Necessary and sufficient condition for strong relativization of higher effective Borel classes

Identify a natural condition that is necessary and sufficient for a real \(f\in N\) to strongly relativize \(\Delta^0_{1+\alpha}\) at Baire space \(N\), for each computable ordinal \(\alpha>0\).

Background

For computable ordinals $\alpha>0$, the paper proves that $\Delta^0_{1+\alpha}$-domination is sufficient for strong relativization at $N$, but also constructs reals that strongly relativize $\Delta^0_{1+\alpha}$ without satisfying this domination condition. Thus the sufficient condition is not necessary, and the paper explicitly leaves open the search for a natural exact characterization.