Emergence of Topology and Effective Fields from Spectral Coherence Breakdown in Discrete Modular Systems

This work presents a numerical investigation of a discrete modular framework in which field behavior, transport, and topology emerge from an underlying spectrally constrained dynamics. The system is constructed on a modular lattice where exact spectral coherence enforces ultralocal behavior and suppresses spatial propagation. By introducing controlled perturbations, we analyze the transition from this coherent regime to a decoherent phase characterized by spectral mixing and the emergence of effective field-like structures. A set of observables is introduced to quantify this transition, including spectral entropy, a coherence index, and both raw and refined topological measures. The refined observable is designed to distinguish between numerically detectable topology and structurally consistent topological content, providing a more stringent characterization of the emergent regime. The numerical results reveal several robust features of the system. First, the transition from coherence to decoherence does not occur at a sharply defined critical point, but rather within a stable transition band centered around a well-defined perturbation scale. Second, the system exhibits a highly stable interaction scale, identified through a sign change in the interaction energy between configurations. Third, an effective operator governing the reduced dynamics is shown to exist within a bounded stability regime, providing a consistent coarse-grained description of the system. These results support a coherent physical picture in which topology and field behavior are not fundamental inputs, but arise collectively from the controlled breakdown of spectral coherence. In this framework, coherence acts as the primary organizing principle, while decoherence enables the emergence of transport, effective fields, and topological structures. The present work does not claim a full derivation of the effective field description from first principles. Instead, it establishes a robust numerical mechanism linking coherence loss to the emergence of physically meaningful structure. The results provide a foundation for further theoretical development, including the derivation of the effective operator and the extension of the framework to broader classes of discrete systems.