Hyperbolic dynamics from a cyclotomic integer:Quasicrystalline projections and spectral arithmetic of J = 1 + ζ₅²

This preprint presents a rigorously proved mathematical construction in arithmetic dynamics arising from the cyclotomic unitJ = 1 + ζ₅² ∈ ℤ[ζ₅].The main text develops an explicit bridge from algebraic number theory to hyperbolic toral dynamics, quasicrystalline cut‑and‑project sets, and adelic norm arithmetic. All results in Sections 1–12 are theorems and stand independently of any physical interpretation.A clearly separated appendix proposes a conjectural physical Ansatz identifying the contracting and expanding eigenspaces with frequency and action, respectively. This appendix is explicitly speculative and is included to explore possible physical interpretations suggested by the mathematical structure.The mathematical core contains no free parameters and is fully determined by the algebraic properties of ℚ(ζ₅).Related to Thorn, A. M. (2026). Twisted Units, Celestial Symmetries and Entropic Motion: Adelic computations, dilogarithmic identities and hyperbolic dynamics in Q(ζ5). Zenodo. https://doi.org/10.5281/zenodo.18311965