QHSX Entanglement–Energy Landscapes: Phase Structure, Correlation Scaling, and Tail-Min Energy Signatures

This preprint presents a reproducible empirical study of entanglement–energy landscapes generated by the QHSX/MEGA ELITE pipeline for the configurations (n, d) ∈ {(8,4), (8,6), (10,4), (10,6)}. The analysis investigates statistical dependence between sampled energy and negativity-derived entanglement proxies, the geometry of the joint energy–entanglement landscape, and minimum / tail-minimum energy trends across configurations. Across all four datasets, the study finds strong negative Pearson and Spearman correlations between energy and the principal entanglement proxy, together with non-uniform and phase-like geometric organization in the sampled landscape. The manuscript does not claim a formal thermodynamic phase transition; rather, it reports reproducible empirical evidence of structured regime separation and coherent low-energy behavior within the analyzed range. The work is intended as an empirical contribution to variational quantum optimization, entanglement-aware benchmarking, and exploratory landscape analysis in sampled quantum circuit families.