Geodesic costs on a scalar field over the periodic table predict diatomic bond dissociation energies

We construct a scalar configuration field Φ on the periodic table lattice from z-score-normalized first ionization energy and covalent radius, with a single cou- pling parameter λ fixed a priori. Geodesic costs computed on this field via Dijkstra’s algorithm predict experimental diatomic bond dissociation energies D0 for 201 diatomics at Spearman ρ = −0.325 (95% CI: [−0.462, −0.180], p < 10−5), outperforming both Manhattan and Euclidean distance baselines without molecular orbital theory, fitted regression, or element-pair-specific parameters. On the sparser gradient-magnitude cost field, the correlation strengthens to ρ = −0.633 (p < 10−7, N = 60). The field’s curvature (second differ- ence along atomic number) also correlates with Pearson–Parr chemical hardness at r = −0.830 (95% CI: [−0.947, −0.604], p < 10−9, N = 35). A 16-configuration ablation study confirms robustness across λ ∈ [0.5, 2.0], connectivity, and cost-field choices. We do not propose this framework as a com- petitor to quantum chemical calculations of bond energies; rather, we present it as evidence that the periodic table possesses intrinsic differential-geometric structure from which chemical observables can be recovered without reference to electronic wavefunctions.