## 6 Four-Zero Texture Model Validation
### Predictions for the \(V_{CKM}\) Elements  
The second part of model validation is as follows: With the data obtained from the valid regions for the four free parameters of the 4-zero texture model, \(A_{u}\), \(A_{d}\), \(\phi_{1}\), and \(\phi_{2}\), found with the help of HE-DEPSO, we can evaluate the feasibility of the model according to the latest experimental values of the \(V_{CKM}\) matrix and the Jarlskog invariant. This can be achieved by checking the model's predictive power when predicting the remaining elements of the \(V_{CKM}\) matrix that were not used within the \(\chi^{2}\) fit, i.e., \(|V_{cd}|\), \(|V_{ud}|\), \(|V_{cs}|\), \(|V_{tb}|\), \(|V_{td}|\), and \(|V_{ts}|\). Using Equations (2)-(5), the Chau-Keng parametrization [3], and the solutions found, we can calculate the numerical predictions for those elements. These predictions are then presented using scatter plots. Each graph includes the experimental central value of each element with its associated experimental error value. The points that fall within the region delimited by the experimental values are considered good predictions, while those outside the region are considered poor predictions. In Figure 9, the predictions for the elements \(|V_{ud}|\) and \(|V_{cd}|\) in the first case study are presented. The black, blue, and orange points indicate a precision of \(\chi^{2}<1\), \(\chi^{2}<1\times 10^{-1}\), and \(\chi^{2}<1\times 10^{-2}\), respectively. For \(|V_{cd}|\), the experimental central value (0.22636) is indicated with a solid red vertical line on the corresponding axis, while its experimental error (\(\pm 0.00048\)) is shown with two red dashed vertical lines on either side of the central value. In the case of \(|V_{ud}|\), the experimental central value (0.97401) is represented with a solid blue horizontal line on the \(|V_{ud}|\) axis, and its experimental error (\(\pm 0.00011\)) is indicated with two blue dashed horizontal lines on either side of the central value. The predictions for the two elements show that the data with a precision of \(\chi^{2}<1\) can be outside the experimental error region, suggesting a poor prediction. On the other hand, the predictions with \(\chi^{2}<1\times 10^{-1}\) and \(\chi^{2}<1\times 10^{-2}\) remain within this region, indicating accurate predictions. This pattern is repeated in the rest of the analyzed case studies. The predictions for the elements \(|V_{cs}|\) and \(|V_{tb}|\) are shown in Figure 10 for the first case study. The experimental central value of \(|V_{cs}|\) is represented by a solid red vertical line at the center of the \(|V_{cs}|\) axis at 0.97320, while its experimental error of \(\pm 0.00011\) is indicated by red dashed vertical lines on either side. The experimental central value of \(|V_{tb}|\) is shown by a solid blue horizontal line at the center of the \(|V_{tb}|\) axis at 0.999172, and its experimental error of \(\pm 0.000024\) is represented by blue dashed horizontal lines on either side. According to the predictions shown in Figure 10, the data obtained with \(\chi^{2}<1\) do not represent good predictions, as they are outside the experimental errors. On the other hand, the predictions  
Figure 8: Allowed regions for \(\phi_{1}\) and \(\phi_{2}\) constrained based on current experimental data with different levels of precision: \(\chi^{2}<1\) (black dots), \(\chi^{2}<1\times 10^{-1}\) (blue dots), and \(\chi^{2}<1\times 10^{-2}\) (orange dots). with \(\chi^{2}<1\times 10^{-1}\) and \(\chi^{2}<1\times 10^{-2}\) remain within the error region, indicating that they are good predictions. This pattern is repeated in the four case studies. Finally, in Figure 11, the predictions for the elements \(|V_{td}|\) and \(|V_{ts}|\) are shown in the four case studies. For \(|V_{td}|\), the experimental central value is represented by a solid vertical line at \(|V_{td}|=0.00854\), and its experimental error with two red dashed vertical lines at \(\pm 0.00016\).