Strictness of the termination-provability inclusion

Determine whether the inclusion of termination-provability classes $PT{RGA} \subseteq PT{OGA}$ is strict; equivalently, establish whether some primitive-recursive step function has a totality universal provable in OGA but not in RGA.

Background

The paper proves that every termination claim provable in RGA is also provable in OGA, yielding $PT{RGA} \subseteq PT{OGA}$. It also establishes a common ceiling: some everywhere-hitting step function lies outside both classes.

Although OGA adds the ATI rule and decides every totality question, ATI directly supplies only decidedness judgments rather than affirmative totality proofs. The authors therefore do not produce a function separating the two termination-provability classes and explicitly leave open whether the inclusion is proper.