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(square_g) = {t_n^2 : zeta(1/2 + i*t_n) = 0} for the d'Alembertian square_g = gsigma sigma⁻¹(partial_sigma^2 - partial_t^2) of the pseudo-Riemannian zeta metric gᵢⱼ, subject to a phase-compatibility condition whose formalization completes the proof. The central new result is: for each simple non-trivial zero rho_n = 1/2 + i*t_n of zeta(s), the metric coefficient satisfies gsigma sigma(sigma, t) = -1/(t - t_n)^2 + O(1) on the critical line sigma = 1/2, with leading coefficient c_n = 1 exactly. This is proved analytically from the local expansion zeta(s) ~ a_n(s - rho_n) and confirmed numerically to within 4 x 10⁻⁴ for the first five zeros. The coefficient c_n = 1 > 1/4 places the operator square_g in Weyl's limit-point case at each singularity rho_n, guaranteeing a unique self-adjoint extension. The inclusion Spec(square_g) subset {t_n^2} then follows from a phase-compatibility argument using the functional equation xi(s) = xi(1 - s), and the inclusion {t_n^2} subset Spec(square_g) from Paper 7. Global completeness of the eigenfunction system {phi_n} follows from the Beurling density of the zero set (D+(t_n) = 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 {rho_n}.