Complex systems across domains — clinical, ecological, and economic — frequently fail despite the availability of extensive data, advanced analytics, and well-intentioned interventions. This work proposes that such failures arise not primarily from insufficient information or incorrect values, but from a loss of relational coherence within system structure.
We introduce a minimal, domain-agnostic framework termed the Geometry of Viability, composed of seven primitives: State (X), Constraints (C), Margins (M), Disturbances (D), Perception (P), Regulation (R), and Options (O). These elements are not analyzed in isolation but through their structured relationships, organized into triads corresponding to a minimal closed system represented geometrically by the Fano plane.
The framework is further formalized through a hierarchy of invariants: pairwise compatibility (ω), triadic coherence (N₃), and global viability (I₄). Together, these define necessary conditions for system persistence across scales.
A central contribution of this work is the reframing of mathematics from a predictive tool to a navigational framework, capable of mapping constraints on possible transitions rather than specifying future states. This shift supports a broader paradigm transition from control-oriented intervention to constraint-aware navigation.
Applications are explored in clinical medicine (decision-making under uncertainty and iatrogenic risk), ecology (flow networks and resilience), and economics and governance (optionality, regulation, and structural fragility). Across these domains, a unifying principle emerges:
Systems remain viable not by controlling outcomes, but by navigating the space of possibilities within constraints.
This work provides both a conceptual lens and an operational framework for maintaining viability in complex adaptive systems.
