Spectral Identification of the Zeta D'Alembertian: The Exact Coeficient cn = 1, the Weyl Limit-Point Condition, and Completeness via Beurling Density

We establish the spectral identification Spec (squareg) = tₙ²: zeta (1/2 + i*tₙ) = 0 for the d'Alembertian squareg = gₒ₈₆₌₀ ₒ₈₆₌₀^-1 (partialₛigma² - partialₜ²) of the pseudo-Riemannian zeta metric g₈₉, subject to a phase-compatibility condition whose formalization completes the proof. The central new result is: for each simple non-trivial zero rhoₙ = 1/2 + i*tₙ of zeta (s), the metric coefficient satisfies gₒ₈₆₌₀ ₒ₈₆₌₀ (sigma, t) = -1/ (t - tₙ) ² + O (1) on the critical line sigma = 1/2, with leading coefficient cₙ = 1 exactly. This is proved analytically from the local expansion zeta (s) ~ aₙ (s - rhoₙ) and confirmed numerically to within 4 x 10^-4 for the first five zeros. The coefficient cₙ = 1 > 1/4 places the operator squareg in Weyl's limit-point case at each singularity rhoₙ, guaranteeing a unique self-adjoint extension. The inclusion Spec (squareg) subset tₙ² then follows from a phase-compatibility argument using the functional equation xi (s) = xi (1 - s), and the inclusion tₙ² subset Spec (squareg) from Paper 7. Global completeness of the eigenfunction system phiₙ follows from the Beurling density of the zero set (D+ (tₙ) = limsup N (T) /T = (log T) / (2*pi) -> infinity). The remaining gap is the rigorous formalization of the phase-compatibility condition, which reduces to a boundary-value problem for Fuchsian differential equations with poles at rhoₙ.