Find an experimental signature of exceptional arithmetic moduli

Identify a directly measurable quantity whose value distinguishes the square modulus \(\tau=i\) or the hexagonal modulus \(\tau=\rho\) from a nearby non-exceptional modulus.

Background

The paper proves that the determinant response $\mathcal K=2\pi(\operatorname{Im}\tau)G_\tau^{-1}$ has an isotropic form at the symmetry-fixed deep holes of the square and hexagonal lattices. However, the authors explicitly state that this determinant response is not observable in the proposed laboratory setup. They also explain that the measurable centroid curvature follows from quadratic dispersion and occurs for broader classes of lattices, so it cannot distinguish the exceptional unit groups. The unresolved problem is therefore to find an experimentally accessible observable that detects the arithmetic or enhanced rotational symmetry associated with $\tau=i$ or $\tau=\rho$.