A variational characterization of geodesic scattering on simple disks

We give necessary and sufficient conditions for a boundary map to be the geodesic scattering map of a smooth simple Riemannian metric on a fixed disk with a prescribed boundary metric. The conditions depend only on the proposed map and that boundary metric. Boundary invariant functions first determine a normalized conformal boundary coordinate. In the resulting Euclidean disk, length inequalities define a nonempty convex set of densities whose strong $L^2$ closure has a unique element of least norm. Realizability is equivalent to smooth positivity and the prescribed boundary value of this element, explicit geodesic and Jacobi equation tests, and a scalar area equality. The proof transfers the length inequalities through the closure by integrating over the geodesic family. Strict monotonicity of the area functional then forces equality of boundary distances and hence of scattering maps. The characterization is exact and qualitative; no finite decision procedure or stability estimate is asserted.