From Metaphor to Term: "Equilibrium Attractor" as a Formal Concept for "Folds" in the HPSR Model

The article formalizes a concept previously designated by the metaphor "fold" in the Hierarchical Pulsating System of Realities (HPSR) model. It is shown that the metaphor is imprecise: in mathematics, "fold" is a singularity of a mapping (fold catastrophe), in geology, a deformation of relief. Moreover, it does not reflect that what is at issue is a stable state of a system. A formal definition is proposed through the apparatus of the theory of dynamical systems: an equilibrium attractor is a point attractor (a stable state in the equilibrium potential V(A)) corresponding to an activated set of elements of reality A(S) at which the formation of stable (for 50–100 billion years) space-time is possible. In the HPSR model, each equilibrium attractor corresponds to one of the parallel universes. The choice of term is justified through comparison with alternatives (fold, attractor, potential well, ontological mode). It is shown that the new term is precise, consistent with the apparatus of HPSR (V(A), equilibrium parameters), and does not conflict with existing concepts. The connection with other concepts of the series is discussed: HPSR (restructuring during pulsation), "Universes Here" (attractor as a parallel universe), EFOS (projection at S), S-switching (transition between attractors as a change of S), modulational interaction (modulation of V(A) shifting the attractor). It is important to distinguish between two types of attractors. The "equilibrium attractor" describes the structural aspect of A(S) — which types of elements of reality are manifested. The "evolutionary attractor" (introduced in a separate work) describes the parametric aspect of the same A(S) — which values fundamental constants take within this structure. These are not two different sets of elements, but two levels of description of one and the same A(S). It is proposed to replace "fold" with "equilibrium attractor" throughout the MBM series, including an updated glossary. Open questions are formulated: hierarchy of potentials, connection of attractors at different levels of HPSR, possibility of detecting residual attractors, formal description of transition (bifurcation, catastrophe), connection with catastrophe theory.