Four Forces, Two Axioms: Unification of Source, Not Structure
A Structural Interpretation of U(1) × SU(2) × SU(3), and Why Exact Gauge-Coupling Unification Will Not Occur

Writing Declaration: This paper was independently authored by Han Qin. All intellectual decisions, framework design, and editorial judgments were made by the author.

Abstract

The Standard Model's non-gravitational gauge structure is SU(3)_C × SU(2)_L × U(1)_Y. This paper proposes a structural interpretation, based on the Self-as-an-End (SAE) framework, for why the three non-gravitational forces correspond to the gauge groups U(1), SU(2), and SU(3). The argument rests on two SAE axioms applied to the DD (Dimension Degree) hierarchy: remainder development motivates complex internal amplitudes (unitary structure), and remainder conservation motivates the restriction to relative internal mixing (special unitary structure for n ≥ 2).

The paper does not claim to derive the full Standard Model Lagrangian; it provides a prior geometric explanation for the order pattern 1 → U(1), 2 → SU(2), 3 → SU(3). The argument has three layers. First, if a DD level's internal degrees of freedom are represented by complex amplitudes with Hermitian-norm-preserving transformations, the natural kinematic symmetry is U(n). Second, for n ≥ 2, restricting to relative mixing among internal directions (removing the overall phase) yields SU(n). Third, SAE interprets 1DD, 2DD, 3DD as single-phase, minimal two-state, and three-state internal structures respectively, yielding: 1DD ↝ U(1) (abelian phase sector), 2DD ↝ SU(2) (weak force), 3DD ↝ SU(3) (strong force).

The paper further proposes that remainder conservation manifests differently at different DD levels: as local gauge sectors at 1DD–3DD, and as a global bookkeeping constraint E₁ + E₂ = 0 across dual 4DDs. This explains why gravity is structurally different from the other three forces.

Keywords: Self-as-an-End, Standard Model, gauge symmetry, U(1), SU(2), SU(3), DD hierarchy, unification, structural interpretation, four forces