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 (= 6/), and the Scaling Law (characterized by the criticality parameter = 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 -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 (1. 72) significantly closer to the predicted theoretical attractor (1. 412) compared to the fragmented sedated state (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/ 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.