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, where solutions do not behave as standard vectors, but as spinors. These spinors exhibit a periodicity and an intrinsic phase sensitivity that breaks time-reversal symmetry, splitting the dynamics into two conjugate, asymmetric modes: a co-rotating and a counter-rotating wave function. This work establishes the formal mathematical scaffolding of this continuous extension and demonstrates that the resulting topology is isomorphic to well-established physical phenomena, ranging from the macroscopic precession of the Foucault Pendulum and optical chirality in the Sagnac effect to the protected edge modes of Topological Superconductors. Building on these isomorphisms, we bridge this theoretical framework to cutting-edge quantum technologies through the concept of "Fibonacci Moiré Printing". We demonstrate how 2D periodic networks in rotated substrates can geometrically induce robust quasicrystalline topology, bypassing the fragility of fine-tuned "magic angles" in twistronics. Furthermore, parity inversion is introduced as a non-Hermitian control mechanism that purges disorder and thermal noise via the Skin Effect, effectively protecting the quantum coherence of Majorana Bound States and correlated moiré phases essential for advanced computing.