A Noble-Gas-Centered Coordinate for Within-Period Ionization Energy Trends and Subshell Anomalies

We introduce a single dimensionless landscape function Jchem(ρ) = cosh(ρ lnφ)−1, with φ = (1+ √ 5)/2, on the noble-gas-centred coordinate ρ = d/Lp ∈ [0, 1), and show that it organizes the four central atomic observables, first ionization energy IE1, electron affinity EA, Mulliken electronegativity χM, and Pearson chemical hardness η, on a single coordinate axis of the periodic table. Two closed-form landscape steps deliver the four observables: the outward step ∆J+ chem assigned to IE1, and the