Contact set localization for nonlocal obstacle problems near order two

We study zero exterior-value obstacle problems whose integral operators have a fixed smooth even angular density and are normalized by 1−s. Suppose that the zero extension of the local solution belongs to H^{1+τ} for some 0 < τ < 1/2 and that its nonnegative L² reaction is supported on the contact set. At a fixed Lipschitz patch of the local free boundary with quadratic gap growth and strictly positive local obstacle forcing, we prove phase trapping and a localized Hausdorff rate O((1−s)^β), where β = γ/(n+2γ) > 0 and γ is an interior Hölder exponent uniform in s. The proof combines an O(1−s) multiplier residual in H^{−s}, variational stability in H^s, and a two-cutoff clearing estimate that separates the local gap error from the remote tail contribution (1−s)M. The geometric conclusion controls the location of all interface sheets in the patch. A complementary family of scalar gaps with smooth oscillating graphs has uniform function estimates and persistent normal variation, identifying the additional graph control needed for convergence of normals.