Authors : Ahmed Mohammed Al-Fahdi

Volume/Issue : Volume 10 - 2025, Issue 2 - February

Google Scholar : https://tinyurl.com/2s4hbjex

Scribd : https://tinyurl.com/29vkuuex

DOI : https://doi.org/10.38124/ijisrt/25feb1287

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Abstract : This paper is built upon a previous paper entitled as “Modular Square One-Way Function& Square Root Algorithm: Analyzing the algorithm for randomness, regularity schematic (codec system) and vector normalization “. In that paper the modular square one-way function was analyzed yields the quadratic residue pattern numeric analyzation in the result section. Analyzing the integer factorization results leads to un expected schematic regularity regarding the irrational part of the remainder (decimal expansion) of nonperfect square root. Such regularity was surprising as the expected results assumed to be random. Rounding such rational numbers and normalizing it yield to what is innovatively called modular factor symbol similar to Legendre symbol. Such codec pattern has characteristics of Hilbert envelope, skewness around perfect root pattern with Hann window. In GPU, such calculations could be computed fastly using IEEE- 754 [1] standard for rounding irrational part of the nonperfect square (decimal expansion) with floating point as what mentioned in inverse square root [1]. All above, illuminating an idea of the statistical analyzation for the root mean square error (RMSE). RSME is a powerful estimator of the prediction models used in the artificial intelligence AI especially for the reinforcement learning (RL). As a new approach in AI Google DeepMind Researchers looking through regression analysis algorithm tuning and representing the numerical values as discrete tokens for large language model (LLM). Such data set tokenization and tuning algorithm are helpful for the speed and the predictability of the model as it hase been recognized in the Deep Seek.[4] . Up on all above and considering AI as a new evaluation approach, this paper will discuss the implementation aspects of such innovative results in sampling, tokenizing, clustering and compressing the base model of the GPT a long with fine tuning Neural Network (NN) reasoning of the Reinforcement model.

Keywords : AI, tokenization, Riemann Integral, Hilbert Envelope, Hann Window, Skewness Generative Pre-Trained Transformer (GPT), Auto-Regressive Transformer, Base Model, Reward Modeling, RPE, Vision-Based Estimation, SL, Sequence-Based Estimation, Temporal Difference (TD), Root Mean Square Error (RMSE), Statistical Analysis ,Numerical Analysis, Regression Analysis, Pigeonhole Clustering, Matryoshka Structure, Reinforcement Learning RL, RSME Estimator, Floating Point IEEE- 754,Auto-Regression Tuning, Parameter Tuning Of Decoder Unit, Vector Normalization, Normalized Token, Minimum Value Estimator (MVE), Statistical Estimation (Mean, Median),Histogram Based Regression.

References :

1. Ahmed ALfahdi, “Modular Square one-way function and square root algorithm” , June 2024
2. DeepSeek-AI ” DeepSeek-V2: A Strong, Economical, and Efficient Mixture-of-Experts Language Model, June 2024
3. DeepSeek-AI “Incentivizing Reasoning Capability in LLMs via Reinforcement Learning” ,Jan 2025
4.  Qwen Team, Alibaba Group “Qwen 2.5-VL Technical Report” Qwen, China ,Feb 2025
5. Google DeepMind “Decoding-based Regression” Song, Bahri :Equal cont , 31 Jan 2025.
6. Google DeepMind “Matryoshka Quantization” Nair, Datta, Dean,Jian,Kusupati :Equal cont, Feb 2025.
7.  Stanford University”The pigeonhole principle lecures 7&8” https//webs.stanford.edu/class/archive/
8. UC Davis Math “The Riemann integral “ https: // www.math.ucdavis.edu & Riemann Sum and Riemann Integral Explained ,Jan 2020.
9. Chai ,Draxler “Root mean square error (RMSE) or mean absolute error (MAE) argument a gainst avoiding RMSE in the literature” May 2014.
10. Andrei Seymour-Howell,Fast inverse square-root program,2021
11. Olumide “Root Mean Square Error (RMSE) in AI: What You Need To Know” August 2023.