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Ontological Uniqueness of Quantum Mechanics:\\ A Pre-Quantum Classification Theorem

Guillermo C. Barraza

Published Jan 5, 2026• 0 citations
Publisher

Abstract

Title Ontological Uniqueness of Quantum Mechanics: A Pre-Quantum Classification Theorem Description This work addresses one of the most fundamental questions in theoretical physics: Why does nature employ quantum mechanical structure rather than alternative mathematical frameworks? We prove a rigorous classification theorem showing that quantum mechanics is not one possible theory among many, but the unique mathematical structure compatible with ontological continuity—the principle that "being cannot cease to be." Unlike previous approaches that either interpret quantum mechanics within its existing formalism (many-worlds, Bohmian mechanics) or derive it from operational measurement axioms (Gleason, Hardy, Chiribella-D'Ariano-Perinotti), this work operates at a pre-quantum logical level: we classify which geometric structures are ontologically admissible before introducing any quantum formalism. Main Result: Any theory satisfying minimal ontological requirements—continuity of being, compositional independence of subsystems, and reversibility of closed-system evolution—must describe the space of identity-classes as complex projective space ℂℙⁿ equipped with the Fubini–Study metric. Complex Hilbert space, unitary dynamics, the Schrödinger equation, and the Born rule emerge as necessary geometric consequences, not as postulates. Alternative structures (real or quaternionic projective spaces, non-homogeneous manifolds) are proven ontologically inadmissible through rigorous exclusion arguments. Key Contributions: Rigorous derivation showing that ontological composition forces a symplectic structure, which combined with homogeneity and isotropy uniquely determines Kähler geometry, selecting ℂℙⁿ as the only admissible space. Complete defense against circularity concerns, with detailed appendices showing that the axioms assume no quantum structure—they are purely ontological requirements for coherent continuous being. Systematic distinction from Gleason/Kochen-Specker theorems and operational axiomatizations, clarifying that this work classifies geometries before quantum mechanics, not constraints within it. Testable predictions in cosmology (Hubble tension resolution, matter fluctuation scaling) and quantum foundations (decoherence rates in molecular interferometry), while rigorously separating structural necessity from empirical determination. Significance: This work transforms quantum mechanics from an empirically successful but conceptually mysterious framework into an ontologically inevitable structure—comparable to Einstein's geometrization of gravity. It shows that quantum structure has no alternative: it is the only mathematical form compatible with coherent, continuous, compositional being. The manuscript includes comprehensive technical appendices with full geometric proofs, axiom justifications, and comparative analysis with existing foundational approaches. Suitable for researchers in quantum foundations, mathematical physics, philosophy of physics, and differential geometry. Document Status: Preprint submitted to Foundations of Physics (January 2026). Related Work: This work is part of the Law of Continuity of Being (LCS) research program, which derives physical and ethical structures from fundamental ontological principles. See author's ORCID profile for related publications. Open Access: This preprint is made available under CC BY 4.0 license to facilitate discussion and feedback from the research community prior to peer review.

