Characterization of maximizing stellar constellations for large boson numbers

Characterize the geometrically inequivalent Majorana stellar constellations of larger numbers of two-level bosons that satisfy the maximal-incompatibility condition and attain the upper bound for the quantifier E_N.

Background

For N two-level bosons, the paper derives the upper bound (E_N)_max = (2/3)N/(N-1) for N >= 3. The maximizing configurations are determined by the isotropy condition M = (N/3)I_3, where M is the second-moment matrix of the Majorana Bloch vectors.

The authors explicitly note that, beyond the already analyzed N = 4 case, progressively richer and potentially inequivalent stellar configurations may satisfy this condition. Determining and classifying these configurations would clarify the geometric structure of states exhibiting maximal property-attribution incompatibility for larger systems.

References

Several questions remain open for future investigation. For larger numbers of two-level bosons, the characterization of the stellar constellations satisfying the maximal-incompatibility condition may reveal a richer structure of geometrically inequivalent configurations, yet the maximal amount of incompatibility ($E_N)_{\max}$ will decrease progressively as the system's size increases.

— Geometric quantifier of the incompatibility of single-particle property attribution in indistinguishable boson systems  (2609.17256 - Céspedes et al., 15 Sep 2026) in Section Final remarks