Effective-Charge Coordinate Q(Z) for the Periodic Law: Cardano-Newton Analytic Inversion, Kantorovich Bounds, Isoelectronic Extension, and Fourier Decomposition

Source abstract

We construct and rigorously analyse a real-analytic periodic-law coordinate Q: [0,∞) → R built from the smooth extension of the Hakala (1952) formula for the atomic number of the noble gas closing period P. Setting the trigonometric correction to zero yields a cubic solved by the Cardano formula for a zeroth-order approximation Q0 (Z); one Newton step provides a closed-form correction Q1 (Z). We prove existence, uniqueness and real-analyticity of Q0 on the physical domain, axiomatic uniqueness of the chosen smooth extension within a minimal ansatz, certified Newton convergence bounds, and a refined asymptotic expansion consistent at leading order with the Thomas-Fermi-Tietz limit. Empirically, Q restores periodic alignment for several tabulated properties but is not by itself a universal predictor. A structured Q model combining trend(Q), circular functions of frac Q, and MJK orbital variables substantially improves leave-one-period-out prediction for IP 1, chi, rcov, and rvdW, and improves the Z ≤ 86 → 87−118 extrapolation for IP1 and radii. An MJK-only ablation (Table 8) further shows that the added value of Q itself, over the MJK combinatorial features alone, is concentrated on IP1 and the radii — most clearly on rcov — while for chi, Tm and EA the improvement is largely inherited from the MJK variables. Electron affinity remains the weakest property and is reported separately as a data-quality-sensitive endpoint. A rank comparison with the Pettifor chemical scale indicates complementary rather than interchangeable ordering.