544 Figure 3: Example of GPR simulated realizations of PC1 and PC2 projected onto the #!–## space. Figure 4 (left) shows GPR interpolation in PC space (PC1 vs. PC2, blue) overlaid with the observed PC scores (red). We can back-transform the model interpolation (posterior) to the actual Y space (N! vs. N#) where we can see that by honoring the data, the model yields variability-consistent estimates (N¶! and N¶#) in the gaps, as shown in Figure 4 (right). Now, one can identify the efficient frontier for N! and N# to optimize the phosphate flotation process (Figure 5, top). Each point along this frontier represents a Pareto-optimal trade-off, which can be mapped back to the predictor space, similar to the example presented in Figure 5 (bottom), to determine the corresponding molecular descriptors for the optimized reagent to enhance the separation efficiency of fluorapatite from dolomite. In practice, we aim to maximize the reagent affinity for fluorapatite while minimizing it for dolomite in order to have a selective flotation separation. Therefore, we can select an optimal trade-off point on the efficient frontier (green marker) and then obtain the corresponding process settings (red marker) as (!(7)= 0.680144, (#(æk1)= 0.221114, (Z(ÑÇïÄ)= 0.0167424, and (Y (α) = 250.343 after inverse min–max scaling. This indicates a candidate reagent profile expected to maximize adsorption on fluorapatite while minimizing adsorption on dolomite, thereby improving the selective flotation separation in practice. A close chemical match in the screened set is 6-methylheptanoic acid (C16H32O4), for which the model predicts adsorption energies of −361 kJ/mol (fluorapatite) and −168 kJ/mol (dolomite), compared to DFT values of −290 kJ/mol and −167 kJ/mol, respectively, demonstrating strong agreement for dolomite rejection and a conservatively stronger (more negative) fluorapatite affinity prediction, consistent with the model’s goal of prioritizing selective candidates for targeted follow-up simulations or lab tests to deliver the most value. The corresponding adsorption configurations on both mineral surfaces are shown in Figure 6. Figure 4: Left: GPR interpolation in PC space. Right: in actual Y space after back-transformation.
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