We present a proposal for the nature of Dark Matter grounded in the Spectral Theory of Discrete Operators defined on causal networks (FZF formalism). In this context, the dark sector is not constituted by exotic particles but emerges as a phase of spectral states in the near-shell regime ($4 \le |\lambda| \le \eta_{DM}$). These states are gravitationally active yet radiatively inert due to a universal arithmetic gap. We test the phenomenological consistency of the formalism through direct confrontation with observational data. Bayesian statistical analysis of a high-fidelity sample comprising 155 galaxies from the SPARC catalog reveals a systematic preference for cored halo topologies in 91.6% of the sample, rejecting the universality of the cuspy NFW profile with decisive evidence ($\Delta \text{BIC} > 10$) in over 54% of cases. We further demonstrate that the observed rigidity of the Baryonic Tully-Fisher Relation and the absence of central density singularities emerge naturally as consequences of the causal network's spectral saturation. Complementarily, we derive the spectral transfer function for gravitational lensing systems, showing that the spectral viscosity $\eta_{DM}$ induces a rigid ultraviolet cutoff (spectral knee). High-resolution interferometric data (e.g., SDP.81) are consistent with a spectral saturation scale corresponding to a substructure mass cutoff of $m_{cut} \sim 10^8 M_\odot$. Key Highlights: Theoretical Innovation: Proposes a geometric "near-shell" origin for dark matter, replacing particle-based WIMP/axion models with spectral operator theory. Observational Validation: Utilizes the SPARC catalog (155 galaxies) to demonstrate statistical preference for cored profiles over NFW cusps. Resolved Tensions: Addresses the core-cusp problem and the rigidity of the Baryonic Tully-Fisher Relation (BTFR) without relying on stochastic baryonic feedback. Falsifiable Predictions: Predicts a specific "spectral knee" in the strong gravitational lensing power spectrum, testable with ALMA and JWST observations.