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‑FCI_th = 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%. This work constitutes the complete mathematical foundation of the A7-HBM-ΩΦ framework, complementing the computational simulations presented in [1], the sleep microstate analysis in [2], and the preliminary theoretical formulation in [3]. The theoretical derivations presented here have been experimentally validated using simultaneous EEG‑ECG recordings from healthy, epileptic, and cardiac patients [4], confirming the predictive power of the eleven theorems. In this updated and expanded version, we further integrate a unified causal framework that links multiscale self‑similarity [5,6], self‑organized criticality [7,8], hierarchical oscillations [10,11,15], and optimization constraints to the emergence of Φ as the optimal solution, and we present the full experimental validation across seven independent datasets. The model unifies geometry, physics, and biology, demonstrating that the brain's hierarchical organization follows from geometric principle.