The Golden Ladder of the 600-Cell Hodge spectra, Galois structure, and the fine-structure constant hierarchy. Critical Planckian Spin Dynamics (CPSD)

The regular polytope {3,3,5} realizes the binary icosahedral group 2I ⊂ SU(2) as its 120 vertices.We present exact spectral results for its simplicial complex and their relation to a numerical hierarchywhose evaluation reproduces the experimental value of α−1.Proved results. (i) The vertex Laplacian ∆0 separates the nine irreducible representations of 2I:each eigenspace is ρ⊕dimρ for a single ρ. (ii) The pseudodeterminants of the four Hodge Laplaciansare computed exactly and factor as A, AB, BC, C through the up-Laplacians, which accountsfor every regularity in their prime factorizations. (iii) The combinatorial Dirac operator carrieseight eigenvalues ±φ±1,±φ±2 whose eigenspaces are exactly the four irreducibles of dimension 2and 3 of 2I. (iv) Two complementary golden identities: R2 = R + I on the extremal sector of∆0, and D2 − D +φ−3I = 0 on the critical sector. (v) The tower of exponents has closed formpn =Fn+d1(G)Fn−1, where d1(G) = 2dmax(G) is the degree of the rst invariant of the Klein ring,by Kostant’s theorem (1984).Status of the hierarchy. Evaluating the series to n = 6 gives 137.035999206000187, within 1.9 ×10−13 of the rubidium determination of Morel et al. (2020). The exponents pn and the odd numeratorscn arise from the same operator via functionals on Z[R]; by contrast we prove that the denominatorsdn cannot belong to that module, and their origin remains open. The connection with α−1 is presentedas a falsi able prediction, not as a rst-principles derivation.