E8‑Root‑Vector Quantization of Hyperbolic Cusp Volumes via φ‑Defects — E8 Intelligence Research

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