Equilibrium-based Deterministic Learning in AI via σ-Regularization

Gradient-based learning methods such as Gradient Descent (GD), Stochastic Gradient Descent (SGD), and Conjugate Gradient Descent (CGD) are widely used in supervised learning and inverse problems. However, when the underlying system is underdetermined, these iterative approaches do not converge to a unique solution; instead, their outcomes depend strongly on initialization, learning rates, numerical precision, and stopping criteria. This study presents a deterministic σ-regularized equilibrium framework, referred to as the Cekirge Method, in which model parameters are obtained through a single closed-form computation rather than iterative optimization. Using a controlled time-indexed dataset, the deterministic equilibrium solution is compared directly with GD, SGD, and CGD under identical experimental conditions. While gradient-based methods follow distinct optimization trajectories and require substantially longer runtimes, the σ-regularized formulation consistently yields a unique and numerically stable solution with minimal computational cost. The results demonstrate that the inability of gradient-based methods to reproduce the deterministic equilibrium in underdetermined systems is not an algorithmic shortcoming, but a structural consequence of trajectory-based optimization in a non-unique solution space. The analysis focuses on formulation-level properties rather than predictive accuracy, emphasizing equilibrium existence, numerical conditioning, parameter stability, and reproducibility. By prioritizing equilibrium recognition over iterative search, the proposed framework highlights deterministic algebraic learning as a complementary paradigm to conventional gradient-based methods, particularly for time-indexed systems where stability and repeatability are critical.