This work develops a unified framework for understanding persistence and failure in complex systems by deriving, rather than assuming, the minimal structures required for relational coherence. Beginning from the requirement that viable systems must resolve interactions beyond pairwise relations, it is shown that triadic closure is the minimal unit of consistency. The unique finite structure satisfying this requirement is the Fano plane, which organizes seven irreducible relational roles into a closed configuration.
When these relations are required to support directed interaction, the structure lifts necessarily to the octonion algebra, introducing non-associativity as a measure of contextual inconsistency. The need to represent structured states leads to the exceptional Jordan algebra , whose cubic norm captures minimal global consistency. Further lifting to the Freudenthal triple system introduces symplectic duality and yields a quartic invariant preserved by the exceptional group , providing the first candidate for a global coherence measure across relational transformations.
To account for the distinction between observable variables and underlying structure, the framework incorporates fiber bundle theory, where measured states are projections of higher-dimensional relational configurations. Sheaf theory and cohomology formalize the transition from local consistency to global coherence, with failure arising as obstruction to the existence of a global section. This yields a structural interpretation of early warning signals as the accumulation of unresolved inconsistencies prior to observable collapse.
The resulting framework is shown to apply across domains. In physics, it aligns with relational interpretations of quantum mechanics and entanglement. In medicine, disease is reinterpreted as loss of relational coherence preceding measurable dysfunction. In ecology, collapse emerges from breakdown of interaction networks before changes in indicators. In economics, crises reflect incoherence between financial and real systems. In governance, policy failure arises from optimizing projections rather than preserving structural integrity.
The central result is that viability is not a property of components but of the coherence of their relations, and that this coherence is governed by invariant structures arising from minimal mathematical constraints. Action within such systems must therefore shift from control of variables to preservation of relational coherence under constraint.
