Analytical determination of the maximal spherical t-design strength

Determine analytically the maximal value of t achievable by a spherical t-design as a function of the number N of points.

Background

The paper identifies stealthy hyperuniform point patterns on the sphere with spherical t-designs: imposing vanishing structure factors S_l for all degrees l up to t yields exact integration for spherical polynomials of degree at most t. A Maxwell-type degree-of-freedom count predicts a realizability threshold near chi = 1, corresponding to a critical relation between the number of points and the largest constrained harmonic degree. The numerical results support this threshold, but the paper explicitly notes that an analytical argument establishing the maximal t as a function of N is still unavailable.

References

This constitutes conclusive evidence regarding the ongoing debate on the maximal value of $t$ in a $t$-design as a function of $N$, which to the best of our knowledge is not settled by an analytical argument.

— Fast generation of spectrally-shaped disorder, on the sphere  (2608.24867 - Casiulis et al., 25 Aug 2026) in Section 3.1, Stealthy Hyperuniformity