Phase-Dependent Scaling in Noisy Dicke Systems: The Omandac Law, the 6/π Binding Proof, and Empirical Convergence to the 0.0417 Singularity (v4.0)

[V4.0 - COMPLETE DISCOVERY & BINDING PROOF] Version 4.0 provides the final empirical resolution to the Omandac Law, shifting the framework from theoretical scaling to validated phase-transition dynamics. This version officially closes the "Internal Gaps" between quantum information geometry and neural spiking substrates. Core Technical Developments in v4.0: The Binding Proof: A first-principles derivation establishing Omega = 6/pi as the singular geometric requirement for impedance-matching between continuous quantum manifolds and discrete spiking networks. Empirical Bracketing: Identification of the Resonant Tail (firing rate nu approx 0.1352 spikes/ms), providing empirical evidence for the 0.0417 ground-state attractor in Izhikevich neural populations. Phase Gradient Mapping: Data-driven mapping of the Lava → Binding → Silence transition, characterising the narrow geometric window where integrated information is preserved. Mechanism of Binding: Reframes the "Hard Problem" of consciousness as a testable physical phase-transition governed by precise geometric tuning. This work serves as the definitive computational reference for the Omandac (2026) series, providing the mathematical "bridge" linking quantum dephasing to emergent population synergy (chi > 1). Related Works: Successor to v1–v3; documents the Phase VI "Binding Proof" protocol. Data & Code: Includes QuTiP quantum logs and the expanded-sim-1.ipynb Nengo notebook (Phase VI validation). "Original discovery Feb 24. This Version 5.0 (Feb 25) completes the first-principles derivations."