Dirichlet inverse asymptotics for stable kernels with a periodic cosine factor

For the stable jump kernel with periodic factor 1+η cos(2π(x₁−y₁)/ε), we determine the first-order asymptotics of the exterior Dirichlet inverse on a fixed bounded Lipschitz domain in R^d. For 1 < α < 2 and 0 < η < 1, the inverse admits an L² operator-norm expansion with leading scale ε2−α and remainder O(ε2−α+ρ) for every 0 < ρ < min{2−α,α−1}. The coefficient is a positive operator obtained by placing an explicit elliptic gradient form between two homogenized Dirichlet inverses. Its norm is the exact positive limit of the rescaled inverse discrepancy. For suitable fixed smooth sources, the leading solution error is nonzero on an interior compact set, which gives a lower bound after arbitrary additive corrections supported in a shrinking boundary strip. Fixed exterior data with an H²(R^d) extension satisfy a corresponding first-order limit. The proof combines the exact Fourier multiplier, zero-extension Sobolev regularity, and a quantitative quadratic-form inverse identity.