Single- and Double-$Λ$ Hypernuclear Correlations Calibrate $ΛΛ$ Interaction Energies

Double-$\Lambda$ hypernuclei are essential for probing the $\Lambda\Lambda$ interaction in the double-strangeness $S=-2$ sector, yet the scarcity of experimental data severely limits systematic predictions. We present an evaluation framework based on nuclear many-body theory that exploits the intrinsic structural similarity between single-$\Lambda$ and double-$\Lambda$ systems to transfer empirical constraints from the well-mapped $S = -1$ sector to the $S = -2$ sector. By analyzing theoretical deviations of binding energies in light single- and double-$\Lambda$ hypernuclei, we identify a robust linear correlation between two sectors. This correlation enables a statistical evaluation of double-$\Lambda$ separation energies ($B_{\Lambda\Lambda}$) and $\Lambda\Lambda$ interaction energies ($\Delta B_{\Lambda\Lambda}$) for heavier double-$\Lambda$ hypernuclei, by drawing on a wealth of empirical data from the single-$\Lambda$ sector with quantified uncertainties. Our results show that evaluated $\Delta B_{\Lambda\Lambda}$ values, while consistent with existing data, are systematically larger than direct relativistic density functional predictions constrained only by the NAGARA event. This discrepancy suggests that standard mean-field-based extrapolations may underestimate $\Lambda\Lambda$ correlations and other many-body effects, motivating an evaluation-based correction that offers crucial benchmarks for future $S = -2$ experiments at facilities such as HIAF and J-PARC.