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Logic Realism Theory: Non-Decomposability, Entanglement, and Bell's Theorem Derived from A_Ω = L₃(I∞)

James (JD) Longmire

Published Mar 11, 2026• 0 citations

Abstract

This working paper derives quantum non-decomposability, Bell's theorem, and Kochen-Specker contextuality from Logic Realism Theory's single foundational equation A_Ω = L₃(I∞), where I∞ is the Infinite Information Space, L₃ is the Three Fundamental Laws of Logic acting as a selection functional, and A_Ω is the actualized domain. The paper establishes that LRT is a one-equation theory: all results are downstream of the core equation with no independent primitives introduced. Key results: (1) the Unique Necessary Successor (UNS) constraint is derived as a theorem of L₃ via Identity and Excluded Middle, tightened by Leibniz Identity of Indiscernibles; (2) quantum probability is reinterpreted as epistemic underdetermination of state description, with the Born rule as the unique consistent measure over incomplete descriptions via Gleason; (3) the Joint Resolution Condition (JRC) is proved equivalent to Schmidt rank > 1, establishing that entangled states are logically forced by tensor product structure; (4) the Non-Decomposability Theorem shows that decomposing a JRC-satisfying configuration into local sub-descriptions necessarily violates Excluded Middle; (5) Bell's theorem and Kochen-Specker contextuality follow as downstream consequences. LRT is distinguished from hidden variable programs (Bohm, 't Hooft) by its grounding direction: from logic to physics, not from physics to supplemented physics. Three open formalizations for Lean 4 are identified.

DOI: 10.5281/zenodo.18950180

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AIP conference proceedings•Jan 1, 2012•1 citation

This working paper derives quantum non-decomposability, Bell's theorem, and Kochen-Specker contextuality from Logic Realism Theory's single foundational equation A_Ω = L₃(I∞), where I∞ is the Infinite Information Space, L₃ is the Three Fundamental Laws of Logic acting as a selection functional, and A_Ω is the actualized domain. The paper establishes that LRT is a one-equation theory: all results are downstream of the core equation with no independent primitives introduced. Key results: (1) the Unique Necessary Successor (UNS) constraint is derived as a theorem of L₃ via Identity and Excluded Middle, tightened by Leibniz Identity of Indiscernibles; (2) quantum probability is reinterpreted as epistemic underdetermination of state description, with the Born rule as the unique consistent measure over incomplete descriptions via Gleason; (3) the Joint Resolution Condition (JRC) is proved equivalent to Schmidt rank > 1, establishing that entangled states are logically forced by tensor product structure; (4) the Non-Decomposability Theorem shows that decomposing a JRC-satisfying configuration into local sub-descriptions necessarily violates Excluded Middle; (5) Bell's theorem and Kochen-Specker contextuality follow as downstream consequences. LRT is distinguished from hidden variable programs (Bohm, 't Hooft) by its grounding direction: from logic to physics, not from physics to supplemented physics. Three open formalizations for Lean 4 are identified.

Logic Realism Theory (LRT) proposes that the Three Fundamental Laws of Logic (3FLL)—Identity, Non-Contradiction, and Excluded Middle—are not merely rules of reasoning but ontological constraints constitutive of physical distinguishability. The core evidence is an asymmetry: we can conceive of 3FLL violations, yet physical reality never produces them. This asymmetry indicates that 3FLL constrain reality, not merely cognition. This paper develops the philosophical foundations of this claim. We argue that: (1) 3FLL have a dual character, functioning both epistemically and ontologically; (2) quantum mechanics emerges as the unique "interface" structure mediating between non-Boolean possibility and Boolean actuality; (3) this framework resolves longstanding interpretive puzzles while maintaining scientific realism. The result is a form of structural realism grounded in logical necessity rather than contingent physical law.

Every physical measurement ever conducted has yielded exactly one outcome: self-identical, non-contradictory, determinate. This paper argues that the Three Fundamental Laws of Logic (Identity, Non-Contradiction, and Excluded Middle) are not cognitive conventions but ontological constraints constitutive of physical distinguishability. From this thesis, quantum mechanics emerges as the unique stable structure at the interface between non-Boolean possibility and Boolean actuality. The framework derives rather than postulates: complex Hilbert space structure follows from interface constraints combined with the physical requirement of compositional consistency (local tomography); the Born rule follows from Gleason's theorem; unitary dynamics follows from information preservation requirements. One structural prediction, that quantum mechanics requires complex amplitudes, has been experimentally confirmed (Renou et al., 2021). The framework specifies twelve explicit conditions that would falsify it. Logic Realism Theory provides unified explanation for seventeen quantum phenomena that other interpretations treat as independent postulates, while maintaining realism, locality, and parsimony.

This companion paper provides the rigorous mathematical constructions underlying Logic Realism Theory. We prove three key results: (1) the Hardy kernel construction derives inner product structure directly from the distinguishability metric D without presupposing Hilbert space or the Born rule; (2) LRT axioms imply all five Masanes-Müller axioms, including MM5 (entanglement connectivity) via the Lee-Selby theorem applied to CBP-enforced purification uniqueness; (3) complex quantum mechanics is the unique theory satisfying LRT axioms. The derivation chain from logical constraints (3FLL) to quantum mechanics is complete—no conditional hedges or irreducible gaps remain.

We present a derivation of the Born rule from the ontological constraints imposed by the Three Fundamental Laws of Logic (L₃) within the Logic Realism Theory (LRT) framework. The central argument proceeds through five stages: (1) L₃ constraints at the vehicle level require that physical systems possess determinate dispositions toward measurement outcomes; (2) these dispositions must satisfy vehicle‑weight invariance—probability assignments cannot depend on the descriptive decomposition chosen; (3) vehicle‑weight invariance, combined with finite additivity, yields the premises of Gleason‑type theorems; (4) Gleason’s theorem then forces the trace form μ(P) = Tr(ρP) for all projectors; (5) for pure states, this reduces to the Born rule p = |⟨φ|ψ⟩|².[Attachment] We provide both an intuitive derivation chain and a formal axiom presentation (A1–A7) suitable for rigorous development. The derivation is non‑circular: L₃ constraints are conceptually and logically prior to the probability measure they entail. We situate this result within the quantum reconstruction literature and discuss implications for foundational physics.

Logic Realism Theory (LRT) derives non-relativistic quantum mechanics from the claim that the Three Fundamental Laws of Logic are ontological constraints constitutive of physical distinguishability. This paper extends LRT to quantum field theory through a tiered derivation structure with explicit physical inputs at each level. We establish three principal results: (1) Fock space structure is derived from LRT axioms plus Lorentz invariance and locality; (2) the spin-statistics connection follows from Lorentz invariance and microcausality via the standard Pauli argument; (3) the uniqueness of complex QFT is established. The paper distinguishes derivations from interpretations. Cluster decomposition, vacuum uniqueness, and the spectral gap are physical inputs (Tier 4), not consequences of logical constraints. Renormalization is interpreted as 3FLL-restoration at UV scales, but this does not derive renormalization group equations. Four predictions and six strong falsifiers are identified.

Modal quantum theory (MQT) is a simplified cousin of ordinary Hilbert space quantum theory. We show that two important theorems of actual quantum theory, the Kochen-Specker theorem excluding non-contextual hidden variables and the Conway-Kochen "free will theorem" about entangled systems, have direct analogues in MQT. The structure of possible measurement results for an entangled system in MQT cannot be represented by probability assignments satisfying the no-signaling principle, such as those given by ordinary quantum theory.