Molecular Periodic Table: Unified Chemical Theory via Φ-Open-Branch Undertaking

This paper presents the complete construction of the Molecular Periodic Table, from a qualitative framework to a quantitative tool, within the theoretical foundation of constraint network dynamics. The physical basis of the Molecular Periodic Table is that the constraint overflow direction distribution Φ of a nuclear node determines how many open undertaking branches each element possesses and the spatial orientation of these branches. A molecule is the stable product formed when these open branches undertake one another and become mutually saturated. This paper establishes a precise mathematical expression for the undertaking tightness S: S = (1 − d̂₁·d̂₂) · f(ΔS), where d̂₁·d̂₂ is the directional dot product of the Φ-open-branches on both sides, and f(ΔS) is the correction factor for the sealing field strength difference. The S-value unifies covalent, ionic, metallic, hydrogen, and van der Waals bonds as the output of a single function under different parameter intervals. Six benchmark molecules (H₂, N₂, HF, O₂, CO, F₂) complete the parameter calibration. On this basis, a complete table of diatomic molecules, a table of typical polyatomic molecule entries, a conductivity and hardness table for giant covalent networks, a list of variable branch numbers for transition metal d-orbitals, and the quantitative relationship between catalytic S-value and catalytic efficiency are constructed. All function parameters are calibrated from publicly available NIST data, and all entries are marked with their data status. Molecular geometries are derived directly from the directionality of Φ-open-branches, with predicted bond angles deviating from measured values by less than two degrees. Catalysis is unified as an undertaking operation at a low-S interface. 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.