The Unified Theory of Segmented Linear Regression Models (SLRM): Deterministic Geometric Deduction from 1D to n-Dimensional Hyperspaces

Traditional Artificial Neural Networks (ANNs) rely on stochastic optimization (SGD) to approximate non-linear manifolds, resulting in high computational entropy and "black-box" opacity. We present the Unified Segmented Linear Regression Model (SLRM) framework, comprising three distinct yet integrated algorithms: Logos (1D sequential compression), Nexus (High-density hyper-cube folding), and Lumin (Sparse hyperspace simplex sectoring). We demonstrate that function approximation can be treated as a deterministic geometric task rather than a stochastic learning process. By bridging these models with the Universal ReLU Equation, we achieve 0.0 error rates in n-dimensional linear sectors with microsecond latency, effectively bypassing the "Curse of Dimensionality."