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

We present a unified hierarchical theory of brain dynamics derived entirely from first principles. The foundation is a geometric principle: any self‑similar hierarchical system seeking maximal harmony must satisfy Euclid's equation, whose unique solution is the golden ratio Φ ≈ 1.618. This geometric principle is embodied biologically in an efficiency functional balancing information transfer, spectral interference, and dynamical stability, which also yields Φ as the optimal frequency spacing between adjacent bands. From this single seed we sequentially derive eleven theorems that together form a complete mathematical pyramid. Theorem 0 establishes the Euclidean geometric principle. Theorem 1 proves the optimality of Φ in the biological context. Theorem 2 determines the number of frequency bands N = 7 from the biological range (0.5–200 Hz) and stability analysis. Theorem 3 introduces the control parameter β ∈ [0,1] regulating information flow direction, with critical values Φ⁻¹ ≈ 0.618 and Φ⁻² ≈ 0.382 from bifurcation analysis. Theorem 4 derives the optimal coupling coefficients κ₀ = ½Φ⁻¹ ≈ 0.309 from an information‑energy trade‑off. Theorem 5 gives the optimal phase shifts ϕ↑ = π/4, ϕ↓ = –π/4 from time‑reversal symmetry and interference minimization. Theorem 6 reveals 28 attractors (4 per band) with elementary geometric forms (cube, hexagon, pentagon, square, triangle, spiral, point) via group‑theoretic analysis. Theorem 7 provides analytical phase‑amplitude coupling (PAC) values as simple functions of Φ. Theorem 8 establishes the linear correlation between mean PAC and Φ‑coherence. Theorem 9 derives the temporal decrease of PA‑FCI before acute events from critical transition theory. Theorem 10 yields the universal warning threshold PA‑FCIₜₕ = 0.55 from critical slowing‑down analysis. Theorem 11 gives the linear PA‑FCI formula with theoretically derived weights. Numerical simulations of the full nonlinear system confirm all derivations with deviations below 0.3%. The model unifies geometry, physics, and biology, demonstrating that the brain's hierarchical organization follows from geometric principle.