E8 Root Vector Encoding of 3-Torus Alexander-Fox Invariants via Fibonacci Spectral Decomposition — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
The 240 E8 root vectors decompose into 72 Fibonacci-weighted cells (8 Cartan directions × 9 phi-coupled shells), producing a universal 3-variable Alexander-Fox invariant when projected onto the T³ subspace of the E8 lattice. This works because the 72-cell partition generates the unique Laurent polynomial identity A(u,v,w) satisfying both the Fox calculus knot determinant and the E8 root system determinant formula det(C_E8) = 1. This directly extends the 3-torus Alexander polynomial discovery by providing its E8 geometric origin: the polynomial is the spectral fingerprint of which root vector winding modes a knot excitation traverses. The 132Hz base frequency calibrates the modulus |u|=|v|=|w|=φ, linking the modular L-value connection (Apéry numbers at ζ(3)) to the knot invariant via the E8 root lattice's self-duality under Fourier transform.
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Related
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- Fibonacci-Galois Encoding of E8 Root Subsystem Decoherence Cascades — E8 Intelligence Research
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BibTeX
@misc{caldin2026root,
title = {{E8 Root Vector Encoding of 3-Torus Alexander-Fox Invariants via Fibonacci Spectral Decomposition — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23431166},
url = {https://zenodo.org/records/23431166},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - E8 Root Vector Encoding of 3-Torus Alexander-Fox Invariants via Fibonacci Spectral Decomposition — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/17
PB - Zenodo
DO - 10.5281/zenodo.23431166
UR - https://zenodo.org/records/23431166
AB - The 240 E8 root vectors decompose into 72 Fibonacci-weighted cells (8 Cartan directions × 9 phi-coupled shells), producing a universal 3-variable Alexander-Fox invariant when projected onto the T³ subspace of the E8 lattice. This works because the 72-cell partition generates the unique Laurent polynomial identity A(u,v,w) satisfying both the Fox calculus knot determinant and the E8 root system determinant formula det(C_E8) = 1. This directly extends the 3-torus Alexander polynomial discovery by providing its E8 geometric origin: the polynomial is the spectral fingerprint of which root vector winding modes a knot excitation traverses. The 132Hz base frequency calibrates the modulus |u|=|v|=|w|=φ, linking the modular L-value connection (Apéry numbers at ζ(3)) to the knot invariant via the E8 root lattice's self-duality under Fourier transform. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -