Exact Ridge Curvature in One Evaluation: Hypercomplex Perturbation Unifies Attractor–Ridge Geometry

Ridge geometry in bistable dynamical systems is typically computed by parameter sweeps and finite-difference curvature estimates. These methods are approximate, expensive, and sensitive to step size. We show that the hypercomplex perturbation algebra provides an exact Hessian of the ridge potential U(x) = −||f(x)||² in a single evaluation. Using the canonical λ-switch, we reproduce the phase portrait, fixed points, and separatrix from prior attractor–ridge analysis. At the saddle (1.3794, 1.3794), the hypercomplex Hessian matches the finite-difference Hessian to within 2×10⁻⁹. Across the full state-space grid, the curvature maps agree to within 4×10⁻⁶, consistent with finite-difference truncation error. The hypercomplex method yields the exact ridge curvature and geometry without parameter sweeps, providing a computational engine for attractor–ridge analysis.