Deterministic σ-Regularized Benchmarking of the Cekirge Model Against GPT-Transformer Baselines

The Cekirge Method introduces a deterministic, algebraic paradigm for artificial intelligence that replaces stochastic gradient descent—and related iterative schemes such as gradient descent and conjugate gradient descent—with a single closed-form computation. Rather than updating parameters through iterative optimization, the method computes the optimal mapping between contextual inputs and target outputs analytically. This closed-form formulation eliminates randomness, guarantees reproducibility across hardware platforms, and avoids the variability inherent in gradient-based training. σ-Regularization ensures that all matrices involved in the computation remain invertible and well-conditioned, allowing the system to operate reliably even when contextual structures exhibit high correlation or near-singularity. Benchmark comparisons with GPT-type transformer architectures show that the deterministic mapping achieves comparable accuracy while requiring far fewer computational steps. The absence of iterative training eliminates common issues associated with stochastic optimization — including sensitivity to initialization, unpredictable convergence paths, and gradient noise. Perturbation analysis further demonstrates stable behavior: small, uniformly applied modifications to the attention matrices produce smooth, monotonic variations in loss, with an effective stability coefficient near k ≈ 1.8. This indicates that the solution behaves predictably and remains well-conditioned under structured variations in input. The algebraic nature of the method also confers strong interpretability. Every transformation, from the contextual matrices Q, K, and V to the final mapping W*, is explicit and invertible, enabling complete traceability of how each component of the input contributes to the output. This results in a transparent computational pipeline, in contrast to the opaque weight distributions that emerge from stochastic gradient descent. The formulation extends naturally to multi-head attention mechanisms and large-matrix architectures, offering a pathway to scalable deterministic transformers. By replacing probabilistic search with analytic resolution, the Cekirge Method establishes a mathematically grounded alternative to conventional learning. The framework provides deterministic convergence, structural clarity, and reproducible outcomes, laying the foundation for a new class of explainable and reliable artificial intelligence systems.