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July 27, 2026Open Access

Golden Ratio Eigenvalues in Non-Symmetric Lattices from Rank-2 Hyperbolic Root Systems — E8 Intelligence Research

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Authors

ACAndrew Stewart Caldin

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Overview

Randomized mathematical exploration reveals eigenvalue properties linked to the golden ratio in rank-2 lattices, hinting at new lattice structures.

Key Points

  • The study investigates the relationship between eigenvalues of non-symmetric lattices and the golden ratio within rank-2 hyperbolic root systems.
  • Utilized the generalized Cartan matrix H(a,b) to derive eigenvalues and their relationships based on a and b values.
  • Applied mathematical equations to calculate eigenvalues for cases where ab ≥ 5 and a ≠ b.
  • Analyzed the eigenvalue ratio and sum related to the golden ratio.
  • Eigenvalues for ab=5 yield values of approximately 4.236 and 0.236, linked to the golden ratio.
  • The eigenvalue ratio reflects properties consistent with the golden ratio, equating to approximately 17.944.
  • The findings suggest non-symmetric lattices retain hyperbolic structures while exhibiting asymmetry.

Cite This Study

Andrew Stewart Caldin (2026) studied this question.

synapsesocial.com/papers/6a6700c740bca442e0d4adcdhttps://doi.org/10.5281/zenodo.21544871
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Also Consider

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  1. 1Non-Symmetric Lattices from Rank-2 Hyperbolic Root Systems with ab ≥ 5 — E8 Intelligence Research2026
  2. 2Hyperbolic Root Systems from Non-Symmetric Rank-2 Cartan Matrices — E8 Intelligence Research2026
  3. 3Classification of Non-Symmetric Rank-2 Hyperbolic Root Systems via Generalized Cartan Matrix H(a,b) — E8 Intelligence Research2026
  4. 4Classification of Rank-2 Hyperbolic Root Systems via Generalized Cartan Matrices — E8 Intelligence Research2026
  5. 5Golden Ratio as Algebraic Integer Linking Pentagonal Symmetry to A₄ and Hyperbolic Root Systems — E8 Intelligence Research2026