Axiomatic Unification of the Foundations of Chemistry: The Undertaking Tightness S Rigorously Unifies All Chemical Bond Types

Since Lewis proposed the electron pair model of the covalent bond in 1916and Pauling established the quantum mechanical theory of the chemical bondin 1939, covalent, ionic, metallic, hydrogen, and van der Waals bonds havebeen regarded as five fundamentally distinct types of interactions. Startingfrom the rigorous mathematical foundations of Constraint Network dynamics—a deterministic discrete dynamical system defined by three axioms and fullyformalized in ZF set theory—this paper proves that these five types ofchemical bonds are in fact different manifestations of a single continuousvariable, the undertaking tightness S, across five natural intervals. S isdefined as S = (1 − d̂₁·d̂₂) · f(ΔS), where d̂₁·d̂₂ is the directional dotproduct of the Φ-open-branches on both sides, and f(ΔS) is the correctionfactor for the sealing field strength difference. This paper rigorouslyproves the existence, uniqueness, and continuity of S (Theorems 1–2), derivesthe analytical functional relationships between the S-value and bond energyand bond length (Theorems 3–4), proves the chemical bond unification theorem(Theorem 5), and proves that the geometry of polyatomic molecules isrigorously determined by the angles between the direction vectors of theΦ-open-branches of the central node (Theorem 6). The predicted values of alltheorems are consistent with experimental data from the NIST database withinchemical accuracy. This result provides chemistry with its first unifiedaxiomatic foundation grounded in a rigorous mathematical framework.