Geometric Thermodynamics of Protein Folding: Sequence-Encoded Riemannian Metrics and Spectral Evidence for Topological Phase Transitions

The prediction of protein folding kinetics remains a fundamental challenge due to the high-dimensional complexity of energy landscapes 1, 2. We present a novel framework, Geometric Thermodynamics, which maps the configuration space onto a Riemannian manifold (M) where the metric tensor (g₈₉) is derived from sequence-specific propensities 3. Our Riemannian Langevin Solver (RLS) achieves a correlation of (R² = 0. 89 0. 03) with experimental folding rates (N=250, ACPro/PFDB datasets) 4, 5 and a (10⁴) speedup over all-atom molecular dynamics (MD) 6. Furthermore, we provide spectral evidence that pH-induced misfolding in () -synuclein is a formal Poincaré-Andronov-Hopf bifurcation, identified by eigenvalue crossing in the flow Jacobian 7, 8.