Two-sided spectral conditioning in the intermediate separation regime

Determine whether arbitrary-cloud spherical Fourier Gram matrices admit uniform two-sided spectral conditioning when the separation satisfies an intermediate scale between (s^{1/3}) and (s^{2/3}), thereby resolving the gap between the known worst-case lower- and upper-frame thresholds.

Background

The paper establishes a sufficient arbitrary-cloud separation condition of order $\kappa\delta_X\gtrsim s^{2/3}$ for two-sided frame bounds on the spherical Fourier atoms, using an absolute coherence row-sum estimate and Gershgorin's theorem. It also constructs grid-restricted counterexamples showing that, in the worst case, the lower-frame bound requires at least the $s^{1/6}$ scale and the upper-frame bound requires at least the $s^{1/3}$ scale.

Consequently, the known results leave an intermediate range between exponents $1/3$ and $2/3$. The unresolved issue is whether uniform two-sided spectral conditioning can be established somewhere in this range, or whether stronger obstructions exist. The authors explicitly distinguish this question from the sharpness of the absolute-row-sum proof method, whose exponent $2/3$ is separately shown to be optimal.