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 + (βA − 1) A on the 600-cell graph (N = 120 vertices, degree k = 12), where βA is derived entirely from invariants of H₄, G₂, and R⁴ polytopes, with zero free parameters. The operator yields αₑff⁻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 ηdisc = N₊ − N₋ = −10 is a topological invariant of the graph, independent of βA. (3) The identity ηdisc − ηcont = −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 + |ηdisc|/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 ηcont = −1 (lower) and ηdisc = −10 (upper). A falsifiable prediction for CODATA 2026: T₆ = 2. 88 × 10⁻¹⁵.
Daniel Zunzunegui (Tue,) studied this question.