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  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.
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