Unified Golden Equation: Complex Extensions, Damped Waves, and Amplified Recurrences — (Addendum 1 of 2 to Golden Equation Papers 1 and 2)

This work establishes the Unified Golden Equation (UGE) as the analytic continuation of the classical Golden Equation to the complex parameter plane k=σ+iθk = σ + iθk=σ+iθ.

The UGE unifies exponential decay, oscillatory, and damped/amplified dynamical regimes through the relation

f(α;σ,θ)=α+α−σe−iθln⁡α−2=0.f(α;σ,θ) = α + α^{−σ} e^{−iθ \ln α} − 2 = 0.f(α;σ,θ)=α+α−σe−iθlnα−2=0.

Existence and uniqueness are proven via the Banach fixed-point theorem, and oscillatory roots are characterized using Rouché’s theorem. The analytic domain D=C∖(−∞,0]D = ℂ \setminus (−∞,0]D=C∖(−∞,0] ensures well-defined principal logarithms.

The framework naturally extends to imaginary Fibonacci and imaginary k-bonacci sequences, providing a complete branch-cut treatment for ψ < 0. Amplified recurrences (k=−mk = −mk=−m) form an independent mathematical family with growth constants β(−m) ≥ 2, proven incompatible with the Golden Equation.

Asymptotic analysis for |θ| → ∞ establishes boundedness and phase perturbation of order O(1/θ). This unification connects discrete recurrence theory, complex dynamics, and analytic continuation in a rigorously defined framework.