We show that every splitting of 0e' in the local structure of the enumeration degrees, 𝔊e, contains at least one low-cuppable member. We apply this new structural property to show that the classes of all 𝓚-pairs in 𝔊e, all downwards properly Σ⁰₂ enumeration degrees and all upwards properly Σ⁰₂ enumeration degrees are first order definable in 𝔊e.
"Cupping and definability in the local structure of the enumeration degrees." J. Symbolic Logic 77 (1) 133 - 158, March 2012. https://doi.org/10.2178/jsl/1327068696