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 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.…