Dual Recursion

This report examines a dual recursion system defined by the paired equations x = 1 - \frac{1}{x} and x = 1 + \frac{1}{x}. The system is structured such that the minus map provides the initial instantiation, the plus map acts as a single entry point, and subsequent iterations of the minus map yield a repeating three-cycle. Mathematically, the plus map converges toward the golden ratio, representing stability and order, while the minus map generates cyclic symmetry, representing periodicity. Together, they form a compact framework that unites convergence and oscillation. This duality has geometric, physical, biological, economic, and computational applications, modeling the transition from harmonic balance to recurring cycles. The systems value lies in its ability to capture complex behaviors through minimal structure, offering both theoretical elegance and practical insight across disciplines.