Closure of the determinantal hierarchy at the resolvent level

Establish whether the determinantal hierarchy of normalized overlap moments q_r closes at the resolvent level of the hierarchical ETH framework.

Background

The random free-fermion ensemble provides an exact hierarchy of same-channel normalized moments q_r, whose fixed-depth limit is governed by squared Gaussian determinants rather than Porter–Thomas statistics. These moments are proposed as targets for a microscopic closure of the fluctuation hierarchy.

However, the paper does not derive a closure connecting this determinantal moment family to the resolvent-level description. Whether such a closure exists remains unresolved.

References

Whether the determinantal hierarchy closes at the resolvent level of the framework is not established here.

— Eigenstate thermalization beyond the envelope: exact two-point overlap statistics in random free fermions  (2609.17037 - Huang, 15 Sep 2026) in Section 6, Discussion, paragraph beginning “It is useful to separate four levels in this benchmark”