An analytical index for predicting miscibility in binary systems of transition metals and lanthanides based on Mendeleev number

An analytical index is proposed for predicting miscibility in binary systems of transition metals and lanthanides: the Mendeleev Analytical Miscibility Index (MAMI), based on two classical physical principles: (i) the incompatibility between rare earths and early transition metals, and (ii) the incompatibility between platinum group metals and the IB/IIB blocks, both encoded via the modified Pettifor scale (Mendeleev number). The index comprises two sigmoid components and ten optimisable parameters. Applied to 813 binary systems, MAMI achieves an accuracy of 92.00% (95% CI: [90.0%, 93.9%]), an MCC of 0.815 and an AUC-ROC of 0.918, demonstrating that miscibility classification is captured by classical physical principles without machine learning. The explicit formula immediately identifies the dominant physical mechanism for any binary pair, an advantage unattainable in black-box neural networks. To quantify the limits of the Hume-Rothery rules within their natural domain, a second index (HRMI) with three descriptors and eight parameters is calibrated on 351 transition metal pairs. Despite its domain advantage, HRMI does not outperform MAMI: accuracy 81.48% versus 87.46%, MCC 0.503 versus 0.678, AUC-ROC 0.741 versus 0.918. The Mendeleev number thus carries more discriminatory signal than atomic radius, crystal structure and valence combined, even within the natural domain of the latter. Comparison with a 770-neuron network achieving near-perfect in-sample classification illustrates the trade-off between memorisation and interpretability: MAMI, with a compression ratio of 81:1, sacrifices approximately 8 percentage points of accuracy in exchange for a physically transparent formula applicable across the full RE–TM–I/II chemical space.