Universal area-decay exponents in K-parametrized non-holomorphic fractal families: a complete classification

We study the family of non-holomorphic complex iterations zn+1 = Kzng(Im(zn))+c, parameterized by K > 0 and a real-valued gate function g. We prove an exact Decoupling Lemma, derive a closed-form two-step stability boundary, and establish that the stable parameter-space area satisfies A(K)∼C(α,g0,R) K−γ with universal exponent determined by the vanishing order αof g at the operating point. This yields three universality classes: Class A (g(0) > 0, γ = 2), Class B (odd zero, γ = 1 with logarithmic correction, exact coefficient CB = 2R), and Class C (infinite-order zero, anomalous scaling A(K)∼2R/√ln K, exact coefficient CC = 2R). We verify the predicted exponent to 0.2% precision across eight functions spanning…