Artificial Intelligence Without Iterative Learning: The Cekirge Deterministic Equilibrium Framework

Modern language models predominantly rely on probabilistic attention mechanisms and iterative training procedures to resolve next-token prediction. In these approaches, query–key (Q–K) interactions are normalized via softmax to produce probability distributions, followed by stochastic sampling or expectation-based selection. While effective in large-scale settings, such formulations inherently depend on training trajectories, random initialization, and repeated parameter updates, leading to variability in outcomes and significant computational cost. This study presents a unified framework that contrasts probabilistic attention with a deterministic allocation methodology, referred to as the Cekirge method, under the same Q–K representation and identical vocabulary. Instead of interpreting Q–K interactions as probabilistic scores, the proposed approach treats them as deterministic constraints and computes model output through a single σ-regularized equilibrium solution of a linear allocation system. No training, softmax normalization, sampling, or initial guess is required. Using an explicit 8-token numerical example, the paper demonstrates that both methodologies operate on the same semantic information yet diverge fundamentally in how constraints are resolved: probabilistic optimization versus deterministic equilibrium recognition. The comparison highlights differences in reproducibility, energy consumption, and interpretability, showing that deterministic allocation yields a unique and stable solution while preserving semantic consistency. The results suggest that probabilistic attention and deterministic equilibrium allocation represent two mathematically coherent but structurally distinct resolutions of the same Q–K framework, opening a path toward energy-efficient, reproducible, and fully interpretable language inference without iterative training. In this work, the term probability distribution is reserved exclusively for softmax-normalized attention outputs, whereas the equilibrium vectors produced by the proposed method are unconstrained allocations with no probabilistic interpretation. No linguistic analysis is intended; the terminology is used strictly in a structural and mathematical sense.