We propose a unified relational dynamical framework for the emergence of structure across physical, biological, cognitive, and technological systems. The starting point is the notion of minimal difference as the simplest deviation from a trivial regime in which relations are globally reducible and generate no structural effects. The key mechanism is the transition from minimal difference to returning difference, understood as an effect that does not vanish along closed trajectories. We show that…
Read moreWe propose a unified relational dynamical framework for the emergence of structure across physical, biological, cognitive, and technological systems. The starting point is the notion of minimal difference as the simplest deviation from a trivial regime in which relations are globally reducible and generate no structural effects. The key mechanism is the transition from minimal difference to returning difference, understood as an effect that does not vanish along closed trajectories. We show that a returning difference constitutes the minimal form of feedback, enabling the accumulation of effects and the onset of nontrivial dynamics.
A central role in the proposed framework is played by recursion, understood not as mere iteration of a rule, but as a structural property of dynamics in which transformations act on the results of previous transformations. Recursion is one of the fundamental mechanisms underlying the generation of structure: it enables the formation of feedback loops, hierarchical organization, and multi-level regulatory systems. In this sense, returning difference can be viewed as the elementary form of recursion grounded in relational dynamics.
We formalize relational dynamics in terms of local relations and their non-closure (curvature), which, in the presence of appropriate operators, generate iterated feedback. The introduction of selection and projection mechanisms leads to invariant sets and concentration of trajectories, interpreted as the formation of stable structures. Extending the model to the statistical level allows for the description of probabilistic dynamics, metastability, and the role of fluctuations.
In the continuous limit, the framework leads to effective field descriptions, and when coupled with geometry, to the co-evolution of fields and spatial structure. At a global level, entropy, information, and elements of geometric structure can be understood as emergent properties arising from relational feedback dynamics.
The proposed approach is structural and effective: it does not replace existing physical theories, but identifies common mechanisms of organization across domains. In particular, we argue that the transition from minimal difference to returning difference—as the elementary form of recursion and feedback—defines a universal threshold at which the emergence of complex structures becomes possible.