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

#Affine Observables, Mode Collapse, and Arithmetic Rigidity Author: Tak Heung Sze Official Website: recursivesequence.netlify.app DOI: 10.5281/zenodo.18269678 Research Summary This release contains the full paper investigating second-order linear recurrences under constrained affine observations.The study identifies a mechanism where arithmetic restrictions force a structural collapse of dynamical features. Core System Model: x_{n+1} = ax_{n} + bx_{n-1}, a, b ∈ ℤ Key Contributions ***Mode Collapse: Oscillatory components become unobservable due to spectral cancellation forced by integrality. **Dimensional Reduction: Admissible trajectories are restricted to invariant lattices. ***Arithmetic Rigidity**: Dynamics reduce to simple translations on discrete sets with modular locking. Canonical Example For the recurrence x_{n+1} = x_{n} + 2x_{n-1}: ***Affine Observable: H(x,y) = 2x + y/5 on section y ≡ 0 (mod 5). ***Resulting Lattice: Λ = { (x_n, x_{n-1}) : x_n ≡ -2^n (mod 15) }. For interactive demonstrations and visual tools, please visit the official project website.