The Riemann hypothesis remains the most formidable challenge in analytic number theory. In our preceding work, we proposed a foundational topological framework where the Riemann zeta function is a spectral manifestation of a 59-dimensional manifold. This paper constructs the explicit algebraic operator H₅₉ to bridge the gap between that topological architecture and quantitative spectral stability. A critical pillar of this synthesis is the observed behavior of eigenvalues in high-dimensional configurations. We integrate the Spectral Concentration Theorem for real parts in non-Hermitian matrices, which dictates the variance of the spectral distribution through the fundamental relation: equation _^2 = (n-2) (n+2) (n-1) Var₇ equation where Var₇ = tr (H^2/n - (tr (H) /n) ^2). This variance equation serves as our primary analytical bridge; it demonstrates that as the system approaches the 59-dimensional limit, the eigenvalues concentrate sharply around the trace ratio, providing a rigorous statistical basis for our geometric model. By demonstrating that the condition Re (ₜ) = 1/2 emerges as the special case where Var₇ = 0, we establish that the critical line is not merely a geometric assumption but a necessary consequence of the system's spectral concentration.
Abdelilah AHMOURI (Sun,) studied this question.