A Ψ-Spectral Recursive Operator System for Emergent Structure

This work introduces a new recursive operator system in which the evolution of a discrete field is governed by state-dependent spectral geometry rather than by spatial diffusion. The core contribution is the construction of a closed feedback loop in which a global spectral functional, Ψₛₚₑ𝚌(Θ), both characterizes the organization of the field and directly determines the operator applied at the next iteration. As a result, the system is governed not by a single fixed operator but by a family of operators indexed by the current state. The framework defines a class of nonlocal, self-referential dynamical systems distinct from diffusion-based, reaction–diffusion, or fixed-operator models. Structure is regulated by the distribution of energy across spectral radii rather than by local spatial interactions alone. The operators are deliberately minimal and controllable. A small number of interpretable parameters govern geometry, feedback sensitivity, modulation, and boundedness, enabling systematic exploration of regimes such as convergence, oscillatory envelopes, metastable plateaus, and abrupt transitions. This work does not claim to solve an existing open problem. Instead, it establishes a new class of recursive operator systems and provides a concrete mathematical and computational framework for studying feedback between global structure and local evolution. Future directions include analytical study of stability and invariants, systematic regime mapping, and extension to higher-dimensional or alternative spectral geometries.