A characterization of the 0-basis homogeneous bounding degrees

We say a countable model A has a 0-basis if the types realized in A are uniformly computable. We say A has a (d-)decidable copy if there exists a model B ∼= A such that the elementary diagram of B is (d-)computable. Goncharov, Millar, and Peretyat’kin independently showed there exists a homogeneous model A 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 A with a 0-basis but no d-decidable copy. A degree d is 0-basis homogeneous bounding if any homogenous A with a 0-basis has a d-decidable copy. In previous work we showed that the nonlow2 ∆2 degrees are 0-basis homogeneous bounding. The result of this paper shows that this is an exact characterization of the 0-basis homogeneous bounding ∆2 degrees.