Molecular Periodic Table: Chemical Unification Based on Φ-Open-Branch Undertaking Rules

This paper constructs a Molecular Periodic Table within the constraint network framework of Energy Ontology, progressing from a qualitative framework to a quantitative tool. The undertaking tightness S is established as the core quantitative variable: S = (1 − d̂₁·d̂₂) · f(ΔS), where d̂₁·d̂₂ is the directional dot product of Φ-open-branches and f(ΔS) corrects for sealing field strength differences. Six benchmark molecules (H₂, N₂, HF, O₂, CO, F₂) calibrate the S-value function, which unifies covalent, ionic, metallic, hydrogen, and van der Waals bonds as outputs across different parameter intervals. Bond angles are derived directly from Φ-open-branch directionality, with predictions deviating from measured values by less than two degrees. Catalysis is unified as an undertaking operation at a low-S interface. All quantitative data—diatomic molecules, polyatomic molecules, giant covalent networks, transition metal d-orbital branch numbers, and catalytic S-value/efficiency relationships—are calibrated from publicly available NIST data and submitted as Supplementary Material, with all entries marked by data status (E, P, I, T). The Molecular Periodic Table upgrades chemistry from a collection of empirical rules to a complete quantitative framework that derives all chemical behavior from first-principle physical laws.