DOI
10.5281/zenodo.18151282

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This work presents a rigorous derivation of quantum mechanics from purely a priori ontological principles, a result not previously achieved within a single self-contained framework. Beginning with the apodictically certain proposition "I am"—immune to all doubt through performative contradiction—we establish through necessary deduction the Principle of Ontological Continuity (POC): Being is continuous and cannot cease or arise ex nihilo. From this single ontological foundation, the complete mathematical structure of quantum mechanics emerges as necessary consequence. We demonstrate that any mathematical representation of ontological continuity must have Hilbert space structure, that ontological conservation necessarily implies unitary evolution, and that the Schrödinger equation follows via Stone's theorem as mathematical necessity, not empirical postulate. The Born rule is derived through three independent converging arguments (Gleason's theorem, direct conservation requirements, and Zurek's envariance), establishing quantum probability as ontological necessity rather than interpretative supplement. Classical foundational problems—wave function collapse, the measurement problem, Bell's non-locality—are resolved without additional assumptions. Collapse emerges as effective description of decoherence; the observer is reconceptualized as self-aware subsystem without ontological dualism; Bell violations confirm ontological unity rather than presenting paradox. Quantum phenomena—superposition as ontological potentiality, tunneling as manifestation of continuity, Pauli exclusion as consequence of ontological structure—are shown to be necessary rather than mysterious. The work establishes quantum mechanics with apodictic certainty mediated through deductive necessity, demonstrating that quantum reality could not have been otherwise. This monograph integrates foundational ontology, mathematical formalism, and resolution of interpretative problems within unified self-contained framework. It deliberately closes at non-relativistic quantum mechanics, establishing stable foundation for future extensions (quantum field theory, general relativity, quantum gravity, normative ethics).
This article is the first part of a three-part series developing Ontological Field Theory (OFT), a unified framework grounding quantum mechanics in foundational ontological principles. The trilogy is structured as follows: Part I — Conceptual Necessity:Why Quantum Mechanics Is Inevitable: Ontological Continuity and the Necessity of Quantum Dynamics establishes that quantum mechanics follows necessarily from minimal requirements on coherent being, independently of specific physical postulates. Part II — Technical Realization:Derivation of the Schrödinger Equation from Ontological Field Theory provides the explicit field-theoretic derivation showing how the Schrödinger equation emerges as the non-relativistic limit of a covariant ontological field. Part III — Critical Context:Why Classical Physics Is an Approximation: Decoherence, Coarse-Graining, and the Limits of Determinism explains why classical determinism and trajectories arise only as emergent, approximate descriptions of fundamentally quantum and relational dynamics. Each article is logically self-contained, but the full argument is distributed across the three contributions. The conceptual analysis motivates the mathematics; the mathematics grounds the physics; and the critical examination situates classical intuition as a limiting case rather than a foundation. The papers are intended to be read as a coherent whole. We argue that quantum mechanics is not an arbitrary empirical framework but an inevitable consequence of minimal ontological requirements. Assuming only that being must exist necessarily, remain consistent, be relational, and admit self-reference, we show that classical deterministic physics is structurally insufficient. Quantum dynamics emerges as the unique low-energy realization of coherent ontological continuity. This work complements explicit field-theoretic derivations by clarifying why quantum mechanics could not have been otherwise.
This article is the second part of a three-paper sequence developing Ontological Field Theory (OFT) as a unified framework grounding quantum mechanics in foundational ontological principles. The trilogy is structured as follows: Part I — Conceptual Necessity:Why Quantum Mechanics Is Inevitable: Ontological Continuity and the Necessity of Quantum Dynamics establishes that quantum mechanics follows necessarily from minimal requirements on coherent being, independently of specific physical postulates. Part II — Technical Realization:Derivation of the Schrödinger Equation from Ontological Field Theory provides the explicit field-theoretic derivation showing how the Schrödinger equation emerges as the non-relativistic limit of a covariant ontological field. Part III — Critical Context:Why Classical Physics Is an Approximation: Decoherence, Coarse-Graining, and the Limits of Determinism explains why classical determinism and trajectories arise only as emergent, approximate descriptions of fundamentally quantum and relational dynamics. Each article is logically self-contained, but the full argument is distributed across the three contributions. The conceptual analysis motivates the mathematics; the mathematics grounds the physics; and the critical examination situates classical intuition as a limiting case rather than a foundation. The papers are intended to be read as a coherent whole. We argue that quantum mechanics is not an arbitrary empirical framework but an inevitable consequence of minimal ontological requirements. Assuming only that being must exist necessarily, remain consistent, be relational, and admit self-reference, we show that classical deterministic physics is structurally insufficient. Quantum dynamics emerges as the unique low-energy realization of coherent ontological continuity. This work complements explicit field-theoretic derivations by clarifying why quantum mechanics could not have been otherwise.
