Topological Asymmetry in Non-Euclidean Geometries: A Generalized Measure-Theoretic Framework and the Flat Limit Theorem

This paper extends the measure-theoretic framework of spatial asymmetry from flat affine spaces to strictly non-Euclidean Riemannian manifolds (M, g). We formalize the non-Euclidean Generalized Geodesic Drift Operator, utilizing the Riemannian logarithmic map to evaluate the exact structural differential between the discrete Fréchet expectation of empirical distributions and the continuous Fréchet mean of the geodesically convex bounding topology. To mathematically guarantee dimensional consistency and isolate pure structural asymmetry, the Riemannian norm of this tangent vector is rigorously normalized against the principal N-th root of the intrinsic topological scale, derived via the Riemannian volume measure. The geometric sensitivity of the resulting Topologically Invariant Divergence Coefficient is analytically evaluated across three fundamental curvature topologies. We mathematically demonstrate geometric suppression under positive sectional curvature (κ > 0) via the Bishop-Gromov volume comparison theorem, exponential structural amplification under negative curvature (κ < 0), and deterministic functional collapse under the periodic boundary conditions of the flat torus. Ultimately, we prove the Flat Limit Theorem, strictly establishing that as sectional curvature asymptotically approaches zero (κ → 0), the generalized non-Euclidean operators identically collapse into the classical affine Euclidean topology. This delivers a universal, curvature-independent diagnostic invariant for continuous and discrete metric geometry.