  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.
