The Spectrum: Fermion-Mass Hierarchy from McKay Spectral Geometry on S³/2I

A 24-entry fermion-mass hierarchy constructed from four ingredients on the Poincaré homology sphere S³/2I: the calibrated vacuum scale μΛ = ρΛ¹ᐟ⁴, geometric phase factors from Kostant polynomials, a hierarchy exponent from McKay graph distance divided by the Coxeter number h(E₈) = 30, and Reidemeister torsion evaluated on vacuum-twisted representations. m(ρ, σ) = μΛ × Cgeom(ρ) × (√ΩΛ)^(dist(ρ)/30) × T²(ρ ⊗ σ) Applied to the 8 nontrivial irreducible representations of the binary icosahedral group 2I across 3 isolated flat connections, the construction produces 24 ranked mass entries. It is presented as a comparison, not a prediction: the observed cosmological constant supplies the mass-scale calibration, the representation-theoretic construction fixes the 24 slots and their subgroup content, and a separately constructed gate system assigns Standard Model quantum-number labels. The corrected comparison gives quantum-number-compatible coverage within a factor of 3 for 5 of the 8 charged fermions beyond the electron calibration cross-check, with 4 surviving sector-first adjudication within that window. The down quark remains assigned but lies outside the window at a factor of 3.23; the up and charm quarks are unassigned; the bottom has compatible coverage outside its structurally named sector. The three neutrino rows are treated as proxy comparisons to an unmeasured lightest mass and the measured solar and atmospheric splitting scales, not as generation assignments. A pre-registered null test on the corrected table gives pₐ = 0.690, showing that the factor-of-3 proximity count is reproduced by random torsion reassignment and therefore supplies no evidence for the specific torsion allocation. What remains established is the internal mathematical structure of the construction: exact algebraic torsion values for all irreducible representations, Galois-pair identities, inverse sector products, the 24-entry organization, and the representation-theoretic gate evaluations. Independent-method reproduction of the corrected torsion calculation remains pending. Latest derivation of this paper can be found at github.com/mass-spectrum