This paper proves exponential convergence of the L²-gradient flow of the residual curvature energy on compact hyperbolic surfaces. The setting is conformal geometry in genus γ ≥ 2, where the reference metric has constant negative curvature and the conformal perturbation is written as g = e^{2ψ}g₀. The main result is that, for sufficiently small initial data in H⁴, the nonlinear fourth-order flow exists globally and converges exponentially to the hyperbolic metric. The proof does not rely on an abstract Łojasiewicz–Simon argument. Instead, it derives a quantitative gradient inequality from explicit perturbation estimates for the linearised residual operator DR(ψ) and its adjoint. The core mechanism is spectral: the Jacobi operator J = −Δ_{g₀} + 2κ₀ has a positive spectral gap on the mean-zero subspace, and this lower bound persists under small nonlinear perturbations. This yields an explicit decay rate for the residual energy, asymptotically approaching the optimal linear rate as the perturbation size tends to zero. The paper closes the local dynamical part of a broader variational–spectral programme on squared-residual energies in conformal geometry. It is aimed at readers in geometric analysis, spectral theory, and nonlinear PDE.