EOSH-Closure in the f -Block: The +3 Dominance, Stability Islands, and Spin-Only Magnetic Moments of the Lanthanide Series from Shell-Hierarchy Arithmetic

The lanthanide series (Z = 57-71) is characterised by the progressive filling of the 4 f subshell behind the outer 5s 2 5p 6 screen. Despite the complexity of these correlated systems, lanthanide chemistry is governed by a single, overwhelming equilibrium point: the +3 oxidation state. In this fourth paper of the EOSH-Closure series, we apply the same closure-core arithmetic (D1-D6) used in Papers I-III to the f-block, extending the extended closure set to C ext ⊃ {54, 61, 68} (the [Xe], [Xe] f 7 , and [Xe] f 14 shells). We show that the +3 dominance is a necessary arithmetic consequence: shedding the outer 6s 2 5d 1 electrons places every lanthanide between the [Xe] and [Xe] f 14 closures, leaving a sheltered 4 f n residue that is magnetically active but bonding-inactive. The four wellknown stability islands-Ce IV (f 0), Eu II and Tb IV (f 7), Yb II (f 14)-emerge as the only charge states at which a lanthanide hits an exact closure, providing a parameter-free arithmetic explanation for deviations from the +3 default. The same f n count feeds all three Hund's rules, yielding the free-ion quantum numbers S, L, J and the Landé g-factor without additional parameters. The resulting Landé moments µ L eff = g J J(J + 1) µ B agree with experimental values to ≤ 0.03 µ B for 13 of 15 ions; the two exceptions (Eu 3+ and Sm 3+) are correctly identified as arising from Van Vleck paramagnetism and J-mixing respectively, both of which require multiplet summation beyond a single ground-state formula. All 15 elements are treated; no empirical parameters are required beyond atomic numbers.