Algebraic Cekirge Method for Deterministic and Energy-efficient Transformer Language Models

This paper introduces a unified deterministic algebraic framework for transformer-style language modeling, extending the σ-Based (Cekirge) methodology toward energy-efficient and interpretable computation. The approach constructs σ-regularized, nonsingular matrices for queries, keys, values, and output weights (Q, K, V, W), enabling attention and decoding to be computed in closed form without iterative stochastic gradient descent. By enforcing σ-regularization, matrix invertibility and numerical stability are guaranteed, allowing direct algebraic determination of weights from paired input–output examples rather than optimization through back propagation. Analytical and numerical experiments, including a controlled five-token model, demonstrate that the deterministic algebraic solution reproduces the predictive behavior of gradient descent while eliminating randomness and reducing computation time by more than sixty-fold. The framework unifies several complementary formulations—the Four-Cluster Deterministic Map, Frozen-Library Forward Training, σ-Matrix Fast Learning, and Inverse Deterministic Energy-Saving Training (IDEST)—each contributing to a closed, energy-saving learning process that transforms optimization into deterministic algebraic resolution. Conceptually, the Cekirge Method parallels perceptual refinement: just as John remains himself while seen through blue eyeglasses but becomes obscured behind a wooden mask. The model’s internal mappings preserve identity under small deterministic perturbations (ε) but lose interpretability under stochastic noise. This analogy captures the method’s central philosophy—learning through controlled perturbation that reveals structure without destroying equilibrium. Finally, the study situates deterministic computation within a sustainability perspective. Global energy consumption, intensified by iterative AI training, has surpassed ecological thresholds. The Cekirge framework reconceives learning as a finite algebraic equilibrium rather than an energy-intensive iterative loop, aligning computational intelligence with thermodynamic and social efficiency. It proposes that future AI systems should pursue mathematical determinism and ecological responsibility in parallel, ensuring progress that is both computationally exact and energetically sustainable. A supplementary real-data experiment confirms that deterministic algebraic decoding achieves comparable accuracy to gradient descent while operating with markedly lower computational energy.