We say a countable model ������ has a 0-basis if the types realized in ������ are uniformly computable. We say ������ has a (d-)decidable copy if there exists a model ������ ≅ ������ such that the elementary diagram of ������ is (d-)computable. Goncharov, Millar, and Peretyat'kin independently showed there exists a homogeneous model ������ with a 0-basis but no decidable copy. We extend this result here. Let d ≤ 0' be any low₂ degree. We show that there exists a homogeneous (...) model ������ with a 0-basis but no d-decidable copy. A degree d is 0-basis homogeneous bounding if any homogenous ������ with a 0-basis has a d-decidable copy. In previous work, we showed that the nonlow₂ $\Delta _{2}^{0}$ degrees are 0-basis homogeneous bounding. The result of this paper shows that this is an exact characterization of the 0-basis homogeneous bounding $\Delta _{2}^{0}$ degrees. (shrink)