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July 27, 2026Open Access

Hyperbolic Root Systems from Non-Symmetric Rank-2 Cartan Matrices — E8 Intelligence Research

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Authors

ACAndrew Stewart Caldin

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Overview

Research explores hyperbolic root systems defined by Cartan matrices, linking to E8 and affine limits.

Key Points

  • The aim is to define and analyze rank-2 hyperbolic root systems characterized by Cartan matrices with integer parameters.
  • Defined generalized Cartan matrix H(a,b) for a,b ∈ ℤ with the condition ab ≥ 5.
  • Discussed the link between non-symmetric cases and E8 through affine limits.
  • Analyzed the behavior of spectral radii corresponding to different values of ab.
  • Rank-2 hyperbolic root systems exist if ab ≥ 5, indicating a strong link to the golden ratio.
  • For ab = 5, the spectral radius approaches φ² ≈ 2.618.
  • The affine cases (ab = 1, 2, 3) demonstrate degeneration to infinite root systems, embedding into the finite E8 root system.

Cite This Study

Andrew Stewart Caldin (2026) studied this question.

synapsesocial.com/papers/6a6700bd40bca442e0d4ab61https://doi.org/10.5281/zenodo.21545084
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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 Rank-2 Hyperbolic Root Systems via Generalized Cartan Matrices — E8 Intelligence Research2026
  2. 2Hyperbolic Root Systems: Extending Cartan Classification via Asymmetric Rank-2 Matrices — E8 Intelligence Research2026
  3. 3Classification of Rank-2 Hyperbolic Root Systems via Integer Pairs (a,b) — E8 Intelligence Research2026
  4. 4Classification of Non-Symmetric Rank-2 Hyperbolic Root Systems via Generalized Cartan Matrix H(a,b) — E8 Intelligence Research2026
  5. 5Non-Symmetric Lattices from Rank-2 Hyperbolic Root Systems with ab ≥ 5 — E8 Intelligence Research2026