A Noble-Gas-Centered Coordinate for Within-Period Atomic Property Trends

We introduce a single dimensionless landscape function $J_{\rm chem}(\rho) = \cosh(\rho \ln \varphi) - 1$, $\varphi = (1+\sqrt{5})/2$, on the noble-gas-centred coordinate $\rho = d/L_p \in [0,1)$, and show that it organizes four central atomic observables: first ionization energy \IE$_1$, electron affinity EA, Mulliken electronegativity $\chi_M$, and Pearson chemical hardness $\eta$, on one periodic-table axis. The outward step $\Delta J_{\rm chem}^{+}$ delivers IE$_1$; the inward gap $\Delta J_{\rm chem}^{-} = J_{\rm chem}(1) - J_{\rm chem}(\rho)$ delivers EA and $\eta$; $\chi_M$ follows by Mulliken's identity. Three results establish the empirical content. (i) The within-period IE$_1$ envelope reproduces the full noble-gas-to-alkali ordering across periods 2--6: of 34 atoms compiled across periods 2-4, 26 lie on the predicted monotone descent and the 8 upward deviations occur exactly at the textbook anomaly sites $\{p^3, d^5, f^7, s^2, d^{10}\}$. (ii) Two golden-ratio identities, ${\rm IE}_1(G_p)/{\rm IE}_1(G_{p+1}) \approx \varphi^{1/4}$ on three heavy noble-gas pairs and ${\rm IE}_1(\text{halogen})/{\rm IE}_1(\text{alkali}) \approx \varphi^2$ on four within-period pairs, agree with NIST data to MAD $\approx 1\%$ and $\approx 5\%$, respectively. (iii) The shared kernel $\Delta J_{\rm chem}^{-}$ provides single-parameter analytical fits to EA across periods 4--6 (MAE $0.3$--$0.4$~eV), to Pearson hardness $\eta$ across periods 2--4 (MAE $\sim 1$~eV on noble-gas maxima up to $10.8$~eV), and to Mulliken $\chi_M$ across a 15-atom four-class benchmark ($R^2 = 0.73$). \edit{At the period-averaged scale level, the shared-kernel relation ${\rm EA}/\eta \approx C^{(p)}_{\mathrm{EA}}/C^{(p)}_{\eta}$ is supported on period-4 NIST data: the empirical nine-atom mean $\overline{{\rm EA}/\eta} = 0.180$ agrees with the predicted constant $0.182$ to better than $1\%$, although individual-atom scatter ($\sigma \approx 0.13$) is much larger.