MBL Crossover: Level-Flow & Phase Diagram

This is the 2D-native companion to the 3D "Level Statistics Across the MBL Transition" simulator. It exactly diagonalizes the same disordered Heisenberg spin chain — the canonical microscopic model for many-body localization — but instead of rendering one snapshot spectrum as a rotating 3D spiral, it sweeps disorder strength W/J across nine values and draws the eigenvalues directly as an energy-vs-disorder level-flow diagram: a plane where avoided crossings mean level repulsion (thermal, Wigner-Dyson) and free crossings mean level clustering (many-body localized, Poisson). Averaging the adjacent gap ratio ⟨r⟩ at each sampled W produces a live phase-transition curve against the two textbook reference values, and a flat 2D histogram shows the full r-distribution at whichever W the slider highlights, benchmarked against the analytic Poisson and GOE probability densities.