The Cekirge σ-Method in AI: Analysis and Broad Applications

Modern machine learning relies predominantly on iterative optimization, assuming that learning must progress by following gradients across a probabilistic loss landscape shaped by Gaussian noise. Such methods depend on learning rates, random initialization, batching strategies, and repeated parameter updates, and their outcomes vary with numerical precision, hardware behavior, and floating-point drift. As dimensionality increases or data matrices become ill-conditioned, these iterative procedures often require extensive tuning and may converge inconsistently or fail altogether. This work presents the Cekirge Global Deterministic σ-Method, a non-iterative learning framework in which model parameters are obtained through a single closed-form computation rather than through iterative descent. Learning is formulated as a deterministic equilibrium problem governed by a σ-regularized equilibrium functional. A behavioral analogy-Cekirge’s Dog-illustrates the operational distinction: a trained working dog runs directly to its target without scanning or correcting its path step by step, whereas iterative algorithms resemble a dog that repeatedly stops, senses, and adjusts direction. Throughout this work, the term equilibrium functiona<i>l</i> is used instead of <i>energy</i>, emphasizing deterministic balance rather than physical or variational interpretations. The proposed framework yields a unique, reproducible solution independent of initialization or hardware, remains stable for ill-conditioned systems, and scales deterministically to large problems through σ-regularized partitioning. A minimal overdetermined example demonstrates that the inefficiency of gradient-based learning is structural rather than dimensional, arising from trajectory-based optimization, while the proposed σ-regularized formulation computes the same equilibrium directly in a single closed-form step.