An Ontological Completion of Geometric Quantum Mechanics

Abstract

Decades of work in geometric quantum mechanics and decoherence theory have clarified how quantum behaviour arises from underlying symplectic and dynamical structures, yet both leave unresolved why measurements yield definite outcomes with |ψ|² frequencies. This paper proposes a deterministic geometric account that retains the strengths of those approaches while eliminating the need for stochastic collapse or multiple-world postulates. The result is a unified description in which observed statistics follow from conserved geometry rather than from probabilistic axioms. Physical reality is represented by a single trajectory ω(t) evolving under Hamiltonian flow on a compact symplectic manifold Σ. A measurement context supplies a physical interaction and readout whose deterministic dynamics carry the microstate into one context-indexed outcome sector, and outcome definiteness arises when that readout is fixed in a stable physical record. The Born weights are not posited as an axiom: unitary symmetry fixes the Fubini–Study reference measure on complex projective state space, and the toric moment map pushes it forward to the uniform measure on the probability simplex, so that the state-dependent barycentric regions have volumes exactly equal to the squared amplitudes of the state in the measurement basis. The framework further predicts a quantitative correspondence between interference visibility V and the dynamical separation I of outcome regions. In the macroscopic limit, V ≈ 1 − I, so high separation (I → 1) corresponds to classical definiteness and negligible remaining visibility. The exact functional form, together with parameter-free calculations and proposed experimental tests, is developed in Paper C and a companion note (2025–2026). The paper establishes a complete ontological basis for deterministic quantum measurement, defines how isolation and re-isolation govern the appearance of classicality, and clarifies compliance with Bell, Kochen–Specker, Fine, and PBR constraints. It shows that a finite, measure-preserving geometry can reproduce the statistical structure of quantum theory while remaining open to empirical refutation. This framework, termed Constraint-Surface Dynamics, thus offers a deterministic and falsifiable alternative foundation for quantum mechanics.