We construct and rigorously analyse a real-analytic periodic-law coordinate \(Q\): {[}0,\(\infty\)) \(\rightarrow\) 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 \( Q _{0}\left( Z \right)\); one Newton step provides a closed-form correction \( Q _{1}\left( Z \right)\). We prove existence, uniqueness and real-analyticity of \( Q _{0}\) 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, \( r _{\mathrm{cov}}\), and \( r _{\mathrm{vdW}}\), and improves the \( Z \leq 86\ \rightarrow \ 87-118\) extrapolation for \({IP}_{1}\) 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 \({IP}_{1}\) and the radii --- most clearly on \( r _{\mathrm{cov}}\) --- while for chi, \( T _{m}\) 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.