Integers: irreducible guides in the search for a unified theory

Abstract

The notion of final theory results from a contrasting understanding of physical reality. Currently, different approaches aim to unify the four forces of nature and discuss whether a final theory may be possible. A key feature of a final theory is irreducibility, however this property has not been seriously exploited. In the paper we present an irreducible mathematical theory that describes physical systems in terms of formation processes of integer relations. The theory has integers and integer relations as the basic elements and is irreducible, because the formation processes are completely controlled by arithmetic. We suggest properties of the formation processes as irreducible guides in the search for a unified theory.

On irreducible description of complex systems

“…The figure shows that for states [2] (t ) > g [2] (t ), t 0 < t < t 8 , the third integrals are not equal f [3] (t 8 ) = g [3] (t 8 ). Thus, the states x and x 0 share two of the quantities and C(x,x 0 ) ¼ 2.…”

“….,Q k )] into the system of k Diophantine equations, as its solution, produces a system of k integer identities or relations. It is shown that the system of integer relations can be associated with a number of hierarchical structures, which can be interpreted as a result of self-organization processes of prime integer relations [1,[3][4][5].…”

On principles in engineering of distributed computing systems

“…In the geometric form the nonlocal correlations are represented by two-dimensional patterns that may characterize the dynamics of the parts in a strong scale covariant form. Significantly, based on the integers and controlled by arithmetic only, the self-organization processes can specify complex systems by information not requiring further explanations (Korotkikh 1999(Korotkikh , 2005.…”

“…To address the situation we suggest an approach based on the description of complex systems in terms of selforganization processes of prime integer relations (Korotkikh 1999(Korotkikh , 2005. To measure the complexity of a system in terms of the partial order of the processes the concept of structural complexity is introduced.…”

Description of Complex Systems in Terms of Self-Organization Processes of Prime Integer Relations

“…1: The figure shows that for states x = +1 − 1 + 1 − 1 − 1+1+1+1 and x ′ = −1−1+1+1+1+1+1−1 the first integrals are equal f [1] (t8) = g [1] (t8), where f = ρ mεδ (x), g = ρ mεδ (x ′ ) and m = 0, ε = 1, δ = 1. It turns out that the second integrals are also equal f [2] (t8) = g [2] (t8), but the third integrals are not f [3] (t8) = g [3] (t8). Thus C(x, x ′ ) = 2.…”