From States to Transitions: An Operator-Based Model of Semantic Order

This paper develops an operator-based geometric model of order that reconstructs the emergence, articulation, stabilization, and withdrawal of structure as a sequence of directed transitions. Instead of treating order as a fixed configuration or a dialectical synthesis of states, the model describes a four-phase architecture (RO, RQ, SO, SQ) connected by a family of operators Ω₁–Ω₄ that regulate visualization, articulation, separation, and re-homogenization. A central result is the formulation of a spiral logic of order. Re-homogenization does not constitute a return to an initial state, but produces a uniform distribution over a historically transformed space of possibilities. Irreversibility is not located primarily in temporal succession, but in the irreversible modification of the configuration space itself. The paper clarifies the role of imaginary degrees of freedom in the separation of principles and shows that higher-dimensional structure, once activated, is not retracted but remains as a latent carrier of operative relevance across subsequent cycles. The resulting architecture offers a geometric and teleodynamic alternative to dialectical, cyclical, and emergentist models of order, relocating synthesis from the level of states to the level of transformations.