The Fibonacci Spectral Ladder: Hopf Fibre Decomposition of the Zeta Spectrum on S³/2I

The Poincaré homology sphere S³/2I admits a Seifert fibration with three singular fibre types: vertex fibres (wrapping 5 times, stabiliser Z₅), face fibres (wrapping 3 times, stabiliser Z₃), and edge fibres (wrapping 2 times, stabiliser Z₂). We decompose the spectral multiplicity function of S³/2I into explicit contributions from each fibre type and show that the vertex contribution — the golden component — follows a period-10 cycle whose values are {2/5, 0, 1/5, 0, 0, 0, −1/5, 0, −2/5, 0}. The character recursion generating this cycle has coefficient 2cos(2π/5) = 1/φ, which is the Fibonacci recursion. Consequently, the eigenvalue thresholds at which irreducible representations of 2I first appear in the…