Modular-cut subextension existence

Construct, for every modular cut M' contained in a modular cut M that determines a single-element extension of an oriented matroid, a single-element extension corresponding to M'.

Background

The paper proposes this conjecture as a tool for extending the rank-four mutation argument to arbitrary ranks and for selecting extensions that pass through prescribed cocircuits. The statement concerns the existence of an oriented-matroid extension associated with a sub-modular-cut.

References

If there is an extension $\mathcal{O} \cup p$ corresponding to a modular cut $\mathcal{M}$ of the corresponding matroid, and if $\mathcal{M}' \subseteq \mathcal{M}$ is also a modular cut, then there exist an extension $\mathcal{O} \cup p'$ corresponding to $\mathcal{M}'$.

— Mutations and (Non-)Euclideaness in oriented matroids  (2501.12951 - Wilhelmi, 22 Jan 2025) in Section Euclideaness and mutations, immediately after Theorem UniformIP2FourAdjacentMutations