Extracting effective scaling exponents in finite-size hyperuniform systems

Hyperuniform systems strongly suppress long-wavelength density fluctuations, which is quantitatively characterized by the small-wavenumber scaling. In finite samples, however, accurately estimating the hyperuniformity exponent {\alpha} can be challenging. The inferred value depends strongly on the range of length scales accessible in the measurement, finite-size effects, and the specific characterization method employed, whether based on Fourier-space structure factors, real-space density fluctuations, or dynamical probes such as diffusion spreadability. In particular, the structure-factor method provides the most direct estimate of {\alpha}, but is sensitive to empirical low-k fitting cutoffs. The number-variance method offers a real-space Class-like diagnosis, but contributes a numerical exponent only when the finite-size data retain Class III-like scaling information. The spreadability method provides a smoother dynamic estimate and reduces configuration-level fluctuations, but requires a physically admissible long-time fitting window. Here, we develop a practical method-aware protocol for robust estimation of the effective scaling exponent {\alpha} in finite-size hyperuniform point configurations, combining three complementary methods with distinct roles. Our protocol summarizes the method-specific estimates through a joint empirical estimator and reports the internal dispersion among the participating methods to determine the optimal estimate.