A 3D fullerene-like polymer's compensation behaviors and phase transitions are studied using the Monte Carlo method. The ground-state phase diagrams and magnetization profiles are presented, and the conditions for the occurrence of the compensation temperature Tcomp are obtained. N-type and P-type magnetization curves are identified according to Néel theory. The phase diagrams are also presented to explore the changes in the transition temperature TC and the compensation temperature Tcomp induced by physical parameters. The critical exponents of the system are calculated, and a good agreement is achieved by comparing the results with others'. Triple hysteresis loops are discovered. Fullerene, discovered in 1985, has been widely studied due to its unique cage structure and excellent stability. It has been extensively used in various fields, including material science, physics, chemistry, and medical science. Fullerene-like structures can be prepared by various methods, such as direct arc current discharge, laser evaporation, and organic synthesis. Although intrinsic fullerene is non-magnetic, it is common to substitute metal atoms to make it magnetic in experiments, thereby broadening the scope of application in the field of magnetic recording devices. The successful preparation of fullerene-like structures provides the experimental basis and model for a multitude of theoretical studies. Various methods, such as the effective-field theory, mean-field approximation, and Monte Carlo method, have been employed to further investigate the phase transitions and hysteresis phenomena of various fullerene-like cage structures. The Monte Carlo method is frequently applied to explore the magnetic properties of fullerene-like cage structures. The Ising model is an effective model used to understand the magnetic and thermodynamic behaviors. In this paper, a 3D fullerene-like mixed-spin (2, 5/2) Ising model is fabricated using the most prevalent ordered binary alloys as the prototype. The model is composed of two kinds of fullerene-like cages, where red and blue spheres represent atoms a and b with different spin values. The interactions formed between atoms are mainly divided into two types: one is the interaction formed between the same type of atoms on one cage (Ja and Jb), and the other is the interaction formed by connecting different atoms between different cages (Jab). Other physical parameters such as the single-ion anisotropies of atoms a (Da) and atoms b (Db), and the external magnetic field (h) are also considered. The Hamiltonian corresponding to this system is given. The Monte Carlo method within the Metropolis algorithm is applied to simulate such 3D fullerene-like polymer. The free boundary condition is applied in all directions. Data are calculated by using 2×10^4 Monte Carlo steps after discarding the previous 8×10^4 Monte Carlo steps. The error of the calculated physical quantity is determined by the Jackknife statistical approach. The physical quantities such as magnetization, susceptibility, and internal energy can be obtained by the following formulasA 3D fullerene-like polymer's compensation behaviors and phase transitions are studied using the Monte Carlo method. The ground-state phase diagrams and magnetization profiles are presented, and the conditions for the occurrence of the compensation temperature Tcomp are obtained. N-type and P-type magnetization curves are identified according to Néel theory. The phase diagrams are also presented to explore the changes in the transition temperature TC and the compensation temperature Tcomp induced by physical parameters. The critical exponents of the system are calculated, and a good agreement is achieved by comparing the results with others'. Triple hysteresis loops are discovered. Fullerene, discovered in 1985, has been widely studied due to its unique cage structure and excellent stability. It has been extensively used in various fields, including material science, physics, chemistry, and medical science. Fullerene-like structures can be prepared by various methods, such as direct arc current discharge, laser evaporation, and organic synthesis. Although intrinsic fullerene is non-magnetic, it is common to substitute metal atoms to make it magnetic in experiments, thereby broadening the scope of application in the field of magnetic recording devices. The successful preparation of fullerene-like structures provides the experimental basis and model for a multitude of theoretical studies. Various methods, such as the effective-field theory, mean-field approximation, and Monte Carlo method, have been employed to further investigate the phase transitions and hysteresis phenomena of various fullerene-like cage structures. The Monte Carlo method is frequently applied to explore the magnetic properties of fullerene-like cage structures. The Ising model is an effective model used to understand the magnetic and thermodynamic behaviors. In this paper, a 3D fullerene-like mixed-spin (2, 5/2) Ising model is fabricated using the most prevalent ordered binary alloys as the prototype. The model is composed of two kinds of fullerene-like cages, where red and blue spheres represent atoms a and b with different spin values. The interactions formed between atoms are mainly divided into two types: one is the interaction formed between the same type of atoms on one cage (Ja and Jb), and the other is the interaction formed by connecting different atoms between different cages (Jab). Other physical parameters such as the single-ion anisotropies of atoms a (Da) and atoms b (Db), and the external magnetic field (h) are also considered. The Hamiltonian corresponding to this system is given. The Monte Carlo method within the Metropolis algorithm is applied to simulate such 3D fullerene-like polymer. The free boundary condition is applied in all directions. Data are calculated by using 2×10^4 Monte Carlo steps after discarding the previous 8×10^4 Monte Carlo steps. The error of the calculated physical quantity is determined by the Jackknife statistical approach. The physical quantities such as magnetization, susceptibility, and internal energy can be obtained by the following formulas