Absence of Structural Phase Transition but Emergence of Heavy-Tailed Statistics in Spin Glass Landscapes Controlled by Kurtosis

This repository contains the full dataset, code, and analysis supporting the study: "Absence of Universal Structural Scaling in Spin Glass Energy Landscapes: A Heavy-Tail Perspective" We investigate whether the universal structural law observed in Graph Laplacian Geometry (M18) extends to spin glass systems. Using a large ensemble of randomly generated spin glass instances with controlled kurtosis (κ), we analyze multiple structural observables of the energy landscape, including: - Effective dimension (dG) via PCA- Number of local minima- Energy landscape roughness (standard deviation of energies) - Energy gaps between lowest minima Our results show that, unlike in GLG systems, kurtosis does not induce a global structural reorganization of the landscape. No universal phase transition or sigmoid collapse is observed in any structural metric. Instead, kurtosis affects the distribution of energy gaps, leading to the emergence of heavy-tailed behavior. A detailed tail analysis (power law vs log-normal) indicates that: - Low κ regimes are consistent with log-normal tails- High κ regimes show signatures compatible with power-law behavior (α ≈ 2. 3–2. 6) This establishes spin glass systems as a structural counterexample to the universality observed in M18, thereby delimiting the domain of validity of the proposed universal law. All experiments are fully reproducible. The repository includes raw data, processing scripts, and analysis pipelines.