Through the application of the geometrical collective model, a number of nuclei have been subjected to an exhaustive investigation. A complete investigation has been conducted on the nuclei, which have a range of 2.60 ≤ R 4/2 ≤ 2.80. A comparative investigation was carried out in order to make a determination regarding the characteristics of the low-lying collective structure. A comparison was made between the energy levels that were measured for each individual nucleus and three distinct sets of theoretical calculations that were derived from the X(3), X(4), and X (5) models. For the purpose of this analysis, a comparison was made between the theoretical calculations and the experimentally measured energy levels. The nucleus that exhibits features that are consistent with the X(4) model is the target that is being sought after during the detection operation. For the entirety of this study, a certain set of criteria has been developed and put through a comprehensive analysis. There was a limited number of nuclei that were able to fulfill these requirements entirely.