In his thesis  Sy Friedman showed that for any inadmissible ordinal β there is an easily definable β-r.e. degree, 01/2, with β-degree strictly between 0 and 0′. In this paper we define, for some weakly admissible β, a β-r.e. degree 03/4 intermediate between 01/2 and 0′. This definition is given using the machinery developed by Maass ,  to study the β-r.e. degrees.
A weak jump for β-recursion theory, the ½-jump, was first defined by Friedman  and has been studied by Sacks and Homer . The degree 03/4 is of interest because of the way it interacts with the half-jump. In particular 03/4 forms the boundary between the half-jump of “hyperregular” and “nonhyperregular” β-r.e. degrees less than 01/2. One consequence of these results is that there are weakly inadmissible β for which either the jump theorem or the density theorem fails.
Much of this paper is based on earlier work in this area by Sy Friedman and Wolfgang Maass. I would like to thank Gerald Sacks for many helpful conversations and comments concerning this work.