Finite-Size Scaling of Structural Coherence in Critical Spin Networks

We introduce the Structural Coherence Index (SCI), a network-based observable designed to distinguish genuine structural coherence from fragmentation-induced false positives in modular systems. The framework combines modularity, network density, and largest connected component persistence into a single observable: SCI = Q × ρ × (LCC / N) We investigate the behavior of SCI in the two-dimensional Ising model near criticality using Monte Carlo simulations, finite-size scaling analysis, ensemble averaging, and randomized controls. The results demonstrate that SCI consistently separates real physical structures from randomized counterparts, remains robust against fragmentation artifacts, and exhibits nontrivial finite-size scaling behavior: SCI(L) ~ L^{-α} with temperature-dependent effective exponents near the Onsager critical point. The study suggests that SCI behaves as a residual structural observable sensitive to multiscale coherence and critical fluctuations in complex systems.