Abstract

This paper introduces the Dynamic Fractal Theorem of Relation (DFTR), a unifying framework that describes how coherence arises and endures across physical, biological, cognitive, and social systems. DFTR posits that existence is constituted not by static entities but by a recursive triad of rupture, repair, and care—the minimal operations through which relational fields sustain integrity over time. Drawing from thermodynamics, systems theory, neuroscience, and process philosophy, the theorem formalizes these operations as interdependent parameters in a dynamic equation of persistence, expressed as dC/dt = (α × Rₚ) – (β × Rᵤ) + (γ × κ). Here, coherence (C) evolves through the continual conversion of rupture (Rᵤ) into repair (Rₚ) under the stabilizing condition of care (κ). This structure reveals communication—not substance—as the universal mechanism of persistence. The paper situates DFTR in dialogue with relational physics, autopoiesis, the free-energy principle, and agential realism, demonstrating its capacity to unify diverse models of self-organization under a single relational grammar. It concludes by exploring the theorem’s implications for ethics and epistemology, proposing care as the structural condition for continuity in both living systems and the scientific enterprise itself.