Sign of the Stokes Constant, Écalle Alien Calculus for Class B, and Analytic |S₁ᴬ(α)| (Paper 13 in the Non-Holomorphic Fractal Series)

We resolve the three open problems stated in Paper 11, §5.2 for the non-holomorphic Class A iteration z_{n+1} = Kz|Im z|^α + c. (1) Sign of S₁ᴬ: Picard–Lefschetz analysis of the half-thimble J₁ at the endpoint saddle t=1 determines sgn(S₁ᴬ) = −1, verified computationally by watson_borel_alien.py at α ∈ {1/4, 1/2, 3/4}. (2) Écalle alien calculus for Class B: the alien-derivative structure Δ̇_ω Φ = S₁ᴮ Φ′ governs the doubly-iterated resurgence with natural variable u = 1/ln ln K. (3) Analytic |S₁ᴬ(α)|: the three Paper 11 anchors lie on a universal curve |S₁ᴬ(α)| ≈ α^{1/3}(1 + c(1−α)), with exact exponent β = 1/3 from cubic turning-point geometry and leading correction c ≈ −0.073. Sets the stage for the full…