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

We prove that no intrinsic spectral construction in the admissible class on S³/2I (the Poincaré homology sphere, quotient by the binary icosahedral group of order 120) constrains individual L-function zeros in the sense of a precise zero-constraint definition. Every vanishing condition reduces to one of four exhaustive obstruction layers: coincidence condition, encoding degeneracy, framework mismatch, or character completeness. The paper establishes two named results: Curvature Duality (Section VI) identifies the structural mechanism: the positive Ricci curvature Ric = 2/R² simultaneously produces the Yang-Mills mass gap (Weitzenböck) and the Pochhammer eigenvalue shift that locks torsion selectivity to s = 0, placing mass realization and spectral zero access in structural opposition. Five supporting results are proved independently of the main theorem: torsion factorization into four Dirichlet L-function special values (two paths, 79 digits); E₈ McKay character selection (12 of 16 killed); scalar Laplacian obstruction at general s; Dirac factorization at all s; and S¹ bridge non-existence. The manifold reads L-function structure with arbitrary precision; including the Riemann zeta function itself through the C8 equivariant eta; but cannot constrain individual zeros. The reading capacity and writing incapacity are two faces of the same geometric fact. Latest derivation of this paper can be found at github.com/the-mirror