Resurgence Corrections for Non-Holomorphic Fractal Families: Exact Class B′ Hypergeometric Identification and Transfer Lemma

Paper 8 asserted a universal square-root Borel branch point at s = 1 for all pure-power-law classes (A, B′, C) of the Bird non-holomorphic fractal classification, and left three structural gaps: no uniform proof mechanism, an unverified Class B′ hypergeometric identification, and no independent analysis of Class A’s asymptotic variable. This paper closes the first two gaps and provides a structural observation on the third. Class B′ (main result). The Poincaré coefficient sequence D_nB′ from Paper 8 has generating function GB′(s) = (π / α sin(π/α)) · ₂F₁(1/2, 1/α; 1; s), an exact Gauss hypergeometric identity proved by Pochhammer expansion. The local exponent at s = 1 is μ(α) = 1/2 − 1/α, which varies continuously with α: algebraic branch point (1−s)μ(α) for 1 < α < 2, logarithmic at α = 2, analytic (convergent) for α > 2. This corrects and sharply refines Paper 8’s universal square-root claim for Class B′. Transfer Lemma (Lemma 3.1). A single ratio-test mechanism covers Class C exactly (c_n^C ≡ 1, recovering G^C(s) = (1−s)−1/2) and provides the analytic frame for any class satisfying the ratio condition. Class B′ is handled by the exact ₂F₁ identification. Class A (Observation 5.1). The natural expansion variable for the Class A area integral is v = K−1/(1+α), identified by the exact substitution y = W(K)·t which brings the Borel–Laplace dual variable to unity. The full asymptotic expansion of A(K) in v and its Borel consequences are deferred to Paper 10.Together with Papers 5–8, these results constitute a corrected and extended (not completed) Borel atlas for the Bird fractal families. The Borel atlas is extended, not completed. The remaining gaps for Classes A and B are explicitly bounded and scheduled for Paper 10.