Minimal Conditions for Geometric Admissibility in Finite Relational Systems: Functional Connectivity, Spectral Collapse, and the Boundary of Diffusion Geometry

We formulate minimal conditions under which a finite weighted relational system admits a diffusion-geometric interpretation. The framework is deliberately operational: geometry is not assumed as a background structure, but is treated as an admissible description only when the system supports coherent propagation. For a finite connected weighted graph with Laplacian L, we identify the first nonzero eigenvalue λ1 as the controlling spectral diagnostic for global diffusion coherence. A positive spectral gap implies finite diffusion mixing scale and stable propagation across the relational support, while collapse of the gap marks loss of diffusiongeometric admissibility. We distinguish topological connectivity from functional diffusion connectivity, emphasizing that a graph may remain connected while failing to support coherent global propagation at the relevant scale. The resulting criterion provides a minimal operator-level foundation for subsequent analysis of failure modes, recovery conditions, and post-geometric phase structure in finite relational systems.