Spectral Unification and Supersymmetric Dynamics of Pathological Transitions in Multivariable Biological Systems: A Network Fokker-Planck Framework

Abstract

We present a rigorous mathematical framework based on Supersymmetric Quantum Mechanics (SUSY QM) and Network Fokker-Planck theory to model stochastic transitions from homeo- static to pathological states in complex biological systems. By embedding non-linear reaction- clearance dynamics directly into an effective free-energy landscape Ueff(Q), we derive the Net- work Fokker-Planck Equation (NFPE) on discretized graph structures. We analytically prove that this formulation guarantees strict total probability conservation via the fundamental topo- logical property of the graph Laplacian. To resolve the essential self-adjointness of the position- dependent diffusion operator, we execute a simultaneous Cole-Hopf gauge and Liouville coordi- nate transformation, mapping the system into a canonical Hermitian tensor Hamiltonian. We de- rive the exact canonical potential Vcan including the metric cross-coupling term−D′∂QUeff/(2kBT). By applying Sears’ Theorem, we prove that Veff = Ω(|Q|α+2+2β) strictly dominates both the cross-coupling and geometric corrections, placing the operator in the infinite Weyl Limit-Point regime under sub-quadratic diffusion bounds (α ≤ 2). Furthermore, we couple the proba- bility evolution to a cooperative sigmoidal Hill-type structural degeneration of the network. We demonstrate that the collapse of the global spectral gap (λglobal 1 →0) under differential topological fragmentation serves as a quantitative spectral biomarker for systemic cognitive dis- connection. Finally, we validate the framework via an N=1 computational proof-of-concept using an empirical 84 ×84 DTI connectome (Desikan-Killiany atlas). We simulate pathological progression by selectively degrading the entorhinal-hippocampal cut-set (Braak staging), and empirically confirm the collapse of the Fiedler value (λ2: 1.92 →0.30, an 84.15% reduction) and demonstrate Pathological De-coherence via Kuramoto dynamics (Kc: 0.52 →3.28), establishing a foundation for predictive biophysical Digital Twins.