Φ-Optimal Hierarchical Brain Oscillations and β-Controlled Cognitive Dynamics: First-Principles Mathematical Foundations of the A7-HBM-ΩΦ Model

We present the mathematical foundations of the A7-HBM-ΩΦ framework, unifying spectral hierarchy, neural geometry, and cognitive control. Through an optimality principle, we derive the golden ratio (Φ ≈ 1.618) as the optimal spectral spacing parameter that minimizes interference while maximizing communication efficiency. Biological constraints (0.5–200 Hz) naturally yield seven hierarchical frequency levels (δ, θ, α, σ, β, γ, Ω), each Φ-scaled from adjacent bands. We introduce a control parameter β ∈ [0,1] regulating information flow direction, with critical transitions at β = Φ⁻² ≈ 0.382 and β = Φ⁻¹ ≈ 0.618. The complete A7-HBM-ΩΦ-Bidi system comprises seven nonlinearly coupled Stuart-Landau oscillators with tri-diagonal β-dependent coupling. These mathematical foundations provide a rigorous basis for modeling hierarchical brain dynamics, enabling future investigations of microstate transitions, stability analysis, and clinical applications within a unified framework.