Fourth-level quantifier-pattern classification

Determine the many-one and dm-equivalence classes of Sigma-zero-four and Pi-zero-four quantifier patterns, including the number of classes at each level.

Background

The paper gives complete realizability-theoretic classifications through the third arithmetical level and states that the fourth-level classifications remain in progress.

References

Determine the equivalence classes of quantifier patterns at the fourth level of the arithmetical hierarchy: (1) How many $\equiv_m$-equivalence classes of $\Sigma0_4$ quantifier patterns are there? (2) How many $\equiv_m$-equivalence classes of $\Pi0_4$ quantifier patterns are there? (3) How many $\equiv_{dm}$-equivalence classes of $\Sigma0_4$ quantifier patterns are there?

— Open Problems in Mathematical Logic  (2608.26628 - Barmpalias et al., 27 Aug 2026) in Section 3, after the classification of the third level