Variational Geometry of Horizontal Newton-Raphson Isoclines, Quarter-Wave Bifurcations, and Dyadic Ceiling Asymptotics of k-Bonacci-Gaussian Exponential Trinomials

We study the complex solutions, parameter thresholds, and root-finding geometry of a family of exponential trinomial equations weighted by the characteristic constants of generalized multi-step Fibonacci sequences. Unlike simpler exponential equations solvable by Lambert functions, these equations couple quadratic Gaussian and linear exponential growth simultaneously. First, we show that as the sequence step order grows toward infinity and the characteristic constant approaches two, the complex roots converge exponentially to a limiting ceiling equation governed by a deficit-halving law. Second, we prove that a near-imaginary root branch crosses from the left half-plane into the right half-plane at a quarter-wave point on the vertical axis at an exact critical parameter of approximately 1.6943, explaining why only the classical two-step Fibonacci equation places this root in the second quadrant while all higher-order sequences place it in the first quadrant. Third, we analyze the curve of points where the complex Newton-Raphson root-finding method takes purely horizontal steps, proving via the Cauchy-Riemann equations that every simple root sits at a flat horizontal vertex of this curve, deriving its sixth-order polynomial expansion, and finding an exact trigonometric equation for its vertical-axis intercept. All analytical perturbation formulas and root tables are verified to fifty decimal digits of precision.