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

This paper extends the generalized measure-theoretic framework of spatialasymmetry into the domain of complex analysis, specifically evaluating thetopological distribution of non-trivial zeros within the critical strip. By redefiningthe critical domain as a smooth Riemannian manifold endowed with a hyperbolicmetric tensor (𝜅 < 0), we mathematically neutralize the metric degenerationinherently caused by logarithmic asymptotic scaling. We formalize the complexGeodesic Drift Operator, utilizing the Riemannian logarithmic map to evaluate thestructural differential between the discrete Fréchet expectation of empirical zerodistributions and the continuous symmetric baseline dictated by the analyticfunctional equation. Through the dimensional normalization of this tangent vector,we derive a strictly dimensionless topological invariant. Ultimately, wedemonstrate that uncompensated spatial deviations from the axis of symmetrymathematically induce infinite exponential divergence within the hyperbolictangent space, strictly contradicting the holomorphic constraints of the analyticcontinuation. This framework establishes a pure differential geometry approach toevaluating measure-theoretic equilibrium in analytic number theory.