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 Dₙ^B′ from Paper 8 has generating function G^B′ (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 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 (cₙC ≡ 1, recovering GC (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.
Michael Bird (Wed,) studied this question.