In the paper we present a description of complex systems in terms of self-organization processes of prime integer relations. A prime integer relation is an indivisible element made up of integers as the basic constituents following a single organizing principle. The prime integer relations control correlation structures of complex systems and may describe complex systems in a strong scale covariant form. It is possible to geometrize the prime integer relations as two-dimensional patterns and isomorphically express the self-organization processes through transformations of the geometric patterns. As a result, prime integer relations can be measured by corresponding geometric patterns specifying the dynamics of complex systems. Determined by arithmetic only, the self-organization processes of prime integer relations can describe complex systems by information not requiring further explanations. This gives the possibility to develop an irreducible theory of complex systems.
The aim of the article is to present the description of complex systems in terms of self-organization processes of prime integer relations and illustrate its main properties. Based on the integers and controlled by arithmetic only, the processes can characterize complex systems by information not requiring further simplification. This raises the possibility to develop an irreducible theory of complex systems.
In the paper we present results to develop an irreducible theory of complex systems in terms of selforganization processes of prime integer relations. Based on the integers and controlled by arithmetic only the self-organization processes can describe complex systems by information not requiring further explanations. Important properties of the description are revealed. It points to a special type of correlations that do not depend on the distances between parts, local times and physical signals and thus proposes a perspective on quantum entanglement. Through a concept of structural complexity the description also computationally suggests the possibility of a general optimality condition of complex systems. The computational experiments indicate that the performance of a complex system may behave as a concave function of the structural complexity. A connection between the optimality condition and the majorization principle in quantum algorithms is identified. A global symmetry of complex systems belonging to the system as a whole, but not necessarily applying to its embedded parts is presented. As arithmetic fully determines the breaking of the global symmetry, there is no further need to explain why the resulting gauge forces exist the way they do and not even slightly different.
Engineering of distributed computing systems requires understanding of principles of complex systems, which have not been yet identified. To address the situation we use a concept of structural complexity and present results of computational experiments suggesting the possibility of a general optimality condition of complex systems. The optimality condition introduces the structural complexity of a system as a key to its optimization.
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