Non-cuppable enumeration degrees via finite injury

We exhibit finite injury constructions of a high Σ02 enumeration degree incomparable with all intermediate Δ02 enumeration degrees, as also of both an upwards properly Σ02 high and a low non-cuppable Σ02 enumeration degree.We also outline how to apply the same methods to prove that, for every Σ02 enumeration degree b there exists a non-cuppable degree a such that b' ≤a' and a" ≤ b", thus showing that there exist non-cuppable Σ02 enumeration degrees at every possible level of the high/low jump hierarchy.