An entire function theory is established for the U-space iteration. Through thetransformation w = ln u, the U-space iteration is equivalently converted intothe iteration of the entire function f(w) = a w + b e^w, where a = 1/s,b = -π/s², and s ∈ ℂ\{0,1}. In the w-coordinate, the critical points arew_c = ln(s/π) + 2πik (k ∈ ℤ), the fixed point is w* = -W₀(π/(s(s-1))),and the multiplier is λ(s) = [1-(s-1)W₀(π/(s(s-1)))]/s. The existence andattractivity of the period-2 orbit {±iπ/2} at s = -2 are proved analytically,and it is shown that this is the unique parameter value admitting a period-2orbit of this pure imaginary form. For |s| 1, where critical orbits may escape despite |a| < 1.Since f(w) = a w + b e^w is an entire function of finite order, the Eremenko-Lyubich theorem excludes wandering domains for all parameters. The U-space andS-space are connected through the U-S relation, linking the transcendentaldynamics of the U-space to the algebraic dynamics of the S-space.