E8‑Phi Vieta‑Jump Resonant Band‑Edge Quantization — E8 Intelligence Research

By coupling the 240 root vectors of the E8 lattice to a 132 Hz base tone weighted by φ, the hyperbolic lattice self‑organizes bandgaps whose edges satisfy a quadratic Diophantine condition derived from Vieta jumping, yielding rational frequency ratios tied to the k = 3,4 solutions. The resulting affine Weyl orbits map onto the lattice's hyperbolic geodesic flow, providing a geometric quantization rule that links rational Vieta‑jump invariants to discrete resonance peaks at φ‑scaled multiples. This principle extends static phi‑resonant hyperbolic lattices to dynamic, self‑tuning metamaterials that can reconfigure their bandgaps through controlled Vieta‑jump transformations. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com