The Spectral Dark Sector: Dark Matter as a Near-Shell Phase in Causal Lattices

Abstract We present a proposal for the nature of Dark Matter grounded in the Spectral Theory of Discrete Operators defined on causal lattices (FZF formalism). In this framework, the dark sector is not constituted by exotic particles but emerges as a phase of near-shell spectral states (4 ≤ |λ| ≤ ηDM), which 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 halo topologies with a saturated core in 91. 6% of the sample, rejecting the universality of the cuspy NFW profile with decisive evidence (ΔBIC > 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 spectral saturation of the causal lattice. Complementarily, we derive the spectral transfer function for gravitational lensing systems, showing that the spectral viscosity ηDM induces a rigid ultraviolet cutoff (spectral knee). High-resolution interferometric data (e. g. , SDP. 81) are shown to be consistent with a spectral saturation scale corresponding to a substructure mass cutoff of mcut ~ 10⁸ Mₛun. 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. • Solved 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 power spectrum of strong gravitational lenses, testable with ALMA and JWST observations.