GUE Statistics from the Spectral Gap: Why the Riemann Zeros Look Random

We construct spectral Dirichlet series Z_Γ(t) = Σ m(l) λ_l^{−½} cos(t log λ_l) for compact spherical space forms S³/Γ, where m(l) are exact Laplacian multiplicities and λ_l = l(l+2) the eigenvalues, and compute the spacing statistics of their zeros. These functions are completely deterministic — no randomness, no ensemble, no chaos — yet their zeros exhibit the level repulsion characteristic of random matrix ensembles. The key finding is that the spectral gap determines the symmetry class. Artificially imposing a spectral gap on the abelian cyclic group Z₁₂₀ (removing modes below l = 4) switches its zero spacings from GOE to GUE (fit ratio drops from 2.13 to 0.60). Conversely, adding just two low-frequency modes to the binary icosahedral group 2I (filling its natural gap) switches it from GUE to GOE (ratio rises from 0.67 to 1.54). The mechanism is the drum/tune decomposition: low-frequency modes act as an incoherent "drum" that smears level repulsion toward GOE, while the high-frequency "tune" — the Fibonacci spectral ladder with incommensurate golden periods — produces GUE statistics through structured interference. The spectral gap silences the drum. The Poincaré homology sphere S³/2I has the largest gap (λ₁ = 168) among all binary polyhedral space forms, producing zero spacings with variance 0.117 — tighter than GUE universality (0.286). The spectral gap thus serves triple duty: it makes the integers sharp, confines the zeta zeros to the critical line, and produces their GUE statistics. Three consequences. One gap. One number: 168.