Phase-Dependent Collective Contrast in Finite Open Dicke Systems: First-Principles Derivation of the Omandac Law and Attractor Dynamics (v5.0)

Description: Version 5.0: First-Principles Complete Milestone. This version completes the theoretical foundation of the Omandac Law by replacing previously empirical claims with rigorous mathematical derivations. Key updates in this version include: * Information-Action Identity: A full derivation of χ ≈ exp(ln(Ω) · η) from directed transfer entropy and mean-field projection. * Thermal Coherence Floor: A derivation from Bose-Einstein statistics establishing fc = 4.49 THz as the required frequency for quantum coherence at 310K. * Izhikevich Binding Coordinate: High-resolution confirmation of the ν^* = 0.0417 spikes/ms attractor at β = 17.30. * HH Hopf Topology: An explanation of the Hodgkin-Huxley "phase-cliff" (fₘᵢₙ ≈ 0.052 spikes/ms) using subcritical Hopf bifurcation theory and Jacobian eigenvalue analysis. * Omandac Excitability Law: A classification of the 0.0417 attractor as a Type I (SNIC) fixed point, determining its accessibility across different neural substrates. This work provides the mathematical "bridge" between quantum dissipative dynamics (Ω = 6/π) and emergent population synergy in neuromorphic substrates. Related Datasets: Includes three new notebooks: binding-point-final.ipynb, solving-the-decoherence-gap.ipynb, and solving-the-substrate.ipynb. "Original discovery Feb 24. This Version 5.0 (Feb 25) completes the first-principles derivations."