Affine Observables, Mode Collapse, and Arithmetic Rigidity in Low-Dimensional Linear Discrete Dynamical Systems

This research investigates low-dimensional linear discrete dynamical systems under constrained affine observations. Specifically, it analyzes second-order linear recurrences xn+1=axn+bxn−1 where a,b∈Z. The paper establishes a general mechanism with three coupled consequences: Mode Collapse: Oscillatory components become unobservable due to spectral cancellation. Dimensional Reduction: Admissible trajectories are confined to discrete invariant lattices. Arithmetic Rigidity: Observed dynamics reduce to translations on discrete sets, exhibiting modular locking. The work proves that spectral cancellation is not a free design choice but a structural necessity compatible with integrality and invariance. It includes a canonical realization for the system xn+1=xn+2xn−1, demonstrating collapse onto a maximal invariant lattice Λ={(xn,xn−1):xn≡−2n(mod15)}.