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

Huseyin Murat Cekirge

doi:doi:10.11648/j.ajai.20250902.25

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

Huseyin Murat Cekirge^*

Published in American Journal of Artificial Intelligence (Volume 9, Issue 2)

Received: 25 October 2025 Accepted: 5 November 2025 Published: 22 November 2025

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Abstract

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.

Published in	American Journal of Artificial Intelligence (Volume 9, Issue 2)
DOI	10.11648/j.ajai.20250902.25
Page(s)	258-271
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2025. Published by Science Publishing Group

Keywords

Deterministic Learning, σ-regularization, Algebraic AI, Energy-efficient Computation, Transformer Models, Cekirge Method, Closed-form Learning, Sustainable Machine Intelligence

References

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Cekirge, H. M. (2025). Algebraic Cekirge Method for Deterministic and Energy-efficient Transformer Language Models. American Journal of Artificial Intelligence, 9(2), 258-271. https://doi.org/10.11648/j.ajai.20250902.25

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ACS Style

Cekirge, H. M. Algebraic Cekirge Method for Deterministic and Energy-efficient Transformer Language Models. Am. J. Artif. Intell. 2025, 9(2), 258-271. doi: 10.11648/j.ajai.20250902.25

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AMA Style

Cekirge HM. Algebraic Cekirge Method for Deterministic and Energy-efficient Transformer Language Models. Am J Artif Intell. 2025;9(2):258-271. doi: 10.11648/j.ajai.20250902.25

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@article{10.11648/j.ajai.20250902.25,
  author = {Huseyin Murat Cekirge},
  title = {Algebraic Cekirge Method for Deterministic and Energy-efficient Transformer Language Models
},
  journal = {American Journal of Artificial Intelligence},
  volume = {9},
  number = {2},
  pages = {258-271},
  doi = {10.11648/j.ajai.20250902.25},
  url = {https://doi.org/10.11648/j.ajai.20250902.25},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajai.20250902.25},
  abstract = {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.
},
 year = {2025}
}

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TY  - JOUR
T1  - Algebraic Cekirge Method for Deterministic and Energy-efficient Transformer Language Models

AU  - Huseyin Murat Cekirge
Y1  - 2025/11/22
PY  - 2025
N1  - https://doi.org/10.11648/j.ajai.20250902.25
DO  - 10.11648/j.ajai.20250902.25
T2  - American Journal of Artificial Intelligence
JF  - American Journal of Artificial Intelligence
JO  - American Journal of Artificial Intelligence
SP  - 258
EP  - 271
PB  - Science Publishing Group
SN  - 2639-9733
UR  - https://doi.org/10.11648/j.ajai.20250902.25
AB  - 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.

VL  - 9
IS  - 2
ER  -

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Huseyin Murat Cekirge

The City College of New York, City University of New York (CUNY), New York, USA

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http://orcid.org/0000-0001-8075-2306

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Cekirge, H. M. (2025). Algebraic Cekirge Method for Deterministic and Energy-efficient Transformer Language Models. American Journal of Artificial Intelligence, 9(2), 258-271. https://doi.org/10.11648/j.ajai.20250902.25

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ACS Style

Cekirge, H. M. Algebraic Cekirge Method for Deterministic and Energy-efficient Transformer Language Models. Am. J. Artif. Intell. 2025, 9(2), 258-271. doi: 10.11648/j.ajai.20250902.25

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AMA Style

Cekirge HM. Algebraic Cekirge Method for Deterministic and Energy-efficient Transformer Language Models. Am J Artif Intell. 2025;9(2):258-271. doi: 10.11648/j.ajai.20250902.25

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@article{10.11648/j.ajai.20250902.25,
  author = {Huseyin Murat Cekirge},
  title = {Algebraic Cekirge Method for Deterministic and Energy-efficient Transformer Language Models
},
  journal = {American Journal of Artificial Intelligence},
  volume = {9},
  number = {2},
  pages = {258-271},
  doi = {10.11648/j.ajai.20250902.25},
  url = {https://doi.org/10.11648/j.ajai.20250902.25},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajai.20250902.25},
  abstract = {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.
},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - Algebraic Cekirge Method for Deterministic and Energy-efficient Transformer Language Models

AU  - Huseyin Murat Cekirge
Y1  - 2025/11/22
PY  - 2025
N1  - https://doi.org/10.11648/j.ajai.20250902.25
DO  - 10.11648/j.ajai.20250902.25
T2  - American Journal of Artificial Intelligence
JF  - American Journal of Artificial Intelligence
JO  - American Journal of Artificial Intelligence
SP  - 258
EP  - 271
PB  - Science Publishing Group
SN  - 2639-9733
UR  - https://doi.org/10.11648/j.ajai.20250902.25
AB  - 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.

VL  - 9
IS  - 2
ER  -

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