E8‑Phi Vieta‑Jump Resonant Band‑Edge Quantization — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
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
Related
- E8 Phi‑Resonant Vieta Spectrum — E8 Intelligence Research
- E8 Vieta-Fermion Coupling: Weyl Reflections Set Mass Hierarchies — E8 Intelligence Research
- Trigolden Vieta Jumping Resonance in Phi‑Scaled E8 Metamaterials — E8 Intelligence Research
- Vieta Jumping as Affine Weyl Group Orbits on Quadratic Forms — E8 Intelligence Research
- Vieta Jumping Reveals Affine Weyl Orbits in Quadratic Diophantine Equations — E8 Intelligence Research
BibTeX
@misc{caldin2026e8phi,
title = {{E8‑Phi Vieta‑Jump Resonant Band‑Edge Quantization — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23447048},
url = {https://zenodo.org/records/23447048},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - E8‑Phi Vieta‑Jump Resonant Band‑Edge Quantization — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/18
PB - Zenodo
DO - 10.5281/zenodo.23447048
UR - https://zenodo.org/records/23447048
AB - 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
ER -