The Golden Transition Constant: A Universal Threshold Between Chaotic and Harmonic Growth in Complex Systems

This paper proposes the existence of a universal constant describing the transition between disordered exponential growth and self-regulated proportional growth.The constant is defined as the ratio between Euler’s number and the square of the Golden Ratio:\Gamma = \frac{e}{\varphi^2} \approx 1.038305—or, in additive form—\Delta = e - \varphi^2 \approx 0.10025This small difference (~3.8 %) represents the stabilization threshold where a system moves from chaotic expansion to coherent organization.Through a variational and informational model, Γ is interpreted as the Golden Transition Constant (GTC) — the critical point where the natural growth rate (Euler’s e) meets harmonic proportion (Golden Ratio φ).The GTC may describe the universal threshold between entropy and self-organization in biological, economic, and cosmological phenomena.