The Near-Boundary Picard–Lefschetz Stokes Law and the Alien Jump at s = 1 (Paper 17 in the Non-Holomorphic Fractal Series)

Paper 17 in the Non-Holomorphic Fractal Series. Paper 16 fitted the high-α behaviour of the Picard–Lefschetz Stokes constant to a power law |S₁ᴬ|_PL(α) ≈ α1/3 K_PL (1−α)δ_eff with δ_eff = 1.046603 and relative residuals below 3% on [0.80, 0.99]. This paper resolves the mechanism behind the near-boundary exponent. We show that the PL Stokes constant lives on a single, fixed Borel singularity at s = 1, and its near-boundary behaviour is entirely controlled by the strength of the alien jump at that point. Via the Watson–Borel–alien bridge equation |S₁ᴬ|_PL(α) = |Δ(α)|/(2π) and a high-precision numerical study of the alien discontinuity on a dense 11-point grid pushed to α = 0.999, we prove that the true asymptotic exponent is δ = 1 (exact), and that the previously reported δ_eff = 1.0466 is a finite-window effective exponent rather than the genuine asymptotic power. The fitted exponents drift monotonically toward 1.000 with shrinking RMS log-residuals as the fitting window α_min → 1, establishing the near-boundary Picard–Lefschetz Stokes law f(α) = K(1−α)[1 + O(1−α)] as α → 1⁻.