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July 27, 2026Open Access

Classification of Non-Symmetric Rank-2 Hyperbolic Root Systems via Generalized Cartan Matrix H(a,b) — E8 Intelligence Research

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Authors

ACAndrew Stewart Caldin

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Overview

Randomized trial classifies non-symmetric rank-2 hyperbolic root systems, implicating algebraic extensions in geometry.

Key Points

  • The aim is to classify non-symmetric rank-2 hyperbolic root systems using the generalized Cartan matrix H(a,b).
  • Utilized generalized Cartan matrix H(a,b) with integer entries a, b where ab ≥ 5.
  • Investigated root length ratios sqrt(a/b) for conditions where a ≠ b.
  • Analyzed the implications for hyperbolic Kac-Moody algebras based on root length and angles.
  • Found distinct root length ratios for non-symmetric cases with specified integer conditions.
  • Established that the ratios approach but do not match the golden ratio, with specific roots calculated.
  • Excluded classical finite and affine systems, confirming placement in the hyperbolic domain.

Cite This Study

Andrew Stewart Caldin (2026) studied this question.

synapsesocial.com/papers/6a6700bd40bca442e0d4ac5bhttps://doi.org/10.5281/zenodo.21544843
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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 — E8 Intelligence Research2026
  2. 2Classification of Rank-2 Hyperbolic Root Systems via Generalized Cartan Matrices — E8 Intelligence Research2026
  3. 3Hyperbolic Root Systems: Extending Cartan Classification via Asymmetric Rank-2 Matrices — E8 Intelligence Research2026
  4. 4Classification of Rank-2 Hyperbolic Root Systems via Integer Pairs (a,b) — E8 Intelligence Research2026
  5. 5Hyperbolic Root Systems from Non-Symmetric Rank-2 Cartan Matrices — E8 Intelligence Research2026