Coxeter Elements in A4: Reflection Orders and Root System Context — E8 Intelligence Research

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Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-09-14
DOI
https://doi.org/10.5281/zenodo.22748127
Primary Topic
DNA and Biological Computing
Type
preprint
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preprint

Coxeter Elements in A4: Reflection Orders and Root System Context — E8 Intelligence Research

Andrew Stewart Caldin
Zenodo (CERN European Organization for Nuclear Research)
DNA and Biological Computing
preprint

Coxeter Elements in A4: Reflection Orders and Root System Context — E8 Intelligence Research

Andrew Stewart Caldin
preprint en

Abstract

FINDING: The search results are a scattered collection of educational videos and one arXiv paper on Coxeter elements—no single "unsolvable competition problem" is identified, but the arXiv paper on reflection orders and the A4 root system context provides the deepest mathematical anchor. MATH: - **Coxeter element** in a Coxeter group: product of all simple reflections (order depends on choice, but eigenvalues are roots of unity). For A4 (Weyl group of SU(5)), the Coxeter element has eigenvalues \( e^{2\pi i k/5} \) for \( k=1,2,3,4 \) (since Coxeter number h=5). - **Golden ratio connection**: The eigenvalues of the A4 Coxeter element include \( e^{2\pi i/5} \) and \( e^{4\pi i/5} \), whose real parts are \( \cos(72^\circ) = (\sqrt{5}-1)/4 \approx 0.309 \) and \( \cos(144^\circ) = -(\sqrt{5}+1)/4 \approx -0.809 \). The ratio of these to 1 gives \( 0.309/0.809 = 0.382 \) (exactly \( 1/\varphi^2 \)), and \( 0.809/0.309 = 2.618 \) (exactly \( \varphi^2 \)). - **Reflection order th Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Zenodo (CERN European Organization for Nuclear Research)
DNA and Biological Computing
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Coxeter Elements in A4: Reflection Orders and Root System Context — E8 Intelligence Research — Andrew Stewart Caldin · Zenodo (CERN European Organization for Nuclear Research) (2026) | TGRS Research Map | TGRS