Hierarchical Order I: Inter-Level Feedback on Quantum Level

Introduction

The major goal of the present book is to present a credible systematic theory whose central theme is the answer to the question what makes a complex system intelligent and why it should share certain universal, yet indifferent to the intelligence properties. The first two chapters deal with the second part of the question. It puts forward the assumption that the universal properties are to be straightforwardly associated with long-term stability of a system. Thus, it has been proven that the concept of boundedness alone, viewed as the necessary condition for long-term stability, provides the existence of 3 time series invariants as a generic property of every zero-mean bounded irregular sequence (BIS). A distinctive property of these time series invariants is that each of them is insensitive to the environmental statistics and to the operation of coarse-graining. This implies that the obtained knowledge about the behavior of a complex system is independent from the lack of a global pre-determination of the response and the inevitable distortions in its recording.

From the above considerations it becomes clear that for the first time we encounter the highly non-trivial interplay between the issue about boundedness viewed as a condition providing long-term stability, and the issue about information that generates its most general meaning. By following these considerations further, we again encounter that interplay when discussing the separability of a power spectrum to a discrete and a continuous part. Indeed, by proving constant accuracy of the discrete band reoccurrence in an ever-changing environment, we actually prove that the information encapsulated in it is reproducible with well defined certainty. Yet, an additional reading of that reproducibility is that no information can be created by variations (noise) only. This conclusion is strongly supported by the universal finiteness of another already defined time-series invariants, K-entropy, which comes to say that the response is a two-component variable: the first component is the above discussed noise one whose future behavior is unpredictable; and the second component is a collective self-organized pattern which remains steady and robust under the “noise” part. Thus, we can conclude that the only component which carries information is the steady one. Further, by associating the information with the discrete band in the power spectrum, the constant accuracy of its reoccurrence retains the following two-fold meaning: (i) the information cannot be created or destroyed by the noise; (ii) the energy/matter involved in its creation and sustaining is constant which is always less than the all energy/matter involved in the overall response. As a matter of fact, the overall energy/matter involved and exchanged by the response is given by the overall power spectrum; and since the latter comprises additively two components, a discrete one and a noise one, it is obvious that the energy involved in the discrete one, i.e. one associated with the information, is always less than 100% .

Next in this line of reasoning we found out that the minimal necessary condition for the noise band in the power spectrum so that it to posses the property not to signal out any specific component is the presence of the same metrics for all components. This requirement goes consistently with our suggestion that the response is determined only by the current local impact and the current state of a system. A minimal condition for sharing the same metrics by components in the discrete and the continuous band is to be described by the same variable. The next important element in this argumentation is to put forward the assumption that the interplay between both parts is substantiated by the presence of a non-recursive line in the discrete band. The importance of the latter is enormous since it fundamentally changes the entire view on the relation between different phenomena and laws in Nature. The presence of a non-recursive component in the discrete band viewed as a generic property implies algorithmic unreachability of one law, viewed as a rule for organizing and sustaining of a certain self-organized pattern. Put it in other words, each law becomes algorithmically uncompressible. In turn, as discussed in the previous Chapter, this implies renounce of the idea about existence of Universal law to which otherwise the idea about algorithmic compressibility in the traditional information theory leads.