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July 26, 2026Open Access

Classification of Non-Symmetric Rank 2 Hyperbolic Root Systems — E8 Intelligence Research

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Authors

ACAndrew Stewart Caldin

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Overview

Randomized trial classifies hyperbolic root systems with non-symmetric ratios, suggesting new algebraic structures.

Key Points

  • Classifying rank 2 hyperbolic root systems defined by non-symmetric root length ratios.
  • Classified root systems with a generalized Cartan matrix H(a,b) for integers a and b with ab ≥ 5.
  • Analyzed root length ratios √(a/b) for non-symmetric cases when a ≠ b.
  • Explored the implications of hyperbolic versus finite and affine classifications.
  • Identified non-symmetric root length ratios indicating hyperbolic root systems.
  • Excluded classical root ratios such as the golden ratio, establishing distinct properties.
  • Highlighted the crystallographic nature of root systems due to integer Cartan integers.

Cite This Study

Andrew Stewart Caldin (2026) studied this question.

synapsesocial.com/papers/6a65a962d3aea3239cd7933chttps://doi.org/10.5281/zenodo.21525156
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Also Consider

Synapse has enriched 5 closely related papers on similar clinical questions. Consider them for comparative context:

  1. 1Classification of Non-Symmetric Rank-2 Hyperbolic Root Systems via Generalized Cartan Matrix H(a,b) — E8 Intelligence Research2026
  2. 2Classification of Rank-2 Hyperbolic Root Systems via Integer Pairs (a,b) — E8 Intelligence Research2026
  3. 3Classification of Rank-2 Hyperbolic Root Systems via Generalized Cartan Matrices — E8 Intelligence Research2026
  4. 4Hyperbolic Root Systems: Extending Cartan Classification via Asymmetric Rank-2 Matrices — E8 Intelligence Research2026
  5. 5Hyperbolic Root Systems from Non-Symmetric Rank-2 Cartan Matrices — E8 Intelligence Research2026