Modular Square One-Way Function & Square Root Algorithm (Part-2): AI practical approaches for applying the paper results in GPT

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.