We present a phenomenological study of $t\bar{t}t\bar{t}$ production in proton-proton collisions at $\sqrt{s} = 13$~TeV, using a Hyper-Graph Neural Network (H-GNN) to discriminate multilepton signal events from the dominant SM backgrounds, namely $t\bar{t}W$, $t\bar{t}Z$, $t\bar{t}H$, $t\bar{t}VV$, single-top associated production, and diboson and triboson processes. In the H-GNN architecture each event is represented as a hypergraph whose nodes correspond to reconstructed jets and leptons and whose hyperedges encode higher-order correlations among arbitrary subsets of these objects, allowing the network to learn the many-body kinematic structures that characterize the $t\bar{t}t\bar{t}$ final state. Combining same-sign di-lepton, tri-lepton, and four-lepton channels following a CMS-like event selection, the H-GNN attains an area under the ROC curve of $0.951$ for the $t\bar{t}t\bar{t}$ signal and yields a statistical significance of $Z = 9.11$ at an integrated luminosity of $\mathcal{L} = 140~\mathrm{fb}^{-1}$, to be compared with $Z = 8.62$ for a SPANet baseline, $Z = 7.37$ for a Particle Transformer baseline, and $Z = 5.13$ obtained by the ATLAS analysis, evaluated under identical event selection. We exploit the improved signal extraction to derive one- and two-parameter $95\%$ confidence level limits on the Wilson coefficients of the dimension-six operators $\mathcal{O}_{Φu}$, $\mathcal{O}^{(1)}_{tt}$, $\mathcal{O}^{(1)}_{qq}$, $\mathcal{O}^{(1)}_{qt}$, and $\mathcal{O}^{(8)}_{qt}$, and we project the expected sensitivity at the HL-LHC integrated luminosities of $1000~\mathrm{fb}^{-1}$ and $3000~\mathrm{fb}^{-1}$ with $50\%$ uncertainty on the background estimation.