## 1 Introduction  
Broadly speaking, physics can be classified into three areas: the theoretical part, which, through mathematical formalism, explains and understands the physical phenomena; the experimental part, which groups disciplines related to the acquisition of data, the methods of data acquisition, and the design and performance of experiments; and finally, the aspect that links these two, known as phenomenology, which deals with (1) validating theoretical models, (2) investigating how to measure their parameters, (3) investigating how to distinguish one model from another, and (4) studying the experimental consequences of these models. This paper addresses a phenomenological approach by providing a solution to the validation of a theoretical model of physics, more specifically, from particle physics, to unveil the mechanisms of fermion mass generation and to reproduce the elements of the \(V_{CKM}\) (Cabibbo-Kobayashi-Maskawa unitary matrix), which describes the mixing of different types of quark during weak interactions by encapsulating the probabilities of transitions between these quark types (see, for example, pertinent introductory reviews [1; 2; 3]), known as the mixing matrix. The beginning of these models goes back to the first years of the 1970s, shortly after the establishment of the standard model (SM) of particle physics. Since then, different approaches have been developed in the context of theoretical and phenomenological models, which can be broadly classified as follows: Radiative mechanisms, where fermionmasses come from contributions of processes generated by new particles added to SM [4; 5]; Textures, where the mass matrix has zeros in some entries [6; 7; 8]; Symmetries between families, where a mathematical group symmetry is added to the theory [9; 10]; and Seesaw mechanisms, an elegant way of generating small masses for neutrinos [11; 12; 13]. These approaches are phenomenological interrelated; for example, when we add an extra group symmetry to the theory it is common to arrive at texture structure to mass matrices [14; 15], and in linear and inverse Seesaw models a texture structure is used to obtain the neutrino masses [16]. The texture formalism was born by considering that certain entries of the mass matrix are zero, such that we can compute analytically the matrix that diagonalizes it and hence the matrix \(V_{CKM}\). In 1977, [PERSON] created this formalism by using 6-zero hermetic textures as a viable model [17]; in 2005, with experimental data from that year, he found that 4-zero hermetic textures were viable to generate the quark masses and the mixing matrix [18]. The analytical approach is very complex, so certain considerations are made to simplify the model. However, the results obtained are not accurate, which is why numerical approaches have recently been used to obtain more accurate solutions. Given the precision of the current experimental data, it is worthwhile assessing the feasibility of these texture models with early studies on the 4-zero mass matrix [19], further exploring their potential to align with experimental data. There are numerical works on 4-zero texture models; however, the detailed numerical descriptions of the techniques and algorithms used are not described by their authors [20]. The numerical analysis of texture models requires a \(\chi^{2}\) criterion that establishes when the theoretical part of the model agrees with the experimental part [21]. That is, to validate the model, a function (which we will call \(\chi^{2}(X)\)) is built, and permissible values of the free parameters \(X\) of the model must be found, such that this function takes the minimum value possible greater than 0 but less than 1. In other words, it is necessary to optimize said function under a certain threshold.