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 \(φ_t\) and a conserved measure \(μ\). A physical experiment corresponds to evolving an initial region \(Ω₀⊂ S\) into a disjoint union of macroscopically distinguishable outcome regions \(\{Ωᵢ\}\), each defined by both dynamical separation and observational distinguishability. We show that for almost every microstate in \(Ω₀\), repeated experiments yield long-run frequencies matching the ratios \(μ(Ωᵢ)/μ(Ω₀)\). 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 \(|Ψ|^2\) in quantum mechanics.