Derivation of the CMB Spectral Index from Dodecahedral Foam Geometry

We derive the scalar spectral index of primordial perturbations from the eigenmode structure of the Poincaré dodecahedral space (PDS) and the icosahedral symmetry group A5. The formula ns = 1 − 2/[ln(ℓmax/ℓmin) × |A5|/(2π)] yields ns = 0.9675 using only three inputs: the first non-trivial PDS eigenmode (ℓmin = 4 from the Molien series of the binary icosahedral group 2I), the CMB damping scale (ℓmax = 2500), and the order of the icosahedral rotation group (|A5| = 60). This value lies 0.6σ from the Planck 2018 measurement of 0.9649 ± 0.0042. When refined through self-consistent Hodge-decomposed optimization on the dodecahedral face complex, the prediction improves to ns = 0.9640 (0.2σ). We additionally show that the PDS eigenmode structure naturally explains two persistent CMB anomalies: the low quadrupole power (D2 = 226 μK2 versus the ΛCDM expectation of ~1200 μK2) and the low-ℓ power deficit, both of which follow from the mode sparsity of PDS at low wavenumber k. These results require zero free parameters beyond the topology of the dodecahedral cell complex itself.