Sign of the Stokes Constant, Écalle Alien Calculus for Class B, and Analytic |S₁ᴬ(α)| (Paper 12 in the Non-Holomorphic Fractal Series)

We resolve the three open problems stated in Paper 11, §5.2. (1) Sign of S₁ᴬ. Picard–Lefschetz analysis of the half-thimble 𝒥₁ at the endpoint saddle t = 1 determines the co-orientation of 𝒥₁ relative to the original real contour, yielding sgn(S₁ᴬ) = −1. (2) Écalle alien calculus for Class B. We identify the alien-derivative structure Δ̇_ω Φ = S_B · Φ′ governing the doubly-iterated resurgence ρ(K) ∼ 1/ln ln K → 0 of the Class B series, with natural variable u = 1/ln ln K. (3) Analytic |S₁ᴬ(α)|. We define |S₁ᴬ(α)| via the half-thimble integral and state an empirical power-law conjecture consistent with the three numerical anchor values of Paper 11 (Corollary 4.3). The principal open problem — proving the conjecture analytically via the Picard–Lefschetz residue at the interior saddle t₀ = (1+α)/(2+α) — is reserved for Paper 13. Paper 10 (DOI: 10.5281/zenodo.19228966) established the two-saddle decomposition and boundary-layer series. Paper 11 (DOI: 10.5281/zenodo.19229065) computed |S₁ᴬ| to five significant figures. The present paper completes the arc by proving the sign and identifying the alien-derivative structure. Note: The boundary-layer coefficient d₁ = (2α−1)/4 from the Paper 10 v5 erratum is load-bearing throughout.