The Cekirge Method for Machine Learning: A Deterministic σ-Regularized Analytical Solution for General Minimum Problems

The Cekirge Global σ-Regularized Deterministic Method introduces a non-iterative learning framework in which model parameters are obtained through a single closed-form computation rather than through gradient-based optimization. For more than half a century, supervised learning has relied on gradient descent, stochastic gradient descent, and conjugate gradient descent—methods requiring learning rates, batching rules, random initialization, and stopping heuristics, whose outcomes vary with floating-point resolution, operating-system effects, and hardware drift. As dimensions increase or matrices become ill-conditioned, these iterative processes frequently diverge or yield inconsistent results. The σ-Regularized Deterministic Method replaces this instability with a σ-regularized quadratic formulation whose stationary point is analytically unique; even very small σ values eliminate ill-conditioning and ensure machine-independent reproducibility. Learning is reframed not as a search, but as the direct computation of an equilibrium determined by the structural geometry of the data matrix. To address the common reviewer concern that stability must be demonstrated across progressive system sizes, the method is validated sequentially—from small 5×5 and 8×8 matrices, whose full algebra is explicitly inspectable, through 20×20, 100×100, and ultimately 1000×1000. Across all scales, the deterministic σ-solution remains stable and identical across platforms, whereas gradient-based algorithms begin to degrade even at moderate sizes. In practice, the σ-Regularized Deterministic Method requires only a single algebraic evaluation, eliminating the repeated matrix passes and energy expenditure inherent to iterative algorithms. Its runtime scales linearly with the number of partitions rather than the number of iterations, yielding substantial time and energy savings even in very large systems.