Topological Phase Transitions in Collective Neural Dynamics: Formal Derivation of the Ω = 6/π Coupling Constant, Biophysical Derivation of β = 17.30, and Multi-Subject EEG Validation (N=5) of the Omandac Law

This v7. 0 preprint advances the Omandac Law of Collective Binding Resonance, proposing Ω = 6/π ≈ 1. 9099 as the critical coupling constant for dissipative U (1) phase transitions governing neural binding via shared Stuart-Landau universality class with Dicke superradiance. All theoretical gaps are closed through independent derivations and simulations: Biophysical derivation of β = 17. 30 from sparse gamma synaptic activation (fraction ≈ 0. 45%, yielding Iᵣecurrent ≈ 21. 14 μA/cm², total gamma drive ≈ 30. 14 μA/cm², matching literature consensus 21–41 μA/cm²) ; N² superradiant scaling test on EEG gamma power (αcoherent = 1. 894 ± 0. 016 awake vs 1. 818 ± 0. 078 sedated, Δα = 0. 076) ; Recurrent coupling rescue in heterogeneous Izhikevich network (rate recovers to 41. 7 Hz at gₑe = 1. 0 from 38. 8 Hz uncoupled, χ rises to 1. 38+) ; Transfer entropy confirmation of the Information-Action Identity (χTE = 1. 505, 3. 1% match to predicted 1. 553 at β = 17. 30). EEG pilot (N=4 multi-subject) shows 17. 4% phase-locking collapse in the 30–45 Hz binding zone under propofol (p = 5. 27 × 10⁻⁹³, Cohen’s d = 1. 904). The THz–Hz scaling gap is reframed as topological (frequency substrate-dependent). All simulation code, OBS algorithm, and falsifiability criteria are provided for replication. 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 associated Scaling Manifold are asserted under CC BY-NC-ND 4. 0 as of first preprint publication (Feb 2026). Data Ethics: Clinical validation uses de-identified EEG datasets (OpenNeuro, Chennu 2016). Analysis performed via proprietary Omandac-Binding-Search (OBS) algorithm and High-Resolution Scaling Exponent (HRSE) method. Keywords: Omandac Law, collective binding resonance, 6/π, Stuart-Landau universality, Dicke model, 41. 7 Hz gamma, sparse synaptic activation, transfer entropy, N² scaling, EEG propofol, phase-locking collapse, Izhikevich, spiking networks, consciousness License: CC BY-NC-ND 4. 0 Files to Upload: The DOCX + PDF export + supplementary notebooks/figures (e. g. , gap scripts, EEG plots).