Intermediate fragment between CM and CM + GWO with weaker strength but comparable utility

Identify a theory fragment extending CM and strictly weaker in proof‑theoretic strength than CM + GWO that nonetheless proves the principal mathematical results obtained via the global well‑ordering axioms, such as Zorn‑type maximality arguments and basis existence in vector spaces.

Background

The paper shows that the full extension CM + GWO reaches the strength of BI, while certain restricted fragments can be interpreted in weaker classical systems such as ATR. However, the author notes a gap: an intermediate extension that preserves much of the mathematical utility developed (e.g., Zorn’s lemma analogues and basis results) yet falls strictly below CM + GWO in proof‑theoretic strength has not been identified.

Resolving this would sharpen the landscape of predicative extensions of CM by balancing strength and applicability, potentially isolating a robust but weaker system that still handles the intended mathematics.