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 theFibonacci sequence, as purely discrete mathematical constructs relegated to Number Theory. By extending thesesystems to the continuous time domain, a hidden Complex Wave Mechanics is revealed, where solutions do notbehave as standard vectors, but as spinors. These spinors exhibit a 4 periodicity and an intrinsic phase sensitivitythat breaks time-reversal symmetry, splitting the dynamics into two conjugate, asymmetric modes: a co-rotatingand a counter-rotating wave function. This work establishes the formal mathematical scaffolding of this continuousextension 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 tothe protected edge modes of Topological Superconductors. Building on these isomorphisms, we bridge thistheoretical 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 quasicrystallinetopology, bypassing the fragility of fine-tuned "magic angles" in twistronics. Furthermore, parity inversion isintroduced 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 foradvanced computing.