Volume-Based Probability: Outcome Frequencies from Deterministic Geometry

In standard quantum mechanics, the Born rule is introduced as a postulate: outcome probabilities equal the squared amplitude of the wavefunction. This paper proposes a deterministic alternative based on the geometry of a constrained state space. We consider a smooth, finite-dimensional, Hausdorff manifold \(S\), equipped with a volume-preserving flow \(\varphi_t\) and a conserved measure \(\mu\). A physical experiment corresponds to evolving an initial region \(\Omega_{0}\subset S\) into a disjoint union of macroscopically distinguishable outcome regions \(\{\Omega_{i}\}\), each defined by both dynamical separation and observational distinguishability. We show that for almost every microstate in \(\Omega_{0}\), repeated experiments yield long-run frequencies matching the ratios \(\mu(\Omega_{i})/\mu(\Omega_{0})\). This result requires no probability postulate, wavefunction, or stochastic process, only deterministic dynamics and geometric structure. This result lays the foundation for Paper B, which shows why this becomes \(|\Psi|^2\) in quantum mechanics.