Non-cuppable enumeration degrees via finite injury

We exhibit finite injury constructions of a high Σ2 enumeration degree incomparable with all intermediate ∆2 enumeration degrees, as also of both an upwards properly Σ2 high and a low2 noncuppable Σ 0 2 enumeration degree†. We also outline how to apply the same methods to prove that, for every Σ2 enumeration degree b there exists a noncuppable degree a such that b′ ≤ a′ and a′′ ≤ b′′, thus showing that there exist noncuppable Σ2 enumeration degrees at every possible level of the high/low jump hierarchy.