EOSH-Closure in the d-Block: Full-Graph Derivation of d n Counts, Ligand Donation, and the 18-Electron Rule

The EOSH-Closure framework,[1] introduced for main-group structure prediction, applies a single arithmetic operation to every atom in the molecular graph: n val (i) = (Z i − OS i) − max{c ∈ C ext : c ≤ Z i − OS i }. Here we show that applying this operation simultaneously to the metal centre and all ligand donor atoms in a coordination compound derives-without external input-the d-electron count (d n), the ligand electron donation (n lig), and the 18-electron rule as a noble-gas closure condition. The covalent-bond-classification (CBC) lookup table for n lig emerges as a consequence of EOSH, not a prerequisite. A 23-complex arithmetic panel (spanning carbonyls, halides, aqua, ammine, and cyano ligands across 3d/4d/5d metals) validates all three derived quantities without exception. Scope boundaries are stated precisely: σ-donor atoms yield n lig by direct D5/D6 application; π-ring systems (Cp − , arene) require the Hückel-count correction where the donor pair is not Lewis-localisable on a single atom. The non-innocent-ligand case is handled by branch-enumeration: all integer oxidationstate assignments consistent with the molecular charge are generated; charge-consistent branches converge to a unique answer (C2 confidence); genuinely divergent branches report alternative outputs (C1). RHF/def2-SVP calculations confirm that QM partial charges carry the correct sign relative to EOSH oxidation states in 14 of 14 atoms across 7 molecules (ionicity ratio λ = 0.06-0.42), validating the sign and ordering of the ionic partition without overclaiming quantitative charge magnitudes.