Informational Unification of Fundamental Interactions: An Operator-Based Framework with Emergent Spectral Hierarchies

We present an exploratory, operator-based framework in which the variety of fundamental interactions is approached from an informational perspective. The framework is built around a single functional acting on a relational substrate, combining local, non-local, and hierarchical contributions together with an irreversible cost term. Rather than treating gauge symmetries as fundamental inputs, the approach investigates whether structured interaction sectors can be associated with distinct stability properties of this operator. The physical content of the proposal is examined through the spectral properties of the Hessian of the informational functional. In this setting, the organisation of eigenvalues is used as a diagnostic tool to characterise the presence or absence of hierarchical structures that may be interpreted as interaction sectors. No attempt is made to derive specific gauge groups or coupling constants. A sequence of computational toy models is introduced to probe the internal consistency of the framework. These models are analysed through systematic comparisons with carefully defined null constructions, using spectral entropy, gap-based measures, and concentration indicators. The results show that non-trivial spectral hierarchies arise only when the full informational structure is present, and are suppressed when key terms are removed. Robustness tests across scales, system sizes, and coupling regimes suggest that these features are not solely attributable to fine-tuned parameter choices. The framework is intended as a pre-unificatory, information-theoretic investigation rather than a complete unification scheme. Its purpose is to clarify how interaction-like structures might emerge from a single operator, and to provide a transparent computational setting in which such ideas can be tested and refined.