Extracting effective scaling exponents in finite-size hyperuniform systems.

Hyperuniform systems strongly suppress long-wavelength density fluctuations, quantitatively characterized by the small-wavenumber power law Sk∼kα. In finite samples, extracting the scaling exponent α is complicated by limited accessible length scales, finite-size effects, and method-dependent fitting windows. Here, we develop a practical, method-aware protocol for obtaining reproducible finite-size effective scaling exponents from hyperuniform point configurations. These estimates are finite-size descriptors obtained under a specified analysis protocol rather than thermodynamic-limit extrapolations. Regularized low-k window screening provides the structure-factor estimate, while a diffusion-length-constrained plateau-window analysis provides the spreadability estimate. The two numerical estimates are combined using an equal-weight default, and sensitivity to the weighting choice is assessed separately. Their separation and repetition-level sampling uncertainty are quantified using method dispersion and paired-bootstrap intervals. Number variance is reported separately as an independent real-space Class-like diagnosis and, only for Class III-like behavior, as an approximate exponent reference. We benchmark the protocol on two-dimensional targeted hyperuniform configurations with prescribed exponents of αtheory= 0.3-4.0. Regularized Sk fitting reduces cutoff sensitivity, the number-variance results follow the expected Class-like behavior in most repetitions, and the plateau-window spreadability protocol avoids inappropriate early-time fits and suppresses configuration-dependent window drift. Across the benchmark set, equal-weight joint estimates remain close to the prescribed exponents and provide reproducible finite-size summaries. This method-aware framework provides a standardized basis for comparing finite hyperuniform samples and diagnosing where individual analysis routes retain or lose exponent information.