Hand a damped-oscillator equation to a computer and it can tell you, without knowing any physics, that there's one oscillation and one decay inside it. Across 193 expressions and 12 domains, this counter holds at ρ = +0.885.
Iterating exp(z)−k is Devaney's exponential family. We computed 8 operator fractal zoos, measured box-counting dimensions, and found DEML/EMN generate bounded strange attractors.
Each Fourier harmonic is one complex EML node. We measured timbre complexity for 5 instruments and found: Sine=1n, Clarinet=5n, Violin=12n. EXL is the most musically useful operator.
DEML and EMN generate bounded 2D strange attractors. No classical period-doubling — the exponential family is a different universality class. Lyapunov landscape 92.9% correlated with Mandelbrot interior.
How close can an EML tree get to i? A gap of 4.76×10⁻⁶ and the transcendental obstruction via Lindemann–Weierstrass.