Characterization and abundance of strongly relativizing reals for higher analytical pointclasses

Characterize, perhaps under Projective Determinacy, the reals that strongly relativize the higher analytical pointclasses \(\Delta^1_n\) for \(n\geq 1\), and determine whether uncountably many such reals exist.

Background

The paper completely characterizes the reals strongly relativizing $\Delta^1_1$ as the hyperlow reals. For higher analytical self-dual pointclasses $\Delta^1_n$, the paper establishes only conditional results under Projective Determinacy and notes that $\Delta^1_n$-singletons provide examples. It remains unresolved whether these examples can be substantially extended or whether there are uncountably many strongly relativizing reals.