On a question of sacks: a partial solution on the positive side

Let us say that a c.e. operator E is degree invariant on any given Turing degree a if X,Y ∈ a → E(X) ≡  T E(Y) . In [4] we construct a c.e. operator E such that ∀X [ X <  T E(X) <  T X′] . While we are unable to produce degree invariance everywhere, we are able to ensure that for every degree a there exists b such that a ∨ 0′ = b ∨ 0′ and E is degree invariant on b . What appears here is an abbreviated version of the material from that paper, stopping short of most technical details.