What question did this study set out to answer?

The aim is to derive non-Abelian gauge symmetry from axioms within a generative mathematical framework.

June 18, 2026Open Access

Non-Abelian Emergence: A Generative Derivation from Axioms to Gauge Field Theory

Key Points

The aim is to derive non-Abelian gauge symmetry from axioms within a generative mathematical framework.
Demonstrated the Non-Abelian Emergence Theorem through proofs involving pixel independence and several lemmas.
Analyzed the self-evolution of a Hexad in multi-component phase fields under varied parameter configurations.
Illustrated the connection between the axioms and emergent gauge symmetries from U(1) to SU(n).
Established the necessity of the Non-Abelian structure constants under certain conditions of differentiability.
Proven that the non-Abelian gauge symmetries are products of the Hexad's self-evolution under different setups.
Demonstrated that the Abelian unification from earlier models is a projection of this higher-layer, non-Abelian framework.

Abstract

Abstract This paper is the capstone of the L3 layer of the generative mathematics system. Its core contribution is the Non-Abelian Emergence Theorem (Theorem 3. 1): within the framework of Axioms 1–5, the self-evolution of a single-body Hexad in a multi-component phase field necessarily gives rise to non-Abelian gauge symmetry. The proof chain starts from pixel independence, proceeds through the inevitability of distinguishability (Lemma 2. 1), the fourfold equivalence (Lemma 2. 2), and the cross-coupling mechanism, and locks the non-Abelian structure constants fₐbc under emergent smoothness. The complete spectrum of gauge symmetries from U (1) to SU (n) are products of the self-evolution of the same Hexad under different parameter configurations. The Non-Abelian Emergence Theorem simultaneously proves that the Abelian unification of the L2 layer is the degenerate projection of the L3 layer in the single-component limit—the upward path and the downward path close here. The fundamental transcendence of generativism over the traditional non-Abelian paradigm lies in this: traditional theory treats non-Abelian structures as experimental presuppositions or material special cases; generativism treats them as axiomatic necessities. The SU (3) × SU (2) × U (1) of the Standard Model is not "God's choice, " but the inevitable product of the self-evolution of the Hexad; the non-Abelian anyons of topological quantum computation are not a fortuitous discovery under extreme conditions, but an inevitable emergence in any system satisfying the condition of distinguishability. From Axioms 1–5 to non-Abelian emergence, from non-Abelian emergence to the gauge group spectrum, from the gauge group spectrum to the completeness of the Hexad, and from the completeness of the Hexad to the fourfold unification—generative mathematics has completed the complete derivation chain from axioms to gauge field theory. Keywords: Non-Abelian Emergence Theorem; Hexad; Axioms 1–5; gauge field theory; inevitability of distinguishability; fourfold equivalence; self-referential minimality; generative unification

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