Topological Asymmetry in Analytic Number Theory: The Generalized Geodesic Dispersion Operator and the Symmetric Constraints of the Critical Strip

Update Note (Version 2.0): This version introduces a rigorous topological refinement to the evaluation of symmetric spatial constraints. In Sections 2 and 3, the analysis of off-axis perturbations was transitioned from the first spatial moment (the Geodesic Drift Operator) to the second spatial moment (the Geodesic Dispersion Operator, or Fréchet Variance). Due to the strict reflection isometry of the Riemann functional equation, symmetric off-axis zeros inherently lock the discrete Fréchet expectation to the critical axis. By substituting the Drift Operator with the Geodesic Dispersion Operator, the framework successfully quantifies the absolute internal metric tension. The proof by contradiction demonstrates that uncompensated transverse perturbations from the critical axis induce infinite exponential dispersion under the negative curvature of the hyperbolic metric, strictly violating holomorphic constraints. Abstract: This paper extends the generalized measure-theoretic framework of spatial asymmetry into the domain of complex analysis, specifically evaluating the topological distribution of non-trivial zeros within the critical strip. By redefining the critical domain as a smooth Riemannian manifold endowed with a hyperbolic metric tensor (k < 0), we mathematically neutralize the metric degeneration inherently caused by logarithmic asymptotic scaling. We formalize the complex Geodesic Dispersion Operator to evaluate the absolute internal metric tension (Fréchet variance) of empirical zero-distributions relative to the continuous symmetric baseline dictated by the analytic functional equation. Ultimately, we demonstrate that uncompensated transverse spatial deviations from the axis of symmetry mathematically induce infinite exponential geometric dispersion under the hyperbolic metric, strictly contradicting the holomorphic constraints of the analytic continuation. This framework establishes a pure differential geometry approach to evaluating measure-theoretic equilibrium in analytic number theory.