Sharp Planar Anisotropic Capillarity Outside Convex Obstacles and Two-Point Rigidity of Transported Boundary Images

For a convex surface tension $φ$ and the wall-energy density $σ(n)=-λ\,φ(-n)$, we prove sharp capillary inequalities outside arbitrary convex planar obstacles for -1<λ≤0 and every finite-perimeter competitor of finite positive volume, including non-even and crystalline tensions. Equality holds precisely for translated dilates of Wulff caps in minimum-volume directions, with the entire truncating facet resting on a flat supporting face. A uniform quantitative estimate controls component concentration, shape up to translation, and directional cap-volume excess. The proof combines exact filling, planar completion, and quantitative Wulff stability. For the transport $DΦ$, Φ=φ²/2, the zero-adhesion boundary-image estimate characterizes centered ellipsoids among even anisotropies with Φ∈ C³ away from the origin and positive definite Hessian. Two-point slab tests with arbitrary boundary data suffice: their projection balances yield a cubic identity forcing $Φ$ to be quadratic. Facets and smooth perturbations provide counterexamples to the transported estimate, including failures at prescribed adhesion levels in $(-1,1)$. Strict finite deficits persist under Hausdorff perturbations of the Wulff body. The classification concerns an unrestricted boundary-data property, stronger than the image estimate needed only for traces of auxiliary Neumann solutions.