Realizability degrees of partial-order obstructions

Determine the realizability-theoretic many-one and Wadge relationships among IFPO, Anti, NWPO, and BadPO, including the specified width-restricted comparisons.

Background

Classically, ill-foundedness, infinite antichains, failure of well-partial-order, and bad sequences describe related properties. Their witness structures differ under realizability-theoretic reducibility, motivating the requested comparisons.

References

Determine the realizability-theoretic many-one and Wadge relationships among the partial-order problems above. In particular: (1) Is $\mathrm{IFPO}\equiv_m\mathrm{BadPO}$? (2) For which $m,n\geq 2$ does $\mathrm{BadPO}_n\leq_m\mathrm{IFPO}_m$ hold? (3) For which $m,n\geq 2$ does $\mathrm{IFPO}_n\nleq_m\mathrm{BadPO}_m$ hold?

— Open Problems in Mathematical Logic  (2608.26628 - Barmpalias et al., 27 Aug 2026) in Section 3, after the definitions of IFPO, Anti, NWPO, and BadPO