The self-organization under boundedness is considered as an operational protocol, which describes the highly non-trivial interplay between the inter-level feedback and the spatio-temporal pattern that describes the corresponding homeostasis. A generic property of the feedback is that it sets metrics in the state space and defines the admissible transitions from any given state. Further, the state space is partitioned into basins-of-attraction so that each of them is tangent to the point called accumulation point. A distinctive property of the basins-of-attraction is that the discrete band of the power spectrum appears as intra-basin invariant and thus turns as appropriate candidate for a “letter.” The accumulation point is associated with the notion of a “space bar.” Then, since the motion in a bounded attractor is orbital, it is appropriate to associate a “word” with an orbit. The latter open the door for assigning a specific non-mechanical engine to each and every orbit. In turn the functional irreversibility of any engine substantiates the sensitivity to permutations of every semantic unit.
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The theory developed in the present book considers hierarchical order as the major implement for strengthening the response through its specification that happens at different hierarchical levels. Thus, on the one hand, the hierarchical structuring serves as an implement for enhancing stability of a complex system. On the other hand, the specification of the response viewed as response to different external stimuli at different hierarchical levels poses the question whether the variety of the responses must obey a general rule for its organization. Moreover the following question arises: how to specify the differences in the response at different hierarchical levels.
It is obvious that a multi-level hierarchical organization implies generic insensitivity of the level response to the variables by which the response is described at any other level. This consideration strongly suggests formation of collective variables for every level of hierarchical organization. On the other hand, the very idea of hierarchical organization implies that each level “feels” the others, i.e. the levels are interconnected by feedbacks. If otherwise, the system would not be a single complex structure but would fall apart into several simple disconnected systems. These considerations suggest that the presence of inter-level feedbacks is a necessary part of the hierarchical organization of any complex system. Thus the question that arises is how to take into account the inter-level feedbacks so that even though each of them commences at a different hierarchical level, it is described by the variables specific for a given level.
The boundedness gives a new prospective to the inter-level feedbacks: they serve to control the amplification of the local fluctuations of the collective variables that describe a given level. As we have considered in the previous Chapter, the amplification of the local fluctuations is a generic property of all systems made of atoms and molecules, i.e. a generic property of systems whose constituents are allowed to exert free motion, even when its velocity is bounded. This fact initiates a systematic study of the roots of the problem and they were found at the quantum level. The root of the problem turns out to be the so far over-looked interplay between the random and the potential motion of the constituents which results in non-unitary interactions. Then, the consideration of the non-unitary interactions under the concept of boundedness provides their exclusive property to be metric-free. The latter implies that their characteristics are not specified by metric properties such as position, velocity etc. The property of being metric-free is justified by the properties set by obeying the operation of coarse-graining instead of that of linear superposition. To remind, the operation of coarse-graining, introduced in Chapter 1, is a non-linear operation which acts non-linearly and non-homogeneously on a bounded irregular sequence so that to preserve its boundedness; on the other hand, the linear superposition allows accumulation of arbitrary amount of matter/energy locally and thus is inconsistent with the concept of boundedness. Although the non-unitary interactions are metric-free, their intensity is controlled by the local concentration of the species.
Thus, we become able to outline the feedback: the non-unitary interactions control local fluctuations of the concentration by permanently keeping the size and the rate of development of each fluctuation bounded. Alongside, the local concentration permanently controls the intensity of the non-unitary interactions. Thus, we come to the conclusion that the local concentration permanently varies within specific margins. Yet, the following questions remain to be clarified: what is the role of the unitary interactions and how does the interplay between them contributes to the hierarchical organization?