Uniqueness of the β′ sequence for higher-order spacings under superposition

Establish the uniqueness of the sequence of modified Dyson indices β′ determined by the equality P^{(k)}(s, β, m) = P(s, β′) for higher-order spacings in superposed circular ensembles: for fixed number m of superposed spectra and fixed Dyson index β ∈ {1,2,4}, show that the sequence β′(k) is unique as k varies; equivalently, for fixed k and β, show that the sequence β′(m) is unique as m varies.

Background

The paper studies higher-order spacings (HOS) in superposed spectra of Dyson’s circular ensembles and introduces a modified Dyson index β′ defined via the generalized Wigner–Dyson identification P^{(k)}(s, β, m) = P(s, β′). Numerical evidence is tabulated for many values of k and m across COE, CUE, and CSE.

The conjectured uniqueness of the sequence β′(k) (or β′(m)) would enable systematic characterization of symmetry structure directly from superposed spectra without desymmetrization, by matching k-th order spacing distributions to nearest-neighbor distributions with parameter β′.

Related uniqueness for spacing ratios was previously conjectured in the literature for circular ensembles; the present work advances an analogous conjecture for spacings and documents extensive numerical support.