced by allowing mathematics to reveal structures not obvious to ordinary perception. Relativity required a new geometry of spacetime. Quantum mechanics required a formalism in which probability amplitudes, operators, Hilbert space, and measurement outcomes replaced older mechanical pictures. Gauge theories, curved manifolds, path integrals, and field equations have all reinforced the same lesson: intuition alone cannot govern fundamental physics. The reverse error is, however, equally dangerous. A mathematical structure does not become physically real merely because it can be written down. Formal systems are permissive; physical reality is constrained. A proposed physical theory must do more than resolve an isolated equation or render a single anomaly formally tractable. It must remain compatible with the observed structure of the world. It must preserve conservation behavior, causal ordering, entropy, memory, measurement stability, and non-signaling unless it provides a precise and empirically adequate account of why those structures are modified. A theory that explains one phenomenon by introducing consequences that are nowhere observed has not solved the problem. It has relocated the problem into formalism. This distinction is particularly important in quantum foundations. Quantum entanglement presents a genuine explanatory challenge. The observed correlations between entangled systems are not reproducible through ordinary local separability. Bell-type results establish that the classical combination of locality and certain realist assumptions cannot recover quantum predictions. The natural theoretical response is to introduce hidden structure: branching worlds, nonlocal pilot waves, retrocausal influence, extra temporal dimensions, observer-relative facts, or higher-dimensional geometry. Some of these proposals are mathematically sophisticated. Some may prove partially correct. But all must satisfy the same basic requirement: they must account for the quantum anomaly without producing unobserved consequences across the rest of observable physics. This paper calls that requirement the Structural Reality Compatibility Test (SRCT). The test may be stated in its simplest form as follows: "A proposed physical theory must not merely be mathematically expressible. It must preserve the observed stability of physical reality unless it provides a precise, testable, and structurally coherent account of why that stability continues to appear preserved."
This series presents a structured and logically sequential formulation of a discrete informational framework governed by global coherence constraints. Although it originates from the same research program developed in the earlier exploratory preprint series (P-series), the works presented here constitute an autonomous and self-contained series, organized in a progressively axiomatic form and focused on the explicit derivation of the framework’s effective structures. The goal of the construction is to investigate whether structures commonly associated with quantum mechanics, thermodynamics, and geometry may emerge from a discrete relational substrate without being introduced as fundamental postulates. Each article addresses a specific level of the theoretical derivation and forms part of a progressive logical structure based on results established in the preceding works of the series. This article develops the emergence of Hilbert-space structure and Schrödinger-type effective dynamics from the coherent two-field sector established in Paper 02. The work demonstrates that the global coherence constraints imposed on the discrete informational substrate naturally generate an effective complex representation endowed with linear superposition properties and unitary-preserving evolution. The analysis shows that the two-component coherent sector admits a mathematically stable reformulation equivalent to a Hilbert-space structure, and that the effective evolution of the coherent amplitudes leads to a Schrödinger-type dynamical equation without introducing quantum mechanics as an initial assumption. The article further derives the probabilistic interpretation associated with the effective amplitudes and clarifies how quantum-like statistical behavior emerges from the internal coherence conditions of the underlying discrete system. No quantum postulates are assumed at the foundational level. The purpose of the article is to demonstrate that the formal structures associated with quantum mechanics can emerge as effective consequences of coherence-preserving informational dynamics. The article does not assume quantum mechanics as a foundational theory and does not postulate the Hilbert-space formalism a priori. Instead, the work demonstrates that, within the internal hypotheses and coherence constraints of the proposed discrete informational framework, structures mathematically equivalent to Hilbert-space representations and Schrödinger-type evolution can arise as effective emergent descriptions. The purpose of the derivation is therefore not to reproduce standard quantum mechanics by assumption, but to show that quantum-like formal structures may consistently emerge from deeper coherence-preserving informational dynamics.