Fractional thermodynamics resolves the Stokes-Einstein breakdown in supercooled water

The breakdown of the Stokes-Einstein relation in supercooled water, where the product [Formula: see text] increases by 35% approaching the homogeneous nucleation limit, has resisted theoretical explanation for decades. We develop a two-state fractional thermodynamics framework that quantitatively resolves this anomaly with 1.0% experimental agreement. The key insight is that translational and rotational degrees of freedom exhibit dramatically different fractional dynamics in the tetrahedral low-density liquid phase: translation approaches ballistic motion ([Formula: see text]) through coherent inter-cage jumps while rotation remains strongly subdiffusive ([Formula: see text]) due to cooperative hydrogen-bond network constraints. This yields a decoupling ratio [Formula: see text] that, combined with percolative transport and critical fluctuations near the Widom line, explains the observed breakdown. We derive modified critical exponents from fractional Landau theory and present testable predictions for neutron scattering and molecular dynamics simulations. This framework establishes fractional calculus as essential for understanding anomalous transport in hydrogen-bonded liquids.