Complex adaptive systems across domains — including biological organisms, ecological communities, financial networks, and geopolitical institutions — exhibit a common pattern of sudden collapse following extended periods of apparent stability. Traditional analyses often focus on individual variables within these systems, yet such variables frequently fail to capture the structural conditions that determine persistence under disturbance.
This paper proposes a minimal relational framework for analyzing viability in complex adaptive systems. The framework identifies seven informational roles — constraints, margins, system state, disturbances, perception, regulation, and optionality — that together form the minimal architecture required for persistence. These roles interact through a set of seven triadic relations that correspond to the unique Steiner triple system , represented by the Fano plane.
This relational grammar generates a geometric representation of system dynamics in which persistence corresponds to trajectories remaining within a viable region of state space defined by constraints and margins. Collapse occurs when margins erode and optional future trajectories disappear. Empirical examples from clinical medicine, coral reef ecology, and financial crises illustrate how these dynamics manifest across domains.
The resulting framework provides a unified perspective on fragility, resilience, and systemic collapse. The appearance of the Fano combinatorial structure suggests deeper connections with exceptional algebraic systems such as the octonions and the Freudenthal triple system associated with the exceptional Lie group . While these mathematical correspondences are presented primarily as scaffolding for future research, they highlight the possibility that persistence in complex adaptive systems may depend on maintaining coherence within a minimal relational architecture.
By identifying the structural conditions that sustain viability, the proposed framework offers a foundation for analyzing resilience across disciplines and for designing institutions and policies that preserve the life-supporting systems upon which human societies depend.
