Deterministic Geometric Preparation and the Emergence of the Born Rule in Two-Path Interference

This paper presents a deterministic geometric derivation of the standard two-path interference law \(P(\Delta)=\cos^2(\Delta/2)\). The model is defined on the unit circle and uses deterministic preparation and projection rather than deriving the interference law from complex amplitude superposition together with the Born rule. A hidden directional variable is uniformly distributed before preparation and then biased according to geometric alignment with a preparation axis. Measurement is modeled deterministically by the sign of directional projection onto a measurement axis. The preparation-weighted geometric ensemble yields the correlation \(E(\Delta)=\cos\Delta\), which reproduces the standard two-path interference probability exactly. In this construction, the harmonic phase dependence appears as an overlap identity on \(S^1\) rather than as a consequence of probabilistic collapse or intrinsic complex amplitude addition. The construction is intentionally minimal. It does not attempt to reconstruct the full Hilbert space structure of quantum theory or provide a complete physical hidden-variable theory. It demonstrates only that the Born interference law for two coherent paths is consistent with deterministic geometric preparation and projection on a continuous hidden-direction space. This paper belongs to the Quantum Foundations area of the Coherence Geometry research corpus and is structurally paired with CGI-RSR-000021, "Geometric Substrate Models and Bell–CHSH Correlations: A Structural Analysis of Assumption Relaxation." Internal reference: CGI-RSR-000022.