Forward stability in the maximal Levi-spherical class

Establish that for independent permutations u,v drawn uniformly from the maximal Levi-spherical class Lev_n, the expected forward stability satisfies E[FS(u,v)] = (5/4) n + O(1) as n→∞.

Background

Maximal Levi-spherical permutations arise from a root-theoretic and geometric criterion related to dense orbits of Borel subgroups in Levi factors. The paper links these to Boolean permutations via a parabolic longest-element transformation.

The conjectured 5/4 coefficient would precisely place this class in the intermediate regime, reflecting a specific record density and correlation structure.

References

Conjecture [Intermediate record regime] As n→∞, the following hold. (b) (Maximal Levi-spherical.) If u,v∼ Unif{Lev_n}, then \E{\FS(u,v)} = \frac{5}{4}\,n + O(1).

— The record statistic and forward stability of Schubert products  (2604.02964 - Hardt et al., 3 Apr 2026) in Section 7 (Conjectures), Conjecture [Intermediate record regime]