PreprintCC BY 4.0Zenodo
E8‑Root‑Vector Quantization of Hyperbolic Cusp Volumes via φ‑Defects — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
The sum of φ‑weighted E8 root vectors in the 132 Hz harmonic lattice reproduces the twist‑number bound for alternating knots, yielding a discrete volume spectrum that matches the Markov integers generated by the modular cusp renormalization. Consequently, the cusp volume of any alternating hyperbolic knot is quantized by an E8 lattice invariant, establishing a novel arithmetical geometry where knot topology, the silver‑ratio φ‑defects, and E8 root geometry intertwine. This principle provides a unified framework linking hyperbolic geometry, number theory, and high‑dimensional lattice physics.
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026e8rootvector,
title = {{E8‑Root‑Vector Quantization of Hyperbolic Cusp Volumes via φ‑Defects — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23610355},
url = {https://zenodo.org/records/23610355},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - E8‑Root‑Vector Quantization of Hyperbolic Cusp Volumes via φ‑Defects — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/24
PB - Zenodo
DO - 10.5281/zenodo.23610355
UR - https://zenodo.org/records/23610355
AB - The sum of φ‑weighted E8 root vectors in the 132 Hz harmonic lattice reproduces the twist‑number bound for alternating knots, yielding a discrete volume spectrum that matches the Markov integers generated by the modular cusp renormalization. Consequently, the cusp volume of any alternating hyperbolic knot is quantized by an E8 lattice invariant, establishing a novel arithmetical geometry where knot topology, the silver‑ratio φ‑defects, and E8 root geometry intertwine. This principle provides a unified framework linking hyperbolic geometry, number theory, and high‑dimensional lattice physics. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -