A Formal Mathematical Extension of the Maximum Cluster Diameter Criteria for Topologically Bounded Anomaly Detection in R^3

The identification of anomalous points within bounded geometric clusters is a fundamental problem in topological data analysis. Previous work established a highly optimized heuristic for identifying anomalies in two-dimensional Euclidean space using the maximum diameter criteria of elliptical boundaries. In this paper, we extend this theoretical framework into a three-dimensional Euclidean manifold, R3. By mapping the boundaries of a tri-axial ellipsoid and defining non-linear distance scalar fields relative to the ellipsoid's extreme vertices, we present a pure mathematical formulation of the 3D anomaly detection heuristic. We introduce the Spherical Manifold Intersection Lemma and provide rigorous proofs leveraging constrained optimization and spectral decomposition, establishing the exact topological bounds where the heuristic holds absolute mathematical certainty.