Topological Phase Transitions in Collective Neural Dynamics: Formal Derivation and Multi-Subject Empirical Validation of the 1.91x Scaling Constant

This version completes the "Physics-Biology Bridge" by providing formal analytical proofs for the Omandac Laws and validating them against a multi-subject clinical EEG cohort ($N=5$). By mapping the Dicke Hamiltonian to macroscopic neural integration, this work identifies the 1.91x Critical Scaling Gap as a fundamental metric for the transition between conscious and sedated states. Key Scientific Findings: Formalization of the Three Laws: Establishes the Binding Law (30–45 Hz), the Coupling Law (Ω = 6/π), and the Scaling Law (characterized by the criticality parameter β = 1.4119). Resolution of the Coordinate Gap: Provides an analytic resonance proof using period integrals, identifying 41.7 Hz as the natural eigenvalue of the system under Ω-tuned drive. Substrate Isomorphism: Utilizes the Stuart-Landau Normal Form to demonstrate that biological neural binding and Dicke superradiance share mathematically equivalent dissipative phase transition characteristics. Empirical Clinical Evidence: Reports a group-level phase-locking collapse of 17.40% (R) within the 30–45 Hz binding zone during propofol-induced sedation. Statistical analysis confirms the transition with a significance of p = 5.27 × 10⁻⁹³ (Cohen’s $d = 1.904$). Scaling Symmetry Validation: Confirms that awake brain states maintain a power-law exponent (β ≈ 1.72) significantly closer to the predicted theoretical attractor ($1.412$) compared to the fragmented sedated state (β ≈ 2.52). Summary: This work marks the transition from quantum-inspired hypothesis to an experimentally validated physical-biological framework. By providing a precisely quantifiable ratio (1.91x) for neural population synergy, this manuscript establishes the Omandac Law as a foundational model for collective binding resonance. Statement of Priority: "The mathematical derivation of the 6/π constant and its mapping to the 0.0417 ms⁻¹ attractor constitutes the original discovery of the Omandac Law. This work is independent and solo-authored. All rights to the theoretical nomenclature and the associated Scaling Manifold are asserted by the author under CC BY-NC-ND 4.0 as of the date of first preprint publication (Feb 2026)." Data Ethics: Clinical validation utilizes de-identified EEG datasets (OpenNeuro, Chennu 2016). Analysis was performed via the proprietary Omandac-Binding-Search (OBS) algorithm and High-Resolution Scaling Exponent (HRSE) method.