Geometric Derivation of the Fine-Structure Constant from the Binary Icosahedral Group 2I and the 600-cell: Spectral Operator, Eta-Invariant, and Index Theory

We present new results within the Critical Planckian Spin Dynamics (CPSD) framework for the geometric derivation of the fine-structure constant α⁻¹ = 137.035999206. Building on the previously published Series S₆ (DOI: 10.5281/zenodo.15077311), we introduce a spectral operator D = kI + (β_A − 1)A on the 600-cell graph (N = 120 vertices, degree k = 12), where β_A is derived entirely from invariants of H₄, G₂, and R⁴ polytopes, with zero free parameters. The operator yields α_eff⁻¹ = 137.0366, |Δα| = 6.16 × 10⁻⁴. New results established in this work: (1) The spectrum of D realizes the regular representation of 2I, with multiplicities d_ρ² for each irreducible representation ρ. (2) The discrete eta-invariant η_disc = N₊ − N₋ = −10 is a topological invariant of the graph, independent of β_A. (3) The identity η_disc − η_cont = −9 = −(d₃)² connects the discrete and continuum (APS) eta-invariants through the critical irrep dimension d₃ = 3, predicting 98.5% of the discretization gap via 9/N². (4) Index(D) = (d₃)² + |η_disc|/2 = 9 + 5 = 14 = dim(G₂). (5) The first two coefficients of S₆ satisfy T₁ = |2I|·d₃/φ² and T₂ = −d₂/φ³, showing that both results share a common algebraic alphabet: the dimensions of the irreps of 2I. Four open problems are identified, with search bounds for F2 determined by η_cont = −1 (lower) and η_disc = −10 (upper). A falsifiable prediction for CODATA 2026: T₆ = 2.88 × 10⁻¹⁵.