Toward a Dynamical Formalization of the Dynamic Harmony Cycle: State Space, Hysteresis, and Recursive Stage Structure

Dynamic Harmony Mathematical Formalization Program.

This paper develops a dynamical systems formalization of the five-phase process grammar proposed by Dynamic Harmony: Destabilization, Binding, Stabilization, Articulation, and Collapse. The work attempts to derive the grammar from first principles in dynamical systems theory rather than from empirical pattern recognition across domains.

The analysis defines a class of coordination systems characterized by bounded state space, feedback dynamics, coordination dependence, accumulating perturbation load, topological mutability, and structural sufficiency of graph invariants. Within this class, regime transitions are examined using tools from dynamical systems theory, bifurcation theory, and network science.

The paper develops a graph-dynamical representation of coordination systems and derives a directed acyclic regime structure describing admissible transitions between coordination regimes. The analysis connects the five phases to recognized dynamical structures including attractor bifurcation, unstable manifold traversal, attractor convergence, secondary bifurcation, and network fragmentation.

This paper is a standalone companion to the Dynamic Harmony Structural Stress-Test Series but is not part of that sequence. While the stress-test program evaluates the framework through adversarial empirical analysis across scientific domains, this paper attempts a formal mathematical derivation of the grammar from dynamical systems principles.