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: The Shatto Theorem (Lemma 8) proves by four independent methods that no natural map exists between the phase position Θ and the spectral parameter s on S³/2I, permanently closing the Phase–log Ω path. The 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. Part of the Mode Identity Theory research program.
Blake Shatto (2026) studied this question.