Optimal regularity for stable obstacle problems with integrable angular densities

We prove a global $C^{1,s}$ estimate for the finite-energy obstacle problem of a symmetric stable operator whose angular density is only integrable. The constant depends on the order, dimension, angular mass bound, and quadratic nondegeneracy bound, without stronger angular integrability or positivity in every direction. The classification step uses the convex primitive: a derivative associated with a boundary ray of a convex contact cone has an upper bound of order $t$, whereas a spectral-measure Hopf lemma gives a lower bound of order $t^s$. We establish the compactness needed when angular densities concentrate, classify convex global solutions, and combine a normalized profile approximation with the density-kernel flatness estimate of Ros-Oton and Weidner. The endpoint obstacle estimate is stated for angular densities; the auxiliary classification allows general finite spectral measures.