On Extensions of Linear Metric Projections: Stability, Planar Anomalies, and Restricted Subspaces (Part 2)

In this extended version, we establish a comprehensive development of the isometric characterization of inner product spaces introduced in our previous framework. We investigate the structural behavior of metric projections $P_V$ across three foundational boundaries: 1. We relax the strict linearity constraint to an \varepsilon-approximate framework, proving that near-linear metric projections force the underlying Banach space to satisfy a perturbed version of the classical parallelogram law. 2. We analyze the anomalous geometric regimes in two-dimensional Minkowski spaces ($\dim X = 2$), providing explicit structural classifications where non-Euclidean configurations governed by Radon curves inherently support linear metric projections. 3. We establish a topological reduction theorem, demonstrating that enforcing linearity exclusively on hyperplanes of co-dimension 1 is sufficient to preserve Hilbertian isometry for dimensions $\dim X \ge 3$. The theoretical results are accompanied by a geometric profile visualization mapping the Radon indicatrix against Euclidean spheres.