Dynamic Group Theory: From Phase Closure to Modular Automorphisms
- 1Independent Researcher, Chongqing, China
Abstract
Traditional group theory takes its four axioms—closure, associativity, identity element, and inverse element—as an unquestionable starting point. Physics treats group theory as the fundamental language for describing symmetries. Yet two fundamental questions have always remained unanswered: Where do the four axioms of group theory come from? And under what mechanisms do physical symmetries themselves arise, persist, and change? This paper systematically answers both questions within the framework of generative mathematics, establishing a “dynamic structure” theory of group theory. The core discovery is that Axiom 4 simultaneously generates two things: the discrete group structure (the seed of algebra) and the discrete coupling rules (the seed of dynamics). The group structure unfolds into modular automorphisms under emergent continuity, the coupling rules unfold into gradient flow, and the two converge into the operator flow equation—the equation is not written a priori, but is the inevitable differential form that modular automorphisms unfold into under emergent continuity. Based on this overarching principle, this paper derives the four axioms of group theory purely algebraically from Axiom 4, establishes modular automorphisms as the dynamical unfolding of groups, proves that the operator flow equation is the quantitative projection of modular automorphisms, and determines the ontological status of group theory within the generative system—group theory is a “property” rather than a “component”: remove it, and it will return, because Axiom 4 remains unchanged. Core conclusion: Group theory does not come from physics; physics comes from group theory; group theory comes from Axiom 4. Equations are quantitative projections of modular automorphisms—algebra is the skeleton of equations, and equations are the flesh of algebra. Keywords: dynamic group theory; Axiom 4; phase closure condition; modular automorphism; operator flow unfolding; contraction of symmetry degrees of freedom; derivation of group axioms; property vs. component
Related
- Unipotent automorphisms of modular group algebras and the homology of the Lazard Lie algebra
- The Dynamic Topology-Algebra Correspondence Principle: From Phase Closure to Galois Theory
- Automorphisms of general modular curves
- Composing group automorphisms with colour-preserving automorphisms of Cayley graphs
- The group of strong automorphisms of the differential field of transseries
BibTeX
@misc{jun2026dynamic,
title = {{Dynamic Group Theory: From Phase Closure to Modular Automorphisms}},
author = {Zhao Jun},
howpublished = {Zenodo (CERN European Organization for Nuclear Research)},
year = {2026},
month = jun,
doi = {10.5281/zenodo.20685196},
url = {https://doi.org/10.5281/zenodo.20685196},
note = {CC BY},
}
RIS
TY - UNPB
TI - Dynamic Group Theory: From Phase Closure to Modular Automorphisms
AU - Zhao Jun
PY - 2026
DA - 2026/06/14
PB - Zenodo (CERN European Organization for Nuclear Research)
DO - 10.5281/zenodo.20685196
UR - https://doi.org/10.5281/zenodo.20685196
AB - Traditional group theory takes its four axioms—closure, associativity, identity element, and inverse element—as an unquestionable starting point. Physics treats group theory as the fundamental language for describing symmetries. Yet two fundamental questions have always remained unanswered: Where do the four axioms of group theory come from? And under what mechanisms do physical symmetries themselves arise, persist, and change? This paper systematically answers both questions within the framework of generative mathematics, establishing a “dynamic structure” theory of group theory. The core discovery is that Axiom 4 simultaneously generates two things: the discrete group structure (the seed of algebra) and the discrete coupling rules (the seed of dynamics). The group structure unfolds into modular automorphisms under emergent continuity, the coupling rules unfold into gradient flow, and the two converge into the operator flow equation—the equation is not written a priori, but is the inevitable differential form that modular automorphisms unfold into under emergent continuity. Based on this overarching principle, this paper derives the four axioms of group theory purely algebraically from Axiom 4, establishes modular automorphisms as the dynamical unfolding of groups, proves that the operator flow equation is the quantitative projection of modular automorphisms, and determines the ontological status of group theory within the generative system—group theory is a “property” rather than a “component”: remove it, and it will return, because Axiom 4 remains unchanged. Core conclusion: Group theory does not come from physics; physics comes from group theory; group theory comes from Axiom 4. Equations are quantitative projections of modular automorphisms—algebra is the skeleton of equations, and equations are the flesh of algebra. Keywords: dynamic group theory; Axiom 4; phase closure condition; modular automorphism; operator flow unfolding; contraction of symmetry degrees of freedom; derivation of group axioms; property vs. component
ER -