Clarifying the Chaotic Range in Systemic Tau: The Intermediate Volatility Zone (|τₛ| < 0.41) and Its Implications for Complex Chaotic Systems

This technical note formally defines and characterizes the chaotic range in Systemic Tau (τₛ), corresponding to the intermediate volatility zone where |τₛ| < 0.41. Previously implicit in validations across ecological, physical, and fractional-order chaotic systems, this regime represents the region of maximal dynamical volatility during active bifurcations. Grounded in Kendall’s tau ordinal correlations and Feigenbaum universality (δ ≈ 4.669, α ≈ 2.502), the chaotic range exhibits weakened ordinal agreement, extreme sensitivity to initial conditions, and robust noise tolerance (up to 15%). Simulations confirm τₛ ≈ 0.036 in fully developed chaos beyond the Feigenbaum point, with variance constrained by σ² ≤ 1/N. By explicitly delineating the boundaries at ±0.41, this note strengthens the predictive capacity of τₛ for early detection of critical transitions. Applications span ecology, climate modeling, artificial intelligence, financial systems, physical attractors, cardiology, materials engineering, social network resilience, and strategic forecasting under uncertainty.