Uniqueness of the modified Dyson index sequence β′ for higher‑order spacings in superposed circular ensembles

Establish the uniqueness of the sequence of modified Dyson indices β′ obtained by fitting the k‑th order spacing distribution P^{(k)}(s,β,m) of m superposed spectra from a circular ensemble with Dyson index β to the nearest‑neighbor spacing distribution P(s,β′), as a function of k (for fixed m and β) or as a function of m (for fixed k and β). Concretely, determine whether the mapping k↦β′ (or m↦β′) is unique for each choice of β∈{1,2,4} when the spectra are drawn from COE, CUE, or CSE and superposed with equal block dimensions.

Background

The paper numerically studies higher‑order spacings (HOS) in superposed spectra of circular ensembles (COE, CUE, CSE) and, for each (m,k,β), identifies a modified Dyson index β′ such that the k‑th order spacing distribution matches the nearest‑neighbor spacing distribution with parameter β′.

Prior work on higher‑order spacing ratios (HOSR) in superposed spectra conjectured uniqueness of the β′ sequence as a function of k or m for given β. This paper extends the investigation to HOS and explicitly conjectures the same uniqueness property for β′, providing extensive tabulations and numerical evidence but leaving the formal proof open.