Resurgent Borel Structure Across Fractal Families: Area-Decay Laws in Classes A, B, and C

The Bird classification partitions non-holomorphic fractal families z_{n+1} = K z_n g(Im(z_n)) + c into classes (A, B, C) by the vanishing order α of the gate function g at the origin. Papers 1–7 completed a resurgent analysis of the Class C Aether family (g(y) = e^y), establishing Borel transforms, Stokes constants, and a Stokes-line hierarchy encoding the Feigenbaum RG spectrum. Here we extend the resurgence programme to Class A (0 < α < 1), Class B (α = 1, logarithmic), and Class B′ (α > 1) families. We prove that the Poincaré asymptotic series for A(K) in every pure-power-law class has a Borel transform with a square-root branch point at s = 1, making the branch-point type a class invariant. The Stokes constant S₁ = C√π carries the class-dependent information. Numerical verification with ten gate functions — covering five classes including the OP-3 shifted-tanh interpolation and the Class C Gaussian/Ricker family — confirms the universal γ-exponents to better than 4% across all power-law classes, and the Class B logarithmic prefactor C_B = 2R analytically via Corollary 6 of. Together with Papers 5–7, these results constitute a complete Borel atlas for the Bird fractal families.