Normal distance distribution in modular Heisenberg groups

Prove that the distance distribution in the higher-dimensional modular Heisenberg group H_{2d+1}(Z/mZ) is asymptotically approximately normal.

Background

The paper notes that distance distributions in finite abelian groups are asymptotically normal and presents computational evidence suggesting the same behavior for higher-dimensional modular Heisenberg groups.

References

The distribution of distances in the higher-dimensional modular Heisenberg group $H_{2d+1}(Z/mZ)$ is approximately normal.

— CayleyPy Growth: Efficient growth computations and hundreds of new conjectures on Cayley graphs (Brief version)  (2509.19162 - Chervov et al., 23 Sep 2025) in Section 10, subsection “Distribution of distances in finite nilpotent groups”