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
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