Based on the recursive nesting postulate and the optimal filling theorem of Qi theory, this paper presents the first complete derivation of the geometric origin of the periodic table from first principles. Core contributions include: (1) Derivation of the theoretical magic number skeleton 2, 8, 20, 34, 54, 88, 142 from the optimal filling theorem; (2) Establishment of the proton-neutron two-component recursion rule N^ (k) = ⌊Φ Z^ (k) ⌋, explaining the evolution of the N/Z ratio in stable nuclides; (3) Proof of a geometric phase transition in the heavy nuclear region (A≥20) from discrete clusters to continuous spherical shells, identifying the critical point at Ne-20; (4) Demonstration of the mathematical isomorphism between Qi theory and quantum mechanical spin-orbit coupling, independently deriving the intruder orbital sequence and experimental magic numbers 28, 50, 82, 126 from the covariant derivative of spherical tangential fields; (5) Recursively closed expressions for the radial mean field (Woods-Saxon potential) and the global nuclear force parameter γ₀, with all parameters uniquely determined by the hydrogen atom anchor, the nucleon Compton wavelength (one-time experimental calibration), and the recursive scaling law, without any independent fitting for magic numbers or specific elements; (6) Establishment of a geometric computation algorithm for individual elements, systematically computing all main-group elements of the periodic table with an average deviation <8% and key element deviation <5%; (7) Extrapolation based on recursive rules, predicting the island of stability for superheavy elements at Z≈114, N≈184. This work marks a fundamental transformation of the periodic table from empirical induction to a mathematical deductive system. A Chinese version of this paper is also available.
Lin Hao (Mon,) studied this question.