Failure Modes of Diffusion-Geometric Admissibility in Finite Relational Systems: Distinct Dynamical Pathways to Spectral Collapse

We study dynamical mechanisms by which finite weighted relational systems approach the boundary of diffusion-geometric admissibility defined by spectral gap collapse, λ1 → 0. We consider time-evolving weighted graphs and identify two distinct mechanisms: (i) uniform degradation, in which all edge weights decay exponentially, and (ii) cumulative capacity-driven overload, in which local excess load generates persistent stress that suppresses edge weights over time. For the degradation mechanism, we derive an exact solution showing exponential collapse of the spectrum while preserving connectivity and eigenstructure. For the overload mechanism, we define a stress-accumulation model and demonstrate numerically that spectral collapse occurs rapidly, accompanied by near-complete localization of the Fiedler eigenvector. In both cases, topological connectivity is preserved. These results establish that spectral collapse is not dynamically unique: distinct mechanisms share common invariants while exhibiting sharply different spectral and localization signatures.