The Octet Rule as Structural Necessity: Arithmetic Inversion Symmetry and the 4.5 Barycenter

The chemical octet rule — the empirical observation that atoms achieve stability with eight valence electrons — has been treated since Lewis (1916) as an experimental fact explained a posteriori by quantum-mechanical orbital filling (s²p⁶). This paper derives the octet as a theorem of arithmetic. The involution φ(x) = 9 − x on {1, …, 9} partitions {1, …, 8} into exactly four pairs summing to 9, with unique fixed point μ = 4.5. The digit 9 acts as closure membrane and cannot participate in any pair. We prove that this structure is universal: for any integer a ≥ 1, the digital root cycle {DR(a), DR(2a), …, DR(9a)} contains 9 as closure, and the remaining eight elements fold into four inversion pairs summing to 9 (Universal Fold Theorem). We extend this to the Fibonacci digital root cycle (period 24), which folds into 12 + 12 elements with perfect inversion at every position. Applications to the periodic table, the geometric constant π = 22/7, and the Fibonacci growth cycle are derived without appeal to empirical data.MSC 2020: 00A79, 92E10, 11A63Keywords: Octet rule, periodic table, inversion symmetry, digital roots, barycenter, universal fold, Fibonacci cycle, ARC framework.