Spectral Geometry of the Poincaré Homology Sphere and L-Function Structure

We report on a systematic investigation of the spectral geometry of S³/2I (the Poincaré homology sphere) and its relationship to L-function structure. Starting from the anti-periodic boundary condition on S¹ = ∂(Möbius), which produces (1 − 2⁻ˢ)ζ(s), we trace spectral consequences through five phases. Five results stand independent of the program's central open question: (1) The Reidemeister torsion of S³/2I factors exactly into four Dirichlet L-function special values, confirmed by two independent derivation paths (Reidemeister and spectral/Kummer/Gauss) to 79-80 digits. (2) The E₈ McKay symmetries kill 12 of 16 Dirichlet characters mod 60, selecting exactly the primes dividing |2I| = 120. (3) The scalar Laplacian's spectral zeta does not factor at general s due to the Ricci curvature shift l(l+2) = (l+1)² − 1, while the Dirac operator's spectral zeta factors finitely at all s. (4) The canonical localized spectral object is the equivariant zeta indexed by conjugacy class, arising from the Selberg trace formula. (5) The positivity structure needed to constrain L-function zeros from spectral data does not exist on this manifold. The wall is precisely characterized. The paper identifies what the spectral geometry of S³/2I produces, what it cannot produce, and why. Part of the Mode Identity Theory research program.