# On an irreducible theory of complex systems

In the paper we present results to develop an irreducible theory of complex systems in terms of self-organization 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.