Symmetry Analysis of Compact Tetraquark States and Implications for the Level Ordering of the Fully Charmed Candidates $X(6600)$, $X(6900)$, and $X(7100)$

Motivated by recent experimental observations, we investigate the $J^P$ distribution of low-energy compact tetraquark states. Assuming that two quarks and two antiquarks are arranged in either a tetrahedral or a square configuration, we employ the restricted representations of the permutation group $S_4$ onto $S_2\times S_2$ to derive the inherent nodal structure of the $qq\bar{q}\bar{q}$ system from that of the $qqqq$ system for orbital angular momentum $L\le 3$. Based on this framework, we determine the distribution of accessible $J^P$ states and find that low-energy compact tetraquark states are likely to favor $J^P=2^{+}$. Our analysis yields two observations that further support the dynamical robustness of symmetry-based classifications in exotic hadron spectroscopy. First, the symmetry-induced $J^P$ distribution of compact tetraquark states closely resembles that obtained for the three-flavor four-quark system. Second, the location of the distribution peak remains unchanged when chromomagnetic interaction (CMI) effects are incorporated. Together, these results suggest that the dominant features of the low-lying spectrum are governed primarily by symmetry constraints rather than by the details of the underlying dynamics. These findings further imply that the fully charmed tetraquark candidates $X(6600)$, $X(6900)$, and $X(7100)$ may occupy relatively low-lying levels of the fully charmed tetraquark spectrum. They also indicate that mechanisms beyond CMI dynamics are likely involved, potentially mitigating or competing with the effects of CMI in compact fully charmed tetraquark states.