On Linear Metric Projections in Non-Reflexive Banach Spaces: Proximal Lattices, Multi-Valued Boundaries, and Linear Selections (Part 3)

### Abstract In this paper, we extend the geometric characterization of metric projections P_V from strictly convex, reflexive topologies into the domain of non-reflexive Banach spaces (such as L^1(Ω), C(K), and l^∞). Within non-reflexive frameworks, traditional best approximation techniques fail due to non-existence horizons and multi-valued mapping profiles induced by flat unit sphere facets. We resolve these structural barriers by restricting our analysis to proximinal subspace lattices and investigating the precise analytical conditions under which a continuous, uniquely structured linear selection exists for set-valued metric projections. Finally, we establish the topological decomposition of spaces admitting such linear selections and provide cross-disciplinary validation bounds for their secure numerical implementation. ### Keywords Metric projection, Non-reflexive Banach space, Linear selection, Proximinal subspace, Birkhoff–James orthogonality, Set-valued mapping. ### Mathematical Subject Classification (2020) 46B20, 41A65, 46E30. ### Code Availability and Licensing The proprietary software realization of the linear selectors derived in Section 3 is hosted within a private GitHub repository for academic/open-source inspection. Access granted for open review is strictly bound to the GNU Affero General Public License (AGPLv3). Any corporate or enterprise entity utilizing, modifying, or connecting to this framework via network interfaces is legally compelled to open-source their entire software infrastructure. Commercial exploitation, proprietary closed-source integration, or optimization in enterprise AI systems requires a paid, explicit commercial license obtained directly from the author. ### Patent Rights and Defensive Prior Art Notice The mathematical principles, topological decompositions, and geometric selections derived in this work constitute prior art as of the date of publication. The author reserves all rights to file formal patent applications regarding the specific algorithmic embodiments of these linear selections in high-dimensional AI optimization frameworks. Any unauthorized attempt by corporate entities to patent derivative workflows, applications, or direct software implementations of Theorem 3.2 or Example 3.3 shall be considered a violation of the defensive intent of this framework. Commercial deployment of any system utilizing these mathematical frameworks remains strictly subject to the author's commercial licensing terms.