Volume-Based Probability: Outcome Frequencies from Deterministic Geometry

This paper presents a geometric derivation of outcome frequencies in deterministic quantum-like systems without assuming the Born rule.

By analyzing systems that evolve under volume-preserving deterministic dynamics, we show that the relative frequencies of distinct outcomes converge, for typical initial states, to volume ratios over the branching structure of state space. The result is a general "volume-ratio rule" that mirrors the role played by the Born rule in quantum mechanics but arises from first principles.

It sets the stage for Paper B, which derives the quantum probability measure μ ∝ ∣ψ∣2 from symmetry and invariance constraints thus completing the derivation of the Born rule from geometric principles.