arXiv:2610.27814v1 Announce Type: new Abstract: Schumaker's lower bound incorporates local geometric corrections and provides insight into the dimensions of bivariate spline spaces. Building on this bound, we study bivariate $C^2$ spline dimensions from three perspectives: universal attainment, generic attainment, and geometric jumps. We prove universal attainment in degree seven for every legal realization of every triangulation. We prove generic attainment in degrees five and six for every fixed combinatorial triangulation and give explicit polynomial conditions certifying generic failure on fixed triangulations in degrees three and four. For geometric jumps with matching local direction types at corresponding interior vertices, we prove nonexistence in degree seven and construct examples in degrees three and five. In degree six, universal attainment and the existence of such geometric jumps remain open questions.