From Regime Boundaries to Chemical Gauge Group: The Inevitable Emergence of the Periodic Law and the Cross-Disciplinary Unification of the Factor Hierarchy Law ——Derivation of the Chemical Gauge Group Theorem, the Chemical Counterpart of the Higgs Mechanism, and the Establishment of the Five-Layer Axiomatic Structure (III)

Abstract

Why are there exactly four types of atomic orbitals—s, p, d, and f? Why does the periodic table have an end? What is the relationship between the Factor Hierarchy Law, quantum mechanics, and gauge field theory? Building on the empirical discoveries of the two preceding papers, this paper completes three core theoretical derivations that elevate the periodic law from an empirical regularity to an inevitable deduction from first principles. First derivation: the Chemical Gauge Group Theorem. The meta-regime factor valence_electrons possesses n=8 mutually exclusive regime states (outermost electron numbers = 1, 2, …, 8), and its moderating effects cover all executive factors with irreducible interaction effects. Under the symmetry constraint of the three-dimensional rotation group SO(3), the unique group-theoretic correspondence of the eight regime states is the l=0, 1, 2, 3 irreducible representations of SO(3)—namely, the s, p, d, and f orbitals. The sum of the dimensions of these irreducible representations is 1+3+5+7=16, exactly corresponding to eight spin-degenerate regime states. No regime state with l≥4 exists, because the l=4 representation is 9-dimensional, corresponding to 18 spin-orbitals, far exceeding the capacity of eight regime states, and the required nuclear charge number would exceed the nuclear stability limit. Second derivation: the division of labor and subsumption between the Factor Hierarchy Law and quantum mechanics. The Factor Hierarchy Law provides the regime-state structure (the eight mutually exclusive regime states of valence_electrons), and quantum mechanics provides the symmetry constraint (the underlying symmetry of electron configuration is SO(3)). Together, they derive the inevitability of s, p, d, and f orbitals. This is not “dependence” on quantum mechanics but rather the demotion of quantum mechanics from a “fundamental theory” to “a provider of symmetry constraints as physical input”—completely isomorphic in logical structure to the “demotion of gauge field theory” in Tang (2026q). Third derivation: the end of the periodic table. The topological constraints of transition bandwidths determine the upper limit of the number of regime centers. As the orbital angular momentum l increases, the spacing between regime centers expands exponentially. The entrance to the g-block (l=4) at Z≈121 is the last detectable regime center. The nuclear charge number required for the entrance to the h-block (l=5) exceeds the nuclear stability limit—the fission barrier height drops below 1 MeV, and any heavier nuclide would fission at the moment of formation. The end of the periodic table is not an arbitrary human stipulation but the inevitable joint consequence of topological constraints from regime centers and the physical limits of nuclear stability. Integrating these three derivations, this paper establishes the five-layer axiomatic structure of the regime-switching theory of the periodic law—Law Layer (three axioms) → Norm Layer (six functional dimensions) → Algorithm Layer (three-step MRSD logic) → Theorem and Proposition Layer (four quantitative laws + the Chemical Gauge Group Theorem) → Theory Layer (nine cross-disciplinary corollaries). This structure is completely isomorphic to the five-layer structure established by Tang (2026q) in particle physics. The Factor Hierarchy Law emerged from financial data, was verified in stellar atmospheres, obtained its mathematical foundation in statistical physics, derived gauge group structures in particle physics, and completed the demotion of fundamental laws in chemistry—it is not the law of a single discipline but the universal logic of the functional stratification of variables in any complex system based on objective data. Research Paradigm Statement: The core methodology, research direction, and final decisions were independently directed by the author. DeepSeek assisted with code implementation, data presentation, and text drafting. The author takes full academic responsibility for the final content.