  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.
