## 3 Results and Discussion
### In-Depth Analysis of Correlations between Dielectric Strength and Various Core Factors  
Notably, the above-discussed equation-based prediction protocols have various core factors, (i.e., polarizability, IE, MW, and novel factors) in their equational forms. Consequently, one may wonder how strongly the core factors are correlated with the predicted dielectric strength. Furthermore, to obtain a deeper understanding of the developed prediction protocols, it should be determined whether higher accuracy is achieved by incorporating core factors with higher correlation degrees. Therefore, the correlation degrees between the core factors and the predicted dielectric strengths of the 168 PFC and non-PFC compounds were examined for each equation-based protocol to associate the correlation degree with the prediction accuracy.  
First, the original core factors (polarizability, IE, and MW) were correlated with the predicted dielectric strengths of the 168 PFC and non-PFC compounds (Figures 3d, 4c and 5c; Table 2). Two of these core factors (i.e., polarizability and MW) exhibited linear correlations with the PIM-/re-PIM-/PIMN-predicted dielectric strengths (\(R^{2}\) > 0.8). It should be noted that the \(R^{2}\) value describes the degree of deviation of a data set from a linear regression line. Thus, a compound with a higher polarizability or IE would have a higher dielectric strength. In contrast, the IE was maintained within a range of 8-14 eV for all the compounds, indicating that the dielectric strength is insensitive to this core factor. _This observation also highlights that the IE should be within the aforementioned range regardless of the type of compound, which is a key condition for achieving reliable dielectric strength predications._ This may account for the unexpected decrease in the accuracy of the dielectric strength prediction with the OPT, which did not include the IE as a core factor, as compared to PIMN (_via HOMO-LUMO gap_).  
Novel core factors introduced in the PIMN and OPT protocols were further correlated with the predicted dielectric strengths of the 168 PFC and non-PFC compounds (Figures 5d-f and 6c,d; Table 2). Notably, the predicted dielectric strengths exhibited significant correlations with the number of electrons, molar volume, and formation energy (\(R^{2}\) > 0.7). In particular, among the novel factors, the number of electrons had the strongest correlation with the dielectric strength (\(R^{2}\) = 0.94), indicating its high contribution to determining the dielectric strength of a compound. In contrast, the HOMO-LUMO gap had a weak correlation with the dielectric strength (\(R^{2}\) = 0.03), implying a negligible contribution to determining the dielectric strength. Importantly, as discussed for the IE, the reliable prediction of dielectric strength strongly relies on the HOMO-LUMO gap, which should be maintained within the range of 5-13 eV for all the compounds. Consequently, constraints on the values of these critical factors (IE and HOMO-LUMO gap) should be incorporated into the equation-based protocol.  
Figure 7: (**a**) Comparison of PIM-, re-PIM-, PIMN-, and OPT-predicted dielectric strengths with experimental dielectric strengths and (**b**) distributions of their relative errors. In (**b**), the distribution for each prediction protocol has blocks of 0–10, 10–20, 20–30, 30–40, 40–50, 50–60, and 60–70.