Riemannian Tensor Analysis Identifies a Bifurcation State in the Single-Cell Transcriptomic Landscape of Parkinson's Disease

The transition from healthy aging to Parkinsons disease (PD) involves highly volatile transcriptomic rewiring that remains invisible to conventional mean-based analyses. To decode this critical tipping point, we integrated single-cell RNA-sequencing with a novel Log-Euclidean Riemannian tensor framework. By conceptualizing distinct transcriptomic states as symmetric positive definite (SPD) covariance tensors, we bypassed Euclidean geometric artifacts to accurately map the macroscopic network architecture of the human prefrontal cortex. Our analysis identified a highly unstable, intermediate bifurcation (BIF) state. Thermodynamic and topological validation demonstrated that this BIF state operates as a definitive geometric saddle point, characterized by maximal von Neumann entropy and positioned perfectly equidistant between the healthy (HC) and pathological (PD) Frechet means on the non-Euclidean manifold. Furthermore, spectral deconstruction of differential covariance networks [Formula] isolated the exact topological drivers of this transition--revealing the catastrophic structural collapse of core synaptic and electrophysiological hubs prior to overt pathological commitment. Ultimately, this ab initio geometric framework fundamentally redefines PD progression, providing a quantitative blueprint to intercept neurodegenerative trajectories during their final reversible window.