Recursive Collapse: A Formal Theory of Coherence, Entropy, and Structural Instability in Complex Systems

We present a general theory of collapse in complex systems, formulated in terms
of internal disorder and structural coherence. A system’s stability is quantified by the
ratio λ(t) = S(t)\(Ealign(t)+ε ), where S(t) measures entropy—interpreted as internal variability, uncertainty, or miscoordination—and Ealign(t) represents the degree of functional coupling among system components.

When λ(t) exceeds a critical threshold, the system enters a regime of instability
where coordination deteriorates, cascading failures occur, and recovery may become
infeasible. We formally define conditions for collapse, hysteresis, and partial recov-
ery, and derive the differential behavior of λ(t) under dynamic entropy growth and
alignment decay.

This framework generalizes failure dynamics across domains including neural net-
works, ecological systems, infrastructure grids, and financial markets. It provides pre-
dictive criteria for system fragility and a quantitative basis for designing early warning
indicators and targeted interventions in high-dimensional, adaptive systems.