Subdivision surfaces have been widely adopted in modeling in part because they introduce a separation between the surface and the underlying basis functions. Such a separation allows for simple schemes that work on general topology surfaces. Multiresolution representations based on subdivision, however, incongruently return to continuous functional spaces in their construction and analysis. In this paper, we investigate a discrete approach to multiresolution construction for a variety of subdivision schemes, based only on the subdivision rules. Noting that a compact representation can only afford to store a subset of the detail information, our construction enforces a constraint between locally adjacent detail terms. In this way, all detail information is recoverable for reconstruction, and a decomposition approach is implied by the constraint. The construction is demonstrated with case studies in DynLevin-Gregory curves and Catmull-Clark surfaces, each of which our method produces results as good as earlier methods. It is further shown that our construction can be interpreted as biorthogonal wavelet systems.