Coherence-Induced Obstruction and Emergent Hopf Structures in a Discrete Mod-9 Lattice

This work investigates a structural obstruction governing the relation between temporal coherence and spatial propagation in discrete linear systems defined over modular lattices. We identify that exact spectral coherence enforces mono-frequency dynamics and leads to ultralocal behavior, suppressing spatial transport. Spatial propagation emerges only after the breakdown of coherence, through spectral mixing and the generation of effective phase gradients, giving rise to a coarse-grained Laplacian operator. Within this emergent regime, the system admits a consistent field interpretation in which geometry and topology arise dynamically. In particular, configurations consistent with Hopf-type solitons appear as attractors without being imposed at the continuum level. These results establish a concrete mechanism linking coherence, decoherence, and the emergence of effective field structure in discrete systems. The analysis is restricted to the class of modular lattice models considered.