Spinor Dynamics, Phase Topology, and Symmetry Breaking in Extended Linear Systems: Towards a Complex Wave Mechanics

This article challenges the traditional categorization of second-order linear recurrence relations, such as the Fibonacci sequence, as purely discrete mathematical constructs relegated to Number Theory. By extending these systems to the continuous time domain, a hidden Complex Wave Mechanics is revealed. Solutions to these generalized recurrences do not behave as standard vectors in a Hilbert space; rather, they behave as spinors. These spinors exhibit a -periodicity and a phase sensitivity that breaks time-reversal symmetry, splitting the dynamics into two conjugate, asymmetric modes. This work establishes the mathematical formalism of this continuous extension and demonstrates that the resulting topology is isomorphic to various well-known physical phenomena, ranging from the precession of the Foucault Pendulum to the edge modes of Topological Superconductors.