Coherence-Gated Dimensional Access: Derivation and Physical Validation of D_eff = D_max × Φ(C)

This paper derives a coherence-gated dimensional access relationship, D_eff = D_max × Φ(C), from the Participation Ratio formalism of Bell and Dean (1970) and Thouless (1974). The coherence function Φ(C) is defined as the ratio of the system's effective Participation Ratio to its maximum possible value, producing a nonlinear scaling relationship between coherence state and accessible dimensionality. Purity 𝒫 = Tr(ρ²) is distinguished from framework coherence C throughout. Physical validation is presented against NOAA HRRR atmospheric data during polar vortex coherence loss preceding the February 22–24, 2026 nor'easter, with the lowest-coherence center tracking from Virginia to Maine over 36 hours. Convergent structural evidence from quantum error correction, fluid dynamics, color perception geometry (Bujack et al., 2022), and market microstructure is documented. The central question posed: whether the derivation is mathematically sound, and whether its cross-domain recurrence constitutes a known result in information geometry or an uncharted structural law.