Parity-based exclusions for Borel levels of L(I)

Determine whether, for uncountable Polish spaces X, the following hold: (a) if α is odd and I ∈ Π^0_{α+1} \ Σ^0_{α+1}, then L_X(I) ≠ Π^0_α(X); and (b) if α is even and J ∈ Σ^0_{α+1} \ Π^0_{α+1}, then L_X(J) ≠ Σ^0_α(X).

Background

Theorem 10.12 constructs ideals whose L(I) attain certain Borel classes at high levels, and Remark 10.13 discusses boundary cases where parity matters. However, for α≥3 these parity-dependent exclusions remain unsettled.

The authors highlight that a positive resolution of their main coanalytic dichotomy (Question 10.9) would imply affirmative answers for α=3, linking these parity questions to the broader classification program.