Arithmetic Attractors and Identity Persistence: A Discrete Dynamical Model of Keith Sequences and Informational Stability

Keith numbers form a rare class of integers defined by a digit-generated linear recurrence in which the original number reappears within its own sequence. Although known for several decades, their structural properties and the mechanisms underlying their extreme sparsity remain poorly understood.In this work, we introduce a dynamical reformulation of Keith sequences by embedding the digit recurrence into a discrete state-space system governed by a companion matrix. Within this framework, the recurrence trajectory can be interpreted as an orbit of a finite-dimensional linear dynamical system. This representation enables the introduction of trajectory observables—including informational inertia, an admissibility field, and a stability functional—which characterize the evolution of the sequence.Using this formulation, we analyze the spectral structure of the recurrence operator and show that the reappearance of the original integer corresponds to a transient intersection between the expanding trajectory and a fixed identity hyperplane in state space. Representative numerical scans over increasing integer ranges confirm that such identity-return events are extremely rare and occur only under tightly constrained dynamical conditions.These results suggest that Keith numbers can be interpreted as non-generic return events in a linear dynamical system determined by digit-based initial conditions. The proposed framework provides a dynamical explanation for their empirical sparsity and offers a basis for studying digit recurrences using tools from dynamical systems, spectral analysis, and computational number theory.