Four mechanisms of emergence of the golden ratio in the Hurwitz tower and the Galois asymmetry of the golden modules of the 600-cell

The golden ratio φ = (1+√5)/2 emerges in the four levels of the Hurwitz tower ℝ ⊂ ℂ ⊂ ℍ ⊂ 𝕆, with distinct mechanisms: dynamical in ℝ, algebro-geometric in ℂ, spectral in ℍ, categorical in 𝕆. At the level ℍ, the combinatorial Dirac operator D of the 600-cell carries the golden ladder: the eight eigenvalues {±φ^±1, ±φ^±2} with exact multiplicities 9,9,9,9,4,4,4,4 (Theorem 5.1). The spectrum of D admits a complete characterisation over ℚ(√5) with exactly 274 eigenvalues (Theorem 7.1). The characteristic polynomial factors over ℤ[x] into 34 irreducible factors; equivalently, the absolute Galois group acts on the roots of χ_D with 34 orbits (Theorem 8.7). All factors have solvable Galois group (Theorem 8.11). The full symmetry group H₄ (order 14400) commutes with D (Proposition 8.5); the subgroup 2I is chosen as reference for its quaternionic interpretation. For the four golden Galois pairs, the eigenspaces E_λ and E_{σ(λ)} are not isomorphic as 2I-modules (Theorem 8.1): this is the relevant Galois asymmetry, distinct from the non-realisation by group elements, which is trivial given equivariance. The analysis extends to a spectral progression of eight regular polytopes in dimensions 2, 3 and 4, with constructive spectral duality for the pair {600, 120} and numerical verification for the pairs {8, 16} and {cube, octahedron}. **Keywords**: golden ratio, Hurwitz tower, regular pentagon, Dirac operator, 600-cell, binary icosahedral group, G₂(1), quantum dimension, symbolic factorisation, Galois symmetry. AI use disclosure. The author used generative AI tools (large language models) for language editing, structural review, and consistency checking of the manuscript and the accompanying reproducibility package. All mathematical content, theorem statements, proofs, computational results, and scientific conclusions were conceived, verified, and are the sole responsibility of the author. No AI tool is listed as author or contributor. The author has reviewed and takes full responsibility for all AI-assisted output.