Initial segments of the Σ0 2 enumeration degrees

Using properties of κ-pairs of sets, we show that every nonzero enumeration degree a bounds a nontrivial initial segment of enumeration degrees whose nonzero elements have all the same jump as a. Some consequences of this fact are derived, that hold in the local structure of the enumeration degrees, including: There is an initial segment of enumeration degrees, whose nonzero elements are all high; there is a nonsplitting high enumeration degree; every noncappable enumeration degree is high; every nonzero low enumeration degree can be capped by degrees of any possible local jump (i.e., any jump that can be realized by enumeration degrees of the local structure); every enumeration degree that bounds a nonzero element of strictly smaller jump, is bounding; every low enumeration degree below a non low enumeration degree a can be capped below a.