Limit-level separation for Σλ vs. Σλ+1 generically c.e. copies

Determine whether there exists, for a limit ordinal λ < ω1, a countable linear ordering that has a Σλ-generically computably enumerable (c.e.) copy but does not have a Σλ+1-generically c.e. copy.

Background

In Section 3.7, the authors construct linear orderings Z(A) that, for a given limit ordinal λ, have Σα-generically computable copies for all α < λ but provably do not have Σλ-generically computable copies (nor Σλ-coarsely c.e. copies) for appropriate choices of A. This leaves open whether a strict separation between Σλ and Σλ+1 can be witnessed at the limit level.

Establishing such a separation would clarify the precise behavior of generically c.e. copies at limit stages and the successor of limit stages in the computable Σ-hierarchy, complementing the earlier completeness and separation results proved for successor levels throughout the paper.