This article is the third and final part of a three-paper sequence developing Ontological Field Theory (OFT) as a unified framework grounding quantum mechanics in foundational ontological principles. The trilogy is structured as follows: Part I — Conceptual Necessity:Why Quantum Mechanics Is Inevitable: Ontological Continuity and the Necessity of Quantum Dynamics establishes that quantum mechanics follows necessarily from minimal requirements on coherent being, independently of specific physical postulates. Part II — Technical Realization:Derivation of the Schrödinger Equation from Ontological Field Theory provides the explicit field-theoretic derivation showing how the Schrödinger equation emerges as the non-relativistic limit of a covariant ontological field. Part III — Critical Context:Why Classical Physics Is an Approximation: Decoherence, Coarse-Graining, and the Limits of Determinism explains why classical determinism and trajectories arise only as emergent, approximate descriptions of fundamentally quantum and relational dynamics. Each article is logically self-contained, but the full argument is distributed across the three contributions. The conceptual analysis motivates the mathematics; the mathematics grounds the physics; and the critical examination situates classical intuition as a limiting case rather than a foundation. The papers are intended to be read as a coherent whole. We argue that quantum mechanics is not an arbitrary empirical framework but an inevitable consequence of minimal ontological requirements. Assuming only that being must exist necessarily, remain consistent, be relational, and admit self-reference, we show that classical deterministic physics is structurally insufficient. Quantum dynamics emerges as the unique low-energy realization of coherent ontological continuity. This work completes the ontological grounding of quantum mechanics by clarifying why it could not have been otherwise.
This work proposes a minimal ontological framework for physical reality grounded in the principle of structural distinguishability. The foundational axiom is the non-zero distinguishability of states, from which constraints, fixation, temporal directionality, spatial structure, and stability conditions of physical configurations are systematically derived. Space is interpreted as a structure of compatibility between alternatives, while time is understood as an ordering of fixation events. The geometrization of physics is treated as a secondary descriptive language of stable regimes of distinguishability. The limits and applicability of geometric representation are analyzed in relation to quantum mechanics and gravitation. The framework does not introduce new physical equations but aims to provide an ontological reconciliation of existing formalisms in quantum theory and general relativity. The work is conceptual and structural in nature and is intended for researchers in philosophy of physics and foundations of theoretical physics.
The persistent challenge of unifying quantum mechanics with general relativity is often traced to a series of apparent structural tensions: complex versus real ontology, probabilistic versus deterministic dynamics, linear versus nonlinear evolution, and spectral discreteness versus geometric continuity. We argue that these tensions do not signal a fundamental incompatibility between the theories, but rather originate from surplus structure in the conventional formulation of quantum mechanics. Beginning from the physically indispensable requirement of matter stability, we prove that the Hamiltonian spectrum can always be shifted to become non-negative. The consequent positivity of energy enforces one-sided temporal frequency support for all physical states, selecting the Hardy space of analytic signals as the admissible solution space. In this space, the real and imaginary parts of the wave function are not independent; they are linked uniquely via the temporal Hilbert transform. Hence, the complex structure of quantum mechanics emerges as a representational artifact of an underlying real operator algebra. Building on our earlier work on Unified Quantum Mechanics and the Unified Real Wave Equation, we establish a rigorous proof of exact equivalence between the real-field evolution and the Schrödinger equation on the physical subspace. Probabilistic quantities arise naturally as quadratic invariants of analytic signals, rather than as independent postulates. The resulting framework provides a realfield formulation of quantum mechanics and gravitation, with stability and causality implemented as foundational principles, and demonstrates that the Einstein field equations are compatible with a unified field-theoretic description of spacetime and matter. Extending the same analyticity principle to spacetime geometry, we propose an analytic formulation of the Einstein field equations, from which classical general relativity is recovered exactly as the real projection.