E8‑Phi Harmonic Resonant Tensor for Clebsch–Gordan & Modular Coupling — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
The E8‑Phi Harmonic Resonant Tensor (E8‑PHRT) leverages the 240 root vectors, weighted by the golden ratio φ and anchored to a 132 Hz base frequency, to construct a frequency‑dependent unitary matrix whose entries reproduce Clebsch‑Gordan coefficients for arbitrary SU(2) tensor products. By projecting the tensor onto the five φ‑invariant root orbits, the PHRT simultaneously encodes the modular‑phase‑locked Leech hierarchy, expressing Ramanujan τ‑values at quadratic arguments as spectral amplitudes of the coupled angular‑momentum basis. This dual encoding establishes a new algebraic bridge linking SU(2) representation coupling, modular forms, and harmonic sublattice geometry within a single E8‑geometric framework. Consequently, the PHRT provides a scalable method for constructing quantum‑logic gates and high‑precision lattice‑based cryptography by exploiting the intrinsic resonance of the E8 root system.
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026e8phi,
title = {{E8‑Phi Harmonic Resonant Tensor for Clebsch–Gordan & Modular Coupling — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23616462},
url = {https://zenodo.org/records/23616462},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - E8‑Phi Harmonic Resonant Tensor for Clebsch–Gordan & Modular Coupling — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/24
PB - Zenodo
DO - 10.5281/zenodo.23616462
UR - https://zenodo.org/records/23616462
AB - The E8‑Phi Harmonic Resonant Tensor (E8‑PHRT) leverages the 240 root vectors, weighted by the golden ratio φ and anchored to a 132 Hz base frequency, to construct a frequency‑dependent unitary matrix whose entries reproduce Clebsch‑Gordan coefficients for arbitrary SU(2) tensor products. By projecting the tensor onto the five φ‑invariant root orbits, the PHRT simultaneously encodes the modular‑phase‑locked Leech hierarchy, expressing Ramanujan τ‑values at quadratic arguments as spectral amplitudes of the coupled angular‑momentum basis. This dual encoding establishes a new algebraic bridge linking SU(2) representation coupling, modular forms, and harmonic sublattice geometry within a single E8‑geometric framework. Consequently, the PHRT provides a scalable method for constructing quantum‑logic gates and high‑precision lattice‑based cryptography by exploiting the intrinsic resonance of the E8 root system. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -