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

Title Ontological Uniqueness of Quantum Mechanics: A Pre-Quantum Classification Theorem Description Version 2.0 — Revised and Improved This work addresses a foundational question in theoretical physics: Why does nature employ quantum mechanical structure rather than alternative mathematical frameworks? Unlike interpretational approaches (many-worlds, Bohmian mechanics) or operational axiomatizations (Hardy, Chiribella-D'Ariano-Perinotti), we explore a geometric classification approach. Main Result: We prove a rigorous classification theorem showing that any physical theory satisfying minimal compositional requirements—metric structure over states, 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 inadmissible through rigorous exclusion arguments. Key Technical Contributions: Rigorous derivation showing that compositional structure forces a symplectic form, 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 demonstrating that axioms assume no quantum structure—they are purely compositional requirements for coherent physical theories. Systematic distinction from Gleason/Kochen-Specker theorems and operational axiomatizations, clarifying that this work classifies geometries before introducing quantum mechanics, not constraints within it. Empirical directions in cosmology (parameter relationships) and decoherence (complexity scaling). These are presented as phenomenological conjectures requiring further development, not as fully derived predictions. We explicitly acknowledge that a key measure of ontological complexity (𝒞_Ω) is proposed as a phenomenological ansatz consistent with geometric structure, not yet derived from first principles. What's New in Version 2.0: Reframed introduction positioning the work as an "alternative classificatory approach" alongside existing foundational programs, rather than claiming to supersede them Explicit acknowledgment that cosmological predictions are preliminary conjectures Honest presentation of decoherence predictions: the complexity measure 𝒞_Ω is identified as a phenomenological ansatz, with formal derivation deferred to future work Complete separation of speculative philosophical extensions (consciousness, ethics) into a dedicated appendix with strong disclaimer, making clear these are not part of the physical theory Significance: This work transforms the status of quantum mechanics from an empirically successful but conceptually mysterious framework into a geometrically inevitable structure—comparable to Einstein's geometrization of gravity. The classification theorem demonstrates that quantum structure is not contingent but necessary: it is the only mathematical form compatible with coherent compositional structure. 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: Version 2.0 preprint prepared for submission to Foundations of Physics (January 2026). This version addresses strategic presentation concerns while preserving all technical rigor. Related Work: This work is part of the Law of Continuity of Being (LCS) research program. 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.